Thermal Structure of Flames in Metal Particle Suspensions

Michael Soo

Department of Mechanical Engineering

McGill University

Montreal, Quebec

December 2016

Supervisors: Professors Jeﬀrey Bergthorson and David Frost

A thesis submitted to McGill University in partial fulﬁllment of the requirements of the degree of Doctor of Philosophy.

Abstract

Understanding the combustion behavior of reactive particle suspensions is central to the goal of predicting and tailoring the performance of metalized explosives and propellants. The mass fraction of metal in these energetic compositions can exceed 20%, leading to the metal burning as a dense suspension in the gaseous products of a hydrocarbon fuel matrix. The combustion characteristics in dense particle suspensions are often extrapolated from those measured from isolated, single particles or dilute suspensions. However, these characteristics depend on the temperatures and composition of the gas which are altered during the reaction of the particle suspension. A simple model is explored to demonstrate how particle concentration may affect combustion time, reaction regime, and thermal structure in the suspension for a zero-dimensional reactor and for one-dimensional flame propagation. Unlike the majority of models previously developed for particle combustion, no external parameters are imposed, such as particle ignition temperature, combustion time, or the assumption of either kinetic- or diffusion-limited particle combustion regimes. Instead, it is demonstrated that these characteristics can be functions of the particle mass concentration, and that the a priori imposition of these characteristics from single-particle combustion data may result in erroneous predictions. There are very few experimental techniques developed to determine combustion characteristics in dense suspensions. The determination of key parameters such as reaction regime, and burning time in suspension combustion is difficult due to the inability to isolate and observe individual particles. Instead, the particle combustion characteristics must be determined from diagnostics on the bulk flame. A flat flame in a suspension of micron-sized aluminum fuel and gaseous oxidizer is stabilized on a counterflow burner to provide a one- dimensional geometry for use with line-of-sight optical diagnostics. An imaging emission spectroscopy and a broadband-laser absorption spectroscopy technique are developed and applied to the flat flame to assess the regime of particle combustion in the flame. These techniques provide a method for determining the combustion time of particles in the suspension.

i Abr´eg´e

Comprendre la combustion de suspensions de particules réactives est essentiel pour prédir et adapter la performance des explosifs et des propergols métallisés. La fraction massique du métal dans ces mélanges énergetiques peut excéder 20% et ce dernier brûle ainsi en tant que suspension dense dans les produits gazeux de la matrice d’hydrocarbures du carburant. Les charactéristiques de ces suspensions denses de particules sont souvent extrapoléesa ` partir de celles mesurées sur des particules isolées ou de suspensions diluées. Cependant, ces charactéristiques dépendent des températures et de la composition du gaz qui sont modifiées pendant la réaction de la suspension des particules. Un modéle simple, développé dans cette thése, explore et démontre comment la concentration des particules peut affecter le temps de combustion, le régime de réaction et la structure thermique d’une suspension pour un réacteur de dimension zéro ainsi que dans la propagation d’une flamme unidimensionnelle. A la différence de la plupart des modles de combustion de particules actuels, ce modéle n’impose aucun paramétre externe comme la température d’allumage, le temps de combustion ou la supposition d’un régime-limite cinétique ou de diffusion. Au contraire, les résultats démontrent que ces charactéristiques peuvent être des fonctions de la concentration massique des particules et qu’une imposition a priori de ces charactéristiques à partir de la combustion de particules uniques peut résulter en prédictions erronées. Très peu de techniques expérimentales permettent la mesure de charactéristiques de combustion des suspensions denses. La détermination de paramètres-clés tels que le régime de réaction ou le temps de combustion de la suspension est difficile car présentement, on est incapable d’isoler et d’observer les particules individuellement. Les charactéristiques de combustion des particules doivent donc être déterminées grâce aux diagnostics de lensem- ble de la flamme. Une flamme plate dans la suspension de micro-particules d’aluminium et d’oxydant gazeux est stabilisée sur un brûleura ` contre-courant, résultant en une géometrie unidimensionnelle qui permet une portée optique pour les diagnostics. Des techniques de spectrométrie de formation d’images d’émission et de spectrométrie d’absorptiona ` bandes larges ont étédéveloppées et appliquées sur la flamme plate pour déterminer le régime de

ii combustion des particules dans la ﬂamme.

iii Acknowledgments

I would like to thank Prof. Jeffrey Bergthorson and Prof. David Frost for their guidance in the completion of the PhD project and for their countless hours editing and critiquing my work. I would also like to thank Dr. Samuel Goroshin whose unique vision and understanding of metal combustion provided inspiration and guidance for the work in this thesis. The technical advice of Prof. Andrew Higgins and Prof. Nick Glumac was invaluable for presentations and in the completion of the diagnostic setups. My colleagues and co-authors, Philippe Julien, Jan Palecka, James Vickery, and Keishi Kumashiro, deserve a special thanks for their comradery and research efforts. I additionally owe a debt of gratitude to the rest of the official and unofficial metal flames students for keeping things interesting during my time at McGill and University of Illinois Urbana- Champaign. My thanks also go to my family: Jake, Pip, Sheila, and David.

iv Contributions of the Author

Section 1.4 of the Introduction: I developed the original analysis of the heterogeneous ignition theory using the steady thermal states presented here.

Chapter 2 Publication [1]: Soo, M., Goroshin, S., Bergthorson, J.M., Frost, D.L. Reaction of a particle suspension in a rapidly-heated oxidizing gas. Propellants, Explosives, Pyrotechnics, 2015. 40(4):604-612.

I performed the analysis for the understanding of heterogeneous ignition. I wrote the derivation to the governing equations and the assumptions. The coding was completed with assistance from Keishi Kumashiro during a summer undergraduate research project whom I supervised. I performed the analysis of the model and wrote the paper. David Frost, Jeﬀrey Bergthorson, and Samuel Goroshin provided research guidance and editorial review.

Chapter 3 Publication [2]: Soo, M.J., Kumashiro, K., Goroshin, S., Frost, D.L., Bergthorson, J.M. Thermal structure of ﬂames in non-volatile fuel suspensions. Proceedings of the Combustion Institute, 2017. 36(2):2351-2358.

I derived the set of governing equations and assumptions. The coding of the model was completed with assistance from Keishi Kumashiro during his undergraduate thesis project. We analyzed the data together. I wrote the paper. David Frost, Jeﬀrey Bergthorson, and Samuel Goroshin provided research guidance and editorial review.

Chapter 4 Publication [3]: Soo, M.J., Goroshin, S., Glumac, N., Kumashiro, K., Vickery, J., Frost, D.L., Bergthor- son, J.M. Emission and laser absorption spectroscopy of ﬂat ﬂames in aluminum suspensions. Combustion and Flame, 2017. 180:230-238.

v I developed the diagnostic setup and wrote the spectral processing codes. Experiments were run with help from Keishi Kumashiro and James Vickery. James Vickery wrote the codes to analyze reference burning velocity from the Particle Image Velocimetry technique. I analyzed the data and wrote the paper. David Frost, Jeﬀrey Bergthorson, Samuel Goroshin, and Nick Glumac provided research guidance and editorial review.

vi Appendix D: Laser Absorption Diagnostic [4]: Soo, M., Glumac, N. Ultraviolet absorption spectroscopy in optically dense ﬁreballs using broadband second-harmonic generation of a pulsed modeless dye laser. Applied Spectroscopy, 2014. 68(5):517-24.

In the process of constructing the laser absorption setup used for the publication in Chapter 4, I developed a method to extend the visible broadband dye laser technique to produce to broadband, ultraviolet (UV) laser light. I performed the experiments, developed the spectral models, and wrote the paper. Nick Glumac provided research guidance and editorial review. While the publication provides an overview of the diagnostic used in [3] and the novel UV technique may be important for future diagnostic developments, the subject of the paper is considered peripheral to the main body of the thesis. It is, therefore, included as supplemental material.

vii List of Figures

1.1 Combustion in condensed-phase fuel suspensions, classified according the fuel volatility...... 5 1.2 Classifications of combustion mode based on thermodynamics equilibrium cal- culationsfromtheanalysisof[5]...... 7 1.3Theexpectedmodesofmetalparticlecombustionfrom[5]...... 8 1.4 Thermal structure for the two limiting kinetic and diffusion regimes for heterogeneous and vapor-phase modes of combustion. A) the kinetic limit, B) the diffusion limit for a heterogeneously burning particle, and C) the diffusion limit for a vapor-phase burning particle...... 13 1.5GeneralS-curveformforaparticle...... 14 1.6 S-curves for a range of particle sizes. At a critical size, ignition degenerates, and the particle reaction is limited to the kinetic regime...... 15 1.7 S-curves simulating particle burnout at a set gas phase temperature.(Left) Particle ignites and transitions critically to the kinetic regime during burnout. (Right) Particle ignites and transitions smoothly to the kinetic regime during burnout...... 17

2.1 Temperature and oxidizer concentration proﬁles adjacent to the particle sur- facefordiﬀerentcombustionregimes...... 28 2.2 (Left) Semenov diagram of the single particle ignition and extinction. (Right)

dependence of the particle temperature (Ts) on gas temperature (Tg) illustrating disappearance of the critical ignition and extinction points below some critical particle size (unstable regimes are marked with dashed lines). The

temperatures are scaled by the activation temperature Ta = Ea/Ru...... 30 2.3 Dependence of the ignition temperature on particle size. Note that below a critical value of the particle radius of 1.9 μm ignition is impossible...... 31

viii 2.4 Time histories of the temperature (scaled to the activation temperature Ta), radius and normalized Damköhler parameter for a single particle with different sizes injected into a hot oxidizing gas flow illustrating the different possible reaction regimes: a) a small particle (1 μm radius) that oxidizes within the kinetic regime, b) an intermediate sized (7 μm) particle that ignites and quenches, and c) a large particle (20 μm) that ignites and burns within the diffusiveregime...... 32 2.5 Reaction of an instantaneously heated suspension of solid fuel particles with a solid fuel concentration of 200 g/m3 with three different particles sizes, resulting in combustion regimes that are a) kinetically-limited, b) intermediate, and c) diffusively-limited...... 32 2.6 Dependence of combustion time on particle size at different solid fuel concentrations...... 35 2.7 Dependence of combustion time on concentration of the solid fuel for particle sizes of 4 and 10 μm...... 36 2.8 The transformation of the range of particle sizes and concentrations (shaded area) where 95% of the particle radius is burned within 500 ms with a 20% and 40% increase of the cooling rate...... 37

3.1 Modiﬁed Semenov diagram illustrating particle ignition (I) at some critical T r temperature ( g0 for 2) and transition to near the diﬀusion limit (D). Ex- tinction (E) occurs as the particle shrinks and undergoes the reverse transition to the kinetic limit (K). Stable states are shown by (•) while unstable states are denoted by (◦). The heat production curve is solid, and the heat loss curve is dashed. Inset: ignition temperature as a function of particle size showing critical radius below which ignition is impossible [1]...... 46 3.2 Particle temperature and radius history of a particle injected into a hot oxdiz-

ing gas at temperature Tg burning in a predominantly diﬀusive combustion

regime (r0 rcrit), an intermediate regime (r0 ∼ rcrit), and a kinetic combus-

tion regime (r0

(φ =0.5). Top: gas and particle temperature non-dimensionalized by Ta =

Ea/Ru; Middle: normalized oxidizer mass fraction, Y , and particle radius, ∗ r/r0; Bottom: reaction heat release rate,q ˙,andDa. The arrow indicates direction of propagation...... 49

ix 3.4 Flame structure proﬁles for three diﬀerent particle sizes in a fuel-rich mixture (φ =1.5).ThesamevariablesareshownasinFig.3.3...... 50

3.5 Burning velocity, ub, as a function of particle size for different equivalence ratios. Hatched area corresponds to regions where unstable flame propagation isobserved...... 51 3.6 Burning velocity as a function of equivalence ratio for different initial particle sizes. The expanded view shows the intersections of burning velocities for lean equivalenceratiosforvariousparticlesizes...... 52 3.7 Instantenous profiles of gas and particle temperatures and Da∗ at different instances in time over a single oscillation period of an oscillating flame. The units of t are seconds. The arrow indicates the direction of propagation. . . . 52 3.8 Comparison of computed burning velocity to semi-empirical asymptotic Zel- dovich and Frank-Kamenetskii (kinetic) and modified Mallard-Le Chatelier (diffusive) flame models. The semi-empirical models are normalized to the

numerical model at r0 =1μm for the kinetic model and r0 =25μmforthe diﬀusivemodel...... 53

4.1 Stabilized Bunsen flames in methane-air and metal-air suspensions from [5]. . 56 4.2 a) Photograph of a flat flame in an aluminum-air suspension, b) schematic of the counterflow flame burner and side view of the laser sheet and dust concentration laser probe and c) top-view schematic of the imaging-emission and laser-absorption spectroscopy setups including simultaneous imaging and concentrationmeasurement...... 64 4.3 Particle size distribution and SEM image of Valimet H-2 Aluminum. The particlesizingresultsareprovidedbythemanufacturer...... 65 4.4 (Left) Greyscale-inverted spectrogram of the aluminum atomic lines from aluminum-air flame with a (non-inverted) blown-up region of the self-reversal region in the inset. (Right) Intensity profiles of the spectrum taken at three different locations: (1) post-flame zone, (2) reaction zone, (3) pre-heat zone. The comparison of the emission line thickness to the instrument resolution functionisalsoshown...... 67 4.5 (Left) Greyscale-inverted spectrogram of the aluminum atomic lines from the aluminum-methane-air flame showing no self-reversal. (Right) Intensity profiles of the spectrum taken at three different locations: (1) post-flame zone, (2) reaction zone, (3) pre-heat zone. The comparison of the emission line thicknesstotheinstrumentresolutionfunctionisalsoshown...... 68

x 4.6 Absorption spectrograms of the Al I atomic lines for aluminum-air and aluminum- methaneairflames...... 69 4.7 (Left) inverted spectrogram of the AlO B−XΔν = −1 band sequence across the flame. (Right) Spectra from two locations in the flame showing the opti- callythickandopticallythinregions...... 71 4.8 Flame temperatures measured using AlO molecular spectra (• and ◦)incom- parison to equilibrium calculations (lines) at different initial aluminum particle concentrations. The predicted mole fraction of gaseous aluminum vapor in equilibrium is also shown...... 73 4.9 (Top) schematic of the micro-scale emission and absorption of aluminum vapor from the flat flame that is integrated along the line of sight to the spectrometer.

(Bottom) Schematic of self-reversal. Hot zone emission of linewidth w1 is

partially absorbed by cold zone with linewidth of w2 where w1 >w2...... 74

C.1 Temperature profiles of the simplified Zeldovich and Barenblatt model. . . . 99 C.2 Relative concentration profiles of the simplified Zeldovich and Barenblatt model.100 C.3 Relative difference between solutions computed at different spatial step-sizes. 101

D.1 Schematic of the entire setup for the absorption measurement. A pulsed Nd:YAG (355 nm) pumps a dye cell in a grazing incidence configuration. The modeless dye beam is then focused onto a BBO-I crystal, and the resulting UV beam is collimated and separated from fundamental and passed through thecombustionchamber...... 106 D.2 Spectral bandwidth of the second harmonic for different focal lengths of the lens used to focus the fundamental beam onto the nonlinear crystal...... 107 D.3 (top) Spectral bandwidth of modeless dye beam with Coumarin 450 as the dye averaged over several pulses. (below) The spectral profile of the second harmonic...... 108 D.4 NO spectrum at low concentration from continuously pulsed laser (20 Hz) integrated over 1 second compared to spectrum simulated by LIFBASE at T =300K...... 110 D.5 NO spectrum under optically thick conditions from a single pulse compared to spectrum simulated by LIFBASE at T =300K...... 111 D.6 Al atomic line absorption in EBWs taken in a series of experiments at different times after bridge wire initiation...... 112 D.7 The absorption spectrum of MgF in flash powder combustion fit with a 3000 Ksimulation...... 113

xi D.8 The absorption spectrum of AlF in ﬂash powder combustion ﬁt with a 2800 K simulation...... 114 D.9 Second order spectrum of AlF with a simulation at 2800 K...... 115

xii List of Tables

2.1 Numerical values of gas and solid-fuel parameters...... 29

3.1 Numerical values of gas and solid-fuel parameters...... 49

C.1 Eﬀect of spatial step-size on calculated burning velocity...... 101

xiii List of symbols

Description Dimension* α Thermal diffusivity of the gas L2T−1 β Convective mass transfer coefficient of a particle LT−1 γ Mass of fuel consumed per unit mass of oxidizer consumed κ Pre-exponential factor of surface reaction rate LT−1 λ Thermal conductivity of the gas MLT−3Θ−1 −3 −1 λI Thermal conductivity of the gas at the solid-gas interface MLT Θ −3 ρg Gas phase density ML −3 ρs Mass density of a particle ML φ Fuel-air equivalence ratio ξ Flame-fixed coordinate L ω C /ρ −3 −1 ˙ Rate of depletion of oxidizer divided by u g0 ML T −3 −1 ω˙ f Rate of depletion of fuel per unit volume ML T −1 ω˙ p Rate of depletion of oxidizer per particle MT −2 −1 ω˙ eff Rate of depletion of fuel per unit surface area ML T τr Reaction time T A Surface area of a particle L2 B Particle, or fuel, concentration ML−3 C Oxidizer concentration in the bulk gas – suspension case ML−3 −3 Cs Oxidizer concentration at the particle surface ML 2 −2 −1 cg Specific heat capacity of the gas phase L T Θ 2 −2 −1 cs Specific heat capacity of the solid phase L T Θ D Mass diffusivity of oxidizer L2T−1 2 −1 DI Mass diffusivity of oxidizer at solid-gas interface L T 2 −2 Ea Activation energy ML T Da Damköhler number Da∗ Normalized Damköhler number G Mass burning flux ML−2T−1 h Convective heat transfer coefficient of a particle MT−3Θ−1 j Oxidizer diffusion flux to the particle surface ML−2T−1 k Rate of surface reaction LT−1 −1 keff Effective rate of reaction LT r Reaction zone thickness L L Characteristic length L

*L: length, T: time, M: mass, Θ: temperature, N: moles.

xiv Description Dimension Le Lewis number ms Mass of a particle M N Number density of particles L−3 Nu Nusselt number −1 −2 pg Gas phase pressure ML T q Heat of reaction per unit mass of fuel L2T−2 ˙ 2 −3 −3 QR Rate of heat production per particle/unit surface area ML T /MT ˙ 2 −3 −3 QL Rate of heat loss per particle/unit surface area ML T /MT q˙ Normalized heat production profile across the flame −1 −3 q˙I Volumetric rate of heat transfer between the phases ML T −1 −3 q˙r Volumetric heat production rate ML T R Specific gas constant L2T−2Θ−1 2 −2 −1 −1 Ru Universal gas constant M L T Θ N r Particle radius L rcrit Critical particle radius for ignition L rminMinimum particle radius for reaction L Sh Sherwood number −1 Sf Burning velocity LT Ta Activation temperature Θ Tad Adiabatic flame temperature Θ Tg Gas phase temperature Θ Tig Ignition temperature Θ Ts Solid phase temperature Θ t Time T tb Combustion time (of a single particle) T u Flow velocity in the laboratory reference frame LT−1 −1 ub Burning velocity LT v Flow velocity in the flame-fixed reference frame LT−1 w Rate of oxidizer concentration per particle per unit surface area ML−2T−1 x Density-weighted coordinate L x Spatial coordinate L xf Flame front position in the density-weighted coordinate space L Y Normalized oxidizer mass fraction z Generic Arrhenius pre-exponential factor Reaction dependent

xv Subscripts

Description 0 Initial or at a ﬁxed value stoich At stoichiometry u In the unburned mixture (for ﬂame study) ox Oxidizer I At the interface between the particle and gas

xvi Contents

Abstract i

Abr´eg´e ii

Acknowledgements iv

Contributions of the Author v

List of Figures viii

List of Tables xiii

List of Symbols xiv

1 Introduction 1 1.1MotivationsfortheStudyofMetalCombustion...... 1 1.1.1 HistoricalPerspective...... 1 1.1.2 Modern Metal Combustion Applications ...... 3 1.2ClassiﬁcationofCombustioninMetalFuelSuspensions...... 4 1.3ModesofMetalParticleCombustion...... 5 1.4ParticleIgnitionandReactionRegimes...... 8 1.4.1 Deﬁning Ignition for Particles ...... 8 1.4.2 HeterogeneousHeatofReactionandHeatLossRates...... 10 1.4.3 Steady-StateS-CurveAnalysis...... 11 1.4.4 ParticlesBurnoutinMixedRegimes...... 16 1.5ScopeofThesis...... 18

2 Reaction Onset and Combustion in a Particle Fuel Suspension 20 2.1Introduction...... 20

xvii 2.2 Publication: Publication: Reaction of a Particle Suspension in a Rapidly- Heated Oxidizing Gas, Propellants, Explosives, Pyrotechnics [1]...... 24 2.2.1 Introduction...... 24 2.2.2 ModelFormulationandResults...... 26 2.2.3 Conclusion...... 38

3 Flame Propagation in Particle Suspensions 39 3.1Introduction...... 39 3.2 Publication: Thermal Structure and Burning Velocity of Flames in Non- volatile Fuel Suspensions, Proceedings of the Combustion Institute [2].... 42 3.2.1 Introduction...... 42 3.2.2 ModelFormulation...... 44 3.2.3 ResultsandDiscussion...... 48 3.2.4 Conclusion...... 54

4 Combustion Diagnostics in Particle Fuel Suspensions 56 4.1Introduction...... 56 4.2 Publication: Emission and Laser Absorption Spectroscopy of Flat Flames in Aluminum Suspensions, Combustion and Flame [3]...... 60 4.2.1 Introduction...... 61 4.2.2 ExperimentalMethods...... 63 4.2.3 GeneralFeaturesoftheEmissionandAbsorptionSpectra...... 66 4.2.4 Discussion...... 72 4.2.5 Conclusion...... 78

5 Conclusions 79 5.1 Synopsis of Contributions ...... 79 5.2 Direction of Future Research ...... 81

A Zero-dimensional Induction Combustion in a Fuel Suspension: Governing Equations 84 A.1ParticleEnergyEquation...... 84 A.2ParticleMass...... 86 A.3BulkGasOxidizerContinuity...... 86 A.4GasThermalEnergy...... 86

xviii B One-dimensional Flame Propagation in a Fuel Suspension: Governing Equations 88 B.1EquationofState...... 88 B.2MomentumConservation...... 88 B.3GasPhaseContinuity...... 89 B.4 Density-Weighted Coordinate ...... 89 B.5TemperatureDependenceofGasandSolidPhaseProperties...... 90 B.6GasPhaseThermalEnergy...... 91 B.7OxidizerContinuity...... 93 B.8SolidPhaseContinuity...... 94 B.9SolidPhaseThermalEnergy...... 96

C Validation of Numerical Methods 98 C.1CodeVeriﬁcation...... 98 C.2DemonstrationofGrid-Independence...... 101

D Laser Absorption Diagnostic 102 D.1Introduction...... 102 D.2DiagnosticSetup...... 105 D.2.1DyeLaserConﬁguration...... 105 D.2.2SecondHarmonicGeneration...... 105 D.3ResultsandDiscussion...... 109 D.3.1StationaryNOGas...... 109 D.3.2 Time Resolution in Exploding Bridge Wires ...... 110 D.3.3Metal-TeﬂonFlashPowders...... 111 D.3.4SpectroscopicModelforAlFandMgF...... 111 D.3.5MgFDetectioninDispersedFlashPowderCombustion...... 113 D.3.6AlFDetectioninDispersedFlashPowderCombustion...... 113 D.4 Summary and Conclusions ...... 115

Bibliography 116

xix Chapter 1

Introduction

1.1 Motivations for the Study of Metal Combustion

1.1.1 Historical Perspective

The scientific study of the combustion of metal powders in gaseous suspensions has been historically motivated by two practical applications: energetic materials and industrial safety. In energetic materials applications, the high volumetric energy density of metal powder fuels makes them attractive to add to explosive and propellant formulations to increase the overall energy output. Typically, for propellant formulations, metal powders are added in high mass loadings up to nearly 20% by weight [6]. This leads to the suspension of metal particles burning in the gaseous products of the hydrocarbon fuel matrix. For explosives, similar to propellants, metal fuels are added to increase both the blast energy and, in some cases, to enhance the blast effect. Blast enhancement is accomplished by explosively dispersing the metal particles into the surrounding air, which burn behind the blast wave enhancing the local pressure [6]. Influential developments in the science of metal combustion for energetic materials began during the mid 20th century. After World War II, the Office of Naval Research (ONR) in the United States, realizing the importance of aerospace superiority, began to fund the development of advanced propulsion systems, including jet engines, rockets, and ramjets. The effort, known as Project SQUID, consisted of a primary contract between the U.S. Navy and Princeton University [7, 8, 9]. Princeton was then tasked with awarding subcontracts to other participating groups and universities. After the 1950s, the scope of Project SQUID shifted away from applied research on propulsion system development to focus more on basic research problems involving the conversion of energy to thrust. It was during this period of Project SQUID that several fields of modern combustion science were developed, including

1 the field of metal combustion. Notable research efforts in metal combustion during Project SQUID and other similar programs were carried out by researchers such as Brzustowski and Glassman [10], Friedman and Maˇcek [11], and Markstein [12]. Many of the theories and concepts developed in those research efforts form the basis of our understanding of metal combustion today. Perhaps the most well known of these efforts is the understanding of modes of combustion for metal particles developed by Glassman [13]. Parts of his analysis are covered in this thesis in the section entitled, Modes of Metal Particle Combustion. It is also important to note the contributions of the Soviet scientists from the same era. Information about the development of Soviet metal combustion research appears to be available in closed literature [14]. However, deducing from the open translated literature, it is likely that intense research efforts began sometime after U.S. efforts, in the 1960s at the Semenov Institute of Chemical Physics [15]. It is clear that much of the basic understanding of metal combustion from the Soviet perspective was formed out of concepts from reactor theory and heterogeneous combustion concepts developed by prominent researchers such as Zel’dovich [16], Frank-Kamenetskii [17], and Semenov [18], just to name a few. Despite the significant overlap of the science in metal combustion developed in both the U.S.A. and the U.S.S.R., certain core concepts developed in the Soviet literature are often omitted from modern English texts when covering metal combustion. Particularly absent from modern literature is a comprehensive analysis of heterogeneous particle ignition and combustion as it relates to metal combustion. To the author’s knowledge, there is no single text that fully covers these concepts. The basic principles are outlined in several fundamental works by Vulis [19], Frank-Kamenetskii [17], Khaikin [20], and Merzhanov [21], and partial reproductions of the theory can be found scattered throughout textbooks on heterogeneous combustion [22, 23]. It is for this reason that a section, entitled Particle Ignition and Reaction Regimes, devoted to the often omitted key concepts, is included in this introduction. This theory also forms the basis of the models for combustion in particle suspensions presented in this thesis. The other prominent application of metal combustion research is industrial safety. Even before the development of metal fuels for energetic materials, it was well known that dusts from the process industries suspended in air (plastics, foods, metals, etc.) could cause highly energetic, uncontrolled explosions. A survey of dust explosions from 1900-1956, published by the National Fire Protection Association, found over a thousand instances of dust explosions over several industries [24]. Almost one tenth of those explosions resulted from the combustion of metal dusts. Much of the understanding of metal combustion encountered in the process industries is

2 centered around the idea that combustion in metal dusts should be treated as a reaction wave propagation phenomenon. This is in contrast to much of fundamental research in metal combustion for energetic materials, which has mainly focused on the combustion characteristics of large, isolated individual particles [25]. A large body of research devoted to understanding dust explosions has focused on studies of flames in dust suspensions [26, 27, 28]. The first fundamental study of reaction wave propagation in metal suspensions was mainly through the efforts of Hans Cassel at the U.S. Bureau of Mines, where he found that a flame could be stabilized in flowing metal suspensions similar to hydrocarbon fuels [29]. Cassel recognized that, in the suspension, the collection of closely spaced particles could cause self-heating of the suspension and thereby accelerate the rates of reaction for the individual particles. He termed this phenomenon the “cooperative mechanism”, realizing that the combustion characteristics of the particles in the suspension would change compared to the combustion of a single, isolated particle [30]. This was also understood by Soviet scientists, and can be seen in the early fundamental theoretical work on combustion in suspensions by Rumanov and Khaikin [31]. This led to the pursuit of metal combustion science in the context of the basic principles that were developed for flame propagation and combustion in reactors similar to studies of gas-phase fuel combustion [29, 26]. To date, the bodies of research from Goroshin et al. [32, 33, 34, 35] and King [36] represent some of the most comprehensive studies of flames in metal suspensions since Cassel’s initial research. The basic principles of combustion in metal suspensions outlined in these studies are reviewed in the introductions of Chapters 2 and 3 presented in this thesis.

1.1.2 Modern Metal Combustion Applications

After the initial push from Project SQUID, there were occasional bursts of research in metal combustion for materials synthesis [37], underwater propulsion [38], as well as continued eﬀorts to understand metal combustion for propellants, explosives and industrial safety. More recently, interest in metal combustion was revived during the 1990s and 2000s as a result of advances in nanotechnology [39]. The accessibility of nano-powders and nano- material synthesis opened up several new research possibilities. It was predicted that the high speciﬁc surface area of nano-sized metals would lead to higher reactivity and enhance combustion characteristics of energetic materials [39]. Several new ideas for applications of metal combustion emerged during this period. Re- search from underwater propulsion using metal-water suspensions was revisited and led to the creation of environmentally friendly, nano-aluminum/ice rocket propellants [40, 41]. As a prototype demonstration, a small rocket using aluminum-ice (ALICE) propellant was suc-

3 cessfully launched in 2009 [42]. During the late 2000s, It was also proposed that metal nano-particles could be used as clean energy carriers for sustainable energy generation [43]. The combustion of metals with air produces no carbon emissions, and metal fuels have high energy and power densities. The solid oxide products can be collected and recycled back into metal fuels. Beach et al. initially proposed injecting powder suspensions into internal combustion engines (ICEs) to produce power [43]. However, this study did not consider that solid metal oxide products would be incompatible with internal combustion systems similar to coal [44]. Later, it was proposed that ﬂames in metal suspensions could be used to provide sustained heat to external combustion engines for power generation and transportation applications [5]. A similar concept was also proposed for combustion of metal-water suspensions [45]. Despite over 50 years of sporadic research eﬀorts, the progress in the science of metal combustion has lagged far behind in comparison to the understanding of gaseous and liquid hydrocarbon fuel combustion. This can be partially attributed to the numerous, often disconnected, interpretations of the physics that govern combustion in metal particle suspensions. This is especially true of the connection between experimental and theoretical studies of single particles versus suspensions of particles. The goal of this introduction is to bring together some of the important core concepts from the literature which serve as the background knowledge required to understand results in the thesis.

1.2 Classiﬁcation of Combustion in Metal Fuel Suspen- sions

The overall combustion behavior in condensed-phase fuel suspensions can be classiﬁed according to the ability of the fuel to volatilize and premix with the gaseous oxidizer. Fuels can be placed on a scale according to the predominant combustion behavior in the droplet or particle suspension, ranging from purely homogeneous combustion (similar to gaseous fuels) to purely heterogeneous combustion, as qualitatively shown in Fig. 1.1. Highly volatile droplet or particle fuels such as plastics, organic dusts, and liquid hydrocarbons will be able to evaporate after some relatively small pre-heating, before combustion can occur. The gasi- ﬁed fuel can then premix with the oxidizing gas, leading to a fuel system where homogeneous combustion phenomena dominate. For fuels like coals or slurry fuels (e.g. liquid hydrocarbons mixed with metal powders [46]), where part of the fuel material will volatilize and premix with the oxidizing gas, there are both heterogeneous and homogeneous components to the combustion behavior, leading to a more or less “hybrid” fuel system [47, 48, 49]

4 The least volatile fuels are generally metal and metalloid powders. For the purposes of brevity in this thesis, the term metals is also meant to include metalloids. Metals, in general, have relatively high volatilization temperatures, and the reactions of metal gases with several oxidizers are nearly unactivated [50, 51]. Accordingly, metal vapor is not able to appreciably premix with the bulk oxidizer before reacting, and the combustion of the fuel system will be more heterogeneous in nature. These classiﬁcations are important for understanding the overall general treatment of combustion of metal suspensions both from a theoretical perspective and for designing experimental approaches.

Figure 1.1: Combustion in condensed-phase fuel suspensions, classiﬁed according the fuel volatility.

The overall more heterogeneous nature of combustion is what separates metal suspensions from other condensed-phase fuels. As an example, most ﬂames in liquid hydrocarbon sprays exhibit typical behaviors of homogeneous combustion. This is due to the fact that the liquid droplets rapidly vaporize in the preheat zone of the ﬂame leading to mostly gas phase combustion in the reaction zone [52].

1.3 Modes of Metal Particle Combustion

Despite the overall heterogeneous behavior of combustion in metal suspensions, there can be similarities to hydrocarbon fuels in the way (i.e. the mode) some metal particles burn. The differences in the physical properties of the metals, such as the volatility of the metal and its oxide, the heat of reaction, and the oxidizer properties, can lead to a variety of modes of particle combustion. This is why, as will be explained, the different metal fuels listed in Fig. 1.1 are not placed in the same location on the scale of combustion classification. The modes of combustion can be established when the kinetic reaction rate of the metal with the oxidizer is fast in comparison to diffusive transport, leading to the combustion of the particle being limited by diffusion. This is generally true for very large particles but, as will be shown in the following section, this assumption does not necessarily hold for small particles. In the case of purely diffusion-limited combustion, the highest temperature that can be achieved is the stoichiometric adiabatic flame temperature of the metal with the oxidizing

5 gas [53]. In one of the early studies of metal combustion for solid propellant research, Glassman [13] realized that many burning metals with non-volatile oxides as products would have a flame temperature limited to the evaporation or dissociation point of the condensed- phase oxides, since the heat of gasification of the oxide exceeded the heat released in the reaction. This basic idea has historically served as a basis for the classification of modes of metal particle combustion [54]. In Glassman’s analysis, the boiling points of the metal fuel are compared to the “dissociation point” or “boiling point” of the oxide (with the underlying assumption that this is the flame temperature) to determine the mode of combustion. In the case that the dissociation point exceeds the boiling point of the metal, the metal droplet can begin to boil, and the evaporating metal burns in a micro-diffusion flame enveloping the droplet, similar to the combustion of a droplet of hydrocarbon fuel [53]. This concept is known as Glassman’s criterion for the vapor-phase combustion of metals. In the case where the dissociation point is less than the boiling point of the metal, combustion is expected to proceed heterogeneously on the surface of the particle. However, the definition for the “dissociation point” of the oxide is vague, since at any given temperature, there may be some partial pressure of dissociated sub-oxide gases in equilibrium with the condensed-phase oxide. Furthermore, different initial conditions (initial temperature, oxygen concentration, pressure, etc.) can change the so-called dissociation point. In the 1990s Steinberg et al. [55] published a sensible argument that Glassman’s approach was only appropriate if the oxide undergoes a first-order phase transition. The limitation of the definition for a dissociation point was acceptable at the time of Glassman’s analysis because it predated the existence of robust, numerical thermodynamic equilibrium solvers. A more modern analysis of the mode of combustion for metal particles [55, 5] can be accomplished through the use of tools such as the NASA chemical equilibrium code (CEA) [56]. The author agrees with the approach of Steinberg et al., however, in a series of comments and responses by both Glassman [57] and Stienburg et al. [58], Glassman defended his original analysis and refused to concede. Steinberg et al. reasoned that the maximum temperature for the reaction of the particle and oxidizer is the stoichiometric adiabatic flame temperature [55]. Comparing the boiling point of the metal to the flame temperature, and the partial pressure of the gaseous oxides predicted in equilibrium, can provide a more complete analysis of the combustion mode [5]. The partial pressure of the gaseous oxides predicted from equilibrium calculations in the NASA CEA code are plotted against the ratio of the flame temperature to metal boiling temperature in Fig. 1.2. The calculations were performed using the constant enthalpy and pressure assumption at atmospheric pressure in air for stoichiometric mixtures of different

6 metals. Stoichiometry is deﬁned as the quantity of reactants required to create products without any excess of reactant. It should be noted that the notion of a stoichiometric mixture is loosely deﬁned. The product species are a result of equilibrium itself and, in general, cannot be known a priori. Here, stoichiometry is calculated based on the assumption of complete oxidation of the metal to the stable metal oxide indicated in Fig. 1.2.

Figure 1.2: Classiﬁcations of combustion mode based on thermodynamics equilibrium calculations from the analysis of [5].

When the ratio of flame temperature to the metal boiling temperature is above unity, the so-called Glassman criterion is met and it is predicted that the particle will burn in a lifted micro-diffusion flame similar to a hydrocarbon droplet. A schematic of this type of vapor-phase combustion is shown as mode A in Fig. 1.3. According to the analysis in Fig. 1.2, metals such as magnesium or aluminum are predicted to burn in this mode. When the flame temperature is less than, or close to, the boiling point of the metal, it is likely that the metal will not readily evaporate and combustion must proceed heterogeneously. There are two modes associated with heterogeneous combustion of the particle. In the case where the reaction forms gaseous oxides or sub-oxides, the oxides will then diffuse away from the reaction zone at the particle surface and condense (if they are able) in a zone lifted from the particle as shown as mode B in Fig. 1.3. Due to the high partial pressure of oxides predicted in equilibrium, a metal like boron is predicted to burn in this mode. The second mode of heterogeneous combustion can occur when the partial pressure of gaseous oxides is low and only condensed-phase oxides can form. The oxide will then form a shell on the particle surface as shown as in mode C on Fig. 1.3. When the density of the oxide is lower than the density of the metal, this shell will be porous according to the Pilling-Bedworth law [59], and diffusion of oxygen through the oxide layer will not significantly inhibit the

7 combustion of the particle. As shown in Fig. 1.2, iron is expected to have only condensed phase oxides, leading to combustion mode C.

Figure 1.3: The expected modes of metal particle combustion from [5].

The existence of multiple modes of combustion that depend on the properties of the particle fuel and gaseous oxidizer is the reason that the metal fuels listed in Fig. 1.1, though having more heterogeneous combustion behavior overall, have been separated on the combustion behavior scale. In air at atmospheric temperature and pressure, metals like aluminum and magnesium can burn in the vapor-phase, and metals like iron will undergo purely heterogeneous combustion. This can be controlled by varying the oxidizing environment which controls the ﬂame temperature. The mode of combustion is important for developing theoretical models of diﬀerent fuel types and, as will be explored in Chapter 4, interpreting results from experimental investigations.

1.4 Particle Ignition and Reaction Regimes

1.4.1 Deﬁning Ignition for Particles

In Glassman’s analysis, the mode of combustion is established after reaction rates become fast compared to diffusive transport. In order to reach this state, the rate of heat produced in the reaction must first be sufficient to overcome the heat loss to the surroundings (due to molecular diffusion, radiation, etc.), which requires some initial heating of the particle. When the heat production rate from an activated reaction is sufficient to overcome heat losses, the thermal state of the particle can experience a criticality known as “ignition”. The particle heat production goes into thermal runaway as the heat production rates become fast compared to loss rates, allowing the transition to a stable, diffusion controlled rate of

8 reaction. After ignition occurs, the predicted modes of particle combustion, discussed in the previous section, can be established. The word “ignition” is often used colloquially to describe the initiation of a reaction leading to self-sustained combustion in an energetic material or particle fuel cloud. This leads to the use of terms like “minimum ignition energy”, “ignition temperature” or “ignition delay”, often loosely defined, to describe the initiation of a reaction [24, 60]. However, in the context of the combustion of particles, “ignition” has a historical and precise definition that is often lumped together with these general, colloquial terms for reaction onset. As will be made apparent in the work of this thesis, the consistent use of terminology is imperative for correctly describing and understanding the phenomena that occur in the combustion of single, isolated particles and in suspensions. The overall non-volatile nature of combustion in metal suspensions means that ignition will occur heterogeneously due to the reaction at the particle-gas interface. This differs from the type of ignition resulting from the reaction of volatilized components at some distance away from the condensed-phase surface, known as homogeneous ignition. Homogeneous ignition is more applicable to hydrocarbon droplets and fuels with more volatile components, like coal [22]. While classic heterogeneous ignition and combustion theory does not precisely describe the ignition and combustion of some metal particles, especially particles with a protective oxide layer (e.g. aluminum [20]), or more volatile fuels (e.g. sulfur [61]), it does provide an understanding of the combustion physics for metal particles as a minimalistic approach. The simplicity of the model makes it a valuable tool for developing an intuition of the expected changes in combustion behavior as a function of the various physical and thermodynamic properties of the fuel and oxidizing gas. Heterogeneous ignition theory is an extension of non-adiabatic Semenov thermal explosion theory [18], and was first applied by Vulis [19] and Frank-Kamenetskii [17] to study coal char particle combustion. The results of their analysis are carefully repeated in Yarin and Hestroni [22] and pieces of the heterogeneous ignition analysis can be found in several books [23, 62]. The primary appeal of the theory is the possibility of understanding ignition, extinction, and the steady thermal combustion regimes of a particle by examining the heat generation and heat loss functions, without examining the transient processes. As will be shown, the ignition and extinction phenomena for a heterogeneously burning particle is analogous to the critical ignition and extinction phenomena that are associated with the classic analysis of a well-stirred reactor [19, 63]. The goal of this section is to present some of the key elements of the theory, related to the ignition, extinction, and the changing regimes of combustion, that are important to understand the thermal structure in the combustion of

9 particle suspensions.

1.4.2 Heterogeneous Heat of Reaction and Heat Loss Rates

Following the approach outlined by Vulis [19] and Frank-Kamenetskii [17], the overall reaction rate of a heterogeneously burning single particle in an oxidizing gas is governed by the competition between oxidizer transport to the particle surface and the kinetic reaction rate at the particle surface. For simplicity, the following assumptions are made: (i) the particle is spherical, (ii) the Biot number is small, (iii) the contributions of Stefan flow are negligible, and (iv) the particle is always in quasi-static equilibrium where the reaction rate is equal to the rate at which oxidizer reaches the surface. This approach assumes that the gaseous diffusion field between the reacting particle surface and the surrounding gas, in the boundary layer surrounding the particle, adjusts to the changing boundary conditions at the particle surface and within the bulk gas instantaneously. As will be shown later in Chapter 2, in the case where the reaction kinetics satisfy a first order equation, the effective reaction rate can be expressed in terms of the combined rates of the diffusion and kinetics as

kβ ω˙ = C (1.1) eﬀ k + β 0 whereω ˙ eﬀ is the mass of particle reacting per unit surface area per unit time, and C0 is the mass concentration of oxidizer in the bulk gas far from the particle surface. The kinetic term, k, is given by the Arrhenius expression,

k = κ0 exp(−Ea/RuTs) (1.2) where κ0 is the pre-exponential factor, Ea is the activation energy, Ru is the universal gas

constant, and Ts is the particle temperature. The term β is the mass transfer coefficient between a particle and the gas. In the absence of Stefan flow, the mass transfer coefficient takes the simple form of

ShD β = 0 (1.3) 2r

Here D0 denotes the oxidizer diffusivity at the particle-gas interface, and r is the particle radius. For a spherical particle that is stationary relative to the gas, the exact solution to the steady-state spherical diffusion equation yields a Sherwood number (Sh) equal to 2. The resulting expression for the heat release rate, assuming a stoichiometric mass coefficient of unity for the particle fuel and oxidizer, is

10 kβ Q˙ = qAω˙ = qA C (1.4) R eﬀ k + β 0 where q is the heat of reaction and A is the particle surface area. For simplicity, molecular conductivity is assumed to be the dominant mechanism of heat loss from the particle, and radiative losses are ignored. For small Biot numbers, the heat loss rate is described by the simple conductive cooling rate expression

˙ QL = hA(Ts − Tg) (1.5) where Tg is the bulk gas temperature, and the heat transfer coeﬃcient for a spherical particle

is h =Nuλ0/2r where λ0 is the interfacial thermal conductivity and the Nusselt number (Nu) is equal to 2 for conditions of a spherical particle that is stationary relative to the gas. The time dependent energy equation for a particle can be written in terms of the sum of the heat-generation source and heat-loss sink terms:

d(mT ) c s = Q˙ − Q˙ (1.6) s dt R L Similar to the analysis of a well-stirred reactor, these heat source and sink terms can be analyzed to find the steady regimes of combustion for the particle. The time derivative in ˙ ˙ Eq. 1.6 is set to zero, and the heat generation rate QR is set equal to the heat loss rate QL.It is important to note that this analysis requires the assumption of steady particle mass. The changing mass in the reaction of the particle affects the generation and loss terms through the changing radius, and, naturally, steady combustion states do not truly exist when the particle is being consumed in the reaction. Nonetheless, the assumption that the mass of the particle changes slowly compared to the time to reach a predicted state provides a way to analyze potential states of particle combustion. It is potential in the sense that a burning particle must transiently shift from its current state towards the predicted steady state, and, during that transition, the particle mass change may be sufficient such that the predicted state is never reached. However, as will be later shown in Chapter 2, the time dependent solutions taking into account particle burn out match the qualitative predictions from the simple steady-state analysis.

1.4.3 Steady-State S-Curve Analysis

One particularly illuminating variation on the analysis of heat generation and heat loss rates is the determination of steady-state particle temperature as a function of the initial temperature of the gas. This is of practical value for realistic fuel systems where dispersed

11 particles may experience a range of gas temperatures. In experimental practice, the analysis is valuable because particle combustion is most often studied by providing an initial gas temperature source such as a flame or shock tube [11, 64, 65], and particle and gas temperatures can be measured from a variety of diagnostics as outlined in Chapter 4. Equating Eq. 1.4 and Eq. 1.5, the equation can be rearranged to find the difference

between the particle temperature Ts and the gas temperature Tg: D0 k(Ts) qC0 Ts − Tg = (1.7) α k(Ts)+β(r) cgρg

Here, the expression is altered to include the thermal diﬀusivity of the gas α = λ0/ρgcg

where ρg and cg are the density and the heat capacity of the gas respectively. For simplicity of the analysis, the Lewis number Le=α/D0 is assumed to be unity. Written in this way, it is clear that the relative values of the particle-temperature- dependent k term and particle-size-dependent β term will alter the steady state temperature of the particle compared to the gas temperature. The competition between these two terms admits two asymptotic temperature states. As diﬀusive rates become fast compared to kinetic rates (β k), the temperature separation of the particle and gas approaches zero and the particle and gas will assume the same temperature in what is known as the kinetic limit. As kinetic rates become fast compared to diﬀusive rates (k β), the separation of temperatures approaches the following expression:

qC0 Ts − Tg = (1.8) cgρg The right hand side term in Eq. 1.8 is equivalent to the temperature rise in the gas if all the oxidizer in the gas is consumed in the reaction [17]. This implies that the upper limit on the particle temperature, assuming a Lewis number of unity, is the stoichiometric, adiabatic ﬂame temperature as previously stated in the section on combustion modes. This is known as the diﬀusion limit. In the case of the heterogeneously burning particle, the maximum temperature at the particle surface is expressed as

qC0 Tstoich = Tg + (1.9) cgρg The local thermal structure of the particle will depend on both the reaction regime and the mode of particle combustion. A particle burning in the kinetic limit will not appreciably exceed the gas phase temperature as shown in Fig. 1.4A, and therefore, the predicted vapor-phase modes of particle combustion discussed in the previous section cannot develop. Particles burning in the kinetic limit must burn heterogeneously.

12 In the case where the diffusion limit is reached, the predicted modes of combustion can be established. A heterogeneously burning particle will have a thermal structure similar to the schematic of Fig. 1.4B, where the particle temperature assumes the stoichiometric adiabatic flame temperature. The theoretical treatment of ignition and combustion for a vapor-phase burning particle are discussed elsewhere [63, 22] and are beyond the scope of this work. In order make a comparison to heterogeneous combustion, however, the general thermal structure of a metal particle burning in the vapor-phase diffusion limit is shown in Fig. 1.4C. Similar to heterogeneous combustion, the maximum temperature reached is the stoichiometric adiabatic flame temperature. This temperature occurs at the diffusion flame lifted from the particle surface due to the evaporating metal vapor from the boiling droplet.

Figure 1.4: Thermal structure for the two limiting kinetic and diffusion regimes for heterogeneous and vapor-phase modes of combustion. A) the kinetic limit, B) the diffusion limit for a heterogeneously burning particle, and C) the diffusion limit for a vapor-phase burning particle.

Equation 1.7 can then be solved to map the steady-state particle temperature as a function of the initial gas temperature as shown for a general case in Fig. 1.5. The temperatures are normalized to the stoichiometric, adiabatic temperature in Eq. 1.9. The interplay between the kinetic and diﬀusion rates produces the classic S-curve forms observed in the analysis of the well-stirred reactor [63, 19]. A change in gas temperature adjusts the heat loss rates for the particle, similar to the way the Damk¨ohler number adjusts the relative rates

13 of heat release and flow rates of cold reactants into the reactor. The particle temperature state is similar to the final temperature state of the reactor. Higher reaction rates push the combustion regime closer to the diffusion limit for the particle or to the intensely burning branch for the well-stirred reactor [63].

1.0

diffusion limit T = T

s stoich

0.9

r egime n

diffusio

0.8

0.7

0.6 2

0.5 stoich extinction T / s 0.4

kinetic limit

ignition T T

0.3

0.2

kinetic r egime

0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5

T /T

g stoi ch

Figure 1.5: General S-curve form for a particle.

The upper branch of the S-curve, close to the diffusion limit on Fig. 1.5, is termed the “diffusion regime”. The lower branch, close to the kinetic limit, is termed the “kinetic regime”. The S-curve analysis shows that at some critical gas temperature, labeled point 1 on Fig. 1.5, the stable temperature state of the particle shifts discontinuously from the kinetic regime up to a stable point in the diffusion regime, and is what is classically defined as particle “ignition”. This is marked by a large increase in particle temperature compared to the gas. The critical gas temperature at which ignition can occur is termed the “ignition temperature”. Ignition is defined as the discontinuous transition from a stable state in the kinetic regime to a stable state in the diffusion regime at a critical temperature. The extinction point, labeled point 2 on Fig. 1.5, is the critical point at which the reverse transition occurs. It is noted that the extinction temperature will always be lower than the ignition temperature, which implies that, after particle ignition, the burning rate is somewhat resistant to changes in the temperature of the surrounding medium. In many experimental studies that focus on the combustion of single particles, a heat source, such as a flame [11] or the reflected shock in a shock tube [64], is provided to induce ignition of the particles. The underlying assumption in many of these experiments is that,

14 after ignition, the gas temperature of the medium has little effect on the combustion rates of the particles. The degree to which this assumption is true is a function of the various physical and chemical properties of the particle and gas mixture. In particular, it is of interest to determine how particle size affects this assumption, as it is an adjustable parameter in practical fuel systems. The importance of particle size is reflected in the extensive literature devoted to the experimental investigation of the ignition and combustion characteristics of particles of different sizes [25]. The qualitative S-curves for a large range of initial particle sizes are plotted in Fig. 1.6 and provide a way to assess the expected burning behavior and the validity of the assumption that, after ignition, the particle will burn essentially independently from the surrounding gas. For larger particles shown on curve 1 of Fig. 1.6, the gas temperature at which ignition occurs is lower than smaller particles, the maximum temperature is closer to the diffusive limit, and the extinction points may not be physical*. This implies that, for a large enough particle, a significant portion of the combustion occurs near the diffusion limit, and it is more likely that the particle will not be able to reach the extinction point, until very close to burnout. After ignition, a particle burning in the asymptotic diffusion limit has a reaction rate essentially independent of any changes to surrounding gas temperature.

1.0

diffusion r egime

0.9 large r 1

0.8

3 kinetic regime

0.7 no r eal

extinction extinction 4

point point 0.6

small r ignition/extinction

0.5

degener ation stoich

no criticality T / s 0.4

T =

0.3

ignition

points

0.2

0.1

regime

kinetic

0.0

0.0 0.1 0.2 0.3 0.4 0.5

T /T

g stoi ch

Figure 1.6: S-curves for a range of particle sizes. At a critical size, ignition degenerates, and the particle reaction is limited to the kinetic regime.

As initial particle sizes decreases, as shown for curve 2 on Fig. 1.6, the ignition and ex-

*A non-physical solution would occur when the extinction temperature is a negative value in contrast to an imaginary value.

15 tinction temperature points begin to move closer together. This suggests that, after ignition, a changing gas temperature can possibly alter the burning state during combustion, shifting the particle from the diffusion regime back to the kinetic regime. It is also important to note that even after ignition and transition to the diffusion regime, the maximum particle temperature begins to depart significantly from the purely diffusive limit. Particles in this size range will burn transiently in mixed regimes of combustion (often referred to as the transition regime [66]) as both the kinetic and the diffusive rates are important in shaping the burning behavior. For very small particles shown for curves 3 and 4 on Fig. 1.6, the ignition and extinction points can coincide, leading to the degeneration of the ignition phenomenon. At some critical particle size, the criticalities of ignition and extinction, leading to discontinuous stable-state transitions, will disappear, and the particle temperature will be a continuous function of the gas temperature. In this case, the boundaries between the diffusion and kinetic regimes becomes ill-defined. After the degeneration of ignition, the particle will no longer burn as a micro-diffusion reactor with a reaction rate isolated from changes in the gas temperature. The reaction rate becomes a continuously varying function of both the gas and particle temperatures. These behaviors are, overall, more consistent with the description of the kinetic regime than the diffusion regime. Therefore, after ignition degeneration occurs, the entire “stretched” S-curve is considered to be in the kinetic regime as shown for curves 3 and 4 on Fig. 1.6. Even in this redefined kinetic regime, it is important to note that the temperature of the particle can still overshoot the gas temperature by some degree. However, this temperature separation will be a continuous function of the gas temperature, and no critical gas temperature exists.

1.4.4 Particles Burnout in Mixed Regimes

While the combustion of particles that burn in the diﬀusive or kinetic limit is relatively straight forward, the burning dynamics of a particle in the mixed regimes is not as clear. As previously stated, the steady state analysis can only provide information about potential states, since particle burnout is not taken into account. However, the S-curve analysis can still provide a picture of how a particle in the transition regime may burn by examining the predicted stable states of decreasing initial particle sizes at a set gas-phase temperature. This, in essence, simulates particle burnout at a given gas-phase temperature. Combustion in mixed regimes can occur in two scenarios. The ﬁrst scenario is shown in the left hand side of Fig. 1.7, and demonstrates a situation where at time t1 the initial gas-

16 1 1

t = t t = t

r = r r = r 1 0 1 0

1 0 1 0

ignition and transition ignition and

to diffusion regime transition to

diffusion

regime

T = T

g 0

0 0

1 1

t = t + t t = t + t

2 1 2 1

r < r r < r

2 1 2 1

mixed regime

combustion stoich stoich

combustion T T / / s s T T

0 0

1 1

t = t + t t = t + t

3 2 3 2

r < r

3 2

r < r

3 2

extinction and transition non-critical transition

to kinetic regime to kinetic regime

0 0

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

T /T T /T

g stoi ch g stoi ch

Figure 1.7: S-curves simulating particle burnout at a set gas phase temperature.(Left) Par- ticle ignites and transitions critically to the kinetic regime during burnout. (Right) Particle ignites and transitions smoothly to the kinetic regime during burnout. phase temperature is relatively low, on the order of the ignition temperature. The particle can ignite and begins to burn in the diffusion regime. As the particle begins to burn out at t2, the final temperature departs from the diffusion limit as the nontrivial dependence on kinetic rates becomes more pronounced. Finally, at t3 where the particle size has further reduced, the extinction point is reached and the particle sharply transitions to the kinetic regime and the remainder of the burnout occurs in this regime. In the second scenario shown in the right hand side of Fig. 1.7, the particle temperature is initially set high compared to the ignition temperature. Here, the particle is able to ignite and begins to burn in the diffusion regime. As the particle size decreases, the relatively high gas temperature prevents the extinction point from being reached and the transition to the kinetic regime will occur smoothly during burnout. As will be shown in Chapter 2, the solutions to the transient problem including particle burn out will yield similar results. This analysis can provide a qualitative understanding of how the particle behaves in induced temperature fields or in the self-induced temperature field resulting from the cooperative mechanism in the reacting suspension.

17 One disadvantage of the steady-state analysis is that it does not provide a feeling for the relative reaction rates at different particle sizes. The thermal S-curve analysis gives the impression that particles that ignite experience faster burning rates due to the higher temperatures realized in the diffusion limit. The real picture is more complicated due to interplay between particle burnout and the regimes of combustion and is best analyzed through the transient problem discussed in Chapter 2. The other important aspect about the steady state analysis, discussed briefly in the context of potential states above, is that the predicted stable states are reached over an infinite time scale. From a practical standpoint, most applications and experiments require ignition and combustion to occur in a relatively constrained timeframe. As such, the trends predicted in this analysis (e.g. ignition temperature as a function of particle size) may not reflect the trends observed in practice. The time dependent problem is more suitable for the examination of combustion within a set timeframe. Overall, this makes the steady state analysis less useful as a predictive tool and more important as a method of understanding the expected regimes of particle combustion.

1.5 Scope of Thesis

The papers in this manuscript-based thesis are presented in their logical order. The basic theoretical principles of heterogeneous combustion are established in simple models, then experimental techniques are developed to verify the theory. The first paper in Chapter 2 develops a model of particle suspension combustion in an adiabatic, zero-dimensional reactor. This extends the understanding of heterogeneous ignition theory explained in the introduction to a time dependent model that examines combustion of both single particles and suspensions. The second paper in Chapter 3 extends the reactor model to one-dimensional flame propagation in the suspension. The two modeling papers included in Chapters 2 and 3 have introductions which overview heterogeneous particle ignition in context of the basic Semenov-type analysis used in the classic texts of Vulis [19] and Frank-Kamenetskii [17]. However, after these papers were written, it was clear that the results of the models and experiments did not readily correlate with the classic heterogeneous ignition explanation. As the models and experiments focus on the thermal structure of the flame, it became necessary to explain heterogeneous ignition in context of steady thermal states as opposed to reaction states (See Section 2.2.2 and 3.2.2). It is for this reason that Section 1.4 includes the basic theoretical principles of heterogeneous particle combustion explained in context of a steady thermal regime analysis. The final paper in Chapter 4 applies the understanding developed in the theoretical

18 papers to the development of experimental diagnostics to examine the structure of particle flames. Key combustion characteristics are extracted using novel diagnostic approaches. The paper also highlights some of the fundamental difficulties encountered in diagnostic development for heterogeneous combustion. The laser diagnostic used in Chapter 4 is outlined in the original work in Appendix D. In the course of constructing the broadband laser technique for the paper in Chapter 4, a novel experimental method to produce broadband, ultraviolet laser light was developed, and a paper was published. This technique may be important for future diagnostic developments in particle flames as discussed in the conclusions. However, the subject of the paper is peripheral to the main subject of the thesis. It is, therefore, included as a supplemental material. The final chapter presents the conclusions and summarizes the key contributions of the thesis.

19 Chapter 2

Reaction Onset and Combustion in a Particle Fuel Suspension

2.1 Introduction

In dense suspensions of particles, the particle reaction rate can be accelerated by self-heating of the mixture. This phenomenon was originally termed the “cooperative mechanism” by Cassel [30], and subsequent literature uses terms such as “collective” effects [34, 67] or “cloud” combustion [36] to describe the same phenomenon. This thesis will refer to the self- heating phenomenon as “collective” effects for disambiguation. It is important to distinguish the collective effect from “group combustion” (unfortunately, also sometimes called “cloud combustion”), which refers to the phenomenon where burning suspensions of volatile droplets or particles form a single flame around the whole group of particles or around clusters of particles [68, 63, 69]. In this scenario, diffusion of oxidizer into the suspension dominates the combustion behavior. The basic principle of the collective effect is that heat released in the reaction is transferred from the particles to the gaseous medium in the suspension. In turn, the rising gas temperature increases the particle reaction rates, leading to faster heat release rates. The feedback mechanism causes the reaction to proceed explosively, after some delay time, due to the exponential dependence of Arrhenius kinetics on temperature. After this onset of reaction, the fuel or oxidizer is rapidly consumed. This is the same mechanism by which a premixed gaseous fuel mixture undergoes thermal explosion. The primary difference between the thermal explosion in a gaseous fuel mixture and a particle fuel suspension is the heat and mass transport processes between the particles and gas that affect the combustion behavior. There is an unfortunate naming convention for gaseous fuels to call the process of thermal

20 explosion “ignition” or “self-ignition” [16]. This can be confusing when referring to ignition of individual particles, defined in the previous chapter, which can also take place in the suspension. To avoid confusion, the thermal explosion of a particle suspension resulting from self-heating is referred to here as “reaction onset”. Two types of combustion in particle suspension flows are examined in this thesis. The first type of combustion is termed induction combustion [16]. Conceptually, this type of combustion can occur in a uniformly-heated, stationary fuel suspension within a reaction vessel. It also describes a case where a uniform particle-oxidizer flow (assuming negligible velocity slip) is moving and reacting at sufficiently high rates such that molecular heat transfer upstream of the flow can be neglected. In the laboratory reference frame, if the flow is heated instantaneously at some point x = 0, a reaction front will stabilize at some distance downstream. In a stationary coordinate system relative to a moving section of the particle flow, the reaction occurs strictly due to self-heating and particle ignition, and heat does not have time to be transported from other layers of the flow. The initial conditions of the suspension must allow the reaction to proceed to completion after a relatively short period of time. This would require that the mixture be rapidly pre-heated. For the sake of simplicity in the explanation, the gas dynamic effects are ignored. The subject of gas dynamic interaction leading to a detonation wave in the suspension is beyond the scope of this thesis and is discussed elsewhere in the literature [70]. Induction combustion neglects spatially dependent transport which greatly simplifies the theoretical treatment of the problem. In the stationary reference frame of a section of moving suspension flow, this effectively reduces to an adiabatic, zero-dimensional reactor problem. The closest practical realization of the inductive mode of combustion for particle suspensions is in metalized explosives. After detonation, the metal particle suspension is rapidly dispersed into the hot detonation product of the hydrocarbon matrix. If the onset of reaction occurs fast enough, conductive heat transfer to other parts of the dispersed suspension may be relatively inconsequential. This can also be realized by the rapid dispersal and combustion of particles behind a shock as examined in shock tube experiments [66, 65]. The induction regime of combustion is likely to also have some applicability to combustion that occurs in rocket motors or ramjet type propulsion. However, in most applications, turbulent or laminar diffusion processes in the flow are non-negligible. Heat from the reaction zone, initiated at some point in the flow, is diffused upstream into the fresh fuel mixture causing a reaction wave to propagate in the suspension. Reaction wave propagation driven by diffusion, or flame propagation, is the second type of combustion that can be realized in a particle suspension and is discussed in Chapter 3. The additional transport processes in flame propagation complicate the theoretical treat-

21 ment of the problem. It follows logically that the first steps to a theoretical understanding of flames in suspensions should be through the analysis of the simpler induction mode of combustion. Induction combustion for a heterogeneously burning particle suspension was theoretically investigated by Soviet scientists in several classic papers. The primary purpose of those papers was to examine the effects of particle size and particle concentration on the critical conditions for reaction onset [67, 71, 72, 73]. The parameters of particle size and concentration were considered of special importance since these parameters are easily adjusted in practical fuel systems. The Soviet work focused on how reaction onset was affected by the interaction of heat transport between the reacting particle and gas and the effect of self-heating from the suspension. However, absent from the literature is a clear understanding of how the particles in the suspension burn after reaction onset. The interplay between the physics of particle ignition and collective effects leads to changes in the burning behavior and thermal structure of particles during burnout. These changes in burning behavior are functions of particle size and concentration. The dynamics of particle ignition can lead to a shift in the regime of combustion as the particle size decreases during burnout, as demonstrated with the steady state analysis in Chapter 1. In the diffusion regime of combustion, the relative insensitiv- ity of the particle burning rate to changes in the gas temperature also can also lead to a non-intuitive response of the burning suspension to heat loss. The lack of understanding of particles burning in the intermediate regime in suspensions is the primary motivation for the paper included in the following section. Within the context of the simple physics of heterogeneously burning particles in a suspension, the basic reaction behaviors expected in metal suspensions can be understood. In the paper, a time dependent theoretical model is employed to examine a stationary particle fuel suspension rapidly introduced into a uniform, high temperature oxidizing gas. The theoretical treatment of the reactive suspension is analogous to the adiabatic explosion problem from classic Semenov theory for reactive gases. The model tracks the particle and gas energy and the mass of particles and oxidizer as they are consumed in the reaction. The paper is meant to be an extension of the understanding provided in the early work by the Soviet scientists. While the early studies often employed a set of simplifications meant to ease the analytical treatment of the problem, the application of analytical tools to treat the fully coupled set of time dependent governing equations is limited. Moreover, the physical intuition is often obscured by analytical approximations (e.g. [72]). For simplicity in this thesis, the equations are solved strictly by numerical methods. Despite this, it is important to distinguish this model as a “minimalistic model” as it does not predict any particular fuel system. It is, instead, meant to demonstrate the basic combustion physics when including

22 particle burnout in heterogeneously burning suspensions. A derivation of the governing equations is provided in Appendix A.

23 2.2 Publication: Publication: Reaction of a Particle Suspension in a Rapidly-Heated Oxidizing Gas, Pro- pellants, Explosives, Pyrotechnics [1]

Michael Soo, Samuel Goroshin, Jeﬀrey M. Bergthorson, and David L. Frost

Abstract

The reaction of a suspension of solid particles in a rapidly-heated oxidizing gas is relevant to metalized explosives and propellants, as well as to combustion of solid fuel-particle suspensions in premixed-gaseous-fuel clouds encountered in accidents within the mining and process industries. A simplified model is considered, using a constant-volume approximation, which assumes that non-volatile particles react heterogeneously via a one-step surface reaction. The resulting unified particle reaction rate includes both kinetic and diffusive reaction resistances. It is shown that the onset of the chemical reaction in a rapidly heated particulate suspension may occur by two different physical mechanisms. The first mechanism, realized in a dilute suspension of particles, is defined by the ignition of a single particle, i.e., by the critical phenomenon associated with the rapid transition from a kinetically- to diffusively-limited reaction regime. The second mechanism dominates the reaction onset in a dense particulate suspension and occurs in a similar manner to the reaction onset in a rapidly-heated homogeneous gas mixture, where the highly-activated reaction occurs in an explosion-like manner after some time delay and preheating. Unlike the single particle ignition phenomenon, the second mechanism lacks criticality and is not limited to particles above a certain size. The interplay between these two reaction-onset mechanisms leads to a nontrivial dependence of the total reaction time on the particle size and solid-fuel concentration within the suspension.

Keywords

Particle Combustion, Ignition, Burning Time, Metalized Explosive

2.2.1 Introduction

A suspension of reactive solid fuel particles dispersed within hot oxidizing gases may be formed, for example, following the detonation or deﬂagration of a metalized energetic material or during an accidental explosion in a coal mine or the process industries [24]. In the

24 first example, the reaction time for the energetic material is typically much shorter than the characteristic time for heat exchange between the particles and the combustion products, hence the particle suspension may be treated as if it were suddenly exposed to a hot oxidizing atmosphere. Measures of the performance of the energetic material, such as the work done by the expanding products and blast wave for an explosive, or the specific impulse of a propellant, will depend critically on the subsequent reaction delay and combustion time of the solid-fuel suspension. The overall system performance may be tailored by choosing appropriate particle sizes and other system parameters as guided by the available experimental data and predictions from theoretical models. Historically, most of the information on solid-fuel reactivity has been obtained from laboratory experiments using relatively large single particles, with sizes ranging from tens to hundreds of microns. In the experiments, the particles are typically either injected into an oxidizing flow produced by a hydrocarbon flame [11], or heated by the gas behind a reflected shock within a shock tube [74]. The experimental results are then used to construct models that are extended to predict the combustion behavior of much smaller particles and of dense particulate suspensions that are common in practical particle-fuel systems. The underlying assumption made when extrapolating to practical energetic materials is that the combustion physics on which the models are built do not change with particle size or concentration. However, the validity of these assumptions is theoretically questionable and difficult to justify, especially in light of recent experiments using nano-sized particles [75, 76] and with large particle concentrations [48, 47, 77]. Experiments in which a small amount of aluminum powder was injected into a shock tube and heated by a reflected shock have indicated that the aluminum particle combustion regime shifts from being predominantly diffusion controlled to a kinetically-controlled reaction regime when the particle size is reduced to a value below about ten microns [78]. In contrast with isolated single particles, the reaction of a particle suspension modifies the ambient gas environment by reducing the concentration of oxidizer and increasing the temperature, which in turn influences the particle combustion rate [30]. The combustion regime for a particle suspension with a given initial particle size may change as the particles burn out and may also differ from the combustion regime of an isolated particle with the same size. Recent experimental observations of flames in hybrid combustible gas-solid fuel mixtures have shown that the combustion physics of the solid suspension depends critically on the initial mass concentration of the solid fuel [47]. For the case of aluminum particles, the particle combustion regime within a methane-air-aluminum flame changes rapidly when the particle concentration reaches a critical threshold value of about 150 g/m3 [47], shifting from

25 a relatively slow oxidation regime, characteristic of an isolated single particle within the same gaseous environment, to a relatively fast reaction regime associated with a flame front. An analogous transition is also observed in suspensions of iron powder in methane-air mixtures indicating that the phenomenon is of a general nature and is not fuel specific [77]. These results also suggest that the common practice of introducing externally-defined parameters, such as ignition temperature, ignition delay,andburn time, obtained from single-particle experiments into reaction models for metalized energetic compositions may be inadequate for cases in which a dense particle suspension is formed. The goal of the present paper is to use a simple, transparent model to describe the characteristics of the combustion of fuel particle suspensions and to illustrate how distinct combustion phenomena may arise that do not occur during the combustion of a single particle. The model does not represent a comprehensive description of the combustion of a specific fuel, but rather demonstrates, in general terms, how the mass concentration of a solid fuel can influence the reaction behavior and combustion regime of the particle suspension with different particle sizes. To accomplish this, it is assumed that the particles undergo a purely heterogeneous reaction with a gaseous oxidizer through one-step surface Arrhenius kinetics which avoids the complications introduced by phase transitions, multistage kinetics, and the complex thermodynamics typical of “real” systems [79]. Through this analysis, it is theoretically shown that the general terms ignition temperature, combustion time,andig- nition delay, adapted from combustion models for large single particles, may be inadequate to describe the complex phenomena associated with the combustion of particle suspensions and for small particle sizes. The presented work illustrates the diverse combustion characteristics associated with adiabatic, constant-volume combustion of a stationary particle suspension for a particular choice of values for the reaction rate and other physical parameters associated with the solid and gaseous components. The effect of applying a rate of heat loss is also examined to illustrate how the combustion of a suspension might behave in a real system subject to, for example, expansion cooling. However, systematic parametric analysis of possible combustion phenomena that may arise in different experimental situations following detonation and deflagration events is beyond the scope of the present work.

2.2.2 Model Formulation and Results

Kinetic-Diﬀusion Reaction Rate

The general treatment of heterogeneous reactions governed by the competition between transport and kinetic rates, and the associated thermal regimes and critical nature of regime

26 change, is thoroughly investigated in the classical work of Frank-Kamenetskii [17] and further analyzed in Vulis [19]. A similar approach is applied here in a simplified form to analyze and interpret different reaction regimes for the case of a single particle and a particulate suspension. As stated above, it is assumed that oxidizer consumption rate by chemical reaction on the particle surface is described by a single-step first-order Arrhenius reaction, i.e.,

dm = C Ak (2.1) dt s

−Ea k= κ0 exp (2.2) RuTs

Here, Cs is the concentration of the oxidizer at the particle surface, A is the particle surface area, and k is the kinetic reaction rate, which depends on the activation energy Ea

and the particle temperature Ts. Following the assumption made by Frank-Kamenetskii

of quasi static equilibrium [17], the concentration of the oxidizer at the particle surface Cs is found from the balance between the rate of consumption of the oxidizer by the chemical reaction in Eq. 2.1 and the rate of transport of the oxidizer to the particle surface by molecular diffusion and convection, i.e., β ACk = Aβ(C − C ) → C = C (2.3) s 0 s s 0 k + β where β is the mass-transfer coefficient between a particle and the gas, per surface area, and C0 is the volumetric concentration of the oxidizer in the flow. By introducing the expression for Cs from Eq. 2.3 into Eq. 2.1, the unified reaction rate, taking into account both the kinetic reaction and diffusion “resistances”, may be written as

dm = ACβ Da∗ (2.4) dt 0 where Da∗ = k/(k + β). The normalized Damköhler parameter Da∗ is related to the traditional Damköhler number, Da= k/β, by the expression Da∗ =Da/(1 + Da). The value of Da∗ is close to unity when the reaction rate is controlled by the diffusion of the oxidizer (i.e., the diffusion regime where k β) and approaches zero when the rate of particle oxidation is determined by Arrhenius surface reaction rate (i.e., the kinetic regime where β k). For the sake of simplicity, the contribution of the Stefan convective flow [17] to the heat and mass transfer between the particle and the gas is neglected. The Stefan flow is negligible when the concentration of the oxidizer in the flow is low and/or the volume of

27 reacted gas oxidizing on particle surface is closely matched by the volume of the gaseous reaction products formed. In the absence of the Stefan ﬂow, the mass and heat exchange coeﬃcients between a particle and the gas take the following simple form:

ShD β = 0 (2.5) 2r

Nuλ h = 0 (2.6) 2r

Figure 2.1: Temperature and oxidizer concentration proﬁles adjacent to the particle surface for diﬀerent combustion regimes.

where the Nusselt (Nu) and Sherwood (Sh) numbers are calculated assuming spherical particles which are stationary relative to the gas (Nu=Sh=2). At Lewis numbers close to unity, and under the particle lumped-capacitance assumption, the profiles of the temperature field and oxidizer concentration around a particle are similar in all oxidation regimes, as showninFig.2.1. In the diffusion-limited stationary reaction regime, the concentration of the oxidizer at the particle surface is close to zero, and the particle temperature is close to the maximum adiabatic flame temperature of the stoichiometric mixture [19]. In the kinetically-limited regime, where diffusion is much faster than the rate of chemical reaction, the oxidizer concentration at the particle surface is close to the concentration within the bulk flow and the temperature difference between the particle and gas does not exceed one characteristic T ≤ R T 2 /E temperature interval, Δ u g0 a [19, 16].

28 Ignition of a Single Particle

The transition from the kinetically-limited to the diffusion-limited reaction regime occurs as a discontinuous jump at a critical particle temperature and is the classic definition of the ignition of a particle. The corresponding critical gas temperature is denoted the ignition temperature. The reverse discontinuous transition from diffusion-limited to kinetically-limited reaction is called extinction. The Semenov thermal ignition diagram, shown in Fig. 2.2(left) adapted for the kinetic/diffusive particle reaction rate, qualitatively illustrates both the critical ignition and extinction points for a single particle and the subsequent transition into stable regimes. It is important to note that once ignition occurs (i.e., the jump to the diffusion-limited burning regime), the particle will also undergo extinction and return to the kinetically-limited burning regime when it reaches a critical size due to particle burnout. This is not reflected in the classical particle ignition theory which is only formulated to determine ignition temperatures and stationary states and ignores the changing particle mass with time [22]. With the inclusion of time-dependent particle burnout, for large initial particle sizes, the mass of the extinguished particle is negligibly small in comparison to the initial mass. However, for decreasing initial particle sizes, extinction occurs more rapidly after ignition, leading to a non-negligible mass of the extinguished particle which burns in the kinetically-limited regime.

Property Value Property Value 1 −1 −1 λ0 0.0238 W m K T0 300 K 1 −1 −1 1 −1 −1 cg 1010.08 J kg K cs 978.56 J kg K −5 2 −1 1 −1 D0 2.01 × 10 m s κ0 70 m s ρ 1 −3 ρ 1 −3 g0 1.172 kg m s 2700 kg m Tg0 2100 K γ 1.12 1 −3 7 1 −1 C0 0.27 kg m q 3.10×10 J kg −5 2 −1 α0 2.01 × 10 m s Ea/Ru 15 000 K Table 2.1: Numerical values of gas and solid-fuel parameters.

There is a critical particle size for which the ignition and extinction temperatures coincide, illustrated in Fig. 2.2(right), leading to the existence of a limit where the particle is unable to ignite at any gas temperature. The breakdown of the ignition criticality is derived from the particle ignition conditions and reflects the dependence of the particle temperature on the gas temperature for the classical stationary model for particle ignition. Below this critical size, there is no ignition temperature as the particle temperature is a monotonically increasing function of the gas-phase temperature and is not significantly different from the gas temperature which is expected in a purely kinetically-limited combustion regime.

29 The particular set of numerical values for the thermodynamic and transport constants of the gas and solid phase used to calculate the curves in Fig. 2.2(right) are shown in Table 2.1 and are also used in the subsequent examples in this paper. The values of the thermodynamic parameters are typical for metal fuels, and with the particular values of the reaction constants chosen, both the diffusion and kinetic combustion regimes occur with micron-sized particles, where the molecular mean free path is still much smaller than the particle radius and, therefore, Knudsen diffusion effects can be neglected.

Figure 2.2: (Left) Semenov diagram of the single particle ignition and extinction. (Right) dependence of the particle temperature (Ts) on gas temperature (Tg) illustrating disappearance of the critical ignition and extinction points below some critical particle size (unstable regimes are marked with dashed lines). The temperatures are scaled by the activation temperature Ta = Ea/Ru.

The ignition temperature calculated from the balance between the rates of heat production and heat loss and their derivatives is plotted in Fig. 2.3 as a function of particle size. It is evident that the particle ignition temperature increases with decreasing particle size, reflecting the fact that the heat loss per unit particle surface area increases proportionally to 1/r. For the set of numerical parameters chosen, the heat loss rate exceeds the heat production rate for any particle temperature when the particle size is below a critical size of about 1.9 μm, and ignition, or a transition to diffusion-limited combustion, becomes impossible. If a particle below this limiting size is injected into a hot oxidizing gas it will heat slightly above the gas temperature and will undergo oxidation within the kinetic reaction regime without igniting. The time history of the temperature and radius of a single particle is plotted in Fig. 2.4 for three different particle sizes corresponding to different combustion regimes. The smallest particle undergoes a stable kinetically-limited oxidation process (Fig. 2.4A). The largest particle ignites leading to a stable diffusive combustion regime with

30 Figure 2.3: Dependence of the ignition temperature on particle size. Note that below a critical value of the particle radius of 1.9 μm ignition is impossible.

Da∗ near unity, which then quenches when the particle size is less than 10% of the initial value (Fig. 2.4C). For an intermediate particle diameter, an intermediate transient regime occurs in which the particle ignites, but extinguishes prior to reaching a fully-developed diffusive-combustion regime (Fig. 2.4B). The length of the kinetically-limited reaction “tail” depends exponentially on the ambient gas temperature and, thus, is usually very long for single particles or for a dilute suspension in which the self-heating of the mixture is negligible. In the transient combustion regime observed in Fig. 2.4B, the time for complete oxidation of the particle in the kinetic regime is usually much longer than for larger particles. It should be noted that the particle size defining the border between the diffusion- and kinetically-limited combustion regimes is not only a function of the reaction parameters but also depends on the heat exchange coefficient between the particles and gas. As an example, the detonation of metalized explosives typically results in much larger gas-particle slip velocities, in comparison with metalized propellants that undergo deflagration, resulting in a shift of the border between the two regimes towards larger particle sizes [80].

Reaction of a Dense Particulate Suspension

The dynamics of the solid-fuel reaction change completely if the initial concentration of the solid fuel is sufficient to significantly influence the gas temperature through the particle heating and subsequent heat release by the particle reaction. The effective reaction rate derived in the previous section is given in Eq. 2.7. The algebraically reduced conservation

of energy equations describing the change of the particle temperature Ts by heat from the reaction and convective heat loss to the gas is given in Eq. 2.8 and its changing radius r

31 Figure 2.4: Time histories of the tempera- Figure 2.5: Reaction of an instanta- ture (scaled to the activation temperature neously heated suspension of solid fuel Ta), radius and normalized Damköhler particles with a solid fuel concentration parameter for a single particle with differ- of 200 g/m3 with three different parti- ent sizes injected into a hot oxidizing gas cles sizes, resulting in combustion regimes flow illustrating the different possible re- that are a) kinetically-limited, b) interme- action regimes: a) a small particle (1 μm diate, and c) diffusively-limited. radius) that oxidizes within the kinetic regime, b) an intermediate sized (7 μm) particle that ignites and quenches, and c) a large particle (20 μm) that ignites and burns within the diffusive regime. 32 due to mass consumption is given in Eq. 2.9. These must be supplemented with equations describing the time dependent temperature of the gas Tg heated convectively by the particles shown in Eq. 2.10 and the depletion of the oxidizer mass concentration C due to the reaction shown in Eq. 2.11.

D0 −Ea kβ Sh κ0 exp( ) k 2r RuTs eﬀ = = − (2.7) k + β κ exp( Ea )+ShD0 0 RuTs 2r dT 3γ C T dr 3 λ T − T s = k − 3 s − 0 Nu s g (2.8) eﬀ 2 dt csρs r r dt 2 csρs r

dr γ = − keﬀC (2.9) dt ρs

dT 3 Bα g = 0 Nu(T − T )r (2.10) t ρ r3 s g d 2 s 0

dC 3B = − k r2C (2.11) t ρ r3 eff d s 0 The governing equations are written with a constant volume approximation and the assumption that the pressure increase does not change the reaction kinetics or the heat and mass transfer constants. The exclusion of the gas-dynamic effects that are specific to a particular experimental situation allows us to focus primarily on the effect of particle mass concentration B in the combustion dynamics of the suspension. The time histories of the particle temperature, radius, concentration and the normalized Damköhler parameter are determined by numerically integrating equations 2.7—2.11 with the assumption that the initially cold particle suspension is suddenly exposed to a hot gas. The results are shown in Fig. 2.5 for the same three particle sizes considered for the reaction of a single particle in Fig. 2.4. For the purposes of this illustration, we assume that the fuel suspension is lean with a mass concentration of 200 kg/m3, corresponding to about two-thirds of the stoichiometric value and with an initial gas temperature of 2100 K, which is characteristic of hydrocarbon flames. As seen from Fig. 2.5A for a small particle size, in contrast with a single particle, the dense suspension of kinetically-reacting particles undergoes a rapid reaction onset and a corresponding sharp temperature increase. Although this behavior has the appearance of an ignition event, the rapid rise of the reaction rate of the small particles within the suspension lacks criticality as the value of the Damköhler parameter does not exceed 0.2 and, thus, the combustion process remains close to the kinetically-limited reaction regime. The nature of

33 the sharp reaction onset in this case is analogous to that of a reactive gas mixture heated under adiabatic conditions. After a certain delay time during which the mixture slowly oxidizes, the reaction proceeds rapidly in an explosion-like manner reflecting the self accel- erating nature of Arrhenius kinetics at high activation energies and large heats of reaction. The adiabatic reaction delay time in this case can be estimated using the same well-known formula for gaseous mixtures if the reaction delay time is significantly longer than the particle heating time [16]. For large particles, which have the ability to ignite and burn within the diffusion-limited regime, there is a complex interplay between the onset of the reaction due to the particle cloud effect and the individual particle ignition (Fig. 2.5C). The calculations indicate that, for any given particle concentration, the cloud effect leading to a reduction in the reaction delay diminishes with increasing initial particle size, where eventually the suspension combustion will behave in a similar fashion to the single particle. For suspensions of intermediate-sized particles, the diffusive-combustion regime is followed by extinction and kinetically-controlled combustion of the remaining fuel similar to the case of a single particle (Fig. 2.5B). However, unlike a single particle, the transition to the kinetically burning branch of the suspension does not result in a long kinetic-reaction particle-burnout “tail” but proceeds rapidly due to much higher gas temperatures. An important parameter for practical particle combustion applications is the sum of the delay and reaction times which indicates how rapidly the solid fuel energy is released for a particular energetic system. This is denoted for sake of brevity as the “combustion time”. The combustion time calculation is limited to the point when 95% of the particle radius is consumed in order to exclude the disproportional contribution of the long, kinetic-reaction “tail” during which a relatively small amount of fuel is consumed after particle extinction. The dependence of the combustion time on particle size is plotted in Fig. 2.6 for different solid fuel concentrations. From Fig. 2.6, the overall combustion time of a single particle initially rises with particle radius, but then sharply drops after the initial particle size crosses a critical threshold at which point the particle burns primarily in a diffusion-limited regime. After this drop, the combustion time slowly increases for increasing particle sizes although this is not readily apparent on the semi-log vertical scale on Fig. 2.6. The fraction of the particle mass that burns in the fast diffusive regime increases with particle size within this size range, compensating for both the increased heating and reaction times. With an increase in fuel concentration within the suspension (and therefore gas temperature), the advantage of the diffusion-combustion regime diminishes and, for high particle concentrations, the overall combustion time increases monotonically with particle size.

34 Figure 2.6: Dependence of combustion time on particle size at diﬀerent solid fuel concentrations.

The same interplay between kinetically- and diffusively-limited combustion regimes at different fuel concentrations explains why the combustion time dependence with respect to concentration intersects for two different particle sizes, as shown in Fig. 2.7. At low fuel concentrations and, thus, lower gas temperatures, the large particles have an advantage. They can ignite and burn faster in the diffusion regime than kinetically-reacting particles at lower temperatures. At high fuel concentrations and, thus, high gas temperatures, the advantage of the diffusive combustion disappears and small particles react more rapidly due to their higher specific surface area. Figure 2.7 also exhibits an initial sharp drop in the overall combustion time with an increase in the fuel concentration for both particle sizes. The combustion time for a suspension of 4 μm particles reacting primarily in the kinetic mode decreases by about an order of magnitude in contrast with a single particle. The exponential dependence of the reaction delay time on the gas temperature, which is proportional to the fuel concentration, is a clear demonstration of the “cloud” or “collective” effect on the combustion process. The drop in combustion time will be even stronger if the increase of the heat and mass transfer coefficients with temperature and pressure are taken into account. Accounting for the cloud effect may explain the extremely low combustion times observed experimentally for aluminum particles burning within the products of metalized explosives that are difficult to reconcile with the much longer times derived from experiments with single particles [81]. The recently proposed hypothesis [82, 83] that the short reaction times are due to the cracking of the

35 Figure 2.7: Dependence of combustion time on concentration of the solid fuel for particle sizes of 4 and 10 μm. protective alumina shell by a shock wave resulting in faster oxidation kinetics cannot explain why combustion times of the same order are observed during the detonation of aluminum suspensions in air where the shock pressure is lower by almost three orders of magnitude [84]. In addition, it is well known that protective oxide ﬁlms can swiftly re-establish on the metal surface if the conditions are not optimum for the particle ignition [85].

Reaction of a Suspension in a Cooling Gas

In a real experimental situation, the system is non-adiabatic and the temperature of the gas decreases with time due to heat loss or may fall sharply due to the expansion of the gas products as in the case of a detonation event. Modeling the complex ﬂow dynamics of explosive formulations is outside the scope of this paper but, in the context of the present model, the general considerations of what may occur during the combustion of a suspension in such a system can be examined through the addition of a simple convective cooling rate of the gas phase as shown in Eq. 2.12 which is a modiﬁed form of Eq. 2.10, i.e.,

dT 3 Bα g = − 0 Nu(T − T )r − h (T − T ) (2.12) t ρ r3 s g w g 0 d 2 s 0

Here, the hw is an arbitrary heat transfer coeﬃcient, and T0 is the ambient temperature. The addition of the cooling rate in this simple form is not connected to any particular experimental situation, but serves to qualitatively demonstrate the type of analysis that can

36 be performed to find optimum particle sizes for a given particle loading. The introduction of the cooling rate affects the ability of the particles to react to completion before the gas temperature drops below the point at which the reaction rate is negligible. The cooling of the gas may also lead to a longer reaction onset time in which case the particles will never react within the given time frame. Among several parameters, including the initial gas temperature and reaction constants, the major factors affecting the reaction completeness within a given time are the particle size and mass concentration of the fuel within the suspension. Although the particle size has long been recognized as a critical parameter, the fuel concentration within the suspension is typically neglected. To analyze this situation, a criterion is set such that if burnout of 95% of the particle radius does not occur within the assigned time frame, then it is considered to be unburned. The resulting combinations of particle size and concentration for which particle burnout occurs (shaded regions) are plotted in coordinates of particle size vs. mass concentration in Fig. 2.7 for three different cooling rates. Particle sizes below 1 μm were not considered in the calculation.

Figure 2.8: The transformation of the range of particle sizes and concentrations (shaded area) where 95% of the particle radius is burned within 500 ms with a 20% and 40% increase of the cooling rate.

This is by no means a comprehensive analysis of the model as the set reaction time window and cooling rates are arbitrary, but it does show that the outcomes are not necessarily

37 intuitive. The resulting particle size/concentration combinations for both the lowest and intermediate cooling rates indicate that at lower concentrations, it is actually more beneﬁcial to use larger particle sizes since the smaller, more kinetically-limited particles do not burn to completion. In the intermediate case, at low concentrations, there is an isolated “island” of large particle sizes that burnout. This island continually shrinks with increasing cooling rate until, for a certain range of concentrations, there will only be a very small range of optimum particle sizes. In the case of the highest cooling rate, it is seen that only relatively small particle sizes have the ability to react to completion.

2.2.3 Conclusion

The primary conclusion of the present work is that the onset of the chemical reaction in a rapidly heated particulate suspension may occur by two different physical mechanisms. The first mechanism, realized in a dilute suspension of particles, is defined by the ignition of a single particle, i.e., by the critical phenomenon associated with the rapid transition from a kinetically- to diffusively-limited reaction regime. The second mechanism dominates the reaction onset in a dense particulate suspension and occurs in a similar manner to the reaction onset in a rapidly-heated homogeneous gas mixture, where the highly-activated reaction occurs in an explosion-like manner after some delay and self-heating. Unlike the ignition phenomenon, the second mechanism lacks criticality and is not limited to particles above a certain size. The interplay between these two reaction onset mechanisms leads to a nontrivial dependence of the total reaction time on the particle size and fuel concentration within the suspension.

Acknowledgments

Special thanks to Keishi Kumashiro for his help in the formulation of the governing equations. Support for this work was provided by the Defense Threat Reduction Agency under contract HDTRA1-11-1-0014 (program manager Suhithi Peiris), the NSERC CREATE on Clean Combustion Engines, and a McGill Engineering Doctoral Award.

38 Chapter 3

Flame Propagation in Particle Suspensions

3.1 Introduction

With the possible exception of metalized explosives, a direct implementation of the purely induction mode of combustion is not likely to be encountered in practical applications especially where controlled energy release is required. Although the induction mode of combustion for gaseous fuels was found to be hydrodynamically stable [16], from an engineering perspective, it is technically impractical. To maintain a stable reaction front in the induction regime for high speed flows, the reaction onset in the suspension must carefully controlled. The exponentially dependent nature of Arrhenius kinetics on the initial temperature demands precise regulation of the initial conditions to maintain a steady reaction front. In practice, this would be difficult to maintain. The most encountered type of combustion in metal suspensions will be a transport driven reaction wave, or flame. The transport of heat from the reaction zone into the fresh mixture permits the propagation of the flame even if the mixture is initially cold. It should be noted that there is also the possibility of an intermediate regime between the induction and the flame propagation types of combustion. The intermediate regime was proposed by Zel’dovich for gaseous fuels, and a similar principle may apply to combustion of heterogeneous fuel suspensions [16]. However, this is beyond the scope of the thesis. For gaseous fuel flames, the most typical method of transport is turbulent or laminar diffusion. This also applies to flames in suspensions of particles [26, 86]. Additionally, for particle flames, radiative transport can also play a role in flame propagation. Unlike gaseous fuel flames which only emit and absorb radiation at discrete wavelengths, condensed-phase

39 species can emit and absorb broadband radiation. When the radiation absorption path length becomes comparable to the diffusion driven flame thickness, then the role of radiation in driving the flame cannot be neglected [31]. However, such conditions are unlikely to occur at the typical stoichiometric concentrations, pressures, and particle sizes typically encountered in metal flames at the laboratory scale. The radiation absorption path length approaches tens of centimeters while the flame thickness is on the order of millimeters [86, 87, 88]. This implies that for small-scale flames, radiation acts as a heat loss mechanism. In large-scale flames, such as those investigated in [86], the flame and radiative transport scales can be separated, and the influence of radiation is to preheat the fresh fuel mixture ahead of the diffusion driven flame. In this sense, despite some possible disagreement in the literature [89, 90], radiation should only play a secondary role and is neglected in this thesis. For flame propagation, one of the most valuable measurable parameters to characterize the combustion is the laminar burning velocity. The laminar burning velocity results from the combination of reaction onset, transport between the particle and gas, and diffusive transport in the suspension flow. The laminar burning velocity is somewhat of an idealized case as it is experimentally difficult to remove the influence of effects such as turbulence, flame stretch, and heat loss [88]. Nevertheless, the trends in laminar burning velocity as a function of the various thermodynamic and physical parameters provide an insight into the fundamental nature of self-sustained combustion in particle fuel systems. Flames in particle suspensions have an important secondary parameter that gaseous fuel flames lack. The role of transport between the particles and gas is not generally reflected by typical combustion characteristics such as burning velocity or flame thickness. However, as demonstrated in the introduction and in Chapter 2, the dynamics of transport lead to different thermal regimes of combustion. The kinetic and diffusive regimes are functions of the heat and mass transport rates between the particles and gas and are characterized by changes in the temperature state of the particles relative to the gaseous medium. This indicates that the thermal structure is also an important measurable parameter as it is a direct consequence of the regime of combustion. It is for this reason that the paper in the following section focuses not only on the burning velocity but also the thermal structure of flames in suspensions. An introduction into the experimental study of thermal structure in particle flames is covered in Chapter 4. In many ways, the results of the zero-dimensional model in Chapter 2 provide insight into the behavior of flame propagation. Similar to gaseous flames, the burning velocity in the suspension is expected to a follow a simple relation, obtained from dimensional analysis, formulated as Sf ∝ α/τr (3.1)

40 Where Sf is the burning velocity, α is the thermal diffusivity and τr is the characteristic reaction time. From the model in Chapter 2, it is clear that the characteristic reaction time changes with both concentration and particle size. For this reason, it is expected that the interplay between the kinetic and diffusive regimes of combustion seen in the burning time behavior should also manifest similarly in the flame behavior. There is no clear understanding in the literature of the fundamental behavior of flame propagation in suspensions of particles that can burn in mixed regimes of combustion. The dynamics between the particle reaction regimes and the diffusion of heat and gaseous species in the suspension flow can lead to non-intuitive behavior of the thermal structure and burning velocity of the flame. This lack of understanding motivates the need for a simple model of flame propagation in heterogeneously burning suspensions. In the paper presented in the next section, the inductive model of combustion from Chapter 2 is extended to model a time dependent, adiabatic, one-dimensional laminar flame including the dynamics of heterogeneous ignition and combustion. The governing equations are again treated numerically. As stated previously, the lack of asymptotic analysis or analytical solution should not diminish the importance of the physical understanding that can be obtained from this simple model. A derivation of the governing equations is provided in Appendix B. The flame study is not meant to be parametric and focuses mainly on the effects of initial particle size and concentration. The model also bears some resemblance to previous models of flame propagation in coal suspensions in the literature [26, 91]. The models use the principles of heterogeneous combustion of particles, outlined in Section 1.4, as the basis of the particle reaction. However, those models add several assumptions in an attempt to simulate combustion of coal, specifically. In the results of these models, the general physics resulting from the heterogeneous reaction and self-heating of the mixture is obscured. The simple model presented here is compared to the classic Zel’dovich expression for burning velocity [16] for a kinetically limited reaction and the Goroshin expression for burning velocity [32] in a diffusion limited particle suspension. The model is meant to bridge these two asymptotic models to demonstrate the behavior of particles burning in the intermediate regime within aflame.

41 3.2 Publication: Thermal Structure and Burning Ve- locity of Flames in Non-volatile Fuel Suspensions, Proceedings of the Combustion Institute [2]

Michael J. Soo, Keishi Kumashiro, Samuel Goroshin, David L. Frost, Jeﬀrey M. Bergthorson

Abstract

Flame propagation through a non-volatile solid-fuel suspension is studied using a simplified, time-dependent numerical model that considers the influence of both diffusional and kinetic rates on the particle combustion process. It is assumed that particles react via a single- step, first-order Arrhenius surface reaction with an oxidizer delivered to the particle surface through gas diffusion. Unlike the majority of models previously developed for flames in suspensions, no external parameters are imposed, such as particle ignition temperature, combustion time, or the assumption of either kinetic- or diffusion-limited particle combustion regimes. Instead, it is demonstrated that these parameters are characteristic values of the flame propagation problem that must be solved together with the burning velocity, and that the a priori imposition of these parameters from single-particle combustion data may result in erroneous predictions. It is also shown that both diffusive and kinetic reaction regimes can alternate within the same flame and that their interaction may result in non-trivial flame behavior. In fuel-lean mixtures, it is demonstrated that this interaction leads to certain particle size ranges where burning velocity increases with increasing particle size, opposite to the expected trend. For even leaner mixtures, the interplay between kinetic and diffusive reaction rates leads to the appearance of a new type of flame instability where kinetic and diffusive regimes alternate in time, resulting in a pulsating regime of flame propagation.

Keywords

Two-phase ﬂames, particle combustion regime, ignition, extinction, instability

3.2.1 Introduction

Understanding the mechanisms controlling ﬂame propagation in suspensions of non-volatile particles is crucial to obtaining eﬃcient combustion of metallized propellants, slurry fuels,

42 pulverized coal, and powdered metals as carbon-free chemical energy carriers [5]. It is also necessary for the mitigation of catastrophic explosions in coal mines or in process industries that involve handling metallic dusts and other combustible solid powders. Like gas flames, flames in particulate suspensions at the laboratory scale are primarily driven by molecular heat diffusion and have comparable burning velocities [5]. Nevertheless, they exhibit several significant differences in their structure and behavior from homogeneous flames due to their multiphase nature. The main distinctive feature of a flame in a solid suspension is the ability of particles to ignite–that is, to transition from a combustion regime limited by reaction kinetics to a regime limited by diffusion of the oxidizing gas towards the particle surface, or in the case of evaporating particles, towards the micro-flame enveloping each individual particle. After ignition, the temperature of the particle or micro-flame can exceed the gas temperature by several hundred degrees, often surpassing the adiabatic flame temperature for fuel-lean mixtures. The particle combustion rate in the diffusion combustion regime is a weak, non-Arrhenius, function of gas temperature. Unlike gas flames, the width of the flame reaction zone in particle suspensions can span a large temperature range and can be comparable to, or even exceed, that of the preheat zone [32]. The existence of diffusion micro-flames within a global flame-front (in effect, flames within the flame), which are insensitive to the bulk gas temperature, makes dust flames resistant to heat loss [5, 92, 17, 19] and also serves to maintain a constant burning velocity with increasing fuel concentration in fuel-rich mixtures [33]. The ability of particles to ignite, together with low ignition temperatures, may result in much wider flame propagation limits for particle suspensions than for gaseous fuels. Despite the overall qualitative understanding of the crucial role of the particle combustion regime on burning velocity and thermal structure, the theoretical description of flames in particulate suspensions has been limited to simple semi-empirical models that postulate either purely diffusion or purely kinetic modes of particle combustion [33, 52]. The diffusive combustion models presume that particles within the suspension ignite and transit to the diffusion regime instantaneously when they reach the ignition temperature of a single, isolated particle. The common assumption is that, after ignition, the particle within the suspension will have a combustion time equal to that of an isolated single particle. Using this approach, particle ignition temperature and combustion time are considered to be external parameters that are independent of the flame-propagation mechanism. As such, they often are taken from experiments with individual particles or calculated using theoretical models for single particle combustion. These assumptions are useful for estimation, but are, in general, not justified and may lead to erroneous predictions. In reality, the particle ignition temperature, particle reaction time, and the actual regime of particle combustion are all

43 characteristic values of the flame propagation problem directly linked to the burning velocity eigenvalue. Moreover, the particle combustion regime may alternate throughout the flame. For example, the particle may start to react in a kinetically limited regime, then transit to a diffusion-dominated combustion mode, before returning to a kinetics-dominated mode [92]. As a result, a non-negligible fraction of the particle mass may be consumed during both diffusive and kinetic combustion, leading to a complex dependence of the burning velocity on particle size and concentration, as demonstrated in this study. In this paper, the thermal structure of a flame in a particulate suspension is investigated using a simplified, transparent model that assumes that non-volatile solid fuel particles react via a single-step Arrhenius surface reaction with gaseous oxidizer delivered to the particle surface by diffusion. Besides incorporating heterogeneous reaction kinetics, the model does not impose the particle combustion mode or any other external combustion parameters. The flame propagation problem is solved numerically in a non-stationary formulation developed by Spalding [93]. This approach avoids the difficulties inherent in a steady-state formulation and permits the investigation of flame stability, which has led to the discovery of a new type of oscillating flame in heterogeneous mixtures.

3.2.2 Model Formulation

Combined Kinetic-Diﬀusive Reaction Rate

Following the quasi-stationary approach of Frank-Kamenetskii [17], the overall reaction rate per unit surface area of a particle in an oxidizing gas, accounting for both kinetic and diffusion “resistances” can be written as kβ ω˙ = γ C (3.2) eff k + β 0 whereω ˙eff is the particle mass consumption per unit surface area, γ is the stoichiometric coef-

ﬁcient, and C0 is the concentration of oxidizer in the bulk gas far from the particle surface [1].

The kinetic term, k, is the overall Arrhenius surface reaction rate, k = κ0 exp (−Ea/RuTs), where κ0 is the pre-exponential factor, Ea is the activation energy, and Ru is the universal gas constant. β is the mass transfer coefficient between a particle and the gas. For simplicity, the contribution of Stefan flow to the heat and mass transfer between the particle and the surrounding gas is assumed to be negligible. Stefan flow is small when the molecular weight of the reaction products is close to that of the consumed oxidizer or when the initial oxidizer concentration is relatively low [17]. In the absence of Stefan flow, the mass transfer coefficient takes the simple form, β =ShDI/2r. Here, DI denotes the oxidizer diffusivity at the particle-gas interface, and r, the particle radius. For a spherical particle

44 that is stationary relative to the gas, the exact solution to the steady-state diffusion equation yields a Sherwood number, Sh, equal to 2 [17]. The reaction rate in Eq. 3.2 can then be written in terms of the normalized Damköhler number, Da∗, which is related to the traditional Damköhler number, Da = k/β,byDa∗ = Da/(1 + Da) [1]. The resulting expression for the heat release rate per unit particle surface area is: ∗ QR = qω˙ eff = qγβDa C0 (3.3) where q is the heat of reaction. This formulation naturally incorporates the two limiting kinetic and diffusive regimes. In a kinetically controlled regime (β k), Da∗ approaches zero. In the diffusion-limited regime (k β), the reaction rate is primarily limited by diffusive transport of oxidizer, and Da∗ approaches unity.

Ignition and Combustion of a Single Particle

The reaction of a suspension of particles within the flame is fundamentally different from the reaction of an isolated single particle [1]. However, mapping the reaction regimes for a single particle is crucial to understanding this difference and interpreting the flame structure in a suspension. The interplay between kinetic and diffusion reaction rates of heterogeneous chemical reactions leading to the processes of ignition and extinction was first investigated by Frank-Kamenetskii [17] and then analyzed further by Vulis [19]. Their analyses are adapted here in a simplified form to interpret the reaction behavior of a single particle injected into a hot oxidizing gas. The modified Semenov diagram shown in Fig. 3.1 plots the reaction heat release rate that accounts for both kinetic and diffusive rates as a function of the particle temperature,

Ts. At low temperatures, the reaction rate increases exponentially with Ts due to Arrhenius kinetics. At high temperatures, Da∗ approaches unity and the reaction rate in Eq. 3.3 becomes practically independent of temperature. If the heat loss from radiation is negligible, as is the case for moderate temperatures and small particles, then the heat loss rate is proportional to the temperature diﬀerence between the particle and gas and is plotted in

Fig. 3.1 as straight lines: QL = h(Ts − Tg). Here h is the heat transfer coeﬃcient, given by h =NuλI/2r,whereλI is the thermal conductivity of the gas at the gas-solid interface. The Nusselt Number, Nu, is equal to 2 for a spherical particle that is stationary relative to the gas [17]. At some critical gas temperature, known as the ignition temperature, the heat loss and the heat release curves become tangent (point I on Fig. 3.1). The particle starts to accu-

45 r

1 r

diffusion limit

i Heat production/loss Heat

I K

r Q cri t

0 r

g0 T

Figure 3.1: Modified Semenov diagram illustrating particle ignition (I) at some critical tem- T r perature ( g0 for 2) and transition to near the diffusion limit (D). Extinction (E) occurs as the particle shrinks and undergoes the reverse transition to the kinetic limit (K). Stable states are shown by (•) while unstable states are denoted by (◦). The heat production curve is solid, and the heat loss curve is dashed. Inset: ignition temperature as a function of particle size showing critical radius below which ignition is impossible [1]. mulate heat and promptly transitions from a predominantly kinetic combustion regime to a predominantly diffusive regime with temperatures close to the adiabatic temperature of the stoichiometric mixture (point D). Once ignited, the burning particle inevitably extinguishes when its radius reduces to the point (r1) where the heat loss and heat release curves become tangent again, albeit at a higher particle temperature (point E). After extinction, the particle will transit to a kinetic burning regime (point K). For large particles, the extinction radius is small in comparison to the initial particle size, and, thus, the residual particle mass at extinction is negligible since m ∼ r3. For smaller particles, the mass of the extinguished particle may not be negligible relative to the initial particle mass. If the initial particle size is reduced even further, the slope of the heat loss curve becomes so steep that tangency of the heat release and heat loss curves becomes impossible. Below this critical radius, the particle cannot ignite at any gas temperature (see inset in Fig. 3.1) [1]. The particle temperature history for three different initial sizes injected into a hot oxidizing gas is illustrated in Fig. 3.2. The first case (r0 rcrit) corresponds to a large particle that undergoes heating, then ignites and reacts to completion almost entirely in the diffusive regime. The second case (r0 ∼ rcrit) corresponds to an intermediate-sized particle that ignites but extinguishes soon afterwards without achieving full-fledged diffusive combustion. After extinction, the particle continues to react in the kinetic regime at a temperature close

to the gas temperature. The third case (r0

46 T r >> r

0 crit

r ~ r

0 crit

r < r

0 crit

Figure 3.2: Particle temperature and radius history of a particle injected into a hot oxdizing gas at temperature Tg burning in a predominantly diﬀusive combustion regime (r0 rcrit), an intermediate regime (r0 ∼ rcrit), and a kinetic combustion regime (r0

This analysis shows that the particle reaction regime is a strong function of the particle radius as well as the surrounding temperature and oxidizer concentration. In a flame, a reacting particle with constantly changing radius is exposed to rising temperature and falling oxidizer concentration fields that are functions of the burning velocity. Therefore, postu- lation of the particle reaction regime and “ignition temperature” and “combustion time” parameters is, generally, unjustified in flame modeling. As demonstrated in this paper, only by solving the flame propagation problem as a whole can the particle combustion regime and other fundamental flame parameters be determined.

Governing Equations and Numerical Method

The flame propagation problem is cast in a time-dependent formulation, based on the method developed by Spalding [93], and solved numerically. This method has been used to analyze gas-phase flames [94]. Smoot et al. also used this method to model flames in coal dust-air mixtures [26], but no attempt was made to draw general conclusions beyond the specifics of coal combustion. The model presented here employs several simplifying assumptions: (i) the solid fuel can be modelled using the continuum approach described in [95, 96], (ii) the velocity slip between the particles and gas is negligible, (iii) the molecular weight and heat capacity of the products and oxidizer are similar, and (iv) the solid particles do not undergo any phase transitions [97]. All of these are second-order effects. The governing equations for the flame are simplified by introducing a density-weighted x x x x ρ x¯ /ρ dx¯ coordinate, , which is related to the physical coordinate, ,by = 0 [ g( ) g0 ] [93].

47 In this coordinate system, the continuity equation is automatically satisfied, and advection effects induced by thermal expansion of the gas are absent. Once found, the solution in x-space can be transformed into x-space, as described by Margolis [98]. For simplicity, the temperature dependences of transport and kinetic properties are chosen to eliminate thermal expansion effects. With these assumptions, the governing equations can be reduced to the following expressions for the gas-phase temperature (Eq. 3.4), particle

temperature (Eq. 3.5), particle mass, ms (Eq. 3.6)), and normalized oxidizer mass fraction,

Y = Yox/Yox,0 (Eq. 3.7). The combined kinetic-diffusion reaction rate,ω ˙ eff,isgivenby Eq. 3.2. ∂T ∂2T ρ c g = λ g + N Ah(T − T ) (3.4) g0 g ∂t 0 ∂x2 0 s g ∂ c (m T )=Aqω˙ Y − Ah(T − T ) (3.5) s ∂t s s eff s g ∂m s = −Aω˙ Y (3.6) ∂t eff ∂Y ∂2Y N ω˙ A = D − 0 eff Y (3.7) ∂t 0 ∂x2 γρ g0

Here, A is the instantaneous particle surface area, cs and cg are the solid and gas-phase specific heats, D0 and λ0 are the bulk mass diffusivity and thermal conductivity of the gas, ρ N respectively, g0 is the initial gas density, and 0 is the initial particle number density. The numerical values for the thermodynamic, transport and reaction properties used are given in Table 1. These values are similar to those for an aluminum particle in air, but the model is not meant to describe aluminum combustion in particular. The reaction parameters are chosen such that the switch from kinetics- to diffusion-dominated particle combustion can be observed in the micrometer particle size range. The equations were numerically integrated in MATLAB by the method of lines using a second-order finite difference formulation and a multi-step, variable order, implicit time integration scheme [99]. The integration was performed over an effectively semi-infinite domain with an adiabatic and impermeable hot wall condition [98]. Because long-term solutions were of interest, the exact initial conditions used in this study turned out to be immaterial [16].

3.2.3 Results and Discussion

Flame Thermal Structure

After an initial transient period, a steady-state ﬂame, propagating at a constant speed,

typically develops. As discussed below, for some values of φ and r0, no steady-state solution is

48 1 −1 −1 λ0, λi 0.02 W m K Ea/Ru 15 000 K 1 −1 −1 1 −1 −1 cg 1010 J kg K cs 979 J kg K −5 2 −1 1 −1 D0, DI 2.0 × 10 m s κ0 70 m s ρ 1 −3 ρ 1 −3 g0 1.17 kg m s 2700 kg m γ 1.12 q 3.10×107 J1kg−1 1 −3 C0 0.27 kg m Table 3.1: Numerical values of gas and solid-fuel parameters.

observed. Steady-state flame profiles of gas and particle temperatures, oxidizer mass fraction and particle radius are shown for the fuel-lean case (φ =0.5) in Fig. 3.3 for three different

initial particle sizes corresponding to overall kinetic (r0 =1μm), diﬀusive (r0 =10μm), and intermediate (r0 =4μm) particle combustion regimes.

0.6 0.6 0.6

r = 1 m r = 10 m r = 4 m

0.5 0.5 0.5 0 0 0

0.4 0.4 0.4

a 0.3 0.3 0.3

/ T

0.2 0.2 0.2 T

0.1 0.1 0.1 T

0.0 0.0 0.0

1.0 1.0 1.0 Y

0.5 0.5 0.5

and r

0 r /

r Y

0.0 0.0 0.0

q 1.0 1.0 1.0

Da*

0.5 0.5 0.5 q

0.0 0.0 0.0 Da* and

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

x (mm) x (mm) x (mm)

Figure 3.3: Flame structure profiles for three different particle sizes in a fuel-lean flame (φ =0.5). Top: gas and particle temperature non-dimensionalized by Ta = Ea/Ru; Middle: normalized oxidizer mass fraction, Y , and particle radius, r/r0; Bottom: reaction heat release rate,q ˙,andDa∗. The arrow indicates direction of propagation.

For small particle sizes, the flame structure is similar to that of a homogeneous gaseous flame. The gas and particle temperatures remain close to each other throughout the flame, and the reaction zone is thin and located near the point where the temperature reaches the adiabatic flame temperature of the mixture. In contrast, the flame structure for large particles is significantly different. After ignition, the particles attain a temperature close to the adiabatic flame temperature of the stoichiometric mixture, which greatly exceeds that of the fuel-lean mixture. In addition, the reaction zone is wide, encompassing a large gas temperature range. There is also a slight lag of the particle temperature profile behind the gas temperature in the flame preheat zone. In the case of intermediate particle sizes, particles ignite, but then extinguish before

49 achieving full-fledged diffusion-limited combustion. Thus, the particle temperature surpasses the adiabatic flame temperature of the mixture, but does not reach that of the stoichiometric mixture. In contrast to the reaction of a single particle of the same size, the kinetic reaction time after the transition back to the kinetic combustion mode is relatively short due to the high flame temperatures and, thus, does not significantly increase the overall combustion time. Flames in fuel-rich suspensions, as shown in Fig. 3.4, have a different structure to that of fuel-lean mixtures, since the limiting reactant changes from the solid fuel, which has a negligible mass diffusivity, to the oxidizing gas, which readily diffuses across the flame. The primary difference is that the temperature of the particles after ignition does not exceed the adiabatic flame temperature of the mixture. The reason for this is that the temperature separation between particles and gas in the diffusive combustion regime is proportional to the concentration of the oxidizer in a particular location, which falls to zero as the temperature rises. Due to complete oxidizer consumption at the end of the combustion zone and, therefore, a negligibly small reaction rate, the value of Da∗ remains constant after the oxidizer is depleted.

0.6 0.6 0.6

r = 1 m r = 4 m r = 10 m

0.5 0.5 0.5

0 0 0

0.4 0.4 0.4 a 0.3 0.3 0.3

T / T

T 0.2 0.2 0.2

T 0.1 0.1 0.1

0.0 0.0 0.0

1.0 1.0 1.0 Y

0.5 0.5 0.5

and r

0 r /

r Y

0.0 0.0 0.0

1.0 1.0 1.0

Da*

0.5 0.5 0.5 q

0.0 0.0 0.0 Da* and

-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4

x (mm) x (mm) x (mm)

Figure 3.4: Flame structure proﬁles for three diﬀerent particle sizes in a fuel-rich mixture (φ =1.5). The same variables are shown as in Fig. 3.3.

Burning Velocity

The burning velocity is calculated based on the ﬂame-front displacement as a function of time. The burning velocity is plotted in Fig. 3.5 as a function of initial particle size for diﬀerent equivalence ratios. The general trend for stoichiometric, rich, and somewhat lean mixtures is that the burning velocity increases monotonically with decreasing particle size.

50 This is simply due to the increasing speciﬁc reaction surface area with decreasing particle size. In contrast, this dependence becomes non-monotonic for very lean mixtures. For equivalence ratios below φ =0.4, there is also a region where the burning velocity curves split into two separate branches between which no steady solution exists.

4 (cm/s) b u

0 2 4 6 8 10

r ( m)

Figure 3.5: Burning velocity, ub, as a function of particle size for diﬀerent equivalence ratios. Hatched area corresponds to regions where unstable ﬂame propagation is observed.

The nontrivial dependence of the burning velocity on particle size in lean mixtures is the result of the intricate interplay between the kinetic and diffusive particle combustion modes. Because of the relatively low flame temperatures in lean mixtures, the combustion time of a larger particle reacting diffusively can be shorter than that of a smaller particle reacting kinetically. This behavior is counter-intuitive, but is a direct consequence of the ability of large particles to ignite and burn with temperatures much higher than the adiabatic flame temperature of the lean mixture. The appearance of “flames within the flame”, results in higher reaction rates for suspensions of larger particles that offset, and potentially even overcome, the reduced specific reaction surface area as compared to finer suspensions. This behavior can be seen first as a plateau in the dependence of burning velocity on particle radius (Fig. 3.5, φ =0.4) and then a decline as particle size decreases (Fig. 3.5, φ<0.4). It can also be seen in Fig. 3.6 as intersections of the burning velocity versus equivalence ratio curves for different particle sizes. As the particle size is increased, eventually the effect of decreasing specific reaction surface area again dominates, and the burning velocity decreases with increasing particle size. These trends have been observed experimentally in the combustion of coal suspensions [26] and liquid sprays [100], although no clear physical explanation was given at the time.

51 4

r = 1 m

r = 3 m

r = 6 m

0 (cm/s)

2 b

r = 10 m

0 u

r = 1 m

1 0

0.30 0.32 0.34 0.36 0.38 0.40

5 (cm/s) b u

r = 10 m

2 4 6 8 10

Figure 3.6: Burning velocity as a function of equivalence ratio for diﬀerent initial particle sizes. The expanded view shows the intersections of burning velocities for lean equivalence ratios for various particle sizes.

Kinetic-Diﬀusive Instabilities

For equivalence ratios below φ =0.4, there is a region where no steady-state flame propagation occurs (see hatched area of Fig. 3.5). Instead, these flames oscillate both in terms of burning velocity and flame structure. An example of instantaneous profiles of particle and gas temperature, and Da∗ at different instances over a single oscillation period are shown in

Fig. 3.7 for an equivalence ratio of φ =0.4 and initial particle size of r0 =4μm.

= 0.325 r = 4 m

0.4 T

0.3 a T / 0.2 T

ad T

t = 0.70

t = 0.80 t = 0.63

4 2

0.1

t = 0.97

5 t = 0.45

0.0

0.4 *

4 0.2 t

t Da

4 8 12 16 20

x (mm)

Figure 3.7: Instantenous profiles of gas and particle temperatures and Da∗ at different instances in time over a single oscillation period of an oscillating flame. The units of t are seconds. The arrow indicates the direction of propagation.

The gas temperature is observed to oscillate about the adiabatic ﬂame temperature, while the particle temperature oscillates with a much greater amplitude than that of the gas. At

52 t3, the combustion regime is more diffusive, as indicated by the large temperature separation ∗ and Da , while at t1 and t5, the regime is more kinetic, as indicated by the low values of temperature separation and Da∗. At first glance, these oscillations resemble the well-known thermo-diffusive flame instability often observed in homogeneous gas [101] and quasi-homogeneous solid mixtures [102], and also recently observed in aluminum dust flames [103]. The physical mechanisms underlying the thermo-diffusive instability and the heterogeneous pulsating flame shown here are, however, fundamentally different. In homogeneous mixtures, pulsations are induced by changes in the gas molecular transport properties (Lewis number), activation energy, and overall heat production, as required by the thermo-diffusive instability criteria formulated by Matkowsky and coworkers [104, 105]. In the particle flame, the fuel has zero diffusivity, and, for a fixed equivalence ratio, the gas-phase transport, kinetic parameters, and overall heat production are fixed. In particular, they do not change with particle size. Instead, the cause of the observed instability is evidently rooted in the processes of particle ignition and extinction that are, indeed, very sensitive to particle size. Thus, it may be called a kinetic-diffusive instability, reflecting the fact that the particle combustion regime in the pulsating flame oscillates between diffusive and kinetic combustion modes. This interesting new phenomenon will be studied in detail in subsequent work.

Comparison to Semi-Empirical Flame Models

Most existing models for ﬂames in suspensions are semi-empirical and postulate either a purely kinetic or purely diﬀusive regime of particle combustion. Plots of the normalized burning velocity versus particle size predicted by such models are compared with the result of the current numerical simulations in Fig. 3.8 for the lean mixture with an equivalence ratio of φ =0.3.

2.5

current model

kinetic regime 0.30

2.0

dif f usive regime

1.5

(cm/s) 1.0 b u

0.5

0.0

5 10 15 20 25

r ( m)

0 Figure 3.8: Comparison of computed burning velocity to semi-empirical asymptotic Zeldovich and Frank-Kamenetskii (kinetic) and modified Mallard-Le Chatelier (diffusive) flame models. The semi-empirical models are normalized to the numerical model at r0 =1μm for the kinetic model and r0 =25μm for the diffusive model.

53 For the case of the purely kinetic regime, the burning velocity is calculated using an analytical expression formulated by Zeldovich and Frank-Kamenetskii, modified for a combustible mixture with a single-step, first order heterogeneous surface reaction [16]. The result is normalized to match that of numerical simulations for the smallest particles, which burn in the kinetic regime. The burning velocity in the case of the purely diffusive regime is calculated using an expression given in [32, 33] that requires two externally defined combustion parameters: the ignition temperature and combustion time of the single particle. Ignition temperatures at different particle sizes are calculated using the Semenov ignition criterion as shown in Fig. 3.1 and have a dependence with particle size similar to that shown in the inset of Fig. 3.1. The particle combustion time is calculated using a simple diffusive reaction model that predicts t ∼ r2 a c 0 dependence [106]. The result of the purely diffusive model is normalized to match that of the numerical simulations for the largest particles, which burn in the diffusive regime. As can be seen in Fig. 3.8, these semi-empirical models only qualitatively predict the asymptotic behavior of burning velocity on particle size where the combustion regimes are either purely kinetic or diffusive. Over a wide particle size range, where diffusion and kinetic rates are comparable, these models cannot capture the physics of the flame propagation and yield erroneous results.

3.2.4 Conclusion

The comprehensive analysis of flame propagation in non-volatile particle suspensions presented in this paper has a historical parallel. It can be compared with the replacement of the semi-empirical Mallard-Le Chatelier flame model, based on the notion of ignition temperature and reaction time as external parameters, to flame models developed in the 20th century based on the Arrhenius reaction law. Here, it is similarly shown that the particle ignition temperature, particle reaction time, and regime of particle combustion are, in general, characteristic values of the flame problem that must be solved for together with the burning velocity. Moreover, this analysis demonstrates that the thermal structure of flames in suspensions is defined primarily by the interplay between kinetic and diffusive reaction modes that also yields non-trivial dependence of the burning velocity on particle size. This interplay leads to the emergence of a new type of pulsating instability that is related to the process of ignition and extinction of particles within the flame.

54 Acknowledgments

Support for this work was provided by the Defense Threat Reduction Agency under contract HDTRA1-11- 1-0014, the NSERC CREATE on Clean Combustion Engines, and a McGill Engineering Doctoral Award. Additional funding was provided by the Panda Faculty Schol- arship in Sustainable Engineering & Design and a William Dawson Scholarship.

55 Chapter 4

Combustion Diagnostics in Particle Fuel Suspensions

4.1 Introduction

As outlined in Chapters 2 and 3, there are several similarities between combustion in metal suspensions and gaseous fuels. For example, a stationary reaction wave can be established in a flowing suspension of metal particles similar to a gaseous fuel flow. A visual comparison of Bunsen flames in different metal fuels compared to a hydrocarbon fuel is shown in Fig. 4.1 and reveals the similarities of their structure. It logically follows that the experimental study of combustion in metal fuels should be similar to the study of gaseous fuels.

Figure 4.1: Stabilized Bunsen ﬂames in methane-air and metal-air suspensions from [5].

The measurement of parameters such as burning velocity, suspension reaction time, and thermal structure of the ﬂame is essential to understanding metalized fuel systems. Experi- mental approaches which investigate the combustion of single particles can only probe a small

56 portion of the physics that govern the burning behavior in dense fuel loadings. This requires experimental approaches to study either the induction type or flame type of combustion in suspensions. Induction type combustion of metal particles has been previously studied at low [107] and moderate particle concentrations [65] in optically accessible shock tubes. Induction combustion has also been studied by introducing suspension flows into furnaces [30]. These types of experiments are useful for determining reaction onset time or critical reaction onset temperature (in the presence of heat loss). Typically, the luminosity from the hot particles, or resonant emission from a gaseous species involved in the combustion, is used as an indication of reaction onset. The radiance of emission in both cases should exhibit a strong dependence on temperature [78]. The reaction times for single particles (or dilute suspensions) also can be measured in shock tubes [108]. The duration of luminous intensity is a generally acceptable measurement for combustion time if the particles burn primarily in the diffusion regime. However, if extinction and transition back to the kinetic regime occurs, as shown in the model of Chapter 2, the subsequent decrease in luminosity may falsely register as particle burnout. For induction combustion of dense suspensions, the measurement of reaction time is more difficult in practice. To perform systematic experiments, care must be taken to maintain a uniform dispersion of particles and uniform heating of the suspension. The flame type of combustion in suspensions is more suitable for the investigation of reaction time and, generally, other combustion parameters. Unlike induction combustion, where the initial heating must be precisely regulated, only a uniform suspension flow is required for a flame. The initiation of the reaction in the suspension flow will establish a reaction wave that can propagate automatically on its own. However, even for flames, the direct measurement of combustion characteristics is not a trivial task. The laminar burning velocity is difficult to measure, even in gaseous fuel flames, as effects from flame stretch and heat losses are difficult to overcome experimentally. Moreover, the variation in the particle size and morphology, as well as effects from agglomeration, can greatly affect burning velocity measurements, making data even between different batches of powder difficult to compare. This issue was addressed in a recent publication where burning velocities in suspensions of a single batch of aluminum powder were measured using different methodologies [109]. The burning velocities measured from counterflow flat flames [109], Bunsen flames [33], spherically expanding flames [103], and flames propagating in tubes [33] exhibited significant scatter. These results reflect the still rudi- mentary understanding of flames in heterogeneously burning suspensions, and more research is required to assess the effects of the flame geometry.

57 The overall flame thickness and reaction zone thickness are also important parameters which describe the fundamental length scales of the combustion phenomenon [63]. Similar to gaseous fuel flames, it was reasoned that a measurement of quenching distance could provide an estimate of the characteristic flame thickness [88]. This led to the development of a methodology for determining quenching distance in dust flames [33]. However, the overall flame thickness represents both the reaction and diffusion processes, while the reaction zone thickness is related only to the characteristic reaction time of the suspension. From

dimensional analysis, the suspension reaction time τr should be related to the burning velocity

Sf and the characteristic reaction zone thickness r, by the simple relation:

τr ∝ r/Sf (4.1)

A measurement of the reaction zone requires an additional diagnostic to measure the zone of the majority of heat release within the flame. In gaseous flames, the reaction zone thickness has been measured by laser diagnostic techniques to a fairly high degree of accuracy [110]. However, no such diagnostics have yet been applied to flames in particle suspensions. The state-of-the-art for measurements of thermal structure in particle flames is perhaps just as primitive as the measurements of burning velocity and flame thickness. The use of intrusive diagnostics like thermocouples are often ineffective because they disrupt the suspension flow [88]. Moreover, the multiple temperature environment cannot be probed in this manner. The preferred techniques are non-intrusive optical diagnostics which are more suitable to the combustion environment in suspensions. The high luminosities of condensed phase combustion lend to the prevalence of emission- based techniques [111, 112, 34]. Laser absorption techniques also appear to be suitable since a probe beam of sufficient power should be able to pass through the high scattering environment [113], although the measurement is path averaged. There is also some tendency to try to apply off-the-shelf diagnostics developed for gaseous combustion directly to metal fuel combustion. However, most advanced techniques developed for gaseous flames are often difficult, if not impossible, to apply to in suspensions due to the presence of condensed phase material. For example, the application of standard Coherent anti-Stokes Raman Spec- troscopy (CARS) to metalized propellants requires significant modifications in the presence of particles [114]. Overall, there is a lack of suitable experimental diagnostic techniques to accurately determine many of the combustion characteristics of flames in particle suspensions. Moreover, the utility of common diagnostic techniques applied to particle flames has not yet been fully assessed. These current experimental limitations serve as the primary motivation for the paper in the following section.

58 The goal of the paper is to examine the implementation of emission and laser absorption diagnostics to stabilized, flat flames in aluminum suspensions. The stationary flame is convenient as it avoids the difficulties of implementing optical and spectral diagnostics to a non-stationary reaction wave or for highly transient induction combustion. The paper also highlights some the key difficulties in the interpretation of data from these measurements. The experimental techniques are used to resolve the multiple temperature nature of the flame expected for combustion in the diffusion regime. Additionally, the study attempts to use the laser absorption and emission spectroscopy technique to determine the reaction zone thickness. This measurement, combined with other measured parameters, is used to extract the reaction time of the suspension. A more thorough explanation of the absorption technique used in the paper is presented in Appendix D.

59 4.2 Publication: Emission and Laser Absorption Spec- troscopy of Flat Flames in Aluminum Suspensions, Combustion and Flame [3]

Michael Soo, Samuel Goroshin, Nick Glumac, Keishi Kumashiro, James Vickery, David L. Frost, Jeﬀrey M. Bergthorson

Abstract

Imaging emission spectroscopy, spatially resolved laser-absorption spectroscopy, and particle image velocimetry (PIV) are applied to a flat flame stabilized in a suspension of micron-sized aluminum. The results from the combination of diagnostics are used to infer the combustion regime of the particles and to estimate the characteristic combustion time of the suspension. It is observed that the reaction zone of the flame in stoichiometric aluminum-air suspensions exhibits strong self-reversal of the atomic aluminum emission lines. These lines also exhibit high optical depths in both emission and absorption spectroscopy. The strong self-reversal and high optical depths indicate high concentrations of aluminum vapor within the reaction zone of the flame at multiple temperatures. These features provide evidence of the formation of vapor-phase micro-diffusion flames around the individual particles in the suspension. In aluminum-methane-air flames, the lack of self-reversal and lower optical depths of the aluminum atomic lines indicate the absence of vapor-phase micro-diffusion flames, and point to a more heterogeneous, and likely kinetically-controlled, particle combustion regime. The reaction zone thickness is estimated from the spatially resolved profiles of aluminum resonance lines in both absorption and emission through the flame. The emission measurements yield a reaction zone thickness on the order of 1.7±0.3 mm in aluminum-air flames, and the absorption measurements yield a thickness on the order of 2.3±0.5. It is demonstrated that the combination of the combustion zone thickness measurement, flame temperatures determined from molecular AlO emission spectra, and particle velocity measurements from the PIV diagnostic permits an estimation of the burning time in the suspension. The burning time in stoichiometric aluminum-air suspensions using the suite of diagnostics is estimated to be on the order of 0.7 milliseconds.

Keywords

aluminum; suspensions; spectroscopy; ﬂames

60 4.2.1 Introduction

Micron-sized metal powders, particularly aluminum, are common energetic additives to propellants and explosives. The mass fraction of metal in these energetic compositions can exceed 20%, meaning that the metal burns as a dense suspension in the gaseous products of a hydrocarbon fuel matrix. Understanding the combustion behavior of metal particles in a dense suspension is central to the goal of predicting and tailoring the performance of metalized explosives and propellants. Metals are also increasingly being investigated for use as recyclable fuels [5, 115, 45], which would be burned in dense suspensions to achieve high power densities and ensure flame stability. The combustion physics of a dense suspension of micron-sized particles can differ considerably from the combustion of large, individual, isolated particles, which have been the dominant subject of experimental work in metal combustion [25]. While large particles burn primarily in a diffusion-controlled regime of combustion, combustion of micron- and submicron-sized particles can be limited by heterogeneous kinetics or they can burn in a transitional diffusion-to-kinetic regime [78]. The combustion characteristics measured from single particle studies, such as combustion time and combustion regime, are often extrapolated to particles in the dense suspensions encountered in the majority of practical fuel systems. However, these characteristics are also functions of ambient temperature and oxidizer concentration and, therefore, will depend on the fuel particle concentration [2, 1]. The extrapolation of characteristic combustion data from single-particle studies to dense metal suspensions, where temperatures can exceed 3000 K, will, in general, result in erroneous predictions. Thus, it is imperative to have experimental techniques to determine combustion characteristics of bulk reacting suspensions. In our previous work, stabilized Bunsen-type flames were used to study combustion characteristics of aluminum and iron suspensions in oxygen and in the products of hydrocarbon flames [48, 109, 116, 33]. Other studies have used freely-propagating flames in tubes [117] or, more recently, transparent latex balloons [86], to study flame propagation speeds, laminar burning velocities, and the formation of instabilities. The burning velocities measured in these studies permit a more realistic estimation of the characteristic combustion time of the suspension compared to experiments with individual particles. However, the determination of key combustion characteristics in suspensions of metal particles is difficult due to the inability to isolate and observe individual particles in the cloud. Optical diagnostic techniques to extract these parameters must be tailored to the specific nature of metal suspension combustion which can be optically dense, contain multiple temperatures, have intense luminosity, and cause strong multiple scattering. These phenomena are not often encountered in hydrocarbon combustion systems and largely prevent the use

61 of “off-the-shelf” approaches to the study of combustion in suspensions. Tailoring standard diagnostic techniques to metalized suspensions is often not straight forward. For example, recent studies show that coherent anti-Stokes Raman scattering (CARS) for gas temperature measurements in heavily metalized propellant flames may only be fea- sible with the use of femtosecond and picosecond laser pulses [114]. Using a technique like planar laser induced florescence (PLIF) to study the combustion of individual burning aluminum particles [118] is not realistic for quantitative measurements in suspensions due to the high particle number density. However, PLIF may still be useful as an imaging technique in these multi-phase environments [119]. Emission spectroscopy was previously applied to aluminum Bunsen flames to compare temperatures from gaseous AlO band spectra and condensed-phase continuum emitters, in an attempt to diagnose the particle combustion regime in the suspension and to compare temperatures to equilibrium calculations [34]. Line-of-sight integrated emission measurements from the cylindrically symmetric Bunsen flame cones required the use of a reverse Abel transform to reconstruct the radial profiles of spectral emission intensity to determine the thermal structure of the flame and the width of the combustion zone. The high optical thickness of both the condensed phase spectra and the AlO molecular bands, however, rendered the Abel reconstruction performed in [34] unreliable. The experiment was further complicated by the presence of strong multiple scattering of light by aluminum and aluminum oxide particles, which contaminated local emission spectra with the radiation from different regions of the flame. The accuracy of temperature measurements using continuum and molecular AlO spectra was also found to be insufficient to resolve the temperature difference between diffusively burning particles and the bulk flame temperature in the fuel-rich flames [34]. Due to these difficulties, the study was not able to draw definite conclusions about the regime of particle combustion in the suspension. In the present study, a counterflow dust burner is used to create a flat aluminum dust flame with a one dimensional flame geometry that eliminates the need for spatial reconstruction using the Abel transform. Using the flat flame, an emission diagnostic technique that permits the direct qualitative determination of the regime of aluminum particle combustion in dense suspensions is demonstrated. The technique is based on the spectroscopic detection of self-reversal of the atomic aluminum emission lines that occurs due to inhomogeneous metal vapor temperatures near the micro-diffusion flames that surround the aluminum particles and in the bulk gas of the suspension. The diagnostic technique is also used to measure the combustion zone thickness. The length of the region where the lines are self-reversed serves as a measurement of the reaction zone and largely avoids the difficulties of interpreting the emission spectra in the presence of multiple scattering.

62 These measurements are compared to those obtained using a complementary diagnostic technique of spatially resolved absorption spectroscopy to detect the aluminum resonant lines. Using a broadband laser as a spectral source, the eﬀects of multiple scattering are greatly reduced, and the length of the region where aluminum is detected serves as a measurement of the reaction zone. Combined with a particle image velocimetry (PIV) technique for estimating the particle residence time in the combustion zone and temperatures measured from AlO molecular spectra, the suite of diagnostics permits the characteristic combustion time of a dense aluminum suspension to be estimated.

4.2.2 Experimental Methods

Counterﬂow Burner and Aluminum/Gas Mixtures

The spectroscopic study is performed on flat aluminum dust flames stabilized using a counterflow dust burner (see Fig. 4.2a). The design of the apparatus is described in [109], where it was used to measure burning velocities in aluminum clouds using a PIV technique. For this study, the apparatus is modified to increase the flame planarity. The coaxial flow nozzle on the opposing jet in [109] is removed in order to eliminate recirculation of the combustion products on the flame periphery because it interferes with concentration and spectral measurements. The air in the coaxial flow around the bottom nozzle is also replaced with inert nitrogen to prevent formation of the diffusion flame often enveloping flat flames at fuel-rich conditions. Concentration of the particles is monitored by a laser attenuation probe described in previous work [47]. In the present study, the laser beam probe passes just above the exit of the bottom nozzle as shown in Fig. 4.2. An attempt was made to keep the concentration of aluminum in the suspension stable at around 300 g/m3 for all measurements. Considering the accuracy of the dust concentration measurements, the actual concentration varies between 270 and 330 g/m3. The combustion in the aluminum suspension is studied in two gaseous environments: air and a stoichiometric methane-air mixture. The gas flow rates are monitored using factory calibrated electronic flow meters and rotameters. The oxygen concentration in aluminum- methane-air mixtures is monitored by an in-situ oxygen analyzer (Oxigraf) to determine the equivalence ratio.

Aluminum Powder

Unlike our previous studies of stationary aluminum dust ﬂames that used Ampal 637 aluminum (Ampal, Inc.) with nodular shaped particles with a size around 6 μm [48, 109, 33],

63 Figure 4.2: a) Photograph of a flat flame in an aluminum-air suspension, b) schematic of the counterflow flame burner and side view of the laser sheet and dust concentration laser probe and c) top-view schematic of the imaging-emission and laser-absorption spectroscopy setups including simultaneous imaging and concentration measurement. the present study employs H-2 aluminum powder produced by Valimet Inc. (Stockton, CA) with spherical particles and a narrow particle size distribution. The particle size distribution in the H-2 powder from laser diffraction measurements is provided by Valimet and shown in

Fig. 4.3. The arithmetic mean diameter (d10) in the H-2 aluminum powder is reported to be around 4.2 μm.

Emission Imaging Spectroscopy Setup

The general schematic of the imaging spectroscopy setup is shown in Fig. 4.2c. The setup consists of a 250 mm focal length spherical lens that images the ﬂame onto the entrance slit of an imaging spectrometer with a focal length of 0.3 m equipped with a CCD array. Due to the internally recessed focal plane of the instrument, the CCD array detector is coupled to the spectrometer by a de-magnifying image relay consisting of two imaging lenses (50 mm Nikkor f /1.4 and 35 mm Nikkor f /2). The detector is a CCD camera with a 4032 (w)×2688 (h) pixel chip (Starlight Xpress SXVR H35) and 9 μm square pixels. Only a 2006×1336 pixel segment of the CCD array is used as it aﬀords faster image readout. The spectrometer is installed with a 3600 gr/mm UV holographic grating. The spectrometer entrance slit width is set to 50 μm permitting an overall spectral dispersion of

64 Figure 4.3: Particle size distribution and SEM image of Valimet H-2 Aluminum. The particle sizing results are provided by the manufacturer.

0.004 nm/pixel covering a spectral bandwidth of about 8 nm with an instrument resolution of about 0.03 nm. For the measurements of the AlO molecular band spectrum, the holographic grating was replaced by a 1200 gr/mm ruled grating. This permits a spectral range of about 40 nm with an overall spectral dispersion of about 0.02 nm/pixel and an instrument resolution of about 0.16 nm.

Absorption Spectroscopy Setup

In order to perform absorption spectroscopy of the atomic aluminum vapor, a Q-switched Nd:YAG laser at the 3rd harmonic (Quantel Ultra) is used to pump a modeless dye laser to produce a beam with broadband spectral output as outlined in previous publications [113, 4]. The dye used is a mixture of Exalite 398 and Exalite 389 (Exciton) at concentrations of 0.36 g/l and 0.17 g/l, respectively, dissolved in reagent grade acetone. The pulse energy used in the present experiments is measured to be less than 0.5 mJ per pulse with a broadband spectral output of nearly 8 nm centered near 395 nm. The broadband dye cell output covers 2 → 2 o both of the Al I doublet ( S1/2 P1/2,3/2) emission lines at 394.4 and 396.15 nm. The beam is expanded (focal length f = −25 mm), collimated (f = 50 mm), and then focused into a narrow “sheet” using a cylindrical lens (f = 300 mm). The sheet is passed through the flame and is focused several inches past the center of the flame in order to reduce the light collected from the flame emitters. The laser sheet is refocused, using a cylindrical lens, onto the slit of a customized spectrometer with a 0.45 m focal length installed with an 1200 gr/mm grating. The spectrometer is coupled to a CCD array by a de-magnifying image relay constructed from aspheric lens (f = 50 mm) and a UV imaging lens (f =25mm).The CCD is an EM-CCD (Rolera EM-C2) camera with a sensor size of 1004×1002 pixels and

65 a pixel size of about 8 μm. This permits a dispersion of 0.02 nm/px and an approximate instrument resolution of 0.12 nm covering a bandwidth of about 16 nm. The camera trigger is synchronized to the laser pulse and the integration time of the camera is minimized to gate the laser pulse signal and minimize collection of the ﬂame emission.

Particle Image Velocimetry (PIV) Setup

A particle image velocimetry technique is used to determine the velocity of the aluminum particles as they enter the flame preheat zone. The details of the PIV technique and the specific difficulties for dust flames are discussed in a previous publication [109]. The beam from a 532 nm wavelength, 5 W continuous-wave laser (Dragon Lasers) is formed into a narrow light sheet using the same set of optical elements as in the laser absorption setup. The illuminated particles in the sheet are filmed at a frame rate of approximately 15000 fps by a Photron SA-5 camera through a macro lens with a 532±2 nm narrow bandpass filter (Andover).

4.2.3 General Features of the Emission and Absorption Spectra

As practically no intrusive probes can be used for measurements inside flames of particulate suspensions, only optical methods, and specifically optical spectroscopy, can provide both qualitative and quantitative information on the flame structure. The present paper focuses primarily on the qualitative analysis of the spatially resolved aluminum atomic spectra that permits determination of the particle combustion regime in the flame and the estimation of the characteristic reaction time of suspension. Analysis of the other flame spectral characteristics is left to future work. This study also highlights some of the specific difficulties encountered in optical diagnostics of flames in metal fuel suspensions, such as multiple light scattering by the condensed- phase media and large optical flame thickness. These effects, rarely encountered in gas-phase combustion, can easily be overlooked when applying similar spectroscopic methods to combustion of condensed-phase fuels.

Aluminum Atomic Line Emission Spectra

A sample emission spectrogram of the two strong atomic Al I lines at 394.4 nm and 396.15 nm emitted from the aluminum-air flame is shown in Fig. 4.4(left). The spectral intensity profiles at three flame locations are shown in Fig. 4.4(right). Two weak atomic lines observed near 393 nm are identified on the spectrogram as neutral Fe lines that are present due to iron impurities in the aluminum powder.

66 Figure 4.4: (Left) Greyscale-inverted spectrogram of the aluminum atomic lines from aluminum-air flame with a (non-inverted) blown-up region of the self-reversal region in the inset. (Right) Intensity profiles of the spectrum taken at three different locations: (1) post-flame zone, (2) reaction zone, (3) pre-heat zone. The comparison of the emission line thickness to the instrument resolution function is also shown.

The Al atomic lines in the flame reaction zone appear to be strongly broadened. The observed broadening beyond the instrument resolution considerably exceeds the expected linewidth from Doppler-, collisional-, etc., broadening mechanisms and can only be attributed to strong self-absorption. The broadening effect of self-absorption is synonymous with large optical thickness and indicates high concentrations of the emitting/absorbing aluminum vapor species. In addition to the line broadening by self-absorption, the Al lines also demonstrate self-reversal in the flame reaction zone (see zone 2 in Fig. 4.4). The phenomenon of line self-reversal can occur only in the presence of a temperature gradient of the emitting/absorbing media along the line of sight [120], and is often observed in spectroscopic studies of plasmas [121] and widely used in astrophysics to infer the structure of stellar atmospheres [122]. Self-reversal is often observed in multi-temperature media due to the dependence of the atomic line broadening on temperature. The intensity profile of a resonance line, in both emission and absorption, from the hot atomic vapor will have a wider half-width than the colder regions of atomic vapor. The difference in line half-widths as well as emission intensity at different temperatures causes the observed atomic line on a spectrograph to be more strongly absorbed, by the colder atoms, in the center of the line than on the wings. The resulting profile of the emission line on a spectrometer is “caved-in” at the center, as shown for zone 2 of Fig. 4.4. The observation of line self-reversal, in general, requires not only temperature gradients in the emitting/absorbing media but will also only be apparent

67 at sufficiently high optical depths with sufficiently high resolution of the spectral instrument. The conditions for the observation of line self-reversal have several other nuances which are outlined and discussed in [120]. In stark contrast to the results for the aluminum-air flame shown in Fig. 4.4, the results for aluminum-methane-air flames, shown in Fig. 4.5, indicate the absence of self-reversal of the aluminum lines in any region of the flame. The aluminum lines are also much thinner than in aluminum-air flames, with the line half-width nearly the same as that of the instrument function.

Figure 4.5: (Left) Greyscale-inverted spectrogram of the aluminum atomic lines from the aluminum-methane-air flame showing no self-reversal. (Right) Intensity profiles of the spectrum taken at three different locations: (1) post-flame zone, (2) reaction zone, (3) pre-heat zone. The comparison of the emission line thickness to the instrument resolution function is also shown.

Taking into account the similar flame geometries and light path lengths through the flames in both mixtures, the observed differences in line-widths suggest a much higher optical thickness of aluminum lines in aluminum-air suspensions compared to the aluminum- methane-air mixture which, in turn, is indicative of a much larger concentration of aluminum vapor in aluminum-air flames. The plausible explanation for why line self-reversal is, or is not, observed in aluminum-air and aluminum-methane-air flames is further discussed in the Discussion section.

Aluminum Atomic Line Absorption Spectra

A typical image of the absorption spectrogram through the ﬂame is shown in Fig. 4.6 for both air and methane-air oxidizing mixtures. The spectral proﬁle of the modeless dye laser

68 exhibits a relatively strong noise level, as was observed in previous studies [113]. There are also variations in intensity along the spatial axis due to the non-uniform beam profile of the laser. Variable attenuation from particles in the medium and small imperfections in the spectrometer slit also create striations in the laser intensity along the spatial coordinate. Despite this non-uniformity, the laser sheet technique can still yield a relatively clear picture of the line-of-sight spatial profiles of Al absorption lines, permitting a reconstruction of spatial distribution of aluminum vapor concentration in the flame. The emission spectrograms in Fig. 4.4 and Fig. 4.5 reveal long zones of aluminum vapor emission along the spatial coordinate compared to the absorption spectrograms in Fig. 4.6. The absorption measurements reveal a zone of aluminum vapor less than a couple of millimeters in length while the emission measurements indicate that aluminum vapor can apparently be observed well above and below the flame zone even near the cold, dispersion nozzle. The observed difference can be explained by the strong multiple scattering of light emitted from the combustion zone by both the unburned aluminum particles in the preheat zone and nano- sized oxides in the post-flame zone. In contrast to the emission measurement, the highly directional laser absorption measurements shown in Fig. 4.6 are not strongly affected by the scattered light and, thus, provides a more realistic measurement of the location of aluminum vapor in the flame.

Figure 4.6: Absorption spectrograms of the Al I atomic lines for aluminum-air and aluminum- methane air ﬂames.

The spatial variations of the laser beam intensity in Fig. 4.6 also reveal an interesting

69 phenomenon showing that light at wavelengths near the resonance wavelength bends upward or downward along the spatial axis in the aluminum-air flame. This can also be observed, albeit to a lesser extent, in the aluminum-methane-air spectrogram. The bending observed here is qualitatively similar to the effect observed in the combined spectroscopic and interferometric technique known as the Hook Method which exploits anomalous dispersion near the resonance lines to measure concentration of species [123]. In the present study, the observed light bending is also due to anomalous dispersion but without the interference fringes seen in the hook method. In these spectrograms, the combustion of the aluminum particles in the flame produces a strong spatial gradient of aluminum vapor. At high aluminum vapor concentrations, anomalous dispersion causes pronounced deviations in the refractive index of the medium at wavelengths near the aluminum resonance lines. The strong spatial concentration gradients, and corresponding gradients in refractive index, lead to a pronounced lensing effect where the flame essentially behaves as a gradient index lens for the laser light frequencies around resonance lines. The lensing effect stretches the spatial appearance of aluminum lines and makes the interpretation of the spatial intensity profiles of absorption spectra difficult. Nevertheless, the degree of lensing that occurs and the width of the absorption line give direct qualitative indication of the amount of aluminum vapor in the reaction zone. It can be easily seen that the spectrogram for aluminum-air flames exhibits a stronger lensing effect, as well as much wider absorption lines, compared to aluminum-methane-air flames. This confirms the results from emission spectroscopy, where considerably higher concentrations of aluminum vapor are observed in aluminum-air flames in comparison to aluminum-methane- air flames.

Molecular Emission Spectra of AlO

The previous spectral study of the Bunsen aluminum dust flame attempted to use the AlO emission spectrum and the condensed phase continuum spectrum to derive spatial distribu- tions of temperatures in the flame [34]. The present results, however, indicate that strong multiple scattering of the light in two-phase metal flames makes spatial interpretation of the emission spectra unreliable. At best, the emission measurements can only indicate the maximum temperature in the flame, since emission intensity is strongly weighted to higher temperatures [124]. The measurement of condensed-phase temperatures, by fitting the continuum spectrum to a Planck distribution, relies on the choice of an appropriate spectral emissivity function of the aluminum oxide emitters, which, in general, is a strong function of wavelength, temperature, and the optical depth [125, 126]. As shown in the literature, a small deviation in emissivity

70 data can lead to large scattering in temperatures, often reaching several hundred degrees [108]. Due to these difficulties, we have opted to derive gas temperatures only from AlO molecular spectra in this study. As in previous work [34], the temperature is found by fitting a simulated spectral intensity distribution in the AlO B−XΔν = −1 band sequence to the experimentally measured spectral intensity. The AlO B−XΔν = −1 band sequence emitted by the aluminum-air flame is shown in Fig. 4.7. Similar to the Al atomic lines, the AlO molecular lines become optically thick in the flame zone (Fig. 4.7(right), zone 1). Away from the flame, the spectrum is optically thin, and, thus, fits reasonably well with a simulation based on optically thin approximations commonly used in literature [107]. In the optically thick zone, a temperature fitting of the AlO band sequence becomes much less accurate. The effect of optical thickness is to enhance the weak lines at the expense of the stronger features. This effect typically skews fits to higher temperature as shown in Fig. 4.7(right) (zone 1). Here the observed spectrum is compared to optically thin spectra simulated using the ExoMol line list for the AlO B−X transition simulated in the PGopher software [127, 128]. It appears that even when optical thickness is accounted for in the model, the relative line intensities become less sensitive to temperature leading to considerable uncertainty in measurements.

Figure 4.7: (Left) inverted spectrogram of the AlO B−XΔν = −1 band sequence across the ﬂame. (Right) Spectra from two locations in the ﬂame showing the optically thick and optically thin regions.

The light emission observed from the cold, pre-ﬂame zone (below zone 3 on Fig. 4.4 and Fig. 4.5) is due to multiple scattering of light from the reaction zone in ﬂame. Scattered light

71 from the flame, in general, will have traveled a smaller distance through the reaction zone in comparison to the light from the flame traveling along the optical axis and is, consequently, an optically thinner version of the spectrum from the flame zone. Due to the strong bias of resonant emission to high temperatures, measurement using scattered light generally can only indicate the maximum temperature of AlO in the flame. In the present experiment, measurements of temperatures at different locations on the spectrogram outside of the combustion zone yield nearly the same temperatures regardless of the location, even in the cold zone upstream of the flame.

Comparison of Flame Temperatures Derived from AlO Spectra with Equilibrium Calculations

The resulting temperatures derived from the optically thin AlO spectra are plotted in Fig. 4.8 together with adiabatic flame temperature, and equilibrium concentrations of aluminum vapor calculated with the NASA CEA code [56]. Equilibrium calculations predict close to 400 K temperature difference between aluminum- air and aluminum-methane-air flames, while the actual measured temperature difference is considerably smaller. The average temperature measured in aluminum-air flames is about 3370 K while the average temperature measured in aluminum-methane-air flames is found to be about 3170 K, with an estimated measurement accuracy of about ±150 K in both cases [107]. It needs to be noted that equilibrium calculations predict higher concentrations of aluminum vapor in aluminum-methane-air mixtures in comparison to aluminum-air mixtures. The emission and absorption spectroscopy results show the exact opposite behavior. The possible explanation for the difference between the measurements and equilibrium calculations is discussed in the Discussion section.

4.2.4 Discussion

Al Emission Self-reversal and the Regime of Combustion in the Flame

In the majority of models in the literature [129], diffusion limited combustion of aluminum particles is similar to hydrocarbon droplet combustion during which aluminum vapor generated by the evaporating aluminum droplet reacts with oxygen in a thin gas-phase flame enveloping the droplet at the distance of several droplet radii. The temperature of such micro-diffusion flames, lifted from the particle surface, is close to the adiabatic temperature of the stoichiometric aluminum mixture with oxidizer. It can be several hundred degrees

72 3800 0.08

aluminum-air

aluminum-methane-air

3600

3400

0.06

3200

3000

0.04

2800 Al (g) mole fraction

2600 Adiabatic Temperature (K) Temperature Adiabatic 0.02

2400

2200

0.00

100 200 300 400

Al particle concentration (g/m )

Figure 4.8: Flame temperatures measured using AlO molecular spectra (• and ◦) in comparison to equilibrium calculations (lines) at different initial aluminum particle concentrations. The predicted mole fraction of gaseous aluminum vapor in equilibrium is also shown. higher than the temperature of the evaporating aluminum droplet and the average temperature of the gas between the reacting particles. The existence of diffusion micro-flames around the particles leads to local micro-gradients of aluminum vapor concentration and temperature within the suspension. The high reactivity of aluminum vapor with various oxidizers [50, 130], generally prevents the formation of a premixed aluminum-oxidizer mixture, such that the aluminum vapor can only exist in the bulk gas of a fuel-rich mixture once the oxidizer has been nearly consumed. For nearly stoichiometric mixtures, a small amount of aluminum vapor forms due to natural dissociation process of aluminum sub-oxides at high temperatures. Traversal of the light emitted by micro-diffusion flames through colder aluminum vapor in the space between the flame and the particle surface, and in the bulk gas, leads to the self-reversal (“caving”) of aluminum lines. This self-reversal results from the different half-width of the Al lines at different temperatures, as illustrated in Fig. 4.9. The existence of diffusion micro-flames around the particles leads to local micro-gradients of aluminum vapor concentration and temperature within the suspension. The high reactivity of aluminum vapor with various oxidizers [50, 130], generally prevents the formation of a premixed aluminum-oxidizer mixture, such that the aluminum vapor can only exist in the bulk gas of a fuel-rich mixture once the oxidizer has been nearly consumed. For nearly stoichiometric mixtures, a small amount of aluminum vapor forms due to natural dissociation process of aluminum sub-oxides at high temperatures. Traversal of the light emitted by micro-diffusion flames through colder aluminum vapor in the space between the

73 flame and the particle surface, and in the bulk gas, leads to the self-reversal (“caving”) of aluminum lines. This self-reversal results from the different half-width of the Al lines at different temperatures, as illustrated in Fig. 4.9.

Figure 4.9: (Top) schematic of the micro-scale emission and absorption of aluminum vapor from the ﬂat ﬂame that is integrated along the line of sight to the spectrometer. (Bottom) Schematic of self-reversal. Hot zone emission of linewidth w1 is partially absorbed by cold zone with linewidth of w2 where w1 >w2.

Self-reversal of aluminum atomic lines was also observed in our previous work on spectroscopy of aluminum Bunsen-type conical flames [34], where the lines were also often caved- in below the intensity of the continuous spectrum. This lineshape was believed to be caused by the macro-temperature gradients along the line-of-sight due to the cylindrical flame geometry, where light emitted from the flame front travels through cooling combustion products containing residual aluminum vapors due to the fuel-rich combustion. The use of the flat flame in the present work precludes the existence of macro-scale temperature gradients along the line-of-sight, suggesting that the observed strong self-reversal of aluminum lines in aluminum-air flames is the result of the micro-scale temperature gradients of the aluminum vapors inside the lifted micro-diffusion flames that surround burning aluminum droplets. It is interesting to note that the self-reversal of aluminum and magnesium atomic lines was first noticed during a spectroscopic study of the combustion of burning magnesium ribbons and aluminum foils, which was also interpreted as evidence of a diffusion-dominated combustion regime due to the presence of multiple temperatures [10]. The diffusion regime of combustion causes high concentrations of aluminum vapors in the space between the aluminum droplet and the micro-diffusion flame, and these vapors

74 are insulated by the flame from the oxidizers within the bulk gas. The observed large optical depths of Al emission and absorption lines indeed indicate very high aluminum vapor concentrations in aluminum-air flames and point to a diffusion-limited flame mode. In this regime of particle combustion, there is no macro-equilibrium state for a given flame cross section, and the temperature and integrated concentrations of gaseous aluminum in the flame can exceed the predicted equilibrium values of the bulk mixture. A similar conclusion was also made by in the study of laminar diffusion flames in aluminum suspensions, where the high optical depth of the atomic Al lines was also attributed to the existence of micro- diffusion flames [131]. In contrast to aluminum-air flames, no line self-reversal was observed in aluminum- methane-air flames. In addition, the atomic Al lines emitted by these flames are generally optically thin, with the line half-width on the order of the instrument resolution. The absence of the line self-reversal, and lower concentration of aluminum vapors indicated by the weak broadening of Al lines, suggest that, unlike in aluminum-air flames, lifted diffusion micro-flames do not form in aluminum-methane-air suspensions. This observation points to a heterogeneous, and most likely kinetically-limited, combustion mode of aluminum in the hydrocarbon combustion products. The disparity between the equilibrium calculations that predict similar concentrations of aluminum vapor in both mixtures, shown in Fig. 4.8, and the large difference in the spectroscopically measured concentrations is explained by the different modes of combustion in each mixture. A shift to kinetically dominated combustion in methane-air mixtures is consistent with the conclusions of our previous hybrid aluminum- methane Bunsen flame studies [48, 47]. While a detailed analysis as to why the change in combustion regime occurs is beyond the scope of the present work, it is not surprising tak-

ing into account the slower rates of heterogeneous aluminum reactions with CO2, CO, and

H2O [25], as well as higher rates of molecular heat transfer in the media containing water molecules and the hydrogen produced by the aluminum-water reaction.

The Width of the Flame Reaction Zone

The thickness of the combustion zone in the flame can be estimated from the spatially resolved spectrogram of aluminum vapor lines in both emission and absorption. As mentioned previously, the strong multiple scattering of the light by unreacted aluminum particles and nano-sized oxide products makes the determination of the location of the aluminum vapor from the intensity of the emitted light unreliable. This causes the luminous zone of the flame to be much larger than the reaction zone thickness. It is possible, however, to use a spatially resolved measurement of the self-reversed line, when it is present, as a method to estimate the flame thickness. The presence of self-reversal on the emission spectrogram is not as

75 affected by multiple scattering since, in order for self-reversal to occur, light must traverse a significant path along the optical axis through the flame zone. Therefore, self-reversal is practically not present in the scattered light. A self-reversed line appears as a double-peaked spectral feature with a local minimum at the central resonance wavelength. The ratio of the intensity at the local minimum at the line center to the peak intensity on either side of the line is a measure of the relative degree of self-reversal. When this ratio is unity, self-reversal is weak either due to low optical depth or small temperature gradients within the medium. When this ratio is close to zero, the self- reversal phenomenon is strong, indicating temperature gradients and typically high optical depths. The value of this spectral intensity ratio subtracted from unity measured along the spatial axis of the spectrogram is a peaked function with a half-width that serves as an estimate for the flame thickness. The average “self-reversal width” measured for the 396.15 nm Al I line in aluminum-air mixtures over several spectrograms indicates a combustion zone thickness on the order of 1.7±0.3 mm. The reported error represents the standard deviation over the measurements. The intensity of absorption at the central line wavelength was measured in relation to the unabsorbed and unperturbed baseline laser spectrum intensity, taken far from the flame zone and away from wavelengths demonstrating the lensing effect. The width of the resulting spatial distribution of intensity in the atomic line center is used as an estimate of the combustion zone thickness. Using this method, a combustion zone thickness in aluminum-air flames, averaged from five spectrograms, is estimated to be about 2.3±0.5 mm, i.e. about 25% larger than the estimate given by the method of the line self-reversal but not significantly different given the uncertainties in each measurement. The estimated length of aluminum vapor in aluminum-methane-air flames of about 1.5±0.7 mm using this method may reflect only part of the combustion zone length where the particles burn in the intermediate diffusion-to-kinetic regime as discussed in recent works [2, 1]. Aluminum absorption lines could also be an indication of hot aluminum-vapor in the equilibrated bulk gas predicted by thermodynamics (see Fig. 4.8). This observation would suggest that the length of aluminum vapor lines reflects cooling of the products and has no correlation with the particle combustion zone.

Estimation of the Suspension Combustion Time

The characteristic reaction time of the suspension can be estimated by dividing the length of the flame reaction zone by the average velocity of the burning particles traversing it. The velocity at which the particles enter the flame is determined from the axial velocity profiles of the particles measured by the PIV technique, where aluminum particles also play the role

76 of the flow tracers. The particles slow down as they approach the stagnation surface and their velocity reaches a minimum as they enter the preheat zone of the flame [132]. After this point, the gas expands due to heating and particles accelerate (see [109]). Due to the very small velocity slip estimated between the H-2 aluminum particles and the gas, the measured particle minimum velocity practically coincides with the velocity of gas and, therefore, is the burning velocity of the flame. Indeed, the measured values of the minimum particle speed in stoichiometric methane-air flames using low loadings of H-2 aluminum particles exhibited values very close to the methane-air burning velocities reported in the literature [133], indicating that the slip between the H-2 particles and the gas is small. The PIV technique cannot track the reacting particles in the flame as they are rapidly obscured by nano-sized oxide combustion products. As a result, the particle velocity at the end of the combustion zone was estimated by multiplying the entrance speed by the gas expansion coefficient calculated by a ratio of the flame temperature and room temperature. The average minimum particle speed and, therefore, burning velocity in aluminum-air flames is measured to be about 48±6 cm/s. The measured burning velocity for H-2 Valimet powder is considerably higher than the measured burning velocities of 6 μm Ampal-637 powder of about 35 cm/s [109]. This observation corresponds to the expected, inversely proportional, dependence of the flame burning velocity on particle size [2]. The particles are assumed to accelerate constantly to their maximum velocity set by the expansion ratio, which is determined from the average flame temperatures measured by the AlO spectra (Fig. 4.8). The average particle velocity is determined to be ≈295 cm/s for flames in aluminum-air suspensions. This gives an estimate for the characteristic reaction time for a nearly stoichiometric aluminum-air suspension of about 0.6 ms using the flame thickness determined by the line self-reversal method and about 0.8 ms using the flame thickness profiles found by spectral absorption. As mentioned previously, the length of aluminum lines measured in aluminum-methane- air flames are not directly related to the flame combustion zone and, therefore, some other markers that can indicate heterogeneous and kinetically-controlled aluminum combustion will need to be developed in future studies. It should also be noted that the methods of flame thickness measurement in the aluminum- air flames are biased toward the combustion of larger particles which would be expected to produce more aluminum vapor from the micro-diffusion flames and to have comparatively longer burning times. The zone of the majority of heat release may be much smaller in comparison to the zone of aluminum vapor and, thus, it is likely that the flame thickness and combustion time are shorter than currently estimated.

77 4.2.5 Conclusion

Flat flames in aluminum-air and aluminum-methane-air suspensions, stabilized using a counterflow burner, are investigated using imaging emission spectroscopy, laser-absorption spectroscopy, and PIV. It is demonstrated that the combination of these optical techniques can be used to extract the combustion characteristics of particles in dense suspensions where isolating and observing individual particles is nearly impossible. The emission diagnostics show that aluminum-air flames exhibit strong self-reversal of atomic aluminum resonance lines in the combustion zone as well as high optical depths in both emission and absorption spectra. Both of these observations are attributed to aluminum particles burning in a diffusion-controlled combustion regime with the formation of lifted micro-diffusion flames around the particles. In contrast, the aluminum resonance lines in the aluminum-methane-air flames show no self-reversal and are more optically thin in both emission and absorption spectra, indicating more heterogeneous, and likely kinetically-controlled, aluminum combustion. The combination of combustion zone thickness measurements from spatially resolved emission and absorption spectra, the flame temperatures derived from AlO molecular spectra, and the particle velocities measured by PIV provide the necessary information to estimate the combustion time of the particle suspension. The estimated characteristic combustion time of the aluminum-air suspension is found to be in the range of 0.6 to 0.8 ms. The diagnostic approach outlined in this paper provides a methodology for studying particle combustion at more realistic particle loadings encountered in practical fuel systems. These diagnostics are important for understanding burning behavior and the contribution of particle energy release to the flow field in dense suspensions.

Acknowledgments

Support for this work was provided by the Defense Threat Reduction Agency under contract HDTRA1-11- 1-0014 and a McGill Engineering Doctoral Award. Additional funding was provided by the Panda Faculty Scholarship in Sustainable Engineering & Design and a William Dawson Scholarship.

78 Chapter 5

Conclusions

5.1 Synopsis of Contributions

The goals of scientific research in metal combustion should be focused on understanding the combustion behavior of metal fuel systems. In most practical scenarios, metal combustion occurs in dense suspensions of powders. Nevertheless, many experimental and theoretical studies tend to focus on the combustion of single, isolated particles. It is often assumed that the combustion characteristics of single particles can be used to anticipate the combustion behavior at dense fuel loadings. As demonstrated in this thesis, however, the combustion behavior of individual particles may not be similar to combustion in suspensions. It is for this reason that both experimental and theoretical approaches to metal combustion need to examine the effect of particle concentration on the combustion behavior. In the first part of this thesis, combustion in a suspension of heterogeneously burning particles is investigated using a simple, transparent theoretical model. It is argued that the general underlying physical behaviors of metal combustion are driven by the heterogeneous nature of the particle reaction and the fundamental mechanisms of mass and heat transport. A simple theoretical treatment is used to examine combustion of single particles, induction combustion in the suspension, and one-dimensional flame propagation in the suspension. Several important scientific contributions to the field of metal combustion are presented in the context of this simple theoretical approach. The fundamental theory of heterogeneous ignition and extinction is explained using the thermal states analysis presented in Section 1.4. This simple analytical treatment of heterogeneous particle ignition is illuminating because it analytically provides an intuition of the effect of parameters like particle size and initial temperature on the regimes of combustion. The analysis shows how ignition, i.e., the critical- state transition from the kinetic regime to the diffusion regime, changes the thermal regimes of the particle combustion. The analysis also demonstrates how particles that are able to

79 ignite and burn in the diffusion regime are resistant to heat loss in the gaseous medium. For particles that burn in mixed regimes between the kinetic and diffusion limits, the effects of particle burnout and changes in the regime of combustion are made apparent from the analysis of potential stable states. This method of explaining heterogeneous ignition theory provides insight into the features of the intermediate regime of combustion for particles beyond the simple analysis often included in textbooks [22, 23]. One critical aspect of this analysis is that it shows a critical particle size, below which the process of ignition degenerates and the particle must burn in the kinetic regime. The model of induction combustion in a suspension in Chapter 2 builds on the analysis of Section 1.4 and provides a physical intuition of the effect of particle concentration on combustion. The primary result of this model is that the combustion behavior of the suspension is governed by two physical mechanisms. The first mechanism is the particle ignition phenomenon which can occur even in dilute suspensions. The second mechanism dominates the reaction behavior in dense suspensions and occurs due to the collective effect of the particles, where reaction proceeds in an explosion-like manner, after some delay, through self-heating. The interplay between these two mechanisms leads to a nontrivial dependence of the reaction time on the particle size and particle concentration. The extension of the induction combustion model to the one-dimensional flame in Chapter 3 provides insight into the basic behavior of heterogeneous flames. The model is meant to replace the asymptotic, semi-empirical models of particle flames based on the notion of ignition temperature and reaction time as external parameters. It is shown that the particle ignition temperature, particle reaction time, and regime of particle combustion are characteristic parameters of the flame comparable to the burning velocity eigenvalue. In this sense, they cannot be determined independently. Similar to the induction combustion analysis, the thermal structure of flames in suspensions is defined primarily by the interplay between kinetic and diffusion reaction regimes that yield a non-trivial dependence of the burning velocity on particle size. This interplay between reaction onset due to self-heating and the criticality of the ignition process leads to the prediction of a new type of pulsating instability that is related to subsequent ignition and extinction processes of particles within the flame. In light of the understanding from the simple heterogeneous combustion models, it is clear that combustion characteristics change not only as a function of particle size but also particle concentration. The second part of the thesis experimentally examines stabilized flames in aluminum suspensions in order to determine the fundamental combustion parameters. It is demonstrated that a combination of diagnostic techniques can be used to extract the combustion characteristics of the suspension where isolating and observing individual

80 particles is nearly impossible. The techniques developed in this study overcome some of the specific problems associated with multiple scattering and optically thick environments encountered in dust flames. The diagnostic study contributes an application of spatially resolved emission and laser absorption spectroscopy to determine the presence of the diffusion regime of combustion in the suspension. This technique is based on the detection of atomic aluminum line self- reversal which occurs due to the presence of multiple temperatures in the medium due to the lifted vapor-phase diffusion-flames around the aluminum particles. The spatially resolved emission and absorption technique also permits a novel measurement of the combustion zone thickness. Combined with the flame temperatures derived from AlO molecular spectra, and the particle velocities measured by PIV, the combustion time of the particle suspension is estimated. These diagnostics are an important first step for understanding burning behavior and the contribution of particle energy release to the flow field in dense suspensions.

5.2 Direction of Future Research

The simple formulation for the heterogeneously reacting particle suspension can provide basic physical intuition of the effect of various parameters on the combustion characteristics. While the focus in this thesis was to examine the effect of particle size and concentration, a parametric study is also possible. The minimalistic model can be extended to include the effects encountered in realistic systems. For instance, the effects of expansion cooling, initial temperature, and pressure may be of significant importance in the combustion behavior of practical metal fuel systems. A more thorough examination of the flame instability discussed in Chapter 3 is also required to determine the conditions of manifestation. This type of model, despite having no apparent analytical solution, is relatively easy to implement in practice and requires little computational power. This makes it a valuable tool for studying fundamental physics of combustion in suspensions. In the literature, there is some tendency for models of heterogeneous combustion to claim quantitative prediction capability [79, 26]. This type of mentality likely stems from the impression that the field of heterogeneous/metal combustion is as developed as the fields of computational fluid dynamics and gaseous combustion physics. It is true that, over the years, several advanced computational tools have been developed to model flames in gaseous fuels [134, 135]. The inclusion of experimentally verified chemistry and physics into the flame model permits an understanding of the flame structure as a function of various operating conditions. The degree to which these models match experimental conditions is a testament to the decades of experimental and theoretical research efforts in gaseous fuel combustion

81 that form the basis of these predictive codes. It is the author’s opinion that quantitatively predictive models of reactive flow in solid fuel suspensions are not yet within reach. Despite the advanced computational tools available, the physical complexity of real heterogeneous fuel combustion, compared to gaseous fuels, requires several theoretical and experimental considerations beyond what has currently been studied. Fundamental investigations are still required to determine the important physics of the problem, how the physics affect the flame structure, and how the flame structure varies as a function of different operating conditions. The minimalistic model presented in this thesis is a single step in the direction towards understanding realistic particle fuels. It is based on the fundamental considerations of flame propagation and the heterogeneous nature of the reaction and heat transfer. However, there are several neglected physics which may not be negligible for real fuels. For example, phase changes, the dynamics of heat and mass exchange between the particle and gas, and contributions of Stefan flow may affect the combustion behavior significantly. It is also of significant interest to understand the combustion behavior of a polydisperse suspension, or a suspension of agglomerates, as encountered in realistic systems. The diagnostic approaches proposed in this thesis also provide direction for future experimental research in flows of reactive suspensions. While more qualitative measurements are reported, the measurements are less important than the proposed direction and methodology. Future diagnostic developments for combustion in suspensions must address fundamental is- sues, such as the presence of multiple temperatures, strongly scattering media, and high optical depths. This requires a paradigm shift in the way flames in particle suspensions are studied experimentally compared to gaseous flames. The laser absorption technique used in this thesis has several possible extensions. As discussed in Chapter 4, emission measurements tend to represent the highest temperatures which are more closely related to particle or micro-flame temperatures. There is currently a lack of a diagnostic which can probe the ambient gas temperature that is also suitable for dust combustion environments. It may be possible to measure the gas temperature in absorption by probing a molecular species that should be present mostly in the bulk gas. This prospect is discussed in Appendix D in the discussion of the laser diagnostic technique. The “hooking” effect in the laser absorption spectrogram observed at high optical depths in Chapter 4 can also be extended to a quantitative measurement. This effect, though perhaps detrimental to the measurements in the paper, can be utilized in a different manner to determine the concentration gradient of aluminum vapor in the flame [136]. This can possibly provide another method to determine flame thickness that can verify the measurements in absorption and the self-reversal profiles.

82 The spatially resolved laser absorption technique is able to work around the effect of scattering to determine the profiles of aluminum vapor in the flame. However, scattering may still affect the measurement to some degree and may not be negligible at larger scales. A few possible ways to remove scattering include ballistic imaging techniques [137] and X-ray radiography [138]. These proposed diagnostics are just a few of several possible solutions that may be able to overcome the specific problems associated with dust flames. Continued efforts are required to model the appropriate physics, combined with experimental techniques and diagnostics to verify the predictions. With continued scientific developments, the field of metal combustion, and heterogeneous combustion in general, may one day reach the level of understanding of conventional hydrocarbon fuels.

83 Appendix A

Zero-dimensional Induction Combustion in a Fuel Suspension: Governing Equations

This section presents the governing equations for adiabatic, induction combustion (or a zero- dimensional reactor) of a mono-sized fuel suspension. The equations are written in the frame of reference of a uniform, stationary suspension and are only functions of time, t. Here, the theoretical treatment of induction combustion in a heterogeneously burning fuel suspension is meant to be analogous to the problem of adiabatic thermal explosion in a gaseous fuel mixture from classic Semenov theory. It is assumed that the gas is heated instantly to some temperature at t = 0. The kinetic and thermodynamic parameters are assumed to be pressure/temperature independent for simplicity. These simpliﬁcations focus the model on the behavior of the suspension combustion due to the particle concentration and particle burnout alone.

A.1 Particle Energy Equation

The model assumes the particles have no radial variations of temperature within the particle (i.e. treated as a lumped capacitance). This implies a high rate of heat transfer inside the particle compared to the heating rate at the surface due to convection. The validity of this condition is determined by examining the Biot number Bi=h /λp where h is the heat transfer coeﬃcient, is the length scale (r in the case of a spherical particle) and λp is the conductivity of the particle. For a spherical particle of radius r in a stationary ﬂow, h can

be estimated by h = λ0/r where λ0 is the conductivity of the gas. The Biot number then

84 takes the form of Bi = λ0/λp. For most metals and oxidizers, Bi 1, and therefore, this assumption is appropriate. Because all particles in the suspension are the same, only the energy equation for a single particle is required and takes the form

d c (mT )=Q˙ − Q˙ (A.1) s dt s R L ˙ ˙ where Ts is the particle temperature and QR and QL are heat gained due to the surface reaction and heat lost due to convection to the gas, respectively. The heat capacity cs is m 4 πr3ρ assumed to be constant, but the mass = 3 s changes during particle burnout. Deﬁning the heat of reaction per unit mass of fuel as q, the rate of heat released may be expressed as: ˙ QR = qAγω˙ eﬀ (A.2) where γ is the stoichiometric ratio (i.e. the mass of fuel consumed per unit mass of oxidizer

consumed), A is the instantaneous surface area of the particle, andω ˙ eﬀ is the particle reaction rate per unit surface area combining both the kinetic and diﬀusive terms, as discussed in Chapter 1 and 2. The reaction rate in this model takes the form of

kβ ω˙ = k C = C (A.3) eff eff k + β where k is defined by Eq. 2.3 and β is defined by Eq. 2.5, and C is the instantaneous oxidizer concentration in the bulk gas. This formulation assumes quasi-stationary conditions where the reaction rate on the particle surface must be equal to the rate at which oxidizer is diffused to the surface [17]. It is also assumed that the contributions from Stefan flow are negligible. This assumption is reasonable when the concentration of the oxidizer in the flow is low or the volume of oxidizer reacting on the particle surface is similar to the volume of products formed. ˙ The term QL takes the same form as Eq. 2.6. The final form of the solid phase energy conservation equation is

d c (mT )=qAγω˙ − hA(T − T )(A.4) s dt s eﬀ s g h λ / r m 4 πr3ρ where =Nu 0 2 , Nu= 2, and = 3 s. Diﬀerentiating the left hand side of Eq. A.4 using the product rule, and using the expression found in Eq. A.5, Eq. A.4 reduces to the expression in Eq. 2.8.

85 A.2 Particle Mass

The particle mass consumed in the reaction is removed from the particle and the resulting oxide is assumed to be gaseous. This assumption simpliﬁes the treatment of the particle mass equation. In this case, the continuity equation for the particle is

dm = −γAω˙ (A.5) dt eﬀ Equation A.5 reduces to Eq. 2.9 shown in Chapter 2. The particle is allowed to shrink −6 until a size where rmin/r0 =1× 10 . This condition prevents the solution of the equation set from breakdown at r =0.

A.3 Bulk Gas Oxidizer Continuity

It is assumed that the molecular weight of the gaseous oxide products are close to that of the oxidizer. Hence, there is no net generation or consumption of the total gas mass, since the oxide replaces the oxidizer species as it is consumed in the reaction. The equation for the consumption of oxidizer takes the form

dC = −NAω˙ (A.6) dt eﬀ where N is the number density of particles and is related to the particle concentration B by

N N B = = (A.7) m 4 πr3ρ 0 3 0 s where r0 is the initial particle size. The expression in Eq. A.6 algebraically reduces to the expression in Eq. 2.11.

A.4 Gas Thermal Energy

The particles are assumed to have a negligible volume contribution and the volume of gas can be treated independently from the particles. The validity of this assumption can be determined by examining the ratio of the volume of particles to the volume of gas which

is given by the expression B/ρs. This volume ratio is typically less than 1% for metals at stoichiometric concentrations. In the adiabatic system, the rate of change of the energy of the gas is equal to the rate of energy transferred by convection to/from the well-mixed solid-phase. Thus, the equation

86 for the gas energy takes the form

dT ρ c g = NQ˙ = NhA(T − T )(A.8) g0 g dt L s g

where ρg0 is the gas density and cg is the speciﬁc heat of the gas. Using the relation in Eq. A.7, Eq. A.8 algebraically reduces to Eq. 2.10.

87 Appendix B

One-dimensional Flame Propagation in a Fuel Suspension: Governing Equations

This section presents the governing equations for a ﬂame in a mono-sized particle suspension propagating in a single spatial coordinate, x, and time, t. The model is analyzed in the paper of Chapter 3. The model consists of one equation of state and six conservation equations for 1) the momentum, 2) the overall continuity of the gas, 3) the thermal energy of the gas phase, 4) the continuity of the oxidizer, 5) the continuity of the solid fuel, and 6) the thermal energy of the solid phase. Several simpliﬁcations and assumptions, outlined in this appendix, are applied to the equations in order to simplify the solution.

B.1 Equation of State

The gas phase is treated as an ideal gas mixture comprised of an oxidizer and an inert component. Hence,

pg = ρgRTg (B.1)

where pg is the gas phase pressure; ρg is the gas phase density; Tg is the gas phase temperature; and R is the speciﬁc gas constant.

B.2 Momentum Conservation

The model only considers deﬂagrations, and as such, the Mach numbers encountered are small. In addition, viscous forces are assumed to be negligible. In this limit, the momentum

88 equation can be integrated to yield simply [139]:

pg ≈ const. (B.2)

Combining this with the equation of state, it is found that

1 ρg ∝ . (B.3) Tg

In addition, it is assumed that the particles experience negligible velocity slip with respect to the gas. Thus the particle number density, N, changes according to the gas density. Hence,

1 N ∝ . (B.4) Tg

B.3 Gas Phase Continuity

It is assumed that the molecular weight of the products are close to that of the oxidizer. Hence, there is no net generation or consumption of the gas phase, so that the equation of overall continuity of the gas phase is as follows:

∂ρ ∂ g + (uρ )=0 (B.5) ∂t ∂x g where u is the x-component of the ﬂow velocity.

B.4 Density-Weighted Coordinate

The governing equations are simplified by introducing a density-weighted coordinate, x, defined as follows: x ρ (x¯,t) x(x,t)= g dx¯. (B.6) ρ 0 g0 where ρg0 is the gas density at the unburned temperature, Tu. The dimension of x,aswith x, is length. By definition, ∂x ρg = (B.7) ∂x ρg0 and ∂x ρ ρ − g u g u . = + x=0 (B.8) ∂t ρg0 ρg0

89 where x = 0 identiﬁes the boundary of the semi-inﬁnite domain, which is in contact with the burnt gas. One can always choose a reference frame that is stationary with respect to u the burnt gas such that x=0 = 0, and, ∂x ρ = − g u. (B.9) ∂t ρg0

It then follows that the t and x derivatives transform as follows ∂ ρ ∂ → g (B.10) ∂x ρg0 ∂x ∂ ρ ∂ ∂ →− g u + . (B.11) ∂t ρg0 ∂x ∂t

With this transformation, proposed initially by Spalding [93], the equation of continuity of the gas phase is automatically satisﬁed, and advection eﬀects due to the thermal expansion of the gas are eliminated.

B.5 Temperature Dependence of Gas and Solid Phase Properties

In order to completely remove the effects of thermal expansion from the governing equations, a number of simplifying assumptions regarding the temperature dependence of certain properties are made. The bulk mass diffusivity and thermal conductivity of the gas are assumed D ∝ T 2 λ ∝ T to have temperature dependences of g and g, respectively, while the mass diffusivity and thermal conductivity at the particle-gas interface are assumed to have temperature dependences of DI ∝ Tg and λI = constant, respectively. In addition, the pre-exponential factor is assumed to have a temperature dependence of κ ∝ Tg. As explained in B.2, the gas density and particle number density, N, are inversely proportional to the gas temperature. Hence, the temperature dependence of the following combined parameters cancel out such that:

λρg = λ0ρg0 (B.12) Dρ2 D ρ2 g = 0 g0 (B.13)

κ/DI = κ0/DI0 (B.14)

Nκ = N0κ0 (B.15)

κρg = κ0ρg0 (B.16)

90 where the subscript 0 denotes evaluation at the initial—or unburned—temperature, Tu.

B.6 Gas Phase Thermal Energy

It is assumed that, within the gas phase, Fourier’s law of heat conduction applies, and that heat transfer due to radiation, inter-diffusion, and the Dufour effect are all negligible. The gas phase thermal energy balance for a differential control volume can then be stated as: ∂ ∂ ∂ ∂T (c ρ T )+ (uc ρ T )= λ g +˙q . (B.17) ∂t g g g ∂x g g g ∂x ∂x I where cg is the specific heat capacity of the gas. Because the gas and solid phases are, in general, not in thermal equilibrium, heat transfer occurs between the phases. This is accounted for byq ˙I, which is the volumetric heat transfer rate between the phases. Further assuming that the gas phase is calorically perfect, this equation, after expanding the first two derivative terms, becomes: ∂ρ ∂ρ ∂u ∂T ∂T ∂ ∂T c T g + c T u g + c T ρ + c ρ g + c ρ u g = λ g +˙q (B.18) g g ∂t g g ∂x g g g ∂x g g ∂t g g ∂x ∂x ∂x I which may be re-arranged as ∂ρ ∂ρ ∂u ∂T ∂T ∂ ∂T c T g + u g + ρ + c ρ g + c ρ u g = λ g +˙q . (B.19) g g ∂t ∂x g ∂x g g ∂t g g ∂x ∂x ∂x I

The expression enclosed by the square brackets is identically zero by Eq. B.5, and, thus, the energy balance simpliﬁes to: ∂T ∂T ∂ ∂T c ρ g + c ρ u g = λ g +˙q . (B.20) g g ∂t g g ∂x ∂x ∂x I

Applying the transformation to the density-weighted coordinate space, x, Eq. B.20 becomes: ∂Tg ρg ∂ ρg ∂Tg cgρg = λ +˙qI. (B.21) ∂t ρg0 ∂x ρg0 ∂x

Invoking Eq. B.12, this simpliﬁes to:

∂T λ ∂2T q˙ c g = 0 g + I . (B.22) g 2 ∂t ρg0 ∂x ρg

91 Deﬁning α0 = λ0/(cgρg0), ∂T ∂2T q˙ g = α g + I . (B.23) 0 2 ∂t ∂x ρgcg

The interphase heat transfer term,q ˙I, is simply the heat transfer by convection between a single particle and the surrounding gas multiplied by the number density of particles:

q˙I = NhA(Ts − Tg). (B.24)

Here, h is the convective heat transfer coeﬃcient between the particle and gas, and A is the surface area of the particle. Since the particles are spherical, A =4πr2,wherer is the instantaneous particle radius. Denoting the mass per particle as m and the particle concentration as B, the number density can be re-written as follows:

B N = . (B.25) m

By dimensional analysis, h may be given by:

hd Nu = (B.26) λI

provided eﬀects due to Stefan ﬂow are negligible [17]. Here, the particle diameter, d,has been taken as the characteristic length. For a spherical particle that is stationary relative to the surrounding gas, it can be shown that Nu = 2. Thus,

λ h = I . (B.27) r

Invoking the invariance of N/ρg, the eﬀect of thermal expansion may be eliminated from the interphase heat transfer term:

∂T ∂2T N g = α g + 0 2πλ r(T − T ). (B.28) 0 2 I s g ∂t ∂x cgρg0

In accordance with Eq. B.25, B0 N0 = (B.29) m0

where B0 is the initial mass concentration of the solid phase; and m0 is the initial mass per

particle. Assuming that, initially, all particles have the same radius, r0,

3B N = 0 (B.30) 0 πρ r3 4 s 0

92 where ρs is the density of the solid, here assumed constant. Thus, the ﬁnal reduced form of the gas-phase thermal energy equation is:

∂T ∂2T 3B λ g = α g + 0 I r(T − T ). (B.31) ∂t 0 ∂x2 ρ c ρ r3 s g 2 s g g0 0

B.7 Oxidizer Continuity

The oxidizer is treated as a gas that diﬀuses through the bulk gas phase according to Fick’s law. Hence, ∂ ∂ ∂Y (ρ Y )+ ρ uY − ρ D = −ω.˙ (B.32) ∂t g ∂x g g ∂x where Y is the normalized mass fraction of oxidizer, andω ˙ is the rate of depletion of oxidizer per unit volume normalized with Cu/ρg0. The oxidizer continuity is formulated in terms of Y because it is independent of gas expansion, unlike the oxidizer concentration, C. Y is related to C by Y = ρg0C/ρgCu,whereCu is the unburned oxidizer concentration. Expanding the derivatives, ∂ρ ∂ ∂Y ∂Y ∂ ∂Y Y g + Y (ρ u)+ρ + ρ u − ρ D = −ω˙ (B.33) ∂t ∂x g g ∂t g ∂x ∂x g ∂x

and rearranging, ∂ρ ∂ ∂Y ∂Y ∂ ∂Y Y g + (ρ u) + ρ + ρ u − ρ D = −ω.˙ (B.34) ∂t ∂x g g ∂t g ∂x ∂x g ∂x

The expression in the square brackets vanishes by Eq. B.5. Hence, the equation for oxidizer continuity simpliﬁes to: ∂Y ∂Y ∂ ∂Y ρ + ρ u = ρ D − ω.˙ (B.35) g ∂t g ∂x ∂x g ∂x

Introducing the density-weighted coordinate, x, ρ2 ∂Y ρg ∂ g ∂Y ρg = D − ω.˙ (B.36) ∂t ρg0 ∂x ρg0 ∂x

Invoking Eq. B.13, and rearranging,

∂Y ∂2Y ω˙ = D − . (B.37) 0 2 ∂t ∂x ρg

93 The rate of depletion of oxidizer per unit volume,ω ˙ ,isrelatedtotherateofdepletionof

oxidizer per particle,ω ˙ p,by: ρg0 ω˙ = N ω˙ p. (B.38) Cu

whereω ˙ p = AkeffCu where Cu is the concentration of oxidizer in the unburned gas, which, expressed in terms of Y ,is: ρg ω˙ p = AkeffCu Y (B.39) ρg0 Therefore, ω˙ = NAkeffY. (B.40) ρg

The effective rate of reaction per unit particle surface area, keff can be written as −1 κr Ea keff = κ +exp . (B.41) DI RuTs

Finally, invoking Eqs. B.14 and B.15, and noting that A =4πr2, the equation of oxidizer conservation reduces to: ∂Y ∂2Y κ r E −1 = D − 4πN κ r2 0 +exp a Y. (B.42) 0 2 0 0 ∂t ∂x DI0 RuTs

B.8 Solid Phase Continuity

A fundamental assumption of the model is that the particle, or fuel, concentration is suﬃ- ciently high so that the solid phase behaves as a continuum in the classical sense. However, no bulk diﬀusive transport takes place within the solid phase. Thus, the species balance for the solid phase takes the following form:

∂B ∂ + (uB)=−ω˙ . (B.43) ∂t ∂x f

whereω ˙ f is the rate of depletion of fuel per unit volume. It is related to the rate of depletion of oxidizer per particle,ω ˙ p,by:

ω˙ f = Nγω˙ p (B.44)

where γ is the mass-based stoichiometric ratio—that is, the mass of fuel consumed per unit mass of oxidizer consumed. Invoking Eq. B.25,

∂ ∂ (Nm)+ (uNm)=−Nγω˙ . (B.45) ∂t ∂x p

94 Expanding the derivatives,

∂N ∂N ∂u ∂m ∂m m + mu + mN + N + Nu = −Nγω˙ . (B.46) ∂t ∂x ∂x ∂t ∂x p

Invoking the invariance of N/ρg,

N ∂ρ N ∂ρ N ∂u ∂m ∂m m 0 g m 0 u g m 0 ρ N Nu −Nγω + + g + + = ˙ p (B.47) ρg0 ∂t ρg0 ∂x ρg0 ∂x ∂t ∂x

Rearranging, N ∂ρ ∂ρ ∂u ∂m ∂m m 0 g u g ρ N Nu −Nγω . + + g + + = ˙ p (B.48) ρg0 ∂t ∂x ∂x ∂t ∂x

The expression enclosed by the square brackets is identically zero by Eq. B.5, and the solid phase continuity equation simpliﬁes to:

∂m ∂m N + Nu = −Nγω˙ . (B.49) ∂t ∂x p

Dividing through by N, ∂m ∂m + u = −γω˙ . (B.50) ∂t ∂x p Introducing the density-weighted coordinate, x,

∂m = −γω˙ (B.51) ∂t p or ∂m ρ κ r E −1 2 g 0 a = −γ4πr Cuκ +exp Y. (B.52) ∂t ρg0 DI0 RuTs m 4 πr3ρ As the particles are spherical, = 3 s. Hence, ∂m ∂r =4πr2ρ . (B.53) ∂t s ∂t

Finally, combining this equation with Eq. B.16, the equation of solid phase continuity simpliﬁes to: ∂r γC κ κ r E −1 = − u 0 0 +exp a Y. (B.54) ∂t ρs DI0 RuTs

95 B.9 Solid Phase Thermal Energy

Starting from the assumption that the solid phase is a continuum with no bulk diﬀusive transport, the thermal energy balance of solid phase for a diﬀerential control volume is:

∂ ∂ (c BT )+ (uc BT )=q ˙ − q˙ (B.55) ∂t s s ∂x s s r I

whereq ˙I is the same interphase heat transfer term as in the gas phase energy equation; and

q˙r is the heat released by the reaction per unit volume. Invoking Eq. B.25, ∂ ∂ (c NmT )+ (uc NmT )=q ˙ − q˙ . (B.56) ∂t s s ∂x s s r I

Assuming that the solid phase heat capacity is constant and expanding the derivatives,

∂N ∂ ∂ ∂ c mT + c mT (uN)+c N (mT )+c uN (mT )=q ˙ − q˙ . (B.57) s s ∂t s s ∂x s ∂t s s ∂x s r I

Invoking the invariance of N/ρg and rearranging, N ∂ρ ∂ ∂ ∂ c mT 0 g ρ u c N mT c uN mT q − q . s s + ( g ) + s ( s)+ s ( s)= ˙r ˙I (B.58) ρg0 ∂t ∂x ∂t ∂x

The expression in the square brackets vanishes by Eq. B.5, and thus,

∂ ∂ c N (mT )+c uN (mT )=q ˙ − q˙ . (B.59) s ∂t s s ∂x s r I

Deﬁning the heat of reaction per unit mass of fuel as q, the rate of heat released per unit volume may be expressed as:

q˙r = Nqγω˙p. (B.60)

Combining equations B.24, B.59, and B.60,

∂ ∂ c N (mT )+c uN (mT )=Nqγω˙ − NhA(T − T ). (B.61) s ∂t s s ∂x s p s g

As with the derivation for the solid phase continuity equation, the number density turns out to be a common factor for all the terms of the solid phase energy equation. Therefore,

∂ ∂ c (mT )+c u (mT )=qγω˙ − hA(T − T ). (B.62) s ∂t s s ∂x s p s g

96 Introducing the density-weighted coordinate, x,

∂ c (mT )=qγω˙ − hA(T − T ). (B.63) s ∂t s p s g

Expanding the derivative,

∂T ∂m c m s + c T = qγω˙ − hA(T − T ). (B.64) s ∂t s s ∂t p s g

Using Eq. B.51, ∂T c m s = γω˙ (q + c T ) − hA(T − T ). (B.65) s ∂t p s s s g

−1 Cu κr Ea 4 3 2 λI Recalling thatω ˙ p = Aκρg +exp Y , m = πρsr , A =4πr ,andh = , ρg0 DI RuTs 3 r −1 1 ∂Ts Cu κr Ea λI csρsr = γρgκ +exp (q + csTs) − (Ts − Tg). (B.66) 3 ∂t ρg0 DI RuTs r

Finally, invoking Eq. B.16, and rearranging, the equation of solid phase thermal energy conservation simpliﬁes to:

− ∂T 3γ κ r E 1 q Y λ T − T s = κ C 0 +exp a + T − 3 I s g . (B.67) 0 u s 2 ∂t ρs DI0 RuTs cs r csρs r

97 Appendix C

Validation of Numerical Methods

The governing equations of the models in Chapters 2 and 3 are integrated using MATLAB. The set of governing equations in Chapter 2 is integrated using MATLAB’s native ordinary differential equation (ODE) solvers. The flame model is integrated using the method of lines, which is a general technique for the solution of time-dependent partial differential equations [99]. In this method, the spatial derivatives are replaced with algebraic approximations, and this results in a system of initial value ordinary differential equations. The resulting system can then be integrated using the native ODE solvers in MATLAB. The governing equations considered here were integrated using a second-order finite difference formulation together with a a multi-step, variable-order, implicit time integration method based on the code provided in [99]. In order to validate this method, the solutions are compared to benchmark studies in the literature that use a similar time dependent flame models.

C.1 Code Veriﬁcation

To verify the code, a classical flame propagation problem with a well known solution was fed into it. The problem is that of one-dimensional, laminar flame propagation through a homogeneous gaseous mixture characterized by the concentration of a single deficient species with no heat losses. The solution produced by the current numerical method is compared to that found by Zeldovich and Barenblatt [94]. The governing equations of this model are as follows:

∂n ∂2n − = −znexp (−E /R T )(C.1) ∂t ∂x2 a u ∂T α ∂2T Q − = znexp (−E /R T )(C.2) ∂t D ∂x2 c a u

98 2500 t = 8 t = 10 2000 t = 12 t = 14

1500 C] ◦ [ T 1000

500

0 5 10152025 x (a) Output of code used for thesis. (b) Solution presented in [94].

Figure C.1: Temperature proﬁles of the simpliﬁed Zeldovich and Barenblatt model.

where n is the concentration of the deficient species, normalized with the initial concen- T α D tration; is the temperature; is the thermal√ diffusivity of the mixture; is the mass diffusivity of the deficient species; x = y/ D,wherey is the coordinate normal to the

flame front; z is the pre-exponential factor; Ea is the activation energy; Ru is the universal gas constant; Q is the heat of reaction; and c is the specific heat of the mixture. Like the governing equations presented in this thesis, this is a system of so-called reaction-diffusion equations. Thus this problem is mathematically similar and serves as a useful benchmark for the numerical method used. The evolution of the solution is observed over a semi-infinite domain (0 ≤ x<∞)with the following auxiliary conditions:

for x>0,t=0,n=1andT = T0

for x =0,t≥ 0,n=0andT = T0 + Q/c.

In accordance with [94], the equations are numerically integrated with the following values ◦ 3 ◦ 4 4 for the mixture properties: T0 = 300 C; Q/c =2× 10 C; Ea/Ru =1.5 × 10 K; z =10 s−1;andα/D = 2. The results are shown in Figs. C.1 and C.2. The corresponding ﬂame proﬁles presented in [94] are juxtaposed for comparison. It is readily seen that the code satisfactorily reproduces the solution presented by Zeldovich and Barenblatt.

99 1

0.9 t = 8 t = 10 0.8 t = 12 t = 14 0.7

0.6

[-] 0.5 n 0.4

0.3

0.2

0.1

0 5 10152025 x (a) Output of code used for thesis. (b) Solution presented in [94].

Figure C.2: Relative concentration proﬁles of the simpliﬁed Zeldovich and Barenblatt model.

100 0.09 0.15 Solid phase temperature 0.08 Solid phase temperature Gas phase temperature Gas phase temperature Oxidizer concentration 0.07 Oxidizer concentration

0.06 0.1 erence 0.05 erence ﬀ ﬀ

0.04 Relative di 0.03 Relative di 0.05

0.02

0.01

0 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.005 0.01 0.015 0.02 0.025 0.03 x [m] x [m] (a) Fuel-lean case. (b) Fuel-rich case.

Figure C.3: Relative diﬀerence between solutions computed at diﬀerent spatial step-sizes.

C.2 Demonstration of Grid-Independence

For the grid-convergence study, the effect of the spatial step-size on the flame profile and burning velocity was found for representative fuel-lean (φ =0.5) and fuel-rich (φ =1.5) cases. The spatial step-size used for all solutions was 10−4 m. The time integration method used was adaptive and, thus, did not allow parametric variation of the temporal step-size. The relative difference between solutions for step-sizes of 10−4 mand5×10−5 m is plotted in Fig. C.3 for both the lean and rich cases. The plots show that halving the spatial step-size yields a relative difference of less than 9 % in the lean case and less than 15 % in the rich case. This result is satisfactory because only qualitative results are of interest in this study. Table C.1 summarizes the results for the burning velocity. The relative difference in both cases is less than 0.5 %. Thus, in terms of burning velocity, the step-size used is more than satisfactory.

φ Spatial step-size ub Relative diﬀerence 0.5 10−4 m 3.7916 cm/s 0.5 5 × 10−4 m 3.7921 cm/s 0.014 % 1.5 10−4 m 4.65 cm/s 1.5 5 × 10−4 m 4.67 cm/s 0.4 %

Table C.1: Eﬀect of spatial step-size on calculated burning velocity.

101 Appendix D

Laser Absorption Diagnostic

This section outlines the broadband absorption technique utilized in Chapter 4 of the thesis. While only broadband visible light was required for the measurements in that paper, an extension of the diagnostic technique to produce broadband ultraviolet (UV) light is required for future work as discussed in the Direction of Future Research section of the conclusion. The original work in the following section was published by the present author in [4] and outlines both the construction and characteristics of the broadband laser system developed for the diagnostics used in Chapter 4 and a novel system to extend the technique to the UV.

Ultraviolet (UV) Absorption Spectroscopy in Optically Dense Fireballs Using Broadband Second Harmonic Generation of a Pulsed Modeless Dye Laser, Applied Spectroscopy [4]

D.1 Introduction

Combustion measurements within the ﬁreballs generated by explosives and other energetic materials have several diagnostic hurdles owing to the multiphase dynamics, the high temperatures, short time scales, and high optical depths of the combustion event. Explosives are often enhanced by the addition of solid phase fuels like metal particles which create high temperature environments of particulate and gas phases within the ﬁreball. The relevant diagnostic time scales within explosives systems are typically on the order of several microseconds necessitating diagnostics which can be performed with this time resolution. One of the key diagnostic capabilities is the characterization of temperatures within

102 the fireball and the ability to monitor chemical species that evolve during the combustion. Optical techniques are more robust within the high temperature, multiphase environments where mechanical probes methods are difficult to implement due to temperature limitations and slow response time. In fireballs containing reactive particles, it is possible that the temperatures of the solid phase and gas phase may be different due to kinetically limited and diffusion limited particle reactions. Spectroscopic diagnostics may be able to analyze the possible separation of particle and gas temperatures by examining the gas phase components of the flame thereby demonstrating the regimes of particle combustion. The high luminosities of fireballs of energetic materials lend to the prevalence of emission based diagnostics. The condensed phase temperature is often determined by using optical pyrometry or by fitting the continuum from the condensed phase to a Planck distribution, and this technique has been used as a temperature diagnostic in energetic materials in many studies [108, 34]. There is also precedent for using metal salt tracers to introduce vaporized atomic species into flames to determine electronic excitation temperatures which can be extended to diagnostics of explosives [111]. Lewis et al. have shown that by using atomic tracer elements seeded into explosives that a gas phase temperature can be measured from the relative intensities of the resulting atomic lines in the emission spectrum of the fireball [140]. Emission from the diatomic species present in the combustion of energetic materials (e.g. AlO) can also be fit to theoretical transitions to determine temperatures and concentrations [124]. Emission spectroscopy measurements probe upper (excited) electronic states, and, in many cases, especially when the upper state is strongly populated by exothermic reactions, the population distribution in the upper state is not representative of the more abundant ground state molecules [111]. In general, a temperature derived from an emission measurement should be checked for rotational/vibrational equilibrium or by comparing to an absorption measurement. In the particular case of optically thick fireballs, it has been shown that the detonation products of gram scale and higher experiments that the attenuation lengths are on the order of a few centimeters implying that the temperatures derived from the emission measurements may be only representative of the outer edge of the fireball [141]. The preferred measurement, in these cases, would be to probe the ground state in absorption which provides quantifiable information on species concentration and temperature, though the measurements are path averaged. However, absorption measurements in fireballs are difficult owing to the intense luminosity of the fireball and the high intensity and coherence required from a broadband source to penetrate the fireball in order to perform the measurement. The limitations of emission spectroscopy have prompted several experiments to probe

103 of the inside of explosive fireballs. Lewis et al. found no significant difference in apparent temperatures measured by emission spectroscopy of the interior and the outer edge of an RDX explosive cloud revealing the possibility of temperature uniformity throughout the fireball at certain times during the combustion [142]. Carney et al. demonstrated the use of a booster optical amplifier to measure the absorption of water vapor in the near-infrared (NIR) and obtained a line-of-sight temperature measurement in a high explosive [143]. In previous work, Glumac was able to show proof-of-concept experiments using a modeless dye laser as a spectral source for absorption spectroscopy measurements in optically thick fireballs [113]. Modeless dye lasers offer a widely tunable, relatively high bandwidth (≈1- 10 nm) coherent source and have moderate divergence and high power. In optically thick regions, there is difficulty in passing sufficient signal to a detector from other broadband sources due to the scattering of the light. The dye laser offers an intense, directional beam allowing the strong rejection of the fireball luminosity and sufficient signal at the detector even at 98% attenuation. Dye beam wavelengths are limited to above the near-ultraviolet (UV) due to the nature of the available dyes. A great number of important diatomic and atomic transitions occur below these wavelengths necessitating broadband UV spectral sources that can access the transitions. The advantage of dye lasers is that almost any visible wavelength can be reached by the appropriate selection of dye. By utilizing broadband second harmonic generation (SHG) of the dye laser beam using a nonlinear crystal, the wavelength range can be extended into the deep UV. Absorption measurements of several more atomic and diatomic species become possible for diagnostics ranging from the transitions found in hydrocarbon flames (e.g. OH, CN, NO) to those associated with the combustion of energetic materials (e.g. Ni, Fe, BO, AlCl). In this work, energetic systems are examined, and a stationary gas of NO is used to calibrate the diagnostic. Time resolution of the diagnostic is demonstrated by probing Al atomic line absorptions in exploding bridge wires (EBW). Finally, the diagnostic is used to obtain spectra from diatomic species of metal-fluorocarbon reactions (MgF and AlF) in dispersed flash powders which are of recent interest to enhance combustion performance in energetic materials [144, 145]. We fit the spectra to a model in order to confirm the species and yield path averaged temperatures.

104 D.2 Diagnostic Setup

D.2.1 Dye Laser Conﬁguration

The schematic of the entire setup is shown in Fig. D.1. A frequency tripled, compact neodymium-doped yttrium aluminum garnet (Nd:YAG) laser (Quantel) at 355 nm with 5 ns pulses at 20 Hz is used as the pump for the dye laser. The maximum power available from the laser is approximately 60 mJ per pulse. The built in flash-lamp/Q-switch delay on the laser is used to adjust the power to pump the dye with 2-4 mJ per pulse which produces sufficient dye laser energy for the experimental setup. The custom dye laser configuration is demonstrated in previous work by Glumac to produce a modeless beam with about 5% efficiency [113]. The 355 nm Nd:YAG beam is passed through a cylindrical lens to produce a horizontally spread beam and then focused down in the vertical direction by a second cylindrical lens onto a quartz dye cuvette 40 mm in length. The cuvette is tilted at a 20 degree angle to minimize etaloning effects. A full reflecting mirror is placed at one end of the laser cavity approximately 10 cm from the cuvette, though for a modeless laser this distance may vary without much consequence. The laser has no output coupler, and the front end is empty aside from an iris to prevent stray light and an achromatic collimating lens with a focal length of 175 mm to counteract the moderate divergence of the beam. In cases where higher power and better beam shape is desired, Glumac shows that a partially reflecting mirror may be used at the front end of the cavity as long as the cavity is designed with very low finesse [113]. Modeless dye lasers designed for broadband coherent anti-Stokes Raman spectroscopy (CARS) utilize Bethune dye cells and pass the initial dye beam through an amplifier to change the beam shape and divergence [146]. The power of the pump beam may be increased by using a flowing dye cell since stationary dyes tend to degrade quickly under high pump powers. The advantage of the modeless configuration is the broadband character which is found to have a much smoother spectral distribution of intensity [113]. The power of the dye beam produced by an empty cavity front end and a stationary dye cell is found to be sufficient for the systems examined in this work. The Coumarin 450, 460, and 540 (Exciton) dye laser pulse energies are measured to be 0.1 0.2 mJ per pulse for a pump beam energy of 2-4 mJ.

D.2.2 Second Harmonic Generation

The wavelengths produced by dye lasers are limited to wavelengths above 330 nm for Nd:YAG pumped dyes. To extend deeper into UV spectrum to access a much greater number of

105 Figure D.1: Schematic of the entire setup for the absorption measurement. A pulsed Nd:YAG (355 nm) pumps a dye cell in a grazing incidence configuration. The modeless dye beam is then focused onto a BBO-I crystal, and the resulting UV beam is collimated and separated from fundamental and passed through the combustion chamber. species, the beam can be passed through a non-linear crystal for SHG. Critical phase matching to the non-linear crystal allows for efficient SHG of a monochromatic beam by simply rotating the crystal to the optimum angle determined by the crystal cut. Wavelengths in a broadband beam which are outside the phase-matching bandwidth, determined by the thickness of the crystal and the diameter of the beam incident on the crystal, will typically have very low SHG efficiencies. The beta barium borate (BBO) type I, 7mm crystal with a cut angle of 52 degrees (Radiant Dyes) used in this study theoretically has less than 0.1 nm of phase matching bandwidth at the desired wavelengths for NO absorption. Shorter crystals will theoretically provide higher spectral bandwidth at the expense of conversion efficiency. The broadband nature of ultra-short laser pulses has led to the development of several techniques for increasing the phase matching bandwidth in crystals using chromatic dispersion of the fundamental beam [147, 148, 149, 150]. Due to the relatively low power of the dye beam in this experimental setup, the use of any method of increasing the bandwidth which requires a significant loss in power is generally not desired. Dispersive elements can cause significant losses due to reflections of the required polarization for SHG in the nonlinear

106 crystal at the surfaces of the dispersive element. Unlike the large bandwidth of ultra-short pulses (typically around 100 nm), the bandwidth of the second harmonic pulse in the present experiments is only few nanometers in wavelength since the bandwidth of the fundamental dye beam has only around 10 nm of bandwidth at maximum. Previous studies indicate that by simply creating a strong focusing condition of the fundamental beam into the crystal, a significant bandwidth in the frequency doubled beam can be achieved at a fraction of the signal loss compared to dispersive methods [151, 152]. To demonstrate this in the current experiment, the spectral bandwidth of the SHG of a Coumarin 540 dye beam under different focusing conditions is plotted in Fig. D.2. The crystal is first rotated to the central phase matching angle of the dye beam, and the lenses are varied without changing the crystal angle. The tight focus decreases the effective interaction length thereby increasing the spectral bandwidth.

Figure D.2: Spectral bandwidth of the second harmonic for diﬀerent focal lengths of the lens used to focus the fundamental beam onto the nonlinear crystal.

Without spectral dispersion, the fundamental dye beam is focused onto the crystal with a 50 mm focal length achromatic lens. The crystal is then rotated until the second harmonic appears and adjusted to the desired spectral position. The combined fundamental and second harmonic beams are then re-collimated using a UV fused silica (UVFS) 25 mm focal length plano-concave lens. It is found in previous studies under strong focusing condition, the UV beam emerges nearly collimated in the horizontal direction [148]. The re-collimating lens allows the collimation of the beam in the vertical direction but causes divergence in the horizontal direction necessitating cylindrical correction optics to shape the beam in the horizontal direction (see Fig. D.1). The combined fundamental and second harmonic emerging from the crystal are ﬁrst passed through a UVFS dispersion prism with an apex

107 angle of 60 degrees to separate the fundamental from the second harmonic and then shaped with a UVFS cylindrical inverse telescope consisting of a 75 mm and 25 mm focal length UVFS plano-convex cylindrical lens. The resulting beam is approximately 5 mm in diameter as seen on a UV viewing card. The spectral proﬁles of an example dye beam (Courmarin 450 at 920 μmol/L concentration in ethanol) and the second harmonic are shown in Fig. D.3. The pulse energy from the second harmonic of the Coumarin 450, 460, and 540 dye beam is determined to be 1-5 μJ per pulse.

Figure D.3: (top) Spectral bandwidth of modeless dye beam with Coumarin 450 as the dye averaged over several pulses. (below) The spectral proﬁle of the second harmonic.

After passing through the combustion chamber, the UV beam is focused onto the 50 μm slit of a customized 1260 mm focal length Czerny-Turner SPEX 1250 spectrometer with deviation angle of 11 degrees and a 3600 grooves/mm UV holographic grating (Optomet- rics). The detector used for NO absorption is a GARRY 3000 which is a CCD line array of 3000 pixels each 7×200 μm. The approximate resolution achieved by this spectrometer is approximately 0.01 nm full-width-half-maximum (FWHM) determined by the FWHM of an elemental calibration line over a range of about 4 nm on the detector. The detector used in ﬂash powder tests is a Hamamatsu C7041 CCD array with 1044×255 pixels each 25×25 μm with an approximate resolution of 0.013 nm FWHM over about a 4 nm range. The spectrum

108 is binned over the entire chip. Even at low dye beam powers, the generated SHG UV beam is suﬃcient to saturate the spectrometer detectors with a single pulse.

D.3 Results and Discussion

D.3.1 Stationary NO Gas

The diagnostic is first calibrated with the NO A←X (0,0) transition in the 226 nm to 227 nm range, the fundamental dye beam is created by pumping a solution of Coumarin 460 (Exciton) in ethanol at 0.005 mol/L. The resulting spectral bandwidth of the UV beam generated from the strong focusing of the dye beam onto the crystal is approximately 1.5 nm at FWHM. The spectral position on the detector is calibrated using a Ni-Ne hollow cathode lamp (Azzota) and checked against the position of the strong NO transitions. A test system of NO is generated in a sealed chamber with fused silica windows using a continuous electric arc discharge in air. The electrodes are mounted within a 5 cm wide sealed chamber with quartz windows. The amount of NO produced in the chamber is correlated to the duration of the arc. The laser is set to pulse continuously (20 Hz), and integration time is set to one second on the detector in order to obtain an averaged spectrum over the 20 pulses. A reference spectrum is first taken in the empty chamber. An optically thin spectrum of NO is obtained by producing an arc for five seconds or until only a slight absorption feature is qualitatively observed in the raw spectrum. The resulting spectrum and comparison to the LIFBASE simulation[153] is shown in Fig. D.4. Good agreement is shown between the simulation and resulting averaged spectrum from several laser shots in the optically thin limit. A single pulse measurement is then taken in a high concentration of NO to simulate conditions in an optically dense fireball medium. The detector is set to a two second integration time and a reference pulse averaged over ten pulses taken in the empty chamber. An arc is then produced in the chamber for 30 seconds until the absorption signal is noticeably visible over the noise from the laser, and a second pulse is recorded. The transmittance from the single pulse measurement is plotted in Fig. D.5. A simulation of the NO absorption spectrum is provided by LIFBASE for comparison using the resolution of 0.013 nm obtained by spectrometer calibration and set to room temperature at 300 K. The single pulse absorption measurement produces a high signal to noise spectrum across a 1 nm region with approximately 30 well resolved rotational lines. A comparison of the optically thick with an optically thin LIFBASE spectrum shows clearly the effects of optical depth in the spectrum, which enhance weaker lines at the expense of stronger ones.

109 Figure D.4: NO spectrum at low concentration from continuously pulsed laser (20 Hz) integrated over 1 second compared to spectrum simulated by LIFBASE at T = 300 K.

D.3.2 Time Resolution in Exploding Bridge Wires

The ability to time resolve species absorption in the UV is demonstrated in the vapor cloud produced by an aluminum EBW. Previous work [113] showed that Al concentration in an exploding bridgewire cloud varies strongly over the first millisecond after breakdown. The same dye mixture used in the stationary NO gas measurement is used to probe the spectral range from 226 nm to 227 nm. To create aluminum bridge wires, aluminum foil is cut into thin strips approximately 0.5 mm wide × 25 μm thick. The thin strips are then clamped between two electrodes set approximately 1 cm apart. The center of the UV beam is then positioned approximately 1 cm above the bridgewire. The electrodes are then connected to a Teledyne RISI FS-43 fireset synchronized to the laser pulse output by a delay generator (Quantum) to allow for measurements at different times after the bridgewire initiation. An optical beam shutter (Thor Labs) is placed in front of the combustion chamber and connected to the pulse generator. Several single shot experiments are conducted with the absorption probe beam passing through the cloud at different times after breakdown. Before each wire is initiated, a reference pulse averaged over ten pulses is taken. The resulting spectra for aluminum absorption lines at 200 s increments after the initiation of the bridge wire are shown in Fig. D.6. Two atomic absorption lines, the Al I 2 o →2 2 o →2 ( P1/2 D3/2) transition at the lower wavelength and the Al I ( P3/2 D5/2) transition at the upper wavelength,[154] are apparent in the spectrum. It is seen initially that after the explosion, the absorption lines are extremely broad while gradually thinning out at later times. The broadening is attributed to the increase in equivalent width due to the optically

110 Figure D.5: NO spectrum under optically thick conditions from a single pulse compared to spectrum simulated by LIFBASE at T = 300 K. thick spectrum. The observed linewidth far exceeds the expected pressure, Doppler, and Stark broadened linewidths in the bridgewire environment.

D.3.3 Metal-Teﬂon Flash Powders

Flash powders mixed with PTFE powder are used to examine ﬂuorinated metal species in absorption in a solid fuel combustion environment. The spectrum of AlF is examined in a

flash powder a powder of containing KClO4/PTFE/Al (57/10/33 by wt. %). The spectrum of MgF is examined in a flash powder containing KClO4/PTFE/Mg (40/10/50 by wt. %). An EBW is used to disperse and ignite the powders. A tungsten bridgewire with 0.005 cm diameter (McMaster Carr) is attached between the electrodes in the combustion chamber. Testing the tungsten wire in the same manner as described in the previous section revealed no absorption features in the regions of interest at any time after the initiation of the bridgewire making it an ideal candidate for dispersing powders. A piece of non-conductive tape is placed directly under the bridgewire to support a small pile of flash powder (5 mg) that rests on top of the wire. The bridge wires are initiated and spectrum recorded in the same manner as described in the previous section.

D.3.4 Spectroscopic Model for AlF and MgF

The procedure for calculation of line positions and intensities used in this study is outlined in Arnold et al. utilizing the approximation to the Voigt proﬁle provided in the reference [155]. The spectroscopic constants for the aluminum AlF A1Π←X1Σ+ transition are taken

111 Figure D.6: Al atomic line absorption in EBWs taken in a series of experiments at diﬀerent times after bridge wire initiation.

from Rowlinson and Barrow [156]. The Franck-Condon factors for the first five vibrational levels of the Δν=0 transitions are taken from Liszt and Smith [157]. Due to the nearly vertical nature of the transition, factors for five subsequent vibrational levels are estimated to be unity. Spectroscopic constants for the magnesium MgF B2Σ+←X2Σ+ transition are taken from Huber and Herzberg [158]. Frank-Condon Factors for the first five levels of the the Δν=0 transition were taken from Maheshwari et al. [159]. Expressions for Hönl-London factors for both molecules are taken from Kovács [160]. The calculation of the absorption coefficients is outlined by Luque and Crosley [153]. The classical Beer’s law expression is used to describe the absorption. There are five adjustable parameters for the simulation of absorption spectrum taking into account the optical thickness: the temperature, a factor of non-resonant absorption/scattering, a factor of the amount of light that passes through the absorbers, an effective product of the number density and path length, and the effective resolution. The slit function of the spectrometer setup is approximated as a Lorentzian function. The transmittance is plotted as a function of the wavelength. The fitting parameters are the same as described in previous work in modeling the AlO absorption spectrum [107]. The model is meant to be primarily used to verify the species being probed. The quantitative accuracy of the temperature measurement is not assessed in this work.

112 D.3.5 MgF Detection in Dispersed Flash Powder Combustion

The MgF B2Σ+←X2Σ+ Δν=0 bandhead is observed at 268.85 nm. In order to reach this transition, a Coumarin 540 dye at 0.0035 mol/L concentration is used to produce the fundamental beam. The UV pulse energy is measured to be 3 μJ. The spectral position on the detector is calibrated using the atomic lines produced by a Fe-Ne hollow cathode lamp (Analyte). The spectral bandwidth after doubling is approximately 2 nm with a profile similar to the bandwidth in Fig. D.2. The combustion of the dispersed magnesium flash powder provides moderate optical thickness of the MgF spectrum at times 500-600 s after the initiation of the tungsten EBW. The resulting spectrum is shown in Fig. D.7 along with the simulation at 3000 K with an effective resolution of 0.015 nm. Despite the path averaged nature of the measurement, a fair fit to the spectrum is obtained using a single temperature.

Figure D.7: The absorption spectrum of MgF in ﬂash powder combustion ﬁt with a 3000 K simulation.

D.3.6 AlF Detection in Dispersed Flash Powder Combustion

The AlF A1Π←X1Σ+ Δν=0 transition is observed with a strong absorption feature with a peak at 227.4 nm. The same Courmarin 460 dye used for the aluminum bridge wire tests is used to produce the fundamental beam. The crystal is angle adjusted to center the beam in the 227 to 228 nm region of interest for AlF. The resulting pulse energy is approximately 1 μJ. The spectral position is calibrated by the Ni-Ne hollow cathode lamp. Measurements of AlF are taken approximately 800-1000 μs after the initiation of the EBW, and produce noticeably optically thick spectrum. There is considerable diﬃculty in obtaining an optically

113 thin spectrum since the highly optically thick signal rapidly decays to negligible signal. An example spectrum of the AlF transition is shown in Fig. D.8 with a simulation at 2800 K with an eﬀective resolution of 0.015 nm. There is disparity of the model in the blue-degraded absorption feature at 227.4 nm possibly due to an unaccounted source of broadening or from the relative uncertainty of the spectroscopic constants in the literature.

Figure D.8: The absorption spectrum of AlF in ﬂash powder combustion ﬁt with a 2800 K simulation.

The intensity and coherence of the UV beam allows for easy alignment of the spectrometer in the second order of the grating. In the second order, the dispersion and resolution are more than doubled. The theoretical instrument resolution becomes 0.0026 nm with a range of 1.4 nm. The second order spectrum is shown in Fig. D.9. It is observed that pulse noise is increased. Although the majority of the rotational lines are too closely spaced to be resolved, a new feature appears in the main spectrum at about 227.5 nm which also appears in the simulated spectrum. The second order spectrum is again ﬁt to a temperature of 2800 K.

Absorption Saturation

High pulse intensities can optically pump the absorption medium so that sufficient population in the in excited state may cause non-linear absorption effects. Full saturation is considered when the excited state population is driven by the laser radiation to match the ground state. In this scenario, the medium becomes effectively transparent to the laser light and no absorption will take place. The limiting pulse energy is determined for NO A2Σ←X2Π(0,0) absorption as an example. The spectral range of the laser is considered from 226 nm to 229 nm. A beam radius of 2.5

114 Figure D.9: Second order spectrum of AlF with a simulation at 2800 K. mm and a pulse width of 5 ns are assumed. The calculated maximum pulse energy at which the excited state population reaches 25% is found to be approximately 0.4 mJ. In cases of higher pulse energy, an increase in the beam radius can reduce the overall beam intensity allowing for greater pulse energy at a small expense of spatial resolution in the measurement.

D.4 Summary and Conclusions

The use of a modeless dye laser as a spectral source for an absorption measurement is extended into the UV beyond the limitations of pumped dye beams by using broadband SHG. Absorption measurements of the NO A2Σ←X2Π(0,0) transition and atomic lines in aluminum EBWs are used to calibrate the diagnostic showing convergence in the optically thin limit over multiple pulses and to demonstrate microsecond time resolution in a series of experiments. The diagnostic is used to probe the fireballs of flash powders dispersed by EBWs to examine the optically thick AlF A1Π←X1Σ+ transition, and the MgF B2Σ+←X2Σ+ transition. A single temperature is derived by fitting simulations of the diatomic transitions to the experimental data. A simple analysis of the laser intensity where saturation effects may not be negligible is given for a two level system for wavelengths near NO and AlF transitions. It is found in this setup that the pulse noise and pulse variation do not degrade the absorption measurement significantly. The average standard deviation in amplitude over the laser pulse spectrum for a sample of ten pulses is found to be 15-20%. The noise level of a single pulse, however, is estimated to be 5-10%. Improvements in the measurement can be made by splitting the beam before the combustion chamber and acquiring a simultaneous reference measurement using an array detector as demonstrated by Glumac [113]. Other broadband laser sources such as femtosecond lasers may offer superior noise characteristics,

115 though currently, these tend to be more expensive than the dye based approach. A strong focusing condition of the fundamental dye beam onto the crystal is found to be sufficient to produce approximately 1-3 nm of bandwidth. The SHG beam is able to saturate the detector in a single pulse and provides enough signal to be used for a single shot measurement in the second order of a grating. This diagnostic provides a simple tool for obtaining a path averaged measurement and overcoming the difficulties of absorption spectroscopy in explosive fireballs and other optically thick combustion systems as shown in previous studies using the visible spectrum of dyes [113]. Several atomic and diatomic species can be accessed for measurement by careful selection of dyes and crystals. Following the assumption of approximately a 2 cm attenuation length from previous studies of optical depth in fireballs [141] and assuming a pulse energy around 10 mJ max, the technique is estimated to be able to scale up to a 20 cm path length.

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