A study of strained extinction for applications in natural gas combustion modeling by Alan Everett Long B.S., Case Western Reserve University (2015) S.M., Massachusetts Institute of Technology (2017) Submitted to the Department of Chemical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Engineering Practice at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 c Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Chemical Engineering May 04, 2020

Certified by ...... William H. Green Hoyt C. Hottel Professor of Chemical Engineering Thesis Supervisor

Accepted by...... Patrick S. Doyle Robert T. Haslam (1911) Professor of Chemical Engineering Chairman, Department Committee on Graduate Theses 2 A study of strained extinction for applications in natural gas combustion modeling by Alan Everett Long

Submitted to the Department of Chemical Engineering on May 04, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Engineering Practice

Abstract Resistance to extinction by stretch is a key property of any flame, and recent work has shown that this property controls the overall structure of several important methane, the principal component of natural gas, based turbulent flames. This work gives an introduction to the parameter that quantifies resistance to extinction by stretch, Extinction Strain Rate (ESR), discussing how it is typically measured experimentally and calculated numerically. The primary limitations of ESR, given its historical definition and computation methods, are: (1) it depends on the dimensions of the experimental apparatus used to measure it, (2) it is often too computationally difficult to calculate using large kinetic models, #species > 500, and (3) under elevated pressures relevant to gas turbines and internal combustion engines, 20-40 atm, it is difficult to measure. Work addressing, in some manner, these three issues is presented. Subsequent work then uses ESR as a validation parameter for producing a kinetic model. To address the first issue, a method is proposed for translating experimental measure- ments for stretch-induced extinction into an unambiguous and apparatus-independent quan- tity (ESR∞) by extrapolating to infinite opposing burner separation distance. The unique- ness of the flame at extinction is shown numerically and supported experimentally for twin premixed, single premixed, and diffusion flames at Lewis numbers greater than and less than one. A method for deriving ESR∞ from finite-boundary experimental studies is proposed and demonstrated for methane and propane diffusion and premixed single flame data. The values agree within the range of differences typically observed between experimental measurements and simulation results for the traditional ESR definition. To address issue two, Ember, a new open-source code for efficiently performing ESR∞ calculations using large, detailed chemical kinetic models is presented. Ember outperforms other standard software, such as Chemkin, in computation time by leveraging rebalanced Strang operator splitting which does not suffer the steady-state inaccuracies of most split- ting methods. Ember is then able to improve computational performance primarily though parallelization and use of quasi-steady-state kinetics integrations at each independent spa- tial discretization point. Ember is validated for computation of ESR and the benefits of its computation techniques are demonstrated.

3 With respect to the third issue, Ember is used to explore ESR∞ pressure trends and kinetic model sensitivities in the current absence of experimental methods to probe the relevant parameter space. ESR shows opposing trends with pressure under lean and rich conditions for methane-air flames. Ultimately, ESR decreases at higher pressures for lean conditions (φ = 0.7) and increases with pressure for rich conditions (φ = 1.3). Under both conditions, the ESR trends are non-monotonic. ESR reaction sensitivities are observed to generally mirror those of laminar flame speed calculations. This suggests that there is limited added value in more efficient methods of ESR reaction sensitivity calculation since efficient adjoint methods already exist for laminar flame speed. Strong species transport parameter sensitivities are observed for the fuel, oxidizer, and bath gas, with the limiting reactant showing the strongest sensitivity. Current levels of uncertainty in species enthalpy suggest little impact of enthalpy errors on ESR predictions for the conditions studied. Using the prior ESR improvements and analysis, an ESR validated kinetic model is pro- duced. The model uses a validation data set that includes relevant ignition delay, laminar flame speed, and extinction strain rate data. The work builds off recent works by Hashemi and coworkers for high pressure oxidation of small hydrocarbons. Prediction of the selected validation data set is improved primarily through sensitivity analysis and comparison with kinetic rate constants from other works. Additionally, to support nitrogen chemistry predic- tions, the full nitrogen subset from the recent review by Glarborg and coworkers is appended to the core model produced. The Reaction Mechanism Generator (RMG) software has re- cently been used to generate the rich chemistry relevant for partial oxidation of methane up to one and two ring aromatics. This rich chemistry subset is added to produce the final kinetic model. An important portion of the rich chemistry included within the produced kinetic model is the route from cyclopentadienenyl radical recombination to naphthalene. Since this work seeks performance at elevated pressures of relevance to gas turbines and internal combustion engines, the pathway is re-computed here with explicit consideration of all relevant pressure dependent pathways on the C10H10 and C10H9 surfaces. Lumped, single-step kinetics, often used to describe the net reaction to naphthalene, are observed to be insufficient. Specifically, the C10H10 intermediate species is observed to live long enough to undergo bimolecular reaction to enter the C10H9 surface before proceeding on to naphthalene. Rate expressions for the full network are produced and included in the kinetic model generated.

Thesis Supervisor: William H. Green Title: Hoyt C. Hottel Professor of Chemical Engineering

4 Acknowledgments

My PhD has been an exceptional journey and I would not have made it to this point without help, support, and guidance from a number of incredible people along the way. My first thanks goes to my thesis advisor Prof. Bill Green. Without his consistent mentorship throughout, this thesis would never have come together. Whenever I’ve run into a road block along the way, Bill always seems to have that one key insight, question, or contact to get the wheels turning again. In addition, I’d like to thank my committee members, Prof. Paul Barton and Prof. Ahmed Ghoniem. Our discussions during committee meetings have been an important factor in shaping and guiding my work. Your willingness to invest additional time in meeting with me one on one has been greatly appreciated.

Particular thanks is also due to Dr. Ray Speth whose prior work laid the groundwork for much of this thesis. Additionally, Ray was routinely an essential source of collaboration and guidance throughout my project.

A special thanks to all my colleagues in the Green Research Group; your insights and perspectives have been instrumental and have made my time in the group an enjoyable expe- rience. In particular, I’d like to thank Colin Grambow for assisting with all my programming and Quantum Chemistry questions over the years, Adeel Jamal, Te-Chun (Jim) Chu, and Alon Grinberg-Dana for working with me on the Exxon project, Mengjie (Max) Liu, Mark Goldman, Kehang Han, and Matt Johnson for always lending a helping hand with under- standing the many facets of RMG, and also to our group social chairs Alon Grinberg-Dana, Mark Payne, Lagnajit (Lucky) Pattanaik, and Mica Smith, for all their efforts in organizing group events; I think Boda-borg is still my favorite!

Thank you to my experimental collaborators from USC, Hugo Burbano, Roe Burell, and Prof. Fokion Egolfopoulos. Many of our early conversations about extinction strain rate helped to build my understanding and set much of the groundwork for the results presented in this thesis. I look forward to future collaborations.

I am also very appreciative of financial support for this work by ExxonMobil. Frequent discussions with our project sponsors, Jonathan Saathoff, Bryan Chapman, and Walt Weiss- man were always a great opportunity to synthesize what we’d learned, chart our next steps,

5 and hear an outside perspective on the work and progress. Next, I’d like to thank Prof. Alan Hatton, and my Practice School directors Dr. Barry Johnston and Dr. Robert Fisher. Working with Takeda in Osaka, Japan was a wonderful experience; the opportunity to work abroad as part of the MIT Practice School was invalu- able. I hope that early in my career I am able to further extend my foreign exposure. Thank you to all of my P.S. team members: Morgan, Emma, Weike, Daniela, and Falco. I’ll never forget our soggy and cold trek up Mt. Fuji or the gorgeous sunrise from the summit we earned the following morning. Thank you to the MIT Energy Club for giving me the opportunity to help lead the 2018 MIT Energy Conference alongside Ryan Shaw. I’m incredibly grateful for all of the hard work from our team: Aditya, Hans, Jason, Jezze, Srimayi, Tatiana, Taylor T., and Taylor V. There were some moments of worry coming during the final weeks, but it was awesome to see it all come together! Beyond the lab and classroom, I’m thankful for all of those who helped me escape the studies and de-stress for a bit. Thank you to the ChemE IM basketball team: Joe, Andrew, Andy, Ian, McLain, Neil, and Mike and the ChemE IM softball team: Alex, Mike, Nick, Nathan, Leia, Troy, Moo Sun, Cache, Katharine, and Tony. Also, to those who shared my love of a good evening powerlifting session: Theo, Chris, and Sean. I couldn’t have asked for a better first year crew to take on the core classes alongside and begin our time here at MIT. Though my Hanabi probably could have done with a bit more practice, its truly been a pleasure. Thank you Colin, Max, Ki-Joo, Falco, Christina, Kim, Sam, and Will. A personal thanks is also due to two important mentors I had before arriving at MIT: Prof. Rohan Akolkar and Dr. Adam Malofsky. The opportunities and guidance each of you have given me throughout the years made this possible. Finally, last, but most certainly not least, thank you to my parents and my little brother. Knowing I can always count on you keeps me moving forward, and it doesn’t hurt that I always have some crisp mountain air and snow beneath my feet to look forward to during our family ski trips as well. THANK YOU ALL!

6 Contents

1 Introduction 23 1.1 General motivation ...... 23 1.1.1 Outlook for natural gas ...... 23 1.1.2 Kinetic modeling ...... 26 1.1.3 Overview of prior methane kinetic modeling studies ...... 27 1.2 Model validation with extinction behavior ...... 29 1.2.1 Conventional definition of ESR ...... 30 1.2.2 Conventional ESR calculation approach ...... 32 1.2.3 Experimental studies of ESR ...... 33 1.2.4 Summary of ESR challenges ...... 34 1.3 Thesis outline ...... 35

2 An apparatus-independent extinction strain rate in counterflow flames 37 2.1 Summary ...... 37 2.2 Background ...... 38 2.3 Methods ...... 41 2.3.1 Computational ...... 41 2.3.2 Experimental ...... 42 2.4 Results & Discussion ...... 43 2.4.1 Extinction flame independence from BSD ...... 43

2.4.2 Experimental method for ESR∞ ...... 46 2.5 Conclusions ...... 49

7 3 Ember: An open-source, transient solver for 1D reacting flow using large kinetic models, applied to strained extinction. 51 3.1 Summary ...... 51 3.2 Background ...... 52 3.3 Methods ...... 56 3.3.1 Governing Equations ...... 56 3.3.2 Spatial & Temporal Discretization ...... 58 3.3.3 Integration of the Split Equations ...... 61 3.4 Results & Discussion ...... 63 3.4.1 Validation of ESR calculation with Ember ...... 63 3.4.2 Ember computational efficiency evaluation ...... 65 3.4.3 Initial studies with Ember ...... 66 3.5 Conclusions ...... 76

4 Numerical investigation of strained extinction at engine-relevant pressures: pressure dependence and sensitivity to chemical and physical parameters 79 4.1 Summary ...... 79 4.2 Background ...... 80 4.3 Methods ...... 83 4.3.1 LFS and ESR pressure profiles ...... 83 4.3.2 Parameter sensitivities ...... 83 4.4 Results & Discussion ...... 86 4.4.1 Pressure profiles ...... 86 4.4.2 Reaction rate constant sensitivity ...... 90 4.4.3 Transport parameter sensitivity ...... 91 4.4.4 Thermodynamic parameter sensitivity ...... 96 4.4.5 Pressure dependent rate coefficients, collision efficiencies, and minor fuel species ...... 100 4.5 Conclusions ...... 105

8 5 An ESR validated kinetic model for applications in natural gas combustion at elevated pressures 107 5.1 Summary ...... 107 5.2 Background ...... 108 5.3 Methods ...... 109 5.3.1 Ignition delay (Autoignition) data ...... 109 5.3.2 Laminar flame speed data ...... 110 5.3.3 Extinction strain rate data ...... 111 5.3.4 Selected validation data ...... 114 5.4 Results and discussion ...... 115 5.4.1 Core model ...... 116 5.4.2 Nitrogen chemistry subset ...... 121 5.4.3 Rich chemistry subset ...... 122 5.5 Conclusions ...... 123

6 Pressure dependent kinetic analysis of pathways to naphthalene from cy- clopentadienyl recombination 131 6.1 Summary ...... 131 6.2 Background ...... 132 6.3 Methodology ...... 136 6.3.1 Quantum mechanical calculation of PES ...... 136 6.3.2 Rate coefficient calculations ...... 136 6.4 Results and discussion ...... 138

6.4.1 C10H10 PES to Fulvalanyl and Azulanyl Radicals ...... 138

6.4.2 C10H9 PES to Azulene and Naphthalene ...... 144 6.5 Conclusions ...... 146

7 Conclusions and future work 149 7.1 Chapter 2 ...... 149 7.2 Chapter 3 ...... 150 7.3 Chapter 4 ...... 152

9 7.4 Chapter 5 ...... 153 7.5 Chapter 6 ...... 154

8 [CAPSTONE] Assessing investment in Natural Gas for commercial truck- ing applications using financial, strategic, and game theoretic frameworks157 8.1 Summary ...... 157 8.2 Background ...... 158 8.3 Financial Analysis ...... 160 8.4 Strategic analysis ...... 165 8.4.1 Threat of new entrants ...... 166 8.4.2 Bargaining power of buyers ...... 167 8.4.3 Threat of substitute products or services ...... 167 8.4.4 Bargaining power of suppliers ...... 168 8.4.5 Rivalry among existing competitors ...... 168 8.5 Game Theory: new entrant vs current market ...... 169 8.6 Conclusions ...... 175 8.7 Acknowledgments ...... 176

References 179

10 List of Figures

1-1 Life-cycle GHG emission per MJ of fuel produced and combusted for both 100-year and 20-year time horizons. Reproduced from [7] with permission from ACS Publications ...... 24

1-2 US shale plays used for the domestic production of natural gas...... 25

1-3 Flame shapes for different methane/hydrogen mixtures plotted as a function of extinction-strain rate. The flow direction is from right to left. Reproduced from [29] with permission from Elsevier ...... 30

1-4 Characteristic flame time versus characteristic outer recirculation zone (ORZ) flow time. Each data point corresponds to a transition of the flame to the ORZ. Reproduced from [32] with permission from Elsevier ...... 31

1-5 (LEFT) Diagram showing the typical opposed flow configuration used when studying strained extinction. (RIGHT) Example axial velocity profile show-

ing the conventional method of determining ESR (Kchar immediately before extinction)...... 31

2-1 Characteristic axial velocity profile obtained when simulating the the coun- terflow jets in one dimension for twin flames with symmetry at the stagnation plane...... 40

11 2-2 (a) Computed ESR vs BSD for twin (φ = 1 from both jets) and single (φ = 1

from one jet and N2 from the opposing jet) premixed methane flames and a methane/air diffusion flame (pure methane opposing air). (b–d) Computed axial velocity profiles immediately before extinction for each of the three flame types at large and small BSD. Displays the consistency of the flame at extinc- tion regardless of BSD...... 43

2-3 (a) Computed ESR vs BSD for twin methane flames at lean (φ = 0.7, Le < 1) and rich (φ = 1.3, Le > 1) conditions. (b,c) Computed velocity profiles dis- playing the the consistency of the flame/hot region at extinction independent of BSD. Inset plots show the rise or fall of peak heat release rate as function

of SRchar for the different Le regimes...... 44 2-4 (a) ESR vs BSD for stoichiometric twin methane flames (φ = 1), compared

to the value calculated for strained potential flow at infinite BSD (ESR∞ = 1935s−1). (b) Axial velocity profiles for selected BSD values. (c) Magnified view of (b) to better highlight the overlap in the hot region...... 45

2-5 Experimental axial velocity profiles for the three different flame types at ex- tinction: twin, single, and diffusion for lean methane (Le < 1) and lean propane (Le > 1). Experimental conditions given in Table 2.1. Profiles demonstrate the consistency of the flame region and reference flame speed at extinction, supporting independence of the near-extinction flame from BSD as was numerically observed in Fig 2-2...... 46

2-6 Computed radial velocity over radial position (strain rate) plotted as a func- tion of axial position for twin, single, and diffusion flames. Curves correspond to data points of Fig. 2-2a. The strain rates in the cold flow regime intersect at a single value...... 47

2-7 Experimental proof of concept of the extrapolation procedure proposed in Fig. 2-6 using the experimental data from Fig. 2-5. Good agreement between the

calculated ESR∞ value and the experimental intersection point is observed. 48

3-1 Strained flame configurations simulated by Ember...... 56

12 3-2 Comparison of extinction profiles generated using Ember and Chemkin for twin methane flames at an equivalence ratio of 0.7 using the Stanford FFCM1 kinetic model [25]. A BSD of 5 cm is used in Chemkin to approximate the infinitely separated potential flow formulation used in Ember. QSS chemistry integration and approximate transport methods are used...... 64

3-3 Characteristic flame types used for studying extinction phenomena. Simu- lations for strains set at 300s−1 on the flame side of the stagnation plane. Stoichiometric equivalence ratios used for single and twin flames. Opposed flow is nitrogen at room temperature for the single premixed flame. Pure methane opposes air for the diffusion flame...... 65

3-4 Parallel efficiency (single CPU time / n-CPU time / n) plotted against num- ber of physical CPUs used to observe the parallel performance of Ember. Perfect parallelism would yield a constant value of 1 as the computational load is evenly spread over the available CPUs. QSS chemistry solver and the transport approximation method are used...... 67

3-5 Methane twin disc flames. φ = 0.7. Initial temperature 298K, atmospheric pressure. Comparison of simulation performance of rebalanced Strang split- ting with traditional Strang splitting for varying global integration step sizes. Inset axes show total computation run times for each of the global steps using both methods...... 68

3-6 Methane twin disc flames. φ = 0.7. Initial temp. 298K, atmospheric pressure. Parallelized over 4 CPUs. Fit to t = AsB where t is time and s is number of species. (a) Computation times for a strain rate of 500 s−1 demonstrat- ing the performance of Ember compared with Chemkin and the benefits of implementing the QSS chemistry integrator and mixture averaged transport approximation. (b) Comparison of Ember and Chemkin ESR computation times for varying kinetic model sizes...... 68

13 3-7 An investigation of Lewis number effects using complex chemistry simulations through elimination of fuel and oxidizer species diffusivities or elimination of energy conduction (λ = 0). This analysis was done using stoichiometric methane flames with a strain of 500 s−1 in Ember. FFCM1 is used as the kinetic model. Perturbations are achieved by first solving to steady state, then setting the specified diffusion parameter to 0 and allowing the system to progress in time to the new steady state...... 70

3-8 Extinction progression for a propane mixture of equivalence ratio 0.8, yielding a Lewis number greater than 1. AramcoMech 2.0 used...... 73

3-9 Extinction progression for a propane mixture of equivalence ratio 1.5, yielding a Lewis number less than 1. AramcoMech 2.0 used...... 74

3-10 Equilibrium mixture for starting equivalence ratios of propane/air gas mix- tures. Nitrogen and minor species are omitted for clarity...... 75

3-11 Theoretical Lewis number predictions included in Combustion Physics [35] for peak reactivity (Left) and progression to extinction (Right). Reproduced with permission from Cambridge University Press...... 75

4-1 A-B) Comparison of laminar flame speed and extinction strain rate pressure profiles for commonly used kinetic models under lean (φ = 0.7) conditions. Experimental data from [118]. All models agree ESR is significantly non- monotonic with pressure at these conditions. C-D) Fractional deviations rel- ative to the FFCM1 mechanism showing the similarities in behavior between LFS and ESR predictions of various models...... 88

4-2 A-B) Comparison of laminar flame speed and extinction strain rate pressure profiles for commonly used kinetic models under rich (φ = 1.3) conditions. Ex- perimental data from [118] C-D) Fractional deviations relative to the FFCM1 model showing the similarities in behavior between LFS and ESR predictions of various models...... 89

14 4-3 Hashemi et al. C2H6 kinetic model reaction sensitivities. Intermediate pres- sure points selected based on the relative minima and maxima observed in initial pressure screens under lean and rich conditions (Fig. 4-1 & 4-2). AB) ESR sensitivities determined using a brute force perturbation of 1.5x. CD) LFS sensitivities determined using the Cantera adjoint method implementation. 92

4-4 Hashemi et al. C2H6 kinetic model transport parameter sensitivities. Deter- mined using a brute force perturbation of 1.2x. A & B contain subfigures showing LFS sensitivity at 1 atm using the same brute force perturbation . 97

4-5 Hashemi et al. C2H6 kinetic model species thermodynamic data pseudo sen- sitivities sensitivities. Determined using a constant shift of 5 kJ/mole for enthalpy, 10 J/molK for entropy, and 5 J/molK for heat capacity...... 100

4-6 Lean (φ = 0.7) methane flames. A) Laminar flame speed pressure profiles showing the impact of completely eliminating pressure dependence from a simple methane kinetic model (17 species, 58 reactions). FFCM1 profile is included for validation of trend given by the simple model. B) Extinction strain rate pressure profiles showing the impact of completely eliminating pressure dependence from a simple methane kinetic model. Again, FFCM1 used to support trend validity. C) Elimination of pressure dependence for each pressure dependent reaction in the simple model independently. D) Laminar flame speed sensitivities for the simple kinetic model...... 103

4-7 Comparison of ESR pressure profiles to assess the impact of assuming unequal collision efficiencies within the kinetic model. A) Lean (φ = 0.7) B) Rich (φ = 1.3) ...... 104

4-8 A-B) Extinction strain rate pressure profiles for the impact of C2H6, CO, and

H2 on predictions. AramcoMech1.3 used...... 105

5-1 Cantera OH species sensitivity directly before ignition compared with igni- tion delay brute force sensitivity values for the FFCM1 kinetic model. Both sensitivity sets are normalized such that the maximum value is ±1. Initial conditions are 1029K and 39atm for the methane comparison simulation. . . 111

15 5-2 Comparison of adjoint laminar flame speed method implemented in Cantera with a brute force perturbation routine. The simple methane kinetic model of 17 species and 58 reactions used previously in Fig. 4-6 is used here. Initial conditions are 298K and an equivalence ratio of 0.7...... 112

5-3 GRIMech3.0 kinetic model performance for the the selected validation data . 116

5-4 FFCM1 kinetic model performance for the the selected validation data . . . 117

5-5 Hashemi C2H6 kinetic model performance for the the selected validation data 117

5-6 Hashemi C2H6 kinetic model sensitivity for ethane laminar flame speed data. Ethane/air at an equivalence ratio of 0.9 ...... 118

5-7 Hashemi C2H6 kinetic model performance after the modification to include the sensitive reaction rates for ethane laminar flame speeds from FFCM1:

CO + OH CO2 + H, 2CH3 C2H5 + H, and C2H4 + H C2H5 ..... 119

5-8 Final, modified Hashemi C2H6 kinetic model performance after various rate updates to improve agreement with the experimental validation data set. . . 121

5-9 Nitrogen chemistry predictions when (LEFT) using GRIMech3.0 and (RIGHT) appending the nitrogen chemistry from the Glarborg review [28] with the core model produced here. Experimental data from [136] ...... 122

5-10 Plug flow reactor data for the rich oxidation of methane at varying temper- atures using the original kinetic model used in [65] which leveraged FFCM1 for the core chemistry. Experimental data from [137]...... 124

5-11 Plug flow reactor data for the rich oxidation of methane at varying tempera- tures using the core model from this work with the rich chemistry from [65] appended. Experimental data from [137]...... 125

6-1 Primary structures of this study. An ending of -yl added to a species indicates a radical of that species formed by hydrogen loss ...... 136

16 6-2 C10H10 surface schematic. The azulanes fall slightly lower in energy than the fulvalanes with an isomerization barrier separating the two groups of roughly equal height to the entrance channel. Due to the number of species, the surface has been broken into several figures as indicated for display purposes, but note that the calculations are performed on the surface as a whole. . . . 139 6-3 CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) for the C10H10 surface initiated by the recombination of two CPDyl radicals. All enthalpies are relative to the adduct T1b which connects to the other portion of the surface. This network was first reported by Melius et al. and in its complete form by Cavallotti et al. [182, 188] ...... 140 6-4 CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) on the C10H10 surface leading to the formation of Azulanyl radicals. All enthalpies are relative to component T1b which represents the dominant entrance channel. Pathway first shown by Cavallotti et al.[188]. Additional symmetry allowed azulane species were included in this analysis but are not shown in this figure for simplicity reasons as they are only accessible via a 170 kJ/mol barrier from T4a...... 142 6-5 (A: Left) Computed pressure dependent rate coefficients for the recombination of two CPDyl radicals to form various products in N2 at 1100 K. The corre- sponding PES is shown in Fig. 6-3 and Fig. 6-4. (B: Right) Rate constant matrix depicting rate constant magnitudes for all well-skipping and direct re- actions between isomers of the C10H10 PES. Reacting species shown on the y axis with the product species along the x axis...... 143 6-6 Rate of C10H9 formation comparison between hydrogen emission from the CPDyl dimer, chemically activated hydrogen emission from the CPDyl + CPDyl reaction, and radical hydrogen abstraction from the CPDyl dimer. At moderate to low temperatures, radical abstraction dominates while at high temperatures the activated path leads formation of C10H9 ...... 145 6-7 CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) on the C10H9 surface leading to the formation of naphthalene and azulene. All enthalpies are relative to component N1b ...... 146

17 6-8 (A: Left) Computed pressure dependent rate coefficients for the species N1b, the primary entrance channel to the C10H9 surface to form various products at 1100K in N2. The corresponding PES is shown in Fig. 6-7. (B: Right) Rate constant matrix for well-skipping and direct reactions among the isomers of the C10H9 PES. Reacting species shown on the y axis with the product species along the x axis. [Flvln = fulvalene, Azln = azulene, Nphln = naphthalene] . 147

8-1 Example scatter plot used for determining βequity according to the CAPM model using monthly returns of the S&P 500 to represent the overall market along with the monthly returns of the company’s stock...... 161

8-2 Historical market prices for diesel fuel and compressed natural gas (CNG) [239]. Extrapolations used to estimate the future price differential necessary for the financial model...... 162

8-3 Calculation showing the break-even average price differential between CNG and Diesel required to turn a profit on the natural gas truck given current engine and storage prices premiums along with lesser efficiency...... 164

8-4 Net present value scenario analysis examining several potential improvements to the value proposition of investing in a natural gas powered trucking fleet. 165

8-5 The five forces that govern marketplace competition according to Michael Porter. Reproduced from [232] ...... 166

8-6 Economic surplus model for a commodity good. All suppliers have the same cost of production and therefore the entire economic surplus goes to the buy- ers/consumers...... 167

8-7 Simultaneous game for investment in NG trucking by an new entrant and an existing competitor. Value ”c” represents the marginal additional investment cost of the NG trucks. Value ”B” is the expected profit without competition calculated according to 8-8. Value ”R” is the expected revenue to be cap- tured by the competitor, here assumed to be larger than ”c” given the cost advantage...... 171

18 8-8 Chance tree summarizing the expected marginal return for investment in NG trucks today. The high price difference is represented by the forecast predic- tions shown in Fig. 8-2 while the low price assumes the the current natural price differential of 40 cents holds constant into the future. For the carbon tax, the same assumption is made as in Fig. 8-4...... 172 8-9 Sequential game representing the NG trucking investment decisions of exist- ing competitors in the market. In all scenarios, the first moving competitor is assumed to have a first mover advantage of 1 year and able to generate profits that year without competition, regardless of whether or not the other competitor chooses to invest. It is also assumed that when both competitors have equivalent cost advantages, price competition will devolve the market to marginal cost pricing given the largely undifferentiated good. Backward induction suggests that current market players are not fiscally incentivised to invest in a NG trucking fleet...... 175

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20 List of Tables

1.1 Selection of kinetic models relevant to natural gas combustion produced since the conclusion of the development of GRIMech...... 29

2.1 Experimental conditions for Fig 2-5. Ambient temperature and pressure, 298K and 1atm...... 47

2.2 Result summary from test of concept for ESR∞ determination shown in Fig. 2-7...... 48

3.1 Combustion kinetic models used for Ember computation speed evaluation. . 67

4.1 Combustion kinetic models used for LFS and ESR pressure dependence eval- uation...... 86

5.1 Selected ignition delay validation data (PART A) ...... 126 5.2 Selected ignition delay validation data (PART B) ...... 127 5.3 Selected laminar flame speed validation data ...... 128 5.4 Selected extinction strain rate validation data. If no second composition is given, it is assumed that the data point is for a twin flame arrangement and the same compositions is used for each jet...... 129

6.1 Selected data set used for the rate analysis of C10H9 formation routes as shown in Fig. 6-6 ...... 144

8.1 Summary of key parameters used for the calculation of the net present value (NPV) of investing in NG trucking. Note that ’MT’ stands for ’metric tonnes’ and ’DGE’ stand for ’diesel gallon equivalent.’ ...... 163

21 8.2 Asset beta and profitability information for leading publicly traded US truck- ing companies. Profitability information given as the 5yr average 2014-2018. 169 8.3 Summary Porter’s Five Forces analysis of the US trucking industry. corresponds to strongly favorable and corresponds to very unfavorable...... 169

22 1 Chapter 1

2 Introduction

3 1.1 General motivation

4 1.1.1 Outlook for natural gas

5 With the most significant impacts of global warming looming ever closer, the need for sus-

6 tainable energy becomes more dire [1, 2]. Renewables, such as wind and solar power, present

7 some of the most promising options for greenhouse gas emission reductions. However, these

8 technologies are not yet ready to dominate world energy production, most notably because

9 of their intermittent nature and the current lack of economical, large scale energy storage

10 solutions [3]. As a result, there exists a need for even incremental improvements to cur-

11 rent energy production and transportation fuels. Natural gas (NG), which is roughly 90%

12 methane, has been leveraged in recent years to this end.

13 NG is advantageous when compared to other means of energy production (coal) and

14 transportation fuels (diesel and gasoline) because of its significantly higher hydrogen to

15 carbon ratio. NG has a H:C ratio of about 3.7:1 as compared to that of diesel fuel at about

16 1.8:1 and coal at about 0.75:1 [4]. This higher H:C ratio leads directly to lower CO2 emissions

17 as more energy is produced from the production of water during combustion than by CO2,

18 leading to a higher energy to mass ratio for methane [5]. This benefit can be directly seen

19 in the green bars of Fig. 1-1. As an added benefit, NG sources tend to contain substantially

20 lower amounts of sulfur which is important for avoiding the production of sulfur dioxide gas

23 Environmental Science & Technology POLICY ANALYSIS

Figure 2. Life-cycle GHG emissions per MJ of fuel produced and combusted for both 100-year and 20-year time horizons. Figure 1-1: Life-cycle GHG emission per MJ of fuel produced and combusted for both 100- year and 20-year time horizons. Reproduced from [7] with permission from ACS Publications impacts of the fuels in specific applications. For NG power plants, to CO2. When comparing the emission impacts of different we estimated the base-case efficiency for a conventional NG boiler fuels one must choose a time frame for comparison, as the to be 33.1%, with a range from 33.0 to 33.5%,13 while a NG com- IPCC calculates GWPs for multiple time horizons such as 21binedduring cycle (NGCC) combustion power plant which would haveleads an efficiencyto acid of rain 47%, among20-, other 100-, and negative 500-year environmental timeframes. The IPCC factors. recommends In with a range from 39 to 55%.40 For coal power plants, we estimated using GWPs for a 100-year time horizon when calculating 22the efficiency[6] the for sulfur a conventional emissions pulverized from coal NG boiler resources to be 34.1%, are observedGHG emissions to be for roughly evaluating variousone seventh climate change of oil mitigation or with a range from 33.5 to 34.4%,13 while a supercritical boiler would policies. When using a 20-year time frame the effects of methane 23havecoal an efficiency resources. of 41.5%, with a range from 39.0 to 44.0%.41 are amplified as it has a relatively short perturbation lifetime For a passenger car, we assumed a fuel economy of 29 miles (12 years). Howarth et al.4 use results from a recent study by 45 24per gallonAn gasoline-equivalent important issue (mpgge) with or 8.11 using L per 100 NG km as and a moreShindell environmentally et al. that suggest a higher friendly GWP for fuel methane source due to is direct that a CNG car with similar performance would have a fuel and indirect aerosol effects. We have chosen to use the current IPCC 46 25economymethane penalty leakage of 5% (on an during mpgge-basis) production primarily because and supply, of published as indicated results. bySupporting the blue Information bars in Table Fig. S13 1-1. presents the weight penalty of on-board CNG storage cylinders. We GWPs that we used in comparison to those Howarth et al.4 used. 26developedThis a impact distribution stems function from for the methane relative fuel itself economy being of a greenhouse gas, roughly 20 times more potent the CNG car as compared to the gasoline car around this value, 3. RESULTS 27withthan the high CO estimate2. Fortunately, that the CNG car methane will have the has same a fuel relatively short lifespan in the atmosphere (only economy, while the low estimate that it will have a 10% reduction With the parametric assumptions incorporated into the 28as comparedapproximately to the gasoline 12 years). car. This, in part, yields theGREET significant model, differences we produced life-cycle between GHG the emissions 100 and for the Several studies have examined the fuel consumption of NG four energy pathways with three functional units. With the 29and20 diesel year transit impact buses and projections[7]. have found that CNG-fueled However, transit since thedistribution majority functions of the negative developedproduction in this study and impact those from buses on average have a fuel economy 20% lower than diesel- 13,14 À previous studies for other key activities such as recovery and 30fueledcomes buses.42 from44 Theseresultsareduetothelowthermale leaks, it is feasible that thisfficiency issue couldrefining be efficiencies, improved we used by GREET regulation stochastic and modeling taxes cap- of a spark-ignited engine (when compared with a compression- ability to generate results with distributions. 31ignitionon diesel emissions engine) operating which at in low turn speed make and load. the43 However, investment3.1. in more GHG Emissions reliable for equipment Fuel Pathways economically per MJ of Fuel ffi fi Produced and Burned. it has been argued that the fuel e ciency bene t of diesel buses Figure 2 presents life-cycle CO2 equiva- 32 hasjustified. been reduced due to the emission control equipment and lent (CO2e) emissions of CO2,CH4,andN2O (which is primarily strategies used to meet the EPA 2010 heavy-duty engine emis- from fuel combustion) of gasoline from conventional crude and 33sion standards.US natural According gas to Cummins-Westport, production and its availability 8.1-L ISL G hasoil risensands, natural steeply gas in from recent conventional years and [8]. shale This gas, rise and coal NG engine achieves fuel efficiencies much closer to a diesel from underground and surface mining. In addition, detailed 34enginehas and been depending predominantly on the duty cycle, fueled the engine by advances can either in frackingbreakdowns technology, of GHG emissions allowing from the much fuel production greater and match an equivalent diesel fuel economy or have a fuel economy infrastructure stages are provided to show the relative impor- 44 35thataccess is 10% lower. to shaleFor our gas analysis, throughout we assumed the that country, for our low (Fig.tance 1-2). of CH The4 and primary CO2 venting advance and flaring in emissions. technology Each bar estimate CNG transit buses would have a 20% reduction in fuel represents the estimate for our base case, the error line on each 36economyleading and for to our far high greater estimate shale a 10% gas reduction, accessibility while using is thatbar representsof horizontal the range drilling. for the probability Horizontal of 10% drilling and 90% (P10 the mean value as our base case. A summary of our end use efficiency and P90) values. Results for GWPs with both 100- and 20-year 37assumptionsallows is a shown single in well Supporting to access Information a greater Table S12. portion oftime the horizon shale are due presented, to its high though length the 20-year to height horizon is 2.2.5. Global Warming Potentials of Greenhouse Gases. intended for comparison with Howarth et al.4 38GWPratio is an [9]. attempt As to a provide result a of simple these measure technological to compare advances,Fuel combustion the US is accounts projected for a large to have portion roughly of total GHG the relative radiative effects of various GHG emissions. The index emissions for all pathways, while the fuel production stage has a fi is defined as the cumulative radiative forcing between the time a 24 signi cant amount of GHG emissions for all pathways except for unit of gas is emitted and a given time horizon, expressed relative surface mined coal. The second largest GHG emission source for

623 dx.doi.org/10.1021/es201942m |Environ. Sci. Technol. 2012, 46, 619–627 Figure 1-2: US shale plays used for the domestic production of natural gas.

39 100 years of accessible NG when considering the current consumption rate [10].

40 While the US might only have a century of NG left in the ground, another impor-

41 tant opportunity for NG production are renewable pathways, such as anaerobic digestion of

42 biomass [11]. Biomass could be relatively easily produced expressly for the purpose of gener-

43 ating methane, or more advantageously, current waste streams could be utilized. There is a

44 significant amount of lignocellulosic agricultural waste that is already an available feedstock

45 for anaerobic digestion [12]. The bio-methane formation pathway primarily forms acetate

46 which is then converted to methane and CO2. An important consideration for these routes is

47 the optimization of selectivity toward methane over the energy loss when producing CO2.A

48 recent review by Chandra et al. examined the sustainable production of methane from bio-

49 waste and compared it with the production of ethanol, another promising sustainable fuel.

50 Results show methane to be the optimal route for maize and wheat wastes, while ethanol is

51 currently more advantageous for rice and sugarcane wastes on a per unit energy basis [12].

25 52 1.1.2 Kinetic modeling

53 During combustion processes, a fuel is mixed with an oxygen source, and the two are re-

54 acted to form predominantly carbon dioxide and water; each are more stable, lower-energy

55 species. The drop in energy is released as heat which can then be harnessed to achieve some

56 task, be that cooking food, generating electricity, or driving the cylinders of a combustion

57 engine. However, though the combustion process is often summarized as the simple net

58 reaction of the fuel and oxidizer to stoichiometric amounts of CO2 and H2O, in reality, the

59 fuel/oxidizer mixture is converted through hundreds to thousands of elementary reaction

60 steps with thousands to tens of thousands of side reactions, depending on the complexity of

61 the fuel/oxidizer. To fundamentally understand this process, and more importantly predict

62 its behavior across a wide range of conditions without testing each individually, one needs to

63 consider and know the rate constants of all of these unique reactions. Even in the simplest

64 cases, it requires three parameters to describe the rate constant of a given reaction. To

65 add even more complexity, these combustion models must also include detailed parameters

66 to describe the thermodynamic properties (enthalpy, entropy, and heat capacity) as well as

67 transport properties (diffusion constants, thermal conductivity, and viscosity).

68 Looking at a simple model that only considers 25 species, if it is assumed each species can

69 react with each other species, there will be 300 reactions, each with at least 3 parameters.

70 Each species will also have 14 parameters to describe the thermodynamic properties and 5

71 to describe the transport properties. Together, even in this very simple model, it adds up to

72 1375 parameters to be known. For a model with 500 species, the same assumptions yields

73 383,750 total parameters to be specified. Accurately determining all of these parameters

74 is no small task and has kept the combustion community busy for many decades as more

75 complete models are constructed and the parameters are continually refined. Beyond the

76 difficulties of accurately determining all these parameters, performing numerical simulations

77 with large fully detailed kinetic models can become too computationally cumbersome and

78 time consuming, such that its not feasible to simply simulate all combustion processes using a

79 single, community agreed upon, set of parameters that encompass all relevant fuels and is ever

80 expanding. This is commonly an issue for computational fluid dynamics (CFD) simulations.

26 81 Therefore, automated methods of efficiently constructing these kinetic models to focus on

82 only certain specific conditions and species are commonly utilized. These softwares utilize

83 measured or calculated parameters when available, and estimate based on similar reactions

84 or species when not. One such software is the Reaction Mechanism Generator, which is

85 open-source and freely available to the community [13].

86 1.1.3 Overview of prior methane kinetic modeling studies

87 Given the numerous parameters and opportunity for each researcher to select the subset of

88 all possible reactions that they believe to be most important, it follows that a multitude of

89 kinetic models have been published over the years. Even for the same primary species and

90 conditions, there exist many published kinetic models, each unique in their own way. Here,

91 a subset of the kinetic models commonly used to describe methane based combustion will

92 be described to give some scope to the task at hand.

93 One of the most influential kinetic models produced is GRI Mech, versions one through

94 three. The work was sponsored by the Gas Research Institute which combined with the

95 Institute of Gas Technology in 2000 to create the Gas Technology Institute that exists to-

96 day. The research to develop GRI Mech took place primarily at University of California

97 at Berkeley, Stanford University, University of Texas at Austin, and SRI International and

98 sought to produce the best possible kinetic model to describe NG flame behavior and ig-

99 nition. The general creation procedure first defined the what was believe to be the range

100 of possible and relevant elementary reactions. Then the GRI team filled in the model with

101 measured values for kinetic rates whenever possible and used estimates when not. Finally,

102 a set of target and test data were used to optimize the numerous kinetic model parameters

103 within the estimated error ranges to best match the observable data such as laminar flame

104 speeds and ignition delays. While this process gives a good balance of fundamental kinetic

105 processes and predictive capabilities, over-fitting is a hazard with so many parameters to

106 tune. Therefore mechanisms produced in this manner are generally not easily extensible

107 adding new kinetic routes. To add reactions or refine the uncertainty in certain reaction

108 rates, requires a re-optimization of the entire kinetic model. As a result, and with the im-

109 provement of current ab initio kinetics computations, there have been some thrusts in the

27 110 direction of un-optimized, more fundamental kinetic models that are easily extensible and

111 reflect the best understanding of the kinetic rates available. However, with so many param-

112 eters, accurate predictive capabilities are sometimes a challenge for these models. Yet, the

113 advantage is that these more fundamentally based kinetic models should be more extensible

114 to conditions outside the range of currently available experimental data.

115 Funding and further development of GRI Mech ended in 2000 when the two entities

116 merged. Since then, a number new updated NG models with a variety of primary goals

117 have been produced by research groups around the world. A selection of some of the most

118 prominent of these is shown in table 1.1. Rather than striving for completeness, the UCSD

119 kinetic models aim for keeping computation times and overall model parameter uncertainties

120 down by only focusing on the most important and sensitive kinetic pathways. Built from a

121 base of GRIMech and various other optimized models focusing on H2/CO combustion up to

122 butadiene oxidation, USC-II is designed as a general purpose model for H2/CO/C1-C4 com-

123 pounds. The AramcoMechs produced at NUI Galway are designed for even greater breadth

124 of combustion mixtures and focus on C1-C4 hydrocarbon and oxygenated fuels. The inclu-

125 sive chemistry of these models is often advantageous when considering the impact of minor

126 species on flame properties or more diverse fuel blends that more accurately represent natural

127 gas than methane as is typically used. The Chernov model then expands capabilities by con-

128 sidering kinetic routes toward polycyclic aromatic hydrocarbons (PAHs) through inclusion of

129 global, non-elementary reactions to naphthalene, indene, and on to higher order PAHs. The

130 routes are initiated primarily by cyclopentadienyl radical reactions. FFCM-1 (Foundational

131 Fuel Chemistry Model) is produced with the same general goal and optimization procedure

132 as GRIMech, but with more rigorous routines and updated rates and uncertainties. The

133 Hashemi models and the Glarborg model diverge from global parameter optimization pro-

134 cess, aiming instead for more fundamental and easily extensible models. The Hashemi CH4

135 model is designed and validated for high pressure methane oxidation processes using ignition

136 delay, laminar flame speed, and species profile data. The model relies heavily on ab inito rate

137 calculations by Klippenstein and coworkers. Hashemi C2H6 extends the methane model to

138 incorporate relevant chemistry for high pressure oxidation of ethane as well. The Glarborg

139 model then builds on the Hashemi models by adding in currently known nitrogen chemical

28 Table 1.1: Selection of kinetic models relevant to natural gas combustion produced since the conclusion of the development of GRIMech. Model # Species # Reactions Year Reference GRI3.0 53 325 2000 [14] UCSD-1 40 189 2001 [15] UCSD-18 57 268 2016 [15] USCII 111 784 2007 [16] AramcoMech1.3 253 1542 2013 [17] AramcoMech2.0 493 2716 2016 [18–23] Chernov PAH 102 831 2014 [24] FFCM-1 38 278 2016 [25] Hashemi CH4 68 631 2016 [26] Hashemi C2H6 68 663 2017 [27] Glarborg NOx 148 1320 2018 [28]

140 routes of relevance to small molecule oxidation.

141 1.2 Model validation with extinction behavior

142 Flame extinction by deformation, Extinction Strain Rate (ESR), has long been recognized

143 as a important flame property, but it is only rarely included as a validation parameter dur-

144 ing kinetic model generation. Yet, recent works have exposed new value in the realm of

145 turbulent combustion for this fundamental flame parameter [29–33] . In 2015, Shanbhogue

146 et al. [29] experimentally observed that flame shape and instability were well characterized

147 by the ESR of the combustion mixture in the case of turbulent, swirl-stabilized, premixed

148 CH4/H2 combustion. These flames were studied at constant Reynolds number and swirl an-

149 gle. Figure 1-3, reproduced from the work, demonstrates the primary finding. A subsequent

150 paper expands this analysis to varying Reynolds numbers and swirl angles, finding that the

151 point at which the flame propagates into the outer recirculation zone of the swirl stabilized

152 flame correlates directly with the ESR. This observation was consistent for varying hydrogen

153 percentage as well [32]. The correlation is reproduced in Fig. 1-4. Another work by Wantabe

154 et al. uses ESR to describe differences in turbulent flame structure for CH4/O2/CO2 and

155 CH4/air flames. The shift in behavior is not captured by the trends in laminar flame speed

156 for the two flame types, but is in fact captured by differences in ESR. These works, while

29 504 S.J. Shanbhogue et al. / Combustion and Flame 163 (2016) 494–507

Fig. 14. Flame shapes for different methane/hydrogen mixtures plotted as a function of extinction-strain rate. The flow direction is from right to left. Figure 1-3: Flame shapes for different methane/hydrogen mixtures plotted as a function of extinction-strain rate. The flow direction is from right to left. Reproduced from [29] with

permission from Elsevier to the 180 Hz mode. This is seen only when the percentage of hy- drogen in the fuel blend exceeds 40% by volume and can also be explained in terms of the extinction strain rate. This is because, a κ ext D ∼ value of U ∞ 18 corresponds to an extinction strain rate of approx- 157 constrained to methane based fuel mixtures, suggestimately that 40 0 0 ESR s −1 . Mixtures plays a criticalbelow 40% role hydrogen in deter- cannot achieve this extinction strain rate if the reactants are at a temperature of 300 K. 158 mining the behavior of turbulent flames, making it important for consideration when creating 5. Concluding remarks and suggestions for future work 159 kinetic models for real systems where these processes are in play.

The goal of this paper was to examine the connection between the flame shape and the observed transitions between stable and unsta- ble combustion, and whether a strain-based flame extinction mecha- nism can explain the data. From the literature, we reviewed a number 160 1.2.1 Conventional definition of ESR of papers that presented some evidence related to this fact. Some re- ported that during transitions between stable and unstable combus- tion, flame shapes were observed to change. Others described sensi- 161 Extinction strain rate (ESR) is the maximum strain or stretch rate that a flame can sustain tive dependence on initial conditions such as minor changes in fuel

Fig. 15. Overall sound pressure levels (data from Fig. 6) plotted as a function of ex- composition or ambient conditions which also hinted at changes due tinction strain162 rate. beforeThe x-axisit is normalized is extinguished by hub diameter [34]. of Forthe swirler axisymmetric and the opposing jets of reacting flow, ESR is the plug-flow velocity of the mixture prior to entering the swirler. to flame stabilization. 163 maximum derivative of axial velocity with respectTo to assess position the influence that can of the be flame sustained shape and imme- minimize the influ- ence of other possible mechanism that might contribute to the tran- mixture. The164 latterdiately is simply upstream a surrogate of the for the flame actual before stretch it rate extinguishes. sition in the Numerically, flame shape, we for constructed axisymmetric an experiment jets, in which we experienced by the flame. varied the fuel composition by keeping the Reynolds and swirl num- First, and165 in Fig.the 14 derivative , we reorganize of axialthe different velocity flame relates shapes directly seen tober the constant radial but velocity, changing givingthe methane/hydrogen the connection ratio in the fuel. for different fuel compositions at different equivalence ratios, but We observed that all mixtures show a consistent trend under 166 to flame ”stretch” [35]. As the flame strain increases, the flame region thins and the radial now as a function of the extinction strain rate. The data shows that acoustically coupled conditions : when burned at low equivalence if the equivalence167 velocityratio and increases,fuel composition causingare shortersuch that gasthey residencere- ratios timesthey inare thestable, flameand regionat some andcritical lowerequivalence tem- ratio, the −1 sult in mixtures with κext < 256 s then we see a bubble flame. If combustor transitions to exhibit harmonic oscillations. For the latter, the equivalence-ratio168 peratures and amount that eventually of hydrogen over in the inhibit mixture the are primary there radical is always formation one dominant reaction frequency of combustion accompanied by harmonics. −1 −1 such that 265 s < κext < 522 s then we see an ISL flame. For strain Mixtures with a higher percentage of hydrogen transition at lower 169 [34]. Schematics−1 for this characteristic setup considering the twin flame approach where the rates beyond κext ∼= 572 s we always see an ORZ stabilized flame. equivalence ratios. We also note that the way we arrange the data, there seems to be To explain the origin of the frequencies observed, we used a 1-D 170 same premixed fuel/oxidizer mixture is input from each jet are shown in Fig. 1-5. an apparent disconnect, i.e. the numerical value for the end of the lumped element acoustic model which incorporated unsteady heat- first group is 256 s −1 but the starting value for the second group is addition from the flame. We showed that all frequencies that we ob- 265 s −1 . This is due, in part to the resolution of our equivalence ratio 30 served are supported by the acoustic model, suggesting that the in- measurements of the raw data in Fig. 8 , which is φ ∼= ±0 . 01 . stabilities are of an acoustic origin. We now also re-plot the pressure amplitude data in the acousti- Chemiluminescence images at different equivalence ratios and cally coupled case from Fig. 6 as a function of the extinction strain fuel composition showed that the flame transitions from one shape rate. This is shown in Fig. 15 . To keep the axes non-dimensional, to another, before finally settling to a shape which shows it stabi- we normalize the extinction strain rates with the inlet diameter and lizes along the outer shear layer and in the ORZ. It is when the flow approach flow velocity. The figure shows that all transitions now reaches this shape or macrostructure that combustion transitions to scale as a function of extinction strain rate highlighting that these harmonic oscillations (in the acoustically coupled case), otherwise transitions are indeed linked to the flame shape and thus flame- the combustor is stable. κ ext D ∼ stabilization. Second, note that at U ∞ 18 there is a second tran- To quantify the influence of burning in the ORZ, we computed the sition, where the combustor shifts frequency from the 110 Hz mode extinction strain rates using an opposed-flow configuration. This was Figure 1-4: Characteristic flame time versus characteristic outer recirculation zone (ORZ) flow time. Each data point corresponds to a transition of the flame to the ORZ. Reproduced from [32] with permission from Elsevier

Premixed Fuel + Air

Flame Streamlines

Stagnation Plane

Premixed Fuel + Air

Figure 1-5: (LEFT) Diagram showing the typical opposed flow configuration used when studying strained extinction. (RIGHT) Example axial velocity profile showing the conven- tional method of determining ESR (Kchar immediately before extinction).

31 171 1.2.2 Conventional ESR calculation approach

172 The conventional mathematical formulation for ESR calculation was first presented by Kee

173 et al. in 1989, and has been modified to include additional physical phenomena such as

174 radiation and thermal diffusion (Soret effect)[36, 37]. The current standard method for ESR

175 calculation uses Chemkin software originally produced at Sandia National Lab and now

176 marketed by ANSYS Inc. The open source combustion software, Cantera, which is currently

177 predominantly developed by MIT’s Ray Speth, is also capable of making ESR calculations,

178 but the implementation lacks the pseudo arc length continuation methods implemented in

179 Chemkin [38, 39]. The more in-depth derivation will be left to the 1989 Kee et al. paper,

180 but the governing equations, as well as a summary of the current Chemkin solver method

181 for the boundary value problem (BVP), are given.

182 Defined for mathematical simplification:

−ρν ρν 1 ∂p G = r F = x H = (1.1) r 2 r ∂r

183 Defined for reduction of derivative order:

dF G − = 0 (1.2) dx

184 Constant derived from axial Cauchy momentum eq:

dH = 0 (1.3) dx

185 Radial Cauchy momentum equation:

d FG! 3G2 d d G!! H − 2 + + µ = 0 (1.4) dx ρ ρ dx dx ρ

186 Enthalpy conservation equation:

! " # ˙ dT 1 d dT ρ X dT 1 X Qrad 2F − λ + cp,kYkVk + [hkw˙k] + = 0 (1.5) dx cp dx dx cp k dx cp k cp

32 187 Species conservation equations:

Y d 2F k + (ρY V ) − w˙ W = 0 ∀k (1.6) dx dx k k k k

188 where ρ is density, νr is radial velocity, νx is axial velocity, r is radial position, p is pressure,

189 µ is viscosity, λ is thermal conductivity, cp is heat capacity at constant pressure, T is temper-

190 ature, Yk are species mass fractions, Vk are species diffusion velocities defined according to

191 the transport formulation used (typically mixture averaged or multicomponent formulation),

192 hk are species enthalpies,w ˙ k are the species net creation/destruction rates, and Wk are the

193 species molecular weights.

194 Current Chemkin implementations then solve the BVP by discretizing the system com-

195 pletely and solving using a damped Newton method [37]. Additionally, Chemkin implements

196 the pseudo-arclength continuation method described by Nishioka et al. in 1998, which uses

197 the specification of internal temperature boundary condition(s) to smoothly navigate the

198 numerical turning point where the flame extinguishes [39]. This approach allows the inlet

199 jet velocities to be solved as dependent variables while the internal temperature boundary

200 condition is slowly stepped down to effectively navigate the numerical turning point of ex-

201 tinction. However, the issue with the complete discretization approach is that the system

202 can quickly become too large for the solver to handle. For example, if the domain contains

203 roughly 1000 discretization points and the mechanism has 500 species similar to the 2016

5 5 204 release of AramcoMech2.0, then the Jacobian would be a 5x10 by 5x10 square, though sig-

205 nificantly sparse. This scaling often leads researchers to pursue skeletal reduced mechanisms

206 to calculate ESR rather than complete and detailed mechanisms, as would be desired for

207 wider applications of the resulting mechanisms that are developed [40, 41].

208 1.2.3 Experimental studies of ESR

209 Study of flame extinction came into focus in the 70’s, but at that time, the main focus was

210 on extinction of diffusion flames [42]. Study began to shift more towards premixed flames

211 with Law et al.’s 1988 paper that reported experimental data for twin premixed methane

212 air flames at a range of stoichiometric ratios at atmospheric pressures [43]. Further efforts

33 213 in calculation and measurement of ESR have been made by Egolfopoulos and coworkers,

214 one of Law’s former students. These efforts have expanded ESR characterization across a

215 wide range of fuels including, ethanol [44], dimethyl ether [44, 45], C5-C12 alkanes [46],

216 cyclopentadiene [47], JP-5 [40], and more [48–51]. Many other significant efforts toward

217 expanding the ESR database have been made by Niemann and coworkers [52–59] and other

218 researchers [60–64]. Some recent works have begun to experimentally characterize ESR

219 at elevated pressures thanks to new, enclosed counterflow apparatuses and refined seeding

220 procedures [51, 58, 64].

221 1.2.4 Summary of ESR challenges

222 If ESR is as important in describing flame behavior as discussed, then the question becomes:

223 why has is it not already widely used in validation of kinetic modeling studies? Extinction

224 characterization as conventionally performed suffers from several important drawbacks.

225 1. Using the conventional definition and measurement approach, ESR depends on the

226 experimental apparatus, more specifically the separation distance between the burners.

227 This prevents general conclusions that a higher ESR corresponds to a stronger flame

228 as would be ideally concluded.

229 2. The conventional approach as implemented in Chemkin takes prohibitively long to

230 calculate extinction when using mechanisms greater than a few hundred species. As

231 models push toward more complex alternative fuel blends or in the case of NG to

232 potential performance enhancing additives, large, detailed models are necessary to

233 understand the fundamental flame behavior.

234 3. Conventional experimental measurement procedures struggle with flame instabilities

235 which have particularly limited the availability of elevated pressure data. In general,

236 extinction behavior is not well-known at pressures of relevance to gas turbines and

237 internal combustion engines. Furthermore, conventional computational methods also

238 struggle for higher pressures (10s of atmospheres) since the flame region becomes very

239 thin and steep and more difficult to solve using Newton methods, even with the use of

240 adaptive meshing.

34 241 1.3 Thesis outline

242 Given the importance of extinction strain rate in studies of turbulent flame behavior, this

243 thesis seeks to revisit ESR computations to resolve several challenges and promote use of

244 ESR in validation, creation, and refinement of detailed chemical kinetic models.

245 Chapter 2 analyzes the ESR dependence on burner separation distance using the tradi-

246 tional approach. It is observed that, for numerical simulations, the strained flame at ex-

247 tinction is identical and independent of the burner separation distance. This observation is

248 further validated with experimental data at several different separation distances. Given the

249 flame at extinction is independent of burner separation distance even though the traditional

250 ESR value does vary, it would be ideal to instead have a single independent parameter. This

251 chapter further presents a method of using experimental measurements at several separation

252 distances to extrapolate to ESR at infinite burner separation distance.

253 Chapter 3 presents a new, open-source code approach to calculating ESR for infinite

254 separation efficiently and for large kinetic models. The traditional numerical approach avail-

255 able in conventional software such as Chemkin scales ESR computation time with number

256 of species to the third power whereas the new code scales nearly linearly.

257 Chapter 4 uses the new code for calculating ESR, Ember, to explore the pressure depen-

258 dence of extinction behavior under lean and rich conditions, up to pressures of relevance to

259 internal combustion engines and gas turbines. These pressures are otherwise unachievable

260 given current experimental approaches to measuring ESR. Sensitivity analysis identifies the

261 key kinetic model parameters for accurate ESR predictions at these pressures.

262 Chapter 5 takes an initial step toward producing an ESR validated kinetic model for

263 high pressure combustion of natural gas. The core model made through modification of a

264 starting model taken from Hashemi et al. [26] and then further expanded with a nitrogen

265 chemistry from Glarborg [28]. Rich partial oxidation chemistry generated by RMG in [65]

266 has also been added to the validated core model.

267 Chapter 6 examines in greater detail an important pathway for the formation of naph-

268 thalene with a focus on incorporating relevant rate constant pressure dependence. The

269 pathway goes from cyclopentadienyl radical recombination to naphthalene through multi-

35 270 ple unimolecular rearrangements and two hydrogen loss reactions. Lifetimes on the C10H10

271 and C10H9 surfaces are long enough for bimolecular reactions are observed. The relevant

272 rate constant parameters from this study are included in the rich chemistry subset of the

273 mechanism produced in this work.

274 Chapter 7 summarizes conclusions of this work and indicates areas of for future study.

275 In particular, efforts toward indirect measurement of ESR from turbulent flames is expected

276 to be a strong opportunity area for investigation of strained extinction at higher pressures.

277 Chapter 8 [will be added during the Sloan portion of the CEP program]

36 278 Chapter 2

279 An apparatus-independent extinction

280 strain rate in counterflow flames

281 2.1 Summary

282 Resistance to extinction by stretch is a key property of any flame, and recent work has

283 shown that this property controls the overall structure of several important types of turbu-

284 lent flames. Multiple definitions of the critical strain rate at extinction (ESR) have been

285 presented in the literature. However, even if the same definition is used, different experi-

286 ments report different extinction strain rates for flames burning the same fuel-air mixture

287 at very similar temperatures using similarly constructed opposed-flow instruments. Here we

288 show that at extinction, all these flames are essentially identical, so one would expect that

289 each would be assigned the same value of a parameter representing its intrinsic resistance-

290 to-stretch-induced-extinction, regardless of the specifics of the experimental apparatus. A

291 similar situation arises in laminar flame speed measurements since different apparatuses

292 could result in different strain rate distributions. In this instance, the community has agreed

293 to report the unstretched laminar flame speed, and methods have been developed to translate

294 the experimental (stretched) flame speed into a universal unstretched laminar flame speed.

295 We propose an analogous method for translating experimental measurements for stretch-

296 induced extinction into an unambiguous and apparatus-independent quantity (ESR∞) by

297 extrapolating to infinite opposing burner separation distance. The uniqueness of the flame

37 298 at extinction is shown numerically and supported experimentally for twin premixed, single

299 premixed, and diffusion flames at Lewis numbers greater than and less than one. A method

300 for deriving ESR∞ from finite-boundary experimental studies is proposed and demonstrated

301 for methane and propane experimental diffusion and premixed single flame data. The two

302 values agree within the range of ESR differences typically observed between experimental

303 measurements and simulation results for the traditional ESR definition.

304 This chapter is principally a reproduction with permission of the results included in [66].

305 Collaborators for these results are: Ray Speth and William Green from MIT and Hugo

306 Burbano, Ashkan Movaghar, and Fokion Egolfopoulos from USC. All experimental results

307 presented here for the convenience of the reader were produced by the USC collaborators

308 using their experimental apparatus.

309 2.2 Background

310 In many combustion environments, such as internal combustion engines and gas turbines,

311 flames are exposed to high rates of surface deformation (stretch rate). Therefore, it is

312 important to have combustion models that can accurately predict how flames will behave

313 in these highly-stretched environments. Flames can be extinguished due to stretch or heat

314 loss, or a coupling of the two. Extinction conditions have been investigated both numerically

315 and experimentally for premixed and diffusion flames in the counterflow configuration [43,

316 67]. The two opposed impinging jets can be represented by a quasi-one dimensional model

317 and the flame stretch can be directly related to the strain rate in the hydrodynamic zone.

318 The strain rate at which extinction occurs is defined as the extinction strain rate (ESR)

319 [34]. This experimental configuration allows for the three following flame configurations to

320 be studied:

321 1. Twin premixed flames with symmetry about the stagnation plane, e.g. [43]

322 2. A single premixed flame opposing an inert flow, e.g. [46]

323 3. A diffusion flame with oxidizer and fuel mixture supplied from opposing jets, e.g. [51]

38 324 For the same equivalence ratio, single premixed flames exhibit a lower ESR than twin

325 flames, as additional thermal energy is lost to the cold, inert, opposed jet. Diffusion flames

326 typically are more prone to extinction since they are more susceptible to reactant leakage

327 given that they cannot propagate and thus adjust to changes in the flow, and the flame is

328 also losing heat to both the fuel and the oxidizer jets. Figure 2-1 shows the corresponding

329 numerically-calculated 1D axial velocity profile for half of the symmetric domain. In exper-

dvaxial 330 iments, a characteristic strain rate SRchar is measured as the maximum value of | dz | in

331 the hydrodynamic zone and its value at extinction corresponds to the ESR that is typically

332 reported; it should be noted however that in stagnation type flows the rigorously derived

dvaxial 333 strain rate is κ = 2vr/r = − dz (see supplementary material for additional details) [35].

334 Additionally, a global strain rate (GSR), originally proposed by Seshadri and Williams [68],

335 is widely used in extinction studies, and it can be computed from the exit velocity and burner

336 separation distance (BSD). However, it is only valid for high Reynolds numbers and when

337 the exit strain rate is negligible. In practical devices, a non-zero strain rate at the exit of

338 the burner is always present, and especially for premixed flames it can have a considerable

339 effect [46, 69]. This makes the ESR formulation a more accurate approach than GSR, since

340 the exit strain rate can be easily accounted for in the ESR formulation. In one of the first

341 experimental studies of ESR, Law et al. [43], used high-contraction nozzles in an impinging

342 counterflow flame configuration to characterize ESR for twin premixed methane and propane

343 flames over a range of equivalence ratios. Modern efforts in ESR experimental measurement

344 have investigated ESR for ethanol [44], dimethyl ether [44, 45], C5-C12 alkanes [46], other

345 fuels [40, 47–51] and fuel mixtures [61, 62]. In addition, experimental capabilities for char-

346 acterizing ESR at elevated pressures have been developed for diffusion flames [51, 57, 58,

347 64].

348 Recent studies have demonstrated the importance of ESR in governing structure and be-

349 havior of turbulent flames [29, 32, 70]. More specifically, it was shown that for premixed, lean

350 CH4/H2 combustion at varying equivalence ratios, it is possible to ‘collapse’ data from vary-

351 ing fuel compositions on the basis of the mixture’s simulated ESR. Using a swirl stabilized

352 combustor, Shanbhogue et al. [29] found ESR to describe well the transition between the

353 observed average flame shapes when burning at constant Reynolds number and swirl angle.

39 354 Taamallah et al. [32] used simulated ESR to generate a modified Karlovitz number, which in

355 turn was used to describe the point at which the turbulent flame begins to propagate in the

356 the outer recirculation zone. This study allowed swirl angle and Reynolds number to vary.

357 The most recent study by Michaels et al. [70] expands beyond swirl stabilized flames to bluff

358 body-stabilized turbulent flames. It was shown that the recirculation zone length after the

359 bluff body correlates with the ESR value of the inlet mixture. In all of these studies, ESR

360 is computed numerically for twin flames assuming an ideal apparatus (no strain rate at the

361 burner exits) and an arbitrary BSD.

362 An unfortunate characteristic of current ESR measurement methods is that the ESR

363 value depends strongly on the experimental apparatus [67]. Principally, it is BSD that

364 has the largest impact on the measured ESR. However, the strain rate boundary condition

365 at the burner exits also impacts the observed ESR by effectively extending the BSD (see Fuel + Air 366 supplementary material). Ina. an ideal experimental setup, the strain rate at the burner exit

367 is negligible, but this is often impossible to achieve. These two factors together effectively

368 mean that for the same fuel/oxidizerFlame mixture, the ESR values are expected to differ for

369 different BSD values, which could complicateStagnation the Plane assessmentStreamlines of the extinction resistance of

370 reacting mixtures under otherwise identical thermodynamic conditions. This is a significant

371 drawback to the use of ESR for fuel reactivity characterization and as a kinetic model

372 validation target. Ideally, a quantity representing resistance to extinction would be uniquely Fuel + Air

2.5 4 Flame Region 3.5 Characteristic Heat 2 Strain: 3 = release cm [kJ rate

𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 2.5 1.5 𝑺𝑺𝑺𝑺 𝒅𝒅𝒗𝒗𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒎𝒎𝒎𝒎𝒎𝒎 𝒅𝒅𝒅𝒅 2

1 1.5 Flow Direction Axial velocity [m/s] Axial velocity - 3

1 s - 1

0.5 ] Reference 0.5 Flame Speed Stagnation Point 0 0 0 2 4 6 8 10 Distance from jet exit [mm]

Figure 2-1: Characteristic axial velocity profile obtained when simulating the the counterflow jets in one dimension for twin flames with symmetry at the stagnation plane.

40 373 defined by the temperature, pressure, and reactant composition of the opposing flows, but

374 would be decoupled from the experimental apparatus.

375 In the present study, an approach is given for converting ESR measured by any apparatus

376 to a single universal value, characteristic of the flame but independent of the apparatus. To

377 achieve this, a method was developed for the extrapolation of ESR to strained potential

378 flow in the infinite BSD limit, ESR∞. The new methodology was applied numerically and

379 experimentally for lean methane and propane twin and single premixed flames and diffusion

380 flames. This approach is conceptually similar to the way the laminar flame speed is reported

381 by extrapolating the value measured in a specific apparatus to a hypothetical zero strain

382 rate [71, 72]. It is suggested that this alternative formulation will make ESR studies more

383 transferable and valuable. Given a fuel chemistry model, ESR∞ is relatively easy to compute

384 using new freely available software [73] which opens the door for routine use of ESR∞ as a

385 validation target for kinetic modeling in much the same way that laminar flame speed and

386 ignition delay are currently used.

387 2.3 Methods

388 2.3.1 Computational

389 ESRs were computed for both finite and infinite burner separation boundary conditions.

390 ESRs for finite burner separation were calculated with Cantera [38], using custom scripts to

391 facilitate computations for many BSDs and conditions. This approach has been validated

392 against the software typically used for ESR calculations, Chemkin-Pro [74]. The general

393 formulation of the governing equations used by Cantera and Chemkin-Pro was derived by

394 Kee et al. [75]. For infinite BSD conditions, the formulation implemented in Ember [73] is

395 used. Ember has been validated for ESR calculation in a recent work [76]. For additional

396 detail on Ember, see the software webpage. The FFCM1 [77] and AramcoMech1.3 [17]

397 kinetic models were used for methane and propane flames respectively. The mixture averaged

398 transport formulation was used except when comparing to experimental data where the

399 multicomponent transport formulation was used for improved accuracy.

41 400 2.3.2 Experimental

401 Experiments by the USC collaborators were carried out in the counterflow configuration

402 at atmospheric pressure and unburned reactant temperature, Tu = 298 K. Following the

403 recommendations by Burrell et al. [69], straight tube burners with a set of flow conditioning

404 screens at the exit were used to prevent the uneven pressure distribution at the stagnation

405 plane from affecting the flow field upstream of the burner exit. Uniform exit axial velocity

406 profiles were obtained with this configuration, which guaranteed the measurement of the

407 actual ESR at the centerline of the flow field. A burner diameter, D = 21 mm, was chosen to

408 do experiments at BSD = 25.0 - 21.0 - 17.0 - 13.5 mm, for which the counterflow configuration

409 assumptions hold. Stable flames at relatively low ESRs, below 100 1/s, were obtained,

410 which allowed well-resolved velocity measurements especially close to the flame region. The

411 relevance of this detail to the main objective of this investigation will be given in the following

412 section.

413 Lean CH4 and C3H8 premixed flames were established in the symmetric twin-flame con-

414 figuration by counterflowing two fuel/oxidizer jets against each other, and in the single pre-

415 mixed flame configuration by counterflowing a fuel/oxidizer jet against a N2 jet. CH4 and

416 C3H8 diffusion flames were established by counterflowing a fuel/N2 jet against an oxidizer

417 jet. For all cases, the jet momenta were balanced. The reactant compositions were chosen

418 to achieve extinction at the highest BSD and highest Reynolds number (∼ 1000 based on

419 burner diameter and kinematic viscosity of the reactants) for which the flames were stable.

420 Subsequently, extinction conditions were measured at lower BSDs, which required lower flow

421 rates. Table 2.1 presents the reactant compositions of all cases studied.

422 The velocity profiles immediately before extinction were measured in both premixed and

423 diffusion flames by first establishing a near-limit flame and then slightly reducing the fuel flow

424 rate by up to 0.5% to achieve extinction (e.g., [41, 46, 67, 78, 79]). The velocity fields were

425 measured throughout the domain at the same time, to achieve more accurate measurements

426 near the flame region, using particle image velocimetry (PIV). High-efficiency nebulizers

427 (MeinhardÂő HEN-170-A0.3) and high-accuracy syringe pumps (ChemyxÂő Nexus 6000)

428 were used to generate submicron silicon oil droplets to seed the flow. ESR was determined

42 429 as the maximum magnitude of the axial velocity gradient along the stagnation streamline

430 in the hydrodynamic zone. For more experimental details see [79]. The uncertainties of

431 the reported data were quantified as the Âś1σ standard deviation from the mean value.

432 Uncertainty in the flow composition has been shown to be less than 0.5% [79].

433 2.4 Results & Discussion

434 2.4.1 Extinction flame independence from BSD

435 In Fig. 2-2a, computed ESR is plotted as a function of BSD for methane/air flames in

436 premixed twin, premixed single, and diffusion configurations. For all flame types, ESR falls

437 as the flame is more constrained by a shorter BSD. An important similarity between the

438 solution profiles at any BSD is observed in Figs. 2-2b–d. Here it is shown that for all

Twin Flames 3.0 a. b. 1800 Twin Flames Sep. = 6 mm Single Premixed Flame Sep. = 50 mm Diffusion Flame 2.5 1600 2.0 ]

1 1400 s [

1.5 R S E 500 1.0 Temp. 400 Axial velocity [m/s] 0.5 300 0.0 0 20 40 60 80 100 3 2 1 0 BSD [mm] Dist. from stagnation plane [mm] Single Premixed Flame Diffusion Flame 2 2 c. Sep. = 6 mm d. Sep. = 8 mm Sep. = 50 mm Sep. = 50 mm 1 1

0 0

1 1 Temp. Temp. Axial velocity [m/s] Axial velocity [m/s]

2 2 2 0 2 2 0 2 Dist. from stagnation plane [mm] Dist. from stagnation plane [mm]

Figure 2-2: (a) Computed ESR vs BSD for twin (φ = 1 from both jets) and single (φ = 1 from one jet and N2 from the opposing jet) premixed methane flames and a methane/air diffusion flame (pure methane opposing air). (b–d) Computed axial velocity profiles immediately before extinction for each of the three flame types at large and small BSD. Displays the consistency of the flame at extinction regardless of BSD.

43 439 three flame types, regardless of BSD, the axial position of the flame just before extinction is

440 constant. Moreover, the structure of the flame at extinction is observed to be independent

441 of the BSD, as shown from the temperature profiles in the insets of Figs. 2-2b–d. The flames

442 are the same, but the ESRs are significantly different.

443 Flames with different Lewis numbers (Le) have been shown to exhibit very different

444 extinction behavior when subjected to increasing strain rates (e.g., [35]). For Le < 1, the

445 burning intensity increases with the strain rate because the enthalpy coming from the mass

446 gained through diffusion is greater than the heat loss from the flame. Thus, extinction is

447 achieved when the strain rate is high enough to move the flame on the stagnation plane

448 and the residence time is reduced to allow reactant leakage. On the contrary, when the

449 strain rate increases less intense burning is observed when Le > 1 due to flame temperature

450 and reactivity reduction. For Le > 1 mixtures, extinction occurs while the flame is away

451 from the stagnation plane. In the case of single flames, heat loss to the opposing N2 jet

452 also plays a role in the extinction process. Though the two Lewis number regimes exhibit

453 different behavior while progressing to extinction, in both regimes near-extinction flame and

454 hot region are independent of BSD, Fig. 2-3.

455 Going a step beyond simple comparison of large and small BSDs, ESR and the corre-

456 sponding axial velocity profile are calculated using a strained flow formulation at infinite

Le < 1 (Lean) Le > 1 (Rich) 1e9 1000 Sep. = 6 mm a. 1.4 b. 1.4 c. Sep. = 50 mm 900 1.2 1.2

800 Q (J/s) 1.0 1.0 ]

1 700 Position

s 0.8 0.8 [

R 1e9

S 600

E 0.6 0.6

500 Axial velocity [m/s] Axial velocity [m/s] 0.4 0.4 Q (J/s) 400 0.2 0.2 Le < 1 (Lean) Sep. = 6 mm 300 Le > 1 (Rich) Sep. = 50 mm Posistion 0.0 0.0 10 20 30 40 50 3 2 1 0 3 2 1 0 BSD [mm] Dist. from stagnation plane [mm] Dist. from stagnation plane [mm]

Figure 2-3: (a) Computed ESR vs BSD for twin methane flames at lean (φ = 0.7, Le < 1) and rich (φ = 1.3, Le > 1) conditions. (b,c) Computed velocity profiles displaying the the consistency of the flame/hot region at extinction independent of BSD. Inset plots show the rise or fall of peak heat release rate as function of SRchar for the different Le regimes.

44 457 BSD. The ESR∞ result can then be directly compared with the finite BSD formulation re-

458 sults previously shown. This is done for stoichiometric, twin, premixed methane flames in

459 Fig. 2-4. The overlap of the near-extinction flame position and hot region holds for even

460 infinite burner separation with the only differences between the three solutions shown in

461 Fig. 2-4 appearing in the velocity profiles of the cold flow before the flame. This result

462 suggests that ESR at infinite BSD, ESR∞, is a suitable general description of the extinction

463 behavior for a given fuel mixture, similar to how laminar flame speeds are usefully reported

464 by convention as a single value at zero strain.

465 In Fig. 2-5 the numerically-observed independence of the near-extinction flame from

466 BSD is tested experimentally. Velocity profiles along the centerline were measured at four

467 BSDs (each within the range of experimental separations used previously e.g.[43]) for CH4

468 and C3H8 twin and single premixed flames, to test both Le regimes, and CH4 and C3H8

469 diffusion flames. The experimental conditions are summarized in Table 2.1. The individual

470 absolute values of the experimental axial velocities are centered with respect to one another

471 by minimizing vertical separation error between the velocity profiles. PIV measurements

472 were possible until silicone oil droplets (boiling point 200C) vaporized close to the flame

473 region. At this temperature, the temperature gradients are not sufficiently large to cause

474 measurable thermophoretic effects. The uncertainty of the PIV data in the flame region

2000 30 a. b. Sep. = mm c. 1.4 Sep. = 50 mm 1900 Sep. = 6 mm 25 Jet Exit 1.2 1800

20 1.0 1700 ] 1

s 0.8 [

1600 15 R S E 0.6 1500 10 Axial velocity [m/s] Axial velocity [m/s]

1400 0.4

5 1300 0.2 ESR Finite boundary ESR 1200 0 0.0 0 10 20 30 40 50 60 25 20 15 10 5 0 1.0 0.8 0.6 0.4 0.2 0.0 BSD [mm] Dist. from stagnation plane [mm] Dist. from stagnation plane [mm]

Figure 2-4: (a) ESR vs BSD for stoichiometric twin methane flames (φ = 1), compared to −1 the value calculated for strained potential flow at infinite BSD (ESR∞ = 1935s ). (b) Axial velocity profiles for selected BSD values. (c) Magnified view of (b) to better highlight the overlap in the hot region.

45 475 is affected by the luminosity of the flame and also hindered by the low velocity in this

476 region when compared to the velocity at the exit of the burner. The minimum pre-flame

477 velocity, typically referred to as the reference flame speed for premixed flames (e.g., [43]),

478 was measured with sufficient accuracy based on detailed uncertainty analysis. For the case

479 of diffusion and single premixed flames, the velocity measurements carried out on the side

480 where a flame is not present had a slightly increased uncertainty due to experimental noise

481 in the vicinity of the stagnation plane where the velocity approaches zero. However, those

482 unavoidable uncertainties are not large enough to alter the conclusions of this study.

483 2.4.2 Experimental method for ESR∞

484 If ESR∞ is to be used as a more general extinction parameter, the question then becomes how

485 to determine it from experimental results where a finite BSD is unavoidable. When using

486 1D approximation to model strained, reacting flows, all velocity gradients in all directions

487 are computed. Plotting computed vr/r (easily available from the numerical solution and

Twin Single Diffusion 8 8 4.5 7 7 4.0 6 6 3.5 5 5 3.0 4 2.5 4

Methane 3 2.0 3 2 1.5 2 1 1.0 Fuel + Oxid. Fuel + Oxid. Fuel + Oxid. Inert 1 Fuel Oxid. 0 10 5 0 5 10 10 5 0 5 10 10 5 0 5 10

8 5.0 8 7 4.5 7 6 4.0 6 5 3.5 5 4 3.0 4 3 Propane 2.5 3 2 L = 13.5mm

2.0 Axial Velocity [m/s] 2 L = 17mm 1 1.5 L = 21mm 1 L = 25mm 0 1.0 10 5 0 5 10 10 5 0 5 10 10 5 0 5 10 Axial position [mm]

Figure 2-5: Experimental axial velocity profiles for the three different flame types at ex- tinction: twin, single, and diffusion for lean methane (Le < 1) and lean propane (Le > 1). Experimental conditions given in Table 2.1. Profiles demonstrate the consistency of the flame region and reference flame speed at extinction, supporting independence of the near- extinction flame from BSD as was numerically observed in Fig 2-2.

46 Table 2.1: Experimental conditions for Fig 2-5. Ambient temperature and pressure, 298K and 1atm. Fuel Type φ Oxidizer

CH4 Twin 0.55 19:81 O2:N2 CH4 Single 0.72 20:80 O2:N2 16.3:83.7 CH4:N2 Diff. - 25:75 O2:N2 C3H8 Twin 0.65 19:81 O2:N2 C3H8 Single 0.70 20:80 O2:N2 8.8:91.2 C3H8:N2 Diff. - 25:75 O2:N2

488 equivalent to half the strain rate) profiles in Fig. 2-6, an important observation can be

489 made. Specifically, when the cold flow vr/r profiles are linearly projected into the flame

490 region, they are all shown to intersect at a single point. Since the strain rate is constant in

491 the cold flow region for the infinite separation potential flow solution, the intersection point

492 must be at ESR∞ [76]. Given that the local strain rate is equal to twice vr/r, the observed

493 intersection allows for the potential experimental determination of ESR∞ by measuring axial

494 velocity profiles directly before extinction for a number of different BSDs. No correlation

495 was found between the intersection of the strain rates with expected parameters such as

496 peak radical concentration, heat release rate, and H+O2=O+OH reaction rate.

497 The diffusion and premixed single flame data were used as an initial test of the proposed

498 method for determining ESR∞ through the intersection of vr/r lines. The results of this

Twin Single Diffusion 2000 600 600 a. b. c.

1800 500 500

1600

] 400 400 s /

1 1400 [

r / r v 1200 300 300 Decreasing Jet Separation 1000 200 200

800 100 100 1.0 0.8 0.6 0.4 0.2 0.0 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Dist. from Stagnation [mm] Dist. from Stagnation [mm] Dist. from Stagnation [mm]

Figure 2-6: Computed radial velocity over radial position (strain rate) plotted as a function of axial position for twin, single, and diffusion flames. Curves correspond to data points of Fig. 2-2a. The strain rates in the cold flow regime intersect at a single value.

47 Single Diffusion 150 150

100 100

50 50

Methane 0 0 50

10 5 0 5 10 10 5 0 5 10

CALC ESR /2 150 100 EXP (L =13.5mm) EXP (L =17mm) ]

1 100 EXP (L =21mm) EXP (L =25mm) 50 s [

r 50 / r v Propane 0 0

10 5 0 5 10 10 5 0 5 10 Axial position [mm]

Figure 2-7: Experimental proof of concept of the extrapolation procedure proposed in Fig. 2-6 using the experimental data from Fig. 2-5. Good agreement between the calculated ESR∞ value and the experimental intersection point is observed.

Table 2.2: Result summary from test of concept for ESR∞ determination shown in Fig. 2-7 Fuel Type Exp. Calc. % Diff. −1 −1 CH4 Single 120s 169s 41 −1 −1 CH4 Diff. 114s 137s 20 −1 −1 C3H8 Single 161s 168s 4 −1 −1 C3H8 Diff. 122s 138s 13

499 analysis are shown in Fig. 2-7 and summarized in Table 2.2. The velocity profiles on the

500 flame side of the domain were horizontally translated such that the respective minimum

501 velocity points all occurred at the same position. In order to plot the experimental vr/r, a

1 dvz 502 finite difference approach was used to calculate 2 dz which was then linearly fit in the cold

503 flow regions. Finally, an optimization routine was used to minimize the distance from a single

504 point to all of the vr/r lines in order to quantitatively identify the intersection point and thus

505 the experimental value for ESR∞. The theoretical value of ESR∞ was calculated for all flames

506 using multicomponent transport formulation, as discussed earlier. Overall, this procedure

507 yields clean intersections for both diffusion and single premixed flames based on the velocity

508 profiles from the four BSDs considered; the reported differences between experimental and

509 computed ESR∞ in table 2.2 attributed to kinetic model uncertainty since finite boundary

48 510 ESR calculations for the experimental data also ranged from 3-40% difference. In closing, it

511 should be emphasized that the reliable experimental ESR∞ determination using the proposed

512 approach requires sufficient spatial resolution of the minimum pre-flame velocity region as it

513 is essential to accurately align the velocity profiles at different BSDs.

514 2.5 Conclusions

515 In this study, an approach was developed for the determination of extinction strain rates

516 in stagnation-type flames that are independent of the burner separation distance. The ap-

517 proach involves extrapolations to infinite burner separation distance that are analogous to

518 extrapolations to zero strain rate for laminar flame speed determination. By eliminating

519 dependence on the experimental apparatus, this approach makes extinction strain rate a

520 viable target for kinetic model validation in addition to use as a governing parameter for

521 turbulent combustion behavior. The independence of the near-extinction flame structure

522 from the burner separation distance was shown first numerically and then validated with

523 experimental velocity measurements. In order to determine experimentally extinction strain

524 rates at the limit of infinite burner separation distance, it is proposed to use the experimen-

525 tally measured axial velocity profiles for near-extinction flames at a few separation distances

526 in order to determine the intersection of the respective strain rate lines. This approach

527 was validated for methane and propane premixed and diffusion flames through experiments

528 that resulted in clear intersection points and reasonable percent differences in ESR∞ when

529 compared with 1D model predictions. Based on these encouraging results, the extinction

530 strain rate at at the limit of infinite burner separation distance could be viewed as a proper

531 parameter for characterizing extinction behavior, and further investigation and examination

532 of the experimental method for determining this parameter is warranted.

49 THIS PAGE INTENTIONALLY LEFT BLANK

50 533 Chapter 3

534 Ember: An open-source, transient

535 solver for 1D reacting flow using large

536 kinetic models, applied to strained

537 extinction.

538 3.1 Summary

539 Simulation of quasi one-dimensional reacting flow is a standard in many combustion studies.

540 Here Ember, a new open-source code for efficiently performing these calculations using large,

541 detailed chemical kinetic models is presented. Ember outperforms other standard software,

542 such as Chemkin, in computation time by leveraging rebalanced Strang operator splitting

543 which does not suffer the steady-state inaccuracies of most splitting methods. The splitting

544 approach and implementation used in Ember is described. Ember is validated for computa-

545 tion of flame extinction through imposed strain, extinction strain rate (ESR), and shown to

546 be capable of modeling three typical experimental strained flame configurations: premixed

547 twin flames, premixed single flames opposing inert, and diffusion flames. As further demon-

548 stration, Ember is used to investigate Lewis number effects on ESR using a detailed chemical

549 kinetic model with 500 species for simulation of strained extinction of lean (Le > 1) and

51 550 rich (Le < 1) propane/air flames. Primary trends predicted by Law [35] using asymptotic

551 theories of strained flames are accurately reproduced with the large, detailed chemical ki-

552 netic model. However, the complicated chemistry introduces some subtle phenomena not

553 seen with single-step models. The Ember software is open-source and freely available to any

554 user online.

555 This chapter is principally a reproduction with permission of the results included in [76].

556 Collaborators for these results are: Ray Speth and William Green. Ember was principally

557 developed and written by Ray Speth primarily for the calculation of time-dependent strained

558 flames. The author of this work has extended Ember for calculation of cylindrically impinging

559 jets and computation of extinction strain rate at infinite burner separation distance.

560 3.2 Background

561 Much of our understanding of combustion is derived from one-dimensional (1D) systems

562 and models. Asymptotic analysis of flame models using simplistic chemistry has proven to

563 be extremely useful in understanding a variety of combustion phenomena; examples include

564 several key contributions by Law [35, 80]. One-dimensional numerical simulations of flat,

565 cylindrically symmetric, or radially symmetric flames using detailed chemistry models are

566 used to compute a variety of important fuel combustion properties including flame speeds,

567 extinction strain rates, and to understand formation of pollutants by different fuels. They are

568 also used in interpretation of flat flame data, including molecular beam mass spectrometry

569 experiments. Often multiple 1D simulations need to be run, e.g., to build up or validate

570 flamelet libraries, or to cover the range of flame conditions that will be encountered in

571 a combustion device. Accurate fuel chemistry models typically include hundreds of species

572 and thousands of reactions, so fast, robust numerical solvers for 1D reacting flows are needed.

573 Currently-available 1D flame solvers including Chemkin, Cantera, and FlameMaster rely on

574 discretized axial domains and damped Newton methods [38, 74, 81]. This means that the

575 system Jacobian bandwidth grows approximately linearly with number of species. However,

576 computation time for factorization of the Jacobian (necessary when using direct Newton

577 methods) scales according to the Jacobian bandwidth cubed [82], causing these solvers to

52 578 slow down significantly as the number of species increases. By leveraging sparsity within the

579 bandwidth, some improvement over this cubic scaling is attainable [83, 84]. New software

580 tools implementing algorithms that scale better for large numbers of species are needed [85].

581 Extinction Strain Rate (ESR) has long been recognized as a particularly important flame

582 property. Recent works have exposed new value in the realm of turbulent combustion for

583 this fundamental flame parameter [29, 32, 70] which has been studied heavily by Law and

584 co-workers [39, 43, 86–90]. In 2015, Shanbhogue et al. [29] observed that flame shape

585 and instability were well characterized by the ESR of the combustion mixture in the case

586 of turbulent, swirl-stabilized, premixed CH4/H2 combustion. These flames were studied at

587 constant Reynolds number and swirl angle. A subsequent paper expands this analysis to

588 varying Reynolds numbers and swirl angles through use of a modified Karlovitz number,

589 defined as the ratio of the outer recirculation zone (ORZ) frequency to the mixture ESR at

590 the point the flame expands into the ORZ. The modified Karlovitz number used was found

591 to be constant for the swirled, turbulent, methane flames studied. This observation was

592 consistent for varying hydrogen percentage as well [32]. The two works, while constrained to

593 methane based fuel mixtures, suggest that ESR plays a critical role in determining the extent

594 and edge behavior of turbulent flames. A third study further highlights the significance of

595 ESR for turbulent flames by examining such flames stabilized by flow around a bluff body.

596 The work finds that the flame offset distance from the bluff body correlates with the ESR

597 value of the fuel/oxidizer mixture. This study again focuses on CH4/H2 flames [70].

598 Law’s early work on flame extinction focused on systems involving droplet vaporization

599 leading to diffusion flames where asymptotic analyses were employed [80, 86]. In a series of

600 experimental works, Law and coworkers detailed extinction of propane/air mixtures in both

601 stagnation flow against a smooth surface and in opposed axisymmetric jet flow [87, 91]. The

602 first work observed the importance of velocity gradient, strain, in governing flame extinc-

603 tion [91]. The second and third works investigated the impact of downstream heat-loss and

604 preferential diffusion on the extinction of premixed flames by using both impinging jets and

605 stagnation flow against a surface. The work found downstream heat loss and/or incomplete

606 combustion, rather than stretch alone, to be necessary to achieve extinction for fuel mixtures

607 in which the deficient reactant had a larger diffusion rate [87]. Additionally, it was observed

53 608 that flame-front instability due to preferential diffusion occurred only under rich conditions

609 for the propane/air mixtures studied [87]. A numerical study examining high-stretch extinc-

610 tion limits by Law and coworkers concludes that radiative losses have little impact on ESR

611 [88]. A 1986 paper by Law et al. provided the first full experimental characterization of ESR

612 as a function of equivalence ratio, doing so for twin/double, impinging, premixed methane

613 and propane flames [43]. In a subsequent work, the importance of Lewis number with regard

614 to flame extinction was demonstrated [92]. Law and coworkers also produced several works

615 on non-steady extinction effects by sinusoidal variations in opposed jet velocities. For high

616 oscillation frequencies, it was shown that the steady ESR might be surpassed without extinc-

617 tion if a more favorable strain rate is once again imposed in a time scale sorter than that of

618 extinction, confirming that extinction is not an instantaneous process [90]. This observation

619 was later expanded to strained premixed flames [93, 94] and to oscillations in the context of

620 spherical flame extinction [95, 96].

621 Further efforts in calculation and measurement of ESR have been made by Egolfopoulos

622 and coworkers, one of Law’s former students. These efforts have expanded ESR characteri-

623 zation across a wide range of fuels including, ethanol [44], dimethyl ether [44, 45], C5-C12

624 alkanes [46], cyclopentadiene [47], JP-5 [40], and more [48–51]. Many other significant efforts

625 toward expanding the ESR database have been made by Niemann and coworkers [52–59] and

626 other researchers [60–64]. Some recent works have begun to experimentally characterize ESR

627 at elevated pressures thanks to new, enclosed counterflow apparatuses and refined seeding

628 procedures [51, 58, 64].

629 Current computations of ESR typically rely on the use of the Chemkin software suite first

630 published by Robert Kee and coworkers in 1980 and later expanded to include a module for

631 the calculation of ESR [97–99]. The ESR computation module in Chemkin is based on a 1988

632 article by Kee et al. which developed the boundary value formulation for 1D axisymmetric

633 opposed jet reacting flow. The plug flow boundary value problem (BVP) formulation pub-

634 lished by Kee et al. [75] offered significantly improved agreement with experimental measure-

635 ments of ESR when compared with previous formulations which assumed infinitely separated

636 jets producing purely potential flow. This formulation was later extended by Nishioka, Law,

637 and Takeno through the introduction of internal temperature boundary conditions which al-

54 638 low for traversing the extinction turning point of the flame S-curve through pseudo-arclength

639 continuation preventing the Jacobian of the numerical system from becoming singular at the

640 extinction point [39]. This method has since been added to the Chemkin ESR module [74].

641 The Nishioka et al. method works well for many smaller kinetic models, and has been used in

642 several recent works for comparisons with experimental results [46, 100]. Unfortunately, the

643 Nishioka et al. method implemented in Chemkin takes prohibitively long when attempting

644 to run calculations for larger more detailed mechanisms, even when using the ChemkinPro

645 solver which takes advantage of Jacobian sparsity (Figure 3-6b). At the same time, many

646 modern kinetic model production efforts are aimed at producing more complete models for

647 increasingly complex fuels and fuel blends [56, 101]. For example, models from NUI Galway,

648 AramcoMech1.3 and AramcoMech2.0, which seek to characterize C1-C4 chemistry, possess

649 253 and 493 species respectively [17–23]. Furthermore, current automated model generation

650 techniques such as the Reaction Mechanism Generator (RMG) also produce kinetic models

651 with large numbers of species [13, 102].

652 In this work, we introduce Ember, a new, freely-available, open-source code for the

653 simulation of dynamic, 1D strained flames which, among other capabilities including laminar

654 and strained flame speeds, provides superior performance for simulation of extinction strain

655 rate with large kinetic models [73]. The code is C++ based for computational efficiency and

656 wrapped in Python for convenience. It achieves improved performance by using rebalanced

657 Strang splitting to reformulate the system into more efficiently solvable ODEs [103]. Further

658 details are discussed in the Methods section of this work. The source code is available to the

659 community on GitHub (https://github.com/speth/ember) under the MIT License, and a

660 set of installation instructions and a dependency list are available in the documentation linked

661 to from GitHub. To demonstrate the new simulation capabilities available using Ember, the

662 time-dependent solver is used to investigate artificial perturbations in Lewis number and

663 track the impact on the flame, as well as calculating ESR for rich propane flames using a

664 large, detailed mechanism. Progression to extinction is examined for Le > 1 and Le < 1.

55 1

Planar Flames: = , = Cylindrical Flame: = , = Disc Flames: = , = 2 x x 𝜶𝜶 𝟎𝟎 𝜷𝜷 𝟏𝟏 𝜶𝜶 𝟏𝟏 𝜷𝜷 𝟏𝟏 𝜶𝜶x 𝟎𝟎 𝜷𝜷 z z z

Figure 3-1: Strained flame configurations simulated by Ember.

665 3.3 Methods

666 The governing equations solved by Ember are for a time-dependent, strained, 1D flame 4

667 with the strain rate imposed as a function of time. We introduce a unified formulation

668 which permits the solution of planar, disc, and cylindrical flames, as depicted in Figure 3-1.

669 In this section, we describe the governing equations for the flame, the spatial and temporal

670 discretization schemes, and the numerical methods used to describe the discretized equations.

671 3.3.1 Governing Equations

672 The governing equations used here are based on the general forms derived by Kee et al.

673 [34]. A unified formulation for the different strained flame geometries can be developed by

674 introducing two parameters, α and β, where α = 0 for planar and disc flames and α = 1 for

675 cylindrical flames, and β = 1 for planar and cylindrical flames and β = 2 for disc flames.

676 The coordinate and velocity normal to the flame are denoted x and v, respectively and the

677 coordinate and velocity tangential to the flame (in which the flame is stretched) are denoted

678 z and u, respectively. The potential flow velocity field in the nonreacting flow (designated

679 with the subscript ∞) can be written as

az a |R |Rα ! u = ; v = s s − x (3.1) ∞ β ∞ α + 1 xα

680 where a is the time-varying strain rate. In the case of the cylindrical flame, the constant

681 Rs represents a line source/sink at the axis of symmetry that allows the stagnation surface

682 to be a cylinder of radius Rs (which may be negative), permitting simulation of flames with

56 683 arbitrary curvature. The stretch rate κ for a flame at radius Rf is

α dR κ = a + f (3.2) Rf dt

684 When the flame is stationary, the stretch rate reduces to κ = a and thus curvature does not

685 contribute to the stretch rate for stationary flames in this configuration.

686 The governing equation for momentum in the z-direction can be written, after removing

687 terms that are either identically zero or neglected by the boundary layer approximation, as

∂u ∂u ∂u ∂p 1 ∂ " ∂u# ρ + ρu + ρv = − + xαµ (3.3) ∂t ∂z ∂x ∂z xα ∂x ∂x

688 The pressure gradient outside the boundary layer can be found by substituting Equation 3.1

689 into Equation 3.3 ∂p ρ z da ρ a2z = − ∞ + ∞ (3.4) ∂z β dt β2

690 where ρ∞ is the density of the unburned mixture. Introducing the notation U ≡ ua/(βu∞) =

691 u/z, substituting Equation 3.4 into Equation 3.3, and dividing by z, we obtain the momentum

692 equation along the stagnation streamline

∂U ∂U ρ da ρ 1 ∂ " ∂U # ρ + ρU 2 + ρv = ∞ + ∞ a2 + xαµ (3.5) ∂t ∂x β dt β2 xα ∂x ∂x

693 The mass continuity equation can be written as

∂ρ 1 ∂ ∂(ρu) + (xαρv) + = 0 (3.6) ∂t xα ∂x ∂z

694 With the substitution for U, this becomes

∂ρ 1 ∂ + (xαρv) + βρU = 0 (3.7) ∂t xα ∂x

695 The conservation equations for the species mass fractions Yk are written as

∂Y ∂Y 1 ∂ ρ k + ρv k = − [xαj ] +ω ˙ W (3.8) ∂t ∂x xα ∂x k k k 57 696 whereω ˙ k are the molar production rates, Wk are the molecular weights. jk are the diffusive

697 mass fluxes, calculated as

∂Y DT ∂T j = −ρD k − k + Y j0 (3.9) k km ∂x T ∂x k

698 where Dkm are the mixture averaged diffusion coefficients relating mass fluxes to mass frac- T 699 tion gradients, Dk are the thermal diffusion coefficients (included when the multicomponent 0 700 transport model is enabled), and j is a correction term defined to enforce the requirement P 0 ∗ 701 jk = 0. We find j by first calculating the uncorrected fluxes jk using Equation 3.9 with 0 0 P ∗ 702 j = 0 and then calculating j = − jk.

703 The energy conservation equation is written neglecting compressibility and viscous effects.

704 Along the stagnation streamline, the energy equation can then be written in terms of the

705 temperature as

K K " # ∂T ∂T 1 X ˆ 1 X ∂T 1 1 ∂ α ∂T ρ + ρv + hkω˙ k + jkcp,k = α x λ (3.10) ∂t ∂x cp k=1 cp k=1 ∂x cp x ∂x ∂x

ˆ 706 where cp is the mixture specific heat capacity, hk are the species molar enthalpies, cp,k are

707 the species specific heat capacities, and λ is the mixture thermal conductivity.

708 3.3.2 Spatial & Temporal Discretization

709 Equations 3.5, 3.7, 3.8, and 3.10 comprise the governing equations for the strained flame.

710 These equations are spatially discretized on an adaptive one-dimensional grid, using first-

711 order upwinded differences for convective terms, and second-order differences for other spatial

712 derivatives. This produces a system of ODEs for T , U and Yk with an algebraic constraint

713 imposed by v. Boundary values or zero-gradient conditions are applied to T , U, and Yk at

714 each end of the domain, and the value of v is specified at one point (which may be internal to

715 the domain). To simulate unstrained laminar flames, an internal fixed temperature condition

716 is applied and used to calculate v at that point.

717 To efficiently integrate the governing equations in time, we employ the rebalanced split-

718 ting method [103] so that reaction, diffusion, and convection terms can be integrated sep-

58 719 arately. The rebalanced splitting method is applied recursively, first splitting the reaction

720 and transport operators and then splitting the convection and diffusion operators within the

721 transport operator. In order to make the split equations easy to integrate in parallel, we also

722 split terms which couple multiple solution components at multiple grid points as a separate

∗ 723 “cross” term. We denote the balanced reaction, diffusion, and convection operators as R ,

∗ ∗ 724 D , and C , respectively; the corresponding splitting constants as R˜, D˜, and C˜; the cross

725 operator as X; and the components associated with the energy, species, and momentum

726 equations with the subscripts T , Yk, and U, respectively. The split governing equations may now be written as

∂T ∗ ∗ ∗ 727 = R + C + D + X (3.11) ∂t T T T T ∂Yk ∗ ∗ ∗ 728 = R + C + D + XY (3.12) ∂t Yk Yk Yk k ∂U ∗ ∗ 729 = CU + DU (3.13) 730 ∂t

where the individual split terms are

K ∗ 1 X ˆ ˜ 731 RT = − hkω˙ k + RT (3.14) ρcp k=1 ∗ ω˙ kWk ˜ 732 R = + RY (3.15) Yk ρ k

∗ ∂T 733 C = −v + C˜ (3.16) T ∂x T ∗ ∂Yk ˜ 734 C = −v + CY (3.17) Yk ∂x k ∗ 2 ∂U ρ∞ da ρ∞ 2 735 C = −U − v + + a + C˜ (3.18) U ∂x ρβ dt ρβ2 U " # ∗ 1 ∂ α ∂T 736 ˜ DT = α x λ + DT (3.19) ρcpx ∂x ∂x " # ∗ 1 ∂ α ∂Yk ˜ 737 D = x ρDkm + DY (3.20) Yk ρxα ∂x ∂x k " # ∗ 1 ∂ α ∂U 738 D = x µ + D˜ (3.21) U ρxα ∂x ∂x U

739 (3.22) 740

59 K 1 X ∂T 741 XT = jkcp,k (3.23) ρcp k=1 ∂x " T !# 1 ∂ α Dk ∂T 0 742 X = x − Y j (3.24) Yk α k 743 ρx ∂x T ∂x

744 The integration of the system state vector yn over a “global” timestep h to find yn+1 is

745 performed by integrating the following ODEs starting at time tn with ξ1, . . . , ξ7 representing the intermediate states of the split solver

dξ1 ∗ 746 = D (ξ ); ξ (t ) = y (3.25) dt 1 1 n n dξ2 ∗ 747 = C (ξ ); ξ (t ) = ξ (t + h/4) (3.26) dt 2 2 n 1 n dξ3 ∗ 748 = D (z ); ξ (t + h/4) = ξ (t + h/2) (3.27) dt 3 3 n 2 n dξ4 ∗ 749 = R (z ); ξ (t ) = ξ (t + h/2) (3.28) dt 4 4 n 3 n dξ5 ∗ 750 = D (z ); ξ (t + h/2) = ξ (t + h) (3.29) dt 5 5 n 4 n dξ6 ∗ 751 = C (z ); ξ (t + h/2) = ξ (t + 3h/4) (3.30) dt 6 6 n 5 n dξ7 ∗ 752 = D (z ); ξ (t + 3h/4) = ξ (t + h) (3.31) dt 7 7 n 6 n

753 yn+1 = ξ7(tn + h) (3.32) 754

755 To compute the splitting constants for the next timestep, we first calculate the following

756 quantities, which can be thought of as average estimates of the reaction, convection, and diffusion terms over the global timestep

1  tn+h ∗∗ ˜ 757 R = ξ4 −R (3.33) h tn 1  tn+h/2 tn+h  ∗∗ ˜ 758 C = ξ2 + ξ6 − C (3.34) h tn tn+h/2 1  tn+h/4 tn+h/2 tn+3h/4 tn+h  ∗∗ ˜ 759 D = ξ1 + ξ3 + ξ5 + ξ7 − D (3.35) 760 h tn tn+h/4 tn+h/2 tn+3h/4

b 761 where the notation ξi|a indicates the difference ξi(b) − ξi(a). The split constants to be used

60 for the next timestep are then computed as

˜ 1 ∗∗ 1 ∗∗ 1 ∗∗ 1 762 R = − 2 R + 2 C + 2 D + 2 X (3.36) ˜ 1 ∗∗ 3 ∗∗ 1 ∗∗ 1 763 C = 4 R − 4 C + 4 D + 4 X (3.37) 1 ∗∗ 1 ∗∗ 1 ∗∗ 1 764 D˜ = R + C − D + X (3.38) 765 4 4 4 4

766 3.3.3 Integration of the Split Equations

767 The reaction terms, Equations 3.14 and 3.15, are independent at each grid point, and thus are

768 integrated separately and in parallel. The equations at each grid point are integrated using

769 either the CVODE [104] which implements the backward differentiation formulas (BDF), or

770 the CHEMEQ2 solver which implements a quasi-steady-state (QSS) algorithm [105]. The

771 latter solver is of note because it does not rely on constructing and factorizing the system

3 772 Jacobian, thus avoiding the O(K ) scaling associated with fully implicit dense solvers.

773 The convection terms, Equations 3.16, 3.17, and 3.18, are integrated together with the

774 continuity equation 3.7. The time derivative of the density appearing in the continuity

∗∗ ∗∗ 775 equation is evaluated at the start of each global timestep based on the values of R , C ,

∗∗ 776 and D . As written, these equations are all coupled by the density. We can eliminate this

777 coupling by first using the ideal gas law to replace the density, ρ = pW/(RT ) where W is the

778 mixture molecular weight and R is the molar gas constant. The mixture molecular weight

779 is defined as 1 Y = X k (3.39) W Wk

780 Differentiating this with respect to time yields

∂W 1 ∂Y = −W 2 X k (3.40) ∂t Wk ∂t

781 Replacing ∂Yk/∂t with Equation 3.17 yields a conservation equation for W

∂W ∂W C˜ = −v − W 2 X Yk (3.41) ∂t ∂x Wk

782 Equations 3.7, 3.16, 3.18, and 3.41 then form a complete system which may be integrated

61 783 over a split timestep by first calculating v using Equation 3.7 given values for T , U, and

784 W , followed by evaluating the time derivatives for the latter three variables. This system of

785 equations is not stiff, and so is integrated using the Adams-Moulton formulas as implemented

786 by CVODE [104]. Once the values of v have been calculated for the split timestep, the

787 individual species equations can be integrated independently and in parallel according to

788 Equation 3.17.

789 Assuming the mixture transport properties to be constant during each global timestep,

790 the diffusion terms for each solution component are independent and can be integrated in

791 parallel. This assumption also means that the ODE for each component is linear with

792 constant coefficients, allowing a straightforward implementation of the second-order BDF

793 integrator to be used.

794 Since the contributions from the cross terms X are small, these terms are calculated at

795 the start of each global time-step and held constant through the split integration stages.

796 Thermodynamic, transport, and kinetic parameters needed in each equation are com-

797 puted using Cantera [38]. We modify the normal rules for computing the mixture-averaged

2 798 diffusion coefficients and viscosity to eliminate the O(K ) scaling that these calculations

799 usually entail. To do this, we apply a threshold condition on the species mole fractions Xk −5 800 and include the contributions of species k on these mixture-averaged properties if Xk > 10 .

801 This approximation has negligible impact on the accuracy of the solutions, but can reduce

802 computational time significantly for large mechanisms.

803 In summary, Ember outperforms conventional solvers principally by solving time depen-

804 dent solution using rebalanced Strang splitting to avoid a large stiff Jacobian and allowing for

805 the stiff chemistry integrations to be performed in parallel, separately at each grid point, and

806 using the significantly more efficient QSS method. The species diffusion integrations are also

807 performed in parallel, separately for each species. The remaining convection terms are decou-

808 pled and integrated in parallel by introducing a surrogate transport equation for the mixture

809 molecular weight. By separating integration of the reaction, diffusion, and convection terms,

810 Ember is a good candidate for implementing further improvements in computation time

811 through use of GPU processing or other algorithmic improvements that can be applied to

812 any of the individual problems, rather than being limited to methods that can be applied

62 813 to the coupled reacting flow problem. As an open-source code, Ember is also amenable to

814 extensions from the community to enable other flame configurations where 1D approxima-

815 tions hold, such as transient, spherically expanding flames and the finite separation distance

816 burner.

817 3.4 Results & Discussion

818 3.4.1 Validation of ESR calculation with Ember

819 As a first step in validation of Ember for ESR calculations a comparison is made between

820 strain progression curves to extinction points calculated by both Chemkin and Ember. A

821 characteristic comparison is shown in Figure 3-2 for a premixed, lean, methane, twin flame

−1 822 extinction. Excellent agreement is observed between Chemkin (ESR = 987s ) and Ember

−1 823 (ESR = 990s ). The QSS integrator and approximate transport methods are observed

824 to have negligible impact on accuracy. ESR for the Chemkin simulation is determined as

dvaxial 825 the maximum of | dz | before the flame using the pseudo arclength continuation method

826 of Nishioka et al. [39] to traverse and identify the maximal strain rate turning point. For

827 Ember, ESR is characterized as the maximum value of the strain parameter, a, that yields a

828 burning solution. Adaptive step sizes for a are used in progressing to extinction. A minimum

829 increase factor of 1.00001 is specified. Each stable flame is determined to have converged

1 dT √ 830 to the steady state solution once || T dt ||/ npoints falls below a tolerance of 10. Ember

831 also supports steady state termination based on the change in heat release rate with time.

832 Initially this may appear an unexpected result since, as discussed in the Methods section of

833 this work, Ember employs a potential flow, or infinitely separated opposed jets, formulation

834 of the 1D reacting flow while Chemkin uses the fixed BVP formulation from Kee et al. [75].

835 However, in this case, the Chemkin simulation has been calculated for a large opposed flow

836 burner separation distance (BSD) (5 cm) allowing it to behave nearly equivalently to the

837 potential flow formulation [106].

838 Ember successfully computes all three types of flames typically considered in extinction

839 studies, namely twin premixed flames, single premixed flames opposing an inert, and diffusion

63 840 flames. Figure 3-3 shows Ember simulations for these three flame types at consistent strains

−1 841 of 300 s . The premixed flames are computed using a stoichiometric equivalence ratio. Note

842 that the stagnation plane is set at x = 0 and the twin flame simulation makes use of the

843 symmetry of the domain. In this comparison, it is clear how the single flame system leads

844 to lower extinction strain rates when compared with twin premixed flames since thermal

845 energy diffuses into the inert opposing jet, thus reducing the strength of the flame. The twin

846 flame achieves a peak temperature of 2170 K while the single flame peaks at 2060K. With a

847 fuel jet of pure methane, the diffusion flame produces the lowest maximum temperature at

848 1900K. Following the general trend of peak flame temperature, Ember calculates the ESR

−1 −1 849 values of these three conditions to be 1920s for twin flames, 590s for the single flame,

−1 850 and 480s for the diffusion flame. The drastic reduction in ESR caused by using an inert

851 opposed flow is often practically necessary from an experimental point of view to prevent

852 turbulence and maintain laminar opposed flow since high initial flow velocities are required

853 to impose high strains.

Twin Methane Flames ( = 0.7)

1800 ] K [

e r 1750 u t a r e p

m 1700 e T

m u 1650 m i x a M 1600

Ember: ESR = 990.33 Chemkin: ESR = 986.98 1550 200 400 600 800 1000 Strain rate [s 1]

Figure 3-2: Comparison of extinction profiles generated using Ember and Chemkin for twin methane flames at an equivalence ratio of 0.7 using the Stanford FFCM1 kinetic model [25]. A BSD of 5 cm is used in Chemkin to approximate the infinitely separated potential flow formulation used in Ember. QSS chemistry integration and approximate transport methods are used.

64 Twin Flames Single Flame Diffusion Flame 2250 100 4 5.0 200 4 5.0 100 4

0 H H H C C C

2000

% % % 1800 2.5 2.5 50 s 2000 s s s s 75 s 25 a a a M 0.0 150 M 0.0 1750 M 0 1600 0.0 0.5 1750 0.5 0.0 0.2 0.2 50 Pos. [cm] Pos. [cm] 50 Pos. [cm] 1500 1400 1500 100 75 25 1200 1250 1250 100 0 50 1000 1000 1000 125 25 800 Temperature [K] Axial Velocity [cm/s] 0 750 150 750 50 600

175 500 500 50 75 400

200 250 250 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.6 0.4 0.2 0.0 0.2 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Distance from stagnation plane [cm] Distance from stagnation plane [cm] Distance from stagnation plane [cm]

Figure 3-3: Characteristic flame types used for studying extinction phenomena. Simulations for strains set at 300s−1 on the flame side of the stagnation plane. Stoichiometric equivalence ratios used for single and twin flames. Opposed flow is nitrogen at room temperature for the single premixed flame. Pure methane opposes air for the diffusion flame.

854 3.4.2 Ember computational efficiency evaluation

855 To assess the magnitude of computational improvement attained by the Ember, a set of

856 strained flame computations were performed for methane flames using a selection of well-

857 established kinetic models of varying sizes,Table 3.1. Due to solver issues (for both Em-

858 ber and Chemkin) caused by reactions with unphysical reverse rates at low temperatures

859 [107], the reaction IC4H9O2+C2H4 IC4H9O2H+C2H3 is removed from AramcoMech 1.3 and

860 4C3H2(S)+M C3H2+M, C4H71-3+C2H3COCH3 C8H131-5,3,TAO, and C4H63,1-2OH C4H612+OH

861 are removed from AramcoMech 2.0. As a characteristic test case, a lean (φ = 0.7) twin

862 methane flame arrangement at standard temperature and pressure for the unburned gas and

−1 863 a strain rate of 500 s is used. For adaptive griding, the Chemkin default parameters are

864 used, 0.1 for gradient and 0.5 for curvature based re-griding. The parallel performance of

865 Ember is shown in Figure 3-4. Ideal parallelism would yield an inverse dependence on the

866 number of cores used since each additional core could perfectly share the computational load.

867 This corresponds to a parallel efficiency of 1 as depicted in Figure 3-4. However, because

868 Ember is only parallelizable down to the grid point for chemistry integrations and requires

869 integration in series for each split step, some deviation ideal behavior is expected. Next, Fig-

870 ure 3-5 demonstrates the value of using rebalanced Strang splitting as opposed to traditional

871 Strang splitting. The principal benefit of ensuring convergence to the correct steady state

65 872 is clear, but as an additional benefit, rebalanced splitting is observed to yield significantly

873 improved computation time. This is attributed to the large steady state offset in the case of

874 traditional Strang splitting which in turn requires more computationally intensive sub-steps

875 as the system continually overshoots the true steady state value. The default global step

876 size in Ember is 20 µs and is used for all other Ember calculations.

877 In Figure 3-6, Ember computation times are compared with those of Chemkin. For

878 these calculations, Chemkin-Pro 17.0 [74] is used and all calculations are constrained to

879 parallelization over four CPUs. For Figure 3-6b the ESR computation time is scaled to the

−1 −1 880 strain rate range of 500s to 1000s . Computation times are fit to number of species

b 881 power laws of the form y = Ax . The power law fitting is based on the factorization of

882 the Jacobian, which scales according to dimension cubed for a dense matrix or bandwidth

883 for a sparse matrix. Figure 3-6a additionally compares the computation time impacts of

884 applying the QSS chemistry integrator and the transport approximation discussed in the

885 Methods section. The most dramatic increase in efficiency comes from the implementation

886 of the QSS chemistry method. When using the internal boundary condition formulation

887 for progression to extinction in Chemkin, the ESR computation time points are observed

888 to scale approximately according as species cubed. The data point for AramcoMech 2.0

889 has been extrapolated to the set ESR range used in Figure 3-6 from a partial progression

890 to extinction. This is a significant loss in computational efficiency when compared to the

891 boundary velocity formulation (Fig. 3-6a) where Chemkin computation times are observed

892 to scale with number of species squared. This suggests that sparse linear algebra methods

893 have been implemented when solving the velocity boundary condition routine, but not for

894 the internal temperature boundary condition routine used in the ESR module. Ember on

895 the other hand, maintains nearly the same ideal scaling observed when computing the test

−1 1.2 896 flame at a strain rate of 500 s , (species ).

897 3.4.3 Initial studies with Ember

898 Analysis of stretched and strained flames in the literature often focuses on the impact of

899 Lewis number, the dimensionless number expressing the ratio of the energy conduction rate

900 to the species diffusion rate typically characterized using the diffusivity of the limiting re-

66 Table 3.1: Combustion kinetic models used for Ember computation speed evaluation. Model # Species # Reactions Reference FFCM1 38 278 [25] GRI3.0 53 325 [14] USCII 111 784 [16] AramcoMech1.3 253 1542 [17] AramcoMech2.0 493 2716 [18–23]

901 actant [35, 87]. Here the newly available time dependent solve capabilities of Ember are

902 leveraged for an investigation into the impact of Lewis number limit behavior on strained

−1 903 flames. This is achieved using twin methane flames simulated at a strain rate of 500s

904 and systematically and instantaneously perturbing thermal conductivity, λ, or individual

905 species diffusion coefficients, artificially forcing the Lewis number to extremes. The diffusion

906 coefficients are modified within Ember such that the mixture averaged values are calculated

907 normally and then the diffusion coefficient for the species of interest is set to 0. The FFCM1

908 kinetic model is used for this analysis [25]. A summary of these results for lean, stoichio-

1.2

1.0

0.8

0.6

0.4

Parallel Efficiency [-] Theoretical Limit FFCM1 GRI3.0 0.2 USC II Aramco1.3 Aramco2.0 0.0 0 2 4 6 8 10 12 14 CPU count [#]

Figure 3-4: Parallel efficiency (single CPU time / n-CPU time / n) plotted against number of physical CPUs used to observe the parallel performance of Ember. Perfect parallelism would yield a constant value of 1 as the computational load is evenly spread over the available CPUs. QSS chemistry solver and the transport approximation method are used.

67 Strang Splitting Rebalanced Strang Splitting

tstep = 5 s tstep = 5 s 1840 tstep = 10 s 1840 tstep = 10 s

tstep = 20 s tstep = 20 s 1820 t = 40 s t = 40 s ] step ] step K K

[ tstep = 80 s [ tstep = 80 s 1820 e 1800 e r r u u t t a a

r 1780 Decreasing Step Size r e e

p p Decreasing 1800 Step Size m m

e 1760 e T T

m m

u 1740 u

m 40 m 1780

i i 15 x x

a 1720 a 10

M 20 M 5 1700 CPU time [s] 1760 CPU time [s] 0 0 25 50 75 25 50 75 t [ s] t [ s] 1680 step step 0.000 0.002 0.004 0.006 0.008 0.010 0.000 0.002 0.004 0.006 0.008 0.010 Simulation time [s] Simulation time [s]

Figure 3-5: Methane twin disc flames. φ = 0.7. Initial temperature 298K, atmospheric pres- sure. Comparison of simulation performance of rebalanced Strang splitting with traditional Strang splitting for varying global integration step sizes. Inset axes show total computation run times for each of the global steps using both methods.

106 (a) (b) 104

105

103

104

2

10 ESR run time [s]

Single strain run time [s] 103 Chemkin (s1.9) CVODE+Mix (s2.5) QSS+Mix (s1.2) Chemkin (s2.9) 1 10 QSS+Approx. (s1.1) Ember, QSS+Approx. (s1.2) 102 100 200 300 400 500 100 200 300 400 500 Mechanism species [#] Mechanism Species [#]

Figure 3-6: Methane twin disc flames. φ = 0.7. Initial temp. 298K, atmospheric pressure. Parallelized over 4 CPUs. Fit to t = AsB where t is time and s is number of species. (a) Computation times for a strain rate of 500 s−1 demonstrating the performance of Ember compared with Chemkin and the benefits of implementing the QSS chemistry integrator and mixture averaged transport approximation. (b) Comparison of Ember and Chemkin ESR computation times for varying kinetic model sizes.

68 909 metric, and rich conditions is shown in Figure 3-7. Since Ember is a transient solver, the

910 convergence to the new steady state in each subplot is shown as a time progression from

911 the initial steady state where all diffusion parameters have their normal value before the

912 given diffusion parameter was perturbed. A first observation is that even without energy

913 conduction, a steady burning flame is be obtained at all three equivalence ratios.This is

914 attributed to the upstream diffusion of radicals and other reactive species. Yet, in each

915 case the steady flame is pushed significantly closer to the stagnation plane in the absence of

916 of energy conduction, indicating a weakening of the flame. As suggested by defining Lewis

917 number based on the limiting reactant, under lean conditions the flame extinguishes with the

918 elimination of methane diffusion and under rich conditions the flame extinguishes with the

919 elimination of oxygen diffusion. Interestingly, under stoichiometric conditions, the elimina-

920 tion of oxygen diffusion extinguishes the flame, but the elimination of methane diffusion does

921 not, though it does show a strong weakening effect on the flame. This observation is ascribed

922 to the importance of O2 in the key branching reaction H + O2 O + OH and the steeper

923 concentration gradient when going from inlet concentration to no remaining O2. Though

924 not shown in Figure 3-7, increases in λ are observed to extinguish the flame as too much

925 thermal energy is lost and eliminating the diffusion of the major products, CO2 and H2O,

926 is observed to have only small impact on the strained flame. However, a more interesting

927 result is obtained when eliminating hydrogen radical diffusion leading to a strong weakening

928 effect in both stoichiometric and rich conditions. While the observed impact of removing

929 energy conduction established its importance in governing the propagation of the flame, this

930 result demonstrates an importance of similar magnitude for the upstream propagation of key

931 radicals. Energy conduction alone does not govern the propagation of the flame.

932 Across all equivalence ratios, methane flames are predicted to have Lewis numbers close

933 to 1 with limited variability [108] making it a non-ideal fuel for further Lewis number inves-

934 tigations which do not artificially manipulate Le as was done in the study summarized in

935 Figure 3-7. Therefore, propane, a convenient fuel used in many studies by Law and cowork-

936 ers [43, 87, 91], is used instead. In the lean limit, propane tends to a Lewis number of 1.83

937 and in the rich limit to 0.93 [35]. To exhibit the capabilities of Ember with complex kinetic

938 models, AramcoMech 1.3 is used here for analysis of lean and rich, strained propane flames.

69 Lean ( = 0.7) Stoich. ( = 1) Rich ( = 1.3) 0 Temp. [K] 0 0 2000 2000 2000 20 20 1000 50 1000 1000 40 40 60 = 0 100 60 80 80 100 150 Axial Velocity [cm/s] Initial State 100 120 Steady State 200 120 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 Dist. from stag. plane [cm]

0 0 2000 0 2000 1500 25 20 1000 1000 1000 50 50 500 40

75 60 DCH4 = 0 100 100 80

125 100 150 150 120 175 200 140 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3

0 0 2000 0 2000 1500 25 20 1000 50 1000 1000 500 50 40 100 75 DO = 0 2 60 100 150 80 125

150 100 200 175 120 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3

Figure 3-7: An investigation of Lewis number effects using complex chemistry simulations through elimination of fuel and oxidizer species diffusivities or elimination of energy conduc- tion (λ = 0). This analysis was done using stoichiometric methane flames with a strain of 500 s−1 in Ember. FFCM1 is used as the kinetic model. Perturbations are achieved by first solving to steady state, then setting the specified diffusion parameter to 0 and allowing the system to progress in time to the new steady state.

70 939 Figure 3-8 shows the results for Le > 1 where energy conduction exceeds propane diffusion

940 toward the flame. As strain is systematically increased toward extinction, the flame moves

941 closer to the stagnation plane, the peak flame temperature falls, and the peak reaction rate

942 falls. The mass fractions of both CO2 and H2O at the stagnation plane trend down from

943 their equilibrium values, signifying incomplete reactions. These trends are in agreement with

944 the trends shown by Law (Figure 3-11) [35, 109]. However, while the CO2 and H2O mass

945 fractions seem to indicate an incomplete reaction, Law characterizes this as an extinction

946 with complete reaction. Looking at the reaction rate curves in Figure 3-8 the spike in reac-

947 tion heat release is observed to be complete before the stagnation plane even at extinction,

948 in accord with the prediction of Law [35], but secondary reactions have clearly not yet con-

949 cluded their progression towards equilibrium. This is an example where multistep chemistry

950 gives a somewhat different result than simple one-step-chemistry flame models.

951 For the rich conditions, Figure 3-9, results in accord with predictions by Law [35] are

952 also observed. For very low strain rates, max temperature rises above the adiabatic flame

953 temperature for the inlet composition and the peak reactivity also rises with increases in

954 hydrodynamic strain. Very near the extinction point, the boundary limitation of the stagna-

955 tion plane and incompleteness of reaction take over and ultimately force the peak reactivity

956 of the strained flame to decrease immediately before extinction. What preliminarily appears

957 to be an nonphysical result is observed for the mass fraction of CO2 which rises with strain

958 to a value well above its predicted equilibrium value based on the composition of the ini-

959 tial mixture. However, when this result is observed in connection with the full equilibrium

960 profiles as functions of equivalence ratio shown in Figure 3-10, an explanation arises that

961 increasing strain enhances the preferential diffusion of oxygen with respect to the rate of

962 energy conduction/propagation of the flame effectively pushing the equivalence ratio back

963 toward 1 from the starting value of 1.5. Law explains these results using a theoretical model

964 of a streamline tube which expands in diameter as it approaches the stagnation plane. Law

965 argues that since thermal and species diffusions are purely perpendicular to the stagnation

966 plane, the change in cross-sectional area of the stream tube allows for additional supply of

967 fuel/oxidizer from the surrounding flow when Le < 1 and further loss of thermal energy

968 when Le > 1 [35].

71 969 The computed ESR values can be compared with the experimental values obtained by

970 Law et al. [43]. The experimental value for finite boundary ESR at an equivalence ratio of

−1 971 0.8 is interpolated to be about 830 s while at an equivalence ratio of 1.5 the experimental

−1 972 value is 1090 s . The infinite boundary ESR values calculated by Ember (Fig.3-8 & 3-

−1 −1 973 9) are 1120 s and 1220 s respectively. Calculating percent difference with respect to

974 the experimental value, the lean value differs by 35% and rich by 12%. The exact burner

975 separation distances (BSDs) are not specified in [43]. However, an approximate range of

976 BSDs are reported as 7 mm to 30 mm. In the previous chapter, it is observed that the

977 critical flame at extinction is independent of the the simulation BSD [106]. Solving Eq. 3.5

978 analytically for invicid and non-reacting flow before the flame yields

2 a 2 vaxial = vax,0 − z (3.42) 4vax,0

979 assuming an ideal plug flow burner without an initial velocity gradient. The strain constant

980 is a, the initial plug flow velocity is vax,0, and the distance from the burner outlet is z.

981 Knowing the value of a at extinction and an axial velocity point before the flame from the

982 Ember solution, it is possible to use Eq. 3.42 to solve for the theoretical plug flow initial

983 velocity, and thus approximate the ESR value for a finite boundary solution. Using the

984 minimum value of the reported BSD range, the finite boundary ESR approximation for the

−1 −1 985 lean propane is 929 s and rich propane becomes 1150 s . Both are overestimates of the

986 corresponding experimental values, but are in significantly better agreement with the lean

987 value now differing by only 12% from simulated to experimental value and the rich value by

988 5.6%. The remaining differences are likely attributed to uncertainties in the the kinetic model

989 parameters and to a lesser degree error introduced by using the more computationally efficient

990 mixture averaged transport formulation as opposed to the full multicomponent formulation

991 including the thermal diffusion coefficients. Non-idealities in the experimental flow field

992 might also contribute to the remaining differences.

72 (a) (b) 0.14

2000

0.12 1900

0.10 1800 Mass Fraction [-] Max Temperature [K] 0.08 1700 CO2 at stagnation plane

Equlibrium CO2

Steady Flame Curve H2O at stagnation plane Equlibrium H O Adiabatic Flame Temp 0.06 2 101 102 103 101 102 103 Strain [s 1] Strain [s 1]

1e9 0 (c) (d) (e) 2000 3.5

3.0 50 1750

1500 2.5 100 1250 2.0

150 1.5 Increasing 1000

Strain Temperature [K] Axial Velocity [cm/s]

Heat Release Rate [J/s] 1.0 750 200

0.5 500

250 0.0 250 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Distance from stagnation plane [cm] Distance from stagnation plane [cm] Distance from stagnation plane [cm]

Figure 3-8: Extinction progression for a propane mixture of equivalence ratio 0.8, yielding a Lewis number greater than 1. AramcoMech 2.0 used.

73 0.115 (a) CO2 at stagnation plane (b) Equlibrium CO2 1980 H2O at stagnation plane

0.110 Equlibrium H2O 1960 0.105 1940

0.100 1920

1900 0.095 Mass Fraction [-] Max Temperature [K] 1880 0.090

1860 Steady Flame Curve 0.085 Adiabatic Flame Temp

101 102 103 101 102 103 Strain [s 1] Strain [s 1]

1e9 2.00 0 (c) 2000 (d) (e)

1.75 1750 50 1.50 1500 1.25 100 1250 1.00 150 Increasing 1000 Strain 0.75 Temperature [K] Axial Velocity [cm/s]

200 750 Heat Release Rate [J/s] 0.50

500 0.25 250 0.00 250 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Distance from stagnation plane [cm] Distance from stagnation plane [cm] Distance from stagnation plane [cm]

Figure 3-9: Extinction progression for a propane mixture of equivalence ratio 1.5, yielding a Lewis number less than 1. AramcoMech 2.0 used.

74 412 Aerodynamics of Laminar Flames

Extinction Le > 1 (complete reaction)

Increasing stretch

Extinction Le = 1 e

t (incomplete reaction) a R

0.20 n o i t c a 0.15 e R

O2

H2O ExtinctionCO Le < 1 0.10 C O(incomplete2 reaction)

Equilibrium Mass Fraction 0.05

0.00

0.0 0.5 1.0 1.5 2.0 2.5 Equivalence Ratio, Distance from Stagnation Surface Figure 10.3.2. Effects of nonequidiffusion on the reaction rate with increasing stretch, especially Figure 3-10: Equilibrium mixturethe for completeness starting of equivalence reactant consumption ratios at of extinction. propane/air gas mixtures. Nitrogen and minor species are omitted for clarity. leading eventually to extinction, with the flame in direct contact with the stagnation 412surface, Aerodynamics as shown of in Laminar Figure Flames10.3.4a. For the diffusionally neutral case of Le = 1, the flame temperature and reaction rate remain unchanged until the reaction zone reaches the surface. With further Extinction increaseLe > 1 in the stretch rate, incomplete reaction and eventually extinction occurs. (complete reaction) Thus incomplete reaction is essential in causing flame extinction for Le ≤ 1 for an Increasing adiabatic and impermeable surface. stretch

Le < 1 Extinction Le = 1 e

t (incomplete reaction) Le = 1 a R

n

o Le > 1 i t c a e R Complete reaction Extinction Incomplete reaction Le < 1 Maximum Flame Temperature (incomplete reaction) Increasing Stretch

Figure 10.3.3. Effects of stretch, nonequidiffusion, and completeness of reaction, on the extinction turning point behavior.

Distance from Stagnation Surface

Figure 10.3.2. Effects of nonequidiffusion on the reaction rate with increasing stretch, especially the completenessFigure of reactant 3-11: consumption Theoretical at extinction. Lewis number predictions included in Combustion Physics [35] for peak reactivity (Left) and progression to extinction (Right). Reproduced with permission leading eventuallyfrom to Cambridge extinction, with University the flame Press. in direct contact with the stagnation surface, as shown in Figure 10.3.4a. For the diffusionally neutral case of Le = 1, the flame temperature and reaction rate remain unchanged until the reaction zone reaches the surface.75 With further increase in the stretch rate, incomplete reaction and eventually extinction occurs. Thus incomplete reaction is essential in causing flame extinction for Le ≤ 1 for an adiabatic and impermeable surface.

Le < 1

Le = 1

Le > 1

Complete reaction Incomplete reaction Maximum Flame Temperature

Increasing Stretch

Figure 10.3.3. Effects of stretch, nonequidiffusion, and completeness of reaction, on the extinction turning point behavior. 993 3.5 Conclusions

994 In this work, Ember, a new, open-source, transient solver, has been introduced for the

995 simulation of one-dimensional, strained flames. Through use of rebalanced Strang splitting,

996 tailored differential equation solvers for each split calculation, and parallelization, Ember

997 significantly outperforms conventional software such as Chemkin, typically used for such

998 combustion simulations in terms of computation time. In the determination of extinction

999 strain rate (ESR), Ember’s computation time scales according to number of species to the

1000 1.2 power while Chemkin-Pro 17.0 is observed to scale with species to the 2.9 power. This

1001 significant improvement in scaling of computation time allows for practical ESR calculations

1002 using large kinetic models such as AramcoMech 2.0 (493 species) which would otherwise not

1003 have been feasible. This is an important capability as increasingly complex fuel blends are

1004 investigated, requiring large, detailed kinetic models. For demonstration of the capabilities

1005 of Ember, it has been used to analyze the Lewis number trends predicted by asymptotic

1006 analyses of strained flames [35]. The transient solver capabilities of Ember were leveraged for

1007 an analysis of initially steady methane flames under lean, stoichiometric, and rich conditions

1008 for which the the energy conduction and individual species diffusions could instantaneously

1009 be perturbed and the strained flame allowed to relax to the new steady state. Of particular

1010 interest was the observed steady flame in the complete absence of energy conduction (Le=0),

1011 propagating through radical and species diffusion alone. Calculations with AramcoMech 2.0

1012 demonstrated the capabilities of Ember with large detailed kinetic models and confirmed

1013 the Lewis number trends predicted by Law’s asymptotic analyses hold even when the fuel

1014 chemistry is extremely complicated. A Lewis number less than one produced a growing

1015 peak reaction rate with increasing strain as additional fuel/oxidizer is diffusively supplied

1016 faster than the thermal propagation of the flame. Alternatively, a Lewis number greater

1017 than one yielded a flame with a steadily decreasing peak reaction rate as increasing strain

1018 enhances thermal energy losses from the flame. Though capable of very rapidly calculating

1019 extinction strain rate under ideal potential flow conditions as demonstrated in this work,

1020 Ember, as of yet cannot produce extinction strain rate predictions for conditions under which

1021 the finite boundary has significant impact. However, a new method of approximating the

76 1022 finite boundary ESR is introduced and demonstrated. Using this method, the simulated rich

1023 propane/air ESR differs by only 6% when compared with a previously reported experimental

1024 value.

77 THIS PAGE INTENTIONALLY LEFT BLANK

78 1025 Chapter 4

1026 Numerical investigation of strained

1027 extinction at engine-relevant

1028 pressures: pressure dependence and

1029 sensitivity to chemical and physical

1030 parameters

1031 4.1 Summary

1032 Resistance to extinction by stretch and and laminar flame speed are important properties

1033 of any combustible mixture. Recent work has shown that extinction by stretch controls

1034 the overall structure of several important types of methane based turbulent flames. The

1035 parameter used to quantify this phenomena, Extinction Strain Rate (ESR), is numerically

1036 studied here for methane based flames across a range of pressures relevant to gas turbine

1037 and internal combustion engines, 1-40atm. The pressure trends are compared with those of

1038 laminar flame speed which is historically better studied. Current kinetic models are found to

1039 differ significantly in their predictions of ESR, particularly at higher pressures. However, the

1040 models agree that ESR of lean flames is a non-monotonic function of pressure and that ESR

79 1041 of rich flames increases significantly with pressure. Pressure dependent kinetics are shown

1042 to be vital to determining ESR pressure trends as are molecular collision efficiencies. Small

1043 amounts of C2H6 and H2 (5% by mol) in the fuel are observed to have significant impact

1044 on ESR at higher pressures. Kinetic model sensitivity analysis is performed using a recently

1045 published high pressure ethane oxidation model. Sensitive reactions largely mirror those

1046 reactions that are sensitive for laminar flame speed calculations. For transport parameters,

1047 Lennard Jones diameter is the most sensitive parameter. Fuel, oxidizer and bath gas show

1048 the greatest transport sensitivity followed by other intermediate species such as CO and H2.

1049 ESR sensitivities to thermodynamic parameters, and in particular species entropy, varied

1050 significantly with pressure. This study informs and motivates further efforts to understand

1051 the phenomena of flame extinction by stretch at elevated pressures.

1052 Collaborators for this work are: Ray Speth, William Green, and Hugo Burbano. Much of

1053 the author’s understanding of the experimental challenges of extinction strain rate calcula-

1054 tions at elevated pressures is a result of discussions with Hugo Burbano, an experimentalist

1055 from USC.

1056 4.2 Background

1057 In many combustion environments, such as internal combustion engines and gas turbines,

1058 flames are exposed to high rates of surface deformation (stretch rate). Therefore, it is

1059 important to have combustion models that can accurately predict how flames will behave in

1060 these highly-stretched environments. Flames can be extinguished due to stretch or heat loss,

1061 or a coupling of the two. Extinction conditions have been investigated both numerically and

1062 experimentally for premixed and diffusion flames in the counterflow configuration, e.g, [43,

1063 67]. The two opposed, impinging jets can be represented by a quasi-one dimensional model

1064 and the flame stretch can be directly related to the strain rate in the hydrodynamic zone [75].

1065 The strain rate at which extinction occurs is defined as the extinction strain rate (ESR) [34].

1066 This experimental configuration facilitates the study of three primary flame configurations:

1067 (1) twin premixed flames, e.g, [43], (2) a single premixed flame opposing an inert flow, e.g,

1068 [46], and (3) A diffusion flame with oxidizer and fuel mixture supplied from opposing jets,

80 1069 e.g, [51].

1070 Recent studies have demonstrated the importance of ESR in determining structure and

1071 behavior of actively burning turbulent flames [29, 30, 32, 33]. Using a swirl stabilized

1072 combustor, Shanbhogue et al. [29] observed ESR to describe the transition between the

1073 time-averaged flame shapes when burning at constant Reynolds number and swirl angle for

1074 various CH4/H2 mixtures. Taamallah et al. [32] used simulated ESR to generate a modified

1075 Karlovitz number, which in turn was used to describe the point at which the turbulent

1076 flame begins to propagate in the the outer recirculation zone. This study allowed swirl

1077 angle and Reynolds number to vary and again considered CH4/H2 mixtures. Wantanabe

1078 et al. [30] and Chakroun et al. [33] examined the instabilities and behavior of turbulent

1079 oxy-combustion. Their works showed ESR to be a better descriptor of differences in flame

1080 stability when compared to LFS and other parameters. In all of these studies, ESR is

1081 computed numerically for twin flames assuming an ideal apparatus (no strain rate at the

1082 burner exits) and an arbitrary burner separation distance.

1083 One of the first experimental measurements explored ESR equivalence ratio dependence

1084 for methane and propane twin, premixed flames [43]. Since that study, many works on

1085 the calculation and experimental measurement of ESR have been made by Egolfopoulos

1086 and coworkers. Their efforts have expanded ESR characterization across a wide range of

1087 fuels including, ethanol [44], dimethyl ether [44, 45], C5-C12 alkanes [46], cyclopentadiene

1088 [47], JP-5 [40], and more [48–51]. Many other significant efforts toward expanding the ESR

1089 database have been made by Niemann and coworkers [52–59] and other researchers [60–64].

1090 Some recent works have begun to experimentally characterize ESR at elevated pressures

1091 thanks to new, enclosed counterflow apparatuses and refined seeding procedures [51, 58, 64].

1092 Unfortunately, even with the enclosed counterflow apparatuses, experimentalists have en-

1093 countered some difficulties in achieving ESR measurements well into the range of pressures

1094 relevant to internal combustion engines and gas turbines. All of the elevated pressure studies

1095 published have used diffusion flames which tend to be the most stable of the three flame

1096 types typically studied in counterflow [51, 58, 64, 110]. In addition, the study by Niemann

1097 et al. that pushes extinction to the highest pressures, nearly to 20 atm, encounters some

1098 radial streaking flame instability starting at around 11 atm [58]. The counterflow measure-

81 1099 ment method in general suffers an inherent issue of the onset of turbulence when seeking to

1100 extinguish flames at higher pressure. Reynolds number is defined as:

ρvd Re = (4.1) µ

1101 where ρ is density, v is average fluid velocity, d is characteristic length, and µ is the fluid

1102 kinematic viscosity. Flows are strictly laminar for Re less than 2300 [34]. Experimentalists

1103 have been able to construct counter flow apparatuses with down to 10 mm diameter opposed

1104 jets [51]. The viscosity for premixed methane-air at 298K and equivalence ratios from 0.5 to

1105 2.0 ranges from 1.82 to 1.74 Pa-s and at atmospheric pressure the density ranges from 1.15

3 1106 to 1.08 kg/m [25, 38]. To achieve extinction, typical opposed jet initial velocities are on the

1107 order of meters per second [66]. Using average values for the viscosity and density ranges

1108 and a conservative velocity of 1 m/s gives a Reynolds number of 620 at atmospheric pressure.

1109 Given the direct correlation with pressure, at 40 atm Reynolds number would be 25,000 and

1110 the flow would be definitively turbulent. In the Niemann et al. work this issue is in part

1111 addressed by replacing nitrogen in the air mixture with helium. Since helium is 1/7th the

1112 mass of nitrogen, and nitrogen makes up about 70% by mole of a stoichiometric methane-air

1113 mixture, this corresponds to roughly a factor of 5 reduction in Reynolds number. This is

1114 very helpful, but clearly does not entirely address the issue. In addition to turbulence limits,

1115 experimentalists also face challenges when seeding the flow. for LDV and PIV measurements

1116 at these higher pressures, the fine silicon oil droplets employed tend to agglomerate and

1117 generate flow non-idealities and instabilities in the flames.

1118 In response to the inherent difficulties for current methods of assessing extinction strain

1119 rate experimentally at pressures of relevance to internal combustion engines and gas turbines,

1120 this study seeks to first explore the premixed flame ESR pressure trends numerically using

1121 a variety of current kinetic models. The goal is to assess how predictions compare and what

1122 trends we can expect across the relevant range of pressures. Beyond comparison of pressure

1123 trends, this study also seeks to perform an in-depth examination of the various parameters

1124 included within a kinetic model for their impact on ESR predictions. The goal is to inform

1125 future experimental efforts that may seek to validate these trends and efforts that aim to

82 1126 refine and further study the key parameters identified for predictions of ESR. A final point

1127 of analysis looks at the impact of key species on ESR predictions.

1128 4.3 Methods

1129 4.3.1 LFS and ESR pressure profiles

1130 This work uses the open-source software Ember [73, 76] to make calculations of extinction

1131 strain rate (ESR) at infinite burner separation distance (BSD) [66] across a range of pressures

1132 relevant to gas turbines and engines, 1-40atm. Recent additions to the Ember code simplify

1133 the process for determining ESR for any combustible mixture. When making laminar flame

1134 speed (LFS) calculations, the open-source software Cantera is used [38]. Input/configuration

1135 files are included in the supplementary materials for reference. Governing equations for the

1136 ESR calculations are described in [76] and for LFS in [34]. For ESR calculations, the twin,

1137 premixed, opposed flame geometry is used. Initial gas temperatures are 298K for both ESR

1138 and LFS calculations. Since natural gas is the primary mixture of interest, methane is used

1139 as the hydrocarbon fuel species and the oxidizer is assumed to be ”air”, O2 :N2 1:3.76.

1140 A number of commonly used combustion kinetic models are used to assess differences

1141 in their predictions, Table 4.1. The reaction IC4H9O2 + C2H4 IC4H9O2H + C2H3 is

1142 removed as mentioned in [76] when using AramcoMech 1.3, since it leads to solver in-

1143 stabilities caused by reactions with unphysical reverse rates at low temperatures [107].

1144 Similarly, in the Hashemi et al. mechanisms, reactions CH3 + M CH + H2 + M and

1145 CH3 + M CH2 + H + M are re-fit with a collision limit constraint in the reverse direc-

1146 tion to modified Arrhenius three body reaction kinetics to avoid the same issue. After

1147 re-fitting of the Hashemi reactions, no ESR or LFS sensitivity to either reaction is observed.

1148 4.3.2 Parameter sensitivities

1149 ESR is numerically defined as the singularity point corresponding to the the maximum strain

1150 rate that can be sustained by the flame. Since Ember leverages time dependent integration

1151 to steady state [76], efficient methods for sensitivity analysis , e.g, [111, 112], are difficult

83 1152 to apply to this system. Therefore, in this work, brute force sensitivities are calculated for

1153 ESR. Uncertainties in rate constants are often greater than a factor of two [25]. Therefore,

1154 for reaction sensitivities, a conservative factor of 1.5 is used to perturb the given reaction

1155 rate constant. The normalized sensitivity is then calculated according to:

∂ln[ESR] Ar ∂[ESR] Ar ∆ESR 1 ∆ESR si = = = = (4.2) ∂ln[Ar] ESR ∂[Ar] ESR ∆Ar PF − 1 ESR

1156 where ESR is the unperturbed extinction strain rate, Ar is the reaction Arrhenius prefactor,

1157 and PF is the perturbation factor (here PF=1.5). Given that there are hundreds to thou-

1158 sands of reactions in most modern kinetic models, some method of isolating the important

1159 reactions to perturb for brute force sensitivity analysis is necessary. Observing that laminar

1160 flame speed sensitivity reaction ordering generally agrees with those obtained for brute force

1161 ESR sensitivity calculations the laminar flame speed sensitivities are first computed using

1162 Cantera and then the top 50 sensitive reactions are selected for ESR sensitivity calcula-

1163 tions. The LFS sensitivities are calculated using the efficient adjoint method implemented

1164 within Cantera. For a small, simple kinetic model the Cantera method was tested against

1165 brute force LFS sensitivity calculations and showed excellent agreement, see supplementary

1166 information.

1167 For ESR sensitivity to transport parameters, Lennard-Jones potential well-depth and

1168 collision diameter were perturbed by a factor of 1.2. Equation 4.2 is used except with the

1169 given transport parameter replacing the Arrhenius prefactor. The dipole moment, polariz-

1170 ability, and rotational relaxation collision number were observed to have negligible extinction

1171 sensitivity, see supplementary information.

1172 For thermodynamic sensitivity analysis, factor change methods are not particularly mean-

1173 ingful due to enthalpy being defined as a relative parameter, i.e. the the amount of uncer-

1174 tainty does not necessarily scale with the magnitude of the value. Thus, the thermodynamic

1175 parameters enthalpy, entropy, and heat capacity at constant pressure, which are described

1176 for each species by NASA polynomials [113], are perturbed by constant offset. Since the

1177 normalized sensitivity equation (Eq. 4.2) is not valid for this type of perturbation, a sim-

ESRperturbed−ESRinitial 1178 ple change ratio ( ) is used for the thermodynamic parameters instead. ESRinitial

84 1179 NASA parameters are converted to thermodynamic properties according to:

C p = a + a T + a T 2 + a T 3 + a T 4 (4.3) R 1 2 3 4 5

Ho(T ) T T 2 T 3 T 4 1 = a + a + a + a + a + a (4.4) RT 1 2 2 3 3 4 4 5 5 6 T

S T 2 T 3 T 4 = a ln(T ) + a T + a + a + a + a (4.5) R 1 2 3 2 4 3 5 4 7

1180 Enthalpies are perturbed by adding 5 kJ/mol (addition of 601.39 to NASA parameter

1181 a6) and entropies by adding 10 J/mol K (adding 1.2027 to NASA parameter a7) using the

1182 uncertainties estimated in [114]. For many of the small molecule species involved in methane

1183 combustion, actual uncertainties are expected to be much less, particularly in the case of

1184 enthalpy [115], but for the brute force method, it is helpful to use worst case values to

1185 ensure sufficient differentiation in ESR. For heat capacity, a perturbation equal to half that

1186 of entropy is used through addition of 0.60139 to parameter a1. Since both enthalpy and

1187 entropy are integrals of heat capacity [113]:

Ho(T ) b R Co(T )dT = 1 + p (4.6) RT T RT

So(T ) Z Co(T ) = b + p dT (4.7) R 2 RT

1188 The perturbation of heat capacity perturbs both enthalpy and entropy by a value that varies

1189 with temperature. Therefore, to maintain consistent reference enthalpy and entropy at 298K

1190 when perturbing heat capacity, 0.60139/298 is subtracted from a6 and 0.60139 ∗ ln(298) is

1191 subtracted from a7.

85 Table 4.1: Combustion kinetic models used for LFS and ESR pressure dependence evaluation. Model # Species # Reactions Reference FFCM1 38 278 [25] GRI3.0 53 325 [14] UCSD (2005) 40 180 [15] UCSD (2016) 57 268 [15] USCII 111 784 [16] AramcoMech1.3 253 1542 [17] Hashemi CH4 68 631 [26] Hashemi C2H6 68 663 [27]

1192 4.4 Results & Discussion

1193 4.4.1 Pressure profiles

1194 In Fig. 4-1AB laminar flame speed (LFS) trends with pressure are compared to extinction

1195 strain rate (ESR) trends with pressure for fuel lean (φ = 0.7) conditions. These profiles are

1196 calculated for a number of commonly used kinetic models (Table 4.1). Visual inspection

1197 of the LFS data shows what appears to be generally good agreement between the kinetic

1198 models. The steady decrease in observed LFS with increase in pressure is attributed to the

1199 increasing density dominating any increases in combustion rate. To account for the changes

1200 in density, sometimes laminar burning flux (LBF = LFS x density) is plotted instead [35].

1201 Laminar burning flux is synonymous with mass burning rate which is also often used. Use

1202 of LBF simply results in a consistently increasing curve instead, either way a monotonic

1203 dependence on pressure. In contrast with the LFS curves, most ESR pressure profiles are

1204 not monotonic; they display an initial increase followed by a subsequent decrease. This trend

1205 is believed to result from an initial dominance of the concentration effect, increasing reaction

1206 rates and thus the strength of the flame, which is then overcome by the impact of pressure

1207 dependent kinetics at higher pressures which favor less reactive species over radicals, eg.

1208 HO2 over H + OH. This concept is later supported in Fig. 4-6. Along this same vein of

1209 reasoning, the extinction Karlovitz number, the ratio of chemical time to flow time (taken

1210 to be the inverse of strain rate) is observed to approach unity at extinction [116, 117], thus

1211 pairing extinction to overall reaction rate trends.

86 1212 To generate further insights about the relative differences between the kinetic models, the

1213 ratio of a given mechanism’s prediction at a particular pressure to a reference mechanism

1214 prediction at the same pressure is plotted. The reference mechanism is semi-arbitrarily

1215 chosen to be FFCM1 because it tends to fall roughly around the middle of the spread of

1216 predictions in Fig. 4-1. Interestingly, the relative difference profiles are very similar between

1217 LFS and ESR, suggesting that, although the overall pressure trends are significantly different

1218 between LFS and ESR, the kinetic impacts on the flame parameters are largely the same.

1219 This lends credence to the concept of using LFS sensitivity values as a preliminary screen

1220 before calculating true brute force ESR sensitivities. While the difference ratio profiles are

1221 quite similar, they are not exactly the same. ESR ratio profiles tend to deviate from the

1222 FFCM1 standard by greater magnitudes. Also note the turning and crossover points are

1223 shifted toward higher pressures for the ESR curves.

1224 Figure 4-2 gives the same pressure profile analysis as was shown for lean flames, but now

1225 under fuel rich conditions, φ = 1.3. For LFS, a similar, strictly decreasing trend is observed

1226 under rich conditions though the initial descent is somewhat faster with respect to pressure.

1227 On the other hand, ESR predictions follow a range of different trends. ESR predictions for all

1228 kinetic models increase nearly linearly above approximately 10 atm, though the slopes vary

1229 from model to model. Some models, such as FFCM1 and AramcoMech1.3 predict relative

1230 maxima around 3 atm, while others, such as GRI3.0, simply increase over the full range of

1231 pressures. The universally increasing trends at higher pressures differ from lean conditions

1232 where ESR decreased at higher pressures for most models, starting around 7atm. Under the

1233 pressures studied here, lean ESR tends to be higher than its rich counterpart except near the

1234 upper limit of the pressure range studied. For applications at high pressures, this difference

1235 in trend could give burning rich an advantage in terms of flame resistance to extinction.

1236 When looking at the difference ratios (Fig. 4-2CD), again similar trends are observed for

1237 both LFS and ESR, but the magnitude of differences between the two is more pronounced

1238 at the higher end of the pressure range.

87 0.200 FFCM-1 FFCM-1 GRI 3.0 2200 GRI 3.0 UCSD (2005) UCSD (2005) 0.175 UCSD (2016) 2000 UCSD (2016) USC II USC II 0.150 AramcoMech 1.3 AramcoMech 1.3 Hashemi (CH4) 1800 Hashemi (CH4)

Hashemi (C2H6) ] Hashemi (C2H6) s

0.125 / Exp. Rozenchan (2002) 1600 1 [

0.100 R 1400 S LFS [m/s] E 0.075 1200

0.050 1000

0.025 800 (a)

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Pressure [atm] Pressure [atm]

(c) (d) 1.6 1.6

1.4 1.4

1.2 1.2

1.0 1.0 LFS relative difference ESR relative difference 0.8 0.8

0.6 0.6 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Pressure [atm] Pressure [atm]

Figure 4-1: A-B) Comparison of laminar flame speed and extinction strain rate pressure profiles for commonly used kinetic models under lean (φ = 0.7) conditions. Experimental data from [118]. All models agree ESR is significantly non-monotonic with pressure at these conditions. C-D) Fractional deviations relative to the FFCM1 mechanism showing the similarities in behavior between LFS and ESR predictions of various models.

88 FFCM-1 FFCM-1 GRI 3.0 GRI 3.0 0.25 3000 UCSD (2005) UCSD (2005) UCSD (2016) UCSD (2016) USC II USC II AramcoMech 1.3 AramcoMech 1.3 0.20 2500 Hashemi (CH4) Hashemi (CH4)

Hashemi (C2H6) ] Hashemi (C2H6) s Exp. Rozenchan (2002) / 1 2000 [ 0.15 R S LFS [m/s] E 1500 0.10

1000

0.05 (a) 500 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Pressure [atm] Pressure [atm]

2.50 2.50 (c) (d)

2.25 2.25

2.00 2.00

1.75 1.75

1.50 1.50

1.25 1.25 LFS relative difference 1.00 ESR relative difference 1.00

0.75 0.75

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Pressure [atm] Pressure [atm]

Figure 4-2: A-B) Comparison of laminar flame speed and extinction strain rate pressure pro- files for commonly used kinetic models under rich (φ = 1.3) conditions. Experimental data from [118] C-D) Fractional deviations relative to the FFCM1 model showing the similarities in behavior between LFS and ESR predictions of various models.

89 1239 4.4.2 Reaction rate constant sensitivity

1240 To find the primary reactions of interest for both the lean and rich conditions, kinetic sen-

1241 sitivity analysis is performed. The Hashemi et al. ethane kinetic model is selected for this

1242 and subsequent sensitivity analyses since it includes kinetic updates for chemistry impacting

1243 methane, is designed for elevated pressure conditions, and represents a good balance of com-

1244 plexity to computation time needed for brute force calculations. The Hashemi et al. model

1245 was validated using flow reactor data up to 100 bar, ignition delay data up to 80 bar and

1246 LFS data up to 10 bar. The top 15 sensitive reactions are shown under each condition (Fig.

1247 4-3), but the full 50 ESR sensitivities calculated and LFS sensitivity results are included

1248 in the supplementary materials. Comparing the the ESR and LFS sensitivity results, Fig.

1249 4-3, shows the good agreement between the LFS and ESR sensitivity rankings on which the

1250 preliminary LFS sensitivity screen is based. Under both lean and rich conditions the sets

1251 of the top 15 most sensitive reactions only differ by one reaction each (Lean: R86 & R54,

1252 Rich: R90 & R212) and the difference occurs near the bottom of the list where sensitivities

1253 are very low in either case. Furthermore, the maximum delta in sensitivity ranking for both

1254 conditions is 3 positions. It is worth noting that the LFS sensitivity values tend to be lower

1255 in magnitude than those calculated for ESR. This is consistent with the more significant

1256 differences between the various kinetic models observed in Fig. 4-1 & 4-2. Furthermore, this

1257 result is consistent with prior works that have shown LFS to scale with the square root of

1258 overall reaction rate [119] while ESR is shown to scale linearly [116, 117].

1259 Within the kinetic model, reactions R1, R29, R59, R67, R83, and R86 are defined using

1260 pressure dependent kinetics. As expected, primary sensitivity is observed to the key branch-

1261 ing reaction, H + O2 O + OH under both lean and rich conditions. Rearranging Eq. 4.2, a

1262 factor of two increase in a given rate will roughly be expected to result in a fractional change

1263 in ESR equal to the sensitivity value. A factor of two reduction in ESR is then predicted

1264 to fractionally change ESR equal to half of the sensitivity value. Since rate constants are

1265 commonly assessed to not be much more accurate than a factor of two, see FFCM1 estimated

1266 rate uncertainties [25], predictions will benefit from increased accuracy of the majority of

1267 the top sensitive reactions. Yet, the majority of these reactions have been recently stud-

90 1268 ied either experimentally or using modern high accuracy ab initio methods. Therefore, a

1269 non-trivial degree of uncertainty in ESR kinetic model predictions appears inevitable given

1270 current methods for measuring or calculating rate coefficients. Examining the references

1271 and rate expressions in the Hashemi et al. model for the top sensitive reactions with a focus

1272 on pressure dependent rates, H + O2(+M) HO2(+M) (R2) was last investigated in 2000

1273 by Troe et al. [120] and could potentially be further refined with current levels of theory

1274 and experimental methods. This reaction is estimated to have an uncertainty factor of two

1275 according to the FFCM1 authors. Additionally, 2CH3 C2H5 + H (R95) is likely pressure

1276 dependent as a part of the 2CH3 C2H6 C2H5 + H network, but it is not currently in-

1277 cluded in Hashemi et al. in a pressure dependent format. Even at atmospheric pressure, the

1278 FFCM1 authors assign this reaction an uncertainty factor of two; it is even less certain at

1279 higher pressures. Another key area of study for kinetic model rate refinement is in pressure

1280 dependent reaction collision efficiencies which will be discussed later in this work.

1281 4.4.3 Transport parameter sensitivity

1282 In 1D flame simulations such as those used for calculation of ESR and LFS, a flame front is

1283 simulated to be propagating within a discretized axial domain. Diffusion and mass transfer to

1284 and from the flame region are predicted using the following transport parameters: Lennard

1285 Jones collision diameter, Lennard Jones potential well-depth, dipole moment, polarizabil-

1286 ity, and rotational relaxation collision number [37]. An initial study using a small simple

1287 example mechanism provided with Chemkin [74], showed no significant ESR sensitivity to

1288 dipole moment, polarizability, and rotational relaxation collision number, see supplementary

1289 materials. Accordingly, Fig. 4-4 focuses on perturbations of L.J. diameter and well-depth.

1290 The transport parameters are used to compute viscosity, ηk, and binary diffusion coefficients,

1291 Dkj according to the relations: √ 5 πmkkbT ηk = 2 (2,2)∗ (4.8) 16 πσkΩ

1292 where mk is the species mass, kb is Boltzman’s constant, T is the temperature, σk is the L.J. (2,2)∗ 1293 diameter, and Ω is the collision integral which is a function of reduced temperature and

1294 reduced dipole moment (See [74] theory documentation for additional details) and

91 Lean ( = 0.7) Rich ( = 1.3)

(a) (b) H + O2 <=> O + OH (R2) H + O2 <=> O + OH (R2)

H + O2 (+M) <=> HO2 (+M) (R1) CH3 + H (+M) <=> CH4 (+M) (R67)

CO + OH <=> CO2 + H (R29) CH3 + HO2 <=> CH3O + OH (R89)

CH3 + HO2 <=> CH3O + OH (R89) CH4 + O2 <=> CH3 + HO2 (R72)

HO2 + OH <=> H2O + O2 (R19) 2 CH3 <=> C2H5 + H (R95)

HCO (+M) <=> CO + H (+M) (R59) CO + OH <=> CO2 + H (R29)

CH4 + O2 <=> CH3 + HO2 (R72) CH3 + O2 <=> CH2O + OH (R91)

CH3 + OH <=> CH3OH (R83) CH4 + OH <=> CH3 + H2O (R70)

CH3 + H (+M) <=> CH4 (+M) (R67) CH4 + H <=> CH3 + H2 (R68)

HCO + O2 <=> CO + HO2 (R64) H + HO2 <=> H2 + O2 (R14) ESR Sensitivity

CH3 + O2 <=> CH2O + OH (R91) CH3 + OH <=> CH2OH + H (R86)

HO2 + O <=> O2 + OH (R17) H + O2 (+M) <=> HO2 (+M) (R1)

2 OH <=> H2O + O (R6) CH3 + O2 <=> CH3O + O (R90)

CH3 + OH <=> CH2OH + H (R86) H + H2O2 <=> H2 + HO2 (R24) 30 atm 30 atm CH3 + O2 <=> CH3O + O (R90) 7 atm H2 + OH <=> H + H2O (R5) 3 atm 1 atm 1 atm

0.5 0.0 0.5 1.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Normalized Sensitivity ( ln[ESR] ) Normalized Sensitivity ( ln[ESR] ) ln[Ai] ln[Ai]

(c) (d) H + O2 <=> O + OH (R2) H + O2 <=> O + OH (R2)

H + O2 (+M) <=> HO2 (+M) (R1) CH3 + H (+M) <=> CH4 (+M) (R67)

CH3 + HO2 <=> CH3O + OH (R89) CH3 + HO2 <=> CH3O + OH (R89)

HO2 + OH <=> H2O + O2 (R19) CH4 + O2 <=> CH3 + HO2 (R72)

CH4 + O2 <=> CH3 + HO2 (R72) CH4 + OH <=> CH3 + H2O (R70)

CO + OH <=> CO2 + H (R29) CO + OH <=> CO2 + H (R29)

HCO (+M) <=> CO + H (+M) (R59) 2 CH3 <=> C2H5 + H (R95)

CH3 + OH <=> CH3OH (R83) CH4 + H <=> CH3 + H2 (R68)

CH3 + O2 <=> CH2O + OH (R91) CH3 + O2 <=> CH2O + OH (R91)

HCO + O2 <=> CO + HO2 (R64) H + HO2 <=> H2 + O2 (R14) LFS Sensitivity

CH3 + H (+M) <=> CH4 (+M) (R67) H2 + OH <=> H + H2O (R5)

HO2 + O <=> O2 + OH (R17) H + O2 (+M) <=> HO2 (+M) (R1)

CH3 + O2 <=> CH3O + O (R90) 2 CH3 (+M) <=> C2H6 (+M) (R212)

2 OH <=> H2O + O (R6) CH3 + OH <=> CH2OH + H (R86)

CH2O + OH <=> H2O + HCO (R54) H + H2O2 <=> H2 + HO2 (R24)

0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 0.6 Normalized Sensitivity ( ln[LFS] ) Normalized Sensitivity ( ln[LFS] ) ln[Ai] ln[Ai]

Figure 4-3: Hashemi et al. C2H6 kinetic model reaction sensitivities. Intermediate pressure points selected based on the relative minima and maxima observed in initial pressure screens under lean and rich conditions (Fig. 4-1 & 4-2). AB) ESR sensitivities determined using a brute force perturbation of 1.5x. CD) LFS sensitivities determined using the Cantera adjoint method implementation.

92 q 3 3 3 2πkb T /mjk Dkj = 2 (1,1)∗ (4.9) 16 P πσjkΩ

1295 where mjk is the reduced molecular mass, P is the pressure, and σjk is the average colli-

1296 sion diameter. The mixture-averaged diffusion formulation [76] is then used to convert the

1297 binary diffusion coefficients into species diffusion coefficients used in the numerical system

1298 to calculate diffusion velocities. An increase in any species’ collision diameter will lead to

1299 a reduction in the diffusion velocity of that species, and, if in significant concentration, a

1300 non-trivial reduction in the viscosity as well. This will impact the viscous dissipation term in

1301 the momentum governing equation. Furthermore, the thermal conductivity of the gas which

1302 governs the thermal diffusion/dissipation depends linearly on the viscosity when using the

1303 parameter formulations in Chemkin and Cantera [38, 74].

1304 To assess and isolate the impact of perturbing the only the overall mixture viscosity, the

1305 source code of Ember was hard modified (flamesolver.cpp, line 683, ”mu[j] = gas.getViscosity()

1306 * PF;”). Using this method for an example methane twin flame arrangement at an equiva-

1307 lence ratio of φ = 0.7, perturbing viscosity by a factor of 2 increased ESR by 2.7% from 1004

−1 −1 −1 1308 s to 1031 s . Dividing viscosity by 2 lowered ESR by 1.6% to 988 s . Removing viscos-

−1 1309 ity entirely drops ESR to 980 s but also has the added impact of increasing computation

1310 time considerably. In Fig. 4-4 the more sensitive L.J. diameters are perturbed by a factor

1311 of 1.2. Even in the scenario in which the perturbed species entirely determines the viscosity,

1312 this only corresponds to a 30% reduction in viscosity as observed in Eq. 4.8. Therefore, the

1313 viscosity changes are not significant for the transport sensitivity results.

1314 On the other hand, a similar source code modification to perturb only the overall mixture

1315 thermal conductivity, (flamesolver.cpp, line 679, ”lambda[j] = gas.getThermalConductivity()

1316 * PF;”), reveals this parameter to be quite important to the calculated ESR. For the same

1317 lean flame as above, a 30% decrease in thermal conductivity yields a 57% increase in ESR to

−1 −1 1318 1580 s and a 30% increase yields a 36% decrease to 640 s . Still, only major species will

1319 have a significant contribution to the overall thermal conductivity. The changes in thermal

1320 conductivity have strong impacts on ESR because the strength of a flame correlates strongly

1321 with its peak temperature and Lewis number, the ratio of thermal diffusion to diffusion of

93 1322 the limiting reactant. By decreasing the gas thermal conductivity, the thermal energy is

1323 better concentrated, yielding a higher temperature flame, and more of the limiting reactant

1324 is able to diffuse into the flame for every unit of thermal energy diffusing away. Without the

1325 hard perturbation of thermal conductivity, the test flame burns at 1760 K for a strain rate

−1 1326 of 500 s , but after a 30% reduction in thermal conductivity the peak temperature rises

1327 to 1840 K for the otherwise same conditions. An opposite dependence is observed for LFS

1328 which Law [35] p. 245 shows to have a linear dependence on thermal conductivity.

1329 In Fig. 4-4, both L.J. diameter and L.J well depth show species ordering and pressure

1330 trends that match for the top four most sensitive species under both lean and rich conditions.

1331 However, L.J. well depth is about an order of magnitude less sensitive than L.J. diameter.

1332 ESR is more sensitive to L.J. diameter of the important species than it is to the rate coefficient

1333 of the key branching reaction, H + O2 O + OH, which is shown to attain a maximum

1334 value of only around 1.5 under rich conditions in Fig. 4-3. Unsurprisingly, under lean

1335 conditions, transport properties for the fuel, methane have the highest sensitives, followed by

1336 nitrogen, and then the oxidizer, oxygen. Similarly, under rich conditions, ESR is especially

1337 sensitive to LJ diameter of molecular oxygen, followed by methane, and then nitrogen.

1338 In both, the LJ diameter of the limiting reactant shows a negative sensitivity. Since a

1339 reduction of L.J. diameter would increase the diffusion coefficient of the species allowing it

1340 to diffuse more quickly into the flame front, the flame will be pushed closer to stoichiometric

1341 conditions and strengthened. Correspondingly, the opposite is true for the excess species,

1342 increasing its diffusion would weaken the flame by pushing it to be effectively more lean or

1343 rich. Since nitrogen is an nonreactive species, its sensitivity stems from its impact on species

1344 diffusing through it and on thermal conductivity as the most abundant species. The 1.2

1345 factor perturbation of nitrogen LJ diameter reduces overall diffusion rates/velocities for all

1346 species by roughly 15%. This overall reduction in diffusion rates reduces ESR under both lean

1347 and rich conditions through a reduction in the diffusion of the limiting reactant as evidenced

1348 by the limiting reactant sensitivities in Fig. 4-4. However, for nitrogen, the reduction in

1349 ESR from lowering the diffusion of the limiting reactant is overcome by the strengthening

1350 effect of reducing the thermal conductivity. This explains the positive sensitivities shown for

1351 nitrogen under both lean and rich conditions. Though significantly less sensitive than the

94 1352 top three species, transport properties of the primary products, H2O and CO2 also appear in

1353 the top most sensitive species list under lean and rich conditions. The consistently negative

1354 sensitivities for the hot products are attributed to their role in carrying thermal energy

1355 upstream. A higher diffusion rate will allow them to more quickly carry thermal energy back

1356 toward the reactants.

1357 Notably the L.J. diameter sensitivities for ESR and LFS at the otherwise same conditions,

1358 Fig. 4-4AB, are significantly different. A relatively strong and comparable sensitivity to

1359 nitrogen is observed for both ESR and LFS though the sign of the sensitivity is opposite.

1360 While constraining the spreading of thermal energy makes for a more strain resistant flame,

1361 thermal energy diffusion is the primary mode of flame propagation [76]. Therefore, decreasing

1362 the thermal conductivity by perturbing nitrogen’s LJ diameter reduces the flame speed. For

1363 LFS, the sensitivity to the diffusion of the limiting reactant is also reversed. This likely is a

1364 result of forcing the flame to move toward the fuel at a faster rate when its diffusion coefficient

1365 is reduced as opposed to the fuel diffusing toward the flame. When tested independently for

1366 LFS, viscosity again has minimal impact on its own. Overall, the LFS transport sensitivities

1367 are significantly lower in magnitude compared with those for ESR and show a significantly

1368 different ordering for most sensitive species.

1369 Considering the strong sensitivity to transport parameters for the fuel, oxidizer, and bath

1370 gas, importance of a full multicomponent formulation for extinction calculations is evident

1371 [121]. As depicted by Ji et al. [46] using the multicomponent formulation typically shifts the

1372 ESR vs equivalence ratio curve to the left. However, due to computational cost considerations

1373 (current multicomponent transport implementations scale with number of species to the third

1374 power) the mixture averaged formulation is used in this study. A more efficient method is

1375 needed. Since only a few species are ever in significant concentration, it may be possible

1376 to efficiently and accurately estimate the multicomponent transport formulation results by

1377 only considering the primary mixture components. Minor and much less sensitive species

1378 could then be estimated using the more efficient mixture averaged formulation. A similar

1379 approximation of only considering the major species is implemented in Ember and works

1380 well [76].

1381 Beyond more accurately including the mixture effects for diffusion coefficients, works

95 1382 by Paul [122, 123] and Middha [124] et al. have also shown that the current calculation

1383 models for even the pure species transport parameters based on the five transport parameters

1384 mentioned previously are insufficient. In particular, Middha et al. shows discrepancies in

1385 high temperature diffusion coefficients, where reliable experimental data is not available, up

1386 to around 20%. At room temperature, 298 K, transport models are within the roughly 3%

1387 diffusion coefficient uncertainties given in [125] for primary fuels such as methane, ethane, and

1388 propane, but this error is expected to grow with temperature. Also of particular relevance is

1389 Paul’s discussion of mixture thermal conductivity treatment for which the methods currently

1390 implemented in Cantera and Chemkin (so called (1,-1) rules) are inaccurate [122]. A 2011

1391 review of transport properties for combustion by Brown et al. [126] further examines current

1392 limitations of transport handling and provides additional recommendations for improvement.

1393 This is highly relevant given the sensitivity to thermal conductivity observed through the

1394 nitrogen transport sensitivity. Given the extreme sensitivity of ESR to transport properties,

1395 Fig. 4-4, implementation of more accurate methods for computing transport properties is

1396 pressing for accurate understanding of extinction behavior at elevated pressures.

1397 4.4.4 Thermodynamic parameter sensitivity

1398 When examining the sensitivities to species thermodynamic properties, an increase in en-

1399 thalpy makes a species less favored at equilibrium. Given that kinetic model reverse rates

1400 are determined based on equilibrium constants, an increase in enthalpy of a species will also

1401 increase the reverse reaction rate if the species is a product and decrease the reverse reaction

1402 rate if the species is a reactant. A change in species enthalpy will additionally impact the

1403 net energy release for the reaction which for certain reactions may impact the peak flame

1404 temperature, a key value related to flame strength and ESR. An increase in entropy causes

1405 a species to be more favored at equilibrium and this favoring effect increases with tempera-

1406 ture based on the formula for Gibbs free energy of a reaction (∆Grxn = ∆Hrxn − T ∆Srxn).

1407 Changes in entropy will not contribute to the energy released by the reaction. Modifications

1408 to heat capacity will increase both enthalpy and entropy of a species. Fortunately enthalpy

1409 and entropy have opposing impacts so there is a partial cancellation of errors in heat capaci-

1410 ties, but its change will have a net impact on the Gibbs free energy toward favoring the given

96 Lean ( = 0.7) Rich ( = 1.3)

CH4 O2 N2 CH4

O2 LFS sensitivity (1atm) N2 LFS sensitivity (1atm)

CO N2 H2 N2 CO O2 CO2 H2O CO2 H OH H2 OH H2 H2O H2O H2 O CO CH4 O2 CO

L.J. Diameter H2O CH4 C2H4 OH H CH3 O CO2 C2H6 HO2 H 0.5 0.0 (a) HO2 0.5 0.0 0.5 (b) 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3

CH4 O2 N2 CH4 O2 N2 CO H2 H2O CO2 CO2 HOCO CH3OO HCOH OCHO CH3O L.J. Well Depth 30 atm 30 atm HOCHO 7 atm CH2OH 3 atm H 1 atm (c) CH3OH 1 atm (d) 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.4 0.2 0.0 0.2 0.4 Normalized Sensitivity ( ln[ESR] ) Normalized Sensitivity ( ln[ESR] ) ln[LJi] ln[LJi]

Figure 4-4: Hashemi et al. C2H6 kinetic model transport parameter sensitivities. Determined using a brute force perturbation of 1.2x. A & B contain subfigures showing LFS sensitivity at 1 atm using the same brute force perturbation

97 1411 species. For minor species not expected to have much impact on the average heat capacity

1412 of the mixture, it is expected that the heat capacity sensitivity results will mostly follow the

1413 entropy sensitivity results.

1414 ESR has a strong positive sensitivity to ∆Hf (O2) under both lean and rich conditions,

1415 Fig. 4-5. This strengthening of the flame by making oxygen a less favored species follows

1416 from the key branching reaction (H + O2 O + OH) since the right hand side radicals will

1417 now be more favored and also the overall heat of combustion will be increased, raising

1418 the flame temperature. Similar logic applies to CH4 which produces the initial hydrogen

1419 radicals that participate in the key branching reaction. Under both lean and rich conditions,

1420 the primary combustion products, CO2 and H2O, show significant negative sensitivities to

1421 enthalpy since increasing the enthalpy of either species will lower the total energy released

1422 during combustion while also making reactive propagation to both products less favored.

1423 The negative sensitivities to the primary radicals, H and OH, also follow intuition since

1424 making either less favorable will reduce its peak concentration and production. An interesting

1425 difference between the sensitivity results at lean and rich conditions is the shift of CH3

1426 from negative sensitivity at lean conditions to positive sensitivity under rich conditions.

1427 Presumably, under lean conditions, it is important for CH3 to be favored since it is important

1428 for the sensitive reaction that produces the initial hydrogen radicals (H + CH3 CH4) but

1429 under rich conditions where CH4 is in excess, making it less favored will increase its reactivity

1430 with other species, helping to drive the reactivity of the overall combustion process.

1431 The Hashemi et al. kinetic model employs thermodynamic NASA polynomials from

1432 the Third Millennium Thermochemical Database [127] and Active Thermochemical Tables

1433 (ATcT) [128, 129]. A recent study by Klippenstein et al. [115] examined the enthalpy values

1434 and uncertainties of a number of key combustion species from the ATcT, comparing with

1435 values from high level ab initio simulations. For the sensitive species observed in Fig. 4-5, the

1436 ab initio calculations agreed with ATcT values within about 0.2 kJ/mol or less, substantially

1437 better than the 5 kJ/mol perturbation used for perturbation analysis. The only exception

1438 was CO2 which gave a deviation of about 1.2 kJ/mol. Using these observed differences as

1439 effective uncertainties in the enthalpy values, there appears to be limited need from an ESR

1440 prediction perspective to refine these values further. The 1.2 kJ/mol difference for CO2

98 1441 would only correspond to a roughly 2% change in ESR prediction.

1442 Entropy perturbations show stronger impacts on ESR and more drastic shifts in sensi-

1443 tivity as functions of pressure. Overall, the trends are less intuitive and more difficult to

1444 describe since the perturbation impact grows with temperature. Along the same vein, one

1445 of the impacts of pressure on the flame is to lead to a more dense, thinner, and often hotter

−1 1446 flame. For example, at the lean conditions studied here for a strain rate of 400 s , the

1447 peak flame temperature is 1780 K at atmospheric conditions but 1860 K at 30 atm, thus

1448 the entropy perturbations will tend to be more emphasized at higher pressures. While there

1449 have been extensive efforts to quantify and minimize enthalpy uncertainty, entropy uncer-

1450 tainty remains a somewhat lesser studied area in the realm of combustion modeling. Entropy

1451 values for atoms, such as hydrogen and oxygen radicals, are known to near perfect precision

1452 since only translational degrees of freedom contribute to the species’ entropy. However, when

1453 vibrational modes, and more importantly hindered rotational degrees of freedom are incor-

1454 porated in the species’ chemical structure, the entropic uncertainty grows. Nevertheless, it

1455 is expected that for most of the sensitive species shown, the accuracy of current ab initio

1456 methods is sufficient for these simple molecules to make their uncertainty in entropy rela-

1457 tively insignificant though, given very strong impact on ESR, some additional verification of

1458 OH entropy may be merited. Additionally, a recent work by Bross et al [130] demonstrates

1459 that, while typically well verified at low temperature, high temperature errors in anhar-

1460 monic partition functions can be significant even for small species of relevance here such as

1461 H2O2, CH4, andCH2.

1462 The heat capacity sensitivity results are largely as expected. Minor species predominantly

1463 mirror the sensitivity results observed for the entropy perturbation. However, major species,

1464 such as nitrogen and oxygen in the lean scenario, show significant negative sensitivities

1465 which are attributed to their contribution to the overall heat capacity of the gas mixture.

1466 The higher heat capacity requires more energy for a given increase in flame temperature. As

1467 an example, for the lean flame at 30 atm the temperature at extinction is initially 1810 K,

1468 but, after the perturbation of nitrogen heat capacity, the temperature at extinction falls to

1469 1710 K. Taking nitrogen as a reference, the diluent gas heat capacity needs be known within

1470 roughly 0.2 J/molK if an approximately linear relation is assumed.

99 kJ Entropy [S] ( 10 J ) Heat Capcity [C ] ( 5 J ) Enthalpy [H] ( 5molK ) molK p molK

O2 H N2 CH4 OH H2O

) H CH4 H 7 .

0 OH CH3 OH

= H2O H2O CH4 (

CO2 H2 O2 n

a CH3 CO CH3 e

L HO2 H2O2 CO 30 atm H2 7 atm HO2 CO2 O (a) 1 atm O (b) H2 (c) 0.2 0.1 0.0 0.1 0.2 0.2 0.0 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0.0 0.2 0.4

O2 OH N2 H H H

) CH4 O2 H2O 3 .

1 H2O CH2 H2

= CO2 H2O O2 (

CH3 HO2 CO

h H2 CH4 CH3 c i

R HO2 H2O2 HO2 30 atm OH 3 atm H2 OH CH2 (d) 1 atm CO (e) CH2 (f) 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.6 0.4 0.2 0.0 0.2 Fractional Change ( [ESR] ) Fractional Change ( [ESR] ) Fractional Change ( [ESR] ) ESRi ESRi ESRi

Figure 4-5: Hashemi et al. C2H6 kinetic model species thermodynamic data pseudo sensitiv- ities sensitivities. Determined using a constant shift of 5 kJ/mole for enthalpy, 10 J/molK for entropy, and 5 J/molK for heat capacity.

1471 4.4.5 Pressure dependent rate coefficients, collision efficiencies,

1472 and minor fuel species

1473 Figure 4-6 investigates the basic importance of including pressure dependent kinetics in

1474 kinetic models for describing the overall ESR pressure trends. Here the impact of impact

1475 of increased concentrations at higher pressures is decoupled from that of actual kinetic rate

1476 changes with pressure by means of complete elimination of pressure dependence or rate

1477 coefficients from a simplified kinetic model for methane chemistry. The model for methane

1478 combustion is included as an example file with the Chemkin simulation software and contains

1479 17 species and 58 reactions [74]. Since a minimal mechanism is being used, this analysis

1480 is only shown for lean flames where simplified chemistry is typically most accurate. The

1481 pressure dependent reactions within the simple model are described using third body reaction

1482 kinetics where the typical modified Arrhenius description of the rate constant is used and

1483 the total gas concentration is multiplied effectively as an additional species to incorporate

100 1484 pressure. As an example, for A + M B + C + M, the reaction rate is given as:

n −Ea RA = −ArT e RT [A][M] (4.10)

1485 where RA is the consumption rate of species A, Ar is the Arrhenius prefactor, n is the

1486 modified Arrhenius temperature exponent, Ea is the Arrhenius activation energy parameter,

1487 R is the ideal gas constant, T is temperature, [A] is the concentration of species A, and [M]

1488 is the concentration of the overall gas mixture at the given pressure. To remove the pressure

1489 dependence of a given reaction within the simple model, the Ar factor is multiplied by the

1490 gas concentration at 1 atm, and the reaction is now written as A B + C. This ensures that

1491 the modified and original model will still give the same prediction at atmospheric conditions

1492 and only begin to deviate from one another as pressure increases.

1493 In Fig. 4-6AB the LFS and ESR pressure trends are shown for the simple kinetic model

1494 before and after removal of all pressure dependent rate coefficients. In addition, the FFCM1

1495 predictions from Fig. 4-1 are included to validate the general trends shown by the simple

1496 kinetic model. Interestingly, the LFS predictions are much higher after removing pressure

1497 dependence, but still follow the same monotonically decreasing trend. On the other hand, the

1498 ESR prediction trend shifts to one of constant increase, losing the relative maximum around

1499 7 atm. In fact, the ESR rises to a point at which it becomes too large for Ember to continue

1500 to converge the flame at a higher strain rate. This clearly shows a dominating behavior of

1501 the pressure dependent kinetics at higher pressures over the ESR increase expected from

1502 the concentration effect for the current conditions. As observed in Fig. 4-2B, under rich

1503 conditions, the concentration effect appears to dominate at higher pressures. Figure 4-6C

1504 then breaks down the impact of removing the pressure dependence of each pressure dependent

1505 reaction included in the simple model. While modifying most of the reactions has little

1506 impact, the pressure dependence of the rate coefficients of three reactions have significant

1507 impact on the calculated ESR. Eliminating the pressure dependence of CH3 + H CH4 or

1508 H + O2 HO2 independently yields the same sharp increase in ESR observed for the overall

1509 removal of pressure dependence. This makes sense since both of these reactions remove H

1510 atoms important in chain branching. In particular, increasing the second reaction rate will

101 1511 shift the primary branching reaction (H + O2) away from the far more reactive O + OH

1512 product channel in favor of HO2. Conversely, the elimination of pressure dependence of

1513 reaction HCO CO + H leads to a reduction in predicted ESR since the increase in this

1514 rate coefficient with pressure yields additional H atoms, vital for the primary branching

1515 reaction. Fig. 4-6D shows the laminar flame speed sensitivity analysis for the simple kinetic

1516 model. Importantly, these key reactions whose pressure dependence has a significant impact

1517 on the ESR predictions show up near the top of the list of sensitive reactions and are the

1518 only pressure dependent reactions in the top 10 shown. Overall, the inclusion of pressure

1519 dependent rate coefficients is vitally important to predictions of ESR as a function of pressure,

1520 even more so than for predictions of LFS.

1521 Pressure dependent kinetics are principally influenced by both the frequency of molecular

1522 collisions and the amount of energy transfered during each collision. Given each molecule’s

1523 unique size and structure, it is then intuitive to expect that their energy transfer behavior

1524 when colliding with another species will also tend to be unique. To correct for this variation

1525 in energy transfer, pressure dependent reactions typically include some set of collision fac-

1526 tors to modify the given pressure dependent rate constant based on the composition of the

1527 surrounding gas. The issue is that these collision efficiencies are typically not well studied

1528 and are frequently simply estimated as integer values which often taken from the historical

1529 values that were fit/estimated when creating GRIMech [14]. However, Jasper and cowork-

1530 ers have introduced fundamental, ab initio methods for determining the impact of various

1531 colliders on specific reaction rates [131–133]. Yet, these methods are computationally in-

1532 tensive, which has slowed widespread application. Furthermore, kinetic modeling softwares,

1533 such as Chemkin and Cantera, have not yet been modified to address a primary finding by

1534 Jasper and coworkers, that the collision efficiencies are temperature dependent. Addition-

1535 ally, a recent work by Burke and Song [134] finds that, beyond neglecting the collisional

1536 temperature dependence, the linear combination procedure based on the overall mixture

1537 composition is insufficient, with errors up to roughly 90% for the reaction they studied,

1538 H + O2(+M) HO2(+M). In Fig. 4-7, the general importance of these collision efficiencies

1539 is demonstrated by setting them all to unity for every pressure depending reaction and ob-

1540 serving the impact on ESR predictions using the FFCM1 model. For lean conditions, the

102 FFCM-1 7000 (b) FFCM-1 0.25 Minimal Mech. Minimal Mech. Minimal Mech (no Pdep.) Minimal Mech (no Pdep.) 6000

0.20 5000

0.15 4000 ESR [1/s] LFS [m/s] 0.10 3000

2000 0.05

1000 (a)

0 10 20 30 40 0 10 20 30 40 Pressure [atm] Pressure [atm]

LFS Sensitivities = 0.7, Ti = 298 K (c) Minimal Mech. [1] CH3 + H + M <=> CH4 + M H + O2 <=> O + OH (41) 5000 [28] CH2O + M <=> H + HCO + M CO + OH <=> CO2 + H (36) [31] HCO + M <=> CO + H + M [35] CO + O + M <=> CO2 + M H + O2 + M <=> HO2 + M (43) [43] H + O2 + M <=> HO2 + M 4000 [48] 2 H + M <=> H2 + M HCO + M <=> CO + H + M (31) [52] H + OH + M <=> H2O + M [53] H + O + M <=> OH + M CH3 + H + M <=> CH4 + M (1) 3000 [56] H2O2 + M <=> 2 OH + M CH3 + O <=> CH2O + H (6) ESR [1/s] H2 + OH <=> H + H2O (40) 2000 P = 1atm H + HCO <=> CO + H2 (32) P = 5atm P = 10atm HCO + O2 <=> CO + HO2 (34) 1000 P = 20atm P = 30atm CH4 + H <=> CH3 + H2 (3) (d) P = 40atm

0 10 20 30 40 0.2 0.0 0.2 0.4 0.6 Pressure [atm] Normalized Rxn Sensitivity

Figure 4-6: Lean (φ = 0.7) methane flames. A) Laminar flame speed pressure profiles showing the impact of completely eliminating pressure dependence from a simple methane kinetic model (17 species, 58 reactions). FFCM1 profile is included for validation of trend given by the simple model. B) Extinction strain rate pressure profiles showing the impact of completely eliminating pressure dependence from a simple methane kinetic model. Again, FFCM1 used to support trend validity. C) Elimination of pressure dependence for each pressure dependent reaction in the simple model independently. D) Laminar flame speed sensitivities for the simple kinetic model.

103 (a) FFCM1 (b) 2500 FFCM1 (No Eff.) 1400

2250 1200

2000

1000 1750 ESR [1/s] ESR [1/s] 1500 800

1250 600 1000

0 5 10 15 20 25 30 35 40 0 10 20 30 40 Pressure [atm] Pressure [atm]

Figure 4-7: Comparison of ESR pressure profiles to assess the impact of assuming unequal collision efficiencies within the kinetic model. A) Lean (φ = 0.7) B) Rich (φ = 1.3)

1541 importance below 5 atm is small, but beyond this point a significantly higher ESR prediction

1542 is observed for the kinetic model with unity collision efficiencies. For rich conditions, the

1543 impact is clear immediately from atmospheric conditions and the low pressure region loses its

1544 relative maximum. To accurately understand ESR pressure trends, clearly an efficient means

1545 of accurately calculating, measuring, or estimating collision efficiencies for pressure depen-

1546 dent reactions is needed along with software implementations in Chemkin, Cantera, etc. that

1547 allow for temperature dependence and leveraging non-linear reduced pressure mixing rules

1548 proposed by Burke and Song.

1549 In addition to understanding the sensitivities of ESR pressure trends to underlying ki-

1550 netic model parameters, it is also of interest to understand the ESR sensitivity to potential

1551 impurities or additives in the fuel mixtures. In Fig. 4-8AB the ESR impacts of 5% by

1552 mol addition of hydrogen, ethane, and carbon monoxide are tested. AramcoMech 1.3 was

1553 used since, of the kinetic models tested in this study, it contains the largest number or

1554 reactions and thus is expected to have the highest probability of including the necessary

1555 chemistry for reactions with the added species. Under both lean and rich conditions, all

1556 three added species are observed to strengthen the flame as pressure increases. However, un-

1557 der lean conditions hydrogen has the most significant impact, while ethane’s impact is most

1558 significant under rich conditions. When examining reaction fluxes, the additional hydro-

104 Lean ( = 0.7) Rich ( = 1.3) 2000 (a) AramcoMech 1.3 AramcoMech 1.3 (5% C2H6) 1600

AramcoMech 1.3 (5% H2) 1800 AramcoMech 1.3 (5% CO)

1400 1600

1400 1200 ESR [1/s] ESR [1/s] 1200 1000

1000

800 800 (b)

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Pressure [atm] Pressure [atm]

Figure 4-8: A-B) Extinction strain rate pressure profiles for the impact of C2H6, CO, and H2 on predictions. AramcoMech1.3 used.

1559 gen helps strengthen the flame by allowing for early thermal energy production by reaction

1560 to water, H2 + OH H + H2O, and production of additional hydrogen radicals necessary

1561 to drive the primary branching reaction. Additionally, the addition of of molecular hydro-

1562 gen minimizes H2 + OH H + H2O running in the reverse direction in the early stages of

1563 the flame. The impact is likely diminished under rich conditions where there is an excess

1564 of hydrocarbon fuel to help supply hydrogen radicals. Somewhat similarly, ethane addi-

1565 tion leads to a notable early increase in the reaction fluxes for C2H6 + H C2H5 + H2 and

1566 C2H6 + OH C2H5 + H2O in the rich flame. Carbon monoxide has a limited impact under

1567 both scenarios. The minor increase in ESR in both cases may be attributed to early reaction

1568 to CO2 and contribution of thermal energy ahead of the primary reaction zone.

1569 4.5 Conclusions

1570 In this study, current kinetic models are found to differ significantly in their predictions of

1571 extinction strain rate (ESR) particularly at higher pressures for lean (φ = 0.7) and rich

1572 (φ = 1.3) conditions. However, the variations in predictions of ESR between the kinetics

1573 models are observed to be generally consistent with the variations in prediction of laminar

1574 flame speed (LFS). This suggests that the same kinetic model parameters affect LFS and

1575 ESR. Consistency between sensitivity of LFS and ESR to rate coefficients supports this

105 1576 observation. Transport parameter sensitivity analysis shows a very strong sensitivity of ESR

1577 to the Lennard Jones diameter of the fuel, oxidizer, and bath gas. For ESR predictions,

1578 improved kinetic transport descriptions as suggested Middha [124], Paul [122, 123], and

1579 Brown [126] are needed. ESR is less sensitive to thermochemical property uncertainties.

1580 Thermodynamic sensitivity analysis shows limited need for more accurate enthalpy values in

1581 terms of ESR predictions. Entropy uncertainties may merit some further investigation, but

1582 are also not expected to be significant for the small species that ESR is most sensitive to under

1583 the conditions studied here. The heat capacity of the bath gas, nitrogen for the flames in this

1584 study, has a strong negative sensitivity. Requiring more energy to increase the temperature

1585 correspondingly weakens the flame. Additionally, pressure dependent kinetics are shown to

1586 be vital to determining ESR pressure trends, as are molecular collision efficiencies. Further

1587 refinement of kinetic model collision efficiency descriptions in line with the work of Jasper

1588 and coworkers [131, 133] and Burke and Song [134] is necessary for accurate ESR predictions

1589 under the conditions of this study. Small amounts of C2H6 and H2 (5% by mol) in the fuel

1590 are observed to have significant impact on ESR at higher pressures. Given the wide variation

1591 in predictions by current kinetic models and uncertainty in many sensitive parameters, it is

1592 important that future efforts seek to study and analyze the phenomena of flame extinction

1593 by stretch at elevated pressures experimentally.

106 1594 Chapter 5

1595 An ESR validated kinetic model for

1596 applications in natural gas

1597 combustion at elevated pressures

1598 5.1 Summary

1599 A basic tool for benchmarking kinetic model performance against a user selected experi-

1600 mental data set is produced. This tool allows for validation based on ignition delay, also

1601 sometimes referred to as autoignition, laminar flame speed, and extinction strain rate data.

1602 Current kinetic models are commonly validated using ignition delay and laminar flame speed

1603 data, but extinction strain rate data is currently much less often used during kinetic model

1604 generation. The tool is then used to produce a kinetic model relevant for natural gas com-

1605 bustion at pressures of relevance to gas turbines and internal combustion engines. The

1606 validation data set used focuses on available CH4,H2, CO, C2H6,C2H4, and C2H2 data.

1607 A recently published model for the high pressure oxidation of ethane is used as a starting

1608 point. After improving the base chemistry agreement with the experimental data, a rich

1609 chemistry submechanism up to the formulation of naphthalene is appended, as well as a

1610 recently published submechanism for nitrogen chemistry. The kinetic model is available in

1611 Chemkin and Cantera format in the supplementary materials.

107 1612 Collaborators for these results are: Te-Chun (Jim) Chu, Alon Grinberg-Dana, and William

1613 Green. Te-Chun Chu ran the calculations that compared the model results with those from

1614 his work in [65] while Alon Grinberg-Dana supported with discussion of available nitrogen

1615 chemistry submechanisms.

1616 5.2 Background

1617 Kinetic model creation for describing natural gas (NG) combustion behavior was primarily

1618 advanced by the Gas Research Institute modeling efforts [14]. This significant, collaborative

1619 effort ultimately led to the production of GRIMech3.0, a standard in combustion simula-

1620 tions since 2000. However, after the production of GRIMech3.0 funding for the collabora-

1621 tion ended, and subsequent modeling efforts diverged, generally focusing on more specific

1622 conditions of interest to a current study or application. Further details on a number of these

1623 more currently produced mechanisms are available in the introduction of this work.

1624 In response to the enormous efforts required to produce detailed kinetic models for even

1625 the simplest of hydrocarbons, automatic kinetic model generation methods have been pro-

1626 duced. Perhaps the most prominent of these is the Reaction Mechanism Generator (RMG)

1627 [13]. The result of multiple decades of academic development at MIT, RMG uses a flux

1628 based algorithm to systematically add relevant species to the kinetic model along with all

1629 corresponding reactions. When a measured or high level ab-initio calculation of a needed

1630 rate constant is not available, RMG uses a database to estimate a given rate based on known

1631 kinetics of similar reactions. Similarly, when the thermodynamics of a potential species are

1632 not known, group additivity methods are used to approximate the enthalpy, entropy, and

1633 heat capacity. Genesys [135] is another automatic generation tool. It differs from the RMG

1634 method by simply adding all species and reactions based on a set of user defined rules.

1635 It also employs a group additivity based method for estimating rates. Avoiding numerous

1636 simulations to determine relevant fluxes makes Genesys quite fast when compared to RMG.

1637 Because of their use of estimated values, and no optimization with regard to experimen-

1638 tal data, automatic generation methods tend to perform somewhat worse for well-studied

1639 chemistry, but are a significant advantage when examining kinetic routes too complex to

108 1640 be measured, calculated, and generally studied individually. Given this drawback and since

1641 small hydrocarbon chemistry has already been fairly extensively studied, the current work

1642 will primarily build and improve by hand from existing models and then leverage recently

1643 automatic generation results for to append a sub-mechanism for rich chemistry.

1644 The goal of this work is to produce a kinetic model for NG combustion at pressures

1645 of relevance to gas turbines and internal combustion engines. The unique contribution of

1646 this work is do so while considering the accuracy of ESR predictions along with the more

1647 traditional parameters, ignition delay and laminar flame speed, since ESR has recently been

1648 shown to be of strong importance when describing and predicting the behavior of turbulent

1649 flames present in the gas turbine and internal combustion engine systems. Submechanisms

1650 for nitrogen and rich chemistry up to naphthalene will be added to improve the versatility of

1651 the final mechanism while allowing one to avoid an overly computationally intensive kinetic

1652 model for applications where nitrogen or rich chemistry are not of particular relevance.

1653 5.3 Methods

1654 A series of functions have been written in Python to allow for the evaluation of a given mech-

1655 anism against a user selected data set. The Python files are included in the supplementary

1656 materials. Base Cantera [38] functions are used for autoignition data and for laminar flame

1657 speed, while Ember [73, 76] is used for extinction strain rate calculations. Sensitivity rou-

1658 tines are available to help identify reactions for further investigation when an experimental

1659 data point is not reproduced well by the kinetic model.

1660 5.3.1 Ignition delay (Autoignition) data

1661 Ignition delay data are simulated using the Cantera ideal gas reactor at constant volume.

1662 Ignition is defined as the time until the maximum change in pressure with respect to time

dP 1663 ( dt ) is attained. Error scores assigned to each experimental data point according to:

E.S. = logx(IDcalc) − logx(IDexp(1 ± uncertainty)) (5.1)

109 1664 The experimental uncertainty is added when the calculated ignition delay time is larger

1665 than the experimental value and subtracted when smaller. The base of the logarithm can

1666 be specified by the user.

1667 If the error score is larger than a user-defined cutoff, the code will then calculate reaction

1668 rate sensitivity values. For ignition delay results, OH radical reaction rate sensitivities are

1669 used in place of true ID sensitivities since efficient methods for ID sensitivity are not available

1670 in Cantera and sensitivities would need to be done by brute force analysis. In Fig. 5-1 a

1671 comparison of the Cantera OH sensitivity values and those obtained by a brute force analysis

1672 for the FFCM1 kinetic model, table 1.1, is given to show the general agreement between

1673 the two sensitivity measures. Unfortunately, even when using the Cantera OH sensitivity

1674 method, the simulation typically takes at least order of magnitude more time. Fortunately,

1675 the base simulations typically only take on the order of seconds to reach ignition. Once the

1676 sensitivity values are available, the largest sensitivities are plotted and saved. Additionally,

1677 a second plot is made only showing the top sensitive reactions that are not present in some

1678 user-specified reference mechanism. This aids in the identification of important missing

1679 chemistry within a given mechanism.

1680 5.3.2 Laminar flame speed data

1681 Laminar flame speed data are evaluated using the FreeFlame Cantera object. The mixture

1682 averaged transport formulation is used for calculating diffusion coefficients. The more accu-

1683 rate multicomponent formulation originally introduced by Dixon-Lewis [121] is not used for

1684 the sake of computational efficiency. The multicomponent formulation scales approximately

1685 with number of species considered in the kinetic model to the third power, making it an

1686 issue for larger, more detailed models. Differences in LFS from using the multicomponent

1687 formulation are typically much less than the uncertainty in a given experimental LFS value.

1688 The same error score routine is used for LFS data, Eq. 5.1. When a given data point’s

1689 error score is larger than the limit for sensitivity analysis, the efficient Cantera adjoint sen-

1690 sitivity method is used. Comparing with LFS brute force sensitivity values shows excellent

1691 agreement, Fig. 5-2. After the reaction sensitivities have been calculated, the same sensitiv-

1692 ity plotting routine is used as was done for ignition delay data.

110 Figure 5-1: Cantera OH species sensitivity directly before ignition compared with ignition delay brute force sensitivity values for the FFCM1 kinetic model. Both sensitivity sets are normalized such that the maximum value is ±1. Initial conditions are 1029K and 39atm for the methane comparison simulation.

1693 5.3.3 Extinction strain rate data

1694 Though most experimental data is reported using finite boundary characterization methods,

1695 ESR calculations are made here for infinite separation using Ember for computational effi-

1696 ciency as discussed in chapter 3. However, the finite boundary results are then approximated

1697 using the analytical solution to the inviscid momentum equation.

1698 Following Kee et al. [34], we assume steady, inviscid, potential flow in cylindrical coor-

1699 dinates, simplifying the Cauchy momentum equation in the radial direction to:

∂v ∂v ∂p ρv r + ρv z = − (5.2) z ∂z r ∂r ∂r

1700 where vz and vr are the fluid velocities in the axial and radial directions, ρ is the density,

1701 and ∂p/∂r is the pressure gradient.

1702 Using the previously stated flow assumptions, the following are defined:

v ∂U −ρv ∂p −ρa2 U ≡ ρ z ; ≡ r ; ≡ (5.3) 2 ∂z r ∂r 4 111 H + O2 <=> O + OH (R41)

H + O2 + M <=> HO2 + M (R43)

CO + OH <=> CO2 + H (R36)

CH3 + H + M <=> CH4 + M (R1)

HCO + M <=> CO + H + M (R31)

HCO + O2 <=> CO + HO2 (R34)

H2 + OH <=> H + H2O (R40)

CH4 + H <=> CH3 + H2 (R3)

CH3 + OH <=> CH2 + H2O (R8)

CH3 + O <=> CH2O + H (R6)

H + HCO <=> CO + H2 (R32)

H + HO2 <=> 2 OH (R45)

CH4 + OH <=> CH3 + H2O (R5)

H + OH + M <=> H2O + M (R52)

H + HO2 <=> H2 + O2 (R54)

CH2O + H <=> H2 + HCO (R27)

CH2O + OH <=> H2O + HCO (R26)

CH2 + O2 <=> CO2 + 2 H (R20)

H2 + O <=> H + OH (R42)

HO2 + OH <=> H2O + O2 (R44) CTRA Adjoint CTRA 1.5x B.F.

0.2 0.0 0.2 0.4 Normalized Sensitivity ( ln[LFS] ) ln[Ai]

Figure 5-2: Comparison of adjoint laminar flame speed method implemented in Cantera with a brute force perturbation routine. The simple methane kinetic model of 17 species and 58 reactions used previously in Fig. 4-6 is used here. Initial conditions are 298K and an equivalence ratio of 0.7.

112 1703 where a is the constant strain parameter for the flow.

1704 This allows the radial momentum equation to be rewritten as:

∂2U ∂U !2 ρa2 2U − + = 0 (5.4) ∂z2 ∂z 4

1705 Greater detail on the definitions used to make this simplification is available in Kee et al.

1706 [34].

1707 Integrating the momentum equation gives:

2 2 2 ∂U 4C1 − ρ a = z + C1 (5.5) ∂z 16C2

1708 2 2 2 4C1 − ρ a 2 U = z + C1z + C2 (5.6) 16C2

1709 where C1 and C2 are integration constants.

∂vz 1710 If the axial velocity vz and its gradient ∂z at the jet outlet z = 0 have the values vz,0

∂vz 1711 and ∂z 0, respectively, then the integration constants can be eliminated and the equation for

1712 the axial velocity becomes:

2 ∂vz 2 ∂z 0 − a 2 ∂vz vz = z + z + vz,0 (5.7) 4vz,0 ∂z 0

1713 With this equation for finite boundary axial velocity as a function of position, the finite

1714 boundary ESR value can be approximated by:

1715 • Selecting a matching point before the flame, (zpt, vpt), from the Ember solution domain

1716 • Shifting zpt to zmod which equals the BSD/2 - (initial distance to stagnation plane)

1717 • Using the quadratic formula to solve for v0 using (zmod, vpt) and Eq. 5.7.

1718 • Taking the analytical derivative of Eq. 5.7 and using (zmod, vpt) and v0 to calculate

dvz 1719 | dz | which is the finite boundary ESR approximation using the conventional definition

1720 of ESR

1721 Unfortunately, this approximation method tends to be less accurate for very small BSDs, on

113 1722 the order of 5mm, and also has non-trivial sensitivity to the matching point selected before

dvz 1723 the flame. For this work, the matching point is selected as the point at which | dz | has

1724 decreased to 95% of its initial, cold flow boundary value from the Ember simulation results.

1725 While many extinction studies use the conventional ESR definition of the maximum value

dvz 1726 of | dz | before the flame, a number of works instead choose to characterize the extinction

1727 point based on the jet exit velocity and BSD, ESRalt = 4v0/BSD. Fortunately, the given

1728 approximation procedure for the conventional ESR experimental definition also yields the v0

1729 value needed to compute an approximation for the alternate method.

1730 When reaction sensitivity values are needed for ESR points that do not agree well with

1731 experimental values, the Cantera adjoint LFS sensitivity method is used, since ESR and LFS

1732 reaction sensitivities typically only differ slightly and the LFS values are able to be attained

1733 much more computationally efficiently (see chapter 4). For premixed ESR results the pre-

1734 mixed gas conditions are directly used. For diffusion flame ESR results, the composition for

1735 the LFS simulation is determined as a 50-50 molar average of the two opposed jets.

1736 Error scores and sensitivity plotting are performed in the same manner as for ID and

1737 LFS data as described previously.

1738 5.3.4 Selected validation data

1739 Validation data is selected to primarily cover methane along with carbon monoxide, hydro-

1740 gen, ethane, ethylene, and acetylene when data is available. Data for fuel mixtures of these

1741 species are also considered when possible. Data are generally chosen to focus on pressures

1742 up to 40 atm and equivalence ratios in the range 0.7 to 1.1. Unless clearly available in the

1743 source, an uncertainty of 10% is used for all experimental values.

1744 The ignition delay data (tables 5.1 & 5.2) covers pressures from 2-64 atm and all of

1745 the prior mentioned species except for ethylene and acetylene. Fuel mixtures of methane

1746 with hydrogen and methane with ethane are also included. Since ignition delay data from

1747 rapid compression machines (RCMs) is generally a non-trivial function of the compression

1748 profile, and the profile data is often not readily available with the published works, only

1749 shock tube data is used in this study. Typically plotted on log scales, it is not uncommon

1750 for kinetic models to deviate significantly from experimental ignition delay data. Laminar

114 1751 flame speed data (table 5.3) covers up to 20 atm for methane and up to 5 atm for ethane and

1752 ethylene. LFS data is included for all of the species considered. As one of the most often

1753 characterized parameters used in kinetic model validation, LFS data is generally expected to

1754 be reproduced well by current kinetic models. Extinction strain rate data (table 5.4) covers

1755 up to 20 atm for diffusion flames of methane, ethylene, and acetylene. However, elevated

1756 pressure data for premixed flame extinction is scarce at this time. Premixed flames typically

1757 have significantly higher ESRs than diffusion flames and so at higher pressures have more

1758 issues with turbulence. There is a general need for higher pressure LFS and ESR data as

1759 well as for additional data for fuel mixtures.

1760 The nitrogen and rich chemistry sub-mechanisms that are appended to the core model

1761 produced in this study are only preliminarily validated using one set of experimental data

1762 each. The nitrogen chemistry is validated using the NO concentrations measured as a func-

1763 tion of pressure and equivalence ratio downstream of a burner stabilized flame [136]. The

1764 rich chemistry is validated using the same flow reactor species data from [137] as was used

1765 when producing the original mechanism [65].

1766 5.4 Results and discussion

1767 As an initial baseline, the validation routine is run with the selected data for a variety of

1768 current kinetic models. As a subset of the current kinetic model performance for this data

1769 set, validation figures are shown for GRIMech3.0 (Fig. 5-3), FFCM1 (Fig. 5-4), and Hashemi

1770 C2H6 (Fig. 5-5). The oldest mechanism, GRIMech3.0 performs the worst of the three with a

1771 total error score of 82 across all data points. FFCM1 comes in second with a total error score

1772 of 71 while the Hashemi model performs the best at 61. These results are as expected since

1773 the Hashemi model was specifically constructed with an emphasis on the elevated pressures

1774 of interest in this work. In general, performance for the selected ignition delay data is poor,

1775 with summed error scores on the order of 50 for all three mechanisms. GRIMech3.0 and

1776 FFCM1 perform similarly at 57 and 58 respectively while the Hashemi model comes in at

1777 46. As expected, the LFS data is predicted best across all three models. GRIMech3.0 is

1778 observed to struggle with ethylene and acetylene data along with lean methane data points.

115 1779 FFCM1 also shows its largest errors for the lean methane data points, but reproduces the

1780 remaining LFS data well. Hashemi model also performs well for the majority of the LFS data

1781 with a modest but consistent over prediction of the ethane data. FFCM1 has the lowest error

1782 score at 0.8 followed by Hashemi at 1.3 and GRIMech3.0 at 3.3. For all three kinetic models,

1783 methane ESR data is generally well predicted while ethane, ethylene, and syngas mixtures

1784 are over predicted and hydrogen flames are under predicted. Again, FFCM1 performs best

1785 a score of 12, followed by Hashemi at 14, and GRI at 21. Though FFCM1 performs slightly

1786 better for two of the three parameters evaluated, the Hashemi model is selected for further

1787 refinement and model development. Hashemi is chosen for its overall superior error score

1788 as well as for it original construction with an emphasis on inclusion of accurate pressure

1789 dependent kinetics.

1790 5.4.1 Core model

1791 As a first step in refinement of the base Hashemi model, the consistent error for the ethane

1792 laminar flame speed data is investigated. sensitivity results for one of the ethane data points

1793 are shown in Fig. 5-6. The typically expected primary branching reaction H + O2 O + OH

Figure 5-3: GRIMech3.0 kinetic model performance for the the selected validation data

116 Figure 5-4: FFCM1 kinetic model performance for the the selected validation data

Figure 5-5: Hashemi C2H6 kinetic model performance for the the selected validation data

117 Figure 5-6: Hashemi C2H6 kinetic model sensitivity for ethane laminar flame speed data. Ethane/air at an equivalence ratio of 0.9

1794 appears with the top sensitivity. However, since this reaction is highly sensitive for nearly

1795 all data points, it is not a good candidate for correcting the systematic over prediction error

1796 observed for ethane laminar flame speed. On the other hand, the following three reactions,

1797 CO + OH CO2 + H, 2CH3 C2H5 + H, and C2H4 + H C2H5 do end up being much

1798 better candidates for modification. Comparing the rate constants from Hashemi to those of

1799 FFCM1 which accurately predicted the ethane LFS data, the Hashemi rate is observed to

1800 be consistently higher for CO + OH CO2 + H and C2H4 + H C2H5 while consistently

1801 lower for 2CH3 C2H5 + H. Therefore, based on the sign of the sensitivity for these three

1802 reactions, switching to the rate constant parameters used in FFCM1 is expected to improve

1803 the predictions. The subsequent three sensitive reactions show the opposite trend such that

1804 switching to the FFCM1 rate would be expected to worsen predictions.

1805 After modification of these three reaction rates, the modified Hashemi model now gives

1806 top performance across all three categories of validation data. The ID data is slightly worse

1807 at 47 from an initial 46. LFS and ESR data improve to the best overall values at 0.07 and

1808 10 respectively. In fact, the LFS data is nearly perfectly represented within the 10% error

1809 bounds.

118 Figure 5-7: Hashemi C2H6 kinetic model performance after the modification to in- clude the sensitive reaction rates for ethane laminar flame speeds from FFCM1: CO + OH CO2 + H, 2CH3 C2H5 + H, and C2H4 + H C2H5

1810 Looking next toward improvement of ID predictions, the works of Huang et al. are

1811 utilized heavily [138–140]. These works are sources of the majority of the ID data used

1812 in validation and include sensitivity and kinetic model analysis produced when improv-

1813 ing predictions of the data from a starting kinetic model based on GRIMech1.2. From

1814 the Huang et al. work focusing on pure methane fuel ignition delays [138], the reaction

1815 rate HO2 + CH4 CH3 + H2O2 is substituted into the modified Hashemi model for im-

1816 proved ID predictions as suggested. The other three important reactions modified in [138],

1817 CH3 + O2 CH3OO, CH3OO + HO2 CH3OOH + O2, and 2CH3OO O2 + 2CH3O had

1818 little impact or worsened predictions when substituted into the modified model.

1819 Sensitivity results and kinetic analyses in the Huang et al. work that considered methane

1820 fuel blends with ethane and propane [140] led to the modification of three additional reac-

1821 tions and the addition of two reactions not previously included within the Hashemi model.

1822 The three reactions modified to use the Huang et al. rates are CH3OO + CH3 2CH3O,

1823 CH3OO + CH4 CH3OOH + CH3, and CH3OO + C2H6 CH3OOH + C2H5. The first of

1824 the three generates a significant improvement in model ID predictions, cutting the ID error

119 1825 score roughly in half. The second generates a further minor improvement for data points

1826 with methane based fuels, and the third gives a modest improvement for the ethane fuels.

1827 To further improve agreement with ethane ignition delay data, the third rate is increased

1828 by a factor of 5. More detailed ab initio calculation of this rate is warranted as is addi-

1829 tional investigation of similar but missing reactions. Found to not be present in the Hashemi

1830 model initially, C2H5 + O2 CH3CH2O + O and C2H4 + CH3OO cy − C2H4O + CH3O

1831 are added and observed to contribute small improvements to the ethane ID data while show-

1832 ing not noticeable impact on the other validation data points.

1833 Hydrogen and syngas sensitivity analysis showed dominant sensitivity to the primary

1834 branching reactions H + O2 O + OH and H + O2 HO2 Since the radical branching re-

1835 action of the two is generally accepted to be well known, the second reaction was perturbed

1836 by a factor of 0.85 to improve predictions. This modification significantly improved hydrogen

1837 ID data and also improved hydrogen ESR data, but at the slight cost of increasing the LFS

1838 error score to 0.33 from 0.07, still very minor compared to the overall errors in ID and ESR.

1839 Given that ESR results show largely the same reaction sensitivities as LFS (shown in

1840 chapter 4) perturbations for the improvement of ESR data focused on transport parame-

1841 ters. Given the consistently observed under prediction of ESR for ethane and ethylene, their

1842 Lennard Jones diameters are increased by 5%. For the consistent under prediction of hydro-

1843 gen ESR, the L.J. diameter of H2 is decreased by 5% to promote the strength of the flame.

1844 Ultimately these modifications yield only a minor improvement to the ESR error score.

1845 With an enormous number of parameters within the kinetic model, over 3500, refinement

1846 efforts can continue almost indefinitely. This makes automated optimization routines such

1847 as those employed by the FFCM1 authors or PrIMe important for further overall improve-

1848 ment of kinetic models since even current best ab initio methods often leave significant and

1849 prediction-relevant uncertainties in calculated rate constants. However, for the purposes of

1850 this work, the current level of agreement of the kinetic model is accepted and the final results

1851 are shown in Fig. 5-8. The ID error score has been reduced by 65% to 16, the LFS error score

1852 is reduced by 75% to 0.33, and the ESR error score has been reduced by 35% to 9. There are

1853 no clear outliers for the predictions fo the validation data and ID and ESR while showing

1854 relatively large error scores, are considered to be roughly within the uncertainty expected

120 Figure 5-8: Final, modified Hashemi C2H6 kinetic model performance after various rate updates to improve agreement with the experimental validation data set.

1855 from the experimental method and in the case of ESR, the computational approximation

1856 method as well.

1857 5.4.2 Nitrogen chemistry subset

1858 The nitrogen chemistry subset is taken directly from a recent review by Glarborg and cowork-

1859 ers [28]. It is appended to the core kinetic model produced in the prior section. To briefly

1860 validate its performance, a subset of the experimental data from [136] is reproduced here.

1861 Overall, with compared with results from GRIMech3.0, the new kinetic model shows im-

1862 proved NO prediction performance for the lean to stoichiometric conditions (Fig. 5-9) that

1863 were the primary focus during generation, but shows a steep under prediction of NO forma-

1864 tion under rich conditions. This discrepancy merits further investigation and might reason-

1865 ably be resolved with comparison of sensitive reactions with GRIMech3.0 which is observed

1866 to give superior performance under rich conditions but under predict for the lean to stoichio-

1867 metric conditions (Fig. 5-9). The inclusion of Nitrogen chemistry within the kinetic model is

1868 observed to shift ESR predictions by less than a percent for a test with GRIMech3.0. These

1869 results included in the supplementary materials.

121 250 3.05 atm (exp) 6.1 atm (exp) 300 3.05 atm (exp) 14.6 atm (exp) )

3 200 6.1 atm (exp) 3.05 atm (calc) 14.6 atm (exp)

) 250 m 3

c 6.1 atm (calc)

( 3.05 atm (calc)

3 m

1 14.6 atm (calc) c 6.1 atm (calc) (

3 0

150 1 200 14.6 atm (calc) 1

0 x

1

y t x i

s y

t 150 n i s e n

d 100

e r d

e r b e 100 b m u m u n

n

50 O O N 50 N

0 0

0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Equivalence ratio, Equivalence ratio,

Figure 5-9: Nitrogen chemistry predictions when (LEFT) using GRIMech3.0 and (RIGHT) appending the nitrogen chemistry from the Glarborg review [28] with the core model pro- duced here. Experimental data from [136]

1870 5.4.3 Rich chemistry subset

1871 The rich chemistry subset has been produced for another work [65] using the automatic Re-

1872 action Mechanism Generator (RMG). That work leveraged FFCM1 for the core chemistry

1873 model. However, the core chemistry is now replaced with the core model produced here. This

1874 is achieved using the ANSYS Chemkin Reaction workbench software to efficiently merge the

1875 models and replace the overlapping chemistry. To confirm that the impact of modifying the

1876 chemistry is not radically detrimental to the accuracy of the rich chemistry, a simulation from

1877 [65] is reproduced here with the new model. Comparing the original model results, Fig. 5-10,

1878 with the current model results, Fig. 5-11, primary products are generally consistent between

1879 the two models and show good agreement with the experimental data. Notably, the new

1880 model produced in this work is observed to show a slightly earlier onset of reactivity ( 20K).

1881 This difference is made more pronounced by the model modifications made to decrease igni-

1882 tion delay time predictions to better agree with experimental data. However, the bulk of the

1883 difference comes from switching to the Hashemi et al. base chemistry. When comparing the

1884 C2 predictions between the two models, the original model over-predicts acetylene (C2H2)

1885 formation while the current model slightly under predicts it. It then follows that the cur-

1886 rent model further under predicts benzene and naphthalene formation. Another important

1887 difference is that the current model over predicts CH2O formation by a factor of two; this

1888 is again likely attributed to the modifications of the peroxy ignition chemistry, suggesting

122 1889 that kinetic ignition routes merit further investigation. It is also pertinent to note that an

1890 important route for the production of naphthalene from cyclopentadieneyl recombination

1891 has been calculated at a high level considering pressure dependent effects in the following

1892 chapter. This network is included in [65] and in the rich chemistry submechanism here.

1893 5.5 Conclusions

1894 A validation tool for kinetic model benchmarking and subsequent improvement has been

1895 produced. The tool allows users to benchmark against ignition delay (autoignition), laminar

1896 flame speed, and extinction strain rate data. Using a data set focusing on CH4,H2, CO,

1897 C2H6,C2H4, and C2H2 at pressures of relevance to gas turbines and internal combustion

1898 engines when available, a kinetic model is produced that represents the selected validation

1899 data significantly better than currently available models. This is achieved by starting with

1900 the high pressure oxidation model from Hashemi and coworkers and subsequently replacing

1901 and modifying sensitive reactions according to uncertainty values and reaction rate sensitiv-

1902 ities. This ultimately improves the overall error score for the Hashemi model by 57% from

1903 61 to 26. Nitrogen chemistry and rich chemistry sub-mechanisms from other prior works are

1904 subsequently added to the core model to allow for use over a greater range of applications.

1905 Further efforts should consider expanding the validation data set through new experimen-

1906 tal data, improving the computational efficiency of the ignition delay sensitivity data, and

1907 leveraging overall optimization procedures similar to those used by the FFCM1 authors. For

1908 such optimization procedures, accurate uncertainty quantification for any sensitive kinetic

1909 parameters is vital but unfortunately not currently available.

123 Figure 5-10: Plug flow reactor data for the rich oxidation of methane at varying temperatures using the original kinetic model used in [65] which leveraged FFCM1 for the core chemistry. Experimental data from [137].

124 Figure 5-11: Plug flow reactor data for the rich oxidation of methane at varying temper- atures using the core model from this work with the rich chemistry from [65] appended. Experimental data from [137].

125 Table 5.1: Selected ignition delay validation data (PART A) T [K] P [bar] Molar composition Exp. [µs] Ref. 1029 38.7 CH4:0.068, O2:0.196, N2:0.736 1548 [138] 1114 39.1 CH4:0.068, O2:0.196, N2:0.736 1050 [138] 1329 39.1 CH4:0.068, O2:0.196, N2:0.736 534 [138] 1143 16.0 CH4:0.068, O2:0.196, N2:0.736 2022 [138] 1208 16.8 CH4:0.068, O2:0.196, N2:0.736 1158 [138] 1304 15.7 CH4:0.068, O2:0.196, N2:0.736 690 [138] 1024 37.5 CH4:0.095, O2:0.192, N2:0.71 2712 [138] 1116 39.5 CH4:0.095, O2:0.192, N2:0.71 978 [138] 1213 40.3 CH4:0.095, O2:0.192, N2:0.71 612 [138] 1032 26.1 CH4:0.095, O2:0.192, N2:0.71 2343 [138] 1187 25.2 CH4:0.095, O2:0.192, N2:0.71 1149 [138] 1309 25.2 CH4:0.095, O2:0.192, N2:0.71 489 [138] 1084 16.2 CH4:0.095, O2:0.192, N2:0.71 1998 [138] 1226 17.0 CH4:0.095, O2:0.192, N2:0.71 1056 [138] 1343 17.9 CH4:0.095, O2:0.192, N2:0.71 444 [138] 1297 36.4 H2:1.6, CH4:9.0, O2:18.8, N2:70.6 332 [139] 1028 37.5 H2:1.6, CH4:9.0, O2:18.8, N2:70.6 1614 [139] 1307 17.0 H2:1.6, CH4:9.0, O2:18.8, N2:70.6 462 [139] 1096 18.8 H2:1.6, CH4:9.0, O2:18.8, N2:70.6 1676 [139] 1318 37.4 H2:4.2, CH4:8.2, O2:18.4, N2:69.2 158 [139] 1035 39.2 H2:4.2, CH4:8.2, O2:18.4, N2:69.2 2166 [139] 1318 17.0 H2:4.2, CH4:8.2, O2:18.4, N2:69.2 260 [139] 1019 16.0 H2:4.2, CH4:8.2, O2:18.4, N2:69.2 3250 [139] 1249 38.2 C2H6:0.34, CH4:8.93, O2:19.05, N2:71.68 343 [140] 960 336.6 C2H6:0.34, CH4:8.93, O2:19.05, N2:71.68 3081 [140] 1271 15.8 C2H6:0.34, CH4:8.93, O2:19.05, N2:71.68 528 [140] 983 14.8 C2H6:0.34, CH4:8.93, O2:19.05, N2:71.68 4800 [140] 983 40.0 C2H6:0.63, CH4:8.44, O2:19.1, N2:71.83 720 [140] 1100 40.0 C2H6:0.63, CH4:8.44, O2:19.1, N2:71.83 1730 [140] 1102 40.2 C2H6:0.89, CH4:8.01, O2:19.13, N2:71.97 1740 [140] 911 39.1 C2H6:0.89, CH4:8.01, O2:19.13, N2:71.97 4900 [140] 1239 17.1 C2H6:0.89, CH4:8.01, O2:19.13, N2:71.97 365 [140] 1057 16.8 C2H6:0.89, CH4:8.01, O2:19.13, N2:71.97 1564 [140]

126 Table 5.2: Selected ignition delay validation data (PART B) T [K] P [bar] Molar composition Exp. [µs] Ref. 1183 2 H2:2.0, O2:1.0, N2:7.52 22 [141] 1086 2 H2:2.0, O2:1.0, N2:7.52 48 [141] 1016 2 H2:2.0, O2:1.0, N2:7.52 280 [141] 980 2 H2:2.0, O2:1.0, N2:7.52 636 [141] 1299 33 H2:2.0, O2:1.0, AR:97.0 13 [141] 1185 33 H2:2.0, O2:1.0, AR:97.0 374 [141] 1557 64 H2:0.1, O2:0.05, AR:99.85 48 [141] 1360 64 H2:0.1, O2:0.05, AR:99.85 170 [141] 1718 3 H2:1.0, O2:1.0, AR:98.0 52 [141] 1254 3 H2:1.0, O2:1.0, AR:98.0 286 [141] 1385 12 CO:0.9, H2:0.1, O2:1.0, AR:98.0 82 [142] 1178 12 CO:0.9, H2:0.1, O2:1.0, AR:98.0 381 [142] 1329 30 CO:0.9, H2:0.1, O2:1.0, AR:98.0 77 [142] 1192 30 CO:0.9, H2:0.1, O2:1.0, AR:98.0 985 [142]

127 Table 5.3: Selected laminar flame speed validation data T [K] P [bar] Molar composition Exp. [cm/s] Ref. 298 1.0 CH4:0.7, O2:2, N2:7.52 15 [118] 298 1.0 CH4:0.9, O2:2, N2:7.52 33 [118] 298 1.0 CH4:1.1, O2:2, N2:7.52 38 [118] 298 10.0 CH4:0.7, O2:2, N2:7.52 4.0 [118] 298 10.0 CH4:0.9, O2:2, N2:7.52 11 [118] 298 10.0 CH4:1.1, O2:2, N2:7.52 14 [118] 298 20.0 CH4:0.8, O2:2, N2:7.52 4.7 [118] 298 20.0 CH4:0.9, O2:2, N2:7.52 8.0 [118] 298 20.0 CH4:1.1, O2:2, N2:7.52 9.1 [118] 298 1.0 C2H6:0.7, O2:3.5, N2:13.16 20 [143] 298 1.0 C2H6:0.9, O2:3.5, N2:13.16 34 [143] 298 1.0 C2H6:1.1, O2:3.5, N2:13.16 38 [143] 298 5.0 C2H6:0.7, O2:3.5, N2:13.16 13 [143] 298 5.0 C2H6:1.0, O2:3.5, N2:13.16 26 [143] 298 5.0 C2H6:1.1, O2:3.5, N2:13.16 28 [143] 298 1.0 C2H4:0.6, O2:3, N2:11.28 23 [143] 298 1.0 C2H4:1.0, O2:3, N2:11.28 63 [143] 298 1.0 C2H4:1.3, O2:3, N2:11.28 60 [143] 298 5.0 C2H4:0.6, O2:3, N2:11.28 12 [143] 298 5.0 C2H4:1.0, O2:3, N2:11.28 44 [143] 298 5.0 C2H4:1.2, O2:3, N2:11.28 49 [143] 298 1.0 C2H2:0.7, O2:2.5, N2:9.4 78 [143] 298 1.0 C2H2:0.9, O2:2.5, N2:9.4 109 [143] 298 1.0 C2H2:1.1, O2:2.5, N2:9.4 128 [143] 298 1.0 H2:0.7, O2:0.5, N2:1.88 124 [142] 298 1.0 H2:1.0, O2:0.5, N2:1.88 218 [142] 298 1.0 H2:1.7, O2:0.5, N2:1.88 284 [142] 298 1.0 H2:5.0, O2:0.5, N2:1.88 110 [142] 298 1.0 CO:0.5, H2:0.5, O2:1.0, N2:3.76 31 [142] 298 1.0 CO:0.8, H2:0.8, O2:1.0, N2:3.76 83 [142] 298 1.0 CO:1.2, H2:1.2, O2:1.0, N2:3.76 137 [142]

128 Table 5.4: Selected extinction strain rate validation data. If no second composition is given, it is assumed that the data point is for a twin flame arrangement and the same compositions is used for each jet. T [K] P [bar] Molar comp. A Molar comp. B Exp. [1/s] Ref. 298 1 CH4:0.7, O2:2, N2:7.52 - 731 [43] 298 1 CH4:0.82, O2:2, N2:7.52 - 1336 [43] 298 1 CH4:0.94, O2:2, N2:7.52 - 1784 [43] 298 1 CH4:1.12, O2:2, N2:7.52 - 1060 [43] 298 1.7 CH4:25, N2:75 O2:2, N2:7.52 187 [58] 298 8.8 CH4:25, N2:75 O2:2, N2:7.52 170 [58] 298 19.8 CH4:25, N2:75 O2:2, N2:7.52 106 [58] 298 1.0 C2H6:11.3, N2:88.7 O2:2, N2:7.52 112 [58] 298 4.1 C2H6:11.3, N2:88.7 O2:2, N2:7.52 201 [58] 298 19.0 C2H6:11.3, N2:88.7 O2:2, N2:7.52 120 [58] 298 2.9 C2H4:0.09,N2:0.91 O2:2, N2:7.52 147 [58] 298 9.6 C2H4:0.09,N2:0.91 O2:2, N2:7.52 166 [58] 298 19.9 C2H4:0.09,N2:0.91 O2:2, N2:7.52 104 [58] 298 1.0 C2H6:0.92, O2:3.5, N2:15.42 N2:1 195 [144] 298 1.0 C2H6:1.04, O2:3.5, N2:15.42 N2:1 299 [144] 298 1.0 C2H6:1.14, O2:3.5, N2:15.42 N2:1 340 [144] 298 1.0 C2H4:0.74, O2:3.0, N2:13.21 N2:1 250 [144] 298 1.0 C2H4:0.90, O2:3.0, N2:13.21 N2:1 653 [144] 298 1.0 C2H4:1.34, O2:3.0, N2:13.21 N2:1 625 [144] 298 1.0 C2H4:0.90, O2:3.0, N2:14.54 N2:1 414 [144] 298 1.0 C2H4:1.00, O2:3.0, N2:14.54 N2:1 595 [144] 298 1.0 C2H4:1.12, O2:3.0, N2:14.54 N2:1 696 [144] 298 1.0 H2:0.29, O2:0.5, N2:1.88 N2:1 200 [51] 298 1.0 H2:0.35, O2:0.5, N2:1.88 N2:1 750 [51] 298 1.0 H2:0.18, N2:0.82 O2:2, N2:7.52 775 [51] 298 4.0 H2:0.15, N2:0.85 O2:2, N2:7.52 510 [51] 298 7.0 H2:0.145, N2:0.855 O2:2, N2:7.52 260 [51] 298 1 CO:0.123, H2:0.017, O2:0.18, N2:0.68 - 193 [49] 298 1 CO:0.096, H2:0.044, O2:0.18, N2:0.68 - 765 [49] 298 1 CO:0.0717, H2:0.048, O2:0.185, N2:0.695 - 573 [49] 298 1 CO:0.134, CH4:0.012, O2:0.179, N2:0.675 - 244 [49] 298 1 CO:0.113, CH4:0.033, O2:0.179, N2:0.675 - 750 [49] 298 1 CO:0.060, CH4:0.040, O2:0.189, N2:0.711 - 564 [49]

129 THIS PAGE INTENTIONALLY LEFT BLANK

130 1910 Chapter 6

1911 Pressure dependent kinetic analysis of

1912 pathways to naphthalene from

1913 cyclopentadienyl recombination

1914 6.1 Summary

1915 Cyclopentadiene (CPD) and cyclopentadienyl radical (CPDyl) reactions are known to pro-

1916 vide fast routes to naphthalene and other polycyclic aromatic hydrocarbon (PAH) precursors

1917 in many systems. In this work, we combine literature quantum chemical pathways for the

1918 CPDyl + CPDyl recombination reaction and provide pressure dependent rate coefficient

1919 calculations and analysis. We find that the simplified 1-step global reaction leading to naph-

1920 thalene and two H atoms used in many kinetic models is not an adequate description of this

1921 chemistry at conditions of relevance to pyrolysis and steam cracking. The C10H10 species is

1922 observed to live long enough to undergo H abstraction reactions to enter the C10H9 potential

1923 energy surface (PES). Rate coefficient expressions as functions of T and P are reported in

1924 Chemkin format for future use in kinetic modeling.

1925 This chapter is principally a reproduction with permission of the results included in

1926 [145]. Collaborators for these results are: Shamel Merchant, Aaron Vandeputte, Hans-

1927 Heinrich Carstensen, Alexander Vervust, Guy Marin, Kevin Van Geem, and William Green.

131 1928 Notably, Shamel Merchant conducted the initial investigation of the pressure dependence of

1929 the kinetic pathways included in this work and Aaron Vandeputte produced the majority of

1930 the quantum mechanical species calculations using Gaussian.

1931 6.2 Background

1932 Polycyclic aromatic hydrocarbons (PAHs) are a class of molecules comprised of hundreds

1933 of chemicals produced from various anthropogenic sources, such as the incomplete combus-

1934 tion of fuels in boilers and heating devices, oil refining processes, and the combustion of

1935 transportation fuels (in particular diesel). Many PAHs are known to be carcinogenic or

1936 mutagenic as well as important precursors to soot [146–150]. Increasing energy demands

1937 and more stringent emission regulations have spurred research into various new technologies

1938 that reduce PAH emissions [151–153]. Despite the attention that these PAHs have attracted,

1939 many questions related to their formation remain not fully understood, i.e. what are the

1940 key reactions responsible for the formation of PAHs and what are the main PAH precursors?

1941 Researchers have developed extensive kinetic mechanisms describing low and high tempera-

1942 ture fuel oxidation, but few exist that are able to predict the PAH growth and subsequent

1943 soot formation [154–156]. Some key efforts toward the latter have come from Frenklach,

1944 Appel et al., and Richter and Howard [157–160]. However, these studies focused primarily

1945 on the chemistry at high temperature flame conditions, at which the HACA mechanisms

1946 first proposed by Frenklach in 1991 as well various similar pathways summarized in the

1947 review article by Richter and Howard [157, 159] dominate. The radicals involved in these

1948 reactions are highly reactive or energetic and thus not present in sufficient concentrations

1949 at lower temperatures. Given their increased stability, resonantly stabilized radicals such as

1950 allyl, propargyl and cyclopentadienyl (CPDyl) have been suggested as possible precursors of

1951 aromatic products at low or moderate temperatures [161–164]. However, most of these reac-

1952 tions lead to mono-aromatic hydrocarbons. Several experimental pyrolysis studies at lower

1953 temperatures detected CPD as an important product and Melton et al. showed that PAH

1954 formation was most sensitive to the CPD concentration [165–168]. Since CPD is easily con-

1955 verted to CPDyl this suggests that CPDyl radicals play a crucial role in PAH formation. In

132 1956 this work, the focus is on the development of a detailed kinetic network for cyclopentadienyl

1957 recombination that is relevant at pyrolytic and low-temperature combustion conditions to

1958 describe the formation of naphthalene, the simplest PAH and an important PAH precursor.

1959 As briefly indicated above, a number of researchers have explored the thermal decom-

1960 position of CPD experimentally. In an attempt to obtain bond dissociation energies, the

1961 decomposition of CPD under pyrolytic conditions was studied by Szwarc in 1950 [169]. This

1962 led to a complicated decomposition spectrum, containing H2, CH4, C2 hydrocarbons and

1963 other species hinting at the complexity of the CPD reaction network. In 1972 Spielman

1964 and Cramers first proposed a potential role of CPD in the formation of the initial aromat-

1965 ics, observing products such as styrene, indene, and naphthalene under pyrolytic conditions

1966 [170]. This was then validated by studies of the pyrolysis and hydrogenolysis of phenol

1967 [171–173]. Burcat et al. studied the high-temperature decomposition reactions of CPD in

1968 a shock-tube in 1996, and modeled it with a simple model of 36 reactions. Roy et al. also

1969 studied CPD reactions using shock tube experiments and reported rate coefficients for the

1970 C-H bond scission and the reaction between CPD and a hydrogen atom [174, 175]. While the

1971 Spielman study and others showed qualitatively that naphthalene appears to be produced

1972 from CPD, in 2003, Murakami et al. first experimentally derived a rate expression for a

1973 net reaction from CPDyl to naphthalene, assuming CPD initially forms its relatively stable

1974 radical [176]. More recently, experimental efforts to understand CPD pyrolysis chemistry

1975 have shifted toward flow reactor studies [177, 178]. Djokic et al. and Kim et al. identified

1976 and quantified numerous pyrolytic products from CPD up to anthracene, phenanthrene and

1977 fluorene supporting the important role of CPD and CPDyl in PAH formation. The crucial

1978 role of CPDyl radicals in PAH formation is additionally confirmed by pyrolysis studies of

1979 anisole (which forms CPDyl by methyl loss and carbon monoxide ejection) and dimethylfu-

1980 ran [165, 179, 180]. More recently, in 2015, Knyazev and Popov experimentally investigated

1981 the total self-reaction rate of CPDyl, finding it to show significantly faster kinetics when

1982 compared with other similar radical self-reactions. Naphthalene and Azulene were detected

1983 as final products with the former being the major product [181].

1984 The first quantum chemical exploration of the potential energy surface (PES) for CPDyl

1985 recombination was reported by Melius et al. in 1996. This study used the BAC-MP4 and

133 1986 MP2 levels of theory to characterize pathways in which two CPDyl radicals recombine,

1987 subsequently lose an H atom to form a fulvalanyl radical (Fig. 6-1), which then isomerizes

1988 and ejects an additional H atom to yield naphthalene. The importance of hydrogen atom

1989 mobility for PAH formation was emphasized by this work as was the role of resonantly

1990 stabilized intermediates [182]. A 2006 paper by Wang et al. extended the surface using

1991 the B3LYP level to incorporate pathways to benzene and indene as well as naphthalene.

1992 However, that work focused on initial reactions between CPD and CPDyl rather than two

1993 CPDyl radicals. It was observed that C-C Κ scission routes tended to be favored at high

1994 temperatures [183]. A 2007 paper by Kislov and Mebel used the RCCSD(T)/6-311G(d,p)

1995 level to investigate the C10H9 potential energy surface. Routes to naphthalene, azulene,

1996 and fulvalene were compared. Assuming unimolecular reactions in the high pressure limit,

1997 fulvalene was computed to be the dominant product above 1500 K, while naphthalene was the

1998 primary species at lower temperatures, and azulene was always a minor product. Kislov and

1999 Mebel extended their work in 2008 to incorporate the relevant portion of the C10H11 surface

2000 consisting principally of routes to indene calculated at the G3 level [184]. Subsequent work by

2001 Kislov and Mebel elucidated the poor likelihood of reaction pathways that would produce

2002 molecular hydrogen along the way to naphthalene from two CPDyl radicals as had been

2003 proposed as a global step for the pathway and incorporated into some kinetic mechansims

2004 [185, 186]. A 2012 paper by Cavallotti et al. revisited routes from the reaction of CPD

2005 with CPDyl using the B3LYP level for most calculations and ROCBS-QB3 for important

2006 flux determining steps along the pathways analyzed. Additional routes were located for the

2007 formation of indene, benzene, vinylfulvene, and phenyl butadiene [187]. In 2013 Cavallotti

2008 et al. analyzed the C10H10 surface at the CBS-QB3 level, focusing on the formation of the

2009 fulvalanyl and azulanyl radicals. Routes to the azulanyl radical were reported to dominate up

2010 to 1450K [188]. This result in turn added relevance of the spiran and methyl walk pathways

2011 from azulene to naphthalene shown by Alder et al. in 2003 [189]. Some additional discussion

2012 of many of these quantum chemical studies can be found in a recent paper by Mebel et al.

2013 [190].

2014 Rate coefficients are needed to quantitatively model PAH formation kinetics. A number

2015 of the previously noted studies presented high pressure limit rate coefficients and global

134 2016 step rate constants. Wang et al. reported only high-pressure limit Arrhenius expressions

2017 for the primary reaction pathway branching reactions they studied [183]. The 2007 and

2018 2008 Kislov and Mebel papers reported rate coefficients and equilibrium constants for all

2019 important reactions they located. These were reported at various relevant temperatures in

2020 table format [184, 191]. Global overall Arrhenius expressions were reported by Cavallotti et

2021 al. in 2012 and 2013 for the formation of the terminal species of interest in those studies [187,

2022 188]. However, none of these studies incorporated pressure dependence in their reported rate

2023 expressions. While the C10 species of interest are relatively large molecules, they are not

2024 large enough to be sure their unimolecular reactions are at the high pressure limit at pyrolysis

2025 and combustion conditions. This follows from a 2003 paper by Wong et al. which developed

2026 convenient criteria for determining the relevance of pressure dependent kinetics for a system

2027 [192]. Following this criteria, a recent model of propene pyrolysis includes a set of pressure

2028 dependent rate expressions for the progression of CPDyl to naphthalene first described by

2029 Melius et al. Rather than calculating the individual rates for the network directly, they are

2030 assigned based on analogy to similar reaction whose rate constants readily available [164].

2031 A brief comparison with this estimated network is presented in the supporting information

2032 of this work.

2033 The objective of the current study is two-fold. First it aims to combine the most promising

2034 pathways previously studied and use this information to calculate a set of pressure-dependent

2035 rate expressions that will be easily used in future kinetic mechanisms. Specifically, generation

2036 of rate expressions in this format is important for database development and improved

2037 accuracy of automatic mechanism generating software such as RMG [13]. Secondly, the

2038 importance of the direct channel

CP Dyl + CP Dyl → naphthalene + 2H (∆H298 = 48 kJ/mol) (6.1)

2039 will be explored. Note that as written, the recombination of CPDyl radicals generates

2040 naphthalene in a single step, preventing any bimolecular chemistry of intermediates in this

2041 obviously complex reaction sequence. Also, it increases reactivity by converting two low-

2042 reactivity radicals into more reactive H atoms.

135 Fulvalene Cyclopentadiene Azulene

+H-shift isomers +H-shift isomers Naphthalene Fulvalanes Azulanes

Figure 6-1: Primary structures of this study. An ending of -yl added to a species indicates a radical of that species formed by hydrogen loss

2043 6.3 Methodology

2044 6.3.1 Quantum mechanical calculation of PES

2045 Even though all PES used in this study have already been investigated previously by other

2046 groups, the electronic structure calculations are repeated here to obtain a consistent data

2047 set for the kinetic analysis. Thermodynamic properties of all species including transition

2048 states are calculated using quantum mechanical computations at the CBS-QB3 level as

2049 implemented in Gaussian 03 and 09 [193–195]. Bond additivity and spin orbit corrections

2050 are included [196, 197]. Most vibrations are approximated as harmonic oscillators with

2051 frequencies computed at the B3LYP/6-311G(2d,d,p) level, but torsional vibrations of key

2052 components are treated as 1D hindered rotors[198]. The calculated CBS-QB3 enthalpy of

2053 formation of CPDyl at 298K is 259 kJ/mol in good agreement with the experimental value

2054 [175]. The optimized geometries for all species studied here are readily available in previous

2055 PES exploratory studies as referenced.

2056 6.3.2 Rate coefficient calculations

2057 For pressure dependent calculations, potential energy surfaces for C10H10, and C10H9 are

2058 modeled using a one-dimensional master equation, accounting for rotational degrees of free-

2059 dom. Microcanonical rate coefficients are computed using the classical RRKM theory and

136 2060 include Eckart tunneling contributions [199–202]. Densities of states are calculated via in-

2061 verse Laplace transform of the partition function using the method of steepest descents [203–

2062 205]. Phenomenological rate coefficients (which give the dynamics of the total isomer pop-

2063 ulations xi(t)) are computed from the conventional master equation model (which give the

2064 detailed dynamics of the isomer population distributions pi(En,t)) using the modified strong

2065 collision (MSC) approximation [206, 207]. Parameters for the MSC calculation are obtained

2066 using the exponential down model < ∆Edown > for the average energy transferred in a col-

2067 lision, which is subsequently converted to the required ∂Eall parameter [206]. The average

2068 downward energy transfer per collision is calculated according to the following temperature

2069 dependent formulation:

n −1 < ∆Edown >=< ∆Edown >300 (T/300K) cm (6.2)

2070

2071 with < ∆Edown>300 = 295 cm-1, and n = 0.7 for N2. These values are adopted based

2072 on azulene collisional energy transfer parameters [208]. The collision frequency is computed

2073 by assuming a Lennard-Jones potential between the bath gas and the species of interest.

2074 Lennard-Jones parameters are approximated by first estimating the critical temperature

2075 and pressure using a group additivity method devised by Joback, and then using the equa-

2076 tions for a Lennard-Jones gas implemented by Harper et al. in RMG [209–211]. All master

2077 equation calculations are performed using the open source CANTHERM software package

2078 [212]. Additionally, barrierless hydrogen loss reactions are assumed fast and temperature

14 3 2079 independent with high pressure limit rate coefficients of 1 ∗ 10 (cm )/(mols) n the recombi-

2080 nation direction for the purposes of this study. This estimate is in agreement with the 2005

2081 Harding et al. work computing the rates of a variety of hydrogen atom-hydrocarbon radical

2082 recombination rates. These rates were shown to vary less than one order of magnitude from

2083 the value assumed here [213]. For the CPDyl + CPDyl association rate, the high pressure

2084 limit rate expression from Knyazev and Popov is used. This value was determined experi-

2085 mentally for a range of 300-600K [181]. Sensitivities for these assumed rates are included in

2086 the supporting information.

137 2087 6.4 Results and discussion

2088 6.4.1 C10H10 PES to Fulvalanyl and Azulanyl Radicals

2089 A schematic of the overall C10H10 surface examined in this study is shown in Fig 6-2.

2090 More detailed enthalpy diagrams will be shown in later figures which give the individual

2091 structures, but divide the PES into two parts due to the large number of species. When

2092 entering the isomerization network through the combination of two CPDyl radicals, the

2093 initial adduct has a number of options to further react. First, it can immediately stabilize

2094 via collisions and energy transfer to surrounding molecules, thermalizing it to one of the

2095 fulvalane species. Based on the energies and barriers, this is expected to be the dominant

2096 channel at sufficiently high pressure. Secondly, if the combination of CPDyl radicals produces

2097 an adduct with sufficient excess energy, it may be able to rearrange and pass over the high

2098 barrier separating the fulvalanes and azulanes before being collision stabilized. Entering

2099 the PES at even higher energy levels allows for skipping over the wells of the C10H10 PES

2100 entirely, emitting a hydrogen radical and proceeding immediately to the C10H9 PES related

2101 to fulvalanyl or azulanyl. While a larger degree of atomic rearrangement is necessary to

2102 proceed directly to azulanyl, those species are of lower energy than the fulvalanyl isomers.

2103 It may thus be possible at certain conditions to generate preferentially azulanyl radicals.

2104 Once a fulvalanyl or azulanyl radical has been formed, the C10H9 species can isomerize or

2105 eliminate an H atom to form naphthalene, fulvalene, or azulene, Fig. 6-7.

2106 Fulvalanyl radical formation: The CPDyl radical recombination reaction and subse-

2107 quent C10H10 surface was first examined by the 1996 work by Melius et al., but that work

2108 considered only three of the six potential C10H10 H-shift isomers and neglected one of the

2109 three fulvalanyl radical isomers [182]. The complete set of isomers was later shown by the

2110 2013 Cavallotti et al. work [188]. The results of our study show that all wells (isomers)

2111 are important and isomerization pathways must be included in the kinetic analysis since

2112 missing low-barrier sigmatropic hydrogen shift reactions and isomers will under-predict the

2113 total yield of C10H10 isomers formed in the recombination reaction. A reduced build-up of

2114 C10H10 isomers will lead to an underestimation of naphthalene and other species produced

2115 through H loss pathways from these isomers.

138 grsa niae o ipa upss u oeta h acltosaepromdon performed are calculations the that several note into broken whole. but been a purposes, has as surface display surface the the the for species, than to indicated of height energy number equal as in the roughly figures to lower of Due groups slightly two channel. fall the entrance separating azulanes the barrier The isomerization an schematic. with surface fulvalanes C10H10 6-2: Figure CPDyl CPDyl +

Boltzmann Energy Distribution Figure 3 Figure (+M) Fulvalanes - [H] - [H] 139 (+M) Fulvalanyl (+M) Azulanes Figure 4 Figure - [H] Figure 7 Figure - [H] (+M) Azulanyl Naphthalene Figure 6-3: CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) for the C10H10 surface initiated by the recombination of two CPDyl radicals. All enthalpies are relative to the adduct T1b which connects to the other portion of the surface. This network was first reported by Melius et al. and in its complete form by Cavallotti et al. [182, 188]

2116 The fulvalane/fulvalanyl portion of the network is shown in Fig. 6-3. The six fulvalane

2117 isomers (T1a to T1f) can easily interconvert due to the relatively low barriers for the sigma-

2118 tropic H shifts in these components. The CBS-QB3 method predicts barriers that are in the

2119 range of 90-120 kJ/mol. This agrees well with previous computational results[182, 188]. The

2120 isomers with co-planar conjugated rings (T1c, T1d and T1e) are significantly more stable

2121 than other isomers. The lowest barrier hydrogen emission occurs between T1b and N1b at

2122 270 kJ/mol. In general, the C-H bond dissociation energies (BDE) of T1a to T1f range

2123 between 270 and 330 kJ/mol, much smaller than C-H bond energies in alkanes which are

2124 around 400 kJ/mol and still lower than the allylic C-H bond energy in propene (about 370

2125 kJ/mol) [214].

2126 Azulanyl radical formation: The pathway used for this study (Fig. 6-4) was first

2127 discussed by Cavallotti et al. as being important at lower temperatures[43]. This is primarily

2128 due to the increased stability of the azulane species (T4a to T4d). Possible reaction pathways

2129 to azulanyl radical were also explored by Kislov and Mebel, though their work examined

140 2130 channels on the C10H9 surface [184, 188].

2131 The portion of the surface relevant for azulanyl radical formation is shown in Fig. 6-4

2132 and begins with species T1b from Fig. 6-3. Similar to the formation of fulvalanyl the path

2133 is relatively simple, passing through a tricyclic species (T2) followed by a ring opening (T3)

2134 and closing to form the azulane species. The azulane species are interconnected by various

2135 hydrogen shift reactions. The H-shifts barriers for the azulane species range from around

2136 100-150 kJ/mol indicating a greater resistance to interconversion than is observed for the

2137 fulvalanyl species. The lowest energy route to the C10H9 surface requires an increase in

2138 energy of 270 kJ/mol from a fulvalane isomer, about 5 kJ/mol less than any path from

2139 an azulane isomer. Though the separating barrier between the fulvalanes and azulanes

2140 is relatively high, it is still about 15 kJ/mol lower than the CPDyl + CPDyl entrance

2141 channel. Therefore, isomerization to the more stable azulane isomers is energetically feasible,

2142 suggesting that rather high azulane concentrations can build up at lower temperatures and

2143 thus subsequent azulane pathways need to be included. Three other symmetry allowed

2144 azulane species are included in the analysis and calculations for the C10H10 surface but are

2145 omitted from Fig. 6-3. since they are only accessible by traversing a 170 kJ/mol higher

2146 barrier from T4a.

2147 Some authors have treated this surface as if the chemically activated routes dominate,

2148 going even so far as to use a single rate expression to describe the direct conversion from two

2149 CPDyl radicals directly to naphthalene and two hydrogen atoms [176]. Here evidence against

2150 the validity of this assumption is presented through calculation of the pressure dependent

2151 rate constants for this network. The resulting rate coefficients for the various reaction path-

2152 ways of the recombination of two CPDyl radicals at 1100 K can be found in Fig. 6-5a as

2153 functions of pressure. H atom forming channels are only of minor importance (< 1%) at

2154 all pressures investigated. Instead, collision stabilization leads to the formation of various

2155 fulvalane isomers. At pressures above atmospheric conditions (P > 1 bar), the recombi-

2156 nation reaction of the two CPDyl radicals results primarily in the formation of T1a and

2157 well-skipping reactions are suppressed. Around 1 bar and below, well-skipping reactions

2158 become important and T1c becomes the dominant product. Therefore, at typical conditions

2159 the intermediate formed after recombination of two CPDyl radicals only slowly stabilizes,

141 Figure 6-4: CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) on the C10H10 surface leading to the formation of Azulanyl radicals. All enthalpies are relative to component T1b which represents the dominant entrance channel. Pathway first shown by Cavallotti et al.[188]. Additional symmetry allowed azulane species were included in this analysis but are not shown in this figure for simplicity reasons as they are only accessible via a 170 kJ/mol barrier from T4a.

2160 allowing the hydrogen atoms to freely move around in the molecule.

2161 The direct and well-skipping rate coefficients, kA→B(T,P), are displayed in matrix form

2162 in Fig. 6-5b. The bottleneck is the conversion of the fulvalanes into species T2, which occurs

2163 on a millisecond timescale at 1100 K. Equilibration between the fulvalane species prior to

2164 the bottle neck occurs on a much faster microsecond time scale. While slower than the

2165 fulvalanes, the azulane species also interconvert rapidly.

2166 With pressure dependent rate analysis confirming that species are stabilized on the

2167 C10H10 surface and do not predominantly proceed directly to C10H9, it is thus possible

2168 that C10H9 formation occurs through three different paths: chemically activated hydrogen

2169 loss from the CPDyl reaction (2CPDyl→C10H9+H), hydrogen emission from the C10H10

2170 species (C10H10→C10H9+H), and radical hydrogen abstraction from the C10H10 species

2171 (C10H10+R.→C10H9+RH). To see how the three pathways relate to one another, C10H9

2172 formation rates are calculated for 1 and 100 bar, Fig. 6-6. This is done by using the fastest

2173 rates to approximate the chemically activated route (2CPDylâĘŠN1b+H) and the emis-

2174 sion route (T1b → N1b+H), while estimating the rate of the hydrogen abstraction route

142

T T T T T T T T T T T T

s

T T mol

T cm T

T s

T k T T

Log

T

T T T 1 11 1 11 1 bar T

Figure 6-5: (A: Left) Computed pressure dependent rate coefficients for the recombination of two CPDyl radicals to form various products in N2 at 1100 K. The corresponding PES is shown in Fig. 6-3 and Fig. 6-4. (B: Right) Rate constant matrix depicting rate constant magnitudes for all well-skipping and direct reactions between isomers of the C10H10 PES. Reacting species shown on the y axis with the product species along the x axis.

2175 as T1b + CPDyl âĘŠ N1b+CPD using the Reaction Mechanism Generator (RMG) and its

−9 3 2176 corresponding database A = 1.1∗10 m /(mols) ,n=4.3,Ea=48 kJ/mol) [13]. For concentra-

2177 tions, CPDyl is assumed in equilibrium with the C10H10 isomers since CPDyl is the starting

2178 species and the radical recombination reaction is fast. The total concentration of CPDyl +

−9 3 2179 2 C10H10 is constrained to 2 ∗ 10 mol/cm for all conditions to simplify the comparison.

2180 This concentration is obtained using simulation of CPD pyrolysis using the POLIMI 1311

2181 mechanism at the low dilution conditions of Djokic et al. at 1000K [178]. See Table 1 for

2182 data used to generate Fig. 6-6. For high temperatures, the chemically activated route to

2183 C10H9 + H plays a major role. Although C10H10 is the dominant product of recombination

2184 of the two CPDyl radicals, the subsequent thermal decomposition of the C10H10 isomers to

2185 C10H9 + H is predicted to play essentially no role in the formation of naphthalene. This is

2186 because C10H10 → CPDyl + CPDyl is energetically favored over C10H10→C10H9+H (Fig.

2187 6-2).

2188 At 1 bar the radical abstraction route from C10H10 to C10H9 becomes the dominant

2189 pathway below 1020K. The crossover temperature rises with pressure to 1230K at 100 bar.

143 Table 6.1: Selected data set used for the rate analysis of C10H9 formation routes as shown in Fig. 6-6 Pressure (bar) 1 100 Temperature (K) 800 1000 1200 800 1000 1200

cm3 kCPDyl+C10H10→C10H9+CPD ( mol·s ) 3.5E+06 3.9E+07 2.2E+08 3.5E+06 3.9E+07 2.2E+08 cm3 k2CPDyl→C10H9+H ( mol·s ) 2.4E+06 5.1E+07 4.5E+08 3.2E+03 1.6E+05 3.0E+06 −1 kC10H10→C10H9+H (s ) 9.2E-10 2.8E-06 5.0E-04 1.2E-09 4.3E-06 9.8E-04 mol [CPDyl] ( cm3 ) 1.0E-11 4.3E-10 1.9E-09 1.0E-11 4.3E-10 1.9E-09 mol [C10H10]( cm3 ) 9.9E-10 7.9E-10 6.0E-11 9.9E-10 7.9E-10 6.0E-11 mol [CPDyl] + 2[C10H10]( cm3 ) 2.0E-09 2.0E-09 2.0E-09 2.0E-09 2.0E-09 2.0E-09

2190 Therefore, radical hydrogen abstraction is particularly relevant at the conditions of pyrolytic

2191 experimental studies of CPD. Additionally, the range of relevance is expected to extend

2192 to higher temperatures if the full system radical pool is taken into account as possible

2193 abstractors rather than just CPDyl as is done here. The marked shifts in slope for each of

2194 the three curves is attributed to a shift in equilibrium to favor CPDyl over C10H10 as seen

2195 in Table 1. For temperatures above this point, CPDyl is the major species while C10H10

2196 becomes the minor species and its concentration begins to fall at an increased rate. A figure

2197 showing this trend is included in supporting information. An important consequence of the

2198 observed importance of radical H abstraction is a shift in net ‘radical’ balance through the

2199 formation of naphthalene from CPDyl at these lower pyrolytic temperatures: ”Two radicals

2200 in, two H atoms out” (when the chemically activated channel dominates) becomes ”three

2201 radicals in, one H atom out” at lower temperatures, causing potentially a marked reduction

2202 in system reactivity.

2203 6.4.2 C10H9 PES to Azulene and Naphthalene

2204 The direct spiran path from the fulvalanyl radical to naphthalene used here (Fig. 6-7) is

2205 consistent with the one originally shown by Melius et al. The surface was subsequently

2206 re-calculated at the B3LYP/6-31G(d,p) level of theory by Wang et al. as well as Kislov

2207 and Mebel who include this route as part of a larger study [183, 184]. In the current study,

2208 CBS-QB3 based energies are used. Based on the discussion above, the low-lying C10H9

2209 species N1b is formed most rapidly. However, H atom scrambling in species N1b is fast

144 00 0 0

0 0

00 0 00 0

00 0 00 0 0 0

s 0 s 0

0 0 mol cm mol cm

0 0

0 0

0 0 0 0

00 00 0 0 0 0 0 0 0 0 000K 000K

Figure 6-6: Rate of C10H9 formation comparison between hydrogen emission from the CPDyl dimer, chemically activated hydrogen emission from the CPDyl + CPDyl reaction, and radical hydrogen abstraction from the CPDyl dimer. At moderate to low temperatures, radical abstraction dominates while at high temperatures the activated path leads formation of C10H9

2210 and leads to the formation of its isomers N1a and N1c. Species N1c can undergo a 1,3-

2211 cycloaddition leading to species N2. β-scission of this component leads to the spirane-type

2212 species N3. This component then reacts via a 1,3-cycloaddition and β-scission to species N5,

2213 which yields naphthalene by β-scission of a C-H bond.

2214 Though the azulane species are not generally observed to be the dominant product on

2215 the C10H10 PES at the conditions discussed in this study, the Cavallotti et al. publication

2216 adds relevance for azulanyl radical conversion to naphthalene in addition to the route from

2217 fulvalanyl radicals. Routes between azulene and naphthalene were previously explored by

2218 Alder et al. who found two dominant options: the spiran path, and the methylene walk

2219 route [189]. Entrance channels for the methylene walk path were observed to be significantly

2220 higher than for the spiran route, and thus the methylene walk route has been neglected here.

2221 This is in agreement with the expectations of Cavallotti et al. who proposed the spiran route

2222 to follow from the azulanyl formation shown in their study. As might be expected, the Alder

2223 spiran path merges with the Melius spiran route from fulvalanyl to naphthalene. This occurs

2224 at species N4 and the combined surfaces are shown in Fig. 6-7.

145 Figure 6-7: CBS-QB3 enthalpy diagram H(0 K) (kJ/mol) on the C10H9 surface leading to the formation of naphthalene and azulene. All enthalpies are relative to component N1b

2225 Considering the C10H9 PES shown in Fig. 6-7, the pressure-dependent rates shown in

2226 Fig. 6-8 are computed. The primary entrance species, N1b, equilibrates with the other

2227 fulvalanyl isomers N1a and N1c on a sub-microsecond timescale. The bottleneck on this

2228 PES is the ring opening reaction N2→N3, passing over a 90 kJ/mol barrier. Interestingly,

2229 after passing through species N2, the C10H9 species is predicted to pass through the azulanyl

2230 species before proceeding on to naphthalene. Since bimolecular reactions are observed to be

2231 of relevance for C10H10, the lifetime of the C10H9 species is determined by simulating the

2232 C10H9 sub-mechanism using Chemkin. When starting with an initial concentration of N1b,

2233 the formation of C10H8 (overwhelmingly naphthalene) follows an exponential curve. This

2234 gives C10H9 half-lives of about 20 µs at 800 K, and 20 Âţs at 1200 K, for a pressure of 1 bar.

2235 At these lifetimes bimolecular reaction rates between C10H9 and other species are expected

2236 to be relevant, diverting away from naphthalene formation and on to higher carbon number

2237 and other species. For example, in combustion systems C10H9 + O2 reactions are expected to

2238 be especially relevant. Rate based automated mechanism generators such as RMG consider

2239 all feasible elementary reactions and will - if necessary - include automatically important

2240 bimolecular reactions for C10H9 during mechanism generation.

2241 6.5 Conclusions

2242 In this study, a pressure-dependence analysis of the CPDyl + CPDyl recombination reaction

2243 and subsequent reactions leading to naphthalene is provided. At all low to intermediate

146 Product N1a N1b N1c N2 N3 N4 N5 N6 N7a N7b N7c Flvln+HAzln+HNphln+H N1a

N1b

s N1c N2 mol

N3 s cm

N4 k

N5

N6 Log Isomer Reactant N7a N7b

N7c Pressure =1 bar Temperature = 1100K

1 11 1 11 1 bar

Figure 6-8: (A: Left) Computed pressure dependent rate coefficients for the species N1b, the primary entrance channel to the C10H9 surface to form various products at 1100K in N2. The corresponding PES is shown in Fig. 6-7. (B: Right) Rate constant matrix for well-skipping and direct reactions among the isomers of the C10H9 PES. Reacting species shown on the y axis with the product species along the x axis. [Flvln = fulvalene, Azln = azulene, Nphln = naphthalene]

2244 temperature conditions (up to 1020K at 1 bar and 1230K at 100 bar), naphthalene is mainly

2245 formed via a series of reaction steps starting with the formation of C10H10 isomers and not

2246 through the direct chemically activated reaction C5H5+C5H5→C10H8+H+H. The addition

2247 of deeper wells to the C10H10 potential energy surface from the Cavallotti et al. work shifts

2248 the equilibrium of the cyclopentadienyl recombination reaction more to the right and allows

2249 the C10H10 species to accumulate more than predicted from the truncated potential energy

2250 surface used by Melius et al. and Kislov and Mebel [182, 184]. Since C-H scission of any

2251 C10H10 isomer is highly endothermic, the C10H10 species are predicted to accumulate to

2252 high concentrations which enables bimolecular hydrogen abstraction reactions to become

2253 competitive. C10H9 formation through H abstraction is expected to be the preferred route

2254 at pyrolytic conditions at 1 bar and temperatures below 1020K. H abstraction is predicted

2255 to be even more important at higher pressures. The net effect of this analysis is that at low

2256 to intermediate temperatures naphthalene formation via CPDyl recombination consumes

2257 three radicals (two CPDyl and one abstracting radical (or atom)) while just one H atom is

2258 produced. This is in stark contrast to the simplified scheme (C5H5+C5H5→C10H8+H+H),

2259 which implies that naphthalene formation goes along with the conversion of two relatively

147 High + H + H Temperatures +

Intermediate & Low + + R + RH + H Temperatures

2260 stable CPDyl radials to two more reactive H atoms. Therefore the reaction pathways are

2261 summarized as: Isomerizations on the C10H9 surface directly to naphthalene are considered

2262 as well as those that pass through the formation of the azulanyl radical. This work indicates

2263 that C10H9 intermediates will be collisionally stabilized and live for a period of roughly 20 µs

2264 at 1200K and 1 bar before decomposing to C10H8+H. Pressure dependent rate coefficients

2265 in CHEMKIN compatible format are given in Supplementary Materials for use in future

2266 kinetic modeling studies.

2267 The formation of the first PAHs is a complex process that involves many intermediates

2268 and reactions. The growth to larger PAHs, such as anthracene and phenanthrene, would

2269 require many more possible intermediates and reactions and might prove too complex to

2270 study manually. Automated network generators, such as RMG, and automatic reaction

2271 discovery are hence indispensable in order to unravel the important steps in the growth of

2272 PAHs [13, 215]. Studies such as this one help to provide the integral theoretic backbone on

2273 which automatic software operates when predicting more complex chemistries. Future work

2274 should try to address the growth to tri- and tetracyclic PAHs through automated means.

148 2275 Chapter 7

2276 Conclusions and future work

2277 In general, the results in this body of work serve to improve the handling of the fundamental

2278 flame parameter, extinction strain rate (ESR) which characterizes a flame’s resistance to

2279 extinction through deformation. This parameter is of particular interest for high pressure,

2280 turbulent flames relevant to gas turbines and internal combustion engines. This work in-

2281 troduces a more fundamental definition for ESR as well as an open source code capable of

2282 efficiently calculating the parameter for large kinetic models. These contributions are then

2283 leveraged when producing a kinetic model for natural gas combustion at relevant pressures.

2284 In the future, ESR predictions may be used in the study and design on next generation

2285 combustors. For example, accurate knowledge of a given fuel-oxidizer extinction point could

2286 allow for design of flow patterns that prevent the flame from propagating completely to the

2287 wall by means of a strain rate barrier through which the flame cannot propagate. Attempt-

2288 ing to predict such strain rate barriers through direct numerical simulation of characteristic

2289 flow patterns would be an interesting future study.

2290 7.1 Chapter 2

2291 An approach was developed for the determination of extinction strain rates in stagnation-

2292 type flames that is independent of the burner separation distance. The approach involves

2293 extrapolations to infinite burner separation distance that are analogous to extrapolations to

2294 zero strain rate for laminar flame speed determination. By eliminating dependence on the

149 2295 experimental apparatus, this approach makes extinction strain rate a viable target for ki-

2296 netic model validation in addition to use as a governing parameter for turbulent combustion

2297 behavior. The independence of the near-extinction flame structure from the burner sepa-

2298 ration distance was shown first numerically and then validated with experimental velocity

2299 measurements. In order to determine experimentally extinction strain rates at the limit of

2300 infinite burner separation distance, it is proposed to use the experimentally measured axial

2301 velocity profiles for near-extinction flames at a few separation distances in order to deter-

2302 mine the intersection of the respective strain rate lines. This approach was validated for

2303 methane and propane premixed and diffusion flames through experiments that resulted in

2304 clear intersection points and reasonable percent differences in ESR∞ when compared with

2305 1D model predictions. Based on these encouraging results, the extinction strain rate at at

2306 the limit of infinite burner separation distance could be viewed as a proper parameter for

2307 characterizing extinction behavior.

2308 Future work in this area should focus on additional validation and examination of the

2309 experimental method for determining this parameter. Though a method for the extrapolation

2310 to infinite separation extinction strain rate has been demonstrated, there is still opportunity

2311 for a superior and possibly more robust method to be developed. Additionally, for the

2312 method described in this work, a question remains regarding any potential physical relevance

2313 of the axial position of the intersection point used to determine the infinite separation ESR.

2314 Once a standard method has been established by the community, significant experimental

2315 efforts will be needed to build and refine a relevant set of data to be used in understanding

2316 important combustion behaviors and validating kinetic models with respect to extinction.

2317 This will allow the models to be used with confidence in predicting certain turbulent flame

2318 characteristics.

2319 7.2 Chapter 3

2320 Ember, a new, open-source, transient solver, has been introduced for the simulation of

2321 one-dimensional, strained flames. Through use of rebalanced Strang splitting, tailored dif-

2322 ferential equation solvers for each split calculation, and parallelization, Ember significantly

150 2323 outperforms conventional software such as Chemkin, typically used for such combustion sim-

2324 ulations. In the determination of extinction strain rate (ESR), Ember’s computation time

2325 scales according to number of species to the 1.2 power while Chemkin-Pro 17.0 is observed to

2326 scale with species to the 2.9 power. This significant improvement in scaling of computation

2327 time allows for practical ESR calculations using large kinetic models such as AramcoMech

2328 2.0 (493 species) which would otherwise not have been feasible. This is an important capa-

2329 bility as increasingly complex fuel blends are investigated, requiring large, detailed kinetic

2330 models. For demonstration of the capabilities of Ember, it has been used to analyze the

2331 Lewis number trends predicted by asymptotic analyses of strained flames [35]. The tran-

2332 sient solver capabilities of Ember were leveraged for an analysis of initially steady methane

2333 flames under lean, stoichiometric, and rich conditions for which the the energy conduction

2334 and individual species diffusions could instantaneously be perturbed and the strained flame

2335 allowed to relax to the new steady state. Of particular interest was the observed steady

2336 flame in the complete absence of energy conduction (Le=0), propagating through radical

2337 and species diffusion alone. Calculations with AramcoMech 2.0 demonstrated the capabil-

2338 ities of Ember with large detailed kinetic models and confirmed the Lewis number trends

2339 predicted by Law’s asymptotic analyses hold even when the fuel chemistry is extremely

2340 complicated. A Lewis number less than one produced a growing peak reaction rate with

2341 increasing strain as additional fuel/oxidizer is diffusively supplied faster than the thermal

2342 propagation of the flame. Alternatively, a Lewis number greater than one yielded a flame

2343 with a steadily decreasing peak reaction rate as increasing strain enhances thermal energy

2344 losses from the flame. Though capable of very rapidly calculating extinction strain rate

2345 under ideal potential flow conditions as demonstrated in this work, Ember, as of yet cannot

2346 produce extinction strain rate predictions for conditions under which the finite boundary

2347 has significant impact. However, a new method of approximating the finite boundary ESR

2348 is introduced and demonstrated. Using this method, the simulated rich propane/air ESR

2349 differs by only 6% when compared with a previously reported experimental value.

2350 Given the strong sensitivity to the diffusion rate of the limiting reactant, as shown in a

2351 subsequent section of this work, an important future effort toward expanding the capabilities

2352 of Ember is in development of an approximation method for the more accurate multicompo-

151 2353 nent transport formulation [74]. As it currently stands, use of the multicompenent transport

2354 formation in Ember, as implemented in Cantera, voids the improvement in computation time

2355 scaling with number of species achieved through use of QSS reaction handling. However a

2356 similar method to the approximation used for the mixture averaged transport formulation

2357 currently implemented in Ember is believed to be feasible. By implementing the multi-

2358 component formulas, but only considering the major species when calculating each diffusion

2359 coefficient. It should be possible recover the nearly linear computational scaling with number

2360 of species in the kinetic model. Given the lack of sensitivity to the diffusion rates of minor

2361 species, it should be reasonable to maintain the mixture averaged method these species to

2362 generate an overall hybrid method.

2363 Though somewhat separate from extinction behavior, there is also an opportunity to

2364 expand Ember to support spherically expanding (or contracting) flames with accurate time

2365 resolution. This is relevant to many of the recent laminar flame studies at elevated pressures

2366 that have been using experimental spherically expanding flames and extrapolating back to

2367 zero curvature. With a spherical formulation, it might also be interesting to investigate

2368 curvature based extinction phenomena that are largely governed by lewis number effects.

2369 7.3 Chapter 4

2370 Current kinetic models are found to differ significantly in their predictions of extinction strain

2371 rate (ESR) particularly at higher pressures for lean (φ = 0.7) and rich (φ = 1.3) conditions.

2372 However, the variations in predictions of ESR between the kinetics models are observed

2373 to be generally consistent with the variations in prediction of laminar flame speed (LFS).

2374 This suggests that the same kinetic model parameters affect LFS and ESR. Consistency

2375 between sensitivity of LFS and ESR to rate coefficients supports this observation. Transport

2376 parameter sensitivity analysis shows a very strong sensitivity of ESR to the Lennard Jones

2377 diameter of the fuel, oxidizer, and bath gas. For ESR predictions, improved kinetic transport

2378 descriptions as suggested Middha [124], Paul [122, 123], and Brown [126] are needed. ESR is

2379 less sensitive to thermochemical property uncertainties. Thermodynamic sensitivity analysis

2380 shows limited need for more accurate enthalpy values in terms of ESR predictions. Entropy

152 2381 uncertainties may merit some further investigation, but are also not expected to be significant

2382 for the small species that ESR is most sensitive to under the conditions studied here. The

2383 heat capacity of the bath gas, nitrogen for the flames in this study, has a strong negative

2384 sensitivity. Requiring more energy to increase the temperature correspondingly weakens the

2385 flame. Additionally, pressure dependent kinetics are shown to be vital to determining ESR

2386 pressure trends, as are molecular collision efficiencies. Further refinement of kinetic model

2387 collision efficiency descriptions in line with the work of Jasper and coworkers [131, 133] and

2388 Burke and Song [134] is necessary for accurate ESR predictions under the conditions of

2389 this study. Small amounts of C2H6 and H2 (5% by mol) in the fuel are observed to have

2390 significant impact on ESR at higher pressures. Given the wide variation in predictions by

2391 current kinetic models and uncertainty in many sensitive parameters, it is important that

2392 future efforts seek to study and analyze the phenomena of flame extinction by stretch at

2393 elevated pressures experimentally.

2394 Given the wide variation in predictions by current kinetic models and uncertainty in

2395 many sensitive parameters, it is important that future efforts seek to study and analyze the

2396 phenomena of flame extinction by stretch at elevated pressures experimentally. In particular

2397 there may be some opportunity to calibrate at atmospheric conditions and then leverage some

2398 of the turbulent flame behavior ESR correlations observed by the Ghoniem lab to indirectly

2399 probe ESR under conditions that conventional opposed flow apparatuses may likely never

2400 achieve due to density induced turbulence effects. This work also shows an opportunity area

2401 in improved handling of collision efficiencies through incorporation of temperature depen-

2402 dence. Since the Jasper ab initio method is fairly computationally intensive, investigating an

2403 efficient method of estimating collision efficiencies for any reaction would be advantageous

2404 as well.

2405 7.4 Chapter 5

2406 A validation tool for kinetic model benchmarking and subsequent improvement has been

2407 produced. The tool allows users to benchmark against ignition delay (autoignition), laminar

2408 flame speed, and extinction strain rate data. Using a data set focusing on CH4,H2, CO, C2H6,

153 2409 C2H4, and C2H2 at pressures of relevance to gas turbines and internal combustion engines

2410 when available, a kinetic model is produced that represents the selected validation data

2411 significantly better than currently available models. This is achieved by starting with the

2412 high pressure oxidation model from Hashemi and coworkers and subsequently replacing and

2413 modifying sensitive reactions according to uncertainty values and reaction rate sensitivities.

2414 This ultimately improves the overall error score for the Hashemi model by 57% from 61

2415 to 26. Nitrogen chemistry and rich chemistry sub-mechanisms from other prior works are

2416 subsequently added to the core model to allow for use over a greater range of applications.

2417 Future efforts should consider expanding the validation data set through new experimen-

2418 tal data, improving the computational efficiency of the ignition delay sensitivity data, and

2419 leveraging overall optimization procedures similar to those used by the FFCM1 authors. For

2420 such optimization procedures, accurate uncertainty quantification for all kinetic parameters

2421 is vital and not currently available.

2422 7.5 Chapter 6

2423 A pressure-dependence analysis of the CPDyl + CPDyl recombination reaction and sub-

2424 sequent reactions leading to naphthalene has been provided. At all low to intermediate

2425 temperature conditions (up to 1020K at 1 bar and 1230K at 100 bar), naphthalene is mainly

2426 formed via a series of reaction steps starting with the formation of C10H10 isomers and not

2427 through the direct chemically activated reaction C5H5 + C5H5 → C10H8 + H + H. The ad-

2428 dition of deeper wells to the C10H10 potential energy surface from the Cavallotti et al. work

2429 shifts the equilibrium of the cyclopentadienyl recombination reaction more to the right and

2430 allows the C10H10 species to accumulate more than predicted from the truncated potential

2431 energy surface used by Melius et al. and Kislov and Mebel [182, 184]. Since C-H scission of

2432 any C10H10 isomer is highly endothermic, the C10H10 species are predicted to accumulate

2433 to high concentrations which enables bimolecular hydrogen abstraction reactions to become

2434 competitive. C10H9 formation through H abstraction is expected to be the preferred route

2435 at pyrolytic conditions at 1 bar and temperatures below 1020K. H abstraction is predicted to

2436 be even more important at higher pressures. The net effect of this analysis is that at low to

154 2437 intermediate temperatures naphthalene formation via CPDyl recombination consumes three

2438 radicals (two CPDyl and one abstracting radical (or atom)) while just one H atom is pro-

2439 duced. This is in stark contrast to the simplified scheme (C5H5 + C5H5 → C10H8 + H + H),

2440 which implies that naphthalene formation goes along with the conversion of two relatively

2441 stable CPDyl radials to two more reactive H atoms. Isomerizations on the C10H9 surface

2442 directly to naphthalene are considered as well as those that pass through the formation of

2443 the azulanyl radical. This work indicates that C10H9 intermediates will be collisionally sta-

2444 bilized and live for a period of roughly 20 µs at 1200K and 1 bar before decomposing to

2445 C10H8+H. Pressure dependent rate coefficients in CHEMKIN compatible format are given

2446 in Supplementary Materials for use in future kinetic modeling studies.

2447 The formation of the first PAHs is a complex process that involves many intermediates

2448 and reactions. The growth to larger PAHs, such as anthracene and phenanthrene, would

2449 require many more possible intermediates and reactions and might prove too complex to

2450 study manually as was done in this work. Automated network generators, such as RMG,

2451 and automatic reaction discovery are hence indispensable in order to unravel the important

2452 steps in the growth of PAHs [13, 215]. Studies such as this one help to provide the integral

2453 theoretic backbone on which automatic software operates when predicting more complex

2454 chemistries. Future work should try to address the growth to tri- and tetracyclic PAHs

2455 through automated means. Additionally, automated methods allow for efficient additions

2456 of relevant bimolecular reactions with intermediate species along the given surface. For

2457 this particular work, these additions would be particularly relevant for the C10H9 surface

2458 and may help describe formation of higher order PAHs while correcting naphthalene over-

2459 predictions observed when applying this network

155 THIS PAGE INTENTIONALLY LEFT BLANK

156 2460 Chapter 8

2461 [CAPSTONE] Assessing investment in

2462 Natural Gas for commercial trucking

2463 applications using financial, strategic,

2464 and game theoretic frameworks

2465 8.1 Summary

2466 Natural gas, cheap and widely available in the domestic US, is promising as a more envi-

2467 ronmentally friendly ’transition’ fuel before more sustainable alternatives (e.g. battery or

2468 hydrogen fuel cell) become truly viable in heavy duty trucking. However, as of yet, US

2469 trucking remains predominantly a business of diesel engines. This analysis examines the in-

2470 vestment decision for transitioning to NG fleets from financial, strategic, and game theoretic

2471 lenses. Financially, transitioning to NG trucking could provide an upside from $33K NPV

2472 per truck under existing technological and economic trends, or as high as $170K per truck

2473 if a carbon tax is implemented and engine efficiencies are improved to be on par with diesel

2474 engines. Strategically the trucking market is assessed to be fairly average, not particularly

2475 attractive or something to strictly avoid. Unfortunately, financial market pressures to invest

2476 in NG trucking unravel when uncertainties and game theory analysis are considered for the

157 2477 largely undifferentiated service and it becomes clear the current financial drivers are insuffi-

2478 cient for prompting the market to transition, as observed. In consideration of this analysis, it

2479 is recommended that, in order to drive the transition to NG trucking, investments be made

2480 primarily in implementing a carbon tax and advancing NG engine efficiency to be on par

2481 with diesel engines to strengthen the financial incentive and overcome the risks associated

2482 with uncertainty surrounding NG trucking investments.

2483 8.2 Background

2484 Global warming is a growing and ever more pressing challenge, making the need for sus-

2485 tainable energy imperative [1, 2]. True renewable energy sources such as wind and solar

2486 technologies are encouraging, but suffer significant barriers with respect to storage and on-

2487 demand capabilities. In the realm of transportation, electric vehicles have captured the eye

2488 of the market through innovations such as the Toyota Prius and the various Tesla market

2489 offerings. However, the on-board batteries suffer slow re-charge periods and high weight

2490 to energy ratios [216] making them less than optimal for medium and heavy duty trucking

2491 applications which are responsible for 25% of road emissions [217]. These are significant

2492 barriers and, though actively under research [218–220], interim measures are necessary to

2493 take action given the imminent environmental concerns. One promising interim measure is

2494 the use of Natural Gas (NG), which is comprised of roughly 90% methane, to replace the

2495 more carbon intensive diesel fuel.

2496 US natural gas production has risen steeply in recent years [8]. The rise has been largely

2497 fueled by advances in fracking technology, allowing much greater access to shale gas through-

2498 out the country. As a result of fracking, made possible by horizontal drilling [9], the US is

2499 projected to have roughly 100 years of accessible NG at the current consumption rate [10].

2500 This reliable source of NG is important from an economic standpoint of maintaining a low

2501 domestic price for NG.

2502 Because methane has a higher hydrogen to carbon ratio, it produces more energy per

2503 equivalent mass of CO2. NG has a H:C ratio of about 3.7:1 as compared to that of diesel fuel

2504 at about 1.8:1 [4]. Additionally, NG sources naturally contain substantially lower amounts of

158 2505 sulfur which is important for avoiding the production of sulfur dioxide gas during combustion

2506 [221, 222]. In [6] the sulfur emissions from NG resources are observed to be roughly one

2507 seventh of oil or coal resources. Sulfur can be removed from other fossil fuels through the

2508 hydro-cracking process [223], but this process is energy intensive in itself.

2509 Even greater green-house gas (GHG) abatement is attainable when using NG in heavy

2510 duty trucking if renewable sources of NG (RNG) are leveraged. Waste streams otherwise

2511 intended for landfills can often be upgraded to RNG through anaerobic digestion [12] or

2512 pyrolysis pathways [224]. Some researchers even propose the use of excess intermittent

2513 renewables (wind and solar) for a methanation process of CO2 [225].

2514 However, while the case for transition to NG trucking seems favorable at first look (low

2515 NG prices and higher carbon efficiency) the overall market has been slow to transition. In

2516 fact, NG vehicles world-wide total less than 15% of the 1.2 billion motor vehicles globally

2517 [225]. In part, the transition has been hindered by four primary hurdles for NG in trans-

2518 portation: (1) engine efficiencies are currently only about 85% of a typical diesel engine

2519 [226, 227] (2) On-board fuel storage systems are significantly more complex and costly [228],

2520 (3) NG is a potent GHG on its own and current infrastructure does not sufficiently inhibit

2521 leakage [7, 229, 230] (4) the true social cost of carbon usage is not translated to the fuel

2522 decision maker through a carbon tax [231].

2523 This study will apply relevant business perspectives to examine the case for investment

2524 in NG trucking in the United States market to better understand the economic factors in-

2525 hibiting the transition to NG trucking. The case for transition will first be examined via

2526 basic financial analysis. The financial analysis will then be followed by a strategic consider-

2527 ation, leveraging the classic framework of Porter’s Five Forces [232]. Finally, game theory is

2528 applied to rationalize how both potential new entrants to the market as well as existing com-

2529 petitors think about the investment in NG fleets while considering how others might react

2530 to their decision. Insight is derived from this analysis to suggest where investments are best

2531 made from both the perspective of the individual and the larger market with environmental

2532 considerations in mind.

159 2533 8.3 Financial Analysis

2534 In beginning analysis of the financial viability of investment in natural gas (NG) trucking, it

2535 is necessary to first assess the required market return or discount rate. The discount rate is

2536 essential in estimating the present value of any future cash flows that will ideally be greater

2537 than the upfront price premium for a NG truck as opposed to the more standard diesel. Here

2538 the the Capital Asset Pricing Model (CAPM) [233–235] is used to determine the appropriate

2539 discount rate. The CAPM model asserts that the risk induced price premium for any asset

2540 follows the functional form:

r = rriskfree + βx(rmarket − rriskfree) (8.1)

2541 where r is the discount rate, rriskfree is the annual return expected for a zero risk investment

2542 of the same duration, typically assumed to be the rate of US treasury bond, and rmarket is the

2543 expected return for investing in the entire market. The βx parameter then describes what

2544 portion of the market return premium is required for the given company. The β parameter

2545 can be derived for a company’s publicly traded debt or equity, and, using both, one can

2546 derive the return require for an asset of the company:

E D β = β + β (8.2) asset equity E + D debt E + D

2547 where E is the market value of the company’s equity and D is the market value of the

2548 company’s debt, typically assumed to be roughly equal to the book value. Additionally, βdebt

2549 typically ranges from 0.1-0.3 [236] and, for this analysis, is assumed to fall within the range

2550 of uncertainty and is simply set to zero.

2551 Here the equity returns for several publicly traded trucking companies [237] were pulled

2552 along with market capitalization of equity and book value of debt values using the Factset

2553 tool [238]. The monthly return data was then plotted vs the S&P 500 and fit using linear

2554 regression. The slope of the fit (Fig. 8-1) is taken to be βequity which is then converted to

2555 βasset using Eq. 8.2. The βasset values are then averaged and used to compute the discount

2556 rate using Eq. 8.1. This analysis yields a βasset of 0.76 and ultimately a discount rate 9.2%.

160 Universal Logistics Holdings (ULH) 40 30 20 10 0 -20 -15 -10 -5 0 5 10 15 -10 -20 -30 y = 1.2389x + 0.0652 R² = 0.1863 -40

MonthlyMarket Retrun, S&P (%) 500 -50 Monthly ULH Return (%)

Figure 8-1: Example scatter plot used for determining βequity according to the CAPM model using monthly returns of the S&P 500 to represent the overall market along with the monthly returns of the company’s stock.

2557 The βasset value less than one suggests that investing in trucking is, on average, less risky

2558 than the overall market, which seems to make logical sense since one would expect a stable

2559 need for the transportation of goods throughout the US.

2560 With the discount rate established, the next key parameter needed for this analysis is

2561 the expected price differential between CNG and diesel fuel. This price difference will be

2562 assumed to be the sole source of savings for the NG truck over the diesel powered truck. In

2563 order to derive an estimate for the price premium, the historical prices for the past 20 years

2564 [239] are plotted and extrapolated using linear fits. Note that particularly the diesel price

2565 is prone to heavy volatility. The fits, Fig. 8-2 predict roughly an 80-90 cent price premium

2566 for the future 8 years, the assumed lifetime of the truck [240]. Yet, the price premium as of

2567 January 1, 2019 is only observed to be 40 cents.

2568 With the fuel price premium established, a number of other parameters, summarized in

2569 table 8.1, are necessary to assess the net present value (NPV) of investing in NG trucks.

2570 To calculate the NPV, the total fuel savings for each year based on the forecast fuel price

161 $4.00

y = 0.06998x - 138.45545 $3.50 R² = 0.28610

$3.00

$2.50

$2.00 y = 0.05080x - 100.50492 R² = 0.71546 $1.50

$1.00 Cost Cost ($/Diesel Gallon Equivalent)

$0.50 Diesel CNG $0.00 2000 2005 2010 2015 2020 2025 2030 Year

Figure 8-2: Historical market prices for diesel fuel and compressed natural gas (CNG) [239]. Extrapolations used to estimate the future price differential necessary for the financial model.

162 Table 8.1: Summary of key parameters used for the calculation of the net present value (NPV) of investing in NG trucking. Note that ’MT’ stands for ’metric tonnes’ and ’DGE’ stand for ’diesel gallon equivalent.’ Parameter Value Units Reference NG engine price premium 10,000 $ [228] NG fuel system price premium 35,000 $ [228] Diesel fuel efficiency (Trucking) 4 MPG [226] NG engine relative efficiency 85 % [226] Weekly travel distance 2,500 miles [241] Operating weeks per year 50 weeks [241] Truck lifetime 8 (1,000,000) yrs (miles) [240, 241] Social cost of carbon 140 $/MT [231] −3 CO2 density of NG 7.29 ∗ 10 MT/DGE [242] −3 CO2 density of Diesel 10.05 ∗ 10 MT/DGE [242]

2571 differential, total miles traveled that year, and fuel efficiency are calculated. Then, the fuel

2572 savings for each year are discounted to present value using the formula:

N X CFn PV = n (8.3) n=1 (1 + r)

2573 where CFN is the cash flow from year N (here the fuel savings for the given year), and r is

2574 the discount rate determined using the β analysis described prior. Subtracting the current

2575 engine and fuel system price premiums associated with purchasing a NG truck, table 8.1,

2576 yields the expected NPV of buying the NG truck as opposed to purchasing a standard diesel

2577 version.

2578 Acknowledging the variability of the diesel price and thus the fuel price premium for

2579 running the truck on diesel, Fig. 8.3 is constructed the to assess average fuel price premium

2580 necessary to yield a benefit of buying a NG truck. Under the current fuel price premium of

2581 40 cents, buying the NG truck equates to a roughly $50K loss. This value is greater than

2582 the $45K engine and fuel system price premiums due to the lower fuel efficiency of the NG

2583 engine causing the CNG to actually cost more than diesel per mile. As a result, an average

2584 fuel price premium of 66 cents/DGE is required in order to break-even on the investment.

2585 Using the linear forecasts for fuel prices yields a more optimistic case for the NG truck gives

2586 an NPV of $34K per truck for an fuel price premium growing from 79 cents per DGE to 92

163 $200,000

$150,000

$100,000

$50,000 Current Prices (1.1.2019) Prices Current $0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ($50,000)

($100,000) Net Present Net Present Value truck per (NPV)

($150,000) Average Price Differential ($/DGE)

Figure 8-3: Calculation showing the break-even average price differential between CNG and Diesel required to turn a profit on the natural gas truck given current engine and storage prices premiums along with lesser efficiency.

2587 cents per DGE. Given the variability in diesel prices and current low price, it would be a

2588 high-risk gamble to invest in NG trucking based on this base-case analysis.

2589 Though the base case for NG trucking is not particularly financially favorable presently,

2590 there are several pending changes that could make a more valuable proposition.The changes

2591 examined here are NG engine efficiency increases, improving NG on-board storage tech-

2592 nology, and implementation of a carbon tax, Fig. 8-4. By simply improving the engine

2593 technology to be of equal fuel efficiency to that of diesel engines, the NPV is observed to

2594 triple from the base case to $100K. Improving the NG fuel storage technology, such that

2595 price premium is cut in half, has a much weaker impact, only improving the NPV to $50K.

2596 It is assumed that, if a Carbon Tax is implemented, it will be equal to the social cost of

2597 carbon (SCC) [231]. This effectively contributes an additional 40 cents per DGE to the fuel

2598 price premium since NG has more energy per CO2 molecule produced, due to the higher

2599 hydrogen to carbon ratio in NG compared to longer chain diesel fuel. The implementation

2600 of a carbon tax produces an intermediate effect, slightly more than doubling the NPV from

164 $250,000

$200,000

$150,000

$100,000

$50,000

$0 Base Savings Equal Efficiency Improved Carbon Tax Tax + Eq. Eff. Storage

Cost Savings Net New Diesel Truck

Figure 8-4: Net present value scenario analysis examining several potential improvements to the value proposition of investing in a natural gas powered trucking fleet.

2601 the base case to $69K. However, the most promising case for NG trucking is made when the

2602 carbon tax is implemented along with improvements in fuel efficiency. The combined effect

2603 is greater than the sum of the individual impacts of each change and is expected to result

2604 in an NPV of $170K per truck. This would be enough to entirely pay for the truck itself

2605 ($160K / NG truck) [228].

2606 8.4 Strategic analysis

2607 Beyond assessing the basic financial viability of investing in NG trucking, it is important

2608 to assess the overall market competitive forces for the trucking industry. This analysis will

2609 suggest whether or not any fuel savings generated would be defensible and maintained by the

2610 investor in the NG trucks. The classical framework for assessing these competitive market

2611 dynamics is ’Porter’s Five Forces’ [232]. The competitive forces according to Porter are

2612 the threat of new entrants, bargaining power of the buyers, threat of substitute products

2613 or services, bargaining power of the suppliers, and rivalry among existing competitors, Fig.

2614 8-5.

165 The Five Competitive Forces That Shape Strategy

are most important varies by industry.1 In mi- tures, such as up-front advertising or research croprocessors, incumbents such as Intel are and development. While major corporations protected by scale economies in research, chip have the financial resources to invade almost fabrication, and consumer marketing. For lawn any industry, the huge capital requirements in care companies like Scotts Miracle-Gro, the certain fields limit the pool of likely entrants. most important scale economies are found in Conversely, in such fields as tax preparation the supply chain and media advertising. In services or short-haul trucking, capital require- small-package delivery, economies of scale ments are minimal and potential entrants arise in national logistical systems and infor- plentiful. mation technology. It is important not to overstate the degree to 2. Demand-side benefits of scale. These bene- which capital requirements alone deter entry. fits, also known as network effects, arise in in- If industry returns are attractive and are ex- dustries where a buyer’s willingness to pay for pected to remain so, and if capital markets are a company’s product increases with the num- efficient, investors will provide entrants with ber of other buyers who also patronize the the funds they need. For aspiring air carriers, company. Buyers may trust larger companies for instance, financing is available to purchase more for a crucial product: Recall the old expensive aircraft because of their high resale adage that no one ever got fired for buying value, one reason why there have been numer- from IBM (when it was the dominant com- ous new airlines in almost every region. puter maker). Buyers may also value being in a 5. Incumbency advantages independent of “network” with a larger number of fellow cus- size. No matter what their size, incumbents tomers. For instance, online auction partici- may have cost or quality advantages not avail- pants are attracted to eBay because it offers able to potential rivals. These advantages can the most potential trading partners. Demand- stem from such sources as proprietary technol- side benefits of scale discourage entry by limit- ogy, preferential access to the best raw mate- ing the willingness of customers to buy from a rial sources, preemption of the most favorable newcomer and by reducing the price the new- geographic locations, established brand identi- comer can command until it builds up a large ties, or cumulative experience that has allowed base of customers. incumbents to learn how to produce more effi- 3. Customer switching costs. Switching costs ciently. Entrants try to bypass such advantages. are fixed costs that buyers face when they Upstart discounters such as Target and Wal- change suppliers. Such costs may arise because a buyer who switches vendors must, for exam- ple, alter product specifications, retrain em- ployees to use a new product, or modify pro- The Five Forces That Shape Industry Competition cesses or information systems. The larger the switching costs, the harder it will be for an en- trant to gain customers. Enterprise resource Threat planning (ERP) software is an example of a of New product with very high switching costs. Once a Entrants company has installed SAP’s ERP system, for example, the costs of moving to a new vendor are astronomical because of embedded data, the fact that internal processes have been Rivalry adapted to SAP, major retraining needs, and Bargaining Among Bargaining the mission-critical nature of the applications. Power of Existing Power of 4. Capital requirements. The need to invest Suppliers Competitors Buyers large financial resources in order to compete can deter new entrants. Capital may be neces- sary not only for fixed facilities but also to ex- tend customer credit, build inventories, and fund start-up losses. The barrier is particularly Threat of great if the capital is required for unrecover- Substitute Products or able and therefore harder-to-finance expendi- Services harvard business review • january 2008 page 5 Figure 8-5: The five forces that govern marketplace competition according to Michael Porter. This document is authorized for use by Alan Long, from 8/13/2018 to 12/14/2018, in the course: Reproduced fromFall 2018 [232] 15.900 Competitive Strategy, Massachusetts Institute of Technology. Any unauthorized use or reproduction of this document is strictly prohibited*.

2615 8.4.1 Threat of new entrants

2616 Looking through the lens of the threat of new entrants, the US trucking industry appears

2617 to be moderately defensible. Here the barriers to entry are principally the capital cost of

2618 purchasing the initial trucking fleet along with the need to establish customer relationships.

2619 From a cost perspective, each truck costs between 120 and 160K depending on whether or

2620 not it runs on diesel or NG [228], and it is difficult to believe that a fleet of anything less

2621 than 20 trucks would be able to serve even a single small to moderately sized customer.

2622 Therefore, startup costs come in at a minimum of about $2.5M.

2623 Second, from the relationships perspective, the service is relatively uniform in the eyes of

2624 the customers, transport from A to B is transport from A to B. Then, for diesel tucks at least,

2625 the costs are relatively consistent throughout the market meaning that competing on price

2626 is not particularly an option. Therefore, trust becomes the primary currency with customers

2627 for this industry, trust that the products will be delivered on time and un-damaged. Given

2628 that trust is typically based on a history of service, any established customer relationships

2629 can be expected to be relatively defensible.

2630 On the other hand, working in favor of new entrants is the relative lack of returns to scale

2631 for the industry. While scale may give certain benefits in the size of customers for which

166 Price (Cost)

Consumer Surplus

Market Supply Price

Demand

Quantity Total Quantity Sold 4

Figure 8-6: Economic surplus model for a commodity good. All suppliers have the same cost of production and therefore the entire economic surplus goes to the buyers/consumers.

2632 the company can and is motivated to compete for, in general, the capital cost of trucks and

2633 variable costs associated with the actual transportation will not vary any significant amount

2634 as an entrant increases in size. Therefore, the new entrant faces relatively the same cost

2635 structure as the existing competitors.

2636 8.4.2 Bargaining power of buyers

2637 Continuing on to the bargaining power of the customers, the industry appears to be fairly

2638 weak. The trucking service is largely non-differentiated and there are at least 10 significant

2639 publicly traded tucking companies in the US [237]. Therefore, there is a sufficient number of

2640 competitors for customers to choose from, and no particular reason for a customer to select

2641 one over the other. This gives the buyers significant power and makes trucking effectively a

2642 commodity, Fig. 8-6

2643 8.4.3 Threat of substitute products or services

2644 Three primary substitutes exist for domestic trucking. The first is aviation, the second

2645 is cargo shipping, and the third is locomotive. Aviation, while significantly faster than

2646 domestic trucking, suffers from much higher costs, giving trucking a price advantage for

167 2647 deliveries where time pressure is not excessive. Additionally, airports are not as ubiquitous

2648 as roads and therefore, there is a limit to how close aviation can get to delivering the product

2649 to the desired customer. On the other hand, cargo shipping tends to have a cost advantage

2650 over trucking but suffers from long and indirect routes as well as an inability to reach the

2651 final destination, similar to transport by plane. Third is transport by locomotive which has

2652 delivery timing similar to trucking, but is again disadvantaged on delivery to final location.

2653 Therefore, the threat of existing substitutes to conventional trucking is minimal and the

2654 industry is strong in this regard.

2655 While current substitutes are not a significant concern for trucking, two pending ad-

2656 vancements, high speed trains (and hyperloop) along with aviation drone transport, could

2657 present an issue. High speed trains within the US would be able to transport closer to the

2658 speed of aviation, without incurring the excessive transportation costs, which would position

2659 them well to eat into much of the domestic trucking market. Similarly, aviation drones may

2660 be able to compete with trucking for deliveries to the final destination, which none of the

2661 current substitutes can accomplish.

2662 8.4.4 Bargaining power of suppliers

2663 The standard diesel engine and semi truck are old technology with generally expired patents,

2664 though there have been several incremental improvements to fuel efficiency. Given the old

2665 technology and presence of a number of competitors in the market (e.g. Cummins, Mack,

2666 Volvo, Paccar, etc.) the suppliers are not expected to carry much power in the diesel truck

2667 market. Conversely, for the natural gas tuck engines, the technology is much newer and

2668 Cummins is the dominant producer for trucking applications. This means the bargaining

2669 power of the suppliers is currently not a major issue for the overall primarily diesel based

2670 industry, but would be for any NG players.

2671 8.4.5 Rivalry among existing competitors

2672 Rivalry among existing competitors appears to be only moderate based on the non-trivial

2673 variability in profit margins, table 8.2. The factor working against the competitor rivalries is

168 Table 8.2: Asset beta and profitability information for leading publicly traded US trucking companies. Profitability information given as the 5yr average 2014-2018.

Competitor βasset Net Profit Margin (%) LSTR 0.64 4.8 ODFL 0.61 11.7 SAIA 1.15 5.1 JBHT 0.96 7.0 ULH 0.70 3.1 MRTN 0.64 6.9 KNX 0.80 11.1 WERN 0.77 6.2 PTSI 0.61 4.9 Average: 0.76 6.8 Standard Deviation: 0.18 2.9

Table 8.3: Summary Porter’s Five Forces analysis of the US trucking industry. corresponds to strongly favorable and corresponds to very unfavorable. Force Rating Threat of new entrants Bargaining power of buyers Threat of substitute products or services Bargaining power of suppliers Rivalry among existing competitors Overall:

2674 the necessity of building trust and relationships in the industry to bolster customer confidence

2675 in timely and undamaged deliveries. This has allowed some competitors to deliver profit

2676 margins as high as 11% while other only hit 3%.

2677 8.5 Game Theory: new entrant vs current market

2678 Though the financial projections appear favorable in several scenarios, no investment decision

2679 can be made without consideration of the potential actions to be taken other competitors

2680 in the industry. This consideration is typically accomplished through game theory analysis.

2681 Game theory aims to simplify complex social dynamics and arrive at a manageable deci-

2682 sion model. Analyzing the approximated model can then hopefully provide some possible

169 2683 explanation for an observed behavior. The most basic game theory analysis is that of the

2684 simultaneous game, i.e. Fig 8-7.

2685 In simultaneous game analysis two ’players’ are typically restricted to two primary actions

2686 under consideration. Then, the approximate payoffs for each player under each scenario are

2687 estimated and input to the simultaneous play matrix. With playoffs estimated, it is then

2688 possible to specify the optimal play for a given player given each selected action by the

2689 opposing player. A box in the option matrix where these optimal actions overlap is specified

2690 as a Nash equilibrium [243, 244] and is an expected outcome of the simultaneous game. The

2691 classical example of a simultaneous game and Nash equilibrium is the ’Prisoner’s Dilemma’

2692 in which the globally non-optimal outcome results from payoff structure and inability to

2693 coordinate decisions with the other player [245].

2694 Fig. 8-7 shows a simultaneous game model diagram for a potential new NG entrant to

2695 the trucking market and an existing trucking company. The new entrant faces the choice

2696 to enter the market with a fleet of new NG trucks and the existing trucking company faces

2697 the choice to update their current fleet to NG trucks or not. In reality, depending on the

2698 size of the trucking fleet expected to be purchased by the new entrant, the incumbent may

2699 more gradually transition its fleet, waiting to retire old trucks. The resulting temporary

2700 imbalance between the number of NG trucks of the new entrant and incumbent could give

2701 an advantage to the new entrant. However, for the analysis here, the simplifying assumption

2702 is made that the incumbent new purchase rate of trucks would be roughly equal to the number

2703 of trucks the new entrant could afford to purchase. Therefore, this analysis continues on a

2704 per truck financial basis, assuming 1:1 competition. Under the scenario in which both invest

2705 in NG trucking, it is assumed that resulting price competition on the largely undifferentiated

2706 product of transportation will consume any marginal cost benefits obtained by switching to

2707 NG, and thus both companies are expected to lose their sunk cost investment, ’c’. Note

2708 that this model omits the larger market which may provide the opportunity to avoid price

2709 competition by either entity simply shifting to compete with another market player. Yet,

2710 eventually other routes/competitors would be exhausted and price competition would ensue.

2711 Further, given that trucking is a largely un-consolidated industry due to the limited returns

2712 to scale the opportunity for regional players to shift to new markets is expected to be limited.

170 Existing Trucking Company A. x Invest & Compete Do Nothing i. ii. (c) (R) (c) B+R Enter & Compete iii. iv. Nash Eq. B 0 Potential NG Trucking Entrant NG Trucking Potential Do Nothing 0 0

Figure 8-7: Simultaneous game for investment in NG trucking by an new entrant and an existing competitor. Value ”c” represents the marginal additional investment cost of the NG trucks. Value ”B” is the expected profit without competition calculated according to 8-8. Value ”R” is the expected revenue to be captured by the competitor, here assumed to be larger than ”c” given the cost advantage.

2713 In the event that the new player enters with NG and the existing player does not, the new

2714 player’s price advantage will allow them to price slightly lower, capturing revenues and

2715 profits from the existing competitor. In Fig. 8-7, ’R’ the amount of lost profits is assumed

2716 to be larger than the cost of investment. The expected marginal return for investing in

2717 NG at existing market revenue rates is ’B,’ and is summarized in Fig. 8-8. In the event

2718 that the potential new competitor knows the existing company will invest in NG trucking

2719 as well, their best response is to not enter. However, if they know the existing company will

2720 not invest, then their best response is to enter with NG trucking. On the other hand, the

2721 existing tucking company has a consistent best option of investing and competing with any

2722 new entrants. This is a dominant strategy given that it is independent of the other player’s

2723 action. Therefore, the best choice options (Nash strategies) overlap in cell ”iii.” giving the

2724 Nash equilibrium of the existing trucking company investing while the potential new entrant

2725 chooses not to enter the market. Calling back to the Porter’s Five Forces analysis, the result

2726 of the simultaneous game further supports the inference about relative protection of the

2727 market from new entrants.

2728 The simultaneous game analysis relies on a calculation of the incremental expected value

171 NPV/truck Yes $69K High Carbon (50%) Tax Expected (50%) No $34K Return Natural price (50%) differential $6K Yes ($22K) Low Carbon (50%) Tax (50%) No ($58K) (50%)

Figure 8-8: Chance tree summarizing the expected marginal return for investment in NG trucks today. The high price difference is represented by the forecast predictions shown in Fig. 8-2 while the low price assumes the the current natural price differential of 40 cents holds constant into the future. For the carbon tax, the same assumption is made as in Fig. 8-4.

2729 of investing in NG trucking in the absence of competitive pressures. This analysis, Fig.

2730 8-8, is derived from the basic financial calculations completed in section 8.3, but relies on

2731 several chance nodes to arrive at the expected value. The first node reflects the potential

2732 for the price differential between NG and diesel fuel. The ’low’ scenario assumes that the

2733 current dip in price differential to only a 40 cent premium for diesel is maintained for the

2734 foreseeable future. The ’high’ scenario assumes the more optimistic projections shown in

2735 Fig. 8-2. The second chance node considered is the potential for implementation of a carbon

2736 tax. If implemented, the same value is assumed as that of Fig. 8-4. This analysis ignores

2737 the potential for future technological advancements since it is considering a present day

2738 investment. Commodity markets and politics are difficult to predict, so the probabilities of

2739 both chance nodes are trivially assumed to be 50%. At the end of this analysis the expected

2740 incremental value is less than convincing at only $6K NPV per converted truck, though it

2741 is still an expected net benefit.

2742 Given that the simultaneous game between the new NG trucking entrant and the existing

2743 competitor indicates that the existing competitor will invest in NG trucks, why then has there

2744 been minimal conversion to NG fleets? One obvious reason is the lack of an overwhelming

2745 economic driver as shown in Fig. 8-8. For example, if one believes the probability of the price

2746 differential staying at its current low value is more probable (or could be exacerbated, e.g.

172 2747 COVID-19), then the expected return may become negative. Also, beyond the uncertainty

2748 around the expected payoff in absence of competition, the analysis omits concerns regarding

2749 the uncertainty surrounding availability of re-fueling stations on key routes and re-sale value

2750 along with potential need for new service relationships and other logistical concerns. These

2751 concerns compiled likely explain why current trucking companies have not begun to roll-over

2752 their fleets to NG trucks without consideration of using the technology to grow their fleet or

2753 compete better in the market.

2754 However, if one is willing to accept the financial drive is sufficient with the aforementioned

2755 hesitations in mind, another potential contributing explanation arises from consideration of

2756 the game played between existing competitors in the trucking market. Modifying the si-

2757 multaneous game (Fig. 8-7) to reflect this competition can be achieved by setting R, the

2758 stolen profits, to zero. Now, neither competitor has a dominant strategy and two Nash

2759 equilibrium arise, cells ”ii.” and ”iii.” Still, the the simultaneous game indicates that one of

2760 the competitors should be expected to upgrade and the other will not, but it does introduce

2761 the hesitation and indecision between which competitor will be first. To more clearly under-

2762 stand the lack of NG adoption though, the simple simultaneous game can be extended to a

2763 sequential game [245] between the existing direct market competitors that takes into account

2764 the statistical occurrences of price differential, carbon tax, and technological development,

2765 Fig. 8-9.

2766 To build and analyze the resulting decision tree, a number of simplifying assumptions are

2767 made. First it is assumed that, if the initial player chooses not to invest in NG, neither will

2768 their competitor and so neither profits. The recursive nature of simply shifting the initial

2769 decision to invest to the competitor is ignored here. Next, the same chance nodes from

2770 Fig. 8-8 are built in, incorporating the same probabilistic and payoff assumptions. Here

2771 it is assumed that the second competitor will wait to see the outcome of the chance nodes

2772 before making their decision. The game model could be expanded to include a decision node

2773 for the competitor’s investment before and after each chance node. However, this further

2774 complexity is not expected to impact the ultimate conclusion. In the event that the price

2775 differential is low, the initial competitor loses money regardless and therefore it is assumed

2776 that the other competitor will not invest in NG engines. Continuing along the high price

173 2777 differential scenario, additional probability nodes are added to incorporate the potential

2778 for cheap on-truck NG storage/supply systems and increasing the thermal efficiency of NG

2779 engines to be equivalent to diesel engines. The technological developments are assumed to

2780 take place after the initial competitor’s investment decision, and are again, given significant

2781 uncertainty, assumed to occur with an even probability of 50%. The payoffs for the various

2782 decisions that the secondary competitor can make are derived from the values and analysis

2783 in Fig. 8-4. It is assumed that, independent of the scenario, the initial competitor to invest

2784 in NG will experience a 1 year first mover advantage without any competition while the

2785 secondary competitor updates their fleet. In practice, fleet updates would take place over

2786 a number of years as the existing fleets are retired incrementally, but if it is assumed that

2787 the turnover rate is roughly the same for each competitor, then the model should tend

2788 to hold. Additionally, if both competitors have a marginal cost advantage, it is assumed

2789 price competition will ensue until one of them is at marginal cost. Here again the further

2790 assumption is made that the two regional competitors are trapped with one another and that

2791 routes are independent and discounts can be applied without uniformly lowering shipping

2792 prices. Under each scenario the secondary competitor chooses to maximize their outcome

2793 alone, denoted by the dashed boxes. With the secondary competitor’s choices established in

2794 each scenario, backward induction [245] allows for determination of the optimal strategy for

2795 the initial competitor when choosing whether or not to invest in a NG trucking fleet. Here

2796 it is observed that in the sequential game, the expected payoff for investing is a loss of $15K

2797 per truck. Hence, according to this sequential game theory model, the initial competitor is

2798 incentivised not to upgrade to NG trucking, yielding the limited adoption rates of NG in

2799 trucking observed.

2800 Building off the the game theoretic analysis discussed, one might wish to accelerate

2801 the transition to NG for environmental externalities which are not considered here other

2802 than through the potential implementation of the carbon tax. Within the bounds of the

2803 models, to accomplish this, the most controllable step is to implement the carbon tax which

2804 would eliminate the lower profitability branch, bringing the expected return for the initial

2805 competitor up to a loss of $0.25K, still insufficient to drive investment on its own. Yet, if

2806 the expected technological advances can be made prior to the first competitor’s decision as

174 Expected Return: [ ($15K), $28 ]

Yes [ ($26K), $135K ] Yes (50%) C2 Invest in NG Trucks? No Yes (50%) Tech. [ $69K, $0 ] Advance? Yes [ ($26K), ($44K) ] No (50%) C2 Invest in No High Carbon NG Trucks? [ $69K, $0 ] (50%) Tax? Yes Yes (50%) C2 Invest in [ ($32K), $85K ] NG Trucks? No No (50%) Tech. [ $33K, $0 ] Natural price Advance? Yes differential? No (50%) C2 Invest in [ ($32K), ($44K) ] NG Trucks? No Yes Yes [ $33K, $0 ] [ ($22K), $0 ] C1 Invest in Low Carbon (50%) NG Trucks? Tax? (50%) No [ ($58K), $0 ] No (50%)

[ $0, $0 ]

Figure 8-9: Sequential game representing the NG trucking investment decisions of existing competitors in the market. In all scenarios, the first moving competitor is assumed to have a first mover advantage of 1 year and able to generate profits that year without competition, regardless of whether or not the other competitor chooses to invest. It is also assumed that when both competitors have equivalent cost advantages, price competition will devolve the market to marginal cost pricing given the largely undifferentiated good. Backward induction suggests that current market players are not fiscally incentivised to invest in a NG trucking fleet.

2807 well, then the payoffs shift significantly in favor of the first mover for an expected return of

2808 more than $70K per truck. Therefore, it is concluded that the current optimal strategy in

2809 the NG trucking market is not to enter with a fleet of new NG trucks, but rather to focus

2810 investment on the necessary technological advancements to shift the market in favor of swift

2811 adoption while capitalizing on patent protected designs. Additionally, though beyond the

2812 scope of analysis here, the shift to NG would undoubtedly be aided by initiatives to establish

2813 and promise availability of re-fueling stations along with other initiatives such as subsidizing

2814 re-sale prices.

2815 8.6 Conclusions

2816 The analysis presented seeks to answer the question of whether or not transitioning to NG

2817 trucking fleets is economically favored. Basic financial analysis suggests that across a variety

175 2818 of scenarios (e.g. implementation of a carbon tax, advancement to equivalent NG engine

2819 efficiency to diesel, and decreased NG fuel storage system costs) investing in a NG truck is

2820 calculated to be a net financial improvement anywhere from $33-168K more profitable than

2821 a diesel truck. However, this financial analyses does not take into consideration competitive

2822 pressures within the overall trucking market. Therefore, the Porter’s Five Forces framework

2823 is leveraged to examine and assess the favorability of the market for investment. The net

2824 result of this analysis is that the market performs best from the standpoint of the threat

2825 of substitute products and worst from the standpoint of bargaining power of the buyers.

2826 Ultimately the market is assessed to be neither clearly favorable nor clearly unfavorable,

2827 suggesting entrance into the market may be a difficult value proposition, but there is limited

2828 need for existing players in the market to escape. Finally, the analysis leverages the lens of

2829 game theory to quantitatively assess a new NG entrant to the market as well as the invest-

2830 ment in upgrading to NG trucks by existing competitors. In a simultaneous game analysis,

2831 the strong incentive of existing competitors to defend their revenue streams discourages any

2832 new entrant. On the other hand, the sequential game between the existing competitors

2833 yields an expected loss for whichever competitor invests first, contributing to the present

2834 market scenario of limited transition to NG trucking as would otherwise be environmentally

2835 favorable. Also playing a contributing role is the large degree of uncertainty surrounding a

2836 number of aspects of the investment including the diesel-NG future price differential, tech-

2837 nological advancements, public policy changes, and various other logistical questions such

2838 as the availability of re-fueling stations. Ultimately, the recommended investment strategy

2839 derived from this analysis is to (1) invest in R&D to develop and patent protect a NG engine

2840 of equal efficiency to diesel engines, and (2) lobby for a carbon tax to increase the benefit of

2841 switching to NG and force the entire industry to purchase the newly developed engine.

2842 8.7 Acknowledgments

2843 AEL additionally gratefully acknowledges his professors and colleagues at the MIT Sloan

2844 School of Management. Particular thanks regarding the materials of this analysis is given

2845 to Professors Don Sull, Gonzalo Cisternas, and Haoxiang Zhu. AEL further thanks his

176 2846 ’core team’ members Jeff, Nick, Fernanda, Kat, and Zoe for their continued support and

2847 collaboration from the start of MBA core curriculum on through graduation.

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