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At least 73 records · Page 4

Effects of unburnt reaction progress on stretch flame dynamics under elevated temperatures

Flames in practical combustors and engines are inevitably subject to the effects of upstream chemical reaction progress and stretch, induced by elevated thermodynamic conditions and flow non-uniformities, respectively. Recent shock tube experiments and simulation studies on flame propagation have shown that flame speed under engine-relevant conditions can be enhanced with non-negligible upstream chemical reaction progress, especially when low-temperature heat release is involved in the unburnt mixture. On the other hand, depending on the mixture equivalence ratio and diluents, nonequidiffusion (including the non-unity Lewis number effect and the preferential diffusion effect) can couple with the flame stretch to fundamentally affect the flame propagation, which can manifest as either facilitation or suppression. Depending on the transport property of a reacting mixture, there hence can be either inhibition or promotion from unburnt reaction progress and stretch effects on flame propagation. In the current work, through one-dimensional numerical simulations of transient planar and spherical flames of n-heptane/air under elevated thermodynamic conditions, the combined effects of upstream chemical reaction progress and stretch on flame propagation are investigated. Results show that for both lean and rich n-heptane/air mixtures, flame speed can be substantially promoted with reaction progress, while the rich mixtures can exhibit opposite stretch dependence after first-stage ignition. Different definitions of effective Lewis number are adopted to explain the change in Markstein length for spherical flame in lean and rich mixtures reformed by the low temperature chemistry. Further, this work fills an important gap in laminar premixed flame research relevant to practical combustion systems and can provide useful insight into local turbulent flame behaviors and phi-sensitivity in engine combustion.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Simulations of Multi-Mode Combustion Regimes Realizable In a Gasoline Direct Injection Engine

Lean and dilute gasoline compression ignition (GCI) operation in spark ignition (SI) engines are an attractive strategy to attain high fuel efficiency and low NOx levels. However, this combustion mode is often limited to low-load engine conditions due to the challenges associated with autoignition controllability. In order to overcome this constrain, multi-mode (MM) operating strategies, consisting of advanced compression ignition (ACI) at low load and conventional SI at high load, have been proposed. In this three-dimensional computational fluid dynamics study, the concept of multi-mode combustion using two RON98 gasoline fuel blends (Co-Optima Alkylate and E30) in a gasoline direct injection (GDI) engine were explored. To this end, a new reduced mechanism for simulating the kinetics of E30 fuel blend is introduced in this study. To cover the varying engine load demands for multi-mode engines, primary combustion dynamics observed in ACI and SI combustion modes was characterized and validated against experimental measurements. In order to implement part-load conditions, a strategy of mode transition between SI and ACI combustion (i.e., mixed-mode combustion) was then explored numerically by creating a virtual test condition. The results obtained from the mixed-mode simulations highlight an important feature that deflagrative flame propagation regime coexists with ignition-assisted end-gas autoignition. This study also identifies a role of turbulent flow property adjacent to premixed flame front in characterizing the mixed-mode combustion. The employed hybrid combustion model was verified to perform simulations aiming at suitable range of multi-mode engine operations.

Kim, Sayop↗

Towards stabilization of pressurized oxy-coal diffusion flames

Designing an effective burner is vital for the development of pressurized oxy-fuel combustion technologies. Turbulent jet diffusion burners are adopted for the pressurized oxy-combustor in this research and a bluff-body is employed to create a recirculation zone to stabilize the flame in pressurized oxy-combustor. The objective of this numerical work is to perform a systematic analysis of the characteristics of this pressurized non-premixed flame. Specifically, a 15-bar POFC combustor of power 100 kWth is modeled employing the Ansys FLUENT commercial platform, using Reynolds-averaged Navier-Stokes (RANS) modeling. The present computational work focuses on identifying the aerothermodynamic features of the isothermal and reacting flow with a disk-shape bluff-body. Results show that fuel-oxidizer stream momentum ratio has great impacts on the temperature profile of the down-fired, co-axial, pressurized oxy-coal, diffusion flame. Parametric study of the width of the burner tip identifies an optimal range for this pressurized burner.

Li, Lei↗

Impact of a Bluff Body on a Pressurized Oxy-fuel Flame

Designing an effective burner is vital for the development of pressurized oxy-fuel combustion technologies. In the present work, turbulent jet diffusion burners are adopted for a pressurized oxy-combustor, with a bluff-body employed to create a recirculation zone, thereby stabilizing the flame in such a combustor. The objective of this numerical study is to perform a systematic analysis of the characteristics of such a pressurized non-premixed flame. Specifically, a 15-bar pressurized oxy-fuel combustor of power 100 kWth is modeled by means of the Ansys FLUENT commercial platform, using the Reynolds-averaged Navier-Stokes (RANS) approach. The present work focuses on identifying the aerothermodynamic features of the pressurized oxy-fuel burner with a disk-shape bluff-body. It is shown that the fuel-to-oxidizer stream momentum ratio has a great impact on the temperature profile of the down-fired, co-axial, pressurized oxy-coal, diffusion flame. A parametric study of the blockage ratio of the burner identifies an optimal range for this pressurized burner.

Li, Lei↗

Machine-Learned Manifold-Based Models for Large Eddy Simulation of Turbulent Combustion

Reduced-order manifold combustion models are commonly used to lower the cost of reacting Large Eddy Simulations (LES) and may be derived either from physical principles as in flamelet models or from data-driven methods like Principal Component Analysis (PCA). In either case, neural networks are increasingly used as part of these models to provide a nonlinear mapping between a small set of pre-defined variables that parameterize the manifold and outputs of interest, such as reaction rates. In this work, we propose a new manifold-based modeling approach that combines the definition of the manifold-parameterizing variables (linear combinations of species), the nonlinear mapping to the outputs, and closure of filtered quantities for LES into the structure of a single neural network. This allows the process used to train the neural network to simultaneously optimize both the functional form of the model and the identities of the inputs to the model. The new approach can flexibly incorporate thermochemical data from any combustion system; if trained on data from 1D flames it can be interpreted as an optimized flamelet model, but it can also be used to learn models from data from more complex configurations. This work presents a priori evaluations of the new approach in both contexts. Evaluation using data from 1D premixed flames demonstrates the physical interpretability of the manifold variables generated by the new approach. Evaluation using data from direct numerical simulations of turbulent flames shows improved predictions relative to either flamelet or PCA-based models in a more complex configuration.

47 OTHER INSTRUMENTATION↗

Phenalenyl growth reactions and implications for prenucleation chemistry of aromatics in flames

The energetics and kinetics of phenalene and phenalenyl growth reactions were studied theoretically. Rate constants of phenalene and phenalenyl H-abstraction and C 2 H 2 addition to the formed radicals were evaluated through quantum-chemical and rate-theory calculations. The obtained values, assigned to all π radicals, were tested in deterministic and kinetic Monte Carlo simulations of aromatics growth under conditions of laminar premixed flames. Kekulé and non-Kekulé structures of the polycyclic aromatic hydrocarbons (PAHs) evolving in the stochastic simulations were identified by on-the-fly constrained optimization. The numerical results demonstrated an increased PAH growth and qualitatively reproduced experimental observations of Homann and co-workers of non-decaying PAH concentrations with nearly equal abundances of even and odd carbon-atom PAHs. The analysis revealed that the PAH growth proceeds via alternating and sterically diverse acetylene and methyl HACA additions. Furthermore, the rapid and diverse spreading in the PAH population supports a nucleation model as PAH dimerization, assisted by the non-equilibrium phenomena, forming planar aromatics first and then transitioning to the PAH–PAH stacking with size.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DNS of Hydrodynamic Instabilities of Laminar H2/O2/N2 Flames at Elevated Pressure

The atypical combustion properties of hydrogen have long provided an interest for hydrogen-enriched combustion, emphasized in recent years by increasingly stringent emission regulations. Significant work has been focused on the small hydrocarbon(C1-C3)/hydrogen blends, but much less is known about the effect of hydrogen on heavier hydrocarbon combustion, of interest for the aviation and automotive industry. In this work, we perform direct numerical simulations (DNS) of spherically expanding laminar n-dodecane/H2 flames in a constant-volume vessel at elevated pressure and a range of hydrogen seeding levels. A low Mach number model is used to represent the gas and flame dynamics allowing for a temporally varying, spatially homogeneous pressure field, while also enabling numerical evolution of the system numerically at time step sizes governed by the advective CFL limit rather than acoustic processes. We use an adaptive mesh refinement (AMR) approach to tackle the large separation of scales associated with high pressure premixed flame surfaces propagating in a laboratory-scale closed domain. We incorporate detailed chemistry and transport models for the reacting flow and focus the analysis on the effects of the pressure rise on the flame propagation characteristics and morphology as the hydrogen content is increased. The presence of fast diffusive H2 and the pressure rise results in dramatic thinning of the flame and the potential onset of hydrodyanmic and theromdiffusive instabilities. We assess the potential role of these instabilities on the ability to accurately measure laminar burning speed experimentally based on the mean flame surface propagation speed.

high pressure↗

Effects of non-thermal termolecular reactions on detonation development in hydrogen (H 2 )/methane (CH 4 ) - air mixtures

The binary fuel blend of H 2 /CH 4 is one of the most promising hydrogen-enriched hydrocarbon fuels in spark-ignition (SI) engines. Yet, the undesirable phenomenon of super-knock, which can severely and in-stantaneously damage an SI engine, limits its widespread adoption. Moreover, there is still a lack of con-sensus on the precise mechanism by which this phenomenon occurs i.e. via flame acceleration or spon-taneous ignition, despite numerous previous investigations. At the same time, recent studies [M. P. Burke, S. J. Klippenstein, Nat. Chem. 9 (2017) 1078 -1082, Y. Tao, A. W. Jasper, Y. Georgievskii, S. J. Klippenstein, R. Sivaramakrishnan, Proc. Combust. Inst. 38 (2021) 515-522] have demonstrated a high probability of occurrence of non-thermal reactions in premixed flames of such H 2 /CH 4 fuel blends with air due to the presence of non-trivial amounts of highly reactive radicals including H, O and OH apart from O 2 . The present study focuses on the evolution of an initial deflagration front to a detonation wave in H 2 /CH 4 - air mixtures under SI engine relevant conditions through fully resolved, constant volume 1D simulations with and without non-thermal reactivity. Non-thermal reactions were included in the macroscopic kinetics model as chemically termolecular reactions facilitated by the H + CH 3 and H + OH radical-radical recombination and the H + O 2 radical-molecule association reactions. Further, the nonthermal reactions result in a corresponding decrease in the reaction fluxes of the incipient recombination/association reactions. Therefore, an additional set of simulations were performed by applying corrections to the respective incipient recombination/association rate constants using the methodology demonstrated by Tao et al. [Y. Tao, A. W. Jasper, Y. Georgievskii, S. J. Klippenstein, R. Sivaramakrishnan, Proc. Combust. Inst. 38 (2021) 515-522]. Compared to the baseline case, the onset of spontaneous ignition in the end-gas region was observed to be delayed in the presence of non-thermal termolecular reactions. Concurrently, the developing detonation was observed to be significantly stronger. In contrast, applying corrections to the recombi-nation/association rate constants resulted in a completely different behavior. Specifically, detonation was observed to occur due to self acceleration of the primary flame in the absence of spontaneous ignition in the end-gas region. Sensitivity analysis was performed to quantify the effects of non-thermal reactions on the duration of heat release rate and thereby the mechanism of detonation formation. In addition, chemical explosive mode analysis (CEMA) was performed to identify the dominant species/reactions re-sponsible for the observed results.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

The role of cool-flame fluctuations in high-pressure spray flames, studied using high-speed optical diagnostics and Large-Eddy Simulations

Here this work investigates the low- and high-temperature ignition and combustion processes, applied to the Engine Combustion Network Spray A flame, combining advanced optical diagnostics and large-eddy simulations (LES). Simultaneous high-speed (50 kHz) formaldehyde (CH 2 0) planar laser-induced fluorescence (PLIF) and line-of-sight OH* chemiluminescence imaging were used to measure the low- and high-temperature flame, during ignition as well as during quasi-steady combustion. While tracking the cool flame at the laser sheet plane, the present experimental setup allows detection of distinct ignition spots and dynamic fluctuations of the lift-off length over time, which overcomes limitations for flame tracking when using schlieren imaging. After significant development to improve LES prediction of the low-and high-temperature flame position, both during the ignition processes and quasi-steady combustion, the simulations were analyzed to gain understanding of the mixture variance and how this variance affects formation/consumption of CH 2 0. Analysis of the high-temperature ignition period shows that a key improvement in the LES is the ability to predict heterogeneous ignition sites, not only in the head of the jet, but in shear layers at the jet edge close to the position where flame lift-off eventually stabilizes. The LES analysis also shows concentrated pockets of CH 2 0, in the center of jet and at 20 mm downstream of the injector (in regions where the equivalence ratio is greater than 6), that are of similar length scale and frequency as the experiment (approximately 5–6 kHz). The periodic oscillation of CH 2 0 match the frequency of pressure waves generated during auto-ignition and reflected within the constant-volume vessel throughout injection. The ability of LES to capture the periodic appearance and destruction of CH 2 0 is particularly important because these structures travel downstream and become rich premixed flames that affect soot production.

42 ENGINEERING↗

MACHINE LEARNING BASED CHEMICAL EXPLOSIVE MODE ANALYSIS (ML-CEMA)

The software consists of a Machine Learning based Chemical Explosive Mode Analysis (ML-CEMA) tool for advanced computational flame diagnostics. CEMA, originally based on eigen-analysis of the local thermochemical system, is capable of identifying reaction fronts and limit phenomena such as auto-ignition and extinction in practical combustion systems, such as internal combustion engines and gas turbine combustors. However, the original CEMA is computationally expensive for large reaction mechanisms that are typically needed to describe fuel chemistry of practical large-hydrocarbon fuels. This novel ML-CEMA tool employs a ML technique to accelerate the eigen-analysis of the basic CEMA approach by orders of magnitude, thus making it suitable for practical fuels. In ML-CEMA,zero-dimensional (0D) reactors and one-dimensional (1D) premixed flames are first used to generate a large number of data points for neural network based ML training. The trained ML model is then used to perform CEMA prediction. This ML-CEMA tool has been demonstrated in canonical 0D and 1D configurations as well as highly-transient three-dimensional spray flames exhibiting multi-mode turbulent combustion, showing promising results. ML-CEMA, as a standalone tool, can be used for computationally-efficient diagnostics of massive datasets generated from both experiments and simulations. For example, based on spatially resolved measurements of a small set of reactive scalars(such as temperature, hydroxyl radical and formaldehyde), ML-CEMA can effectively identify flame fronts and rare events. ML-CEMA also provides a robust online or offline flame feature detection tool. When used for on-the-fly simulations, ML-CEMA further enables zone-adaptive combustion modeling, in which the predicted eigenvalue is used as a robust mode indicator for judicious assignment of locally-valid combustion models. This ML-CEMA based zone-adaptive model can lead to substantial computational cost savings when used for large-scale simulations of multi-mode combustion systems. Third Party Code Web Page to Download Code Web Page Location of Third Party License

Xu, Chao↗

Oxidation of methylamine

A detailed chemical kinetic model for oxidation of methylamine has been developed, based on theoretical work and a critical evaluation of data from the literature. In this work, the rate coefficients for the reactions of CH 3 NH 2 + O 2 → CH 2 NH 2 / CH 3 NH + HO 2 , CH 3 NH 2 + H → CH 3 + NH 3 , CH 3 NH → CH 2 NH 2 , and CH 3 NH + O 2 → CH 2 NH + HO 2 were calculated from ab initio theory. The mechanism was validated against experimental results from batch reactors, flow reactors, shock tubes, and premixed flames. The model predicts satisfactorily explosion limits for CH 3 NH 2 and its oxidation in a flow reactor. However, oxidation in the presence of nitric oxide, which strongly promotes reaction at lower temperatures, is only described qualitatively. Furthermore, calculated flame speeds are higher than reported experimental values; the model does not capture the inhibiting effect of the NH 2 group in CH 3 NH 2 compared to CH 4 . More work is desirable to confirm the products of the CH 3 NH + NO reaction and to look into possible pathways to NH 3 in methylamine oxidation.

10 SYNTHETIC FUELS↗

Model assessment of synthetic jets for turbulent combustion experiments

Abstract Understanding turbulent premixed flames is essential to predict and optimize advanced combustion strategies, but critical capability gaps exist for collecting and validating measurements such as turbulent flame speed. Here, we evaluate synthetic jets as a new, promising turbulence generation device for constant-volume combustion chambers, quantitatively assessing turbulence intensity and spatial uniformity in a hypothetical 4,189-cm 3 vessel for various premixture conditions. Graphic Abstract

33 ADVANCED PROPULSION SYSTEMS↗

Machine learning assisted modeling of mixing timescale for LES/PDF of high-Karlovitz turbulent premixed combustion

Accurate modeling of mixing in the transported probability density function (PDF) method remains a great challenge, especially for turbulent premixed combustion under extreme conditions such as high Karlovitz number Ka. Recently, a power-law based mixing timescale model was developed for the large-eddy simulations (LES)/PDF modeling of high-Ka number turbulent premixed flames. It is found in this work that the power-law mixing timescale model is highly sensitive to the model parameters. It is thus critically needed to develop accurate calibration of these model parameters. The empirical specification of the model parameters developed in Zhang et. al. is found to be inadequate for accurate modeling of the mixing timescale. Here, machine learning is introduced as an attractive alternative in this work for the specification of the model parameters. A high-Ka number DNS jet flame is used as the training and validation of the machine learning models. The choices of the input parameters are discussed and compared for the machine learning models. The effect of differential molecular diffusion on mixing is examined by including the effect of the Lewis number in the training of the machine learning models. The performance of different machine learning algorithms is compared for the specification of the mixing model parameters. Overall, excellent performance of the machine learning models is observed for assisting the mixing modeling. The feasibility, interpretability, applicability, generality, and portability of using machine learning are discussed in general to provide a perspective on applying data-driven machine learning for turbulent combustion modeling studies.

42 ENGINEERING↗

Turbulence-induced bias in time-averaged laser absorption tomography of correlated concentration and temperature fields with a first-order correction

The influence of correlated scalar fluctuations on time-averaged laser absorption tomography measurements of temperature and species in a piloted turbulent premixed flame was examined using a coupled spectroscopic and fluid-dynamic analysis. To understand bias associated with turbulence, spatio-temporally resolved temperature and species mole fraction profiles predicted by large eddy simulations (LES) were used to synthetically generate time-resolved line-of-sight absorption measurements at short time scales (microsecond) to reflect the unsteady nature of a canonical jet burner across various transverse measurement planes. Inversion methods were employed on the time-averaged line-of-sight data to produce radially-resolved temperature and mole fraction profiles, analogous to those produced by laser absorption tomography performed on a time-averaged axisymmetric flowfield. It is shown that bias in the measurements compared to true time-averaged scalar fields is a function primarily of temperature dependence in absorptivity and non-zero correlation between temperature and species concentration scalars. Finally, a first-order correction to tomography measurements is proposed to account for the bias based on estimated correlations and the known spectroscopic parameters of the probed absorption transitions.

42 ENGINEERING↗

Co-optimized machine-learned manifold models for large eddy simulation of turbulent combustion

Many modeling approaches in large eddy simulation (LES) of turbulent combustion employ a projection of the thermochemical state onto a low-dimensional manifold within state space to reduce the number of transported variables and hence computational cost. Flamelet-generated manifolds (FGM) is an example of a well-established, physics-based approach, but increasingly, principal component analysis (PCA) is being used as a data-driven method for generating manifold models. For both approaches, the nonlinear relationship between the location on the predefined manifold and the outputs of interest, such as reaction rates, can be tabulated or encoded in a neural network. This work proposes a new approach for manifold modeling that extends these existing approaches. A modified neural network structure simultaneously encodes the definition of the manifold variables, the nonlinear mapping, and the subfilter closure for LES. This allows all three of these aspects of the model to be co-optimized, generating a model from any source of combustion thermochemical state data. The manifold parameterizing variables are constrained to be linear combinations of species, as in FGM and PCA-based models, to aid in interpretability and implementation. For LES, subfilter variances of the manifold variables are also included as inputs. Two types of a priori analysis are performed to evaluate the new approach. In the first, the model is trained on data from one-dimensional premixed flames. In this case, the approach recovers the behavior of flamelet-based manifold approaches, and in fact slightly improves performance by identifying an optimized progress variable. The approach is also applied to data from direct numerical simulations of spherical ignition kernels in isotropic turbulence. For any specified manifold dimensionality, the new approach provides substantially lower prediction errors than a PCA-based model developed from the same data set. Additionally, the LES formulation of the new approach can provide accurate predictions for filtered reaction rates across a variety of filter widths.

97 MATHEMATICS AND COMPUTING↗

a priori uncertainty quantification of reacting turbulence closure models using Bayesian neural networks

While many physics-based closure model forms have been posited for the sub-filter scale (SFS) in large eddy simulation (LES), vast amounts of data available from direct numerical simulations (DNS) create opportunities to leverage data-driven modeling techniques. Albeit flexible, data-driven models still depend on the dataset and the functional form of the model chosen. Increased adoption of such models requires reliable uncertainty estimates both in the data-informed and out-of-distribution regimes. Here, in this work, we employ Bayesian neural networks (BNNs) to capture both epistemic and aleatoric uncertainties in a reacting flow model. In particular, we model the filtered progress variable scalar dissipation rate which plays a key role in the dynamics of turbulent premixed flames. We demonstrate that BNN models can provide unique insights about the structure of uncertainty of the data-driven closure models. We also propose a method for the incorporation of out-of-distribution information in a BNN, which can be used for out-of-distribution query detection. The efficacy of the model is demonstrated by a priori evaluation on a dataset consisting of a variety of flame conditions and fuels.

97 MATHEMATICS AND COMPUTING↗

The Identity and Chemistry of C 7 H 7 Radicals Observed during Soot Formation

Here we used aerosol mass spectrometry coupled with tunable synchrotron photoionization to measure radical and closed-shell species associated with particle formation in premixed flames and during pyrolysis of butane, ethylene, and methane. We analyzed photoionization (PI) spectra for the C 7 H 7 radical to identify the isomers present during particle formation. For the combustion and pyrolysis of all three fuels, the PI spectra can be fit reasonably well with contributions from four radical isomers: benzyl, tropyl, vinylcyclopentadienyl, and o-tolyl. Although there are significant experimental uncertainties in the isomeric speciation of C7H 7 , the results clearly demonstrate that the isomeric composition of C 7 H 7 strongly depends on the combustion or pyrolysis conditions and the fuel or precursors. Fits to the PI spectra using reference curves for these isomers suggest that all of these isomers may contribute to m/z 91 in butane and methane flames, but only benzyl and vinylcyclopentadienyl contribute to the C 7 H 7 isomer signal in the ethylene flame. Only tropyl and benzyl appear to play a role during pyrolytic particle formation from ethylene, and only tropyl, vinylcyclopentadienyl, and o-tolyl appear to participate during particle formation from butane pyrolysis. There also seems to be a contribution from an isomer with an ionization energy below 7.5 eV for the flames but not for the pyrolysis conditions. Kinetic models with updated and new reactions and rate coefficients for the C 7 H 7 reaction network predict benzyl, tropyl, vinylcyclopentadienyl, and o-tolyl to be the primary C 7 H 7 isomers and predict negligible contributions from other C 7 H 7 isomers. These updated models provide better agreement with the measurements than the original versions of the models but, nonetheless, underpredict the relative concentrations of tropyl, vinylcyclopentadienyl, and o-tolyl in both flames and pyrolysis and overpredict benzyl in pyrolysis. Our results suggest that there are additional important formation pathways for the vinylcyclopentadienyl, tropyl, and o-tolyl radicals and/or loss pathways for the benzyl radical that are currently unaccounted for in the present models.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Cleavage of an aromatic ring and radical migration

The present study undertakes a theoretical evaluation of thermal decomposition of aromatic-ring radicals. Potential energy surfaces and associated reaction rate coefficients were calculated for 1- and 2-naphthalenyl, acetanaphthylenyl, and pyrenyl radicals. Kinetic Monte-Carlo simulations were performed to examine the rates of ring cleavage in two sooting laminar premixed flames of ethylene. The simulations showed that the thermal aromatic-ring cleavage is comparable in rate to oxyradical decomposition in a heavier-sooting flame. The simulation also revealed, unexpectedly, fast internal ring radical migration, comparable in frequency to reaction events of aromatic growth.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗