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The role of theory in combustion science
Abstract
Prometheus has been identified as the mythological deliverer of combustion science to mankind. Sciences traditionally have been divided into two parts—experiment and theory. A question therefore arises naturally: Was Prometheus more nearly an experimenter or a theoretician? My purpose is to attempt to review for you the history of the contribution of theory to the science of combustion. To do so I need to describe “theory” and “science” and to try to trace combustion from antiquity on into the future. A conclusion will be that today most sciences; and notably that of combustion, must be divided into three rather than the traditional two parts. But first, let's address that burning question raised above.
... Sometime later, thermodiffusive flame instabilities (preferential diffusion) were described by a distinguished limit [7,8]. As previously pointed out by Williams [9], such approaches do not need advanced mathematics, but, instead, an acute physical insight into the problem, as mastered by Zeldovich [2]. By itself, the label "asymptotic" is by no means a guarantee of a good theory. ...
... The purpose of this paper is to present some recent advances made in unsteady combustion waves, following the methodology outlined above. The theoretical analysis for instabilities of premixed laminar flames has been developed during the past two decades, and the results have been reviewed in various papers [9][10][11][12][13]. We focus our attention on the noteworthy experimental work of Searby and coworkers [14][15][16][17][18][19][20][21], carried out in parallel with these theories. ...
The purpose of this paper is to present some advances in the theory of unsteady combustion waves propagating in premixed gases. Many of these results have been obtained during the past two decades through theoretical analyses and carefully controlled experiments at the laboratory scale. Experimental studies on hydrodynamic instability of freely propagating flames, sound emission of turbulent flames, and acoustic instabilities of flames propagating in tubes are presented along with measurements of the Markstein number. Turbulence- (or noise) induced cellular flames are also considered. The last part of the paper is devoted to some recent theoretical results on detonation waves, including initiation, instabilities, and cellular structures.
... We conclude with statements by Forman Williams from his lecture "The Role of Theory in Combustion Science" for the International Symposium on Combustion [162], which is consistent with our understanding of the goals and methods of scientific inquiry. ...
The monograph presents, for the first time comprehensively, the results of the original theory of turbulent premixed flames based on the Kolmogorov hypothesis approach and the model of turbulent premixed and partially premixed combustion, the foundation of which is this theory. This one-dimensional theory, associated with the constant density approximation, describes the internal structure and global properties of flames propagating through a turbulent medium that is described by Kolmogorov's theory. The combustion regimes of laminar flamelet, thickened (microturbulent) flamelet, and distributed preheat zone are considered, as well as different propagation stages corresponding to different statistical states of the instantaneous flame. Generalization of the kinematic equations describing the flame motions to the case of three-dimensional turbulent flows of variable density has led to the balance equations of the TFC combustion model (implemented in the CFD codes Fluent and CFX), which are used in combination with two-parameter Kolmogorov-type turbulence models. The results of validation of the RANS-, LES-, and joint RANS/LES versions of the TFC combustion model against standard experimental data and their practical applications are discussed. It is concluded that the hypothesis-driven approach provides a new framework for understanding turbulent flames, offering potential improvements in combustion efficiency and safety for industrial applications.
... Flames are used in various ways to produce chemical reactions [21,22]. The bead test in analytical chemistry is one example. ...
A fundamental understanding of the stabilization mechanisms of a flame within very small spaces is of both fundamental and practical significance. This study relates to the scale effects on the combustion characteristics and flame stability of a methane-air mixture in continuous flow reactors. Computational fluid dynamics simulations are conducted to gain insights into reactor performance such as species concentrations, temperatures, and flames. The factors affecting combustion characteristics are determined for the continuous flow reactors. Particular focus is placed on determining essential factors that affect the performance of the continuous flow reactors. The results indicate that for large separation distances, the reaction rate is low and reaction spreads out over the burner due to slow heat transfer from the wall where ignition starts towards the centerline where un-combusted mixture exists. For smaller separation distances, reaction is more localized, with greater intensity, causing steeper temperature and composition gradients. When the separation distance is small, the transverse length scale of the fluid is small enough that the transverse heat transfer affects the centerline temperature, and thus, the maximum fluid temperature. In this case, the maximum fluid temperature is significantly lower than that for larger separations and incomplete conversion is observed. There appears to be an optimum plate separation that exhibits the shortest flame location. This distance permits the wall to preheat the inlet fluid fast enough, yet not to reduce significantly the centerline temperature. Burners with small plate-separation distances suffer from axial heat transfer through the walls of the burner, reducing the maximum fluid temperature and flame stability. Burners with large gaps slowly transport heat axially to preheat the feed, resulting in a late flame ignition and possible blowout.
... This is a flow configuration of interest for the high-Reynolds number flows typically encountered in burners, and has been widely used as a canonical problem to represent local flow conditions in strained mixing layers (Peters 1986(Peters , 2000. In particular, the flow structure of laminar counterflow burners has been recently studied and reviewed in theoretical (Weiss, Coenen, Sánchez 2017), numerical (Carpio et al. 2017), and experimental work (Egolfopoulos et al. 2014) and has been used as baseline in many premixed (Libby and Williams 1982, 1987 In this context, many theoretical investigations have been reported based on the application of activation energy asymptotics to an overall irreversible reaction with high activation energy (Klimov 1963, Williams 1992). In the case of premixed flames under strain, these analyses have been carried out for small, moderate and large values of the strain rate, including effects of variable density (Libby and Williams 1982), Lewis number of the deficient reactant (Libby, Liñán, Williams 1983) and nonadiabatic temperatures of the burned gases . ...
We present an asymptotic analysis of a strained premixed flame in the mixing layer between two counterflowing streams: one with fresh reactants at a temperature T u and other with the burned gases at temperature T b , which may be different from the adiabatic combustion temperature T e = qY Fu /c p + T u of the fresh gases. A one-step irreversible Arrhenius reaction model, of high activation energy, is used for the asymptotic analysis, together with the thermal-diffusive approximation of constant density and transport properties-easily generalized to variable density and transport properties with the use of a heat-conduction-weighted coordinate. The analysis for near unity Lewis numbers of the fuel by Libby, Liñán & Williams (1983) is extended here to arbitrary non-unity Lewis numbers, of relevance to a wide variety of applications, ranging from hydrogen-fuelled combustors to heavy fuel systems. In analogy with Liñán's analysis of counterflow diffusion flames, three asymptotic distinguished regimes are identified for premixed flames for large activation energies and the appropriate Damköhler numbers-the ratio of the characteristic diffusion and reaction times. These regimes are: the premixed flame regime, the partial burning regime and the nearly frozen ignition regime. The analytical expressions obtained for these regimes, of the dimensionless reaction rate as a function of the Damköhler number, are seen to describe with good accuracy the results obtained from the numerical integration of the full problem.
... We begin with a quote from Williams' Hottel Plenary Lecture entitled 'The role of theory in combustion science' [1]: "It is relevant to distinguish between the science and the technology of the subject. The march of technology has never hesitated. ...
In this paper, we critically analyzed possibilities of probability density function (PDF) methods for the closed-form description of combustion chemical effects in turbulent premixed flames. We came to the conclusion that the concept of a closed-form description of chemical effects in the classical modeling strategy in the PDF method based on the use of reaction-independent mixing models is not applicable to turbulent flames. The reason for this is the strong dependence of mixing on the combustion reactions due to the thin-reaction-zone nature of turbulent combustion confirmed in recent optical studies and direct numerical simulations. In this case, the chemical effect is caused by coupled reaction-diffusion processes that take place in thin zones of instantaneous combustion. We considered possible alternative modeling strategies in the PDF method that would allow the chemical effects to be described in a closed form and came to the conclusion that this is possible only in a hypothetical case where instantaneous combustion occurs in reaction zones identical to the reaction zone of the undisturbed laminar flame. For turbulent combustion in the laminar flamelet regime, we use an inverse modeling strategy where the model PDF directly contains the characteristics of the laminar flame. For turbulent combustion in the distributed preheat zone regime, we offer an original joint direct/inverse modeling strategy. For turbulent combustion in the thickened flamelet regime, we combine the joint direct/inverse and inverse modeling strategies correspondingly for simulation of the thickened flamelet structure and for the determination of the global characteristics of the turbulent flame.
... We begin with a quote from Williams' Hottel Plenary Lecture entitled 'The role of theory in combustion science' [1]: "It is relevant to distinguish between the science and the technology of the subject. The march of technology has never hesitated. ...
In this paper, we critically analyzed possibilities of probability density function (PDF) methods for the closed-form description of combustion chemical effects in turbulent premixed flames. We came to the conclusion that the concept of a closed-form description of chemical effects in the classical modeling strategy in the PDF method based on the use of reaction-independent mixing models is not applicable to turbulent flames. The reason for this is the strong dependence of mixing on the combustion reactions due to the thin-reaction-zone nature of turbulent combustion confirmed in recent optical studies and direct numerical simulations. In this case, the chemical effect is caused by coupled reaction–diffusion processes that take place in thin zones of instantaneous combustion. We considered possible alternative modeling strategies in the PDF method that would allow the chemical effects to be described in a closed form and came to the conclusion that this is possible only in a hypothetical case where instantaneous combustion occurs in reaction zones identical to the reaction zone of the undisturbed laminar flame. For turbulent combustion in the laminar flamelet regime, we use an inverse modeling strategy where the model PDF directly contains the characteristics of the laminar flame. For turbulent combustion in the distributed preheat zone regime, we offer an original joint direct/inverse modeling strategy. For turbulent combustion in the thickened flamelet regime, we combine the joint direct/inverse and inverse modeling strategies correspondingly for simulation of the thickened flamelet structure and for the determination of the global characteristics of the turbulent flame.
... The combustion process has played a significant role in improving the living standard of humans. Society has utilized combustion for various purposes, ranging from domestic such as cooking and heating to industrial applications such as for refinement and treatment of metals to make instruments and tools [1,2]. Combustion is indeed a fundamental component that extensively adds to our present ways of life in countless manners. ...
The alarming rate at which the available fossil fuel-based resources are depleting has raised grave concerns among all stakeholders about the future of the world. These limited energy resources play a very significant role in conventional energy generation techniques and processes. Traditional energy consumption devices must be made more efficient to prolong the availability of fossil fuels. Along with achieving a substantial increase in the efficiency of these devices, it is the need of the hour to focus on minimizing the hazardous emissions resulting from improper combustion and inefficient design, leading to the ever-growing issue of climate change. Porous burners are one of the few energy consumption devices that are known to facilitate efficient combustion along with low emissions of detrimental combustion products. In the porous burner technology, ceramic-based porous media deliver better performance due to their superior physical, chemical, and thermal characteristics compared to metal-based media. The present paper, therefore, provides an overview of the application of different ceramic-based porous media in burner technology. Advancements, particularly over the last twenty years, in the development and use of novel ceramic composites for porous burner applications, have also been discussed.
... These velocities, which cannot be predicted by the Favre-averaged equations, are at the same time described directly by the twofluid conditional equations. It would be appropriate here to quote a statement made by Forman Williams in his Hottel lecture entitled "The Role of Theory in Combustion Science" [5]: ...
This paper extends a recent theoretical study that was previously presented in the form of a brief communication (Zimont, C&F, 192, 2018, 221-223), in which we proposed a simple splitting method for the derivation of two-fluid conditionally averaged equations of turbulent premixed combustion in the flamelet regime, formulated more conveniently for applications involving unclosed equations without surface-averaged unknowns. This two-fluid conditional averaging paradigm avoids the challenge in the Favre averaging paradigm of modeling the countergradient scalar transport phenomenon and the unusually large velocity fluctuations in a turbulent premixed flame. It is a more suitable conceptual framework that is likely to be more convenient in the long run than the traditional Favre averaging method. In this article, we further develop this paradigm and pay particular attention to the problem of modeling turbulent premixed combustion in the context of a two-fluid approach. We formulate and analyze the unclosed differential equations in terms of the conditions of the Reynolds stresses τij,u , τij,b and the mean chemical source ρW¯ , which are the only modeling unknowns required in our alternative conditionally averaged equations. These equations are necessary for the development of model differential equations for the Reynolds stresses and the chemical source in the advanced modeling and simulation of turbulent premixed combustion. We propose a simpler approach to modeling the conditional Reynolds stresses based on the use of the two-fluid conditional equations of the standard “ k-ε ” turbulence model, which we formulate using the splitting method. The main problem arising here is the appearance in these equations of unknown terms describing the exchange of the turbulent energy k and dissipation rate ε in the unburned and burned gases. We propose an approximate way to avoid this problem. We formulate a simple algebraic expression for the mean chemical source that follows from our previous theoretical analysis of the transient turbulent premixed flame in the intermediate asymptotic stage, in which small-scale wrinkles in the instantaneous flame surface reach statistical equilibrium, while the large-scale wrinkles remain in statistical nonequilibrium.
... The G-equation, proposed by Williams [34], is used as the combustion model in the present study. It uses a level-set approach to describe evolution of the flame front as an interface between the unburned and burned gases. ...
Here, internal combustion engine operating speed effects on combustion cycle-to-cycle variations (CCV) are numerically investigated. The recent study by Ghaderi Masouleh et al. (2018) is extended to higher engine speeds including 560, 800 and 1000 RPM. The 3D scale-resolving simulations are carried out in a spark ignited simplified engine geometry under fuel lean condition. The numerical results include the following main findings. (1) Flow velocity and turbulence levels are noted to increase with RPM. (2) For a fixed spark timing, the combustion duration in CAD time increases with RPM contrasting the respective trend in physical time. (3) The link between early flow conditions around the spark position and the whole cycle combustion rate is demonstrated and explained for all the RPM for the investigated three example cycles. (4) On average, the moderate increase of turbulent flame speed with RPM is not able to compensate the reduced physical time for combustion. Hence, the higher RPM cycles burn typically slower in CAD time. (5) On average, the increased combustion duration in CAD time for higher RPM increases the CAD period, where the spark kernel is highly prone to local turbulence fluctuations. (6) A noted effect of RPM on CCV is the stretched combustion duration in CAD time so that the effect of the initial fluctuations can persist for a longer CAD period. (7) In the present model, the velocity magnitude near the spark largely explains cycle-to-cycle variations in the investigated low RPM range.
... The G-equation model proposed by Williams [55] uses a level-set method to describe the evolution of the flame front as an interface between the unburned and burned gases. A non-reacting passive scalar G is introduced where the isosurface = G G 0 divides the domain into unburned and burned regions with < G G 0 and > G G 0 , respectively. ...
Premixed, spark ignited combustion of lean methane at fuel to air equivalence ratio of 0.58 is numerically investigated in a piston-cylinder assembly. The simplified numerical configuration is tailored to emulate the intake, compression and spark ignition processes in engines. Large-eddy simulation is employed in the core flow along with a zonal hybrid wall treatment in near-wall regions. The G-equation level-set method is used to simulate flame propagation with a detailed chemistry based laminar flame speed correlation developed herein. The main numerical findings of this paper are as follows: (1) Despite the geometrical simplicity, the present set-up is shown to exhibit relatively large cycle-to-cycle variation for the three investigated cycles. (2) The local thermodynamics and fluid dynamic conditions around the spark close to the ignition location initiate the first discrepancies between the cycles. (3) These early variations are then amplified due to the subsequent differences in the early growth of flame area. (4) The cycle-to-cycle variation in the present set-up is shown to be largely a consequence of the local flow fluctuations close to the spark position and timing, while the results indicated a less dominating role of thermal stratification on cycle-to-cycle variation. (5) The asymmetric combustion behavior was explained to be a combined effect of burning rate and convection velocity, while convection velocity proved to be the major contributor. (6) Finally, a numerical test in the present model setup indicated large spark kernels being less prone to cycle-to-cycle variations than small kernels.
... Water is cold and wet, and earth is cold and dry. With the technology evolution at the end of 18th century which witnessed the inventions of the steam turbine, piston engines and gas turbine engines, modern combustion science began with the development of chemical kinetics theory in the mid 19th century [62]. ...
In this thesis, aspects of atomization, combustion and emission were investigated for a range of biodiesels. The selected fuels have different carbon chain lengths and unsaturation degrees and consequently different fuel oxygen contents (FO). Fundamental studies of secondary atomization were conducted using an air cross flow stream. Investigation of spray flames was performed using a hot co-flow burner. Fuel utilization was examined using a well-instrumented common-rail engine.
Secondary atomization was characterized in terms of three fluid elements namely small drops, large objects and ligaments. It was found that at a low We number, a significant change in shape probabilities occurs when moving from a bag break-up regime to higher We numbers and some small differences in liquid shapes are found amongst the tested fuels. However, up to a certain Weber number (such as We = 200), the probability of detection of different shapes is almost independent of the breakup regimes as well as the fuel properties. Comparing performances of flame structures in the hot co-flow burner indicated that the auto-ignition characteristics (e.g. the change in chemiluminescence emission and the growth of reaction zone width) are affected by the fuel-air ratio and also by the fuel molecular structure.
Engine studies confirmed that differences in biofuel molecular profiles significantly affect engine combustion and emission characteristics. The study on engine cycle variability established a link between the cyclic variability and the oxygen ratio, which is a good indicator of stoichiometry. The current research also revealed that a critical key to reducing the total particle mass, particle size, particle number, and black carbon concentration is to increase the FO. However, an increase in the FO leads to a substantial increase in the total particle number per unit of particle mass, the amount of black carbon per unit of particle mass, and the reactive oxygen species concentration.
... Over the past forty years the theory of combustion waves, in both homogeneous and heterogeneous systems, reached rather a high level of conceptual coherence, and to date it is perhaps one of the most elegant areas of classical phenomenology, presenting a graphic example of how much natural phenomena could be deduced from a few fundamental principles. Some of the main achievements are summarized in recent surveys [14,32]. ...
We consider a model of gas–solid combustion with free interface proposed by L. Kagan and G. I. Sivashinsky. Our approach is twofold: (I) We eliminate the front and get to a fully nonlinear system with boundary conditions; (II) We use a fourth-order pseudo-differential equation for the front to achieve asymptotic regimes in rescaled variables. In both cases, we implement a numerical algorithm based on spectral method and represent numerically the evolution of the char. Fingering pattern formation occurs when the planar front is unstable. A series of simulations is presented to demonstrate the evolution of sparse fingers (I) and chaotic fingering (II).
... Would the Navier-Stokes equations, for example, predict the existence of crown fire ab initio or the roles that radiation and convection play in ignition of pine needles? The dependency of theory upon 'detailed numerical treatments' was rejected by Williams (1992), who averred that 'Theory needn't be right to be good, theory needn't be mathematical to be right, and theory needn't be incomprehensible to be mathematical'. ...
We explore the basis of understanding wildland fire behaviour with the intention of stimulating curiosity and promoting fundamental investigations of fire spread problems that persist even in the presence of tremendous modelling advances. Internationally, many fire models have been developed based on a variety of assumptions and expressions for the fundamental heat transfer and combustion processes. The diversity of these assumptions raises the question as to whether the absence of a sound and coherent fire spread theory is partly responsible. We explore the thesis that, without a common understanding of what processes occur and how they occur, model reliability cannot be confirmed. A theory is defined as a collection of logically connected hypotheses that provide a coherent explanation of some aspect of reality. Models implement theory for a particular purpose, including hypotheses of phenomena and practical uses, such as prediction. We emphasise the need for theory and demonstrate the difference between theory and modelling. Increasingly sophisticated fire management requires modelling capabilities well beyond the fundamental basis of current models. These capabilities can only be met with fundamental fire behaviour research. Furthermore, possibilities as well as limitations for modelling may not be known or knowable without first having the theory.
... Williams narrates in one of his publications[152] that the first time he was exposed to the full equations describing the chemically reacting flows, he had said to himself, Surely there is enough there to occupy me for a lifetime.Spray combustion in an acoustically induced oscillating flow field is perhaps one of the toughest problems in thermo-fluids. It therefore, warrants a thorough investigation of each and every process involved and the 'lifetime' observation by Williams certainly appears to be true. ...
Thermoacoustic instabilities in modern high-performance, low-emission gas turbine engines are often observable as large amplitude pressure oscillations and can result in serious performance and structural degradations. These acoustic oscillations can cause oscillations in combustor through-flows and given the right phase conditions, can also drive unsteady heat release. To curb the potential harms caused by the existence of thermoacoustic instabilities, recent efforts have focused on the active suppression of these instabilities. Intuitively, development of effective active combustion control methodologies is strongly dependent on the knowledge of the onset and sustenance of thermoacoustic instabilities. Specially, non-premixed spray combustion environment pose additional challenges due to the inherent unstable dynamics of sprays. The understanding of the manner in which the combustor acoustics affect the spray characteristics, which in turn result in heat release oscillation, is therefore, of paramount importance. The experimental investigations and the modeling studies conducted towards achieving this knowledge have been presented in this dissertation. Experimental efforts comprise both reacting and non-reacting flow studies. Reacting flow experiments were conducted on a overall lean direct injection, swirl-stabilized combustor rig. The investigations spanned combustor characterization and stability mapping over the operating regime. The onset of thermoacoustic instability and the transition of the combustor to two unstable regimes were investigated via phase-locked chemiluminescence imaging and measurement and phase-locked acoustic characterization. It was found that the onset of the thermoacoustic instability is a function of the energy gain of the system, while the sustenance of instability is due to the in-phase relationship between combustor acoustics and unsteady heat release driven by acoustic oscillations. The presence of non-linearities in the system between combustor acoustic and heat release and also between combustor acoustics and air through-flow were found to exist. The impact of high amplitude limit-cycle pressure on droplet breakdown under very low mean airflow and the localized effects of forced primary fuel modulations on heat release were also investigated. The non-reacting flow experiments were conducted to study the spray behavior under the presence of an acoustic field. An isothermal acoustic rig was specially fabricated, where the pressure oscillations were generated using an acoustic driver. Phase Doppler Anemometry was used to measure the droplet velocities and sizes under varying acoustic forcing conditions and spray feed pressures. Measurements made at different locations in the spray were related to these variations in mean and unsteady inputs. The droplet velocities were found to show a second order response to acoustic forcing with the cut-off frequency equal to the relaxation time corresponding to mean droplet size. It was also found that under acoustic forcing the droplets migrate radially away from the spray centerline and show oscillatory excursions in their movement. Modeling efforts were undertaken to gain physical insights of spray dynamics under the influence of acoustic forcing and to explain the experimental findings. The radial migration of droplets and their oscillatory movement were validated. The flame characteristics in the two unstable regimes and the transition between them were explained. It was found that under certain acoustic and mean air-flow condition, bands of high droplet densities were formed which resulted in diffusion type group burning of droplets. It was also shown that very high acoustic amplitudes cause secondary breakup of droplets.
... 14s-~53 Past experimental and theoretical studies on fire whirls were mainly focused at understanding the scale model ('ideal model') phenomena, and little on experiments with scale modeling although the subject is obviously important 12-14.1. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Scale model studies on fire whirls and related phenomena, including atmospheric instabilities in- Table 1. ...
Summarized are recent experimental findings of fire spread phenomena. This review covers flame spread over solids (including melting solids and metals), large-scale spread through discrete fuels (such as fire brands and fire whirls), and scale modeling techniques applied to flame spread study. Emphasis is placed on the importance of observation in experiments which is the source of imagination and successful modeling.
... Dentro de los estudios teóricos destacan los análisis asintóticos basados en cinéticas de una sola etapa con alta energía de activación (Klimov [58], Williams [160]). Estos análisis se han llevado a cabo para valores pequeños, moderados y grandes del ritmo de estiramiento incluyendo el efecto de la densidad variable (Libby y Williams [66]), así como la influencia del número de Lewis (Libby et al [64]) y la no adiabaticidad asociada a temperaturas de los gases quemados distintas de la adiabática (Libby y Williams [67] ...
En esta tesis se analiza la estructura y dinámica de llamas laminares premezcladas y de difusión sometidas a perturbaciones características de los flujos laminares y turbulentos. El análisis. basado en una combinación de métodos analíticos y numéricos, incluye los efectos de los valores distinto de las difusividades técnicas y másicas del combustible y el oxidante. En primer lugar, se presenta un estudio exhaustivo de los regímenes de combustión en la llama premezclada sometida a estiramiento entre dos corrientes opuestas, una de reactantes y otra de gases quemados cuya temperatura puede ser distinta a la temperatura adiabática de la corriente de reactantes, considerando desviaciones importantes del número de Lewis del reactante limitante (típicamente el combustible) respecto a la unidad. El análisis supone una reacción global de tipo Arrhenius con alta energía de activación y emplea la aproximación termodifusiva de densidad y propiedades de transporte constantes. Los resultados proporcionan las condiciones críticas de extinción e ignición de la llama premezclada en términos del número de Damkohler, revelando la existencia de tres regímenes de combustión: el régimen de llama premezclada, el régimen de combustión parcial y el régimen de ignición. A continuación se presenta un análisis original de las perturbaciones inducidas en la llama de difusión por parejas de torbellinos contrarrotatorios y anillos de vorticidad de intensidad suficiente para producir la extinsión local. Se ha investigado el caso de química rápida, en el que el problema se puede reducir a uno de mezcla mediante una formulación basada en escalares pasivos. El análisis, basado en un modelo sencillo para los torbellinos y un valor constante para la densidad y las propiedades de transporte del gas, proporciona un procedimiento para describir la distorsión de la capa de mezcla, así como la evolución espacial y temporal del espesor de la capa de mezcla. Esto permite la evolución de la disipación escalar, que determina las condiciones para la extinción local de la llama y para la propagación de bordes de llama (o llamas triples) a lo largo de la capa de mezcla. Este análisis explica perfectamente los resultados de experimentos llevados a cabo en la Universidad de Yale y en la Universidad de París. Los resultados de este análisis se utilizan posteriormente para estudiar, usando un modelo Langrangiano sencillo. la evolución espacial y temporal de los bordes de llama (o llama triples) que aparecen tras la extinción local de la llama de difusión. El modelo supone que la velocidad de propagación del borde de llama respecto al fluido depende exclusivamente del número de Damkohler local, que se puede obtener del análisis anterior. Para evaluar dicha velocidad se utilizan resultados previos sobre la propagación de frentes de ignición y extinción en capas de mezcla sometidas a estiramiento que tiene en cuenta el efecto de liberación de calor. Los resultados del modelo también reproducen con gran fiabilidad las observaciones experimentales. Por último, se presenta un análisis de la estructura y la dinámica de las capas de mezcla reactivas en contracorriente en el que se incluye el efecto de los cambios de densidad y de las variaciones de las propiedades de transporte con la temperatura. El objetivo es comparar la respuesta de la capa de mezcla perturbada por un torbellino anular con la que se obtiene en el caso de densidad constante, siempre en el caso de química rápida. El análisis muestra que la respuesta de la capa de mezcla es más rápida cuando se retiene los efectos de densidad variable, mientras que la respuesta en amplitud cambia muy poco cuando se retienen las variaciones de densidad. También se discuten brevemente las dificultades que surgen cuando la capa de mezcla constituye además una entrefase de densidad. Para ello, se presenta una solución exacta de las ecuaciones de Euler para la propagación de torbellinos en presencia de entrefases abruptas de densidad que permite aclarar algunos aspectos relevantes de este tipo de flujos.
... To understand the acoustic-flame coupling on a firstprinciples basis, one has to look into the structure of the flame as well as its associated hydrodynamic field. A theoretical framework for this is the largeactivation-energy asymptotics (AEA; Williams 1992). The key physical basis that leads to mathematical simplification is the scale disparity, that is, chemical reaction takes place in an inner layer much thinner than the flame thickness d, so that matched-asymptotic-expansion technique can be used to analyse systematically each of these regions. ...
This paper presents an asymptotic approach to combustion instability in premixed flames under the assumptions of large activation energy and small Mach number. The entire flow consists of four distinct yet fully interactive sub-regions, which accommodate the chemical reaction, heat transport, hydrodynamics and acoustics, respectively. A reduced system was derived to describe the intricate coupling between the flame and acoustics that underlies the combustion instability. The asymptotically reduced system was employed to study the weakly nonlinear interaction between the Darrieus-Landau instability and the longitudinal acoustic mode of the combustion chamber. The general asymptotic formulation includes the influence of enthalpy fluctuation in the oncoming mixture. It is shown that one-dimensional enthalpy fluctuation, through its interaction with flame, produces sound waves, and may cause parametric instability of the flame. The mutual coupling between the sound wave and parametric instability is analysed at the instability thresholds.
We analyze the nature of the known difficulties that arise when trying to apply the kinematic G-equation and the surface average Σ-equation to the Reynolds-Averaged Navier–Stokes modeling of turbulent premixed combustion, based on the use of the Favre average equations of combustion and hydrodynamics. We consider this issue in the context of an analytical formulation that includes an asymptotically closed infinite system of successively derived unclosed moment combustion and hydrodynamics equations of the problem. We show that the cause of these difficulties is the incompatibility of the G-equation and Σ-equation with the other instantaneous and averaged equations of the problem. This leads us to formulate a compatibility principle, i.e., a rule for the identification of misconceptions about the use of instantaneous and averaged equations that are correct by themselves, due to their incompatibility with the equations of the corresponding analytical formulation. Special attention is devoted to a comparison of two approaches to a description of instantaneous and average premixed combustion, based on the use of the progress variable c and the scalar G. We state that these approaches are equivalent. To prove this, we derive the equations missing from these two approaches and show that the following two groups of instantaneous and Favre-averaged equations, formulated in terms of c and G, respectively, are equivalent. The main result is the proposed compatibility principle. The conclusion following from our analysis is that the concepts of the scalar G and flame surface density Σ useful in applications could be considered superfluous from the theoretical viewpoint.
This text describes simulation methods to compute instabilities which appear in flames and more generally in turbulent flows. Most practical flames are turbulent and operate at large Reynolds numbers so that the presentation focuses on recent Large Eddy Simulation (LES) tools used for these reacting flows. The physical instabilities appearing in the flow are discussed as well as the instabilities generated by the numerical methods. Combustion is one of the most CPU consuming field of physics today and none of these computations can be done without massively parallel computers: recent examples of massively parallel LES for gas turbine combustion chambers are provided. Computing turbulent flows on these systems raise additional issues of bifurcation and repeatability which are also presented on canonical turbulent flows as well as real combustors. Finally, the text also discusses numerical instabilities which can appear in the simulation of turbulent flows when low-dissipation, high-precision numerical methods are used.
A microflame, defined in this paper as a diffusion flame established on a burner whose size is less than 1 mm, is used as a small heat source having high heating density. Multiple microflames are often used to achieve a high density and uniform heating performance. However, when the microflames are placed too closely, they merge into a larger flame, which is no longer a microflame. Therefore, the knowledge of the critical bumer-bumer distance at which two microflames touch each other is important when designing such a device. This paper theoretically discusses the critical distance of two identical micro-slot burners, which are slot burners of the slot widths less than 1 mm. Two micro-slot burners are modeled as line-source burners in a uniform flow. The mixture-fraction model is then used to predict flame shapes, based on which the critical distance is derived.
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The topics introduced in this chapter cover a small portion of flame studies that have been accomplished in the past. Some interpretations given here are the author's view and subject to further discussion. The intention of the author is to discuss the fundamentals of combustion and to stimulate readers to think scientifically when they deal with forest fire problems. References are provided for readers who are interested in advancing their understanding in flames and combustion. This chapter intends to help ecologists, forest fire researchers, and fire-fighting strategists understand some of the fundamental aspects of combustion and think scientifically about their forest fire problems. As with any discipline, the best approach to understanding forest fires is to grasp the fundamentals—in this case, the structure and behavior of flames. Explanation and discussion are based on the physical and chemical aspects of combustion, with little emphasis on strict mathematical treatment of equations. The chapter addresses the basic knowledge on the structure of diffusion flames and scaling laws, premixed flames, ignition, diffusion flame extinguishment, spreading flames, and the mechanism of diffusion flame anchoring. Some general background in combustion research is introduced at the beginning of the chapter.
The present paper deals with the description of the interacting multiscale processes governing spray vaporization and combustion downstream from the near-injector atomization region in liquid-fueled burners. One of the main objectives is to emphasize the progress made in the mathematical description and understanding of reactive spray flows by incorporation of rationally derived simplifications based on the disparity of length and time scales present in the problem. In particular, we aim to show how the disparity of the scales that correspond – with increasing values of their orders of magnitude – to the droplet size, interdroplet spacing, and width of the spray jets, ensures the validity of their homogenized description. The two-way coupling associated with exchanges of mass, momentum, and energy between the gas and the liquid phases is dominated by the homogenized exchanges with the gas provided collectively by the droplets, and not by the direct interaction between neighboring droplets. The formulation is used as a basis to address nonpremixed spray diffusion flames in the Burke-Schumann limit of infinitely fast chemical reactions, with the conservation equations written in terms of chemistry-free coupling functions that allow for general nonunity Lewis numbers of the fuel vapor. Laminar canonical problems that have been used in the past to shed light on different aspects of spray-combustion phenomena are also discussed, including spherical spray clouds and structures of counterflow spray flames in mixing layers. The presentation ends with a brief account of some open problems and modeling challenges.
The propagation of a two-dimensional diffusive flame over a combustible material is studied by solving steady-state conservation equations written in the coordinate system fixed on the flame front. The analysis of a two-dimensional problem featuring flame spread rate as an eigenvalue has shown that there is no unique solution relative to the flame spread rate unless some additional condition is involved. A novel approach to the prediction of flame spread rate is proposed using the principles of irreversible thermodynamics. The steady flame propagation is considered as a stationary non-equilibrium thermodynamic state, which can be characterized, according to the formulation of Prigogine, by minimal entropy production. A numerical algorithm for the prediction of flame spread rate has been developed and tested on the investigation of downward flame spread over thin sheets of paper. The adequate physical background of the proposed approach and satisfactory agreement with experimental data have been shown.
Issues associated with modeling the multiscale nature of detonation are reviewed, and potential applications to detonation-driven propulsion systems are discussed. It is suggested that a failure of most existing computations to simultaneously capture the intrinsic microscales of the underlying continuum model along with engineering macroscales could in part explain existing discrepancies between numerical predictions and experimental obser- vation. Mathematical and computational strategies for addressing general problems in multiscale physics are first examined, followed by focus on their application to detonation modeling. Results are given for a simple detonation with one-step kinetics, which undergoes a period-doubling transition to chaos; as activation energy is increased, such a system exhibits larger scales than are commonly considered. In contrast, for systems with detailed kinet- ics, scales finer than are commonly considered are revealed to be present in models typically used for detonation propulsion systems. Some modern computational strategies that have been recently applied to more efficiently cap- ture the multiscale physics of detonation are discussed: intrinsic low-dimensional manifolds for rational filtering of fast chemistry modes, and a wavelet adaptive multilevel representation to filter small-amplitude fine-scale spatial modes. An example that shows the common strategy of relying upon numerical viscosity to filter fine-scale physics induces nonphysical structures downstream of a detonation is given.
The influence of differential diffusion on the statistical behavior of the local displacement speed (Sd) in relation to flame curvature is studied based on three-dimensional compressible direct numerical simulations (DNS) of statistically planar flames with single-step Arrhenius-type chemistry. Three different Lewis number cases (Le=0.8, 1.0, and 1.2) are considered. In order to study the influence of differential diffusion on curvature effects in flame propagation, temperature statistics are presented in terms of standard probability density functions (pdfs) and also joint pdfs with curvature for the nonunity Lewis number cases. Temperature statistics are found to be consistent with previous incompressible combustion DNS studies. It is found that both dilatation and tangential strain rate are negatively correlated with curvature. The relative strength of these two correlations determines the nature of the correlation between surface density function (SDF) (|∇c|) and curvature. It is also found that the variations of temperature and SDF on an isosurface of reaction progress variable (c) have significant influence on local displacement speed behavior. Displacement speed statistics are presented in terms of standard pdfs as well as joint pdfs with curvature for the three Lewis number cases. The curvature response of the displacement speed and its different components is found to be nonlinear which is consistent with previous two-dimensional DNS with detailed chemistry. The observed nonlinear behavior in the present study in the absence of a detailed chemical mechanism is explained through the influence of differential diffusion.
Theoretical analyses are reported on the combustion of spherical alcohol droplets under microgravity conditions. The theory addresses burning rates, water dissolution in the droplet during combustion, possible instabilities that can lead to liquid-phase circulation, and conditions for flame extinction. Comparisons are made between theoretical predictions and results of drop-tower experiments. The primary conclusions are that internal circulation occurs for methanol droplets and that the extinction conditions predicted under this assumption agree with experiment.
Recent theoretical advances in premixed gas combustion are reviewed. The attention is focused on (1) self-acceleration of outward propagating wrinkled flames sustained by the intrinsic flame instability, (2) fragmentation of near-limit cellular flames and formation of self-drifting flame balls, (3) flame acceleration and extinction by large-scale turbulence, (4) multiplicity of detonation regimes in hydraulically resisted flows and the phenomenon of shock-free pressure-driven combustion, and (5) hydraulic resistance as a mechanisms of deflagration-to-detonation transition.
Markstein lengths (defined as the reduction in burning velocity per unit stretch) have been measured for a series of CO/H2/air flames at atmospheric pressure. Values as a function of stoichiometry are reported for three fuels (95%/5% CO/H2, 50%/50% CO/H2, and 100% H2). In addition, the dependence of Markstein length on H2 content has been measured for stoichiometric CO/H2 mixtures.The experimental technique involves expanding spherical flames. A simple expression is fitted to radius/time data, to yield two parameters: flame speed at infinite radius and a flame relaxation parameter, which contains the sum of flame-stretch and flame-thickness effects. Subtracting the latter reveals the influence of stretch. Markstein lengths are referenced to a point within the flame, determined by computer modeling as the position where the mass fluxes of planar, ID, and stationary spherical flames are equal. This occurs at flame temperatures in the range 700–1300 K, depending on CO/H2 ratio and fuel/air stoichiometry.It is found that, at most CO/H2 ratios Markstein lengths are essentially the same as for H2/air flames. We conclude that, in these cases, the hydrogen governs the stretch behavior. But when the hydrogen content is sufficiently low, carbon monoxide becomes dominant. The stretch behavior is then very similar to that of C2H4, with both showing behavior characteristic of fuel with near-unity Lewis number.
This paper presents a theory to predict the extinction limit of laminar jet diffusion microflames, defined as flames established on submillimeter-diameter burners. The classical Burke–Schumann (BS) theory is first extended to include the effect of one-step, finite-rate chemistry. Then, a theory of diffusion-flame extinction is applied using activation-energy asymptotics to predict the extinction limit. The present theory correctly reproduces experimental observations, i.e., uL∼d−2, where uL is the fuel jet velocity at the extinction limit (lower limit) and d the burner diameter. According to the BS theory, the gradient of mixture fraction at the flame-sheet location is infinite at the burner rim, and it decreases with increasing axial distance to the minimal value at the flame tip. Therefore, local extinction initiates at the burner rim, and extinction occurs when the mixture-fraction gradient at the flame tip is greater than a critical value. This view of microflame extinction is supported by the results of experiments and numerical simulations. It is found that the present theory can be applied for various types of fuels, those are, methane, propane, and butane.
Reduced chemical kinetic models have been developed to describe the combustion fundamentals for practical high hydrocarbons fuels over a wide range of experimental conditions. The fuels include n-Butane, Benzene. n-Heptane. Gasoline, Kerosene (JP-8), and n-Hexadecane. The mechanism for each fuel includes a single reaction expression for fuel and oxygen to form formaldehyde (CH2O) and hydrogen (H2) or carbon monoxide (CO)together with a detailed reaction mechanism for CH2O-CHO-CO-H2-O2 oxidation. These kinetic mechanisms will be as generally applicable as possible and can be used in 2-D or 3-D combustion models to understand the practical combustion and emission problems in engines and furnaces. Each mechanism consists of 13 chemical species with 22 elementary reactionsThe present reduced kinetic mechanisms are used in one-dimensional laminar premixed flame model and incorporated detailed representation of transport fluxes to predicted laminar burning velocity and flame structure. These predicted results were compared satisfactorily with the experimental data for each fuel over a wide ranges of equivalence ratio, pressure, and temperature. In addition, the flammability limits for different types of fuels were also examined. A single reaction expression for breakdown each of the above fuels has been driven here, and is used with CH2O-CHO-CO-H2-O2 mechanism to predict satisfactorily the experimental combustion fundamentals for these practical fuels. These mechanisms are only valid from lean to near stoichiometric flames and they also, lead to improve the accuracy of the predicted radical species compared to the past quasi-global model that has assumed CO and H2 as reactions products. An algebraic expression for burning velocity of each fuels is derived, in terms of equivalence ratio, initial pressure and temperature. and can be used in complex models.
Combustion is a complex phenomenon characterized by the interaction and competition of various physical and chemical processes. The correct description of chemical changes requires the application of reaction mechanisms that consist of several hundreds or thousands of elementary reactions. This creates an opportunity for kinetic modeling to play important role in understanding the combustion phenomenon. This chapter provides an overview of combustion modeling, the main aim being to present a comprehensive knowledge base for combustion kinetic modeling. The coupling of fluid dynamics and chemical kinetics using commercial software such as CHEMKIN and FLUENT is also discussed. Moreover, an exemplary approach for reducing complex chemical reaction mechanisms is illustrated with reference to an H2 + O2 mixture in an adiabatic system. This approach involves the identification of redundant species via rate sensitivity analysis, and of redundant reactions, by principal component analysis of the rate sensitivity matrix. An eigenvalue–eigenvector analysis is used to extract meaningful kinetic information from linear sensitivity coefficients computed for all species of chemical mechanism at several time points. The main advantage of this method lies in its ability to reveal those parts of the mechanism, which consist of strongly interacting reactions, and to indicate their importance within the mechanism. By using the above procedures, reduced reaction mechanisms could be developed at different chosen conditions and employed in CFD codes in place of detailed mechanisms, giving due consideration to flow fields.
Keywords:
Combustion;
kinetic modeling;
mechanism reduction;
Jacobian analysis;
time-scale mechanism reduction;
computational fluid dynamics
An aerodynamic structure of a laminar boundary layer over a flat plate with uniform fuel injection from the flat plate and with diffusion flame is investigated numerically. Elliptic type conservation equations are used to take into account the pressure variation within the boundary layer. Velocities and the pressure are solved numerically by SIMPLER algorithm. One step irreversible chemical reaction of methane is assumed. An Arrhenius type chemical reaction rate model is assumed and the pre-exponential factor is varied from 1.0 1012 to 1.0 1030 m3/(kg s) as a parameter of the reactivity in order to elucidate the effect of the reactivity on the structure of the boundary layer. When the chemical reaction is very fast, the leading edge of the reaction zone reaches the flat plate. As the chemical reaction rate is decreased with a decrease in the pre- exponential factor, the leading edge of the reaction zone parts from the flat plate and it shifts downstream. The fuel is injected in front of the leading edge of the reaction zone, where the air is dominant, and the oxygen penetrates into the fuel dominant zone through the region between the leading edge and the flat plate. As a consequence, a premixed gas is formed around the leading edge of the reaction zone. The premixed gas seems to react in the region apart from the main visible flame.
This paper deals with predictive methods of the combustion rates in the case of premixed mixtures. Comparisons are made between two of these predictive methods. The first approach is based on a standard eddy break-up (EBU) model supplemented by a condition on the chemical composition of the mixture. The reaction can develop within a fixed range of richness. The second method uses the concept of fractal surfaces. Moreover a spectral analysis allows us to introduce the role of the small turbulent structures embedded in the preheat zone. By means of the concept of fractal surfaces the real area of the flame surface can be predicted starting from its mean value and from L (integral length scale), u′ and ϑk (Kolmogorov length scale). The location of the flame is deduced from the standard EBU model previously referred to. u′ ànd {ε} are also known along the mean flame surface. The area of the real indented surface can be inferred. The combustion rate can be deduced by associating the area of the flame front with a convenient flame speed accounting for the role of the small structures embedded in the preheat zone. The results found by the two methods do not depart from each other by more than 10%
A discussion on progress and challenges associated with combustion dynamics and control covers basic interactions leading to combustion instabilities; historical perspective; current dynamics and control problems, e.g., premixed mode of combustion in stationary gas turbines; elementary processes; dynamics of unsteady strained flames; flame response to incident inhomogeneities; methane/air premixed flame modulated by acoustic perturbations; couple flame motion under acoustic perturbation; perturbed laminar flames interacting with boundaries; theoretical description of combustion instabilities; large Eddy simulation for turbulent flame dynamics; active control; and operating point control. This is an abstract of a paper presented at the 29th International Symposium on Combustion (Sapporo, Japan 7/21-25/2002).
Laser diagnostics and flow simulation techniques are now providing information that, if available 50 years ago, would have allowed Damköhler to show how turbulence generates flame area. In the absence of this information, many turbulent flame speed models have been created, most based on Kolmogorov concepts that ignore the turbulence vortical structure. Over the last 20 years, the vorticity structure in mixing layers and jets has been shown to determine the entrainment and mixing behavior, and these effects need to be duplicated by combustion models. Turbulence simulations reveal the intense vorticity structure as filaments and simulations of passive flamelet propagation show how this vorticity creates flame area and defines the shape of the expected chemical reaction surface. Understanding how volume expansion interacts with flow structure should improve experimental methods for determining turbulent flame speed. Since the last decade has given us such powerful new tools to create and see turbulent combustion microscopic behavior, it seems that a solution of turbulent combustion within the next decade would not be surprising in the hindsight of 2004.
In honor of the fiftieth anniversary of the Combustion Institute, we are asked to assess accomplishments of theory in combustion over the past fifty years and prospects for the future. The title of our article is chosen to emphasize that development of theory necessarily goes hand-in-hand with specification of a model. Good conceptual models underlie successful mathematical theories. Models and theories are discussed here for deflagrations, detonations, diffusion flames, ignition, propellant combustion, and turbulent combustion. In many of these areas, the genesis of mathematical theories occurred during the past fifty years, and in all of them significant advances are anticipated in the future. Increasing interaction between theory and computation will aid this progress. We hope that, although certainly not complete in topical coverage or reference citation, the presentation may suggest useful directions for future research in combustion theory.
Various experimental and DNS data show that premixed combustion is affected by the differences between the coefficients of molecular transport of fuel, oxidant, and heat not only at weak but also at moderate and high turbulence. In particular, turbulent flame speed increases with decreasing the Lewis number of the deficient reactant, the effect being very strong for lean hydrogen mixtures. Various concepts; flame instability, flame stretch, local extinction, leading point, that aim at describing the effects of molecular transport on turbulent flame propagation and structure are critically discussed and the results of relevant studies of perturbed laminar flames (unstable flames, flame balls, flames in vortex tubes) are reviewed. The crucial role played by extremely curved laminar flamelets in the propagation of moderately and highly turbulent flames is highlighted and the relevant physical mechanisms are discussed.
The aim of this work is to develop numerical methods and software for simulation and optimization of complex processes in catalytic monoliths to achieve better understanding of the physic-chemical processes in catalytic reactors. The fluid dynamics are modelled by the boundary layer equations (BLEs), which are a large system of parabolic partial differential equations (PDEs) with highly nonlinear boundary conditions arising from the coupling of surface processes with the flow field inside the channel. The BLEs are obtained by simplifying the comprehensive model described by the Navier-Stokes equations and applying the boundary approximation theory. The surface and gas-phase chemical reactions are described by detailed models. The PDEs are semi-discretized using the method of lines leading to a structured system of differential-algebraic equations (DAEs). The DAEs are integrated by an implicit method, based on backward differentiation formulas (BDF). We develop a new BDF code with tailored efficient and robust numerical methods by exploiting the structure, and by an appropriate scaling for ill-conditioned iteration matrices, and by computing consistent initial values. Efficient methods for computation of partial derivatives in the framework of automatic differentiation and of finite differences are introduced and compared. Our newly developed simulation tool is more stable than the existing simulation tool, and faster than by a factor of ten to more than 60, depending on the applications. To improve the performance of catalytic reactors (e.g., maximizing gas conversion or selectivity) we can control certain process conditions, such as temperature at the catalyst wall or the ratio of catalytic active surface area to the geometric surface area or the gas composition, the temperature, or the velocity at the inlet of the catalyst. It is the first time that this problem is generally formulated as an optimal control problem constrained by a system of PDEs describing the chemical fluid dynamics process and additional constraints. The direct shooting approach in combination with sequential quadratic programming (SQP) method is used for solving the resulting optimal control problem. An efficient numerical method for computation of the derivatives required by the SQP method is introduced. In addition, error analysis for the numerical Newton method is investigated in detail. We introduce a new error model. Based on our error model and analysis, the limiting accuracy of the solution of nonlinear equations by the numerical Newton method can be obtained. Our newly developed software package for simulation and optimization can be applied to different reaction mechanisms and channel settings with different initial/boundary conditions. This software is applied to two practical applications: catalytic combustion of methane and conversion of ethane to ethylene. The numerical results are presented. The simulation software provides a useful tool for the validation of reactions mechanisms. The software package allows, e.g., for a better design and operation of the conversion of natural gas to higher hydrocarbons or the improvement of exhaust treatment in cars. Ziel dieser Arbeit ist die Entwicklung numerischer Methoden und Programme zur Simulation und Optimierung komplexer Prozesse in katalytischen Monolithen, um die physikalisch-chemischen Prozesse in katalytischen Reaktoren besser verstehen zu können. Die Strömungen werden mittels der Grenzschichtgleichungen modelliert. Sie bilden ein großes System von partiellen Differentialgleichungen (PDEs) mit hochgradig nichtlinearen Randbedingungen, die sich aus der Kopplung der Oberflächenprozesse mit dem Strömungsfeld innerhalb des Kanalsergeben. Die Grenzschichtgleichungen werden abgeleitet, indem das Navier-Stokes-Modell vereinfacht und die Grenzschichtnäherung angewendet wird. Die Beschreibung der Gasphasen- und Oberflächenreaktionen erfolgt durch detaillierte Modelle. Die PDEs werden mit der Hilfe der Linienmethode semi-diskretisiert. Daraus ergibt sich ein differential-algebraisches Gleichungssystem. Das differential-algebraische Gleichungssystem wird durch eine implizite Methode integriert, die auf den "Backward-Differentiation-Formulae" (BDF) beruht. Es wird ein neuer BDF-Code mit speziell zugeschnittenen, effizienten und robusten numerischen Methoden entwickelt, der insbesondere alle Strukturen ausnutzt, schlecht-konditionierte Iterationsmatrizen geeignet skaliert und konsistente Anfangswerte berechnet. Effiziente Methoden zur Berechnung der partiellen Ableitungen im Rahmen der automatischen Differenzierung und der finiten Differenzen werden eingeführt und miteinander gekoppelt. Das neu entwickelte Simulationswerkzeug ist stabiler als das existierende und in Abhängigkeit der Anwendung 10 bis 60-mal schneller. Durch Variation bestimmter Prozessparameter lässt sich das Verhalten katalytischer Reaktoren verbessern (z.B. durch Maximierung von Umsatz oder Selektivität). Dazu zählen die Temperatur an der Katalysatorwand, das Verhältnis von katalytisch aktiver und Gesamtoberfläche sowie die Gaszusammensetzung, die Temperatur und die Geschwindigkeit am Eingang des Katalysators. Dieses Problem wird zum ersten Mal als Optimierungsproblem allgemein formuliert, das durch ein System von PDEs dargestellt wird. Dabei beschreiben die PDEs die reaktive Strömung sowie zusätzliche Bedingungen. Zur Lösung des sich ergebenden Optimierungsproblems wird ein "direktes Schieß verfahren" in Verbindung mit der Methode der sequentiellen quadratischen Programmierung (SQP) benutzt. Eine effiziente Vorgehensweise zur Berechnung der für die SQP-Methode erforderlichen Ableitungen wird dargestellt. Ein neues Modell zur Fehleranalyse der Newton-Methode wird eingeführt. Dadurch lässt sich die maximal erzielbare Genauigkeit der Lösung von nicht-linearen Gleichungen besser abschätzen. Das neu entwickelte Softwarepaket zur Simulation und Optimierung eignet sich für verschiedene Reaktionsmechanismen und Kanäle mit verschiedenen Anfangs- und/oder Randbedingungen. Die Software wird exemplarisch für zwei Anwendungen eingesetzt: katalytische Verbrennung von Methan und Umsetzung von Ethan zu Ethylen. Die numerischen Ergebnisse werden dargestellt. Diese Simulationssoftware ist geeignet zur Validierung von Reaktionsmechanismen. Sie ermöglicht die Optimierung chemischer Prozesse, wie zum Beispiel die Umsetzung von Erdgas in wertvolle Kohlenwasserstoffe oder die Abgasnachbehandlung in Kraftfahrzeugen.
An analysis is carried out of the electric field-induced evaporation of ions from the surface of a polar liquid that is being electrosprayed in a vacuum. The high-field cone-to-jet transition region of the electrospray, where ion evaporation occurs, is studied taking advantage of its small size and neglecting the inertia of the liquid and the space charge around the liquid. Evaporated ions and charged drops coexist in a range of flow rates, which is investigated numerically. The structure of the cone-to-jet transition comprises: a hydrodynamic region where the nearly equipotential surface of the liquid departs from a Taylor cone and becomes a jet; a slender region where the radius of the jet decreases and the electric field increases while the pressure and the viscous stress balance the electric stress at the surface; the ion evaporation region of high, nearly constant field; and a charged, continuously strained jet that will eventually break into drops. Estimates of the ion and drop contributions to the total, conduction-limited current show that the first of these contributions dominates for small flow rates, while most of the mass is still carried by the drops.
The objective of the present paper is to review some developments that have occurred in detonation theory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection of the cell size, dynamical self-quenching, direct (blast) or spontaneous initiation, and transition from deflagration to detonation. These phenomena are all well documented by experiments since the sixties but remained unexplained until recently. In the first part of the paper, the patterns of cellular detonations are described by an asymptotic solution to nonlinear hyperbolic equations (reactive Euler equations) in the form of unsteady (sometime chaotic) and multidimensional traveling-waves. In the second part, turning points of quasi-steady solutions are shown to correspond to critical conditions of fully unsteady problems, either for (direct or spontaneous) initiation or for spontaneous failure (self-quenching). Physical insights are tentatively presented rather than technical aspects. The challenge is to identify the physical mechanisms with their relevant parameters, and more specifically to explain how the length-scales involved in detonation dynamics are larger by two order of magnitude (at least) than the length-scale involved in the steady planar traveling-wave solution (detonation thickness).
Mechanisms and rates of upward spread of turbulent flames along thermally thick vertical sheets are considered for both noncharring and charring fuels. By addressing the time dependence of the rate of mass loss of the burning face of a charring fuel, a linear integral equation of the Volterra type is derived for the spread rate. Measurements of spread rates, of flame heights and of surface temperature histories are reported for polymethylmethacrylate and for Douglas-fir particle board for flames initiated and supported by a line-source gas burner, with various rates of heat release, located at the base of the fuel face. Sustained spread occurs for the synthetic polymer and not for the wood. Comparisons of measurements with theory aid in estimating characteristic parameters for the fuels.
In this paper we have analyzed the steady-state process leading to ignition of a combustible mixture of hydrogen, oxygen and nitrogen by a hot flat plate in a boundary layer flow. For plate temperatures larger than the crossover temperature dictated by the competition between the chain branching reaction H + 02 → OH + O and the chain breaking reaction H + 02 + M → H02 + M, the ignition event corresponds to a typical chain branching explosion with negligible heat release, in a first approximation. The boundary layer equations are solved using the fact that the activation energy of the chain branching reaction H + 02 → OH + O is relatively large, employing the reduced kinetic mechanism appropriate for this regime. The equations reduce to a single integro-differential equation for the concentration of atomic hydrogen. The ignition condition can be assumed to be reached when one of the shuffle reactions reaches partial equilibrium. On the other hand, for low plate temperatures, the ignition event is characterized by a thermal runaway. The governing equations reduce to a single one-parameter integro-differential equation, to be solved numerically.
A flame-spread model is analyzed in which heat release occurs at the planar interface of two media, each of which moves with a different but constant velocity. The steady-state, two-dimensional equations for conservation of energy in each medium are solved subject to a prescribed temperature distribution on the downstream half of the interface and continuity of the normal heat flux on the upstream half. Differing thermal conductivities in normal and streamwise directions are allowed in each medium. The approach involves introduction of Fourier transforms in the streamwise coordinate and use of the Wiener-Hopf technique. The model is shown to be equivalent to that of de Ris with radiant transfer neglected and also may be interpreted in terms of distributed electrical or radiant heating without combustion. Parametric results are obtained for various heat fluxes and for spread rates. The study helps to improve understanding of mechanisms of flame spread under conditions controlled by heat transfer.
The present paper explores whether the power of the computer can be used not only for generating numerical solutions, but also for deriving alternative formulations for the given problems. The author limits the study to stiff systems of ordinary differential equations. The task is to translate the general singular perturbation procedures used by human theoreticians for this class of problems into a programmable set of computations; the output from the computations shall provide both the numerical solutions and the alternative formulation of the given problem.
A minimal chemical-kinetic mechanism for the oxidation of n-heptane is reduced to a global two-step mechanism by the systematic application of partial-equilibrium and steady-state assumptions, with the global reaction rates related to the elementary rates. From this mechanism, the structure of spherically symmetrical diffusion flames around n-heptane droplets is analyzed using rate-ratio asymptotics. The outer transport zones are described by the classical flame-sheet analysis with Lewis numbers of unity. The inner structure consists of a thin fuel-consumption zone on the rich side of the flame and a broader but still thin layer on the lean side where H 2 and CO are oxidized. The theory identifies a scalar dissipation rate, related to the droplet diameter, appropriate for droplet burning. From the analysis, the variations in flame temperature and in species concentrations with the stoichiometric scalar dissipation rate X 11 were obtained. Since extinction occurs where x 51 reaches a maximum, the extinction diameters for n-heptane droplets can be estimated from the results and are given for different pressures and ambient oxygen concentrations.
Modification of a previous (Peters, 1988)analysis of the fractal dimension of turbulent premixed flames gives the result D = 7/3. Though this result is not much different from the previous result D = 2.35, the present analysis has novel dynamical implications. Namely, it is proposed that the flame front is not equivalent to a passively convected interface, explaining why fractal flames are observed in flow fields which do not appear to be fractal.
The structure of a diffusion flame near a vortex is investigated by numerical simulation of the conserved-scalar field and analytical estimates of the convective-reactive-diffusive balance. Over a wide range of Péclet numbers, the stoichiometric contour is only partially wrapped around the vortex forming a flame tongue. At the flame tip the compressive strain rate is tangent to the reaction sheet, significantly diminishing normal gradients. The convection tangential to the sheet then creates a convective-reactive balance of thermal enthalpy, in contrast to the usual diffusive-reactive balance produced by compressive strain normal to the sheet. The tip region is thereby predicted to become the hottest part of the flamelet, which appears to agree with experimental observations.
The characteristics of isenthalpic planar premixed flames in counterflowing, high Reynolds number streams with counter- and corotating swirl are analyzed by activation energy asymptotics for a one-step Arrhenius reaction with small departures from unity Lewis numbers. Density changes resulting from heat release are fully taken into account. For moderate rates of rotation, the situation considered in the present study, there is only one stagnation plane that is coincident with the plane of symmetry separating the inviscid outer flows and that involves a viscous layer with an embedded flame in some circumstances. The analysis of a flame in the viscous layer is relevant to the experimentally observed variation of the lean flammability limit with the rotation rate. Comparison with the existing experimental data is carried out.
A numerical study has been initiated to investigate quantitatively the effects of geometry on the ignition characteristics of a reactive solid exposed to interrupted heating rates. The initial geometry selected was a semi-infinite square corner. Reactant depletion was included. Critical ignition conditions were determined for a wide range of physicochemical parameters by numerical integration of the equations via forward finite difference techniques. Geometry and parameter effects were delineated by performing detailed comparisons to existing one-dimensional results, indicating that 1) the ignition delay was reduced by a factor of 2. 5 to 3. 4 depending upon particular parameter values, 2) critical ignition temperatures were slightly higher for the two-dimensional case, 3) delay time as a function of system parameters was developed and reduced to the same form as the one-dimensional case.
We consider a system of reaction diffusion equations which describe gasless combustion of condensed systems. To analytically describe recent experimental results, we show that a solution exhibiting a periodically pulsating, propagating reaction front arises as a Hopf bifurication from a solution describing a uniformly propagating front. The bifurcation parameter is the product of a nondimensional activation energy and a factor which is a measure of the difference between the nondimensionalized temperature of unburned propellant and the combustion products. We show that the uniformly propagating plant front is stable for parameter values below the critical value. Above the critical value the plane front becomes unstable and perturbations of the system evolve to the bifurcated state, i.e., to the pulsating propagating state. In our nonlinear analysis we calculate the amplitude, frequency and velocity of the propagating pulsating front. In addition we demonstrate analytically that the mean velocity of the oscillatory front is less than the velocity of the uniformly propagating plane front.
This chapter reviews the history of combustion science and technology. The pioneering experiments in combustion research, some 600,000 years ago, were concerned with flame propagation rather than ignition. The initial ignition source was provided by nature in the form of the electrical discharge plasma of a thunderstorm or as volcanic lava, depending on location. The first half-million years of combustion research were characterized by the supposition that there are infinite reservoirs of fuel and repositories for products. The limitations to applying control by electric fields to practical combustion systems have been calculated for all three phases: removal from products, control of residence time in growth zones, and the removal of charged nuclei. The suppression of soot formation and its promotion can be greatly strengthened when electron emission is increased by a negative potential on the emitter and suppressed when electrons are prevented from escaping by the application of a positive field.
Experiments were performed to obtain histories of surface temperatures and rates of upward flame spread for velocity oriented, thermally thick wood slabs exposed to surface fluxes of thermal radiation up to 2.6 W/cm². Above a critical irradiance sustained upward flame spread occurred for Douglas-fir particle board with pilot initiation at the base of the fuel face. Data obtained included temperatures, flame heights, pyrolysis-front heights, combustion duration, and char-layer thickness for various irradiances and preheat times. The measurements were compared with theory.
Scaling properties of the field equation governing propagation of a thin flame front in a turbulent medium are discussed. It is shown that if the turbulent flame velocity uT can be expressed through the turbulence intensity urms and the laminar flame velocity u0 as uT/u0 ∞ (urms/u0), then α → 1 in the scale invariant regime when urms → ∞.
The problem of propagation of turbulent premixed flame is analyzed using the field equation introduced recently by Kerstein, Ashurst and Williams (1987). The dynamic renormalization group method is applied to this equation and the formula for the turbulent flame velocity is derived in the lowest order in the ε-expansion. The formula, which does not include adjustable parameters, agrees well with experimental (Abdel-Gayed et al., 1984) and numerical (Ashurst & Barr 1983) results on flame propagation in high-Reynolds number turbulent media. Ways to design transport and large-eddy (sub-grid) models for simulation of combustion processes, based on the ideas developed in the present paper, are discussed.
A general invariant expression is derived for the stretch experienced by a flame due either to its motion or to the nonuniform flow of the gas through it. This expression is given in terms of the local fluid velocity and the shape of the flame front. Specific examples in which the flame stretch takes a simplified form are discussed. Some remarks are made regarding the relation between the three distinct properties of flames: stretch, speed and temperature.
A short information is given on the results of work carried out at the Branch of the Institute of Chemical Physics, U.S.S.R. Academy of Sciences, since 1967 on studying the combustion processes caused by the interaction of chemical elements in the condensed phase and leading to the formation of refractory compounds. New phenomena and processes are described which are revealed when investigating the combustion of the systems of this class, viz solid-phase combustion.fast combustion in the condensed phase, filtering combustion, combustion in liquid nitrogen, spinning combustion, self-oscillating combustion and repeated combustion. A new direction in employment of combustion processes is discussed, viz. a self-propagating high-temperature synthesis of refractory nitrides, carbides, borides, silicides and other compounds.
In this work we present the exact solution for the rate of creeping flame spread over thermally thin materials based on de Ris' (1969) formulation of the problem. The publication of the present work was prompted by a recent paper by Wichman and Williams (1984). These authors noted that de Ris (1969) obtained theoretical values for flame spread rates which are by a factor of two larger than recent experimental data (Fernandez-Pello et at., 1981). Wichman and Williams (1984) correctly suggested that this discrepancy between the theory and the experiment may have resulted from an approximation by de Ris(1969) of an irrational kernel by a rational function. The substitute kernel used by de Ris (1969) allowed him to obtain an analytical solution of the flame spread rates; however, this approximation may have introduced errors in the calculation of proportionality constants. The exact solution for creeping flame spread rates, which we present in the final analysis, agrees with the experimental data.
In this work, we use essentially the same formulation of the flame spread problem as de Ris (1969) presented earlier. Specifically, we consider flame spread over a thermally thin material in an opposing air flow; flame spread occurs by thermal djffusion of heat upstream of the fire origin.
An asymptotic analysis of large activation energy with regard to ignition or flame initiation of a cold combustible gas stream coming into contact with a hot inert gas stream is presented. Two-step reaction kinetics, that is, a first-order chain-branching reaction and a second-order chain-termination reaction, responsible for ignition is adopted, and the dependence of ignition time (or flame attachment distance) on the second-order recombination reaction is examined. Equations for ignition delay are given by means of elementary functions, which show that ignition time increases with an increase in recombination reaction rate.
The asymptotic structure of a counterflow methane-air diffusion flame is analyzed using a three-step chemical kinetic mechanism, which was deduced in a systematic way through steady state and partial equilibrium assumptions from a detailed chemical kinetic mechanism for oxidation of methane. The rates for the three steps are related to the rates of elementary reactions. First the kinetic scheme is based on the most important (principal) reactions to derive the basic structure. When a number of additional elementary chemical reactions are added the results of the asymptotic analysis are found to be in very good agreement with previous numerical calculations that used a complete kinetic mechanism, as well as with experiments.
A theoretical model is developed for estimating the rate of flame spread under conditions of heat-transfer control with account taken of the fact that the gas velocity is not uniform. It is shown that the functional dependence of the spread rate on parameters such as the external gas velocity is modified from that obtained in the classical study of deRis. The modified formula is capable of lending interpretation to previously anomalous experimental observations.
We present a nonlinear theory for the evolution in the front position of a premixed planar flame held downstream of a coplanar porous flat burner of infinite conductance. The theory assumes a unit Lewis number for the reactant and an Arrhenius law for the overall reaction rate. The problem is solved with the aid of singular perturbation methods for large activation energies and Fourier transform techniques. We show that (i). Before the minimum steady state standoff distance from the burner is reached, the flame front position exhibits slow periodic oscillations, the evolution of which follows a delayed nonlinear first-order differential equation of Hutchinson type (ii). The oscillation-generating time lag results from the travel of temperature disturbances between the burner surface and the flame front (iii). The time-averaged standoff distance is appreciably different from its steady analog.
A review is presented, primarily of work pursued by the author and his co-workers over the past five years in the area of extinction of laminar flames. Most of the emphasis is placed on diffusion flames, but some consideration is given to premixed flames. The objective has been to develop simplified descriptions of extinction involving a few chemical-kinetic parameters that can be measured readily in the laboratory. The results may be applied to extinction of fires if the turbulent flames therein are composed of collections of laminar flamelets.
A model for an edge-cooled flat flame burner is used to obtain expressions for the flame speed, flame temperature, standoff distance as well as the quenching distance for a plane flame front. For a given standoff distance there is a low-temperature as well as a high-temperature solution. It is shown by a linear stability analysis of the plane front that the high-temperature solution is unstable when the Lewis number is sufficiently large and the inflow velocity sufficiently less than the adiabatic flame speed. It is also shown that this instability is the type that will lead to a bifurcating time-periodic solution describing a pulsating flame.
A system of reaction-diffusion equations describing the propagation of combustion waves along a thermally insulated cylindrical sample of solid fuel is considered. Uniform propagation of a plane combustion wave is subjected to a linear stability analysis. It is shown that, if the activation energy is sufficiently high and the diameter of the sample sufficiently large, then the experimentally observed spinning propagation of combustion waves appears as a Hopf-type bifurcation of the solution corresponding to a plane wave. The possibility of similar phenomena in gas combustion is discussed. 8 refs.
Models are considered for diffusion flames involving a fuel consisting of mixtures of carbon monoxide, hydrogen, and nitrogen and an oxidizer consisting of mixtures of oxygen and nitrogen. Initially the kinetic scheme is reduced systematically to the two-step mechanism CO + HâO {r reversible} COâ + Hâ + Oâ {r reversible} 2HâO, the water-gas shift, and hydrogen oxidation. In a model for low strain rates there is a broad region of water-gas equilibrium, bounded on the fuel-lean side by a thin zone of hydrogen oxidation and on the fuel- rich side by a thin zone of sudden water-gas freezing caused by chain-carrier removal. Structures of each of these zones are analyzed with the aid of asymptotic methods, and it is shown that a three-step mechanism is needed to obtain a reasonably accurate description of the water-gas freezing. In these analyses simplified transport descriptions are employed in which only the Lewis number of hydrogen is allowed to differ from unity.
A simplified theoretical model is developed for describing the flame structure and deflagration velocity of steady, planar, adiabatic combustion of nitramines. The model involves exothermic decomposition in a liquid layer, equilibrium vaporization and exothermic combustion in the gas. Judicious selection of energetic and rate parameters on the basis of available experimental information resulted in reasonable agreement between predicted and measured burning rates and pressure and temperature sensitivities.
The combustion of solids and high-density fluids, such as the combustion synthesis of refractory materials or the high-pressure oxidation of organic compounds in supercritical water, is often characterized by a highly temperature-dependent mass diffusivity. In particular, the case of an Arrhenius dependence of this quantity on temperature is considered, where the activation energy of reactant diffusion is of the same order as that of the overall chemical reaction. Exploiting the asymptotic limit of large activation energies, it is shown that reactant diffusion becomes significant in the thin reaction zone, where a balance among convection, reaction, and diffusion is maintained. Relative to the case of either a zero or constant mass diffusion coefficient, this leads to modifications in the steady, planar flame-speed eigenvalue. A new asymptotic model is then derived that predicts corresponding shifts in the neutral stability boundary that marks the transition from steady to nonsteady modes of flame propagation.
An asymptotic model is derived for the nonsteady, nonplanar deflagration of energetic solids, such as nitramines, that experience exothermic reactions in liquid layers at their surfaces. The analysis takes into account the role of interphase heat transfer and the relative motion of gas and liquid in the resulting two-phase region at the surface. A basic solution corresponding to steady, planar burning is obtained, and a linear stability analysis yields a neutral stability boundary beyond which the basic solutionis unstable to pulsating disturbances. In particular, it is found that stability of the basic solution is lost for sufficiently large values of the activation energy and/or sufficiently small values of the interphase heat transfer parameter. In the combined limit that the latter approaches infinity and a velocity-perturbation parameter, which accounts for two-phase viscous and Marangoni effects, approaches zero, the neutral stability boundary is the same as that encountered for separate-phase solid propellants with an intrusive gas flame.
Simplified asymptotic equations for the interaction of high-frequency, small-amplitude waves in a chemically reacting gas are developed and analyzed. The equations for nonlinear acoustic simple waves in a reacting mixture are solved explicitly, and reveal substantial wave amplification through combustion for a wide range of activation energies and heat release. The asymptotic equations for the resonant interaction of almost periodic wave trains in a reacting gas are also developed. One prominent new effect of acoustic resonance documented through numerical experiments is the transfer of energy from cold spots to hot spots through resonant interaction; this leads to more dramatic wave amplification through combustion that in the case of nonlinear simple waves. The asymptotic regime discussed here, involving a balance of small amplitudes and small wavelengths for fixed activation energy, illustrates in a simplified context some of the main new phenomena being analyzed in the current theoretical combustion literature in more complex situations.
Theoretical studies on flame stability have generally been based on one of two approaches: the hydrodynamic model which accounts for thermal expansion due to combustion, but ignores flame structure, and the diffusional-thermal model which considers flame structure in a prescribed constant-density flow, thus ignoring thermal expansion. The present study is based on a model in which both thermal expansion and flame structure are accounted for. The model describes the dynamics of flame fronts including their stability. The stability of plane and curved flames is discussed. In particular, the effect of flame stretch, thermal expansion, Lewis number, and Prandtl number on stability is determined.
The merged regime of an ozone decomposition flame (Linan and Rodriguez, 1985) is investigated theoretically, applying the activation-energy asymptotic approach of Rogg and Wichman (1985). The structure and burning rate of the flame are characterized by carrying the analysis to the second order in the reciprocal of a Zel'dovich number, and numerical results based on the rate data and transport-coefficient correlations of Heimerl and Coffee (1980) are presented in tables. It is found that the two-term expansion reduces the errors of the one-term expansion to a few percent. 5 references.
The objective of the book is twofold, first to present basic material on solid propellant rockets which can be used for classroom instruction and second to carry the reader to the frontiers of research in a number of specific areas of solid propellant rocketry. Although there is some material which might appropriately be used in undergraduate courses, the instructional value of the book lies primarily at the graduate level. An attempt has been made to enhance the educational utility of the monograph by presenting the more elementary aspects of the subject first, before proceeding to detailed and more advanced treatment of specific areas of research. An attempt has also been made to present the research topics in a pedagogic manner, to aid the graduate student or the practising engineer who is not familiar with the subject material.
A numerical procedure is presented for implementing the Green’s function method of sensitivity analysis in chemical kinetics. The procedure is applied to three sets of chemical reactions: the Chapman mechanism for ozone kinetics, a mechanism for methane combustion and a model for formaldehyde oxidation in the presence of carbon monoxide. Whenever possible, comparisons with alternative methods of sensitivity analysis are made. It is shown that carefully analyzed sensitivity profiles can be used in conjunction with experiments and/or models to obtain useful information about chemical kinetic behavior. By using methods from multivariable calculus an entire family of sensitivity coefficients may be derived from the elementary sensitivities obtained by solving differential equations. Each elementary or derived sensitivity coefficient has a unique physical interpretation in terms of an experiment or modeling calculation. A simple nonlinear interpolation formula is suggested for easily estimating higher‐order sensitivity information. Finally the overall computational efficacy of the Green’s function method of sensitivity analysis is assessed.
Two-term expansions for burning velocity in activation-energy asymptotics are developed for four-species, two-reactant, steady, planar, adiabatic laminar flames with irreversible one-step chemistry. General Stefan-Maxwell transport is included with Soret and Dufour effects, pressure gradients, body forces and radiant transport neglected. The results lead to identification of effective Lewis numbers (combinations of Lewis numbers for different pairs of species) that affect the burning velocity, and by exhibiting influences of fully variable transport and thermodynamic properties they provide a basis for achieving improved accuracy in asymptotic analysis of flame propagation in two-reactant systems such as hydrogen-halogen mixtures. When suitably specialized to the more restrictive conditions considered in earlier work, the burning velocities obtained here agree with those derived previously
The characteristics of planar premixed laminar flames are analyzed for general fields of rates of strain that include axisymmetric and two-dimensional counterflowing streams of reactants and products as special cases. Activation energy asymptotics are employed for a one-step Arrhenius reaction with small departures from adiabaticity and from unity values of the Lewis number. Density changes by heat release are fully taken into account. Low to moderate strain rates are considered intitially but high rates are subsequently treated. Attention is directed mainly to the influence on flame structure, extinction and ignition of a parameter ĉ determining the three-dimensional character of the rate of strain field. Axisymmetric flow corresponds to ĉ= 1, two-dimensional flow to ĉ= 0, while there is inflow along one of the coordinates parallel to the reaction sheet if − 1 ĉ≤0. Attention is devoted to the existence of multiple solutions and their features for − 1 ĉ≤0. The boundaries for extinction and ignition in terms of parameters measuring the deviations from adiabaticity and from unity Lewis number are found to be insensitive to ĉ but Damköhler numbers on these boundaries vary with ĉ. The significance of density variations for the special case of ĉ= − 1 is discussed.
An expression is derived for the upstream heat flux at the inception point of a spreading diffusion flame. The spreading flame is modeled, near the inception point, as a quenched diffusion flame. Since quantitative calculations for flame quenching in the flame-spread configuration do not exist, the conditions for quenching of the spreading flame are assumed to be essentially identical to the conditions for quenching of a laminar flame-jet formed, for example, by fuel issuing into an oxidizing atmosphere from a tube or a duct or by a flame in counterflowing streams of fuel and oxidizer.
The one-dimensional stability of an isobaric burner-stabilized premixed flame is investigated for arbitrary Lewis number and stoichiometry in the asymptotic limit of large activation energy. Assuming a one-step irreversible chemical reaction in which fuel and oxidizer react to form a product, a linear stability analysis is Used to calculate the neutral stability boundary in Lewis number-activation energy space as a function of incoming flow velocity (or equivalently, the burned temperature) The major result is that although a steady-state adiabatic flame is likely to be stable for typical parameter values, a value of the incoming flow velocity sufficiently less than the adiabatic flame speed is destabilizing to the extent that the unstable region becomes feasible for many flames. Consequently, if all other parameters are fixed, there exists for such flames a critical value of the incoming flow velocity at which the time-asymptotic solution to the time-dependent problem bifurcates from the nontrivial steady state solution. Time-oscillating burner-stabilized flames are observable experimentally, and realistic numerical calculations of a fuel-rich H2IO2 premixed flame are presented which further verify the existence of such time-periodic solutions for sufficiently small incoming flow velocities. In addition, the numerical results indicate the existence of a secondary instability phenomenon in which a singly periodic pulsating solution abruptly becomes doubly periodic.
The structure of a steady one-dimensional isobaric deflagration is examined for the case of a direct first-order one-step irreversible exothermic unimolecular decomposition under Arrhenius kinetics. In particular, the eigenvalue giving the speed of propagation of the laminar flame into the unburned gas is sought for constant Lewis number of order unity. The method of matched asymptotic expansion is invoked in the physically interesting limit of activation temperature large relative to the adiabatic flame temperature. The leading approximation for the eigenvalue is found to be a generalization of the result given by Zeldovich, Frank-Kamenetski, and Semenov for Lewis number unity. The first two terms in the asymptotic expansion for the eigenvalue yield an expression superior to any previously published.
A formula for the ignition time, that Bradley obtained empirically, is derived analytically by a rigorous asymptotic analysis of the limit of large activation energy. Two terms in the expansion must be retained. The second term reveals the existence—just prior to ignition—of a reactive-diffusive zone at the surface of the solid and a transient-diffusive zone in the interior. The analysis also exhibits a universal correction factor for Bradley's formula, of order unity.
The structure of a steady, planar. premixed flame in a slowly expanding gas flow is analyzed. The reaction kinetics are represented by a mechanism consisting of a chain branching reaction and a chain breaking reaction. An asymptotic analysis is performed in the limit of a large value for the activation energy of the chain branching reaction, with the activation energy for the chain breaking reaction presumed to be zero. The analysis is valid for cases where the mass fraction of the intermediate species is of order unity, An analytical expression is obtained for the change of the mass burning rate with the Karlovitz number, the parameter characterizing flame stretch. This expression shows the role of differential diffusion of heat and the reactant, of differential diffusion of reactant and intermediate species and of enhanced diffusion of intermediates on flame propagation. It is seen that these three different effects may cancel each other. Depending on the parameter range the mass burning rate may be either decreased or increased.
Renormalization group approach to the problem of turbulent flame speed, alternative to that recently adopted by Yakhot is proposed An equation for turbulent flame speed as a function of the turbulent field intensity is derived. The “bending” effect, resembling that known from experiments, is described.
The reaction rate of a heterogeneous, or diffusion-limited, reaction in a solid combustible mixture, sub seq uent to melting of one of the reactants, is proportional to exp[ - m(1- Y)](l- y) exp( - [etilde]/[ttilde]), where Y is the unreacted fraction and r is the temperature. We exploit the largeness of the activation energy Eto derive an asymptotic model for the propagation of a reaction front through such a mixture. The analysis parallels a similar study of homogeneous condensed combustion (Margolis, 1983), in which the reaction rate both before and after melting is proportional to Y exp( - [etilde]/[ttilde]). We derive formulas for the propagation speed of a steady planar reaction front, as well as present the asymptotic model for nonsteady, non planar combustion. This model is identical in form to that obtained from a special limiting case of the homogeneous problem. Thus, the well-known loss of stability of the stead y planar solution for sufficiently large values of a modified activation energy parameter, and the bifuraction to va rious pulsating and spinningmodes of propagation, is essentially unchanged from that predicted for homogeneous condensed combustion (Margolis et al.,1985). Con sequently, this nonsteady instability is attributable to the more highly nonlinear Arrhenius dependence on temperature for large activation energies, and is independent of the functional dependence of the reaction rate on Y.