The analysis of diffusion controlled reactions with non-equal diffusivities of the reactatnts

We show, using as a model problem the description of the reaction of a fuel pocket with the oxygen of the environment, how to derive a system of conservation equations to describe the evolution of the temperature and concentrations of the reactants, when these are not initially mixed, or only partially mixed, and the characteristic reaction time is very short compared with the characteristic diffusion time

Diffusion-controlled combustion

We devote this brief review to some relevant aspects of diffusion-controlled combustion. After a survey of the conservation equations involved, we shall describe the Burke-Schumann limit, which is applicable when the reaction time at the flame is very short compared with the mixing time. Using as a protopypical example the How downstream from a fuel injector in a combustor chamber, we next introduce some phenomena related to finite-rate kinetics. We shall see how the high temperature sensitivity typical of combustion reactions is responsible for the presence near the injector of chemically frozen regions of low temperature where the reactants mix without chemical reaction, these regions being separated by thin premixed flames, with rich and lean branches, from regions of near equilibrium flow, where the reactants coexist only in a thin trailing diffusion flame. The role of these triple llames in the ignition, anchoring, and lift-off processes of diffusion flames will be briefly discussed

Analytical description of chaotic oscillations in a toroidal thermosyphon

Multiple time scales and singular perturbation techniques are used to describe the ordinary and the chaotic oscillations due to natural convection in a fluid loop subject to a known external heat flux. The turbulent flow in the loop is modelled using the hydraulic approximation with a quadratic friction law. No steady solutions exist if the heat is added mainly to the top half and extracted from the bottom half of the loop, and two steady convective solutions may exist if one proceeds otherwise; these convective solutions may loose stability when the heat input is shifted from the side toward the bottom. The instability leads, first, to a periodic convective flow and then, after a period doubling Feigenbaum cascade, to a chaotic motion. An intermittent type transition from limit cycles to chaos is also found in the analysis. The transition to chaos can be described in terms of a non-invertible return map, obtained by singular perturbation techniques for loops with long length, when the system becomes strongly dissipative

Ignition and flame sptead in laminar mixing layers

In order to identify some of the mathematical problems encoun-tered in analyzing the ignition and ñame spread in mixing layéis, \ve shall describe the structure of the laminar mixing layer between two parallel streams of a fu el and air, initially separated by a splitter píate, undergoing an Arrhenius reaction. If the activation energy of the reaction is lower than a critical valué, there is only one steady solution of the problein, showing a transition from nearly frozen mixing to diffusion controlled combustión downstream of the píate. For higher, typical, valúes of the activation energy we may find a multiplicity of solutions, depending on the valué of the Damkohler number D, characterized by the temperature of the hotter of the two streams. For valúes of the Damkohler number lower than a critical valué Dc, there is a solution where a thermal runaway is found to oceur, after an induction length, at a point that serves as the origin of premixed flames that do not propágate upstream. For valúes of D larger than a critical lift-ofif valué D¡(< Dc) we find a solution with diffusion controlled combustión in a diffusion fíame. This fíame is anchored, with a triple-fíame structure, in the near wake of the splitter píate, where upstream heat conduction to the píate plays a dominant role. In the interval D¡ < D < Dc there is a third, unstable, solution. This solution determines where an external ignition source shonld be placed so that, by means of upstream triple-flame propagation to the splitter píate, transition to diffusion controlled combustión can take plac

The effects of displacement induced by thermal perturbations on the structure and stability of boundary-layer flows

The free-interaction influence of a thermal expansión process in boundary-layer gas flow is analyzed using the formalism of triple-deck theory. The physical model considered is the forced convection of a gas flowing over a flat plate subject to a heated slab. Both linearized and full nonlinear solutions are obtained using Fourier transform methods and spectral numericaí techniques. The influence of monochromatic thermal perturbation on boundary-layer stability (lower branch) is studied and first-ordcr correction of the lower branch neutral stability curve for the boundary-layer flow has been obtained. The shift of neutral stability is then computed for different values of the therma! perturbation wave number, making unstable some otherwise stable modes

Mixing layer ignition of hydrogen

A theoretical analysis is given for the high-temperature ignition in a laminar mixing layer between hydrogen and air at the high temperatures characteristic of supersonic combustión. We analyze the most important practical case where the temperature of the air stream is higher than that of the hydrogen stream. In this case, the chemical reactions responsible for ignition occur in the air side of the mixing layer, where the mixture is lean. A simplified reduced mechanism is found to describe the ignition process. The radicáis OH and H follow the steady-state approximation while the radical O is the chain branching species following an autocatalytic reaction with moderately large activation energy. Numerical results of the governing equations for large valúes of the activation energy are presented and from a symplified analysis, we obtain a closed form solution of the ignition distance as a function of the physicochemical parameters

Autoignition in nonpremixed flow

The objective of this investigation has been to improve understanding of autoignition processes in nonpremixed flow fields of the types encountered in Diesel-engine ignition, through theoretical analyses that employ asymptotic methods of applied mathematics. The work was intended to develop formulas and equations that can be used in activities of applied research, such as code development, aimed at providing tools useful for the design of Diesel engines. The formulas may also be used directly for ignition estimates.Characteristic time scales were identified for these ignition problems. Their relative magnitudes were employed to define different regimes of ignition and to obtain simplified partial differential equations that describe ignition in these regimes. Effects of turbulence on ignition were addressed. Special attention was devoted to unsteady mixing layers, involving both variable strain and variable pressure, for which ignition-time formulas were derived. In addition, ignition analyses were completed for variable-volume chambers with arbitrary initial spatial variations of temperature and composition, to determine pressure histories produced by ignition-front propagation. These studies were based on one-step, Arrhenius approximations for the chemical kinetics and were restricted to ignition stages that precede ordinary flame propagation. Additional work considered triple-flame propagation that can odcur in mixing layers after ignition, with this same chemical-kinetic description, and asymptotic analysis of n-heptane ignition on the basis of a four-step, semi-empirical model for the chemical kinetics. In this latter study, the region of negative effective overall activation energy, between 800 K and 1100 K, was identified as exhibiting unusual ignition dynamics, and the asymptotic ignition-time formulas were shown to give good agreement with predictions of numerical integrations. This research has helped to strengthen the foundations of ignition theory for nonuniform media. It provided simplified descriptions of ignition processes that can be employed in studies of Diesel combustion that are oriented more towards development than are the present investigations. The asymptotic methods employed in this work thus appear capable of providing quite useful results

External heating of a flat plate in a convective flow

The steady-state and íransient processes of the exteinal heating of a flat píate under a convective flow is studied in thís paper, with inclusión of the axial heat conduction tíirough the píate. The balance equations reduce to a single integro-differential equation with only one parameter, a, denoting the ratio of the ability of the píate tocarry heat inthestreamwisedirectionto the abilityof the gas tocarry heat outof the píate. The two Iímits of a good conducting plate ([alpha]-»[infinite]) and a bad conducting plate ([alpha]-»[infinite]) are analised through the application of a regular perturbation procedure for the first case and a singular perturbation technique for the latter. The existence of two boundary layers at both edges of the píate is shown and their structure are analysed. The evolution of the temperature of the píate is then obtained for a constant external energy flux input

The structure of diffusion flames

These notes are devoted to the analysis of reacting flows in systems where the reactants are initially unpremixed. We shall deal here with the combustión reactions, encountered very often in applications, which take place in the gas phase, between a fuel and an oxidizer, (typically the oxygen of the air). The combustión reactions are exothermic, so that we shall see large temperature differences in the flow field, and the reaction rates are very sensitively dependent on temperature. Then in regions of the flow field where the temperature is low the reactants will mix without chemical reactions, while in regions of high temperature the flow will be in chemical equilibrium. When the reaction can be modelled by an overall irreversible reaction between the fuel and the oxygen, the chemical equilibrium condition implies that one of the reactants has locally zero concentration, because it has been previously completely consumed by the chemical reaction. The chemical reaction has taken place in thin reaction layers in the form of rich or lean premixed flames, when they sepárate regions of frozen flow from regions of equilibrium flow, or diffusion flames, when they sepárate a región without fuel from a región without oxidizer

El plasma en la atmósfera

La alta atmosfera esta constituida por un gas ionizado (plasma) de débil densidad. que, bajo la acción de los campos electromagneticos genera-dos en la tierra y las radiaciones procedentes del espacio exterior, da lugar a multitud de fenome-nos físicos de extraordinario interés para la ciencia y para la técnic