Nondimensional Activation Energy Research Articles

Laminar flamelet concepts in turbulent combustion

The laminar flamelet concept covers a regime in turbulent combustion where chemistry (as compared to transport processes) is fast such that it occurs in asymptotically thin layers—called flamelets—embedded within the turbulent flow field. This situation occurs in most practical combustion systems including reciprocating engines and gas turbine combustors. The inner structure of the flamelets is one-dimensional and time dependent. This is shown by an asymptotic expansion for the Damköhler number of the rate determining reaction which is assumed to be large. Other non-dimensional chemical parameters such as the nondimensional activation energy or Zeldovich number may also be large and may be related to the Damköhler number by a distinguished asymptoiic limit. Examples of the flamelet structure are presented using onestep model kinetics or a reduced four-step quasi-global mechanism for methane flames. For non-premixed combustion a formal coordinate transformation using the mixture fraction Z as independent variable leads to a universal description. The instantaneous scalar dissipation rate χ of the conserved scalar Z is identified to represent the diffusion time scale that is compared with the chemical time scale in the definition of the Damköhler number. Flame stretch increases the scalar dissipation rate in a turbulent flow field. If it exceeds a critical value χ q the diffusion flamelet will extinguish. Considering the probability density distribution of χ , it is shown how local extinction reduces the number of burnable flamelets and thereby the mean reaction rate. Furthermore, local extinction events may interrupt the connection to burnable flamelets which are not yet reached by an ignition source and will therefore not be ignited. This phenomenon, described by percolation theory, is used to derive criteria for the stability of lifted flames. It is shown how values of ∋ q obtained from laminar experiments scale with turbulent residence times to describe lift-off of turbulent jet diffusion flames. For non-premixed combustion it is concluded that the outer mixing field—by imposing the scalar dissipation rate—dominates the flamelet behaviour because the flamelet is attached to the surface of stoichiometric mixture. The flamelet response may be two-fold: burning or non-burning quasi-stationary states. This is the reason why classical turbulence models readily can be used in the flamelet regime of non-premixed combustion. The extent to which burnable yet non-burning flamelets and unsteady transition events contribute to the overall statistics in turbulent non-premixed flames needs still to be explored further. For premixed combustion the interaction between flamelets and the outer flow is much stronger because the flame front can propagate normal to itself. The chemical time scale and the thermal diffusivity determine the flame thickness and the flame velocity. The flamelet concept is valid if the flame thickness is smaller than the smallest length scale in the turbulent flow, the Kolmogorov scale. Also, if the turbulence intensity v′ is larger than the laminar flame velocity, there is a local interaction between the flame front and the turbulent flow which corrugates the front. A new length scale L G =v F 3 /∈ , the Gibson scale, is introduced which describes the smaller size of the burnt gas pockets of the front. Here v F is the laminar flame velocity and ∈ the dissipation of turbulent kinetic energy in the oncoming flow. Eddies smaller than L G cannot corrugate the flame front due to their smaller circumferential velocity while larger eddies up to the macro length scale will only convect the front within the flow field. Flame stretch effects are the most efficient at the smallest scale L G . If stretch combined with differential diffusion of temperature and the deficient reactant, represented by a Lewis number different from unity, is imposed on the flamelet, its inner structure will respond leading to a change in flame velocity and in some cases to extinction. Transient effects of this response are much more important than for diffusion flamelets. A new mechanism of premixed flamelet extinction, based on the diffusion of radicals out of the reaction zone, is described by Rogg. Recent progress in the Bray-Moss-Libby formulation and the pdf-transport equation approach by Pope are presented. Finally, different approaches to predict the turbulent flame velocity including an argument based on the fractal dimension of the flame front are discussed.

Bifurcation of Pulsating and Spinning Reaction Fronts in Condensed Two-Phase Combustion

Abstract We employ a nonlinear stability analysis to describe the bifurcation of pulsating and spinning modes of combustion in condensed media. We adopt the two-phase model of Margolis (1983) in which the modified nondimensional activation energy Δ of the reaction is large, but finite, and in which the limiting component of the mixture melts during the reaction process, as characterized by a nondimensional melting parameter M. We identify several types of non-steady solution branches which bifurcate from the steady palanar solution and show that they are supercritical and stable only for certain realistic ranges of M. For example, the spinning modes, though supercritical and stable for a range of M > 0, are subcritical and unstable for M = 0.

A generalized Semenov model for thermal ignition in nonuniform temperature systems

Critical properties of the Semenov model which describes thermal ignition in a uniform temperature system have been reanalyzed in terms of independent parameters, nondimensional activation energy, and surface heat loss. The analysis is then extended to the distributed temperature problem by the introduction of a correlation factor which effectively establishes a link between the Semenov model and the Frank-Kamenetskii parameter. This approach is capable of giving satisfactory results for properties such as ignition criteria, ignition times, and temperature profiles for a system with nonuniform temperature distributions. Specific results are presented mostly for the slab, but extensions to the cylinder and the sphere are also discussed.

A Numerical Study of Tunnel Fires

Abstract Abstract-Three theoretical models of plane flames burning on a cooled porous-plug type of flame-holder are reviewed and compared with experimentally observed relationships between stand-off distance, flame speed and temperature. It is shown that for most practical burners their conductance is large and that for near adiabatic conditions, the order of the non-dimensional stand-off distance ceases to be 0(1), but is O(ln θ) where θ is the non-dimensional activation energy.

Influence of Temperature Fluctuations on the Arrhenius Model for Reactionst

Abstract The influence of temperature fluctuations on the Arrhenius model for chemical reactions is estimated for idealized low-level turbulence and large values of non-dimensional activation energy.

Turbulent mean reaction rates in the limit of large activation energies

Closed-form expressions for the turbulent mean reaction rate and its covariance with the temperature are derived for premixed and non-premixed combustion. The limit of large activation energies is exploited for a chemical reaction rate that, by virtue of coupling functions, depends on the mixture fraction and a non-equilibrium progress variable only. The probability density function (p.d.f.) formulation with an assumed shape of the p.d.f. is used; a beta-function distribution is assumed for the progress variable. The mean reaction rate is expressed in terms of the mean and the variance of the temperature and, for non-premixed combustion, of the mixture fraction. The reaction kinetics are represented by the non-dimensional activation energy and the laminar flame velocity. For non-premixed systems the possibility of local extinction by flame stretch is considered.

Ignition of a Reactive Solid Exposed to a Step in Surface Temperature

The title problem is classical in ignition theory. Here it is solved by an asymptotic analysis in which the nondimensional activation energy is treated as a large parameter. Inert and reactive zones are identified, and a temperature peak develops in the latter during a stage of transition to ignition. A criterion of thermal runaway yields a formally correct asymptotic expansion for the ignition time which agrees with results of numerical integrations if the activation energy is sufficiently large.

Propagation of a Pulsating Reaction Front in Solid Fuel Combustion

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 bifurcation 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 temperatures 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. diffusion equations for temperature and concentration, with the diffusion coefficient of the combustible component which limits the chemical reaction taken to be zero. In an interesting recent paper, Merzhanov, Filonenko and Borovinskaya (2) report on various new phenomena in the combustion of condensed systems. Among these phenomena they describe what they refer to as autooscillatory combustion for a mixture of 3Nb + 2B. They find that the velocity of propagation of the reaction front exhibits periodic pulsations. They also noted that the burned samples have a layered structure, normal to the front, with the number of layers equal to the number of pulsations. This work is described in the recent monographs of Novozhilov (3), and Zeldo- vich, Leypunsky and Librovich (4) in which they attribute the cause of these pulsations to the absence of diffusion. The existence of pulsating combustion fronts was first described theoretically by Shkadinsky, Khaikin and Merzhanov (5) who obtained

A theory for spherical and cylindrical laminar premixed flames

A model has been developed which attempts to predict the behavior of the radially symmetric premixed laminar flame governed by a first order reaction. Provisions have been made for the necessary non-adiabatic conditions, ranges of Damkohler number, ranges of non-dimensional activation energy, arbitrary Lewis number, and either cylindrical or spherical geometry. A technique for the solution of the governing equations has been proposed which allows both of the unknown initial derivatives to be accurately determined in a reasonable amount of computation time. Flame structure, conductive heat loss, diffusion potential, flame stand-off distance, and flame thickness have been examined for spherical symmetry for a variety of Damkohler numbers and reduced activation energies.

Use of asymptotic analysis of the large activation- energy limit to compare graphical methods of treating thermogravimetry data

An earlier numerical analysis showed that the second approximate method of Horowitz and Metzger can be rendered exceedingly accurate for reduction of thermogravimetry data. It is demonstrated here that this result can be justified on the basis of an asymptotic expansion with a nondimensional activation energy as the large parameter. The order of magnitude of the error is ascertained for this and two other approximate methods. Higher-order terms in the approximation are developed.

Radiant ignition of a reactive solid with in-depth absorption

An asymptotic analysis of the limit of large activation energy is presented for radiant ignition of asolid that experiences a one-step Arrhenius reaction in the condensed phase. Both constant and time-dependent radiant-energy fluxes are considered, and the complete range of values is covered for the absorption coefficient μ. It is shown that as μ increases, the structure of the transition stage, which follows the inert heat-conduction stage, passes from a thermal explosion without heat conduction, to a single transient heat-conduction zone with distributed chemical heat release, to a two-zone structure composed of a reactive-diffusive-absorptive zone near the surface and a transient-diffusive zone in the interior. For very high values of μ, the reactive-diffusive-absorptive zone further splits into a surface absorption zone and an interior reactive-diffusive zone, thereby reproducing results obtained previously for ignition by a surface-applied energy flux. The analysis shows that contrary to earlier expectation, the nondimensional absorption coefficient must be at least as large as the nondimensional activation energy for in-depth absorption to affect the ignition time negligibly.