Stephen B. Margolis

An Asymptotic Theory of Condensed Two-Phase Flame Propagation

A model is presented for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process. An asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed. A linear stability analysis is then used to show that for sufficiently large values of the activation energy and/or a special group of melting parameters, the steady, planar solution loses stability to various types of planar and nonplanar pulsating modes. The effect of melting is found to be destabilizing in the sense that these pulsating modes occur for lower values of the activation energy than would be the case for strictly solid fuel combustion.

Time-dependent solution of a premixed laminar flame

Abstract The one-dimensional, time-dependent, multicomponent premixed laminar flame is solved via a highly accurate method of lines approach. The neglect of pressure variations and viscous dissipation and the use of a Lagrangian spatial coordinate reduce the problem to a system of parabolic partial differential equations for the species concentrations and the temperature. Introducing an appropriate B-spline (finite element) basis for the spatial variation and imposing collocation and boundary conditions on the time-dependent coefficients produce a stiff ordinary initial value problem which can be solved by standard techniques. Physical results of special interest include the transient and steady-state profiles of fluid velocity, temperature, and species concentrations through the reaction zone and the upstream velocity (flame speed) of the combustible mixture required to asymptotically stabilize the flame. The analysis is illustrated for the case of an ozone decomposition flame and a comparison with other theoretical predictions shows that the use of less accurate methods can result in significant errors in the predicted values of minor species profiles and the flame speed.

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.

Bifurcation Phenomena in Burner-Stabilized Premixed Flames

Abstract 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 nontriv...

Effects of two-phase flow in a model for nitramine deflagration

Methods of asymptotic analysis are employed to extend an earlier model for the deflagration of nitramines to account for the presence of bubbles and droplets in a two-phase layer at the propellant surface during combustion. Two zones are identified in the two-phase region: one, at higher liquid volume fractions, maintains evaporative equilibrium, whereas the other, at lower liquid volume fractions, exhibits nonequilibrium vaporization. By introducing the most reasonable estimates for two-phase behavior of nitramines, the steady burning rates are found to be close to those obtained for models with a sharp liquid-gas interface. Good agreement with measured burning rates and pressure and temperature sensitivities are achieved through reasonable approximations concerning overall chemical-kinetic parameters.

The transition to nonsteady deflagration in gasless combustion

Abstract Inherent in premixed combustion theory is the existence of neutral stability boundaries across which the stability of a steadily propagating deflagration is lost to one or more nonsteady, nonplanar modes of burning as a critical parameter is varied. This phenomenon occurs not only in classical premixed flame propagation, but also in the deflagration of solid and liquid propellants, in chemical reactors, and in the combustion of intermetallic solids. Here, the focus is on the theoretical description of both linear and nonlinear stability in the latter, which is often referred to either as ‘gasless combustion’, or as ‘combustion synthesis’ due to its application in the synthesis of new refractory materials. The theoretical investigation of stability in these systems is being accomplished through the derivation and analysis of approximate models obtained from activation-energy asymptotics, which in turn have the advantage of admitting an explicit representation of the basic solution that is undergoing a change of stability. The resulting nonsteady, multidimensional models are then able to describe primary and higher-order transitions to various nonuniform and even chaotic modes of combustion. The prediction of these new types of combustion waves not only helps to explain recent experimental observations, but also indicates the existence of totally new phenomena not yet documented by experimental studies. In this review, the analysis of nonsteady instability phenomena and the bifurcation of nonuniformly propagating deflagration waves, which are regarded as intermediate modes of propagation in the transition from steady to chaotic (turbulent) burning, is presented as a distinct discipline that arises not only in gasless combustion, but in virtually all premixed combustion systems.

Nonlinear Stability and Bifurcation in the Transition from Laminar to Turbulent Flame Propagation

Abstract Abstract–We review recent analytical results in the theory of transition from laminar to turbulent premixed Rame propagation. We exploit the fact that the overall activation energy is large to formally derive dynamical flame sheet models, which are then used to predict instability thresholds as functions of the various parameters in the problem, at which steps in the transition occur. Employing perturbation techniques, we then describe bifurcations from a steady, planar flame to both pulsating and cellular modes of propagation. These nonsteady, nonplanar propagation modes represent intermediate stages in the evolution from laminar to turbulent combustion.

Influences of two-phase flow in the deflagration of homogeneous solids

Theoretical analyses are developed for the deflagration of solids such as nitramines that experience exothermic reactions in liquid layers at their surfaces. Relative motion of gas and liquid in a two-phase region at the surface is considered, with influences of pressure gradients and of surface-tension gradients taken into account for the drops and bubbles. It is shown that these influences tend to produce gas velocities in excess of liquid velocities. Burning-rate expressions are derived by activation-energy asymptotics, with special attention paid to the role of interphase heat transfer.

An Asymptotic Theory of Heterogeneous Condensed Combustion

Abstract 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)-n 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...

Diffusional/Thermal Coupling and Intrinsic Instability of Solid Propellant Combustion

Abstract Intrinsic instability in the steady, planar deflagration of a homogeneous solid propellant is considered through an asymptotic analysis for large values of nondimensional overall activation energies for the surface pyrolysis and gas-phase combustion processes. It is shown that the previously known pulsating instability is essentially connected with condensed-phase pyrolysis, and that new instability phenomena, which are associated with intrinsic gas-flame instability and which are sensitive to the value of the gas-phase Lewis number and to the distance of the gas flame from the propellant surface, arise. These results are obtained by relaxing the usual assumptions of quasi-steadiness and quasi-planarity for the gas phase, so that the coupling of intrinsic diffusional/thermal instabilities in the gas and solid phases becomes an integral feature of the model. The steady, planar deflagration thus may be unstable not only to pulsating disturbances, but also to (time-independent, nonplanar) cellular p...