Limit Of High Activation Energy Research Articles

Asymptotic study of premixed flames in inert porous media layers of finite width: Parametric analysis of heat recirculation phenomena

In this paper, we present an investigation on super-adiabatic premixed flames in an inert porous medium layer of finite length. The combustion process is modeled by a one-step Arrhenius kinetics in which density variations are taken into account. The asymptotic case of large ratio of thermal conductivities of solid to gas phases and the high activation energy limit are explored. The results obtained for these limiting cases are compared to those for finite values of these parameters. The use of the flame sheet model allows us to obtain steady-state solutions in an analytical form thus facilitating the parametric analysis.The investigation focuses on the phenomenon of multiplicity of steady-state solutions. It is shown that two or three (nontrivial) steady-state solutions are possible, depending on the flow rate intensity. The critical values of the parameters for the existence of multiple solutions are also determined. The stability of the obtained solutions is finally investigated by means of time-dependent simulations.

Analytical study of superadiabatic small-scale combustors with a two-step chain-branching chemistry model: Lean burning below the flammability limit

A study of combustion in small-scale superadiabatic burners with a counter-flow heat exchange segment is presented for a simple two-step chain-branching chemistry model. The work is focused on the possibility of burning below the flammability limit. The investigation is carried out in the high activation energy limit on the basis of analytical solutions. The existence of multiple steady-state regimes is demonstrated and their stability properties are investigated. These analytical results are finally compared with the results of numerical simulations carried out for finite activation energies.

Coupled analytical approach to predict piloted flaming ignition of non-charring polymers

Classical thermal theory of piloted ignition is extended by coupling the heat balance at the exposed sample surface and the finite-rate pyrolysis in the material volume. Approximate analytical solutions for the sample temperature are obtained for an arbitrary sample thickness, with the external radiative heating, surface re-radiation, heat of gasification, and the convective heat flux corrected for blowing taken into account. The volatile mass flux is evaluated by integrating the pyrolysis rate throughout the layer, with the assumption of high activation energy limit. Critical mass flux of combustible volatiles is used as the ignition criterion. This enables the ignition temperature to be evaluated instead of being pre-assumed as is done in the classical thermal theory. Coupled analytical approach proposed in this work is verified by comparisons to the numerical solution obtained by the Pyropolis model for the same problem setup. This approach has also been validated by comparisons to published experimental data (ignition temperatures and times to ignition) for three non-charring thermoplastics: polymethylmethacrylate, polyethylene and polypropylene.

Vortex breakdown in premixed reacting flows with swirl in a finite-length circular open pipe

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb–Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb–Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.

Front instability in a condensed phase combustion model

AbstractWe consider a condensed phase (or solid) combustion model and its linearization around the travelling front solution. We construct an Evans function to characterize the eigenvalues of the linearized problem. We estimate this functional in the high activation energy limit. We deduce the existence of zeros with nonnegative real part for high activation energy, which proves the linear instability of the travelling front solution.

Stability of viscous detonations for Majda’s model

Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of a version of Majda’s scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves, J. Differential Equations 233 (2) (2007) 654–698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of a Majda-type model without a smallness assumption. Notably, our results indicate that, for this simplified, scalar model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to “galloping” or “pulsating” solutions as is observed in the full reactive Navier–Stokes equations. This answers in the negative, for this model, a question posed by Majda as to whether such scalar detonation models capture this aspect of detonation behavior.

Convective combustion in porous media: singular limit of high activation energy

In this paper, we consider a system of degenerate reaction diffusion equations which describes a convective (pressure driven) regime of combustion in porous media. The goal of this paper is to study the behaviour of this system in the limit of high activation energy. We show that the limit solution for this problem in arbitrary spatial dimension solves a parabolic equation with memory term similar to one arising in solid combustion. Moreover, under the additional assumption of the solution being time increasing, we prove that the limit problem coincides with Stefan problem for supercooled water with spatially inhomogeneous coefficients. We also obtain the precise limit problem for (not necessarily planar) travelling waves in any dimension.

Chemical Reactions as $\Gamma$-Limit of Diffusion

We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential $H/\varepsilon$. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simpler system modeling the spatial diffusion of A and B combined with the reaction $A\rightleftharpoons B$. With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, $\Gamma$-convergence.

From Diffusion to Reaction via $\Gamma$-Convergence

We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential $H/\varepsilon$. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simple system modeling the diffusion of A and B, and the reaction $A\rightleftharpoons B$. The aim of this paper is to give a rigorous proof of Kramers's formal derivation and to embed chemical reactions and diffusion processes in a common variational framework which allows one to derive the former as a singular limit of the latter, thus establishing a connection between two worlds often regarded as separate. The singular limit is analyzed by means of $\Gamma$-convergence in the space o...

Existence of a Degenerate Singularity in the High Activation Energy Limit of a Reaction–Diffusion Equation

We consider the singular perturbation problem where , β is a Lipschitz continuous function such that β > 0 in (0, 1), β ≡ 0 outside (0, 1) and . We construct an example exhibiting a degenerate singularity as ε k ↘ 0. More precisely, there is a sequence of solutions u ε k → u as k → ∞, and there exists x 0 ∈ ∂{u > 0} such that Known results suggest that this singularity must be unstable, which makes it hard to capture analytically and numerically. Our result answers a question raised by Jean–Michel Roquejoffre at the FBP'08 in Stockholm.

Pulsating traveling waves in the singular limit of a reaction–diffusion system in solid combustion

We consider a coupled system of parabolic/ODE equations describing solid combustion. For a given rescaling of the reaction term (the high activation energy limit), we show that the limit solution solves a free boundary problem which is to our knowledge new.In the time-increasing case, the limit coincides with the Stefan problem with spatially inhomogeneous coefficients. In general it is a parabolic equation with a memory term.In the first part of our paper we give a characterization of the limit problem in one space dimension. In the second part of the paper, we construct a family of pulsating traveling waves for the limit one phase Stefan problem with periodic coefficients. This corresponds to the assumption of periodic initial concentration of reactant.

Travelling waves for a combustion model coupled with hyperbolic radiation moment models

In this paper we consider a model of a flame propagating in a gas containing inert dust. The combustion model is a model with simple chemistry obtained in the high activation energy limit. The radiative field is modeled either by the $P_1$ isotropic model or by the $M_1$ anisotropic model. Under some restrictions on the parameters we prove the existence of travelling waves for this hierarchy of combustion radiation models by Schauder's fixed point argument on bounded domains and uniform estimates.

A two phase elliptic singular perturbation problem with a forcing term

We study the following two phase elliptic singular perturbation problem: Δ u ε = β ε ( u ε ) + f ε , in Ω ⊂ R N , where ε > 0 , β ε ( s ) = 1 ε β ( s ε ) , with β a Lipschitz function satisfying β > 0 in ( 0 , 1 ) , β ≡ 0 outside ( 0 , 1 ) and ∫ β ( s ) d s = M . The functions u ε and f ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit ( ε → 0 ) and we show that limit functions are solutions to the two phase free boundary problem: Δ u = f χ { u ≢ 0 } in Ω ∖ ∂ { u > 0 } , | ∇ u + | 2 − | ∇ u − | 2 = 2 M on Ω ∩ ∂ { u > 0 } , where f = lim f ε , in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f ε ≡ 0 . The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.

Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u ( x , t ) ≥ 0 , u(x,t)\geq 0, defined in a domain D ⊂ R N × ( 0 , T ) \mathcal {D} \subset {\mathbb {R}}^{N}\times (0,T) and such that \[ Δ u + ∑ a i u x i − u t = 0 in D ∩ { u > 0 } . \Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text {in}\quad \mathcal {D}\cap \{u>0\}. \] We also assume that the interior boundary of the positivity set, D ∩ ∂ { u > 0 } \mathcal {D} \cap \partial \{u> 0\} , so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: \[ u = 0 , − ∂ u / ∂ ν = C . u=0 ,\quad -\partial u/\partial \nu = C. \] Here ν \nu denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of D \mathcal {D} . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.

Steady Deflagration of HMX With Simple Kinetics: A Gas Phase Chain Reaction Model

A new approach is presented for modeling steady combustion of energetic solids, in particular HMX. A simplified, global, gas phase chain reaction kinetic mechanism is employed. Specifically, a zero-order, high activation energy thermal decomposition initiation reaction in the condensed phase followed by a second-order, low activation energy chain reaction in the gas phase is assumed. A closed-form solution is obtained, which is based on the activation energy asymptotics analysis of Lengelle in the condensed phase and the assumption of zero activation energy in the gas phase. Comparisons between the model and a variety of experimental observations over a wide range of pressures and initial temperatures are presented and demonstrate the validity of the approach. The model provides excellent agreement with burning rate data (including sensitivity to pressure and initial temperature) and temperature profile data (in particular the gas phase). This suggests that in the realm of simplified, approximate kinetics modeling of energetic solids, the low gas phase activation energy limit is a more appropriate model than the classical high activation energy limit or heuristic flame sheet models. The model also indicates that the condensed phase reaction zone plays an important role in determining the deflagration rate of HMX, underscoring the need for better understanding of the chemistry in this zone.

Failure and reignition of one-dimensional detonations—The high activation energy limit

The structure of failed one-dimensional detonations is derived using a high activation energy analysis. On a time scale longer than the chemical time based upon the von Neumann temperature, high activation energy one-dimensional detonations break down into a weaker shock, a contact surface separating hot burned gases from a colder, unburned mixture, and an expansion wave. While this leading order solution is unaffected by chemistry, hence self-similar, the perturbation, accounting for the chemistry, which depends upon chemical times, is not. By accounting for the chemistry, the perturbation problem determines the delay until reignition of the detonation occurs. This problem is almost identical to the problem of initiation in the region between a shock and contact surface, which is created by the collision of two shock waves. The main difference between that problem and the current analysis is that the downstream boundary condition now consists of radiating acoustics into hot burned products, at the location of the surface discontinuity. When the Newtonian limit is applied, that is, for a ratio of the specific heats approaching unity, the hot spot at which reignition occurs approaches the location of the contact surface. The time and length to reignition are then found to vary exponentially with the activation energy of the mixture. However, the Newtonian limit is not a very realistic model, because it makes the interval between a Mach number of 1/√γ and 1 disappear: in this range of Mach numbers, adding energy to a steady flow lowers the temperature, hence the reaction rate.

Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem

We consider a free-boundary problem for the heat equation which arises in the description of premixed equi-diffusional flames in the limit of high activation energy. It consists of the heat equation u t = Δu, u > o, posed in an a priori unknown set Ω ⊂ Q T = R N × (0, T) for some T > 0 with boundary conditions on the free lateral boundary τ = ∂Ω∩ Q T (the flame front): u = o and ∂u ∂v = − 1. We impose initial condition u 0( x) ≥ 0 on the known initial domain Ω 0 = Ω ∩ {t = 0} . The paper establishes a theory of existence, uniqueness and regularity for radial symmetric solutions having bounded support. We remark that such solutions vanish in finite time (extinction phenomenon). In the paper we analyze the different types of possible extinction behaviour. We also investigate the focusing behaviour for solutions whose support expands in finite time to fill a hole. In all the cases the asymptotic behaviour is shown to be self-similar.

An asymptotic derivation of the linear stability of the square-wave detonation using the Newtonian limit

The two-dimensional linear stability of a detonation wave characterized by a one-step irreversible Arrhenius reaction is examined by a two-parameter asymptotic approach. The first is the limit of high activation energy in which the underlying steady detona­tion structure tends to the classical square-wave profile. The second is due to Blythe & Crighton ( Proc. R. Soc. Lond . A 426, 189-209 (1989)) and assumes the Newtonian limit in which the ratio of the isotropic sound speed to the isothermal sound speed is close to unity. It is found that under two possible choices of distinguished limit between the two parameters, analytical forms for the perturbation variables in the induction zone and equilibrium zones of the perturbed detonation can be found in normal mode form. A dispersion relation describing the growth rate of the perturba­tions is then obtained in one case through a compatibility condition on the structure of the perturbation eigenfunctions at the flame front, and in the other through a matching of the perturbation variables across the flame front into the equilibrium zone, where an acoustic radiation condition is imposed.

Modes of burning in filtration combustion

We describe and analyze a model of filtration combustion, in which a gas is forced at high pressure into a porous solid matrix so that after ignition and under favourable conditions a combustion wave can propagate through the medium. We consider the case of counter flow, where the gas is forced into the reaction zone through the unreacted part of the porous solid. Relations are derived for the steady state propagation of a planar combustion wave or front in the limit of high-activation energy, from which the propagation speed, reaction temperature, and reacted mass fraction of the solid product can be found in terms of the mass flux of the injected gas, the gas pressure and mass flux on exit from the front, and other physico-chemical parameters describing the system. Two distinct modes of combustion are discussed, corresponding to the reaction being driven to completion by exhaustion of either the gaseous or the solid component, these being referred to as the gas-deficient and solid-deficient modes of burning respectively. For both homogeneous and heterogeneous forms of the reaction rate we find that there is a critical inlet mass flux for the incoming gas below which steady state solutions no longer exist and that there are parameter values for which multiple steady states occur.

On the use of operator-splitting methods for the equations of combustion

A model problem is analyzed to illustrate the operator-splitting method for combustion problems with low Mach-number flows. In the limit of high Damköhler number, it is shown that the solution of the combustion equations can be separated at each time step into two parts, (1) a spatially homogeneous explosion calculation over the initial part of the time step and (2) a transport calculation over the remainder of the time step that redistributes the chemical energy released in (1). The limit of high activation energy (β → ∞) allows approximate analytical solution of the explosion equations. The method is applicable primarily for problems with two time scales, a short chemical time scale and a much longer convection or diffusion time scale. Nevertheless, it can still be profitably used for problems described only by the single chemical scale if certain restrictions are obeyed. The operator-splitting method is then applied to solve the problem of two initially separated coflowing streams of fuel and oxidizer that subsequently mix and react. Many advantages and disadvantages of operator splitting are well represented by this example; some, such as those for confined flames, are not. Error norms are evaluated, CPU times compared, and influences of the Damköhler number D and characteristic time step ratio χ analyzed. Extensions to more general cases are discussed.