Critical initial conditions for spontaneous thermal ignition

Weber, R. O.; Balakrishnan, E.; Wake, G. C.

doi:10.1039/a806516k

Cited by 11 publications

(10 citation statements)

References 2 publications

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“…For a monotonically decreasing sequence, {µ n } = the wavefront approaches a constant speed from below. Similar features to the case m, n ≥ 1 have been studied for reaction-diffusion systems associated with combustion problems (see, for example, Weber, Balakrishnan, and Wake [18], Gray and Wake [4], and Lacey and Wake [7]). In Figure 8.9 wavefronts with different initial data (u 0 = σ = 0.2, 1, 10) can be seen approaching the same constant speed as t → ∞.…”

Section: Numerical Methods and Resultsmentioning

confidence: 64%

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The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. II. The Initial Boundary Value Problem

McCabe¹,

Leach²,

Needham³

2000

SIAM J. Appl. Math.

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In this paper we examine an initial-boundary-value problem for a singular reactiondiffusion equation arising from a model of isothermal chemical autocatalysis with termination. We consider the case where the reaction and termination orders, m and n, respectively, are fractional; that is, 0 < m, n < 1 (and hence the nonlinearities are not Lipschitz continuous). We show that for m < n, the problem develops spatially uniform solutions which grow algebraically in t (time) and approach a nonzero equilibrium state as t → ∞. For m = n, with k > 0 (k being a parameter which measures the relative strength of the termination step to that of the autocatalysis), we show that solutions approach the equilibrium state, 1 − k for k < 1 and 0 for k = 1, uniformly over the reaction domain as t → ∞. In contrast, for k > 1, we demonstrate that the problem exhibits solutions which decay to zero with contracting support in finite t. Finally, for m > n, we show that traveling waves with semi-infinite support occur when 0 < k < kc, where kc varies with m and n, provided that the initial data exceeds a parameter dependent critical threshold, otherwise solutions decay to zero in finite t with contracting support.

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Section: Numerical Methods and Resultsmentioning

confidence: 64%

“…Matching follows directly giving 18) and the matching condition In particular, as the solution is classical, we require G(η 0 ) = G (η 0 ) = 0.…”

Section: = Nmentioning

confidence: 99%

The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. II. The Initial Boundary Value Problem

McCabe¹,

Leach²,

Needham³

2000

SIAM J. Appl. Math.

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“…It is intuitively obvious, however, that A C cr would better serve as the parameter of choice in defining the critical threshold since A exhibits a strong dependence on the value of . Previous cursory numerical investigation has supported the basic preceding conclusions [9], and the intent here is to buttress this argument with additional computational results. Specifically, we construct critical threshold curves for all three principal centrosymmetric solids over a wide range of values for λ and .…”

Section: Parametric Surveymentioning

confidence: 63%

“…The distinct advantage, however, is that it can fully account for the initial assembly conditions and is therefore able to provide accurate estimates for the critical initial temperature. Weber et al [9], for instance, recently conducted a limited computational study of the nonstationary model which clearly demonstrated that the practical critical assembly temperature may, under certain circumstances, be 5-10% lower than the critical temperature obtained from steady-state theory. This has profound industrial implications in the assessment of fire hazards and the definition of fire safety standards where a clear distinction must be drawn between the classic "storage" problem, where only steadystate temperatures are important, and the less recognized "assembly" problem, where the initial temperature threshold for self-ignition is of vital concern.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Determination of Critical Conditions for Thermal Ignition

Luo

Wake²,

Hawk

2009

ANZIAM J.

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Ignition or thermal explosion in an oxidizing porous body of material can be described by a dimensionless reaction-diffusion equation of the form ∂ t u = ∇ 2 u + λe −1/u . Here such equations will be formulated in symmetrically shaped bounded regions , effectively reducing the mathematical formulation to that of one dimension. This is critically re-examined from a modern perspective using numerical methods. A computer algorithm is constructed and used to carry out a broad-ranging evaluation of the watershed critical initial temperature conditions for thermal ignition of nonuniform assemblies. It is then shown how the resulting mathematical structure for the ignition threshold curves can be correlated by a hyperbolic conic section with a high degree of accuracy over the full range of positive ambient temperature values. However, this sometimes over-predicts (which is bad) and sometimes under-predicts (which is good) the critical initial condition. The definition of additional dimensionless parameters is found to generate further simplification, leading to a universal correlating form capable of collapsing the entire solution space onto a single line in the plane of the new variables. In addition, this study considers the physically intuitive conjecture that spatial moments of the initial temperature profile ought to possess a direct mathematical link to the critical ignition threshold. As such, the mth-order spatial moment of the critical total energy content integrals is defined, and an empirical result is derived stating that certain orders of this moment should be insensitive to changes in ambient temperature and initial shape profile and may be considered functionally dependent on the dimensionless eigenvalue only, within some quantifiable error band. Spatial moment integrals, based on computed critical threshold conditions, are found to support this conjecture, with the best accuracy obtained for the second-order moments.2000 Mathematics subject classification: primary 80A25; secondary 35K55.

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No Radial Symmetries in the Arrhenius-Semenov Thermal Explosion Equation

Gustafson

Ash

2007

Functional Analysis and Evolution Equations

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Critical initial conditions for spontaneous thermal ignition

Cited by 11 publications

References 2 publications

The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. II. The Initial Boundary Value Problem

The Evolution of Travelling Waves in Fractional Order Autocatalysis with Decay. II. The Initial Boundary Value Problem

Numerical Determination of Critical Conditions for Thermal Ignition

No Radial Symmetries in the Arrhenius-Semenov Thermal Explosion Equation

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