On the disappearance of criticality in the theory of thermal ignition

A recent article by K. K. Tam in this journal stated results on the values of the activation energy for which a nonlinear eigenvalue problem, which arises in the steady-state theory of thermal ignition, has multiple solutions. This report points to an error contained therein and gives the corrected values of the activation energy for which multiplicity occurs.

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Zusammenfassung Eine vor kurzem erschienene Arbeit in dieser Zeitschrift von K. K. Tam stellte Ergebnisse für die Werte der Aktivierungsenergie fest, für die ein nichtlineares Eigenwert-Problem, das in der stationären Theorie der thermischen Zündung entsteht, vielfache Lösungen hat. Dieser Bericht weist auf einen darin enthaltenen Fehler hin und gibt die korrigierten Werte der Aktivierungsenergie, für die es vielfache Lösungen gibt.

     

The disappearance of criticality in the theory of thermal ignition

Summary In the study of thermal ignition, when reactant consumption is neglected, criticality is usually taken as the point at which there is a large jump in the maximum steady temperature for a small change in the external parameters. If the equations are taken in their exact form, without the Frank-Kamenetskii approximation RT _a E0, criticality vanishes for large. In this paper a model is introduced for which the value of at which criticality vanishes can be determined precisely for geometries of an infinite slab and infinite circular cylinder.

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Zusammenfassung Wird beim Studium thermischer Explosionen der Reaktionsverbrauch vernachlässigt, so spricht man üblicherweise von einem kritischen Punkt, wenn dort bei einer kleinen Änderung der äußeren Parameter ein großer Sprung der stationären Maximaltemperatur auftritt. Bei den exakten Gleichungen (ohne die Frank-Kamenetskii'sche Approximation RT _a E0) gibt es für große keinen kritischen Punkt mehr. In dieser Arbeit wird ein Modell eingeführt, bei dem der Wert von , für welchen der kritische Punkt verschwindet, im Fall der unendlichen Platte und des unendlichen Zylinders genau bestimmt werden kann.

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Nomenclature A pre-exponential factor in Arrhenius law - d length scale (e.g. radius of cylinder and half-thickness of slab) - E Arrhenius activation energy - k thermal conductivity - Q exothermicity per unit mass - R gas constant - S surface of reacting material - T local absolute temperature - T _a surface temperature - V volume occupied by the reacting material - x dimensionless distance coordinate - =E(T-T _a )/RT _a ² dimensionless temperature - solution of Eqs. (6), (7) - maximum value of inV - =(1+)^–1 - maximum value of inV - density of material - =RT _a /E - ₀ smallest value of for which criticality vanishes - =QEAd ² exp (-E/RT _a )/kRT _a ² - _cr critical value of - ^* maximum value of for which positive solutions exist Supported in part by the Victoria Univeristy of Wellington Fellowship Committee and the European Research Office of the U.S. Army.     

Disappearance of criticality in thermal ignition for a simple reactive viscous flow

Conditions for the disappearance of criticality for reactive plane-Poiseuille flow are obtained by means of a perturbation series in the viscous heating parameter; an approximation to the Arrhenius reaction-rate term proposed in this journal by Bazley and Wake is used.

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Résumé On cherche les conditions pour la disparition de la limite critique pour l'écoulement Poiseuille de réactif entre deux plans en utilisant une série de perturbation de termes d'un paramètre que représente les effets thermiques de la viscosité; nous employons une approximation proposée dans ce journal par Bazley et Wake qui simplifie le terme d'Arrhenius dans l'équation de la balance de chaleur.

     

On the stability of bifurcation branches in thermal ignition

Summary A method is given to determine the stability of stationary solutions of the thermal ignition equation for the case ofn-dimensional spherical symmetry, together with the number of unstable modes. For sufficiently high temperature and activation temperature this number is arbitrarily large. Some numerical results on the solutions and their stability are reported.

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Zusammenfassung Ein Verfahren zur Bestimmung der Stabilität stationärer Lösungen der Gleichung der thermischen Selbstzündung imn-dimensionalen kugelsymmetrischen Fall sowie der Anzahl instabiler Eigenfunktionen wird angegeben. Für genügend hohe Temperatur und Aktivierungstemperatur ist diese Anzahl beliebig groß. Einige numerische Ergebnisse über die Lösungen und ihre Stabilität werden berichtet.

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Part of this work has been done during the author's stay at the Applied Physics Laboratory of Johns Hopkins University Laurel Maryland USA     

On criticality problems in energy dependent neutron transport theory

In this paper, we consider the criticality problem for energy dependent neutron transport in an isotropically scattering, homogeneous slab. Under a positivity assumption on the scattering kernel, we can find an expression relating the thickness of the slab to a parameter characterizing production by fission. This is accomplished by exploiting the Perron-Frobenius-Jentsch characterization of positive operators (i.e. those leaving invariant a normal, reproducing cone in a Banach space). We point out that those techniques work for classes of multigroup problems where the Case singular eigenfunction approach is not as feasible as in the one-group theory, which is also analyzed.     

Exact solutions and ignition parameters in the Arrhenius conduction theory of gaseous thermal explosion

Summary Exact solutions, bifurcation diagrams, and critical ignition parameters are obtained for the true Arrhenius equation for conductive gaseous thermal explosion.

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Zusammenfassung Exakte Lösungen, Verzweigungsdiagramme, und kritische Zündungsparameter werden für die genaue Arrhenius-Gleichung für thermische Explosionen in leitenden Gasen erhalten.

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Partially supported by NSF Grant NSF MCS 80-12220.     

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

Criticality in a model for thermal ignition in three or more dimensions

An explicitly resolvable model, which was instroduced in a previous paper (see [1]), is used to obtain exact behaviour of its bifurcation curves. The model closely approximates the true Arrhenius law for a spherical vessel of reacting material undergoing an exothermic reaction in three or more dimensions. For a sequence of values of a parameter, which is the reciprocal of the dimensionless activation energy, the number of the solutions changes for certain values of the eigenparameter Further, there exist solutions for all then is non-zero.

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Zusammenfassung Mit Hilfe eines explizit lösbaren Modells, das in einer früheren Arbeit eingeführt wurde (siehe [1]), erhält man das exakte Verhalten der zugehörigen Verzweigungskurven. Das Modell approximiert gut das Arrheniussche Gesetz für exotherme Reaktionen in einem sphärischen Topf in drei oder mehr Dimensionen. Für eine Folge von Werten des Parameters, welches als Reziproke der dimensionslosen Aktivierungsenergie dient, ändert sich die Anzahl der Lösungen zum Eigenwert, der durch gesteuert wird. Weiter gibt es für alle mindestens eine Lösung, sofern 0 gilt.

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Supported in part by the Deutsche Forschungsgemeinschaft and in part by the Victoria University of Wellington Fellowship Committee.     

On singular diffusion equations with applications to self-organized criticality

We consider solutions of the singular diffusion equation _t, = (u^m?1 u_x)_x, m ≦ 0, associated with the flux boundary condition lim_x→?∞ (u^m?1u_x)_x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L¹ for ?1 < m ≦ 0, but have non-integrable tails for m ≦ ?1. We also show that these traveling waves are the same as those for the scalar conservation law u_t = ?[f(u)]_x + u_xx for a particular nonlinear convection term f(u) = f(u;m, λ). © 1993 John Wiley & Sons, Inc.     

On the hardness of evaluating criticality of activities in a planar network with duration intervals

Complexity results for problems of evaluating the criticality of activities in planar networks with duration time intervals are presented. We show that the problems of asserting whether an activity is possibly critical, and of computing bounds on the float of an activity in these networks are NP-complete and NP-hard, respectively.     

On the latest starting times and criticality of activities in a network with imprecise durations

This paper deals with problems of computing possible values of latest starting times and determining types of criticality for all activities in a network with interval or fuzzy activity durations. Although the problem of computing the latest starting times has been solved, a novel polynomial algorithm which is easy to understand and improves complexity is proposed.     

On a Monte Carlo method for neutron transport criticality computations

^** Email: maire{at}univ-tln.fr^*** Email: denis.talay{at}sophia.inria.fr We give a stochastic representation of the principal eigenvalueof some homogeneous neutron transport operators. Our constructionis based upon the Feynman–Kac formula for integral transportequations, and uses probabilistic techniques only. We developa Monte Carlo method for criticality computations. We numericallytest this method on various homogeneous and inhomogeneous problems,and compare our results with those obtained by standard methods.     

On the sheaf theory

The“Gel’fand sheaf” of a topological algebra is endowed with auniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided.     

On the theory of grossone

A trivial formalization is given for the informal reasonings of a series of papers by Ya. D. Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.