The Thermal Stability of a Partially Insulated Reactive Slab

Herbert, D. M.

doi:10.1093/qjmam/39.2.197

Cited by 6 publications

(4 citation statements)

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“…As L -+ 00 the order €2 shift in Ac vanishes, and the shift is O(€2). This is in disagreement with the results of Adler [1] and Herbert [5], who obtained O(€) and O(€2) shifts respectively for the case L = 00.…”

Section: Introductioncontrasting

confidence: 99%

“…We note that the coefficient of E2 in (7.12) tends to zero as L ~ 00. For this case Adler [1] obtained an O(E) correction to Ao(a o ) = 0.87846, and Herbert [5] Table 4, and its agreement with the numerical results is even poorer than that of (7.12). The analyses of Adler and of Herbert seem to be based upon assumptions about the asymptotic expansion which are contradicted by their subsequent results.…”

Section: Perturbations Of Boundary Conditionsmentioning

confidence: 87%

“…We note that the coefficient of E2 in (7.12) tends to zero as L ~ 00. For this case Adler [1] obtained an O(E) correction to Ao(a o ) = 0.87846, and Herbert [5] obtained an O(E Z ) correction instead. His result is also shown in Table 4, and its agreement with the numerical results is even poorer than that of (7.12).…”

Section: (N2 + S2)mentioning

confidence: 99%

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Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Ward

Keller

1991

Stud Appl Math

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Nonlinear eigenvalue problems are considered for partial differential equations and boundary conditions of the form 6.u+AF(x,u) =0, xED; The problems are perturbed by deleting a small subdomain D. from D and imposing a boundary condition on the surface of the resulting hole, or else by changing the constant b in the boundary condition to a different constant E -1 k on a small part of aD. In both cases the perturbed solution is constructed for E small by the method of matched asymptotic expansions. Particular attention is paid to the calculation of A/E), the critical value of A at which the number of solutions changes by two, for example from two to none. This value of A represents a simple fold point in the bifurcation diagram of Ilull versus A. The analysis is applied to chemical reactors with F(x, u) = e U , in which case Ac is the critical value of the Frank-Kamenetskii parameter. The asymptotic results are compared with exact and numerical results for some special cases, and fair agreement is found. Some previous asymptotic calculations are found to be incorrect.

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Section: Introductioncontrasting

confidence: 99%

Section: Perturbations Of Boundary Conditionsmentioning

confidence: 87%

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Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Ward

Keller

1991

Stud Appl Math

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“…Problems where the surface temperature is specified and is non-uniform are of physical interest, Adler (1975Adler ( , 1978, Adler & Nhamburo (1985). Alternatively the surface may be partly insulated and the body at a constant surface temperature elsewhere (Adler 1984;Zaturska 1984;Greenway & Spence 1985;Herbert 1986). In problems of this type, criticality will depend on the degree of non-uniformity of the surface temperature, or the amount of insulation.…”

Section: Introductionmentioning

confidence: 99%

Criticality in a nearly circular cylinder

1987

Proc. R. Soc. Lond. A

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The problem of determining the critical conditions in a non-circular cylinder of infinite length is examined, in which the reaction is of zero order and where the Frank-Kamenetskii approximation to the reaction rate has been made. The generators are assumed parallel and the boundary of a cross section is given by r = 1 + ε cos θ ( ε ≪ 1), where ( r , θ ) are plane polar coordinates. The surface conditions assumed are (i) uniform temperature and (ii) newtonian cooling. By suitable expansions of the interior temperature and the Frank-Kamenetskii parameter δ , it is shown that the critical value of δ for (ii) is δ crit ( ε, B ) = 8 σ c /(1 + σ c ) 2 exp[ – (4 σ c )/ B (1 + σ c )] x [1 - ε 2 { σ 2 c /1 + σ c ) + (1/ B ) σ c (1 + 3 σ c )/(1 + σ c ) 2 } + O ( ε 4 )], Where σ c = -2/ B + (1 + 4/ B 2 ) ½ and where B is a Biot number proportional to the surface-heat transfer coefficient. For (i), the simple result δ crit ( ε , ∞) = 2 - ε 2 + O ( ε 4 ), is obtained.

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Thermal explosion theory for a slab with partial insulation: A correction to δc(ϵ)

Adler¹

1987

Combustion and Flame

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The Thermal Stability of a Partially Insulated Reactive Slab

Cited by 6 publications

References 0 publications

Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Nonlinear Eigenvalue Problems under Strong Localized Perturbations with Applications to Chemical Reactors

Criticality in a nearly circular cylinder

Thermal explosion theory for a slab with partial insulation: A correction to δc(ϵ)

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