Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. ।. Arbitrary Biot number and general reaction-rate laws

1983 ◽  
Vol 390 (1799) ◽  
pp. 247-264 ◽  
Keyword(s):  
Temperature Dependence ◽  
Biot Number ◽  
Reaction Rate ◽  
Critical Value ◽  
Original Form ◽  
Central Temperature ◽  
Rate Laws ◽  
Temperature Excess ◽  
Uniform Case ◽  
Thermal Explosions

The conductive theory of thermal explosion in its original form (Frank-Kamenetskii, Acta phys. -chim . URSS (1939)) expresses the balance between heat generation and heat conduction in terms of the dimensionless parameter δ = [QEA a 2 0 c n 0 exp ( - E/RT a )]/ K RT 2 a Stability is lost when δ exceeds a critical value which, in this approximation, depends only on the geometry of the system. Matters are usually more complicated than this. First, heat transfer is often impeded both in the interior (by conduction) and at the surface; the relative importance of these impedances is expressed by the Biot number B i = X a o / K - Second, the temperature dependence of reaction rate may not be well enough represented by the 'exponential approximation’ (which simply implies a doubling of rate every so many degrees). The natural and convenient dimensionless measure here is the parameter ϵ = RT a /E. In the present paper, critical values for the parameter δ and for the dimensionless central-temperature excess Ɵ o have been evaluated for the whole range of Biot number from the uniform case (Semenov extreme, B i → 0) to the Frank-Kamenetskii extreme (B i →∞). The procedures can handle any temperature-dependence of rate and are illustrated here for the Arrhenius and 'bimolecular’ forms for which, k∝ exp ( — E /RT ) and k ∝T ½ exp ( - E /R T ) respectively. When E /RT a is not large, criticality is lost at ϵ = ϵ tr ≤ ¼. Such transitional values for the reduced ambient temperature ϵ, for the critical value of δ, and for the dimensionless central temperature excess Ɵ 0 have also been obtained. They are represented both graphically and numerically. The present results are also compared with earlier work.

Thermal explosion and times-to-ignition in systems with distributed temperatures Il. The influence of reactant consumption

1984 ◽  
Vol 391 (1801) ◽  
pp. 269-294 ◽  
Keyword(s):  
Biot Number ◽  
Reaction Rate ◽  
Ignition Time ◽  
Critical Value ◽  
Maximum Temperature ◽  
Infinite Cylinder ◽  
Concentration Dependences ◽  
Temperature Dependences ◽  
Arrhenius Temperature Dependence ◽  
Thermal Explosions

In part I, we studied thermal explosions in systems with distributed temperatures, and more particularly the evolution of reactant temperatures in time. Reactant consumption, however, was ignored. Here we consider the influence of reactant consumption for the whole range of Biot number from the Semenov extreme (β = 0) to the Frank-Kamenetskii extreme (β → ∞). We concentrate on numerical results for the three simplest geometries (infinite slab, infinite cylinder and sphere), but our route is valid for arbitrary geometry. Earlier treatments in this field are extended by considering not only generalized temperature-dependences of reaction rate, f (θ), but also generalized concentration-dependences of reaction rate, g ( w ). An important influence of reactant consumption is to modify the critical value for the Frank-Kamenetskii parameter δ, δ cr /δ 0 = 1 + ϕ( g w /B) ⅔ , where the coefficient ϕ depends on the geometry, g w represents an effective order of reaction (to be evaluated at initial temperature and concentration), B is a dimensionless adiabatic temperature rise and δ 0 is the critical value of δ when reactant consumption is ignored ( B → ∞). The value of the constant ϕ is readily calculated from certain simple integrals. Its dependence on geometry and its variation with Biot number β are presented and discussed. It is shown that for β → ∞, reactant consumption has a smaller effect on the critical behaviour than in the case where β = 0, and the same trend is found for intermediate values of β. The role of diffusion of reactant is examined. It is found that the behaviour, as gauged by the leading-order terms of an asymptotic analysis, falls into one of two classes, so that diffusion either may be entirely ignored or is so rapid that concentrations are uniform. Results for the two classes show a remarkably close correspondence and permit a uniform treatment. Results are given for the time taken by a system either to ignite or, otherwise, to reach a maximum temperature. For important limiting cases simple asymptotic formulae are given for these times; they constitute good approximations over wide ranges. For moderately supercritical systems the time to ignition, t ign , differs by little from the ignition time t ign (B → ∞) for the case of zero reactant consumption: t ign / t ign ( B → ∞) = 1+ G ( g w / B ) (δ-δ 0 /δ) -3/2 + O ( g w / B ) 2 . For the Arrhenius temperature dependence with large activation energy the constant G is about unity for all cases .

Thermal explosions and the disappearance of criticality at small activation energies: exact results for the slab

1979 ◽  
Vol 368 (1735) ◽  
pp. 441-461 ◽  
Keyword(s):  
Critical Value ◽  
Activation Energies ◽  
Critical Conditions ◽  
Infinite Cylinder ◽  
Exothermic Reactions ◽  
Ambient Temperatures ◽  
Temperature Excess ◽  
Infinite Slab ◽  
Temperature Dependences ◽  
Thermal Explosions

For exothermic reactions obeying the Arrhenius equation in circumstances in which heat flow is purely conductive, critical conditions for thermal explosion are satisfied when a dimensionless group = QE »/ kR T attains a critical value £Cr = ca. 0.88, 2 or ca. 3.32 for the infinite slab, infinite cylinder or sphere. This result (Frank-Kamenetskii 1938) of stationary state treatm ent is appropriate so long as activation energies are not too low or ambient temperatures are not too high: E > RTa (or e = R Ta/E 1). Criticality persists for E decreasing or T increasing so long as e is smaller than a transitional value etr. At this transitional value etr, only continuous behaviour is possible: ignition phenomena disappear. Accurate transitional values for the reduced ambient temperature e, for the critical value of 8, and for the reduced central temperature excess 0m, have been calculated by quadrature for the infinite slab. The following results are obtained under Frank-Kamenetskii boundary conditions (α ->∞) for two common temperature dependences.

Criteria for thermal explosions with and without reactant consumption

1977 ◽  
Vol 357 (1691) ◽  
pp. 403-422 ◽  
Keyword(s):  
Differential Equations ◽  
Heat Loss ◽  
Heat Production ◽  
Heat Balance ◽  
Arrhenius Equation ◽  
Critical Value ◽  
Critical Values ◽  
Stationary States ◽  
Uniform Case ◽  
Thermal Explosions

Thermal explosions occur when reactions evolve heat too rapidly for a stable balance between heat production and heat loss to be preserved. Even when reactions are kinetically simple, and obey the Arrhenius equation, the differential equations for heat balance and reactant consumption cannot be solved explicitly to express temperatures and concentrations as functions of time unless strong simplifications are made. This difficulty exists for the spatially uniform (Semenov) as well as for the distributed temperature (Frank-Kamenetskii) case. Solutions become possible if strong simplifications are made (no reactant consumption; approximations to the Arrhenius term). Ignition is then represented by the threshold at which stationary states disappear. A single parameter (see appendix for definitions and symbols) summarizes the criteria for ignition. In the spatially uniform case, the Semenov parameter has the critical value e_1. In the distributed temperature case, the Frank-Kamenetskii parameter has critical values that depend on the geometry.

Thermal explosions with extensive reactant consumption: a new criterion for criticality

1983 ◽  
Vol 390 (1798) ◽  
pp. 13-30 ◽  
Keyword(s):  
Time History ◽  
Initial Reaction Rate ◽  
Critical Value ◽  
Smooth Transition ◽  
Maximum Temperature ◽  
Initial Reaction ◽  
Temperature Excess ◽  
Explosive Reaction ◽  
Thermal Explosions ◽  
New Criterion

In classical treatments of thermal runaway, reactant consumption is commonly ignored. Criticality is readily identified as the disappearance of steady states. The distinction between subcritical and supercritical systems is sharp. For explosive behaviour, infinite excess temperatures ( θ → ∞) are reached in a finite time; non-explosive reaction is characterized by a low stationary state of self-heating( θ ≼ 1). These are divided discontinuously by a critical value for the Semenov number ψ : ψ = QVAc m 0 e - E / RT a / X S ( RT 2 a / E ) ; ψ cr = e -1 ; θ cr = 1. When reactant consumption occurs, this sharpness disappears. Each temperature-time history evolves to a maximum temperature excess ∆ T * or θ * before decaying back to ambient. A new criterion is presented here for thermal ignition in systems with extensive reactant consumption. It is based on the dependence of θ * on the initial reaction rate. We identify criticality with the maximum sensitivity of θ * to ψ , and look for the point of inflexion in the function θ *( ψ ). Such a definition is closely related to that used implicitly by an experimentalist. The critical value of the Semenov number is derived as an integral equation. This is solved numerically to yield ψ cr as a function of the reaction exothermicity B . The calculations lead to values for up to 2 or 3 times greater than the classical value found when reactant consump­tion is ignored. There is a smooth transition between new and old values close to the classical limit (at very large B ). Our results coincide with the predictions from asymptotic analysis for B greater than ca . 50. They peel away significantly as B diminishes through the range 50 > B > 4, a range that can be met in practice for dilute gases or solid masses of low reactivity.

Thermal explosion and times to ignition in systems with distributed temperatures. IV. Behaviour at points of criticality and transition

1987 ◽  
Vol 409 (1836) ◽  
pp. 37-45 ◽  
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Thermal Explosion ◽  
Reaction Rate ◽  
Critical Value ◽  
Analytic Solutions ◽  
Dimensionless Group ◽  
Leading Order ◽  
Temperature Excess ◽  
Reduced Temperature

The conductive theory of thermal explosion in systems with distributed temperatures expresses conditions for the breakdown of stability as the critical value of a single dimensionless group δ , Frank-Kamenetskii’s parameter. For values of δ above critical, there is thermal runaway; for values of δ less than critical, the system is quiescent. Analytic solutions for excess temperatures as a function of position and time are not generally available for these systems, but can be found when δ is close to the critical value, by using the techniques of asymptotic analysis. In this paper we consider two related problems. First, the approach of temperature to the critical, steady-state profile is found when δ takes precisely its critical value. The second concerns systems where criticality is just being lost through the coalescence of two critical points. It establishes how the new ‘transitional’ steady-state profile is then approached. In both circumstances we neglect reactant consumption. We allow for an arbitrary dependence of reaction rate on temperature, and an arbitrary Biot number in the boundary conditions. The solutions are found in terms of leading-order descriptions of the behaviour of the reduced temperature-excess as it moves towards the steady state profile.

Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. II. An asymptotic analysis of critically at the extremes of Biot number (( Bi ) -> 0, ( Bi ) -> ∞ for generalized reaction rate-laws

1984 ◽  
Vol 392 (1803) ◽  
pp. 301-322 ◽  
Keyword(s):  
Biot Number ◽  
Asymptotic Expansions ◽  
Perturbation Methods ◽  
Computer Time ◽  
Infinite Cylinder ◽  
Rate Laws ◽  
Asymptotic Expressions ◽  
Thermal Explosions ◽  
Biot Numbers ◽  
Resistance To Heat

Numerical attacks on the problem of criticality in thermally igniting systems with generalized resistance to heat-transfer are expensive in computer time and particular to the cases studied. We show here how very general circumstances may be treated analytically by straightforward perturbation methods (asymptotic expansions). Asymptotic expressions of considerable precision can be made (i) starting from the Semenov extreme (( Bi = 0) in terms of ( Bi ), and (ii) starting from the Frank-Kamenetskii extreme in terms of ( Bi ) -> ∞) in terms of ( Bi -1 They apply to any geometry and they are presented here for the infinite slab, the infinite cylinder and the sphere as expressions for critical values of the Frank-Kamenetskii or Semenov parameters and for critical centre temperature θ in terms of Biot number. The importance of the method is that it can cope with any temperature- dependence of rate coefficient f(θ) ), and although the asymptotic expansions are strictly valid only at the extremes, for δ they together cover the whole range of Biot numbers. Numerical comparisons are given for the case f(θ) = e θ , for which the results are well known and for the case f(θ) = exp[ θ /( 1+ ε θ )], corresponding to Arrhenius kinetics.

Kinetik der Reaktion von 2.4-Dinitrofluorbenzol mit Imidazol-, SH- und Imidazol-SH-Verbindungen / Kinetics of the Reaction of 2,4-dinitro-1-fluorobenzene with Imidazole-, SH- and Imidazole-SH-compounds

1971 ◽  
Vol 26 (10) ◽  
pp. 1010-1016 ◽  
Author(s):  
Renate Voigt ◽  
Helmut Wenck ◽  
Friedhelm Schneider
Keyword(s):  
Temperature Dependence ◽  
Rate Constants ◽  
Reaction Rate ◽  
Ph Dependence ◽  
First Order ◽  
Molecular Transfer ◽  
Thiolate Anion ◽  
Nucleophilic Reactivity ◽  
Kinetics Of ◽  
Lindley Equation

First order rate constants of the reaction of a series of SH-, imidazole- and imidazole/SH-compounds with FDNB as well as their pH- and temperature dependence were determined. Some of the tested imidazole/SH-compounds exhibit a higher nucleophilic reactivity as is expected on the basis of their pKSH-values. This enhanced reactivity is caused by an activation of the SH-groups by a neighbouring imidazole residue. The pH-independent rate constants were calculated using the Lindley equation.The kinetics of DNP-transfer from DNP-imidazole to SH-compounds were investigated. The pH-dependence of the reaction displays a maximum curve. Donor in this reaction is the DNP-imidazolecation and acceptor the thiolate anion.The reaction rate of FDNB with imidazole derivatives is two to three orders of magnitude slower than with SH-compounds.No inter- or intra-molecular transfer of the DNP-residue from sulfure to imidazole takes place.

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