T. Boddington

Thermal Theory of Spontaneous Ignition: Criticality in Bodies of Arbitrary Shape

For an exothermic reaction to lead to explosion, critical criteria involving reactant geometry, reaction kinetics, heat transfer and temperature have to be satisfied. In favourable cases, the critical conditionstreatments are either confined to idealized geometries, namely, the sphere, infinite cylinder or infinite slab, or require the simplest representations of heat transfer. may be summarized in a single parameter, Frank-Kamenetskii’s δ being the best known, but analytical. In the present paper a general steady-state description is given of the critical conditions for explosion of an exothermic reactant mass of virtually unrestricted geometry in which heat flow is resisted both internally (conductive flow) and at the surface (Newtonian cooling). The description is founded upon the behaviour of stationary-state systems under two extremes of Biot number—that corresponding to Semenov’s case (Bi→0) and that corresponding to Frank-Kamenetskii’s case (Bi→∞) it covers these and intermediate cases. For Semenov’s conditions, the solution is already known, but a fresh interpretation is given in terms of a characteristic dimension—the mean radius Rs. A variety of results for criticality is tabulated. For Frank-Kamenetskii’s conditions, the central result is an approximate general solution for the stationary temperature distribution within any body having a centre. Critical conditions follow naturally. They have the simple form: qσAexp⁡(−E/RTa)kRTa2/ER02er≡δer(R0)=3F(j), where F(j) is close to unity, being a feeble function of shape through a universally defined shape parameter j, and δcr(R0) is Frank-Kamenetskii’s δ evaluated in terms of a universally defined characteristic dimension R0—a harmonic square mean radius weighted in proportion to solid angle:1R02=14π∫∫dωa2. Expressions for the mean radius R0 have been evaluated and are tabulated for a broad range of geometries. The critical values generated for δ are only about 1 % in error for a great diversity of shapes. No adjustable parameters appear in the solution and there is no requirement of an ad hoc treatment of any particular geometric feature, all bodies being treated identically. Critical sizes are evaluated for many different shapes. For arbitrary shape and arbitrary Biot number (0 < < ∞) an empirical criterion is proposed which predicts critical sizes for a great diversity of cases to within a few parts per cent. Rigorous, closely adjacent upper and lower bounds on critical sizes are derived and compared with our results and with previous investigations, and the status of previous approaches is assessed explicitly. For the most part they lack the generality, precision and ease of application of the present approach.

Criteria for thermal explosions with and without reactant consumption

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

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 = QVAcm0 e-E / RTa / XS(RT2a/E); ψcr = e-1; θcr = 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 ψcr 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 explosions and the disappearance of criticality at small activation energies: exact results for the slab

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.

Theory of Thermal Explosions with Simultaneous Parallel Reactions. I. Foundations and the One-Dimensional Case

Many real, exothermic systems involve more than one simultaneous reaction. Even when they are chemically independent, interactions must arise through their several responses to the collective generation of heat. A simple and unifying approach is possible to the behaviour of such systems below and up to criticality. It introduces a communal activation energy E as the basis for dimensionless quantities (θ, δ, ϵ and so on) but otherwise involves only familiar ideas from basic thermal explosion theory. The definition of E is E = RT2 d (In Z)/dT, where Z = ƩZi. Here, Z is the rate of energy release per unit volume (the power density) by the whole system and Zi is the contribution of the constituent i. This enables us to define and use the conventional dimensionless parameter δ for the whole system and for its constituent reactions. We illustrate affairs by considering a pair of concurrent, exothermic reactions; heat is transferred solely by conduction towards the faces (temperature Ta) of an infinite slab of thickness 2a and conductivity k. For a constituent reaction (i = 1, 2 here) δi = (Ea2/k RT2a) Zi(Ta) and for the whole system δ = δ1 + δ2 (+...) for two (or more) reactions. We find that the condition δ > δcr guarantees instability, where δcr is always less than 0.878. The bounds 0.65 < δcr < 0.878 are good enough for a substantial range of relative sizes of activation energy 0.2 < E1/E2 < 5. We also pursue the problem numerically and present solutions for critical δ and critical central temperature excess over the whole composition range for a pair of simultaneous exothermic reactions.

Thermal Explosion and Times-to-Ignition in Systems with Distributed Temperatures. I. Reactant Consumption Ignored

Hitherto the time evolution of temperature profiles within systems capable of reacting exothermically and perhaps of exploding has only been known for the limiting case in which the temperature remains essentially uniform throughout. In this paper we consider the more difficult and general problem in which the temperature profile is not uniform. We concentrate on numerical results for the three simplest geometries (infinite slab, infinite cylinder, sphere) but our solutions are valid for arbitrary geometry. Similarly we concentrate on Arrhenius temperature-dependence; the solutions can again cope readily with general forms. The principal result of the paper is a description of the evolution of the central temperature when circumstances are marginally supercritical. When the value of Frank-Kamenetskii’s reduced reaction rate parameter (δ) slightly exceeds its critical value (δcr), the temperature at first increases rapidly and then moves very slowly for a long while before a final, rapid acceleration leads to ignition. Times to ignition (t) all have the form t/tad = M/(δ/δcr - 1)½ where tad is a known time closely corresponding to the time to ignition under adiabatic conditions. The value of the constant M is readily calculated from certain simple integrals. Its variation with geometry and Biot number β is presented and discussed. The limiting form of our expressions for β → 0 is precisely the well known solution of the problem for Semenov boundary conditions. We also compare our results with an exact (numerical) solution of the time-dependent problem for a sphere with fixed surface temperature. Not only is the agreement close to criticality very good but also it yields a fair approximation up to δ ≈ 2δcr.

Thermal explosion and the theory of its initiation by steady intense light

The initiation of explosion by steady intense light (from lasers or other sources) involves degradation of the radiation to heat, leading to self-heating and thermal runaway. The critical conditions for such a thermal mechanism can still be expressed in terms of the standard dimensionless group δ = qσa02AEexp ( -E/RTa)/k RTa2. When δ attains a critical value δcr, thermal explosion occurs: the critical value depends on the intensity of the radiation. The dependence of δcr on the light intensity β is computed numerically for an Arrhenius temperature-dependence of rate. The corresponding critical values for the reduced central temperature-excesses θ0, cr are also obtained. If the ambient temperature is too high or the activation energy is too low, so that є = RTa/E is not very small, the phenomenon of criticality disappears. Accurate transitional values for the reduced ambient temperature єtr are calculated as a function of the intensity of the light.

Thermal explosion, times to ignition and near-critical behaviour in uniform-temperature systems. Part 2.—Generalized dependences of rates on temperature and concentration

This paper extends the range of earlier but narrower treatments of the time evolution of reactant temperature in exothermic, homogeneous systems near to criticality. It considers (i) generalized temperature dependences of the reaction rate coefficient, ƒ(θ), and (ii) generalized concentration dependences of the reaction rate, g(w). Here θ is the dimensionless temperature excess and text-decoration:overlinew=(c/c0) is the fractional concentration.When reactant consumption can be neglected, the temperature dependence of the rate of reaction is represented by k∝ exp (–E/RTa)ƒ(θ). The Arrhenius form implies ƒ(θ)= exp [θ/(1 +Iµθ)], and the case of Iµ→ 0 corresponds to the exponential approximation. Time to ignition has the form [graphic omitted], where M′={2π2/[ƒ(θ0)ƒθθ(θ0)]}1/2. This correctly generates the value M′= [graphic omitted]2π/e for the exponential approximation.When transition is reached, our results show that M′→∞; therefore tign lengthens without limit even for a fixed degree of supercriticality.A similar systematic approach allows the influence of reactant consumption on the critical value of the Semenov number to be assessed for a very wide variety of reaction rate expressions (and not merely first or nth order). We find the general form: ψcr=ψ0[1 + 2.946(ƒ0/ƒθθ)1/3(gw/B)2/3]. (For an nth-order deceleratory reaction gw=n.)

Thermal explosion, times to ignition and near-critical behaviour in uniform-temperature systems. Part 4.—Effects of programmed ambient temperature

When a deceleratory exothermic reaction proceeds in a closed vessel under conditions of varying ambient temperature, thermal runaway to explosive speeds may occur. The course of events is determined by the rate of change of ambient temperature, the responsiveness of the reaction rate to both temperature and extent of reaction, and the maximum possible extent of self-heating.The problem is investigated systematically, though not by employing ‘large activation energy’ asymptotics. We give a very general analysis for any temperature and concentration dependence (not merely mth-order Arrhenius). The primitive model which neglects reactant consumption and uses a fixed ambient temperature is seen to be important in understanding the full problem. The critical conditions are then expressed in terms of shifts in the Semenov number (Se or ψ) from the primitive value.The treatment also deals with whether it is possible to prevent an ignition in a normally supercritical reaction by external cooling. Times to ignition, or to reach maximum temperature for a subcritical reaction, are given as simple but general formulae for both external heating and cooling.

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

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) = eθ, for which the results are well known and for the case f(θ)= exp[θ /( 1+εθ )], corresponding to Arrhenius kinetics.