Chang-Gen Feng

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.

Thermal Explosion and Times-to-Ignition in Systems with Distributed Temperatures II. The Influence of Reactant Consumption

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, δcr/δ0 = 1 + ϕ(gw/B)⅔, where the coefficient ϕ depends on the geometry, gw represents an effective order of reaction (to be evaluated at initial temperature and concentration), B is a dimensionless adiabatic temperature rise and δ0 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, tign, differs by little from the ignition time tign(B → ∞) for the case of zero reactant consumption: tign/tign (B → ∞) = 1+G(gw/B) (δ-δ0/δ)-3/2 +O(gw/B)2. For the Arrhenius temperature dependence with large activation energy the constant G is about unity for all cases.

Thermal explosion, times to ignition and near-critical behaviour in uniform-temperature systems. Part 3.—Effects of variable heat-transfer coefficients

Although it is common in thermal-explosion theory to neglect the temperature dependence of thermophysical properties, there are circumstances in which this is not justifiable. This paper examines how (i) times-to-ignition and (ii) the Semenov criterion for criticality are affected when the heat-transfer coefficient χ is not constant but depends on temperature, χ/χ0=h(θ). A systematic approach is used that can cope with generalized forms of temperature dependence and concentration dependence of reaction rate.When reactant consumption is unimportant the time-to-ignition depends on the degree of supercriticality according to tign//tad=L//(ψ/ψcr–1)½∝1//(degree of supercriticality)1/2.Values for L are readily calculated; L is commonly of order unity but grows without limit as transition (from criticality to continuous behaviour) is approached.When reactant consumption is considered, the shift of the Semenov criterion ψ from its standard value at criticality ψ0 is given by (ψcr/ψ0)–=Δcrθ–2/3ad(1 –H)–1/3., Values for Δcr and H are again readily calculated.

Thermal explosions, critically and transition in systems with variable thermal conductivity. Distributed temperatures

The conductive theory of thermal explosion introduced by Frank-Kamenetskii neglects any temperature dependence of thermal conductivity κ. In certain circumstances this is too simple a view, and the present paper evaluates the consequences of assuming a realistic dependence on temperature. Two important forms are considered linear: κ/κ0= 1 +α(T–Ta)= 1 + aθ=h1(θ) square root: κ/κ0=(T/Ta)1/2=(1 +Iµθ)=h2(θ) In these equations θ=(T–Ta)/(RT2a/E) denotes the dimensionless excess temperature and Iµ=RTa/E reflects the ambient temperature Ta. The heat-balance equation in the stationary state becomes div [h(θ) grad θ]+δ exp [θ/(1 +Iµθ)]= 0, where δ is the conventional dimensionless measure of the reaction rate. This equation has been solved numerically for the three ‘class A’ geometries (sphere, infinite cylinder and infinite slab) for Iµ≠ 0 (Arrhenius form, general case) and for Iµ→ 0 (exponential approximation) subject to the condition θ= 0 at the boundary. The following features have been established: (a) temperature-position profiles for subcritical and critical circumstances for different a and Iµ, (b) critical values of the parameters δ and θ0 and (c) the disappearance of criticality at high temperatures or low activation energies (transition).The results are compared with their simpler prototypes for the Semenov uniform-temperature case when the heat-transfer coefficient χ is not constant but depends upon temperature.

Thermal explosions, criticality and transition in systems with variable heat-transfer coefficient. Uniform temperatures (semenov case)

Although it is common to neglect the temperature dependence of thermal conductivity and of heat-transfer coefficients in thermal explosion theory, there are certain circumstances where this is not appropriate. This paper examines how criticality and transition are affected in the uniform-temperature case when conventional heat-transfer coefficient χ depends on temperature. The types of dependence examined are chosen partly for their intrinsic importance and partly to highlight similarities and differences between uniform- and distributed-temperature fields.

Times to ignition in systems initiated by light

An infinite slab of an exothermic reactant has one face kept at ambient temperature whilst the other is subjected to a continuous flux of radiant energy. When reactant consumption can be ignored, a stationary temperature distribution is approached that can persist as long as conditions for criticality are not surpassed (δ≦δcr). When conditions are marginally supercritical, the temperature distribution eventually peels away. Times-to-ignition tign are shown to obey the rule tign∝(δ–δcr)–1/2 where δcr depends on the intensity of the illumination, and values for the proportionality constant are calculated.