Frank-Kamenetskii Parameter Research Articles

Combustion hazards associated with the hot stacking of processed materials

Normally materials which are processed at high temperatures are allowed to cool before they are packed for storage or delivery. Occasionally a change of routine may cause the hot product to be packed before it has cooled to an appreciable extent. The risk is that spontaneous ignition may then take place in the packaged material. We analyse this type of problem numerically, with particular regard to the critical stacking temperature, T i,c , of cubes of material. We investigate its dependence both on the external ambient temperature and on the size of the packaging. The numerical calculations to solve the spatial and time dependent energy conservation equation with conductive heat transport using the finite difference method required a three dimensional grid, set up as 20 3 mesh points. The numerical tests were performed using parameters relevant to the exothermic reaction in cellulosic material. The initial condition was posed as a spatially-uniform, excess temperature within each packing case. This contribution differs from previous studies since, commonly, the analytical methods used refer to the critical, initial temperature excess, δ T i,c , within the reactant in dimensionless terms (through the Frank-Kamenetskii parameters ψ and σ) in only one-dimensional geometries. The transposition from the dimensionless solutions to the practical application of T i,c , is not very readily achieved.

Criticality in reactors under domain or external temperature perturbations

The conditions for the onset of thermal runaway in reactors with small non-uniformities is investigated. The reaction is modelled by an Arrhenius heat generation term with a finite activation energy and the dimensionless temperature,u0, is taken to satisfy a nonlinear equation of the form Δu0+λ0F(u0) = 0,x∈D; ∂vu0+bu0= 0.xϵ∂D. We investigate three classes of perturbations of this problem. First, we treat a small temperature variation maintained on the boundary of the domain. Secondly, we consider a small distortion of the boundary of a circular cylindrical domain, and thirdly, we analyse the effect of a small hole in the domain. In each case we derive asymptotic expansions for the critical Frank-Kamenetskii parameter,λc(ϵ), whereϵis a measure of the size of the perturbation. A numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion ofλc. Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem.

Thermal explosion theory with Arrhenius kinetics: homogeneous and inhomogeneous media

The classical thermal explosion problem for a reactive slab with Arrhenius kinetics and small activation energy parameter is considered. Perturbation series expansions are obtained that give an expression for the critical Frank-Kamenetskii parameter. We verify that the often used ‘Frank-Kamenetskii approximation’ is the first-order problem in such an expansion procedure. The analysis is extended to a reactive slab with a random distribution of reactant. The statistical properties are incorporated in an effective thermal diffusivity that may be expressed as a function of its mean and the variance about the mean. Some other expressions are also considered. It is shown that a random distribution decreases the thermal stability when compared with criticality with mean properties.

Finite activation energy reactions Part II: A reactive ridge

The effect is determined of a large,O(ɛ−2),ɛ≪1, activation energy on the thermal stability of a reactant in the form of a two-dimensional ridge, undergoing a steady zero-order exothermic reaction. The reactive ridge decreases in depth at a rate ofO(ɛ2) from a maximum ofO(1). The Biot number of the uniform lower surface of the reactant is taken to be zero and of the upper surface to beO(ɛ−2). The critical Frank-Kamenetskii parameter is determined toO(ɛ2).

Reaction-Diffusion Model for Combustion with Fuel Consumption: I. Dirichlet Boundary Conditions

A model, derived in a previous paper, for the reaction between a gaseous oxidant and a solid porous fuel is analysed further for general Robin boundary conditions. Numerical solutions are obtained and the effects of varying the dimensionless parameters, particularly the Frank-Kamenetskii parameter λ and the Lewis number L, are discussed in detail and compared with results obtained previously when Dirichlet boundary conditions are applied. Analytic solutions are obtained for the small-time development and for large values of λ. This latter solution shows the existence of a propagating reaction-diffusion burning wave, and has features which are qualitatively different to those derived earlier

Effect of autocatalysis on the critical conditions of focal thermal ignition

We examine the problem of focal thermal ignition in a substance capable of autocatalytic conversion. We have derived an expression by the method of joined asymptotic expansions for the critical value of the Frank-Kamenetskii parameter. We have determined the singularities of this expression, associated with autocatalysis. We have studied the limit cases of strong and weak autocatalysis.

Thermal explosion for multidimensional reactants with partial insulation

The thermal stability of reacting masses of varied geometries with partial insulation is examined. Upper and lower bounds for the critical value of the Frank-Kamenetskii parameter are determined, and numerical values for this parameter are obtained when the reactant takes the form of a long rod of square (or rectangular, for one aspect ratio) cross section. This parameter provides a useful criterion for the safe storage of reactive materials.

Critical Heat Losses to Avoid Self-Heating in Coal Piles

Abstract The Frank-Kamenetskii theory of the self-heating of a reactive slab is extended to include volumetric heat losses. The method of non-dimensionalising the heat loss highlights the important role of the Frank-Kamenetskii parameter in the heat dissipation term. Two types of heat loss are considered: firstly that without any dependence on temperature and secondly that with a global (and linear) temperature dependence. It is shown that in either case there is a critical non-dimensional heaf-loss rate beyond which thermal runaway will not occur. This value corresponds to when the actual heat loss is a certain small fraction of the volumetric heat generation. The fraction depends, through the Frank-Kamenetskii parameter, on the constants governing the chemical reaction— in particular the activation energy and heat of reaction. In that such volumetric heat losses are to be expected with the burning of low rank coals with high moisture content, the theory gives some indication of the critical water conten...

Thermal ignition of annular layer and its hydrodynamic analog

1. Critical conditions of thermal ignition of an annular layer have been investigated, with an arbitrary temperature difference at the cylinder surfaces and also with one surface thermostatted and the other heat-insulated. Transcendental equations determining the critical value of the Frank-Kamenetskii parameter are obtained. 2. With the same temperature boundary conditions, dissipative flow of a viscous liquid with an exponential temperature dependence of the viscosity between two rotating cylinders has been considered, with the value of the tangential stress specified at the surface of the rotating cylinder. An analogy is established between thermal ignition of an annular layer and dissipative liquid flow between two rotating cylinders. 3. In the case of one heat-insulated surface, the analogy exists between the ignition of an annular layer from the inner heat-insulated cylinder and dissipative flow with a heat-insulated outer cylinder, and conversely. 4. For these cases, there is a simple relation between the critical values of the Frank-Kamenetskii parameter of hydrodynamic thermal explosion and thermal explosion of chemical explosion; see Eq. (39).

Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures III. Asymptotic analysis of the disappearance of criticality (transition) at the extreme of small Biot number

Although numerical attacks on the problem of transition (loss of criticality) in thermally igniting systems with generalized resistance to heat transfer are always possible, they are particular to the cases studied. Very general circumstances may be treated analytically by perturbation methods (asymptotic expansion). Asymptotic expressions of considerable precision starting from the Semenov extreme (( Bi ) = 0) can be made in terms of Biot number ( Bi ). They apply to any geometry, and they are presented here for the infinite slab, the infinite cylinder and the sphere as expressions in terms of ( Bi ) for transitional values of the Frank-Kamenetskii or Semenov parameters, for transitional values of centre temperatures θ 0 and for transitional values of reduced ambient temperatures є . The method can cope with any temperature-dependence of rate coefficient f ( θ , є ). Numerical comparison is given for the case corresponding to Arrhenius kinetics, namely f ( θ , ε ) = exp [ θ /(1 + єθ )].

Travelling waves in exothermic systems

The classical theoretical problem of thermal ignition and extinction in a reactive slab of infinite extent under conditions near transition to continuous behaviour is revisited. It is assumed that the system is governed by two parameters. The first corresponds to the Frank-Kamenetskii parameter, δ; the second is in some circumstances related to the dimensionless ambient temperature of inverse activation energy ( β = RT a / E ) and in other circumstances to the dimensionless adiabatic temperature rise ( θ ad or B ). The value of the second parameter ( β or B ) is assumed to be close to its transition value, where a ‘cuspoidal’ behaviour of the reacting system appears. A perturbation analysis of the problem shows that additional, spatially distributed states exist in the system in the form of travelling waves of reaction. One of the newly discovered solutions is stable and corresponds to the one-dimensional combustion wave. The second solution is unstable and cannot be related to a real physical situation.

Numerical simulation of the thermal explosions of gaseous methyl isocyanide

A numerical simulation of the gaseous thermal explosions of methyl isocyanide and of mixtures of methyl isocyanide with methyl cyanide, is described. For all spherical vessel sizes from 0.3 to 13 dm3 and for temperatures from 330 to 370 °C, the computed explosion limits agree with the experimental values to within ca.± 0.2 Torr for critical explosion pressures in the range 2–11 Torr. For explosions of undiluted methyl isocyanide, the usual Frank-Kamenetskii critical parameter δc is ca. 3.95, and the reasons for its departure from the standard value of 3.32 are examined. Most important is reactant consumption, followed by gas flow away from the centre of the vessel, and variation of thermal conductivity with temperature; of lesser importance is the variation of the activation energy with pressure or of heat capacity with temperature.

Disappearance of criticality in thermal explosion for reactive viscous flows

This study numerically investigates the mass and heat transfer in a pressure induced flow of a reactive third-grade fluid with Reynolds’ viscosity model through a fixed cylindrical annulus, neglecting material consumption. The coupled nonlinear system of equations governing the flow are obtained in cylindrical coordinates and further cast into dimensionless forms via standard transformation. With graphical illustrations, the influence of salient physical parameters on the axial temperature and flow velocity are studied. Critical conditions are discussed using shooting method. The numerical solutions show very good agreement with those previously obtained from two-term series expansion as a special case. The effects of emerging parameters on the thermal critical characteristics are also discussed in detail. The results obtained are validated with relevant published articles in the literature. The quantitative effects of a state variable and several dimensionless parameters such as pressure gradient, numerical exponent of viscosity, an exponent of the pre-exponential factor, cylinder ratio, viscous heating parameter, and the material parameter accounting for the non-Newtonian nature of the fluid, on the activation energy, Frank-Kamenetskii parameter and Maximum inner cylinder temperature are explored.

Thermal explosion and times to ignition in systems with distributed temperatures. III. The influence of a steadily increasing (ramped) ambient temperature

In classical treatments of thermal explosion, reactant consumption is ignored and ambient temperatures are assumed constant. We turn here to the very important case where the temperature of the surroundings is varying at a steady rate. The influence of this ramping and the importance of reactant consumption are considered for the whole range of Biot numbers from the Semenov extreme (β = 0) to the Frank-Kamenetskii extreme (β→∞) for arbitrary geometry and for any concentration- and temperature-dependence of reaction rate. External heating modifies the critical value for the Frank-Kamenetskii parameter δ ; δ cr /δ 0 = 1 + P[Qg w - α )/B] ⅔ . Here δ 0 is the critical value without reactant consumption, α is the dimensionless rate of change of ambient temperature (and is negative for external cooling), B is the dimensionless adiabatic temperature rise, and g w is the effective order of reaction. The numbers P and Q are of order unity and are determined here for all important circumstances. Illustrative values are presented for the sphere, the infinite cylinder and the infinite slab. Times to ignition are also evaluated and, for limiting cases, simple asymptotic formulae are given which are good approximations over wide ranges. Many of the new results parallel those for constant ambient temperature but a new branch of solutions is found if α > Qg w where thermal runaway becomes inevitable and criticality is lost.

The Critical Conditions in a Reactive Slab with Slowly Varying Surface Temperature

Liouville's non-linear partial differential equation is considered for an infinite rectangular strip domain with a slowly varying boundary condition. The equation describes a layer of chemically reactive material under conditions where the resistance to surface heat transfer is negligible and the ambient temperature varies slowly along the surface. Symmetrical heating by a zero order exothermic reaction is assumed. If ε is a small dimensionless temperature difference between regions where the surface temperature is effectively constant, a perturbation series solution in δ may be determined provided the Frank-Kamenetskii parameter δ satisfies δ ≤ δc(ε). It is shown that a plausible value for the critical parameter is δc(ε) = δc(0) e−e,where δc(0) = 0.878. The corresponding critical temperature distribution is shown to have a dependence on ε different from that for subcritical cases.

Molecular dynamics simulation of thermal ignition in a reacting hard sphere fluid

Thermal ignition behavior of a fluid of hard disk particles capable of collision-induced exothermic reactions has been studied by means of molecular dynamics simulation. Critical values of the Frank-Kamenetskii parameter δ have been determined and compared with numerical solutions to the classical model of thermal explosion. Other properties that have been obtained from the simulation are temperature profile and ignition times. Present results show that a discrete system of several hundred particles can show behavior predicted by macroscopic equation; moreover, the simulation approach can provide information on fluctuations which is not contained in the continuum theories.

The reactive slab with partial insulation: A lower bound to criticality

The problem of determining critical conditions in a chemically reactive slab when the surface is partially insulated has been examined. A lower bound for the critical Frank-Kamenetskii parameter, which is a function of the insulation widthe and the activation energy parameterβ(β=R T 0/E), has been determined. Forβ=0 this bound is identical to that given by Khudyaev's method.

A generalized Semenov model for thermal ignition in nonuniform temperature systems

Critical properties of the Semenov model which describes thermal ignition in a uniform temperature system have been reanalyzed in terms of independent parameters, nondimensional activation energy, and surface heat loss. The analysis is then extended to the distributed temperature problem by the introduction of a correlation factor which effectively establishes a link between the Semenov model and the Frank-Kamenetskii parameter. This approach is capable of giving satisfactory results for properties such as ignition criteria, ignition times, and temperature profiles for a system with nonuniform temperature distributions. Specific results are presented mostly for the slab, but extensions to the cylinder and the sphere are also discussed.

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

Thermal explosion theory for partially insulated reactants

Thermal stability of reacting masses of slab and cylindrical form, having parts of their surfaces insulated and the remainder offering no resistance to heat transfer, is investigated; only symmetrically heated reactants are considered. It is assumed that the ratio of insulation size to slab width or cylindrical radius is small; perturbation expansions are used to determine the critical Frank-Kamenetskii parameter as a series in terms of this ratio.