Critical Frank-Kamenetskii Parameter Research Articles

Gelfand-type problem for turbulent jets

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet introduced in [11]. This model falls into the general class of Gelfand-type problems and constitutes a boundary value problem for a certain semi-linear elliptic equation that depends on two parameters: α characterizing the flow rate and λ (Frank-Kamenetskii parameter) characterizing the strength of the reaction. Similarly to the classical Gelfand problem, this equation admits a solution when the Frank-Kamenetskii parameter λ does not exceed some critical value λ⁎(α) and admits no solutions for larger values of λ. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit (α≫1). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to λ⁎) in this regime.

Numerical Convergence Analysis of the Frank-Kamenetskii Equation.

This work investigates the convergence dynamics of a numerical scheme employed for the approximation and solution of the Frank–Kamenetskii partial differential equation. A framework for computing the critical Frank–Kamenetskii parameter to arbitrary accuracy is presented and used in the subsequent numerical simulations. The numerical method employed is a Crank–Nicolson type implicit scheme coupled with a fourth order spatial discretisation as well as a Newton–Raphson update step which allows for the nonlinear source term to be treated implicitly. This numerical implementation allows for the analysis of the convergence of the transient solution toward the steady-state solution. The choice of termination criteria, numerically dictating this convergence, is interrogated and it is found that the traditional choice for termination is insufficient in the case of the Frank–Kamenetskii partial differential equation which exhibits slow transience as the solution approaches the steady-state. Four measures of convergence are proposed, compared and discussed herein.

Calculation of critical parameters for spontaneous combustion for some complex geometries using an indirect numerical method

In the theory of spontaneous combustion, identifying the critical value of the Frank- Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature. doi:10.1017/S1446181117000578

CALCULATION OF CRITICAL PARAMETERS FOR SPONTANEOUS COMBUSTION FOR SOME COMPLEX GEOMETRIES USING AN INDIRECT NUMERICAL METHOD

In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.

Combined Effect of Two Approximations to Critical Parameters in the Spontaneous Combustion of Cellulosic Materials

In the theory of Frank-Kamenetskii, the values of the critical Frank-Kamenetskii parameter and critical dimensionless temperature were obtained at two approximations. As we care more about the spontaneous of organic materials,we want to know how much effect the two approximations have for spontaneous combustion of organic materials. Cellulosic materials (such as sawdust) are typical organic materials of spontaneous combustion. In this paper, we provide an efficient indirect method to calculate the critical parameters of spontaneous combustion of cellulosic materials at different conditions. We can know that, the effect of dimensionless activation energy and Biot number to critical Frank-Kamenetskii parameter can be almost offset for cellulosic materials. However, the effect of dimensionless activation energy and Biot number to critical dimensionless temperature cannot be ignored. But, as the critical Frank-Kamenetskii parameter is more useful than the dimensionless temperature, the two approximations can be reasonable for spontaneous combustion of cellulosic materials or other materials like them.

Digital simulation of thermal reactions

Abstract Simulations of the equation for thermal expansion of a reacting gas have been carried out, exploring both the (possible) steady states and time-marching solutions. The critical Frank-Kamenetskii parameter δ cr has been evaluated to seven decimal places for the slab, cylinder and spherical geometries and the role of the critical activation parameter ϵ was explored. It was found that there exist one or more mathematical steady states for any δ if ϵ > 0 , the curves for steady temperature at the center of the geometry plotted against δ tending to a straight line at large δ . Critical values of ϵ , the values above which this plot has a single solution for a given δ , have been computed to eight decimals. Time marching simulations showed that the Crank–Nicolson method, applied consistently, produces very accurate results, compared with the implementation in which the nonlinear term is rendered explicit. Where for a given δ there are several mathematical steady states, a time march usually settles on the lowest such state (if it settles at all), regardless of where the simulation is started, within the possible limits. The mathematical multiple steady states are not attained by time marching simulations, and are also physically unlikely.

Approximate critical conditions in thermal explosion theory for a two-step kinetic model

Approximate critical conditions for a thermal explosion problem is developed for a two-step reactions based on theories of Semenov and Frank-Kamenetskii. The aim is to examine the contributions of the radical termination step and the temperature dependent pre-exponential factor on the critical parameters within the framework of classical stationary and non-stationary theories. In the non-stationary case, a more general expression for the critical Semenov parameter (Ψ cr ) and critical temperature (θ cr ) were obtained by asymptotic procedure. In the stationary case, numerical estimates for the critical Frank-Kamenetskii parameter (δ cr ) and the critical temperature (θ cr ) were obtained by variational method technique. It was observed that the Semenov and Frank-Kamenetskii parameters are greatly influenced by the termination step and the variable pre-exponential factor. Apart from elucidating hitherto unknown features in the theory of thermal explosion, the results are more general as some known results are easily recovered.

A note on the thermal stability of a reactive non-Newtonian flow in a cylindrical pipe

The effect of variable viscosity and viscous dissipation on the thermal stability of a one-step exothermic, reactive non-Newtonian flow in a cylindrical pipe was considered assuming negligible reactant consumption. The governing equations; momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by closed-form and approximate techniques respectively. We obtained closed-form solutions for the momentum equation via the Homotopy analysis method(HAM), while the resulting energy equation was analysed for thermal stability via the variational technique. Parametric analysis of the solutions were conducted on the dimensionless velocity ( v( r)), the critical Frank-Kamenetskii parameter( δ cr ) and the critical temperature( θ cr ) and expressed graphically and in tabular forms.

Effect of radiation on the critical Frank – Kamenetskii parameter of thermal ignition in a combustible gas containing fuel droplets.

We investigate the effect radiation on the critical Frank-Kamenetskii parameter of thermal ignition in a combustible gas containing fuel droplets (furnaces, gas turbines and internal combustion engines). Previous works show that radiation delays ignition in a well stirred reactor. In this paper we show that, even in a non-homogeneous reaction, the critical Frank-Kemenetskii parameter increases as the radiation parameter increases when the activation energy is high. This confirms the experimented observation in diesel engines. Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 217-220

On Thermal Explosion of Sensitized Reaction with Variable Heat Loss

This study discusses the influence of heat loss on the critical ignition temperature and Frank-Kamenetskii parameter of a non-linear ordinary differential equation arising in thermal sensitized reaction. The reaction obeys the Arrhenius expression with temperature dependent pre-exponential factor, taking heat exchange between the reacting material and its surrounding into account. The consequences of heat loss are explored within the framework of one dimensional, steady state model. The numerical estimations based on shooting method techniques show the effect of heat loss parameter on the critical values of ignition temperature and Frank-Kamenetskii parameter.

Thermal Explosion of a Plate under Boundary Conditions of the Second and Third Kinds

The critical conditions for a thermal explosion of a plane layer under boundary conditions of the second and third kinds specified on its surfaces are studied. The critical Frank–Kamenetskii parameter is calculated. Approximating functions used to determine the critical parameter are given.

Thermal explosion of a hollow cylinder

The critical conditions for thermal explosions with nonsymmetric boundary conditions of the first kind are studied in terms of the Grishin-Bostandzhiyan model for thermal explosion of a hollow cylinder. The cases in which a hot boundary lies on the inner and outer surfaces of the cylinder are studied. The results of a numerical analysis are presented and approximate functions are given for determining the critical Frank-Kamenetskii parameter.

Laser ignition of A combustible gas: A composite layer steady-state model

A composite layer model for laser ignition has been investigated. It consists of a kaowood mat, an adjacent combustible gas layer and a high intensity beam impinging on the planar interface from the gas side. Some of the simplifying assumptions made include, steady-state conditions, a single-step reaction with negligible reactant consumption within the gas and, initially, a beam of sufficient width so that a one-dimensional process can be considered. The model is later extended to one in which the beam intensity varies with radial distance from its axis; it is shown, however, that this model can in general only be solved numerically. The assumption of a small activation energy parameter, permits the use of suitable expansions for the dimensionless gas temperature and Frank-Kamenetskii parameter whose critical value is to be determined. The dominant term of this critical parameter has been obtained under the assumption that the heat loss on the gas side is principally by radiation. It is shown that the critical interface temperature and Frank-Kamenetskii parameter are monotonic functions of a defined heat loss parameter.

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).

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.

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.

Thermal explosion theory for a slab with partial insulation

Liouville's nonlinear partial differential equation is considered for an infinite rectangular strip domain with mixed boundary conditions. It arises in the determination of the two-dimensional temperature distribution within an exothermically reacting slab having parts of its surface insulated and the remainder offering no resistance to heat transfer. For symmetrical heating, with insulating strips on upper and lower surfaces, the critical Frank-Kamenetskii parameter is found to be δ c(∈)=δ c(0)(1— ∈—0.555 ∈ 3 2 +⋯), where δ c (0) = 0.878 and ϵ, assumed small, is the ratio of the insulation width to the slab thickness.

Thermal explosion theory for a slab with time-periodic surface temperature variation

The problem of the onset of thermal explosion in a slab subject to time-periodic surface temperature variation has been investigated where the slab is symmetrically heated by an exothermic zero-order chemical reaction. The main purpose of the investigation was to obtain the critical Frank-Kamenetskii parameter #c(e, w) as a function of amplitude e and frequency w/2n of the surface temperature oscillation. We have shown that with values of this parameter, such that 0 < e, w) steady temperature oscillations can be maintained within the slab, but that 8 > 8c{e, w) must lead to thermal explosion. For a period of oscillation of 24 h and suitable values for the thermal diffusivity and half-width of the slab, o) 27i, and the parameter range 0 < e ^ 4 covers ambient temperature fluctuations likely to be encountered by hazardous materials, such as in the hold of a ship in tropical seas. The problem has been examined in three different ways, (i) A comparison theorem for partial differential equations has been used to determine an analytical bound on 8,the result shows th at stable oscillations exist for all oj, provided 8^ 8C e~e (8C = 0.878), this represents a lower bound to the stability surface in (8, e, oj) space, (ii) Perturbation theory, for small amplitude, has been used to determine the critical parameter in the form oj) — 8C + e1 281(oj) + ... with £x(w) a function of frequency. Comparison with the exact numerical solution shows that this gives values which differ by less than about 0-2 % for oj^ 2k and 0 ^ e^ 1. (iii) The energy conservation equation has been solved numerically over a rectangular mesh representing the half-width of the slab and one period of steady oscillation. For given e, oj such solution could be found for 8 sufficiently small; it is shown th at the breakdown of the numerical process is associated with criticality, allowing the limiting parameter to be determined. This method has been used to obtain curves of £c(e, oj) versus e for oj = 2 4n, 8tt and the range 0 ^ ^ 4.