Igor Goldfarb

Thermal ignition analysis of a monodisperse spray with radiation

The system of equations describing the effects of heating, evaporation, and combustion of fuel droplets in a monodisperse spray is simplified assuming that the Nusselt and Sherwood numbers are equal to 2. The radiative energy exchange between fuel droplets surface and gas is described by using the P-1 model with Marshak boundary conditions. The chemical term is presented in the Arrhenius form with the pre-exponential factor calculated from the enthalpy equation, using the Shell autoignition model. The resultant, singularly perturbed system of ordinary differential equations is analyzed, based on the geometrical version of the integral manifold method. The ignition process is subdivided into two stages: droplet evaporation and ignition of the gaseous mixture. Results predicted by the analytical solutions are compared with those predicted by the CFD package VECTIS. It is suggested that the analytical solution underpredicts the evaporation time. A considerably better agreement between the evaporation times predicted by VECTIS and the proposed theory is achieved when the gas temperature is assumed to be equal to the local temperature in the vicinity of droplets. The effects of thermal radiation are significant, especially at high temperatures and with large droplets, and cannot be ignored.

Liquid flow in foams

A model equation for describing liquid motion in a foam of polyhedral structure is proposed. A dimensionless parameter characterizing the structure of the foam, namely, the ratio of the volume energy densities of the capillary and gravitational forces, is introduced. When the gravitational forces predominate over the capillary forces, the out-flow process may be regarded as a kinematic wave that can be described by the Burgers equation. In the opposite case, the capillary absorption can be described by a quasilinear parabolic equation.

Thermal radiation effect on thermal explosion in gas containing fuel droplets

The effect of thermal radiation on the dynamics of a thermal explosion of a flammable gas mixture with the addition of volatile fuel droplets is studied. This is based on an original physical model of self-ignition. The thermal radiation energy exchange between the evaporating surface of the fuel droplets and burning gas is described using the P-1 model with Marshak boundary conditions. The original system of equations describing the effects of heating, evaporation and the combustion of fuel droplets is simplified to enable their analysis using asymptotic methods. The mathematical formulation is eventually reduced to a singularly perturbed system of ordinary differential equations. This allows us to apply the advanced geometric asymptotic technique (integral manifold method) for the qualitative analysis of the behaviour of the solution. Possible types of dynamic behaviour of the system are classified and parametric regions of their existence are determined analytically. The main attention is concentrated ...

On thermal explosion of a cool spray in a hot gas

The effect of a flammable spray on thermal explosion in a preheated combustible gas mixture is investigated using a simplified model that contains the essentials of the basic physical processes at work. The study represents a re-examination of the question of the ignition of a spray of droplets from the viewpoint of an explosion problem, in which the droplets are taken to be a source of endothermicity. Use is made of various methods for the qualitative analysis of systems of differential equations in order to examine the dynamics of the system. Possible types of dynamical behavior of the system are looked into and parametric regions of their existence are determined analytically. Peculiarities, of these dynamical regimes are investigated, and their dependence on the physical system parameters are analyzed. In particular, analytical formulas are developed for ignition delay times by exploiting the sensitivity of the process to the chemical activation energy. A qualitative comparison of predicted ignition times with independent experimental measurements from the literature yields good order of magnitude agreement.

Thermal explosion in a combustible gas containing fuel droplets

An original physical model of self-ignition in a combustible gas mixture containing liquid fuel droplets is developed. The droplets are small enough for the gas-droplet mixture to be considered as a fine mist such that individual droplet burning is subsumed into a well-stirred, spatially invariant burning approximation. A classical Semenov-type analysis is used to describe the exothermic reaction, and the endothermic terms involve the use of quasi-steady mass transfer/heat balance and the Clausius-Clapeyron evaporative law. The resulting analysis predicts the ignition delay which is a function of the system parameters. Results are given for typical dynamical regimes. The case of different initial temperatures for droplets and gas is highly relevant to gas turbine lean blow-out and re-ignition.

Thermal explosion in a hot gas mixture with fuel droplets: a two reactant model

We extend previous analyses of thermal explosion in a gas-droplets mixture to permit a more complete description of the chemistry via a single-step two-reactant model of general order, rather than the prior deficient reactant model. A detailed mathematical analysis has been carried out of this new physical model that encompasses oxidizer effects (in both fuel rich and fuel lean situations) on the thermal explosion of a hot combustible mixture of gases and cool evaporating fuel droplets. The closed mathematical formulation involves a singularly perturbed system of four highly non-linear ordinary differential equations. The entire dynamical picture of the system is qualitatively exposed by exploiting the geometrical version of the powerful asymptotic approach known as the method of integral manifolds (MIM). It was found that the systems behaviour can be classified according to the values of nine dimensionless parameters. All possible types of dynamical behaviour of the system were studied and the parametric regions of their existence were delineated, with emphasis on the underlying physico-chemical processes at play. Both conventional explosive and delayed regimes were found to occur, including the freeze delay regime. Whereas this latter important regime had been associated with physically unviable operating conditions in previous deficient reactant models, it was found that the current use of a single-step two-reactant chemical kinetic model renders the freeze delay regime physically plausible. Due to its practical importance the delayed regimes were analysed in detail and explicit analytical formulae for delay and evaporation times were extracted. The predictions were found to agree rather well with the results of direct numerical simulations. It was also found that the stoichiometry of the initial mixture per se does not lead to a natural classification of different sorts of regimes. Rather, the ratio of two key parameters plays the dominant role in defining the relevant fast variables and their associated dynamical regimes, irrespective of the initial mixture stoichiometry.

Heat transfer effect on sound propagation in foam

The influence of heat transfer on the propagation of acoustic disturbances in a gas–liquid foam of polyhedral structure is considered in this paper. The presence of quasiordered structure of microcapillaries (Plateau–Gibbs channels) results in the appearance of a hydrodynamic effect caused by liquid motion along the channels and due to the competition with the contribution of heat transfer in general wave dissipation. The interphase heat transfer process in a gas‐filled foam is considered within the framework of a cell model. Liquid motion through the proposed channels depends on the radial dynamics of the foam cell and on the hydroconductivity of the foam defining a free liquid motion through the channels. The Rayleigh equation analog, which takes into account liquid motion through the channels of the foam and small moisture content, is obtained. The dispersion relation shows that the influence of surface tension can be ignored. The effects of added liquid mass and the viscous interphase mechanism in the...

Novel numerical decomposition approaches for multiscale combustion and kinetic models

A general theoretical procedure of decomposition of original multiscale system of ODEs into fast and slow subsystems is presented. Multiscale systems arise in modelling of chemical, biochemical, mechanical systems. The performed analysis shows that existing algorithms can be interpreted as possible realizations of the general framework for identification of slow invariant manifolds (slow motions). The aim of this framework is a decomposition of the original system of equations into two separate sets - fast and slow sub-systems. The analysis is based on a new concept of a singularly perturbed vector field. General procedures for decomposition of singularly perturbed fields onto fast and slow parts are presented and this permits us to develop the quasi-linearization method for identification of the fast and the slow subprocesses for complicated kinetics and combustion problems. Application of the suggested numerical technique (the quasi-linearization method) demonstrates that a number of the uncovered restrictions on existing numerical procedures are successfully overcome. The proposed numerical procedure is applied to the highly non-linear problems of mathematical theory of combustion and demonstrates an essentially better performance with respect to existing ones.

A study of delayed spontaneous insulation fires

The phenomenon of gaseous thermal explosion within inert insulation material filled with a combustible evaporating fuel is considered. A novel approach to estimate the activation energy is suggested using the available experimental data and the relevant theoretical predictions concerning the delay ignition time.

Thermal explosion in a droplet-gas cloud

The effect of the presence of a spray of liquid fuel on thermal explosion in a combustible droplet-gas cloud is investigated. By ‘thermal explosion’ we refer exclusively to the initial stages of the behaviour of the combustible medium as its temperature begins to rise and various competing physical and chemical processes are called into play. A qualitative analysis of the system of governing equations is carried out using an advanced geometrical asymptotic technique (the integral manifold method). Possible types of dynamical behaviour of the system are classified and parametric regions of their existence are determined analytically. It is demonstrated that the original problem can be decomposed into two subproblems, due to the underlying hierarchical time scale structure. The first subproblem relates to the droplet heat up period, for which a relatively rapid time scale is applicable. The second subproblem begins at the saturation point. For the latter, more significant second stage, it is found that there are five main dynamical regimes: slow regimes, conventional fast explosive regimes, thermal explosion with freeze delay and two different types of thermal explosion with delay (the concentration of the combustible gas decreases or increases). Upper and lower bounds for the delay time are derived analytically and compared with results of numerical simulations, with rather satisfactory agreement.