J. Barry Greenberg

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 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.

Theoretical Model of the Transient Combustion of Organic-Gellant-Based Gel Fuel Droplets

Experimental evidence of the combustion process of an all-organic gel fuel droplet indicates that at a certain time after ignition, evaporation of the liquid fuel results in the formation of an elastic layer of high-viscosity gellant around the droplet, which prevents further vaporization. As a result, constantly expanding vapor bubbles are produced within the droplet. Eventually, the layer ruptures and jets of fuel vapor are released. A theoretical, time-dependent model of organic-gellant-based gel droplet combustion has been developed and numerically solved. The results indicate that the evaporation rate of the liquid fuel from the droplet surface depends on droplet size and strongly affects the thickness of the gellant layer. The tensile stress, applied to the gellant layer during the formation of the bubbles, reaches high levels in short periods of time and causes the droplet to rupture when it exceeds the layer material rupture stress. The stage during which the gellant layer is formed is almost three orders of magnitude longer than the stage of bubble formation and layer rupture.

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.

Spray flame dynamics with oscillating flow and droplet grouping

The co-flow laminar spray diffusion flame in an oscillating flow field is investigated. Mild slip is permitted between the droplets and their host surroundings and droplet grouping resulting from the host flow oscillations is accounted for. The spray is modelled using the sectional approach and a perturbation analysis using a small sectional Stokes number is utilised for solving the liquid phase governing equations. The effect of droplet grouping is described through a specially constructed model for the vaporisation Damkohler number. The large chemical Damkohler number assumption is adopted and a formal analytical solution is developed for Schwab-Zeldovitch parameters through which the dynamics of the spray flame front shapes and thermal fields are deduced. Computed results based on the solutions demonstrate how the phenomenon of droplet grouping can lead to the existence of multiple flame sheets as a result of the dynamic change in the type of the main homogeneous flame from under- to over-ventilated as the flow field oscillates. Concomitant fluctuating thermal fields are also shown to be present indicating a potential impact on undesirable pollutants production.

Linear stability analysis of laminar premixed fuel-rich double-spray flames

This paper considers the stability of a double-spray premixed flame formed when both fuel and oxidizer are initially present in the form of sprays of evaporating liquid droplets. To simplify the inherent complexity that characterizes the analytic solution of multi-phase combustion processes, the analysis is restricted to fuel-rich laminar premixed double-spray flames, and assumes a single-step global chemical reaction mechanism. Steady-state solutions are obtained and the sensitivity of the flame temperature and the flame propagating velocity to the initial liquid fuel and/or oxidizer loads are established. The stability analysis revealed an increased proneness to cellular instability induced by the presence of the two sprays, and for the fuel-rich case considered here the influence of the liquid oxidizer was found to be more pronounced than that of the liquid fuel. Similar effects were noted for the neutral pulsating stability boundaries. The impact of unequal latent heats of vaporization is also investigated and found to be in keeping with the destabilizing influence of heat loss due to droplet evaporation. It should be noted that as far as the authors are aware no experimental evidence is available for (at least) validation of the predictions. However, they do concur in a general and reasonable fashion with independent experimental evidence in the literature of the behavior of single fuel spray laminar premixed flames.

Analysis of radiative heat-loss effects in thermal explosion of a gas using the integral manifold method

Semenovs classical model of thermal explosion in a combustible gas mixture is modified to include radiative (rather than conductive) heat-loss effects and gas-density changes. A geometrical asymptotic technique (the method of integral manifolds - MIM) is exploited to perform a qualitative analysis of the governing equations. The strength of this method lies in the compact, clear geometrical/analytical rendition and classification of all possible dynamical scenarios, in terms of the physico-chemical parameters of the system. It is found that there are two main dynamical regimes of the system: cooling regimes and fast explosive regimes. Peculiarities of these dynamical regimes are investigated and their dependence on physical system parameters is analyzed. A criterion for the occurrence of thermal explosion is disclosed. An estimate for the maximum mixture temperature is also derived analytically. It is found that, under certain operating conditions, the dynamics are such that the initial explosive stage of the process essentially behaves adiabatically before succumbing to the dominance of the radiative heat loss that brings the system down to the ambient temperature.

EDGE FLAMES WITH A FUEL SPRAY AND REACTANTS HAVING DIFFERENT DIFFUSIVITIES

The nature of the edge flame formed downstream of a splitter plate initially separating two co-flows of fuel vapor and droplets and an oxidant is investigated. The way in which both the presence of a spray of fuel droplets and the different diffusivities of the fuel vapor and the oxidant influence the combustion in the mixing layer formed is examined. The spray is described using the sectional approach and for simplicity a mono-sectional model is utilized. The steady state behavior of the system is described using a diffusional-thermal model and the governing non-linear equations are solved numerically. The results illustrate how the initial droplet load, the vaporization Damkohler number and the Lewis numbers of the fuel vapor and oxidant determine the intensity and location of the edge flame and its trailing diffusion flame. These parameters are also shown to possess critical values for the onset of flame oscillations. In particular, altering the oxidant Lewis number (by, say, changing the diluent in which it is supplied) can effectively dampen undesirable spray induced oscillations.

On Blow-Out and Lift-Off of Laminar Jet Spray Diffusion Flames

ABSTRACT A laminar jet spray diffusion flame is analyzed mathematically for the first time using an extension of classical similarity solutions for gaseous jet flames. The analysis enables a comparison to be drawn between conditions for flame stability or flame blow-out for purely gaseous flames and for spray flames. It is found that, in contrast to the Schmidt number criteria relevant to gas flames, spray-related parameters also play a critical role in determining potential flame scenarios. Excellent qualitative agreement for lift-off height is found when comparing predictions of the theory and independent experimental evidence from the literature.

Propagation of a Spherical Flame through a Polydisperse Fuel Spray

A new evolution equation is derived for a laminar flame front propagating into air and a polydisperse liquid fuel spray cloud. The asymptotic analysis employed for developing the equation exploits the usual inverse large activation energy parameter associated with chemical reaction in the flame. It is demonstrated that the droplet size distribution in the fuel cloud can be critical in determining whether propagation or extinction of the flame front occurs, even for different size distributions initially having the same Sauter Mean Diameter (SMD).