Journal of Combustion

Volume 2016, Article ID 1069873, 14 pages

http://dx.doi.org/10.1155/2016/1069873

## On Laminar Rich Premixed Polydisperse Spray Flame Propagation with Heat Loss

Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel

Received 22 November 2015; Revised 4 February 2016; Accepted 7 February 2016

Academic Editor: Ashwani K. Gupta

Copyright © 2016 G. Kats and J. B. Greenberg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A mathematical analysis of laminar premixed spray flame propagation with heat loss is presented. The analysis makes use of a distributed approximation of the Arrhenius exponential term in the reaction rate expression and leads to an implicit expression for the laminar burning velocity dependent on the spray-related parameters for the fuel, gas-related parameters and the intensity of the heat losses. It is shown that the initial droplet load, the value of the evaporation coefficient, and the initial size distribution are the spray-related parameters which exert an influence on the onset of extinction. The combination of these parameters governs the manner in which the spray heat loss is distributed spatially and it is this feature that is the main factor, when taken together with volumetric heat loss, which determines the spray’s impact on flame propagation and extinction.

#### 1. Introduction

Spalding [1] was the first to treat the problem of a laminar gas flame propagating through a combustible premixture in the presence of heat losses, for example, due to heat loss by conduction to the walls of the combustion chamber or radiation. In keeping with experimental evidence it was found that for a given heat loss there exist two possible burning velocities, one stable and the other not. Extinction occurs at the point of traversal from the stable to the unstable mode of propagation. Essentially, this happens when the heat loss is too great. The theory agreed well with experiments for flame propagation and extinction in tubes. Subsequent work [2, 3] also dealt with similar problems of one-dimensional flame propagation and examined different aspects of extinction of these flames. Buckmaster [4] reexamined the aforementioned problem using asymptotic tools and was able to construct the slow and fast waves as well as to predict a simple explicit quenching criterion. Joulin and Clavin [5] considered the stability of laminar premixed flames subject to linear heat loss and found a variety of instabilities for the different regimes (slow and fast waves) examined by previous researchers. Nicoli and Clavin [6] considered the effect of variable heat loss intensities on the dynamics of a premixed flame. Clavin and Nicoli [7] investigated heat loss effects on stability limits of downward propagating premixed flames.

In the context of mathematical analysis of one-dimensional premixed spray flames, some attention was directed to the influence of heat losses [8, 9], although when the stability of such flames was considered heat losses were not accounted for [10–13]. However, in [8, 9] the linear volumetric heat losses were taken as being of order , where is inversely proportional to the activation energy of the assumed global chemical reaction, and were only applied in the region between the onset of droplet evaporation and the flame front. In addition, the sprays were taken to be monodisperse.

As pointed out by Sirignano [14] radiation impacts on* individual* droplet heating and evaporation in several ways. Primarily, droplets may be heated by radiation from high temperature gases. Or, alternatively, radiation may decrease the flame temperature so that radiative (and conductive) heat transfer to droplets will be diminished. In modeling the behavior of single droplets the radiative heating effect is expressed via a modification to the latent heat of evaporation. However, Sazhin [15], in discussing single droplets, argues that taking them as grey opaque bodies “overlooks the fact that droplet radiative heating takes place not at their surface (as in the case of convective heating) but via the absorption of thermal radiation penetrating inside the droplets.” He therefore assumes for modelling purposes that the droplet is semitransparent.

In the current paper we investigate the propagation of an off-stoichiometric rich laminar premixed polydisperse spray flame in the presence of heat loss, for the first time. In this paper we restrict our attention exclusively to the stable branch of propagation down to conditions of extinction. Our ultimate aim is to explore spray flame ignition since during the first moments of application of the igniter the role of heat losses can be rather dominant. It is a well-established fact that modern combustors in aircraft need to satisfy a large number of requirements. Of particular interest is the fact that under extreme conditions they must reignite following flame extinction without any problems and without any external help. The possibility of extinction also exists in cold and wet conditions (e.g., in a hailstorm) as well as at high altitudes due to oxygen starvation. The presence of liquid fuel in the form of a multisized spray of droplets that must first produce a sufficient amount of fuel vapor for successful ignition increases the difficulties. The current work is a prelude to such an ignition study that will be reported in the future.

In a previous publication [16] we modified a nonasymptotic mathematical approach [17, 18] to analyzing gas flame propagation and successfully applied it to examine the propagation of liquid fuel spray flames and double spray flames (i.e., both fuel and oxidizer supplied as a spray of droplets). For propagation studies this approach seems to be a viable alternative to an asymptotic approach. Here we adopt the same methodology.

The structure of the paper is as follows. We present the governing equations and the assumptions upon which they are based. We then explain how they are solved and present their solution. Finally, we examine how the combination of volumetric and spray-related heat losses influences the propagation and extinction conditions of the spray flames.

#### 2. Governing Equations and Problem Definition

##### 2.1. Assumptions

We consider a laminar one-dimensional premixed flame propagating into an off-stoichiometric fresh homogeneous mixture of fuel vapor, liquid fuel droplets, oxygen, and an inert gas. A schematic of the situation considered is shown in Figure 1. The flame is taken to propagate from left to right. The droplets are viewed from a far-field vantage point; that is, their average velocity is equal to that of their host environment. For qualitative purposes this approach has been demonstrated to be quite valid [19]. The spray is taken to be polydisperse; that is, at any point in space and time there is a distinct size distribution of the spray’s droplets. The temperature of the droplets is taken to be that of the surroundings; essentially the droplets heat-up time is small compared to the characteristic time associated with their motion. Droplet evaporation is assumed negligible until a prescribed reference temperature (such as the boiling temperature of the liquid fuel) is attained.