Numerical simulations applying to the analysis of thermal explosion of organic gel fuel in a hot gas Ophir Nave a,⇑ , Vitcheslav Bykov b , Vladimir Gol’dshtein a , Yaron Lehavi c a Department of Mathematics, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel b Institute für Technische Thermodynamics, Universität Karlsruhe, (TH) Kaiserstrabe 12, D-76131 Karlsruhe, Germany c Department of Physics, David Yellin, Jerusalem, Israel article info Article history: Received 25 March 2011 Received in revised form 31 May 2011 Accepted 1 June 2011 Available online 23 June 2011 Keywords: Gel fuel droplet Numerical simulations Method of integral manifolds Singular perturbed system abstract In this work we investigate theoretically and numerically the nature of the evolution of a hot gas mixture containing organic gel fuel droplets of different radii with oscillatory evaporation within the context of thermal explosion theory. The polydisperse spray is modeled using a probability density function (PDF). In this paper we take into account the time evolution of the size distribution due to the evapora- tion process by using the kinetic equation. This new model is in the form of singular perturbed system (SPS) of highly nonlinear of ordinary differential equations. This SPS form of the model and the non- dimensionalization of the equations enable us first to apply the methods of integral manifolds (MIM) and second to identify the parameters that play a major role in determining the dynamical regimes of the considered system. Our analytical and numerical simulations results that based on the MIM show that the dynamical behavior is different than that found with the conventional liquid droplets. An explicit expression of the critical condition for thermal explosion limit is derived analytically and represents a generalization of the critical parameter of the classical Semenov theory. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction There is a growing interest in the use of gel propellants in a vari- ety of propulsion contexts due to their rather favorable high per- formance and safety characteristics. The uniqueness of gels, as opposed to conventional liquid or solid propellants, is compactly captured by Brinker and Scherer [1] in their definition of a gel as a ‘‘substance that contains a continuous solid skeleton enclosing a continuous liquid phase. The continuity of the solid structure gives elasticity to the gel’’, (Chapter 2). A thorough review of the background material on gels utill 2006 is given by Natan and Rahimi [2] and Goldfarb et al. [3]. In our paper we focus on an interesting experimental discovery of Solomon and Natan [4]. They found that organic gel fuel exhibited a pulsating-type of evapora- tion behavior, in contrast to the usual d 2 -law evaporating behavior of inorganically based gel fuel. Observations indicate that in the former case, the internal structure of the droplet alters as heating occurs and an elastic layer of gellant forms around the liquid fuel. The droplet swells and the gellant layer is perforated, eventually releasing evaporated fuel to the surroundings. After the occurrence of this burst of fuel vapor the droplet shrinks, an elastic outer layer is formed, and the cycle repeats itself a number of times until the liquid fuel in the droplet is completely depleted. To describe the phenomenon of thermal explosion in organic gel fuel in a hot gas mathematically, one has to contend with a set of strongly nonlin- ear coupled partial differential equations. In order to disentangle the many physio-chemical processes at play, and to attempt to identify their realms of relevance and importance, it is reasonable to adopt an approach that focuses on singling out the key driving mechanisms. As in our previous work [5,6], the size of distribution of fuel droplets is described in a continuous way. It is a continuous approximation of a discrete object (polydisperse spray), which seems to be reasonable because of the large variety of droplet radii. It is also motivated by an experimental work that uses a classical probability density function (PDF) for the polydisperse spray description [7]. The models of thermal explosion involve different time scales. Therefore, the natural way of modeling these processes is to con- sider them as a Singular Perturbed System (SPS) of ordinary differential equations. In order to investigate such a system, differ- ent asymptotic methods can be applied. In this paper we used the geometric asymptotic method (Method of Invariant Manifolds, MIM) that has been introduced by Bogolubov and Mitropolsky [8], Strygin and Sobolev [9] and Gol’dshtein and Sobolev [10] as a 0016-2361/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.fuel.2011.06.004 ⇑ Corresponding author. Fax: +972 8 647 7648. E-mail addresses: naveof@cs.bgu.ac.il (O. Nave), bykovv@itt.uni-karlsruhe.de (V. Bykov), vladimirg@bgu.ac.il (V. Gol’dshtein). Fuel 90 (2011) 3410–3416 Contents lists available at ScienceDirect Fuel journal homepage: www.elsevier.com/locate/fuel