Advances in Mathematical Physics

Volume 2016 (2016), Article ID 9384541, 6 pages

http://dx.doi.org/10.1155/2016/9384541

## Thermal Stability Investigation in a Reactive Sphere of Combustible Material

Department of Mathematics, Vaal University of Technology, Private Bag Box X021, Vanderbijlpark 1911, South Africa

Received 22 February 2016; Revised 17 June 2016; Accepted 10 July 2016

Academic Editor: Ricardo Weder

Copyright © 2016 R. S. Lebelo. 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

An investigation of thermal stability in a stockpile of combustible material is considered. The combustible material is any carbon containing material that can react with oxygen trapped in a stockpile due to exothermic chemical reaction. The complicated process is modelled in a sphere and one-dimensional energy equation is used to solve the problem. The semi-implicit finite difference method (FDM) is applied to tackle the nonlinear differential equation governing the problem. Graphical solutions are displayed to describe effects of embedded kinetic parameters on the temperature of the system.

#### 1. Introduction

Thermal stability in a stockpile of combustible material has drawn attention of many researchers. The interest is sparked by self-ignited fires that cause hazards to the environment, vegetation, and domestic and wild animals. Some veld fires are caused by stockpiles of combustible materials that have been left without attention for a long time. Some physical factors or parameters that influence the self-ignition process are combustible material particle size, volume-to-surface ratio, porosity, thermal conductivity, density, and heat capacity including convection in the surroundings [1–3]. The three groups, into which the parameters that influence self-ignition in stockpiles of combustible materials are distinguished, are outlined in [4]. Ignition also depends on chemical kinetic factors, and in this paper we study the effects of these embedded parameters on thermal stability of self-ignited processes. Exothermic chemical reaction taking place in a reactive stockpile results in heat generation. Should the heat generation within a stockpile exceed heat loss to the ambient surroundings, thermal instability and runaway leading to self-ignition may occur [5–7]. Thermal explosion criticality concept was investigated by Frank-Kamenetskii and he also developed the steady state theory for combustion of reactive materials due to exothermic chemical reaction [8]. Frank-Kamenetskii’s work enables researchers to evaluate thermal criticality values for steady state exothermic chemical reaction. These criticality values are helpful to study thermal stability in processes of self-ignition of combustible materials due to exothermic chemical reaction. Self-ignited combustion is a complicated process that involves many radicals which are short-lived and their interaction is nonlinear [9, 10]. The complicated processes of combustion due to exothermic chemical reactions in reactive stockpile are modelled mathematically in order to assess materials property thermal stability, climate change indicators, and mitigation strategies [11, 12]. Thermal stability of reactive materials was studied by several authors [13–15] who carried out the investigation in a reactive slab. In this paper, the investigation is carried out in a sphere of reactive material. The nonlinear differential equation governing the problem is tackled numerically using semi-implicit FDM. The paper layout is done as follows; in Section 2, the mathematical formulation is described. The numerical analysis is done in Section 3 and results and discussion are given in Section 4.

#### 2. Mathematical Formulation

The study of thermal stability of a stockpile of reactive material is modelled in a sphere of constant thermal conductivity. It is assumed that the sphere undergoes an th-order oxidation chemical reaction and that no reaction consumption takes place. Figure 1 illustrates the geometry of the problem.