(271e) Approximate Solutions for the Steady States of a Self Heating Model With Reactant Consumption in Simple Geometries and Calculation of the Critical Conditions for Thermal Explosion

Approximate Solutions for the Steady States of a Combustion Model with  Reactant Consumption in Simple Geometries and Calculation of the Critical Conditions for Self Heating

Giovanni Cocchi

Department of Civil, Chemical, Environmental and Material Engineering - University of Bologna, Via Terracini 28, 40131 Bologna, Italy

* Corresponding Author Email Address: giovanni.cocchi2@unibo.it

The bulk storage of substance liable to spontaneous exothermic reactions could pose a relevant fire hazard in some industries, such as those that deal with coal, biomass, organic peroxides and energetic materials. In fact, if the heat that is generated inside the storage piles or inside the packages of the above mentioned materials overcame the heat that is being exchanged with the surrounding environment, thermal runaway could begin and smoldering or flaming combustion could onset [1]. In the case of energetic materials, especially under confinement, the thermal cook off could even lead to deflagrations or detonations. Traditionally, for solid substances, these scenarios have been evaluated by means of models, like those of Frank Kamenetskii [1] and of Gray and Wake [2], that can be used for estimating the critical conditions at which the thermal power generated by the exothermic reactions inside the body overcame the cooling provided by the surrounding ambient. These models take into account the conductive heat transfer inside the body and are formulated in term of a partial differential equation, that for selected geometry can be reduced to an ordinary differential equation, thanks to symmetry. It is well known that, for a given substance, the critical conditions depend on the size, on the geometry and on the heat transfer conditions of the body, as well as on the temperature of the ambient in which the material is being stored. Under supercritical conditions, a steady state solution for the energy balance equation will no longer exist and the onset of criticality can be discussed in term of existence of the above mentioned steady state solutions. These simple models neglect the reactant consumption, in view of the fact that under subcritical conditions the reactant consumption can be assumed to be negligible. In 1998 Shouman published a simplified, yet accurate, solution to the thermal explosion problem in solid substances that is based on the Gray Wake model [3]. The approximated approach proposed by Shouman is based on the assumption that heat generation rate, that rigorously should depends on the temperature according to the Arrhenius law and thus should depend on the position inside the body, can be evaluated at the maximum temperature or at the average temperature of the solid domain. In the present work the same approximations will be used for discussing the steady states and the critical conditions for thermal explosion in a combustion model that takes into account the consumption of a reactant that diffuse through the body and that is being supplied by the external environment [4]. This can be the case of the oxygen that is required for the oxidation of a cellulosic or carbonaceous materials. The heat and mass transfer equations are coupled and different choice can be exploited while applying the approximation strategy, leading to solutions for the steady state and for the criticality conditions that are slightly different each other. In this work the approximate solutions for the steady state of the combustion model with reactant consumption and the calculated criticality conditions for a slab, a sphere and an infinite cylinder will be discussed  and compared with numerical solutions of the model without approximations. Finally, this approximate procedure will be extended to non simple geometries, in which heat and mass transfer can be evaluated by means of Finite Element codes.

References

[1] P.C. Bowes Self Heating: evaluating and controlling the hazards, !984, Amserdam, Elsevier.

[2] Luo, W., Wake, G.C., and Hawk, C.W., Numerical Determination of Critical Conditions

for Thermal Ignition, The ANZIAM Journal, 50, 2009,283-305.

[3] A.R. Shouman, A Very Simple yet Accurate Solution to the Thermal Explosion Problem, Journal of Loss Prevention in the Process Industries, 11,1998, 383–390.

[4] A. A. Shah, G.C. Wake, The Existence of Steady State to a Combustion Model with Internal Heating, Nonlinear Analysis: Real World Applications, 5,2004, 421-439.