On Thermal Stability Analysis for a Reacting Slab

In this chapter, thermal stability analysis of the steady state exothermic chemical reaction in a slab of combustible material is investigated in the present of convective heat loss to the ambient. The nonlinear d ifferential equations governing the system are obtained and solved using perturbation technique together with a special type of Hermite-Padé series summation and improvement method. The effects of various embedded parameters on the temperature profile and thermal stability of the system are p resented graphically and discussed quantitatively. The possibility of thermal runaway phenomena was shown and the corresponding thermal criticality values were obtained and illustrated on a bifurcation diagram. The results reveal the thermal stability crit icality as well as the effects of various embedded parameters on the system.


Introduction
The study of thermal deco mposition of reactive material in a slab is paramount in understanding the heat transfer of engineering processes. It is well known that thermal decomposition of different materials is dependent on size, shape and surface or environ mental temperature as well as the physical properties of the material and environment. In other words for any given geo metry, there is a crit ical size and surface temperature above which the heat generation inside the solid exceeds the heat dissipation to the surroundings [1].
Theoretical study of transient heating in a slab of combustible material due to exothermic chemical react ion plays a significant ro le in many industrial app lications [1]. These include: heavy oil recovery, storage of cellulosic materials, the pyrolysis of bio mass and coal, the co mbustion of solids, waste incineration, coal gasification, etc. Without adequate knowledge of a reacting system, exothermic chemical process can accelerate significantly leading to runaway reaction, possible explosion, economical losses and cause emission of carbon dio xide and of to xic gases, like carbon mono xide through incomplete co mbustion ( [2], [3], [4]). [5] presented a comprehensive rev iew of chemical kinet ic models for the heating-up of combustible materials. Meanwhile, analytical solutions of the highly nonlinear part ial differential equations governing transient heating in a slab of co mbustible material due to exothermic reactions are usually impossible or extremely difficult to obtain. Hence in most cases, a numerical solution approach is adopted( [6], [7], [8], [9], [10], [1]) There are serious challenges of dangerous reactive material which justify the analysis of thermal deco mposition of reactive material in a slab. Because many facilit ies and systems have chemically reactive materials that may have serious consequences if they are not handled, used and stored properly. Previous studies of the thermal deco mposition of reactive material dealt with different dimensional shapes like cylindrical, rectangular parallelepiped, sphere, etc. The model to be introduced in this study is based on studies of [2] and [3]. [2] studied the dynamics of hollow material and investigated the problem of strong exothermic exp losions in cylindrical p ipe, for putting large activation energy and concluded that the procedure reveals accurately the steady thermal criticality condition. [3] studied hydrodynamics and investigated the effect of variab le v iscosity in a thermal decomposable generalized Newtonian fluid subjected to unsteady one dimension shear flow in which the result were presented for some parameters in the problem. It was observed that increasing the non-Newtonian nature of the flu id helps to delay the onset of thermal runaway when co mpared to Newtonian nature of the fluid. [4] studied the thermal decomposition in a slab and found that time-independent solutions for the spatial structure of temperature, considering a slab with isothermal boundaries subjected to exothermic reaction and uniform p lastic heating.
Understanding the heat transfer and thermal stability characteristics of a reacting slab of co mbustive materials is extremely important in order to ensure the safety of its storage, handling and transportation( [5,6]). Moreover, the phenomena of spontaneous ignition due to exothermic chemical kinetics in bulk solids such as coal, grain, hay, wool, etc., can be theoretically described by thermal co mbustion theory developed by [7,8]. One of the most important advantages of theoretical methods is that they can be applied as soon as a kinetic model had been evaluated fro m data fro m laboratory scale kinetic experiments. In part icular, they allow estimation of runaway parameters in the earliest stages of the life cycle o f a reactive material, thus ensuring elimination or significant reduction of the necessity for explosive experiments( [9,10]).
In this present study, we investigate the heat transfer and thermal stability characteristics of a reacting slab of co mbustible material in the presence of heat loss, neglecting the reactant consumption. The associated nonlinear d ifferential equation modelling the problem is tackled analytically using perturbation method together with a special type of Hermite-Padé series summat ion and imp rovement method ( [11,12]). Pertinent results are presented graphically and discussed.

Mathematical Model
Consider the steady-state of an exothermic chemical reaction in a slab of combustible material with possibility of heat loss to the surrounding ambient. Assuming no reactant consumption, the one dimensional heat balance nonlinear partial d ifferential equation governing the system given by [5,9,10] is; with the in itial and boundary conditions as 2, 0, m = − represent numerical exponent for sensitized, Arrhenius and Bimolecu lar kinetics respectively.
We introduce the following dimensionless variables into equations.
and we obtain the dimensionless governing equations as where λ is the Frank-Kameneskii parameter, ε the activation energy parameter, δ the heat loss parameter. The thermal decomposition and stability of the reacting combustible material depend on the parameters in equation.(4), wh ich are of great importance with respect to applications in the area of industrial safety and handling techniques of exp losives.

Perturbation Method
Due to the nonlinearity nature of the temperature variable in equations(4)-(5), it is convenient to seek a solution in the form of a power series expansion in parameter λ , i.e.
Substituting the solution series in equation.(6) into equations. (4) and (5) and collecting the coefficients of like powers of λ , we obtained the follo wings: with ( ) Using a computer symbo lic algeb ra package(MA PLE), the first few terms of the above solution series in Eq.(6) are obtained. We are aware that these power series solutions are valid for very s mall parameter values of λ . Ho wever, using Hermite-Padé series su mmat ion and improvement technique, we have extended the usability o f the solution series beyond small parameter values as illustrated in the following section.

Thermal Stability Analysis
Fro m the application point of v iew, it is very important to determine the appearance of criticality or non-existence of steady-state solution for certain parameter values. In order to achieve this, a special type of Hermite-Padé series summation and improvement method will be utilized( [11,12]). Let be a given part ial sum. We note that equation (14) can be used to approximate any output of the solution of the problem under investigation, by employing Taylor expansion in a given small parameter. For instance, the series for the slab ma ximu m te mperature(i.e.

Results and Discussion
In this section, we validate the above theoretical results using physically realistic values of various embedded parameters in the numerical experiment. We note that increasing parameter value of λ indicates an increase in the rate of exothermic chemical kinetics in the slab. In order to obtain the thermal stability criterion in the react ing slab, the Hermite-Padé appro ximation procedure in section(4) above was applied to the first few terms of the solution series in section(3) and we obtained the results as shown in tables (1) and(2) belo w: The results in table(1) reveal the rapid convergence of Hermite-Padé series summation and improvement procedure with gradual increase in the number of series coefficients utilized in the appro ximants for the thermal crit icality condition. In table (2), it is noteworthy that the magnitude of thermal exp losion crit icality ( ) c λ increases with an increase in the parameter values of 0 β > due to an increase in the heat loss to the surrounding ambient. This invariably will lead to a delay in the develop ment of thermal runaway in the reacting slab. Since the heat is not accumulated in the slab, it enhances the thermal stability of the system. Similar effect of thermal stability enhancement is observed with increasing values of activation energy parameter ( ) ε . Increasing values of ε imp lies that the activation energy of the reacting slab is very low; hence the volatility tendency of the reacting slab is greatly reduced. Furthermore, it is interesting to note from table (2) (2). In figure(4) we observe that the slab temperature decreases with an increase in the heat loss parameter δ . The decrease in the slab temperature with increasing δ can be attributed to the cooling action of heat loss on the slab. Figure(5) show that the slab temperature increases with an increase in the parameter values of λ . As the Frank-Kamenetskii parameter ( ) λ increases, the slab internal heat generation due to exothermic reaction increases, this invariably leads to an elevation in the slab temperature.

Conclusions
The steady state exothermic chemical react ion in a slab of combustible material is considered. The nonlinear ord inary differential equation governing the problem was formulated and solved analytically using the perturbation technique together with a special type of the Hermite-Padé series summation and improvement method. The possibility of thermal runaway phenomena was shown and the corresponding thermal criticality values were obtained and illustrated on a bifurcation diagram. An increase in the parameter value of λ indicates increasing rate of exothermic chemical kinetics. As parameter δ increases in value (δ>0), the rate of heat loss to the ambient increases as well as the thermal explosion crit icality values. This invariab ly enhances the thermal stability of the reacting slab by preventing the occurrence of thermal runaway.