Research Article  Open Access
On the Critical Behaviour of Exothermic Explosions in Class A Geometries
Abstract
The aim of this work is to apply the homotopy perturbation method for solving the steady state equations of the exothermic decomposition of a combustible material obeying Arrhenius, Bimolecular, and Sensitised laws of reaction rates. These equations are formulated on some Class A geometries (an infinite cylinder, an infinite slab, and a sphere). We also investigate the effect of FrankKamenetskii parameter on bifurcation and thermal criticality by means of the DombSykes graphical method.
1. Introduction
The safety in transport and storage of combustible materials is a key issue in pyrotechnic applications. These materials are often subjected to selfignition. This internal heating occurs when an explosive substance is brought to a sufficient temperature so that the process of decomposition begins to produce significant exothermic effects. This involves a thermal runaway phenomenon accompanied with an increase of the temperature producing a rapid thermal decomposition. The understanding of the factors that control this phenomenon is of fundamental importance in many industrial processes.
This phenomenon was first introduced in the 1930s by Semonov, Zeldovith and FrankKamenetskii, and their pioneering contributions were summarized in [1]. Furthermore, FrankKamenetskii developed the steadystate theory of thermal explosion. In this theory, the FrankKamenetskii approximation allows us to determine the critical values, which constitute limit values not to be exceeded to avoid the phenomenon of selfignition. Some studies deal with a chain of several reactions in this phenomenon of selfignition. Several works in the literature have applied this theory to different combustible materials geometries. Boddington et al. [2–4] have considered the special case of twostep parallel exothermic reactions for the infinite slab where solutions by quadrature were possible. GrahamEagle and Wake [5] considered a system of simultaneous exothermic reactions. They extended the investigations of Boddington et al. [2–4] to the other two geometries of the infinite circular cylinder and the sphere. A variational method was used to evaluate the critical values of FrankKamenetskii parameter and the maximum temperature. The same authors [6] extended the treatment of simultaneous reactions to the case where one reaction is exothermic and the other endothermic, leading to the phenomenon of the disappearance of criticality or transition, which can also happen with a single reaction in the case of very low activation energy. In this case, the FrankKamenetskii approximation is no longer valid. They used a variational method [7] to determine numerically the values of the parameter which characterizes the transition in the parameter space.
Recently, Ajadi and Gol’dshtein [8] employed a threestep reaction kinetics model (initiation, propagation, and termination steps). The calculation of the criticality was made by means of a variational method for (infinite slab, infinite cylinder, and sphere) under Arrhenius laws of reaction rates by using an effective activation energy approximation. Balakrishnan et al. [9] calculated the critical values for some nonClass A geometries (infinite square rod and cube). Their critical values were found using the finite difference method.
In applied mathematics or engineering problems, numerical methods commonly used such as finite difference, finite element, or characteristics method, need large size of computational works due to discretization and usually the effect of roundoff error causes loss of accuracy in the results. In addition to this drawback, these methods with limited precision include slow runtimes, numerical instabilities, and difficulties in handling redundant constraints.
Analytical traditional methods commonly used for solving these problems can be very useful, especially for the calibration of numerical calculations. Among these approaches, the classical perturbation method is based on the existence of small parameters but the overwhelming majority of linear and nonlinear problems have no small parameters, at all.
To overcome this shortcoming, the homotopy perturbation method (HPM) was first introduced by He [10–13]. The main idea of this method is to introduce an embedding parameter to construct a homotopy and give solutions of the deformed problem as power series in . When , the system of equations is reduced to a simplified form which admits exact solutions, and, at , the system takes its original form and gives the desired solutions. This method has been extensively employed by numerous authors to solve a large variety of linear and nonlinear problems. We can cite the works [14–17], the list is not exhaustive.
In this paper, we examine the steadystate solutions for the strongly exothermic decomposition of a combustible material of a symmetric Class A geometries, uniformly heating, under Arrhenius, Biomolecular, and Sensitised kinetics, neglecting the consumption of the material.
The contribution of the present work is twofold. First, we calculate the temperature field using the HPM in a symbolic computational language. The second aim is to study the thermal criticality conditions of the problem.
Critical values for different geometries are found by using the DombSykes technique [18] as a useful tool to extract singularities and to perform analytic continuation. This method has shown to be very accurate and simpler to implement, when compared with variational method [6, 8] or HermitePadé approximants method [19–21].
The structure of this paper is as follows. Section 2 presents the boundary value problem governing the ignition of a viscous combustible material for symmetric Class A geometries. In Section 3, we will apply the homotopy perturbation method to this nonlinear boundary value problem. The fourth section is assigned to a brief description of the Domb and Sykes method and its application to calculate the bifurcation points. Section 5 is devoted to the results obtained by the proposed method, and comparison with other works will be performed. Conclusions will appear in Section 6.
2. Mathematical Model
We consider the steadystate solutions for the strongly exothermic decomposition of a viscous combustible material. Neglecting the reactant consumption, the equation for the temperature may be written in terms of physical variables together with the boundary conditions as follow: with the absolute temperature, the wall temperature, the thermal conductivity of the material, the heat of reaction, the rate constant, the activation energy, the universal gas constant, the initial concentration of the reactant species, the Planck’s number, the Boltzmann’s constant, the vibration frequency, the geometry half width, and the radial distance in the normal direction. is the numerical exponent, such that represent numerical exponent for Sensitised Arrhenius and Bimolecular kinetics (Boddington et al. [3], Makinde [20], Bowes [22], Bebernes and Eberly [23], Okoya [24]), and is the geometry factor representing, respectively, slab, cylindrical, and spherical geometries.
The following dimensionless variables and parameters are introduced in (2.1): to obtain the dimensionless governing equation together with the corresponding boundary conditions. We get the following equations: where , represent the FrankKamenetskii and activation energy parameters, respectively. Hereafter, we will suppress the bar symbol for clarity.
3. Homotopy Perturbation Method Solution
In this section, we will apply the homotopy perturbation method (HPM) to nonlinear ordinary differential equation (2.4). According to this method, we can construct a homotopy as follows: where Applying the perturbation technique [25], we can assume that the solution of (3.1) can be expressed as a series in where is an embedding parameter. When , we can obtain the initial guesses; when , (3.1) turns out to be the original one.
Setting , we obtain an approximate solution of (2.4): One has to substitute relation (3.3) into the governing equation (3.1), collect the powers of , and obtain a sequence of differential equations and boundary conditions. The solution for the temperature field for Sensitised, Arrhenius, and Bimolecular reaction rates for different geometries are given as According to (3.4), we get the solution for the temperature field for the three reaction rates.
4. Bifurcation Study by DombSykes Method
The modelling of physical phenomena often results in nonlinear problems for some unknown function, say depending on a parameter . Usually, the problems cannot be solved exactly. The solutions of these nonlinear systems are dominated by their singularities which must be real and positive in order to have a physical sense.
We suppose that up to the point , the solution is analytic in the interval , then one can solve the problem by expanding the solution in a power series If an infinite number of is known, the radius of convergence, which is the distance from the origin to the nearest singularity, limiting the range of validity of the series (4.1), can be calculated using D’Alembert’s ratio.
However, for most nonlinear problems, it is rare to find an unlimited number of terms of the power series (4.1). The nearest singularity cannot therefore be determined precisely.
Using symbolic calculus codes, it is now possible to calculate a sufficient number of terms in the series, to study precisely the solution, and there exist a variety of methods devised for extracting the required information of the singularities from a finite number of series coefficients. The most frequently used methods are the ratiolike methods, such as the DombSykes method [26, 27], NevilleAitken extrapolation [26], and seminumerical approximant methods, such as Padé approximants [28] or HermitePadé approximants [19–21].
Herein, we are concerned with the bifurcation analysis by analytic continuation as well as with the dominant behavior of the solution by using partial sum (3.3). This may be done calculating the nearest singularity to the origin. A useful tool for extracting this singularity is the DombSykes method. This technique is easier to implement than others, and, when it is improved, it gives a precise value of the singularity. This method, successfully applied in various problems [29–32], is presented here.
According to Fuchs (Bender and Orszag [27]), there are two possible forms: The radius of convergence of the series (4.1) may be found by d’Alembert ratio: Of course, only a finite number of coefficients are known, so that it is difficult to obtain precisely the limit. Domb and Sykes have suggested that the inverse ratio has the following expansion: Domb and Sykes have pointed out that it is more reliable to plot versus , that is, to bring to the origin, rather than plotting versus . The plot of versus is known as DombSykes plot. In this plot, the intersection of the straight line with the axis is exactly , and the slope of this line gives the exponent . Unfortunately, is often a slowly converging series. A good way to improve the convergence is to use the Richardson extrapolation (Bender and Orszag [27]), which is appropriate for this kind of sequences and can be defined as follows.
If can be written in the following form: , then the Richardson extrapolation is This new sequence has a quicker convergence to the limit .
5. Results and Discussion
Using a computer algebra system, we obtained the first 30 terms of the solutions series (3.5).
In order to verify numerically whether the proposed methodology leads to high accuracy, we evaluate the numerical solutions using a collocation method proposed by Shampine et al. [33].
In Figures 1, 2, and 3, we have plotted the numerical and the HPM solutions in all Class A geometries and for all numerical exponent . These figures show a perfect agreement of both solutions.
The obtained solutions, in comparison with the numerical solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.
For a further information, Figure 4 illustrates a comparison between the exact solutions known for the special cases () when a closed form is available [8] and the series solutions obtained by using the homotopy perturbation method. The obtained results are found to be in good agreement with the exact solutions. We can notice that the deviation between HPM and exact solution do not exceed 0.2% for both slab and cylindrical geometries.
In the following, we will focus on the calculation of radius of convergence of the series solutions (3.5). Up to the point , the solution is analytic and has a real singularity at the value . It would be interesting to know the exact nature of the singularity and if this singularity corresponds to a bifurcation point. Let the maximum temperature be a characteristic quantity which qualifies the solution. We rearrange in a series of to write where different coefficient for are given in Table 1. Seeing Figure 5, one can say that the ratio in the simple geometry of infinite circular cylinder converges; however, it is extremely difficult to conclude to a precise value of radius of convergence. It is necessary to use some process of convergence acceleration. The Richardson extrapolation is appropriate for this kind of convergence. The results of Table 2 show that the radius of convergence is . Two solution branches are therefore identified with a bifurcation point at . Two solution branches (type I and II) are identified with a bifurcation point at (i.e., turning point) as shown in a sketch of bifurcation diagram in Figure 6.


(a)
(b)
It is possible to calculate the limited series defining by a division procedure. This function is plotted in Figure 7.
The graph of , as shown in Figure 6(a), is not compatible with the expected singularity (4.2). The curve versus should be in the form of Figure 6(b). To analyse the paradox, the inverse function is considered. Its series is given by inverting the series . One can see in Figure 7 that this function has a horizontal tangent for the values in the case , ().
We summarize the results of all Class A geometries in Tables 3, 4, and 5, while is calculated by improving the series (5.1) by a suitable Padé approximant [27].



The critical values ( and ) were computed using the improved DombSykes applied to the temperature field obtained by HPM in all Class A geometries.
In order to verify the accuracy of this approximate method, we compare the critical values obtained using this method with the exact methods [1, 2], variational methods [5, 8], and HermitePadé approach [19–21] for the special case of . The method used in the present work gives results approximately equal to those discussed in the literature, as shown in Table 4.
Tables 5, 6, and 7 depict the variations of and with . In these tables, we observe that the magnitude of thermal criticality conditions for a viscous combustible material with high activation energy () is lower than the one for moderate values of activation energy (, ). This implies that for moderate values of activation energy, the criticality varies from one reaction to another, as shown in Tables 5, 6, and 7. Explosion in Bimolecular reactions seems to occur faster than in Arrhenius and Sensitised reactions.


6. Conclusion
We studied the problem of exothermic explosion of a viscous combustible in Class A geometries under Arrhenius, Bimolecular, and Sensitised laws of reaction rates with the homotopy perturbation method. The results show that this method provides excellent approximation of the solution of this nonlinear system with high accuracy.
A bifurcation study is performed with the DombSykes graphical method to calculate the critical value of this problem. The results show that these critical values increase with the activation energy.
Nomenclature
Geometry half width  
Rate constant  
Initial concentration of the reactant  
Activation energy  
Planck’s number  
Geometry factor  
Thermal conductivity of the material  
Boltzmann’s constant  
Linear operator  
Numerical exponent  
Embedding parameter  
Radial distance  
Dimensionless radial distance  
Heat of reaction  
Universal gas constant  
Absolute temperature  
Wall temperature. 
Greek Symbols
Vibration frequency  
Dimensionless temperature  
Dimensionless activation energy  
FrankKamenetskii parameter  
Exponent. 
Subscripts
cr:  Critical 
max:  Maximum. 
Acknowledgments
This work is supported by the French Ministry of Research and Education through invited professor contract. The authors are highly grateful to “ENSI de Bourges” for providing excellent research environment and facilities and to Professor J. H. He for his valuable comments.
References
 D. A. FrankKamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, NY, USA, 1969.
 T. Boddington, P. Gray, and C. Robinson, “Thermal explosions and the disappearance of criticality at small activation energies: exact results for the slab,” Proceedings of the Royal Society of London, vol. 368, pp. 441–461, 1979. View at: Publisher Site  Google Scholar
 T. Boddington, C.G. Feng, and P. Gray, “Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. I. arbitrary biot number and general reactionrate laws,” Proceedings of the Royal Society of London, vol. 390, pp. 247–264, 1983. View at: Publisher Site  Google Scholar
 T. Boddington, P. Gray, and G. C. Wake, “Theory of thermal explosions with simultaneous parallel reactions. I. foundations and the onedimensional case,” Proceedings of the Royal Society of London, vol. 393, pp. 85–100, 1984. View at: Publisher Site  Google Scholar
 J. G. GrahamEagle and G. C. Wake, “Theory of thermal explosions with simultaneous parallel reactions. II. The two and threedimensional cases and the variational method,” Proceedings of The Royal Society of London A, vol. 401, no. 1820, pp. 195–202, 1985. View at: Publisher Site  Google Scholar
 J. G. GrahamEagle and G. C. Wake, “The theory of thermal explosions with simultaneous parallel reactions.III. Disappearance of critical behaviour with one exothermic and one endothermic reaction,” Proceedings of The Royal Society of London A, vol. 407, pp. 183–198, 1986. View at: Publisher Site  Google Scholar
 I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice Hall, Englewood Cliffs, NJ, USA, 1963.
 S. O. Ajadi and V. Gol'dshtein, “Critical behaviour in a threestep reaction kinetics model,” Combustion Theory and Modelling, vol. 13, no. 1, pp. 1–16, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. Balakrishnan, A. Swift, and G. C. Wake, “Critical values for some nonclass A geometries in thermal ignition theory,” Mathematical and Computer Modelling, vol. 24, pp. 1–10, 1996. View at: Google Scholar
 J. H. He, “A coupling method of a homotopy technique and a perturbation technique for nonlinear problems,” International Journal of NonLinear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. View at: Publisher Site  Google Scholar
 J.H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J.H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters. A, vol. 350, no. 12, pp. 87–88, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J.H. He, “An elementary introduction to the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 410–412, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 P. Donald Ariel, “The homotopy perturbation method and analytical solution of the problem of flow past a rotating risk,” Computers & Mathematics with Applications, vol. 58, no. 1112, pp. 2504–2513, 2009. View at: Publisher Site  Google Scholar
 M. J. Hosseini, M. Gorji, and M. Ghanbarpour, “Solution of temperature distribution in a radiating fin using homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2009, Article ID 831362, 8 pages, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. Rajabi, D. D. Ganji, and H. Taherian, “Application of homotopy perturbation method in nonlinear heat conduction and convection equations,” Physics Letters. A, vol. 360, no. 45, pp. 570–573, 2007. View at: Publisher Site  Google Scholar
 A. M. Siddiqui, A. Zeb, Q. K. Ghori, and A. M. Benharbit, “Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 182–192, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 C. Domb and M. F. Sykes, “On the susceptibility of a ferromagnetic above the Curie point,” Proceedings of The Royal Society of London, vol. 240, pp. 214–228, 1957. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 O. D. Makinde, “Strongly exothermic explosions in a cylindrical pipe: a case study of series summation technique,” Mechanics Research Communications, vol. 32, no. 2, pp. 191–195, 2005. View at: Publisher Site  Google Scholar
 O. D. Makinde, “Exothermic explosions in a slab: a case study of series summation technique,” International Communications in Heat and Mass Transfer, vol. 31, no. 8, pp. 1227–1231, 2004. View at: Publisher Site  Google Scholar
 O. D. Makinde and E. Osalusi, “Exothermic explosions in symmetric geometriesan exploitation of perturbation technique,” Romanian Journal of Physics, vol. 50, no. 56, pp. 621–625, 2005. View at: Google Scholar
 P. C. Bowes, SelfHeating, Evaluating and Controlling the Hazard, Elsevier, Amsterdam, The Netherlands, 1984.
 J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83, Springer, New York, NY, USA, 1989.
 S. S. Okoya, “On the behaviour of Solutions to a system of ordinary differential equations modelling branched chain reaction,” International Communications in Heat and Mass Transfer, vol. 29, no. 8, pp. 1169–1176, 2002. View at: Publisher Site  Google Scholar
 A. H. Nayfeh, Problems in Perturbation, John Wiley & Sons, New York, NY, USA, 1985.
 A. J. Gutmann, “Asymptotic analysis of power series expansions,” in Phase Transitions and Critical Phenomena, C. Domb and J. Lebowitz, Eds., vol. 13, Academic Press, London, UK, 1989. View at: Google Scholar
 C. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw–Hill, New York, NY, USA, 1978.
 G. A. Baker and P. GravesMorris, Padé Approximants, Cambridge University Press, Cambridge, UK, 1996.
 A. Aziz and T. Y. Na, Perturbation Methods in Heat Transfer, Hemisphere Publishing, New York, NY, USA, 1984.
 M. ErRiani and O. SeroGuillaume, “Shapes of liquid drops obtained using symbolic computation,” Journal of Symbolic Computation, vol. 40, no. 6, pp. 1340–1360, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. W. Schwartz and A. K. Whitney, “A semianalytic solution for nonlinear standing waves in deep water,” Journal of Fluid Mechanics, vol. 107, pp. 147–171, 1981. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Koji Ohkitani and John D. Gibbon, “Numerical study of singularity formation in a class of Euler and NavierStokes flows,” Physics of Fluids, vol. 12, no. 12, pp. 3181–3194, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. F. Shampine, M. W. Reichelt, and J. Kierzenka, “Solving boundary value problems for ordinary differential equations in mATLAB with bvp4c,” http://www.mathworks.com/bvp_tutorial. View at: Google Scholar
Copyright
Copyright © 2011 Mustapha ErRiani and Khaled Chetehouna. 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.