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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 Frank-Kamenetskii parameter on bifurcation and thermal criticality by means of the Domb-Sykes graphical method.

The safety in transport and storage of combustible materials is a key issue in pyrotechnic applications. These materials are often subjected to self-ignition. 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 Frank-Kamenetskii, and their pioneering contributions were summarized in [

Recently, Ajadi and Gol’dshtein [

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 round-off 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 [

In this paper, we examine the steady-state 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 Domb-Sykes technique [

The structure of this paper is as follows. Section

We consider the steady-state solutions for the strongly exothermic decomposition of a viscous combustible material. Neglecting the reactant consumption, the equation for the temperature

The following dimensionless variables and parameters are introduced in (

In this section, we will apply the homotopy perturbation method (HPM) to nonlinear ordinary differential equation (

Setting

The modelling of physical phenomena often results in nonlinear problems for some unknown function, say

We suppose that up to the point

However, for most nonlinear problems, it is rare to find an unlimited number of terms of the power series (

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 ratio-like methods, such as the Domb-Sykes method [

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 (

According to Fuchs (Bender and Orszag [

If

Using a computer algebra system, we obtained the first 30 terms of the solutions series (

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. [

In Figures

Comparison of slab numerical and HPM results of the boundary value problem (

Comparison of cylinder numerical and HPM results of the boundary value problem (

Comparison of sphere numerical and HPM results of the boundary value problem (

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

Comparison of HPM results and exact solutions of the boundary value problem (

In the following, we will focus on the calculation of radius of convergence

Coefficients _{max},

1 | 0.25 | 7 | 2.337 | 13 | 1.455 | 19 | 1.290 | 25 | 1.338 |

2 | 4.687 | 8 | 9.588 | 14 | 6.515 | 20 | 5.978 | 26 | 6.310 |

3 | 1.302 | 9 | 4.024 | 15 | 2.939 | 21 | 2.778 | 27 | 2.982 |

4 | 4.272 | 10 | 1.720 | 16 | 1.334 | 22 | 1.296 | 28 | 1.412 |

5 | 1.538 | 11 | 7.465 | 17 | 6.096 | 23 | 6.064 | 29 | 6.700 |

6 | 5.874 | 12 | 3.279 | 18 | 2.798 | 24 | 2.845 | 30 | 3.184 |

Values of the Richardson extrapolation

20 | 0.4994 | 0.4999 | 0.5 | 0.4999 |

21 | 0.4994 | 0.5 | 0.4999 | 0.5 |

22 | 0.4995 | 0.5 | 0.5 | 0.4999 |

23 | 0.4995 | 0.5 | 0.4999 | 0.5 |

24 | 0.4995 | 0.4999 | 0.5 | 0.4999 |

25 | 0.4996 | 0.5 | 0.4999 | 0.5 |

Sykes-Domb plot for

It is possible to calculate the limited series defining

Plot of the function

The graph of

We summarize the results of all Class A geometries in Tables

Computation showing criticality for Sensitised, Arrhenius, and Bimolecular reactions for

0 | 0 | 0.5 | 1 |

1 | 2 | 0.5 | 1 |

2 | 3.32324 | 0.5 | 1.61782 |

Comparison of critical values obtained by different methods.

Exact method | Variational method | Hermite-Padé approach | Present work | |
---|---|---|---|---|

Slab | ||||

Cylinder | — | |||

Sphere | — | — |

Variation of

−2 | 0.01 | 0.90624 | 1.24566 |

0.1 | 1.31389 | 2.22239 | |

0 | 0.01 | 0.8878 | 1 |

0.1 | 0.98819 | 1.52434 | |

0.5 | 0.01 | 0.88331 | 1 |

0.1 | 0.93221 | 1.42024 |

The critical values (

In order to verify the accuracy of this approximate method, we compare the critical values obtained using this method with the exact methods [

Tables

Variation of

−2 | 0.01 | 2.06415 | 1 |

0.1 | 3.01620 | 2.65419 | |

0 | 0.01 | 2.02216 | 1 |

0.1 | 2.26129 | 1.80247 | |

0.5 | 0.01 | 2.01192 | 1 |

0.1 | 2.13219 | 1.677327 |

Variation of

−2 | 0.01 | 3.45641 | 1.67123 |

0.1 | 4.32871 | 2.1177 | |

0 | 0.01 | 3.42112 | 1.62756 |

0.1 | 4.22245 | 2.21856 | |

0.5 | 0.01 | 3.15167 | 1.63331 |

0.1 | 4.04567 | 2.23811 |

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 Domb-Sykes graphical method to calculate the critical value of this problem. The results show that these critical values increase with the activation energy.

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.

Vibration frequency

Dimensionless temperature

Dimensionless activation energy

Frank-Kamenetskii parameter

Exponent.

Critical

Maximum.

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.