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Research ArticleAnalytical Solution of Nonlinear Dynamics ofa Self-Igniting Reaction-Diffusion System UsingModified Adomian Decomposition Method
Felicia Shirly Peace1 Narmatha Sathiyaseelan1 and Lakshmanan Rajendran2
1 Department of Mathematics Lady Doak College Madurai Tamil Nadu 625002 India2Department of Mathematics Madura College Madurai Tamil Nadu 625011 India
Correspondence should be addressed to Lakshmanan Rajendran raj smsrediffmailcom
Received 24 September 2013 Revised 18 December 2013 Accepted 23 January 2014 Published 24 April 2014
Academic Editor Jean-Pierre Corriou
Copyright copy 2014 Felicia Shirly Peace et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A mathematical model of the dynamics of the self-ignition of a reaction-diffusion system is studied in this paper An approximateanalytical method (modified Adomian decomposition method) is used to solve nonlinear differential equations under steady-state condition Analytical expressions for concentrations of the gas reactant and the temperature have been derived for Lewisnumber (Le) and parameters 120573 120574 and 1206012 Furthermore in this work the numerical simulation of the problem is also reportedusing MATLAB program An agreement between analytical and numerical results is noted
1 Introduction
Nonlinear dynamical phenomena in combustion processare an active area of experimental and theoretical researchMathematical models that describe this phenomenon canbe considered as nonlinear dynamical systems The dynamiccharacterization of such models was developed by Continilloet al [1] The detailed numerical simulation of autoigni-tion of coal stockpiles leads to the observation of steadyregimes To better investigate this phenomenon two sim-plified distributed-parameter models were discussed whichincorporate heat conduction mass diffusion and one-stepArrhenius exothermic chemical reaction Both model equa-tions were solved with straight forward finite-differenceschemes [2] The problem of spontaneous ignition of coalstockpiles is challenging for safety implications and for itstheoretical complexity a spontaneous combustion reactiontakes place in a bed of solid fuel while flow driven bynatural convection generated by the onset of temperaturegradients within the pile occurs Coal stockpiles self-ignitewhen reaction of coal with oxygen present in the atmospheregenerates heat that is not efficiently removed toward theexternal ambient [3] Continillo et al [4 5] have analyzed the
self-combustion of coal piles in the absence of natural convec-tion The three main phenomena in the self-ignition of coalstockpile are convection reaction anddiffusionOn the otherhand Continillo et al [6] studied the dynamic behavior of atwo-dimensional coal pile also by accounting for natural con-vection As part of a comprehensive study of self-heating ofcoal stockpiles a simple mathematical model has been devel-oped To our knowledge no rigorous analytical expressionsof gas reactant (Y) and temperature (119879) have been derivedfor all possible values of parameters under steady-stateconditionsThepurpose of this paper is to derive approximateanalytical expressions for gas reactant concentration andtemperature using the modified Adomian decompositionmethod
2 Mathematical Formulation ofthe Boundary Value Problem
The nonlinear differential equations are those of a distribut-ed-parameter dynamic model of heterogeneous reaction in aone-dimensional layerThe gaseous reactant diffuses throughthe reacting medium and a first order one-step exothermic
Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2014 Article ID 825797 8 pageshttpdxdoiorg1011552014825797
2 International Journal of Chemical Engineering
chemical reaction takes place The reaction rate dependson the temperature through the Arrhenius exponentialThe Arrhenius rate equation is a mathematical expressionwhich relates the rate constant of a chemical reaction to theexponential value of the temperature The model nonlinearequations in dimensionless form are [1]
120597119884
120597119905= Le120597
2119884
1205971199092minus 1206012119884 exp(minus
120574
119879)
120597119879
120597119905=1205972119879
1205971199092+ 1205731206012119884 exp(minus
120574
119879)
(1)
where 119884 is the concentration of the gas reactant T is thetemperature Le is the Lewis number (the ratio betweenmass and heat diffusivities) 120573 is the dimensionless heatof reaction 120601 is the thermal Thiele modulus (the ratioof the time scale of the limiting transport mechanism tothe time scale of intrinsic reaction kinetics [7]) and 120574 isthe dimensionless activation energy (minimum amount ofenergy between reactant molecules for effective collisionsbetween them) The boundary conditions are
10038161003816100381610038161003816100381610038161003816119909=1= 0 for 119905 gt 0
(2)
Under steady-state condition the equations become
1198892119884
1198891199092minus1206012
Le119884 exp(minus
120574
119879) = 0
1198892119879
1198891199092+ 1205731206012119884 exp(minus
120574
119879) = 0
(3)
with boundary conditions
119879 = 1 119884 = 1 at 119909 = 0
119879 = 1119889119884
119889119909= 0 at 119909 = 1
(4)
3 Analytical Solution of Nonlinear Dynamicsof a Self-Igniting Reaction-DiffusionSystem under Steady-State Condition UsingModified Adomian Decomposition Method
In the recent years much attention is devoted to the applica-tion of theAdomian decompositionmethod to the solution ofvarious scientific models [8] An efficient modification of thestandard Adomian decomposition method for solving initialvalue problem in the second order partial differential equa-tion yields the MADM The MADM without linearizationperturbation transformation or discretization gives an ana-lytical solution in terms of a rapidly convergent infinite powerseries with easily computable terms The results show thatthe rate of convergence of modified Adomian decompositionmethod is higher than standard Adomian decompositionmethod [9ndash13] Using this method (see Appendix A) we
Figure 1 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 120574 = 136and various values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
obtain an approximate analytical expression of concentrationof gas reactant (119884) and temperature (119879) (see Appendix B) asfollows
119884 (119909) = 1 +1206012
Leexp (minus120574)(119909
2
2minus 119909) +
1206014
Leexp (minus2120574)
times [1
Le(1199094
24minus1199093
6+119909
3) minus
120574120573
2(1199094
12minus1199093
6+119909
6)]
(5)
119879 (119909) = 1 minus 1205731206012 exp (minus120574)(1199092
2minus119909
2) minus 1205731206014 exp (minus2120574)
times [1
Le(1199094
24minus1199093
6+119909
8) minus
120574120573
2(1199094
12minus1199093
6+119909
12)]
(6)
4 Numerical Simulation
Nonlinear diffusion equation (3) for the boundary condition(4) is also solved numerically We have used the functionpdex1 in MATLAB software to solve the initial-boundaryvalue problems for the nonlinear differential equationsnumerically This numerical solution is compared with ouranalytical results in Figures 1ndash4 Upon comparison it givesa satisfactory agreement for all values of the dimensionlessparameters 120573 120574 and 1206012TheMATLAB program is also givenin Algorithm 1
5 Discussion
Equations (5) and (6) represent the simplest form of approx-imate analytical expressions for the concentration of gasreactant and temperature for all values of parameters 120572120573 120574 and 1206012 The Thiele number (thermal) is the ratio oflayer thickness (L) and thermal diffusivity (120572) Equation (5)
International Journal of Chemical Engineering 3
1206012 = 1000
1206012 = 5000
1206012 = 10000
1206012 = 20000
1206012 = 25000
1206012 = 30000
1206012 = 35000
10
1
1005
101
102
103
104
1045
1015
1025
1035
09950807060504030201 09
Dimensionless spatial coordinate (x)
Dim
ensio
nles
s tem
pera
ture
(T)
Figure 2 Dimensionless temperature T versus dimensionless spa-tial coordinate using (6) for Le = 0233 120573 = 4287 120574 =136 andvarious values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
Dim
ensio
nles
s con
cent
ratio
n (Y
)
0 1
1
098
096
094
092
08801 02 03 04 05 06 07 08 09
09
Dimensionless spatial coordinate (x)
120574 = 11
120574 = 10120574 = 12
120574 = 13
120574 = 115
120574 = 105 120574 = 125
120574 = 135
Figure 3 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 1206012 = 1000and various values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
represents the new approximate analytical expression ofconcentration of gas reactant The numerical solution iscompared with the analytical results in Figures 1ndash4 Thesefigures represent the analytical and numerical concentra-tion profiles of gas reactant and temperature for differentvalues of parameters 120573 120574 and 1206012 Figure 1 represents thedimensionless concentration Y versus dimensionless spatialcoordinate 119909 for 1206012 le 35000 From the figure it is inferredthat the value of Y decreases when the value of 1206012 or layerthickness increases Figure 2 illustrates the dimensionlessconcentrationY versus dimensionless spatial coordinate119909 for1206012 le 35000 and we infer that the dimensionless temperatureincreases with the increase in values of layer thicknessFigure 3 represents the dimensionless concentrationY versusdimensionless spatial coordinate 119909 for values of 120574 le 136From the figure it is inferred that the value of Y decreases
120574 = 135
120574 = 125
120574 = 115
120574 = 105
120574 = 13
120574 = 11 120574 = 10
120574 = 12
0 1
1
07
101
102
103
08 090605040302010995
1005
1015
1025
1035
Dim
ensio
nles
s tem
pera
ture
(T)
Dimensionless spatial coordinate (x)
Figure 4 Dimensionless temperature T versus dimensionless spa-tial coordinate x using (6) for Le = 0233 120573 = 4287 1206012 = 1000 andvarious values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
119909 Numerical value of 119884 Analytical value of 119884 deviation0 1 1 002 09661 09713 05304 09398 09489 09706 09216 09329 12308 09113 09233 1321 09082 09201 131
when the value of dimensionless activation energy (120574)increases Figure 4 illustrates the dimensionless temperatureT versus dimensionless spatial coordinate 119909 for the valuesof 120574 le 136 the dimensionless temperature increases withincrease in values of dimensionless activation energy (120574)From Figures 2 and 4 it is evident that the maximum valuefor dimensionless concentration is 1 and that temperatureattains its maximum value when the spatial coordinate 119909 =05 Figure 5 confirms the results given by Figures 1 to4
Our analytical results are compared with the numericalresults for the dimensionless concentration Y in Table 1The maximum relative error between our analytical resultsand simulation results for the concentration Y is 13Also in Table 2 our analytical results are compared withthe numerical results for the dimensionless temperature TSatisfactory agreement is noted The maximum relative errorin this case is 04
6 Conclusion
The system of steady-state nonlinear differential equa-tions in different dynamic modes for the concentration of
Algorithm 1 MATLAB program to find the numerical solution of nonlinear differential equation (1) when the time (119905) is large (or) (3)
Table 2 Comparison between analytical and numerical values ofdimensionless temperature 119879 (Le = 0233 120573 = 4287 120574 = 136 and1206012 = 30000)
119909 Analytical value of 119879 Numerical value of 119879 deviation0 1 1 002 10155 11013 02704 10235 10191 04206 10233 10191 04108 10152 10128 0241 1 1 0
gas reactant and temperature has been solved analyticallyThe model investigated the influence of parameters overthe temperature and concentration of gas reactant in thedynamicmodeThe approximate analytical expression for thesteady-state concentration of gas reactant and temperature
is obtained using the modified Adomian decompositionmethod A satisfactory agreement with the numerical resultis noted The analytical results will be useful to characterizethe model predictions for the various values of parametersThiele number (1206012) Lewis number (Le) layer thickness (L)and thermal diffusivity (120572)
Appendices
A Basic Concept of Modified AdomianDecomposition Method
Consider the nonlinear differential equation in the form
Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574
where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows
119871 = 119909minus1198991198892
1198891199092119909119899119910 (A3)
So problem (A1) can be written as
119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)
The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
chemical reaction takes place The reaction rate dependson the temperature through the Arrhenius exponentialThe Arrhenius rate equation is a mathematical expressionwhich relates the rate constant of a chemical reaction to theexponential value of the temperature The model nonlinearequations in dimensionless form are [1]
120597119884
120597119905= Le120597
2119884
1205971199092minus 1206012119884 exp(minus
120574
119879)
120597119879
120597119905=1205972119879
1205971199092+ 1205731206012119884 exp(minus
120574
119879)
(1)
where 119884 is the concentration of the gas reactant T is thetemperature Le is the Lewis number (the ratio betweenmass and heat diffusivities) 120573 is the dimensionless heatof reaction 120601 is the thermal Thiele modulus (the ratioof the time scale of the limiting transport mechanism tothe time scale of intrinsic reaction kinetics [7]) and 120574 isthe dimensionless activation energy (minimum amount ofenergy between reactant molecules for effective collisionsbetween them) The boundary conditions are
10038161003816100381610038161003816100381610038161003816119909=1= 0 for 119905 gt 0
(2)
Under steady-state condition the equations become
1198892119884
1198891199092minus1206012
Le119884 exp(minus
120574
119879) = 0
1198892119879
1198891199092+ 1205731206012119884 exp(minus
120574
119879) = 0
(3)
with boundary conditions
119879 = 1 119884 = 1 at 119909 = 0
119879 = 1119889119884
119889119909= 0 at 119909 = 1
(4)
3 Analytical Solution of Nonlinear Dynamicsof a Self-Igniting Reaction-DiffusionSystem under Steady-State Condition UsingModified Adomian Decomposition Method
In the recent years much attention is devoted to the applica-tion of theAdomian decompositionmethod to the solution ofvarious scientific models [8] An efficient modification of thestandard Adomian decomposition method for solving initialvalue problem in the second order partial differential equa-tion yields the MADM The MADM without linearizationperturbation transformation or discretization gives an ana-lytical solution in terms of a rapidly convergent infinite powerseries with easily computable terms The results show thatthe rate of convergence of modified Adomian decompositionmethod is higher than standard Adomian decompositionmethod [9ndash13] Using this method (see Appendix A) we
Figure 1 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 120574 = 136and various values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
obtain an approximate analytical expression of concentrationof gas reactant (119884) and temperature (119879) (see Appendix B) asfollows
119884 (119909) = 1 +1206012
Leexp (minus120574)(119909
2
2minus 119909) +
1206014
Leexp (minus2120574)
times [1
Le(1199094
24minus1199093
6+119909
3) minus
120574120573
2(1199094
12minus1199093
6+119909
6)]
(5)
119879 (119909) = 1 minus 1205731206012 exp (minus120574)(1199092
2minus119909
2) minus 1205731206014 exp (minus2120574)
times [1
Le(1199094
24minus1199093
6+119909
8) minus
120574120573
2(1199094
12minus1199093
6+119909
12)]
(6)
4 Numerical Simulation
Nonlinear diffusion equation (3) for the boundary condition(4) is also solved numerically We have used the functionpdex1 in MATLAB software to solve the initial-boundaryvalue problems for the nonlinear differential equationsnumerically This numerical solution is compared with ouranalytical results in Figures 1ndash4 Upon comparison it givesa satisfactory agreement for all values of the dimensionlessparameters 120573 120574 and 1206012TheMATLAB program is also givenin Algorithm 1
5 Discussion
Equations (5) and (6) represent the simplest form of approx-imate analytical expressions for the concentration of gasreactant and temperature for all values of parameters 120572120573 120574 and 1206012 The Thiele number (thermal) is the ratio oflayer thickness (L) and thermal diffusivity (120572) Equation (5)
International Journal of Chemical Engineering 3
1206012 = 1000
1206012 = 5000
1206012 = 10000
1206012 = 20000
1206012 = 25000
1206012 = 30000
1206012 = 35000
10
1
1005
101
102
103
104
1045
1015
1025
1035
09950807060504030201 09
Dimensionless spatial coordinate (x)
Dim
ensio
nles
s tem
pera
ture
(T)
Figure 2 Dimensionless temperature T versus dimensionless spa-tial coordinate using (6) for Le = 0233 120573 = 4287 120574 =136 andvarious values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
Dim
ensio
nles
s con
cent
ratio
n (Y
)
0 1
1
098
096
094
092
08801 02 03 04 05 06 07 08 09
09
Dimensionless spatial coordinate (x)
120574 = 11
120574 = 10120574 = 12
120574 = 13
120574 = 115
120574 = 105 120574 = 125
120574 = 135
Figure 3 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 1206012 = 1000and various values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
represents the new approximate analytical expression ofconcentration of gas reactant The numerical solution iscompared with the analytical results in Figures 1ndash4 Thesefigures represent the analytical and numerical concentra-tion profiles of gas reactant and temperature for differentvalues of parameters 120573 120574 and 1206012 Figure 1 represents thedimensionless concentration Y versus dimensionless spatialcoordinate 119909 for 1206012 le 35000 From the figure it is inferredthat the value of Y decreases when the value of 1206012 or layerthickness increases Figure 2 illustrates the dimensionlessconcentrationY versus dimensionless spatial coordinate119909 for1206012 le 35000 and we infer that the dimensionless temperatureincreases with the increase in values of layer thicknessFigure 3 represents the dimensionless concentrationY versusdimensionless spatial coordinate 119909 for values of 120574 le 136From the figure it is inferred that the value of Y decreases
120574 = 135
120574 = 125
120574 = 115
120574 = 105
120574 = 13
120574 = 11 120574 = 10
120574 = 12
0 1
1
07
101
102
103
08 090605040302010995
1005
1015
1025
1035
Dim
ensio
nles
s tem
pera
ture
(T)
Dimensionless spatial coordinate (x)
Figure 4 Dimensionless temperature T versus dimensionless spa-tial coordinate x using (6) for Le = 0233 120573 = 4287 1206012 = 1000 andvarious values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
119909 Numerical value of 119884 Analytical value of 119884 deviation0 1 1 002 09661 09713 05304 09398 09489 09706 09216 09329 12308 09113 09233 1321 09082 09201 131
when the value of dimensionless activation energy (120574)increases Figure 4 illustrates the dimensionless temperatureT versus dimensionless spatial coordinate 119909 for the valuesof 120574 le 136 the dimensionless temperature increases withincrease in values of dimensionless activation energy (120574)From Figures 2 and 4 it is evident that the maximum valuefor dimensionless concentration is 1 and that temperatureattains its maximum value when the spatial coordinate 119909 =05 Figure 5 confirms the results given by Figures 1 to4
Our analytical results are compared with the numericalresults for the dimensionless concentration Y in Table 1The maximum relative error between our analytical resultsand simulation results for the concentration Y is 13Also in Table 2 our analytical results are compared withthe numerical results for the dimensionless temperature TSatisfactory agreement is noted The maximum relative errorin this case is 04
6 Conclusion
The system of steady-state nonlinear differential equa-tions in different dynamic modes for the concentration of
Algorithm 1 MATLAB program to find the numerical solution of nonlinear differential equation (1) when the time (119905) is large (or) (3)
Table 2 Comparison between analytical and numerical values ofdimensionless temperature 119879 (Le = 0233 120573 = 4287 120574 = 136 and1206012 = 30000)
119909 Analytical value of 119879 Numerical value of 119879 deviation0 1 1 002 10155 11013 02704 10235 10191 04206 10233 10191 04108 10152 10128 0241 1 1 0
gas reactant and temperature has been solved analyticallyThe model investigated the influence of parameters overthe temperature and concentration of gas reactant in thedynamicmodeThe approximate analytical expression for thesteady-state concentration of gas reactant and temperature
is obtained using the modified Adomian decompositionmethod A satisfactory agreement with the numerical resultis noted The analytical results will be useful to characterizethe model predictions for the various values of parametersThiele number (1206012) Lewis number (Le) layer thickness (L)and thermal diffusivity (120572)
Appendices
A Basic Concept of Modified AdomianDecomposition Method
Consider the nonlinear differential equation in the form
Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574
where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows
119871 = 119909minus1198991198892
1198891199092119909119899119910 (A3)
So problem (A1) can be written as
119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)
The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
Figure 2 Dimensionless temperature T versus dimensionless spa-tial coordinate using (6) for Le = 0233 120573 = 4287 120574 =136 andvarious values of 1206012 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
Dim
ensio
nles
s con
cent
ratio
n (Y
)
0 1
1
098
096
094
092
08801 02 03 04 05 06 07 08 09
09
Dimensionless spatial coordinate (x)
120574 = 11
120574 = 10120574 = 12
120574 = 13
120574 = 115
120574 = 105 120574 = 125
120574 = 135
Figure 3 Dimensionless concentration Y versus dimensionlessspatial coordinate x using (5) for Le = 0233 120573 = 4287 1206012 = 1000and various values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
represents the new approximate analytical expression ofconcentration of gas reactant The numerical solution iscompared with the analytical results in Figures 1ndash4 Thesefigures represent the analytical and numerical concentra-tion profiles of gas reactant and temperature for differentvalues of parameters 120573 120574 and 1206012 Figure 1 represents thedimensionless concentration Y versus dimensionless spatialcoordinate 119909 for 1206012 le 35000 From the figure it is inferredthat the value of Y decreases when the value of 1206012 or layerthickness increases Figure 2 illustrates the dimensionlessconcentrationY versus dimensionless spatial coordinate119909 for1206012 le 35000 and we infer that the dimensionless temperatureincreases with the increase in values of layer thicknessFigure 3 represents the dimensionless concentrationY versusdimensionless spatial coordinate 119909 for values of 120574 le 136From the figure it is inferred that the value of Y decreases
120574 = 135
120574 = 125
120574 = 115
120574 = 105
120574 = 13
120574 = 11 120574 = 10
120574 = 12
0 1
1
07
101
102
103
08 090605040302010995
1005
1015
1025
1035
Dim
ensio
nles
s tem
pera
ture
(T)
Dimensionless spatial coordinate (x)
Figure 4 Dimensionless temperature T versus dimensionless spa-tial coordinate x using (6) for Le = 0233 120573 = 4287 1206012 = 1000 andvarious values of 120574 Solid lines represent the analytical solutionwhereas the dotted lines are for the numerical solution
119909 Numerical value of 119884 Analytical value of 119884 deviation0 1 1 002 09661 09713 05304 09398 09489 09706 09216 09329 12308 09113 09233 1321 09082 09201 131
when the value of dimensionless activation energy (120574)increases Figure 4 illustrates the dimensionless temperatureT versus dimensionless spatial coordinate 119909 for the valuesof 120574 le 136 the dimensionless temperature increases withincrease in values of dimensionless activation energy (120574)From Figures 2 and 4 it is evident that the maximum valuefor dimensionless concentration is 1 and that temperatureattains its maximum value when the spatial coordinate 119909 =05 Figure 5 confirms the results given by Figures 1 to4
Our analytical results are compared with the numericalresults for the dimensionless concentration Y in Table 1The maximum relative error between our analytical resultsand simulation results for the concentration Y is 13Also in Table 2 our analytical results are compared withthe numerical results for the dimensionless temperature TSatisfactory agreement is noted The maximum relative errorin this case is 04
6 Conclusion
The system of steady-state nonlinear differential equa-tions in different dynamic modes for the concentration of
Algorithm 1 MATLAB program to find the numerical solution of nonlinear differential equation (1) when the time (119905) is large (or) (3)
Table 2 Comparison between analytical and numerical values ofdimensionless temperature 119879 (Le = 0233 120573 = 4287 120574 = 136 and1206012 = 30000)
119909 Analytical value of 119879 Numerical value of 119879 deviation0 1 1 002 10155 11013 02704 10235 10191 04206 10233 10191 04108 10152 10128 0241 1 1 0
gas reactant and temperature has been solved analyticallyThe model investigated the influence of parameters overthe temperature and concentration of gas reactant in thedynamicmodeThe approximate analytical expression for thesteady-state concentration of gas reactant and temperature
is obtained using the modified Adomian decompositionmethod A satisfactory agreement with the numerical resultis noted The analytical results will be useful to characterizethe model predictions for the various values of parametersThiele number (1206012) Lewis number (Le) layer thickness (L)and thermal diffusivity (120572)
Appendices
A Basic Concept of Modified AdomianDecomposition Method
Consider the nonlinear differential equation in the form
Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574
where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows
119871 = 119909minus1198991198892
1198891199092119909119899119910 (A3)
So problem (A1) can be written as
119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)
The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
Algorithm 1 MATLAB program to find the numerical solution of nonlinear differential equation (1) when the time (119905) is large (or) (3)
Table 2 Comparison between analytical and numerical values ofdimensionless temperature 119879 (Le = 0233 120573 = 4287 120574 = 136 and1206012 = 30000)
119909 Analytical value of 119879 Numerical value of 119879 deviation0 1 1 002 10155 11013 02704 10235 10191 04206 10233 10191 04108 10152 10128 0241 1 1 0
gas reactant and temperature has been solved analyticallyThe model investigated the influence of parameters overthe temperature and concentration of gas reactant in thedynamicmodeThe approximate analytical expression for thesteady-state concentration of gas reactant and temperature
is obtained using the modified Adomian decompositionmethod A satisfactory agreement with the numerical resultis noted The analytical results will be useful to characterizethe model predictions for the various values of parametersThiele number (1206012) Lewis number (Le) layer thickness (L)and thermal diffusivity (120572)
Appendices
A Basic Concept of Modified AdomianDecomposition Method
Consider the nonlinear differential equation in the form
Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574
where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows
119871 = 119909minus1198991198892
1198891199092119909119899119910 (A3)
So problem (A1) can be written as
119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)
The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
Figure 5 (a)The normalized three-dimensional concentration of gas reactantY versus dimensionless spatial coordinate 119909 andThiele number1206012 (b) the normalized three-dimensional concentration of gas reactant Y versus dimensionless spatial coordinate 119909 and dimensionlessactivation energy 120574 (c) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and Thiele number 1206012and (d) the normalized three-dimensional temperature T versus dimensionless spatial coordinate 119909 and dimensionless activation energy 120574
where 119873(119910) is a nonlinear real function 119892(119909) is the givenfunction and 119860 and 119861 are constants We propose the newdifferential operator as follows
119871 = 119909minus1198991198892
1198891199092119909119899119910 (A3)
So problem (A1) can be written as
119871 (119910) = 119892 (119909) minus 119873 (119910) (A4)
The inverse operator 119871minus1 is therefore considered a twofoldintegral operator as follows
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
From (A9) and (A12) we can determine the components119910119899(119909) and hence the series solution of 119910(119909) in (A7) can be
immediately obtained
B Analytical Solution of Nonlinear Reaction-Diffusion Equation (3) Using ModifiedAdomian Decomposition Method
In this appendix we derive the general solution of nonlinearequation (3) by using modified Adomian decompositionmethod We write (3) in the operator form as
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
120574 Dimensionless activation energy 119864(1198770)
Δ119867 Enthalpy of reaction120588 DensityΦ Thiele numberradic119896
01198712120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the University Grant Commission(UGC) Minor Project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal secretary at Madura CollegeBoard and Dr R Murali principal at Madura College(autonomous) Madurai Tamil Nadu India for their con-stant encouragement
References
[1] G Continillo V Faraoni P L Maffettone and S CrescitellildquoNon-linear dynamics of a self-igniting reaction-diffusion sys-temrdquo Chemical Engineering Science vol 55 no 2 pp 303ndash3092000
[2] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behavior of some distrib-uted-parameter systemsrdquo in Proceedings of the 1st National
8 International Journal of Chemical Engineering
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002
Conference Chaos and Fractals in Chemical Engineering pp 218ndash226 World Scientific Singapore May 1994
[3] C Gaetano and G Giovanni Characterization of ChaoticDynamics in the Spontaneous Combustion of Coal Stock-piles Istituto di Ricerchesulla Combustione (IRC) ConsiglioNazionaledelle Ricerche Naples Italy 1996
[4] G Continillo P L Maffettone and S Crescitelli First Con-ference on Chemical and Process Engineering Paper no 22Firenze Italy 1993
[5] G Continillo P L Maffettone and S Crescitelli ldquoOn thenumerical simulation of the chaotic behaviour of somedistributed-parameter systemsrdquo inChaos and Fractals in Chem-ical Engineering G Biardi M Giona and A R Giona Eds pp218ndash226 World Scientific Singapore 1995
[6] G Continillo P L Maffettone and S Crescitelli in ICheaP-2C T Eris Ed vol 1 of AIDIC Conference Series pp 415ndash424Firenze Italy 1995
[7] W B J Zimmerman Microfluidics History Theory amp Applica-tions Springer New York NY USA 2006
[8] G Adomian ldquoConvergent series solution of nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol11 no 2 pp 225ndash230 1984
[9] Y Q Hasan and L M Zhu ldquoModified Adomian decompositionmethod for singular initial value problems in the second-orderordinary differential equationsrdquo Surveys in Mathematics and ItsApplications vol 3 pp 183ndash193 2008
[10] Y Q Hasan and L M Zhu ldquoSolving singular boundary valueproblems of higher-order ordinary differential equations bymodified Adomian decomposition methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 6pp 2592ndash2596 2009
[11] A-M Wazwaz ldquoA reliable modification of Adomian decompo-sitionmethodrdquoAppliedMathematics and Computation vol 102no 1 pp 77ndash86 1999
[12] A-M Wazwaz ldquoAnalytical approximations and pade approx-imants for volterrarsquos population modelrdquo Applied Mathematicsand Computation vol 100 no 1 pp 13ndash25 1999
[13] A-MWazwaz ldquoA newmethod for solving singular initial valueproblems in the second-order ordinary differential equationsrdquoAppliedMathematics and Computation vol 128 no 1 pp 45ndash572002