research article optimal homotopy asymptotic solution for...

Research ArticleOptimal Homotopy Asymptotic Solution forExothermic Reactions Model with Constant Heat Source ina Porous Medium

Fazle Mabood1 and Nopparat Pochai23

1Department of Mathematics University of Peshawar Peshawar Pakistan2Department of Mathematics King Mongkutrsquos Institute of Technology Ladkrabang Bangkok 10520 Thailand3Centre of Excellence in Mathematics CHE Si Ayutthaya Road Bangkok 10400 Thailand

Correspondence should be addressed to Nopparat Pochai nop mathyahoocom

Received 27 May 2015 Accepted 7 June 2015

Academic Editor John D Clayton

Copyright copy 2015 F Mabood and N Pochai 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

The heat flow patterns profiles are required for heat transfer simulation in each type of the thermal insulation The exothermicreaction models in porous medium can prescribe the problems in the form of nonlinear ordinary differential equations In thisresearch the driving force model due to the temperature gradients is considered A governing equation of the model is restrictedinto an energy balance equation that provides the temperature profile in conduction state with constant heat source on the steadystate The proposed optimal homotopy asymptotic method (OHAM) is used to compute the solutions of the exothermic reactionsequation

1 Introduction

In physical systems energy is obtained from chemical bondsIf bonds are broken energy is needed If bonds are formedenergy is released Each type of bond has specific bondenergy It can be predictedwhether a chemical reactionwouldrelease or need heat by using bond energies If there is moreenergy used to form the bonds than to break the bonds heatis given offThis is well known as an exothermic reaction Onthe other hand if a reaction needs an input of energy it is saidto be an endothermic reaction The ability to break bonds isactivated energy

Convection has obtained growth uses in many areas suchas solar energy conversion underground coal gasificationgeothermal energy extraction ground water contaminanttransport and oil reservoir simulationThe exothermic reac-tionmodel is focused on the system inwhich the driving forcewas due to the applied temperature gradients at the boundaryof the system In [1ndash4] they proposed the investigationof Rayleigh-Bernard-type convection They also study theconvective instabilities that arise due to exothermic reactions

model in a porous mediumThe exothermic reactions releasethe heat create density differences within the fluid andinduce natural convection that turn out the rate of reactionaffects [5] The nonuniform flow of convective motion that isgenerated by heat sources is investigated by [6ndash8] In [9ndash13]they propose the two- and three-dimensional models ofnatural convection among different types of porous medium

In this research the optimal homotopy asymptoticmethod for conduction solutions is proposed The modelequation is a steady-state energy balance equation of thetemperature profile in conduction state with constant heatsource

The optimal homotopy asymptotic method is an approx-imate analytical tool that is simple and straightforward anddoes not require the existence of any small or large parameteras does traditional perturbation method As observed byHerisanu and Marinca [14] the most significant featureOHAM is the optimal control of the convergence of solu-tions via a particular convergence-control function 119867 andthis ensures a very fast convergence when its components(known as convergence-control parameters) are optimally

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 825683 4 pageshttpdxdoiorg1011552015825683

2 Advances in Mathematical Physics

determined In the recent paper of Herisanu et al [15] wherethe authors focused on nonlinear dynamical model of apermanent magnet synchronous generator in their studya different way of construction of homotopy is developedto ensure the fast convergence of the OHAM solutionsto the exact one Optimal Homotopy Asymptotic Method(OHAM) has been successfully been applied to linear andnonlinear problems [16 17] This paper is organized asfollows First in Section 2 exothermic reaction model ispresented In Section 3 we described the basic principlesof the optimal homotopy asymptotic method The optimalhomotopy asymptotic method solution of the problem isgiven in Section 4 Section 5 is devoted for the concludingremarks

2 Exothermic Reactions Model

In this section we introduce a pseudohomogeneous modelto express convective driven by an exothermic reaction Thecase of a porous medium wall thickness (0 lt 119911

1015840lt 119871)

is focused The normal assumption in the continuity andmomentum equations in the steady-state energy balancepresents a nondimensional formof a BVP for the temperatureprofile [5 13]

11988921205790

1198891199112+119861120601

2(1minus

1205790119861) exp(

1205741205790120574 + 1205790

) = 0 (1)

Here 1205790is the temperature the parameter 119861 is the maximum

feasible temperature in the absence of natural convection 1206012

is the ratio of the characteristic time for diffusion of heatgenerator and 120574 is the dimensionless activation energy In thecase of the constant heat source (1) can be written as

11988921205790

1198891199112+119861120601

2(1minus

1205790119861) = 0 (2)

subject to boundary condition

1198891205790119889119911

= 0 at 119911 = 0

1205790 = 0 at 119911 = 1(3)

3 Basic Principles of Optimal HomotopyAsymptotic Method

We review the basic principles of the optimal homotopyasymptotic method as follows

(i) Consider the following differential equation

119860 [119906 (119909)] + 119886 (119909) = 0 119909 isin Ω (4)

where Ω is problem domain 119860(119906) = 119871(119906) + 119873(119906) where 119871119873 are linear and nonlinear operators 119906(119909) is an unknownfunction and 119886(119909) is a known function

(ii) Construct an optimal homotopy equation as

(1minus119901) [119871 (120601 (119909 119901)) + 119886 (119909)]

minus119867 (119901) [119860 (120601 (119909 119901)) + 119886 (119909)] = 0(5)

where 0 le 119901 le 1 is an embedding parameter and119867(119901) = sum

119898

119894=1 119901119894119870119894is auxiliary function on which the con-

vergence of the solution greatly dependent Here 119870119895are

the convergence-control parameters The auxiliary function119867(119901) also adjusts the convergence domain and controls theconvergence region

(iii) Expand 120601(119909 119901 119870119895) in Taylorrsquos series about 119901 one has

an approximate solution

120601 (119909 119901 119870119895) = 1199060 (119909) +

infin

sum

119896=1119906119896(119909119870119895) 119901119896

119895 = 1 2 3

(6)

Many researchers have observed that the convergence of theseries equation (6) depends upon 119870

119895 (119895 = 1 2 119898) if it is

convergent then we obtain

V = V0 (119909) +119898

sum

119896=1V119896(119909119870119895) (7)

(iv) Substituting (7) in (4) we have the following residual

119877 (119909119870119895) = 119871 ( (119909 119870

119895)) + 119886 (119909) +119873( (119909119870

119895)) (8)

If119877(119909119870119895) = 0 then Vwill be the exact solution For nonlinear

problems generally this will not be the case For determining119870119895 (119895 = 1 2 119898) collocationmethod Ritz method or the

method of least squares can be used(v) Finally substituting the optimal values of the

convergence-control parameters 119870119895in (7) one can get the

approximate solution

4 Application of OHAM to an ExothermicReaction Model

Applying OHAM on (2) the zeroth first and second orderproblems are

(1minus119901) (12057910158401015840

0 ) minus119867 (119901) (12057910158401015840+119861120601

2(1minus

1205790119861)) = 0 (9)

We consider 1205790119867(119901) in the following manner

120579 = 12057900 +11990112057901 +119901212057902

1198671 (119901) = 1199011198701 +11990121198702

(10)

41 Zeroth Order Problem

12057910158401015840

00 = 0 (11)

with boundary conditions

12057900 (1) = 0

1205791015840

00 (0) = 0(12)

The solution of (11) with boundary condition (12) is

12057900 (119911) = 0 (13)

Advances in Mathematical Physics 3

42 First Order Problem

12057910158401015840

01 minus11987011206012119861 = 0 (14)


12057901 (1) = 0

1205791015840

01 (0) = 0(15)


12057901 (119911 1198701) =1198701120601

2119861

2(119911

2minus 1) (16)

43 Second Order Problem

12057910158401015840

02 (119911 1198701 1198702) = 11987011206012119861+119870

21120601

2119861minus

12119870

21120601

4119861119911

2

+12119870

21120601

4119861+

121198702120601

2119861

(17)


12057902 (1) = 0

1205791015840

02 (0) = 0(18)


12057902 (119911 1198701 1198702) =minus124

1206014119870

21119861119911

4+1212060121198701119861119911

2

+121206012119870

21119861119911

2+141206014119870

21119861119911

2

+1212060121198702119861119911

2minus

524

1206014119870

21119861

minus1212060121198701119861minus

121206012119870

21119861minus

1212060121198702119861

(19)

The final three terms solution via OHAM for 119901 = 1 is

1205790 (119911 1198701 1198702) = 12057900 (119911) + 12057901 (119911 1198701)

+ 12057902 (119911 1198701 1198702) (20)

The method of least squares is used to determine the con-vergence control parameters1198701 and1198702 in (20) In particularcase for 120601 = 1 119861 = 10 the values of the convergencecontrol parameters are 1198701 = minus08337205022 and 1198702 =

minus002092667470By substituting the values of 1198701 and 1198702 in (20) and after

simplification we can obtain the second order approximatesolution via OHAM To check the accuracy of the OHAMsolution a comparison between the solutions determined byOHAMandnumericalmethodswasmade and is presented inTable 1 Graphical representation of the solution using finitedifference technique [5] OHAM and Runge-Kutta Fehlbergfourth fifth order method is shown in Figure 1 an excellent

Table 1 Comparison of 1205790(119911) via OHAM and RKF45 for 120601 = 1 119861 =

10

119885 FDM [5] RKF45 OHAM Percentage error00 3114344 3518277 3518285 000022701 3046176 3485927 3485969 000120402 2911251 3388613 3388675 000182903 2711819 3225339 3225359 000062004 2451166 2994264 2994284 000066705 2133897 2693071 2693037 000126206 1766284 2318441 2318432 000038807 1356680 1866723 1866701 000117808 0915960 1333395 1333311 000629909 0457980 0713042 0713046 000056010 0000000 0000000 0000000 mdash

1 2 3 4 5 6 7 8 9 10 11

Temperature

0051

152

253

354

FDMRKF45OHAM

z

Figure 1 Comparison of analytical and numerical solution

agreement can be observedWe can see that the OHAM givesa better accurate solution than the traditional finite differencetechnique of [5] On the other hand the OHAM gives acontinuity solution but the traditional finite difference tech-nique gives a discrete solution It follows that the solutions ofthe OHAM is easier to implement than the finite differencesolutions

In Figure 2 we exhibit the effect of different values of 120601with fixed value of 119861 on temperature profile

5 Concluding Remarks

In this paper one has described an optimal homotopyasymptotic technique for obtaining the temperature profilesin porous medium We can see that the temperature reducesto the end The OHAM scheme for obtaining the model isconvenient to implement The OHAM gives fourth orderaccurate solutions It follows that the method has no insta-bility problem The model should be considered in the caseof nonconstant heat source


0 02 04 06 08 10

1

2

3

4

51205790(z)

z

120601 = 1

120601 = 2

120601 = 3

120601 = 5

B = 5

Figure 2

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by the Centre of Excellence inMathe-matics the Commission on Higher EducationThailandTheauthors greatly appreciate valuable comments received fromProfessor John D Clayton and their reviewers

References

[1] J L Beck ldquoConvection in a box of porous material saturatedwith fluidrdquo Physics of Fluids vol 15 no 8 pp 1377ndash1383 1972

[2] S HDavis ldquoConvection in a box linear theoryrdquo Journal of FluidMechanics vol 30 no 3 pp 465ndash478 1967

[3] Z Gershuni and E M Zhukovitskii Convective Stability ofIncompressible Fluids vol 4 Israel Program for ScientificTranslations 1976

[4] E R Lapwood ldquoConvection of a fluid in a porous mediumrdquoProceedings of the Cambridge Philosophical Society vol 44 pp508ndash521 1948

[5] N Pochai and J Jaisaardsuetrong ldquoA numerical treatment ofan exothermic reactions model with constant heat source ina porous medium using finite difference methodrdquo AdvancedStudies in Biology vol 4 no 6 pp 287ndash296 2012

[6] D R Jones ldquoThe dynamic stability of confined exothermicallyreacting fluidsrdquo International Journal of Heat andMass Transfervol 16 no 1 pp 157ndash167 1973

[7] M Tveitereid ldquoThermal convection in a horizontal porous layerwith internal heat sourcesrdquo International Journal of Heat andMass Transfer vol 20 no 10 pp 1045ndash1050 1977

[8] J B Bdzil andH L Frisch ldquoChemically driven convectionrdquoTheJournal of Chemical Physics vol 72 no 3 pp 1875ndash1886 1980

[9] H Viljoen and V Hlavacek ldquoChemically driven convection ina porous mediumrdquo AIChE Journal vol 33 no 8 pp 1344ndash13501987

[10] H J Viljoen J E Gatica and H Vladimir ldquoBifurcation analysisof chemically driven convectionrdquoChemical Engineering Sciencevol 45 no 2 pp 503ndash517 1990

[11] WW Farr J F Gabitto D Luss and V Balakotaiah ldquoReaction-driven convection in a porous mediumrdquo AIChE Journal vol 37no 7 pp 963ndash985 1991

[12] K Nandakumar and H J Weinitschke ldquoA bifurcation study ofchemically driven convection in a porous mediumrdquo ChemicalEngineering Science vol 47 no 15-16 pp 4107ndash4120 1992

[13] S Subramanian and V Balakotaiah ldquoConvective instabili-ties induced by exothermic reactions occurring in a porousmediumrdquo Physics of Fluids vol 6 no 9 pp 2907ndash2922 1994

[14] N Herisanu and V Marinca ldquoAccurate analytical solutions tooscillators with discontinuities and fractional-power restoringforce by means of the optimal homotopy asymptotic methodrdquoComputers amp Mathematics with Applications vol 60 no 6 pp1607ndash1615 2010

[15] N Herisanu V Marinca and G Madescu ldquoAn analyticalapproach to non-linear dynamical model of a permanentmagnet synchronous generatorrdquoWind Energy 2014

[16] F Mabood and N Pochai ldquoAsymptotic solution for a waterquality model in a uniform streamrdquo International Journal ofEngineering Mathematics vol 2013 Article ID 135140 4 pages2013

[17] V Marinca and N Herisanu ldquoOptimal homotopy perturbationmethod for strongly nonlinear differential equationsrdquoNonlinearScience Letters A vol 1 no 3 pp 273ndash280 2010

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


determined In the recent paper of Herisanu et al [15] wherethe authors focused on nonlinear dynamical model of apermanent magnet synchronous generator in their studya different way of construction of homotopy is developedto ensure the fast convergence of the OHAM solutionsto the exact one Optimal Homotopy Asymptotic Method(OHAM) has been successfully been applied to linear andnonlinear problems [16 17] This paper is organized asfollows First in Section 2 exothermic reaction model ispresented In Section 3 we described the basic principlesof the optimal homotopy asymptotic method The optimalhomotopy asymptotic method solution of the problem isgiven in Section 4 Section 5 is devoted for the concludingremarks

2 Exothermic Reactions Model

In this section we introduce a pseudohomogeneous modelto express convective driven by an exothermic reaction Thecase of a porous medium wall thickness (0 lt 119911

1015840lt 119871)

is focused The normal assumption in the continuity andmomentum equations in the steady-state energy balancepresents a nondimensional formof a BVP for the temperatureprofile [5 13]

11988921205790

1198891199112+119861120601

2(1minus

1205790119861) exp(

1205741205790120574 + 1205790

) = 0 (1)

Here 1205790is the temperature the parameter 119861 is the maximum

feasible temperature in the absence of natural convection 1206012

is the ratio of the characteristic time for diffusion of heatgenerator and 120574 is the dimensionless activation energy In thecase of the constant heat source (1) can be written as

11988921205790

1198891199112+119861120601

2(1minus

1205790119861) = 0 (2)

subject to boundary condition

1198891205790119889119911

= 0 at 119911 = 0

1205790 = 0 at 119911 = 1(3)

3 Basic Principles of Optimal HomotopyAsymptotic Method

We review the basic principles of the optimal homotopyasymptotic method as follows

(i) Consider the following differential equation

119860 [119906 (119909)] + 119886 (119909) = 0 119909 isin Ω (4)

where Ω is problem domain 119860(119906) = 119871(119906) + 119873(119906) where 119871119873 are linear and nonlinear operators 119906(119909) is an unknownfunction and 119886(119909) is a known function

(ii) Construct an optimal homotopy equation as

(1minus119901) [119871 (120601 (119909 119901)) + 119886 (119909)]

minus119867 (119901) [119860 (120601 (119909 119901)) + 119886 (119909)] = 0(5)

where 0 le 119901 le 1 is an embedding parameter and119867(119901) = sum

119898

119894=1 119901119894119870119894is auxiliary function on which the con-

vergence of the solution greatly dependent Here 119870119895are

the convergence-control parameters The auxiliary function119867(119901) also adjusts the convergence domain and controls theconvergence region

(iii) Expand 120601(119909 119901 119870119895) in Taylorrsquos series about 119901 one has

an approximate solution

120601 (119909 119901 119870119895) = 1199060 (119909) +

infin

sum

119896=1119906119896(119909119870119895) 119901119896

119895 = 1 2 3

(6)

Many researchers have observed that the convergence of theseries equation (6) depends upon 119870

119895 (119895 = 1 2 119898) if it is

convergent then we obtain

V = V0 (119909) +119898

sum

119896=1V119896(119909119870119895) (7)

(iv) Substituting (7) in (4) we have the following residual

119877 (119909119870119895) = 119871 ( (119909 119870

119895)) + 119886 (119909) +119873( (119909119870

119895)) (8)

If119877(119909119870119895) = 0 then Vwill be the exact solution For nonlinear

problems generally this will not be the case For determining119870119895 (119895 = 1 2 119898) collocationmethod Ritz method or the

method of least squares can be used(v) Finally substituting the optimal values of the

convergence-control parameters 119870119895in (7) one can get the

approximate solution

4 Application of OHAM to an ExothermicReaction Model

Applying OHAM on (2) the zeroth first and second orderproblems are

(1minus119901) (12057910158401015840

0 ) minus119867 (119901) (12057910158401015840+119861120601

2(1minus

1205790119861)) = 0 (9)

We consider 1205790119867(119901) in the following manner

120579 = 12057900 +11990112057901 +119901212057902

1198671 (119901) = 1199011198701 +11990121198702

(10)

41 Zeroth Order Problem

12057910158401015840

00 = 0 (11)


12057900 (1) = 0

1205791015840

00 (0) = 0(12)


12057900 (119911) = 0 (13)



12057910158401015840

01 minus11987011206012119861 = 0 (14)


12057901 (1) = 0

1205791015840

01 (0) = 0(15)


12057901 (119911 1198701) =1198701120601

2119861

2(119911

2minus 1) (16)


12057910158401015840

02 (119911 1198701 1198702) = 11987011206012119861+119870

21120601

2119861minus

12119870

21120601

4119861119911

2

+12119870

21120601

4119861+

121198702120601

2119861

(17)


12057902 (1) = 0

1205791015840

02 (0) = 0(18)


12057902 (119911 1198701 1198702) =minus124

1206014119870

21119861119911

4+1212060121198701119861119911

2

+121206012119870

21119861119911

2+141206014119870

21119861119911

2

+1212060121198702119861119911

2minus

524

1206014119870

21119861

minus1212060121198701119861minus

121206012119870

21119861minus

1212060121198702119861

(19)


1205790 (119911 1198701 1198702) = 12057900 (119911) + 12057901 (119911 1198701)

+ 12057902 (119911 1198701 1198702) (20)





10


1 2 3 4 5 6 7 8 9 10 11

Temperature

0051

152

253

354

FDMRKF45OHAM

z







0 02 04 06 08 10

1

2

3

4

51205790(z)

z

120601 = 1

120601 = 2

120601 = 3

120601 = 5

B = 5

Figure 2



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











12057910158401015840

01 minus11987011206012119861 = 0 (14)


12057901 (1) = 0

1205791015840

01 (0) = 0(15)


12057901 (119911 1198701) =1198701120601

2119861

2(119911

2minus 1) (16)


12057910158401015840

02 (119911 1198701 1198702) = 11987011206012119861+119870

21120601

2119861minus

12119870

21120601

4119861119911

2

+12119870

21120601

4119861+

121198702120601

2119861

(17)


12057902 (1) = 0

1205791015840

02 (0) = 0(18)


12057902 (119911 1198701 1198702) =minus124

1206014119870

21119861119911

4+1212060121198701119861119911

2

+121206012119870

21119861119911

2+141206014119870

21119861119911

2

+1212060121198702119861119911

2minus

524

1206014119870

21119861

minus1212060121198701119861minus

121206012119870

21119861minus

1212060121198702119861

(19)


1205790 (119911 1198701 1198702) = 12057900 (119911) + 12057901 (119911 1198701)

+ 12057902 (119911 1198701 1198702) (20)





10


1 2 3 4 5 6 7 8 9 10 11

Temperature

0051

152

253

354

FDMRKF45OHAM

z







0 02 04 06 08 10

1

2

3

4

51205790(z)

z

120601 = 1

120601 = 2

120601 = 3

120601 = 5

B = 5

Figure 2



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

1

2

3

4

51205790(z)

z

120601 = 1

120601 = 2

120601 = 3

120601 = 5

B = 5

Figure 2



Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimal homotopy asymptotic solution for...

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