Hindawi Publishing CorporationISRNThermodynamicsVolume 2013 Article ID 282481 8 pageshttpdxdoiorg1011552013282481

Research ArticleAnalytical Solution of Nonlinear Boundary ValueProblem for Fin Efficiency of Convective Straight Fins withTemperature-Dependent Thermal Conductivity

K Saravanakumar1 V Ananthaswamy1 M Subha2 and L Rajendran1

1 Department of Mathematics The Madura College Madurai 625011 TN India2Madurai Sivakasi Nadars Pioneer Meenakshi Womenrsquos College Poovanthi 630611 Sivaganga District TN India

Correspondence should be addressed to L Rajendran raj smsrediffmailcom

Received 9 July 2013 Accepted 2 August 2013

Academic Editors R R Burnette E Curotto and I Kim

Copyright copy 2013 K Saravanakumar 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

We have employed homotopy analysis method (HAM) to evaluate the approximate analytical solution of the nonlinear equationarising in the convective straight fins with temperature-dependent thermal conductivity problem Solutions are presented for thedimensionless temperature distribution and fin efficiency of the nonlinear equation The analytical results are compared withprevious work and satisfactory agreement is noted

1 Introduction

The discipline of heat transfer typically considered an aspectof mechanical engineering and chemical engineering dealswith specific applied methods by which thermal energy in asystem is generated or converted or transferred to anothersystem Heat transfer includes the mechanisms of heat con-duction thermal radiation and mass transfer The analysisof extended surface heat transfer is extensively presented byKraus et al [1] Arslanturk [2] used decompositionmethod toevaluate the temperature distribution and analytical expres-sion for the fin efficiency

In the study of heat transfer a fin is a surface that extendsfrom an object to increase the rate of heat transfer to orfrom the environment by increasing convection Incroperaand Dewitt [3] presented a series of studies on the topicof heat transfer Mokheimer [4] discussed a series of fin-efficiency curves for annular fins of rectangular constantheat flow area triangular concave parabolic and convexparabolic profiles for a wide range of radius ratios and thedimensionless parameter based on the locally variable heattransfer coefficient The optimum dimensions of circular finswith variable profile and temperature-dependent thermalconductivity have been obtained by Zubair et al [5] A new

approach to calculate thermal performance of a singular finwith variable thermal properties has been presented by Kouet al [6]

Joneidi et al [7] studied an analytical solution of finefficiency of convective straight fins with temperature-dependent thermal conductivity by the DTM In this presentpaper we first apply homotopy analysis method to obtainan approximation of analytical expression of fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityThis problem is comparedwith Joneidi et al [7]

2 Mathematical Formulation of the BoundaryValue Problem

Consider a straight fin with a temperature-dependent ther-mal conductivity arbitrary constant cross-sectional area 119860119862perimeter 119875 and length 119887 (see Figure 1) The fin is attached toa base surface of temperature 119879119887 and extends into a fluid oftemperature 119879119886 and its tip is insulated The one-dimensionalenergy balance equation is given as follows

119860119862

119889

119889119909[119896 (119879)

119889119879

119889119909] minus 119875ℏ (119879

119887minus 119879119886) = 0 (1)

2 ISRNThermodynamics

where 119879 is the temperature 119896(119879) is the temperature-dependent thermal conductivity of the fin material 119875 isthe fin perimeter and ℏ is the heat transfer coefficient Thethermal conductivity of the finmaterial is assumed as follows

119896 (119879) = 119896119886 [1 + 120582 (119879 minus 119879

119886)] (2)

where 119896119886is the thermal conductivity at the ambient fluid

temperature of the fin and 120582 is the parameter describingthe variation of the thermal conductivity The followingdimensionless parameters are introduced [4]

120579 =119879 minus 119879119886

119879119887minus 119879119886

120585 =119909

119887 120573 = 120582 (119879

119887minus 119879119886)

120595 = (ℏ1198751198872

119896119886119860119862

)

12

(3)

Using the previous dimensionless variables the dimension-less form of (1) becomes as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (4)

where 120579 is the dimensionless temperature 120585 is the dimension-less coordinate 120573 is the dimensionless parameter describingthermal conductivity and 120595 is the thermogeometric finparameter The boundary conditions are given as follows

119889120579

119889120585= 0 when 120585 = 0

120579 = 1 when 120585 = 1

(5)

3 Fin Efficiency

The heat transfer rate from the fin is found by using Newtonrsquoslaw of cooling

Consider

119876 = int

119887

0

119875 (119879 minus 119879119886) 119889119909(6)

The ratio of the fin heat transfer rate to the heat transferrate of the fin if the entire fin was at the base temperature iscommonly called as the fin efficiency

120578 =119876

119876ideal=

int

119887

0

119875 (119879 minus 119879119886) 119889119909

119875119887 (119879119887minus 119879119886)

= int

1

120585=0

120579 (120585) 119889120585(7)

4 Approximation of Analytical Solution

41 Homotopy Analysis Method (HAM) Thehomotopy anal-ysis method employs the concept of the homotopy fromtopology to flexibly generate a convergent series solutionfor nonlinear systems The HAM [8ndash13] was devised byShijun Liao which is a powerful analytical method for solving

nonlinear problems The greater generality of this methodoften allows for strong convergence of the solution overlarger spacial and parameter domains This method providesan analytical solution in terms of an infinite power seriesHowever there is a practical need to evaluate this solutionand to obtain numerical values from the infinite power seriesIn order to investigate the accuracy of the homotopy analysismethod (HAM) solution with a finite number of terms thesystem of differential equations was solved

The homotopy analysis method [14] is a good techniquecomparing to another perturbation method The HAM givesexcellent flexibility in the expression of the solution and howthe solution is explicitly obtained Different from all reportedperturbation and nonperturbative techniques the homotopyanalysis method itself provides us with a convenient wayto control and adjust the convergence region and rate ofapproximation series when necessary Briefly speaking thehomotopy analysis method has the following advantages itis valid even if a given nonlinear problem does not containany smalllarge parameters at all it can be employed toefficiently approximate a nonlinear problem by choosingdifferent sets of base functions In this paper we employHAM(see Appendix A) to solve the nonlinear equation

42 Solution of Dimensionless Temperature Using HAM wecan obtain the dimensionless temperature (see Appendix B)as follows

120579 (120585) = (1

cosh (120595)minus

ℎ (ℎ + 2) 120573 cosh (2120595)

3cosh3 (120595)

+2ℎ21205732cosh2 (2120595)

9cosh5 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 2) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(8)

When 120573 rarr 0 from (4) the temperature is as follows

120579 (120585) =cosh (120595120585)

cosh (120595) (9)

This is the exact solution of (4) when 120573 = 0

ISRNThermodynamics 3

43 Determination of Fin Efficiency Using (7) and (8) we canobtain the fin efficiency given as follows

120578 = (1

cosh (120595)+

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)

minusℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595))sinh (120595)

120595

+ℎ (ℎ + 7) 120573 sinh (2120595)

6120595cosh2 (120595)+

ℎ21205732 sinh (3120595)

16120595cosh3 (120595)

+ℎ21205732(120595 cosh (120595) minus sinh (120595))

12120595cosh3 (120595)

minusℎ120573 cosh (2120595) sinh (120595)

3120595cosh3 (120595)

minusℎ21205732 cosh (2120595) sinh (2120595)

9120595cosh4 (120595)

(10)

5 Comparison with Previous Work (Joneidiet al [7])

Joneidi and coworker [7] obtained the closed form of thesolution of (1) using DTM (Tables 1 and 2) as follows

120579 (120585) = 119886 +1205952119886

2 (1 + 120573119886)1205852minus1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)31205854

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)51205856

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

1205858+ sdot sdot sdot

(11)

The value of 119886 can be obtained from the following equation

120579 (1) = 119886 +1205952119886

2 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

+ sdot sdot sdot = 1

(12)

b

x dx Tb

h Ta

Figure 1 The sketch of a straight fin

The solution of fin efficiency is

120578 = 119886 +1205952119886

6 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

120(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

5040(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

362880(1 + 120573119886)7

+ sdot sdot sdot

(13)

6 Discussion

Equation (8) (this work) and (Equation (11)) (previous work)represent the analytical expressions of temperature for allvalues of parameters Among these two expressions our resultis the simplest one

Figure 2 represents dimensionless temperature distribu-tion 120579 versus the dimensionless coordinate 120585 for variousvalues of 120595 using (8) From this figure it is clear that when 120595

increases (ie the fin length 119887 increases or the cross-sectionalarea of the fin 119860

119862decreases) the dimensionless temperature

120579 decreasesFigure 3 illustrates the temperature distribution of the

fin for different values of 120595 and 120573 Figure 3(a) depicts theeffect of the thermal conductivity 120573 on the temperature 120579According to (8) the temperature increases when the thermalconductivity of the fin 120573 increases for all values of 120595 Asan observation in (10) it can be concluded that the finefficiency also increases It is noted that the temperature onfin 120579 increases when 120573 increases (see Figure 3(b)) for the fixedvalue of 120595 = 1

Figure 4 shows the fin efficiency 120578 as a function ofthermogeometric fin parameter 120595 for different values ofthe thermal conductivity parameter 120573 using (10) From thisfigure it can be seen that the fin efficiency 120578 decreases as120595 increases However the fin efficiency 120578 increases as thethermal conductivity 120573 increases

4 ISRNThermodynamics

Table 1 Our work is compared with Joneidi et alrsquos [7] work and exact solution of dimensionless temperature 120579(120585) (linear case)

120585(a) 120573 = 0 120595 = 05 (b) 120573 = 0 120595 = 1

HAM (8)(this work)

DTM [7](11) Exact HAM (8)

(this work)DTM [7]

(11) Exact

0 088681 088681 088681 064805 064805 064805005 088709 088709 088709 064886 064886 064886010 088792 088792 088792 065129 065129 065129015 088931 088931 088931 065535 065535 065535020 089125 089125 089125 066105 066105 066105025 089375 089375 089375 066841 066841 066841030 089681 089681 089681 067743 067743 067743035 090043 090043 090043 068815 068815 068815040 090461 090461 090461 070059 070059 070059045 090936 090936 090936 071478 071478 071478050 091467 091467 091467 073076 073076 073076055 092056 092056 092056 074856 074856 074856060 092702 092702 092702 076824 076824 076824065 093406 093406 093406 078984 078984 078984070 094169 094169 094169 081341 081341 081341075 094990 094990 094990 083902 083902 083902080 095871 095871 095871 086673 086673 086673085 096812 096812 096812 089660 089660 089660090 097813 097813 097813 092871 092871 092871095 098875 098875 098875 096315 096315 096315100 100000 100000 099999 100000 100000 100000

0 01 02 03 04 05 06 07 08 09 1082

084

086

088

09

092

094

096

098

1

102

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

120595 = 05

120595 = 055

120595 = 06

120595 = 065

120573 = 0

Figure 2 Dimensionless temperature 120579 versus dimensionless coor-dinate 120585 for various values of 120595 when 120573 = 0 and ℎ = minus084 Thekey to the graph (solid line) represents our approximate solutionpresented in this work (8) and (+) represents Joneidi et alrsquos [7] work(Equation (11))

7 Conclusions

There are two main goals that we aimed for this work Thefirst is to employ the powerful homotopy analysis method

to investigate nonlinear differential equation arising in con-vective straight fins with temperature-dependent thermalconductivity problem The second is to achieve good resultsin predicting the solution of the heat transfer equationsin engineering The two goals are achieved successfully InHAM we can choose ℎ in an appropriate way which controlsthe convergence of the series

Appendices

A Basic Idea of Liaorsquos [15] HomotopyAnalysis Method

Consider the following differential equation [15]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator 119905 denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin the similar way By means of generalizing the conventionalhomotopymethod Liao constructed the so-called zero-orderdeformation equation as follows

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

ISRNThermodynamics 5

Table 2 Our work is compared with Joneidi et alrsquos [7] work and numerical solution of dimensionless temperature 120579(120585) (nonlinear case)

120585120573 = 04 120595 = 1 120573 = 02 120595 = 05

NS DTM [7](11)

HAM (8)(this work) NS DTM [7]

(11)HAM (8)(this work)

0 071604 071604 071604 090344 090344 090344005 071674 071674 071674 090368 090368 090368010 071883 071883 071883 090440 090440 090440015 072231 072231 072231 090559 090559 090559020 072718 072718 072718 090727 090727 090727025 073346 073346 073346 090942 090942 090942030 074114 074114 074114 091206 091206 091206035 075022 075022 075022 091517 091517 091517040 076072 076072 076072 091877 091877 091877045 077264 077264 077264 092285 092285 092285050 078599 078599 078599 092741 092741 092741055 080077 080077 080077 093246 093246 093246060 081699 081699 081699 093799 093799 093799065 083467 083467 083467 094402 094402 094402070 085381 085381 085381 095053 095053 095053075 087442 087442 087442 095753 095753 095753080 089652 089652 089652 096503 096503 096503085 092011 092011 092009 097302 097302 097302090 094522 094522 094518 098151 098151 098151095 097184 097184 097181 099050 099050 099050100 100000 099999 100000 099999 099999 099999

0 01 02 03 04 05 06 07 08 09 1084

086

088

09

092

094

096

098

1

102

120573 = 03

120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 05

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(a)

0 01 02 03 04 05 06 07 08 09 1055

06

065

07

075

08

085

09

095

1

120573 = 03120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 1

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(b)

Figure 3 Dimensionless temperature 120579 versus dimensionless coordinate 120585 for various values of 120573 when (a) 120595 = 05 ℎ = minus084 and (b)120595 = 1 ℎ = minus084 The key to the graph (solid line) represents our approximate solution presented in this work (8) and (+) representsEquation (11) [7]

of 119906(119905) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and 119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution 119906(119905)

Expanding 120593(119905 119901) in Taylor series with respect to 119901 we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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ISRN Astronomy and Astrophysics

Volume 2013

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Condensed Matter PhysicsAdvances in

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Volume 2013

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Advances in

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GravityJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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AerodynamicsJournal of

Volume 2013

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Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

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High Energy PhysicsAdvances in

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Soft MatterJournal of

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Statistical MechanicsInternational Journal of

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PhotonicsJournal of

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ThermodynamicsJournal of

2 ISRNThermodynamics

where 119879 is the temperature 119896(119879) is the temperature-dependent thermal conductivity of the fin material 119875 isthe fin perimeter and ℏ is the heat transfer coefficient Thethermal conductivity of the finmaterial is assumed as follows

119896 (119879) = 119896119886 [1 + 120582 (119879 minus 119879

119886)] (2)

where 119896119886is the thermal conductivity at the ambient fluid

temperature of the fin and 120582 is the parameter describingthe variation of the thermal conductivity The followingdimensionless parameters are introduced [4]

120579 =119879 minus 119879119886

119879119887minus 119879119886

120585 =119909

119887 120573 = 120582 (119879

119887minus 119879119886)

120595 = (ℏ1198751198872

119896119886119860119862

)

12

(3)

Using the previous dimensionless variables the dimension-less form of (1) becomes as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (4)

where 120579 is the dimensionless temperature 120585 is the dimension-less coordinate 120573 is the dimensionless parameter describingthermal conductivity and 120595 is the thermogeometric finparameter The boundary conditions are given as follows

119889120579

119889120585= 0 when 120585 = 0

120579 = 1 when 120585 = 1

(5)

3 Fin Efficiency

The heat transfer rate from the fin is found by using Newtonrsquoslaw of cooling

Consider

119876 = int

119887

0

119875 (119879 minus 119879119886) 119889119909(6)

The ratio of the fin heat transfer rate to the heat transferrate of the fin if the entire fin was at the base temperature iscommonly called as the fin efficiency

120578 =119876

119876ideal=

int

119887

0

119875 (119879 minus 119879119886) 119889119909

119875119887 (119879119887minus 119879119886)

= int

1

120585=0

120579 (120585) 119889120585(7)

4 Approximation of Analytical Solution

41 Homotopy Analysis Method (HAM) Thehomotopy anal-ysis method employs the concept of the homotopy fromtopology to flexibly generate a convergent series solutionfor nonlinear systems The HAM [8ndash13] was devised byShijun Liao which is a powerful analytical method for solving

nonlinear problems The greater generality of this methodoften allows for strong convergence of the solution overlarger spacial and parameter domains This method providesan analytical solution in terms of an infinite power seriesHowever there is a practical need to evaluate this solutionand to obtain numerical values from the infinite power seriesIn order to investigate the accuracy of the homotopy analysismethod (HAM) solution with a finite number of terms thesystem of differential equations was solved

The homotopy analysis method [14] is a good techniquecomparing to another perturbation method The HAM givesexcellent flexibility in the expression of the solution and howthe solution is explicitly obtained Different from all reportedperturbation and nonperturbative techniques the homotopyanalysis method itself provides us with a convenient wayto control and adjust the convergence region and rate ofapproximation series when necessary Briefly speaking thehomotopy analysis method has the following advantages itis valid even if a given nonlinear problem does not containany smalllarge parameters at all it can be employed toefficiently approximate a nonlinear problem by choosingdifferent sets of base functions In this paper we employHAM(see Appendix A) to solve the nonlinear equation

42 Solution of Dimensionless Temperature Using HAM wecan obtain the dimensionless temperature (see Appendix B)as follows

120579 (120585) = (1

cosh (120595)minus

ℎ (ℎ + 2) 120573 cosh (2120595)

3cosh3 (120595)

+2ℎ21205732cosh2 (2120595)

9cosh5 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 2) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(8)

When 120573 rarr 0 from (4) the temperature is as follows

120579 (120585) =cosh (120595120585)

cosh (120595) (9)

This is the exact solution of (4) when 120573 = 0

ISRNThermodynamics 3

43 Determination of Fin Efficiency Using (7) and (8) we canobtain the fin efficiency given as follows

120578 = (1

cosh (120595)+

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)

minusℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595))sinh (120595)

120595

+ℎ (ℎ + 7) 120573 sinh (2120595)

6120595cosh2 (120595)+

ℎ21205732 sinh (3120595)

16120595cosh3 (120595)

+ℎ21205732(120595 cosh (120595) minus sinh (120595))

12120595cosh3 (120595)

minusℎ120573 cosh (2120595) sinh (120595)

3120595cosh3 (120595)

minusℎ21205732 cosh (2120595) sinh (2120595)

9120595cosh4 (120595)

(10)

5 Comparison with Previous Work (Joneidiet al [7])

Joneidi and coworker [7] obtained the closed form of thesolution of (1) using DTM (Tables 1 and 2) as follows

120579 (120585) = 119886 +1205952119886

2 (1 + 120573119886)1205852minus1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)31205854

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)51205856

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

1205858+ sdot sdot sdot

(11)

The value of 119886 can be obtained from the following equation

120579 (1) = 119886 +1205952119886

2 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

+ sdot sdot sdot = 1

(12)

b

x dx Tb

h Ta

Figure 1 The sketch of a straight fin

The solution of fin efficiency is

120578 = 119886 +1205952119886

6 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

120(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

5040(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

362880(1 + 120573119886)7

+ sdot sdot sdot

(13)

6 Discussion

Equation (8) (this work) and (Equation (11)) (previous work)represent the analytical expressions of temperature for allvalues of parameters Among these two expressions our resultis the simplest one

Figure 2 represents dimensionless temperature distribu-tion 120579 versus the dimensionless coordinate 120585 for variousvalues of 120595 using (8) From this figure it is clear that when 120595

increases (ie the fin length 119887 increases or the cross-sectionalarea of the fin 119860

119862decreases) the dimensionless temperature

120579 decreasesFigure 3 illustrates the temperature distribution of the

fin for different values of 120595 and 120573 Figure 3(a) depicts theeffect of the thermal conductivity 120573 on the temperature 120579According to (8) the temperature increases when the thermalconductivity of the fin 120573 increases for all values of 120595 Asan observation in (10) it can be concluded that the finefficiency also increases It is noted that the temperature onfin 120579 increases when 120573 increases (see Figure 3(b)) for the fixedvalue of 120595 = 1

Figure 4 shows the fin efficiency 120578 as a function ofthermogeometric fin parameter 120595 for different values ofthe thermal conductivity parameter 120573 using (10) From thisfigure it can be seen that the fin efficiency 120578 decreases as120595 increases However the fin efficiency 120578 increases as thethermal conductivity 120573 increases

4 ISRNThermodynamics

Table 1 Our work is compared with Joneidi et alrsquos [7] work and exact solution of dimensionless temperature 120579(120585) (linear case)

120585(a) 120573 = 0 120595 = 05 (b) 120573 = 0 120595 = 1

HAM (8)(this work)

DTM [7](11) Exact HAM (8)

(this work)DTM [7]

(11) Exact

0 088681 088681 088681 064805 064805 064805005 088709 088709 088709 064886 064886 064886010 088792 088792 088792 065129 065129 065129015 088931 088931 088931 065535 065535 065535020 089125 089125 089125 066105 066105 066105025 089375 089375 089375 066841 066841 066841030 089681 089681 089681 067743 067743 067743035 090043 090043 090043 068815 068815 068815040 090461 090461 090461 070059 070059 070059045 090936 090936 090936 071478 071478 071478050 091467 091467 091467 073076 073076 073076055 092056 092056 092056 074856 074856 074856060 092702 092702 092702 076824 076824 076824065 093406 093406 093406 078984 078984 078984070 094169 094169 094169 081341 081341 081341075 094990 094990 094990 083902 083902 083902080 095871 095871 095871 086673 086673 086673085 096812 096812 096812 089660 089660 089660090 097813 097813 097813 092871 092871 092871095 098875 098875 098875 096315 096315 096315100 100000 100000 099999 100000 100000 100000

0 01 02 03 04 05 06 07 08 09 1082

084

086

088

09

092

094

096

098

1

102

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

120595 = 05

120595 = 055

120595 = 06

120595 = 065

120573 = 0

Figure 2 Dimensionless temperature 120579 versus dimensionless coor-dinate 120585 for various values of 120595 when 120573 = 0 and ℎ = minus084 Thekey to the graph (solid line) represents our approximate solutionpresented in this work (8) and (+) represents Joneidi et alrsquos [7] work(Equation (11))

7 Conclusions

There are two main goals that we aimed for this work Thefirst is to employ the powerful homotopy analysis method

to investigate nonlinear differential equation arising in con-vective straight fins with temperature-dependent thermalconductivity problem The second is to achieve good resultsin predicting the solution of the heat transfer equationsin engineering The two goals are achieved successfully InHAM we can choose ℎ in an appropriate way which controlsthe convergence of the series

Appendices

A Basic Idea of Liaorsquos [15] HomotopyAnalysis Method

Consider the following differential equation [15]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator 119905 denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin the similar way By means of generalizing the conventionalhomotopymethod Liao constructed the so-called zero-orderdeformation equation as follows

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

ISRNThermodynamics 5

Table 2 Our work is compared with Joneidi et alrsquos [7] work and numerical solution of dimensionless temperature 120579(120585) (nonlinear case)

120585120573 = 04 120595 = 1 120573 = 02 120595 = 05

NS DTM [7](11)

HAM (8)(this work) NS DTM [7]

(11)HAM (8)(this work)

0 071604 071604 071604 090344 090344 090344005 071674 071674 071674 090368 090368 090368010 071883 071883 071883 090440 090440 090440015 072231 072231 072231 090559 090559 090559020 072718 072718 072718 090727 090727 090727025 073346 073346 073346 090942 090942 090942030 074114 074114 074114 091206 091206 091206035 075022 075022 075022 091517 091517 091517040 076072 076072 076072 091877 091877 091877045 077264 077264 077264 092285 092285 092285050 078599 078599 078599 092741 092741 092741055 080077 080077 080077 093246 093246 093246060 081699 081699 081699 093799 093799 093799065 083467 083467 083467 094402 094402 094402070 085381 085381 085381 095053 095053 095053075 087442 087442 087442 095753 095753 095753080 089652 089652 089652 096503 096503 096503085 092011 092011 092009 097302 097302 097302090 094522 094522 094518 098151 098151 098151095 097184 097184 097181 099050 099050 099050100 100000 099999 100000 099999 099999 099999

0 01 02 03 04 05 06 07 08 09 1084

086

088

09

092

094

096

098

1

102

120573 = 03

120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 05

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(a)

0 01 02 03 04 05 06 07 08 09 1055

06

065

07

075

08

085

09

095

1

120573 = 03120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 1

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(b)

Figure 3 Dimensionless temperature 120579 versus dimensionless coordinate 120585 for various values of 120573 when (a) 120595 = 05 ℎ = minus084 and (b)120595 = 1 ℎ = minus084 The key to the graph (solid line) represents our approximate solution presented in this work (8) and (+) representsEquation (11) [7]

of 119906(119905) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and 119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution 119906(119905)

Expanding 120593(119905 119901) in Taylor series with respect to 119901 we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

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Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

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Volume 2013

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Advances in

Astronomy

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GravityJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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AerodynamicsJournal of

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Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

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ThermodynamicsJournal of

ISRNThermodynamics 3

43 Determination of Fin Efficiency Using (7) and (8) we canobtain the fin efficiency given as follows

120578 = (1

cosh (120595)+

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)

minusℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595))sinh (120595)

120595

+ℎ (ℎ + 7) 120573 sinh (2120595)

6120595cosh2 (120595)+

ℎ21205732 sinh (3120595)

16120595cosh3 (120595)

+ℎ21205732(120595 cosh (120595) minus sinh (120595))

12120595cosh3 (120595)

minusℎ120573 cosh (2120595) sinh (120595)

3120595cosh3 (120595)

minusℎ21205732 cosh (2120595) sinh (2120595)

9120595cosh4 (120595)

(10)

5 Comparison with Previous Work (Joneidiet al [7])

Joneidi and coworker [7] obtained the closed form of thesolution of (1) using DTM (Tables 1 and 2) as follows

120579 (120585) = 119886 +1205952119886

2 (1 + 120573119886)1205852minus1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)31205854

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)51205856

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

1205858+ sdot sdot sdot

(11)

The value of 119886 can be obtained from the following equation

120579 (1) = 119886 +1205952119886

2 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

24(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

720(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

40320(1 + 120573119886)7

+ sdot sdot sdot = 1

(12)

b

x dx Tb

h Ta

Figure 1 The sketch of a straight fin

The solution of fin efficiency is

120578 = 119886 +1205952119886

6 (1 + 120573119886)minus

1205954119886 (2120573119886 minus 1)

120(1 + 120573119886)3

+1205956119886 (2120573119886 minus 1) (14120573119886 minus 1)

5040(1 + 120573119886)5

minus

1205958119886 (2120573119886 minus 1) (25592120573

31198863minus 7152120573

21198862+ 330120573119886 minus 1)

362880(1 + 120573119886)7

+ sdot sdot sdot

(13)

6 Discussion

Equation (8) (this work) and (Equation (11)) (previous work)represent the analytical expressions of temperature for allvalues of parameters Among these two expressions our resultis the simplest one

Figure 2 represents dimensionless temperature distribu-tion 120579 versus the dimensionless coordinate 120585 for variousvalues of 120595 using (8) From this figure it is clear that when 120595

increases (ie the fin length 119887 increases or the cross-sectionalarea of the fin 119860

119862decreases) the dimensionless temperature

120579 decreasesFigure 3 illustrates the temperature distribution of the

fin for different values of 120595 and 120573 Figure 3(a) depicts theeffect of the thermal conductivity 120573 on the temperature 120579According to (8) the temperature increases when the thermalconductivity of the fin 120573 increases for all values of 120595 Asan observation in (10) it can be concluded that the finefficiency also increases It is noted that the temperature onfin 120579 increases when 120573 increases (see Figure 3(b)) for the fixedvalue of 120595 = 1

Figure 4 shows the fin efficiency 120578 as a function ofthermogeometric fin parameter 120595 for different values ofthe thermal conductivity parameter 120573 using (10) From thisfigure it can be seen that the fin efficiency 120578 decreases as120595 increases However the fin efficiency 120578 increases as thethermal conductivity 120573 increases

4 ISRNThermodynamics

Table 1 Our work is compared with Joneidi et alrsquos [7] work and exact solution of dimensionless temperature 120579(120585) (linear case)

120585(a) 120573 = 0 120595 = 05 (b) 120573 = 0 120595 = 1

HAM (8)(this work)

DTM [7](11) Exact HAM (8)

(this work)DTM [7]

(11) Exact

0 088681 088681 088681 064805 064805 064805005 088709 088709 088709 064886 064886 064886010 088792 088792 088792 065129 065129 065129015 088931 088931 088931 065535 065535 065535020 089125 089125 089125 066105 066105 066105025 089375 089375 089375 066841 066841 066841030 089681 089681 089681 067743 067743 067743035 090043 090043 090043 068815 068815 068815040 090461 090461 090461 070059 070059 070059045 090936 090936 090936 071478 071478 071478050 091467 091467 091467 073076 073076 073076055 092056 092056 092056 074856 074856 074856060 092702 092702 092702 076824 076824 076824065 093406 093406 093406 078984 078984 078984070 094169 094169 094169 081341 081341 081341075 094990 094990 094990 083902 083902 083902080 095871 095871 095871 086673 086673 086673085 096812 096812 096812 089660 089660 089660090 097813 097813 097813 092871 092871 092871095 098875 098875 098875 096315 096315 096315100 100000 100000 099999 100000 100000 100000

0 01 02 03 04 05 06 07 08 09 1082

084

086

088

09

092

094

096

098

1

102

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

120595 = 05

120595 = 055

120595 = 06

120595 = 065

120573 = 0

Figure 2 Dimensionless temperature 120579 versus dimensionless coor-dinate 120585 for various values of 120595 when 120573 = 0 and ℎ = minus084 Thekey to the graph (solid line) represents our approximate solutionpresented in this work (8) and (+) represents Joneidi et alrsquos [7] work(Equation (11))

7 Conclusions

There are two main goals that we aimed for this work Thefirst is to employ the powerful homotopy analysis method

to investigate nonlinear differential equation arising in con-vective straight fins with temperature-dependent thermalconductivity problem The second is to achieve good resultsin predicting the solution of the heat transfer equationsin engineering The two goals are achieved successfully InHAM we can choose ℎ in an appropriate way which controlsthe convergence of the series

Appendices

A Basic Idea of Liaorsquos [15] HomotopyAnalysis Method

Consider the following differential equation [15]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator 119905 denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin the similar way By means of generalizing the conventionalhomotopymethod Liao constructed the so-called zero-orderdeformation equation as follows

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

ISRNThermodynamics 5

Table 2 Our work is compared with Joneidi et alrsquos [7] work and numerical solution of dimensionless temperature 120579(120585) (nonlinear case)

120585120573 = 04 120595 = 1 120573 = 02 120595 = 05

NS DTM [7](11)

HAM (8)(this work) NS DTM [7]

(11)HAM (8)(this work)

0 071604 071604 071604 090344 090344 090344005 071674 071674 071674 090368 090368 090368010 071883 071883 071883 090440 090440 090440015 072231 072231 072231 090559 090559 090559020 072718 072718 072718 090727 090727 090727025 073346 073346 073346 090942 090942 090942030 074114 074114 074114 091206 091206 091206035 075022 075022 075022 091517 091517 091517040 076072 076072 076072 091877 091877 091877045 077264 077264 077264 092285 092285 092285050 078599 078599 078599 092741 092741 092741055 080077 080077 080077 093246 093246 093246060 081699 081699 081699 093799 093799 093799065 083467 083467 083467 094402 094402 094402070 085381 085381 085381 095053 095053 095053075 087442 087442 087442 095753 095753 095753080 089652 089652 089652 096503 096503 096503085 092011 092011 092009 097302 097302 097302090 094522 094522 094518 098151 098151 098151095 097184 097184 097181 099050 099050 099050100 100000 099999 100000 099999 099999 099999

0 01 02 03 04 05 06 07 08 09 1084

086

088

09

092

094

096

098

1

102

120573 = 03

120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 05

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(a)

0 01 02 03 04 05 06 07 08 09 1055

06

065

07

075

08

085

09

095

1

120573 = 03120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 1

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(b)

Figure 3 Dimensionless temperature 120579 versus dimensionless coordinate 120585 for various values of 120573 when (a) 120595 = 05 ℎ = minus084 and (b)120595 = 1 ℎ = minus084 The key to the graph (solid line) represents our approximate solution presented in this work (8) and (+) representsEquation (11) [7]

of 119906(119905) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and 119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution 119906(119905)

Expanding 120593(119905 119901) in Taylor series with respect to 119901 we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

4 ISRNThermodynamics

Table 1 Our work is compared with Joneidi et alrsquos [7] work and exact solution of dimensionless temperature 120579(120585) (linear case)

120585(a) 120573 = 0 120595 = 05 (b) 120573 = 0 120595 = 1

HAM (8)(this work)

DTM [7](11) Exact HAM (8)

(this work)DTM [7]

(11) Exact

0 088681 088681 088681 064805 064805 064805005 088709 088709 088709 064886 064886 064886010 088792 088792 088792 065129 065129 065129015 088931 088931 088931 065535 065535 065535020 089125 089125 089125 066105 066105 066105025 089375 089375 089375 066841 066841 066841030 089681 089681 089681 067743 067743 067743035 090043 090043 090043 068815 068815 068815040 090461 090461 090461 070059 070059 070059045 090936 090936 090936 071478 071478 071478050 091467 091467 091467 073076 073076 073076055 092056 092056 092056 074856 074856 074856060 092702 092702 092702 076824 076824 076824065 093406 093406 093406 078984 078984 078984070 094169 094169 094169 081341 081341 081341075 094990 094990 094990 083902 083902 083902080 095871 095871 095871 086673 086673 086673085 096812 096812 096812 089660 089660 089660090 097813 097813 097813 092871 092871 092871095 098875 098875 098875 096315 096315 096315100 100000 100000 099999 100000 100000 100000

0 01 02 03 04 05 06 07 08 09 1082

084

086

088

09

092

094

096

098

1

102

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

120595 = 05

120595 = 055

120595 = 06

120595 = 065

120573 = 0

Figure 2 Dimensionless temperature 120579 versus dimensionless coor-dinate 120585 for various values of 120595 when 120573 = 0 and ℎ = minus084 Thekey to the graph (solid line) represents our approximate solutionpresented in this work (8) and (+) represents Joneidi et alrsquos [7] work(Equation (11))

7 Conclusions

There are two main goals that we aimed for this work Thefirst is to employ the powerful homotopy analysis method

to investigate nonlinear differential equation arising in con-vective straight fins with temperature-dependent thermalconductivity problem The second is to achieve good resultsin predicting the solution of the heat transfer equationsin engineering The two goals are achieved successfully InHAM we can choose ℎ in an appropriate way which controlsthe convergence of the series

Appendices

A Basic Idea of Liaorsquos [15] HomotopyAnalysis Method

Consider the following differential equation [15]

119873[119906 (119905)] = 0 (A1)

where 119873 is a nonlinear operator 119905 denotes an independentvariable and 119906(119905) is an unknown function For simplicity weignore all boundary or initial conditions which can be treatedin the similar way By means of generalizing the conventionalhomotopymethod Liao constructed the so-called zero-orderdeformation equation as follows

(1 minus 119901) 119871 [120593 (119905 119901) minus 1199060 (119905)] = 119901ℎ119867 (119905)119873 [120593 (119905 119901)] (A2)

where 119901 isin [0 1] is the embedding parameter ℎ = 0 is anonzero auxiliary parameter 119867(119905) = 0 is an auxiliary func-tion 119871 is an auxiliary linear operator 119906

0(119905) is an initial guess

ISRNThermodynamics 5

Table 2 Our work is compared with Joneidi et alrsquos [7] work and numerical solution of dimensionless temperature 120579(120585) (nonlinear case)

120585120573 = 04 120595 = 1 120573 = 02 120595 = 05

NS DTM [7](11)

HAM (8)(this work) NS DTM [7]

(11)HAM (8)(this work)

0 071604 071604 071604 090344 090344 090344005 071674 071674 071674 090368 090368 090368010 071883 071883 071883 090440 090440 090440015 072231 072231 072231 090559 090559 090559020 072718 072718 072718 090727 090727 090727025 073346 073346 073346 090942 090942 090942030 074114 074114 074114 091206 091206 091206035 075022 075022 075022 091517 091517 091517040 076072 076072 076072 091877 091877 091877045 077264 077264 077264 092285 092285 092285050 078599 078599 078599 092741 092741 092741055 080077 080077 080077 093246 093246 093246060 081699 081699 081699 093799 093799 093799065 083467 083467 083467 094402 094402 094402070 085381 085381 085381 095053 095053 095053075 087442 087442 087442 095753 095753 095753080 089652 089652 089652 096503 096503 096503085 092011 092011 092009 097302 097302 097302090 094522 094522 094518 098151 098151 098151095 097184 097184 097181 099050 099050 099050100 100000 099999 100000 099999 099999 099999

0 01 02 03 04 05 06 07 08 09 1084

086

088

09

092

094

096

098

1

102

120573 = 03

120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 05

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(a)

0 01 02 03 04 05 06 07 08 09 1055

06

065

07

075

08

085

09

095

1

120573 = 03120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 1

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(b)

Figure 3 Dimensionless temperature 120579 versus dimensionless coordinate 120585 for various values of 120573 when (a) 120595 = 05 ℎ = minus084 and (b)120595 = 1 ℎ = minus084 The key to the graph (solid line) represents our approximate solution presented in this work (8) and (+) representsEquation (11) [7]

of 119906(119905) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and 119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution 119906(119905)

Expanding 120593(119905 119901) in Taylor series with respect to 119901 we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

ISRNThermodynamics 5

Table 2 Our work is compared with Joneidi et alrsquos [7] work and numerical solution of dimensionless temperature 120579(120585) (nonlinear case)

120585120573 = 04 120595 = 1 120573 = 02 120595 = 05

NS DTM [7](11)

HAM (8)(this work) NS DTM [7]

(11)HAM (8)(this work)

0 071604 071604 071604 090344 090344 090344005 071674 071674 071674 090368 090368 090368010 071883 071883 071883 090440 090440 090440015 072231 072231 072231 090559 090559 090559020 072718 072718 072718 090727 090727 090727025 073346 073346 073346 090942 090942 090942030 074114 074114 074114 091206 091206 091206035 075022 075022 075022 091517 091517 091517040 076072 076072 076072 091877 091877 091877045 077264 077264 077264 092285 092285 092285050 078599 078599 078599 092741 092741 092741055 080077 080077 080077 093246 093246 093246060 081699 081699 081699 093799 093799 093799065 083467 083467 083467 094402 094402 094402070 085381 085381 085381 095053 095053 095053075 087442 087442 087442 095753 095753 095753080 089652 089652 089652 096503 096503 096503085 092011 092011 092009 097302 097302 097302090 094522 094522 094518 098151 098151 098151095 097184 097184 097181 099050 099050 099050100 100000 099999 100000 099999 099999 099999

0 01 02 03 04 05 06 07 08 09 1084

086

088

09

092

094

096

098

1

102

120573 = 03

120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 05

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(a)

0 01 02 03 04 05 06 07 08 09 1055

06

065

07

075

08

085

09

095

1

120573 = 03120573 = 01

120573 = minus01

120573 = minus03

120573 = minus02

120595 = 1

Dimensionless coordinate 120585

Dim

ensio

nles

s tem

pera

ture

120579

(b)

Figure 3 Dimensionless temperature 120579 versus dimensionless coordinate 120585 for various values of 120573 when (a) 120595 = 05 ℎ = minus084 and (b)120595 = 1 ℎ = minus084 The key to the graph (solid line) represents our approximate solution presented in this work (8) and (+) representsEquation (11) [7]

of 119906(119905) and 120593(119905 119901) is an unknown function It is importantthat one has great freedom to choose auxiliary unknowns inHAM Obviously when 119901 = 0 and 119901 = 1 it holds

120593 (119905 0) = 1199060 (119905) 120593 (119905 1) = 119906 (119905) (A3)

respectively Thus as 119901 increases from 0 to 1 the solution120593(119905 119901) varies from the initial guess 119906

0(119905) to the solution 119906(119905)

Expanding 120593(119905 119901) in Taylor series with respect to 119901 we have

120593 (119905 119901) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) 119901

119898 (A4)

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

6 ISRNThermodynamics

0 05 1 15 2 25 3 35 402

03

04

05

06

07

08

09

1

120573 = 04120573 = 06

120573 = minus02120573 = 0

120573 = minus04 120573 = 02

120595

120578

Figure 4 Influence of thermal conductivity parameter 120573 on the finefficiency 120578 with the thermogeometric fin parameter 120595

where

119906119898 (119905) =1

119898

120597119898120593 (119905 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(A5)

If the auxiliary linear operator the initial guess the auxiliaryparameter ℎ and the auxiliary function are so properlychosen the series equation (A4) converges at 119901 = 1 thenwe have

119906 (119905) = 1199060 (119905) +

+infin

sum

119898=1

119906119898 (119905) (A6)

Define the vector

119906119899= 1199060 1199061 119906

119899 (A7)

Differentiating (A2) for 119898 times with respect to the embed-ding parameter 119901 then setting 119901 = 0 and finally dividingthembym wewill have the so-called119898th-order deformationequation as follows

119871 [119906119898minus 120594119898119906119898minus1

] = ℎ119867 (119905)R119898 (119898minus1) (A8)

where

R119898(119898minus1

) =1

(119898 minus 1)

120597119898minus1

119873[120593 (119905 119901)]

120597119901119898minus1

100381610038161003816100381610038161003816100381610038161003816119901=0

120594119898

= 0 119898 le 1

1 119898 gt 1

(A9)

Applying 119871minus1 on both sides of (A8) we get

119906119898 (119905) = 120594

119898119906119898minus1 (119905) + ℎ119871

minus1[119867 (119905)R119898 (119898minus1)] (A10)

In this way it is easily to obtain 119906119898for 119898 ge 1 at 119872th order

we have

119906 (119905) =

119872

sum

119898=0

119906119898 (119905) (A11)

When 119872 rarr +infin we get an accurate approximation ofthe original (A1) For the convergence of the above methodwe refer the reader to Liao [15] If (A1) admits uniquesolution then this method will produce the unique solutionIf (A1) does not possess unique solution the HAM will givea solution among many other (possible) solutions

B Solution of (4) Using HomotopyAnalysis Method

In this appendix we indicate how (8) in this paper is derivedThehomotopy analysismethodwas constructed to determinethe solution of (4)-(5) as follows

1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579 = 0 (B1)

In order to solve (B1) bymeans of theHAMwefirst constructthe zeroth-order deformation equation by taking119867(119905) = 1

Consider

(1 minus 119901) [1198892120579

1198891205852minus 1205952120579]

= 119901ℎ[1198892120579

1198891205852+ 120573120579

1198892120579

1198891205852+ 120573(

119889120579

119889120585)

2

minus 1205952120579]

(B2)

The approximate solution of (B2) is as follows

120579 = 1205790+ 1199011205791+ 1199011205792+ sdot sdot sdot (B3)

Substituting (B3) into (B1) and comparing the coefficients oflike powers of p we get

119901011988921205790

1198891205852minus 12059521205790= 0 (B4)

119901111988921205791

1198891205852minus 12059521205791

= (ℎ + 1) [11988921205790

1198891205852minus 12059521205790]

+ ℎ120573[1205790

11988921205790

1198891205852+ (

11988921205790

1198891205852)

2

]

(B5)

119901211988921205792

1198891205852minus 12059521205792

= (ℎ + 1) [11988921205791

1198891205852minus 12059521205791]

+ ℎ120573[1205790

11988921205791

1198891205852+ 1205791

11988921205790

1198891205852+ 2

1198891205790

1198891205852

1198891205791

1198891205852]

(B6)

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

ISRNThermodynamics 7

09982

09982

09982

09982

09982

09982

09982

09982

09982

09982

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579(0

75)

(a)

623

6235

624

6245

625

6255

626

6265

627

6275

628

minus1 minus095 minus09 minus085 minus08 minus075 minus07

h

120573 = 02 120595 = 01

120579998400 (0

75)

times10minus3

(b)

Figure 5 The ℎ curves to indicate the convergence region for 120573 = 02 and 120595 = 01

The boundary conditions are

1198891205790

119889120585= 0 when 120585 = 0 120579

0= 1 when 120585 = 1 (B7)

119889120579119894

119889120585= 0 when 120585 = 0 120579119894 = 0 when 120585 = 1

forall119894 = 1 2 3

(B8)

Now applying the boundary conditions (B7) in (B4) we get

1205790 (120585) =

cosh (120595120585)

cosh (120595) (B9)

Substituting the value of 1205790in (B5) and solving the equation

using the boundary conditions (B8) we obtain the followingresults

1205791 (120585) = minus

ℎ120573 cosh (2120595)

3cosh3 (120595)cosh (120595120585) +

ℎ120573 cosh (2120595120585)

3cosh2 (120595) (B10)

Upon solving (B6) by substituting the values of 1205790and 1205791and

using the boundary conditions equation (B8) we can find thefollowing results

1205792 (120585) = (

2ℎ21205732cosh2 (2120595)

9cosh5 (120595)minus

ℎ (ℎ + 1) 120573 cosh (2120595)

3cosh3 (120595)

minus3ℎ21205732 cosh (3120595)

16cosh4 (120595)minus

ℎ21205732120595 sinh (120595)

12cosh4 (120595)) cosh (120595120585)

+ (ℎ (ℎ + 1) 120573

3cosh2 (120595)minus

2ℎ21205732 cosh (2120595)

9cosh4 (120595)) cosh (2120595120585)

+3ℎ21205732 cosh (3120595120585)

16cosh3 (120595)+

ℎ21205732120595120585 sinh (120595120585)

12cosh3 (120595)

(B11)

After three successive iterations the solutions of 120579 reach thebetter approximation Adding (B9) to (B11) we get (8) in thetext

C Determining the Region of ℎ for Validity

The analytical solution should converge It should be notedthat the auxiliary parameter ℎ controls the convergenceand accuracy of the solution series The analytical solutionrepresented by (8) contains the auxiliary parameter ℎ whichgives the convergence region and rate of approximation forthe homotopy analysis method In order to define the regionsuch that the solution series is independent of ℎ a multipleof ℎ curves are plotted The region where the dimensionlesstemperature 120579(120585) and 120579

1015840(120585) versus ℎ is a horizontal line

known as the convergence region for the correspondingfunctionThe common region among 120579(120585) and its derivativesare known as the overall convergence region To study theinfluence of ℎ on the convergence of solution ℎ-curves of120579(075) and 120579

1015840(075) are plotted in Figures 5(a) and 5(b)

respectively for 120573 = 02 120595 = 01 These figures clearly in-dicate that the valid region of ℎ is about minus086 to minus082Similarly we can find the value of the convergence controlparameter ℎ for different values of the constant parameters

Nomenclature

119860119862 Cross-sectional area of the fin (m2)

119887 Fin length (m)ℏ Heat transfer coefficient (W mminus1 Kminus1)119896 Thermal conductivity of the fin material (W

mminus1 Kminus1)119896119886 Thermal conductivity at the ambient fluid

temperature (W mminus1 Kminus1)119896119887 Thermal conductivity at the base

temperature (W mminus1 Kminus1)

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

8 ISRNThermodynamics

119875 Fin perimeter (m)119876 Heat-transfer rate (W)119879119886 Temperature of surface 119886 (K)

119879119887 Temperature of surface 119887 (K)

119909 Distance measured from the fin tip (m)

Greek Letters

120582 The slope of the thermalconductivity-temperature curve

120573 Dimensionless parameter describingvariation of the thermal conductivity

120578 Fin efficiency120585 Dimensionless coordinate120595 Thermogeometric fin parameter120579 Dimensionless temperature

Acknowledgments

This work is supported by the University Grant Commis-sion (UGC) Minor project no F MRP-412212 (MRPUGC-SERO) Hyderabad Government of India The authors arethankful to Shri S Natanagopal Secretary at the MaduraCollege Board and Dr R Murali Principal at the MaduraCollege (Autonomous)Madurai Tamil Nadu India for theirconstant encouragement

References

[1] A Kraus A Aziz and J Welty Extended Surface Heat TransferJohn Wiley amp Sons New York NY USA 2001

[2] C Arslanturk ldquoA decomposition method for fin efficiency ofconvective straight fins with temperature-dependent thermalconductivityrdquo International Communications in Heat and MassTransfer vol 32 no 6 pp 831ndash841 2005

[3] F P Incropera and D P Dewitt Introduction to Heat TransferJohn Wiley amp Sons New York NY USA 3rd edition 1996

[4] E M A Mokheimer ldquoPerformance of annular fins withdifferent profiles subject to variable heat transfer coefficientrdquoInternational Journal of Heat and Mass Transfer vol 45 no 17pp 3631ndash3642 2002

[5] S M Zubair A Z Al-Garni and J S Nizami ldquoThe opti-mal dimensions of circular fins with variable profile andtemperature-dependent thermal conductivityrdquo InternationalJournal of Heat andMass Transfer vol 39 no 16 pp 3431ndash34391996

[6] H-S Kou J-J Lee and C-Y Lai ldquoThermal analysis of alongitudinal fin with variable thermal properties by recursiveformulationrdquo International Journal of Heat and Mass Transfervol 48 no 11 pp 2266ndash2277 2005

[7] AA Joneidi DDGanji andMBabaelahi ldquoDifferential Trans-formation Method to determine fin efficiency of convectivestraight fins with temperature dependent thermal conductivityrdquoInternational Communications in Heat and Mass Transfer vol36 no 7 pp 757ndash762 2009

[8] S J Liao The proposed homotopy analysis technique for thesolution of non linear problems [PhD thesis] Shanghai JiaoTongUniversity 1992

[9] S-J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[10] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[11] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010

[12] S J Liao The Homotopy Analysis Method in Non LinearDifferential Equations Springer New York NY USA 2012

[13] G Domairry and H Bararnia ldquoAn approximation of theanalytic solution of some nonlinear heat transfer equations asurvey by using homotopy analysis methodrdquo Advanced Studiesin Theoretical Physics vol 2 no 11 pp 507ndash518 2008

[14] H Jafari C Chun and S M Saeidy ldquoAnalytical solution fornonlinear gas dynamic Homotopy analysis methodrdquo AppliedMathematics vol 4 pp 149ndash154 2009

[15] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA1st edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of