Series Solution for Steady Heat Transfer in a Heat-Generating Fin with Convection and Radiation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 806873 7 pageshttpdxdoiorg1011552013806873

Research ArticleSeries Solution for Steady Heat Transfer in a Heat-GeneratingFin with Convection and Radiation

Fazle Mabood1 Waqar A Khan2 and Ahmad Izani Md Ismail1

1 School of Mathematical Sciences Universiti Sains Malaysia 11800 Penang Malaysia2 Department of Engineering Sciences NationalUniversity of Sciences andTechnology PNEngineeringCollege Karachi 75350 Pakistan

Correspondence should be addressed to Fazle Mabood mabood1971yahoocom

Received 27 May 2013 Accepted 7 August 2013

Academic Editor Farzad Khani

Copyright copy 2013 Fazle Mabood et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The steady heat transfer in a heat-generating fin with simultaneous surface convection and radiation is studied analytically usingoptimal homotopy asymptotic method (OHAM)The steady response of the fin depends on the convection-conduction parameterradiation-conduction parameter heat generation parameter and dimensionless sink temperature The heat transfer problem ismodeled using two-point boundary value conditions The results of the dimensionless temperature profile for different values ofconvection-conduction radiation-conduction heat generation and sink temperature parameters are presented graphically and intabular form Comparison of the solution using OHAMwith homotopy analysis method (HAM) and Runge-Kutta-Fehlberg fourth-fifth-order numerical method for various values of controlling parameters is presented The comparison shows that the OHAMresults are in excellent agreement with NM

1 Introduction

Fins (extended surfaces) are widely used to enhance theheat transfer rate between a hot surface and its surroundingfluid Fin applications have included the cooling of computerprocessors air conditioning units refrigerators air-cooledengines and oil carrying pipelines In the past three decadesfins have gained vast recognition for cooling electronic toolsas heat sinks The subject of extended surface heat transferis now a fully developed technology but with continuingcontributions fromnumerous researchers Background infor-mation on heat transfer in extended surfaces may be foundin the books [1 2] where the authors have presented wide-ranging coverage of the various facts of this technology

Numerous mathematical models related to heat transferin fins of various shapes with different boundary conditionsare well documented in the research literature For instancethe mathematical analysis of convective fins was first pro-vided by Gardener [3] based on the assumption of constantconductivity and a uniform coefficient of convective heattransfer along the fin surface Khani et al [4] presented someexact solutions for 1D fin problem with uniform thermalconductivity and heat transfer coefficient Khani et al [5] also

provided a series solution for 1D fin problem with constantheat transfer coefficient and temperature dependent thermalconductivity

A variety of approximate analytical methods have beenused to study the transient response of fins Aziz and Na[6] presented a coordinate perturbation expansion for theresponse of an infinitely long fin due to a step changein the base temperature Chang et al [7] used the meth-ods of optimal linearization and variational embeddingand Campo [8] utilized variational techniques to analyzeradiative-convective fins under unsteady operating condi-tions Solutions for transient heat transfer were constructedfor fins by Onur [9] Aziz and Torabi [10] have presentedthe numerical analysis of transient heat transfer in fin withtemperature dependent heat transfer coefficient

Exact steady-state solutions of 2D models of fin havingconstant thermal conductivity and heat transfer coefficientandwith no internal heat generationwere analyzed in [11ndash17]The addition of internal heat generation function based withspatial dependence is discussed [18ndash20]

In this paper we used a new approximate methodnamely optimal homotopy asymptotic method [21ndash27] forsteady-state heat transfer with internal heat generation fin

2 Mathematical Problems in Engineering

Insulated tip

Cross-sectional

h Tinfin

120576

Tb

x

x = 0

x = b

k 120572 q

Perimeter P

area A

Figure 1 A straight fin with constant cross-sectional area

and investigated numerically the effects of the differentgoverning parameters on dimensionless temperature profilein a nonlinear fin-type problem For comparison purposesthe governing highly nonlinear problem is also solved usingRunge-Kutta-Fehlberg fourth-fifth-order method and homo-topy analysis method (HAM) developed by Liao [28]

The paper is planned as follows in Section 2 we formu-late our nonlinear problem basic principles of OHAM arediscussed in Section 3 solution of the problem via OHAMis presented in Section 4 and Section 5 is reserved for resultsand discussion Conclusions are drawn in Section 6

2 Mathematical Formulation

Consider a straight fin of constant cross-sectional area 119860

(rectangular cylindrical elliptic etc) perimeter of the cross-section 119875 and length 119887 as shown in Figure 1 The fin hasa thermal conductivity 119896 and a thermal diffusivity 120572 Thesurface of the fin behaves as a gray diffuse surface with anemissivity 120576 The fin is deemed to be initially in thermalequilibriumwith the surroundings at temperature119879

119904 Its tip is

insulated A volumetric internal heat generation rate 119902 occursin the fin The fin loses heat by simultaneous convection andradiation to its surroundings at temperature 119879

119904 The same

sink temperature is used for both convection and radiationto avoid the introduction of an additional parameter in theproblem

For one-dimensional steady conduction in the fin theenergy equation may be written as

1205972

119879

1205971199092minus

ℎ119875

119896119860(119879 minus 119879

119904) minus

120576120590119875 (1198794

minus 1198794

119904)

119896119860+

119902

119896= 0 (1)

The initial and boundary conditions are

119879 (119909) = 119879119904

119879 (119887) = 119879119887

119889119879

119889119909(0) = 0

(2)

where 119909 is measured from the tip of the fin with theintroduction of the following definitions

120579 =119879

119879119887

120579119904=

119879119904

119879119887

119883 =119909

119887 ℎ

119887= 119862 (119879

119887minus 119879119904)

119873119888=

ℎ1198871198751198872

119896119860 119873

119903=

1205761205901198751198793

1199041198872

119896119860

119876gen =1199021198872

119896119879119887

(3)

Equations (2) and (3) can be written in dimensionless formas follows

1198892

120579

1198891198832minus

119873119888

(1 minus 120579119904)(120579 minus 120579

119904)2

minus 119873119903(1205794

minus 1205794

119904) + 119876gen = 0 (4)

120579 (119887) = 1 (5)

119889120579

119889119883(0) = 0 (6)

The instantaneous base heat flow is given by

119902119887= 119896119860

119889119879

119889119909(119887) (7)

which may be expressed in dimensionless form as follows

119876119887=

119902119887119887

119896119860119879119887

=119889120579

119889119883(1) (8)

The instantaneous convective heat loss from the fin is givenby

119902119888= 119875int

119887

0

ℎ (119879 minus 119879119904) 119889119909 (9)

or in dimensionless form as

119876119888=

119902119888119887

119896119860119879119887

=119873119888

(1 minus 120579119904)int

1

0

(120579 minus 120579119904)2

119889119883 (10)

Similarly the instantaneous radiative heat loss from the fincan be obtained as

119902119903= 120576120590119875int

119887

0

(1198794

minus 1198794

119904) 119889119909 (11)


119876119903=

119902119903119887

119896119860119879119887

= 119873119903int

1

0

(1205794

minus 1205794

119904) 119889119883 (12)

The instantaneous total surface heat loss in dimensionlessform is the sum of convective and radiative losses given by(11) and (13) that is

119876loss = 119876119888+ 119876119903 (13)

Mathematical Problems in Engineering 3

The instantaneous rate of energy storage in the fin can becalculated from the energy balance as follows

119902stored = 119902119887+ 119902gen minus 119902loss (14)

or in dimensionless form as119876stored = 119876

119887+ 119876gen minus 119876loss (15)

where

119876gen =1199021198872

119896119879119887

(16)

3 Basic Principles of OHAM

We review the basic principles of OHAM as expounded byMarinca et al [21ndash24] as well as other researchers including[25 26]

(i) Let us consider the following differential equationI [v (120596)] + 119886 (120596) = 0 119909 isin Ω (17)

where Ω is problem domain I(v) = 119871(v) + 119873(v)where 119871 and 119873 are linear and nonlinear operatorsv(119909) is an unknown function and 119886(120596) is a knownfunction

(ii) Construct an optimal homotopy equation as(1 minus 119901)[119871 (120601 (120596 119901))+119886 (120596)]minus119867 (119901)[I (120601(120596 119901))+119886 (120596)]=0

(18)

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

119898

119896=1119901119896

119862119896is auxiliary function on which

the convergence of the solution greatly depends Theauxiliary function 119867(119901) also adjusts the convergencedomain and controls the convergence region

(iii) Expand 120601(120596 119901 119862119895) in Taylorrsquos series about 119901 One has

an approximate solution

120601 (120596 119901 119862119895) = v0(120596) +

infin

sum

119896=1

v119896(120596 119862119895) 119901119896

119895 = 1 2 3

(19)

Many researchers have observed that the convergenceof the series equation (19) depends upon 119862

119895 (119895 =

1 2 119898) If it is convergent then we obtain

v = v0(120596) +

119898

sum

119896=1

v119896(120596 119862119895) (20)

(iv) Substituting (20) in (17) we have the following resid-ual

119877 (120596 119862119895) = 119871 (v (120596 119862

119895)) + 119886 (120596) + 119873(v (120596 119862

119895)) (21)

If 119877(120596 119862119895) = 0 then v will be the exact solution

For nonlinear problems generally this will not be thecase For determining119862

119895(119895 = 1 2 119898) Galerkinrsquos

Method Ritz Method or the method of least squarescan be used

(v) Finally substitute these constants in (21) and one canget the approximate solution

4 OHAM Solution for Heat-Generating Fin

According to the OHAM (1) can be written as

(1 minus 119901) (12057910158401015840

) minus 119867 (119901)

times (12057910158401015840

minus119873119888

(1 minus 120579119904)(120579 minus 120579

119904)2

minus 119873119903(1205794

minus 1205794

119904) + 119876119892) = 0

(22)

where prime denotes differentiation with respect to 119883We consider 120579 and 119867(119901) as follows

120579 = 1205790+ 1199011205791+ 1199012

1205792

119867 (119901) = 1199011198621+ 1199012

1198622

(23)

Using (23) in (22) and after some simplifying and rearrangingthe terms based on the powers of 119901 we obtain the zeroth-first- and second-order problems as follows

The zeroth-order problem is

1198892

1205790(119883)

1198891198832= 0 (24)

with boundary conditions

1205790(119887) = 1

1198891205790(0)

119889119883= 0 (25)

Its solution is

1205790(119883) = 1 (26)

The first-order problem is

1198892

1205791(119883 1198621)

1198891198832minus 1198621119876119892+

11986211198731198881205792

119904

1 minus 120579119904

minus 11986211198731199031205794

119904minus

211986211198731198881205791199041205790

1 minus 120579119904

+11986211198731198881205792

0

1 minus 120579119904

+ 11986211198731199031205794

0minus (1 + 119862

1)1198892

1205790(119883)

1198891198832= 0

(27)


1205791(119887) = 0

1198891205791(0)

119889119883= 0 (28)

having solution

1205791(119883 1198621) =

1

2(1198872

1198621119873119888minus 1198832

1198621119873119888+ 1198872

1198621119873119903minus 1198832

1198621119873119903

minus 1198872

1198621119876119892+ 1198832

1198621119876119892minus 1198872

1198621119873119888120579119904

+1198832

1198621119873119888120579119904minus 1198872

11986211198731199031205794

119904+ 1198832

11986211198731199031205794

119904)

(29)


Table 1 Comparison of percentage error between OHAM HAM and NM for temperature at 119873119888= 01 119873

119903= 01 and 120579

119904= 001

119883119876119892= 0 119876

119892= 1

OHAM HAM NM OHAM error HAM error OHAM HAM NM OHAM error HAM error0 09078 09075 09078 0 0033 18085 18081 18085 0 002201 09088 09081 09087 0011 0066 18005 18011 18007 0011 002202 09115 09112 09115 0 0032 17763 17759 17770 0039 006103 09161 09158 09161 0 0032 17358 17352 17375 0098 013204 09226 09220 09224 0021 0043 16792 16788 16821 0172 039805 09308 09302 09306 0021 0042 16065 16062 16105 0248 026606 09410 09413 09407 0032 0063 15175 15181 15224 0321 037407 09529 09522 09526 0031 0041 14124 14111 14176 0366 045808 09668 09671 09665 0031 0062 12911 12909 12958 0362 037809 09825 09818 09822 0031 004 11536 11521 11567 0268 039710 1 1 1 0 0 1 1 1 0 0


119888= 05 and 119876

119892= 2

119883120579119904= 001 120579

119904= 04


The second-order problem is

1198892

1205792(119883 1198621 1198622)

1198891198832minus 1198622119876119892+

11986221198731198881205792

119904

1 minus 120579119904

minus 11986221198731199031205794

119904minus

211986221198731198881205791199041205790

1 minus 120579119904

+11986221198731198881205792

0

1 minus 120579119904

+ 11986221198731199031205794

0minus

211986211198731198881205791199041205791

1 minus 120579119904

+2119862111987311988812057901205791

1 minus 120579119904

+ 411986211198731199031205793

01205791minus 1198622

1198892

1205790(119883)

1198891198832minus (1 + 119862

1)1198892

1205791(119883 1198621)

1198891198832= 0

(30)


1205792(119887) = 0

1198891205792(0)

119889119883= 0 (31)

It is given by

1205792(119883 1198621 1198622)

=1

12(61198872

1198621119873119888minus 61198832

1198621119873119888+ 61198872

1198622

1119873119888minus 61198832

1198622

1119873119888

+ 61198872

1198622119873119888minus 61198832

1198621119873119903minus 61198832

1198622119873119888+ 51198874

1198622

11198732

119888

minus 61198872

1198832

1198622

11198732

119888+ 1198834

1198622

11198732

119888+ 61198872

1198621119873119903+ 151198874

1198622

1119873119888119873119903

+ 61198872

1198622

1119873119903minus 61198832

1198622

1119873119903+ 61198872

1198622119873119903minus 61198832

1198622119873119903

+ 101198874

1198622

11198732

119903minus 121198872

1198832

1198622

11198732

119903minus 181198872

1198832

1198622

1119873119888119873119903

+ 31198834

1198622

1119873119888119873119903minus 61198872

1198622

1119876119892+ 61198832

1198622

1119876119892minus 61198872

1198622119876119892

+ 21198834

1198622

11198732

119903minus 61198872

1198621119876119892+ 61198832

1198621119876119892+ 61198832

1198622119876119892

minus 51198874

1198622

1119873119888119876119892+ 61198872

1198832

1198622

1119873119888119876119892minus 1198834

1198622

1119873119888119876119892

minus 101198874

1198622

1119873119903119876119892+ 121198872

1198832

1198622

1119873119903119876119892minus 21198834

1198622

1119873119903119876119892

minus 61198872

1198621119873119888120579119904+61198832

1198621119873119888120579119904minus61198872

1198622

1119873119888120579119904+61198832

1198622

1119873119888120579119904

minus 61198872

1198622119873119888120579119904+61198832

1198622119873119888120579119904minus1198834

1198622

11198732

119888120579119904minus51198874

1198622

11198732

119888120579119904

+ 61198872

1198832

1198622

11198732

119888120579119904minus101198874

1198622

1119873119888119873119903120579119904+121198872

1198832

1198622

1119873119888119873119903120579119904

minus 61198872

11986221198731199031205794

119904minus 21198834

1198622

1119873119888119873119903120579119904minus 61198872

11986211198731199031205794

119904

+ 61198832

11986211198731199031205794

119904minus 61198872

1198622

11198731199031205794

119904

+ 61198832

1198622

11198731199031205794

119904+ 61198832

11986221198731199031205794

119904minus 51198874

1198622

11198731198881198731199031205794

119904


Table 3 Comparison of OHAM and NM for temperature at 119873119903= 01 120579

119904= 01 and 119876

119892= 0

119883119873119888= 01 119873

119888= 03 119873

119888= 1

OHAM NM error OHAM NM error OHAM NM error0 08172 08187 0183 07623 07668 0586 06409 06465 086601 08190 08190 0 07647 07691 0572 06445 06496 078502 08245 08259 0169 07718 07758 0515 06552 06592 060703 08336 08348 0143 07883 07871 0152 06732 06752 029604 08464 08474 0118 08004 08029 0311 06983 06979 005705 08629 08636 0081 08217 08234 0206 07307 07375 092206 08830 08834 0045 08579 08586 0081 07696 07646 065407 09067 09069 0022 08788 08788 0 08018 08096 096308 09341 09341 0 09145 09138 0076 08693 08632 070709 09652 09651 0010 09548 09541 0073 09291 09263 030210 1 1 0 1 1 0 1 1 0


119904= 03 and 119876

119892= 1

119883119873119903= 01 119873

119903= 03 119873

119903= 05

OHAM NM error OHAM NM error OHAM NM error0 11866 11869 0025 11166 11204 0339 10521 10522 0009

01 11848 11852 0033 11154 11193 0348 10516 10518 0019

02 11792 11801 0076 11119 11160 0367 10501 10503 0019

03 11698 11715 0145 11061 11104 0387 10477 10479 0019

04 11568 11590 0189 10979 11026 0426 10442 10445 0028

05 11400 11435 0306 10874 10923 0448 10398 10400 0019

06 11194 11237 0382 10746 10795 0453 10342 10345 0028

07 10952 10997 0409 10594 10641 0441 10275 10278 0029

08 10672 10713 0382 10419 10458 0373 10196 10198 0019

09 10354 10382 0269 10221 10245 0234 10104 10106 0019

10 1 1 0 1 1 0 1 1 0

+ 61198872

1198832

1198622

11198731198881198731199031205794

119904minus 1198834

1198622

11198731198881198731199031205794

119904minus 101198874

1198622

11198732

1199031205794

119904

+121198872

1198832

1198622

11198732

1199031205794

119904minus 21198834

1198622

11198732

1199031205794

119904)

(32)

The second-order approximate solution by OHAM for 119901 = 1

is

120579 (119883 1198621 1198622) = 1205790(119883) + 120579

1(119883 1198621) + 1205792(119883 1198621 1198622) (33)

We use the method of least squares to obtain 1198621 1198622the

unknown convergent constant in 120579In particular case119862

1= minus000002137 and119862

2= minus079986

for 119873119903= 03 119873

119888= 03 119876

119892= 04 120579

119904= 02 and 119887 = 1

By considering the values of 1198621 1198622in (33) and after

simplifying the second-order approximate analytical OHAMsolution can be obtained

120579 = 1 +1

2(minus0000002982 + 2982 times 10

minus6

1198832

)

+1

12(minus06696 + 0669596119883

2

+ 573 times 10minus11

1198834

)

(34)

5 Results and Discussion

Equation (4) shows that fin temperature is based on fourparameters119873

119903119873119888 120579119904 and119876

119892which govern this highly non-

linear second-order differential equation The effect of eachparameter on fin temperature is tabulated and graphicallypresented for different values of the controlling parameters

In order to validate the accuracy of our approximatesolution via OHAM we have presented a comparative studyof OHAM solution with homotopy analysis method (HAM)and numerical solution (Runge-Kutta-Fehlberg fourth-fifth-ordermethod) Table 1 has been prepared to exhibit the com-parison of dimensionless temperature 120579 obtained by OHAMhomotopy analysis method (HAM) and the numericalmethod (NM) for several values of heat-generating parameter119876119892 when other parameters are fixed It is observed that

with increasing values of internal heat-generating parameter119876119892 the temperature profile gradually increases Clearly the

OHAM solutions are very close to the numerical solution ascompared to HAM This can be seen from the percentageerror in the dimensionless temperature obtained by OHAMHAM and NMThe increase in dimensionless temperature 120579

is also evident in Table 2 in which we have used different val-ues of sink temperature parameter 120579

119904 and other parameters


00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X

Figure 2 Effect of internal heat generation on fin dimensionlesstemperature for fixed values of 119873

119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X

Figure 3 Effect of convection parameter on fin dimensionlesstemperature for fixed values of 119873

119903 120579119904 and 119876

119892

values are predetermined From Tables 1 and 2 it is observedthat our OHAM solutions are more accurate than HAM thisconfirms that OHAM is more consistent with approximateanalytical method than with HAMThemajor factor in HAMis its computational time for finding the ℎ (h curve) whilein OHAM the ensuring convergence of the solution dependson parameters 119862

1 1198622 which are optimally determined

resultantly HAM in more time consuming than OHAMIn Table 3 we show the comparison of dimensionless

temperature 120579 obtained byOHAMand the numericalmethod(NM) for several values of convection parameter 119873

119888 while

other parameters are kept unchanged It is observed that withthe increase of 119873

119888 the temperature profile shows decrease

and the same phenomena of decrease in dimensionlesstemperature 120579 can be observed in Table 4 for different valuesof radiation parameter119873

119903 when the other parameters values

are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X

Figure 4 Effect of radiation parameter on fin dimensionlesstemperature for fixed values of 119873

119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X

Figure 5 Effect of sink temperature parameter on fin dimensionlesstemperature for fixed values of 119873

119888 120579119904 and 119876

119892

In Figures 2 3 4 and 5 we depict the dimensionlesstemperature profile 120579 and its variation for different values ofparameters It is important to note that the dimensionlesstemperature increases with each controlling parameter

6 Conclusion

We have successfully applied the optimal homotopy asymp-totic method for the approximate solution of steady stateof heat-generating fin with simultaneous surfaces convec-tion and radiation The effects of radiation parameter 119873

119903

convection parameter119873119888 internal heat-generating parameter

119876119892 and the sink temperature parameter 120579

119904on temperature

profile in the fin are investigated analytically It is observedthat dimensionless fin temperature profile is dependent onthe four parameters 119873

119903 119873119888 119876119892 and 120579

119904 Comparison for

the dimensionless temperature has been made between the


solutions obtained using OHAM with HAM and Runge-Kutta-Fehlberg fourth-fifth-order method It is found that theOHAM solution is very close to the numerical solution thanHAM which reveals the reliability and efficiency of OHAMApproximate analytical solution to highly nonlinear problemwas achieved without any assumption of linearization andwe can extend this approach to a variety of nonlinear heattransfer problems

References

[1] B Sunden and P J Heggs Recent Advances in Analysis of HeatTransfer for Fin Type Surfaces WIT Press Boston Mass USA2000

[2] A D Kraus A Aziz and J Welty Extended Surface HeatTransfer John Wiley amp Sons New York NY USA 2001

[3] K A Gardener ldquoEfficiency of extended surfacerdquo Journal of HeatTransfer vol 67 pp 621ndash631 1945

[4] F Khani M A Raji and H H Nejad ldquoAnalytical solutionsand efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficientrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 14 no 8 pp 3327ndash3338 2009

[5] F Khani M A Raji and S Hamedi-Nezhad ldquoA series solutionof the fin problem with a temperature-dependent thermal con-ductivityrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3007ndash3017 2009

[6] A Aziz and T Y Na ldquoTransient response of fins by coordinateperturbation expansionrdquo International Journal of Heat andMassTransfer vol 23 no 12 pp 1695ndash1698 1980

[7] Y M Chang C K Chen and JW Cleaver ldquoTransient responseof fins by optimal linearization and variational embeddingmethodsrdquo ASME Journal of Heat Transfer vol V 104 no 4 pp813ndash815 1982

[8] A Campo ldquoVariational techniques applied to radiative-convective fins with steady and unsteady conditionsrdquo Warme-und Stoffubertragung vol 9 no 2 pp 139ndash144 1976

[9] N Onur ldquoA simplified approach to the transient conduction ina two-dimensional finrdquo International Communications in Heatand Mass Transfer vol 23 no 2 pp 225ndash238 1996

[10] A Aziz and M Torabi ldquoConvective-radiative fins with simul-taneous variation of thermal conductivity heat transfer coeffi-cient and surface emissivity with temperaturerdquo Heat Transfer-Asian Research vol 41 no 2 pp 99ndash113 2012

[11] L-T Chen ldquoTwo-dimensional fin efficiency with combinedheat and mass transfer between water-wetted fin surface andmoving moist airstreamrdquo International Journal of Heat andFluid Flow vol 12 no 1 pp 71ndash76 1991

[12] A Aziz and H Nguyen ldquoTwo-dimensional performance ofconvecting-radiating fins of different profile shapesrdquo Warmeund Stoffubertragung vol 28 no 8 pp 481ndash487 1993

[13] H S Kang andDC Look Jr ldquoTwodimensional trapezoidal finsanalysisrdquo Computational Mechanics vol 19 no 3 pp 247ndash2501997

[14] A Aziz and O D Makinde ldquoHeat transfer and entropygeneration in a two-dimensional orthotropic convection pinfinrdquo International Journal of Exergy vol 7 no 5 pp 579ndash5922010

[15] S M Zubair A F M Arif and M H Sharqawy ldquoThermalanalysis and optimization of orthotropic pin fins a closed-form

analytical solutionrdquo Journal of Heat Transfer vol 132 no 3 pp1ndash8 2010

[16] Y Xia and A M Jacobi ldquoAn exact solution to steady heatconduction in a two-dimensional slab on a one-dimensional finapplication to frosted heat exchangersrdquo International Journal ofHeat and Mass Transfer vol 47 no 14-16 pp 3317ndash3326 2004

[17] M YMalik andA Rafiq ldquoTwo-dimensional finwith convectivebase conditionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 1 pp 147ndash154 2010

[18] A Aziz ldquoEffects of internal heat generation anisotropy andbase temperature nonuniformity on heat transfer from a two-dimensional rectangular finrdquoHeat Transfer Engineering vol 14no 2 pp 63ndash70 1993

[19] S Jahangeer M K Ramis and G Jilani ldquoConjugate heat trans-fer analysis of a heat generating vertical platerdquo InternationalJournal of Heat and Mass Transfer vol 50 no 1-2 pp 85ndash932007

[20] M K Ramis G Jilani and S Jahangeer ldquoConjugateconduction-forced convection heat transfer analysis of arectangular nuclear fuel element with non-uniform volumetricenergy generationrdquo International Journal of Heat and MassTransfer vol 51 no 3-4 pp 517ndash525 2008

[21] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic Method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008

[22] V Marinca N Herisanu and I Nemes ldquoOptimal homotopyasymptotic method with application to thin film flowrdquo CentralEuropean Journal of Physics vol 6 no 3 pp 648ndash653 2008

[23] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010

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

[25] S Islam R A Shah I Ali andNM Allah ldquoOptimal homotopyasymptotic solutions of couette and poiseuille flows of a thirdgrade fluid with heat transfer analysisrdquo International Journal ofNonlinear Sciences and Numerical Simulation vol 11 no 6 pp389ndash400 2010

[26] FMaboodWAKhan andA IM Ismail ldquoOptimal homotopyasymptoticmethod for heat transfer in hollow spherewith robinboundary conditionsrdquo Heat Transfer-Asian Research 7 pages2013

[27] F Mabood W A Khan and A I M Ismail ldquoSolution ofSolution of nonlinear boundary layer equation for flat platevia optimal homotopy asymptoticmethodrdquoHeat Transfer-AsianResearch 7 pages 2013

[28] S J Liao On the proposed homotopy analysis technique fornonlinear problems and its applications [PhD thesis] ShanghaiJio Tong University 1992

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013


Mathematical Problems in Engineering


Abstract and Applied Analysis

ISRN Applied Mathematics


Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics


Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences


ISRN Geometry



Discrete Dynamicsin Nature and Society


Volume 2013

Advances in

Mathematical Physics

ISRN Algebra


ProbabilityandStatistics

Journal of


ISRN Mathematical Analysis


Journal ofApplied Mathematics


Advances in

DecisionSciences



Stochastic AnalysisInternational Journal of

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

The Scientific World Journal


ISRN Discrete Mathematics


DifferentialEquations


Volume 2013


Insulated tip

Cross-sectional

h Tinfin

120576

Tb

x

x = 0

x = b

k 120572 q

Perimeter P

area A

Figure 1 A straight fin with constant cross-sectional area

and investigated numerically the effects of the differentgoverning parameters on dimensionless temperature profilein a nonlinear fin-type problem For comparison purposesthe governing highly nonlinear problem is also solved usingRunge-Kutta-Fehlberg fourth-fifth-order method and homo-topy analysis method (HAM) developed by Liao [28]

The paper is planned as follows in Section 2 we formu-late our nonlinear problem basic principles of OHAM arediscussed in Section 3 solution of the problem via OHAMis presented in Section 4 and Section 5 is reserved for resultsand discussion Conclusions are drawn in Section 6

2 Mathematical Formulation

Consider a straight fin of constant cross-sectional area 119860

(rectangular cylindrical elliptic etc) perimeter of the cross-section 119875 and length 119887 as shown in Figure 1 The fin hasa thermal conductivity 119896 and a thermal diffusivity 120572 Thesurface of the fin behaves as a gray diffuse surface with anemissivity 120576 The fin is deemed to be initially in thermalequilibriumwith the surroundings at temperature119879

119904 Its tip is

insulated A volumetric internal heat generation rate 119902 occursin the fin The fin loses heat by simultaneous convection andradiation to its surroundings at temperature 119879

119904 The same

sink temperature is used for both convection and radiationto avoid the introduction of an additional parameter in theproblem

For one-dimensional steady conduction in the fin theenergy equation may be written as

1205972

119879

1205971199092minus

ℎ119875

119896119860(119879 minus 119879

119904) minus

120576120590119875 (1198794

minus 1198794

119904)

119896119860+

119902

119896= 0 (1)

The initial and boundary conditions are

119879 (119909) = 119879119904

119879 (119887) = 119879119887

119889119879

119889119909(0) = 0

(2)

where 119909 is measured from the tip of the fin with theintroduction of the following definitions

120579 =119879

119879119887

120579119904=

119879119904

119879119887

119883 =119909

119887 ℎ

119887= 119862 (119879

119887minus 119879119904)

119873119888=

ℎ1198871198751198872

119896119860 119873

119903=

1205761205901198751198793

1199041198872

119896119860

119876gen =1199021198872

119896119879119887

(3)

Equations (2) and (3) can be written in dimensionless formas follows

1198892

120579

1198891198832minus

119873119888

(1 minus 120579119904)(120579 minus 120579

119904)2

minus 119873119903(1205794

minus 1205794

119904) + 119876gen = 0 (4)

120579 (119887) = 1 (5)

119889120579

119889119883(0) = 0 (6)

The instantaneous base heat flow is given by

119902119887= 119896119860

119889119879

119889119909(119887) (7)

which may be expressed in dimensionless form as follows

119876119887=

119902119887119887

119896119860119879119887

=119889120579

119889119883(1) (8)

The instantaneous convective heat loss from the fin is givenby

119902119888= 119875int

119887

0

ℎ (119879 minus 119879119904) 119889119909 (9)


119876119888=

119902119888119887

119896119860119879119887

=119873119888

(1 minus 120579119904)int

1

0

(120579 minus 120579119904)2

119889119883 (10)

Similarly the instantaneous radiative heat loss from the fincan be obtained as

119902119903= 120576120590119875int

119887

0

(1198794

minus 1198794

119904) 119889119909 (11)


119876119903=

119902119903119887

119896119860119879119887

= 119873119903int

1

0

(1205794

minus 1205794

119904) 119889119883 (12)

The instantaneous total surface heat loss in dimensionlessform is the sum of convective and radiative losses given by(11) and (13) that is

119876loss = 119876119888+ 119876119903 (13)





119887+ 119876gen minus 119876loss (15)

where

119876gen =1199021198872

119896119879119887

(16)






(18)


119898

119896=1119901119896





120601 (120596 119901 119862119895) = v0(120596) +

infin

sum

119896=1

v119896(120596 119862119895) 119901119896

119895 = 1 2 3

(19)


119895 (119895 =


v = v0(120596) +

119898

sum

119896=1

v119896(120596 119862119895) (20)


119877 (120596 119862119895) = 119871 (v (120596 119862

119895)) + 119886 (120596) + 119873(v (120596 119862

119895)) (21)



119895(119895 = 1 2 119898) Galerkinrsquos





(1 minus 119901) (12057910158401015840

) minus 119867 (119901)

times (12057910158401015840

minus119873119888

(1 minus 120579119904)(120579 minus 120579

119904)2

minus 119873119903(1205794

minus 1205794

119904) + 119876119892) = 0

(22)


120579 = 1205790+ 1199011205791+ 1199012

1205792

119867 (119901) = 1199011198621+ 1199012

1198622

(23)



1198892

1205790(119883)

1198891198832= 0 (24)


1205790(119887) = 1

1198891205790(0)

119889119883= 0 (25)

Its solution is

1205790(119883) = 1 (26)


1198892

1205791(119883 1198621)

1198891198832minus 1198621119876119892+

11986211198731198881205792

119904

1 minus 120579119904

minus 11986211198731199031205794

119904minus

211986211198731198881205791199041205790

1 minus 120579119904

+11986211198731198881205792

0

1 minus 120579119904

+ 11986211198731199031205794

0minus (1 + 119862

1)1198892

1205790(119883)

1198891198832= 0

(27)


1205791(119887) = 0

1198891205791(0)

119889119883= 0 (28)

having solution

1205791(119883 1198621) =

1

2(1198872

1198621119873119888minus 1198832

1198621119873119888+ 1198872

1198621119873119903minus 1198832

1198621119873119903

minus 1198872

1198621119876119892+ 1198832

1198621119876119892minus 1198872

1198621119873119888120579119904

+1198832

1198621119873119888120579119904minus 1198872

11986211198731199031205794

119904+ 1198832

11986211198731199031205794

119904)

(29)



119903= 01 and 120579

119904= 001

119883119876119892= 0 119876

119892= 1



119888= 05 and 119876

119892= 2

119883120579119904= 001 120579

119904= 04



1198892

1205792(119883 1198621 1198622)

1198891198832minus 1198622119876119892+

11986221198731198881205792

119904

1 minus 120579119904

minus 11986221198731199031205794

119904minus

211986221198731198881205791199041205790

1 minus 120579119904

+11986221198731198881205792

0

1 minus 120579119904

+ 11986221198731199031205794

0minus

211986211198731198881205791199041205791

1 minus 120579119904

+2119862111987311988812057901205791

1 minus 120579119904

+ 411986211198731199031205793

01205791minus 1198622

1198892

1205790(119883)

1198891198832minus (1 + 119862

1)1198892

1205791(119883 1198621)

1198891198832= 0

(30)


1205792(119887) = 0

1198891205792(0)

119889119883= 0 (31)

It is given by

1205792(119883 1198621 1198622)

=1

12(61198872

1198621119873119888minus 61198832

1198621119873119888+ 61198872

1198622

1119873119888minus 61198832

1198622

1119873119888

+ 61198872

1198622119873119888minus 61198832

1198621119873119903minus 61198832

1198622119873119888+ 51198874

1198622

11198732

119888

minus 61198872

1198832

1198622

11198732

119888+ 1198834

1198622

11198732

119888+ 61198872

1198621119873119903+ 151198874

1198622

1119873119888119873119903

+ 61198872

1198622

1119873119903minus 61198832

1198622

1119873119903+ 61198872

1198622119873119903minus 61198832

1198622119873119903

+ 101198874

1198622

11198732

119903minus 121198872

1198832

1198622

11198732

119903minus 181198872

1198832

1198622

1119873119888119873119903

+ 31198834

1198622

1119873119888119873119903minus 61198872

1198622

1119876119892+ 61198832

1198622

1119876119892minus 61198872

1198622119876119892

+ 21198834

1198622

11198732

119903minus 61198872

1198621119876119892+ 61198832

1198621119876119892+ 61198832

1198622119876119892

minus 51198874

1198622

1119873119888119876119892+ 61198872

1198832

1198622

1119873119888119876119892minus 1198834

1198622

1119873119888119876119892

minus 101198874

1198622

1119873119903119876119892+ 121198872

1198832

1198622

1119873119903119876119892minus 21198834

1198622

1119873119903119876119892

minus 61198872

1198621119873119888120579119904+61198832

1198621119873119888120579119904minus61198872

1198622

1119873119888120579119904+61198832

1198622

1119873119888120579119904

minus 61198872

1198622119873119888120579119904+61198832

1198622119873119888120579119904minus1198834

1198622

11198732

119888120579119904minus51198874

1198622

11198732

119888120579119904

+ 61198872

1198832

1198622

11198732

119888120579119904minus101198874

1198622

1119873119888119873119903120579119904+121198872

1198832

1198622

1119873119888119873119903120579119904

minus 61198872

11986221198731199031205794

119904minus 21198834

1198622

1119873119888119873119903120579119904minus 61198872

11986211198731199031205794

119904

+ 61198832

11986211198731199031205794

119904minus 61198872

1198622

11198731199031205794

119904

+ 61198832

1198622

11198731199031205794

119904+ 61198832

11986221198731199031205794

119904minus 51198874

1198622

11198731198881198731199031205794

119904



119904= 01 and 119876

119892= 0

119883119873119888= 01 119873

119888= 03 119873

119888= 1



119904= 03 and 119876

119892= 1

119883119873119903= 01 119873

119903= 03 119873

119903= 05


01 11848 11852 0033 11154 11193 0348 10516 10518 0019

02 11792 11801 0076 11119 11160 0367 10501 10503 0019

03 11698 11715 0145 11061 11104 0387 10477 10479 0019

04 11568 11590 0189 10979 11026 0426 10442 10445 0028

05 11400 11435 0306 10874 10923 0448 10398 10400 0019

06 11194 11237 0382 10746 10795 0453 10342 10345 0028

07 10952 10997 0409 10594 10641 0441 10275 10278 0029

08 10672 10713 0382 10419 10458 0373 10196 10198 0019

09 10354 10382 0269 10221 10245 0234 10104 10106 0019

10 1 1 0 1 1 0 1 1 0

+ 61198872

1198832

1198622

11198731198881198731199031205794

119904minus 1198834

1198622

11198731198881198731199031205794

119904minus 101198874

1198622

11198732

1199031205794

119904

+121198872

1198832

1198622

11198732

1199031205794

119904minus 21198834

1198622

11198732

1199031205794

119904)

(32)


is

120579 (119883 1198621 1198622) = 1205790(119883) + 120579

1(119883 1198621) + 1205792(119883 1198621 1198622) (33)



1= minus000002137 and119862

2= minus079986

for 119873119903= 03 119873

119888= 03 119876

119892= 04 120579

119904= 02 and 119887 = 1



120579 = 1 +1

2(minus0000002982 + 2982 times 10

minus6

1198832

)

+1

12(minus06696 + 0669596119883

2


1198834

)

(34)



119903119873119888 120579119904 and119876









00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X


119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X


119903 120579119904 and 119876

119892





119888 while





are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X


119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X


119888 120579119904 and 119876

119892


6 Conclusion


119903





119903 119873119888 119876119892 and 120579





References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013





119887+ 119876gen minus 119876loss (15)

where

119876gen =1199021198872

119896119879119887

(16)






(18)


119898

119896=1119901119896





120601 (120596 119901 119862119895) = v0(120596) +

infin

sum

119896=1

v119896(120596 119862119895) 119901119896

119895 = 1 2 3

(19)


119895 (119895 =


v = v0(120596) +

119898

sum

119896=1

v119896(120596 119862119895) (20)


119877 (120596 119862119895) = 119871 (v (120596 119862

119895)) + 119886 (120596) + 119873(v (120596 119862

119895)) (21)



119895(119895 = 1 2 119898) Galerkinrsquos





(1 minus 119901) (12057910158401015840

) minus 119867 (119901)

times (12057910158401015840

minus119873119888

(1 minus 120579119904)(120579 minus 120579

119904)2

minus 119873119903(1205794

minus 1205794

119904) + 119876119892) = 0

(22)


120579 = 1205790+ 1199011205791+ 1199012

1205792

119867 (119901) = 1199011198621+ 1199012

1198622

(23)



1198892

1205790(119883)

1198891198832= 0 (24)


1205790(119887) = 1

1198891205790(0)

119889119883= 0 (25)

Its solution is

1205790(119883) = 1 (26)


1198892

1205791(119883 1198621)

1198891198832minus 1198621119876119892+

11986211198731198881205792

119904

1 minus 120579119904

minus 11986211198731199031205794

119904minus

211986211198731198881205791199041205790

1 minus 120579119904

+11986211198731198881205792

0

1 minus 120579119904

+ 11986211198731199031205794

0minus (1 + 119862

1)1198892

1205790(119883)

1198891198832= 0

(27)


1205791(119887) = 0

1198891205791(0)

119889119883= 0 (28)

having solution

1205791(119883 1198621) =

1

2(1198872

1198621119873119888minus 1198832

1198621119873119888+ 1198872

1198621119873119903minus 1198832

1198621119873119903

minus 1198872

1198621119876119892+ 1198832

1198621119876119892minus 1198872

1198621119873119888120579119904

+1198832

1198621119873119888120579119904minus 1198872

11986211198731199031205794

119904+ 1198832

11986211198731199031205794

119904)

(29)



119903= 01 and 120579

119904= 001

119883119876119892= 0 119876

119892= 1



119888= 05 and 119876

119892= 2

119883120579119904= 001 120579

119904= 04



1198892

1205792(119883 1198621 1198622)

1198891198832minus 1198622119876119892+

11986221198731198881205792

119904

1 minus 120579119904

minus 11986221198731199031205794

119904minus

211986221198731198881205791199041205790

1 minus 120579119904

+11986221198731198881205792

0

1 minus 120579119904

+ 11986221198731199031205794

0minus

211986211198731198881205791199041205791

1 minus 120579119904

+2119862111987311988812057901205791

1 minus 120579119904

+ 411986211198731199031205793

01205791minus 1198622

1198892

1205790(119883)

1198891198832minus (1 + 119862

1)1198892

1205791(119883 1198621)

1198891198832= 0

(30)


1205792(119887) = 0

1198891205792(0)

119889119883= 0 (31)

It is given by

1205792(119883 1198621 1198622)

=1

12(61198872

1198621119873119888minus 61198832

1198621119873119888+ 61198872

1198622

1119873119888minus 61198832

1198622

1119873119888

+ 61198872

1198622119873119888minus 61198832

1198621119873119903minus 61198832

1198622119873119888+ 51198874

1198622

11198732

119888

minus 61198872

1198832

1198622

11198732

119888+ 1198834

1198622

11198732

119888+ 61198872

1198621119873119903+ 151198874

1198622

1119873119888119873119903

+ 61198872

1198622

1119873119903minus 61198832

1198622

1119873119903+ 61198872

1198622119873119903minus 61198832

1198622119873119903

+ 101198874

1198622

11198732

119903minus 121198872

1198832

1198622

11198732

119903minus 181198872

1198832

1198622

1119873119888119873119903

+ 31198834

1198622

1119873119888119873119903minus 61198872

1198622

1119876119892+ 61198832

1198622

1119876119892minus 61198872

1198622119876119892

+ 21198834

1198622

11198732

119903minus 61198872

1198621119876119892+ 61198832

1198621119876119892+ 61198832

1198622119876119892

minus 51198874

1198622

1119873119888119876119892+ 61198872

1198832

1198622

1119873119888119876119892minus 1198834

1198622

1119873119888119876119892

minus 101198874

1198622

1119873119903119876119892+ 121198872

1198832

1198622

1119873119903119876119892minus 21198834

1198622

1119873119903119876119892

minus 61198872

1198621119873119888120579119904+61198832

1198621119873119888120579119904minus61198872

1198622

1119873119888120579119904+61198832

1198622

1119873119888120579119904

minus 61198872

1198622119873119888120579119904+61198832

1198622119873119888120579119904minus1198834

1198622

11198732

119888120579119904minus51198874

1198622

11198732

119888120579119904

+ 61198872

1198832

1198622

11198732

119888120579119904minus101198874

1198622

1119873119888119873119903120579119904+121198872

1198832

1198622

1119873119888119873119903120579119904

minus 61198872

11986221198731199031205794

119904minus 21198834

1198622

1119873119888119873119903120579119904minus 61198872

11986211198731199031205794

119904

+ 61198832

11986211198731199031205794

119904minus 61198872

1198622

11198731199031205794

119904

+ 61198832

1198622

11198731199031205794

119904+ 61198832

11986221198731199031205794

119904minus 51198874

1198622

11198731198881198731199031205794

119904



119904= 01 and 119876

119892= 0

119883119873119888= 01 119873

119888= 03 119873

119888= 1



119904= 03 and 119876

119892= 1

119883119873119903= 01 119873

119903= 03 119873

119903= 05


01 11848 11852 0033 11154 11193 0348 10516 10518 0019

02 11792 11801 0076 11119 11160 0367 10501 10503 0019

03 11698 11715 0145 11061 11104 0387 10477 10479 0019

04 11568 11590 0189 10979 11026 0426 10442 10445 0028

05 11400 11435 0306 10874 10923 0448 10398 10400 0019

06 11194 11237 0382 10746 10795 0453 10342 10345 0028

07 10952 10997 0409 10594 10641 0441 10275 10278 0029

08 10672 10713 0382 10419 10458 0373 10196 10198 0019

09 10354 10382 0269 10221 10245 0234 10104 10106 0019

10 1 1 0 1 1 0 1 1 0

+ 61198872

1198832

1198622

11198731198881198731199031205794

119904minus 1198834

1198622

11198731198881198731199031205794

119904minus 101198874

1198622

11198732

1199031205794

119904

+121198872

1198832

1198622

11198732

1199031205794

119904minus 21198834

1198622

11198732

1199031205794

119904)

(32)


is

120579 (119883 1198621 1198622) = 1205790(119883) + 120579

1(119883 1198621) + 1205792(119883 1198621 1198622) (33)



1= minus000002137 and119862

2= minus079986

for 119873119903= 03 119873

119888= 03 119876

119892= 04 120579

119904= 02 and 119887 = 1



120579 = 1 +1

2(minus0000002982 + 2982 times 10

minus6

1198832

)

+1

12(minus06696 + 0669596119883

2


1198834

)

(34)



119903119873119888 120579119904 and119876









00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X


119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X


119903 120579119904 and 119876

119892





119888 while





are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X


119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X


119888 120579119904 and 119876

119892


6 Conclusion


119903





119903 119873119888 119876119892 and 120579





References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



119903= 01 and 120579

119904= 001

119883119876119892= 0 119876

119892= 1



119888= 05 and 119876

119892= 2

119883120579119904= 001 120579

119904= 04



1198892

1205792(119883 1198621 1198622)

1198891198832minus 1198622119876119892+

11986221198731198881205792

119904

1 minus 120579119904

minus 11986221198731199031205794

119904minus

211986221198731198881205791199041205790

1 minus 120579119904

+11986221198731198881205792

0

1 minus 120579119904

+ 11986221198731199031205794

0minus

211986211198731198881205791199041205791

1 minus 120579119904

+2119862111987311988812057901205791

1 minus 120579119904

+ 411986211198731199031205793

01205791minus 1198622

1198892

1205790(119883)

1198891198832minus (1 + 119862

1)1198892

1205791(119883 1198621)

1198891198832= 0

(30)


1205792(119887) = 0

1198891205792(0)

119889119883= 0 (31)

It is given by

1205792(119883 1198621 1198622)

=1

12(61198872

1198621119873119888minus 61198832

1198621119873119888+ 61198872

1198622

1119873119888minus 61198832

1198622

1119873119888

+ 61198872

1198622119873119888minus 61198832

1198621119873119903minus 61198832

1198622119873119888+ 51198874

1198622

11198732

119888

minus 61198872

1198832

1198622

11198732

119888+ 1198834

1198622

11198732

119888+ 61198872

1198621119873119903+ 151198874

1198622

1119873119888119873119903

+ 61198872

1198622

1119873119903minus 61198832

1198622

1119873119903+ 61198872

1198622119873119903minus 61198832

1198622119873119903

+ 101198874

1198622

11198732

119903minus 121198872

1198832

1198622

11198732

119903minus 181198872

1198832

1198622

1119873119888119873119903

+ 31198834

1198622

1119873119888119873119903minus 61198872

1198622

1119876119892+ 61198832

1198622

1119876119892minus 61198872

1198622119876119892

+ 21198834

1198622

11198732

119903minus 61198872

1198621119876119892+ 61198832

1198621119876119892+ 61198832

1198622119876119892

minus 51198874

1198622

1119873119888119876119892+ 61198872

1198832

1198622

1119873119888119876119892minus 1198834

1198622

1119873119888119876119892

minus 101198874

1198622

1119873119903119876119892+ 121198872

1198832

1198622

1119873119903119876119892minus 21198834

1198622

1119873119903119876119892

minus 61198872

1198621119873119888120579119904+61198832

1198621119873119888120579119904minus61198872

1198622

1119873119888120579119904+61198832

1198622

1119873119888120579119904

minus 61198872

1198622119873119888120579119904+61198832

1198622119873119888120579119904minus1198834

1198622

11198732

119888120579119904minus51198874

1198622

11198732

119888120579119904

+ 61198872

1198832

1198622

11198732

119888120579119904minus101198874

1198622

1119873119888119873119903120579119904+121198872

1198832

1198622

1119873119888119873119903120579119904

minus 61198872

11986221198731199031205794

119904minus 21198834

1198622

1119873119888119873119903120579119904minus 61198872

11986211198731199031205794

119904

+ 61198832

11986211198731199031205794

119904minus 61198872

1198622

11198731199031205794

119904

+ 61198832

1198622

11198731199031205794

119904+ 61198832

11986221198731199031205794

119904minus 51198874

1198622

11198731198881198731199031205794

119904



119904= 01 and 119876

119892= 0

119883119873119888= 01 119873

119888= 03 119873

119888= 1



119904= 03 and 119876

119892= 1

119883119873119903= 01 119873

119903= 03 119873

119903= 05


01 11848 11852 0033 11154 11193 0348 10516 10518 0019

02 11792 11801 0076 11119 11160 0367 10501 10503 0019

03 11698 11715 0145 11061 11104 0387 10477 10479 0019

04 11568 11590 0189 10979 11026 0426 10442 10445 0028

05 11400 11435 0306 10874 10923 0448 10398 10400 0019

06 11194 11237 0382 10746 10795 0453 10342 10345 0028

07 10952 10997 0409 10594 10641 0441 10275 10278 0029

08 10672 10713 0382 10419 10458 0373 10196 10198 0019

09 10354 10382 0269 10221 10245 0234 10104 10106 0019

10 1 1 0 1 1 0 1 1 0

+ 61198872

1198832

1198622

11198731198881198731199031205794

119904minus 1198834

1198622

11198731198881198731199031205794

119904minus 101198874

1198622

11198732

1199031205794

119904

+121198872

1198832

1198622

11198732

1199031205794

119904minus 21198834

1198622

11198732

1199031205794

119904)

(32)


is

120579 (119883 1198621 1198622) = 1205790(119883) + 120579

1(119883 1198621) + 1205792(119883 1198621 1198622) (33)



1= minus000002137 and119862

2= minus079986

for 119873119903= 03 119873

119888= 03 119876

119892= 04 120579

119904= 02 and 119887 = 1



120579 = 1 +1

2(minus0000002982 + 2982 times 10

minus6

1198832

)

+1

12(minus06696 + 0669596119883

2


1198834

)

(34)



119903119873119888 120579119904 and119876









00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X


119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X


119903 120579119904 and 119876

119892





119888 while





are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X


119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X


119888 120579119904 and 119876

119892


6 Conclusion


119903





119903 119873119888 119876119892 and 120579





References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



119904= 01 and 119876

119892= 0

119883119873119888= 01 119873

119888= 03 119873

119888= 1



119904= 03 and 119876

119892= 1

119883119873119903= 01 119873

119903= 03 119873

119903= 05


01 11848 11852 0033 11154 11193 0348 10516 10518 0019

02 11792 11801 0076 11119 11160 0367 10501 10503 0019

03 11698 11715 0145 11061 11104 0387 10477 10479 0019

04 11568 11590 0189 10979 11026 0426 10442 10445 0028

05 11400 11435 0306 10874 10923 0448 10398 10400 0019

06 11194 11237 0382 10746 10795 0453 10342 10345 0028

07 10952 10997 0409 10594 10641 0441 10275 10278 0029

08 10672 10713 0382 10419 10458 0373 10196 10198 0019

09 10354 10382 0269 10221 10245 0234 10104 10106 0019

10 1 1 0 1 1 0 1 1 0

+ 61198872

1198832

1198622

11198731198881198731199031205794

119904minus 1198834

1198622

11198731198881198731199031205794

119904minus 101198874

1198622

11198732

1199031205794

119904

+121198872

1198832

1198622

11198732

1199031205794

119904minus 21198834

1198622

11198732

1199031205794

119904)

(32)


is

120579 (119883 1198621 1198622) = 1205790(119883) + 120579

1(119883 1198621) + 1205792(119883 1198621 1198622) (33)



1= minus000002137 and119862

2= minus079986

for 119873119903= 03 119873

119888= 03 119876

119892= 04 120579

119904= 02 and 119887 = 1



120579 = 1 +1

2(minus0000002982 + 2982 times 10

minus6

1198832

)

+1

12(minus06696 + 0669596119883

2


1198834

)

(34)



119903119873119888 120579119904 and119876









00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X


119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X


119903 120579119904 and 119876

119892





119888 while





are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X


119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X


119888 120579119904 and 119876

119892


6 Conclusion


119903





119903 119873119888 119876119892 and 120579





References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


00 02 04 06 08 10100

105

110

115

120

125

130

135

140

120579

NM

Qg = 1 15 17 2

OHAM

Nr = 05

Nc = 05

120579s = 02

X


119903 119873119888 and 120579

119904

Nc = 01 04 07 1

00 02 04 06 08 10070

075

080

085

090

095

100

Nr = 02

Qg = 0

120579 120579s = 03

NMOHAM

X


119903 120579119904 and 119876

119892





119888 while





are fixed

Nr = 04 06 08 1

00 02 04 06 08 1005

06

07

08

09

10

Nc = 04

Qg = 03120579s = 005

120579

NMOHAM

X


119888 120579119904 and 119876

119892

120579s = 04 03 02 01

00 02 04 06 08 10090

092

094

096

098

100

Nr = 03

Nc = 03

Qg = 04120579

NMOHAM

X


119888 120579119904 and 119876

119892


6 Conclusion


119903





119903 119873119888 119876119892 and 120579





References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013



References































OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013


OperationsResearch

Advances in









Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences











Volume 2013

Series Solution for Steady Heat Transfer in a Heat-Generating Fin with Convection and Radiation

Documents

Transcript of Series Solution for Steady Heat Transfer in a Heat-Generating Fin with Convection and Radiation