Optimization of circular fins with variable thermal parameters

\ PERGAMON

Journal of The Franklin Institute 225B "0888# 66Ð84

Optimization of circular _ns with variable thermal parameters Lien!Tsai Yu\ Cha|o!Kuang Chen Department of Mechanical Engineering\ National Cheng!Kung University\ Tainan\ Taiwan 69090\ Republic of China "Received 03 September 0885^ accepted 14 February 0886#

Abstract The optimization of rectangular pro_le circular _ns with variable thermal conductivity and convective heat transfer coe.cients is discussed[ The linear variation of the thermal conductivity is considered to be of the form k  ka"0¦b"T−Ta##\ and the heat transfer coe.cient is assumed to vary according to an exponential function with the distance from the bore of the form h  hb exp "g"r−rb#:"re−rb##[ The nonlinear conductingÐconvectingÐradiating heat transfer equation is solved by the di}erential transformation method[ The e}ective of convectiveÐ radiative heat transfer at the _n tip is considered[ It is shown that\ considering the thermal conductivity and heat transfer coe.cient are both constant\ for a given _n volume\ the optimum _n length is almost independent of the _n base temperature for pure convection[ However\ for both convectionÐradiation and pure radiation\ the length of the optimum _ns for higher temperatures is shorter than the length of the _ns with lower temperatures[ Þ 0887 The Franklin Institute[ Published by Elsevier Science Ltd

0[ Introduction Extended surfaces are extensively used in various industrial applications ð0Ł[ Fins are employed to enhance the heat transfer between the primary surface and its convective\ radiative or convectiveÐradiative environment[ Naturally\ optimization of the _n shapes is of great interest for many engineering topics[ The optimization problem considered here focuses on _nding the shapes and dimen! sions of the _ns which has laid out two perspectives] one is to minimize the volume or mass for a given amount of heat dissipation\ and the other is to maximize the heat dissipation for a given volume or mass^ in other words\ we want to select a suitable

 Corresponding author[ Department of Mechanical Engineering\ Cheng Shiu Junior College of Tech! nology + Commerce\ Neau!Song\ Kaohsiung\ Taiwan 72294\ R[O[C[ 9905Ð9921:87:, ! see front matter Þ 0887 The Franklin Institute[ Published by Elsevier Science Ltd PII] S 9 9 0 5 Ð 9 9 2 1 " 8 6 # 9 9 9 1 0 Ð 4

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pro_le\ and then to determine the dimensions of the _ns which yield the maximum heat dissipation for a given volume and shape of the _ns ð1Ł[ Optimum shapes for thin _ns with power law temperature distribution rejecting heat by convection and radiation has been considered by Cobble ð2Ł and Wilkins ð3Ł[ Guceri and Maday ð4Ł used the minimum principle to search the least weight of conductingÐconvecting circular cooling _n with constant thermal parameters[ Razani and Ahmadi ð5Ł studied the problem of minimizing the maximum temperature of a purely conducting circular _n with heat generation via a variational technique[ Raz! elos and Imre ð6Ł presented the optimum dimensions of circular _ns with a pro_le of constant slope\ including the e}ect of the linear variation of the thermal conductivity and the heat transfer coe.cient which is assumed to vary according to a power law with distance from the bore[ Heggs\ Ingham and Manzoor ð7Ł used linearly varying heat transfer coe.cient to discuss the e}ects of nonuniform heat transfer from an annular _n of triangular pro_le[ Netrakanti and Huang ð8Ł examined the optimization of annular _ns with variable thermal parameters by Invariant Imbedding method[ In this paper\ we consider the optimization of circular _ns with a rectangular pro_le\ including the e}ect of the variable thermal conductivity "which depends on temperature linearly#[ The e}ect of a variable heat transfer coe.cient in the exponent form of radius is also considered[ It is shown that the heat transfer coe.cient increases towards the _n tip ð6\ 7Ł[ The nonlinear heat transfer equation for conductiveÐ convectiveÐradiative rectangular pro_le circular _ns was solved by the di}erential transformation method[ The e}ects of convectiveÐradiative heat rejecting at the _ns tip were considered[ Therefore\ from a practical point of view\ we feel that maximizing the heat dissipation for a given volume or mass can cause a better consequence[

1[ Differential transformation method 1[0[ Basic theory Let u"x# be an analytic function in a domain D and the Taylor series expansion of the function u"x# about ordinary point x  xi is well known to be of the form "x−xi#s dsu"x# s; s9 dxs

0 1



u"x#  s

[x W D[

"0#

xxi

When xi  9\ "0# is called the Maclaurin series of u"x# and has the form of xs dsu"x# s9 s; dxs 

u"x#  s

0 1

By the de_nition ð09Ł

x9

[x W D[

"1#

L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

U"s# 

Hs dsu"x# s; dxs

0 1

\

68

"2#

x9

and substitute eqn "2# into eqn "1#\ the result becomes 

u"x#  s s9

x s U"s#[ H

01

"3#

Hence\ the di}erential!transform pair in the variable x for the function u"x# are di}erential transform] Hs dsu"x# U"s#  s; dxs

0 1

\

"4#

x9

and inversion formula]  x s u"x#  s U"s#\ s9 H

01

"5#

where H is a constant "domain or subdomain#[ In the real application\ the function u"x# may be expressed in _nite terms[ From eqn "5#\ it becomes n

u"x#  s s9

x s U"s#[ H

01

"6#

s Equation "6# implies S sn¦0 "x:H# U"s# was neglected[

1[1[ Domain split In order to speed up the convergent rate and the accuracy of calculation\ the entire domain of x needs to be divided into subdomains[ To illustrate the domain division\ we consider an example of di}erential equation du"x# ¦u"x#  9\ 9 ¾ x ¾ L[ dx

"7#

By di}erential transformation\ eqn "7# becomes U"s¦0#  −

H U"s#[ s¦0

"8#

Now\ let Hi represent the subdomains interval as shown in Fig[ 0\ where SHi  L\ i  9\ 0\ 1\ [ [ [ [ In the _rst subdomain\ the function u9"x# is represented by

"09#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

Fig[ 0[ Domain split[

n

u9"x#  s s9

n x s x s U9"s#  U9"9#¦ s U9"s#\ H9 s0 H9

0 1

0 1

"00#

where 9 ¾ x ¾ H9 and the given left boundary condition is u9"9# "for the time domain\ it is initial condition#[ Setting x  9 in eqn "00#\ it yields U9"9#  u9"9#[

"01#

The _nial value of the function u9"x# is n

u9"H9#  s U9"s#[

"02#

s9

However\ u9"H9# is also the _rst value "left boundary condition# of function u0"x# in the second subdomain[ In the second subdomain\ the function u0"x# is n

u0"x#  s s9

n x−H9 s x−H9 s U0"s#  U0"9#¦ s U0"s#\ H0 H0 s0

0

1

0

1

H9 ¾ x ¾ H9¦H0[

"03#

In this subdomain\ if we take x¦H9 instead of x in eqn "03#\ it becomes n

u0"x#  s s9

n x s x s U0"s#  U0"9#¦ s U0"s#\ 9 ¾ x ¾ H0[ H0 s0 H0

0 1

From eqns "02# and "04#\ we have

0 1

"04#

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70

u0"9#  U0"9# n

 u9"H9#  s U9"s#[

"05#

s9

Similarly\ the right boundary value of u0"x# is exactly the same as the left boundary value of u1"x#[ As a result\ the formula for the left boundary value of the ith subdomain is ui"9#  Ui"9# n

 ui−0"Hi−0#  s Ui−0"s#[

"06#

s9

We conclude that the left boundary value of ui"x# can always be calculated from the transformed functions obtained in the previous subdomain[ In a similar manner\ the di}erential equation can be subsequently solved from one subdomain to the next subdomain\ by keeping the subdomain interval small enough to ensure a high con! vergent rate for the expansion series and the calculation accuracy[ 1[2[ Accuracy check To ensure the overall accuracy is satisfactory\ an accuracy check is executed after the second subdomain according to the following equation ð09Ł[ n

ui−0"9#  ui"9#¦ s s0

Hi−0 s Ui"s#[ Hi

0 1

"07#

The right hand side of eqn "07# may be calculated[ If the result is not close enough to the value of the left hand side of eqn "07#\ the processes of the transformation are repeated by increasing the number of subdomains[

2[ Analysis of _ns 2[0[ Governin` equation and boundary conditions The steady state energy equation for a conductiveÐconvectiveÐradiative rectangular pro_le circular _n with variable thermal parameters as shown in Fig[ 1 can be written as

0

1

dT 1 1s d k"T#r  h"r#r"T−Ta#¦ r"oT3−aT3e #\ rb ¾ r ¾ re\ dr dr W W

"08#

where k"T# is the thermal conductivity\ W the _n thickness\ h"r# the convective heat transfer coe.cient between the _n surface and the ambient\ r the radius of the _n\ Ta the ambient temperature\ Te the e}ective temperature of radiative surface except the

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

Fig[ 1[ Rectangular pro_le circular _ns[

_n\ a the absorptivity of the _n at Te\ o the emissivity of the _n at T\ and s the StefanÐ Boltzmann constant[ In order to consider the e}ect of possible di}erences in the emitting and absorbing spectra\ the emissivity\ o\ and the absorptivity\ a\ are not taken to be equal ð00Ł[ The thermal conductivity\ k of the _n material is assumed to be a linear function of temperature according to k"T#  ka"0¦b"T−Ta##\

"19#

where ka is the thermal conductivity at the ambient temperature\ Ta\ and b is the parameter describing the variation of thermal conductivity[ The heat transfer coe.cient is shown both theoretically and experimentally ð6\ 7Ł to increase towards the _n tip[ In order to estimate its e}ect on the optimal _n dimensions "at least qualitatively#\ we have assumed h as an exponential function of the _n radius according to h"r#  hb exp "g"r−rb#:"re−rb##\

"10#

where hb is the convective heat transfer coe.cient at the _n base\ rb the bore radius\ re the _n tip radius\ and g the parameter of heat transfer coe.cient\ where g  9 indicates a constant heat transfer coe.cient[ Equation "08# should be solved with the boundary conditions at the bore and the _n tip[ These boundary conditions are assumed as r  r b\ T  T b\ r  re\ −k"T#

dT  h"r#"T−Ta#¦s"oT3−aT3e #[ dr

Now\ we introduce the dimensionless parameters as follows] u

Tb Te r−rb r−rb T \ ub  \ ue  \ R   ^ L  re−rb[ Ta Ta Ta re−rb L

"11# "12#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

Then\ substituting eqns "19# and "10# into the eqn "08#\ the dimensionless form of governing eqn "08# and boundary condition eqns "11# and "12# become L"0−bTa#

du du d1u d1u ¦LbTau ¦rb"0−bTa# 1 ¦rbbTau 1 dR dR dR dR

¦L"0−bTa#R



d1u dR

1

¦LbTaRu

d1u dR

1

¦rbbTa

du 1 du ¦LbTaR dR dR

0 1

1

0 1

1L1rbhb 1L2hb "u exp "gR#−exp "gR##¦ "Ru exp "gR#−R exp "gR## Wka Wka ¦

1L1rbsT2a 3 1L2sT2a "ou −au3e #¦ "oRu3−au3e R#\ 9 ¾ R ¾ 0\ Wka Wka

"13#

R  9\ u  ub\

"14#

du Lhb LsT2a 3  "u−0# exp "gR#¦ "ou −au3e #[ dR ka ka

"15#

R  0\ −"0¦bTau−bTa# 2[1[ Temperature distribution

We divide the unit dimensionless _n length into four equivalent subdomains "one may take more or less#[ Hence\ the subdomains are H9  H0  H1  H2  03 "Fig[ 1#[ Via the inversion formula "5#\ the _rst four terms "one may take more terms# of the power series of the solutions of each domain are 2

u9"R#  s s9 2

u0"R#  s s9 2

u1"R#  s s9 2

u2"R#  s s9

R s U9"s#\ 9 ¾ R ¾ 9[14\ H9

"16#

R s U0"s#\ 9 ¾ R ¾ 9[14\ H0

"17#

R s U1"s#\ 9 ¾ R ¾ 9[14\ H1

"18#

R s U2"s#\ 9 ¾ R ¾ 9[14[ H2

"29#

0 0 0 0

1 1 1 1

Subdomain 0 ð9\ 03Ł] In this subdomain\ from the boundary eqn "14# and solution eqn "16#\ we obtained "Fig[ 1\ point 9#[ U9"9#  u9"9#  ub[ Via di}erential transformation\ the governing eqn "13# becomes

"20#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

L"0−bTa#

s s¦0 l¦0 U9"s¦0#¦LbTa s U9"s−l# U "l¦0# H9 H9 9 l9

"s¦0#"s¦1#

¦rb"0−bTa#

1 9

H s

¦L"0−bTa# s

s

U9"s¦1#¦rbbTa s U9"s−l# l9

"s¦0−l#"s¦1−l# H19

l9 s

l

¦LbTa s U9"s−l# s l9

"l¦0#"l¦1#

U9"l¦1#

U9"s¦1−l#Hd"l−0#

"l¦0−m#"l¦1−m#

m9

H19

H19

U9"l¦1−m#Hd"m−0#

s

"s¦0−l# l¦0 U9"s¦0−l# U9"l¦0# H H9 9 l9

¦rbbTa s s

l "s¦0−l# "l¦0−m# U9"s¦0−l# s U9"l¦0−m#Hd"m−0# H9 h9 l9 m9

¦LbTa s



1L1rbhb s "gH9#l "gH9#s − s U9"s−l# Wka l9 l; s;

0 0

1

¦

l s 1L2hb s "gH9#l−m "gH9#s−l s U9"s−l# s Hd"m−0#− s Hd"l−0# Wka l9 m9 "l−m#; l9 "s−l#;

¦

s l m 1L1rbsT2a o s U9"s−l# s U9"l−m# s U9"m−n#U9"n#−au3e d"s# Wka l9 m9 n9

¦

s l m n 1L2sT2a o s U9"s−l# s U9"l−m# s U9"m−n# s U9"n−q#Hd"q−0# Wka l9 m9 n9 q9

1 1

0

0

1

−au3e Hd"s−0# \ "21#

where d"s# 

0 for s  9

6

9 otherwise[

Setting s  9\ 0\ from the equation "21#\ we may obtain "0¦bTaU9"9#−bTa#U9"1#  −

H9 L "0¦bTaU9"9#−bTa#U9"0# 1rb

0 H19L1hb H19L1sT2a − bTaU19"0#¦ "U9"9#−0#¦ "oU39"9#−au3e #\ 1 Wka Wka "0¦bTaU9"9#−bTa#U9"2#

"22#

L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

− −

74

H9L "0¦bTaU9"9#−bTa#U9"1# 2rb

H 9L H19L1hb bTaU19"0#−bTaU9"0#U9"1#¦ "U9"0#¦gH9U9"9#−gH9# 2rb 2Wka

¦

H29L2hb 3H19L1osT2a 2 "U9"9#−0#¦ U9"9#U9"0# 2rbWka 2Wka

¦

H29L2sT2a "oU39"9#−au3e #[ 2rbWka

"23#

Subdomain 1 ð03\ 01Ł] At the point 0 "Fig[ 1#\ two conditions should be satis_ed\ that is u9"R#=RH9  u0"R#=R9\ and

du9 dR

b



RH9

du0 dR

b

"24#

[

R9

By eqns "16#\ "17# and "24#\ we have 2

U0"9#  s U9"s#\

"25#

s9 2

U0"0#  s sU9"s#[

"26#

s0

In this subdomain uses R¦9[14 instead of R in the eqn "13#\ and then we obtain L"0−bTa#

du0 d1u0 du0 d1u0 ¦r bT u ¦LbTau0 ¦rb"0−bTa# b a 0 dR dR dR1 dR1

¦L"0−bTa#R ¦rbbTa



d1u0

d1u0 L d1u0 L d1u0 "0−bT bT ¦ # ¦LbT Ru ¦ u a a 0 a 0 dR1 3 dR1 dR1 3 dR1

du0 1 du0 1 L du0 ¦LbTaR ¦ bTa dR dR 3 dR

0 1 01

0 1

1

0 1

g g 1L1rbhb 1L2hb exp "u0 exp "gR#−exp "gR##¦ exp Wka 3 Wka 3

0 10

Ru0 exp "gR#

0 1L1rbsT2a 3 0 "ou0−au3e # ¦ u0 exp "gR#−R exp "gR#− exp "gR# ¦ 3 3 Wka

1

¦

1L2sT2a 0 0 oRu30¦ ou30−au3e R¦ Wka 3 3

0

0 11

[

Via di}erential transformation\ the eqn "27# becomes

"27#

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L"0−bTa#

s¦0 U "s¦0# H0 0 s

¦LbTa s U0"s−l# l9 s

¦rbbTa s U0"s−l#

l¦0 "s¦0#"s¦1# U "l¦0#¦rb"0−bTa# U9"s¦1# H0 9 H10 "l¦0#"l¦1# H10

l9 s

¦L"0−bTa# s

U0"l¦1#

"s¦0−l#"s¦1−l# H10

l9

U0"s¦1−l#H0d"l−0#

L "s¦0#"s¦1# ¦ "0−bTa# U0"s¦1# 3 H10 s

l

"l¦0−m#"l¦1−m# U0"l¦1−m#H0d"m−0# m9 H10

¦LbTa s U0"s−l# s l9

s "l¦0#"l¦1# L U0"l¦1# ¦ bTa s U0"s−l# 3 l9 H10 s

"s¦0−l# l¦0 U0"s¦0−l# U "l¦0# H0 H0 0 l9

¦rbbTa s s

l "s¦0−l# "l¦0−m# U0"s¦0−l# s U0"l¦0−m#H0d"m−0# H0 H0 l9 m9

¦LbTa s

s L "s¦0−l# l¦0 ¦ bTa s U0"s¦0−l# U "l¦0# 3 H0 H0 0 l9



g 1L1rbhb exp Wka 3

¦

0 10 0 10

g 1L2hb exp Wka 3

s

s U0"s−l#

l9 s

"gH0#l "gH0#s − l; s;

1

"gH0#l−m H0d"m−0# m9 "l−m#; l

s U0"s−l# s

l9

¦

0 s "gH0#l s "gH0#s−l 0 "gH0#s s U0"s−l# −s H0d"l−0#− 3 l9 l; 3 s; l9 "s−l#;

¦

s l m 1L1rbsT2a o s U0"s−l# s U0"l−m# s U0"m−n#U0"n#−au3e d"s# Wka l9 m9 n9

1

0

1

¦

s l m n 1L2sT2a o s U0"s−l# s U0"l−m# s U0"m−n# s U0"n−q#H0d"q−0# Wka l9 m9 n9 q9

¦

l m o s 0 s U0"s−l# s U0"l−m# s U0"m−n#U0"n#−au3e H0d"s−0#¦ d"s# 3 l9 3 m9 n9

0

0

11

[

"28#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

Settings s  9\ 0\ from eqn "28#\ we get

0

1rb 1 0

H

¦

0

1H10

1

"0−bTa¦bTaU0"9##U0"1#

L "0−bTa¦bTaU0"9##U0"0# H0

− −

L

0

L 1L1rbhb 1 ¦ bT U "0#¦ exp "g:3#"U0"9#−0# 0 a Wka H10 3H10

1

rb

¦

1L1rbsT2a L2hb exp "g:3#"U0"9#−0#¦ "oU30"9#−au3e # 1Wka Wka

¦

L2sT2a "oU30"9#−au3e #\ 1Wka

5rb H10

¦

−

2L 1H10

1

"39#

"0−bTa¦bTaU0"9##U0"2#

3L "0−bTa¦bTaU0"9##U0"1# H0

0

1

−

1L 5rb 2L bTaU10"0#− 1 ¦ 1 bTaU0"0#U0"1# H0 H0 1H0

¦

1L1rbhb exp "g:3#"U0"0#¦gH0"U0"9#−0## Wka

¦

1L2hb 0 0 exp "g:3# H0"U0"9#−0#¦ gH0"U0"9#−0#¦ U0"0# Wka 3 3

¦

7oL1rbsT2a 2 1L2sT2a U0"9#U0"0#¦ "H0"oU30"9#−au3e #¦oU20"9#U0"0##[ Wka Wka

0

1 "30#

Using the similar procedures\ the next two subdomain equations can be derived[ Subdomain 2 ð01\ 23Ł] 2

U1"9#  s U0"s#\

"31#

s9 2

U1"0#  s sU0"s#\ s0

0

1rb 1 1

H

¦

L H11

1

"0−bTa¦bTaU1"9##U1"1#

"32#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

− −

0

5rb 1 1

H

¦

−

L "0−bTa¦bTaU1"9##U1"0# H1

0

rb 1 1

H

¦

L 1H11

1

bTaU11"0#

¦

1L1rbhb exp "g:1#"U1"9#−0# Wka

¦

L2hb 1L1rbsT2a exp "g:1#"U1"9#−0#¦ "oU31"9#−au3e # Wka Wka

¦

L2sT2a "oU31"9#−au3e #\ Wka

2L 1H11

1

"33#

"0−bTa¦bTaU1"9##U1"2#

3L "0−bTa¦bTaU1"9##U1"1# H1

0

1

−

5rb 2L 1L bTaU11"0#− 1 ¦ 1 bTaU1"0#U1"1# H1 H1 1H1

¦

1L1rbhb exp "g:1#"U1"0#¦gH1"U1"9#−0## Wka

¦

1L2hb 0 0 exp "g:1# H1"U1"9#−0#¦ gH1"U1"9#−0#¦ U1"0# Wka 1 1

¦

7oL1rbsT2a 2 1L2sT2a U1"9#U1"0#¦ "H1"oU31"9#−au3e #¦1oU21"9#U1"0##[ Wka Wka

0

1 "34#

Subdomain 3 ð23\ 0Ł] 2

U2"9#  s U1"s#\

"35#

s9 2

U2"0#  s sU1"s#\ s0

0

1rb 1 2

H

¦

2L 1H12

−

1

"0−bTa¦bTaU2"9##U2"1#

L "0−bTa¦bTaU2"9##U2"0# H2

"36#

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L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

0

−

rb

H

1 2

¦

2L 3H

1 2

1 0 1

bTaU12"0#¦

2 1L1rbhb exp g "U2"9#−0# Wka 3

0 1

2 L2hb 2 1L1rbsT2a ¦ = exp g "U2"9#−0#¦ "oU32"9#−au3e # 1 Wka 3 Wka 2 L2sT2a ¦ = "oU32"9#−au3e #\ 1 Wka

0

5rb 1 2

H

¦

8L 1H12

−

1

"37#

"0−bTa¦bTaU2"9##U2"2#

3L "0−bTa¦bTaU2"9##U2"1# H2

0 0 1 0 10

1

−

8L 1L 5rb bTaU12"0#− 1 ¦ 1 bTaU2"0#U2"1# H2 H2 1H2

¦

1L1rbhb 2 exp g "U2"0#¦gH2"U2"9#−0## Wka 3

¦

1L2hb 2 exp g Wka 3

¦

7oL1rbsT2a 2 1L2sT2a U2"9#U2"0#¦ "H2"oU32"9# Wka Wka

1

2 2 H2"U2"9#−0#¦ gH2"U2"9#−0#¦ U2"0# 3 3

−au3e #¦2oU22"9#U2"0##[

"38#

Finally "at point 3#\ the boundary condition eqn "15# of the _n tip becomes R  H2\ −"0−bTa¦bTau2"R##

du2"R# Lhb exp "gH2#"u2−0#  dR ka ¦

LsT2a 3 "ou2−au3e #[ ka

"49#

Substituting eqn "29# into eqn "49#\ we have

0

2

0−bTa¦bTa s U2"s# s9

10

1

0

1

2 0 2 Lhb s sU2"s# ¦ exp "gH2# s U2"s#−0 H2 s0 ka s9

¦

3 2 LsT2a o s u2"s# −au3e  9[ ka s9

00

1

1

"40#

By eqns "20#\ "22#\ "23#\ "25#\ "26#\ "39#Ð"38# and "40#\ we can solve for U9"0#[ After

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substituting U9"0# into eqns "22#Ð"23#\ "25#Ð"26# and "39#Ð"38#\ we then obtain Ui" j#^ i\ j  9\ 0\ 1\ 2[ Finally\ we _nd out the dimensionless temperature distribution of each subdomain u9"R#\ u0"R#\ u1"R# and u2"R# via eqns "16#Ð"29#[ 2[2[ Heat transfer rate The heat dissipation is de_ned by applying the Fourier|s law at the _n base[ q  −kA

dT dr

b

−

rrb

6

1prbWkaTa du9 "0¦bTau9−bTa# L dR

7

[

"41#

R9

Substituting eqn "16# into eqn "41#\ we have q−

1prbWkaTa"0¦bTaU9"9#−bTa#U9"0# [ H9L

"42#

2[3[ Fin optimization The optimization of the cooling properties of _ns can be made by either minimizing the volume "weight# for any required heat dissipation or maximizing the heat dis! sipation for any given _n volume[ The solutions in the _rst group of problems result in _n shapes with curved surfaces[ Such optimum shapes are di.cult and expensive to fabricate[ Therefore\ from a practical point of view\ in this study the second group method is used[ The volume of _n is V  pW"re−rb#"re¦rb#  pWL"L¦1rb#\

"43#

where V is the _n volume and L  re−rb[ From eqn "42#\ the dimensionless heat dissipation rate is de_ned as N

q 1pW"0¦bTaU9"9#−bTa#U9"0# \ − rbkaTa H 9L

"44#

where N the dimensionless parameter of an optimum _n[ The maximum heat dissipation value expressed the optimum _n characteristics and the dimensions are the optimum _n con_guration[ The optimization procedure is to keep the _n volume constant\ and then by eqns "43# and "44# to search the dimensions of the _n thickness\ W\ and the _n length\ L\ for that the maximum value of N reaches[

3[ Results and discussions In order to show the optimization calculations of a conductingÐconvectingÐrad! iating rectangular pro_le circular _n[ The following values are used as an example only] ka  52[8 W:mK\ b  9\ Ta  Te  299 K\ hb  19 W:m1 K\ g  9\

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80

Fig[ 2[ The optimum dimensionless parameter\ N\ as a function of re:rb[

o  a  0\s  4[56×09−7 W:m1 K3\ V  9[999901 m2\ rb  9[91 m[ Figure 2 shows the pure convection optimum dimensionless parameter\ N\ as a function of the value of re:rb\ for three di}erent base temperatures of Tb  349 K\ Tb  599 K and Tb  899 K[ It presents that the optimum _n dimensions are almost independent of the _n base temperature\ Tb\ for pure convection[ Figures 3 and 4 show the pure radiation and convectionÐradiation optimum dimen! sionless parameter\ N\ as a function of the ratio of tip radius to bore radius\ re:rb\ for three di}erent base temperatures of Tb  349 K\ Tb  599 K and Tb  899 K[ It states that the optimum _ns length for higher base temperatures are shorter than the length of the _ns with lower base temperatures[ The curves for optimum dimensionless parameter\ N\ as a function of the value of re:rb are similarly for both pure radiation and convectionÐradiation _ns[ Figure 5 presents the temperature distribution for the optimum _n of the di}erent _n base temperature\ Tb\ for convectingÐradiating heat dissipation[ Figure 6 states the pure radiation optimum dimensionless parameter\ N\ at di}erent e}ective temperature\ Te[ It presents that the optimum _ns length are almost inde! pendent of the e}ective temperature\ Te[ Figure 7 shows the optimum dimensionless parameter\ N\ for di}erent heat transfer coe.cients\ hb\ at Tb  349 K in pure convection heat dissipation[ It states that the optimum _ns length at higher heat transfer coe.cient\ hb\ are shorter than the length of the _ns with lower heat transfer coe.cients[ From the results of the analysis\ it appears that the present method is a simple\ fast and straightforward method[ Hence\ the di}erential transformation method is one of

81

L[!T[ Yu\ C[!K[ Chen:Journal of The Franklin Institute 225B "0888# 66Ð84

Fig[ 3[ The optimum dimensionless parameter\ N\ as a function of re:rb[

Fig[ 4[ The optimum dimensionless parameter\ N\ as a function of re:rb[

the powerful techniques for solving nonlinear heat transfer equations for _ns with nonlinear boundary conditions at the tip[ Although the temperature distribution solution requires numerical calculations\ the solution is a close form series ðsee eqns

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82

Fig[ 5[ The temperature distribution for the optimum _n for di}erent value of base temperatures[

Fig[ 6[ The optimum dimensionless parameter\ N\ as a function of re:rb for di}erent values of the e}ective temperature\ Te[

"16#Ð"29#Ł[ It is particularly advantageous when we want to _nd di}erentiation or integration[ Hence\ it does not require curve _tting[ In _n optimization\ we used Fourier|s law to obtain the maximum heat transfer

83

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Fig[ 7[ The optimum dimensionless parameter\ N\ as a function of re:rb for di}erent values of heat transfer coe.cient\ hb[

rate[ So\ the close form series solution may be directly di}erentiated to get the temperature distribution gradient at the _n base[

4[ Conclusion In pure convection heat dissipation\ the length of the optimum _n is almost inde! pendent of the _ns base temperature\ Tb[ But for a given base temperature\ Tb\ the length of the optimum _ns at higher heat transfer coe.cient\ hb\ is shorter than that with a lower heat transfer coe.cient[ In convectionÐradiation and pure radiation heat rejection\ the optimum _ns length at higher base temperature is shorter than that with a lower base temperature[ For pure radiation\ the length of the optimum _ns is almost independent of di}erent e}ective temperatures\ Te\ at a given base temperature\ Tb[ From the results of the analysis\ it appears that the present method is a simple\ fast and straightforward method[ Hence\ the di}erential transformation method is one of the powerful technique for solving nonlinear heat transfer equations for _ns with nonlinear boundary conditions at the tip[ Although the temperature distribution solution requires numerical calculations\ the solution is a close form series ðsee eqns "16#Ð"29#Ł[ It is particularly advantageous when we want to _nd di}erentiation or integration[ Hence\ it does not require curve _tting[

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84

References ð0Ł Q[D[ Kern\ D[A[\ Kraus\ Extended Surface Heat Transfer[ McGraw!Hill\ New York\ 0861[ ð1Ł A[ Aziz\ Optimum dimensions of extended surfaces operating in a convective environment[ ASME Applied Mechanics Review\ 0881\ 34 "4#[ ð2Ł M[H[ Cobble\ Optimum _n shape[ Journal of the Franklin Institute\ 0860\ 0 "3#[ ð3Ł J[E[ Wilkins\ Jr[\ Optimum shapes for _ns rejecting heat by convection and radiation[ Journal of the Franklin Institute\ 0863\ 186 "0#[ ð4Ł S[ Guceri\ C[J[ Maday\ A least weight circular cooling _n[ ASME Journal of Engineering for Industry\ 0864\ 0089Ð0082[ ð5Ł A[ Razani\ G[ Ahmadi\ On optimization of circular _ns with heat generation[ Journal of the Franklin Institute\ 0866\ 292 "1#[ ð6Ł P[ Razelos\ K[ Imre\ The optimum dimensions of circular _ns with variable thermal parameters[ ASME Journal of Heat Transfer\ 0879\ 091\ 319Ð314[ ð7Ł P[J[\ Heggs\ D[B[ Ingham\ M[ Manzoor\ The e}ects of nonuniform heat transfer from an annular _n of triangular pro_le[ ASME Journal of Heat Transfer\ 0870\ 092\ 073Ð074[ ð8Ł M[N[ Netrakanti\ C[L[D[ Huang\ Optimization of annular _ns with variable thermal parameters by invariant imbedding[ ASME Journal of Heat Transfer\ 0874\ 096\ 855Ð869[ ð09Ł X[ Zhou\ Di}erential transformation and its applications for electrical circuits[ Wuhan China] Huazhong University Press\ in Chinese\ 0875[ ð00Ł A[ Razani\ H[ Zohoor\ Optimum dimensions of convectiveÐradiative spines using a temperature correlated pro_le[ Journal of the Franklin Institute\ 0880\ 217 "3#\ 360Ð375[