Energy Conversion and Management 50 (2009) 2622–2631

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Energy Conversion and Management

journal homepage: www.elsevier .com/locate /enconman

Performance and optimum design of convective–radiative rectangularfin with convective base heating, wall conduction resistance, andcontact resistance between the wall and the fin base

Abdul Aziz a,*, Arlen B. Beers-Green b

a Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, United Statesb ARES Corporation, 851 University Blvd, S, Suite 100, Albuquerque, NM 87106, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 January 2008Received in revised form 17 November 2008Accepted 21 June 2009Available online 12 July 2009

Keywords:Rectangular finSurface convection–radiationConvective base heatingOptimum design

0196-8904/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.enconman.2009.06.017

* Corresponding author. Tel.: +1 509 323 3540; faxE-mail address: aziz@gonzaga.edu (A. Aziz).

This paper investigates the performance and optimum design of a longitudinal rectangular fin attached toa convectively heated wall of finite thickness. The exposed surfaces of the fin lose heat to the environ-mental sink by simultaneous convection and radiation. The tip of the fin is assumed to lose heat by con-vection and radiation to the same sink. The analysis and optimization of the fin is conducted numericallyusing the symbolic algebra package Maple. The temperature distribution, the heat transfer rates, and thefin efficiency data is presented illustrating how the thermal performance of the fin is affected by the con-vection-conduction number, the radiation-conduction number, the base convection Biot number, theconvection and radiation Biot numbers at the tip, and the dimensionless sink temperature. Charts are pre-sented showing the relationship between the optimum convection-conduction number and the optimumradiation-conduction number for different values of the base convection Biot number and dimensionlesssink temperature and fixed values of the convection and radiation Biot numbers at the tip. Unlike the fewother papers which have applied the Adomian’s decomposition and the differential quadrature elementmethod to this problem but give illustrative results for specific fin geometry and thermal variables, thepresent graphical data are generally applicable and can be used by fin designers without delving intothe mathematical details of the computational techniques.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The performance and optimum design of longitudinal fins ofrectangular profile with simultaneous surface convection and radi-ation have been studied in numerous publications because of theimportance of such studies in many applications. A comprehensivereview of the literature on this topic has been provided by Krauset al. [1] and Aziz and Kraus [2] where numerous papers are citedand need not be repeated here. However, with the exception of fewrecent papers, the studies are based on the assumption of a con-stant and known fin base temperature. In many physical situations,the fin is attached to one side of a wall of finite thickness while theother side of the wall is in contact with a hot fluid from which heattransmitted through the wall is ultimately rejected by convectionand radiation from the surface of the fin to the environment (sink).Aziz [3,4] found that the convection resistance of the hot fluid andthe conduction resistance of the primary surface significantly af-fect the performance and optimum design of convecting fins ofrectangular, triangular and concave parabolic profiles. Later Ma

ll rights reserved.

: +1 509 323 5871.

and Chung [5] adopted the same model and used a golden sectionsearch technique to establish optimum dimensions of convective–radiative rectangular fins. Subsequently, Chung and Zhou [6] ex-tended the analysis in [5] to a radial (annular) fin of trapezoidalprofile and included the contact resistance between the wall andthe fin base in addition to the convective resistance of the hot fluidand the wall conduction resistance. More recently, Chiu and Chen[7] utilized Adomian’s decomposition procedure to evaluate theheat transfer characteristics of a convecting–radiating longitudinalfin of rectangular profile. They improved upon the previously citedworks by considering a convective–radiative fin tip (instead of aninsulated fin tip) and allowing the thermal conductivity of the finto vary with temperature. However, no optimization analysis wasperformed. The optimization study omitted by Chiu and Chen [7]was later conducted by Malekzadeh et al. [8] who used the differ-ential quadrature element method (DQEM) in conjunction with thegolden section search for the optimum design. To check the accu-racy of the DQEM, they also used Adomian’s decomposition proce-dure, Taylor transformation technique [9], and the finite differencemethod and claimed the DQEM was computationally superior to fi-nite difference method. Chiu and Chen [7], Malekzadeh et al. [8]and Yu and Chen [9] all illustrate their procedures by citing and

Nomenclature

a thermal conductivity parameter, dimensionlessAc fin cross-sectional area, m2

Bi Biot number for base convection, hL/k, dimensionlessNt, 1 convection Biot number at fin tip, hcL/k, dimensionlessNt, 2 radiation Biot number at fin tip, erLT3

f =k, dimensionless

h lumped heat transfer coefficient, 1hfþ dw

kwþ R00t;c

h i�1

hc convection coefficient of the sink fluid, W/m2 Khf convection coefficient of the base fluid, W/m2 Khr radiation heat transfer coefficient, W/m2 Kk fin, thermal conductivity at temperature T, W/m Kks fin thermal conductivity at temperature Ts, W/m Kkf thermal conductivity of the base fluid, W/m Kkw thermal conductivity of the wall, W/m KL fin length, mNc convection-conduction number, hcPL2/ksAc, dimension-

lessN* linearized radiation-conduction number, hrPL2/ksAc

dimensionless

Nr radiation-conduction number, erPL2T3f =ksAc , dimen-

sionlessP fin perimeter, mq fin heat transfer rate, Wqideal ideal fin heat transfer rate, WQ dimensionless heat transfer rate, qL/ksAcTf

T fin temperature at any location x, KTb fin base temperature, KTf temperature of the base fluid, KTs sink temperature, Kw fin thickness, mx axial distance measured from the base of the fin, mX dimensionless distance, x/Le fin surface emissivity, dimensionlessr Stefan–Boltzmann constant, 5.67 � 10�8 W/m2 K4

d wall thicknessh dimensionless temperature, T/Tf

hs dimensionless sink temperature, Ts/Tf

opt optimum

A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631 2623

discussing the numerical results for a single fin with specifiedmaterial properties, base convection thermal parameters, andenvironment conditions. Because of the specific nature of the re-sults, the designers need to use these computational proceduresfor their own design calculations. Furthermore, none of the paperscited provide any information on the efficiency of the fin which is aquantity of fundamental interest.

The present work therefore has three objectives. The first is tomodify the mathematical model in [7–9] to include the effects ofwall conduction resistance and the contact resistance betweenthe wall and the fin base. The second one is to generate perfor-mance and design charts that are general purpose and readilyusable rather than the limited to one fin example as in [7–9]. Thethird objective is to demonstrate that such information can be eas-ily generated (with a few lines of code) by using the symbolic alge-bra package Maple which offers a highly accurate and extensivelytested fourth–fifth order Runge–Kutta–Fehlberg algorithm withautomatic step size control [10] for solving nonlinear boundary va-lue problems. The same Maple program can be used to derive theoptimum fin design information.

Fig. 1. Convecting–radiating fin with convective base heating.

2. Mathematical model

Consider a rectangular longitudinal fin of length L, thickness w,cross-sectional area Ac, perimeter P, thermal conductivity k andsurface emissivity e attached to a wall of thickness dw and thermalconductivity kw as shown in Fig. 1. The contact resistance betweenthe wall and the fin is R00t;c . The rear side of the wall is in contactwith a fluid at temperature Tf which heats the wall through a con-vective heat transfer coefficient hf. The heat conducted through thebase of the fin is dissipated by convection (characterized by a con-vection heat transfer coefficient hc) and radiation to the environ-ment. For both convective and radiative heat dissipations, thesink temperature is assumed to be the same, i.e. Ts. The tip of thefin also convects and radiates heat to the same sink. The fin is as-sumed to contain no internal heat sources although a source termcan be easily incorporated in the differential equation if necessary.

The thermal conductivity of the fin k is assumed to vary linearlywith temperature, that is,

k ¼ ks½1þ aðT � TsÞ� ð1Þ

where a is a constant and ks is the thermal conductivity at the sinktemperature Ts. To keep reasonable number of variables in the anal-ysis, the convection resistance between the hot fluid and the wall,the conduction resistance of the wall, and the contact resistance be-tween the wall and the fin are lumped together by defining a con-vection heat transfer coefficient h as follows:

h ¼ 1hfþ dw

kwþ R00t;c

� ��1

ð2Þ

For one-dimensional conduction in the fin, the energy equationand the boundary conditions are as follows.

ddx

ksð1þ aðT � TsÞÞdTdx

� �� hcPðT � TsÞ � erPðT4 � T4

s Þ ¼ 0 ð3Þ

x ¼ 0; �kdTdx¼ hðTf � TÞ ð4Þ

x ¼ L; �kdTdx¼ hcðT � TsÞ þ erðT4 � T4

s Þ ð5Þ

The fin heat transfer rate q equals the heat conduction rate atthe base of the fin which is

Fig. 2. Effect of Bi on the temperature distribution in a fin of constant thermalconductivity with an insulated tip. Nc = 1, Nr = 1, and hs = 0.2.

2624 A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631

q ¼ �ks ½1þ aðT � TsÞ�AcdTdx

� �x¼0

ð6Þ

The heat transfer rate also equals the heat transmitted from thehot fluid to the base of the fin which is given by

q ¼ hAcðTf � TbÞ ð7Þ

The ideal fin heat transfer qideal is realized if the entire fin sur-face was at the base temperature and may be found as

qideal ¼ hcPLðTb � TsÞ þ erPLðT4b � T4

s Þ þ hcAcðTb � TsÞ

þ erAcðT4b � T4

s Þ ð8Þ

The fin efficiency g is defined as the ratio of q and qideal.

g ¼ qqideal

ð9Þ

We now introduce the following dimensionless quantities

h¼T=Tf ; hs¼Ts=Tf ; X¼x=L; a¼aTf ; Nc¼hcPL2=ksAc; Nr¼erPL2=ksAc

Bi¼hL=ks; Nt;1¼hcL=ks; Nt;2¼erLT3f =ks; Q ¼qL=AcksTf ;

Qideal¼qidealL=AcksTf

ð10Þ

to convert Eqs. (3)–(8) into the following dimensionless equations.

ddXð1þ aðh� hsÞÞ

dhdX

� �� Ncðh� hsÞ � Nrðh4 � h4

s Þ ¼ 0 ð11Þ

X ¼ 0; �½1þ aðh� hsÞ�dhdX¼ Bið1� hÞ ð12Þ

X ¼ 1; �½1þ aðh� hsÞ�dhdX¼ Nt;1ðh� hsÞ þ N2;tðh4 � h4

s Þ ð13Þ

The dimensionless heat conduction rate at the base of the fin orthe fin heat transfer rate Q may be written using Eq. (6) as

Q ¼ � ½1þ aðh� hsÞ�dhdX

� �X¼0

ð14Þ

The use of Eq. (7) gives the heat transfer rate Q as

Q ¼ Bið1� hbÞ ð15Þ

Similarly, the dimensionless ideal fin heat transfer rate Qideal

may be expressed as

Q ideal ¼ ðNc þ Nt;1Þðhb � hsÞ þ ðNr þ Nt;2Þðh4b � h4

bÞ ð16Þ

Elimination of hb between Eqs. (15) and (16) gives

Q ideal ¼ ðNc þ N1;tÞ 1� QBi� hs

� �þ ðNr

þ Nt;2Þ 1� QBi

� �4

� h4s

" #ð17Þ

The fin efficiency g may now be obtained as

g ¼ Q

ðNc þ Nt;1Þ 1� QBi� hs

� �þ ðNr þ Nt;2Þ 1� Q

Bi

� �4 � h4s

h i ð18Þ

where Q is evaluated from Eq. (14).

3. Fin performance results

Eqs. (11)–(13) constitute a two-point nonlinear boundary valueproblem. Both boundary conditions (12) and (13) are nonlinear andof the third kind. The commercially available computational soft-ware Maple has a powerful and well tested routine based on theRunge–Kutta–Fehlberg fourth–fifth order method for solvingboundary value problems. Appendix A shows how Eqs. (11)–(13)can be implemented in a Maple worksheet. As can be seen, the in-

put consists of a few lines of code including the plot command. Asample output for h, Q, and g with the fin and environment param-eters fixed at a = 0.2, Nc = 1, Nr = 1, Bi = 1, hs = 0.2, Nt, 1 = 0.1, Nt, 2 =0.1 is also included with the worksheet. Both the ratios Nc/Nt, 1

and Nr/Nt, 2 equal the ratio of fin surface area PL and the fincross-sectional Ac, i.e. PL/Ac which is usually of the order of 10(slender fins). The amount of time needed to create the worksheetwas only few minutes. This is a very modest effort compared withthe computational effort needed to code the Adomian’s decompo-sition procedure [7] or the DQEM [8] or Taylor transformation [9].

The accuracy of Maple results was tested against the analyticalsolutions for a pure convection fin of constant thermal conductiv-ity and no tip heat loss. The analytical solutions are [3,4]

h ¼ 1þ Biðhf � 1Þ cosh NcXNc sinh Nc þ Bi cosh Nc

ð19Þ

Q ¼ Biðhf � 1ÞNc sinh Nc

Nc sinh Nc þ Bi cosh Ncð20Þ

The Maple results agreed with the analytical results within 0.2%and hence validated the accuracy of the Maple routine used.

Fig. 2 shows how the temperature distribution in a fin of con-stant thermal conductivity with an insulated tip (a = 0, Nt, 1 = 0,Nt, 2 = 0) is affected by the variation of Bi characterizing the baseconvection process, the wall conduction resistance, and the contactresistance between the wall and the fin base. For this study the val-ues of other variables were kept fixed at Nc = 1. Nr = 1, and hs = 0.2.For the fixed wall and contact resistances, the increase in Bi indi-cates an increase in the value of hf. For a fixed value of hf, the in-crease in Bi indicates a reduction in the wall conductionresistance and/or the contact resistance between the wall andthe fin base. In all three cases, the increase in Bi drives increasedheat flow to the base of the fin which results in the elevation oftemperature throughout the fin. The increase in heat flow withthe increase in Bi is shown in Fig. 3. The fin efficiency however de-creases as Bi increases (Fig. 3). Both curves in Fig. 3 tend to level off

Fig. 3. Effect of Bi on the fin heat transfer rate and fin efficiency. Nc = 1, Nr = 1, hs = 0.2.

A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631 2625

beyond Bi = 10 and asymptotically approach the values for a finwith constant base temperature equal to Tf.

To study the effect of varying the convection-conduction num-ber Nc on the performance of the fin, the other variables were keptfixed at a = 0, Nt, 1 = 0, Nt, 2 = 0, Bi = 1, Nr = 1, hs = 0.2. Fig. 4 showsthe temperature distributions for Nc = 1, 2, 3, and 4. The corre-sponding data for the base heat conduction rate and the fin effi-ciency are provided in Fig. 5. As the surface convection getsstronger, the temperatures in the fin get increasingly depressed(Fig. 4) and the heat flow through the base of the fin increasesbut the efficiency decreases (Fig. 5). The results of Figs. 4 and 5are consistent with the well known performance characteristicsof pure convective fins [11].

Fig. 4. Effect of Nc on the temperature distribution in a fin of constant thermalconductivity with an insulated tip. Bi = 1, Nr = 1, and hs = 0.2.

Figs. 6 and 7 illustrate the effect of varying the radiation-con-duction parameter Nr on the performance of the fin with a = 0, Nt, 1

= 0, Nt, 2 = 0, Bi = 1, hs = 0.2. The pattern is similar to that observedin Figs. 4 and 5. It may, however, be noted that the heat flowthrough the base is not as strongly influenced by the increase insurface radiation strength (Fig. 7) as it is by the increase in surfaceconvection strength (Fig. 5). In order to quantify this observationfurther, the base conduction heat transfer rates were calculatedfor pure convective and pure radiative fins. These results collectedin Table 1 show that if the heat transfer rate produced by a pureconvective fin with a certain value of Nc is to be replicated by apure radiative fin, the radiation number Nr would have to be setapproximately equal to the square of Nc. For example, a dimension-less heat transfer rate of 0.5489 is obtained in a pure convective finwith Nc = 5 but for the same heat transfer rate to be obtained in apure radiative fin, the latter must operate with Nr greater than 25.The relative weakness of radiation as heat transport mechanismcompared with convection has also been confirmed by the experi-mental studies of fins with simultaneous natural convection andradiation. In an experimental study of a pin fin, Mueller andAbu-Mulaweh [12] have found that the radiation heat loss is only15–20% of the total heat loss from a horizontal pin fin with simul-taneous natural convection and radiation. The efficiency data inTable 1 shows that for the same heat dissipation, the efficiencyof a pure radiative fin is consistently lower than that of a pure con-vective fin. The performance of a radiative fin can be augmentedwith the addition of convection but such augmentation is obvi-ously not possible in space applications where convection is absentand the designer must work with the inherently reduced heattransfer performance of the pure radiative fin.

The effect of changing the environment (sink) temperature isshown in Figs. 8 and 9. As the sink temperature increases, boththe base heat flow and the fin efficiency decrease (Fig. 9) but thetemperature at every point in the fin increases (Fig. 8). The largestheat transfer rate occurs when the sink temperature is 0 K.

We now turn to the results that illustrate the effect of temper-ature dependent thermal conductivity of the fin and the effect con-vecting–radiating fin tip. The tip parameters are set at Nt, 1 = 0.1and Nt, 2 = 0.1. Fixing the values of Nc = 1, Nr = 1 and hs = 0.2, calcu-lations were carried for a = ±0.2 which allows for ±25% variation ofthe fin thermal conductivity between its value at the sink temper-ature and its value at the hot fluid temperature. Huang and Shah[13] considered the effect of variable thermal conductivity in con-vective fins and advise that for most metals, this percentage varia-

Fig. 5. Effect of Nc on the fin heat transfer rate and fin efficiency. Bi = 1, Nr = 1, and hs = 0.2.

Fig. 6. Effect of Nr on the temperature distribution in a fin of constant thermalconductivity with an insulated tip. Bi = 1, Nc = 1, and hs = 0.2.

2626 A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631

tion of thermal conductivity of most fin metals is about 12.5% for atemperature difference of 200 �C. Thus the choice of a = ±0.2 morethan adequately covers most practical applications. It can be seenfrom Fig. 3 that he heat transfer rate and fin efficiency are lowerfor a = 0.2 (thermal conductivity increasing with temperature)and higher for a = �0.2 (thermal conductivity decreasing with tem-perature) compared with the case of constant thermal conductivityi.e. a = 0. The results in Fig. 3 show that in the presence of a con-vecting–radiating tip and temperature dependent thermal conduc-tivity the dimensionless heat transfer rate and the fin efficiency canchange by much as 20% compared with the values for a fin of con-stant thermal conductivity and an insulated tip. For comparison,consider Fig. 10 which has been adapted from Malekzadeh et al.

[8]. The thermal conductivity changes from 300 W/m K to 50 W/m K. The value of Tf � Ts equals 300 K which gives a = �0.22 whichis close to our choice of a = �0.2. Reading the same figure gives anincrease in the heat transfer rate of 100(2.05 � 1.72)/1.72 = 20%which is fortuitously close to the increase of roughly 20% observedin our calculations and supports our conclusion that the heat trans-fer rate and the fin efficiency are significantly affected by temper-ature dependent thermal conductivity and a convecting–radiatingtip.

The effect of temperature dependent thermal conductivity andconvecting–radiating tip are also shown in Fig. 5. As in Fig. 3, theheat transfer rate and fin efficiency are lower for a = 0.2 and higherfor a = �0.2 compared with the case of constant thermal conduc-tivity i.e. a = 0. The same conclusion is reached when Figs. 7 and9 are examined.

Although Figs. 2–9 provide a fairly comprehensive picture of thethermal performance of convecting–radiating fins, the reader canuse the Maple worksheet in Appendix A to determine the thermalperformance of the fin from his or her own specific data.

4. Optimum fin design

For fins with specified profile (rectangular in our case), there aretwo approaches to derive optimum dimensions of the fin. One ap-proach is to fix the volume or profile area of the fin and find thelength and thickness so that the heat dissipation from the fin ismaximized. The second approach is to specify the heat dissipationrequirement and then find the dimensions of the fin so that thevolume or the profile area is minimized. The two approaches areequivalent. Here the first approach is adopted.

If the profile area Ap = wL is fixed, then w = Ap/L and Eq. (14) maybe rewritten as

q=Tf ks ¼ �Ap

L2 ½1þ aðh� hsÞ�dhdX

� �X¼0

ð21Þ

where h and its derivative dhdX are functions of the following dimen-

sionless variables expressed in terms of the profile area.

Nc ¼ 2hcL3=ksAp; Nr ¼ 2erL3=ksAp

Bi ¼ hL=ks; Nt;1 ¼ hcL=ks; Nt;2 ¼ erLT3f =ks

ð22Þ

where P � 2 for a slender fin of unit width. For fixed values of h, hc,ks, e, Tf, and Ap, the quantity on the right-hand side of Eq. (21) be-comes a function of L. By varying L, the maximum value of q may befound by repeatedly using the Maple program in Appendix A to ob-

Fig. 7. Effect of Nr on the fin heat transfer rate and fin efficiency. Bi = 1, Nc = 1, and hs = 0.2.

Table 1Heat transfer rates and efficiencies for pure convective and pure radiative fins.

Pure convective fin, Nr = 0 Pure radiative fin, Nc = 0

Nc Q g Nr Q g

1 0.3459 0.7616 1 0.2339 0.68245 0.5489 0.4371 25 0.5030 0.3388

10 0.6073 0.3152 100 0.5945 0.2336

Fig. 8. Effect of hs on the temperature distribution in a fin of constant thermalconductivity with an insulated tip. Bi = 1, Nc = 1, and Nr = 1.

A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631 2627

tain the numerical values of the right-hand side of Eq. (21). Theoptimum value of L thus found is then substituted in dimensionlessvariables defined by Eq. (22) to calculate their optimum values .The

large amount of data thus generated was used to create general pur-pose design charts shown in Figs. 11–14.

The methodology described in the preceding paragraph wastested for a pure convection fin with constant thermal conductivityand no tip heat loss for which an analytical result for the determi-nation of Nc,opt was derived by Aziz [3,4] as follows

4Nc;optsinh2Nc;opt ¼ Bið6Nc;opt � sinh 2Nc;optÞ ð23Þ

The comparison of the Maple generated results and the analyt-ical results predicted by Eq. (23) revealed a maximum discrepancyof only 0.32 percent thus validating the present results.

Fig. 11 shows the curves of Nc,opt versus Nr,opt for different val-ues of Biot number Bi for hs = 0. The solid lines show the results forthe insulated tip, the dashed lines the results for convecting–radi-ating tip with Nt, 1 = 0.1, Nt, 2 = 0.1 and a = 0.2, and the dashed-dotlines the results for convecting–radiating tip with Nt, 1 = 0.1, Nt, 2

= 0.1 and a = �0.2. The scale for the vertical axis was dictated bythe fact that for a pure convecting fin, Nc,opt = 2 as shown in[2,14] for the constant base temperature condition and from theknowledge from [3] that as Bi decreases, Nc,opt decreases. The hor-izontal scale was chosen from 0 to 1 because for a purely radiatingfin, Nr,opt < 1 for all values of hs [14]. Furthermore it was antici-pated from the linearized model of Sparrow and Niewerth [14]that for simultaneous convection and radiation, the values of Nc,opt

and Nr,opt would always be less than those when convection andradiation occur separately. In so far the effects of variable thermalconductivity and active fin tip are concerned, it has been estab-lished that for a pure convective fin that Nc,opt ranges between 0and 2.25 [1]. Thus it was anticipated that the horizontal and ver-tical scales chosen would encompass the entire spectrum of opti-mum data. Indeed that turned out to be the case. Figs. 11–15present the optimum fin design charts for hs = 0.2, 0.4, 0.6, and0.8, respectively.

It is observed from Figs. 11–15 that Nc,opt decreases withincreasing Nr,opt for all values of Bi. In practical design terms itmeans that the optimum fins for combined convection and radia-tion are shorter and thicker (less slender or chubby) comparedwith those for pure convection and pure radiation. It is also seenthat as Bi decreases, both Nc,opt and Nr,opt decrease which again im-ply that as Bi decreases i.e. the base convection becomes progres-sively weaker, the optimum fins become still slender (morechubby). The curves marked Bi =1 correspond to that case of con-stant base temperature. These results match within 6% with the re-sults in [14]. We also used the example data given in [9] and

Fig. 10. Effect of variable thermal conductivity on fin heat transfer rate. Adaptedfrom Malekzadeh et al. [8].

Fig. 9. Effect of hs on the fin heat transfer rate and fin efficiency. Bi = 1, Nc = 1, and Nr = 1.

2628 A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631

determined the optimum fin lengths. The values obtained were4.8 cm, 9.13 cm, and 12.07 cm for Tf = 900 K, 600 K, and 450 K,respectively, which compare very favorably with the values of5 cm, 9 cm and 12 cm reported in [8,9]. Thus the present optimum

Fig. 11. Relationship between Nc,opt

designs and those obtained with DQEM [8] and Taylor transforma-tion method [9] are close. The three approaches seem to validateone another.

The effect of a convective–radiative fin tip is to lower Nc,opt andNr,opt compared with the values for insulated tip. This would truefor all Bi numbers. This means the optimum fins with active tipare less slender than their insulated tip counterparts. This conclu-sion is in line with the optimum results for a pure convecting finfor an active tip versus an insulated tip [1]. For a given convec-tive–radiative tip heat loss, that is, for fixed values of Nt, 1 andNt, 2, Figs. 11–15 show that if the thermal conductivity of the fin de-creases with temperature (a < 0), Nc,opt and Nr,opt both decrease. Theoptimum fins are shorter and thicker than the optimum fins forconstant thermal conductivity. However, if the thermal conductiv-ity of the fin increases with temperature (a > 0), Nc,opt and Nr,opt

both increase indicating that the optimum fins are longer and thin-ner than the optimum fins for constant thermal conductivity. Thisconclusion is consistent with those found in studying the effect oftemperature dependent thermal conductivity on optimum dimen-sions of pure convecting fins [1].

and Nr,opt for various Bi. hs = 0.

Fig. 12. Relationship between Nc,opt and Nr,opt for various Bi. hs = 0.2.

Fig. 13. Relationship between Nc,opt and Nr,opt for various Bi. hs = 0.4.

Fig. 14. Relationship between Nc,opt and Nr,opt for various Bi. hs = 0.6.

A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631 2629

Fig. 15. Relationship between Nc,opt and Nr,opt for various Bi. hs = 0.8.

2630 A. Aziz, A.B. Beers-Green / Energy Conversion and Management 50 (2009) 2622–2631

5. Conclusions

The study has led to the following conclusions.

1. The nonlinear boundary value problem describing the tem-perature distribution in a variable thermal conductivity con-vecting–radiating rectangular fin with convective resistance,wall conduction resistance, and contact resistance at itsbase and a convective–radiative tip can be solved numeri-cally with a program consisting of few lines of code inMaple.

2. The program can be used to generate information about thetemperature distribution in the fin, the fin heat transfer rate,and the fin efficiency for a wide range of thermal parameterscontrolling the thermal performance of the fin.

3. By repeated use of the program, general purpose optimum findesign charts can be created.

4. Compared with the other methods such as Adomian’s decompo-sition method, Taylor transformation method, and differentialquadrature element method, the Maple implementationrequires only modest computational effort and produces resultsof the same accuracy as the other methods.

5. The values of Nc,opt and Nr,opt are always less than those whenconvection and radiation occur separately. The optimum con-vecting–radiating fins are less slender than the optimum finsfor pure convection and pure radiation.

6. The effect a convective–radiative tip is to lower the values ofNc,opt and Nr,opt compared with their values for an insulated tipwhich implies that with an active fin tip the optimum fins areshorter and thicker (less slender) than their insulated tipcounterparts.

7. For fixed convective–radiative tip heat loss, Nc,opt and Nr,opt

both decrease if the thermal conductivity of the fin decreaseswith temperature (a < 0). The optimum fins are shorter andthicker than the optimum fins for constant thermal conduc-tivity. However, if the thermal conductivity of the finincreases with temperature (a > 0), Nc,opt and Nr,opt bothincrease indicating that the optimum fins are longer and thin-ner than the optimum fins for constant thermal conductivity.These conclusions are consistent with those found in studyingthe effects of temperature dependent thermal conductivityand an active tip on optimum dimensions of pure convectingfins [1].

Acknowledgement

The authors gratefully acknowledge the help of Professor RobertLopez, Maple Fellow at Maplesoft Inc, Waterloo, Canada during thisresearch.

Appendix A. Maple worksheet

> restart> Eq1 := diff ((1 + a�(theta(X) � theta[s]))�diff(theta(X),X),X) � N[c]�

(theta(X) � theta[s]) � N[r]�(theta(X)4 � theta[s]4) = 0

Eq1 :¼ ad

dXhðXÞ

� �2

þ ð1þ aðhðXÞ � hsÞÞd2

dX2 hðXÞ !

� NcðhðxÞ � hsÞ � NrðhðXÞ4 � h4s Þ ¼ 0

> bc1 :=�(1 + a�(theta(0)� theta[s]))�D(theta)(0)� Bi�(1� theta(0))

bc1 :¼ �ð1þ aðhð0Þ � hsÞÞDðhÞð0Þ � Bið1� hð0ÞÞ

> bc2 :=�(1 + a�(theta(1)� theta[s]))�D(theta)(1)� N[t,1]�(theta(1)�theta[s]) � N[t,2]�(theta(1)4 � theta[s]4)

bc2 :¼�ð1þaðhð1Þ�hsÞÞDðhÞð1Þ�Nt;1ðhð1Þ�h½s�Þ�Nt;2ðhð1Þ4�h4s Þ

> parameters := [a = 0.2, N[c] = 1, N[r] = 1, theta[s] = 0.2, Bi = 1,N[t,1] = 0.1, N[t,2] = 0.1]

parameters :¼ ½a ¼ 0:2; Nc ¼ 1; Nr ¼ 1; hs ¼ 0:2;Bi ¼ 1; Nt;1 ¼ 0:1; Nt;2 ¼ 0:1�

> Eq2 := eval(Eql, parameters)

Eq2 :¼ 0:2ddx

hðXÞ� �2

þ ð0:96þ 0:2hðXÞÞ d2

dX2 hðXÞ !

� hðXÞ þ 0:2016� hðXÞ4 ¼ 0

> BC1: = eval(bcl, parameters)

BC1 :¼ �ð0:96þ 0:2hð0ÞÞDðhÞð0Þ � 1þ hð0Þ

> BC2 := eval(bc2, parameters)

A. Aziz, A.B. Beers-Green / Energy Conversion

BC2 :¼ �ð0:96þ 0:2hð1ÞÞDðhÞð1Þ � 0:1hð1Þ þ 0:02016� 0:1hð1Þ4

> F := dsolve([Eq2, BC1 = 0, BC2 = 0], theta(X), numeric, approxsoln =[theta(X) = exp(�X)])

F :¼ procðx bvpÞ . . . end proc

> plots[odeplot](F)

> Basetemperature: = rhs(F(0)[2])

Basetemperature :¼ 0:61175619898578315

> Tiptemperature: = rhs(F(1)[2])

Tiptemperature :¼ 0:43723184779453411

> Q: = �rhs(F(0)[3])

Q :¼ 0:35870407383174823

> N[c] := 1; N[r]: = 1; N[t,1]: = 0.1; N[t,2]: = 0.1; theta[s]: = 0.2;

Bi: = 1;

and Management 50 (2009) 2622–2631 2631

Nc :¼ 1Nr :¼ 1Nt;1 :¼ 0:1Nt;2 :¼ 0:1hs :¼ 0:2Bi :¼ 1

> Efficiency: = Q/((N[c]+N[t, 1])� 1� QBi� theta½s�

� �+ (N[r]+N[t, 2])�

1� QBi

4�

� theta½s�4�

)

Efficiency :¼ 0:535607682

References

[1] Kraus AD, Aziz A, Welty JR. Extended surface heat transfer. NY: John Wiley;2001.

[2] Aziz A, Kraus AD. Optimum design of radiating and convecting–radiating fins.Heat Transfer Eng 1996:744–78.

[3] Aziz A. Optimization of rectangular and triangular fins with convectiveboundary conditions. Int Commun Heat Mass Transfer 1985:479–82.

[4] Aziz A. Addendum to optimization of rectangular and triangular fins withconvective boundary conditions. Int Commun Heat Mass Transfer 1985:743–4.

[5] Ma Z, Chung BTF. Optimization of convecting–radiating longitudinalrectangular fins with thermal resistance at base wall. In: Proceedings of the1997 thirty-fifth Heat Transfer and Fluid Mechanics Institute, May 29–301997, Sacramento, CA, California State University; 1997.

[6] Chung BTF, Zhou Y. Optimal design for convecting–radiating annular fins oftrapezoidal profile. In: Proceedings of 2001 national heat transfer conference,June 10–12 2001, Anaheim, CA. NY: ASME; 2001. p. 1383–93.

[7] Chiu C-H, Chen C-K. Application of Adomian’s decomposition procedure to theanalysis of convective–radiative fins. ASME J Heat Transfer 2003;125:312–6.

[8] Malekzadeh P, Rahideh H, Karami G. Optimization of convective radiative finsusing differential quadrature element method. Energy Convers Manage2006;47:1505–14.

[9] Yu L-T, Chen C-K. Application of Taylor transformation to optimize rectangularfins with variable thermal parameter. Appl Math Modelling 1998;22:11–21.

[10] Burden RL, Faires JD. Numerical analysis. 5th ed. Boston, MA: PWS PublishingCo; 1993.

[11] Incropera FP et al. Fundamentals of heat and mass transfer. 6th ed. NewYork: John Wiley; 2007.

[12] Mueller DW, Abu-Mulaweh HI. Prediction of the temperature in a fin cooled bynatural convection and radiation. Appl Therm Eng 2006;26:1662–8.

[13] Huang LJ, Shah RK. Assessment of calculation methods for efficiency of straightfins of rectangular profile. Int J Heat Fluid Flow 1992;13:282–93.

[14] Sparrow EM, Niewerth ER. Radiating, convecting, and conducting fins:numerical and linearized solutions. Int J Heat Mass Transfer 1968;11:377–9.