Transient Heat Transfer in a Heat-Generating Fin with Radiationand Convection with Temperature-Dependent Heat Transfer

Coefficient

Waqar A. Khan1 and A. Aziz2 1Department of Engineering Sciences, National University of Science and Technology,

Karachi, Pakistan 2Department of Mechanical Engineering, School of Engineering and Applied Science,

Gonzaga University, Spokane, WA, USA

The transient heat transfer in a heat-generating fin with simultaneous surfaceconvection and radiation is studied numerically for a step change in base temperature.The convection heat transfer coefficient is assumed to be a power law function of thelocal temperature difference between the fin and its surrounding fluid. The values ofthe power exponent n are chosen to include simulation of natural convection (laminarand turbulent) and nucleate boiling among other convective heat transfer modes. Thefin is assumed to have uniform internal heat generation. The transient response of thefin depends on the convection-conduction parameter, radiation-conduction parameter,heat generation parameter, power exponent, and the dimensionless sink temperature.The instantaneous heat transfer characteristics such as the base heat transfer, surfaceheat loss, and energy stored are reported for a range of values of these parameters.When the internal heat generation exceeds a threshold the fin acts as a heat sink insteadof a heat source. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res, 41(5), 402–417,2012; Published online 14 May 2012 in Wiley Online Library (wileyonlineli-brary.com/journal/htj). DOI 10.1002/htj.21012

Key words: convection-radiation, fin, internal heat generation, power law,transient

1. Introduction

For over a century, extended surfaces (fins) have provided an effective and inexpensive meansof enhancing the heat transfer rate between a hot surface and its surrounding fluid. Traditionalapplications have included internal combustion engines, compressors, heat exchangers, and controlsystems. With the advent of the space age, radiating finned surfaces became common in the designof heat rejection systems for spacecraft. In the last 30 years, the use of fins as heat sinks for coolingelectronic devices has gained enormous popularity. The subject of extended surface heat transfer isnow a mature technology with contributions from a vast number of researchers throughout the world.The book by Kraus, Aziz, and Welty [1] provides comprehensive coverage of the various facets ofthis technology.

© 2012 Wiley Periodicals, Inc.

Heat Transfer—Asian Research, 41 (5), 2012

402

The transient response of fins is of vital interest in studying the performance of heat sinks forelectronics cooling, high-speed aircraft, intermittently operating heat exchangers, control equipment,electric motors, clutches, and solar systems. The earliest work on the transient response of a convectingradial or annular fin was done by Chapman [2]. He applied the method of separation of variables toderive analytical expressions for temperature distribution and base heat flow rate. He also presentedthe information in graphical form to facilitate quick design calculations, particularly those pertainingto the heat dissipation from the finned, air-cooled internal combustion engines. The same type ofanalysis for a convecting straight fin of constant area was performed by Donaldson and Shouman [3].They reported results for the boundary conditions of a step change in base temperature as well as astep change in base heat flux. The analytical results for the step change in base heat flux were alsoverified experimentally. For a convectively-heated fin base, Aziz and Kraus [4] developed a separationof variable-type solution for the transient response of a convecting straight fin when a step change inthe temperature of the hot fluid occurs.

Because the series solutions converge slowly for small values of time, Suryanarayana [5, 6]adopted the Laplace transform method which allowed him to derive a solution that converged rapidlyfor small values of time. In addition to the cases of step change in base temperature, a step change inbase heat flux, and a step change in base fluid temperature, he also gave solutions for the cases ofoscillatory changes in base temperature and base heat flux. The case of a step change in theenvironment temperature has been studied by Papadopoulos, Guzman-Garcia, and Bailey [7]. Maoand Rook [8] considered a convecting straight fin of constant area with tip heat loss and used theLaplace transform method to determine the transient response of the fin under three base conditions:a step change in base temperature; a step change in base heat flux; and a step change in base fluidtemperature. Beck et al. [9] used Green’s functions to study the transient response of convectingstraight fins of constant area.

A variety of approximate analytical methods have been used to study the transient responseof fins. Kim [10] used the Kantorovich method to derive an approximate solution for the transientresponse of a convecting straight fin of constant area. Aziz and Na [11] proposed a coordinateperturbation expansion for the response of an infinitely long fin due to a step change in the basetemperature. They incorporated power-law type surface heat dissipation so that the analysis appliedto cooling due to film boiling, laminar and turbulent natural convection, nucleate boiling, and radiationto space at absolute zero temperature. The methods of optimal linearization and variational embeddingwere used by Chang et al. [12]. Campo [13] utilized variational techniques to analyze radiative-con-vective fins under unsteady operating conditions. Recently, Cole et al. [14] used the method of Green’sfunction to develop an exact analytical solution for the transient response of a convecting straight finof constant area with a step change in base heat flux. A single unified solution was derived for threetip boundary conditions: fixed tip temperature; an insulated tip; and a convective tip. For the case ofthe insulated tip, they also developed a quasi-steady solution which agreed with the exact solution forlarge values of times but gave large errors for small values of time. However, for large values of time,the quasi-steady solution required far less computational effort than the exact solution. The quasi-steady solution, accurate at large values of time, was used to obtain the convective heat transfercoefficient data and was found to match the experimental data for a railroad roller bearing whose outerbearing race was modeled as a transient fin.

403

In circumstances where the transverse Biot number for a fin is not small, the temperature fieldin the fin becomes two-dimensional. The transient response of fins with two-dimensional heatconduction has been investigated by Tseng et al. [15], Chen and Chen [16], Onur [17], Chu et al. [18],Ju et al. [19], Singh et al. [20], Malekzadeh et al. [21], and Yu and Chen [22]. These authors haveemployed techniques such as Laplace transformation, Taylor transformation, differential quadratureelement method, perturbation method, variational method, method of lines, and hybrid analytic/nu-meric methods.

The preceding literature review has brought several points into focus. First, all transient studieshave assumed that the convective heat transfer coefficient over the surface of the fin is a constant. Innatural convection-cooled heat sinks, the convection and radiation are comparable and the heattransfer is a power-law function of the local temperature difference between the fin and its surroundingmedium. For example, the power law exponent n is 0.25 for laminar natural convection and 0.33 forturbulent natural convection [23]. Experimental studies such as Ref. 24 have also confirmed that theconvective heat transfer coefficient decreases from the base to the tip, which is consistent with thepower-law type dependence on local temperature difference. Second, none of the studies haveincluded internal heat generation in the fin which may arise in nuclear applications where the fin isexposed to gamma rays. The presence of internal heat generation can dramatically reduce the abilityof the fin to dissipate heat from its primary surface. Indeed, a situation can arise when the base of thefin can become a heat sink instead of a heat source. Third, the information about how the instantaneoussurface heat dissipation and the instantaneous energy storage in the fin change with time is neverreported. Instead, the sum of these components, that is, the instantaneous base heat flow is reportedwhen the base of the fin experiences a step change in temperature. These shortcomings in the previousstudies are removed and a comprehensive picture of the transient process is provided. Specifically westudy the transient response of a heat-generating straight fin with simultaneous radiation andconvection, the latter being modeled with a temperature-dependent convection heat transfer coeffi-cient. The fin equation together with the initial and boundary conditions is solved numerically. Theresults for the case of zero internal heat generation are of immediate applicability to the transientresponse of natural convection heat sinks. This information is crucial for time-sensitive applicationsand in conditions where the base temperature and/or the flow over the heat sink fluctuate as noted inRef. 25. The numerical data in Ref. 25 was obtained using the Icepack software and pertains only totwo specific heat sinks. The dimensionless data of the present paper are applicable to heat sinks ingeneral as well as other applications where fins experience internal heat generation.

2. Mathematical Formulation

Consider a straight fin of constant cross-sectional area A (rectangular, cylindrical, elliptic,etc.), perimeter of the cross-section P, and length b as shown in Fig. 1. The fin has a thermalconductivity k and a thermal diffusivity α. The surface of the fin behaves as a gray diffuse surfacewith an emissivity ε. The fin is initially in thermal equilibrium with the surroundings at temperatureTs. At time t > 0, the base of the fin undergoes a step change in temperature from Ts to Tb while its tipremains insulated. A volumetric internal heat generation rate q

. occurs in the fin. The fin loses heat by

simultaneous convection and radiation to its surroundings at temperature Ts. The same sink tempera-ture is used for both convection and radiation to avoid the introduction of an additional parameter inthe problem. The convection heat transfer coefficient h at any location on the surface of the fin is

404

assumed to have a power-law type dependence on the temperature difference between the fin and itssurroundings. The specific form is taken as

where C is constant and n is the power law index. The values of n are 0.25, 0.33, and 3 for laminarnatural convection, turbulent natural convection, and nucleate boiling, respectively [26, 27].

For one-dimensional transient conduction in the fin, the energy equation may be written as

The initial and boundary conditions are

where x is measured from the tip of the fin. With the introduction of the following definitions,

Equations (2) to (5) can be written in dimensionless form as follows:

Fig. 1. A straight fin of constant cross-sectional area.

(1)

(2)

(3)

(4)

(5)

(7)

(8)

(9)

(10)

(6)

405

The instantaneous base heat flow is given by

which may be expressed in dimensionless form as follows:

The instantaneous convective heat loss from the fin is given by

or in dimensionless form as

Similarly, the instantaneous radiative heat loss from the fin can be obtained as

or in dimensionless form as

The instantaneous total surface heat loss in dimensionless form is the sum of convective andradiative losses given by Eqs. (14) and (15), that is,

The instantaneous rate of energy storage in the fin can be calculated from the energy balanceas follows:

or in dimensionless form as

where

3. Numerical Procedure

Equation (7) with initial and boundary conditions Eqs. (8) to (10) was solved numericallyusing the algorithm available in Maple 13 for solving parabolic partial differential equations. The

(11)

(12)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(13)

406

procedure can be implemented by calling the command pdsolve. This command can deliver an exactanalytical solution if available or, with the numeric option specified, it can deliver the solution in theform of a module from which the numerical solution can be extracted in the form of numerical dataor as a plot or as an animation. The solutions generated in this paper are obtained by the default methodwhich uses a second-order (in space and time) centered implicit finite difference scheme. The spaceand time steps were ∆X = ∆τ = 0.001. The procedure also offers the choice of the number of pointsfor plotting the data. Numerical experiments revealed that the use of 1000 points was more thanenough to generate smooth graphs. For each value of time, τ, the 1000 point values of θ were used toobtain a least-squares-fit, fourth-order polynomial which was then used in Eqs. (14) and (16) tocompute the values of Qc and Qr, respectively.

The accuracy of the numerical procedure used by Maple and the least-squares polynomial fitwas tested against the exact analytical solutions available for a pure convection fin with no internalheat generation and constant heat transfer coefficient [3, 4]. The numerical results agreed with theanalytical results to at least four decimal places and confirmed the accuracy of our numericalprocedure.

4. Results and Discussion

Equation 7 shows that the transient response of the fin depends on five parameters:Nc, Nr, Qgen, θs, and n. The effect of each parameter on fin temperature, base heat flow, surface heatloss, and heat stored in the fin will now be presented and discussed.

Effect of power law index

The effect of the power law index n on the development of the temperature field in the fin isillustrated in Fig. 2. The fin is assumed to have no internal heat generation. During the initial transienti.e., for the low values of τ, the effect of n is barely noticeable, but as the transient progresses, theeffect becomes increasingly pronounced particularly on the tip temperature. Since the curves for τ =3 very nearly represents the steady state, their trend may be compared, at least qualitatively, with theresults of Cortell [28] who studied the steady-state performance of a pure convection fin withtemperature-dependent heat transfer coefficient in the form of Eq. (1). Just as the present results fora convecting-radiating fin indicate higher fin temperature with higher values of n, so do his resultsfor a pure convecting fin.

Figure 3 shows the instantaneous base heat flow, the instantaneous surface heat loss (convec-tion and radiation), and the instantaneous energy stored in the fin. The values of parameters Nc, Nr,and θs are the same as in Fig. 2 but the values of n are chosen as 0.25 (laminar natural convection)and 2.0. The results for n = 0.25 are of immediate relevance on the transient performance of a naturalconvection air-cooled heat sink where the convection and radiative heat losses are comparable(Nc = Nr). In the early part of the transient, the surface heat loss is small and the bulk of the energyflow from the base of the fin gets stored in the fin. As the time increases, the surface heat loss increasesand the stored component of the energy decreases. When steady state is reached at τ ≈ 1.5, the storageterm vanishes and the base heat flow matches the surface heat loss. A comparison of the results for n= 0.25 and n = 2 shows that the base heat flow and surface heat loss are lower at n = 2 than for n =0.25 but the storage component is relatively insensitive to the value of n. The results of Fig. 3 are

407

Fig. 2. Effect of power law exponent n on temperature evolution in the fin with zero internal heatgeneration. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

Fig. 3. Effect of power law exponent n on base heat flow, surface heat loss, and energy stored.[Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

408

qualitatively consistent with the results for a pure convection fin reported by Cortell [28] whichshowed that the base heat flow decreases as the power law index n increases.

Effect of convection parameter

Figure 4 depicts the effect of convection parameter Nc on the transient temperature distributionin the fin. The fin temperature at every location is higher when it operates at a lower value of Nc. Thedifference in fin temperatures for the two values of Nc is small during the early part of the transientprocess but becomes quite significant as the steady state is approached.

The instantaneous base heat flow, the instantaneous surface heat loss (convection andradiation), and the instantaneous energy stored in the fin are given for two values of Nc in Fig. 5. AsNc is increased from 0.1 to 1.0, the heat flow from the base of the fin is significantly enhanced asexpected. This is accompanied by a significant increase in the surface heat loss (convective andradiative). For both values of Nc, the base heat flow and the surface heat loss become equal when thefin attains a steady state. It may be noted that the dimensionless time to steady state, τss, is about 2when the fin operates at Nc = 0.1 but its value decreases to about 1.4 when the value of Nc is increasedto 1.0. Assume that the change in Nc from 0.1 to 1.0 is due to the increase in fin length from b1 tob2. The use of definition of Nc gives b1

2 / b2

2 = 0.1/1.0 = 0.1. If t1 and t2 are the corresponding dimensionaltimes required to attain steady state, then using the definition of τ gives t1 / t2 = 0.1, τ1 / τ2 = 0.1(2/1.44)= 0.14. Thus, the shorter fin of length b1 takes only 0.14 of the time needed by the longer fin of lengthb2 = 3.16 b1 to attain a steady state. Thus heat sinks made of shorter fins would attain the steady statemuch faster than heat sinks made of longer fins. This conclusion, of course, assumes the heat sinksare made of identical materials, have identical perimeters and cross-sectional areas, and operate underconvective and radiative environments. An interesting perspective on how the heat sink materialaffects the transient response can be obtained as follows. Let the change in Nc from 0.1 to 1.0 be due

Fig. 4. Effect of convection parameter on temperature evolution in the fin with zero internal heatgeneration. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

409

to the decrease in the thermal conductivity of the material. Then, with a little algebraic manipulation,it may be shown that t1 / t2 = 0.14(ρ1c1 / ρ2c2). Thus, a fin made of a material with a smaller productof density and specific heat such as aluminum would have a faster response compared with a fin madeof copper which has a comparatively larger product of density and specific heat. Indeed this has beenshown to be the case in the Icepack simulations of aluminum and copper heat sinks [25] with a stepchange in base heat flux. It should be kept in mind that the factor 0.14 which is based on a thermalconductivity ratio of k1 / k2 = 10 would be much higher (0.98) for aluminum and copper heat sinkswhich gives t1 / t2 = 0.72. The fact that the present value of 0.72 is close to the value of 0.75 reportedin Ref. 25 indicates that the time to attain steady state is not significantly affected by the step boundarycondition (temperature or heat flux). As noted in Ref. 25 and confirmed by the present study that analuminum heat sink is preferable for fast response applications but a copper heat sink may be a betterchoice for high-heat flux applications and in circumstances where a slower variation of temperaturewith time is desirable. The material and manufacturing costs would also influence the final decision.

Effect of radiation parameter

A comparison of Figs. 6 and 7 with Figs. 4 and 5 shows that the effect of changing the radiationparameter Nr is remarkably similar. The temperature levels in the fin are higher for Nr = 0.5 than forNr = 1.0 (Fig. 6). The dimensionless time to steady state is higher for Nr = 0.1 than for Nr = 1.0.Because of the remarkably similar trends, the discussion in the preceding subsection is equallyapplicable here and need not be repeated.

Effect of internal heat generation

The steady-state performance of fins with internal heat generation has been studied in theliterature [1, 29, 30] but the transient study of a heat-generating fin is probably unique to the present

Fig. 5. Effect of convection parameter on base heat flow, surface heat loss, and energy stored.[Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

410

work. First, we present the case where the strength of the internal heat generation is such that the finis still able to remove some heat from its primary surface. Such a case is illustrated in Fig. 8 where,as expected, the internal heat generation gives rise to higher temperature compared with thoseoccurring without internal heat generation. The maximum difference is seen when the steady state isreached.

Fig. 7. Effect of radiation parameter on base heat flow, surface heat loss, and energy stored. [Colorfigure can be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

Fig. 6. Effect of radiation parameter on the temperature evolution in a fin with zero internal heatgeneration. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

411

When the internal energy generation is present, the energy gained by the fin is the sum ofenergy generated within the fin and the energy extracted from the primary surface, i.e.,

The energy stored in the fin is given by

The quantities Qgain, Qloss, and Qstored are plotted in Fig. 8 for Qgen = 0 and Qgen = 0.5. In viewof Eq. (21), the internal heat generation increases the energy gained by the fin. Because of the highertemperatures in the fin occurring due to internal heat generation (see Fig. 8), the surface heat loss ishigher with internal heat generation. When the steady state is reached, Qgain = Qloss and Qstored becomeszero. Figure 9 also shows that internal heat generation has little effect on dimensionless time to steadystate. For both cases, τss ≈ 1.3.

Figure 10 shows the transient temperature profiles in the fin when the internal heat generationis strong. As seen previously in Fig. 8, the temperature levels in the fin rise as the internal heatgeneration increases. At τ = 0.1, the temperature gradients at X = 1 (fin base) for all three values ofQgen are positive indicating that base heat flow is positive. This is also shown by Qbase curves in Fig.11. As τ increases to 0.3, the temperature gradient at the base of the fin decreases, becoming virtuallyzero when Qgen = 2. For Qgen = 3, the temperature gradient at the base of the fin becomes negativeindicating that heat actually flows into the primary surface to which the fin is attached. This conclusioncan also be drawn from Fig. 10 where Qbase becomes negative as the transient progresses. As explainedearlier, the larger the internal heat generation, the higher the fin temperature, and consequently higherthe surface heat loss. The results of Figs. 10 and 11 clearly demonstrate that when the internal heat

Fig. 8. Effect of internal heat generation on temperature evolution in the fin. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

(21)

(22)

412

Fig. 9. Effect of weak internal heat generation on base heat flow, surface heat loss, and energystored. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

Fig. 10. Effect of strong internal heat generation on temperature evolution in the fin. [Color figurecan be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

413

Fig. 12. Effect of sink temperature on temperature evolution in the fin. [Color figure can be viewedin the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

Fig. 11. Effect of strong internal heat generation on base heat flow, surface heat loss, and energystored. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com/journal/htj.]

414

generation exceeds a certain threshold value, the fin loses its ability to extract heat from the primarysurface. Indeed, the primary surface becomes a heat sink rather than a heat source.

Effect of sink temperature

The transient temperature distributions for θs = 0.01 and 0.1 are shown in Fig. 12. As the sinktemperature is lowered from 0.1 to 0.01, the temperature difference between the base of the fin andthe sink gets larger and the temperature gradient at the base increases, and the temperature in the finis lowered. The increase in the base temperature gradient is reflected in the increased base heat flowin Fig. 13.

5. Conclusions

In this paper, the transient heat transfer in a heat generating fin is studied numerically withsimultaneous surface convection and radiation for a step change in base temperature. The effects ofthe power law index n, convection parameter Nc, radiation parameter Nr, and the sink temperature onthe transient temperature field in the fin are investigated numerically. It is shown that the transientresponse of the fin depends on the convection-conduction parameter Nc, radiation-conductionparameter Nr, heat generation parameter Qgen, power exponent n, and the dimensionless sinktemperature, θs. The instantaneous heat transfer characteristics such as the base heat transfer, surfaceheat loss, and energy stored are reported for a range of values of these parameters.

Acknowledgment

The authors would like to thank Professor Robert N. Lopez, Maple Fellow at Maplesoft Inc.,Waterloo, Canada for his assistance during this research.

Fig. 13. Effect of sink temperature on base heat flow, surface heat loss, and energy stored. [Colorfigure can be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

415

Literature Cited

1. Kraus AD, Aziz A, Welty JR. Extended surface heat transfer. Wiley; 2001.2. Chapman AJ. Transient heat conduction in annular fins of uniform thickness. Chem Eng Symp

Ser 1959;55:195–201.3. Donaldson AB, Shouman AR. Unsteady state temperature distribution in a convecting fin of

constant area. Appl Sci Res 1972;26:75–85.4. Aziz A, Kraus AD. Transient heat transfer in extended surfaces. ASME Appl Mech Rev

1995;48:317–350.5. Suryanarayana NV. Transient response of straight fins. ASME J Heat Transf 1975;97:417–423.6. Suryanarayana NV. Transient response of straight fins, Part II. ASME J Heat Transf

1976;98:324–326.7. Papadopoulos KD, Guzman-Gracia AG, Bailey RV. Response of straight and circular fins to

fluid temperature changes. Int Commun Heat Mass Transf 1990;17:2587–2595.8. Mao J, Rook S. Transient analyses of extended surfaces with convective tip. Int Commun Heat

Mass Transf 1993;21:85–94.9. Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B. Heat conduction with Green’s functions.

Hemisphere; 1992.10. Kim RH. The Kantorovich method in the variational formulation to an unsteady heat conduc-

tion. Lett Heat Mass Transf 1976;3:73–80.11. Aziz A, Na TY. Transient response of fins with coordinate perturbation expansion. Int J Heat

Mass Transf 1980;23:1695–1698.12. Chang YM, Chen CK, Cleaver JW. Transient response of fins by optimal linearization and

variational embedding methods. ASME J Heat Transf 1982;104:813–815.13. Campo A. Variational techniques applied to radiative-convective fins with steady and unsteady

conditions. Warme- und Stoffubertragung 1976;9:139–144.14. Cole KD, Tarawneh C, Wilson B. Analysis of flux-base fins for estimation of heat transfer

coefficient. Int J Heat Mass Transf 2009;52:92–99.15. Tseng JK, Shih CL, Wu WJ. Analysis of transient heat transfer in straight fins of various shapes

with its base subjected to a decayed exponential function of time in heat flux. Comput Struct1993;47:289–297.

16. Chen CY, Chen CK. Transient response of annular fins of various shapes subjected to constantbase heat fluxes. J Phys D Appl Phys 1994;27:2302–2306.

17. Onur N. A simplified approach to the transient conduction in a two-dimensional fin. IntCommun Heat Mass Transf 1996;23:225–238.

18. Chu H-S, Wang C-I, Chen C-K. Transient response of a composite straight fin. ASME J HeatTransf 1983;105:307–311.

19. Ju Y, Chou Y, Hsiao C. A new approach to the transient conduction in a 2-D rectangular fin.Int J Heat Mass Transf 1989;32:1657–1661.

20. Singh IV, Sandeep K, Prakash R. Heat transfer analysis of two-dimensional fins using ameshless element free Galerkin method. Numer Heat Transf 2003;44:73–84.

21. Malekzadeh P, Rahideh H, Karami G. A differential quadrature element method for nonlineartransient heat transfer analysis of extended surfaces. Numer Heat Transf 2006;49:511–523.

22. Yu L-T, Chen C-K. Application of the Taylor transformation to the transient response of anannular fin. Heat Transf Eng 1999;20:78–87.

23. Holman JP. Heat Transfer, 10th ed. McGraw Hill; 2010.24. Orzechowski T. Determining the local value of the heat transfer coefficient on a fin surface.

Exp Therm Fluid Sci 2007;31:947–955.

416

25. Heat sink transient response in Q-pedia, a magazine edited by K. Azar and B. Tavassoli andpublished by Advanced Thermal Solution, Inc, Norwood, MA, Vol. III, Issue X, pp. 16–19,November 2009.

26. Lesnic D, Heggs PJ. A decomposition method for power-law fin-type problems. Int CommunHeat Mass Transf 2004;31:673–682.

27. Khani F, Aziz A. Thermal analysis of a longitudinal trapezoidal fin with temperature dependentthermal conductivity and heat transfer coefficient. Commun Nonlinear Sci Numer Simul2010;15:590–601.

28. Cortell R. A numerical analysis to the non-linear fin problem. J Zhejiang Univ Sci A2008;9:648–653.

29. Kundu B, Das PK. Optimum profiles of fins with volumetric heat generation; a unifiedapproach. ASME J Heat Transf 2005;127:945–948.

30. Bouaziz MN. Application of exp/imp finite difference for nonlinear heat transfer fin effect ofnonlinearities of fin efficiency. Adv Theor Appl Mech 2009;2:19–32.

"F F F"

417