Numerical analysis of two dimensional pin fins with non-constant base heat flux

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Energy Conversion and Management 46 (2005) 881–892

Numerical analysis of two dimensional pin fins withnon-constant base heat flux

Yu-Ching Yang, Haw-Long Lee *, Eing-Jer Wei, Jenn-Fa Lee,Tser-Son Wu

Department of Mechanical Engineering, Kun Shan University of Technology, No. 949, Da-Wong Road, Yung-Kong,

Tainan 710-03, Taiwan, Republic of China

Received 16 March 2004; accepted 24 June 2004

Available online 26 August 2004

Abstract

This paper deals with the transient analysis of a two dimensional pin fin that contains tip convection and

is subjected to time and space dependent heat flux at the fin base. A method of combining the Laplacetransformation and finite difference is applied to analyze the problem. The general solution of the governing

equation is first solved in the transform domain. The numerical transient solutions of temperature incre-

ments in the real domain are then obtained by using the matrix similarity transformation and Fourier series

technique. Comparing with existing analytical solutions, the results presented here show high accuracy. The

computational procedures established in this paper are also applicable to other types of boundary condi-

tions, such as variable convection heat transfer coefficient.

2004 Elsevier Ltd. All rights reserved.

Keywords: Laplace transformation; Finite difference; Fourier series technique; Pin fin

1. Introduction

Fins are widely used to increase the surface area and, consequently, to enhance the rate ofheat exchange between a heated surface and a colder ambient fluid, such as in applications of

0196-8904/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.06.009

* Corresponding author. Tel.: +886 620 504 96; fax: +886 620 505 09.

E-mail address: [email protected] (H.-L. Lee).

mailto:[email protected]

Nomenclature

L pin fin lengthR* radius of pin finR dimensionless radius of pin fin (=R*/L)t* timet dimensionless time (=at*/L2)h convective heat transfer coefficient at lateral surfacehT convective heat transfer coefficient at tip surfaceH ratio of convective heat transfer coefficients (=hT/h)k thermal conductivity of pin fina thermal diffusivityBia transversal Biot number (=hR*/k)BiT tip Biot number (=hTL/k)G geometry parameter of pin fin (=L/R*)T 1 ambient fluid temperature

q0 heat flux magnitude at fin baseh* transient temperatureh dimensionless transient temperature increment ½¼ ðh T

1Þ=ðq0 L=kÞm total mesh points in x directionn total mesh points in r direction

882 Y.-C. Yang et al. / Energy Conversion and Management 46 (2005) 881–892

semiconductors, heat exchangers, power generators and electronic components. So far, there havebeen many studies [1–3] focused on steady state analysis of one or two dimensional fins. The re-sults of steady state analysis are suitable for fin design in most real applications. Nevertheless, theanalysis of the transient response of fins is also important in some applications [4–9]. For conven-ience of analysis, most studies in the literature assumed a step change in the base temperature or inthe base heat flux, and the others considered periodic cases. However, in reality, the temperature,or the heat flux, at the fin base might be a function of both time and space simultaneously.

The applications of short pin fins are usually found in compact heat exchangers [10] and aircooled turbine blades [11]. So far, as mentioned above, there have been many investigations deal-ing with the transient analysis of two dimensional pin fins. However, little of the literature is con-cerned with the transient analysis of a two dimensional pin fin that is subjected to a time and spacedependent heat flux at the fin base. In this article, a hybrid numerical method is adopted to inves-tigate the transient response of a two dimensional pin fin that contains tip convection and is sub-jected to a time and space variation base heat flux. The general solution of the governing equationis first solved in the transform domain by the method of combining the Laplace transformationand finite difference. Then, the inverse transformation to the real domain is completed by themethod of matrix similarity transformation and Fourier series technique [12,13]. Comparing withexisting analytical solutions, the results presented by the computational procedures of this paperare accurate and efficient. In addition, the presented procedures allow us to calculate the temper-ature values of specific nodes at a specific time and, therefore, are more economical.

Y.-C. Yang et al. / Energy Conversion and Management 46 (2005) 881–892 883

2. Analysis

The configuration of a two dimensional pin fin is shown in Fig. 1. The analysis is based on thefollowing assumptions:

(a) The pin fin is initially at ambient temperature, T 1.

(b) The material properties of the pin fin, the heat transfer coefficients of the ambient fluid h andhT and the temperature of the ambient fluid T

1 are all assumed to be constant.(c) The temperature distribution varies with x*, r*, and t* in the two dimensional pin fin.(d) There is no heat source or sink in the pin fin.(e) Radiation is neglected in the pin fin.

The governing differential equation for the temperature distribution in the pin fin and its asso-ciated initial and boundary conditions are

ohðx; r; tÞot

¼ a o2hðx; r; tÞ

ox2þ 1

r oh

ðx; r; tÞor

þ o2hðx; r; tÞ

or2

ð1Þ

t ¼ 0 hðx; r; 0Þ ¼ T 1 ð2Þ

t > 0 x ¼ 0 k ohð0; r; tÞox

¼ q0 f ðr; tÞ ð3Þ

x ¼ L k ohðL; r; tÞox

þ hT ½hðL; r; tÞ T 1 ¼ 0 ð4Þ

r ¼ 0ohðx; 0; tÞ

or¼ 0 ð5Þ

h

h

hT

L

2R

r*

x * *

Fig. 1. Configuration of a pin fin.


r ¼ R k ohðx;R; tÞ

orþ h ½hðx;R; tÞ T

1 ¼ 0 ð6Þ

where q0 f ðr; tÞ is the heat flux at the fin base.Define the following non-dimensional variables:

r ¼ r=L x ¼ x=L t ¼ at=L2 h ¼ ðh T 1Þ=ðq0 L=kÞ Bia ¼ hR=k

BiT ¼ hTL=k G ¼ L=R R ¼ R=L ð7Þ

Substitute the non-dimensional quantities given in Eq. (7) into Eqs. (1)–(6), then the governingequation and its associated initial and boundary conditions have the following dimensionlessforms:

ohðx; r; tÞot

¼ o2hðx; r; tÞox2

þ 1

r ohðx; r; tÞ

orþ o

2hðx; r; tÞor2

ð8Þ

t ¼ 0 hðx; r; 0Þ ¼ 0 ð9Þ

t > 0 x ¼ 0 ohðr; 0; tÞox

¼ f ðr; tÞ ð10Þ

x ¼ 1ohðr; 1; tÞ

oxþ BiT hðr; 1; tÞ ¼ 0 ð11Þ

r ¼ 0ohð0; x; tÞ

or¼ 0 ð12Þ

r ¼ RohðR; x; tÞ

orþ G Bia hðR; x; tÞ ¼ 0 ð13Þ

In order to study the effects of the material parameters on the heat transfer of the fin, the tipBiot number BiT can be rewritten as follows:

BiT ¼ hTLk

¼ hTh LR

hR

k¼ H G Bia ð14Þ

whereH = hT/h represents the ratio of the tip convective heat transfer coefficient to the lateral con-vective heat transfer coefficient.

Applying the Laplace transformation with respect to time to the problem of Eqs. (8)–(13), weget the following results:

s hðx; r; sÞ ¼ o2hðx; r; sÞox2

þ 1

r ohðx; r; sÞor

þ o2hðx; r; sÞor2

ð15Þ

x ¼ 0 ohð0; r; sÞox

¼ f ðr; sÞ ð16Þ

x ¼ 1ohð1; r; sÞ

oxþ BiT hðr; 1; sÞ ¼ 0 ð17Þ


r ¼ 0ohðx; 0; sÞ

or¼ 0 ð18Þ

r ¼ Rohðx;R; sÞ

orþ G Bia hðx;R; sÞ ¼ 0 ð19Þ

Then, applying the central finite difference in Eq. (15), we obtain the following discretized equa-tion for r 5 0:

s hi;j ¼hiþ1;j 2 hi;j þ hiþ1;j

Dx2iþ 1

rjhi;jþ1 hi;j1

2 Drjþhi;jþ1 2 hi;j þ hi;j1

Dr2jð20Þ

where Dxi = 1/(m 1) and Drj = R /(n1). It has to be noted that the term involving 1/r in Eq. (15)is not applicable when the denominator r is equal to zero. By LHospitals rule, we can replacesuch term involving 1/r and get the following equation for r = 0:

s hðx; r; sÞ ¼ o2hðx; r; sÞox2

þ 2 o2hðx; r; sÞor2

ð21Þ

The discretized form of Eq. (21) is

s hi;j ¼hiþ1;j 2 hi;j þ hiþ1;j

Dx2iþ 2

hi;jþ1 2 hi;j þ hi;j1

Dr2jð22Þ

Finally, Eqs. (20) and (22) can be rewritten as

Ai;j hi1;j þ ðBi;j sÞ hi;j þ Ci;j hiþ1;j þ Di;j hi;j1 þ Ei;j hi;jþ1 ¼ 0

for i ¼ 1; 2; . . . ;m and j ¼ 1; 2; . . . ; n ð23Þ

where for r = 0 (j = 1):

Ai;j ¼ Ci;j ¼1

Dx2iBi;j ¼ 2

Dx2i 4

Dr2jDi;j ¼ Ei;j ¼

2

Dr2jð24Þ

while for r5 0 (j 5 1):

Ai;j ¼ Ci;j ¼1

Dx2iBi;j ¼ 2

Dx2i 2

Dr2jDi;j ¼

1

2 Dr2j 2 Drj

rj

Ei;j ¼1

2 Dr2j 2þ Drj

rj

ð25Þ

Substituting the boundary conditions of Eqs. (16)–(19) into Eq. (23) and writing it in matrixform, we obtain the following equation:


B1;1 2A1;1 0 2D1;1 0 0

A1;1 A1;1 0 0 A1;1 B1;1 Ei;j

0 . ..

0 0 Am1;n

0 2Am;n Bm;n gxy

2666666664

3777777775 s½I

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

h1;1h2;1

hm1;nhm;n

266666664

377777775¼

Q1;1

0

Q1;j

Q1;n

266666664

377777775

ð26Þ

where gxy = 2 Æ Em,n Æ Drn Æ G Æ Bia + 2 Æ Am,n Æ Dxm Æ BiT and

Q1;j ¼ 2 A1;j Dx1 f jðr; sÞ ð27Þ

Eq. (26) can be rewritten in the following form:

f½M s½Ig fhi;jg ¼ fQi;jg ð28Þ

where the matrices [M], fhi;jg and fQi;jg are the corresponding matrices in Eq. (26). In Eq. (28),[M] is a (m · n) · (m · n) band matrix with real number, fhi;jg is a (m · n) · 1 vector representingthe unknown temperatures and fQi;jg is a (m · n) · 1 complex vector representing the forcingterms. In general, we can obtain the solutions of all the m · n nodal points in the transform do-main by Eq. (28). Then, the inverse Laplace transformation to the time domain can be achieved bya numerical method. However, for the sake of saving computing time, Eq. (28) does not allow usspecifically to calculate some nodal values. Therefore, sometimes, it is not economical to solve forthe nodal values of vector fhi;jg through Eq. (28), especially when we want to compute only somenodal values. In the situation when only specific nodal values at a specific time are required, wecan calculate by matrix operation to save the huge computer time.

Since the (m · n) · (m · n) matrix [M] is a nonsingular real matrix, it is diagonalizable andpossesses a set of (m · n) linearly independent eigenvectors. There exists a nonsingular transitionmatrix [P] such that [P]1[M][P] = diag[M] and the matrices [M] and diag[M] are similar, havingthe same eigenvalues. The matrix diag[M] is defined as

diag½M ¼

k1 k2

. ..

kmn

266664

377775 ð29Þ

where kl (l = 1,2,. . .,m · n) is the eigenvalues of matrix [M].Substituting Eq. (29) into Eq. (28), we obtain the following equation:

f½P 1½M ½P s½P 1½I½P g ½P 1fhi;jg ¼ ½P 1fQi;jg ð30Þ

Eq. (30) can be rewritten as

fdiag½M s½Ig fhi;jg ¼ fQi;jg ð31Þ

where

fhi;jg ¼ ½P 1fhi;jg ð32Þ

Fig. 2

chang


fQi;jg ¼ ½P 1fQi;jg ð33Þ

From Eq. (31), the following solutions can be obtained:

hi;j ¼

Qi;j

kl sð34Þ

By applying the inverse Laplace transformation to Eq. (34), we get the solution hi;j. However,sometimes, it is difficult to find the inverse Laplace transformation of the complicated Eq. (34) bythe residue theorem. In this article, the inverse Laplace transformation is completed by a generalmethod, known as the Fourier series technique (refer to Appendix A). Finally, after we have ob-tained hi;j, we can calculate the solution hi,j by

fhi;jg ¼ ½P fhi;jg ð35Þ

3. Results and discussions

The foregoing theoretical analysis for a pin fin will be illustrated by the following numericalresults in Figs. 2–6. First, to show the accuracy of our analysis, Fig. 2 compares the results bythe presented method with those of analytical solutions in Ref. [10] for a pin fin that is subjectedto a time and space variation base temperature change in such a function as (1 0.1r2)(0.9 +0.1cos(2pt)). In Fig. 2, it is found that the fin surface temperature distributions by those twomethods are in good agreement.

Next, Figs. 3–6 show the numerical results of a pin fin that is subjected to a time andspace dependent heat flux at the base in such a function as f(r,t) = eXt(1 0.4r2) in Eq. (10).

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1Numerical solutionsAnalytical solutions

θ

t = 0.001

t = 0.01

t = 0.1

t = 1.0

Bia= 0.2, H = 1

G = 5, Ω = 0.5

. Comparison of presented numerical solutions with analytical solutions for a fin subjected to a base temperature

e of ((1 0.1r2)(0.9 + 0.1cos(2pt))).

(a)

(b)

0 0.2 0.4 0.6 0.8 1x

0

0.1

0.2

0.3CenterSurface

t = 0.01

t = 0.1

t = 0.5

t = 1.0

0 0.2 0.4 0.6 0.8 1x

0

0.05

0.1

0.15

0.2

0.25CenterSurface

t = 0.01

t = 0.1

t = 0.5

t = 1.0

Fig. 3. Temperature distributions at various t values with G = 5, H = 1, X = 0.5 and (a) Bia = 0.2, (b) Bia = 0.5.


Theoretically, the performance of a two dimensional pin fin depends on the lateral Biot numberBia the geometric parameter G and the ratio of convective heat transfer coefficients H. Fig. 3(a)and (b) show the distributions of the temperature increments at the fin centerline (r = 0) and atthe fin surface (r = R) as a function of x and t for the cases of G = 5, H = 1, X = 0.5 and (a)Bia = 0.2, (b) Bia = 0.5, respectively. It can be found that the temperature increments decreasewith increasing radial distance and the effect of 2D conduction is not negligible, especially at largevalues of time. Moreover, as Bia increases, the 2D effect becomes more pronounced.

Fig. 4 illustrates the effects of time t and the ratio of convective heat transfer coefficients H onthe distributions of the centerline temperature increments for given Bia = 0.2, G = 5 and X = 0.5.It is shown that the variations of the distributions are insensitive to changes inH at small values of

0 0.2 0.4 0.6 0.8 1x

0

0.1

0.2

0.3H = 1.0H = 0.0H = 10.0

t = 0.01

t = 0.1

t = 1.0

θ

Fig. 4. Effects of H on centerline temperature distributions at various t values with Bia = 0.2, G = 5 and X = 0.5.

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6H = 1.0H = 0.0H = 10.0

G = 2

G = 5

G = 10

Fig. 5. Effects of G and H on centerline temperature distributions at t = 1 with Bia = 0.2 and X = 0.5.


time. This is because all the energy transfer at the base will increase the internal energy of the fin atthe beginning, and the energy has not yet penetrated from the base to the tip at small time. Inaddition, the effect of H increases with increasing radial distance. In other words, the effect ofH is more significant near the tip of the pin fin. This phenomenon is reasonable according tothe definition of H. Fig. 5 shows the effects of the geometry parameter G and the ratio of convec-tive heat transfer coefficients H on the distributions of the centerline temperature increments forgiven Bia = 0.2, t = 1 and X = 0.5. It is found that the effect of H on the distributions is less sig-nificant as G increases. The reason for this phenomenon is that, for a fixed value of G, moreenergy will be convected from the tip surface to the surroundings as H becomes larger. Therefore,the temperature increments at the fin centerline decrease when H increases.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3CenterSurface

= 1.0

= 0.2

= 0.5

= 2.0

x

Fig. 6. Effects of X on temperature distributions at t = 1 with Bia = 0.2, G = 5 and H = 1.


Finally, Fig. 6 depicts the temperature increment distributions of the pin fin for various valuesof X with Bia = 0.2, G = 5, H = 1 and t = 1. Since, from the definition of X, a large X value meansthat the heat flux of the exponential function will decay rapidly, the temperature increments of thefin decrease as X increases in Fig. 6. In addition, the 2D effect decreases with the increase of X.

4. Conclusions

A new numerical method of combining the Laplace transformation and the finite difference isapplied to analyze the transient problem of a two dimensional pin fin that contains tip convectionand is subjected to a time and space dependent heat flux at the fin base. The accuracy of the pre-sented method is checked by existing analytical solutions. Other types of boundary conditions,such as a time dependent change in base temperature, can also be solved by the method.

The numerical results show that the effect of 2D conduction is more significant at large time. Aslateral Biot number Bia increases, the 2D effect becomes more pronounced. On the other hand, thevariations of the temperature increment distributions are insensitive to changes in H at smallvalues of time for given Bia = 0.2, G = 1 and X = 0.5. In addition, the effect of the ratio of convec-tive heat transfer coefficients H on the distributions of temperature increments increases as Gdecreases for given Bia = 0.2, t = 1 and X = 0.5.

Appendix A

In this paper, a general method, known as the Fourier series technique [12,13], is employed tocomplete the inverse Laplace transformation. When a function F(t) is given, the Laplace transfor-mation and its inversion formula are defined as


F ðsÞ ¼Z 1

0

F ðtÞest dt ðA:1Þ

F ðtÞ ¼ 1

2pi

Z cþi1

ci1F ðsÞest ds ðA:2Þ

where the selected, positive arbitrary constant, c, should be greater than the real parts of all sin-gularities of F ðsÞ.

By letting s = c + ix Eqs. (A.1) and (A.2) are converted to those of the Fourier transformation

F ðcþ ixÞ ¼Z 1

0

F ðtÞecteixt dt ðA:3Þ

F ðtÞ ¼ ect

2p

Z 1

1F ðcþ ixÞeixt ds ðA:4Þ

The above integral can be approximated by

F ðtÞ ffi ect

2p

X1n¼1

F ðcþ inDxÞeinDxtDx ðA:5Þ

and letting Dx = p/T*, yields

F ðtÞ ffi ect

2T

X1n¼1

F ðcþ inp=T Þeinðp=T Þt ðA:6Þ

Considering the relation

F ðcþ inp=T Þeinðp=T Þt þ F ðc inp=T Þeinðp=T Þt ¼ 2Re F ðcþ inp=T Þeinðp=T Þt ðA:7Þ

and truncating the infinite series at the Nth term, we have the following numerical formula:

F ðtÞ ffi ect

T 1

2F ðcÞ þRe

XNn¼1

F ðcþ inp=T Þeinðp=T Þt

" #; 06 t6 2T ðA:8Þ

From this equation, the numerical value of F(t) at any time t in the interval 0 6 t 6 2T* can beobtained.

At t = T*, Eq. (A.8) becomes

F ðT Þ ffi ecT

T 1

2F ðcÞ þRe

XNn¼1

F ðcþ inp=T Þð1Þn" #

ðA:9Þ

from Eq. (A.9), the numerical value at the time T* can be calculated.

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Numerical analysis of two dimensional pin fins with non-constant base heat flux

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