- Email: [email protected]

Optimization of non-symmetric convective–radiative annular ﬁns by diﬀerential quadrature method P. Malekzadeh a b

a,b,*

, H. Rahideh c, A.R. Setoodeh

d

Department of Mechanical Engineering, School of Engineering, Persian Gulf University, Bushehr 75168, Iran Center of Excellence for Computational Mechanics in Mechanical Engineering, Shiraz University, Shiraz, Iran c Department of Chemical Engineering, School of Engineering, Persian Gulf University, Bushehr 75168, Iran d Department of Mechanical Engineering, Ferdowsi University, Mashhad, Iran Received 27 April 2006; accepted 12 November 2006 Available online 27 December 2006

Abstract The shape optimization of non-symmetric, convective–radiative annular ﬁns is performed based on two-dimensional heat transfer analysis. The formulations are general so that the spatial and temperature dependent geometrical and thermal parameters can easily be implemented. The thermal conductivity of the ﬁn is assumed to vary as a linear function of the temperature. The convective–radiative condition at the external surfaces of the ﬁn and the eﬀective convective condition at the ﬁn base are considered. The diﬀerential quadrature method (DQM), as a simple, accurate and computationally eﬃcient numerical tool, is used to solve the nonlinear two-dimensional heat transfer equation and the related nonlinear boundary conditions in the polar coordinate system. The accuracy of the method is demonstrated by comparing its results with those of the ﬁnite diﬀerence method. It is shown that by using fewer grid points, highly accurate results are obtained. Less computational eﬀort of the method with respect to the ﬁnite diﬀerence method is shown. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Annular ﬁn; Heat transfer; Optimization; Diﬀerential quadrature method

1. Introduction Finned surfaces have been used for a long period as a heat dissipation mechanism to increase the heat transfer rate between a primary surface and the surrounding ﬂuid in heat exchange devices. Therefore, optimization of the design of ﬁns for high performance, light weight and compact heat transfer components is of signiﬁcant importance. For this reason, an extensive review is seen on this topic [1]. The optimization of ﬁns is generally based on two approaches: 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 [2,3]. Extensive literature exists on the above optimization prob*

Corresponding author. Tel.: +98 771 4222150; fax: +98 771 4540376. E-mail addresses: [email protected], [email protected] (P. Malekzadeh). 0196-8904/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.11.002

lem for convective ﬁns. However, most previous studies were based on one-dimensional heat transfer analysis [2– 11]. The one-dimensional approach is convenient, but may be in error for certain physical conditions [2,11]. Based on two-dimensional heat transfer analysis, Lalot et al. [12] developed an expression for the temperature distribution and the eﬃciency of annular ﬁns made of two materials. Arslanturk [13] used two-dimensional analytical approaches for constant volume optimization of rectangular cross section annular ﬁns with non-symmetric convective boundary conditions. In the previous works [12,13], the radiation heat transfer was ignored and constant thermal conductivity was considered. In this paper, as an extension of the previous works [14– 16], the optimization of annular ﬁns with general boundary conditions based on two-dimensional heat transfer analysis is investigated. A two-dimensional diﬀerential quadrature method (DQM) is used to obtain the heat transfer rate.

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Nomenclature Axij ; Arij ﬁrst order weighting coeﬃcients in x and r directions Bxij ; Brij second order weighting coeﬃcients in x and r directions Bif Biot number at annular ﬁn base (hfto/ka) Bil Biot number at lower surface of annular ﬁn tip (haLo/ka) Bit Biot number at annular ﬁn tip (hato/ka) Biu Biot number at top surface of annular ﬁn tip (haLo/ka) ha convection heat transfer coeﬃcient of ambient hf convection heat transfer coeﬃcient of ﬂuid K thermal conductivity ka thermal conductivity at ambient temperature Lo thickness of ﬁn Nx, Nr number of grid points in x and r directions qmax maximum heat dissipation R radial coordinate variable Ro outer radius of annular ﬁn Ri inner radius of annular ﬁn Linearly varying thermal conductivity is considered. Convective boundary conditions at the base and convective– radiative heat transfer at the other boundaries of the ﬁn are assumed.

Consider a typical annular ﬁn as shown in Fig. 1. A combination of a convection–radiation mechanism is considered on the external surfaces of the annular ﬁn. Allowing the thermal conductivity of the annular ﬁn to be a function of temperature, its energy balance equation may be written as KðT Þ

T Ta Tf Ts to Vf X x a b e k1, k 2 h hs r

non-dimensional radial coordinate variable RRi Ro Ri

temperature at arbitrary point of annular ﬁn ambient temperature ﬂuid temperature eﬀective sink temperature of radiation surface of annular ﬁn length of annular ﬁn (Ro Ri) volume of ﬁn axial coordinate variable non-dimensional axial coordinate variable (X/Lo) absorptivity slope of thermal conductivity–temperature divided by intercept ka emissivity ratio of thickness and inner radius to length of annular ﬁn (Lo/to, Ri/to) non-dimensional temperature (T/Ta) non-dimensional surrounding temperature (Ts/Ta) Stefan–Boltzmann constant

KðT Þ ¼ k a ½1 þ bðT T a Þ

2

o T oKðT Þ oT o T oKðT Þ oT KðT Þ oT þ KðT Þ 2 þ þ ¼0 þ oX oX oR oR R oR oX 2 oR ð1Þ

Neglecting the thermal resistance of the primary surfaces at the boundaries, the general form of the external boundary conditions becomes oT At X ¼ 0 : KðT Þ þ hl ðT T a Þ þ rðeT 4 aT 4s Þ ¼ 0 ð2Þ oX oT þ hu ðT T a Þ þ rðeT 4 aT 4s Þ ¼ 0 ð3Þ At X ¼ Lo : KðT Þ oX At R ¼ Ri : either T ðX ; Ri Þ ¼ f ðX Þ or oT þ hf ðT T f Þ ¼ 0 KðT Þ ð4Þ oX oT þ ht ðT T a Þ þ rðeT 4 aT 4s Þ ¼ 0 ð5Þ At R ¼ Ro : KðT Þ oX In the present analysis, without loss of generality, the thermal conductivity is assumed to be a linear function of temperature [17]:

ð6Þ

In order to simplify the parameter studies, the following non-dimensional variables are deﬁned: x ¼ X =Lo ;

2. Governing equations

2

r

r¼

R Ri ; Ro Ri

h¼

T ; Ta

hs ¼

Ts ; Ta

1 hj t o ðLo ; Ri Þ; Bij ¼ ; j ¼ f; t; l; u; to ka rT 3 Lo rT 3 to ðma ; me Þ ¼ a ða; eÞ; ðna ; ne Þ ¼ a ða; eÞ ka ka

c ¼ bT a ;

ðk1 ; k2 Þ ¼

ð7Þ

By using the aforementioned non-dimensional variables, the governing equation and its associated boundary conditions become Eq. (1): 2 2 2 oh k1 oh 2o h ½1 þ cðh 1Þ 2 þ þ k1 2 ox or k2 þ r or " # 2 2 oh oh þc þ k21 ¼0 ox or

ð8Þ

Eqs. (2) and (3): oh þ Bil ðh 1Þ þ me h4 ma h4s ¼ 0 ox ð9Þ oh At x ¼ 1 : ½1 þ cðh 1Þ þ Biu ðh 1Þ þ me h4 ma h4s ¼ 0 ox ð10Þ

At x ¼ 0 : ½1 þ cðh 1Þ

Eq. (4):

P. Malekzadeh et al. / Energy Conversion and Management 48 (2007) 1671–1677

1673

X, x

to

q conv + q rad

Lo R, r

q conv + q rad

Fig. 1. The geometry of the annular ﬁn.

At r ¼ 0 : either hðx; 1Þ ¼ F ðxÞ or oh ½1 þ cðh 1Þ þ Bif ðh hf Þ ¼ 0 or

ð11Þ

for j ¼ 1; . . . ; N r :

Eq. (5): oh At r ¼ 1 : ½1 þ cðh 1Þ þ Bit ðh 1Þ þ ne h4 na h4s ¼ 0 or ð12Þ 3. Discretization of the governing equation and boundary conditions In this section, by using the diﬀerential quadrature (DQ) rules for derivatives, the discretized form of the governing equation and the related boundary conditions are derived. The spatial domain is discretized into Nx and Nr grid points in the x and r directions, respectively. Uniform or non-uniform grid points can be used to discretize the spatial domain, however, based on our experiences in this study and previous DQ works [14–16,18] with the same number of grid points, non-uniform grid points yields results with better accuracy. Therefore, a cosine type grid generation rule is used to discretize the spatial domain as 1 pði 1Þ xi ¼ 1 cos ; 2 ðN x 1Þ 1 pðj 1Þ rj ¼ 1 cos 2 ðN r 1Þ for i ¼ 1; . . . ; N x and j ¼ 1; . . . ; N r

ð13Þ

Using the DQ rules for the spatial derivatives (see Appendix A.), the DQ analogs of the governing equation become " 2 X Nx Nr X k1 x ½1 þ cðhij 1Þ Bim hmj þ Arjn hin k þ r 2 j m¼1 n¼1 2 # !2 !2 3 N N Nr x r X X X þk2 Br hin þ c4 Ax hmj þ k2 Ar hin 5 ¼ 0 1

im

jn

n¼1

where hij = h(xi, rj). In a similar manner, the DQ discretized forms of the external boundary conditions become Eqs. (9) and (10):

m¼1

for i ¼ 2; . . . ; N x 1; j ¼ 2; . . . ; N r 1

1

jn

n¼1

ð14Þ

½1 þ cðhij 1Þ

Nx X

! Axim hmj

þ Bilj ðhij 1Þ

m¼1 4

4

ð15Þ

4

ð16Þ

þ me ðhij Þ ma ðhsj Þ ¼ 0 for i ¼ 1 ! Nx X Axim hmj þ Biuj ðhij 1Þ ½1 þ cðhij 1Þ m¼1 4

þ me ðhij Þ ma ðhsj Þ ¼ 0 for i ¼ N x Eqs. (11) and (12): for i ¼ 2; . . . ; N x 1 : Either hij ¼ F ðxj Þ

or ½1 þ cðhij 1Þ

Nr X

! Arjn hin

n¼1

þ Bifi ðhij hfi Þ ¼ 0 ½1 þ cðhij 1Þ

Nr X

for j ¼ 1 !

Arjn hin

ð17Þ

þ Biti ðhij 1Þ þ ne ðhij Þ4

n¼1 4

na ðhsi Þ ¼ 0

for j ¼ N r

ð18Þ

A Newton–Raphson method is used to solve the resulting nonlinear algebric system of Eqs. (14)–(18). After obtaining the temperature, the heat ﬂux at any section of the annular ﬁn can be obtained using Fourier’s heat transfer law, but for a convective–radiative boundary, it is preferred to use the convective–radiative law to calculate the heat ﬂux at the boundary surfaces. The non-dimensional heat transfer from the annular ﬁn base in the case of a convective condition can be obtained as q¼

Z

Lo

hf ðT f T Þ2pR dX Z 1 r ¼ 2pRi Lo hf T a ðhf hÞ 1 þ dx k2 0 0

ð19Þ

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P. Malekzadeh et al. / Energy Conversion and Management 48 (2007) 1671–1677

Using the trapezoidal rule to calculate the integral in Eq. (19) numerically, the result becomes rn q ¼ ð2pLo to hf T a Þ ðhf h1n Þ 1 þ k2 m¼1 rm þðhf h1m Þ 1 þ ðrn rm Þ; n ¼ m þ 1 k2

k a ¼ 63:9 W=m K; b ¼ 0:000758 K1 ; T a ¼ T s ¼ 300 K; T f ¼ 600 K; hf ¼ 1000 W=m2 K; hl ¼ hu ¼ ht ¼ 20 W=m2 K; a ¼ e ¼ 1:0;

N r 1 X

r ¼ 5:67 108 W=m2 K4 ; V f ¼ 5:89 10

5

m

Ri ¼ 0:025 m;

3

ð20Þ

4. Optimization procedure Optimization of the cooling properties of ﬁns can be made by either minimizing the volume (weight) for a required heat dissipation or maximizing the heat dissipation for any given ﬁn volume (weight). The solution of these problems results in ﬁn shapes with curved surfaces. Such an optimum shape is practically diﬃcult and expensive to fabricate. Thus, for most applications, the rectangular proﬁle is preferred despite the fact that it does not utilize the material most eﬃciently. The maximum heat dissipation is expressed by the optimum ﬁn characteristics and the determined dimensions are the optimum ﬁns conﬁguration. In the present study, the golden section search method [19] is used to obtain the optimum dimensions of the annular ﬁns.

5. Numerical examples In this section, after demonstrating the usefulness of the method, the eﬀects of diﬀerent types of boundary conditions, i.e. convection, radiation, convection–radiation, and the thermal parameters on the characteristics of uniform annular ﬁns will be investigated. In the solved examples, the thermal conductivity is assumed to be a linear function of temperature (see Eq. (6)); and at the ﬁn base, a convective boundary condition is considered. The values of the geometrical parameters, material properties and the other heat transfer parameters used in the solved examples are as follows:

As a ﬁrst example, the accuracy and convergence behavior of the presented DQM for a uniform annular ﬁn with convective–radiative boundary conditions at its external surfaces is investigated. In Table 1, the eﬀects of diﬀerent numbers of grid points on the optimum ratio of inner radius to length (k2) and maximum heat dissipation for a given ﬁn volume are presented. To validate the DQM results, this example is also solved using the ﬁnite diﬀerence method (FDM). A uniform mesh is used for the ﬁnite diﬀerence method. It is obvious that employing the DQM, accurate results can be obtained using only ﬁve grid points in the x and r directions. Faster convergence of the DQM with respect to the FDM is quite evident. In this table, a comparison between the CPU time requirements of the DQM and FDM is also made, which show much less computational eﬀorts of the DQM with respect to FDM. The eﬀect of ﬁn tip heat transfer on the optimum value of k2 and the maximum heat transfer dissipated by the annular ﬁn is investigated as another example. In Figs. 2 and 3, the heat transfer versus k2 for an annular ﬁn with insulated and convective–radiative tip are presented, respectively. It should be mentioned that the insulation assumption at the ﬁn tip simpliﬁes the calculations of the optimum dimensions. This is because, for this situation, in contrast with convection–radiation at the ﬁn tip, only an extreme value exists for heat dissipation, see Figs. 2 and 3. From Fig. 3, it is obvious that for the ﬁn with heat transferred from the tip, ﬁrst a maximum and then a minimum heat dissipation occurred on increasing the ratio of k2. The insulation assumption at the ﬁn tip reduced the local maximum heat dissipation of the ﬁn from qmax = 329 (W) to qmax = 299 (W) and increased the optimum ratio of inner radius to length from 0.58 to 0.79. It is worthwhile to note that there are design restrictions for the optimum

Table 1 Comparisons of computational characteristics of DQM and FDM DQM

FDM

Nx

Nr

k2opt

qmax (W)

Time (s)

Nx

Nr

k2opt

qmax (W)

Time (s)

5 5 7 7 9 11 13

5 7 7 9 11 11 13

0.58289 0.58170 0.58170 0.58170 0.58170 0.58170 0.58170

298.47 298.77 298.76 298.78 298.77 298.77 298.77

0.703 1.141 1.891 2.735 5.579 7.625 13.906

5 5 7 11 11 21 21

5 7 7 11 21 31 41

0.59600 0.61066 0.59072 0.58679 0.59443 0.58677 0.58800

293.47 312.38 295.46 296.88 306.45 300.91 302.42

2.672 2.125 9.063 19.852 65.578 501.81 1394.84

P. Malekzadeh et al. / Energy Conversion and Management 48 (2007) 1671–1677

Conve ction-Radiation Conve ction Radiation

Convection-Radiation Convection

Maximum heat , q (W)

q (W)

275.0

200.0

125.0

50.0 0.25

0.85

λ2

1.45

560.0

410.0

260.0

110.0

2.05

5.0

50.0

95.0

140.0

Heat transfer coefficent (W/m2K)

Fig. 2. Heat dissipated by annular ﬁn with insulated tip versus inner radius to length, k2.

Fig. 4. The variation of maximum heat transfer of annular ﬁn as a function of heat transfer coeﬃcient.

280.0

485.0

Maximum heat , q (W)

Conve c tion-Radiation Cnve c tiono Radiation 395.0

q (W)

1675

305.0

Convection-Radiation Convection Radiation

190.0

100.0

215.0 10.0 25.0

125.0

100.0

175.0

250.0

Temperaure difference (Tf-Ta), (K) 0.3

1.2

2.1

3.0

λ2

Fig. 5. The variation of maximum heat transfer of annular ﬁn as a function of temperature diﬀerence.

Fig. 3. Heat dissipated by annular ﬁn with convective–radiative tip versus inner radius to length, k2.

300. 0

Maximum heat , q (W)

value of k2 with heat transfer from the tip of annular ﬁns, but there always exists an optimum value of inner radius to length of an annular ﬁn with an insulated tip. The same behaviors were observed for longitudinal ﬁns [14]. The eﬀects of convective heat transfer coeﬃcient, ﬂuid and ambient temperature diﬀerence (Tf Ta) and thermal conductivity of the ﬁn on the maximum heat transfer are shown in Figs. 4–6. Also, the eﬀects of the same thermal parameters on the optimum value of k2 are presented in Figs. 7–9. From Figs. 4–6, it is obvious that the maximum heat transfer increased by increasing the aforementioned thermal parameters. However, as is obvious from Fig. 9, the thermal conductivity has an inverse eﬀect on the optimum value of k2. The eﬀect of the temperature dependence of the thermal conductivity, b, on the maximum heat transfer by the ﬁn is shown in Fig. 10. As is obvious from this

Convection-Radiation Convection Radiation

255. 0

210. 0

165. 0 25. 0

130. 0

235. 0

340. 0

Thermal conductivity (W/mK) Fig. 6. The variation of maximum heat transfer of annular ﬁn as a function of thermal conductivity.

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P. Malekzadeh et al. / Energy Conversion and Management 48 (2007) 1671–1677

Convection-Radiation Convection

β =-0. 003 β =0.0 β =0.01

300

λ2

Heat transfer , q (W)

0.79

0. 54

225

0.29 5.0

50. 0

95. 0

140. 0

Heat transfer coefficent (W/m2K) Fig. 7. The variation of optimum value of k2 of annular ﬁn as a function of heat transfer coeﬃcient.

150 0.25

0.50

0.75

1.00

1.25

λ2 Fig. 10. Heat dissipated by annular ﬁn with insulated tip versus k2 for various slopes of the thermal conductivity, b.

0.59

ﬁgure, for positive values of b, the maximum heat transfer by the ﬁn is increased for increasing b. 0.50 λ2

6. Conclusion

0. 41 Convection-Radiation Convection Radiation 0.32 25. 0

100. 0

175. 0

250. 0

Temperaure difference (Tf-Ta), (K) Fig. 8. The variation of optimum value of k2 of annular ﬁn as a function of temperature diﬀerence.

Convection-Radiation Convection Radiation

In this paper, a diﬀerential quadrature optimization of convective–radiative annular ﬁns based on two-dimensional heat transfer analysis was presented. Diﬀerent types of boundary conditions were considered. Well converged, accurate results with only a few grid points were obtained, thus reducing the computational cost. By solving this type of problem, the low computational eﬀorts of the DQM for two-dimensional iterative problems were investigated. Fast convergence and high accuracy of the DQM as a useful and practically important method can be used to solve problems associated with the complex conditions presented by ﬁn boundaries and geometries. The eﬀects of diﬀerent thermal parameters on the maximum heat transfer by the ﬁn and the optimal value of the inner radius to length of the annular ﬁn k2 were investigated, which may be used as a benchmark solution by other researchers.

0.58

λ2

Appendix A. DQ weighting coeﬃcients

0.49

0.40 25. 0

130. 0

235. 0

340. 0

Thermal conductivity (W/m K) Fig. 9. The variation of optimum value of k2 of annular ﬁn as a function of thermal conductivity.

The basic idea of the diﬀerential quadrature method is that the derivative of a function with respect to a space variable at a given sampling point is approximated as a weighted linear sum of the sampling points in the domain of that variable. In order to illustrate the DQ approximation, consider a function f(x, r) having its ﬁeld on a rectangular domain 0 6 x 6 1 and 0 6 r 6 1. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to the DQ method, the sth derivative of a function f(x, r) can be approximated as

P. Malekzadeh et al. / Energy Conversion and Management 48 (2007) 1671–1677

Nx X os f ðx; rÞ xðsÞ ¼ A f ðxm ; rj Þ oxs ðx;rÞ¼ðxi ;rj Þ m¼1 im ¼

Nx X

References

xðsÞ

Aij fmj

m¼1

for i ¼ 1; 2; . . . ; N x and s ¼ 1; 2; . . . ; N x 1

ðA:1Þ

From this equation, one can deduce that the important components of DQ approximations are weighting coeﬃcients and the choice of sampling points. In order to determine the weighting coeﬃcients, a set of test functions should be used in Eq. (A.1). For the polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coeﬃcients for the ﬁrst order derivatives in the x direction are, thus, determined as [20] 8 Mðxi Þ > for i 6¼ j > ðxi xj ÞMðxj Þ > < N x x P i; j ¼ 1; 2; . . . ; N x Aij ¼ ðA:2Þ Axij for i ¼ j > > > : j¼1 i6¼j

where Mðxi Þ ¼

Nx Y

ðxi xj Þ

j¼1;i6¼j

The weighting coeﬃcients of the second order derivative can be obtained as [19] ½Bxij ¼ ½Axij ½Axij ¼ ½Axij

2

ðA:3Þ

In a similar manner, the weighting coeﬃcients for the r direction can be obtained. In numerical computations, Chebyshev–Gauss–Lobatto quadrature points are used, that is [20], xi 1 ði 1Þp ¼ 1 cos ; ðN x 1Þ a 2 rj 1 ðj 1Þp ¼ 1 cos ðN r 1Þ b 2 for i ¼ 1; 2; . . . ; N x and j ¼ 1; 2; . . . ; N r

1677

ðA:4Þ

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