A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity

International Communications in Heat and Mass Transfer 32 (2005) 831 – 841 www.elsevier.com/locate/ichmt

A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivityB Cihat Arslanturk Department of Mechanical Engineering, Ataturk University, 25240 Erzurum, Turkey Available online 17 March 2005

Abstract The Adomian decomposition method (ADM) has been used to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and to determine the temperature distribution within the fin. The method is useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. The ADM provides an analytical solution in the form of an infinite power series. The fin efficiency of the straight fins with temperature-dependent thermal conductivity has been obtained as a function of thermo-geometric fin parameter and the thermal conductivity parameter describing the variation of the thermal conductivity. It has been observed that the thermal conductivity parameter has a strong influence over the fin efficiency. The data from the present solutions has been correlated for a wide range of thermo-geometric fin parameter and the thermal conductivity parameter. The resulting correlation equations can assist thermal design engineers for designing of straight fins with temperature-dependent thermal conductivity. D 2005 Elsevier Ltd. All rights reserved. Keywords: Adomian decomposition method; Variable thermal conductivity; Extended surface; Fin efficiency

1. Introduction Extended surfaces are extensively used in various industrial applications. An extensive review on this topic is presented by Kern and Krause [1]. Fins are employed to enhance the heat transfer between the B

Communicated by J.P. Hartnett and W.J. Minkowycz. E-mail address: [email protected].

0735-1933/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.10.006

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C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

primary surface and its convective, radiating or convective-radiating environment. A considerable amount of research has been conducted into the variable thermal parameters which are associated with fins operating in practical situations. Aziz and Hug [2] used the regular perturbation method and a numerical solution method to compute a closed form solution for a straight convecting fin with temperature dependent thermal conductivity. Razelos and Imre [3] considered the variation of the convective heat transfer coefficient from the base of a convecting fin to its tip. A method of temperaturecorrelated profiles is used to obtain the solution of optimum convective fin when the thermal conductivity and heat transfer coefficient are functions of temperature [4]. Laor and Kalman [5] examined straight, spine and annular fins governed by power law-type temperature dependence of the heat transfer coefficient. Yu and Chen [6] investigated the optimal fin length of a convective–radiative straight fin with rectangular profile under convective boundary conditions and variable thermal conductivity. Yu and Chen [7] assumed that the linear variation of the thermal conductivity and exponential function with the distance of the heat transfer coefficient and then, solved the nonlinear conducting–convecting–radiating heat transfer equation by the differential transformation method. Bouaziz et al. [8] presented the efficiency of longitudinal fins with temperature-dependent thermophysical properties. ADM is used for solving various nonlinear heat transfer problems [9–11]. Chiu and Chen [12] investigated a convective longitudinal fin with temperature-dependent thermal conductivity for optimal geometry of the fin. In another study carried out by Chiu and Chen [13], ADM is used for expressing the thermal stress within an annular fin. In the present paper, the energy balance for a differential fin element is developed. The resulting nonlinear differential equation is solved by ADM to evaluate the temperature distribution within the fin. Using the temperature distribution, the efficiency of the fins is expressed through a term called thermogeometric fin parameter, w and thermal conductivity parameter, b describing the variation of the thermal conductivity. Since the resulting analytical expression for the fin efficiency is complicated, the data from the expression has been correlated for a wide range of thermo-geometric fin parameter and the thermal conductivity parameter. The correlation equations of compact form are useful for designing of the straight fins with variable thermal conductivity.

2. Problem description Consider a straight fin with a temperature-dependent thermal conductivity, arbitrary constant crosssectional area A c, perimeter P and length b (see Fig. 1). The fin is attached to a base surface of temperature T b, extends into a fluid of temperature Ta, and its tip is insulated. The one-dimensional energy balance equation is given 
d dT ð1Þ k ðT Þ  PhðTb  Ta Þ ¼ 0 Ac dx dx The thermal conductivity of the fin material is assumed to be a linear function of temperature according to k ðT Þ ¼ ka ½1 þ kðT  Ta Þ

ð2Þ

C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

833

h,Ta

.T x

b

dx b

Fig. 1. Geometry of a straight fin.

where k a is the thermal conductivity at the ambient fluid temperature of the fin and k is the parameter describing the variation of the thermal conductivity. Employing the following dimensionless parameters h¼

T  Ta Tb  Ta

n¼

x b

 b ¼ kðTb  Ta Þ W ¼

the formulation of the problem reduces to  2 d2 h d2 h dh þ bh þ b  W2 h ¼ 0 2 2 dn Bn Bn

hPb2 ka Ac

1=2 ð3Þ

ð4aÞ

dh ¼ 0 at n ¼ 0 dn

ð4bÞ

h ¼ 1 at n ¼ 1

ð4cÞ

3. The Adomian decomposition method The Adomian decomposition scheme is a method for solving a wide range of problems whose mathematical models yield equation or system of equations involving algebraic, differential, integral and integro-differential equations [14–16]. We consider a general nonlinear equation Lu þ Ru þ Nu ¼ g

ð5Þ

where L is highest order derivative which is assumed to be easily invertible, R the linear differential operator of less order than L, Nu represents the nonlinear terms and g is the source term. Applying the inverse operator L  1 to the both sides of Eq. (5), and using the given conditions we obtain u ¼ f ð xÞ  L1 ðRuÞ  L1 ðNuÞ;

ð6Þ

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C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

where the function f(x) represents the terms arising from integrating the source term g(x), and from the using given conditions, all of which are assumed to be prescribed. For nonlinear differential equations, the nonlinear operator Nu = F(u) is represented by an infinite series of the so-called Adomian polynomials F ðuÞ ¼

l X

ð7Þ

Am

m¼0

The polynomials A m are generated for all kind of nonlinearity so that A 0 depends only on u 0, A 1 depends on u 0 and u 1, and so on. The Adomian method defines the solution u(x) by the series u¼

l X

ð8Þ

um

m¼0

In the case of F(u), the infinite series is a Taylor expansion about u 0. In other words F ðuÞ ¼ F ðu0 Þ þ FVðu0 Þðu  u0 Þ þ FWðu0 Þ

ðu  u0 Þ2 ðu  u0 Þ3 : : : þ Fjðu0 Þ þ 2! 3!

ð9Þ

By rewriting Eq. (8) as u u 0 =u 1 + u 2 + u 3 + . . ., substituting it into Eq. (9) and then equating two expressions for F(u) found in Eq.(9) and Eq. (7) defines formulas for the Adomian polynomials F ðuÞ ¼ A1 þ A2 þ : : : ¼ F ðu0 Þ þ FVðu0 Þðu1 þ u2 þ N Þ þ FWðu0 Þ

ðu1 þ u2 þ : : : Þ2 : : : þ 2!

ð10Þ

By equating terms in Eq. (10), the first few Adomian’s polynomials A 0, A 1, A 2, A 3 and A 4 are given A0 ¼ F ðu0 Þ;

ð11aÞ

A1 ¼ u1 FVðu0 Þ;

ð11bÞ

A2 ¼ u2 FVðu0 Þ þ

1 2 u FWðu0 Þ; 2! 1

1 3 u Fjðu0 Þ 3! 1   1 2 1 1 u2 þ u1 u3 FWðu0 Þ þ u21 u2 Fjðu0 Þ þ u41 F ðivÞ ðu0 Þ A4 ¼ u4 FVðu0 Þ þ 2! 2! 4!

ð11cÞ

A3 ¼ u3 FVðu0 Þ þ u1 u2 FWðu0 Þ þ

ð11dÞ

v

ð11eÞ

Now that the A k are known, Eq. (7) can be substituted in Eq. (6) to specify the terms in the expansion for the solution of Eq.(8).

C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

835

4. Fin temperature distribution Following the Adomian decomposition analysis, the linear operator is defined as: L¼

d2 : dn2

ð12Þ

Consequently, Eq. (4a) can be written as follows:  2 d2 h dh Lh ¼  bh 2 þ  b þ W2 h ¼  bNA  bN B þ W2 h dn Bn

ð13Þ

where NA ¼ h

l X d2 h ¼ Am dn2 m¼0

 NB ¼

dh dn

2 ¼

l X

Bm

ð14aÞ

ð14bÞ

m¼0

are nonlinear terms. Hence, using Eqs. (11a)–(11e) gives A0 ¼ h 0

d2 h0 dn2

ð15aÞ

A1 ¼ h 1

d2 h0 d2 h1 þ h 0 dn2 dn2

ð15bÞ

A2 ¼ h 2

d2 h0 d2 h1 d2 h2 þ h þ h 1 0 dn2 dn2 dn2

ð15cÞ

A3 ¼ h 3

d2 h0 d2 h1 d2 h2 d2 h3 þ h þ h þ h 2 1 0 dn2 dn2 dn2 dn2

ð15dÞ

A4 ¼ h 4

d2 h0 d2 h1 d2 h2 d2 h3 d2 h4 þ h þ h þ h þ h 3 2 1 0 dn2 dn2 dn2 dn2 dn2 ð15eÞ

v and



2

B0 ¼

dh0 dn

B1 ¼ 2

dh0 dh1 dn dn

ð16aÞ

ð16bÞ

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C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

 B2 ¼

dh1 dn

2 þ2

dh0 dh2 dn dn

ð16cÞ

dh1 dh2 dh0 dh3 þ2 dn dn dn dn   dh2 2 dh1 dh3 dh0 dh4 þ2 B4 ¼ þ2 dn dn dn dn dn B3 ¼ 2

ð16dÞ

v

ð16eÞ

Applying the inverse operator L  1 to both sides of Eq. (13), we obtain L1 Lh ¼  bL1 N A  bL1 N B þ W2 L1 h;

ð17Þ

h ¼ h0  bL1 NA  bL1 N B þ W2 L1 h:

ð18Þ

and 1

If L is a second-order operator, L is a twofold indefinite integral, i.e. dhð0Þ ; L1 Lh ¼ h  hð0Þ  n dn thus, h0 ¼ hð0Þ þ n

dhð0Þ : dn

ð19Þ

ð20Þ

With the boundary condition given in Eq. (4b), h(0) is any arbitrary constant, C. The next iterates are determined recursively by hmþ1 ¼  bL1 Am  bL1 Bm þ W2 L1 hm

ð21Þ

Therefore, the first four iterates are expressed as: h0 ¼ C h1 ¼

1 Cw2 n2 2

1 1 h2 ¼  bC 2 W2 n2 þ CW4 n4 2 24 h3 ¼

1 2 3 2 2 5 1 b C W n  bC 2 W4 n4 þ CW6 n6 2 24 720

ð22aÞ ð22bÞ ð22cÞ ð22dÞ

1 1 7 1 bC 2 W6 n6 þ CW8 n8 h4 ¼  b3 C 4 W2 n2 þ b2 C 3 W4 n4  2 2 240 40; 320 v

v

ð22eÞ

C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

Upon summing those iterations, the n-term approximation is expressed as l X Unþ1 ¼ hm ¼ h0 þ h1 þ h2 þ : : : þ hn

837

ð23Þ

m¼0

The sum U n+1 may serve as a practical solution and converges very rapidly.  C 1  bC þ b2 C 2  b3 C 3 þ b4 C 4 W2 n2 2   C  C  þ 1  5bC þ 12b2 C 2  22b3 C 3 W4 n4 þ 1  21bC þ 123b2 C 2 W6 n6 24 720 C C ð1  85bC ÞW8 n8 þ W10 n10 þ : : : þ 40; 320 3; 628; 800

h¼Cþ

ð24Þ

The coefficient C representing the temperature at the fin tip, that must lie in the interval (0, 1), can be evaluated from the boundary condition given in Eq. (4c) using the Newton–Raphson method.

5. Fin efficiency The heat transfer rate from the fin is found by using Newton’s law of cooling. Z b Q¼ PðT  Ta Þdx

ð25Þ

0

The ratio of the actual heat transfer from the fin surface to that, that would transfer if the whole fin surface were at the same temperature as the base is commonly called as the fin efficiency. Z b Z 1 PðT  Ta Þdx Q 0 ¼ g¼ ¼ hðnÞdn ð26Þ Qideal PbðTb  Ta Þ n¼0 Integrating Eq. (26), the efficiency of straight fins is obtained as an analytical expression as follows.   C C  g¼C 1  bC þ b2 C 2  b3 C 3 W2 þ 1  5bC þ 12b2 C 2  22b3 C 3 W4 6 120  6 C  C C 2 2 þ 1  21bC þ 123b C W þ ð1  85bC ÞW8 þ W10 þ : : : 5040 362; 880 39; 916; 800 ð27Þ

6. Results and discussion The Adomian decomposition method provides an analytical solution in terms of an infinite power series. The convergence of this series has been established by Cherruault [17] and Cherruault et al. [18] for more general nonlinear problems.

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C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

Table 1 The dimensionless temperature distribution for the case of constant thermal conductivity, i.e. b =0.0 n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M = 0.5

M = 1.0

ADM

Exact

ADM

Exact

0.886819 0.887928 0.891257 0.896815 0.904615 0.914677 0.927026 0.941694 0.958716 0.978135 1.000000

0.886819 0.887928 0.891257 0.896814 0.904614 0.914677 0.927026 0.941693 0.958715 0.978135 1.000000

0.648055 0.651298 0.661059 0.677436 0.700594 0.730763 0.768246 0.813418 0.866731 0.928718 1.000000

0.648054 0.651297 0.661059 0.677436 0.700594 0.730763 0.768246 0.813418 0.866731 0.928718 1.000000

In order to calculate the temperature distribution within the fin for given b and w, it must evaluate the coefficient C from the boundary condition given in Eq.(4c). This calculation is carried out using the Newton–Raphson method. The dimensionless temperature distribution within the fin was calculated by taking the first six terms from the series solution. For the case of constant thermal conductivity, the results of the present analysis 1.00

β = -0.5, 0.5, 0.2

θ 0.95

0.90

0.85

(a) 0.80 0.0

0.2

0.4

0.6

0.8 ξ

1.0

0.8

1.0

1.0

θ

β = -0.5, 0.5, 0.2

0.9 0.8 0.7 0.6

(b) 0.5 0.0

0.2

0.4

0.6

ξ

Fig. 2. Temperature distribution in convective fins with variable thermal conductivity (a) w = 0.5 and (b) w =1.0.

C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841 1.0

839

exact solution

Fin efficiency, η

0.9 β = -0.6, 0.6, 0.2

0.8 0.7 0.6 0.5 0.4 0.3 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Thermo-geometric fin parameter, ψ

Fig. 3. Variation of the fin efficiency with the thermo-geometric fin parameter for different values of the thermal conductivity parameter.

are tabulated against the analytical solution [19] in Table 1. A very good agreement between the results was observed, which confirms the validity of the ADM. The dimensionless temperature distributions along the fin surface with b varying from 0.5 to 0.5 are depicted in Fig. 2a and b for different values of w = 0.5 and w = 1.0, respectively. It will be seen that, if the thermal conductivity of the fin’s material increases with the temperature, the mean temperature increases. Fig. 3 shows the fin efficiency as a function of the thermo-geometric fin parameter for seven different values of the thermal conductivity parameter. From Fig. 3, it can be seen that the results from the decomposition method match with the exact solution for the case of constant thermal conductivity, i.e., b = 0. In order to use the present solutions by thermal design engineers, the fin efficiency expressed as a function of the thermo-geometric fin parameter for an attained thermal conductivity parameter. With this

Table 2 Coefficients in Eq.(28) for the fin efficiency b

a

b

c

d

e

R2

 0.6  0.5  0.4  0.3  0.2  0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

1.000772983 1.000111622 0.999666184 0.999425848 0.999351819 0.999369398 0.999309535 0.999179911 0.998763908 0.998266682 0.998095998 0.998559919 0.999195681

0.639679344 0.486908039 0.358238503 0.253396469 0.172239563 0.114623793 0.084034787 0.064067031 0.053733641 0.015005057  0.09687311  0.28279818  0.47115608

0.634222355 0.491790302 0.368554213 0.265379544 0.183476391 0.124249146 0.093498870 0.074338527 0.067769401 0.032135917 0.08317754 0.28222498 0.48695373

0.869265949 0.758890527 0.661202095 0.576129902 0.503739186 0.444221613 0.398549710 0.364814806 0.343935551 0.331172591 0.318642222 0.298500873 0.272318074

0.027831160 0.032999759 0.036901495 0.039395170 0.040326425 0.039589252 0.036213074 0.034449453 0.034295552 0.044214511 0.072185765 0.114117088 0.151496439

0.9999983729 0.9999995414 0.9999999301 0.9999998978 0.9999997455 0.9999996159 0.9999992939 0.9999986491 0.9999952545 0.9999835148 0.9999576807 0.9999301564 0.9999204932

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C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

correlation equation for the tip temperature, it is assumed that the equation has the following form for an attained thermal conductivity parameter. The coefficients in Eq. (28) are tabulated in Table 2 for the fin efficiency. g¼

a þ cW þ eW2 1 þ bW þ dW2

ð28Þ

7. Conclusions Convective straight fins with temperature-dependent thermal conductivity were analyzed using the Adomian decomposition method. The decomposition method supplies reliable results in the form of analytical approximation converging very rapidly. The nonlinear differential equation, which governs the fin temperature distribution, was solved and then the fin efficiency was calculated. The results are expressed in terms of suitable dimensionless parameters and presented in terms of regression equations obtained by standard statistical techniques. These results can be used for designing straight fins with variable thermal conductivity. Nomenclature cross-sectional area of the fin (m2) Ac Adomian’s polynomials Am b fin length (m) C an integral constant arising in Eq. (20) h heat transfer coefficient (W m 2 K 1) k thermal conductivity of the fin material (W m 1 K 1) thermal conductivity at the ambient fluid temperature (W m 1 K 1) ka thermal conductivity at the base temperature (W m 1 K 1) kb L the higher order derivative L  1 inverse operator of L N nonlinear terms P fin perimeter (m) R remainder of the linear operator Q heat-transfer rate (W) T temperature (K) x distance measured from the fin tip (m) Greek b g n k w h

symbols dimensionless parameter describing variation of the thermal conductivity fin efficiency dimensionless coordinate the slope of the thermal conductivity–temperature curve (K 1) thermo-geometric fin parameter dimensionless temperature

C. Arslanturk / International Communications in Heat and Mass Transfer 32 (2005) 831–841

841

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