A model on the basis of analytics for computing maximum heat transfer in porous fins

International Journal of Heat and Mass Transfer 55 (2012) 7611–7622

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A model on the basis of analytics for computing maximum heat transfer in porous fins Balaram Kundu a,b,⇑, Dipankar Bhanja a,1, Kwan-Soo Lee b a b

Department of Mechanical Engineering, Jadavpur University, Kolkata 700 032, India School of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Sungdong-gu, Seoul 133-791, Republic of Korea

a r t i c l e

i n f o

Article history: Received 14 May 2012 Received in revised form 17 July 2012 Accepted 24 July 2012 Available online 19 August 2012 Keywords: Darcy model Straight fin Natural convection Performance Porous surface Optimization

a b s t r a c t This study presents an analytical work on the performance and optimum design analysis of porous fin of various profiles operating in convection environment. Straight fins of four different profiles, namely, rectangular, convex parabolic and two exponential types are considered for the present investigation. An analytical technique based on the Adomian decomposition method is proposed for the solution methodology as the governing energy equations of porous fins for all the profiles are non-linear. A comparative study has been carried out among the results obtained from the porous and solid fins, and an appreciable difference has been noticed for a range of design conditions. Finally, the result shows that the heat transfer rate in an exponential profile with negative power factor is much higher than the rectangular profile but slightly higher than the convex profile. On the other hand, the fin performance is observed to be better for exponential profiles with positive power factor than other three profiles. A significant increase in heat transfer through porous fins occurs for any geometric fin compared to that of solid fins for a low porosity and high flow parameter. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The high rate of heat transfer with reduced size and cost is in demand for a number of engineering applications such as electronic components, heat exchangers, economizers, superheaters, airplanes, internal combustion engines etc. Extended surfaces (fins) are one of the heat exchanging devices that are employed extensively to increase heat transfer rate. The major heat transfer from fin surfaces to surrounding fluid takes place by convection mode. Under a constant heat transfer coefficient and a given temperature difference between primary surface and ambient condition, the rate of heat transfer depends mostly on the surface area of the fin. Nevertheless, increasing surface area will result in increasing overall dimension of the equipment and thereby increasing material cost and size of the equipment. Consequently, the system becomes bulky and uneconomical. It is also well known that the rate of heat conduction from the fin decreases with the increase of fin length and hence the entire heat transfer surface of a fin may not be equally utilized. Thus, there is a continuous effort by

⇑ Corresponding author at: Department of Mechanical Engineering, Jadavpur University, Kolkata 700 032, India. Tel.: +91 33 2414 6890; fax: +91 33 2414 6890. E-mail addresses: [email protected] (B. Kundu), d_bhanja@rediffmail. com (D. Bhanja), [email protected] (K.-S. Lee). 1 Present address: Department of Mechanical Engineering, Dr. B.C. Roy Engineering College, Durgapur 713 206, India. 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.07.069

the designers to determine the optimum fin profile and shape which will maximize the rate of heat transfer for a specified fin volume or minimize the fin volume for a given heat transfer rate. A variational calculus technique had been used first by Duffin [1] to find out the optimum fin shape. Karlekar and Chao [2] had shown an optimization procedure for achieving maximum dissipation from a longitudinal fin system of trapezoidal profile with mutual irradiation. An overview of the fin optimum shaping issue has been depicted by Snider and Kraus [3]. Jany and Bejan [4] investigated the optimum shape for straight fins with temperature dependent thermal conductivity. Arslanturk [5] prescribed simple correlation equations for optimum design of annular fins with uniform cross section. In the optimization analysis, the fin volume was taken as fixed to determine dimensionless geometrical parameters of the fin with maximum heat transfer rate. The optimum dimensions of circular fins of trapezoidal profile with variable thermal conductivity and heat transfer coefficient had been calculated by Razelos and Imre [6]. They considered a linear variation of the thermal conductivity with the temperature and a power law variation of heat transfer coefficient with the distance from the base. Mokheimer [7] reported the performance of annular fins of different profiles subject to locally variable heat transfer coefficient. The performance of the fin expressed in terms of fin efficiency in the form of curves for different types of fins. These curves have been obtained based on constant convection heat transfer coefficient. Performance and optimum dimensions of longitudinal and

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Nomenclature a Ai, Bi Bi cp F g G h keff kf ks kR K Lx L1 x L n q Q qi Qi

exponential power facctor Adomian Polynomials, i = 0, 1, . . . Biot number, htb =ks specific heat of the fluid passing through porous fin, J kg1 K1 function defined in Eq. (22) gravity constant, m s2 function defined in Eq. (23) heat transfer coefficient over the fin surface, Wm2 K1 effective thermal conductivity defined in Eq. (4), Wm1 K1 thermal conductivity of the fluid, Wm1 K1 thermal conductivity of the fin material, Wm1 K1 thermal conductivity ratio, kf =ks permeability of the porous fin, m2 highest order derivatives inverse operator of Lx, see Eq. (9) length of the fin, m thickness profile index actual heat transfer rate per unit width of the porous fin, W m1 dimensionless actual heat transfer rate per unit width, see Eq. (17) ideal heat transfer rate per unit width, W m1 dimensionless ideal heat transfer rate per unit width, see Eq. (18)

annular fins and spines with a spatial dependent heat transfer coefficient have been presented by Kundu and Das [8]. An analytical solution for straight fin with combined heat and mass transfer had been applied by Kundu [9] and Sharqawy and Zubair [10]. They used four different profiles for the fin and compared the temperature distribution and fin efficiency for them. An analytical solution for fin efficiency of straight fins and spines with different profiles and annular fins of rectangular and constant heat flow area profiles subject to constant heat transfer coefficient had been presented by Gardner [11]. Moradi and Ahmadikia [12] investigated analytically the three different straight fin profiles namely rectangular, convex and exponential that has a temperature dependent thermal conductivity. They predicted that efficiency of the exponential profile is higher than the rectangular and convex profile. A thorough research work had been devoted to determine the optimum fin envelope-shapes of longitudinal, spine and annular fins under different conditions such as, convection, heat generation, condensation, dehumidification, etc. [13–20]. With considering all nonlinearity effects, recently, Kundu and Lee [18–20] analyzed fin performances and optimum fin shapes under dehumidifying conditions by the de of the new analytical models. In manufacturing point of view, the optimum profile shape may be slightly difficult due to its fragile shape at the tip. To avoid this difficulty, different fin shapes like triangular, trapezoidal, recto-trapezoidal, straight fin with step change in thickness, SRC profile, eccentric, elliptic, T-shape, etc. [21–29] had been established for the design purpose. In a restricted design application, superiorities of all these profiled shapes were highlighted in their analyses. The heat transfer performance of fins can be increased by adopting the constructal fin shape. Xie et al. [30] demonstrated numerically the constructal optimization of the twice Y-shaped assemblies of fins with six degrees for the heat transfer performance measured by thermal resistance under various conditions. They indicated that the lowest maximum thermal resistance of the twice Y-shaped assemblies of fins decreases by 36.37% compared with that of once Y-shaped assembly of fins. Kundu and

R1 tb t Ta Tb T U V x X

non-dimensional flow parameter used in Eq. (4), qC p gKbðT b  T a Þ=ðhcÞ thickness at the fin base, m thickness of the fin, m ambient temperature, K fin base temperature, K temperature, K dimensionless volume, see Eq. (20) volume per unit width, (m2) axial length measured from flange tip as shown in Fig. 1, m dimensionless axial distance, x/L

Greek symbols B coefficient of thermal expansion, K1 e effectiveness of the fin x1,x2 parameters used in Eq. (4) x3,x4 notations defined in Eq. (15d) g efficiency of the fin f porosity w base thickness to length ratio, tb/L h dimensionless temperature, ðT  T a Þ=ðT b  T a Þ h0 dimensionless tip temperature c kinematic viscosity, m s2 q density of the fluid, kg m3 / thermo-porosity, 1  f þ f kR

Bhanja [28] investigated an analytical technique to determine the thermal performance and optimum dimensions of constructal T-shape fins under one-dimensional heat conduction subject to variable thermal conductivity and heat transfer coefficient. With two-dimensional heat transfer model, a finite element analysis for the constructal optimization of T-shaped fins had been studied by Xie et al. [31] with adopting the minimization of equivalent thermal resistance based on the entransy dissipation and the minimization of maximum thermal resistance as an optimization objective, respectively. Using analytical method, Xiao et al. [32] optimized an umbrella-shaped assembly of cylindrical fins by taking into consideration of the dimensionless mean thermal resistance as the performance index. By taking minimum entransy dissipation rate as an optimization objective, Feng et al. [33] analyzed an optimization of the constructs of the leaf-like fins based on the constructal theory. Heat transfer in porous medium has gained considerable attention of many researchers in recent years. The basic philosophy behind using porous fins is to increase the effective surface area through which heat convected to ambient fluid. The use of porous fins is an excellent passive method for providing high heat transfer rates for electronic components in a small, light weight, low maintenance and energy free package. This is justified by the fact that porous media play a vital role in different thermal systems. A good insight into the subject is given by Pop and Ingham [34] and Nield and Bejan [35]. The problem of natural convection from a horizontal cylinder with multiple equally spaced high conductivity permeable fins on its outer surface has been investigated by Bassam and Abu-Hijleh [36]. They concluded that porous fins provide much higher heat transfer rate than traditional fins. Varol et al. [37] performed a theoretical study of buoyancy-driven flow and heat transfer in an inclined trapezoidal enclosure filled with a fluid-saturated porous medium heated and cooled from inclined walls. On the other hand natural convection heat transfer and fluid flow in porous triangular enclosures with vertical solid adiabatic thin fin have been numerically analyzed by Varol et al. [38]. Kiwan and Al-Nimr

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[39] numerically investigated the effect of using porous fins to enhance the heat transfer from a given surface. The thermal performance of porous fins is also estimated and compared with that of the conventional solid fins. A simple method has developed by Kiwan [40] to analyze the performance of porous fins in a natural convection environment. Naidu et al. [41] solved numerically the problem of natural convection heat transfer from a vertical cylinder fin to a saturated porous medium in a cylindrical enclosure. A two-dimensional conjugate free convection from a vertical plate fin with a rounded tip embedded in a porous medium is investigated by Vaszi et al. [42]. They have predicted that for a rounded tip with no restricted condition, such as an insulated tip, should be imposed in mathematical formulation. Makinde and Aziz [43] used a numerical approach to study the heat and mass transfer from a vertical plate embedded in a porous medium. Recently Xu et al. [44] analyzed the thermal transport in highly porous, opencelled metallic foams sandwiched between two infinite parallel plates using the Brinkman–Darcy and two-equation models. They highlighted that the comprehensive performance of a metallic foam channel is superior to empty channel in a particular porosity range and optimal pore density range. From the thorough literature survey summarized above, it is apparent that all the works dealing with various profiles were investigated on solid fin. To the best of the author’s knowledge, there has not been any effort on porous fin dealing with tapered profiles. However, tapered fins are always better with respect to the heat transfer rate per unit fin volume. In addition, most of the cases of porous fins, the governing equation was solved numerically. The aim of the present article is to investigate analytically the performance and optimization analysis of straight porous fin of three different profiles and results are compared among these geometries. The results are also tallied with the published value [12] so that a comparison with the solid fin is established. For this study, Adomian decomposition method is applied to solve nonlinear governing energy equations of different profiles for obtaining temperature distribution along the length of the fin. The present approximate analytical technique is a very useful and practical method for solving any class of nonlinear governing equations without adopting linearization or perturbation technique. It provides an analytical solution in the form of power series where the temperature on the fin surface can be expressed explicitly as a function of position along the length of the fin.

2. Mathematical model and assumptions Fig. 1 depicts a straight porous fin of various geometries having length L and thickness t(x). Fin is attached to a vertical isothermal wall of uniform temperature Tb and the fin surface is exposed to a convective environment at constant temperature Ta. As the fin is porous, it allows fluid to penetrate through it, which enhances the convective heat transfer. Although effective thermal conductivity of the porous fin decreases due to removal of solid material, the increase in effective surface area compensated this reduction in heat transfer. Certain practical assumptions given below are made to simplify the present study: 1. Porous medium is homogeneous, isotropic and saturated with a single phase fluid. 2. Physical properties of solid as well as fluid are invariable except density which may affect the buoyancy term where Boussinesq approximation is employed. 3. Fluid and porous mediums are locally thermodynamic equilibrium in the domain. 4. Darcy formulation is used to simulate the interaction between the porous medium and fluid.

tb

base

tb

L x (B)

x

L tb

base

(A)

tb

x L (C)

L x (D)

Fig. 1. Schematic of different straight porous fin profiles: (A) Rectangular; (B) Convex; (C) Exponential a < 0; and (D) Exponential a > 0.

5. Temperature variation inside the fin is one-dimensional. 6. There is no bond resistance at the fin base and there are no heat sources in the fin itself. 7. Heat transferred through the outermost edge of the fin is negligible compared to that passing through the sides. In this study, total convective heat transfer from the porous fin is calculated as the summation of convection due to motion of the fluid passing through the fin pores and that of from the solid surface. By applying an energy balance to the differential segment of the porous fin with considering only convection, mathematically it yields

_ p ðT  T a Þ þ hP Dxð1  fÞðT  T a Þ qðxÞ  qðx þ DxÞ ¼ mc

ð1Þ

_ is the mass flow rate of the fluid passing through the porwhere m _ ¼ qv DxW. The fluid velocity can be estimated from ous material, m Darcy model, v ¼ gKbðT  T a Þ=c [35]. With these expressions, onedimensional energy equation of a straight porous thin fin can be written by using Fourier’s law of heat conduction as

keff

  d dT q cp gKbðT  T a Þ2 tðxÞ   2hð1  fÞ ðT  T a Þ ¼ 0 dx dx c

ð2Þ

Present work studies the straight porous fin of three different geometries that are defined according to variation of the fin thickness along its extended length. All the fin geometries considered in this study start with a thickness tb at the base. Thickness t(x) for different profiles can be defined as follows

tðxÞ ¼ tb ðx=LÞn tðxÞ ¼ tb eað1x=LÞ ;

for rectangular and convex profiles

ð3aÞ

a – 0 for exponential profile

ð3bÞ

It may be noted that from Eq. (3a), rectangular and convex profiles can be obtained by putting index value n ¼ 0 and n ¼ 0:5, respectively. On the other hand, Eq. (3b) predicts the thickness of an exponential profile. For the exponential profile, present work deals with both positive and negative value of exponential power factor a. In order to express Eq. (2) in non-dimensional form, the following dimensionless parameters are defined as

  x tb T  T a ; ; ; L L Tb  Ta   BiR1 2ð1  fÞBi ðx1 ; x2 Þ ¼ ; ; /w2 /w2 keff ¼ 1  f þ f; kR ; /¼ ks   htb qcp gKbðT b  T a Þ kf ðBi; R1 ; kR Þ ¼ ; ; hc ks ks ðX; w; hÞ ¼

ð4Þ

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Thus, using Eqs. (3) and (4), the energy equation of the porous fin (Eq. (2)) for these three profiles are reduced in dimensionless form as

L1 x ¼

Z

Z

X

0

X

ðÞ dXdX

ð9Þ

0

2

dh  x1 h2  x2 h ¼ 0; for rectangular þ nX n1 2 dX dX ðn ¼ 0Þ and convex ðn ¼ 0:5Þ profile

Xn

d h

Now the unknown function hm, m  1 can be decomposed into a sum of components defined by the decomposition series as

ð5aÞ h¼

2

d h

dh  ea ðx1 h2 þ x2 hÞ ¼ 0; eaX 2  aeaX dX dX for exponential profile

1 X hm

ð10Þ

m¼0

ð5bÞ

Therefore, Eq. (8) can be expressed as

8 ! ! ! 1 1 1 X X X > > 1 n 1 n 1 1 > > h þ x1 L x X Am þ x 2 L x X hm  nLx X dhm =dX > 1 < 0 X m¼0 m¼0 m¼0 ! ! ! hm ¼ 1 1 1 > X X X > m¼0 1 1 1 > a aX a aX > h 0 þ e x1 L x e Am þ e x2 Lx e hm þ aLx dhm =dX > : m¼0

m¼0

m¼0

The coordinate X is selected along the length of the fin starting from the tip surface. The boundary conditions for determining temperature distribution are considered as follows:

h ¼ 1; at X ¼ 1

ð6aÞ

and

dh=dX ¼ 0; at X ¼ 0

ð11Þ

ð6bÞ

where Am is the so-called Adomian polynomial corresponding to the nonlinear terms h2. In order to determine the higher order terms Eq. (10) can be written with a recursive relationship as ( n n 1 x1 L1 Am1 Þ þ x2 L1 hm1 Þ  nL1 dhm1 =dXÞ x ðX x ðX x ðX ;m  1 hm ¼ 1 aX 1 aX 1 a a e x1 Lx ðe Am1 Þ þ e x2 Lx ðe hm1 Þ þ aLx ðdhm1 =dXÞ

ð12Þ The practical solution will be the m-terms approximation um to h which is usually written as

2.1. Fin performance The dimensionless temperature distribution of the straight porous fin surface can be obtained by solving the governing differential equation (5) with the boundary conditions (6). In this regard, it may be pointed out that the governing equations for these profiles are non-linear second order differential equations, which may be very difficult to solve analytically by using conventional techniques. In recent time, the Adomian decomposition method [45] is widely used for the analytical solution of many frontier problems involving such type of strong non-linearistic and stochasticity in parameters. The method proved to be powerful and converged very rapidly without need of linearization of the equation. In the present problem, Adomian decomposition method is adopted for the solution. The solution procedure of the given problem can be furnished briefly as follows [45,46]: Writing Eq. (5) in differential operator form as

Lx h ¼ x1 X n h2 þ x2 X n h  nX 1 dh=dX

ð7aÞ

Lx h ¼ ea x1 eaX h2 þ ea x2 eaX h þ adh=dX

ð7bÞ

where Lx is the linear second order differential operator 2 2 ðLx ¼ d h=dX Þ which is invertible. Applying inverse operator ðL1 x Þ on both sides of Eq.(7) yields

h ¼ hð0Þ þ X

  dhð0Þ dh n 2 n þ x1 L1 h Þ þ x2 L1 hÞ  nL1 X 1 x ðX x ðX x dX dX ð8aÞ

h ¼ hð0Þ þ X

  dhð0Þ 1 aX 1 dh aX 2 a þ ea x1 L1 x ðe h Þ þ e x2 Lx ðe hÞ þ aLx dX dX ð8bÞ

where h(0) is the dimensionless tip temperature of the fin which can be denoted as h0. The inverse operator is expressed by twofold indefinite integrals as

um ¼

m 1 X

hi ¼ h0 þ h1 þ h2 þ    þ hm1 ;

mP1

ð13Þ

i¼0

Thus, theoretically um approaches to h a s m ! 1. Practically, few numbers is enough to get the final result as the Adomian decomposition solution becomes rapidly converged. The first step for the Adomian decomposition method is to calculate the Adomian polynomials for nonlinear terms. For a nonlinear operator FðhðXÞÞ, the polynomials [45] are given below:

1 1 0 Fðh0 Þ A0 C B C B C B A1 C B h1 F 0 ðh0 Þ C B C B C B C B 0 00 2 C B A2 C B h2 F ðh0 Þ þ 2!1 h1 F ðh0 Þ C B C¼B C B C B B A3 C B h3 F 0 ðh0 Þ þ h1 h2 F 00 ðh0 Þ þ 1 h3 F 000 ðh0 Þ C C B C B 3! 1 A @ A @ .. .. . . 0

ð14aÞ

The Adomian polynomials for the nonlinear term h2 can be expressed in a matrix form as

1 h20 C B BA C B C 2h1 h0 B 1C B C B C B B A2 C B 2h2 h0 þ h2 C C B C¼B 1 C B C B B A3 C B 2h3 h0 þ 2h2 h1 C C @ A @ A .. .. . . 0A 1 0

0

ð14bÞ

Using Eqs. (10)–(14), the following temperature expressions for the adopted profiles are obtained after integration: (a) Rectangular profile(n = 0)

X2 X4 X6 þ x3 x4 þ x3 ðx24 þ 6x1 x3 Þ 2! 4! 6! X8 2 þ x3 x4 ðx4 þ 36x1 x3 Þ þ    8!

h ¼ h0 þ x3

ð15aÞ

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where

1.0

Present Work Numerical Work Published Work [12]

0.9

x3 ¼ h0 ðh0 x1 þ x2 Þ; x4 ¼ ð2h0 x1 þ x2 Þ

The temperature distribution for different profiles described in aforementioned section is a function of unknown temperature h0. This dimensionless tip temperature h0 can be determined for different profiles, separately by applying boundary condition (6a). For every profile, a transcendental algebraic equation as a function of h0 is obtained as

a=1.5 n=0

θ

0.8 ψ=0.05

0.7

R1=0.1

(a) Rectangular profile (n = 0)

ζ=0.8

0.6

KR=0.0001

a=-1.5

1 1 1  x3 x4  x3 ðx24 þ 6x1 x3 Þ 2! 4! 6! 1  x3 x4 ðx24 þ 36x1 x3 Þ     ¼ 0 8!

1  h0  x3

Bi=0.001

n=0.5

0.5 0.0

0.2

0.4

0.6

0.8

1.0

1 1 1  64x3 x4  256x3 ð7x24 þ 5x1 x3 Þ 3! 6! 9! 1  2048x3 x4 ð35x24 þ 109x1 x3 Þ   ¼ 0 ð16bÞ 12 !

1  h0  4x3

Fig. 2. Comparison of results for temperature distributions predicted by different methods of analysis of various profiles.

(b) Convex profile n ¼ 1=2

X 3=2 X3 X 9=2 þ 64x3 x4 þ 256x3 ð7x24 þ 5x1 x3 Þ 3! 6! 9! X6 2 þ 2048x3 x4 ð35x4 þ 109x1 x3 Þ þ  ð15bÞ 12 !

(c) Exponential profile ðða – 0ÞÞ

h ¼ h0 þ 4x3

1  h0  

(c) Exponential profile ða – 0Þ

h ¼ h0 þ

 þ  þ þ

4a4

x3

36a6

x3

4a6

þ ð4eaX  a2 X 2  4aX  4Þ

þ

ðe2aX  4aXeaX þ 4eaX  2aX  5Þ 2 4

ðx þ 4x1 x3 Þ ðe

3aX

2 4

ðx þ 2x1 x3 Þ ðaXe



 3aX  1Þ

2aX

e

2aX

þ

þ aX þ 1Þ



x3

ð2x24  4x1 x3 Þ ðe2aX  2aX  1 Þ 8a6

x3

aX

2 4

ðx  4x1 x3 ÞðaXe 2a6

x3

4a6

 2e

aX



þ aX þ 2 Þ

x3 2a2

x3 x4 4a4

x3

36a6

x3

þ ð4eaX  a2 X 2  4aX  4Þ

ðe2aX  4aXeaX þ 4eaX  2aX  5Þ

ðx24 þ 4x1 x3 Þ ðe3aX  3aX  1Þ

4a6

ðx24 þ 2x1 x3 Þ ðaXe2aX  e2aX þ aX þ 1Þ

8a6

ð2x24  4x1 x3 Þ ðe2aX  2aX  1 Þ

2a6

ðx24  4x1 x3 Þ ðaXeaX  2eaX þ aX þ 2 Þ

4a6

ð5x24 þ 4x1 x3 Þ ðeaX  aX  1Þ

x3 x3 x3

x1 x23 a6

ða2 X 2 eaX  4aXeaX þ 6eaX  2aX  6Þ þ    ¼ 0 ð16cÞ

ð5x24 þ 4x1 x3 Þ ðeaX  aX  1Þ

x1 x23 a6

ða2 X 2 eaX  4aXeaX þ 6eaX  2aX  6Þ þ    ð15cÞ

1.2

Rectangular Convex Exponential, a=-1.5 Exponential, a=-2.5

1.0

0.8

Eq. (16) can be solved by Newton–Raphson iterative method for determination of unknown tip temperature h0 . The final tip tem-

ζ=0.7 ζ=0.9

0.8

0.6

Rectangular Convex Exponential, a=-1.5 Exponential, a=-2.5

1.0

θ

þ

x3 x4



θ

þ

2a2

Bi=0.001 Bi=0.005

0.6 0.4

ψ=0.05

ψ=0.05

R1=0.1

R1=0.1

Bi=0.001 KR=0.0001

0.4 0.0

0.2

ð16aÞ

(b) Convex profile (n =1/2)

X

x3

ð15dÞ

0.4

0.6

0.8

0.2

1.0

0.0 0.0

Bi=0.001 KR=0.0001

0.2

0.4

0.6

X

X

(A)

(B)

0.8

1.0

Fig. 3. Temperature distribution predicted from various profiles: (A) as a function of f and (B) as a function of Bi.

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perature is obtained after satisfying the necessary convergence criteria. Temperature distribution expressed in Eq. (15) is a spatial function only for a known design condition from which actual heat transfer rate through each profile can be determined by applying Fourier’s law of heat conduction at the base.

Q¼

  q dh ¼ /w ks ðT b  T a Þ dX X¼1

ð17Þ

Ideal heat flow rate can be obtained by considering whole fin surface temperature as the base temperature and that can be expressed as

Qi ¼

qi Bi ¼ ½R1 þ 2 ð1  fÞ  ks ðT b  T a Þ w

search activities have already been devoted to analyze optimization of fins. The result of this optimization analysis provides general guidelines relative to the dimensionless features of a well-designed fin. The optimization of a known geometric fin can be made either by maximization of heat transfer rate for a given fin volume or by minimization of fin volume for a desired heat transfer rate. However, outcome from both the approaches give the same value and thus depending on the design specification, analysis can easily be made. In the present paper, a generalized optimization scheme is adopted on the basis of the aforementioned approaches. The volume of the fin per unit width can be written in a dimensionless form as

ð18Þ 2

U¼

Thus, the fin efficiency can be calculated as

g ¼ Q =Q i

ð19Þ

2.2. Optimization analysis It is obvious that when a fin is attached in a primary surface, the weight and cost of the equipment increases. Thus extensive re-

2

ks

8 2 rectangular profile > < Bi =w ¼ 2Bi2 =3w convex profile > : a 2 ð e  1ÞBi =aw exponential profile

ð20Þ

From the heat transfer and volume expressions, it is clear that the thermophysical parameters Bi and w are dependent upon the optimization analysis. To determine the optimum parameters, the optimum criterion is necessary which can be derived from Euler equation after eliminating the Lagrangian multiplier as

1.0

1.0 Present Result Numerical Result

0.9

ψ=0.05 ζ=0.8

0.8

R1=0.1

η

Solid Fin

0.7

Present Result Numerical Result

0.9

ψ=0.05 ζ=0.8

0.8

η

h V

R1=0.1 Solid Fin

0.7

-4

KR=1x10

0.6

0.6 Porous Fin

0.5

Porous Fin

0.5

-4

KR=1x10

0.4 0.0000

0.0025

0.4 0.0000

0.0050

0.0025

Bi

Bi

(A)

(B)

0.0050

1.0 Present Result Numerical Result

0.9

ψ=0.05 ζ=0.8

a=2.5

0.8

R1=0.1 0.7

η

a=-2.5

0.6 Solid Fin

0.5 Porous Fin

0.4

-4

KR=1x10

0.3 0.0000

0.0025

0.0050

Bi

(C) Fig. 4. Fin efficiency of solid and porous fins of different profiles as a function of Biot number predicted by present and numerical analyses: (A) Rectangular; (B) Convex; and (C) Exponential.

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     @Q @U @Q @U  ¼0 @Bi @w @w @Bi

0.015

ð21Þ

Optimum Point

Eq. (21) can be solved along with the constraint equation on the basis of design requirement. Either the heat transfer duty or the fin volume can be considered as a constraint. The constraint can be formed as follows:

/wðdh=dXÞX¼1  Q

0.012 Convex

Q

( FðBi; wÞ ¼ 0 ¼

Exponential, a=-2.5

ð22Þ

2

Bi =w  U

Rectangular

Eq. (21) is simplified by using Eqs. (17) and (20) and can be expressed by a function as

GðBi; wÞ ¼ Bi

Exponential, a=2.5

0.006

@ @ ðdh=dXÞX¼1 þ 2w ðdh=dXÞX¼1 þ 2ðdh=dXÞX¼1 ¼ 0 @Bi @w ð23Þ

In order to determine the optimum fin dimensions, Eqs. (22) and (23) can be solved simultaneously by using generalized Newton– Raphson method. The approximate root values for Newton–Raphson method for any iteration can be obtained by using just previously iterative values. For the numerical calculation, it is required to sat-

0.009

0.003 0.00

0.05

0.10

ψ Fig. 6. Heat transfer rate as a function of w obtained from various profiles for a design condition U ¼ 1  106 , R1 ¼ 1:0, K R ¼ 1  104 , and f ¼ 0:3.

0.85

0.9 Exponential, a=2.5

0.80

0.8

Exponential, a=2.5

Rectangular

Rectangular Concave

0.7

η

η

0.75

0.70

0.65

0.6

ψ=0.05 ζ=0.8

Concave

R1=0.1

Exponential, a=-2.5

Exponential, a=-2.5 ψ=0.05 -4

Bi=0.001

KR=1x10

0.5

R1=0.1 Bi=0.001

0.60 0.000

0.025

0.4 0.0

0.050

0.5

ζ

KR

(A)

(B) 0.9 Exponential, a=2.5

0.8 Rectangular

0.7

η

Concave

0.6

ψ=0.05 -4

KR=1x10

0.5

ζ=0.8

Bi=0.001 Exponential, a=-2.5

0.4 0.0

0.5

1.0

R1

(C) Fig. 5. Fin performances determined from various profiles: (A) g vs. kR; (B) g vs. f; and (C) g vs. R1.

1.0

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B. Kundu et al. / International Journal of Heat and Mass Transfer 55 (2012) 7611–7622

isfy the necessary and sufficient convergence criteria. To start the solution process, the initial guess values for the roots have been taken cautiously so that the convergence criteria for each iteration have been satisfied. 3. Results and discussion The main objective of the present article is to establish an analytical expression for determination of thermal performance and optimization of straight porous fin with four different profiles. The Adomian decomposition method was used as an analytical tool. Before furnishing outcomes obtained from the present work, it may be required to validate with the previous work as mentioned in the literature. However, it cannot be done directly due to unavailability of the results data in the literature for the variable thickness fin. Therefore, a numerical technique, namely, finite difference method is employed on the present set of equations to ob-

tain the numerical result. The difference equations for individual geometries are formulated from their corresponding energy equation by Taylor series central difference scheme considering the linearization of source terms. These different equations are solved simultaneously by the Gauss–Seidel iterative method. The final result yields after satisfying the convergence criteria for its solution. For a comparative study, the result for temperature distribution in the porous fin of different profiles obtained from the proposed analytical work and the numerical method is depicted in Fig. 2. The results indicate that the numerical data and analytical method are in agreement with each other. A comparative study between the porous fin (present work) and the solid fin [12] has also been prepared by plotting the published result in the same figure. The numerical result from published work is also shown. The numerical values of the published work are exactly matched with the analytical values. Both present and published work has been done for a straight fin of rectangular, convex and exponential profiles. However, it can be

0.018

0.018 Porous Fin Solid Fin

ζ=0.1

0.015

0.015 ζ=0.1

0.012

Q

Q

ζ=0.3

0.012

ζ=0.3

Porous Fin Solid Fin

0.009

0.006 0.00

0.009

Locus of Maximum Heat Transfer Rate

0.05

Locus of Maximum Heat Transfer Rate

0.10

0.006 0.00

0.05

ψ

ψ

(A)

(B)

0.10

0.015 Porous Fin Solid Fin

Q

ζ=0.1

0.010

ζ=0.3

Locus of Maximum Heat Transfer Rate

0.005 0.00

0.05

0.10

ψ (C) Fig. 7. Maximum heat transfer with the variation of porosity among various profiles in a design condition U ¼ 1  106 , R1 ¼ 1:0, and K R ¼ 1  104 : (A) Exponential ða ¼ 2:5Þ; (B) Convex n ¼ 0:5 and (C) Rectangular n ¼ 0:0.

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mentioned once-again that the published model had been analyzed for solid fins whereas the present model is done with porous fins. The results show that irrespective of any profile chosen, dimensionless tip temperature of the porous fin is lower than that of the solid fin. The results also show that the porous fin with exponential profile with a negative exponential power factor ‘‘a’’ provides a minimum tip temperature among these profiles considered. Fig. 3A is presented to show the influence of porosity f on the temperature distribution for these profiles. The fin surface temperature decreases with the increase of porosity parameter for all the profiles. Actually, a high porosity value reduces the effective thermal conductivity of the fin due to the removal of the solid material. At the same time, convective heat transfer increases due to more fluid passing through pores. Combining these two effects result in the decrease in lower tip temperature. This figure predicts that the variation of dimensionless temperature is considerably high for the exponential profile with the value of exponential parameter, a ¼ 2:5. Fig. 3B displays the variation of the fin surface tem-

perature with the conductive-convective parameter Bi. Due to increase in thermal resistance by increasing Bi, the fin surface temperature declines for all the profiles. An ideal temperature variation can be obtained by taking Bi value approaching zero. The variations of tip temperatures among these profiles are similar as that of the variation with porosity. However, the influence of temperature on Bi is significantly high. The variation of fin efficiency with the variation of Biot number Bi is displayed in Fig. 4A, B, and C for rectangular, convex and exponential profiles, respectively. For all the fin geometries the fin performance parameters decreased with the increase of Bi. This variation can be understood from Fig. 3B. With an increase in Bi, the variation of temperature in the whole fin surface increases which declines the fin efficiency as well. For the comparative study, fin efficiency of solid fins [12] is also furnished in this figure. It has been stated earlier that the published article worked on solid fin of same geometries. Thus, from the figure it can be pointed out that the solid fin provides better performance in respect to that of

0.018

0.018 R1=2.0

Porous Fin Solid Fin R1=2.0

0.012

Q

Q

R1=1.0

0.012 R1=1.0

Porous Fin Solid Fin Locus of Maximum Heat Transfer Rate

0.006 0.00

Locus of Maximum Heat Transfer Rate

0.05

0.006 0.00

0.10

0.05

ψ

ψ

(A)

(B)

0.10

0.015 Porous Fin Solid Fin

Q

R1=2.0

0.010

R1=1.0

Locus of Maximum Heat Transfer Rate

0.005 0.00

0.05

0.10

ψ (C) Fig. 8. Maximum heat transfer with the variation of flow parameter R1 among various profiles in a design condition U ¼ 1  106 , f ¼ 0:3, and K R ¼ 1  104 : (A) Exponential ða ¼ 2:5Þ; (B) Convex n ¼ 0:5. and (C) Rectangular n ¼ 0:0.

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porous fins. The figure also shows the porous fin of exponential profile with positive power factor provides better fin performance with respect to the other profiles. This betterment enhances with Bi. On the other hand, a poor fin performance is observed in an exponential fin profile with a negative power factor. For the validation, numerical results are also plotted for solid and porous fins and an exact matching with the analytical values is found. To augment heat transfer rate, the value of design parameters, namely, thermal conductivity ratio, porosity, and flow parameter R1 are highly dependent upon the application in which fin is employed. For different applications, these values may be different. Thus, the knowledge is required about the fin performance with these design variables. Fig. 5 is drawn for the above purpose. The fin efficiency as a function of kR is illustrated in Fig. 5A. With the increase in kR, the fin efficiency increases linearly irrespective of any type of fin geometry. The thermal conductivity kR is the ratio of thermal conductivity of the fluid to that of solid. In general, an increase in kR enhances the effective thermal conductivity of the fin material. Due to increase in effective thermal conductivity, the conductive resistance decreases and therefore the fin efficiency is an increasing function with kR. Fig. 5A also shows that the fin efficiency of the exponential profile with a ¼ 2:5 is more in comparison with other profiles. Next an assignment has been made to determine the fin efficiency as a function of porosity f and fluid flow parameter R1 depicted in Fig. 5B and C, respectively. The respective graph reveals that the fin efficiency is decreasing function of both the porosity and flow parameters. The fin efficiency decreases slowly with f up to f ¼ 0:85 for all profiles and then it drops very sharply. Near f ¼ 1, the fin efficiency does not depend upon the fin shape. However, efficiency drops significantly with variation of R1 which is almost at a constant rate. This fact can be interpreted as follows. An increase in porosity causes increase in conductive resistance due to decrease in effective thermal conductivity of the porous fin, which in turn, reduces the fin efficiency. The thermal resistance is adequately large with porosity after f ¼ 0:85 and consequently fin efficiency diminishes remarkably. With an increase in R1, drop in fin temperature enhances and as a result, fin efficiency declines. Typical result obtained from the optimization study among the aforesaid geometries is shown in Fig. 6 for identical design param-

0.04

eters. Here the fin volume is taken as a constraint. For each profile, the rate of heat transfer initially increases, then reaches to a maximum value and finally starts declining. The peak of the curves indicates an optimum design condition for a specified fin volume. The results show that the straight porous fin with an exponential profile with a ¼ 2:5; have slightly higher heat transfer rate in comparison with convex profile, and an exponential profile with a ¼ 2:5 gives poor heat transfer rate. However, in most industrial applications the rectangular profile is used due to its easy in construction. The results also suggest a high optimum w value for exponential profile with negative power factor in comparison to that of other profiles. The variations in dimensionless heat transfer rates with the variation of f and w plotted for the three profiles is depicted in Fig. 7. A comparative study between the porous and solid fins has also been made. Figure shows that irrespective of any profile chosen porous fins with lower porosity transfer more heat than solid fin. Nevertheless, the solid fin is more effective than the porous fin possessing high porosity value. Thus, it can be highlighted that for a given fin volume, the optimum design condition of a fin is strongly influenced by the porosity parameter. The loci of the maximum heat transfer are also shown in this plot. Figure also shows that the dimensionless heat transfer rate is preeminent for the exponential profile with a negative power factor amongst these profiles. Furthermore, the optimum W value is increased with the increase of porosity parameter. However, this increment is marginal. The influence of the physical flow parameter R1 on the optimum condition for different profiles is illustrated in Fig. 8. The heat transfer rate is enhanced with the parameter R1 whatever the profile has been adopted. The optimum w improves if R1 is increased. For every profile, the locus of optimum design for maximization of heat transfer rate with w and with the variation of R1 may be a straight line. As the parameter R1 depends on permeability of the porous fin, it increases the ability of the working fluid to penetrate more through the fin pores and enhances the heat transfer rate. For the comparison of heat transfer rate among these profiles, this figure shows that exponential profile with a negative power factor transfers more heat with respect to other profiles and suggest high optimum w. In comparison to the solid fin, porous fins transfer more heat rate at a higher R1.

Exponential, a=-2.5 Convex Rectangular Exponential, a=2.5

0.9 Locus of optimum point

0.03 0.6

0.02

η

Q

Locus of maximum -5 U=1x10 heat transfer rate

-5

0.3 0.01

0.05

ψ

(A)

-6

Exponential, a=-2.5 Convex Rectangular Exponential, a=2.5

-6

U=1x10

0.00 0.00

U=1x10 to U=1x10

0.10

0.0 0.00

0.05

0.10

ψ (B)

Fig. 9. Optimum fin design parameters as a function of U and w among various profiles in a design condition R1 ¼ 2, f ¼ 0:3, and K R ¼ 1  104 : (A) Heat transfer rate and (B) Efficiency.

B. Kundu et al. / International Journal of Heat and Mass Transfer 55 (2012) 7611–7622

Another typical result obtained from the optimization study is prepared for the effect of fin volume on the optimum design condition which is envisaged in Fig. 9. The design curves are generated with the variation of W for different fin volumes. Fig. 9A depicts that the heat transfer rate through the porous fin of these profiles as a function of w for different constants U. For each of the variation of the constant fin volume, the rate of heat transfer initially increases, reaches to a maximum value and then decreases with further increase in w. The peak of the curves gives an optimum design condition for a specified U and a design parameter. For particular value of other parameters, the maximum rate of heat transfer increases traditionally when the fin volume is increased. The loci of the optimum w designate the optimum heat transfer rate Q for different fin volumes. The corresponding change in the fin efficiency is also displayed in Fig. 9B, as a function of w and U. Although the fin efficiency decreases with the increase in fin volume, it does not change significantly with the fin volume at an optimum condition as displayed in Fig. 9B. From the figure, it is clear that at the optimum point, the exponential profile is highest or lowest efficient than other profiles irrespective of any fin volume. Thus the analysis of exponential porous fins is of great importance for enhancing heat transfer as well as fin efficiency. 4. Concluding remarks Owing to the gradual decrease in heat conduction rate from the fin-base to fin-tip, effective utilization of fin material may not be possible for the fins of all geometries. For this reason, many researchers had already investigated fins of different profiles. However, among of these a very few one is compatible in manufacturing process. Lots of investigation is still engaged to find out the optimum fin geometry on the basis of enhancement of heat transfer rate as well as ease of fabrication. The present study works on the performance and optimum design analysis of straight porous fins of four different geometries, namely, rectangular, convex, and two exponential types. The following concluding remarks can be drawn from the present study: 1. Temperature distribution in the porous fin is highly dependent upon the fin profiles. Exponential profile gives a lower or higher fin surface temperature. For each profile, the present work for porous fins predicts lower tip temperature than published work considered for solid fins. 2. Increase in thermophysical parameters f and Bi decrease the fin surface temperature for all the profiles considered in the present study. 3. With the increase in thermal conductivity ratio kR, fin efficiency increases linearly for all the profiles. On the other hand, fin efficiency shows a reverse trend with the increase of f and Bi. Moreover, fin performance is better for exponential profile with a positive power factor than other three profiles. 4. The maximum heat transfer rate for exponential profile is slightly higher than convex profile but significantly higher than rectangular profile though the negative value of exponential parameter is deciding term. 5. For all the fin profiles, rate of heat transfer decrease with the increase of porosity, whereas the effect of fin volume and parameter R1 show a reverse trend. The locus of the maximum heat transfer with variation of U and R1 is almost linear in nature noticed for porous fins and is slightly nonlinear with the porosity. The efficiency at the optimum point does not much depend upon the fin volume whereas an exponential fin gives maximum fin efficiency. 6. An increase in fin volume increases both the maximum heat transfer rate and the optimum aspect ratio w .

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