Optimization of rectangular and triangular fins with convective boundary condition

INT. OCMM. HEAT MASS TRANSFER 0735-1933/85 $3.00 + .00 Vol. 12, pp. 479-482, 1985 ©Pergamon Press Ltd. Printed in the United States

OPTIMIZATION OF RECTANGULAR AND TRIANGULAR FINS WITH CONVECTIVE BOU[~)ARY CONDITION

A. Aziz Department of Mechancial Engineering Gonzaga University Spokane, WA 99258 (C~mkm_icated by J.P. Hartnett and W.J. Minkowycz)

Introduction

The analysis for the optimum dimensions of longitudinal convecting fins that is available in the literature e.g. [i] is based on the assumption of known base temperature.

However, in an actual situation, one

knows the temperature of the fluid Tf and the heat transfer coefficient hf on the other side of the primary surface as shown in Fig. i.

Thus, a more

realistic analysis must employ a convective boundary condition and include the conduction resistance of the wall.

While Suryanarayana [2] has

adopted this model to calculate the heat transfer rate from an array of straight rectangular fins, the corresponding optimization study has not been made.

In this letter, we consider the optimization of single

rectangular and triangular fins with convective boundary condition. Ignoring the wall resistance (i.e. 6 = O in Fig. i), the analysis derives the heat transfer rates for both rectangular and triangular fins.

For

each geometry, the optimization procedure leads to a relationship between the fin parameter N and Biot n ~ r optimum condition.

Bi that must be satisfied for the

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A. Aziz

Vol. 12, No. 4

Analysis

For a rectangular fin (Fig. la), the governing equations are

d2.0.- N2(0-1) = 0 dX2 dO dO X = 0, ~-~ = 0; X = 1, ~ = B i ( 0 f - 0 ) where 0 = T/Ta, Of = Tf/Ta, X = x/L, N 2 = 2 h L 2 / k w ,

(1) (2) Bi = hfL/k.

Here,

T = temperature, Ta = t~mperature of the environment surrounding the fin, Tf = temperature of the fluid at the fin base, L = fin length, w = fin thickness, k = fin thermal conductivity, h a = convective heat transfer coefficient on the fin surface, hf = heat transfer coefficient at the fin base, and x = axial distance measured from the fin tip. The solution of (i) subject to (2) is Bi(0f-l) coshNX 8 = I +

NsinhN + BlcoshN

(3)

From (3), the heat transfer rate q can be derived as kwT q =

a

[

L ....

Bi(Of-l) NslnhN NslnhN + BicoshN ]

(4)

The usual optmization procedure is to fix the profile area, wL, express q as a function of w alone, and impose the condition dq = O for dw maximum q. Such a procedure when applied to (4) leads to the following relationship between N and Bi 4 Nslnh2N Bi = 6 N - sinh2N ~:luation (5) is represented graphically in Fig. 2.

(5) For Bi ÷ ~ , equation

(5) reduces to the case of known base temperature given in [i] with N = 1.4192.

Unlike the case of known base temperature, a trial and error

solution is needed for the convective boundary case.

For a desired heat

transfer rate, the design procedure would be as follows. value of L which, together with the other data, fixes Bi. equation

Select a trial Use Fig. 2 or

(5) to obtain the value of N which now fixes the thickness w.

Next, calculate q from (4) and compare with the desired q. match is obtained.

Repeat until a

Vol. 12, No. 4

OPTIMIZATION OF ~

AND ~

FINS

For a triangular fin (Fig. ib), the governing equations are

X d20 + ~d8 - ~ - N2(0-1) ffi 0

(6)

dX2 X---0, ~dO =

O; X = I, -dO = Bi(Of-0) dX

(7)

The solutions for 8 and q

where the symbols are as defined previously. are

(8)

8 = 1 + Bi(8f-l) I0 (2N/X) NII(2N) + Bi I0(2N)

q . k ~ a [ Bi(ef-1) N ~ (2N) L

(9)

NII(2N ) + Bi I0(2N)

In this case, the profile area ½ wL is kept fixed while q is maximized. For the optimum condition, the relationship between N and Bi takes the following form I Bi = 3102 (2N) -

12(2N)

(I0)

210(2N) I 1 (2N) N - 3112 (2N)

In equations (8-10), I0 and I1 stand for the modified Bessel functions of the first kind of zero and first order respectively. sentation of equation (i0) is given in Fig. 2.

A graphical repre-

For B i + =

, equation (I0)

reduces to the case of known base temperature given in [i] with N = 1.3094. As in the case of rectangular fin, a trial and error approach is necessary to fix the optimum dimensions.

~l~i~l~mar~

The foregoing analysis has ignored the thermal resistance of the primary surface ~ i i .

It has been shown in [2] that such an approach

leads to satisfactory results (compared to two-dimensional finitedifference computations)

for q i f ~ -- 0.25.

applies to the present analysis.

Thus, this restruction also

For situations where this restriction is

violated, the analysis should be modified to include the thermal resistance of the ~ I I .

The approach then should be to minimize the sum of the

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A. Aziz

Vol. 12, No. 4

convective resistance at the base, the conduction resistance of the wall, and the resistance of the fin while keeping the profile area of the fin fixed. References i. D. Q. Kern and A. D. Kraus, Extended Surface Heat Transfer, McGraw Hill, pp. 126-131 (1972). 2.

N.V. Suryanarayana, J. Heat Transfer, 9_~9,129 (1977).

eTa,h a

--4

i~

Kw

"Tf,hf

E

I

"Ta'ha

" Tf,hf

J7 T

L X

X

Fig. 1 (a)

Fig. 1 (b) Triangular fin

Bectangular fin 15 1.5

I0 a

- -

'

'

l

01.S

5

I

,

,

,

j

,

1.419'2

t1.0

Tr/ '

1.0

.5

0n

.

,

J

5

!

~

'

Bi-1

!

!

!

I0

Fig. 3 Relationship between N and Bi for optimum ~ s i c m s

,

~ 15