Finite element analysis of triangular fins attached to a thick wall

Pergamon

~nn,,,urrrs & Srrucrures Vol. 51. No. 6. pp. 945-957. 1995 Copyright 8: 1995 Elwier Science Ltd Printed in Great Britain. All rights reserved co45-7949195 S9.50 + 0.00

00457949(95)00105-0

FINITE

ELEMENT ANALYSIS OF TRIANGULAR ATTACHED TO A THICK WALL

FINS

S. Abratet and P. Newnham Mechanical

and Aerospace

Engineering,

Engineering Mechanics Department, Rolla, MO 65401, U.S.A.

University

of Missouri-Rolla,

(Receitled 29 June 1994) Abstract-Heat conduction in an array of triangular fins with an attached wall is modeled using the finite element method. An adaptive mesh refinement technique is developed giving accuracy comparable to uniform mesh refinement and much increased computational efficiency. The effects of wall thickness and fin spacing are examined for various Biot numbers. It is shown that for low Biot numbers (Bi < 0. I), the one-dimensional assumption is valid but for higher Biot numbers (Bi ~0. l), two-dimensional heat conduction must be considered, temperature distributions at the fin root are always non-uniform and the fin is found not to be effective.

of the lower number of nodes required, increased computational efficiency.

INTRODUCTION

of heat transfer in fins traditionally assumes one-dimensional heat conduction [ 11. Recently, twodimensional analyses have been performed using both analytical and numerical methods. Another common simplification in the analysis of fins is assuming a constant temperature at the base of the fin. Previous numbers [l-6] have assumed temperature distributions at the fin root to attempt to give a more accurate representation of heat conduction in the fin. The finite element method allows complicated geometries to be modeled allowing the effect of the wall at the fin to be taken into account. Temperature distributions at the fin root are found for various fin and wall configurations with varying convection boundary conditions on the fin. The effect of mesh refinement on solution accuracy is considered by Aziz and Nguyen [2] for twodimensional heat conduction in a triangular fin. In this case, the grid is refined uniformly giving a large increase in the number of nodes required for a small increase in solution accuracy. For two-dimensional heat conduction in a triangular fin, there are regions where the temperature gradient is relatively steep requiring a fine mesh and regions where the temperature gradient is relatively small so that larger elements can be used. An adaptive mesh refinement technique is developed here, allowing the grid to be refined in regions of high temperature gradient and leaving the grid relatively coarse elsewhere. This approach gives accuracy comparable to the uniform mesh refinement but, because

it gives much

Analyses

FORMULATION

Steady-state heat conduction in isotropic governed by the energy equation: cY=T a=T -gp+y=o, ay

solids is

(1)

where T is the temperature. Following the classical procedure [7], a finite element model was developed using quadrilateral elements with four to eight nodes. The resulting set of algebraic equations:

was solved numerically using the Gaussian elimination procedure. TWO-DIMENSIONAL

HEAT

CONDUCTION

The analysis of heat transfer in fins of various geometries has traditionally assumed one-dimensional heat conduction in the fin. The Biot number is a commonly used non-dimensional ratio relating the heat transfer due to convection and the heat transfer due to conduction: Bi

,hs k

where h is the heat-transfer coefficient, k is the thermal conductivity and s is a characteristic length. Lau and Tan [3] have shown that, for large Biot numbers (Bi ~0. l), the one-dimensional assumption

t Department of Technology, Southern Illinois University at Carbondale, Carbondale, IL 62901, U.S.A. 94s

946

S. Abrate and P. Newnham

is unsatisfactory and two-dimensional heat conduction must be considered for accurate results. Various authors [l-6] have analyzed two-dimensional heat conduction using analytical and numerical techniques for simple fin geometries such as rectangles and triangles. Published results are used then to validate the finite element model developed for this study. The most common triangular fin analyzed has a length, L,, a base thickness 2w, with a constant temperature r, at the fin root and convection on the sloping surfaces with ambient temperature, Q, and heat transfer coefficient, h (Fig. 1). The characteristic length in eqn (3) is taken to be half of the base width, w. Only the top half of the fin is modeled due to symmetry about the x axis and a 625 node uniform grid is used for the analysis (Fig. 2). Temperature distributions are found for fin length to width ratio alpha, CL= 2L,/w = 3 and Biot numbers, Bi = 0.01, 0.1 and 1. The temperatures along the centerline of the fin, for these Biot numbers, match exactly those found by Look [6], using a separation of variables technique, (Fig. 3) and indicates the validity of the finite element code.

Fig. 2. Uniform

Fig. I. Geometry of a symmetric triangular fin with constant temperature at the root and convection on the fin surface.

The temperature distribution throughout the fin with the Biot number of 0.1 is approximately onedimensional as the temperature in the fin varies little in the y direction (Fig. 4a). The two-dimensional effect becomes more apparent at higher Biot numbers (Fig. 4b and c). With a Biot number of 10, the

625 node finite element

mesh for the triangular

--------@---

fin

l --_. Bi =

Fig. 3. Temperature

distribution

along the fin centerline for a triangular numbers of 0.01, 0.1 and 1.0.

0.01

fin with a = 2L,/w

= 3 and Biot

941

Finite element analysis of triangular fins attached to a thick wall 64

1 XL

0

Fig. 4. Temperature

0.5

distribution

1

1.5

2

1.5

2

in a triangular fin with a = 2L,/w (b) 10 and (c) 100.

temperature distribution is clearly two-dimensional with the gradient greatest at the root, particularly towards the top, and less steep towards the tip and varying in both the x and y directions. The effect is even more pronounced for a Biot number of 100 with a small area at the top of the root with a large temperature gradient and the majority of the fin with a very small gradient especially towards the tip. GRID REFINEMENT USING AN ADAPTIVE TECHNIQUE

MESH

The accuracy of finite element solutions is often confirmed by performing a grid refinement study where the mesh is uniformly refined until the solution

2.5

3

a

2.5

= 3 and Biot numbers

3

of (a) 0.1,

converges [2]. Large numbers of nodes are often required to obtained the accuracy desired because of localized areas containing relatively large temperature gradients. Regions with large temperature gradients require a fine mesh for the solution to converge, while in areas with smaller gradients, larger elements could be used. Adaptive mesh techniques refine the mesh only where needed allowing accuracy of solution whilst minimizing the number of elements required hence increasing computational efficiency [8-l 11. The adaptive mesh technique developed here is of the a-posteriori type, where the fin is initially modeled using a coarse grid of quadrilateral elements. The technique can be divided into two stages. The first stage requires deciding which of the elements in the

948

S. Abrate

and P. Newnham

coarse grid needs to be subdivided. This is accomplished by performing the finite element analysis on the coarse grid and calculating the energy stored in each element. The energy in each element is compared to the overall mean energy of the whole mesh. If the energy in an individual element is more than one standard deviation greater than the mean energy, indicating relatively large temperature gradients in the element, the element is subdivided into four elements thus refining the mesh. The second stage of the technique consists of subdividing each of the indicated elements and renumbering the nodes, minimizing the bandwidth of the mesh to reduce the size of the global matrices and allowing the finite element analysis to be performed efficiently. To illustrate the procedure, a triangular fin with the same configuration and boundary conditions used previously, with a Biot number of IO, is considered. The fin is modeled initially with a coarse mesh of 10 elements and 17 nodes (Fig. Sa). The mesh is subse-

quently refined adaptively four times (Fig. She) until the difference in temperatures, at selected reference nodes, between subsequent adaptive runs is less than the specified value, in this case a value of 0.002 was used. The refinement of the mesh in the area around the top of the fin root, where the temperature gradient is greatest, and the lack of refinement towards the tip where the temperature gradient is relatively small can be clearly seen (Fig. Se). Convergence of the adaptive mesh refinement technique is compared to that of uniform mesh refinement of the triangular fin. Starting with the coarse mesh (Fig. Sa), the grid was uniformly refined in four stages until the grid contained 576 elements and 625 nodes. The non-dimensional temperature (T/T,). at the node located at (0.75,O.S) in the fin was used for the study. For that point all temperatures found are compared to the temperature obtained using the most refined uniform grid. The relative error is calculated using the following relation:

(a) Relative error =

abs( T,, - T)

’

Ti,

(b)

(4)

where T,,is the temperature at the node found using the 625 node uniformly refined mesh. The adaptive technique uses less elements for a specified convergence criterion (Fig. 6). With the adaptive technique, the mesh requires 51 nodes to obtain a relative error of 0.0074 as compared to the uniform refinement which requires 169 nodes for the same relative error. Because fewer elements are required, the adaptive technique is significantly more efficient than the uniform grid refinement. The adaptive mesh technique allows complicated problems, having areas with large temperature gradients that are not known ahead of time, to be analyzed using less computational resources.

ANALYSIS OF TRIANGULAR FINS ATTACHED TO A THICK WALL

(4

(e)

In the literature, fins are usually modeled assuming a constant temperature distribution at the fin root [l-6]. This type of analysis does not account for effects, such as heat conduction and convection on the wall connected to the fin, which cause nonconstant temperature distributions along the edge of the fin root affecting the heat flow due to the fin. To more accurately analyze heat conduction in fins, temperature distributions along the fin root that are powers of trigonometric functions are frequently used in the literature [4, S]. An example of an assumed temperature distribution along the fin root given by Look [6] is: ,

Fig. 5. Adaptively refined grids for a triangular fin with Bi = 10 and a = 2L,/w = 3. (a) Initial grid and, (b) first, (c) second, (d) third and (e) fourth, refinements.

where a is a positive or negative constant and m is varied between zero and five. These temperature

Finite element

analysis

of triangular

fins attached

949

to a thick wall

17 -+

uniform mesh refinement

t

adaptive mesh refinement

0.1 tj 2 QJ ._z ;; 2 0.01

0.001

I

1cI

100

x)

number of nodes Fig. 6. Relative

error

for the triangular

(a)

I~......~ Y

(b)

.

fin with uniform

La

and adaptive

mesh refinement.

~

1

W

.. .

c

Toa

h

Fig. 7. Geometry of a multifin array of triangular the outer wall and fin surface. (a) Multifin array

fins with constant base temperature and convection on attached to wall and (b) domain of problem solution.

950

S. Abrate and P. Newnham

distributions are maximum or minimum at the centre of the fin root depending on the sign of the constant a. With m equal to zero, the temperature T, along the fin root is constant and the curve becomes steeper as m is increased. The idea of including a wall attached to the fin root to more accurately model heat conduction in long rectangular fins has been examined by Juca and Prata [12]. This analysis is extended here to cover triangular fins with varying geometries and Biot numbers. The fin and wall are considered to be part of a multiple array with the boundary at the inner wall held at a constant temperature, Tb (Fig. 7a). Convection occurs over the outer wall and fin boundaries with heat convection coefficient, h, and surrounding fluid temperature, 7’,. The fin and wall are modeled as a section of the array due to symmetry, Fig. 7b, with the fin length, wall thickness and spacing between fins related by the non-dimensional parameters G,, G, and G3, respectively.

where w is half of the base width of the fin root, L, is the length of the fin, L, is the wall thickness and L, is half the distance between adjacent fins. Two-dimensional heat conduction effects can be

clearly seen in the temperature distribution of a typical fin with a Biot number of 10 (Fig. 8a). The heat flow is approximately one-dimensional in the area of the wall away from the fin root. The temperature distribution becomes curved in the area of the wall close to the base of the fin. The temperature is greatest at the center of the fin root and lowest at the edges with large variations in temperatures occurring in both x and y directions indicating the two-dimensional character of the heat flow along the fin. With a Biot number of 0.1 the two-dimensional nature of the heat flow in the area around the fin/wall interface can be seen, but is much less pronounced than with a Biot number of IO (Fig. 8b). The temperature contours in the fin away from the root appear to be close to one-dimensional. The temperature increase at the fin root when Bi = 10 indicates that the fin is acting as an insulator, and is not conducting heat away from the wall. Thus, the fin is performing no useful function. The temperature decrease at the fin root of the fin with Bi = 0.1 indicates that heat is being conducted away from the wall and, in contrast to the fin with the higher Biot number, the presence of the fin is effective in conducting heat away from the wall. To examine the effect of varying the Biot number on the temperature distribution along the fin root, and heat flow through the fin and wall, a typical fin

2

(4

1.75 1.5 1.25

B

R

1 0.75 0.5 0.25 0 -1

-0.5

0

0.5

1.5

2

2.5

3

1.5

2

2.5

3

a

-1

Fig. 8. Temperature

-0.5

distributions

0

0.5

‘a

in the triangular fin with the wall attached, Biot numbers of (a) 10 and (b) 0.1.

G, = 3, G, = 1, G, =

1 and

951

Finite element analysis of triangular fins attached to a thick wall

..

0.1

( l-

0

0.1

0.2

0.3

0.4

0.5

0.5

0.7

0.8

0.9

1

1.1

T/Tb Fig. 9. Effect of varying the Biot number of the triangular fin and attached wall on the temperature distribution along the fin root, G, = 3, G, = 3 and G, = 1.

with GI = 3, G2 = 3 and G3 = 1 is modeled with the Biot number varying between 0.01 and 100. The temperature distribution along the y axis of the fin is shown in Fig. 9 with the fin root being located below the dashed line y/w = 1 and the wall above that line. With Biot numbers greater than 10, the temperature distribution is similar to that assumed by Look and Kang [S] with a maximum temperature at the fin centerline falling to a minimum temperature at the edge of the fin, which stays constant along the wall. The temperature distribution along the root becomes

flatter as the Biot number decreases until at a Biot between 0.5 and 1 the temperature is constant along the fin root. With Biot numbers below 0.5, the temperature at the centre of the fin root decreases until the maximum root temperature occurs at the edges of the fin root at the interface of the fin root with the wall. To allow the heat transfer rate Q,, across the wall and fin to be compared they are given per unit length, along the wall and base of the fin, respectively, and non-dimensionalized using eqn (9):

0.1

1

Biot number Fig. 10. Average heat flow across the fin root and attached wall with varying Biot numbers, G, = 3, G, = 3 and G, = 1.

952

S. Abrate and P. Newnham

wall

-

Fig. 11.Effect of varying the thickness of the attached wall on the temperature distribution along the fin root,Bi=lO,G,=3andG,=l.

where q. is the average heat transfer rate per unit length along they axis. The average heat transfer rate qo, is found by summing the heat flow q, calculated in each element across a particular boundary and dividing by the boundary length. For Biot numbers above 1, the heat transfer rate, Qo, across the wall is greater than the heat transfer rate across the fin (Fig. 10). In this case, the fin is acting as an insulator and is not performing any useful function as, under

these conditions, a section of wall without fins will have higher transfer rates than a section of wall with fins. In contrast, as the Biot number is decreased below one, the opposite effect occurs and the fin has a transfer rate greater than the wall. The effect of varying the fin and wall geometry was examined by varying the dimensionless parameter G2 between 0.1 and 5 with G, = 3 and G, = 3. The analysis was performed for Biot numbers of 0.1 and 10 (Figs 11 and 12). The temperature distribution along the fin root with a Biot number of 10 is non-constant for all values of non-dimensional wall

**.* *-..

0.2-

ol”

L..

*... *.... . . . *...

*... *...

*... *.....

O.lS-

. . ...* . . ...*

..*......_

d

-

wall

......

fin

. . . . . . . ...*_ ..*........

8

g

.+........_

O.‘;: 0.05 -

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

G2 Fig. 12. Effect on the temperature distribution along the fin root of varying the thickness of the attached wall, Bi=G,=3andG,=l.

Finite element

analysis

of triangular

fins attached

to a thick wall

953

2 1.81.61.41.2-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T/rb Fig. 13. Average heat flow across the fin root and attached wall with varying wall thickness, Bi = 10, G,=3 and G,=l.

thickness, G, and the temperature distribution along the wall above the fin is constant, apart from a small distance close to the fin root (Fig. 11). As the wall becomes thinner, G2 becomes smaller, and the temperature distribution becomes less parabolic and closer to a constant temperature. With a Biot number of 10, the temperature at the centre of the fin root is less than the temperature at the edge of the fin root for all Biot numbers. The greater heat transfer rate in the fin causes the lower temperatures at the fin root. The temperature distribution along the wall above

the fin is parabolic in shape and is appreciably more curved than with the Biot number of 0.1. The heat transfer rates Q,, per unit length along the y axis, for Biot numbers 10 and 0.1, varying the wall thickness are shown in Figs 13 and 14, respectively. The heat transfer rate for the fin is less than the heat transfer rate for the wall with Bi = 10 for all wall thicknesses. With Bi= 0.1, the heat transfer rate is greater for the fin than the wall for all wall thicknesses. In both cases, as the wall thickness is increased, the heat transfer rate goes down, indicating

2 1.8-

/

1.6-

: fin 0.8 0.6 0.4 0.2 -

0.5

i

G2

j

0.6

0.7

0.8

0.9

T/rb Fig. 14. Average heat flow across the fin root and attached G, = 3 and G, =

wall with varying

I

wall thickness,

Bi =O.l,

954

S. Abrate and P. Newnham 0.06

0.05 -

0

-

tin

. . . . . ..

Wall

1

2

3

4

5

6

I

8

G1 Fig. 15. Effect of varying the fin length on the average heat flow per unit length across the fin root and attached wall, G, = 3, G, = 1 and Bi = 0.01.

that for greater fin efficiency the wall needs to be as thin as possible. The third parameter to be varied is the length of the fin. For G2 = 3 and G, = 1 with Bi = 0.01 and 10, the fin length ratio, G, was varied between one, corresponding to a short fin length, and eight, corresponding to a long fin length. With Bi = 0.01, the heat transfer rates for the fin are greater than those for the wall for all fin lengths (Fig. 15). The heat transfer rate for the wall increases slightly as the fin length is increased, whereas the heat transfer rate increases significantly for the wall, as the fin length is increased, due to the increase in length of the convecting surface along the fin.

Heat flow in the fin with Bi = 10 is less than that across the wall for all fin lengths as the fin is acting as an insulator in this situation (Fig. 16). The heat flow in the wall increases slightly and the heat flow in the fin decreases slightly as G, increases from two to three. As the fin length ratio is increased beyond two, the heat flow in both the wall and fin remains almost constant. A measure of the usefulness of a fin is the fin effectiveness ratio [ 131: fin effectiveness average heat flow with fin = average heat flow without fin

Fig. 16. Effect of varying the fin length on the average heat flow per unit length across the fin root and attached wall, G, = 3, G, = 1 and Bi = 10.

(10)

955

Finite element analysis of triangular fins attached to a thick wall

_

-

Bi=O.Ol

_

...... Bi=lO

2.s E 2

s s ‘J 8

IS_ l-

9

-

3

_

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..a

0.5 -

0

,,,.,,.,,,,,.,,IIII,llll,l.l.,llll,lll~ 0

1

2

3

4

5

6

I

G1 Fig. 17. Effect of varying the fin length on the fin effectivenessratio for the triangular fin and attached wall, G,=3 and Gj= 1. A fin effectiveness ratio above one indicates that the presence of the fin increases the heat flow. The fin effectiveness ratio for the fin described above with Bi = 10 is less than one and stays constant as the fin length is increased (Fig. 17). The fin with Bi =O.Ol has a fin effectiveness ratio above one which increases as the fin length is increased. The rate of increase of fin effectiveness slows as the fin length increases. TEMPERATURE DISTRIBUTION AT THE FIN ROOT

As discussed previously, a commonly assumed temperature distribution at the fin root is that described in eqn (5). The constants a and m in the equation are 64

found from temperature distributions determined using the finite element model, with a least squares curve fitting procedure. Values for the constants a and m are found for a fin with G, = 3, Gz = 3 and G, = 1 and Biot numbers varying between 0.001 and 100 (Fig. 18). Plots of a and m consist of two regions, low Biot numbers (Bi c 0.3) and high Biot numbers (Bi > 0.3). The value of u for low Biot numbers is negative, whereas for high Biot numbers, a is positive and increases as the Biot number increases. Negative values of CIindicate that the temperature at the center of the fin root is less than at the edge. Positive values indicate a higher temperature at the center of the fin

0.14

-0.02 0.001

0

Fig. 18. Effect of varying the fin length on the values of (a) a and (b) m in the equation for the temperature distribution along the root, T = TW+ a cosm(ny/2/). (Conrinued overleuJ)

956

S. Abrate and P. Newnham

(b)

E

Biot number Fig. 18-Coonrinued.

root than at the edge. Relatively constant temperature distributions along the fin root are indicated by small values of a and occur for Biot numbers close to 0.3. As the Biot number increases from 0.001 the value of m stays relatively constant, about 0.6, until the Biot number approaches 0.3 when it decreases sharply. As the Biot number increases above 0.3, the value of m decreases sharply from large values down to about 0.6 again. Apart from the narrow area close to 0.3, where the temperature is relatively constant along the fin root due to the small value of a, m stays very close to 0.6. Values for other fin and wall geometries show similar trends. CONCLUSIONS

Analysis of heat transfer in arrays of triangular fins attached to a wall is performed using finite element analysis. An adaptive grid refinement technique is developed and compared to uniform mesh refinement. Adaptive refinement uses less nodes and elements than uniform refinement to obtain accurate solutions, allowing large and complicated geometries to be modeled using less computational resources. For small Biot numbers (Bi < 0.1) heat conduction is one-dimensional, whereas for large Biot numbers (Bi ~0.1) heat conduction is two-dimensional. Temperature distributions and average heat flow across the fin root are found in triangular fins with a wall attached. It is shown that, for the fin to be effective, the Biot number must be small (W < 0.1). The temperature at the center of the fin root is less than the temperature at the edge of the root for small Biot numbers (Bi < O.l), agreeing with results obtained by Juca and Prata [12] for long rectangular fins. Actual temperature distributions can be approxi-

mated by a simple formula with negative values of a and approximately constant values of m with small Biot numbers (Bi < 0.3). This appears to be the first two-dimensional analysis of triangular fins with an attached wall. Temperature distributions along the root of a triangular fin with the attached wall have not been previously determined. REFERENCES

dimensions of extended surfaces I. A. Aziz, Optimum operating in a convective environment. Appl. Mech. Rev. 45, 155-173 (1992). effects in 2. A. Aziz and H. Nguyen. Two-dimensional a triangular convecting fin. AIAA J. Thermophys. 6, 165-167 (1992). heat 3. W. Lau and C. Tan, Errors in one-dimensional transfer analysis in straight and annular fins. J. Hear Transfer 549-55 1 (1973). of a 4. D. C. Look and H. S. Kang, Optimization thermally non-symmetric fin: preliminary evaluation. Int. J. Heat Mass Transfer 35, 2057-2060 (1992). in 5. D. C. Look and H. S.“Kang; Effects of variation root temperature on heat lost from a thermally nonsymmetric fin. Int. J. Hear Mass Tramfer 34, 1059-1065 (1991). D. C. Look, Separation of variables on a triangular geometry. Q. J. appl. Math. 1, 141-148 (1992). J. N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York (1984). 0. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posferiori error estimates, part 2: error estimates and adaptivity. Int. J. namer. Meth. Engng 33, 1365-1382 (1992). 9 E. J. Probert, 0. Hassan, K. Morgan and J. Peraire, Adautive exnlicit and implicit finite element methods for transient thermal analysis. Int. J. turner. Meth. Engng 35, 655-670 (1992). 10. J. A. Kolodziej and Z. Konczak, A self adaptive h-refinement technique for non-linear heat conduction problems, ZAMM Z. angew. Math. Mech. 72, T526-

T529 (1992).

Finite element analysis of triangular fins attached to a thick wall 11. N. Kikuchi, Adaptive grid-design methods for finite element analysis. Comput. Meth. appl. Mech. Engng 55, 129-160 (1986). 12. P. C. S. JucL and A. T. Prata, Two-dimensional fins

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attached to a thick wall-effect of non-uniform root temperature. ht. J. Heat Mass Transfer 36,233-236 (1992). 13. J. P. Holman, Heat Trunsfer, 7th Edn. McGraw-Hill, New York (1990).