Optimum configuration of a fin for boiling heat transfer

Optimum Conjguration of a Fin for Boiling Heat Transfer by R.

H. YEH

Department of Marine Engineering University, Taiwan, R.O.C. and s.

Technology,

National

Taiwan Ocean

P. LIAW

Department of Marine Engineering, Taiwan. R.O.C.

National

Taiwan Ocean University,

volume, an optimum shape offn is proposed ABSTRACT : Based on the principle qfminimum in this study. The outer appearance is essentially like a cylindrical.fin and the excavated part is in the inner portion qf the@. First, the surf&e heat,flux is assumed to follow a power-law dependence. With the aid of one-dimensional hnalysis, the temperature distributions and the projiles of the excavation hole are catculatedfor various single heat transj2r modes. Secondly, the boiling curve of isopropyl alcohol on copper surface is used as an input to the model. The base temperature in thejilm boiling rqgime is taken and the optimum geometry is presented. Finally, the eficiency based on the volume and heat duty is discussed.

Nomenclature A

Bi h k L m

Q 4 R r T u, v V 4 r e

cross-sectional area of solid fin Biot number = hL/k heat transfer coefficient thermal conductivity length of fin power-law exponent total heat transfer of a fin surface heat flux of fin radius of cylindrical fin or half-thickness of longitudinal radius of excavation for cylindrical fin or half-thickness longitudinal fin temperature transformation parameters volume of fin coordinate in axial direction temperature superheat fin efficiency dimensionless temperature

The Franklinlnst1tutc001&003?;93 $5 OO+O.OO

fin of excavation

for

1.53

R. H. Yeh and S. P. Liar Subscripts, superscripts

b 0 min *

fin base fin tip minimum nondimensional

quantity

I. Introduction Extended surfaces are frequently used in heat exchange devices for the purpose of increasing the heat transfer between a primary surface and the surrounding fluid. In the design of cooling devices on vehicles, especially aircraft, the problem of exchanging the greatest amount of heat with the least amount of weight in the exchanger is of paramount importance. For a specified material, this research leads to the consideration of the minimum volume. For pure conducting fins, a criterion for optimum shape was proposed by Schmidt (1) using the principle of a constant heat flux. Later, Duffin (2) confirmed this result by applying a rigorous variational approach. In their studies, the fin profile is calculated to be a parabola and has a zero thickness at the outer edge. The effect of internal heat generation on the optimum shape was first considered by Minkler and Rouleau (3), but a more rigorous treatment was given by Liu (4), for heat generations which are directly proportional to the temperature. Optimum shape of a purely radiating fin was obtained by Wilkins (5-7) for a variety of geometries. In addition, a great number of recent papers have reported to date in which different aspects of the problem of fin optimization are investigated with many additional conditions (S-11). However, none of them is applicable to boiling heat transfer. In the use of fins in efficient heat exchangers, such as heat transfer with phase change, it is desirable to know the optimum geometry so that a given amount of heat can be transferred in a minimum volume. Haley and Westwater (12) used numerical computation to find out an optimum shape of the spine fin which turns out to resemble a spade on a playing card. Due to the difficulty in fabrication, Cash et al. (13) modified the configuration by using a two-cone assembly attached to a small cylindrical neck. Nevertheless, these structures at the fin base are very weak. the neck of these fins. In addition, “vapor trapped” may occur underneath In this study, based on the requirement of minimum volume, an improved fin with superior heat transfer characteristics is proposed and analysed. The determination of this optimum fin is carried out with the aid of transformation parameters (6, 12). In addition, the heat duties are made to compare with a solid one and a spade-shaped fin (12). II. Theoretical

Formulation

If a fin is immersed in a pool of boiling liquid, regardless of the effect of the generating vapor, the temperature-dependent wall heat flux is usually assumed to be the same as the value of a typical boiling curve which was obtained experimentally from a large isothermal surface (12). Hence, the thermal characteristics

154

Fin Conjgurution

for Boiling Heat Transfer

can be quantified. In the present study, this boiling curve is presumed to be known. The outer appearance of the proposed fin is essentially the same as a cylindrical fin. By excavating part of the mass near the centerline, the weight of the fin is minimized at a given base heat flux. As shown in Fig. 1, an axisymmetric fin is cooled in a pool of liquid. The heat is conducted through the material and finally dissipated to the surroundings with a surface heat flux q(AT), which is a function of the temperature difference between the wall and the ambient liquid. Note that the excavated part is filled with insulation material. Before solving the problem analytically, it is convenient to nondimensionalize the governing equation and boundary conditions. Selecting the following group of dimensionless variables : Q*

=

Z

z* = ~

..__%

’

Ak,ATb

L r* =Y R

k* = ;-, b

L L” = z

tl=&, b

the one-dimensional steady-state inside a pin fin is then expressed

dQ* = ~~~dz*

with the appropriate

boundary

heat conduction as

equation

with no heat generation

1

$ k*(l-r*‘)gi = 4L*q* :

conditions

Q* =0

at

Fin material

z* =0

Fin has 7

(2)

.:‘I :‘. :

4 Excavated/ part

FIG.

in Great Bntain

1.

Schematic

cross-sectional

view of the proposed

fin.

R. H. Yeh and S. P. Liar

O=l Next, we introduce

the following

at

z*=l.

transformation

(3)

parameters

:

(4)

u=- 6 ub

‘I s0

k*q*2 d0,

(5)

where

s 1

ur, = 6

k*q*’

d0

(6)

0

III. Cylindrical

Differentiating

Substituting

Fins

Eq. (4) with respect to u, gives

into Eq. (l), yields d-” d0

Employing

1

Q;

2L*q*

3 z

,;,dcdu (8)

( du ) do’

Eqs (l), (4) and (5), the above equation

becomes

(9)

In the following derivation, the volume of a fin is minimized. Defining the dimensionless volume as the ratio of the volume of the proposed fin to a solid one, the volume is written as f/* =

’ (, -r*2)&*. s0

(10)

With the aid of Eqs (8) and (9), it takes the form

where ZL(, is an integration of Eq. (5) from zero to the superheat hand side of the above equation can be rearranged to read

156

at fin tip. The right-

Journal

of the Frmkhn Pergamon

lnstitutc Press Ltd

Fin Configuration for Boiling Heat Transjtir

v*

Apparently,

=

the minimum

6Lq;:irh[~~(~-l~du+l+u”] (12) volume

of the fin is obtained v;,,

=

~

Qb

as

*3

6L*‘u,

(13)

by setting u0 = 0 and u = v. This shows that the temperature at the insulated end will drop to the ambient liquid temperature. Subsequently, the configuration of the proposed fin is thus found to be (14)

At fin tip, u0 = 0, thus r* is equal to unity. The temperature distribution can be obtained by coupling Eqs (5) and (14). Since k* depends on the material of fin only, and q* was assumed to be attainable from a boiling curve in case the temperature at fin base is known. Once Qc and L* are prescribed, the volume as well as the geometry of the proposed fin can be determined. Fin efficiency, y, is defined as the ratio of total heat transfer to the ideal heat transfer of an isothermal fin at the base temperature. The fin efficiency is obtained as

Qh* y = 4L*qb*.

(16)

Fin subject to a power-law wall heatjux This problem can be simplified if the wall heat flux can be approximated by a power dependence. Referring to the base condition, the expression of heat transfer function may be written as q* = (Bi),,d”‘,

(17)

where (Bi)b iS equal to hbL/kb, and hb represents the surface heat transfer coefficient at fin base temperature. According to boundary condition (2), there is no heat transfer at the fin tip. Hence the present model is applicable only for a positive m. Besides, in some situations, it can be represented by the appropriate choice of nz. For instance, the exponent m may take the values of 0.75, 1, 3.5 and 4 when the fin is cooled due to the film boiling or film condensation, convection, nucleate boiling and radiation, respectively. By assuming constant thermal conductivity, we use the expression in Eq. (17) to compute the length of a fin as L*

=

.exPcL 2(B&(m+2)

Equations

(13)-( 16) are then simplified

Vol. 330. No I. pp. 153-163. Prmted m Great Britain

1993

(18)

to 157

R. H. Yeh and S. P. Liaw

(19) (20) (21) and m+2

v-4

y = 2(2m+1)’

Longitudinal,fins The above analysis is also applicable to a longitudinal fin with an infinite width. As shown in Fig. 1, the height of the fin can be considered as 2R and the excavation at the center part is equivalent to 2r. The minimum volume of this fin is then calculated as

g$.b

V$” = The relation

between

the axial position

and temperature

L*ub

excavation

L*Ub it is obtained

(24)

becomes

r*=l_2e*b2 As for the fin efficiency,

is found to be

0 /&mm l:jq* dQ, s0

?* - 2% The temperature-dependent

(23)

(25)

u’:3q*.

as

(26) Similarly, in the case of a single heat transfer mode, the length and excavation of an optimum longitudinal fin can thus be simplified to L*

_

QiVm+J (Bi)b(mf2)

(mf2) __Qb*z*(5,n+ y* = ] _ ~~3

(27)

I)w+a,

In addition, it is interesting to note that the expressions of V&,, H and yefor both optimum cylindrical fin and longitudinal fin are identical.

158

Journalofthe

Franklm lnslitute Pergamon Press Ltd

Fin C&figuration

“0

for Boiling Heat Transfer

0.5

1

Z* FIG.

2. Temperature profiles.

IV. Results and Discussion From the above analysis, it is clear that the behavior of the proposed optimum cylindrical fin and optimum longitudinal fin are similar. Only the characteristics of optimum pin fin is presented herein. First, a fin subject to a power-law temperaturedependent surface heat flux is investigated. Figure 2 shows the temperature profiles for various exponents. The temperature decreases along the fin axis and subsequently drops to the ambient temperature at fin tip. A linear temperature distribution is observed at m = 1. In addition, the temperature profiles are convex for m > 1 whereas they are concave for m < 1. At the same distance from the fin base, a higher temperature is observed for a larger m, but the corresponding wall heat flux is lower. The variation of surface heat flux along the fin is displayed in Fig. 3. From Eq. (21), it is easily derived that

1

i?

m

y 0.5 I u-

0

FIG. 3. Distribution Vol. 330. No 1. pp 153-163, Pnnted I” Great Bntam

1993

of surface heat flux. 159

R. H. Yeh and S. P. Liaw

is the necessary condition for the existence of the excavating radius. Therefore the present model works only when Eq. (29) is satisfied. The profiles of r* for m = 1 and 4 are shown in Fig. 4. For a larger Qc, it is conceivable that the more solid portion of a fin is needed to achieve a larger amount of heat dissipation. Again, owing to the prescribed adiabatic boundary condition at the fin tip, in the above analysis, only the results of m > 0 are available. A pure copper fin boiling in a pool of saturated isopropyl alcohol served as a typical example in this study. The temperature-dependent heat transfer coefficient is borrowed from Haley (14) and is the same as that used by Biyikli (15) or Liaw and Yeh (16). For.a given fin base area, there exists an upper limit on the total heat load. Certainly, the highest heat transfer rate can be obtained in an infinitely long solid fin, but the space and weight are not allowed. Hence the criterion of minimum volume is adopted. From Eq. (15), the relation between the minimum fin radius and total heat duty is written as

The equality of the above equation is shown graphically in Fig. 5 for Qb = 1000, 500 and 100 W. It is seen that a higher heat transfer rate can be achieved by a larger radius of the fin, especially while the temperature at fin base is low. Nevertheless, at the high base temperature region, R,,,, is merely a function of Qb. It is interesting to note the cases of fin base at film boiling. The idea is to diminish the surface area in film boiling and convection and expand the surface area in transition and nucleate boiling. Thus the thermal resistance at the base of fin is

FIG. 4. Radius of excavated portion in the proposed fin. Journal

160

of the Frankhn Institute Pcrgamon Press Ltd

Fin Configuration for Boiling Heat Transfer

FIG. 5. The minimum outer radius of the optimum fin for Qb = 1000, 500 and 100 W.

increased by reducing the cross-section area causing a large temperature drop in the film boiling region. On the other hand, the enlarged cross-section area slows the temperature drop so that the highly efficient nucleate and transition boiling regimes spread over a large area on the fin surface. Figure 6 shows the temperature

26 13

13

2 > 3 2

0 1

b

i

i 2

i

4.1

(cm>

FIG. 6. The temperature profile, surface heat flux, and configuration (Qb = 500 W, AT, = 150 K). Vol. 330, No. I, pp. 153-163. Prmted in Great Bmain

of the optimum fin

1993

161

R. H. Yeh und S. P. Limit

V

AT,, Reference

(K)

Solid fin (16) Spade fin (12) Present work

100 100 100

Solid fin (16) Spade fin (12) Present work

150 150 150

Qtu'V

LxR

(cm’)

(cm x cm)

(W/cm’)

556 500 500

20.2 13.0 11.0

3..8 x 1.3 4.5x 1.5 3.8 x 1.3

27.5 38.5 45.5

122 500 500

21.8 13.1 11.3

4.1 x 1.3 4.5 x 1.5 4.1 x 1.3

5.6 38.2 44.2

profile, local heat flux and geometry of a fin for its base temperature in film boiling regime. The optimization work is similar to those discussed earlier in Figs 2-4. Larger cross-section area near the fin base is used to delay the temperature drop in this portion ; hence, high heat flux covers most of the surface. As the heat transfer coefficient decreases along the fin length, the fin volume is minimized by tapering to a circle at fin tip. The proposed fin is just like a nozzle. It is seen that a narrow collar is made to cause a large temperature drop until the appearance of transition boiling on the fin surface. Conceivably, the length of the collar will be longer if the base temperature is higher. In Fig. 6. the temperature profile and surface heat flux of a solid fin is also included. From the upper part of Fig. 6, the film boiling region is effectively shrunk by excavation. The multi-type boiling is established in the excavated fin while the solid one is still in inefficient film boiling only. The many differences in heat duty can be observed in this case. The geometries and heat duties of a solid fin, a spade-shaped fin (12), and the present work are listed in Table I for base temperatures at 100 and 150 K. It should be pointed out that the listed R is the value of maximum outer radius. Due to the possession of superior characteristics, the proposed optimum fin has shown to be the most efficient, and thus the least volume is needed. V. Conclusions In this paper, based on the principle of minimum volume optimization, work is done by excavating part of the material in the core of the fin. The results cover both cylindrical and longitudinal fins. For a given base area, there is a restriction imposed on the maximum heat transfer rate. The inner excavation of the proposed fin operating in film boiling is similar to a nozzle; however, the outer appearance is of a cylindrical fin. The proposed fin is attractive because it has shown to be superior to a solid fin or to a spade-shaped fin.

Acknowledgement This work was supported NSCSI-0401-E019-01.

in part by the National

Science Council through

Journal

162

Contract

No.

of the Franklin Institu:e Pcrgamon Press Ltd

Fin Co$quration,for

Boiling

Heut

Transfer

References (1) E. Schmidt, (2) (3) (4) (5) (6)

(7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

“Die Warmeiibertragung durch Rippen”, Z. VDI, Vol. 70, pp. 8855947, 1926. R. J. Duffin. “A variational problem related to cooling fins”, J. Math. Mech., Vol. 8, pp. 47756, 1959. W. S. Minkler and W. T. Rouleau, “The effect of internal heat generation on heat transfer in thin fins”, Nucl. Sci. Enyny, Vol. 7, pp. 400406, 1960. C. Y. Liu, “A variational problem relating to cooling fins with heat generation”, Q. Appl. Math., Vol. 19, pp. 2455251. 1961. J. E. Wilkins Jr, “Minimizing the mass of thin radiating fins”, J. Aerospace Sci., Vol. 27, p. 145, 1960. J. E. Wilkins Jr, “Minimum mass thin fins for space radiators”, Proc. 1960, Heat Transfer and Fluid Mechanics Institute, pp. 2299243, Standford University Press, Standford. CA, 1960. J. E. Wilkins Jr, “Minimum mass thin fins which transfer heat only by radiation to surrounding at absolute zero”, J. Sm. Ind. Appl. Math., Vol. 8, pp. 630-639, 1960. G. Ahmadi and A. Razani. “Some optimization problems relating to cooling fins”, ht. J. Heat Mass Transf>r, Vol. 16, pp. 236992375, 1973. C. J. Maday, “The minimum weight one-dimensional straight fin”, ASME J. Enyny Ind, Vol. 97. pp. 161Il65, 1974. I. Mikk, “Convective fin of minimum mass”, ht. J. Heat Mass Tram@, Vol. 22, pp. 7077711, 1980. P. Razelos and K. Imre, “Minimum mass convective fins with variable heat transfer coefficients”, J. Franklin Inst., Vol. 3 15, pp. 2699282, 1983. K. W. Haley and J. W. Westwater, “Boiling heat transfer from single tins”, Proc. 3rd Int. Heat Transfer Conf., Chicago, IL, Vol. 13, pp. 2455253, 1966. D. R. Cash, G. J. Klein and J. W. Westwater, “Approximate optimum fin design for boiling heat transfer”, J. Elxt Tram@, Vol. 93, pp. 19-24, 1971. K. W. Haley, Design of fins for boiling heat transfer, Ph.D. Thesis, University of Illinois, Urbana, IL, 1965. S. Biyikli, “Optimum use of longitudinal fins of rectangular profile in boiling liquids”, J. Heat Transf>r, Vol. 107, pp. 9688970, 1985. S. P. Liaw and R. H. Yeh, “Analysis of pool boiling heat transfer on a single cylindrical fin”, J. CSME, Vol. I I, pp. 2855292, 1990.

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1993

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