Heat Transfer From Extended Surfaces

Heat and Mass Transfer 39 (2003) 131–138 DOI 10.1007/s00231-002-0338-3 10.1007/s00231-002-0338-3

Heat transfer from extended surfaces subject to variable heat transfer coefficient Esmail M.A. Mokheimer Mokheimer

131

Abstract The present article investigates the effect of locally variable heat transfer coefficient on the performance of extended surfaces (fins) subject to natural convection. Fins of different profiles have been investigated. The fin profiles presently considered are namely; straight and pin fin with rectangular (constant diameter), convex parabolic, triangular (conical) and concave parabolic profiles and radial fins with constant profile with different radius ratios. The local heat transfer coefficient was considered as function of the local temperature and has been obtained using the available correlations of natural convection for each pertinent extended surface considered. The performance of the fin has been expressed in terms of the fin efficiency. Comparisons between the present results for all fins considered and the results obtained for the corresponding fins subject to constant heat transfer coefficient along the fin are presented. Comparisons, i.e. showed an excellent agreement with the experimental results available in the literature. Results show that there is a considerable deviation between the fin efficiency calculated based on constant heat transfer coefficient and that calculated based on variable heat transfer coefficient and this deviation increases with the dimensionless parameter m.

into one fluid, the total surface for heat transfer is thereby  increased. The use of fins in one side of a wall separating two heat-exchanging fluids is exploited most if the fins are attached to or made an integral part of that face on which the thermal resistivity is greatest. In such a case the fin serve the purpose of artificially increasing the surface transmittan transmittance. ce. Thus, Thus, fins find numerous numerous application applicationss in electrical apparatus in which generated heat must be efficiently dissipated, in specialized installations of single and double-pipe heat exchangers, on cylinders of air cooled internal-combustion engines. Recently, finned surfaces are widely used in compact heat exchangers that are used in many applications applications such as air condition conditioners, ers, aircrafts, aircrafts, chemical processing plants, etc … Finned Finned surfaces surfaces are also used in cooling electronic components. The general disposition of fins on the base surface is usually either longitudinal (straight fins) or circumferential (radial fins). Fins may also be disposed in the form of continuous spiral on the base surface or in the form of  individual rods known as pin-fins or spines. The crosssection shape of the extended surface in a plane normal to the base surface is to be referred to as the profile of the fin or spine. Different fin profiles considered in the present study are shown in Fig. 1. 1 Disposition of fins on the base surface results in increase Introduction of the the tota totall surf surfaceareaof aceareaof heat heat tran transfe sfer. r. It migh mightt be expe expect cted ed Extended surface is used specially to enhance the heat that that the the rate rate of heat heat tran transfe sferr per per unit unit of the the base base surf surfac acee area area transfer rate between a solid and an adjoining fluid. Such would increase in direct proportion. However, the average an extended surface is termed a fin. In a conventional heat surface temperature of this strips (fins), by virtue of temexchanger heat is transferred from one fluid to another perature gradient through them, tends to decrease apthrough a metallic wall. The rate of heat transfer is directly  proaching the temperature of the surrounding fluid. So, the proportional to the extent of the wall surface, the heat effective temperature difference is decreased and the net transfer coefficient and to the temperature difference be- increa increase se of heat heat transf transfer er would would not be in directpropo directproporti rtion on to tween one fluid and the adjacent surface. If thin strips the the incr increa ease se of the the surf surfac acee area area and and may may be cons consid ider erabl ablyy less less (fins) of metals are attached to the basic surface, extending than than that that woul would d be anti antici cipa pate ted d on the the basi basiss of the the incr increa ease se of  surface surface area area alone. alone. The ratio of the actual actual heat heat transfe transferr from from the fin surface to that would transfer if the whole fin surface Received: Received: 5 February February 2001 were were at the the same same temp temper erat atur uree as the the base base is comm common only ly call called ed Published online: 10 September 2002 as the fin efficiency. Ó Springer-Verlag 2002 Parsons and Harper [1], derived an equation for the efEsmail M.A. Mokheimer ficiency of straight fins of constant thickness in their inAssistant Professor, Mechanical Engineering Department, vestigation of airplane-engine radiators. Harper and Brown King Fahd University of Petroleum & Minerals, [2], in connection with air-cooled aircraft engines, inP. O. Box: 279, Dhahran 31261, Saudi Arabia E-mail: [email protected] vestig vestigate ated d straig straight ht fins of constan constantt thickn thickness ess,, wedgewedge-sha shaped ped straight fins and annular fins of constant thickness; equaOn Leave from Ain Shams University  tions for the fin efficiency of each type were presented and The author would like to extend his thanks to King Fahd University  the errors involved in certain of the assumptions were of Petroleum and Minerals for the support of this article. The evaluated. Schmidt [3] studied the same three types of fin author also would like to offer his sincere thanks to Prof. H. Z. Barakat due to his valuable discussions during this work. from the material economy point of view. He stated that the

132

Fig. 1a–c. (a) Straight fin profiles and coordinates, (b) Pin fin (spines) profiles and coordinates, (c) Coordinates of Annular fin with rectangular profile

least metal is required for given conditions if the temperature gradient is linear, and showed how thethickness of each type of fin must be varied to produce this result. Finding, in general, that the calculated profiles were impractical to manufacture, Schmidt proceeded to show the optimum dimensions for straight and annular fins of constant thickness and for wedge-shaped straight fins under given operating conditions. The temperature gradient in conical and cylindrical spines was determined by Focke [4]. In this work, Focke, like Schmidt, showed how the spine thickness must be varied in order to keep the material requirement to a minimum; he, too, found that the result impractical and went to determine the optimum cylindrical- and conicalspine dimensions. Murray [5] presented equations for the temperature gradient and the effectiveness of annular fins with constant thickness with a symmetrical temperature distribution around the base of the fin. Carrier and Anderson [6]

discussed straight fins of constant thickness, annular fins of constant thickness and annular fins of constant crosssectional area, presenting equations for fin efficiency of  each. In the latter two cases, the solutions were given in the form of infinite series. Avrami and Little [7] derived equations for the temperature gradient in thick-bar fins and showed under what conditions fins might act as insulators on the basic surface. Approximate equations were also given including, as a special case, that of Harper and Brown. A rather unusual application of Harper and Brown’s equation was made by  Gardner [8], in considering the ligaments between holes in heat-exchanger tube sheets as fins and thereby estimating the temperature distribution in tube sheets. Gardner [9] derived general equations for the temperature gradient and fin efficiency in any extended surface to which a set of  idealized assumptions are applicable. In this regard, Gardner [9] presented analytical solutions for fin efficiency 

for straight fins and spines with different profiles and annular fins of rectangular and constant heat flow area profiles subject to constant heat transfer coefficient. Assuming that the heat transfer coefficient is a power function of the temperature difference of a straight fin of a rectangular profile and that of the ambient, Unal [10] obtained a closed form solution for the one dimensional temperature distribution for different values of the exponent in the power function. An exact solution for the rate of heat transfer from a rectangular fin governed by  a power law-type temperature dependence heat transfer coefficient has been obtained by Sen and Trinh [11]. Rong-Hua Yeh [12] presented the optimum dimensions and heat transfer characteristics of spines with different profiles. In this study, the temperature-dependent heat transfer coefficient is assumed to be a power-law type. Rong-Hua Yeh [12] did not present the fin efficiency of  spines subject to temperature dependent heat transfer coefficient. Performance and optimum dimensions of  longitudinal and annular fins and spines with a temperature dependent heat transfer coefficient have been presented by Laor and Kalman [13]. In this work, Laor and Kalman considered the heat transfer coefficient as a power function of temperature and used exponent values in the power function that represent different heat transfer mechanisms such as free convection, fully developed boiling and radiation. Few studies presented experimental investigation on free convection heat transfer from rectangular fin arrays. Starner and McManus [14] presented average heat transfer coefficient for four fin arrays positioned with vertical, 45 degree, and horizontal base while dissipating heat to room air. Average heat transfer coefficients were found to be strongly affected by the fin array positioning. Average heat transfer coefficients have been also presented by Harahap and Mcmanus [15] for fin array positioned with their base oriented horizontally. Jones and Smith [16] reported experimental average heat transfer coefficients for free convection cooling of arrays of isothermal fins on horizontal surfaces and introduced a simplified correlation. They also suggested an optimum arrangement for maximum heat transfer and a preliminary design method including weight consideration. Sobhan et al. [17] presented an experimental study for free convection heat transfer from fins and fin arrays attached to a heated horizontal base. Local values of heat flux, temperature, heat transfer coefficient, local and overall Nusselt numbers have been obtained for three cases namely, an isothermal vertical flat plate, a single fin attached to a heated horizontal base and a fin array. Correlation was presented relating the overall Nusselt number with the relevant non-dimensional parameters in these cases. Yu¨ncu¨ and Anbar [18] and Gu¨vence and Yu¨ncu¨ [19] presented experimental investigation on performance of fin arrays in free convection on horizontal and vertical base, respectively. These studies reported that for a given baseto-ambient temperature difference, the convection heat transfer rate from fin array takes a maximum value as a function of fin spacing and fin height. Optimization of the ratio of the fin height to the distance between fins in an array of rectangular vertical fins was obtained

experimentally by Welling and Wooldridge [20]. The variation of this ratio with fin temperature was also presented. The effect of fin parameters on the radiation and free convection heat transfer from a finned horizontal cylindrical heater has been studied experimentally by Karaback [21]. The fins used were circular fins. The experimental set-up was capable of  analyzing the effect of fin diameter and spacing on heat transfer. From the thorough literature survey summarized above, the author found that there is no theoretical or experimental work in the literature reported the effect of  temperature-dependent heat transfer coefficient on the fin efficiency of horizontal fins with different profiles subject to natural convection except the work presented by RongHua Yeh [12] and Laor and Kalman [13]. No attention has been given in the literature to the effect of local variations of the heat transfer coefficient on the upper and lower surfaces of horizontal straight fins with different profiles subject to natural convection. The aim of the present article is to present a numerical study for the effect of  temperature-dependent free convection heat transfer coefficient on the fin efficiency for different types of horizontal fins. This type of study would be of direct use by  the heat-transfer equipment designers and rating engineers. 2 Mathematical model and assumptions In some situations, the heat-transfer coefficient undoubtedly does vary from point to point on the fin. For example; for free convection, the heat transfer coefficient is proportional to the temperature difference between the surface and the adjacent fluid raised to the power of (1/4). This proportionality index ranges between 1/7 to 3 for the cases having fully developed boiling and equals to 3 for radiation [22]. The main objective of this paper is to study  the effect of the local heat-transfer coefficient along the fin on the fin performance represented by the fin efficiency for straight fins and spines with different profiles (e.g., constant, convex parabolic, conical, and concave parabolic profiles, i.e., variable cross section area) as well as radial fins of constant thickness for cases with temperature dependent heat transfer coefficient specially if the natural convection is the dominant mode of heat transfer in the fluid surrounding the fin. The fin profile is defined according to the variation of  the fin thickness along its extended length. The general equation of the fin profiles studied during the present article are; Straight fins: The thickness may vary thus x 11ÀÀ2nn  y ¼  yb 1 À L

 

Spines: The circular section diameter may vary thus

 x  y ¼  y 1 À b

1À2n 2Àn

L Annular (radial) fins: The thickness of the radial fins considered in this study will be constant.

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The general partial differential equation governing the steady heat transfer from all fins can be written as:





d  d h ks Ax À P x hx h ¼ 0 dx dx

134

Where: ks is the fin material thermal conductivity which is assumed constant, Ax is the cross-section area perpendicular to the heat flow, P x is the perimeter of that section and hx is the local convection heat transfer coefficient. The Nu k heat transfer coefficient, hx ¼ Dxx  f  Where; k f  is the ambient fluid thermal conductivity, Dx is the local characteristic length and Nux is the local Nusselt number which can be calculated based on the empirical natural convection equations for plates and cylinders, for straight fins and spines, respectively [23]. Straight fins:

Upper surface

Nux ¼ 0:54 Ra1x=4

Lower surface

Nux ¼ 0:27 Ra1x=4

Spines:

8>< 9>= 0:387 Ra Nu ¼ 0:6 þ h >: 1 þ ð0:559=PrÞ i >;

2

1=6 x

x

9=16

8=27

 g bhD3

Where: Rax ¼ ma x Where Dx is the local surface area over the perimeter for the straight fins and the local diameter for the spines. The fin profile exponent n and the form of  the partial differential equation for each type of fin studied will be summarized in the following table 1. These equations will be solved for thermal boundary  conditions of having the base kept at constant and uniform temperature and the fin tip is kept thermally insulated. The above nonlinear ordinary differential equations have been converted to algebraic equations using the finite difference techniques. The final finite difference form of the governing equation is summarized in the following table. 3 Results and discussions The finite difference equations presented have been tested for the effect of mesh size on the accuracy of the solution. The numerical solution for a pin fin with concave paraTable 1. Governing equations of  all types of fin considered

Profile Straight Fins Constant thickness (Rectangular) Convex parabolic Triangular Concave parabolic Spines Constant diameter Convex parabolic Conical Concave parabolic Annular Constant thickness

bolic profile has been obtained via numerical meshes of 5, 10, 15 and 20 grid points. The numerical solution for this case showed independence on the grid size for mesh with grid points of 15 and above. The difference between the fin efficiency that is obtained numerically via a grid of 15 points with respect to that obtained via a grid of 20 points was 0.015%. So, a grid of 15 points has been adopted through out the work. Moreover, the present numerical scheme, the solution algorithm and the solution computer code have been first bench-marked via providing the numerical solution for simple cases that have readily available closed form analytical solution. These cases are namely; straight fins, spines and cylindrical fins with constant profiles with constant heat transfer coefficient along the fin surface. The numerical solution and the analytical solution for the aforesaid cases were almost typical. Such a comparison was a validation for the finite difference scheme, the solution algorithm and the computer code used during the present study. Moreover, the present work has been also validated via a comparison with experimental work of the research group lead by professor Yu¨ncu¨. In their investigation on fin performance of rectangular fins on horizontal base in free convection, Yu¨ncu¨ and Anbar [18] reported the heat transfer rates from a horizontal flat plate as function of the surface and ambient temperature difference. They presented these heat transfer rates as the limiting values of heat transfer rates from vertical fin arrays on horizontal base when the fin heights become very small. Yu¨ncu¨ and Anbar [18] reported the heat transfer rates from a horizontal flat plate with dimensions of 0.25 · 0.10 m with the surface and ambient temperature difference ranges between 20 to 130 °C. An intermediate temperature difference within this range, namely 90 °C, has been selected for comparison. The heat transfer rate from such a plate with temperature difference of 90 °C was found to be 14.23 W as experimentally  reported by Yu¨ncu¨ and Anbar [18]. This is equivalent to 569.231 W/m2 of the plate surface area. A special run of the presently developed computer code has been carried out to calculate the heat transfer rate from a horizontal rectangular fin of the same dimensions mentioned above with a base-toambient temperature difference of 90 °C. The present code calculate the actual heat transfer rate based on variable temperature along the fin and accordingly a variable

n

Governing Equation

1/2

d 2 h dX 2

1/3 0

ðhu þhl Þ 2 À1=2 d h 1 d  h ð1 À X Þ1=2 dX  2 À 2 ð1 À  X Þ dX  À kyb L h ¼ 0 ðhu þhl Þ 2 d 2 h d h ð1 À X Þ dX  2 À dX  À kyb L h ¼ 0 2 2 d  h d h ð1 À X Þ dX 2 À 2ð1 À X Þ dX  À ðhukyþbhl Þ L2 h ¼ 0

±¥ 1/2 0 –1 ±¥ –

À ðhukyþbhl Þ L2 h ¼ 0 2

d 2 h dX 2

À 4kyh X b L2 h ¼ 0 À1=2 d h 4h X  2 d 2 h ð1 À X Þ1=2 dX  2 À ð1 À  X Þ dX  À kyb L h ¼ 0 4h Xu 2 d 2 h d h ð1 À X Þ dX  2 À 2 dX  À ky L h ¼ 0 b 2 4h X  2 d  h d h ð1 À X Þ2 dX  2 À 4ð1 À  X Þ dX  À ky L h ¼ 0 b d 2 h dX 2

ðhu þhl Þ 2 1 d h À  X  dX  À kyb L h ¼ 0

temperature-dependent natural convection heat transfer coefficient. It also calculates the maximum possible heat transfer rate from the fin if it were kept at the maximum possible surface to ambient temperature as that of the base while the heat transfer coefficient varies as function of the local temperature as well as if it is taken constant as that of  the base. The actual heat transfer per unit surface area of the horizontal rectangular (0.25 · 0.10 m) fin with base temperature of 90 °C as calculated from the present code was 515.049 W/m2 which is less than the experimental value reported by Yu¨ncu¨ and Anbar [18] by 9.52%. This is attributed to the fact that during the experiment, the flat horizontal plate was kept isothermal at a constant temperature. Accordingly, this isothermal plate would have a uniform natural convection heat transfer coefficient along its surface. So, for the sake of comparison, the author found that the maximum possible heat transfer rate from the fin is the most appropriate value to be compared with the only  available experimental results mentioned above. This maximum possible heat transfer from the fin is obtained if  its surface acquires the maximum possible temperature and is subjected to the maximum possible heat transfer coefficient as that of the base. This maximum possible heat transfer from the (0.25 · 0.10 m) fin with base to ambient temperature difference of 90 °C has been calculated by the present code and was found to be 542.07 W/m 2 of the fin surface area. This is less than the experimental value reported by Yu¨ncu¨ and Anbar [18] by 4.77%. This deviation might be attributed to the difference between the heat transfer coefficient calculated by the code as function of  temperature using the correlation given by [23] and presented earlier in this article and that calculated and used by  Yu¨ncu¨ and Anbar [18]. It is worth mentioning that the maximum heat transfer coefficient used for this particular run calculated from the correlation [23] base on a based temperature to ambient temperature of 90 °C was 6.02 (W/ m2 Æ K) while the heat transfer coefficient during the experimental work as reported by Yu¨ncu¨ and Anbar [18] ranged between 5.889 and 7.361 (W/m2 Æ K). This also might be attributed to measurement accuracy and approximations

Table 2. Finite difference representation of the governing equations for all types of fin considered

Profile Straight Fins Constant thickness (Rectangular) Convex parabolic Triangular Concave parabolic Spines Constant diameter Convex parabolic Conical Concave parabolic

Subject to boundary conditions: at X  = 0, h = 0 and at X  = 1, d h dX  ¼ 0

Annular Constant thickness

in calculating the heat transfer by free convection from the plate during the experiment. It is worth mentioning here that the heat transfer by free convection from the plate was calculated as reported by Yu¨ncu¨ and Anbar [18] by subtracting the estimated heat transfer by radiation from the measured total heat transfer from the plate. The radiation heat transfer from the plate was estimated using a relation that includes an experimentally evaluated parameters. It is worth mentioning also that, according to Yu¨ncu¨ and Anbar [18], the deviation between their experimentally obtained Nusselt number for the free convection from a horizontal plate and that obtained from McAdams correlation [24] was 9%. So, a deviation of 4.77% between the experimentally  obtained heat transfer rate and that obtained numerically  from the present code for the same operating conditions is within the numerical and experimental errors. This comparison reveals an excellent agreement between the present theoretical results obtained numerically via the presently  developed code and the pertinent experimental results reported by Yu¨ncu¨ and Anbar [18]. After the validation of the numerical model and the computer code as summarized above, the code has been used to solve the heat transfer governing equation for the three considered types of fins subject to variable heat transfer coefficient that varies as a function of the local temperature along the fin surface. The program is used to solve the finite difference equations for all cases under study that are summarized in table 2 to get the temperature distribution along the fin. To solve these equations, one needs to evaluate the local values of the dimensionless parameter m which is function of the local heat transfer coefficient which is in turn is a function of the local temperature. Hence, the solution had to be of iterative nature. So, a special computer code has been designed and developed to solve the governing equations iteratively and obtain the local temperature distribution along the fin. This temperature distribution is then used to calculate the actual local heat transfer rate along the fin. This local heat transfer is numerically integrated to calculate the overall actual heat transfer rate through the whole fin surface. The maximum

m

Finite Difference Form

q  ffiffi ffi ffi ffi ffi ffi q  ffiffi ffi ffi ffi ffi ffi L q   ffi ffi ffi ffi ffi ffi ffi L q  ffiffi ffi ffi ffi ffi ffi L q   ffi ffi ffi L q  ffiffi ffi L q  ffiffi ffi L q  ffiffi ffi L q  ffiffi ffi ffi ffi ffi ffi L L

ðhu þhl Þ kyb ðhu þhl Þ kyb ðhu þhl Þ kyb ðhu þhl Þ kyb

iþ1 hi ¼ 2hþiÀm12þðh X  Þ2

D

ð1À X  ÞÀ1=2 D X 

hi ¼ hi ¼ hi ¼

i ð1À X i Þ1=2 ðhiþ1 þhiÀ1 ÞÀ ðhiþ1 ÀhiÀ1 Þ 4 1=2 2 2ð1À X i Þ þm ðD X Þ2 ð1À X i Þðhiþ1 þhiÀ1 ÞÀD2 X ðhiþ1 ÀhiÀ1 Þ 2ð1À X i Þþm2 ðD X Þ2 2 ð1À X i Þ ðhiþ1 þhiÀ1 ÞÀð1À X i ÞD X ðhiþ1 ÀhiÀ1 Þ 2ð1À X i Þ2 þm2 ðD X Þ2

4h kyb

iþ1 hi ¼ 2hþiÀm12þðDh X  Þ2

4h kyb

hi ¼

4h kyb

hi ¼

4h kyb

hi ¼

ðhu þhl Þ kyb

hi ¼

ð1À X  ÞÀ1=2 D X 

i ð1À X i Þ1=2 ðhiþ1 þhiÀ1 ÞÀ ðhiþ1 ÀhiÀ1 Þ 2 2ð1À X i Þ1=2 þm2 ðD X Þ2 ð1À X i Þðhiþ1 þhiÀ1 ÞÀD X ðhiþ1 ÀhiÀ1 Þ 2ð1À X i Þþm2 ðD X Þ2 ð1À X i Þ2 ðhiþ1 þhiÀ1 ÞÀ2ð1À X i ÞD X ðhiþ1 ÀhiÀ1 Þ 2ð1À X i Þ2 þm2 ðD X Þ2 D X  ðhiþ1 þhiÀ1 ÞÀ2 X  ðhiþ1 ÀhiÀ1 Þ

i

2þm2 ðD X Þ2

135

136

Fig. 2. Comparison of straight fin efficiencies for different profiles, (1) Rectangular profile, ———— its analytical solution, (2) Convex parabolic profile, (3) Triangular profile, (4) Concave parabolic profile

possible heat transfer rate is also calculated locally based on the local heat transfer while the temperature was considered as if it were constant as that of the base. This local maximum possible heat transfer rate is integrated numerically  to calculate the total maximum possible heat transfer rate through the fin. The ratio of the total actual heat transfer rate and the total maximum possible heat transfer rate was used during the present study as the fin efficiency, as used by Gardner [9] and all heat transfer textbooks. The fin efficiency is then plotted against the dimensionless parameter m that is given in table 2 and averaged along the fin. Results obtained for fins subject to variable heat transfer coefficient are presented in Fig. 2 for straight fin with different profile, Fig. 3 for spines with different profiles and Fig. 4 for radial fins with rectangular profile for different radius ratio. In all of these three figures (Figs. 2, 3, 4), the available analytical solution has been plotted as dotted lines to illustrate the deviation between the fin efficiency based on the constant heat transfer coefficient and that is based on the variable heat transfer coefficient as a function of the local temperature along the fin. Moreover the fin efficiency calculated using constant heat transfer coefficient along the fin (as given by Gardener [9] and most of the heat transfer textbooks) have been compared with the efficiency calculated through the present work based on thevariable heat transfer coefficient along the fin as function of the temperature, for selected values of the dimensionless parameter m, is summarized in tables (3, 4, 5) for straight fins, spines with different fin profiles and radial fins with rectangular profile anddifferent radius ratio. These results show that the assumption of constant heat transfer coefficient along the fin in heat transfer situations that is dominated by natural convection mode, would lead to a real underestimation of the fin efficiency. Thus, the use of the fin efficiency predicted by the present study 

Fig. 3. Comparison of pin fin efficiency for different profiles; (1) Constant diameter profile, ———— its analytical solution, (2) Convex parabolic profile, (3) Conical profile, (4) Concave parabolic

Fig. 4. Comparison of radial fin efficiencies with rectangular profile for different radius ratio, ———— analytical solution for r o/ri = 1

based on variable heat transfer coefficient as function of  the local temperature along the fin would result in a considerable reduction of the fin material since the surface area required would be reduced. This can be simply shown by using the equation of heat transfer from fins;

q f  ¼ g f  Ahhb It is clear from this equation that the fin surface area required to transfer a specific amount of heat under certain

Table 3. Comparison of the fin efficiency for straight fins

Table 4. Comparison of the fin efficiency for pin fins

Profile

n

Rectangular Convex Parabolic Triangular Concave Parabolic

1/2 1/3 0 ±¥

Rectangular Convex Parabolic Triangular Concave Parabolic

1/2 1/3 0 ±¥

Rectangular Convex Parabolic Triangular Concave Parabolic

1/2 1/3 0 ±¥

Rectangular Convex Parabolic Triangular Concave Parabolic

1/2 1/3 0 ±¥

Rectangular Convex Parabolic Triangular Concave Parabolic

1/2 1/3 0 ±¥

m =1 0.762 0.700 0.735 0.618 m=2 0.484 0.458 0.432 0.389 m=3 0.332 0.316 0.300 0.279 m=4 0.250 0.242 0.232 0.219 m=5 0.200 0.189 – 0.179

Profile

n

Gardener [9]

Constant diameter Convex Parabolic Conical Concave Parabolic

1/2 0 –1 ±¥

Constant diameter Convex Parabolic Conical Concave Parabolic

1/2 0 –1 ±¥

Constant diameter Convex Parabolic Conical Concave Parabolic

1/2 0 –1 ±¥

Constant diameter Convex Parabolic Conical Concave Parabolic

1/2 0 –1 ±¥

Constant diameter Convex Parabolic Conical Concave Parabolic

1/2 0 –1 ±¥

operating conditions is inversely proportional to the fin efficiency. So, if the designer used the above equation to estimate the area of a fin subject to variable heat transfer coefficient would obtain less values for the area if he used the fin efficiency calculated in the present paper based on variable heat transfer coefficient than that he would obtain if  he used the fin efficiency given in heat transfer text books that is calculated based on constant heat transfer coefficient.

Gardener [9]

m =1 0.6280 0.7180 0.7780 0.8421 m =2 0.3510 0.4536 0.5288 0.6368 m=3 0.2356 0.320 0.389 0.500 m=4 0.1770 0.2470 – 0.4105 m=5 0.1420 0.2000 – 0.3470

Present

Difference %

0.7790 0.7540 0.7235 0.6537

2.18 7.162 –1.548 5.492

0.5190 0.4970 0.4750 0.4430

6.743 7.868 9.136 12.099

0.3786 0.3673 0.3580 0.3468

12.414 14.040 16.201 6.788

0.3050 0.2999 0.2961 0.2943

18.090 19.273 21.816 25.609

0.2622 0.2608 0.2606 0.2629

31.1 27.47 – 31.95

Present

Difference %

0.6573 0.7340 0.7780 0.8540

4.4576 2.1798 0 1.3946

0.4059 0.5007 0.5542 0.6699

13.378 9.4060 4.5830 4.9410

0.3006 0.3811 0.4290 0.5461

21.623 16.032 9.3240 8.4460

0.2538 0.3187 0.3587 0.4642

30.2600 22.4976 – 11.5683

0.2302 0.2847 0.3172 0.4111

38.3145 29.7506 – 15.5923

The results show also that the deviation between the fin efficiency calculated based on constant heat transfer coefficient and that calculated based on variable heat transfer coefficient increases with the increase of the dimensionless parameter m. This deviation reaches, at m = 5, a value of 32% for straight fins, 38% for spines with constant profile and 39% for radial fins with rectangular profile and radius ratio of 4.

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4. Focke, R., Die Nadel als Kuhelemente, Forschung auf dem Gebiete des Ingenieurwesens, Vol. 13, 1942, pp. 34–42 5. Murray, W. M., Heat Dissipation Through an annular Disk or Fin Profile Gardener [9] Present Difference % of Uniform Thickness, Journal of applied Mechanics, Trans. ASME, Vol. 60, 1938, p. A-78 Radius Ratio m =1 6. Carrier, W. H. and Anderson, S. W., The Resistance to Heat Flow  1 0.7615 0.7792 2.274 through Finned Tubing, Heating, Piping and air conditioning, Vol. 10, 1944, pp. 304–320 2 0.6920 0.7243 4.460 7. Avrami Melvin and Little, J. B., Diffusion of Heat Through a 3 0.6420 0.6883 6.731 Rectangular Bar and the cooling and insulating Effect of Fins, I. 4 0.6105 0.6622 7.802 The Steady State, Journal of applied Physics, Vol. 13, 1942, m =2 pp. 225–264 1 0.4820 0.5190 7.130 8. Gardner, K. A., Heat Exchanger Tube Sheet Temperature, Refiner 2 0.3915 0.4452 12.069 and Natural Gasoline Manufacturer, Vol. 21, 1942, pp. 71–77 3 0.3320 0.4015 17.319 9. Gardner, K. A., Efficiency of Extended Surface, Trans. ASME, 4 0.3115 0.3714 16.119 J. Heat Transfer, Vol 67, 1945, pp. 621–631 m=3 10. Unal, H. C., Determination of the Temperature Distribution in an 1 0.3310 0.3787 12.585 Extended Surface with a non-Uniform Heat Transfer Coefficient, 2 0.2560 0.3132 18.263 International Journal of Heat and Mass Transfer, Vol. 28, No. 12, 3 0.2142 0.2751 22.129 1985, pp. 2270–2284 4 0.1895 0.2493 23.985 11. Sen, A. K. and Trinh, S., An exact Solution for the Rate of Heat Transfer From Rectangular Fin Governed by a Power Law-Type m=4 Temperature Dependence, Transactions of ASME, Journal of  1 0.2498 0.3050 18.008 Heat Transfer, Vol. 108, 1986, pp. 457–459 2 0.1873 0.2485 24.633 12. Rong-Hua Yeh, On Optimum Spines, Journal of Thermo3 0.1560 0.2156 27.631 dynamics and Heat Transfer, Vol. 9, No. 2, 1995, pp. 359–362 4 0.1316 0.1934 31.932 13. Laor, K. and Kalman, H., Performance and Optimum Dimensions m=5 of Different Cooling Fins with a Temperature Dependent Heat 1 0.2000 0.2622 23.719 Transfer Coefficient, International Journal of Heat and Mass 2 0.1445 0.2133 32.265 Transfer, Vol. 39, No. 9, 1996, pp. 1993–2003 3 0.1189 0.1843 35.469 14. Starner, K. E. and McManus, H. N., JR., An Experimental In4 0.1000 0.1644 39.167 vestigation of Free-Convection Heat Transfer From RectangularFin Arrays, Journal of Heat Transfer, Transactions of the ASME, August 1963, pp. 273–278 15. Harahap, F. and McManus, H. N., JR., Natural Convection Heat 4 Transfer From Horizontal Rectangular Fin Arrays, Journal of  Conclusion Heat Transfer, Transactions of the ASME, February 1967, Heat transfer from extended surfaces subject to locally  pp. 32–38 variable heat transfer coefficient has been studied. The 16. Jones, C.D. and Smith, L. F., Optimization of Rectangular Fins on local heat transfer coefficient as function of the local Horizontal Surfaces for Free Convection Heat Transfer, Journal of  Heat Transfer, Transactions of the ASME, February 1970, temperature has been obtained using the available correpp. 6–10 lations of natural convection for each pertinent extended 17. Sobhan, C. B., Venkateshan, S. P. and Seetharamu, K. N., Exsurface considered. The results showed that the assumpperimental Studies on Steady Free Convection Heat Transfer tion of constant heat transfer coefficient along the fin in from Fins and Fin Arrays, Wa¨rme-und Stoffu¨bertragun (Heat and Mass Transfer), Vol. 25, 1990, pp. 345–352 such cases leads to a significant underestimation of the fin ¨ ncu¨, H. and Anbar, G., An Experimental Investigation on 18. Yu efficiency. The deviation between the fin efficiency calcuPerformance of Rectangular Fins on a Horizontal Base in Free lated based on constant heat transfer coefficient and that Convection Heat Transfer, Heat and Mass Transfer, Vol. 33, 1998, calculated based on variable heat transfer coefficient inpp. 507–514 creases with the dimensionless parameter m. The use of  19. Gu¨vence, A. and Yu¨ncu¨, H., An Experimental Investigation on Performance of Fins on a Horizontal Base in Free Convection the present results by the designer of heat transfer Heat Transfer, Heat and Mass Transfer, Vol. 37, 2001, pp. 409– equipment that involve extended surface subject to natural 416 convection heat transfer mode would result in a con20. Welling, J. R. and Wooldridge, C. B., Free Convection Heat siderable reduction in the extended surface area and hence Transfer Coefficients from Rectangular Vertical Fins, Journal of  Heat Transfer, Transactions of the ASME, November 1965, a significant reduction in the weight and size of the heat pp. 439–444 transfer equipment. 21. Karaback, R., The Effect of Fin Parameter on the Radiation and Free Convection from a Finned Horizontal Cylindrical Heater, References Energy Convers. Mgmt., Vol. 33, No. 11, 1992, pp. 997–1005 1. Parsons, S. R. and Harper, D. R., Radiators for Aircraft Engines, 22. Holman, J. P., Heat transfer (SI Metric Edn). McGraw-Hill, New  U. S. Bureau of Standards, Technical Paper no. 211, 1922, pp. York, 1989 327–330 23. Incropera, F. P. and Dewitt, D.P., Introduction to Heat Transfer, 2. Harper, D. R. and Brown W. B., Mathematical Equations for Heat John Wiley & Sons, 1996 Conduction in the Fins of Air-Cooled Engines, National Advisory  24. McAdams, W. H., Heat Transfer, 3 rd ed., McGraw Hill, 1954, New  Committee for Aeronautics, report no. 158, 1922 York 3. Schmidt, E., Die Warmeubertrgung durch Rippen, Zeit. V. D. I., Vol. 70, 1926, pp. 885–889, and 947–951 Table 5. Comparison of the fin efficiency for annular fins with rectangular profile

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