Heat transfer from extended surfaces subject to va

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Heat	transfer	from	extended	surfaces	subject	to
variable	heat	transfer	coefficient
Article		in		Heat	and	Mass	Transfer	·	January	2003
DOI:	10.1007/s00231-002-0338-3
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Heat transfer from extended surfaces subject to variable
heat transfer coefficient
Esmail M.A. Mokheimer
Abstract The present article investigates the effect of lo-
cally 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 ra-
tios. 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 perfor-
mance 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 corre-
sponding 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.
1
Introduction
Extended surface is used specially to enhance the heat
transfer rate between a solid and an adjoining fluid. Such
an extended surface is termed a fin. In a conventional heat
exchanger heat is transferred from one fluid to another
through a metallic wall. The rate of heat transfer is directly
proportional to the extent of the wall surface, the heat
transfer coefficient and to the temperature difference be-
tween one fluid and the adjacent surface. If thin strips
(fins) of metals are attached to the basic surface, extending
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
transmittance. Thus, fins find numerous applications in
electrical apparatus in which generated heat must be effi-
ciently 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 such as air conditioners, aircrafts,
chemical processing plants, etc… Finned surfaces are also
used in cooling electronic components.
The general disposition of fins on the base surface is
usually either longitudinal (straight fins) or circumfer-
ential (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 cross-
section 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.
Disposition of fins on the base surface results in increase
of the total surface area of heat transfer. It might be expected
that the rate of heat transfer per unit of the base surface area
would increase in direct proportion. However, the average
surface temperature of this strips (fins), by virtue of tem-
perature gradient through them, tends to decrease ap-
proaching the temperature of the surrounding fluid. So, the
effective temperature difference is decreased and the net
increase of heat transfer would not be in direct proportion to
the increase of the surface area and may be considerably less
than that would be anticipated on the basis of the increase of
surface area alone. The ratio of the actual heat transfer from
the fin surface to that would transfer if the whole fin surface
were at the same temperature as the base is commonly called
as the fin efficiency.
Parsons and Harper [1], derived an equation for the ef-
ficiency of straight fins of constant thickness in their in-
vestigation of airplane-engine radiators. Harper and Brown
[2], in connection with air-cooled aircraft engines, in-
vestigated straight fins of constant thickness, wedge-shaped
straight fins and annular fins of constant thickness; equa-
tions for the fin efficiency of each type were presented and
the errors involved in certain of the assumptions were
evaluated. Schmidt [3] studied the same three types of fin
from the material economy point of view. He stated that the
Received: 5 February 2001
Published online: 10 September 2002
	 Springer-Verlag 2002
Esmail M.A. Mokheimer
Assistant Professor, Mechanical Engineering Department,
King Fahd University of Petroleum & Minerals,
P. O. Box: 279, Dhahran 31261, Saudi Arabia
E-mail: esmailm@kfupm.edu.sa
On Leave from Ain Shams University
The author would like to extend his thanks to King Fahd University
of Petroleum and Minerals for the support of this article. The
author also would like to offer his sincere thanks to Prof.
H. Z. Barakat due to his valuable discussions during this work.
Heat and Mass Transfer 39 (2003) 131–138
DOI 10.1007/s00231-002-0338-3
131
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least metal is required for given conditions if the tempera-
ture gradient is linear, and showed how the thickness 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 di-
mensions for straight and annular fins of constant thickness
and for wedge-shaped straight fins under given operating
conditions. The temperature gradient in conical and cy-
lindrical 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 conical-
spine 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 cross-
sectional 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 tem-
perature 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
Fig. 1a–c. (a) Straight fin profiles and coordinates, (b) Pin fin
(spines) profiles and coordinates, (c) Coordinates of Annular fin with
rectangular profile
132
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 efficiencyof
spines subject to temperature dependent heat transfer
coefficient. Performance and optimum dimensions of
longitudinal and annular fins and spines with a tempera-
ture 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 hor-
izontal 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 base-
to-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 Rong-
Hua 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 ar-
ticle 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 hor-
izontal fins. This type of study would be of direct use by
the heat-transfer equipment designers and rating en-
gineers.
2
Mathematical model and assumptions
In some situations, the heat-transfer coefficient un-
doubtedly 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., con-
stant, 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 de-
pendent 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
y ¼ yb 1 � x
L
� �1�2n
1�n
Spines: The circular section diameter may vary thus
y ¼ yb 1 � x
L
� �1�2n
2�n
Annular (radial) fins: The thickness of the radial fins
considered in this study will be constant.
133
The general partial differential equation governing the
steady heat transfer from all fins can be written as:
d
dx
ksAx
dh
dx
� �
� Pxhxh ¼ 0
Where: ks is the fin material thermal conductivity which is
assumed constant, Ax is the cross-section area perpendi-
cular to the heat flow, Px is the perimeter of that section
and hx is the local convection heat transfer coefficient. The
heat transfer coefficient, hx ¼ NuxkfDx Where; kf is the ambi-
ent 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 Ra1=4x
Lower surface Nux ¼ 0:27 Ra1=4x
Spines:
Nux ¼ 0:6 þ 0:387 Ra
1=6
x
1 þ ð0:559=PrÞ9=16
h i8=27
8><
>:
9>=
>;
2
Where: Rax ¼ gbhD
3
x
ma 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 gov-
erning 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 para-
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 nu-
merical solution for simple cases that have readily avail-
able 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 numericalsolution and the
analytical solution for the aforesaid cases were almost ty-
pical. Such a comparison was a validation for the finite
difference scheme, the solution algorithm and the com-
puter 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 per-
formance 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 pre-
sented 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 dif-
ference 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-to-
ambient 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
Table 1. Governing equations of
all types of fin considered Profile n Governing Equation
Straight Fins
Constant thickness
(Rectangular)
1/2 d
2h
dX2 � ðhuþhlÞkyb L2h ¼ 0
Convex parabolic 1/3 ð1 � XÞ1=2 d2hdX2 � 12 ð1 � XÞ�1=2 dhdX � ðhuþhlÞkyb L2h ¼ 0
Triangular 0 ð1 � XÞ d2hdX2 � dhdX � ðhuþhlÞkyb L2h ¼ 0
Concave parabolic ±¥ ð1 � XÞ2 d2hdX2 � 2ð1 � XÞ dhdX � ðhuþhlÞkyb L2h ¼ 0
Spines
Constant diameter 1/2 d
2h
dX2 � 4hXkyb L2h ¼ 0
Convex parabolic 0 ð1 � XÞ1=2 d2hdX2 � ð1 � XÞ�1=2 dhdX � 4hXkyb L2h ¼ 0
Conical –1 ð1 � XÞ d2hdX2 � 2 dhdX � 4hXukyb L2h ¼ 0
Concave parabolic ±¥ ð1 � XÞ2 d2hdX2 � 4ð1 � XÞ dhdX � 4hXkyb L2h ¼ 0
Annular
Constant thickness – d
2h
dX2 � 1X dhdX � ðhuþhlÞkyb L2h ¼ 0
134
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 tem-
perature 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 at-
tributed to the fact that during the experiment, the flat
horizontal plate was kept isothermal at a constant tem-
perature. 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 coeffi-
cient 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/m2 of the fin
surface area. This is less than the experimental value re-
ported 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 pre-
sented 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 ex-
perimental 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
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 sub-
tracting 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 com-
parison reveals an excellent agreement between the present
theoretical results obtained numerically via the presently
developed code and the pertinent experimental results re-
ported 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 tem-
perature. Hence, the solution had to be of iterative nature.
So, a special computer code has been designed and devel-
oped to solve the governing equations iteratively and obtain
the local temperature distribution along the fin. This tem-
perature 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
Table 2. Finite difference re-
presentation of the governing
equations for all types of fin
considered
Subject to boundary conditions:
at X = 0, h = 0 and at X = 1,
dh
dX ¼ 0
Profile m Finite Difference Form
Straight Fins
Constant thickness
(Rectangular)
L
ffiffiffiffiffiffiffiffiffiffiffiffi
ðhuþhlÞ
kyb
q
hi ¼ hi�1þhiþ12þm2ðDXÞ2
Convex parabolic L
ffiffiffiffiffiffiffiffiffiffiffiffi
ðhuþhlÞ
kyb
q
hi ¼ ð1�XiÞ
1=2ðhiþ1þhi�1Þ�ð1�XiÞ
�1=2DX
4 ðhiþ1�hi�1Þ
2ð1�XiÞ1=2þm2ðDXÞ2
Triangular L
ffiffiffiffiffiffiffiffiffiffiffiffi
ðhuþhlÞ
kyb
q
hi ¼ ð1�XiÞðhiþ1þhi�1Þ�
DX
2 ðhiþ1�hi�1Þ
2ð1�XiÞþm2ðDXÞ2
Concave parabolic L
ffiffiffiffiffiffiffiffiffiffiffiffi
ðhuþhlÞ
kyb
q
hi ¼ ð1�XiÞ
2ðhiþ1þhi�1Þ�ð1�XiÞDXðhiþ1�hi�1Þ
2ð1�XiÞ2þm2ðDXÞ2
Spines
Constant diameter L
ffiffiffiffiffi
4h
kyb
q
hi ¼ hi�1þhiþ12þm2ðDXÞ2
Convex parabolic L
ffiffiffiffiffi
4h
kyb
qhi ¼ ð1�XiÞ
1=2ðhiþ1þhi�1Þ�ð1�XiÞ
�1=2DX
2 ðhiþ1�hi�1Þ
2ð1�XiÞ1=2þm2ðDXÞ2
Conical L
ffiffiffiffiffi
4h
kyb
q
hi ¼ ð1�XiÞðhiþ1þhi�1Þ�DXðhiþ1�hi�1Þ2ð1�XiÞþm2ðDXÞ2
Concave parabolic L
ffiffiffiffiffi
4h
kyb
q
hi ¼ ð1�XiÞ
2ðhiþ1þhi�1Þ�2ð1�XiÞDXðhiþ1�hi�1Þ
2ð1�XiÞ2þm2ðDXÞ2
Annular
Constant thickness L
ffiffiffiffiffiffiffiffiffiffiffiffi
ðhuþhlÞ
kyb
q
hi ¼
ðhiþ1þhi�1Þ�DX2Xiðhiþ1�hi�1Þ
2þm2ðDXÞ2
135
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 max-
imum 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 ef-
ficiency 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 pro-
files 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 ef-
ficiency 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 the variable 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 and different 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
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;
qf ¼ gf Ahhb
It is clear from this equation that the fin surface area re-
quired to transfer a specific amount of heat under certain
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
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 ro/ri = 1
136
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.
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 di-
mensionless 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.
Table 3. Comparison of the fin
efficiency for straight fins Profile n Gardener [9] Present Difference %
m =1
Rectangular 1/2 0.762 0.7790 2.18
Convex Parabolic 1/3 0.700 0.7540 7.162
Triangular 0 0.735 0.7235 –1.548
Concave Parabolic ±¥ 0.618 0.6537 5.492
m = 2
Rectangular 1/2 0.484 0.5190 6.743
Convex Parabolic 1/3 0.458 0.4970 7.868
Triangular 0 0.432 0.4750 9.136
Concave Parabolic ±¥ 0.389 0.4430 12.099
m = 3
Rectangular 1/2 0.332 0.3786 12.414
Convex Parabolic 1/3 0.316 0.3673 14.040
Triangular 0 0.300 0.3580 16.201
Concave Parabolic ±¥ 0.279 0.3468 6.788
m = 4
Rectangular 1/2 0.250 0.3050 18.090
Convex Parabolic 1/3 0.242 0.2999 19.273
Triangular 0 0.232 0.2961 21.816
Concave Parabolic ±¥ 0.219 0.2943 25.609
m = 5
Rectangular 1/2 0.200 0.2622 31.1
Convex Parabolic 1/3 0.189 0.2608 27.47
Triangular 0 – 0.2606 –
Concave Parabolic ±¥ 0.179 0.2629 31.95
Table 4. Comparison of the fin
efficiency for pin fins Profile n Gardener [9] Present Difference %
m =1
Constant diameter 1/2 0.6280 0.6573 4.4576
Convex Parabolic 0 0.7180 0.7340 2.1798
Conical –1 0.7780 0.7780 0
Concave Parabolic ±¥ 0.8421 0.8540 1.3946
m =2
Constant diameter 1/2 0.3510 0.4059 13.378
Convex Parabolic 0 0.4536 0.5007 9.4060
Conical –1 0.5288 0.5542 4.5830
Concave Parabolic ±¥ 0.6368 0.6699 4.9410
m = 3
Constant diameter 1/2 0.2356 0.3006 21.623
Convex Parabolic 0 0.320 0.3811 16.032
Conical –1 0.389 0.4290 9.3240
Concave Parabolic ±¥ 0.500 0.5461 8.4460
m = 4
Constant diameter 1/2 0.1770 0.2538 30.2600
Convex Parabolic 0 0.2470 0.3187 22.4976
Conical –1 – 0.3587 –
Concave Parabolic ±¥ 0.4105 0.4642 11.5683
m = 5
Constant diameter 1/2 0.1420 0.2302 38.3145
Convex Parabolic 0 0.2000 0.2847 29.7506
Conical –1 – 0.3172 –
Concave Parabolic ±¥ 0.3470 0.4111 15.5923
137
4
Conclusion
Heat transfer from extended surfaces subject to locally
variable heat transfer coefficient has been studied. The
local heat transfer coefficient as function of the local
temperature has been obtained using the available corre-
lations of natural convection for each pertinent extended
surface considered. The results showed that the assump-
tion of constant heat transfer coefficient along the fin in
such cases leads to a significant underestimation of the fin
efficiency. The deviation between the fin efficiency calcu-
lated based on constant heat transfer coefficient and that
calculated based on variable heat transfer coefficient in-
creases with the dimensionless parameter m. The use of
the present results by the designer of heat transfer
equipment that involve extended surface subject to natural
convection heat transfer mode would result in a con-
siderable reduction in the extended surface area and hence
a significant reduction in the weight and size of the heat
transfer equipment.
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Table 5. Comparison of the fin efficiency for annular fins with
rectangular profile
Profile Gardener [9] Present Difference %
Radius Ratio m =1
1 0.7615 0.7792 2.274
2 0.6920 0.7243 4.460
3 0.6420 0.6883 6.731
4 0.6105 0.6622 7.802
m =2
1 0.4820 0.5190 7.130
2 0.3915 0.4452 12.069
3 0.3320 0.4015 17.319
4 0.3115 0.3714 16.119
m = 3
1 0.3310 0.3787 12.585
2 0.2560 0.3132 18.263
3 0.2142 0.2751 22.129
4 0.1895 0.2493 23.985
m = 4
1 0.2498 0.3050 18.008
2 0.1873 0.2485 24.633
3 0.1560 0.2156 27.631
4 0.1316 0.1934 31.932
m = 5
1 0.2000 0.2622 23.719
2 0.1445 0.2133 32.265
3 0.1189 0.1843 35.469
4 0.1000 0.1644 39.167
138
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