APROXIMATE ANALYTIC ANALYSIS OF ANNULAR FINS WITH
UNIFORM THICKNESS BY WAY OF THE MEAN VALUE THEOREM
FOR INTEGRAL TO AVOID MODIFIED BESSEL FUNCTIONS
Antonio Acosta-Iborra*
Departamento de Ingeniería Térmica y de Fluidos,
Universidad Carlos III de Madrid,
Avda. Universidad 30,
Leganés 28911,
Madrid, Spain
Antonio Campo
Department of Mechanical Engineering,
The University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249, USA
* Corresponding author
Phone: (34) 916248465
Fax: (34) 916249430
E-mail: aacosta@ing.uc3m.es
Abstract
In the analysis of annular fins of uniform thickness, the main obstacle is without question,
the variable coefficient 1 r multiplying the first order derivative, dT dr in the quasi onedimensional heat conduction equation. A good-natured manipulation of the problematic variable
coefficient 1 r is the principal objective of the present paper on engineering education.
Specifically, we seek to apply the mean value theorem for integration to the variable coefficient
1 r viewed as an auxiliary function, the proper fin domain extending from the inner radius r1 to the
outer radius r2 . It is demonstrated in a convincing manner that approximate analytic temperature
profiles of good quality are easy to obtain without resorting to the exact analytic temperature
Page-1/26
profile embodying modified Bessel functions. Instructors and students in heat transfer courses may
be the beneficiaries because of the easiness in computing the temperatures and heat transfer rates
for realistic combinations of the two controlling parameters: the normalized radii ratio c and the
thermo-geometric fin parameter.
Keywords: annular fin with uniform thickness; mean value theorem for integrals; simple
approximate temperature solution.
Nomenclature
ht
k
Bi
transversal Biot number,
c
normalized radii ratio,
EBi
enlarged transversal Biot number, EBi 
E
relative error for the fin efficiency 
Et
relative error for the dimensionless tip temperature  (1)
h
mean convection coefficient ………………………………………………
Iv
modified Bessel function of first kind and order v
k
thermal conductivity ………………………………………………………..
Kv
modified Bessel function of second kind and order v
L
length, r2  r1 ………………………………………………………………..
MR
mean value of the auxiliary function f (R) =
Q
actual heat transfer …………………………………………………………..
W
Qi
ideal heat transfer …………………………………………………………….
W
r
radial coordinate ……………………………………………….…………......
m
r1
inner radius ……………………………………………………..…………….
m
r1
r2
ht  r2  r1 


k  t 
2
W m-2 K-1
W m-1 K-1
m
1
in [ c , 1]
R
Page-2/26
r2
outer radius ……………………………………………………..…………….
R
normalized radial coordinate,
S
exposed surface………………………………………………………..….
t
semithickness …………………..……………………………………………
m
T
temperature …………………………………………………………………...
K
Tb
base temperature …………………………………………………..……..…...
K
Tf
fluid temperature ………………………………………………..….…………
K
m
r
r2
m2
Greek letters
h
…………………....………………..………..
kt
β2
thermogeometric parameter,

dimensionless group,
η
fin efficiency or dimensionless heat transfer,
θ
normalized dimensionless temperature,
1,2
roots of the auxiliary equation (13)

dimensionless thermo-geometric parameter, L

fin effectiveness

1 c
Q
Qi
T  Tf
Tb  T f
Subscripts
b
base
i
ideal
f
fluid
t
tip
Page-3/26
h
kt
m-2
1. Introduction
One passive method for augmenting heat transfer between hot solid bodies and
surrounding cold fluids increases the surface area of the solid body in contact with the fluid by
attaching thin strips of material, called extended surfaces or fins. The enlargement in surface area
may be morph in the form of spines, straight fins or annular fins with various cross-sections.
Obviously, the problem of determining the heat flow in a fin bundle requires prior knowledge of
the temperature profile in a single fin.
There are two fin shapes of paramount importance in engineering applications, one is the
straight fin of uniform thickness and the other is the annular fin of uniform thickness. In the great
majority of textbooks on heat transfer, the section devoted to fin heat transfer begins with a
mathematical analysis of the straight fin of uniform thickness that leads to exact expressions in
terms of exponentials or equivalent hyperbolic functions for calculating a) the temperature
profile, b) the heat transfer rate and c) the fin efficiency. However, this is not the case with the
annular fin of uniform thickness where the cross-sectional area and the surface area are functions
of the radial coordinate. In view of the impending difficulty, most textbooks skip the
mathematical analysis and present only the fin efficiency diagram to facilitates the calculation of
the heat transfer rate. However, there are exceptions in the old textbooks by Boelter et al. [1] and
Jakob [2] and the new textbooks by Mills [3] and Incropera and DeWitt [4], who do explain the
mathematical analysis that eventually supplies exact analytic expressions for a) the temperature
profile, b) the heat transfer rate and c) the fin efficiency. Moreover, it is worth adding that the
efficiency diagram is included in [4], but not in [1-3]. From a historical perspective, several exact
solutions to the problem of heat conduction in an annular fin of constant thickness have been
developed by Harper and Brown [5], Murray [6], Carrier and Anderson [7] and Gardner [8].
These exact solutions are based upon the standard assumptions of quasi one-dimensional
conduction in the radial direction.
Under the quasi one-dimensional formulation, the temperature descend along an annular
fin with uniform cross-section is governed by a differential equation of second order with one
variable coefficient 1 r that multiplies the first order derivative dT dr . The homogeneous
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version of the differential equation is named the modified Bessel equation of zero order wherein
the variable coefficient 1 r is troublesome. A review of the heat conduction literature reveals no
previous efforts aimed at solving this modified Bessel equation by means of approximate
analytic procedures.
The present study addresses an elementary analytic avenue for the treatment of annular fins
of uniform thickness in an approximate manner. The central idea is to replace the cumbersome
variable coefficient 1 r by an approximate constant coefficient. Invoking the mean value theorem
for integration, one viable avenue is to substitute 1 r , viewed as an auxiliary function, in the proper
fin domain [ r1 , r2 ] by its mean value. A beneficial consequence of this approach is that the
transformed fin equation holds constant coefficients. Herein, the two controlling parameters are the
normalized radii ratio and the thermo-geometric parameter.
It is envisioned that the simple computational procedure to be delineated in the paper on
engineering education may facilitate the quick determination of approximate analytic temperature
profiles and heat transfer rates for annular fins of uniform thickness without the intervention of
modified Bessel functions.
2. Modeling and Quantities of Engineering Interest
An annular fin of uniform thickness dissipating heat by convection from a round tube or
circular rod to a surrounding fluid is sketched in Fig. 1. The three fin dimensions are: uniform
thickness 2t , inner radius r1 and outer radius r2 . In the modeling, the classical Murray-Gardner
assumptions (Murray [6], Gardner [8]) are adopted: steadiness in heat flow; constant thermal
conductivity k ; uniform heat transfer coefficient h ; unvarying fluid temperature Tf; prescribed fin
base temperature Tb; preponderance of radial temperature gradients over those in the transversal
direction; negligible heat transfer at the outermost fin section (i.e., adiabatic fin tip); and null heat
sources or sinks. Correspondingly, the resulting quasi one-dimensional fin equation framed in
cylindrical coordinates is
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d  dT  h
r
  r (T  T f ) = 0
dr  dr  kt
in r1  r  r2
(1)
or in expanded form
2
d T + 1 dT   2 (T  T ) = 0
f
dr 2 r dr
in r1  r  r2
(2)
where  2  h tk is the thermo-geometric parameter [1]. As far as the classification is concerned,
Eq. (2) is a non-homogeneous differential equation of second order with variable coefficients,
wherein the variable coefficient 1 r is of intricate form.
The proper boundary conditions implying prescribed temperature at the fin base r1 and zero
heat loss at the fin tip r2 are
T (r1 ) = Tb
and
dT (r2 )
=0
dr
(3a, 3b)
Fundamentally speaking, when a body surface is extended by a protruding fin of any
shape, the external convective resistance decreases because the surface area is increased, but on
the other hand the internal conductive resistance increases because heat is conducted through the
fin before being convected to the fluid.
Calculation of the heat transfer from an annular fin of uniform thickness can be carried out
directly in two ways:
(1) differentiating T(r) at the fin base r = r1:
Q1  kAb
dT (r1 )
dT (r1 )
 4 k t r1
dr
dr
(4a)
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or (2) integrating T(r) over the fin surface S:
r2
Q2   h (T  T f ) dS  4 h  (T  T f ) r dr
S
(4b)
r1
Clearly, Eq. (4a) is easier to implement.
Upon introducing the normalized dimensionless temperature T  T f  Tb  T f  and the
normalized dimensionless radial coordinate R  r r2 , the parameter c  r1 r2 emerges as the
normalized radii ratio. Thereby, Eq. (1b) is transformed to
2
2
d  1 d
+

 =0
d R 2 R dR (1  c)2
in c  R  1
(5)
along with the boundary conditions
 (c ) = 1
and
d (1)
=0
dR
(6a,6b)
Eq. (5) is classified as the modified Bessel equation of zero order (Polyanin and Zaitsev [9]).
Despite that the transversal Biot number Bi  ht k is the natural reference parameter in fin
analysis, the enlarged Biot number
ht  r  r 
EBi   2 1 
k  t 
2
(or  2 for short) is another viable parameter. Herein, the fin slenderness ratio is defined as the
length L  r2  r1 divided by the semi-thickness t.
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Conversely, the heat transfer Q from any fin can be estimated indirectly using the concept
of dimensionless heat transfer or fin efficiency   Q Qi as proposed by Gardner [8]. In here, Qi
is an ideal heat transfer from an identical reference fin maintained at the base temperature Tb
(equivalent to a material with infinite thermal conductivity k  ). For the specific case of
annular fins of uniform thickness, the ideal heat transfer is given by
Qi  2 (r2  r1 ) h (Tb  T f )
2
2
As noted before the computation of  may be carried out in two ways:
(1) by differentiation of  ( R ) at the fin base leading to
1 = 
1  2c 1  c   d  (c)


 2  1  c  dR
(7a)
or (2) by integration of  ( R ) over the fin surface, leading to
 2  1
 ( R ) R dR
2  c
 1 c 
2 = 
(7b)
At the end, the magnitude of heat transfer Q is obtainable with
Q   Qi   2 (r2  r1 ) h (Tb  T f )
2
2
(7c)
where η comes from Eq. (7a) or (7b).
3. Exact Calculation Procedure
The exact analytic solution of Eqs. (5) and (6) gives way to the exact dimensionless
temperature profile  ( R ) found in [3]:
Page-8/26
 (R) =
I1(  )K 0 (  R )  I 0 (  R)K1 ( )
I1(  )K 0 (  c)  I 0 (  c)K1(  )
(8)
where I v (*) and Kv (*) are the modified Bessel functions of first and second kind, both of order v
and  stands for the dimensionless group  /(1  c ) (see Nomenclature).
The safe-touch temperature of hot bodies is an important issue for the safety of technical
personnel in plant environments (Arthur and Anderson [10]). In this regard, the fin tips of
annular fins of uniform thickness are prone to be touched accidentally. Because of this, the fin tip
temperature T(r2 ) is considered by design engineers as a “parameter of relevance”. Therefore,
the exact dimensionless tip temperature θ(1) follows from Eq. (8). That is
 (1) 
I1 ( ) K 0 ( )  I 0 ( ) K1 ( )
I1 ( ) K 0 ( c)  I 0 ( c) K1 ( )
(9)
Further, the two  avenues in Eqs. (7a) and (7b) coalesce into the exact fin efficiency
1  2c   I1(  )K1(  c)  I1(  c) K1(  ) 
= 
  1  c   I1(  )K0 (  c) + I 0 (  c) K1(  ) 
(10)
Numerical evaluations of the sequence of equations (8)–(10) are elaborate even with contemporary
symbolic algebra codes, like Mathematica, Maple and Matlab.
4. Approximate Calculation Procedure
The idea behind the mean value theorem for integrals boils down to replacing an
auxiliary function in a certain closed interval by an equivalent number. From Differential
Calculus (Stewart [11]), the mean value theorem for integrals can be stated as follows. Let f(x)
be a continuous function on [a, b] and the mean (or average) value of f (x) is:
Page-9/26
f 
1 b
f ( x) dx
b  a a
Why not extend this idea to replace a variable coefficient appearing in a differential
equation by an equivalent number so that the variable coefficient becomes a number i.e., a
constant coefficient? The derived benefit is self-evident, because differential equations with
constant coefficients are easier to solve than differential equations with variable coefficients.
The disturbing variable coefficient 1 r in Eq. (5) may be viewed as a continuous function
f ( R)  1 R outlining a hyperbola segment on the closed R–interval [c, 1]. Upon applying the
mean value theorem for integrals to f (R ) , the result is
MR 
1 1 dR  ln c

1  c c R
1 c
(11)
Next, replacing the variable coefficient 1 r with the constant coefficient MR in Eq. (5), it is
converted to
2
2
d  + M d  
 =0
R
d R2
dR (1  c)2
in c  R  1
(12)
This is a differential equation of second order with constant coefficients and homogeneous. Hence,
the general solution of Eq. (Error! Reference source not found.) is (Boyce and DiPrima [12]):
 ( R)  C1 e R  C2 e R
1
(7)
2
The two distinct roots of the auxiliary equation are
1 , 2 
 M R  M R2 
4 2
(1  c) 2
(8)
2
Page-10/26
The combination of Eqs. (6), (7), and (14) culminates in the particular solution of Eq. (12). In other
words, the approximate dimensionless temperature profile
2e  R 1  1e  R 1
 ( R) 
2e  c 1  1e ( c 1)
1
2
1
2
(9)
At this juncture, it is reasonable to pause for a moment to contrast the complex structure of the
exact temperature profile in Eq. (8) involving four modified Bessel functions against the simple
exponential structure of the approximate temperature profile in Eq. (9) involving four exponential
functions. Consequently, Eq. (15) being of ultra compact form, constitutes the centerpiece of the
present work.
Additionally, by virtue of Eq. (9), the approximate dimensionless tip temperature θ(1) is
 (1) 
2 e
2  1
 1e ( c 1)
1 ( c 1)
(16)
2
In the fin efficiency diagram for annular fins of uniform cross-section in [4], the family of
 curves is parameterized by the radii ratio
r2
varying from 1 (straight fin of uniform thickness)
r1
up to a maximum of 5. In terms of the normalized radii ratio c, this ample span is synonymous with
the reduced c-interval 0.2 ≤ c ≤ 1. In passing, it should be mentioned that the latter format was
chosen in the fin efficiency diagram in the textbook by Chapman [13]. In this sense, realistic
numbers for the emerging MR in terms of c values of practical significance are listed in Table 1.
Returning to Eq. (15), the two approximate fin efficiencies can be generated through the
tandem of Eqs. (7a) and (7b). That is,
(1) by differentiation of  ( R ) at the fin base:
Page-11/26
1 
 c 1
 c 1
2c  e 1    e 2   


1  c 2  2 e1  c 1  1e2  c 1 
(17a)
or (2) by integration of  ( R ) over the fin surface:
2
2 
1  c2
 23 1  1  1  c1  e1  c 1   13 2  1  1  c2  e2  c 1  
 





1  c 1
2  c 1
2 2

1 2 2 e
 1e





(17b)
It should be expected that the differentiation approach in the short Eq. (17a) could produce
numbers that are different than those related to the integral approach in the large Eq. (17b). The
explanation for this disparity is that the approximate temperature profile in Eq. (9) does not satisfy
exactly the governing fin equation (5). From physical grounds, the heat by conduction entering the
fin at the base and the heat by convection dissipated along the surface of the fin could be unequal.
This is the reason why Arpaci [14] recommended that whenever  ( R ) is approximate, the
integration approach η2 in Eq. (17b) must be preferred over the differentiation approach η1 in Eq.
(17a).
5. Presentation of Results
Inspection of the fin efficiency diagram found in [4] reveals that the smallest radii ratio is
c  0.2 (corresponding to r2  5r1 ). This radii ratio was deliberately selected here as a critical test
case in order to analyze the totality of the results.
The exact dimensionless temperature profiles calculated with Eq. (4) are compared against
the approximate temperature profiles in Eq. (9) deduced in this work in Fig. 2. Combined with
c  0.2 , three temperature curves for a small  = 0.5, an intermediate  = 2 and a large  = 10
are plotted in the figure. The comparison for the three  values reveals satisfactory quality
between the approximate and exact temperature profiles. Interestingly, the approximate
temperature profiles do not degenerate for the large  = 10 because Eq. (9) is physically
consistent. In other words, the approximate temperature profile tends rapidly to zero whenever
Page-12/26
 .
Using the approximate analytic temperature of Eq. (15) for both the smallest c = 0.2 and
the largest 0.8, the fin efficiencies estimated via the differential approach 1 in Eq. (16.a) and the
integral approach  2 in Eq. (11.b) are listed in Table 2 for the trio  = 0.5, 1.5 and 3. The exact fin
efficiencies  computed from Eq. (10) range from 0.1720 for the pair c = 0.2 and  = 3 to 0.9160
for the pair c = 0.8 and  = 0.5. In Table 2, the relative error in the efficiency E is defined as:
E 
approx.  exact
exact
(18)
First, for the smallest c = 0.2, the relative errors E for the differential-based 1 vary from -1.62e-1
when  = 3 to -1.77e-1 when  = 0.5. Second, for the largest c = 0.8, the relative errors E for the
differential-based 1 vary from -2.88e-3 when  = 3 to -4.08e-1 when  = 0.5. Third, for the
smallest c = 0.2, the relative errors E for the integral-based  2 vary from 3.5e-2 when  = 3 to 4.37e-3 when  = 0.5. Fourth, for the largest c = 0.8, the relative errors E for the integral-based
 2 vary from -8.43e-4 when  = 3 to 4.08e-5 when  = 0.5. As may be seen, all relative errors
E are insignificant. Moreover, the fin efficiency conveyed through the integral-based  furnishes
more accurate results than the alternate derivative-based  . This statement is in harmony with the
recommendations made in [Error! Reference source not found.].
As the numbers listed in Table 2 demonstrate, the differences between the efficiency
results based on the integral and derivative approach diminish for large values of c . In fact, in the
limiting case corresponding to c  1 , the approximate and the exact predictions coincide, both
collapsing to:
c 1 
tanh( )
(13)

This expression can be easily deduced from the approximate Eqs. (11) taking into account that the
Page-13/26
roots confirm that 1,2 (1  c)   whenever c  1 . It should be noted that Eq. (13) is the fin
efficiency for a longitudinal fin of uniform thickness [3,4] and same  , which is a logical
similitude owing to the null curvature in the annular fin when c tends to unity and L is maintained
constant.
Fig. 4 depicts the tip temperature as a function of the dimensionless thermo-geometric
parameter  and the radii ratio c. For the smallest c = 0.2 and the largest 0.8, and the trio  = 0.5,
1.5 and 3, the exact dimensionless tip temperatures range from (1) range from 0.879 for the
combination c = 0.8 and  = 0.5 to 0.0559 for the combination of c = 0.2 and  = 3. The latter
having a nearly zero value is representative of an infinite annular fin.
As before, the relative errors Et for the dimensionless tip fin temperature calculated with
Et 
 (1)approx.   (1)exact
 (1)exact
(19)
are listed in Table 3. First, for the smallest c = 0.2, the relative errors Et vary from -8.35e-3 when
 = 0.5 to -3.65e-1 when  = 3. Second, for the largest c = 0.8, the relative errors Et vary from 1.50e-5 when  = 0.5 to -4.08e-1 when  = 0.5. All Et are very small.
If c tends to unity while maintaining L constant, the approximate equation (10) for the tip
temperature simplifies to
 c 1 (1) 
1
cosh( )
(15)
In general, the approximate results for both  (1) and η deteriorate when the radii ratio c
decreases because the differences between the constant mean values MR
coefficient 1/R of the descriptive fin equation are higher.
Page-14/26
and the variable
Finally, in the event that the instructor or design engineer decides to maximize the heat
transfer from a single annular fin of uniform thickness, pertinent information about the three
optimal fin dimensions r1, r2 and t was first published in the textbook by Jakob [2] and years later
in the articles by Brown [15], Ullmann and Kalman [16] and Arslanturk [17].
6. Conclusions
In this study on engineering education, concepts from courses on calculus, differential
equations and heat transfer have been blended in a unique way. In calculating the performance of
annular fins of uniform thickness, the use of the mean value theorem for simplifying the fin
equation, i.e., the modified Bessel differential equation, gives way to approximate temperature
solutions endowed with an unsurpassed combination of accuracy and easiness. Differences
between the analytic temperature approximations developed in the present work and the exact
analytic temperature profiles relying on Bessel functions are probably below the level of
inaccuracy introduced by the Murray-Gardner assumptions on both exact and approximate
temperatures.
References
1. Boelter, L. M. K., Cherry, V. H., Johnson, H. A. and Martinelli, R. C., Heat Transfer Notes,
pp. IIB18-IIB19, University of California Press, Berkeley and Los Angeles, CA, 1946.
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3. Mills, A. F., Heat and Mass Transfer, 2nd ed., pp. 90-93, Prentice-Hall, Upper Saddle River, NJ,
1999.
4. Incropera, F.P. and DeWitt, D.P., Introduction to Heat Transfer, 4th edition, pp. 139-140, John
Wiley, New York, NY, 2002.
Page-15/26
5. Harper, D. R. and Brown, W. B., Mathematical Equations for Heat Conduction in the Fins of
Air-Cooled Engines, NACA Report No. 158, 1922.
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Heating, Piping, and Air Conditioning, Vol. 10, 1944, pp. 304-320.
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CRC Press, Boca Raton, FL, 1995.
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Problems, 7th edition, John Wiley, New York, NY, 2001.
13. Chapman, A. J., Fundamentals of Heat Transfer, 5th edition, MacMillan, New York, NY, 1987.
14. Arpaci, V., Conduction Heat Transfer, Addison-Wesley, Reading, MA, 1966.
15. Brown, A., Optimum dimensions of uniform annular fins, Int. J. Heat Mass Transfer, Vol. 8,
pp. 655-662, 1965.
16. Ullmann, A. and Kalman, H., Efficiency and optimized dimensions of annular fins of
Page-16/26
different cross-section shapes, Int. J. Heat Mass Transfer, Vol. 32, pp. 1105-1110, 1989.
17. Arslanturk, C., Simple correlation equations for optimum design of annular fins with
uniform thickness, Applied Thermal Engineering, Vol. 25, pp. 2463-2468, 2005.
Page-17/26
List of Figures:
Fig. 1. Sketch of an annular fin of uniform thickness
Fig. 2. Comparison between the approximate and exact dimensionless temperature profiles for a
fixed normalized radii ratio c  0.2 combined with three different fin parameters  .
Fig. 3. Comparison between the approximate and exact fin efficiencies as a function of the
dimensionless fin parameter  for different radii ratios c .
Fig. 4. Comparison between the approximate and exact tip temperatures as a function of the
dimensionless fin parameter  for different radii ratios c .
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List of Tables:
Table 1. Functional mean MR in terms of the normalized radii ratio c
Table 2. Comparison of the fin efficiencies 
Table 3. Comparison of the dimensionless fin tip temperatures θ (1)
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FIGURE-1
r
2t
r1
L
r2
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FIGURE-2
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FIGURE-3
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FIGURE-4
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Table 1
Functional mean MR for
typical radii ratios c
c
MR
0.2
2.012
0.4
1.527
0.6
1.277
0.8
1.116
1.0
1.000
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Table 2
Comparison of the fin efficiencies 
Procedure
c

Relative
error
E (%)
Exact
derivative
0.2
3
-1.62e-1
0.1720
integral
0.2
3
3.50e-2
0.1720
derivative
0.2
1.5
-1.83e-1
0.4020
integral
0.2
1.5
-1.25e-3
0.4020
derivative
0.2
0.5
-1.77e-1
0.8470
integral
0.2
0.5
-4.37e-3
0.8470
derivative
0.8
3
-2.88e-3
0.3068
integral
0.8
3
8.43e-4
0.3068
derivative
0.8
1.5
-3.69e-3
0.5760
integral
0.8
1.5
3.34e-4
0.5760
derivative
0.8
0.5
-4.08e-3
0.9160
integral
0.8
0.5
4.08e-5
0.9160
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Table 3
Comparison of the dimensionless tip temperatures  (1)
c

Relative
error Et
Exact
(%)
0.2
3
-3.65e-2
0.0559
0.2
1.5
-3.04e-2
0.2918
0.2
0.5
-8.35e-3
0.8159
0.8
3
-1.02e-4
0.0921
0.8
1.5
-7.18e-5
0.4061
0.8
0.5
-1.50e-5
0.8790
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