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Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
1
Aproximate Analytic Analysis Of Annular Fins With Uniform Thickness By Way Of The Mean Value Theorem For Integration That Avoids Modified
Bessel Functions
Antonio Acosta−Iborra1, Antonio Campo2
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 temperature dT dr in the governing quasi one-dimensional 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 1 r , viewed as an auxiliary function in the annular
fin domain extending from the inner radius 1r to the outer radius2r . 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 profile embodying four modified Bessel functions. Surely, instructors and students in heat transfer courses will be the beneficiaries of this finding because of the easiness in calculating the temperatures and heat transfer rates for realistic combinations of the two controlling parameters: the normalized radii ratio and the thermo-geometric fin parameter. Keywords - annular fin with uniform thickness; mean value theorem for integration; approximate temperature distribution of simple exponential form.
Nomenclature
Bi transversal Biot number, k
ht
c normalized radii ratio, 2
1
r
r
EBi enlarged transversal Biot number, 2
2 1r rhtEBi
k t
− =
Eη relative error for the fin efficiency η
1Departamento de Ingenieria Termica y de Fluidos, Universidad Carlos III de Madrid, Madrid, Spain. aacosta@ing.uc3m.es
2 Departament of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX. USA. antonio.campo@utsa.edu Note. The manuscript for this paper was submitted for review and possible publication on November 9th, 2012; accepted on February 25th, 2013. This paper is part of the Latin American and Caribbean Journal of Engineering Education, Vol. 7, No. 1, 2013. © LACCEI, ISSN 1935-0295.
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
2
tE relative error for the dimensionless tip temperature (1)θ
h mean convection coefficient ……………………………………………… W m-2 K-1
vI modified Bessel function of first kind and order v
k thermal conductivity ……………………………………………………….. W m-1 K-1
vK modified Bessel function of second kind and order v
L length, 2 1r r− ……………………………………………………………….. m
MR mean value of the auxiliary function f (R) = R
1 in [ c , 1]
Q actual heat transfer ………………………………………………………….. W
iQ ideal heat transfer ……………………………………………………………. W
r radial coordinate ……………………………………………….…………...... m
1r inner radius ……………………………………………………..……………. m
2r outer radius ……………………………………………………..……………. m
R normalized radial coordinate, 2r
r
S exposed surface………………………………………………………..…. m2
t semi−thickness …………………..…………………………………………… m
T temperature …………………………………………………………………... K
bT base temperature …………………………………………………..……..…... K
Tf fluid temperature ………………………………………………..….………… K
Greek letters
β2 thermo−geometric parameter,
kt
h…………………....………………..……….. m-2
γ dimensionless group, c−1
ξ
η fin efficiency or dimensionless heat transfer, iQ
Q
θ normalized dimensionless temperature, fb
f
TT
TT
−−
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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1,2λ roots of the auxiliary equation (13)
ξ dimensionless thermo-geometric parameter, kt
hL
Subscripts
b base
i ideal
f fluid
t tip
1. INTRODUCTION
One traditional 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 of the body may be morph in the
form of spines, straight fins or annular fins with
various cross-sections. Conceptually, the
problem of determining the total heat flow in a
fin bundle attached to a solid body 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 on heat transfer skip the
mathematical analysis and present only the fin
efficiency diagram to facilitate 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 in-depth 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,
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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it is worth adding that the fin efficiency diagram
is included in [4], but not in [1-3]. From a
historical perspective, several exact solutions to
the 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]. This collection of exact
solutions is based upon the standard
assumptions of quasi one-dimensional
conduction in the radial direction of the annular
fin.
Under the prevalent 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 r1 that multiplies the first order
temperature derivative drdT . The
homogeneous version of the differential
equation is named the modified Bessel equation
of zero order, wherein the variable coefficient
r1 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 r1
by an approximate constant coefficient. Invoking
the mean value theorem for integration, one
viable avenue is to substitute r1 , viewed as an
auxiliary function in the proper fin domain
[ 1r , 2r ] by the mean value of the function. A
beneficial consequence of this approach is that
the transformed quasi one-dimensional fin
equation now holds constant coefficients. Herein,
the two controlling parameters are the
normalized radii ratio and the thermo-geometric
parameter.
It is envisioned that the analytic
approximate 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, such
as (*)vI and (*)vK
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
thickness2t , inner radius 1r and outer radius 2r .
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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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 transversal temperature gradients; negligible
heat transfer at the outermost fin section (i.e.,
adiabatic fin tip); and null heat sources or sinks.
Accordingly, the governing quasi one-
dimensional fin equation framed in cylindrical
coordinates is
21)( rrrin0 = TT rkt
h
dr
dTr
dr
d f ≤≤−−
(1)
or in an alternate expanded form
where tkh=2β is called the thermo-geometric
parameter [1]. As far as the classification of Eq.
(2) is concerned, it is a non-homogeneous
differential equation of second order with
variable coefficients, wherein the variable
coefficient r1 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
Fundamentally speaking, when a body
surface is extended by a protruding fin of any
shape, the external convective resistance Rc
decreases because the surface area is increased,
but on the other hand the internal conductive
resistance Rk 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 to a surrounding fluid
can be carried out directly in two ways:
(1) differentiating T(r) at the fin base r = r1:
dr
rdTrtk
dr
rdTkAQ b
)(4
)( 11
11 π−=−=
(4a)
or (2) integrating T(r) over the fin surface S:
∫∫ −=−= 2
1
)(4)(2
r
r fS f drrTThdSTThQ π
(4b)
212
2)( rrrin0 = TT
dr
dT
r
1 +
dr
Td f
2
≤≤−− β
(2)
0 = dr
rdT and T = rT b
)()( 2
1
(3a, 3b)
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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Clearly, Eq. (4a) is easier to implement.
Upon introducing the normalized
dimensionless temperature ( ) ( )fbf TTTT −− and
the normalized dimensionless radial
coordinate 2rrR = , the parameter 21 rrc =
emerges as the normalized radii ratio. Thereby,
Eq. (2) is transformed to
2
21
(1 )
2
2
1 dd + = 0 in c Rd R dR cR
θ θ ξ θ− ≤ ≤−
(5)
along with the boundary conditions
(1)( )
d c = 1 and = 0
dR
θθ
(6a,6b)
Eq. (5) is named the modified Bessel equation of
zero order (Polyanin and Zaitsev [9]).
Despite that the transversal Biot number khtBi = is the natural reference parameter in fin heat transfer analysis
2
2 1r rhtEBi
k t
− =
(or 2ξ for short) seems to be a better parameter
for the annular fin of uniform thickness under
study. Herein, the fin slenderness ratio is defined
as the length 2 1L r r= − divided by the semi-
thickness t.
Conversely, the heat transfer Q from any
fin can be estimated indirectly using the concept
of dimensionless heat transfer or fin efficiency
iQQ=η as proposed by Gardner [8]. In here,
iQ is an ideal heat transfer from an identical
reference fin maintained at the base temperature
Tb (equivalent to an ideal material with infinite
thermal conductivity k →∞ ). For the specific
case of annular fins of uniform thickness, the
ideal heat transfer is given by
)()(2 21
22 fbi TThrrQ −−= π
As noted before the computation of η may be
carried out in two different ways:
(1) by differentiation of ( )Rθ at the fin
base leading to
( )1 2
2 11 ( )
1
c c d c =
c dR
θηξ
− − +
or (2) by integration of ( )Rθ over the fin
surface, resulting in
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
7
2 2
2( )
11
c = R R dRc
η θ ∫ −
(7b)
At the end, the magnitude of heat transfer Q is
obtainable with the expression
[ ])()(2 21
22 fbi TThrrQQ −−== πηη
(7c)
where η comes from either 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 ( )Rθ as found in [3]:
1 0 0 1
1 0 0 1
) (I ( )K ( R I ( R)K ) (R) =
I ( )K ( c) I ( c)K ( )
γ γ γ γθγ γ γ γ
++
(8)
where (*)vI and (*)vK are the modified Bessel
functions of first and second kind, both of order
v and γ stands for the dimensionless group
)1/( c−ξ (see Nomenclature).
On the other hand, the safe-touch
temperature of hot bodies is an important issue
for the safety of technical personnel working 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 )r(T 2 is considered by design
engineers as a “parameter of relevance”.
Therefore, the exact dimensionless tip
temperature θ(1) follows from Eq. (8),
1 0 0 1
1 0 0 1
( ) ( ) ( ) ( )(1)
( ) ( ) ( ) ( )
I K I K
I K c I c K
γ γ γ γθγ γ γ γ
+=+
(9)
Further, the two η −avenues in Eqs. (7a)
and (7b) coalesce into the exact fin efficiency
1 1 1 1
1 0 0 1
1 2
1
I ( )K ( c) I ( c) K ( )c =
c I ( )K ( c) + I ( c) K ( )
γ γ γ γηξ γ γ γ γ
− +
(10)
Numerical evaluations of the sequence of Eqs.
(8)–(10) are elaborate and time-consuming, even
with contemporary symbolic algebra codes, like
Mathematica, Maple and Matlab.
4. Approximate Calculation Procedure
The idea behind the mean value theorem
for integration boils down to replacing an
auxiliary function in a certain closed interval by
an equivalent representative number. From
Differential Calculus (Stewart [11]), the mean
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
8
value theorem for integration 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:
∫−=
b
adxxf
abf )(
1
Why not trying to extend this idea to
replace a variable coefficient appearing in a
differential equation by an equivalent number so
that the variable coefficient is channeled
through a number i.e., a constant coefficient?
The derived benefit is self-evident, because
ordinary differential equations with constant
coefficients are easier to solve than those with
variable coefficients.
The disturbing variable coefficient R1 in
Eq. (5) may be viewed as a continuous function
RRf 1)( = outlining a hyperbola segment on
the closed R–interval [c, 1]. Upon applying the
mean value theorem for integration to )(Rf , the
end result is
c
cdR
RcM
cR −−=
−= ∫ 1
ln11
1 1
(11)
Next, replacing the variable coefficient R1 with
the constant coefficient MR in Eq. (5), this
equation is converted to
1)1( 2
2
≤≤−
− Rcin0 = c
dR
d M+
Rdd R2
2
θξθθ
(12)
That 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])
1 21 2( ) R RR C e C eλ λθ = +
(7)
The two distinct roots of the auxiliary equation
are
2
)1(4
,2
22
21
cMM RR −
+±−=
ξ
λλ
(8)
The combination of Eqs. (12)–(14) culminates in
the particular solution
( ) ( )
( )
1 2
1 2
1 12 1
1 ( 1)2 1
( )R R
c c
e eR
e e
λ λ
λ λ
λ λθλ λ
− −
− −
−=−
(9)
In other words, the approximate dimensionless
temperature profile.
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
9
It is reasonable to pause for a moment at
this juncture to contrast the complex structure of
the exact temperature profile in Eq. (8) involving
four modified Bessel functions I0 (*), I1 (*), K0 (*)
and K1 (*), against the simple structure of the
approximate temperature profile in Eq. (9) with
four exponential functions. Consequently, Eq.
(15) being of ultra compact form, constitutes the
centerpiece of the present work.
Moreover, by virtue of Eq. (9), the
approximate dimensionless tip temperature θ(1)
is
computed from
1 2
2 1( 1) ( 1)
2 1
(1)c ce eλ λλ λθ
λ λ− −
−=−
(16)
Turning our attention to the fin efficiency
diagram for annular fins of uniform cross-section
in [4], the family of η -curves is parameterized
by the radii ratio 1
2
r
r varying from 1 (straight fin
of uniform thickness) up to a maximum of 5. It
should be mentioned that this is the format used
for the fin efficiency diagram in the textbook by
Chapman [13]. In terms of the normalized radii
ratio c, the large span 1 ≤ 1
2
r
r ≤ 5 is analogous to
the reduced c-interval 0.2 ≤ c ≤ 1. In this sense,
realistic numbers for the emerging MR in terms of
practical c values are listed in Table 1.
Returning to Eq. (15) momentarily, the
two approximate fin efficiencies can be
generated through the tandem of Eqs. (7a) and
(7b), i.e.,
(1) by differentiation of ( )Rθ at the fin
base:
( ) ( )
( ) ( )
1 2
1 2
1 1
1 2 1 12 1
2
1
c c
c c
c e e
c e e
λ λ
λ ληλ λ
− −
− −
−= − −
or (2) by integration of ( )Rθ over the fin
surface:
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
10
( ) ( ) ( ) ( )
( ) ( )
1 2
1 2
1 13 32 1 1 1 2 2
2 2 1 12 21 2 2 1
1 1 1 12
1
c c
c c
c e c e
c e e
λ λ
λ λ
λ λ λ λ λ λη
λ λ λ λ
− −
− −
− + − − − + − = − −
(17b)
Incidentally, it should be expected that the
differentiation approach in the short Eq. (17a)
could produce numbers that are slightly 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 the governing fin equation
(12) exactly. 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
( )Rθ 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
in [4] reveals that the smallest radii ratio is
0.2c = (corresponding to2 15r r= ). This
particular radii ratio 0.2 was deliberately selected
here as a critical test case in order to analyze the
totality of the numerical results.
The exact dimensionless temperature
profiles calculated with Eq. (8) are compared
against the approximate dimensionless
temperature profiles in Eq. (9) deduced in this
work in Fig. 2. Combined with 0.2c = , three
dimensionless temperature curves for a small ξ
= 0.5, an intermediate ξ = 2 and a large ξ = 10
are plotted in the figure. The comparison for the
three ξ values reveals satisfactory quality
between the approximate and exact
dimensionless temperature profiles. Interestingly,
the approximate dimensionless temperature
profiles do not degenerate for the large ξ = 10
because Eq. (9) is physically consistent. In other
words, the approximate dimensionless
temperature profile tends rapidly to zero,
whenever ξ → ∞
Using the approximate dimensionless
analytic temperature of Eq. (15) for the tandem
of the smallest c = 0.2 and the largest 0.8, the fin
efficiencies estimated via the differential
approach 1η in Eq. (16.a) and the integral
approach 2η in Eq. (11.b) are listed in Table 2 for
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
11
the trio of ξ = 0.5, 1.5 and 3. The exact fin
efficiencies η computed from Eq. (10) range
from 0.1720 for the pair c = 0.2 and ξ = 3 to
0.916 for the pair c = 0.8 and ξ = 0.5. In Table 2,
the relative error in the efficiency Eη is defined
as:
.approx exact
exact
Eη
η ηη
−=
(18)
First, for the smallest c = 0.2, the relative errors
Eη for the differential-based 1η vary from -1.62e-
1 when ξ = 3 to -1.77e-1 when ξ = 0.5. Second,
for the largest c = 0.8, the relative errors Eη for
the differential-based 1η vary from -2.88e-3
when ξ = 3 to -4.08e-1 when ξ = 0.5. Third, for
the smallest c = 0.2, the relative errors Eη for the
integral-based 2η vary from 3.5e-2 when ξ = 3
to -4.37e-3 when ξ = 0.5. Fourth, for the largest
c = 0.8, the relative errors Eη for the integral-
based 2η vary from -8.43e-4 when ξ = 3 to
4.08e-5 when ξ = 0.5. As may be seen, all
relative errors Eη are insignificant. Moreover, the
fin efficiency conveyed through the integral-
based η 2 furnishes more accurate results than the
alternate derivative-based η 1. Again, 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 fin
efficiency results based on the integral approach
and the derivative approach diminish for large
values of c . In fact, in the limiting case
corresponding to 1c = , the approximate and the
exact predictions coincide, both magnitudes
collapsing into:
1
tanh( )c
ξηξ= =
This expression can be easily deduced from the
approximate Eqs. (11) taking into account that
the roots confirm that 1,2(1 )cλ ξ− → ± whenever
1c → . It should also be noted that Eq. (13)
stands for the fin efficiency for a longitudinal fin
of uniform thickness [3,4] and same ξ , which is
a logical similitude owing to the null curvature in
the annular fin when c tends to unity and L is
maintained constant.
Figure 3 reveals a perfect matching
between the exact fin efficiencies η varying with
the dimensionless fin parameter ξ utilizing Eq.
(10) and the two approximate fin efficiencies η1
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
12
and η2 for c = 0.4 and 0.8 employing Eqs. (17a)
and (17b). However, for the smallest c = 0.2, η1
deteriorates, whereas η2 being more robust
provides good results.
Fig. 4 depicts the dimensionless tip
temperature as a function of the dimensionless
thermo-geometric parameter ξ and the radii ratio
c. For the smallest c = 0.2 and the largest c = 0.8,
and the trio ξ = 0.5, 1.5 and 3, the exact
dimensionless tip temperatures range from θ(1)
range from 0.879 for the combination c = 0.8 and
ξ = 0.5 to 0.0559 for the combination of c = 0.2
and ξ = 3. The latter having a nearly zero value is
representative of an infinite annular fin.
Also, the relative errors tE for the
dimensionless tip fin temperature calculated with
.(1) (1)
(1)approx exact
texact
Eθ θ
θ−
=
(19)
are listed in Table 3. First, for the smallest c =
0.2, the relative errors tE vary from -8.35e-3
when ξ = 0.5 to -3.65e-1 when ξ = 3. Second,
for the largest c = 0.8, the relative errors tE vary
from -1.50e-5 when ξ = 0.5 to -1.02e-1 when ξ
= 0.5. It is noticeable that all tE are very small.
If c tends to unity while maintaining L
constant, the approximate equation (10) for the
estimation of the dimensionless tip temperature
simplifies to
1
1(1)
cosh( )cθξ= =
In general, the approximate results for both
(1)θ and η deteriorate when the radii ratio c
decreases because the differences between the
constant mean values MR and the dimensionless
variable coefficient 1/R of the descriptive fin
equation are higher.
Finally, in the event that the instructor
decides to maximize the heat transfer from a
single annular fin of uniform thickness, the
required information about the three optimal fin
dimensions r1, r2 and t was made available in
the textbook by Jakob [2] by means of a simple
nomogram and many ears later in a sequence of
articles by Brown [15], Ullmann and Kalman
[16] and Arslanturk [17], all using combination
of figures.
6. Conclusions
In this study on engineering education,
various concepts from courses on calculus,
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
13
ordinary differential equations and heat transfer
have been blended in a unique way. In analyzing
annular fins of uniform thickness, the mean value
theorem for integration is used for simplifying
the descriptive quasi one-dimensional fin
equation, namely the modified Bessel differential
equation. This gives way to the 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 classical exact analytic
temperature profiles relying on four modified
Bessel functions are probably below the level of
inaccuracy introduced by the Murray-Gardner
assumptions.
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, CA and Los Angeles, CA, 1946.
2. Jakob, M., Heat Transfer, Vol. 1, pp. 232-234,
John Wiley, New York, NY, 1949.
3. Mills, A. F., Heat and Mass Transfer, 2nd
edition, 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.
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.
6. Murray, W. M., Heat dissipation through an
annular disk or fin of uniform thickness, Journal
of Applied Mechanics, Transactions of ASME,
Vol. 60, p. A-78, 1938.
7. Carrier, W. H. and Anderson, S. W., The
resistance to heat flow through finned tubing,
Heating, Piping, and Air Conditioning, Vol. 10,
pp. 304-320, 1944.
8. Gardner, K. A., Efficiency of extended
surfaces, Transactions of ASME, Vol. 67, pp.
621-631, 1945.
9. Polyanin, A. D. and Zaitsev, V. F., Handbook
of Exact Solutions for Differential Equations,
CRC Press, Boca Raton, FL, 1995.
10. Arthur, K. and Anderson, A., Too hot to
handle?: An investigation into safe touch
temperatures. In Proceedings of the ASME
International Mechanical Engineering Congress
and Exposition (IMECE), pp. 11-17, Anaheim,
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CA, 2004.
11. Stewart, J., Single Variable Calculus, 3th
edition, Brooks/Cole, Pacific Groove, CA, 2002.
12. Boyce, W. E. and DiPrima R. C., Elementary
Differential Equations and Boundary Value
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, International Journal of Heat and
Mass Transfer, Vol. 8, pp. 655-662, 1965.
16. Ullmann, A. and Kalman, H., Efficiency and
optimized dimensions of annular fins of
different cross-section shapes, International
Journal of Heat and 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.
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
15
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 0.2c = when combined with three different fin parameters ξ .
Fig. 3. Comparison between the approximate and exact fin efficiencies as a function of the dimensionless
fin parameter ξ for different normalized radii ratios c .
Fig. 4. Comparison between the approximate and exact tip temperatures as a function of the dimensionless
fin parameter ξ for different normalized radii ratios c .
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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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 η
Table 3. Comparison of the dimensionless fin tip temperatures θ (1)
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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FIGURE-1
2t
r1
r2
L
r
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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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
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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Table 2
Comparison of the fin efficiencies η
Procedure c ξ Relative error
Eη (%)
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
Latin American and Caribbean Journal of Engineering Education Vol. 7(1), 2013
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Table 3
Comparison of the dimensionless tip temperatures )1(θ
c ξ Relative error tE (%)
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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