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Hybrid differential transformation and finite difference method to annular fin
with temperature-dependent thermal conductivity
Huan-Sen Peng, Chieh-Li Chen
Department of Aeronautics and Astronautics, National Cheng Kung University Tainan 70101, Taiwan
a r t i c l e i n f o
Article history:Received 5 September 2010
Received in revised form 14 January 2011
Accepted 14 January 2011
Available online 25 February 2011
Keywords:
Differential transformation
Finite difference
Annular fin
Heat transfer
a b s t r a c t
A hybrid numerical technique which combines the differential transformation and finite differencemethod is utilized to investigate the annular fin with temperature-dependent thermal conductivity.
The exposed surfaces of the fin dissipate heat to the surroundings by convection and radiation. The influ-
ences of the convective heat transfer coefficient, absorptivity, emissivity and thermal conductivity
parameter on the temperature distribution are examined. The results show that the convective heat
transfer plays a dominant role for heat dissipation under the convectionradiation condition. The opti-
mum radii ratio of fin which maximizes the heat transfer rate and fin efficiency is also discussed.
2011 Elsevier Ltd. All rights reserved.
1. Introduction
In a variety of engineering applications, extended surfaces are
frequently adopted to enhance the rate of heat dissipation between
the system and the surroundings. The heat transfer mechanism of
fin is to conduct heat from heat source to the fin surface via
conduction, and then dissipate heat to the surrounding fluid via
convection, radiation, or simultaneous convectionradiation. In or-
der to design a practical fin, it is necessary to realize a fins dynamic
temperature response. In the case of constant thermal conductiv-
ity, the analytical solution can be easily obtained. In fact, a consid-
erable amount of research has been conducted into the variable
thermal parameters which associated with fins operating in practi-
cal situations [1]. In such case, the governing equation of fin will be
nonlinear and a numerical treatment-with suitable algorithms.
Yu and Chen [2] proposed the Taylor transformation and finite-
difference approximation to analyze the nonlinear transient heat
transfer problem of the rectangular profile annular fin, where con-vectionradiation fin tip and step temperature change occurring in
fin base were considered. The effects of the heat transfer from the
fin tip to the surroundings, the absorptivity and the emissivity
were discussed. The results demonstrated that the Taylor transfor-
mation is a useful technique to the solution of nonlinear fin
problem. Yang and Chu [3] studied the transient coupled thermo-
elasticity of an annular fin with its base suddenly subjected to a
heat flux of a decayed exponential function of time. The thermo-
mechanical coupling effect was taken into account in the govern-
ing equations of heat conduction. The transient distributions of
temperature increments and thermal stresses in the real domain
were presented numerically. The results showed that the coupling
effect in the annular fin can reduce the heat transfer rates and
cause a lag in the increments of the temperature. Chiu and Chen
[4] used Adomians double decomposition method to solve the
conductionconvectionradiation heat transfer equation of the
circular fins subjected to the nonlinear boundary conditions. The
results indicated that the thermal conductivity dependence on
the temperature should be considered for the convectionradiation
or pure convection conditions. And it is reasonable to assume the
thermal conductivity constant under the pure radiation heat trans-
fer if the temperature difference between the base and the ambient
is not significant. A convectiveradiative longitudinal fin with var-
iable thermal conductivity was also investigated by Chiu and Chen
[5] using the Adomian decomposition method. Different heat dissi-
pation mechanisms were considered, namely pure convection,
pure radiation, and simultaneous convection and radiation. The re-sults showed that the Adomian decomposition method is an effec-
tive and efficient tool in the solution of nonlinear problems
associated with complex conditions. The optimization of annular
fins with non-symmetric boundary conditions was investigated
by Arslanturk [6], where the two-dimensional heat diffusion equa-
tion was solved analytically for temperature distribution and heat
transfer rate. Yang [7] proposed a sequential method to estimate
the periodic boundary conditions on the non-Fourier fin problem.
The inverse solution at each time step is solved by a modified New-
tonRaphson method. The results showed that their method is able
to find both direct and inverse solution of the non-Fourier fin under
the periodic thermal conditions. Naphon [8] investigated the heat
0017-9310/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2011.02.019
Corresponding author. Tel.: +886 6 2757575x63678; fax: +886 6 2389940.
E-mail address: [email protected](C.-L. Chen).
International Journal of Heat and Mass Transfer 54 (2011) 24272433
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.02.019mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.02.019http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.02.019mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.02.0197/28/2019 1-s2.0-S0017931011000743-main
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transfer characteristics of the annular fin under dry-surface, par-
tially wet-surface, and fully wet-surface conditions. The theoretical
results of the heat transfer characteristics and the efficiency of the
annular fins were presented. The effects of inlet condition of work-
ing fluids and the fin dimensions were examined. The results
showed that the various parameters have significant effect on the
temperature distribution. Chen et al. [9] utilized the conjugate gra-
dient method based on an inverse algorithm to estimate the con-
vection heat transfer coefficient of an annular fin. Results showed
that the technique proposed herein can accurately estimate the
convection heat transfer coefficient, temperature distributions
and thermal stress distribution for all the cases considered in this
study. Joneidi et al. [10] used the differential transformation meth-
od to investigate fin efficiency of convective straight fins with tem-perature-dependent thermal conductivity. They analyzed the
effects of thermo-geometric fin parameter and thermal conductiv-
ity parameter.
The modelling of the heat transfer process of the fin reduce the
experimental cost and gives insight into the process. In this study,
the differential transformation method is employed to solve the
heat transfer equation. This method was first introduced by Zhou
[11] to solve linear and non-linear initial value problems of electri-
cal circuits. It has been applied to various applications such as
vibration problems [12], StrumLiouville problem [13], advec-
tivedispersive transport problem [14], nonlinear or hyperbolic
heat conduction [15], two-boundary-value problems [16], etc. In
this paper, a hybrid numerical technique which combines the dif-
ferential transformation and finite difference approximation is em-
ployed to study the fins heat transfer problem. The influences of
the convective heat transfer coefficient (ha), absorptivity (a), emis-sivity (e) and thermal conductivity parameter (b) on the tempera-ture distribution are examined. The validations of numerical
results are also demonstrated.
2. Differential transformation method
Differential transformation is one of the useful methods to solve
the ordinary/partial differential equations with high convergent
rate and accuracy of calculation. The basic concept of the differen-
tial transformation is available in the literatures [17,18].
By definition, the one-dimensional differential transformation
of a function x(t) can be introduced as follows. If x(t) is analyticin the time domain T, let
/t; k dkxt
dtk
8t2 T; 1
where k belongs to the set of nonnegative integer denoted as the K
domain. For t= ti, /(t, k) = /(ti,k)
Xik /ti; k dkxt
dtk
" #tti
8k 2 K; 2
where Xi(k) is called the spectrum of x(t) at t= ti in the domain. If
x(t) is analytic then x(t) can be represented as
xt
X1
k0
t tik
k!Xk: 3
The above equation is known as the inverse transformation of X(k).
If X(k) is defined as
Xk Mkdkqtxt
dtk
" #tt0
; where k 0;1;2;3; . . . ; 1 4
Then the function x(t) can be described as
xt 1
qt
X1k0
t t0k
k!
Xk
Mk; 5
where M(k)0, q(t)0. M(k) is called the weighting factor and q(t) is
regarded as a kernel corresponding to x(t). If M(k) = 1 and q(t) = 1
then Eqs. (2) and (4) are equivalent and Eq. (3) is a special case of
Eq. (5).In this study, the transformation with M(k) = Hk/k! and q(t) = 1
is applied, where H is the time horizon of interest. Then Eq. (4)
becomes
Xk Hk
k!
dkxt
dtk
" #tt0
; where k 0;1;2;3; . . . ; 1: 6
Using the differential transformation, a differential equation in the
time domain can be transformed to be an algebraic equation in
the K domain and x(t) can be obtained by finite-term Taylor series
plus a remainder, as
xt 1
qt XN1
k0
t t0k
k!
Xk
Mk RNt; 7
where
Nomenclature
A cross-sectional area (m2)cp specific heat (J/kgK)ha convective heat transfer coefficient (W/m
2K)H time spank nonnegative integer denoted as the K domain
ka conductivity (W/mK)L fin length (m)M(k) weight factorq heat flux (W/m2)q(t) a kernel corresponding to x(t)r radius (m)re tip radius (m)R dimensionless radiust time (s)T temperature (K)Ta ambient temperature (K)U(x, k) differential transform ofTW fin thickness (m)
x(t) original functionX(k) transformed function
Greek symbolsa absorptivityd thickness of the inner walle emissivityg fin efficiencyh dimensionless temperatureq density (kg/m3)r StefanBoltzmann constants dimensionless time
Subscriptsb basee effective
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Assumed that the conductive thermal resistance of the inner
surface wall, i.e., the thickness of the inner wall d = 0, is ignored.
Then the initial condition and boundary conditions of the fin can
be expressed as follows.
T Ta; t 0; 16
T Tb; r rb; 17
KTdT
dr haT Ta r eT
4
aT4
e
; r re: 18
In order to make the equations dimensionless, the following
non-dimensional parameters are introduced:
h T
Ta; he
TeTa
; hb TbTa
; R r rbre rb
r rbL
; s kat
qcpL2
;
Introducing the above non-dimensional parameters into the equa-
tions, it yields
d2h
dR2
n1hd2h
dR2
n1dh
dR
2
L
LR rb
dh
dR
L
LR rbn1h
dh
dR
n2h 1 n3h4 n4 n5
dh
ds; 19
h 1 s 0; 20
h hb; R 0; 21
m1dh
dR m2h
dh
dR m3h m4h
4 m3 m5; R 1; 22
where
n1 bTa
1 bTan2
2haL2
Wka
1
1 bTan3
2rT3aL2e
Wka
1
1 bTa
n4 2rT3aL
2aWka
h4e1 bTa
n5 1
1 bTa
m1 1 bTa; m2 bTa; m3 haLka
; m4 rT3
aLeka
; m5 rT3
aLah4
e
ka:
Take differential transform of the non-dimensional governing equa-
tion (Eq. (19)) with respect to time. We obtain
n5k1HUR; k 1 d
2UR;k
dR2 n1UR; k
d2UR;k
dR2
n1dUR;k
dR dUR;k
dR L
LRrb
dUR;kdR
n1L
LRrbUR; k dUR;k
dR
n2UR;k dk n3UR; k UR; k UR; k UR; k n4dk
23
where the symbol denotes the convolution operation in K do-
main, Dftgt FkGk Pk
l0FlGk l. Similarly, the initial
condition (Eq. (20)) and boundary conditions (Eqs. (21) and (22))
are transformed to be:
UR;0 1 24
U0; k hb k 0
0 kP 1
25
m1dU1; k
dR m2U1; k
dU1;k
dR m3U1;k m4U1;kt
U1; k U1; k U1; k m3dk m5dk 26
Take second-order accurate central finite difference approximation
with respect to R, Eq. (23) becomes:
n5k1HUik 1
Ui1k2UikUi1k
DR2
n1Uik Ui1k2UikUi1k
DR2
h in1
Ui1kUi1k
2DR
h i
Ui1kUi1k
2DR
h i L
LRirb
Ui1kUi1k
2DR
h in1
LLRirb
Uik Ui1kUi1k
2DR
h in2Uik dk n3Uik Uik Uik Uik n4dk
27
and the initial condition and boundary conditions are:
Ui0 1 28
U1k hb k 0
0 kP 1
29
m13Uimaxk 4Uimax1k Uimax2k
2DR
m2Uimaxk
3Uimaxk 4Uimax1k Uimax2k
2DR
m3Uimaxk
m4Uimaxk Uimaxk Uimaxk Uimaxk
m3dk m5dk; 30
where i max is the total number of finite difference segments.
The heat transfer rate and fin efficiency are the important infor-
mation in the performance of the fin. The heat transfer rate is de-
fined as:
q KTAdT
drrrb
31
The fin efficiency is the ratio of the actual heat being transferred
from the fin surface to the surrounding fluid to the heat which
would be transferred if the entire fin area were at the base temper-
ature. The fin efficiency is defined by:
g q
qideal32
where qideal haAtotal surfaceTb Ta rAtotal surface eT4b aT
4e
4. Results and discussion
In this study, the temperature distributions of annular fin with
temperature-dependent thermal conductivity are investigated. The
assumption of dimensions of annular fin and the surrounding con-ditions are the same as in [4]. The effects of the convective heat
transfer coefficient (ha), absorptivity (a), emissivity (e) and thermalconductivity parameter (b) on the temperature distribution are
examined. The governing equation (Eq. (27)) with initial and
Table 1
Node temperatures of the fin.
Heat t ransf er mechanism Methodology r (m)
0.03 0.04 0.05 0.06
Convectionradiation Adomians double decomposition [4] 570.917 554.373 545.663 542.499
Present method 571.05 554.67 546.00 542.82
Pure convection Adomians double decomposition [4] 577.070 563.724 556.613 554.022
Present method 576.99 563.88 556.86 554.31
Pure radiation Adomians double decomposition [4] 591.342 586.323 583.636 582.652
Present method 591.27 586.29 583.62 582.63
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boundary conditions (Eqs. (28)(30)) are utilized to solve the prob-
lem. The ambient temperature Ta and base temperature Tb are as-sumed to be 300 K and 600 K, respectively. The parameters used in
r (m)
400
450
500
550
600
T(K)
(a)
r (m)
400
450
500
550
600
T(K)
(b)
0.02 0.03 0.04 0.05 0.06
0.02 0.03 0.04 0.05 0.06
0.02 0.03 0.04 0.05 0.06
r (m)
400
450
500
550
600
T(K)
(c)
= 0.00018
= 0
= -0.00018
= 0.00018
= 0
= -0.00018
= 0.00018
= 0
= -0.00018
Fig. 3. Transient temperature distribution of the fin: (a) for convectionradiation
heat transfer (ha = 50 W/m2 K, a = e = 0.8), (b) for pure convection heat transfer
(ha = 50 W/m2 K, a = e = 0), (c) for pure radiation heat transfer (ha = 0, a = e = 0.8).
0.02 0.03 0.04 0.05 0.06
r (m)
400
450
500
550
600
T(K
)
= = 0.4
= = 0.8
t=steady state
t=30s
t=20s
t=10s
Fig. 4. Effect ofa and e on the temperature distribution in a fin with pure radiationheat transfer (ha = 0, b = 0).
0.02 0.03 0.04 0.05 0.06
r (m)
350
400
450
500
550
600
T(K)
ha=50W/m2K
ha=100W/m2K
ha=150W/m2K
ha=200W/m2K
ha=250W/m2
K
ha=300W/m2K
Fig. 5. Effect of ha on the temperature distribution in a fin with convection
radiation heat transfer (a = e = 0.8, b = 0, steady state).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a = e
0.7
0.75
0.8
0.85
0.9
250
300
350
400
q(W)
= 0.00018
= 0
= -0.00018
Fig. 6. Effect ofa and e on the fin heat transfer rate and fin efficiency ( ha = 50 W/m2 K, steady state).
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present study include the convective heat transfer coefficient
(ha = 50300), absorptivity (a = 0.4, 0.8), emissivity (e = 0.4,0.8)and thermal conductivity parameter (b = 0.00018,0,0.00018).
In order to make a comparison with the known results and to
check the accuracy of the present method, the present results are
compared with the results by the Adomian double decomposition
method [4] as shown in Table 1. The comparisons are made with
constant thermal conductivity for convectionradiation, pureconvection and pure radiation cases, respectively. It demonstrates
that the present results have good agreements with the result of
[4].
Fig. 3(a) shows the transient temperature distribution of the fin
with convectionradiation heat transfer. It reveals that the fin tem-
perature always decreases monotonically from the base of the fin
towards the tip, and the temperature decays sharply in the begin-
ning. This is because all the energy has not yet transferred from the
base to the tip in a very short time. The temperature distribution
increases as the time increases and approaches steady state even-
tually. The result also shows the influence of thermal conductivity
on the temperature distribution. If the thermal conductivity of the
fin increases with temperature (b > 0), the temperature distribu-
tion increases. On the contrary, if the thermal conductivity de-
creases with temperature (b < 0), the temperature distribution
decreases. This is an effect of the nonlinearity due to tempera-
ture-dependent thermal conductivity.
Fig. 3(b) and (c) show the transient temperature distribution of
the fin with pure convection and pure radiation heat transfer,
respectively. The trends of the temperature distributions are simi-
lar to convectionradiation heat transfer ones. As time goes by, the
temperature distributions rise gradually. It can be noted the tem-
perature of the fin with pure radiation rises much faster than other
conditions as time increases. This is due to the fact that heat is dis-
sipated by only radiation. The heat transfer rate from the fin sur-
face is low. In the steady state, the temperature distribution of
the fin with pure radiation heat transfer is higher than other con-
ditions. The previous results also reveal that convection heat trans-
fer is the main effective heat dissipation mechanism at thissituation. In addition, the difference in the temperature distribu-
tion with varying thermal conductivity appeared to be minimal
for pure radiation heat transfer in the steady state. The effect of
the thermal conductivity of the fin is unapparent.
Fig. 4 depicts the effect ofa and e on the temperature distribu-tion in a fin with pure radiation heat transfer. It can be found that
the temperature distribution rises gradually as time goes by. At
t= 10 s, the difference of temperature distribution with the differ-
ent a and e appeared to be minimal. As time increases, the fin tem-perature rises, the magnitude of radiative heat transfer become
larger and the effects ofa and e become more significant. The dif-ference of temperature between a = e = 0.4 and a = e = 0.8 are moreobvious near the tip of the fin.
Fig. 5 shows the effect of ha on the temperature distribution ina fin with pure convection heat transfer. As can be seen in Fig. 5,
the temperature distribution decreases when ha increased. A
higher ha, i.e. the effects of the convective cooling on the fin be-
come more significant, will enhance external cooling and cause
the temperature distribution of the fin to decrease gradually. Be-
sides, the decrement of temperature decreases gradually with
increasing ha.
The effect ofa and e on the fin heat transfer rate and fin effi-ciency is illustrated in Fig. 6. It demonstrates that the heat transfer
rate increases as a and e increases, but the fin efficiency decreaseswith the increase ofa and e. The heat transfer rate and fin effi-ciency are higher for b = 0.00018 (thermal conductivity increasing
with temperature) and lower for b = 0.00018 (thermal conductiv-
ity decreasing with temperature) compared with the case of con-stant thermal conductivity.
Fig. 7 shows the effect ofha on the fin heat transfer rate and fin
efficiency. The fin surface convection becomes stronger with the
increasing of convective heat transfer coefficient. A strong surface
convection will decrease the temperature of fin, and lead to a
increasing in the heat flow from the fin base to the tip. The heat
transfer rate increase. But the fin efficiency decreases with the in-
crease of convective heat transfer coefficient.
Fig. 8 illustrates the heat transfer rate and fin efficiency as afunction of the fin radii ratio. The fin radii ratio increases as the
cross section area of the fin decreases under the constraint that
the fin volume is equal to a given value. The radii ratio increases
will enlarge the heat convection surfaces, i.e. the heat transfer rates
increase, but the heat conduction from the fin base decreases be-
cause of the decreasing of cross section area of the fin. A suitable
value (optimum value) of radii ratio will maximize the heat trans-
fer rate. In addition, the optimum value of radii ratio decreases as
ha increases. The trends of fin efficiency curves are similar to fin
heat transfer rate curves. The optimum fin efficiency occurs at a
certain value of radii ratio. The optimum value of radii ratio de-
creases as ha increases, too.
0 100 200 300 400 500ha (W/m
2K)
0.4
0.5
0.6
0.7
0.8
0.9
1
0
400
800
1200
1600
q(W)
= 0.00018
= -0.00018
= 0
Fig. 7. Effect ofha on the fin heat transfer rate and fin efficiency (a = e = 0.8, steadystate).
0 2 4 6 8 10re/rb
0
200
400
600
800
1000
1200
1400
q(W)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1b = 0.00018
b = 0
b = -0.00018
ha=50W/m2K
ha=150W/m2K
ha=100W/m2K
Fig. 8. Effect of radii ratio on the fin heat transfer rate and fin efficiency for differentvalues of ha (a = e = 0.8, steady state).
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5. Conclusions
The hybrid numerical method of differential transformation and
finite difference approximation is employed to study the nonlinear
heat transfer problem. The influences of convective heat transfer
coefficient, absorptivity and emissivity on the temperature distri-
bution, heat transfer rate and fin efficiency are discussed. The opti-
mum radii ratio of fin which maximizes the heat transfer rate andfin efficiency is also examined. The results reveal that convective
heat transfer is the main effective heat dissipation mechanism un-
der the convectionradiation condition. The heat transfer rate in-
creases as ha increases, but the fin efficiency decreases with the
increase of ha. The optimum radii ratio for heat transfer rate and
fin efficiency decreases as ha increases. The results show that the
hybrid method provides effective and efficient procedure to study
nonlinear heat transfer problem.
Acknowledgements
The support of the National Science Council of Taiwan under
the Grant No. NSC 98-2221-E-006-209-MY2 and National Cheng
Kung University under the Grant No. D98-1500 is gratefullyacknowledged.
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