1-s2.0-S0017931011000743-main

7/28/2019 1-s2.0-S0017931011000743-main

1/7

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.2011.02.019

7/28/2019 1-s2.0-S0017931011000743-main

2/7

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

2428 H.-S. Peng, C.-L. Chen / International Journal of Heat and Mass Transfer 54 (2011) 24272433

7/28/2019 1-s2.0-S0017931011000743-main

3/7

7/28/2019 1-s2.0-S0017931011000743-main

4/7

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

http://-/?-http://-/?-

7/28/2019 1-s2.0-S0017931011000743-main

5/7

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).

H.-S. Peng, C.-L. Chen / International Journal of Heat and Mass Transfer 54 (2011) 24272433 2431

7/28/2019 1-s2.0-S0017931011000743-main

6/7

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).


7/28/2019 1-s2.0-S0017931011000743-main

7/7

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.

References

[1] Q.D. Kern, D.A. Draus, Extended Surface Heat Transfer, McGraw-Hill, New York,1972.

[2] L.T. Yu, C.K. Chen, Application of the taylor transformation to the transienttemperature response of an annular fin, Heat Transfer Engineering 20 (1)(1999) 7887.

[3] Y.C. Yang, S.S. Chu, Transient coupled thermoelastic analysis of an annular fin,International Communications in Heat and Mass Transfer 28 (8) (2001) 11031114.

[4] C.H. Chiu, C.K. Chen, Application of the decomposition method to thermalstresses in isotropic circular fins with temperature-dependent thermalconductivity, Acta Mechanica 157 (2002) 147158.

[5] C.H. Chiu, C.K. Chen, Application of Adomians decomposition procedure to theanalysis of convectiveradiative fins, Journal of Heat Transfer 125 (2003) 312316.

[6] C. Arslanturk, Performance analysis and optimization of a thermally non-symmetric annular fin, International Communications in Heat and MassTransfer 31 (8) (2004) 11431153.

[7] C.Y. Yang, Estimation of the periodic thermal conditions on the non-fourier fin

problem, International Journal of Heat and Mass Transfer 48 (2005) 35063515.

[8] P. Naphon, Study on the heat transfer characteristics of the annular fin underdry-surface, partially wet-surface, and fully wet-surface conditions,International Communications in Heat and Mass Transfer 33 (2006) 112121.

[9] W.L. Chen, Y.C. Yang, H.L. Lee, Inverse problem in determining convection heattransfer coefficient of an annular fin, Energy Conversion and Management 48(2007) 10811088.

[10] A.A. Joneidi, D.D. Ganji, M. Babaelahi, Differential transformation method todetermine fin efficiency of convective straight fins with temperaturedependent thermal conductivity, International communications in Heat andMass Transfer 36 (2009) 757762.

[11] J.K. Zhou, Differential Transformation and its Applications for ElectricalCircuits, Huazhong University Press, Wuhan, China, 1986.

[12] S.H. Ho, C.K. Chen, Analysis of general elastically end restrained non-uniformbeams using differential transform, Applied Mathematical Modelling 22(1998) 219234.

[13] C.K. Chen, S.H. Ho, Application of differential transformation to eigenvalueproblems, Applied Mathematics and Computation 79 (1996) 173188.

[14] C.K. Chen, S.P. Ju, Application of differential transformation to transientadvectivedispersive transport equation, Applied Mathematics andComputation 155 (2004) 2538.

[15] C.Y. Lo, B.Y. Chen, Application of the hybrid differential transform/control-volume method to hyperbolic heat conduction problems, Numerical HeatTransfer, Part B: Fundamentals 55 (3) (2009) 219231.

[16] C.L. Chen, Y.C. Liu, Solution of two-boundary-value problems using thedifferential transformation method, Journal of Optimization Theory andApplication 99 (1) (1998) 2335.

[17] M.J. Jang, C.L. Chen, Y.C. Liu, On solving the initial-value problems using thedifferential transformation method, Applied Mathematics and Computation115 (23) (2000) 145160.

[18] M.J. Jang, C.L. Chen, Y.C. Liu, Two-dimensional differential transformationmethod for partial differential equations, Applied Mathematics andComputation 121 (2001) 261270.

H.-S. Peng, C.-L. Chen / International Journal of Heat and Mass Transfer 54 (2011) 24272433 2433

Documents

1-s2.0-S0017931011000743-main