Research ArticleAnalytical Solution of Heat Conduction forHollow Cylinders with Time-Dependent Boundary Conditionand Time-Dependent Heat Transfer Coefficient

Te-Wen Tu1 and Sen-Yung Lee2

1Department of Mechanical Engineering Air Force Institute of Technology No 198 Jieshou W Road Gangshan TownshipKaohsiung 820 Taiwan2Department of Mechanical Engineering National Cheng Kung University No 1 University Road Tainan 701 Taiwan

Correspondence should be addressed to Sen-Yung Lee syleemailnckuedutw

Received 5 May 2015 Accepted 21 July 2015

Academic Editor Assunta Andreozzi

Copyright copy 2015 T-W Tu and S-Y Lee This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An analytical solution for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heattransfer coefficient at different surfaces is developed for the first time The methodology is an extension of the shifting functionmethod By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions thesystem is transformed into a partial differential equation with homogenous boundary conditions only The transformed system isthus solved by series expansion theorem Limiting cases of the solution are studied and numerical results are compared with thosein the literature The convergence rate of the present solution is fast and the analytical solution is simple and accurate Also theinfluence of physical parameters on the temperature distribution of a hollow cylinder along the radial direction is investigated

1 Introduction

The problems of transient heat flow in hollow cylinders areimportant in many engineering applications Heat exchangertubes solidification of metal tube casting cannon barrelstime variation heating on walls of circular structure andheat treatment on hollow cylinders are typical examples Itis well known that if the temperature andor the heat fluxare prescribed at the boundary surface then the heat transfersystem includes heat conduction coefficient only on the otherhand if the boundary surface dissipates heat by convectionon the basis of Newtonrsquos law of cooling the heat transfercoefficient will be included in the boundary term

For the problem of heat conduction in hollow cylinderswith time-dependent boundary conditions of any kinds atinner and outer surfaces the associated governing differentialequation is a second-order Bessel differential equation withconstant coefficients After conducting a Hankel transforma-tion the analytical solutions can be obtained and found in thetextbook by Ozisik [1]

For the heat transfer in hollow cylinders with mixedtype boundary condition and time-dependent heat transfercoefficient simultaneously the problem cannot be solved byany analytical methods such as the method of separationof variable Laplace transform and Hankel transform Fewstudies in Cartesian coordinate system can be found and var-ious approximated and numerical methods were proposedBy introducing a new variable Ivanov and Salomatov [2 3]together with Postolrsquonik [4] transformed the linear governingequation into a nonlinear form After ignoring the nonlin-ear term they developed an approximated solution whichwas claimed to be valid for the system with Biot numberbeing less than 025 Moreover Kozlov [5] used Laplacetransformation to study the problems with Biot function ina rational combination of sines cosines polynomials andexponentials Even though it is possible to obtain the exactseries solution of a specified transformed system the problemis the computation of the inverse Laplace transformationwhich generally requires integration in the complex planeBecker et al [6] took finite difference method and Laplace

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 203404 9 pageshttpdxdoiorg1011552015203404

2 Journal of Applied Mathematics

transformation method to study the heating of the rockadjacent to water flowing through a crevice Recently Chenand his colleagues [7] proposed an analytical solution byusing the shifting function method for the heat conductionin a slab with time-dependent heat transfer coefficient at oneend Yatskiv et al [8] studied the thermostressed state ofcylinder with thin near-surface layer having time-dependentthermophysical properties They reduced the problem to anintegrodifferential equation with variable coefficients andsolved it by the spline approximation

In addition different approximation methods such as theiterative perturbation method [9] the time-varying eigen-function expansion method with finite integral transforms[10 11] generalized integral transforms [12] and the Liepoint symmetry analysis [13] were used to study this kind ofproblems Various inverse schemes for determining the time-dependent heat transfer coefficient were developed by someresearchers [14ndash20]

According to the literature because of the complexityand difficulty of the methodology none of any analyticalsolutions for the heat conduction in a hollow cylinder withtime-dependent boundary condition and time-dependentheat transfer coefficient existed This work extends themethodology of shifting function method [7 21 22] todevelop an analytical solution with closed form for the heattransfer in hollow cylinders with time-dependent bound-ary condition and time-dependent heat transfer coefficientsimultaneously By setting the Biot function in a particularform and introducing two specially chosen shifting functionsthe system is transformed into a partial differential equationwith homogenous boundary conditions and can be solved byseries expansion theorem Examples are given to demonstratethe methodology and numerical results are compared withthose in the literature And last but not least the influence ofphysical parameters on the temperature profile is studied

2 Mathematical Modeling

Consider the transient heat conduction in heat exchangertubes as shown in Figure 1 A fluid with time-varyingtemperature is running inside the hollow cylinder and theheat is dissipated by the time-dependent convection at theouter surface into an environment of zero temperature Thegoverning differential equation of the system is

1205972119879

1205971199032+1119903

120597119879

120597119903=

1120572

120597119879

120597119905 119886 lt 119903 lt 119887 119905 gt 0 (1)

where 119879 is the temperature 119903 is the space variable 120572 is thethermal diffusivity 119905 is the time variable and 119886 and 119887 denoteinner and outer radii respectively The boundary and initialconditions of the boundary value problem are

119879 = 119867 (119905) at 119903 = 119886

minus 119896120597119879

120597119903= ℎ (119905) 119879 at 119903 = 119887

119879 (119903) = 1198790(119903) when 119905 = 0

(2)

r

a

b

Initially at minusk120597T

120597r= h(t)T

T = H(t)

T = T0(r)

Figure 1 Hollow cylinders with time-dependent temperature andtime-dependent heat transfer coefficient at inner and outer surfaces

Here 119867(119905) is a time-dependent temperature function at theinner surface 119896 is the thermal conductivity ℎ(119905) is a time-dependent heat transfer coefficient function and 1198790(119903) isan initial temperature function For consistence in initialtemperature field 119867(0) must be equal to 1198790(119886) The aboveproblem can be normalized by defining

120579 =119879

119879119903

119877 =119903

119887

120591 =120572119905

1198872

119903 =119886

119887

120595 (120591) =119867 (119905)

119879119903

Bi (120591) = ℎ (119905) 119887

119896

1205790 (119877) =1198790 (119903)

119879119903

(3)

where 119879119903is a constant reference temperature and the dimen-

sionless boundary value problem will then become

1205972120579

1205971198772 +1119877

120597120579

120597119877=120597120579

120597120591 119903 lt 119877 lt 1 120591 gt 0 (4)

120579 = 120595 (120591) at 119877 = 119903 (5)

120597120579

120597119877+Bi (120591) 120579 = 0 at 119877 = 1 (6)

120579 (119877 0) = 1205790 (119877) when 120591 = 0 (7)

Journal of Applied Mathematics 3

To keep the boundary condition of the third kind at outersurface in the following analysis one sets the Biot functionBi(120591) in the form of

Bi (120591) = 120575 +119865 (120591) (8)

where 120575 and 119865(120591) are defined as

120575 = Bi (0)

119865 (120591) = Bi (120591) minusBi (0) (9)

It is obvious that 119865(0) = 0 and the boundary condition at119877 = 1 can be rewritten as

120597120579

120597119877+ 120575120579 = minus119865 (120591) 120579 at 119877 = 1 (10)

3 The Shifting Function Method

31 Change of Variable To find the solution for the second-order differential equation with time-dependent boundarycondition and time-dependent heat transfer coefficient atdifferent surfaces the shifting functionmethod [7 21 22] wasextended by taking

120579 (119877 120591) = V (119877 120591) + 1198921 (119877) 1198911 (120591) + 1198922 (119877) 1198912 (120591) (11)

where V(119877 120591) is called transformed function 119892119894(119877) (119894 = 1 2)

are two shifting functions to be specified and 119891119894(120591) (119894 = 1 2)

are the auxiliary time functions defined as

1198911 (120591) = 120595 (120591)

1198912 (120591) = minus119865 (120591) 120579 (1 120591) (12)

Substituting (11) into (4) (5) (10) and (7) one has thefollowing equation

1205972V (119877 120591)1205971198772 +

1119877

120597V (119877 120591)120597119877

=120597V (119877 120591)

120597120591+ 1198921 (119877)

1198891198911 (120591)

119889120591

minus [11988921198921 (119877)

1198891198772 +1119877

1198891198921 (119877)

119889119877]1198911 (120591)

+ 1198922 (119877)1198891198912 (120591)

119889120591

minus [11988921198922 (119877)

1198891198772 +1119877

1198891198922 (119877)

119889119877]1198912 (120591)

(13)

and the associated boundary and initial conditions now are

V (119903 120591) + 1198921 (119903) 1198911 (120591) + 1198922 (119903) 1198912 (120591) = 1198911 (120591)

at 119877 = 119903

120597V (1 120591)120597119877

+ 120575V (1 120591) + [1198891198921 (1)119889119877

+ 1205751198921 (1)] 1198911 (120591)

+ [1198891198922 (1)119889119877

+ 1205751198922 (1)] 1198912 (120591) = 1198912 (120591) at 119877 = 1

V (119877 0) + 1198921 (119877) 1198911 (0) + 1198922 (119877) 1198912 (0) = 1205790 (119877)

(14)

Something worthy to mention is that (13) contains threefunctions that is V(119877 120591) and 119892

119894(119877) (119894 = 1 2) and hence it

cannot be solved directly

32 The Shifting Functions For convenience in the analysisthe two shifting functions are specifically chosen in order tosatisfy the following conditions

1198921 (119903) = 1

1198922 (119903) = 0

at 119877 = 119903

1198921 (1) = 0

1198922 (1) = 0

1198891198921 (1)119889119877

= 0

1198891198922 (1)119889119877

= 1

at 119877 = 1

(15)

Consequently the shifting functions can be easily determinedas

1198921 (119877) = (1 minus 1198771 minus 119903

)

2

1198922 (119877) =(119877 minus 1) (119877 minus 119903)

1 minus 119903

(16)

Substituting these shifting functions and auxiliary time func-tions into (11) yields

120579 (119877 120591) = V (119877 120591) + (1 minus 1198771 minus 119903

)

2120595 (120591)

minus [(119877 minus 1) (119877 minus 119903)

1 minus 119903]119865 (120591) 120579 (1 120591)

(17)

When setting 119877 = 1 in the equation above one has therelation

120579 (1 120591) = V (1 120591) (18)

Therefore two functions in governing differential equation(13) are integrated to one With (16) and (18) (13) can berewritten in terms of the function V(119877 120591) as

120597V (119877 120591)120597120591

minus1205972V (119877 120591)1205971198772 minus

1119877

120597V (119877 120591)120597119877

minus1198922 (119877)119889

119889120591[119865 (120591) V (1 120591)] + 1198892 (119877) 119865 (120591) V (1 120591)

= minus 1198921 (119877) (120591) + 1198891 (119877) 120595 (120591)

(19)

where 119889119894(119877) (119894 = 1 2) are defined as

1198891 (119877) =4119877 minus 2

119877 (1 minus 119903)2

1198892 (119877) =4119877 minus (1 + 119903)119877 (1 minus 119903)

(20)

4 Journal of Applied Mathematics

Meanwhile the associated boundary conditions of the trans-formed function turn to homogeneous ones as follows

V (119903 120591) = 0

120597V (1 120591)120597119877

+ 120575V (1 120591) = 0(21)

Since 1198912(0) = minus119865(0)120579(1 0) and 119865(0) = 0 hence theassociated initial condition can be simplified as

V (119877 0) = 1205790 (119877) minus 1198921 (119877) 120595 (0) = V0 (119877) (22)

33 Series Expansion To find the solution for the boundaryvalue problem of heat conduction that is (19)ndash(22) one usesthe method of series expansion with try functions

120601119899(119877) = 1198690 (120582119899119877) minus

1198690 (120582119899119903)

1198840 (120582119899119903)1198840 (120582119899119877)

119899 = 1 2 3

(23)

satisfying the boundary conditions (21) Here the character-istic values 120582

119899are the roots of the transcendental equation

1198690 (120582119899119903) 1198841 (120582119899) minus 1198691 (120582119899) 1198840 (120582119899119903)

1198690 (120582119899119903) 1198840 (120582119899) minus 1198690 (120582119899) 1198840 (120582119899119903)=

120575

120582119899

(24)

The try functions have the following orthogonal property

int

1

119903

119877120601119898 (119877) 120601119899 (119877) 119889119877 =

0 119898 = 119899

119873119899 119898 = 119899

(25)

where the norms119873119899are

119873119899=

121205822119899

[(1205822119899+ 120575

2) 120601

2119899(1) minus 4

120587211988420 (120582119899119903)

] (26)

Now one can assume that the solution of the physicalproblem takes the form of

V (119877 120591) =infin

sum

119899=1120601119899(119877) 119902119899(120591) 119899 = 1 2 3 (27)

where 119902119899(120591) (119899 = 1 2 3 ) are time-dependent generalized

coordinates Substituting solution from (27) into differentialequation (19) leads to

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus 120601

10158401015840

119899(119877) 119902119899(120591) minus

11198771206011015840

119899(119877) 119902119899(120591)

minus 1198922 (119877) 120601119899 (1) [ (120591) 119902119899 (120591) + 119865 (120591) 119902119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) = minus1198921 (119877) (120591)

+ 1198891 (119877) 120595 (120591)

(28)

Expanding 1198921(119877) and 1198891(119877) on the right hand side of (28) inseries forms we obtain

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899(120591) + (120591) 119902

119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) =infin

sum

119899=1[minus120574119899120601119899 (119877) (120591)

+ 120574119899120601119899(119877) 120595 (120591)]

(29)

where 120574119899and 120574119899are

120574119899=int1119903119877120601119899(119877) 1198921 (119877) 119889119877

119873119899

=1

(1 minus 119903)2[1205851198691 minus 21205851198692

+ 1205851198693 minus

1198690 (120582119899119903)

1198840 (120582119899119903)(1205851198841 minus 21205851198842 + 1205851198843)]

120574119899=int1119903119877120601119899(119877) 1198891 (119877) 119889119877

119873119899

=2

(1 minus 119903)2[21205851198691 minus 1205851198690

minus1198690 (120582119899119903)

1198840 (120582119899119903)(21205851198841 minus 1205851198840)]

(30)

in which 120585119882119894

(119894 = 0 1 2 3119882 = 119869 119884) are given as

1205851198820 =

int11199031198820 (120582119899119877) 119889119877

119873119899

1205851198822 =

int111990311987721198820 (120582119899119877) 119889119877

119873119899

1205851198821 =

int11199031198771198820 (120582119899119877) 119889119877

119873119899

=1

120582119899119873119899

[1198821 (120582119899)

minus 1199031198821 (120582119899119903)]

1205851198823 =

int111990311987731198820 (120582119899119877) 119889119877

119873119899

=1

1205823119899119873119899

[21205821198991198820 (120582119899)

+ (1205822119899minus 4)1198821 (120582119899) minus 2120582119899119903

21198820 (120582119899119903)

minus 119903 (12058221198991199032minus 4)1198821 (120582119899119903)]

(31)

From (29) one can let

120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899 (120591) + (120591) 119902119899 (120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591)

= minus 120574119899120601119899(119877) (120591) + 120574

119899120601119899(119877) 120595 (120591)

(32)

Journal of Applied Mathematics 5

After taking the inner product with try function 119877120601119899(119877) and

integrating from 119903 to 1 the resulting differential equation nowis

119902119899(120591) +

1205822119899minus 120573119899 (120591) + 120573

119899119865 (120591)

1 minus 120573119899119865 (120591)

119902119899(120591) = 120585

119899(120591) (33)

where 120573119899and 120573

119899are

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198922 (119877) 119889119877

119873119899

=120601119899(1)

1 minus 1199031205851198693

minus (1+ 119903) 1205851198692 + 1199031205851198691

minus1198690 (120582119899119903)

1198840 (120582119899119903)[1205851198843 minus (1+ 119903) 1205851198842 + 1199031205851198841]

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198892 (119877) 119889119877

119873119899

=120601119899 (1)1 minus 119903

41205851198691

minus (1+ 119903) 1205851198690 minus1198690 (120582119899119903)

1198840 (120582119899119903)[41205851198841 minus (1+ 119903) 1205851198840]

(34)

and 120585119899(120591) is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 120573119899119865 (120591)

(35)

The associated initial condition is

119902119899 (0) =

int1119903119877120601119899(119877) V0 (119877) 119889119877119873119899

(36)

As a result the complete solution of the ordinary differentialequation (33) subject to the initial condition (36) is

119902119899(120591) = 119876

119899(120591) [119902

119899(0) +int

120591

0

120585119899(120593)

119876119899(120593)

119889120593] (37)

where 119876119899(120591) is

119876119899 (120591) = 119890

minusint120591

0 ((1205822119899minus120573119899(120577)+120573

119899119865(120577))(1minus120573

119899119865(120577)))119889120577

(38)

After substituting (16) (18) (23) and (27) back to (11) oneobtains the analytical solution of the physical problem

120579 (119877 120591) =

infin

sum

119899=1[120601119899(119877) minus 120601

119899(1) 1198922 (119877) 119865 (120591)] 119902119899 (120591)

+ 1198921 (119877) 120595 (120591)

(39)

where the summation is taken over all eigenvalues 120582119899of the

problem

34 Constant Heat Transfer Coefficient at 119877 = 1 When theheat transfer coefficient ℎ at 119877 = 1 is time-independent theBiot function is a constant 120575 and 119865(120591) = 0 The infinite seriessolution (39) is reduced to

120579 (119877 120591) =

infin

sum

119899=1120601119899 (119877) 119902119899 (120591) + 120574119899120601119899 (119877) 120595 (120591) (40)

where the generalized coordinates 119902119899(120591) are

119902119899(120591) = 119890

minus1205822119899120591119902119899(0) +int

120591

0[minus120574119899 (120593) + 120574

119899120595 (120593)]

sdot 1198901205822119899120593119889120593

(41)

The 119902119899(0)rsquos for the problem under consideration are

119902119899 (0) =

int1119903119877120601119899(119877) [1205790 (119877) minus 120595 (0) 1198921 (119877)] 119889119877

119873119899

(42)

Introducing (42) in (41) and performing integration by partswe can get

119902119899 (120591) =

119890minus120582

2119899120591

119873119899

[int

1

119903

119877120601119899 (119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]minus 120574

119899120595 (120591)

(43)

Substituting (43) into (40) yields the temperature distribu-tion

120579 (119877 120591) =

infin

sum

119899=1

119890minus120582

2119899120591

119873119899

120601119899(119877) [int

1

119903

119877120601119899(119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]

(44)

This solution is the same as that obtained via the integraltransform method by Ozisik [1]

4 Verification and Example

To illustrate the previous analysis and the accuracy of thethree-term approximation solution one examines the follow-ing case

The time-dependent boundary condition 120595(120591) consid-ered at 119877 = 119903 is taken as

120595 (120591) = 1198861 minus 1198871119890minus1199041120591 cos1205961120591 (45)

and differentiating it with respect to 120591 leads to

(120591) = 1198871119890minus1199041120591 (1199041 cos1205961120591 +1205961 sin1205961120591) (46)

where 1198861 and 1198871 are two arbitrary constants and 1199041 and 1205961 aretwo parameters

The Biot function considered at boundary 119877 = 1 is

Bi (120591) = 1198862 minus 1198872119890minus1199042120591 cos1205962120591 (47)

where 1198862 and 1198872 are two arbitrary constants and 1199042 and 1205962 aretwo parameters According to (8)-(9) we obtain

120575 = 1198862 minus 1198872

119865 (120591) = 1198872 (1minus 119890minus1199042120591 cos1205962120591)

(120591) = 1198872119890minus1199042120591 (1199042 cos1205962120591 +1205962 sin1205962120591)

(48)

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

transformation method to study the heating of the rockadjacent to water flowing through a crevice Recently Chenand his colleagues [7] proposed an analytical solution byusing the shifting function method for the heat conductionin a slab with time-dependent heat transfer coefficient at oneend Yatskiv et al [8] studied the thermostressed state ofcylinder with thin near-surface layer having time-dependentthermophysical properties They reduced the problem to anintegrodifferential equation with variable coefficients andsolved it by the spline approximation

In addition different approximation methods such as theiterative perturbation method [9] the time-varying eigen-function expansion method with finite integral transforms[10 11] generalized integral transforms [12] and the Liepoint symmetry analysis [13] were used to study this kind ofproblems Various inverse schemes for determining the time-dependent heat transfer coefficient were developed by someresearchers [14ndash20]

According to the literature because of the complexityand difficulty of the methodology none of any analyticalsolutions for the heat conduction in a hollow cylinder withtime-dependent boundary condition and time-dependentheat transfer coefficient existed This work extends themethodology of shifting function method [7 21 22] todevelop an analytical solution with closed form for the heattransfer in hollow cylinders with time-dependent bound-ary condition and time-dependent heat transfer coefficientsimultaneously By setting the Biot function in a particularform and introducing two specially chosen shifting functionsthe system is transformed into a partial differential equationwith homogenous boundary conditions and can be solved byseries expansion theorem Examples are given to demonstratethe methodology and numerical results are compared withthose in the literature And last but not least the influence ofphysical parameters on the temperature profile is studied

2 Mathematical Modeling

Consider the transient heat conduction in heat exchangertubes as shown in Figure 1 A fluid with time-varyingtemperature is running inside the hollow cylinder and theheat is dissipated by the time-dependent convection at theouter surface into an environment of zero temperature Thegoverning differential equation of the system is

1205972119879

1205971199032+1119903

120597119879

120597119903=

1120572

120597119879

120597119905 119886 lt 119903 lt 119887 119905 gt 0 (1)

where 119879 is the temperature 119903 is the space variable 120572 is thethermal diffusivity 119905 is the time variable and 119886 and 119887 denoteinner and outer radii respectively The boundary and initialconditions of the boundary value problem are

119879 = 119867 (119905) at 119903 = 119886

minus 119896120597119879

120597119903= ℎ (119905) 119879 at 119903 = 119887

119879 (119903) = 1198790(119903) when 119905 = 0

(2)

r

a

b

Initially at minusk120597T

120597r= h(t)T

T = H(t)

T = T0(r)

Figure 1 Hollow cylinders with time-dependent temperature andtime-dependent heat transfer coefficient at inner and outer surfaces

Here 119867(119905) is a time-dependent temperature function at theinner surface 119896 is the thermal conductivity ℎ(119905) is a time-dependent heat transfer coefficient function and 1198790(119903) isan initial temperature function For consistence in initialtemperature field 119867(0) must be equal to 1198790(119886) The aboveproblem can be normalized by defining

120579 =119879

119879119903

119877 =119903

119887

120591 =120572119905

1198872

119903 =119886

119887

120595 (120591) =119867 (119905)

119879119903

Bi (120591) = ℎ (119905) 119887

119896

1205790 (119877) =1198790 (119903)

119879119903

(3)

where 119879119903is a constant reference temperature and the dimen-

sionless boundary value problem will then become

1205972120579

1205971198772 +1119877

120597120579

120597119877=120597120579

120597120591 119903 lt 119877 lt 1 120591 gt 0 (4)

120579 = 120595 (120591) at 119877 = 119903 (5)

120597120579

120597119877+Bi (120591) 120579 = 0 at 119877 = 1 (6)

120579 (119877 0) = 1205790 (119877) when 120591 = 0 (7)

Journal of Applied Mathematics 3

To keep the boundary condition of the third kind at outersurface in the following analysis one sets the Biot functionBi(120591) in the form of

Bi (120591) = 120575 +119865 (120591) (8)

where 120575 and 119865(120591) are defined as

120575 = Bi (0)

119865 (120591) = Bi (120591) minusBi (0) (9)

It is obvious that 119865(0) = 0 and the boundary condition at119877 = 1 can be rewritten as

120597120579

120597119877+ 120575120579 = minus119865 (120591) 120579 at 119877 = 1 (10)

3 The Shifting Function Method

31 Change of Variable To find the solution for the second-order differential equation with time-dependent boundarycondition and time-dependent heat transfer coefficient atdifferent surfaces the shifting functionmethod [7 21 22] wasextended by taking

120579 (119877 120591) = V (119877 120591) + 1198921 (119877) 1198911 (120591) + 1198922 (119877) 1198912 (120591) (11)

where V(119877 120591) is called transformed function 119892119894(119877) (119894 = 1 2)

are two shifting functions to be specified and 119891119894(120591) (119894 = 1 2)

are the auxiliary time functions defined as

1198911 (120591) = 120595 (120591)

1198912 (120591) = minus119865 (120591) 120579 (1 120591) (12)

Substituting (11) into (4) (5) (10) and (7) one has thefollowing equation

1205972V (119877 120591)1205971198772 +

1119877

120597V (119877 120591)120597119877

=120597V (119877 120591)

120597120591+ 1198921 (119877)

1198891198911 (120591)

119889120591

minus [11988921198921 (119877)

1198891198772 +1119877

1198891198921 (119877)

119889119877]1198911 (120591)

+ 1198922 (119877)1198891198912 (120591)

119889120591

minus [11988921198922 (119877)

1198891198772 +1119877

1198891198922 (119877)

119889119877]1198912 (120591)

(13)

and the associated boundary and initial conditions now are

V (119903 120591) + 1198921 (119903) 1198911 (120591) + 1198922 (119903) 1198912 (120591) = 1198911 (120591)

at 119877 = 119903

120597V (1 120591)120597119877

+ 120575V (1 120591) + [1198891198921 (1)119889119877

+ 1205751198921 (1)] 1198911 (120591)

+ [1198891198922 (1)119889119877

+ 1205751198922 (1)] 1198912 (120591) = 1198912 (120591) at 119877 = 1

V (119877 0) + 1198921 (119877) 1198911 (0) + 1198922 (119877) 1198912 (0) = 1205790 (119877)

(14)

Something worthy to mention is that (13) contains threefunctions that is V(119877 120591) and 119892

119894(119877) (119894 = 1 2) and hence it

cannot be solved directly

32 The Shifting Functions For convenience in the analysisthe two shifting functions are specifically chosen in order tosatisfy the following conditions

1198921 (119903) = 1

1198922 (119903) = 0

at 119877 = 119903

1198921 (1) = 0

1198922 (1) = 0

1198891198921 (1)119889119877

= 0

1198891198922 (1)119889119877

= 1

at 119877 = 1

(15)

Consequently the shifting functions can be easily determinedas

1198921 (119877) = (1 minus 1198771 minus 119903

)

2

1198922 (119877) =(119877 minus 1) (119877 minus 119903)

1 minus 119903

(16)

Substituting these shifting functions and auxiliary time func-tions into (11) yields

120579 (119877 120591) = V (119877 120591) + (1 minus 1198771 minus 119903

)

2120595 (120591)

minus [(119877 minus 1) (119877 minus 119903)

1 minus 119903]119865 (120591) 120579 (1 120591)

(17)

When setting 119877 = 1 in the equation above one has therelation

120579 (1 120591) = V (1 120591) (18)

Therefore two functions in governing differential equation(13) are integrated to one With (16) and (18) (13) can berewritten in terms of the function V(119877 120591) as

120597V (119877 120591)120597120591

minus1205972V (119877 120591)1205971198772 minus

1119877

120597V (119877 120591)120597119877

minus1198922 (119877)119889

119889120591[119865 (120591) V (1 120591)] + 1198892 (119877) 119865 (120591) V (1 120591)

= minus 1198921 (119877) (120591) + 1198891 (119877) 120595 (120591)

(19)

where 119889119894(119877) (119894 = 1 2) are defined as

1198891 (119877) =4119877 minus 2

119877 (1 minus 119903)2

1198892 (119877) =4119877 minus (1 + 119903)119877 (1 minus 119903)

(20)

4 Journal of Applied Mathematics

Meanwhile the associated boundary conditions of the trans-formed function turn to homogeneous ones as follows

V (119903 120591) = 0

120597V (1 120591)120597119877

+ 120575V (1 120591) = 0(21)

Since 1198912(0) = minus119865(0)120579(1 0) and 119865(0) = 0 hence theassociated initial condition can be simplified as

V (119877 0) = 1205790 (119877) minus 1198921 (119877) 120595 (0) = V0 (119877) (22)

33 Series Expansion To find the solution for the boundaryvalue problem of heat conduction that is (19)ndash(22) one usesthe method of series expansion with try functions

120601119899(119877) = 1198690 (120582119899119877) minus

1198690 (120582119899119903)

1198840 (120582119899119903)1198840 (120582119899119877)

119899 = 1 2 3

(23)

satisfying the boundary conditions (21) Here the character-istic values 120582

119899are the roots of the transcendental equation

1198690 (120582119899119903) 1198841 (120582119899) minus 1198691 (120582119899) 1198840 (120582119899119903)

1198690 (120582119899119903) 1198840 (120582119899) minus 1198690 (120582119899) 1198840 (120582119899119903)=

120575

120582119899

(24)

The try functions have the following orthogonal property

int

1

119903

119877120601119898 (119877) 120601119899 (119877) 119889119877 =

0 119898 = 119899

119873119899 119898 = 119899

(25)

where the norms119873119899are

119873119899=

121205822119899

[(1205822119899+ 120575

2) 120601

2119899(1) minus 4

120587211988420 (120582119899119903)

] (26)

Now one can assume that the solution of the physicalproblem takes the form of

V (119877 120591) =infin

sum

119899=1120601119899(119877) 119902119899(120591) 119899 = 1 2 3 (27)

where 119902119899(120591) (119899 = 1 2 3 ) are time-dependent generalized

coordinates Substituting solution from (27) into differentialequation (19) leads to

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus 120601

10158401015840

119899(119877) 119902119899(120591) minus

11198771206011015840

119899(119877) 119902119899(120591)

minus 1198922 (119877) 120601119899 (1) [ (120591) 119902119899 (120591) + 119865 (120591) 119902119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) = minus1198921 (119877) (120591)

+ 1198891 (119877) 120595 (120591)

(28)

Expanding 1198921(119877) and 1198891(119877) on the right hand side of (28) inseries forms we obtain

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899(120591) + (120591) 119902

119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) =infin

sum

119899=1[minus120574119899120601119899 (119877) (120591)

+ 120574119899120601119899(119877) 120595 (120591)]

(29)

where 120574119899and 120574119899are

120574119899=int1119903119877120601119899(119877) 1198921 (119877) 119889119877

119873119899

=1

(1 minus 119903)2[1205851198691 minus 21205851198692

+ 1205851198693 minus

1198690 (120582119899119903)

1198840 (120582119899119903)(1205851198841 minus 21205851198842 + 1205851198843)]

120574119899=int1119903119877120601119899(119877) 1198891 (119877) 119889119877

119873119899

=2

(1 minus 119903)2[21205851198691 minus 1205851198690

minus1198690 (120582119899119903)

1198840 (120582119899119903)(21205851198841 minus 1205851198840)]

(30)

in which 120585119882119894

(119894 = 0 1 2 3119882 = 119869 119884) are given as

1205851198820 =

int11199031198820 (120582119899119877) 119889119877

119873119899

1205851198822 =

int111990311987721198820 (120582119899119877) 119889119877

119873119899

1205851198821 =

int11199031198771198820 (120582119899119877) 119889119877

119873119899

=1

120582119899119873119899

[1198821 (120582119899)

minus 1199031198821 (120582119899119903)]

1205851198823 =

int111990311987731198820 (120582119899119877) 119889119877

119873119899

=1

1205823119899119873119899

[21205821198991198820 (120582119899)

+ (1205822119899minus 4)1198821 (120582119899) minus 2120582119899119903

21198820 (120582119899119903)

minus 119903 (12058221198991199032minus 4)1198821 (120582119899119903)]

(31)

From (29) one can let

120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899 (120591) + (120591) 119902119899 (120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591)

= minus 120574119899120601119899(119877) (120591) + 120574

119899120601119899(119877) 120595 (120591)

(32)

Journal of Applied Mathematics 5

After taking the inner product with try function 119877120601119899(119877) and

integrating from 119903 to 1 the resulting differential equation nowis

119902119899(120591) +

1205822119899minus 120573119899 (120591) + 120573

119899119865 (120591)

1 minus 120573119899119865 (120591)

119902119899(120591) = 120585

119899(120591) (33)

where 120573119899and 120573

119899are

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198922 (119877) 119889119877

119873119899

=120601119899(1)

1 minus 1199031205851198693

minus (1+ 119903) 1205851198692 + 1199031205851198691

minus1198690 (120582119899119903)

1198840 (120582119899119903)[1205851198843 minus (1+ 119903) 1205851198842 + 1199031205851198841]

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198892 (119877) 119889119877

119873119899

=120601119899 (1)1 minus 119903

41205851198691

minus (1+ 119903) 1205851198690 minus1198690 (120582119899119903)

1198840 (120582119899119903)[41205851198841 minus (1+ 119903) 1205851198840]

(34)

and 120585119899(120591) is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 120573119899119865 (120591)

(35)

The associated initial condition is

119902119899 (0) =

int1119903119877120601119899(119877) V0 (119877) 119889119877119873119899

(36)

As a result the complete solution of the ordinary differentialequation (33) subject to the initial condition (36) is

119902119899(120591) = 119876

119899(120591) [119902

119899(0) +int

120591

0

120585119899(120593)

119876119899(120593)

119889120593] (37)

where 119876119899(120591) is

119876119899 (120591) = 119890

minusint120591

0 ((1205822119899minus120573119899(120577)+120573

119899119865(120577))(1minus120573

119899119865(120577)))119889120577

(38)

After substituting (16) (18) (23) and (27) back to (11) oneobtains the analytical solution of the physical problem

120579 (119877 120591) =

infin

sum

119899=1[120601119899(119877) minus 120601

119899(1) 1198922 (119877) 119865 (120591)] 119902119899 (120591)

+ 1198921 (119877) 120595 (120591)

(39)

where the summation is taken over all eigenvalues 120582119899of the

problem

34 Constant Heat Transfer Coefficient at 119877 = 1 When theheat transfer coefficient ℎ at 119877 = 1 is time-independent theBiot function is a constant 120575 and 119865(120591) = 0 The infinite seriessolution (39) is reduced to

120579 (119877 120591) =

infin

sum

119899=1120601119899 (119877) 119902119899 (120591) + 120574119899120601119899 (119877) 120595 (120591) (40)

where the generalized coordinates 119902119899(120591) are

119902119899(120591) = 119890

minus1205822119899120591119902119899(0) +int

120591

0[minus120574119899 (120593) + 120574

119899120595 (120593)]

sdot 1198901205822119899120593119889120593

(41)

The 119902119899(0)rsquos for the problem under consideration are

119902119899 (0) =

int1119903119877120601119899(119877) [1205790 (119877) minus 120595 (0) 1198921 (119877)] 119889119877

119873119899

(42)

Introducing (42) in (41) and performing integration by partswe can get

119902119899 (120591) =

119890minus120582

2119899120591

119873119899

[int

1

119903

119877120601119899 (119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]minus 120574

119899120595 (120591)

(43)

Substituting (43) into (40) yields the temperature distribu-tion

120579 (119877 120591) =

infin

sum

119899=1

119890minus120582

2119899120591

119873119899

120601119899(119877) [int

1

119903

119877120601119899(119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]

(44)

This solution is the same as that obtained via the integraltransform method by Ozisik [1]

4 Verification and Example

To illustrate the previous analysis and the accuracy of thethree-term approximation solution one examines the follow-ing case

The time-dependent boundary condition 120595(120591) consid-ered at 119877 = 119903 is taken as

120595 (120591) = 1198861 minus 1198871119890minus1199041120591 cos1205961120591 (45)

and differentiating it with respect to 120591 leads to

(120591) = 1198871119890minus1199041120591 (1199041 cos1205961120591 +1205961 sin1205961120591) (46)

where 1198861 and 1198871 are two arbitrary constants and 1199041 and 1205961 aretwo parameters

The Biot function considered at boundary 119877 = 1 is

Bi (120591) = 1198862 minus 1198872119890minus1199042120591 cos1205962120591 (47)

where 1198862 and 1198872 are two arbitrary constants and 1199042 and 1205962 aretwo parameters According to (8)-(9) we obtain

120575 = 1198862 minus 1198872

119865 (120591) = 1198872 (1minus 119890minus1199042120591 cos1205962120591)

(120591) = 1198872119890minus1199042120591 (1199042 cos1205962120591 +1205962 sin1205962120591)

(48)

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

To keep the boundary condition of the third kind at outersurface in the following analysis one sets the Biot functionBi(120591) in the form of

Bi (120591) = 120575 +119865 (120591) (8)

where 120575 and 119865(120591) are defined as

120575 = Bi (0)

119865 (120591) = Bi (120591) minusBi (0) (9)

It is obvious that 119865(0) = 0 and the boundary condition at119877 = 1 can be rewritten as

120597120579

120597119877+ 120575120579 = minus119865 (120591) 120579 at 119877 = 1 (10)

3 The Shifting Function Method

31 Change of Variable To find the solution for the second-order differential equation with time-dependent boundarycondition and time-dependent heat transfer coefficient atdifferent surfaces the shifting functionmethod [7 21 22] wasextended by taking

120579 (119877 120591) = V (119877 120591) + 1198921 (119877) 1198911 (120591) + 1198922 (119877) 1198912 (120591) (11)

where V(119877 120591) is called transformed function 119892119894(119877) (119894 = 1 2)

are two shifting functions to be specified and 119891119894(120591) (119894 = 1 2)

are the auxiliary time functions defined as

1198911 (120591) = 120595 (120591)

1198912 (120591) = minus119865 (120591) 120579 (1 120591) (12)

Substituting (11) into (4) (5) (10) and (7) one has thefollowing equation

1205972V (119877 120591)1205971198772 +

1119877

120597V (119877 120591)120597119877

=120597V (119877 120591)

120597120591+ 1198921 (119877)

1198891198911 (120591)

119889120591

minus [11988921198921 (119877)

1198891198772 +1119877

1198891198921 (119877)

119889119877]1198911 (120591)

+ 1198922 (119877)1198891198912 (120591)

119889120591

minus [11988921198922 (119877)

1198891198772 +1119877

1198891198922 (119877)

119889119877]1198912 (120591)

(13)

and the associated boundary and initial conditions now are

V (119903 120591) + 1198921 (119903) 1198911 (120591) + 1198922 (119903) 1198912 (120591) = 1198911 (120591)

at 119877 = 119903

120597V (1 120591)120597119877

+ 120575V (1 120591) + [1198891198921 (1)119889119877

+ 1205751198921 (1)] 1198911 (120591)

+ [1198891198922 (1)119889119877

+ 1205751198922 (1)] 1198912 (120591) = 1198912 (120591) at 119877 = 1

V (119877 0) + 1198921 (119877) 1198911 (0) + 1198922 (119877) 1198912 (0) = 1205790 (119877)

(14)

Something worthy to mention is that (13) contains threefunctions that is V(119877 120591) and 119892

119894(119877) (119894 = 1 2) and hence it

cannot be solved directly

32 The Shifting Functions For convenience in the analysisthe two shifting functions are specifically chosen in order tosatisfy the following conditions

1198921 (119903) = 1

1198922 (119903) = 0

at 119877 = 119903

1198921 (1) = 0

1198922 (1) = 0

1198891198921 (1)119889119877

= 0

1198891198922 (1)119889119877

= 1

at 119877 = 1

(15)

Consequently the shifting functions can be easily determinedas

1198921 (119877) = (1 minus 1198771 minus 119903

)

2

1198922 (119877) =(119877 minus 1) (119877 minus 119903)

1 minus 119903

(16)

Substituting these shifting functions and auxiliary time func-tions into (11) yields

120579 (119877 120591) = V (119877 120591) + (1 minus 1198771 minus 119903

)

2120595 (120591)

minus [(119877 minus 1) (119877 minus 119903)

1 minus 119903]119865 (120591) 120579 (1 120591)

(17)

When setting 119877 = 1 in the equation above one has therelation

120579 (1 120591) = V (1 120591) (18)

Therefore two functions in governing differential equation(13) are integrated to one With (16) and (18) (13) can berewritten in terms of the function V(119877 120591) as

120597V (119877 120591)120597120591

minus1205972V (119877 120591)1205971198772 minus

1119877

120597V (119877 120591)120597119877

minus1198922 (119877)119889

119889120591[119865 (120591) V (1 120591)] + 1198892 (119877) 119865 (120591) V (1 120591)

= minus 1198921 (119877) (120591) + 1198891 (119877) 120595 (120591)

(19)

where 119889119894(119877) (119894 = 1 2) are defined as

1198891 (119877) =4119877 minus 2

119877 (1 minus 119903)2

1198892 (119877) =4119877 minus (1 + 119903)119877 (1 minus 119903)

(20)

4 Journal of Applied Mathematics

Meanwhile the associated boundary conditions of the trans-formed function turn to homogeneous ones as follows

V (119903 120591) = 0

120597V (1 120591)120597119877

+ 120575V (1 120591) = 0(21)

Since 1198912(0) = minus119865(0)120579(1 0) and 119865(0) = 0 hence theassociated initial condition can be simplified as

V (119877 0) = 1205790 (119877) minus 1198921 (119877) 120595 (0) = V0 (119877) (22)

33 Series Expansion To find the solution for the boundaryvalue problem of heat conduction that is (19)ndash(22) one usesthe method of series expansion with try functions

120601119899(119877) = 1198690 (120582119899119877) minus

1198690 (120582119899119903)

1198840 (120582119899119903)1198840 (120582119899119877)

119899 = 1 2 3

(23)

satisfying the boundary conditions (21) Here the character-istic values 120582

119899are the roots of the transcendental equation

1198690 (120582119899119903) 1198841 (120582119899) minus 1198691 (120582119899) 1198840 (120582119899119903)

1198690 (120582119899119903) 1198840 (120582119899) minus 1198690 (120582119899) 1198840 (120582119899119903)=

120575

120582119899

(24)

The try functions have the following orthogonal property

int

1

119903

119877120601119898 (119877) 120601119899 (119877) 119889119877 =

0 119898 = 119899

119873119899 119898 = 119899

(25)

where the norms119873119899are

119873119899=

121205822119899

[(1205822119899+ 120575

2) 120601

2119899(1) minus 4

120587211988420 (120582119899119903)

] (26)

Now one can assume that the solution of the physicalproblem takes the form of

V (119877 120591) =infin

sum

119899=1120601119899(119877) 119902119899(120591) 119899 = 1 2 3 (27)

where 119902119899(120591) (119899 = 1 2 3 ) are time-dependent generalized

coordinates Substituting solution from (27) into differentialequation (19) leads to

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus 120601

10158401015840

119899(119877) 119902119899(120591) minus

11198771206011015840

119899(119877) 119902119899(120591)

minus 1198922 (119877) 120601119899 (1) [ (120591) 119902119899 (120591) + 119865 (120591) 119902119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) = minus1198921 (119877) (120591)

+ 1198891 (119877) 120595 (120591)

(28)

Expanding 1198921(119877) and 1198891(119877) on the right hand side of (28) inseries forms we obtain

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899(120591) + (120591) 119902

119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) =infin

sum

119899=1[minus120574119899120601119899 (119877) (120591)

+ 120574119899120601119899(119877) 120595 (120591)]

(29)

where 120574119899and 120574119899are

120574119899=int1119903119877120601119899(119877) 1198921 (119877) 119889119877

119873119899

=1

(1 minus 119903)2[1205851198691 minus 21205851198692

+ 1205851198693 minus

1198690 (120582119899119903)

1198840 (120582119899119903)(1205851198841 minus 21205851198842 + 1205851198843)]

120574119899=int1119903119877120601119899(119877) 1198891 (119877) 119889119877

119873119899

=2

(1 minus 119903)2[21205851198691 minus 1205851198690

minus1198690 (120582119899119903)

1198840 (120582119899119903)(21205851198841 minus 1205851198840)]

(30)

in which 120585119882119894

(119894 = 0 1 2 3119882 = 119869 119884) are given as

1205851198820 =

int11199031198820 (120582119899119877) 119889119877

119873119899

1205851198822 =

int111990311987721198820 (120582119899119877) 119889119877

119873119899

1205851198821 =

int11199031198771198820 (120582119899119877) 119889119877

119873119899

=1

120582119899119873119899

[1198821 (120582119899)

minus 1199031198821 (120582119899119903)]

1205851198823 =

int111990311987731198820 (120582119899119877) 119889119877

119873119899

=1

1205823119899119873119899

[21205821198991198820 (120582119899)

+ (1205822119899minus 4)1198821 (120582119899) minus 2120582119899119903

21198820 (120582119899119903)

minus 119903 (12058221198991199032minus 4)1198821 (120582119899119903)]

(31)

From (29) one can let

120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899 (120591) + (120591) 119902119899 (120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591)

= minus 120574119899120601119899(119877) (120591) + 120574

119899120601119899(119877) 120595 (120591)

(32)

Journal of Applied Mathematics 5

After taking the inner product with try function 119877120601119899(119877) and

integrating from 119903 to 1 the resulting differential equation nowis

119902119899(120591) +

1205822119899minus 120573119899 (120591) + 120573

119899119865 (120591)

1 minus 120573119899119865 (120591)

119902119899(120591) = 120585

119899(120591) (33)

where 120573119899and 120573

119899are

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198922 (119877) 119889119877

119873119899

=120601119899(1)

1 minus 1199031205851198693

minus (1+ 119903) 1205851198692 + 1199031205851198691

minus1198690 (120582119899119903)

1198840 (120582119899119903)[1205851198843 minus (1+ 119903) 1205851198842 + 1199031205851198841]

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198892 (119877) 119889119877

119873119899

=120601119899 (1)1 minus 119903

41205851198691

minus (1+ 119903) 1205851198690 minus1198690 (120582119899119903)

1198840 (120582119899119903)[41205851198841 minus (1+ 119903) 1205851198840]

(34)

and 120585119899(120591) is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 120573119899119865 (120591)

(35)

The associated initial condition is

119902119899 (0) =

int1119903119877120601119899(119877) V0 (119877) 119889119877119873119899

(36)

As a result the complete solution of the ordinary differentialequation (33) subject to the initial condition (36) is

119902119899(120591) = 119876

119899(120591) [119902

119899(0) +int

120591

0

120585119899(120593)

119876119899(120593)

119889120593] (37)

where 119876119899(120591) is

119876119899 (120591) = 119890

minusint120591

0 ((1205822119899minus120573119899(120577)+120573

119899119865(120577))(1minus120573

119899119865(120577)))119889120577

(38)

After substituting (16) (18) (23) and (27) back to (11) oneobtains the analytical solution of the physical problem

120579 (119877 120591) =

infin

sum

119899=1[120601119899(119877) minus 120601

119899(1) 1198922 (119877) 119865 (120591)] 119902119899 (120591)

+ 1198921 (119877) 120595 (120591)

(39)

where the summation is taken over all eigenvalues 120582119899of the

problem

34 Constant Heat Transfer Coefficient at 119877 = 1 When theheat transfer coefficient ℎ at 119877 = 1 is time-independent theBiot function is a constant 120575 and 119865(120591) = 0 The infinite seriessolution (39) is reduced to

120579 (119877 120591) =

infin

sum

119899=1120601119899 (119877) 119902119899 (120591) + 120574119899120601119899 (119877) 120595 (120591) (40)

where the generalized coordinates 119902119899(120591) are

119902119899(120591) = 119890

minus1205822119899120591119902119899(0) +int

120591

0[minus120574119899 (120593) + 120574

119899120595 (120593)]

sdot 1198901205822119899120593119889120593

(41)

The 119902119899(0)rsquos for the problem under consideration are

119902119899 (0) =

int1119903119877120601119899(119877) [1205790 (119877) minus 120595 (0) 1198921 (119877)] 119889119877

119873119899

(42)

Introducing (42) in (41) and performing integration by partswe can get

119902119899 (120591) =

119890minus120582

2119899120591

119873119899

[int

1

119903

119877120601119899 (119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]minus 120574

119899120595 (120591)

(43)

Substituting (43) into (40) yields the temperature distribu-tion

120579 (119877 120591) =

infin

sum

119899=1

119890minus120582

2119899120591

119873119899

120601119899(119877) [int

1

119903

119877120601119899(119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]

(44)

This solution is the same as that obtained via the integraltransform method by Ozisik [1]

4 Verification and Example

To illustrate the previous analysis and the accuracy of thethree-term approximation solution one examines the follow-ing case

The time-dependent boundary condition 120595(120591) consid-ered at 119877 = 119903 is taken as

120595 (120591) = 1198861 minus 1198871119890minus1199041120591 cos1205961120591 (45)

and differentiating it with respect to 120591 leads to

(120591) = 1198871119890minus1199041120591 (1199041 cos1205961120591 +1205961 sin1205961120591) (46)

where 1198861 and 1198871 are two arbitrary constants and 1199041 and 1205961 aretwo parameters

The Biot function considered at boundary 119877 = 1 is

Bi (120591) = 1198862 minus 1198872119890minus1199042120591 cos1205962120591 (47)

where 1198862 and 1198872 are two arbitrary constants and 1199042 and 1205962 aretwo parameters According to (8)-(9) we obtain

120575 = 1198862 minus 1198872

119865 (120591) = 1198872 (1minus 119890minus1199042120591 cos1205962120591)

(120591) = 1198872119890minus1199042120591 (1199042 cos1205962120591 +1205962 sin1205962120591)

(48)

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Meanwhile the associated boundary conditions of the trans-formed function turn to homogeneous ones as follows

V (119903 120591) = 0

120597V (1 120591)120597119877

+ 120575V (1 120591) = 0(21)

Since 1198912(0) = minus119865(0)120579(1 0) and 119865(0) = 0 hence theassociated initial condition can be simplified as

V (119877 0) = 1205790 (119877) minus 1198921 (119877) 120595 (0) = V0 (119877) (22)

33 Series Expansion To find the solution for the boundaryvalue problem of heat conduction that is (19)ndash(22) one usesthe method of series expansion with try functions

120601119899(119877) = 1198690 (120582119899119877) minus

1198690 (120582119899119903)

1198840 (120582119899119903)1198840 (120582119899119877)

119899 = 1 2 3

(23)

satisfying the boundary conditions (21) Here the character-istic values 120582

119899are the roots of the transcendental equation

1198690 (120582119899119903) 1198841 (120582119899) minus 1198691 (120582119899) 1198840 (120582119899119903)

1198690 (120582119899119903) 1198840 (120582119899) minus 1198690 (120582119899) 1198840 (120582119899119903)=

120575

120582119899

(24)

The try functions have the following orthogonal property

int

1

119903

119877120601119898 (119877) 120601119899 (119877) 119889119877 =

0 119898 = 119899

119873119899 119898 = 119899

(25)

where the norms119873119899are

119873119899=

121205822119899

[(1205822119899+ 120575

2) 120601

2119899(1) minus 4

120587211988420 (120582119899119903)

] (26)

Now one can assume that the solution of the physicalproblem takes the form of

V (119877 120591) =infin

sum

119899=1120601119899(119877) 119902119899(120591) 119899 = 1 2 3 (27)

where 119902119899(120591) (119899 = 1 2 3 ) are time-dependent generalized

coordinates Substituting solution from (27) into differentialequation (19) leads to

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus 120601

10158401015840

119899(119877) 119902119899(120591) minus

11198771206011015840

119899(119877) 119902119899(120591)

minus 1198922 (119877) 120601119899 (1) [ (120591) 119902119899 (120591) + 119865 (120591) 119902119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) = minus1198921 (119877) (120591)

+ 1198891 (119877) 120595 (120591)

(28)

Expanding 1198921(119877) and 1198891(119877) on the right hand side of (28) inseries forms we obtain

infin

sum

119899=1120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899(120591) + (120591) 119902

119899(120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591) =infin

sum

119899=1[minus120574119899120601119899 (119877) (120591)

+ 120574119899120601119899(119877) 120595 (120591)]

(29)

where 120574119899and 120574119899are

120574119899=int1119903119877120601119899(119877) 1198921 (119877) 119889119877

119873119899

=1

(1 minus 119903)2[1205851198691 minus 21205851198692

+ 1205851198693 minus

1198690 (120582119899119903)

1198840 (120582119899119903)(1205851198841 minus 21205851198842 + 1205851198843)]

120574119899=int1119903119877120601119899(119877) 1198891 (119877) 119889119877

119873119899

=2

(1 minus 119903)2[21205851198691 minus 1205851198690

minus1198690 (120582119899119903)

1198840 (120582119899119903)(21205851198841 minus 1205851198840)]

(30)

in which 120585119882119894

(119894 = 0 1 2 3119882 = 119869 119884) are given as

1205851198820 =

int11199031198820 (120582119899119877) 119889119877

119873119899

1205851198822 =

int111990311987721198820 (120582119899119877) 119889119877

119873119899

1205851198821 =

int11199031198771198820 (120582119899119877) 119889119877

119873119899

=1

120582119899119873119899

[1198821 (120582119899)

minus 1199031198821 (120582119899119903)]

1205851198823 =

int111990311987731198820 (120582119899119877) 119889119877

119873119899

=1

1205823119899119873119899

[21205821198991198820 (120582119899)

+ (1205822119899minus 4)1198821 (120582119899) minus 2120582119899119903

21198820 (120582119899119903)

minus 119903 (12058221198991199032minus 4)1198821 (120582119899119903)]

(31)

From (29) one can let

120601119899(119877) 119902119899(120591) minus [120601

10158401015840

119899(119877) +

11198771206011015840

119899(119877)] 119902

119899(120591)

minus 1198922 (119877) 120601119899 (1) [119865 (120591) 119902119899 (120591) + (120591) 119902119899 (120591)]

+ 1198892 (119877) 120601119899 (1) 119865 (120591) 119902119899 (120591)

= minus 120574119899120601119899(119877) (120591) + 120574

119899120601119899(119877) 120595 (120591)

(32)

Journal of Applied Mathematics 5

After taking the inner product with try function 119877120601119899(119877) and

integrating from 119903 to 1 the resulting differential equation nowis

119902119899(120591) +

1205822119899minus 120573119899 (120591) + 120573

119899119865 (120591)

1 minus 120573119899119865 (120591)

119902119899(120591) = 120585

119899(120591) (33)

where 120573119899and 120573

119899are

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198922 (119877) 119889119877

119873119899

=120601119899(1)

1 minus 1199031205851198693

minus (1+ 119903) 1205851198692 + 1199031205851198691

minus1198690 (120582119899119903)

1198840 (120582119899119903)[1205851198843 minus (1+ 119903) 1205851198842 + 1199031205851198841]

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198892 (119877) 119889119877

119873119899

=120601119899 (1)1 minus 119903

41205851198691

minus (1+ 119903) 1205851198690 minus1198690 (120582119899119903)

1198840 (120582119899119903)[41205851198841 minus (1+ 119903) 1205851198840]

(34)

and 120585119899(120591) is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 120573119899119865 (120591)

(35)

The associated initial condition is

119902119899 (0) =

int1119903119877120601119899(119877) V0 (119877) 119889119877119873119899

(36)

As a result the complete solution of the ordinary differentialequation (33) subject to the initial condition (36) is

119902119899(120591) = 119876

119899(120591) [119902

119899(0) +int

120591

0

120585119899(120593)

119876119899(120593)

119889120593] (37)

where 119876119899(120591) is

119876119899 (120591) = 119890

minusint120591

0 ((1205822119899minus120573119899(120577)+120573

119899119865(120577))(1minus120573

119899119865(120577)))119889120577

(38)

After substituting (16) (18) (23) and (27) back to (11) oneobtains the analytical solution of the physical problem

120579 (119877 120591) =

infin

sum

119899=1[120601119899(119877) minus 120601

119899(1) 1198922 (119877) 119865 (120591)] 119902119899 (120591)

+ 1198921 (119877) 120595 (120591)

(39)

where the summation is taken over all eigenvalues 120582119899of the

problem

34 Constant Heat Transfer Coefficient at 119877 = 1 When theheat transfer coefficient ℎ at 119877 = 1 is time-independent theBiot function is a constant 120575 and 119865(120591) = 0 The infinite seriessolution (39) is reduced to

120579 (119877 120591) =

infin

sum

119899=1120601119899 (119877) 119902119899 (120591) + 120574119899120601119899 (119877) 120595 (120591) (40)

where the generalized coordinates 119902119899(120591) are

119902119899(120591) = 119890

minus1205822119899120591119902119899(0) +int

120591

0[minus120574119899 (120593) + 120574

119899120595 (120593)]

sdot 1198901205822119899120593119889120593

(41)

The 119902119899(0)rsquos for the problem under consideration are

119902119899 (0) =

int1119903119877120601119899(119877) [1205790 (119877) minus 120595 (0) 1198921 (119877)] 119889119877

119873119899

(42)

Introducing (42) in (41) and performing integration by partswe can get

119902119899 (120591) =

119890minus120582

2119899120591

119873119899

[int

1

119903

119877120601119899 (119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]minus 120574

119899120595 (120591)

(43)

Substituting (43) into (40) yields the temperature distribu-tion

120579 (119877 120591) =

infin

sum

119899=1

119890minus120582

2119899120591

119873119899

120601119899(119877) [int

1

119903

119877120601119899(119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]

(44)

This solution is the same as that obtained via the integraltransform method by Ozisik [1]

4 Verification and Example

To illustrate the previous analysis and the accuracy of thethree-term approximation solution one examines the follow-ing case

The time-dependent boundary condition 120595(120591) consid-ered at 119877 = 119903 is taken as

120595 (120591) = 1198861 minus 1198871119890minus1199041120591 cos1205961120591 (45)

and differentiating it with respect to 120591 leads to

(120591) = 1198871119890minus1199041120591 (1199041 cos1205961120591 +1205961 sin1205961120591) (46)

where 1198861 and 1198871 are two arbitrary constants and 1199041 and 1205961 aretwo parameters

The Biot function considered at boundary 119877 = 1 is

Bi (120591) = 1198862 minus 1198872119890minus1199042120591 cos1205962120591 (47)

where 1198862 and 1198872 are two arbitrary constants and 1199042 and 1205962 aretwo parameters According to (8)-(9) we obtain

120575 = 1198862 minus 1198872

119865 (120591) = 1198872 (1minus 119890minus1199042120591 cos1205962120591)

(120591) = 1198872119890minus1199042120591 (1199042 cos1205962120591 +1205962 sin1205962120591)

(48)

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

After taking the inner product with try function 119877120601119899(119877) and

integrating from 119903 to 1 the resulting differential equation nowis

119902119899(120591) +

1205822119899minus 120573119899 (120591) + 120573

119899119865 (120591)

1 minus 120573119899119865 (120591)

119902119899(120591) = 120585

119899(120591) (33)

where 120573119899and 120573

119899are

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198922 (119877) 119889119877

119873119899

=120601119899(1)

1 minus 1199031205851198693

minus (1+ 119903) 1205851198692 + 1199031205851198691

minus1198690 (120582119899119903)

1198840 (120582119899119903)[1205851198843 minus (1+ 119903) 1205851198842 + 1199031205851198841]

120573119899= 120601119899(1)

int1119903119877120601119899 (119877) 1198892 (119877) 119889119877

119873119899

=120601119899 (1)1 minus 119903

41205851198691

minus (1+ 119903) 1205851198690 minus1198690 (120582119899119903)

1198840 (120582119899119903)[41205851198841 minus (1+ 119903) 1205851198840]

(34)

and 120585119899(120591) is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 120573119899119865 (120591)

(35)

The associated initial condition is

119902119899 (0) =

int1119903119877120601119899(119877) V0 (119877) 119889119877119873119899

(36)

As a result the complete solution of the ordinary differentialequation (33) subject to the initial condition (36) is

119902119899(120591) = 119876

119899(120591) [119902

119899(0) +int

120591

0

120585119899(120593)

119876119899(120593)

119889120593] (37)

where 119876119899(120591) is

119876119899 (120591) = 119890

minusint120591

0 ((1205822119899minus120573119899(120577)+120573

119899119865(120577))(1minus120573

119899119865(120577)))119889120577

(38)

After substituting (16) (18) (23) and (27) back to (11) oneobtains the analytical solution of the physical problem

120579 (119877 120591) =

infin

sum

119899=1[120601119899(119877) minus 120601

119899(1) 1198922 (119877) 119865 (120591)] 119902119899 (120591)

+ 1198921 (119877) 120595 (120591)

(39)

where the summation is taken over all eigenvalues 120582119899of the

problem

34 Constant Heat Transfer Coefficient at 119877 = 1 When theheat transfer coefficient ℎ at 119877 = 1 is time-independent theBiot function is a constant 120575 and 119865(120591) = 0 The infinite seriessolution (39) is reduced to

120579 (119877 120591) =

infin

sum

119899=1120601119899 (119877) 119902119899 (120591) + 120574119899120601119899 (119877) 120595 (120591) (40)

where the generalized coordinates 119902119899(120591) are

119902119899(120591) = 119890

minus1205822119899120591119902119899(0) +int

120591

0[minus120574119899 (120593) + 120574

119899120595 (120593)]

sdot 1198901205822119899120593119889120593

(41)

The 119902119899(0)rsquos for the problem under consideration are

119902119899 (0) =

int1119903119877120601119899(119877) [1205790 (119877) minus 120595 (0) 1198921 (119877)] 119889119877

119873119899

(42)

Introducing (42) in (41) and performing integration by partswe can get

119902119899 (120591) =

119890minus120582

2119899120591

119873119899

[int

1

119903

119877120601119899 (119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]minus 120574

119899120595 (120591)

(43)

Substituting (43) into (40) yields the temperature distribu-tion

120579 (119877 120591) =

infin

sum

119899=1

119890minus120582

2119899120591

119873119899

120601119899(119877) [int

1

119903

119877120601119899(119877) 1205790 (119877) 119889119877

minus2

1205871198840 (120582119899119903)int

120591

01198901205822119899120593120595 (120593) 119889120593]

(44)

This solution is the same as that obtained via the integraltransform method by Ozisik [1]

4 Verification and Example

To illustrate the previous analysis and the accuracy of thethree-term approximation solution one examines the follow-ing case

The time-dependent boundary condition 120595(120591) consid-ered at 119877 = 119903 is taken as

120595 (120591) = 1198861 minus 1198871119890minus1199041120591 cos1205961120591 (45)

and differentiating it with respect to 120591 leads to

(120591) = 1198871119890minus1199041120591 (1199041 cos1205961120591 +1205961 sin1205961120591) (46)

where 1198861 and 1198871 are two arbitrary constants and 1199041 and 1205961 aretwo parameters

The Biot function considered at boundary 119877 = 1 is

Bi (120591) = 1198862 minus 1198872119890minus1199042120591 cos1205962120591 (47)

where 1198862 and 1198872 are two arbitrary constants and 1199042 and 1205962 aretwo parameters According to (8)-(9) we obtain

120575 = 1198862 minus 1198872

119865 (120591) = 1198872 (1minus 119890minus1199042120591 cos1205962120591)

(120591) = 1198872119890minus1199042120591 (1199042 cos1205962120591 +1205962 sin1205962120591)

(48)

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Consequently the temperature distribution in the hollowcylinder is

120579 (119877 120591) =

infin

sum

119899=1119902119899(120591) [120601

119899(119877)

minus(119877 minus 1) (119877 minus 119903)

1 minus 119903120601119899 (1) 1198872 (1minus 119890

minus1199042120591 cos1205962120591)]

+(1 minus 1198771 minus 119903

)

2(1198861 minus 1198871119890

minus1199041120591 cos1205961120591)

(49)

where the 119902119899(120591)rsquos are defined in (37)The associated 120585

119899(120591) now

is

120585119899(120591) =

minus120574119899 (120591) + 120574

119899120595 (120591)

1 minus 1205731198991198872 (1 minus 119890minus1199042120591 cos1205962120591)

(50)

To avoid numerical instability that occurred in computing119902119899(120591) (37) is rewritten as

119902119899(120591) = 119876

119899(120591) 119902119899(0) +int

120591

0120585119899(120593) sdot

119876119899(120591)

119876119899(120593)

119889120593 (51)

Since the initial conditions cannot have effect on thesteady-state response we consider only the heat conductionin a hollow cylinderwith constant initial temperature 1205790(119883) =1205790 as prescribed in the previous sections The 119902

119899(0)rsquos are now

computed as

119902119899(0) = [120585

1198691 minus1198690 (120582119899119903)

1198840 (120582119899119903)] 1205790 minus 120574119899120595 (0) (52)

For consistence in the temperature field the constant 1205790 istaken as zero in the following examples

In comparisonwith the literature the example of constantBiot function is studied first Bi(120591) = 1 and time-dependenttemperature function 120595(120591) = 1 minus 119890

minus120591 are chosen in thecase In Table 1 we find that the convergence of the presentsolution is faster than that of Ozisik [1] The error of three-term approximation in present study is less than 04 on thecontrary at least twenty-term approximation is required toget the same accuracy in Ozisikrsquos [1] cases

In the case of time-dependent boundary condition andtime-dependent heat transfer coefficient at both surfaces weconsider the time-dependent temperature function 120595(120591) =

1 minus 119890minus120591 cos 120591 and the Biot function Bi(120591) = 2 minus 119890

minus120591 FromTable 2 one can find that the error of three-term approxima-tion is less than 04 Because of large values of Bi(120591) theinternal conductance of the hollow cylinder is small whereasthe heat transfer coefficient at the surface is large In turnthe fact implies that the temperature distribution within thehollow cylinder is nonuniform Therefore we find that thelarger the Biot function that is when 120591 approaches to 10 inTable 2 the more the iteration numbers

Figure 2 depicts the temperature profiles along the radialof the hollow cylinder at different times 120591 = 01 and 120591 = 1We find that the temperature at 119877 = 06 is higher than thetemperature at 119877 = 1 and the temperature profile decreasesat the negative slope for every case It is clear since the heat

06 07 08 09 1000

02

04

06

08

10

R

s1 = 2 120591 = 10

s1 = 1 120591 = 10

s1 = 2 120591 = 01

s1 = 1 120591 = 01

120579(R120591)

Figure 2 Temperature distribution and variation along the radialof the hollow cylinder with different parameter 1199041 of temperaturefunction 120595(120591) [120595(120591) = 1 minus 119890minus1199041120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

Table 1 Temperatures of the hollow cylinder at outer surface and atvarious times [120595(120591) = 1 minus 119890minus120591 Bi(120591) = 1 1205790 = 0]

120591

120579 (1 120591)2 terms 3 terms 20 terms

A B A B A B01 00263 00155 00261 00337 00262 0025005 02287 01841 02296 02610 02297 022481 03981 03263 03998 04502 03998 039195 06541 05416 06591 07364 06598 0644910 06578 05456 06675 07417 06690 06495A present solution (39) B Ozisik [1] (44)

Table 2 Temperatures of the hollow cylinder at outer surface andat various times [120595(120591) = 1 minus 119890minus120591 cos 120591 Bi(120591) = 2 minus 119890minus120591 1205790 = 0]

120591120579(1 120591)

2 terms 3 terms 10 terms 20 terms01 00267 00265 00266 0026605 02398 02408 02408 024081 04163 04185 04184 041855 04900 04951 04951 0495810 04889 04987 04989 05001

source comes to the hollow cylinder from inner surface 119877 =

06 and the heat dissipates from 119877 = 1 to the surroundingenvironment

Variable heat source versus variable Biot function isdrawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures3(a) and 3(b) respectively Two cases of Biot function Bi(120591) =2 minus 119890minus120591 (solid lines) and Bi(120591) = 2 minus 119890

minus2120591 (dash lines) areconsidered Due to the fact that the function 119890

minus2120591 severelydecays as time goes therefore in the same temperature

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

0 1 2 3 4 500

02

04

06

08

120591

120579(08120591)

(a) At 119877 = 08

0 1 2 3 4 500

01

02

03

04

05

06

s1 = 2 s2 = 1

s1 = 2 s2 = 2

s1 = 1 s2 = 1

s1 = 1 s2 = 2

120579(10120591)

120591

(b) At 119877 = 1

Figure 3 Influence of function parameters 1199041and 1199042 on the temperatures of the hollow cylinder at middle and right surfaces [120595(120591) = 1minus119890minus1199041120591

Bi(120591) = 2 minus 119890minus1199042120591 1205790= 0]

06 07 08 09 1000

04

08

12

16

R

1205961 = 2 120591 = 10

1205961 = 1 120591 = 10

1205961 = 2 120591 = 04

1205961 = 1 120591 = 04

120579(R120591)

Figure 4 Temperature distribution and variation along the radial of the hollow cylinder with different parameter 1205961 of temperature function120595(120591) [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus120591 120579

0= 0]

function 120595(120591) the temperature in Bi(120591) = 2 minus 119890minus2120591 is less thanthat in Bi(120591) = 2 minus 119890

minus120591 as 120591 proceeds That is to say moreheat will be dissipated into the surrounding environment forBi(120591) = 2 minus 119890minus2120591 as 120591 goes

Figure 4 depicts the effect of the parameter 1205961 of tem-perature function120595(120591) upon the temperature variation of thehollow cylinder It is found that in the same temperaturefunction120595(120591) the temperature for 120591 = 04 is less than that for120591 = 10 Besides as 120591 increases from 04 to 10 the differencebetween temperatures at 1205961 = 1 and at 1205961 = 2 becomessignificant

Periodical heat source versus time-varying Biot functionis drawn to show the temperature variation of the hollowcylinder at 119877 = 08 and 119877 = 10 with respect to 120591 in Figures5(a) and 5(b) respectively Two cases of heat source 120595(120591) =1 minus cos 120591 (solid lines) and 120595(120591) = 1 minus cos 2120591 (dash lines) areconsidered At the same 120591 the temperature of Bi(120591) = 2minus119890minus120591 isless than that of Bi(120591) = 2minus119890minus01120591 for constant120595(120591)The reasonis that more heat has been dissipated into the surroundingenvironment at the case of Bi(120591) = 2minus 119890minus120591 It can be observedthat as 120591 proceeds in the beginning the temperatures arenonsensitive with 1205961 parameters as shown in Figure 5

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 1 2 3 4 500

04

08

12

16

20

24

120579(08120591)

120591

1205961 = 1 s2 = 01

1205961 = 1 s2 = 10

1205961 = 2 s2 = 01

1205961 = 2 s2 = 10

(a) At 119877 = 08

0 1 2 3 4 500

04

08

12

16

20

120591

120579(10120591)

1205961 = 1 s2 = 01

1205961 = 1 s2 = 1

1205961 = 2 s2 = 01

1205961 = 2 s2 = 1

(b) At 119877 = 1

Figure 5 Influence of 1205961 parameter and 1199042 parameter on thetemperature variation of the hollow cylinder at middle and rightsurfaces [120595(120591) = 1 minus cos1205961120591 Bi(120591) = 2 minus 119890minus1199042120591 120579

0= 0]

5 Conclusion

An analytical solution for the heat conduction in a hollowcylinder with time-dependent boundary conditions of dif-ferent kinds at both surfaces was developed for the firsttime The surface is subject to a time-dependent temperaturefield at inner surface whereas the heat is dissipated by time-dependent convection from outer surface into a surroundingenvironment at zero temperature The methodology is anextension of the shifting function method and the presentresults are identical to those in the literature when constantBiot function is considered Since the methodology doesnot use integral transform it has a proven result Theproposed method can also be easily extended to various

heat conduction problems of hollow cylinders with time-dependent boundary conditions of different kinds at bothsurfaces

Nomenclatures

119886 119887 Inner and outer radii (m)1198861 1198871 1198862 1198872 Arbitrary constants used to express

temperature and Biot functionsBi Biot function1198891 1198892 Auxiliary functions1198911 1198912 Auxiliary time functions119865 Biot function minus a constant1198921 1198922 Shifting functions

ℎ Time-dependent heat transfer coefficientat outer surface (Wsdotmminus2sdotKminus1)

119867 Variable temperature function at innersurface (K)

1198690 Bessel function of order zero of the firstkind

119896 Thermal conductivity (Wsdotmminus1sdotKminus1)119873119899 Norm of try functions

119902119899 Time-dependent generalized coordinates

119903 Space variable (m)119903 Ratio of inner radius over outer radius119877 Dimensionless radius1199041 1199042 Parameters used to express temperature

and Biot functions119905 Time variable (sec)119879 Temperature (K)119879119903 Constant reference temperature (K)

1198790 Initial temperature (K)V Auxiliary function1198840 Bessel function of order zero of the

second kind

Greek Symbols

120572 Thermal diffusivity (m2sdotsminus1)120573119899 120573119899 Auxiliary functions

120575 Initial value of Biot function120601119899 Eigenfunctions

120593 Auxiliary integration variable120574119899 120574119899 Auxiliary functions

120582119899 Eigenvalues

120579 Dimensionless temperature1205790 Dimensionless initial temperature120591 Dimensionless time variable1205961 1205962 Parameters for temperature and Biot functions120585119899 Auxiliary function

120585119882119894 Auxiliary functions to express integration

terms of Bessel functions120595 Dimensionless time-dependent temperature

function120577 Auxiliary integration variable

Subscripts

0 1 2 119898 119899 119869 119884 Indices

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

It is gratefully acknowledged that this work was supported bythe National Science Council of Taiwan under Grants NSC103-2221-E-006-048 and NSC 95-2221-E-344-001

References

[1] M N Ozisik ldquoHeat conduction in the cylindrical coordinatesystemrdquo in Boundary Value Problems of Heat Conduction chap-ter 3 pp 125ndash193 International Textbook Company ScrantonPa USA 1st edition 1968

[2] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics and Thermophysics vol 9 no 1pp 63ndash64 1965

[3] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics andThermophysics vol 11 no 2 pp 151ndash152 1966

[4] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics andThermophysics vol 18 no 2 pp 316ndash322 1970

[5] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsand Thermophysics vol 18 no 1 pp 133ndash138 1970

[6] N M Becker R L Bivins Y C Hsu H D Murphy AB White Jr and G M Wing ldquoHeat diffusion with time-dependent convective boundary conditionrdquo International Jour-nal for Numerical Methods in Engineering vol 19 no 12 pp1871ndash1880 1983

[7] H T Chen S L Sun H C Huang and S Y Lee ldquoAnalyticclosed solution for the heat conduction with time dependentheat convection coefficient at one boundaryrdquo Computer Model-ing in Engineering amp Sciences vol 59 no 2 pp 107ndash126 2010

[8] O I Yatskiv R M Shvetsrsquo and B Y Bobyk ldquoThermostressedstate of a cylinder with thin near-surface layer having time-dependent thermophysical propertiesrdquo Journal of MathematicalSciences vol 187 no 5 pp 647ndash666 2012

[9] Z J Holy ldquoTemperature and stresses in reactor fuel elementsdue to time- and space-dependent heat-transfer coefficientsrdquoNuclear Engineering and Design vol 18 no 1 pp 145ndash197 1972

[10] M N Ozisik and R L Murray ldquoOn the solution of linear dif-fusion problems with variable boundary condition parametersrdquoASME Journal of Heat Transfer vol 96 no 1 pp 48ndash51 1974

[11] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344ndash345 1975

[12] R M Cotta and C A C Santos ldquoNonsteady diffusion withvariable coefficients in the boundary conditionsrdquo Journal ofEngineering Physics and Thermophysics vol 61 no 5 pp 1411ndash1418 1991

[13] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering vol 2008Article ID 347568 9 pages 2008

[14] S Chantasiriwan ldquoInverse heat conduction problem of deter-mining time-dependent heat transfer coefficientrdquo InternationalJournal ofHeat andMass Transfer vol 42 no 23 pp 4275ndash42851999

[15] J Su and G F Hewitt ldquoInverse heat conduction problem ofestimating time-varying heat transfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004

[16] J Zueco F Alhama and C F G Fernandez ldquoInverse problemof estimating time-dependent heat transfer coefficient with thenetwork simulation methodrdquo Communications in NumericalMethods in Engineering vol 21 no 1 pp 39ndash48 2005

[17] D Słota ldquoUsing genetic algorithms for the determination of anheat transfer coefficient in three-phase inverse Stefan problemrdquoInternational Communications in Heat and Mass Transfer vol35 no 2 pp 149ndash156 2008

[18] H-T Chen and X-Y Wu ldquoInvestigation of heat transfercoefficient in two-dimensional transient inverse heat conduc-tion problems using the hybrid inverse schemerdquo InternationalJournal for Numerical Methods in Engineering vol 73 no 1 pp107ndash122 2008

[19] T T M Onyango D B Ingham D Lesnic and M SlodickaldquoDetermination of a time-dependent heat transfer coefficientfromnon-standard boundarymeasurementsrdquoMathematics andComputers in Simulation vol 79 no 5 pp 1577ndash1584 2009

[20] J Sladek V Sladek P H Wen and Y C Hon ldquoThe inverseproblem of determining heat transfer coefficients by the mesh-less local Petrov-Galerkin methodrdquo Computer Modeling inEngineering amp Sciences vol 48 no 2 pp 191ndash218 2009

[21] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquoASME Jour-nal of Applied Mechanics vol 63 no 2 pp 474ndash478 1996

[22] S Y Lee S Y Lu Y T Liu and H C Huang ldquoExact largedeflection of Timoshenko beams with nonlinear boundaryconditionsrdquo Computer Modeling in Engineering amp Sciences vol33 no 3 pp 293ndash312 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of