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Research Article An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient Ryoichi Chiba Department of Mechanical Systems Engineering, National Institute of Technology, Asahikawa College, 2-2-1-6 Shunkodai, Asahikawa 071-8142, Japan Correspondence should be addressed to Ryoichi Chiba; [email protected] Received 17 December 2017; Accepted 12 March 2018; Published 19 April 2018 Academic Editor: Filippo de Monte Copyright © 2018 Ryoichi Chiba. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. e composite slab, which has thermal contact resistance at −1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shiſting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. e solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. e numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs. 1. Introduction Generally, heat transfer coefficients (HTCs) vary with time, and their variations can be random or, in certain cases, peri- odic. For example, because of irregular fluid motion around turbine blades, the HTC of the blade surfaces fluctuates [1]. e surface HTCs of the fuel elements in a boiling water reactor and a pebble bed reactor also vary over time [2]. e same holds for the HTCs between a casting and its metal moulds [3], on diesel fuel droplets subjected to transient heating [4], and on solids enveloped by pulsating flows of liquid or gas in internal combustion engines [5]. In addition, during the thermal processing of foods, the pattern of air circulation may be altered around them, resulting in changes of the surface HTC [6]. e time variation in the surface aspects of objects is also one of the possible reasons (e.g., oxidation, dust contamination, and fissuring) [7]. Such a temporal change in the HTC leads to alterations in the object’s temperature. Transient heat conduction with boundary conditions including a time-dependent HTC has been investigated since the late 1960s. Studies using (semi)numerical methods include the combined use of the finite difference and Laplace transform methods [8] and the parameter-group transforma- tion followed by the Runge-Kutta shooting method [9]. Early studies using approximate methods include those of Ivanov et al. [10–12], which transformed the governing equation into a nonlinear equation using the change of variables and omitted the nonlinear term from the derived equation by restricting the research object to thin bodies. Kozlov [13] reduced the original problem to finding a solution to an infinite number of simultaneous ordinary differential equations, approximating it by a finite number. In addition, the application of the Laplace-Carson integral transform [7], finite integral transform methods with time-dependent eigenvalues [14, 15], and Lie point symmetry analysis [16] were reported. Further, an analytical method based on the Laplace transform and bifrequency transfer function was presented, but the inverse transform is extremely difficult [17]. An analytical solution to the heat conduction problem with arbitrary time-dependent HTC at the boundary had not been obtained for a long period because of the difficulty Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4707860, 11 pages https://doi.org/10.1155/2018/4707860

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Research ArticleAn Analytical Solution for Transient HeatConduction in a Composite Slab with Time-DependentHeat Transfer Coefficient

Ryoichi Chiba

Department of Mechanical Systems Engineering National Institute of Technology Asahikawa College 2-2-1-6 ShunkodaiAsahikawa 071-8142 Japan

Correspondence should be addressed to Ryoichi Chiba chibaasahikawa-nctacjp

Received 17 December 2017 Accepted 12 March 2018 Published 19 April 2018

Academic Editor Filippo de Monte

Copyright copy 2018 Ryoichi ChibaThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of 119899 layers whoseheat transfer coefficient on an external boundary is an arbitrary function of time The composite slab which has thermal contactresistance at 119899 minus 1 interfaces as well as an arbitrary initial temperature distribution and internal heat generation convectivelyexchanges heat at the external boundaries with two different time-varying surroundings To obtain the analytical solution theshifting functionmethod is first used which yields new partial differential equations under conventional types of external boundaryconditions The solution for the derived differential equations is then obtained by means of an orthogonal expansion techniqueNumerical calculations are performed for two composite slabs whose heat transfer coefficient on the heated surface is either anexponential or a trigonometric function of time The numerical results demonstrate the effects of temporal variations in the heattransfer coefficient on the transient temperature field of composite slabs

1 Introduction

Generally heat transfer coefficients (HTCs) vary with timeand their variations can be random or in certain cases peri-odic For example because of irregular fluid motion aroundturbine blades the HTC of the blade surfaces fluctuates [1]The surface HTCs of the fuel elements in a boiling waterreactor and a pebble bed reactor also vary over time [2] Thesame holds for the HTCs between a casting and its metalmoulds [3] on diesel fuel droplets subjected to transientheating [4] and on solids enveloped by pulsating flows ofliquid or gas in internal combustion engines [5] In additionduring the thermal processing of foods the pattern of aircirculation may be altered around them resulting in changesof the surface HTC [6] The time variation in the surfaceaspects of objects is also one of the possible reasons (egoxidation dust contamination and fissuring) [7] Such atemporal change in theHTC leads to alterations in the objectrsquostemperature

Transient heat conduction with boundary conditionsincluding a time-dependent HTC has been investigated

since the late 1960s Studies using (semi)numerical methodsinclude the combined use of the finite difference and Laplacetransformmethods [8] and the parameter-group transforma-tion followed by the Runge-Kutta shooting method [9] Earlystudies using approximate methods include those of Ivanovet al [10ndash12] which transformed the governing equationinto a nonlinear equation using the change of variablesand omitted the nonlinear term from the derived equationby restricting the research object to thin bodies Kozlov[13] reduced the original problem to finding a solution toan infinite number of simultaneous ordinary differentialequations approximating it by a finite number In additionthe application of the Laplace-Carson integral transform[7] finite integral transform methods with time-dependenteigenvalues [14 15] and Lie point symmetry analysis [16]were reported Further an analytical method based on theLaplace transform and bifrequency transfer function waspresented but the inverse transform is extremely difficult[17] An analytical solution to the heat conduction problemwith arbitrary time-dependent HTC at the boundary hadnot been obtained for a long period because of the difficulty

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4707860 11 pageshttpsdoiorg10115520184707860

2 Mathematical Problems in Engineering

posed by eigenfunctions and eigenvalues that depend ontime However Lee and colleagues [18] resolved this situationin 2010 they successfully derived explicit-form solutions bymeans of the shifting function method [19 20]

On the other hand there have been few studies withcomposite media Ilrsquochenko [21] obtained an approximatesolution to the heat conduction problem of a two-layeredplate with time-dependent HTC by replacing the continu-ous time function of the HTC with a piecewise functionPrikhodrsquoko [22] derived a transient temperature solution ofa two-layer plate using a special series expansion of Greenrsquosfunction while assuming that the temperature is uniformacross the thickness in one layer Yener and Ozisik [23]extended an analytical method based on a finite integraltransform [14] to solve the problem for finite compositemedia with variable boundary condition parameters All ofthem are however approximate methods leading to lessrigorous analysis

Composite (or multiregion) media have a wide appli-cation in many industrial environmental and biologicalfields (eg in turbines heat exchangers fuel cells and brakesystems) Thus it is industrially important to study this kindof heat conduction problems for composite media To bespecific the need to analyse multiregion problems includingcontact resistances has been underlined owing to the canningof fuel elements in a reactor [24] and design of thermalinsulation systems [21 22] In those systems ambient tem-peraturemay vary with time as well as HTC In studying suchengineering problems analytical solutions are highly valuableas they provide greater insight into the solution behaviourwhich is typically missing in any numerical solution Theycan be used to benchmark numerical solutions as well Theforegoing facts motivate the present study

In this paper we address the one-dimensional transientheat conduction problem of an infinite composite slab withgeneral time-dependent HTC at one external boundary Theshifting function method developed by Chen et al [18] andan orthogonal expansion method [25 26] are used to obtainan analytical solution for the multiregion heat conductioncaused by time-varying ambient temperatures Main differ-ences from the approach for single layered slab [20] arethe initial shifting of solution in each layer the additionof interface conditions and the eigenfunction expansionof functions based on the Vodicka type of orthogonalityrelationship which would be the novelty of the presentanalytical method Numerical calculations are performed toverify the solution for several test cases Subsequently twonumerical examples are given to quantify the effects of thetemporal variations in the HTC on the temperature profilesin the slabs

2 Problem Formulation

Let us consider a composite slab consisting of 119899 layers asshown in Figure 1 which has finite thickness 119867 but extendsto infinity in the other two dimensions The composite slabhas an arbitrary initial temperature distribution 119879in(119911) andis subjected to two different arbitrary time-varying ambienttemperatures 1199101(119905) and 1199102(119905) at its external boundaries

z0 = 0

z1

z2

ziminus1

zi

zi+1

znminus1

zn = H

y2(t)

ℎ1

z

ℎ2(t)

1 1 Q1(z t)

2 2 Q2(z t)

r1

i i Qi(z t)

i+1 i+1 Qi+1(z t)

n n Qn(z t)rnminus1

r2

riminus1

ri

ri+1

InitiallyT = TＣＨ(z)

x

Layer 1

Layer 2

Layer i

Layer i + 1

Layer n

y1(t)

Figure 1 Schematic of an infinite composite slab with a time-dependent heat transfer coefficient on one external boundary

The boundaries have different HTCs ℎ1 and ℎ2(119905) one ofwhich depends on time At the 119899 minus 1 interfacial boundariesheat is transferred through different constant conductancecoefficients 119903119894 under the condition of heat flux continuity Atime- and space-dependent internal heat generation 119876119894(119911 119905)is also considered in each layer The material of each layeris isotropic and homogeneous with constant thermophysicalproperties

The temperature distribution in the 119894th layer of 119899 solidlyjoined layers is given by the diffusion equation

1120581119894120597119879119894120597119905 = 12059721198791198941205971199112 + 119876119894 (119911 119905)120582119894 for 119894 = 1 2 119899 (1)

This composite slab is subject to the following initial andexternal boundary conditions

119879119894 = 119879in (119911)at 119905 = 0 for 119894 = 1 2 119899

1205821 1205971198791120597119911 minus ℎ1 [1198791 minus 1199101 (119905)] = 0 at 119911 = 0120582119899 120597119879119899120597119911 + ℎ2 (119905) [119879119899 minus 1199102 (119905)] = 0 at 119911 = 119867

(2)

At the internal boundaries the slab is subject to the followinginterface conditions

120582119894 120597119879119894120597119911 + 119903119894 (119879119894 minus 119879119894+1) = 0at 119911 = 119911119894 for 119894 = 1 2 119899 minus 1

(3)

Mathematical Problems in Engineering 3

120582119894 120597119879119894120597119911 = 120582119894+1 120597119879119894+1120597119911at 119911 = 119911119894 for 119894 = 1 2 119899 minus 1

(4)

where (3) implies the discontinuity of temperature at theinterfaces and (4) states that the heat flux is continuous at theinterfaces When 119903119894 rarrinfin (3) leads to 119879119894 minus119879119894+1 = 0 implyingthe continuity of temperature or perfect thermal contact atthe interfaces For convenience we nondimensionalise (1) asfollows

1120581119894120597120579119894120597120591 = 12059721205791198941205971205852 + 119876119894 (120585 120591) for 119894 = 1 2 119899 (5)

and the initial boundary and interface conditions may bestated as

120579119894 = 120579in (120585)at 120591 = 0 for 119894 = 1 2 119899 (6)

1205971205791120597120585 minus 1198611 [1205791 minus 1198841 (120591)] = 0 at 120585 = 0 (7)

120597120579119899120597120585 + 1198612 (120591) 120579119899 = 120595 (120591) at 120585 = 1 (8)

120597120579119894120597120585 + 119877119894 (120579119894 minus 120579119894+1) = 0at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(9)

120582119894 120597120579119894120597120585 = 120582119894+1 120597120579119894+1120597120585at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(10)

where the dimensionless quantities are defined as follows

1198611 = ℎ11198671205821 1198612 (120591) = ℎ2 (119905)119867120582119899

119876119894 (120585 120591) = 119876119894 (119911 119905)11986721205821198941198790 119877119894 = 119903119894119867120582119894

119884119895 (120591) = 119910119895 (119905)1198790 for 119895 = 1 2120579119894 (120585 120591) = 119879119894 (119911 119905)1198790 120579in (120585) = 119879in (119911)1198790

120581119894 = 1205811198941205810 120582119894 = 1205821198941205820 120585 = 119911119867120585119894 = 119911119894119867120591 = 12058101199051198672

120595 (120591) = 1198612 (120591) 1198842 (120591) (11)

For very large values of 119877119894 at the interfacial boundaries theproblem is reduced to that of temperature continuity at theinterfaces

To make the following analysis manageable 1198612(120591) isseparated in the form of

1198612 (120591) = 120575 + 119865 (120591) (12)

where 120575 is the initial value of the Biot number that is 1198612(0)Substituting (12) into (8) yields

120597120579119899120597120585 + 120575120579119899 = minus119865 (120591) 120579119899 + 120595 (120591) at 120585 = 1 (13)

3 Analysis

31 Shifting FunctionMethod To solve the system of (5)ndash(10)we first introduce the substitution [20]

120579119894 (120585 120591) = V119894 (120585 120591) + 119891 (120591) 119892119894 (120585) for 119894 = 1 2 119899 (14)

where V119894(120585 120591) is the transformed function 119892119894(120585) is a shiftingfunction to be determined and 119891(120591) is the auxiliary functiondefined as

119891 (120591) = minus119865 (120591) 120579119899 (1 120591) + 120595 (120591) (15)

Substituting (14) into (5)ndash(7) (9) (10) and (13) yields

1120581119894 [120597V119894 (120585 120591)120597120591 + d119891 (120591)

d120591 119892119894 (120585)] = 1205972V119894 (120585 120591)1205971205852+ 119891 (120591) d2119892119894 (120585)

d1205852 + 119876119894 (120585 120591)for 119894 = 1 2 119899

(16)

V119894 (120585 0) + 119891 (0) 119892119894 (120585) = 120579in (120585) for 119894 = 1 2 119899 (17)

120597V1 (0 120591)120597120585 + 119891 (120591) d1198921 (0)d120585 minus 1198611 [V1 (0 120591)

+ 119891 (120591) 1198921 (0) minus 1198841 (120591)] = 0(18)

4 Mathematical Problems in Engineering

120597V119899 (1 120591)120597120585 + 119891 (120591) d119892119899 (1)d120585 + 120575 [V119899 (1 120591)

+ 119891 (120591) 119892119899 (1)] = 119891 (120591) (19)

120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 + 119877119894 [V119894 (120585119894 120591)

+ 119891 (120591) 119892119894 (120585119894) minus V119894+1 (120585119894 120591) minus 119891 (120591) 119892119894+1 (120585119894)] = 0for 119894 = 1 2 119899 minus 1

(20)

120582119894 [120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 ] = 120582119894+1 [120597V119894+1 (120585119894 120591)120597120585

+ 119891 (120591) d119892119894+1 (120585119894)d120585 ] for 119894 = 1 2 119899 minus 1

(21)

Following the procedure of the shifting function methodwe suitably assume the shifting function119892119894(120585) to be a functionsatisfying the following system of equations

d2119892119894 (120585)d1205852 = 119886

for 119894 = 1 2 119899(22)

d1198921 (0)d120585 minus 11986111198921 (0) = 0 (23)

d119892119899 (1)d120585 = 1 (24)

119892119899 (1) = 0 (25)

d119892119894 (120585119894)d120585 + 119877119894 [119892119894 (120585119894) minus 119892119894+1 (120585119894)] = 0

for 119894 = 1 2 119899 minus 1(26)

120582119894 d119892119894 (120585119894)d120585 = 120582119894+1 d119892119894+1 (120585119894)d120585for 119894 = 1 2 119899 minus 1

(27)

where 119886 is an unknown constantThe general solution of (22)is given by

119892119894 (120585) = 11988621205852 + 119887119894120585 + 119888119894 (28)

where 119887119894 and 119888119894 (119894 = 1 2 119899) are additional unknownconstants A total of (2119899 + 1) unknown constants are readilydetermined from (23)ndash(27)

Substituting (25) into (14) yields

120579119899 (1 120591) = V119899 (1 120591) (29)

The substitution of (22) and (25) into (16) results in thefollowing partial differential equation

120597V119894 (120585 120591)120597120591 minus 120581119894 1205972V119894 (120585 120591)1205971205852 minus 119892119894 (120585) 119865 (120591) V119899 (1 120591)+ [minus119892119894 (120585) (120591) + 120581119894119886119865 (120591)] V119899 (1 120591)

= minus119892119894 (120585) (120591) + 120581119894119886120595 (120591) + 120581119894119876119894 (120585 120591)for 119894 = 1 2 119899

(30)

where the ldquoover-dotrdquo denotes the time derivative Consider-ing (23)ndash(27) the associated initial boundary and interfaceconditions are obtained as follows

V119894 (120585 0) = 120579in (120585) minus 120595 (0) 119892119894 (120585) for 119894 = 1 2 119899 (31)

120597V1 (0 120591)120597120585 minus 1198611 [V1 (0 120591) minus 1198841 (120591)] = 0 (32)

120597V119899 (1 120591)120597120585 + 120575V119899 (1 120591) = 0 (33)

120597V119894 (120585119894 120591)120597120585 + 119877119894 [V119894 (120585119894 120591) minus V119894+1 (120585119894 120591)] = 0for 119894 = 1 2 119899 minus 1

(34)

120582119894 120597V119894 (120585119894 120591)120597120585 = 120582119894+1 120597V119894+1 (120585119894 120591)120597120585for 119894 = 1 2 119899 minus 1

(35)

32 Orthogonal Expansion Technique The next step is tofind a solution to the multiregion heat conduction problemexpressed by (30)ndash(35) In the literature various analyticalmethods are available for transient heat conduction in com-posite media such as the orthogonal expansion techniqueLaplace transformmethodmethod of separation of variablesGreenrsquos function method and finite integral transform tech-nique In the present work Vodickarsquos method [25] whichhas been proven effective in analysing multiregion problemswith general space- and time-dependent heat sources andnonhomogeneous boundary conditions is applied to solvethe system of (30)ndash(35)

The solution to (30)ndash(35) is assumed to be

V119894 (120585 120591) = infinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 119871 119894 (120585) 1198841 (120591)for 119894 = 1 2 119899

(36)

with the following function

119871 119894 (120585) = 119862119894120585 + 119863119894 for 119894 = 1 2 119899 (37)

The constants 119862119894 and 119863119894 are obtained from the followingrelationships

d1198711 (0)d120585 minus 11986111198711 (0) = 1198611d119871119899 (1)d120585 + 120575119871119899 (1) = 0

d119871 119894 (120585119894)d120585 + 119877119894 [119871 119894 (120585119894) minus 119871 119894+1 (120585119894)] = 0

Mathematical Problems in Engineering 5

120582119894 d119871 119894 (120585119894)d120585 = 120582119894+1 d119871 119894+1 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(38)

The function 119883119894119898(120585) is the solution to the eigenvalueproblem corresponding to (30)ndash(35) and is given as follows

119883119894119898 (120585) = 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585) (39)

where 119889119894119898 = 120574119898radic120581119894 and 120574119898 is an eigenvalue The unknownconstants 119860 119894119898 and 119861119894119898 are determined from the followingconditions which are obtained by substituting (36)ndash(39) intothe boundary and interface conditions (32)ndash(35)

d1198831119898 (0)d120585 minus 11986111198831119898 (0) = 0

d119883119899119898 (1)d120585 + 120575119883119899119898 (1) = 0

d119883119894119898 (120585119894)d120585 + 119877119894 [119883119894119898 (120585119894) minus 119883(119894+1)119898 (120585119894)] = 0

120582119894 d119883119894119898 (120585119894)d120585 = 120582119894+1 d119883(119894+1)119898 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(40)

In other words a system of 2119899 linear homogeneous equationsis obtained for the constants 119860 119894119898 and 119861119894119898 The eigenvalues120574119898 (119898 = 1 2 ) are obtained from the condition underwhich this system of equations has a nontrivial solution andare therefore positive roots of the following transcendentalequation in matrix notation

G sdot E1 sdot E2 sdot sdot sdotE119899minus1 sdot u = 0 (41)

whereG = [minus1198611 1198891119898] E119894 = J119894 sdot K119894+1J119894 = [120582119894119889119894119898 cos (119889119894119898120585119894) minus119889119894119898 cos (119889119894119898120585119894) minus 119877119894 sin (119889119894119898120585119894)

120582119894119889119894119898 sin (119889119894119898120585119894) minus119889119894119898 sin (119889119894119898120585119894) + 119877119894 cos (119889119894119898120585119894)] K119894+1

= [ 119877119894 cos [119889(119894+1)119898120585119894] 119877119894 sin [119889(119894+1)119898120585119894]minus120582119894+1119889(119894+1)119898 sin [119889(119894+1)119898120585119894] 120582119894+1119889(119894+1)119898 cos [119889(119894+1)119898120585119894]]

u = [119889119899119898 cos (119889119899119898) + 120575 sin (119889119899119898)119889119899119898 sin (119889119899119898) minus 120575 cos (119889119899119898)]

(42)

By substituting (36) into (31) the following equation isobtained

119882119894 (120585) = infinsum119898=1

120601119898 (0)119883119894119898 (120585)= 120579in (120585) minus 120595 (0) 119892119894 (120585) + 119871 119894 (120585) 1198841 (0)

(43)

The eigenfunction 119883119894119898(120585) given by (39) has an orthogonalrelationship with discontinuous weighting function it isexpressed as follows [26]

119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119883119894119897 (120585)119883119894119896 (120585) d120585 =

const (119897 = 119896)0 (119897 = 119896) (44)

Functions 119892119894(120585)119882119894(120585) 119871 119894(120585) and 120581119894119876119894(120585 120591) and a constant 120581119894can be expanded in an infinite series of119883119894119898(120585)

119892119894 (120585) = infinsum119898=1

120572119898119883119894119898 (120585)

119882119894 (120585) = infinsum119898=1

119908119898119883119894119898 (120585)

119871 119894 (120585) = infinsum119898=1

119897119898119883119894119898 (120585)

120581119894119876119894 (120585 120591) = infinsum119898=1

119902119898 (120591)119883119894119898 (120585)

120581119894 = infinsum119898=1

120578119898119883119894119898 (120585)

(45)

where the expansion coefficients are given by

120572119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119892119894 (120585)119883119894119898 (120585) d120585

119908119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119882119894 (120585)119883119894119898 (120585) d120585

119897119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119871 119894 (120585)119883119894119898 (120585) d120585

119902119898 (120591) = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119876119894 (120585 120591)119883119894119898 (120585) d120585

120578119898 = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119883119894119898 (120585) d120585

(46)

in which the norm119873119898 is119873119898 = 119899sum

119894=1

120582119894120581119894 int120585119894

120585119894minus1[119883119894119898 (120585)]2 d120585 (47)

The substitution of (36) and (45) into (30) yields thefollowing equationinfinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 1198971198981 (120591)119883119894119898 (120585)+ 1205742119898120601119898 (120591)119883119894119898 (120585) minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)+ 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 2: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

2 Mathematical Problems in Engineering

posed by eigenfunctions and eigenvalues that depend ontime However Lee and colleagues [18] resolved this situationin 2010 they successfully derived explicit-form solutions bymeans of the shifting function method [19 20]

On the other hand there have been few studies withcomposite media Ilrsquochenko [21] obtained an approximatesolution to the heat conduction problem of a two-layeredplate with time-dependent HTC by replacing the continu-ous time function of the HTC with a piecewise functionPrikhodrsquoko [22] derived a transient temperature solution ofa two-layer plate using a special series expansion of Greenrsquosfunction while assuming that the temperature is uniformacross the thickness in one layer Yener and Ozisik [23]extended an analytical method based on a finite integraltransform [14] to solve the problem for finite compositemedia with variable boundary condition parameters All ofthem are however approximate methods leading to lessrigorous analysis

Composite (or multiregion) media have a wide appli-cation in many industrial environmental and biologicalfields (eg in turbines heat exchangers fuel cells and brakesystems) Thus it is industrially important to study this kindof heat conduction problems for composite media To bespecific the need to analyse multiregion problems includingcontact resistances has been underlined owing to the canningof fuel elements in a reactor [24] and design of thermalinsulation systems [21 22] In those systems ambient tem-peraturemay vary with time as well as HTC In studying suchengineering problems analytical solutions are highly valuableas they provide greater insight into the solution behaviourwhich is typically missing in any numerical solution Theycan be used to benchmark numerical solutions as well Theforegoing facts motivate the present study

In this paper we address the one-dimensional transientheat conduction problem of an infinite composite slab withgeneral time-dependent HTC at one external boundary Theshifting function method developed by Chen et al [18] andan orthogonal expansion method [25 26] are used to obtainan analytical solution for the multiregion heat conductioncaused by time-varying ambient temperatures Main differ-ences from the approach for single layered slab [20] arethe initial shifting of solution in each layer the additionof interface conditions and the eigenfunction expansionof functions based on the Vodicka type of orthogonalityrelationship which would be the novelty of the presentanalytical method Numerical calculations are performed toverify the solution for several test cases Subsequently twonumerical examples are given to quantify the effects of thetemporal variations in the HTC on the temperature profilesin the slabs

2 Problem Formulation

Let us consider a composite slab consisting of 119899 layers asshown in Figure 1 which has finite thickness 119867 but extendsto infinity in the other two dimensions The composite slabhas an arbitrary initial temperature distribution 119879in(119911) andis subjected to two different arbitrary time-varying ambienttemperatures 1199101(119905) and 1199102(119905) at its external boundaries

z0 = 0

z1

z2

ziminus1

zi

zi+1

znminus1

zn = H

y2(t)

ℎ1

z

ℎ2(t)

1 1 Q1(z t)

2 2 Q2(z t)

r1

i i Qi(z t)

i+1 i+1 Qi+1(z t)

n n Qn(z t)rnminus1

r2

riminus1

ri

ri+1

InitiallyT = TＣＨ(z)

x

Layer 1

Layer 2

Layer i

Layer i + 1

Layer n

y1(t)

Figure 1 Schematic of an infinite composite slab with a time-dependent heat transfer coefficient on one external boundary

The boundaries have different HTCs ℎ1 and ℎ2(119905) one ofwhich depends on time At the 119899 minus 1 interfacial boundariesheat is transferred through different constant conductancecoefficients 119903119894 under the condition of heat flux continuity Atime- and space-dependent internal heat generation 119876119894(119911 119905)is also considered in each layer The material of each layeris isotropic and homogeneous with constant thermophysicalproperties

The temperature distribution in the 119894th layer of 119899 solidlyjoined layers is given by the diffusion equation

1120581119894120597119879119894120597119905 = 12059721198791198941205971199112 + 119876119894 (119911 119905)120582119894 for 119894 = 1 2 119899 (1)

This composite slab is subject to the following initial andexternal boundary conditions

119879119894 = 119879in (119911)at 119905 = 0 for 119894 = 1 2 119899

1205821 1205971198791120597119911 minus ℎ1 [1198791 minus 1199101 (119905)] = 0 at 119911 = 0120582119899 120597119879119899120597119911 + ℎ2 (119905) [119879119899 minus 1199102 (119905)] = 0 at 119911 = 119867

(2)

At the internal boundaries the slab is subject to the followinginterface conditions

120582119894 120597119879119894120597119911 + 119903119894 (119879119894 minus 119879119894+1) = 0at 119911 = 119911119894 for 119894 = 1 2 119899 minus 1

(3)

Mathematical Problems in Engineering 3

120582119894 120597119879119894120597119911 = 120582119894+1 120597119879119894+1120597119911at 119911 = 119911119894 for 119894 = 1 2 119899 minus 1

(4)

where (3) implies the discontinuity of temperature at theinterfaces and (4) states that the heat flux is continuous at theinterfaces When 119903119894 rarrinfin (3) leads to 119879119894 minus119879119894+1 = 0 implyingthe continuity of temperature or perfect thermal contact atthe interfaces For convenience we nondimensionalise (1) asfollows

1120581119894120597120579119894120597120591 = 12059721205791198941205971205852 + 119876119894 (120585 120591) for 119894 = 1 2 119899 (5)

and the initial boundary and interface conditions may bestated as

120579119894 = 120579in (120585)at 120591 = 0 for 119894 = 1 2 119899 (6)

1205971205791120597120585 minus 1198611 [1205791 minus 1198841 (120591)] = 0 at 120585 = 0 (7)

120597120579119899120597120585 + 1198612 (120591) 120579119899 = 120595 (120591) at 120585 = 1 (8)

120597120579119894120597120585 + 119877119894 (120579119894 minus 120579119894+1) = 0at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(9)

120582119894 120597120579119894120597120585 = 120582119894+1 120597120579119894+1120597120585at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(10)

where the dimensionless quantities are defined as follows

1198611 = ℎ11198671205821 1198612 (120591) = ℎ2 (119905)119867120582119899

119876119894 (120585 120591) = 119876119894 (119911 119905)11986721205821198941198790 119877119894 = 119903119894119867120582119894

119884119895 (120591) = 119910119895 (119905)1198790 for 119895 = 1 2120579119894 (120585 120591) = 119879119894 (119911 119905)1198790 120579in (120585) = 119879in (119911)1198790

120581119894 = 1205811198941205810 120582119894 = 1205821198941205820 120585 = 119911119867120585119894 = 119911119894119867120591 = 12058101199051198672

120595 (120591) = 1198612 (120591) 1198842 (120591) (11)

For very large values of 119877119894 at the interfacial boundaries theproblem is reduced to that of temperature continuity at theinterfaces

To make the following analysis manageable 1198612(120591) isseparated in the form of

1198612 (120591) = 120575 + 119865 (120591) (12)

where 120575 is the initial value of the Biot number that is 1198612(0)Substituting (12) into (8) yields

120597120579119899120597120585 + 120575120579119899 = minus119865 (120591) 120579119899 + 120595 (120591) at 120585 = 1 (13)

3 Analysis

31 Shifting FunctionMethod To solve the system of (5)ndash(10)we first introduce the substitution [20]

120579119894 (120585 120591) = V119894 (120585 120591) + 119891 (120591) 119892119894 (120585) for 119894 = 1 2 119899 (14)

where V119894(120585 120591) is the transformed function 119892119894(120585) is a shiftingfunction to be determined and 119891(120591) is the auxiliary functiondefined as

119891 (120591) = minus119865 (120591) 120579119899 (1 120591) + 120595 (120591) (15)

Substituting (14) into (5)ndash(7) (9) (10) and (13) yields

1120581119894 [120597V119894 (120585 120591)120597120591 + d119891 (120591)

d120591 119892119894 (120585)] = 1205972V119894 (120585 120591)1205971205852+ 119891 (120591) d2119892119894 (120585)

d1205852 + 119876119894 (120585 120591)for 119894 = 1 2 119899

(16)

V119894 (120585 0) + 119891 (0) 119892119894 (120585) = 120579in (120585) for 119894 = 1 2 119899 (17)

120597V1 (0 120591)120597120585 + 119891 (120591) d1198921 (0)d120585 minus 1198611 [V1 (0 120591)

+ 119891 (120591) 1198921 (0) minus 1198841 (120591)] = 0(18)

4 Mathematical Problems in Engineering

120597V119899 (1 120591)120597120585 + 119891 (120591) d119892119899 (1)d120585 + 120575 [V119899 (1 120591)

+ 119891 (120591) 119892119899 (1)] = 119891 (120591) (19)

120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 + 119877119894 [V119894 (120585119894 120591)

+ 119891 (120591) 119892119894 (120585119894) minus V119894+1 (120585119894 120591) minus 119891 (120591) 119892119894+1 (120585119894)] = 0for 119894 = 1 2 119899 minus 1

(20)

120582119894 [120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 ] = 120582119894+1 [120597V119894+1 (120585119894 120591)120597120585

+ 119891 (120591) d119892119894+1 (120585119894)d120585 ] for 119894 = 1 2 119899 minus 1

(21)

Following the procedure of the shifting function methodwe suitably assume the shifting function119892119894(120585) to be a functionsatisfying the following system of equations

d2119892119894 (120585)d1205852 = 119886

for 119894 = 1 2 119899(22)

d1198921 (0)d120585 minus 11986111198921 (0) = 0 (23)

d119892119899 (1)d120585 = 1 (24)

119892119899 (1) = 0 (25)

d119892119894 (120585119894)d120585 + 119877119894 [119892119894 (120585119894) minus 119892119894+1 (120585119894)] = 0

for 119894 = 1 2 119899 minus 1(26)

120582119894 d119892119894 (120585119894)d120585 = 120582119894+1 d119892119894+1 (120585119894)d120585for 119894 = 1 2 119899 minus 1

(27)

where 119886 is an unknown constantThe general solution of (22)is given by

119892119894 (120585) = 11988621205852 + 119887119894120585 + 119888119894 (28)

where 119887119894 and 119888119894 (119894 = 1 2 119899) are additional unknownconstants A total of (2119899 + 1) unknown constants are readilydetermined from (23)ndash(27)

Substituting (25) into (14) yields

120579119899 (1 120591) = V119899 (1 120591) (29)

The substitution of (22) and (25) into (16) results in thefollowing partial differential equation

120597V119894 (120585 120591)120597120591 minus 120581119894 1205972V119894 (120585 120591)1205971205852 minus 119892119894 (120585) 119865 (120591) V119899 (1 120591)+ [minus119892119894 (120585) (120591) + 120581119894119886119865 (120591)] V119899 (1 120591)

= minus119892119894 (120585) (120591) + 120581119894119886120595 (120591) + 120581119894119876119894 (120585 120591)for 119894 = 1 2 119899

(30)

where the ldquoover-dotrdquo denotes the time derivative Consider-ing (23)ndash(27) the associated initial boundary and interfaceconditions are obtained as follows

V119894 (120585 0) = 120579in (120585) minus 120595 (0) 119892119894 (120585) for 119894 = 1 2 119899 (31)

120597V1 (0 120591)120597120585 minus 1198611 [V1 (0 120591) minus 1198841 (120591)] = 0 (32)

120597V119899 (1 120591)120597120585 + 120575V119899 (1 120591) = 0 (33)

120597V119894 (120585119894 120591)120597120585 + 119877119894 [V119894 (120585119894 120591) minus V119894+1 (120585119894 120591)] = 0for 119894 = 1 2 119899 minus 1

(34)

120582119894 120597V119894 (120585119894 120591)120597120585 = 120582119894+1 120597V119894+1 (120585119894 120591)120597120585for 119894 = 1 2 119899 minus 1

(35)

32 Orthogonal Expansion Technique The next step is tofind a solution to the multiregion heat conduction problemexpressed by (30)ndash(35) In the literature various analyticalmethods are available for transient heat conduction in com-posite media such as the orthogonal expansion techniqueLaplace transformmethodmethod of separation of variablesGreenrsquos function method and finite integral transform tech-nique In the present work Vodickarsquos method [25] whichhas been proven effective in analysing multiregion problemswith general space- and time-dependent heat sources andnonhomogeneous boundary conditions is applied to solvethe system of (30)ndash(35)

The solution to (30)ndash(35) is assumed to be

V119894 (120585 120591) = infinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 119871 119894 (120585) 1198841 (120591)for 119894 = 1 2 119899

(36)

with the following function

119871 119894 (120585) = 119862119894120585 + 119863119894 for 119894 = 1 2 119899 (37)

The constants 119862119894 and 119863119894 are obtained from the followingrelationships

d1198711 (0)d120585 minus 11986111198711 (0) = 1198611d119871119899 (1)d120585 + 120575119871119899 (1) = 0

d119871 119894 (120585119894)d120585 + 119877119894 [119871 119894 (120585119894) minus 119871 119894+1 (120585119894)] = 0

Mathematical Problems in Engineering 5

120582119894 d119871 119894 (120585119894)d120585 = 120582119894+1 d119871 119894+1 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(38)

The function 119883119894119898(120585) is the solution to the eigenvalueproblem corresponding to (30)ndash(35) and is given as follows

119883119894119898 (120585) = 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585) (39)

where 119889119894119898 = 120574119898radic120581119894 and 120574119898 is an eigenvalue The unknownconstants 119860 119894119898 and 119861119894119898 are determined from the followingconditions which are obtained by substituting (36)ndash(39) intothe boundary and interface conditions (32)ndash(35)

d1198831119898 (0)d120585 minus 11986111198831119898 (0) = 0

d119883119899119898 (1)d120585 + 120575119883119899119898 (1) = 0

d119883119894119898 (120585119894)d120585 + 119877119894 [119883119894119898 (120585119894) minus 119883(119894+1)119898 (120585119894)] = 0

120582119894 d119883119894119898 (120585119894)d120585 = 120582119894+1 d119883(119894+1)119898 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(40)

In other words a system of 2119899 linear homogeneous equationsis obtained for the constants 119860 119894119898 and 119861119894119898 The eigenvalues120574119898 (119898 = 1 2 ) are obtained from the condition underwhich this system of equations has a nontrivial solution andare therefore positive roots of the following transcendentalequation in matrix notation

G sdot E1 sdot E2 sdot sdot sdotE119899minus1 sdot u = 0 (41)

whereG = [minus1198611 1198891119898] E119894 = J119894 sdot K119894+1J119894 = [120582119894119889119894119898 cos (119889119894119898120585119894) minus119889119894119898 cos (119889119894119898120585119894) minus 119877119894 sin (119889119894119898120585119894)

120582119894119889119894119898 sin (119889119894119898120585119894) minus119889119894119898 sin (119889119894119898120585119894) + 119877119894 cos (119889119894119898120585119894)] K119894+1

= [ 119877119894 cos [119889(119894+1)119898120585119894] 119877119894 sin [119889(119894+1)119898120585119894]minus120582119894+1119889(119894+1)119898 sin [119889(119894+1)119898120585119894] 120582119894+1119889(119894+1)119898 cos [119889(119894+1)119898120585119894]]

u = [119889119899119898 cos (119889119899119898) + 120575 sin (119889119899119898)119889119899119898 sin (119889119899119898) minus 120575 cos (119889119899119898)]

(42)

By substituting (36) into (31) the following equation isobtained

119882119894 (120585) = infinsum119898=1

120601119898 (0)119883119894119898 (120585)= 120579in (120585) minus 120595 (0) 119892119894 (120585) + 119871 119894 (120585) 1198841 (0)

(43)

The eigenfunction 119883119894119898(120585) given by (39) has an orthogonalrelationship with discontinuous weighting function it isexpressed as follows [26]

119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119883119894119897 (120585)119883119894119896 (120585) d120585 =

const (119897 = 119896)0 (119897 = 119896) (44)

Functions 119892119894(120585)119882119894(120585) 119871 119894(120585) and 120581119894119876119894(120585 120591) and a constant 120581119894can be expanded in an infinite series of119883119894119898(120585)

119892119894 (120585) = infinsum119898=1

120572119898119883119894119898 (120585)

119882119894 (120585) = infinsum119898=1

119908119898119883119894119898 (120585)

119871 119894 (120585) = infinsum119898=1

119897119898119883119894119898 (120585)

120581119894119876119894 (120585 120591) = infinsum119898=1

119902119898 (120591)119883119894119898 (120585)

120581119894 = infinsum119898=1

120578119898119883119894119898 (120585)

(45)

where the expansion coefficients are given by

120572119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119892119894 (120585)119883119894119898 (120585) d120585

119908119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119882119894 (120585)119883119894119898 (120585) d120585

119897119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119871 119894 (120585)119883119894119898 (120585) d120585

119902119898 (120591) = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119876119894 (120585 120591)119883119894119898 (120585) d120585

120578119898 = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119883119894119898 (120585) d120585

(46)

in which the norm119873119898 is119873119898 = 119899sum

119894=1

120582119894120581119894 int120585119894

120585119894minus1[119883119894119898 (120585)]2 d120585 (47)

The substitution of (36) and (45) into (30) yields thefollowing equationinfinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 1198971198981 (120591)119883119894119898 (120585)+ 1205742119898120601119898 (120591)119883119894119898 (120585) minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)+ 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 3: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Mathematical Problems in Engineering 3

120582119894 120597119879119894120597119911 = 120582119894+1 120597119879119894+1120597119911at 119911 = 119911119894 for 119894 = 1 2 119899 minus 1

(4)

where (3) implies the discontinuity of temperature at theinterfaces and (4) states that the heat flux is continuous at theinterfaces When 119903119894 rarrinfin (3) leads to 119879119894 minus119879119894+1 = 0 implyingthe continuity of temperature or perfect thermal contact atthe interfaces For convenience we nondimensionalise (1) asfollows

1120581119894120597120579119894120597120591 = 12059721205791198941205971205852 + 119876119894 (120585 120591) for 119894 = 1 2 119899 (5)

and the initial boundary and interface conditions may bestated as

120579119894 = 120579in (120585)at 120591 = 0 for 119894 = 1 2 119899 (6)

1205971205791120597120585 minus 1198611 [1205791 minus 1198841 (120591)] = 0 at 120585 = 0 (7)

120597120579119899120597120585 + 1198612 (120591) 120579119899 = 120595 (120591) at 120585 = 1 (8)

120597120579119894120597120585 + 119877119894 (120579119894 minus 120579119894+1) = 0at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(9)

120582119894 120597120579119894120597120585 = 120582119894+1 120597120579119894+1120597120585at 120585 = 120585119894 for 119894 = 1 2 119899 minus 1

(10)

where the dimensionless quantities are defined as follows

1198611 = ℎ11198671205821 1198612 (120591) = ℎ2 (119905)119867120582119899

119876119894 (120585 120591) = 119876119894 (119911 119905)11986721205821198941198790 119877119894 = 119903119894119867120582119894

119884119895 (120591) = 119910119895 (119905)1198790 for 119895 = 1 2120579119894 (120585 120591) = 119879119894 (119911 119905)1198790 120579in (120585) = 119879in (119911)1198790

120581119894 = 1205811198941205810 120582119894 = 1205821198941205820 120585 = 119911119867120585119894 = 119911119894119867120591 = 12058101199051198672

120595 (120591) = 1198612 (120591) 1198842 (120591) (11)

For very large values of 119877119894 at the interfacial boundaries theproblem is reduced to that of temperature continuity at theinterfaces

To make the following analysis manageable 1198612(120591) isseparated in the form of

1198612 (120591) = 120575 + 119865 (120591) (12)

where 120575 is the initial value of the Biot number that is 1198612(0)Substituting (12) into (8) yields

120597120579119899120597120585 + 120575120579119899 = minus119865 (120591) 120579119899 + 120595 (120591) at 120585 = 1 (13)

3 Analysis

31 Shifting FunctionMethod To solve the system of (5)ndash(10)we first introduce the substitution [20]

120579119894 (120585 120591) = V119894 (120585 120591) + 119891 (120591) 119892119894 (120585) for 119894 = 1 2 119899 (14)

where V119894(120585 120591) is the transformed function 119892119894(120585) is a shiftingfunction to be determined and 119891(120591) is the auxiliary functiondefined as

119891 (120591) = minus119865 (120591) 120579119899 (1 120591) + 120595 (120591) (15)

Substituting (14) into (5)ndash(7) (9) (10) and (13) yields

1120581119894 [120597V119894 (120585 120591)120597120591 + d119891 (120591)

d120591 119892119894 (120585)] = 1205972V119894 (120585 120591)1205971205852+ 119891 (120591) d2119892119894 (120585)

d1205852 + 119876119894 (120585 120591)for 119894 = 1 2 119899

(16)

V119894 (120585 0) + 119891 (0) 119892119894 (120585) = 120579in (120585) for 119894 = 1 2 119899 (17)

120597V1 (0 120591)120597120585 + 119891 (120591) d1198921 (0)d120585 minus 1198611 [V1 (0 120591)

+ 119891 (120591) 1198921 (0) minus 1198841 (120591)] = 0(18)

4 Mathematical Problems in Engineering

120597V119899 (1 120591)120597120585 + 119891 (120591) d119892119899 (1)d120585 + 120575 [V119899 (1 120591)

+ 119891 (120591) 119892119899 (1)] = 119891 (120591) (19)

120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 + 119877119894 [V119894 (120585119894 120591)

+ 119891 (120591) 119892119894 (120585119894) minus V119894+1 (120585119894 120591) minus 119891 (120591) 119892119894+1 (120585119894)] = 0for 119894 = 1 2 119899 minus 1

(20)

120582119894 [120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 ] = 120582119894+1 [120597V119894+1 (120585119894 120591)120597120585

+ 119891 (120591) d119892119894+1 (120585119894)d120585 ] for 119894 = 1 2 119899 minus 1

(21)

Following the procedure of the shifting function methodwe suitably assume the shifting function119892119894(120585) to be a functionsatisfying the following system of equations

d2119892119894 (120585)d1205852 = 119886

for 119894 = 1 2 119899(22)

d1198921 (0)d120585 minus 11986111198921 (0) = 0 (23)

d119892119899 (1)d120585 = 1 (24)

119892119899 (1) = 0 (25)

d119892119894 (120585119894)d120585 + 119877119894 [119892119894 (120585119894) minus 119892119894+1 (120585119894)] = 0

for 119894 = 1 2 119899 minus 1(26)

120582119894 d119892119894 (120585119894)d120585 = 120582119894+1 d119892119894+1 (120585119894)d120585for 119894 = 1 2 119899 minus 1

(27)

where 119886 is an unknown constantThe general solution of (22)is given by

119892119894 (120585) = 11988621205852 + 119887119894120585 + 119888119894 (28)

where 119887119894 and 119888119894 (119894 = 1 2 119899) are additional unknownconstants A total of (2119899 + 1) unknown constants are readilydetermined from (23)ndash(27)

Substituting (25) into (14) yields

120579119899 (1 120591) = V119899 (1 120591) (29)

The substitution of (22) and (25) into (16) results in thefollowing partial differential equation

120597V119894 (120585 120591)120597120591 minus 120581119894 1205972V119894 (120585 120591)1205971205852 minus 119892119894 (120585) 119865 (120591) V119899 (1 120591)+ [minus119892119894 (120585) (120591) + 120581119894119886119865 (120591)] V119899 (1 120591)

= minus119892119894 (120585) (120591) + 120581119894119886120595 (120591) + 120581119894119876119894 (120585 120591)for 119894 = 1 2 119899

(30)

where the ldquoover-dotrdquo denotes the time derivative Consider-ing (23)ndash(27) the associated initial boundary and interfaceconditions are obtained as follows

V119894 (120585 0) = 120579in (120585) minus 120595 (0) 119892119894 (120585) for 119894 = 1 2 119899 (31)

120597V1 (0 120591)120597120585 minus 1198611 [V1 (0 120591) minus 1198841 (120591)] = 0 (32)

120597V119899 (1 120591)120597120585 + 120575V119899 (1 120591) = 0 (33)

120597V119894 (120585119894 120591)120597120585 + 119877119894 [V119894 (120585119894 120591) minus V119894+1 (120585119894 120591)] = 0for 119894 = 1 2 119899 minus 1

(34)

120582119894 120597V119894 (120585119894 120591)120597120585 = 120582119894+1 120597V119894+1 (120585119894 120591)120597120585for 119894 = 1 2 119899 minus 1

(35)

32 Orthogonal Expansion Technique The next step is tofind a solution to the multiregion heat conduction problemexpressed by (30)ndash(35) In the literature various analyticalmethods are available for transient heat conduction in com-posite media such as the orthogonal expansion techniqueLaplace transformmethodmethod of separation of variablesGreenrsquos function method and finite integral transform tech-nique In the present work Vodickarsquos method [25] whichhas been proven effective in analysing multiregion problemswith general space- and time-dependent heat sources andnonhomogeneous boundary conditions is applied to solvethe system of (30)ndash(35)

The solution to (30)ndash(35) is assumed to be

V119894 (120585 120591) = infinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 119871 119894 (120585) 1198841 (120591)for 119894 = 1 2 119899

(36)

with the following function

119871 119894 (120585) = 119862119894120585 + 119863119894 for 119894 = 1 2 119899 (37)

The constants 119862119894 and 119863119894 are obtained from the followingrelationships

d1198711 (0)d120585 minus 11986111198711 (0) = 1198611d119871119899 (1)d120585 + 120575119871119899 (1) = 0

d119871 119894 (120585119894)d120585 + 119877119894 [119871 119894 (120585119894) minus 119871 119894+1 (120585119894)] = 0

Mathematical Problems in Engineering 5

120582119894 d119871 119894 (120585119894)d120585 = 120582119894+1 d119871 119894+1 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(38)

The function 119883119894119898(120585) is the solution to the eigenvalueproblem corresponding to (30)ndash(35) and is given as follows

119883119894119898 (120585) = 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585) (39)

where 119889119894119898 = 120574119898radic120581119894 and 120574119898 is an eigenvalue The unknownconstants 119860 119894119898 and 119861119894119898 are determined from the followingconditions which are obtained by substituting (36)ndash(39) intothe boundary and interface conditions (32)ndash(35)

d1198831119898 (0)d120585 minus 11986111198831119898 (0) = 0

d119883119899119898 (1)d120585 + 120575119883119899119898 (1) = 0

d119883119894119898 (120585119894)d120585 + 119877119894 [119883119894119898 (120585119894) minus 119883(119894+1)119898 (120585119894)] = 0

120582119894 d119883119894119898 (120585119894)d120585 = 120582119894+1 d119883(119894+1)119898 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(40)

In other words a system of 2119899 linear homogeneous equationsis obtained for the constants 119860 119894119898 and 119861119894119898 The eigenvalues120574119898 (119898 = 1 2 ) are obtained from the condition underwhich this system of equations has a nontrivial solution andare therefore positive roots of the following transcendentalequation in matrix notation

G sdot E1 sdot E2 sdot sdot sdotE119899minus1 sdot u = 0 (41)

whereG = [minus1198611 1198891119898] E119894 = J119894 sdot K119894+1J119894 = [120582119894119889119894119898 cos (119889119894119898120585119894) minus119889119894119898 cos (119889119894119898120585119894) minus 119877119894 sin (119889119894119898120585119894)

120582119894119889119894119898 sin (119889119894119898120585119894) minus119889119894119898 sin (119889119894119898120585119894) + 119877119894 cos (119889119894119898120585119894)] K119894+1

= [ 119877119894 cos [119889(119894+1)119898120585119894] 119877119894 sin [119889(119894+1)119898120585119894]minus120582119894+1119889(119894+1)119898 sin [119889(119894+1)119898120585119894] 120582119894+1119889(119894+1)119898 cos [119889(119894+1)119898120585119894]]

u = [119889119899119898 cos (119889119899119898) + 120575 sin (119889119899119898)119889119899119898 sin (119889119899119898) minus 120575 cos (119889119899119898)]

(42)

By substituting (36) into (31) the following equation isobtained

119882119894 (120585) = infinsum119898=1

120601119898 (0)119883119894119898 (120585)= 120579in (120585) minus 120595 (0) 119892119894 (120585) + 119871 119894 (120585) 1198841 (0)

(43)

The eigenfunction 119883119894119898(120585) given by (39) has an orthogonalrelationship with discontinuous weighting function it isexpressed as follows [26]

119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119883119894119897 (120585)119883119894119896 (120585) d120585 =

const (119897 = 119896)0 (119897 = 119896) (44)

Functions 119892119894(120585)119882119894(120585) 119871 119894(120585) and 120581119894119876119894(120585 120591) and a constant 120581119894can be expanded in an infinite series of119883119894119898(120585)

119892119894 (120585) = infinsum119898=1

120572119898119883119894119898 (120585)

119882119894 (120585) = infinsum119898=1

119908119898119883119894119898 (120585)

119871 119894 (120585) = infinsum119898=1

119897119898119883119894119898 (120585)

120581119894119876119894 (120585 120591) = infinsum119898=1

119902119898 (120591)119883119894119898 (120585)

120581119894 = infinsum119898=1

120578119898119883119894119898 (120585)

(45)

where the expansion coefficients are given by

120572119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119892119894 (120585)119883119894119898 (120585) d120585

119908119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119882119894 (120585)119883119894119898 (120585) d120585

119897119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119871 119894 (120585)119883119894119898 (120585) d120585

119902119898 (120591) = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119876119894 (120585 120591)119883119894119898 (120585) d120585

120578119898 = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119883119894119898 (120585) d120585

(46)

in which the norm119873119898 is119873119898 = 119899sum

119894=1

120582119894120581119894 int120585119894

120585119894minus1[119883119894119898 (120585)]2 d120585 (47)

The substitution of (36) and (45) into (30) yields thefollowing equationinfinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 1198971198981 (120591)119883119894119898 (120585)+ 1205742119898120601119898 (120591)119883119894119898 (120585) minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)+ 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 4: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

4 Mathematical Problems in Engineering

120597V119899 (1 120591)120597120585 + 119891 (120591) d119892119899 (1)d120585 + 120575 [V119899 (1 120591)

+ 119891 (120591) 119892119899 (1)] = 119891 (120591) (19)

120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 + 119877119894 [V119894 (120585119894 120591)

+ 119891 (120591) 119892119894 (120585119894) minus V119894+1 (120585119894 120591) minus 119891 (120591) 119892119894+1 (120585119894)] = 0for 119894 = 1 2 119899 minus 1

(20)

120582119894 [120597V119894 (120585119894 120591)120597120585 + 119891 (120591) d119892119894 (120585119894)d120585 ] = 120582119894+1 [120597V119894+1 (120585119894 120591)120597120585

+ 119891 (120591) d119892119894+1 (120585119894)d120585 ] for 119894 = 1 2 119899 minus 1

(21)

Following the procedure of the shifting function methodwe suitably assume the shifting function119892119894(120585) to be a functionsatisfying the following system of equations

d2119892119894 (120585)d1205852 = 119886

for 119894 = 1 2 119899(22)

d1198921 (0)d120585 minus 11986111198921 (0) = 0 (23)

d119892119899 (1)d120585 = 1 (24)

119892119899 (1) = 0 (25)

d119892119894 (120585119894)d120585 + 119877119894 [119892119894 (120585119894) minus 119892119894+1 (120585119894)] = 0

for 119894 = 1 2 119899 minus 1(26)

120582119894 d119892119894 (120585119894)d120585 = 120582119894+1 d119892119894+1 (120585119894)d120585for 119894 = 1 2 119899 minus 1

(27)

where 119886 is an unknown constantThe general solution of (22)is given by

119892119894 (120585) = 11988621205852 + 119887119894120585 + 119888119894 (28)

where 119887119894 and 119888119894 (119894 = 1 2 119899) are additional unknownconstants A total of (2119899 + 1) unknown constants are readilydetermined from (23)ndash(27)

Substituting (25) into (14) yields

120579119899 (1 120591) = V119899 (1 120591) (29)

The substitution of (22) and (25) into (16) results in thefollowing partial differential equation

120597V119894 (120585 120591)120597120591 minus 120581119894 1205972V119894 (120585 120591)1205971205852 minus 119892119894 (120585) 119865 (120591) V119899 (1 120591)+ [minus119892119894 (120585) (120591) + 120581119894119886119865 (120591)] V119899 (1 120591)

= minus119892119894 (120585) (120591) + 120581119894119886120595 (120591) + 120581119894119876119894 (120585 120591)for 119894 = 1 2 119899

(30)

where the ldquoover-dotrdquo denotes the time derivative Consider-ing (23)ndash(27) the associated initial boundary and interfaceconditions are obtained as follows

V119894 (120585 0) = 120579in (120585) minus 120595 (0) 119892119894 (120585) for 119894 = 1 2 119899 (31)

120597V1 (0 120591)120597120585 minus 1198611 [V1 (0 120591) minus 1198841 (120591)] = 0 (32)

120597V119899 (1 120591)120597120585 + 120575V119899 (1 120591) = 0 (33)

120597V119894 (120585119894 120591)120597120585 + 119877119894 [V119894 (120585119894 120591) minus V119894+1 (120585119894 120591)] = 0for 119894 = 1 2 119899 minus 1

(34)

120582119894 120597V119894 (120585119894 120591)120597120585 = 120582119894+1 120597V119894+1 (120585119894 120591)120597120585for 119894 = 1 2 119899 minus 1

(35)

32 Orthogonal Expansion Technique The next step is tofind a solution to the multiregion heat conduction problemexpressed by (30)ndash(35) In the literature various analyticalmethods are available for transient heat conduction in com-posite media such as the orthogonal expansion techniqueLaplace transformmethodmethod of separation of variablesGreenrsquos function method and finite integral transform tech-nique In the present work Vodickarsquos method [25] whichhas been proven effective in analysing multiregion problemswith general space- and time-dependent heat sources andnonhomogeneous boundary conditions is applied to solvethe system of (30)ndash(35)

The solution to (30)ndash(35) is assumed to be

V119894 (120585 120591) = infinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 119871 119894 (120585) 1198841 (120591)for 119894 = 1 2 119899

(36)

with the following function

119871 119894 (120585) = 119862119894120585 + 119863119894 for 119894 = 1 2 119899 (37)

The constants 119862119894 and 119863119894 are obtained from the followingrelationships

d1198711 (0)d120585 minus 11986111198711 (0) = 1198611d119871119899 (1)d120585 + 120575119871119899 (1) = 0

d119871 119894 (120585119894)d120585 + 119877119894 [119871 119894 (120585119894) minus 119871 119894+1 (120585119894)] = 0

Mathematical Problems in Engineering 5

120582119894 d119871 119894 (120585119894)d120585 = 120582119894+1 d119871 119894+1 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(38)

The function 119883119894119898(120585) is the solution to the eigenvalueproblem corresponding to (30)ndash(35) and is given as follows

119883119894119898 (120585) = 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585) (39)

where 119889119894119898 = 120574119898radic120581119894 and 120574119898 is an eigenvalue The unknownconstants 119860 119894119898 and 119861119894119898 are determined from the followingconditions which are obtained by substituting (36)ndash(39) intothe boundary and interface conditions (32)ndash(35)

d1198831119898 (0)d120585 minus 11986111198831119898 (0) = 0

d119883119899119898 (1)d120585 + 120575119883119899119898 (1) = 0

d119883119894119898 (120585119894)d120585 + 119877119894 [119883119894119898 (120585119894) minus 119883(119894+1)119898 (120585119894)] = 0

120582119894 d119883119894119898 (120585119894)d120585 = 120582119894+1 d119883(119894+1)119898 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(40)

In other words a system of 2119899 linear homogeneous equationsis obtained for the constants 119860 119894119898 and 119861119894119898 The eigenvalues120574119898 (119898 = 1 2 ) are obtained from the condition underwhich this system of equations has a nontrivial solution andare therefore positive roots of the following transcendentalequation in matrix notation

G sdot E1 sdot E2 sdot sdot sdotE119899minus1 sdot u = 0 (41)

whereG = [minus1198611 1198891119898] E119894 = J119894 sdot K119894+1J119894 = [120582119894119889119894119898 cos (119889119894119898120585119894) minus119889119894119898 cos (119889119894119898120585119894) minus 119877119894 sin (119889119894119898120585119894)

120582119894119889119894119898 sin (119889119894119898120585119894) minus119889119894119898 sin (119889119894119898120585119894) + 119877119894 cos (119889119894119898120585119894)] K119894+1

= [ 119877119894 cos [119889(119894+1)119898120585119894] 119877119894 sin [119889(119894+1)119898120585119894]minus120582119894+1119889(119894+1)119898 sin [119889(119894+1)119898120585119894] 120582119894+1119889(119894+1)119898 cos [119889(119894+1)119898120585119894]]

u = [119889119899119898 cos (119889119899119898) + 120575 sin (119889119899119898)119889119899119898 sin (119889119899119898) minus 120575 cos (119889119899119898)]

(42)

By substituting (36) into (31) the following equation isobtained

119882119894 (120585) = infinsum119898=1

120601119898 (0)119883119894119898 (120585)= 120579in (120585) minus 120595 (0) 119892119894 (120585) + 119871 119894 (120585) 1198841 (0)

(43)

The eigenfunction 119883119894119898(120585) given by (39) has an orthogonalrelationship with discontinuous weighting function it isexpressed as follows [26]

119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119883119894119897 (120585)119883119894119896 (120585) d120585 =

const (119897 = 119896)0 (119897 = 119896) (44)

Functions 119892119894(120585)119882119894(120585) 119871 119894(120585) and 120581119894119876119894(120585 120591) and a constant 120581119894can be expanded in an infinite series of119883119894119898(120585)

119892119894 (120585) = infinsum119898=1

120572119898119883119894119898 (120585)

119882119894 (120585) = infinsum119898=1

119908119898119883119894119898 (120585)

119871 119894 (120585) = infinsum119898=1

119897119898119883119894119898 (120585)

120581119894119876119894 (120585 120591) = infinsum119898=1

119902119898 (120591)119883119894119898 (120585)

120581119894 = infinsum119898=1

120578119898119883119894119898 (120585)

(45)

where the expansion coefficients are given by

120572119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119892119894 (120585)119883119894119898 (120585) d120585

119908119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119882119894 (120585)119883119894119898 (120585) d120585

119897119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119871 119894 (120585)119883119894119898 (120585) d120585

119902119898 (120591) = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119876119894 (120585 120591)119883119894119898 (120585) d120585

120578119898 = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119883119894119898 (120585) d120585

(46)

in which the norm119873119898 is119873119898 = 119899sum

119894=1

120582119894120581119894 int120585119894

120585119894minus1[119883119894119898 (120585)]2 d120585 (47)

The substitution of (36) and (45) into (30) yields thefollowing equationinfinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 1198971198981 (120591)119883119894119898 (120585)+ 1205742119898120601119898 (120591)119883119894119898 (120585) minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)+ 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 5: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Mathematical Problems in Engineering 5

120582119894 d119871 119894 (120585119894)d120585 = 120582119894+1 d119871 119894+1 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(38)

The function 119883119894119898(120585) is the solution to the eigenvalueproblem corresponding to (30)ndash(35) and is given as follows

119883119894119898 (120585) = 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585) (39)

where 119889119894119898 = 120574119898radic120581119894 and 120574119898 is an eigenvalue The unknownconstants 119860 119894119898 and 119861119894119898 are determined from the followingconditions which are obtained by substituting (36)ndash(39) intothe boundary and interface conditions (32)ndash(35)

d1198831119898 (0)d120585 minus 11986111198831119898 (0) = 0

d119883119899119898 (1)d120585 + 120575119883119899119898 (1) = 0

d119883119894119898 (120585119894)d120585 + 119877119894 [119883119894119898 (120585119894) minus 119883(119894+1)119898 (120585119894)] = 0

120582119894 d119883119894119898 (120585119894)d120585 = 120582119894+1 d119883(119894+1)119898 (120585119894)d120585 for 119894 = 1 2 119899 minus 1

(40)

In other words a system of 2119899 linear homogeneous equationsis obtained for the constants 119860 119894119898 and 119861119894119898 The eigenvalues120574119898 (119898 = 1 2 ) are obtained from the condition underwhich this system of equations has a nontrivial solution andare therefore positive roots of the following transcendentalequation in matrix notation

G sdot E1 sdot E2 sdot sdot sdotE119899minus1 sdot u = 0 (41)

whereG = [minus1198611 1198891119898] E119894 = J119894 sdot K119894+1J119894 = [120582119894119889119894119898 cos (119889119894119898120585119894) minus119889119894119898 cos (119889119894119898120585119894) minus 119877119894 sin (119889119894119898120585119894)

120582119894119889119894119898 sin (119889119894119898120585119894) minus119889119894119898 sin (119889119894119898120585119894) + 119877119894 cos (119889119894119898120585119894)] K119894+1

= [ 119877119894 cos [119889(119894+1)119898120585119894] 119877119894 sin [119889(119894+1)119898120585119894]minus120582119894+1119889(119894+1)119898 sin [119889(119894+1)119898120585119894] 120582119894+1119889(119894+1)119898 cos [119889(119894+1)119898120585119894]]

u = [119889119899119898 cos (119889119899119898) + 120575 sin (119889119899119898)119889119899119898 sin (119889119899119898) minus 120575 cos (119889119899119898)]

(42)

By substituting (36) into (31) the following equation isobtained

119882119894 (120585) = infinsum119898=1

120601119898 (0)119883119894119898 (120585)= 120579in (120585) minus 120595 (0) 119892119894 (120585) + 119871 119894 (120585) 1198841 (0)

(43)

The eigenfunction 119883119894119898(120585) given by (39) has an orthogonalrelationship with discontinuous weighting function it isexpressed as follows [26]

119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119883119894119897 (120585)119883119894119896 (120585) d120585 =

const (119897 = 119896)0 (119897 = 119896) (44)

Functions 119892119894(120585)119882119894(120585) 119871 119894(120585) and 120581119894119876119894(120585 120591) and a constant 120581119894can be expanded in an infinite series of119883119894119898(120585)

119892119894 (120585) = infinsum119898=1

120572119898119883119894119898 (120585)

119882119894 (120585) = infinsum119898=1

119908119898119883119894119898 (120585)

119871 119894 (120585) = infinsum119898=1

119897119898119883119894119898 (120585)

120581119894119876119894 (120585 120591) = infinsum119898=1

119902119898 (120591)119883119894119898 (120585)

120581119894 = infinsum119898=1

120578119898119883119894119898 (120585)

(45)

where the expansion coefficients are given by

120572119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119892119894 (120585)119883119894119898 (120585) d120585

119908119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119882119894 (120585)119883119894119898 (120585) d120585

119897119898 = 1119873119898119899sum119894=1

120582119894120581119894 int120585119894

120585119894minus1119871 119894 (120585)119883119894119898 (120585) d120585

119902119898 (120591) = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119876119894 (120585 120591)119883119894119898 (120585) d120585

120578119898 = 1119873119898119899sum119894=1120582119894 int120585119894120585119894minus1

119883119894119898 (120585) d120585

(46)

in which the norm119873119898 is119873119898 = 119899sum

119894=1

120582119894120581119894 int120585119894

120585119894minus1[119883119894119898 (120585)]2 d120585 (47)

The substitution of (36) and (45) into (30) yields thefollowing equationinfinsum119898=1

120601119898 (120591)119883119894119898 (120585) minus 1198971198981 (120591)119883119894119898 (120585)+ 1205742119898120601119898 (120591)119883119894119898 (120585) minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)+ 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 6: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

6 Mathematical Problems in Engineering

minus 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585) + 120572119898 (120591)119883119894119898 (120585)minus 119886120595 (120591) 120578119898119883119894119898 (120585) minus 119902119898 (120591)119883119894119898 (120585) = 0

(48)

Equation (48) is satisfied if the expression in the braces is zerothat is

120601119898 (120591)119883119894119898 (120585) + 1205742119898120601119898 (120591)119883119894119898 (120585)minus 119892119894 (120585) 119865 (120591) 120601119898 (120591)119883119899119898 (1)+ [minus119892119894 (120585) (120591) + 119886119865 (120591)] 120601119898 (120591)119883119899119898 (1)

= 1198971198981 (120591)119883119894119898 (120585)minus 120572119898119871119899 (1)119883119894119898 (120585) [119865 (120591) 1 (120591) + (120591) 1198841 (120591)]+ 119886119865 (120591) 119871119899 (1) 1198841 (120591) 120578119898119883119894119898 (120585)

minus 120572119898 (120591)119883119894119898 (120585) + 119886120595 (120591) 120578119898119883119894119898 (120585)+ 119902119898 (120591)119883119894119898 (120585)

(49)

Equation (49) is multiplied by (120582119894120581119894)119883119894119898(120585) integrated overthe region [120585119894minus1 120585119894] and then summed up for 119894 = 1 2 119899 toyield

120601119898 (120591) + 1205742119898 minus 120573119898 (120591) + 119886120594119898119865 (120591)1 minus 120573119898119865 (120591) 120601119898 (120591) = 120589119898 (120591) (50)

where 120573119898 and 120594119898 are given by

120573119898 = 119883119899119898 (1) sdot 120572119898120594119898 = 119883119899119898 (1) sdot 120578119898 (51)

and 120577119898(120591) is defined as

120589119898 (120591) = 1 (120591) [119897119898 minus 119865 (120591) 119871119899 (1) 120572119898] + 119871119899 (1) 1198841 (120591) [119886119865 (120591) 120578119898 minus 120572119898 (120591)] minus (120591) 120572119898 + 119886120595 (120591) 120578119898 + 119902119898 (120591)1 minus 120573119898119865 (120591) (52)

By solving (50) with the condition 120601119898(0) = 119908119898 which isobtained from the comparison between (43) and (45)2 weobtain 120601119898(120591) as120601119898 (120591) = 119890minusint1205910 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904 [119908119898

+ int1205910120577119898 (120583) sdot 119890int1205830 ((1205742119898minus120573119898(119904)+119886120594119898119865(119904))(1minus120573119898119865(119904)))d119904d120583]

(53)

After substituting (28) (29) (36) (37) and (39) backto (14) the dimensionless temperature in the 119894th layer 120585 isin[120585119894minus1 120585119894] of the composite slab is derived as

120579119894 (120585 120591) = infinsum119898=1

120601119898 (120591) 119860 119894119898 cos (119889119894119898120585) + 119861119894119898 sin (119889119894119898120585)minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591)sdot [119860119899119898 cos (119889119899119898) + 119861119899119898 sin (119889119899119898)] minus 1198841 (120591) [119862119894120585+ 119863119894 minus (11988621205852 + 119887119894120585 + 119888119894)119865 (120591) (119862119899 + 119863119899)] + (11988621205852+ 119887119894120585 + 119888119894)120595 (120591)

(54)

In the methodology for deriving the temperature solutionmain differences from the counterpart for single layer [20] arethe following (i) the solution is first shifted in each layer byusing not a single shifting function across the total thicknessbut shifting functions that differ for each layer (see (14)) (ii)since there exist interfaces the interface conditions (see (20)and (21)) are added and (iii) various functions are expanded

by using eigenfunctions that fulfil an orthogonal relationshipwith discontinuous weighting function (see (44))

4 Verification of Analytical Solution

To illustrate the applicability of the solution derived first asingle-layered heat conduction problem whose solution waspreviously known [20] is modelled as a multilayer problemwith adjacent layers having identical physical propertiesThis problem (hereinafter called Case A) involves a time-dependent HTC (or Biot number) on one boundary surfaceof the slab The following numerical parameters are adopted[20] 120579in = 0336 1198611 = 0 1198612(120591) = 12 minus exp(minus2120591) 1198842(120591) = 15 minus05 exp(minus120591) and 119876119894 = 0 For the numerical computation weuse the analytical solution (see (54)) taking 119899 = 5 and 120585119894 (119894 =1 2 3 4) evenly spaced between 0 and 1 assuming perfectcontact at the interfaces and uniform thermal conductivityand diffusivity within the entire region that is 119877119894 = infin120582119894 = 1 and 120581119894 = 1

Second we consider another test case with a constant Biotnumber to verify our analytical solution for multiregion heatconduction This test case (called conveniently Case B here)represents transient diffusion in a composite slab consistingof 119899 = 10 layers with evenly spaced 120585119894Thematerial propertiesare taken as 120582119894 = 120581119894 = 1 for 119894 = 1 3 5 7 9 and 120582119894 = 120581119894 = 01for 119894 = 2 4 6 8 10 [27]The constants appearing in the initialand boundary conditions are 120579in = 0 1198611 = infin 1198612 = 0 and1198841 = 1 Moreover imperfect thermal contact is applied at theinterfaces (119877119894 = 05120581119894 119894 = 1 2 9)

Figures 2 and 3 plot the analytical solution (see (54))and numerical results previously published for Cases A andB respectively The derived analytical solution is computedusing the first 20 terms in the infinite series In both cases the

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

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Page 7: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Mathematical Problems in Engineering 7

= 4

= 1

0

05

1

15

(

)

02 04 06 08 10

Figure 2 Solution verification for Case A the derived analyticalsolution is plotted as a solid line and the solution presented in [20]is plotted as circles

analytical solution is in good agreement with the solutions ofthe existing literature Tables 1 and 2 show the relative errordefined by

error = max119894=121198991003816100381610038161003816120579119894 (120585119894 120591) minus Θ (120585119894 120591)1003816100381610038161003816

max119894=121198991003816100381610038161003816120579119894 (120585119894 120591)1003816100381610038161003816 (55)

where 120579119894 is the derived solution andΘ is the solution obtainedusing the analytical method [20] for Case A and one of eitherthe semianalytical method [27] or Fokas method [28] forCase B As shown in the tables the relative error is withinapproximately 02

5 Illustrative Examples

In this section we apply the analytical solution described inSection 3 to two industrial applications Because we focus onthe effect of time-dependentHTC at the heated surface on thetransient temperature distribution of composite media onlythe cases with constant ambient temperature and no internalheat generation are considered here

51 Bilayered Plate with Exponentially Varying HTC Con-sider the case of a bilayered plate of aluminium and leadwhose initial temperature is zero Equal thicknesses for bothlayers are used which leads to 1205851 = 12 with perfect thermalcontact For 120591 gt 0 the surrounding medium on the side oflead is raised to a temperature of1198842 = 1while the other surfaceof the plate is insulated (1198611 = 0) For the Biot number at 120585 =1 we choose 1198612(120591) = 1199011 minus 1199012 exp(minus119904120591) where 1199011 1199012 and119904 are arbitrary constants The following material propertieshave been used in the calculations [26] 1205821 = 5850 1205822 = 11205811 = 4096 and 1205812 = 1

The resulting temperature distributions are given inFigures 4 and 5 for various values of location 120585 and Fourier

2

0007

= 10

02 04 06 08 10

0

02

04

06

08

1

(

)

Figure 3 Solution verification for Case B the derived analyticalsolution is plotted as a solid line and the solution presented in [27]is plotted as circles

Table 1 Relative error in Case A

Fourier number (120591) 1 4Error 208 times 10minus3 139 times 10minus4

number 120591 In Figure 4 the temporal variation in 1198612 is alsoshown in the top part which ranges from 02 to 12 Inaddition Table 3 lists the temperatures at both the insulatedand heated surfaces for different number of terms in theseries to show the convergence behaviour of the presentanalytical solution From Table 3 it can be seen that theconvergence of the present solution is very fast When theFourier number is larger than 02 the error for the one-term approximation solution is less than 03 This fastconvergence of the solution is one of the notable featuresthat the analytical solutions obtained through the shiftingfunction method have [18 20] Note that all the graphicalrepresentations shown in Section 5 are based on the resultsobtained from the four-term approximation

As shown in Figure 4 the temporal variations in thetemperatures correspond to the variation in 1198612 being themost remarkable for 119904 = 2 for which the rate of increase in1198612 is the highestThe time-dependent Biot number 1198612 greatlyaffects the temperature of the heated surface for a small valueof 120591 having a lagged influence on the temperature at theinsulated boundary It is worth noting that the influence onthe insulated boundary is not negligible Figure 5 shows thatdifferent 119904 values make ameasurable difference in the profilesof 120579 throughout the thickness52 Three-Layered Composite with Periodically Varying HTCConsider the case of a three-layered composite made of tinaluminium and lead with interfaces at 1205851 = 13 and 1205852 = 23[26] Again we assume perfect thermal contact The initialand boundary conditions are the same as those in Section 51except that1198612(120591) = 1199011minus1199012 cos(120596120591) where120596 is a constantThe

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 8: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

8 Mathematical Problems in Engineering

1

05

s = 2

s = 2

1

05

0

05

1

(0

)

0

12

B2()

1 1001

(a)

1

05

s = 2

s = 2

1

05

0

05

1

(1

)

0

12

B2()

1 1001

(b)

Figure 4 Influence of parameter 119904 on the temperature variation of the bilayered plate at two locations (a) 120585 = 0 and (b) 120585 = 1 for 1199011 = 12and 1199012 = 1

Table 2 Relative error in Case B

Fourier number (120591) 0007 2 10Error (semianalytical) 156 times 10minus3 564 times 10minus4 862 times 10minus4

Error (Fokas method) 532 times 10minus5 504 times 10minus5 501 times 10minus5

Table 3 Dimensionless temperature 120579 for different number of terms in the series of (54) for s = 1 1199011 = 12 and 1199012 = 1

120591 120579(0 120591) 120579(1 120591)1 term 2 terms 4 terms 10 terms 1 term 2 terms 4 terms 10 terms

01 minus00064 minus00024 minus00024 minus00024 01116 01006 01006 0100602 00110 00119 00119 00119 01590 01563 01563 0156303 00344 00345 00345 00345 02051 02045 02045 0204504 00622 00622 00622 00622 02498 02497 02497 0249705 00934 00935 00935 00935 02928 02928 02928 029281 02723 02723 02723 02723 04807 04807 04807 0480715 04461 04461 04461 04461 06241 06241 06241 062412 05903 05903 05903 05903 07300 07300 07300 073003 07853 07853 07853 07853 08624 08624 08624 08624

following thermal conductivities and diffusivities have beenused [26] 1205821 = 1900 1205822 = 5850 1205823 = 1 1205811 = 1633 1205812 =4096 and 1205813 = 1

Figure 6 illustrates the temperature variations at twoexternal boundaries for different 120596 values The periodicallyvarying 1198612 ranges from 02 to 22 as shown in the toppart of the figures Visible oscillations are observed in bothtemperatures for 120596 = 2 and 4 for which the change cycle of1198612 is short Moreover unlike the case in Section 51 thereare time slots during which the magnitude relation of theBiot numbers is not necessarily consistent with that of thetemperatures for different 120596 values The temperature profilesof the composite at 120591 = 1 and 4 are plotted in Figure 7which are all monotonic functions of 120585 with discontinuousgradient at the interfaces It can be observed that the effects

of the temporal variations in 1198612 propagate through the wholethickness of the three-layered composite

6 Conclusions

An analytical method based on the shifting function methodhas been presented for solving the one-dimensional tran-sient heat conduction problem with time-dependent HTCfor a composite slab consisting of an arbitrary number oflayers Applying the shifting function method yields newpartial differential equations under conventional types ofinitial boundary and interfacial conditions In this paperthe derived equations were solved by using an orthogonalexpansion based on the Vodicka type of orthogonality rela-tionship Numerical calculations were performed for two

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 9: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Mathematical Problems in Engineering 9

Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

s = 2= 1= 05

02 04 06 08 10

Figure 5 Temperature distributions along the thickness of the bilayered plate at 120591 = 1 and 4 with different 119904 values for 1199011 = 12 and 1199012 = 1

12

12 = 4

= 4

0

05

1

(0

)

0

3

B2()

1 2 3 4 5 60

(a)

12

12 = 4

= 4

0

05

1

(1

)

0

3

B2()

1 2 3 4 5 60

(b)

Figure 6 Influence of parameter 120596 on the temperature variation of the three-layered composite at two locations (a) 120585 = 0 and (b) 120585 = 1 for1199011 = 12 and 1199012 = 1

representative composite slabsThenumerical results demon-strated the effects of temporal variations in the HTC on thetransient temperature field in particular on the temperaturesat two external boundaries

This paper considers the temporal variation in the HTCat only one external boundary The solution needs to beextended so that the time-dependency of the HTCs atboth boundaries can be considered This might possibly beachieved by introducing two different shifting functions [1929] The present analytical method can easily be extended to

other simple geometries it changes nothing beyond the formsof the eigenfunction shifting function function 119871 119894(120585) andthe orthogonal relationship of (44) The analytical solutionderived is applicable to nonhomogeneous materials such asfunctionally graded materials under the laminate approxi-mation theory [30]

The presented analytical method is available for transientheat conduction problems with arbitrary time-dependentHTC However currently it cannot be applied to those inwhich the HTC is a function of position or of temperature

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 10: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

10 Mathematical Problems in Engineering

Tin Aluminium Lead0

02

04

06

08

1

(

)

= 4

= 1

= 4= 2= 1

02 04 06 08 10

Figure 7 Temperature distributions along the thickness of thethree-layered composite at 120591 = 1 and 4 with different 120596 values for1199011 = 12 and 1199012 = 1

Nomenclature

119886 Constant in 119892119894(120585)119860 119894119898 119861119894119898 Constants in119883119894119898(120585)1198611 1198612(120591) Biot numbers119887119894 119888119894 Constants in 119892119894(120585)119862119894119863119894 Constants in 119871 119894(120585)119889119894119898 Constant (= 120574119898radic120581119894)119891(120591) Auxiliary time function defined by (15)119865(120591) Nonstationary part of 1198612(120591)119892119894(120585) Shifting function defined by (14)ℎ1 ℎ2(119905) Heat transfer coefficients119867 Thickness119894 Positive integer119897119898 Expansion coefficient defined by (46)119871 119894(120585) Function of 120585 defined by (36)119898 119899 Integers119873119898 Norm defined by (47)1199011 1199012 Constants in 1198612(120591)119902119898(120591) Function of time defined by (46)119876119894(120585 120591) Distributed source in the 119894th region119903119894 Conductance coefficient between 119894thand (119894 + 1)th regions119877119894 Dimensionless conductance coefficient119904 Constant119905 Time119879119894(119909 119905) Temperature in the 119894th region

V119894(120585 120591) Transformed function of 120579119894(120585 120591)119908119898 Expansion coefficient defined by (46)119882119894(120585) Derived function of 120585 defined by (43)119883119894119898(120585) Eigenfunction1199101(119905) 1199102(119905) Temperatures of surroundings

1198841(120591) 1198842(120591) Dimensionless temperatures ofsurroundings119911 Spatial coordinate

Greek Symbols

120572119898 Expansion coefficient defined by (46)120573119898 Constant (= 119883119899119898(1) sdot 120572119898)120594119898 Constant (= 119883119899119898(1) sdot 120578119898)120575 Initial value of 1198612(120591)120601119898(120591) Function of time defined by (36)120574119898 Eigenvalue120578119898 Expansion coefficient defined by (46)120595(120591) Function of time (= 1198612(120591) sdot 1198842(120591))120581119894 Thermal diffusivity in the 119894th region120581119894 Ratio of thermal diffusivities120582119894 Thermal conductivity in the 119894th region120582119894 Ratio of thermal conductivities120579119894(120585 120591) Dimensionless temperature in the 119894th region120591 Fourier number120596 Constant120577119898(120591) Function of time defined by (52)120585 Dimensionless spatial coordinate

Subscripts

in Initial value0 Reference value

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The author would like to thankDrs N E Sheils TW Tu andE J Carr for technical assistance in verifying the analyticalsolution

References

[1] M Mori and M Kondo ldquoTemperature and Thermal StressAnalysis in a Structure with Uncertain Heat Transfer BoundaryConditionsrdquo Transactions of the Japan Society of MechanicalEngineers Series A vol 59 no 562 pp 1514ndash1518 1993

[2] J J Thompson and Z J Holy ldquoAxisymmetric thermal responseproblems for a spherical fuel element with time dependent heattransfer coefficientsrdquoNuclear Engineering and Design vol 9 no1 pp 29ndash44 1969

[3] T G Kim and Z H Lee ldquoTime-varying heat transfer coeffi-cients between tube-shaped casting and metal moldrdquo Interna-tional Journal ofHeat andMass Transfer vol 40 no 15 pp 3513ndash3525 1997

[4] S S Sazhin P A KrutitskiiW A Abdelghaffar et al ldquoTransientheating of diesel fuel dropletsrdquo International Journal of Heat andMass Transfer vol 47 no 14-16 pp 3327ndash3340 2004

[5] I M Fedotkin A M Aizen and I A Goloshchuk ldquoApplicationof integral equations to heat conduction problems in which the

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 11: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Mathematical Problems in Engineering 11

heat transfer coefficient variesrdquo Journal of Engineering Physicsvol 28 no 3 pp 392ndash395 1975

[6] B Nicolai and J De Baerdemaeker ldquoSimulation of heat transferin foods with stochastic initial and boundary conditionsrdquo TransIChemE 70C pp 78ndash82 1992

[7] B Y Lyubov and N I Yalovoi ldquoHeat conductivity of a bodywith variable heat exchange coefficientrdquo Journal of EngineeringPhysics vol 17 no 4 pp 1264ndash1270 1972

[8] 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

[9] M B Abd-el-Malek and M M Helal ldquoGroup method solutionfor solving nonlinear heat diffusion problemsrdquo Applied Mathe-matical Modelling vol 30 no 9 pp 930ndash940 2006

[10] V V Ivanov andV V Salomatov ldquoOn the calculation of the tem-perature field in solids with variable heat-transfer coefficientsrdquoJournal of Engineering Physics vol 9 no 1 pp 63-64 1967

[11] V V Ivanov and V V Salomatov ldquoUnsteady temperature fieldin solid bodies with variable heat transfer coefficientrdquo Journalof Engineering Physics vol 11 no 2 pp 151-152 1969

[12] Y S Postolrsquonik ldquoOne-dimensional convective heating with atime-dependent heat-transfer coefficientrdquo Journal of Engineer-ing Physics vol 18 no 2 pp 233ndash238 1972

[13] V N Kozlov ldquoTemperature field of an infinite plate in the caseof a variable heat-exchange coefficientrdquo Journal of EngineeringPhysics vol 16 no 1 pp 95ndash97 1972

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

[15] M D Mikhailov ldquoOn the solution of the heat equation withtime dependent coefficientrdquo International Journal of Heat andMass Transfer vol 18 no 2 pp 344-345 1975

[16] R J Moitsheki ldquoTransient heat diffusion with temperature-dependent conductivity and time-dependent heat transfercoefficientrdquo Mathematical Problems in Engineering Article ID347568 9 pages 2008

[17] V N Kozlov ldquoSolution of heat-conduction problem with vari-able heat-exchange coefficientrdquo Journal of Engineering Physicsvol 18 no 1 pp 100ndash104 1972

[18] 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

[19] T W Tu and S Y Lee ldquoAnalytical solution of heat conductionfor hollow cylinders with time-dependent boundary conditionand time-dependent heat transfer coefficientrdquo Journal of AppliedMathematics Article ID 203404 9 pages 2015

[20] T W Tu and S Y Lee ldquoExact temperature field in a slab withtime varying ambient temperature and time-dependent heattransfer coefficientrdquo International Journal of Thermal Sciencesvol 116 pp 82ndash90 2017

[21] O T Ilrsquochenko ldquoTemperature field of a two-layered plate withtime-varying heat-transfer conditionsrdquo Journal of EngineeringPhysics vol 19 no 6 pp 1567ndash1570 1973

[22] I M Prikhodrsquoko ldquoThermal conductivity of a two-layer wall fora time-varying heat-transfer coefficient and ambient tempera-turerdquo Journal of Engineering Physics vol 18 no 2 pp 239ndash2421972

[23] Y Yener and M N Ozisik ldquoOn the solution of unsteady heatconduction in multi-region finite media with time dependentheat transfer coefficientrdquo in Proceedings of the 5th InternationalHeat Transfer Conference Tokyo pp 188ndash192 1974

[24] 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

[25] V Vodicka ldquoEindimensionale Warmeleitung in geschichtetenKorpernrdquoMathematische Nachrichten vol 14 pp 47ndash55 1955

[26] G P Mulholland and M H Cobble ldquoDiffusion through com-posite mediardquo International Journal of Heat and Mass Transfervol 15 no 1 pp 147ndash160 1972

[27] E J Carr and I W Turner ldquoA semi-analytical solution formultilayer diffusion in a compositemediumconsisting of a largenumber of layersrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 15-16 pp 7034ndash7050 2016

[28] N E Sheils ldquoMultilayer diffusion in a composite medium withimperfect contactrdquoAppliedMathematicalModelling Simulationand Computation for Engineering and Environmental Systemsvol 46 pp 450ndash464 2017

[29] S Y Lee and C C Huang ldquoAnalytic solutions for heat con-duction in functionally graded circular hollow cylinders withtime-dependent boundary conditionsrdquoMathematical Problemsin Engineering vol 2013 Article ID 816385 8 pages 2013

[30] H M Wang ldquoAn effective approach for transient thermalanalysis in a functionally graded hollow cylinderrdquo InternationalJournal of Heat and Mass Transfer vol 67 pp 499ndash505 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 12: An Analytical Solution for Transient Heat Conduction in a ...downloads.hindawi.com › journals › mpe › 2018 › 4707860.pdf · An Analytical Solution for Transient Heat Conduction

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


C12 - Transient Thermal Conduction Example

C12 - Transient Thermal Conduction Example

Transient Heat Conduction ppt

Transient Heat Conduction ppt

Applications of Transient Conduction

Applications of Transient Conduction

Kuliah 9 Transient Heat Conduction

Kuliah 9 Transient Heat Conduction

TRANSIENT HEAT CONDUCTION · TRANSIENT HEAT CONDUCTION Course Contents 4.1 Introduction 4.2 Transient Conduction in Solids with Infinite Thermal Conductivity k ∞ (Lumped Parameter

TRANSIENT HEAT CONDUCTION · TRANSIENT HEAT CONDUCTION Course Contents 4.1 Introduction 4.2 Transient Conduction in Solids with Infinite Thermal Conductivity k ∞ (Lumped Parameter

Transient Conduction: Spatial Effects and the Role of Analytical Solutions Chapter 5 Sections 5.4 to 5.7 Lecture 10.

Transient Conduction: Spatial Effects and the Role of Analytical Solutions Chapter 5 Sections 5.4 to 5.7 Lecture 10.

Equivalent circuit for transient conduction and convection ...

Equivalent circuit for transient conduction and convection ...

ANALYSIS OF TRANSIENT HEAT CONDUCTION IN DIFFERENT GEOMETRIESethesis.nitrkl.ac.in/1486/1/Analysis_of_transient_heat_conduction... · analysis of transient heat conduction in different

ANALYSIS OF TRANSIENT HEAT CONDUCTION IN DIFFERENT GEOMETRIESethesis.nitrkl.ac.in/1486/1/Analysis_of_transient_heat_conduction... · analysis of transient heat conduction in different

SOLVING TRANSIENT CONDUCTION AND RADIATION USING … · Solving Transient Conduction And Radiation Using Finite ... The effect of heat conduction during ... Solving Transient Conduction

SOLVING TRANSIENT CONDUCTION AND RADIATION USING … · Solving Transient Conduction And Radiation Using Finite ... The effect of heat conduction during ... Solving Transient Conduction

06. Transient Heat Conduction+SemiInfinite

06. Transient Heat Conduction+SemiInfinite

TRANSIENT CONDUCTION Lumped Thermal Capacitance Method Analytical Method: Separation of Variables Semi-Infinite Solid: Similarity Solution Numerical Method:

TRANSIENT CONDUCTION Lumped Thermal Capacitance Method Analytical Method: Separation of Variables Semi-Infinite Solid: Similarity Solution Numerical Method:

(9 10) Transient Heat Conduction

(9 10) Transient Heat Conduction

Transient Conduction: Lumped Capacitanceeng.utah.edu/~whitty/chen3453/Lecture 11 - Transient... · 2014-09-22 · Transient Conduction: Lumped Capacitance ... • If Bi < 0.1 then

Transient Conduction: Lumped Capacitanceeng.utah.edu/~whitty/chen3453/Lecture 11 - Transient... · 2014-09-22 · Transient Conduction: Lumped Capacitance ... • If Bi < 0.1 then

Nonlinear, Transient Conduction Heat Transfer …sanders/ThesisMain.pdfNonlinear, Transient Conduction Heat Transfer Using A Discontinuous Galerkin Hierarchical Finite Element Method

Nonlinear, Transient Conduction Heat Transfer …sanders/ThesisMain.pdfNonlinear, Transient Conduction Heat Transfer Using A Discontinuous Galerkin Hierarchical Finite Element Method

Chapter 5 : Transient Conduction

Chapter 5 : Transient Conduction

2.5D Transient Heat Transfer (Conduction and Convection ...

2.5D Transient Heat Transfer (Conduction and Convection ...

TRANSIENT HEAT CONDUCTION T - Landing

TRANSIENT HEAT CONDUCTION T - Landing

Numerical methods in Transient-heat-conduction

Numerical methods in Transient-heat-conduction

Transient Heat Conduction in Capillary Porous Bodiescdn.intechopen.com/pdfs/22251/InTech-Transient... · Transient Heat Conduction in Capillary Porous Bodies 151 needed for the melting

Transient Heat Conduction in Capillary Porous Bodiescdn.intechopen.com/pdfs/22251/InTech-Transient... · Transient Heat Conduction in Capillary Porous Bodies 151 needed for the melting

REINFORCING CONCEPTS OF TRANSIENT HEAT CONDUCTION …

REINFORCING CONCEPTS OF TRANSIENT HEAT CONDUCTION …