Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-Layered Media

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Analytical exact solutions of heat conduction problems

for anisotropic multi-layered media

Chien-Ching Ma

*

_{, Shin-Wen Chang}

Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC Received 20 February 2003; received in revised form 31 October 2003

Abstract

Analytical exact solutions of a fundamental heat conduction problem in anisotropic multi-layered media are pre-sented in this study. The steady-state temperature and heat flux fields in multi-layered media with anisotropic properties in each layer subjected to prescribed temperature on the surfaces are analyzed in detail. Investigations on anisotropic heat conduction problems are tedious due to the presence of many material constants and the complex form of the governing partial differential equation. It is desirable to reduce the dependence on material constants in advance of the analysis of a given boundary value problem. One of the objectives of this study is to develop an effective analytical method to construct full-field solutions in anisotropic multi-layered media. A linear coordinate transformation is introduced to simplify the problem. The linear coordinate transformation reduces the anisotropic multi-layered heat conduction problem to an equivalent isotropic ones without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, explicit closed-form solutions of the specific problems are presented in series forms. The numerical results of the temperature and heat flux distributions for anisotropic multi-layered media are provided in full-field configurations.

Keywords: Heat conduction; Anisotropic media; Multi-layered; Coordinate transformation

1. Introduction

Many materials in which the thermal conductivity varies with direction are called anisotropic materials. As a result of interesting usage of anisotropic materials in engineering applications, the development of heat con-duction in anisotropic media has grown considerably in recent years. To date, few reported results of tempera-ture distribution or heat ﬂux ﬁelds in anisotropic media have appeared in the open literature. A number of standard text books (Carslaw and Jaeger [1], Ozisik [2]) have devoted a considerable portion of their contents to heat conduction problems in anisotropic bodies. Most of the earlier works for heat conduction in anisotropic

materials have been limited to one-dimensional prob-lems in crystal physics [3,4]. Tauchert and Akoz [5] solved the temperature fields of a two-dimensional anisotropic slab using complex conjugate quantities. Mulholland and Gupta [6] investigated a three-dimen-sional anisotropic body of arbitrary shape by using coordinate transformations to principal axes. Chang [7] solved the heat conduction problem in a three-dimen-sional configuration by conventional Fourier transfor-mation. Poon [8] first surveyed the transformation of heat conduction problems in layered composites from anisotropic to orthotropic. Poon et al. [9] extended coordinate transformation of the anisotropic heat con-duction problem to isotropic one. Zhang [10] developed a partition-matching technique to solve a two-dimen-sional anisotropic strip with prescribed temperature on the boundary.

In earlier papers, analytical solutions of anisotropic heat conduction problems have been limited to simple or

*

Corresponding author. Tel.: 2-2365-9996; fax: +886-2-2363-1755.

E-mail address:ccma@ntu.edu.tw(C.-C. Ma).

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special cases [2]. In conventional studies of a multi-dimensional anisotropic medium subjected to distrib-uted temperature or heat flux in or on the media, the analytical solution was obtained by Fourier transfor-mation. It is unlikely to find in most cases the general solutions with respect to each of the spatial variables to satisfy partial differential equations of anisotropic heat conduction equations and boundary conditions. The work of Yan et al. [11] studied two-layered isotropic bodies with homogeneous form of the conduction equation and the Green function solution is used to incorporate the effects of the internal heat source and non-homogeneous boundary conditions. They obtained the series solutions for three-dimensional temperature distribution by Fourier transformation, Laplace trans-formation and eigenvalue methods. Consequently, it is more difficult to get general analytic solutions satisfying all the boundary conditions for multi-layered aniso-tropic heat conduction problem because of the conti-nuity of temperature and heat flux on the interface and the cross-derivatives in the governing equation. There-fore, the cross term, which is the crux in solving the anisotropic heat conduction problem, is very trouble-some to analyze when one uses conventional solution techniques to solve isotropic heat conduction problems. Due to the mathematical difficulties of the problem, only few solutions for heat conduction in anisotropic media have appeared in the literature and much more work remains to be done.

Exact solutions for heat conduction problems in multi-layered media is of interest in electronic systems and composite materials in a wide variety of modern

engineering applications. Consequently, the thermal problems of heat dissipation from devices and systems have become extremely important. The inherent aniso-tropic nature of layered composites make the analysis more involved than that of isotropic counterpart. However, it may be pointed out that the exact and complete solution for multi-layered bodies of even iso-tropic media has not been reported to date because of the mathematical difficulties. The mathematical diffi-culties for heat conduction problem in multi-layered media are caused by the complex form of the governing partial differential equation and by the boundary and continuity conditions associated with the problem. Hsieh and Ma [12] used a linear coordinate transfor-mation to solve the heat conduction problem for a thin-layer medium with anisotropic properties. Exact closed-form solutions of temperature and heat flux fields were obtained by them.

In this study, a two-dimensional heat conduction problem for anisotropic multi-layered media subjected to prescribed temperature on the surfaces is investi-gated. The number of the layer is arbitrary, the thermal conductivities and the thickness are different in each layer. One of the objectives of this study is to develop an effective methodology to construct the analytical full-field solution for this problem. Investigations on anisotropic heat conduction problems are tedious due to the presence of many material constants and the cross-derivative term of the governing equation. It is desirable to reduce the dependence on material constants in ad-vance of the analysis of a given boundary value prob-lem. A special linear coordinate transformation is Nomenclature

a half region of the prescribed temperature in

the top surface

b half region of the prescribed temperature in

the bottom surface

ðcj; djÞ undetermined coeﬃcients

D the shifted distance for the position of the

concentrated temperature

fðxÞ, gðxÞ arbitrary functions

FðxÞ, GðxÞ Fourier transforms of f ðxÞ, gðxÞ

Fa

k geometrical dependent function on the

thickness of the layer

Gj=jþ1 relative matrix for the coeﬃcients of the

adjacent layer

hj vertical distance of the interface for the jth

layer from the top surface

Hj vertical distance of the interface for the jth

layer from the top surface after the coordi-nate transformation

Hn the thickness of the multi-layered medium

after transformation

k non-dimensional thermal conductivity

kij thermal conductivity

M1

k material dependent function on the

refrac-tion and reflecrefrac-tion coefficients ðqx; qyÞ heat flux

rj=jþ1 refraction coeﬃcient

tj=jþ1 reﬂection coeﬃcient

T temperature

ðx; yÞ coordinates

ðX ; Y Þ coordinates after transformation

Greek symbols

a, b coordinate transform coeﬃcients

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introduced in this study to simplify the governing heat conduction equation without complicating the conti-nuity and boundary conditions of the problem. Based on this transformation, the original anisotropic multi-layered problem is converted to an equivalent isotropic problem with a similar geometrical configuration. Ex-plicit closed-form solutions for the temperature and heat flux are expressed in a series form. Numerical results of the full-field distribution for temperature and heat flux are presented in graphic form and are discussed in detail.

2. Basic formulation and linear coordinate transformation Consider an anisotropic material that is homoge-neous and has constant thermo-physical properties. The governing partial diﬀerential equation for the heat con-duction problem in a two-dimensional Cartesian coor-dinate system is given by

k11 o2_T ox2þ 2k12 o2_T ox oyþ k22 o2_T oy2 ¼ 0; ð1Þ

where k11, k12 and k22 are thermal conductivity

coeffi-cients, and T is the temperature field. The corresponding heat fluxes are given as

qx¼ k11 oT ox k12 oT oy; qy¼ k12 oT ox k22 oT oy: ð2Þ

Based on irreversible thermo-dynamics, it can be shown that k11k22> k212 and the coeﬃcients k11 and k22

are positive. The governing equation expressed in Eq. (1) is a general homogeneous second order partial differ-ential equation with constant coefficients. Such a linear partial differential equation can be transformed into the Laplace equation by a linear coordinate transformation. A special linear coordinate transformation is introduced as X Y ¼ 1 a 0 b x y ; ð3Þ where a¼ k12 k22, b¼ k k22and k¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k11k22 k212 p . After the coordinate transformation, Eq. (1) can be rewritten as

the standard Laplace equation in theðX ; Y Þ coordinate

system k o 2_T oX2 þo 2_T oY2 ¼ 0: ð4Þ

It is interesting to note that the mixed derivative is eliminated from Eq. (1). The relationships between the heat ﬂux in the two coordinate systems are given by

qy¼ koT_oY¼ qY; qx¼ akoYoT bk oT oX¼ bqX aqY; or qY ¼ k oT oY; qX ¼ koToX: ð5Þ In a mathematical sense, Eqs. (1) and (2) are trans-formed to Eqs. (4) and (5) by the linear coordinate transformation expressed in Eq. (3), or in a physical sense, the governing equation (1) and the heat ﬂux and temperature relation (2) of an anisotropic heat conduc-tion problem are converted into an equivalent isotropic problem by properly changing the geometry of the body using the linear coordinate transformation, Eq. (3). The coordinate transformation in Eq. (3) has the following characteristics: (a) it is linear and continuous, (b) an anisotropic problem is converted to an isotropic prob-lem after the transformation, (c) there is no stretch and rotation in the horizontal direction. These important features oﬀer advantages in dealing with straight boundaries and interfaces in the multi-layered system discussed in the present study.

The linear coordinate transformation described by Eq. (3) can be used to solve the anisotropic heat con-duction problem for only a single material. However, for a multi-layered anisotropic medium with straight inter-faces as shown in Fig. 1, a modiﬁcation of the linear coordinate transformation will be introduced in the following section to transform the multi-layered aniso-tropic problem to an equivalent multi-layered isoaniso-tropic problem.

Fig. 1. Conﬁguration and coordinates system of an anisotropic multi-layered medium (a) and after the linear coordinate transformation (b).

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3. Full-ﬁeld solutions for anisotropic multi-layered media subjected to concentrated temperature

In this section, the full-ﬁeld solutions for the heat conduction problem of an anisotropic n-layered medium

subjected to a concentrated temperature T0 applied on

the top surface, as depicted in Fig. 1(a), will be analyzed. The number of the layer is arbitrary, the thermal con-ductivities and thickness in each layer are diﬀerent. The steady-state heat conduction equation in each layer is expressed as k11ðjÞ o2_TðjÞ ox2 þ 2k ðjÞ 12 o2_TðjÞ ox oyþ k ðjÞ 22 o2_TðjÞ oy2 ¼ 0; j¼ 1; 2; . . . ; n: ð6Þ

The boundary conditions on the top and bottom surfaces of the layered medium are

Tð1Þj_y¼0¼ T0dðxÞ; TðnÞjy¼hn ¼ 0; ð7Þ

where dð Þ is the delta function. The perfect thermal contact condition is assumed for the adjacent layer. The temperature and heat ﬂux continuity conditions at the

interface between the jth and jþ 1th layer yield

TðjÞ_j y¼hj¼ T ðjþ1Þ_j y¼hj; qðjÞ y jy¼hj¼ q ðjþ1Þ y jy¼hj; j¼ 1; 2; . . . ; n 1: ð8Þ

In order to maintain the geometry of the layered conﬁguration, the linear coordinate transformation de-scribed in Eq. (3) is modiﬁed for each layer as follows:

X Y ¼ 1 aj 0 bj " # x y þX j1 k¼1 hk ak akþ1 b_k bkþ1 ; j¼ 1; 2; . . . ; n; ð9Þ where aj¼ kðjÞ₁₂ kðjÞ₂₂, bj¼ kj kðjÞ₂₂ and kj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k11ðjÞk ðjÞ 22 k ðjÞ2 12 q . Comparing with Eq. (3), the first term in the right-hand side of Eq. (9) retains exactly the same form while the second term with a summation becomes the modified term. The new coordinate transformation possesses the following characteristics: (a) no gaps or overlaps are generated along the interface, (b) no sliding and mis-matches occur along the interface. The geometric

con-ﬁguration in the transformedðX ; Y Þ coordinate is shown

in Fig. 1(b). Note that while the thickness of each layer is changed, the interfaces are parallel to the x-axis. The new geometric conﬁguration after the coordinate trans-formation is similar to the original problem.

The heat conduction equations in the transformed coordinate for each layer are governed by the standard Laplace equation kj o2_TðjÞ oX2 þo 2_TðjÞ oY2 ¼ 0: ð10Þ

Furthermore, the temperature T and the heat ﬂux qY

are still continuous along the interfaces in the trans-formed coordinates, TðjÞ_j Y¼Hj¼ T ðjþ1Þ_j Y¼Hj; qðjÞY jY¼Hj¼ q ðjþ1Þ Y jY¼Hj; j¼ 1; 2; . . . ; n 1; ð11Þ

where Hj¼ bjhjþPj1_k¼1ðbk bkþ1Þhk. The top and

bot-tom boundary conditions are expressed as

Tð1ÞjY¼0¼ T0dðX Þ; TðnÞjY¼Hn ¼ 0: ð12Þ

The relations between heat ﬂux ﬁeld and temperature

ﬁeld expressed in the ðX ; Y Þ coordinates within each

layer become qðjÞ_X ðX ; Y Þ ¼ kj oTðjÞ_{ðX ; Y Þ} oX ; qðjÞ_Y ðX ; Y Þ ¼ kjoT ðjÞ_{ðX ; Y Þ} oY ; j¼ 1; 2; . . . ; n 1: ð13Þ

The boundary value problem described by Eqs. (10)– (13) is similar to the multi-layered problem for an isotropic material. Hence, the linear coordinate trans-formation presented in Eq. (9) changes the original complicated anisotropic multi-layered problem to the corresponding isotropic multi-layered problem with a similar geometric conﬁguration and boundary condi-tions.

The boundary value problem will be solved by the Fourier transform technique. Take the Fourier trans-form pairs deﬁned as

e T Tðx; Y Þ ¼ Z 1 1 TðX ; Y ÞeixX_{dX ;} TðX ; Y Þ ¼ 1 2p Z 1 1 e T Tðx; Y ÞeixX_dx; ð14Þ

where an overtilde denotes the transformed quantity, x is the transform variable, and i¼pffiffiffiffiffiffiffi1. By applying the Fourier transformation to the governing partial differ-ential equation (10), the equation in transformed do-main will be an ordinary differential equation of order two as follows: d2TTeðjÞ_{ðx; Y Þ} dY2 x 2_e T TðjÞðx; Y Þ ¼ 0: ð15Þ

The general solutions of the temperature and heat ﬂux can be presented in the matrix form as

e T TðjÞ ~ q qðjÞ_Y ¼ e xY _exY kjxexY kjxexY cj dj : ð16Þ

Here cj and dj are undetermined coeﬃcients for each

layer and can be obtained from the proper boundary and continuity conditions. It is noted that the variable x in the above equation is regarded as a parameter.

By using the continuity conditions at the interfaces, the relation for the coeﬃcients of the adjacent layer can be expressed as

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cj dj ¼ 1 rj=jþ1 Gj=jþ1 cjþ1 djþ1 ; j¼ 1; 2; . . . ; n 1; ð17Þ where rj=jþ1¼ 2kj kjþ kjþ1 ; Gj=jþ1¼ 1 tj=jþ1e2xHj tj=jþ1e2xHj 1 " # ; tj=jþ1¼ kj kjþ1 kjþ kjþ1 :

Here rj=jþ1 and tj=jþ1 are called the refraction and the

reflection coefficients, respectively. Therefore, the rela-tion between the coefficients of the jth layer and the nth layer can be expressed as

cj dj ¼ Y n1 k¼j 1 rk=kþ1 Gk=kþ1 " # cn dn ; ð18Þ where Yn k¼1 ak¼ a1 a2. . . an:

By setting j¼ 1 in Eq. (18) and applying the

boundary conditions as indicated in Eq. (12), the

coef-ﬁcients cnand dnin the nth layer are obtained explicitly

as follows: cn dn ¼ T0 A1þ A2 e2xHn 1 ; ð19Þ

where A1and A2are expressed in a matrix form as

A1 A2 ¼ Y n1 k¼1 1 rk=kþ1 Gk=kþ1 " # e2xHn 1 : ð20Þ

The undetermined constants cjand djfor each layer

are determined with the aid of the recurrence relations given in Eqs. (18) and (19). After substituting the

coef-ﬁcients cjand djinto Eq. (16), the full-ﬁeld solutions for

each layer are completely determined in the transformed domain. The solutions of temperature and heat ﬂux in the transformed domain for each layer are ﬁnally ex-pressed as e T TðjÞ ~ q qðjÞ_Y " # ¼ T0 A1þ A2 exY _exY kjxexY kjxexY Y n1 k¼j 1 rk=kþ1 Gk=kþ1 " # e2xHn 1 : ð21Þ

Since the solutions in Fourier transformed domain have been constructed, to inverse the solutions will be the next step. Because of the denominators in Eq. (21), it is impossible to inverse the Fourier transform directly. By examining the structure of the denominator of Eq.

(21), both the numerator and denominator are

multi-plied by a constant S¼Qn1_k¼1rk=kþ1. Then it becomes,

e T TðjÞ ~ q qðjÞY " # ¼ ST0 SðA1þ A2Þ exY _exY kjxexY kjxexY Y n1 k¼j 1 rk=kþ1 Gk=kþ1 " # e2xHn 1 : ð22Þ

The denominator in Eq. (22), SðA1þ A2Þ, can be

decomposed into the form of ð1 pÞ where

p¼ 1 SðA1þ A2Þ. It can be shown that p < 1 for

x >0 . By a series expansion, we obtain 1

1p¼

P1

l¼0plso

that Eq. (22) can be rewritten as e T TðjÞ ~ q qðjÞY " # ¼ ST0 exY _exY kjxexY kjxexY Y n1 k¼j 1 rk=kþ1 Gk=kþ1 " # e2xHn 1 X 1 l¼0 pl_: ð23Þ Since the solutions in the transformed domain ex-pressed in Eq. (23) are mainly exponential functions of x, the inverse Fourier transformation can be performed term by term. By omitting the lengthy algebraic deri-vation, the explicit solutions for temperature and heat ﬂux are obtained as follows:

TðjÞðX ; Y Þ ¼ T0 X1 l¼0 XN k¼1 M1 k Yþ Fa k X2_{þ ðY þ F}a kÞ 2 þ Yþ F b k X2_{þ ðY þ F}b kÞ 2 ! ; ð24Þ qðjÞY ðX ; Y Þ ¼ T0kj X1 l¼0 XN k¼1 M_k1 X 2_{ðY þ F}a kÞ 2 X2_{þ ðY þ F}a kÞ 2 2 0 B @ þ X 2_{ðY þ F}b kÞ 2 X2_{þ ðY þ F}b kÞ 2 2 1 C A; ð25Þ qðjÞ_X ðX ; Y Þ ¼ 2T0kj X1 l¼0 XN k¼1 M1 k XðY þ Fa kÞ X2_{þ ðY þ F}a kÞ 2 2 0 B @ þ XðY þ F b kÞ X2_{þ ðY þ F}b kÞ 2 2 1 C A; ð26Þ where N ¼ ð2nj_Þð2n_1Þl

. Here n is the number of lay-ers, and j is the jth layer where the solution is required.

The terms M1

k , Fka and Fkb in Eq. (24)–(26) are deﬁned

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a1¼ 1; f1A1¼ 2Hn; f1A2 ¼ 0; aiþ2k1¼ aitnk=nkþ1; fA1 iþ2k1¼ ðf A1 i þ 2Hnþ 2HnkÞ; fA2 iþ2k1¼ ðf A2 i þ 2Hn 2HnkÞ; 8 > > > < > > > : k¼ 1; 2; . . . n 1; i¼ 1; 2; . . . 2k1_; ð27aÞ rpi ¼ ai; rp₂n1_þi¼ ai; fip¼ f A2 i ; f p 2n1_þi¼ f A1 i ; i¼ 1; 2; . . . 2n1_; ð27bÞ rl k¼ Ql o¼1 rp_i o; gl k¼ Pl o¼1 fipo; 8 > > < > > : i1; i2; i3; . . . il¼ 1; 2; 3; . . . 2n 1; k¼X l1 o¼1 ðio 1Þð2n 1Þloþ il; ð27cÞ for l¼ 0, r0 i0 ¼ 1 and g 0 i0 ¼ 0 M1 ði1Þð2n_1Þl_þk¼ 1 2p Q j1 o¼1 ro=oþ1 P 2nj i¼1 P ð2n_1Þl k¼1 airlk; Fa ði1Þð2n_1Þl_þk¼ g_klþ f A1 i ; Fb ði1Þð2n_1Þl_þk¼ gklþ f A2 i ; 8 > > > > < > > > > : k¼ 1; 2; . . . ð2n_1Þl ; i¼ 1; 2; . . . 2nj_: ð27dÞ Finally, by substituting X and Y defined in Eq. (9) into Eqs. (24)–(26) and using Eq. (5), the explicit expressions of temperature and heat flux fields for anisotropic multi-layered media subjected to a

pre-scribed concentrated temperature T0 on the top surface

are presented as follows: TðjÞðx; yÞ ¼ T0 X1 l¼0 XN k¼1 M1 k Yþ Fa k X2_{þ ðY þ F}a kÞ 2 þ Yþ F b k X2_{þ ðY þ F}b kÞ 2 ! ; ð28Þ qðjÞ y ðx; yÞ ¼ T0kj X1 l¼0 XN k¼1 M1 k X2_{ðY þ F}a kÞ 2 X2_{þ ðY þ F}a kÞ 2 2 0 B @ þ X 2_{ðY þ F}b kÞ 2 X2_{þ ðY þ F}b kÞ 2 2 1 C A; ð29Þ qðjÞ x ðx; yÞ ¼ T0kj X1 l¼0 XN k¼1 M1 k

X½bjðY þ FkaÞ ajX þ ðY þ FkaÞ½bX ajðY þ FkaÞ X2_{þ ðY þ F}a kÞ 2 2 0 B @ þX½bjðY þ F b kÞ ajX þ ðY þ FkbÞ½bX ajðY þ FkbÞ X2_{þ ðY þ F}b kÞ 2 2 1 C A; ð30Þ X Y ¼ 1₀ a_bj j x y þX j1 k¼1 hk ak akþ1 b_k bkþ1 :

It is interesting to note that Fa

k and F

b

k are dependent

on the thickness of the layer, i.e., Hj, and Mk1 depends

only on the refraction and reﬂection coeﬃcients, i.e., rj=jþ1and tj=jþ1.

If the concentrated temperature is applied on the bottom surface of the anisotropic multi-layered medium, the boundary conditions become

Tð1Þj_y¼0¼ 0; TðnÞj_y¼h

n ¼ T0dðxÞ: ð31Þ

By using the similar method indicated previously, the solutions in the Fourier transformed domain are ob-tained as follows: e T TðjÞ ~ q qðjÞ_Y " # ¼ e ixDxHn B1þ B2e2xHn exY _exY kjxexY kjxexY Y j1 k¼1 1 rkþ1=k Gnkþ1=nk " # 1 1 ; ð32Þ where B1 B2 " # ¼ Y n1 k¼1 1 rkþ1=k Gnkþ1=nk " # 1 1 " # ; D¼ anhnþ Xn1 k¼1 ðak akþ1Þhk; Hn¼ bnhnþ Xn1 k¼1 ðbk bkþ1Þhk:

Note that D is a shifted amount in the horizontal

direction of the concentrated temperature and Hn is the

total thickness of the multi-layered medium after applying the linear coordinate transformation as indi-cated in Eq. (9).

By using the series expansion technique and the in-verse Fourier transformation, the explicit solutions can be expressed as follows: TðjÞðX ; Y Þ ¼ T0 X1 l¼0 XN k¼1 M2 k Yþ Fc k ðX DÞ2þ ðY þ Fc kÞ 2 þ Yþ F d k ðX DÞ2þ ðY þ Fd kÞ 2 ! ; ð33Þ qðjÞY ðX ; Y Þ ¼ T0kj X1 l¼0 XN k¼1 M_k2 ðX DÞ 2 ðY þ Fc kÞ 2 ðX DÞ2_{þ ðY þ F}c kÞ 2 2 0 B @ þ ðX DÞ 2 ðY þ Fd kÞ 2 ðX DÞ2þ ðY þ Fd kÞ 2 2 1 C A; ð34Þ

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qðjÞ_X ðX ; Y Þ ¼ 2T0kj X1 l¼0 XN k¼1 M2 k ðX DÞðY þ Fc kÞ ðX DÞ2þ ðY þ Fc kÞ 2 2 0 B @ þ ðX DÞðY þ F d kÞ ðX DÞ2þ ðY þ Fd kÞ 2 2 1 C A; ð35Þ where b1¼ 1f1B¼ 0; biþ2k1¼ bitkþ1=k; fB iþ2k1¼ ðfiBþ 2HkÞ; 8 > < > : k¼ 1; 2; . . . m 1; i¼ 1; 2; . . . 2k1_; ð36aÞ rip¼ bi; r2pn1_þi¼ bi; fip¼ fiB; f p 2n1_þi¼ fiB 2Hn; i¼ 1; 2; . . . 2n1_; ð36bÞ rl k¼ Ql o¼1 rp_i o; gl k¼ Pl o¼1 fipo; 8 > > < > > : i1; i2; i3; . . . il¼ 1; 2; . . . 2n 1; k¼X l1 o¼1 ðio 1Þð2n 1Þloþ il; ð36cÞ for l¼ 0, r0 i0¼ 1 and g 0 i0¼ 0, M2 ði1Þð2n_1Þl_þk¼_2p1 Q n1 o¼1 roþ1=o P 2j1 i¼1 P ð2n_1Þl k¼1 birlk; Fc ði1Þð2n_1Þl_þk¼ gl_kþ f_iB Hn; Fd ði1Þð2n_1Þl_þk¼ g l k f B i 3Hn; 8 > > > > < > > > > : k¼ 1; 2; . . . ð2n_1Þl ; i¼ 1; 2; . . . 2j1_; ð36dÞ in which, tjþ1=j¼ kjþ1 kj kjþ1þ kj ; rjþ1=j¼ 2kjþ1 kjþ kjþ1 :

4. Explicit solutions for distributed temperature on surfaces

The full-ﬁeld solutions of anisotropic multi-layered media subjected to concentrated temperature on the surfaces are obtained in the previous section. In this section, the solutions of temperature and heat ﬂux for multi-layered media subjected to distributed tempera-ture on the surfaces will be discussed.

The deﬁnition of convolution and the convolution property of Fourier transform are as follows:

fðxÞ gðxÞ ¼ Z 1 1 fðsÞgðx sÞ ds; Iðf ðxÞ gðxÞÞ ¼ eFFðxÞ eGGðxÞ: ð37Þ

By using the convolution property of the Fourier transform and the Green’s function in the transformed domain, it is easy to construct solutions for distributed temperature applied in the surfaces. Now consider the

case that the top surface on y¼ 0, jxj < a is under the

action of a prescribed uniformly distributed tempera-ture, that is, the boundary condition is replaced by Tð1Þ_j

y¼0¼ T0fH ðx þ aÞ H ðx aÞg; ð38Þ

where HðÞ is the Heaviside function. The boundary

condition in the transformed domain is eTTð1Þ¼2T0sin ax

x .

It is easy to write down the complete solution in the Fourier transformed domain as follows:

e T TðjÞ ~ q qðjÞY " # ¼ 2T0sin ax xðA1þ A2Þ exY _exY kjxexY kjxexY Y n1 k¼j 1 rk=kþ1 Gk=kþ1 " # e2xHn 1 : ð39Þ

The explicit solutions of temperature and heat ﬂux for multi-layered media subjected to a uniformly

dis-tributed temperature T0 in the region 2a on the top

surface are expressed as follows: TðjÞðX ; Y Þ ¼ T0 X1 l¼0 XN k¼1 M_k1 tan1 Xþ a Yþ Fa k tan1 X a Yþ Fa k þ tan1 Xþ a Yþ Fb k tan1 X a Yþ Fb k ; ð40Þ qðjÞYðX ; Y Þ ¼ T0kj X1 l¼0 XN k¼1 M1 k Xþ a ðX þ aÞ2þ ðY þ Fa kÞ 2 X a ðX aÞ2þ ðY þ Fa kÞ 2 þ Xþ a ðX þ aÞ2þ ðY þ Fb kÞ 2 X a ðX aÞ2þ ðY þ Fb kÞ 2 0 B B B B @ 1 C C C C A; ð41Þ qðjÞXðX ; Y Þ ¼ T0kj X1 l¼0 XN k¼1 M1 k Yþ Fa k ðX þ aÞ2_{þ ðY þ F}a kÞ 2 Yþ Fa k ðX aÞ2_{þ ðY þ F}a kÞ 2 þ Yþ F b k ðX þ aÞ2þ ðY þ Fb kÞ 2 Yþ Fb k ðX aÞ2þ ðY þ Fb kÞ 2 0 B B B @ 1 C C C A; ð42Þ where M1 k, F a k and F b

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Similarly, the solutions of anisotropic multi-layered media subjected to a uniformly distributed temperature

T0 in the regionjxj 6 b on the bottom surface are

ob-tained as follows: TðjÞðX ; Y Þ ¼ T0 X1 l¼0 XN k¼1 M_k2 tan1X D þ b Yþ Fc k tan1X D b Yþ Fc k þ tan1X D þ b Yþ Fd k tan1X D b Yþ Fd k 0 B B B @ 1 C C C A; ð43Þ qðjÞYðX ; Y Þ ¼ T0kj X1 l¼0 XN k¼1 M2 k X D þ b ðX D þ bÞ2þ ðY þ Fc kÞ 2 X D b ðX D bÞ2þ ðY þ Fc kÞ 2 þ X D þ b ðX D þ bÞ2þ ðY þ Fd kÞ 2 X D b ðX D bÞ2þ ðY þ Fd kÞ 2 0 B B B @ 1 C C C A; ð44Þ qðjÞXðX ;Y Þ ¼ T0kj X1 l¼0 XN k¼1 M2 k Yþ Fc k ðX D þ bÞ2þ ðY þ Fc kÞ 2 Yþ Fc k ðX D bÞ2þ ðY þ Fc kÞ 2 þ Yþ F d k ðX D þ bÞ2 þ ðY þ Fd kÞ 2 Yþ Fd k ðX D bÞ2 þ ðY þ Fd kÞ 2 0 B B B @ 1 C C C A; ð45Þ where M2 k, F c k and F d

k are given in Eqs. (36a)–(36d).

5. Numerical results

By using the analytical explicit solutions developed in the previous sections, numerical calculations of tem-perature and heat ﬂux are obtained for anisotropic multi-layered media via a computational program. The full-ﬁeld analysis for the anisotropic layered medium consisting of 10 layers subjected to prescribed tempera-ture on surfaces will be discussed in detail. The thermal conductivities for each layer are listed in Table 1.

Figs. 2–4 show the full-ﬁeld distributions of temper-ature and heat ﬂuxes for prescribed uniformly

distrib-uted temperature T0 on the top surfaceh 6 x 6 h, the

thickness for each layer is the same and equal to h. In the full-ﬁeld distribution contours, solid lines and dot lines are used to indicate positive and negative values, respectively. In anisotropic multi-layered media, the

Table 1

The thermal conductivities for the anisotropic ten-layered medium

Layer Thermal conductivity (W/m K)

k11 k12 k22 1 44.01 11.91 85.28 2 76.56 20.63 52.73 3 30.65 3.37 28.82 4 83.61 18.12 20.84 5 33.67 0 33.67 6 52.73 20.63 76.56 7 28.82 3.37 30.65 8 83.61 18.12 20.84 9 33.67 0 33.67 10 85.28 11.91 44.01

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symmetry for the temperature and heat flux fields that is found in the isotropic material is distorted due to the material anisotropy. It is shown in the figures that

the temperature and heat ﬂux qy are continuous at the

interfaces. This also indicates that the convergence and accuracy for the numerical calculation are satisﬁed.

However, the heat ﬂux qxis discontinuous at the

inter-faces and the values are small except at the first layer. Next, the full-field analysis of anisotropic multi-lay-ered media with different layer thickness for each layer is considered. The full-field distributions of temperature and heat flux in the y-direction for prescribed uniformly

distributed temperature 2T0 at two regions ð2h 6 x 6

h; h 6 x 6 2hÞ on the top surface and constant

tem-perature T0 on the entire bottom surface are shown in

Figs. 5 and 6, respectively. Fig. 7 shows the temperature

ﬁeld for prescribed constant temperature 2T0 at

h 6 x 6 h on the top surface and constant temperature

T0 at2h 6 x 6 2h on the bottom surface.

The use of composite materials in a wide variety of modern engineering applications has been rapidly increasing over the past few decades. The increasing use of composite materials in the automotive and aerospace industries has motivated research into solution methods to investigate the thermal properties of these materials. Numerical calculations for layered composites of 12 Fig. 4. Full-ﬁeld heat ﬂux qxdistribution for prescribed uniformly distributed temperature T0on the top surfaceh 6 x 6 h.

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ﬁber-reinforced layers will be considered. The ﬁber angle, h, is measured counterclockwise from the positive x-axis

to the ﬁber direction. A [0°/30°/60°/90°/120°/150°]2

lam-inated composite is considered ﬁrst. By regarding each layer as being homogeneous and anistotropic, the gross

thermal conductivities k11, k12, k22¼ 30:65, 3.37, 28.82

W/m K in the material coordinates of the layer are used. The gross thermal conductivities in the structured coor-dinates for a given ﬁber orientation h of the layer can be determined via the tensor transformation equation. The numerical result of the temperature distribution for

pre-scribed temperature 2T0 in a region2h 6 x 6 2h on the

Fig. 5. Full-ﬁeld temperature distribution for prescribed uniformly distributed temperature 2T0 at two regions ð2h 6 x 6 h;

h 6 x 62hÞ on the top surface and constant temperature T0on the bottom surface.

Fig. 6. Full-ﬁeld heat ﬂux qy distribution for prescribed uniformly distributed temperature 2T0 at two regionsð2h 6 x 6 h;

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top surface is shown in Fig. 8. Next, a composite layered

medium with stacking sequence [0°/60°/)60°]2S is

inves-tigated and the result is shown in Fig. 9.

Finally, we consider the prescribed surface tempera-ture as a function in the form

Tð1Þj_y¼0¼ T0 1þ cosp_hx jxj 6 h 0 jxj > h: ð46Þ Figs. 10 and 11 indicate the full-field distributions of temperature and heat flux in the y-direction for an Fig. 7. Full-field temperature distribution for prescribed uniformly distributed temperature 2T0ath 6 x 6 h on the top surface and

constant temperature T0at2h 6 x 6 2h on the bottom surface.

Fig. 8. Full-ﬁeld temperature distribution of a [30°/60°/90°/120°/150°]2 laminated composite for prescribed uniformly distributed

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anisotropic layered medium consisting of 10 layers. The thickness for each layer is diﬀerent and the thermal conductivities are presented in Table 1.

6. Summary and conclusions

A two-dimensional steady-state thermal conduction problem of anisotropic multi-layered media is

consi-dered in this study. A linear coordinate transformation for multi-layered media is introduced to simplify the governing heat conduction equation without compli-cating the boundary and interface conditions. The linear coordinate transformation introduced in this study substantially reduces the dependence of the solution on thermal conductivities and the original anisotropic multi-layered heat conduction problem is reduced to an equivalent isotropic problem. By using the Fourier Fig. 9. Full-ﬁeld temperature distribution of a [0°/60°/)60°]2Slaminated composite for prescribed uniformly distributed temperature

2T0at2h 6 x 6 2h on the top surface.

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transform technique and a series expansion, exact ana-lytical solutions for the full-field distribution of tem-perature and heat flux are presented in explicit series forms. The solutions are easy to handle in numerical computation. The numerical results for the full-field distribution for different boundary conditions are pre-sented and are discussed in detail. Solutions for other cases of boundary temperature distribution can be constructed from the basic solution obtained in this study by superposition. The analytical method provided in this study can also be extended to solve the aniso-tropic heat conduction problem in multi-layered media with embedded heat sources and the results will be given in a follow-up paper.

Acknowledgements

The ﬁnancial support of the authors from the Na-tional Science Council, People’s Republic of China, through grant NSC 89-2212-E002-018 to National Taiwan University is gratefully acknowledged.

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