Composite Structures 106 (2013) 288–295

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Composite Structures

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A general exact solution for heat conduction in multilayer sphericalcomposite laminates

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.06.005

⇑ Corresponding author. Tel.: +98 9123726933; fax: +98 2733300258.E-mail addresses: mnorouzi@shahroodut.ac.ir (M. Norouzi), a.a.delouei@gmail.

com (A. Amiri Delouei), ms.mohsenseilsepour@gmail.com (M. Seilsepour).

M. Norouzi ⇑, A. Amiri Delouei, M. SeilsepourMechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran

a r t i c l e i n f o

Article history:Available online 25 June 2013

Keywords:Analytical solutionSpherical composite laminateHeat conductionFourier–Legendre seriesThomas algorithm

a b s t r a c t

In this study, an exact analytical solution for steady conductive heat transfer in multilayer spherical fiberreinforced composite laminates is presented as the first time. Here, the orthotropic temperature distribu-tion of laminate is obtained under the general linear boundary conditions that are suitable for variousconditions including combinations of conduction, convection, and radiation both inside and outside ofthe sphere. The temperature and heat flux continuity is applied between the laminas. In order to obtainthe exact solution, the separation of variables method is used and the set of equations related to the coef-ficient of Fourier–Legendre series of temperature distribution is solved using the recursive Thomas algo-rithm. The capability of the present solution is examined by applying it on two industrial applications fordifferent fiber arrangements of multilayer spherical laminates.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The fiber reinforced multilayer composite materials have inter-ested significantly in modern engineering. This fact is due to vastadvantages of these materials like high strength-to-density ratio,stiffness-to-density ratio, high corrosion resistance and plasticityas compared with most materials. Most of these unique advantagesare because of two properties of these materials i.e. (1) combiningdifferent physical, mechanical and thermal properties of variousmaterials; and (2) ability to change the fibers’ orientations of everylayer to meet the design requirements. Furthermore, increasing themore effective manufacturing technologies of composite materialsaccumulated over the years caused decreasing the cost of thesekind of materials. Today, reinforced composites have been usedenormously in aerospace and marine industries, pressure vessels,fluid reservoirs, pipes and so on. Although, the knowledge ofcomposite materials has a reasonable progressive with the devel-opment of their applications, for example mechanical analysis[1–7], but thermal analysis is an exception. Heat transfer in com-posite laminates is vital for analyzing of thermal stress [8], thermalshock [9], controlling directional heat transfer through laminatesand fiber placement in production processes [10,11].

The problem of heat conduction in multilayer structures can besubdivided based on the coordinates of solutions as: heat conduc-tion in Cartesian coordinate [12–21], cylindrical coordinate in r � z

[22–27], and r � u [28,29] directions; and heat transfer in spheri-cal shapes [30,31].

Blanc and Touratier [12] present a new simple refined compu-tational model to analyze heat conduction in composite laminates.This model was based on an equivalent single layer approachallows to satisfy the continuity of temperatures and the heat fluxbetween the layers, as well as the boundary conditions.

Separation of variables technique was used by Miller andWeaver [13] to predict the temperature distribution through amulti-layered system subject to complex boundary conditions.The system is subjected to both convection and radiation boundaryconditions and results agree well with numerical results under thesame boundary conditions.

Ma et al. [14,15] developed a closed-form solution for heat con-duction in an anisotropic single layer [14] and multi-layered [15]media. A linear coordinate transformation is used to simplify theproblem into an equivalent isotropic one.

Salt [16,17] investigated the response of a 2-D multi-layercomposite slab, to a sudden temperature change. The solution isanalytically examined in two- and three-layer composite slabs.Monte [18–21] developed several analytical solutions for heat con-duction in 2-D composites.

Kayhani et al. [22–24] presented an exact analytical solution foraxisymmetric steady heat conduction in cylindrical multi-layercomposite laminates. The unsteady solution of this problem hasbeen presented by Amiri Delouei et al. [25]. In these studies, anew Fourier transformation has been developed for steady and un-steady cases. Furthermore the Meromorphic function method wasutilized to find the transient temperature distribution in laminate.Also, some studies have been investigated the conductive heat

Fig. 1. Direction of fibers in a spherical laminate.

M. Norouzi et al. / Composite Structures 106 (2013) 288–295 289

transfer in cylinders made from a functional graded material andcomposite laminates [26,27].

The asymmetric steady and transient heat conduction in cylin-drical composite laminates have been studied by Kayhani et al.[28] and Norouzi et al. [29], respectively. Separation of variablesmethod and Laplace Transformation was used to solve the partialdifferential equations. The solution they obtained is only valid forlong pipes and vessels.

Only few studies have considered heat conduction of sphericalmulti-layered materials. Jain et al. [30,31] proposed an analyticalseries solution for heat conduction in r � h spherical coordinates.Although this solution is valid for different kind of boundary con-ditions but materials in each layer have been considered in isotro-pic type. This solution is valid only for multi-layer spheres andcannot apply for multi-layer rein-forced composites spheres.

In this study, an exact analytical solution for steady state heatconduction in spherical composite laminates is presented. Lami-nates are in spherical shape (see Fig. 1) and composed from matrixand fiber materials. Heat conduction is considered in r � h direc-tions where r and h represent radius and cone angle, respectively.Fibers are winded in circumferential direction (Fig. 1). The bound-ary conditions are the general linear boundary conditions whichcan simplified to all mechanisms of heat transfer both inside andoutside of laminate. Governing equation of orthotropic heat con-duction in each layer has been achieved and solved based on theseparation of variables method. Using the separation of variablesmethod, the solution can be reduced to the expansion of an arbi-trary function into a series of Legendre polynomials. Consideringthe thermal boundary conditions inside and outside the cylinder,and applying the continuity of the temperature and the heat fluxbetween the layers, the Fourier–Legendre coefficients are obtained.The Thomas algorithm is used to obtain the solution of the set ofequations related to the temperature distribution coefficients. Toour knowledge, this general analytical solution for sphericalreinforced composite is the first one in this field. The ability of cur-rent solution is examined via two industrial examples consist of acomposite vessel and a composite shell. The effect of compositethermal design parameters, i.e. fiber’s direction and compositematerial of each layer, are investigated in details.

2. Conduction in spherical composites

In this section, the equations of conductive heat transfer inspherical composite materials are presented. The Fourier equation

of orthotropic material in spherical coordinate system is as follows[32]:

qr

qh

q/

0B@

1CA ¼ �

k11 k12 k13

k21 k22 k23

k31 k32 k33

264

375

@T@r1r@T@h1

r sin h@T@/

0B@

1CA ð1Þ

where k is the conductive heat transfer coefficient, T and q are tem-perature and heat flux, respectively. According to thermodynamicreciprocity, the tensor of conductive heat coefficients should besymmetric:

kij ¼ kji ð2aÞ

On the other hand, the second law of thermodynamics caused thatthe diametric elements of this tensor are positive so the followingrelation must be satisfied [32–34]:

kiikjj > kij for i – j ð2bÞ

Using the Clausius_Duhem inequality, the following inequalities forthe conductive coefficients of orthotropic materials are achieved:

kðiiÞ P 0 ð2cÞ12ðkðiiÞkðjjÞ � kðjiÞkðijÞÞP 0 ð2dÞ

eijkkð1jÞkð2jÞkð3jÞP0 ð2eÞ

where kij represents the symmetric part of tensor:

kðijÞ ¼ kðjiÞ ¼kij þ kji

2ð2fÞ

These relations are valid in all coordinate systems. Two separatecoordinate systems must be considered to investigate heat transferproblems in composite laminates: ‘‘on-axis’’ coordinate system (x1,x2, x3) and ‘‘off-axis’’ coordinate system (r, /, h) [35]. The direction ofthe ‘‘on axis’’ coordinate depends on the fibers’ orientation in eachlayer: x1 is parallel to fiber, x2 is perpendicular to fiber in layer andx3 is perpendicular to layer. Since composite materials are generallyfabricated by laying layers on top of each other, the fiber orientationmay differ between layers. We need to define an off axis coordinatesystem to study the physical properties in unique directions. Thus,there is an angular deviation between the on axis and off axis coor-dinates. The Fourier equation in on-axis coordinate will be [36]:

qr

qh

q/

0B@

1CA ¼ �

k11 0 00 k22 00 0 k33

0B@

1CA

@T@r1r@T@h1

r sin h@T@/

0B@

1CA ð3Þ

The off-axis conductivity tensor ½�k� is obtained by applying the rota-tion h to the on-axis conductivity tensor [k]:

½�k� ¼ TðhÞ½k�Tð�hÞ ð4Þ

where T(h) is the rotation tensor [28]:

TðhÞ ¼cosðhÞ � sinðhÞ 0sinðhÞ cosðhÞ 0

0 0 1

264

375 ð5Þ

The heat conduction coefficients can be directly obtained fromexperimental measurements or be calculated based on the theoret-ical models [36–46].

3. Modeling and Formulations

In this section, governing equation of heat conduction in spher-ical composite is presented and general boundary conditions usedin this study are introduced. Fibers’ direction can be varied in eachlayer (see Fig. 1) and (r, /, h) are the off-axis coordinates. Applying

290 M. Norouzi et al. / Composite Structures 106 (2013) 288–295

the balance of energy in element of sphere which has shown inFig. 2, the following equation will be achieved:

@q/dA/

@/d/þ @qhdAh

@hdhþ @qrdAr

@rdr ¼ qcp

@T@t

dv ð7Þ

Surface areas and volume of sphere element are as follows:

dAr ¼ r2 sin hd/dh

dAh ¼ r sin hd/dh

dA/ ¼ rdhdr

dv ¼ r2 sin hdrd/dh

ð8Þ

Quantities q and cp in Eq. (7) are the density and specific heatcapacity at constant pressure, respectively. Substituting Eq. (1)and Eq. (8) into Eq. (7) will be resulted in:

�k111r2

@

@rðr2 @T

@rÞ þ �k22

1r2 sin h

@2T

@U2 þ�k33

1r2 sin h

@

@hðsin h

@T@hÞ

þ ð�k12 þ �k21Þ

r sin h@2T@r@/

þ �k121

r2 sin h@T@Uþ ð

�k13 þ �k31Þr

@2T@r@h

þ �k13

� 1r2

@T@hþ ð�k32 þ �k23Þ

1r2 sin h

@2T@h@U

þ �k31cos hr sin h

@T@r

¼ qCp@T@t

ð9Þ

Here, steady-state conductive heat transfer in the r and h directionsare considered. Thus, Eq. (9) can be simplified to

�k111r2

@

@rðr2 @T

@rÞ þ �k22

1r2 sin h

@

@hsin h

@T@h

� �¼ 0 ð10Þ

The off-axis component of conductivity tensor (�k11 and �k22) are ob-tained by substituting Eq. (5) into Eq. (4):

k11� ¼ k22

k22� ¼ m2

l k11 þ n2l k22; ml ¼ cos Wi; nl ¼ sin Wi

�ð11Þ

where W is the angle between the tangent line to fibers and h direc-tion as shown schematically in Fig. 1. Substituting the determinedoff-axis coefficients (Eq. (11)) into energy equation (Eq. (10))results:

1r2

@

@rr2 @T@r

� �þ 1

l2i

1r2 sin h

@

@hsin h

@T@h

� �¼ 0 ð12Þ

Fig. 2. Schematic of heat fluxes on a spherical element.

where parameter li is given by:

li ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k22

m2l k11 þ n2

l k22

sð13Þ

It is important to mention that li can be changed layer by layer andso the energy equation will be change in each layer; this fact leadsto different temperature distribution in layers. In order to connectthe different temperature distributions in each layer, continuity oftemperature and heat flux in margin of each pair of layers mustbe considered as follows:

TðiÞ � Tðiþ1Þ ¼ 0 ð14aÞ

kðiÞ22@TðiÞ

@r� kðiþ1Þ

22@Tðiþ1Þ

@r¼ 0 ð14bÞ

The general linear boundary conditions inside and outside of thesphere are in the following forms which can covers wide range ofapplicable thermal conditions:

a1Tðr0; hÞ þ b1@T@rðr0; hÞ ¼ f1ðhÞ ð15aÞ

a2Tðrnl; hÞ þ b2@T@rðrnl; hÞ ¼ f2ðhÞ ð15bÞ

Note that f1(h), f2(h) are the arbitrary functions, the constant a1, a2

have the same dimension as convection coefficient and b1, b2 havethe same dimension as conduction coefficient.

4. Analytical solution under general boundary conditions

In this section, the analytical solution of steady temperaturedistribution under generalized linear boundary conditions is pre-sented based on separation of variables method. By applying theseparation of variables method on Eq. (12), the temperature distri-bution could be separated as two independent functions R(r) andH(h):

Tðr; hÞ ¼ RðrÞHðhÞ ð16Þ

Substituting Eq. (16) into the Eq. (12), heat conduction equation hasbeen separated as:

r2 R00

Rþ 2r

R0

R

� �¼ � 1

l2

cos hsin h

_HHþ

€HH

!¼ k ð17Þ

where k is a constant. By supposing x = sinh, the separated equationin h direction can be solved as a Legendre equation:

@

@x1� x2 @H

@x

� �þ nðnþ 1ÞH ¼ 0 ð18Þ

The solution of Eq. (18) is as follows [47]:

HðhÞ ¼X1n¼0

AnPnðcos hÞ ð19Þ

where Pn indicates the Legendre function of degree n and order one,and An is the coefficient of Legendre series. Comparing Eqs. (17) and(18), k will be achieved as follows:

k ¼ nðnþ 1Þl2 ð20Þ

According to Eq. (17), the separated equation in r direction is an Eu-ler equation with the underneath solution:

RnðrÞ ¼ Bnrnl2 þ Cnr

�ðnþ1Þl2 for n P 1

B0 ln r þ C0 for n ¼ 0

(ð21Þ

The temperature distribution in each layer will be:

M. Norouzi et al. / Composite Structures 106 (2013) 288–295 291

TðiÞðr; hÞ ¼ aðiÞ0 lnr

rnl

� �þ bðiÞ0

� �P0ðcos hÞ

þX1n¼1

aðiÞnr

rnl

� � nl2

þ bðiÞnr

rnl

� ��ðnþ1Þl2

!Pnðcos hÞ ð22Þ

where index i refer to the number of layers and the following rela-tions are existed for coefficients of above temperature distribution:

aðiÞv ¼ AðiÞv BðiÞv

bðiÞv ¼ AðiÞv CðiÞvv ¼ 0;n ð23Þ

Finally, by applying the inside and outside boundary conditions inthe direction of r and applying the continuity of temperature andheat flux at the boundary located between layers, the coefficientsa0, b0, an, bn are obtained as follows:

� By applying boundary condition inside and outside of sphereEqs. (15a) and (15b), we have:

a1 lnr0

rnl

� �þ b1

1r0

� �� �að1Þ0 þ a1bð1Þ0

� �P0ðcos hÞ

þ

a1rn

l2m

0 þ b1nl2

0r

nl2

0

�1

0

!að1Þn þ

a2r�ðnþ1Þ

l20

0 þ b2�ðnþ1Þ

l20

r�ðnþ1Þ

l20

�1

0

!bð1Þn

0BBBBBBB@

1CCCCCCCA

Pnðcos hÞ ¼ f1ðhÞ

ð24aÞ

b2

rnlaðnlÞ

0 þ a2bðnlÞ0

� �P0ðcos hÞ

þX1n¼1

a2rn

l2nl

nl þ b2n

l2m

rn

l2nl

�1

nl

!aðnlÞ

n þ

a2r

�ðnþ1Þl2

nlnl þ b2

�ðnþ1Þl2

nl

r

�ðnþ1Þl2

nl

�1

nl

0@

1AbðnlÞ

n

0BBBBBBB@

1CCCCCCCA

Pnðcos hÞ ¼ f2ðhÞ

ð24bÞ

� The following equations could be expressed by applying thetemperature and the heat flux continuity at the boundarylocated between the layer i and i + 1 (Eqs. (14a) and (14b)):

lnri

rnl

� �aðiÞ0 þ bðiÞ0 � ln

ri

rnl

� �aðiþ1Þ

0 þ bðiþ1Þ0

� �P0ðcos hÞ

þX1n¼1

rirnl

� � nl2

i aðiÞn þ rirnl

� ��ðnþ1Þl2

i bðiÞn �

rirnl

� � nl2

iþ1 aðiþ1Þn � ri

rnl

� ��ðnþ1Þl2

iþ1 bðiþ1Þn

0BBB@

1CCCAPnðcos hÞ ¼ 0 ð24cÞ

1rnl

� �ln

ri

rnl

� �aðiÞ0 �

1rnl

� �ln

ri

rnl

� �aðiþ1Þ

0

� �P0ðcos hÞ

þX1n¼1

nl2

i

rirnl

� � nl2

i

�1aðiÞn � ðnþ1Þ

l2i

rirnl

� ��ðnþ1Þl2

i

�1bðiÞn �

nl2

iþ1

rirnl

� � nl2

iþ1

�1aðiþ1Þ

n þ ðnþ1Þl2

iþ1

rirnl

� ��ðnþ1Þl2

iþ1

�1bðiþ1Þ

n

0BBBB@

1CCCCAPnðcos hÞ

¼ 0 ð24dÞ

Using the existing relations for orthogonal Legendre functions[47,48] and rearranging the Eqs. (24a) to (24d), the unknown coef-ficients will be achieved.� Resorting Eq. (24a) results:

a1 lnr0

rnl

� �þb1

1r0

� �� �að1Þ0 þa1bð1Þ0 ¼ F0

0;

a1rn

l2m

0 þb1nl2

0

rnl2

0

�1

0

!að1Þn þ a2r

�ðnþ1Þl2

00 þb2

�ðnþ1Þl2

0

r�ðnþ1Þ

l20

�1

0

!bð1Þn

¼ F0n

ð25aÞ

� Similarly, Eq. (24b) results:

b2

rnlaðnlÞ

0 þ a2bðnlÞ0 ¼ Fnl

0

a2rn

l2m

nl þ b2nl2

mr

nl2

m�1

nl

� �aðnlÞ

n þ ða2r�ðnþ1Þ

l2m

nl

þ b2�ðnþ 1Þ

l2m

r�ðnþ1Þ

l2m�1

nl ÞbðnlÞn ¼ Fnl

n

ð25bÞ

where

F0v ¼

2nþ 12

Z p

0f1ðhÞPvðcosðhÞÞ sinðhÞdh

v ¼ 0;nFnlv ¼

2nþ 12

Z p

0f2ðhÞPvðcosðhÞÞ sinðhÞdh

ð25cÞ

� Also, regarding to the Eq. (24c):

lnri

rnl

� �aðiÞ0 þbðiÞ0 � ln

ri

rnl

� �aðiþ1Þ

0 þbðiþ1Þ0

� �¼0

ri

rnl

� � nl2

i aðiÞn þri

rnl

� ��ðnþ1Þl2

i bðiÞn �ri

rnl

� � nl2

iþ1 aðiþ1Þn � ri

rnl

� ��ðnþ1Þl2

iþ1 bðiþ1Þn ¼0

ð25dÞ

� Rearranging the Eq. (24d) results:

1rnl

� �ln

ri

rnl

� �aðiÞ0 �

1rnl

� �ln

ri

rnl

� �aðiþ1Þ

0

� �¼ 0;

nl2

i

ri

rnl

� � nl2

i

�1

aðiÞn �ðnþ1Þ

l2i

ri

rnl

� ��ðnþ1Þl2

i

�1

bðiÞn �n

l2iþ1

ri

rnl

� � nl2

iþ1

�1

aðiþ1Þn

þðnþ1Þl2

iþ1

ri

rnl

� ��ðnþ1Þl2

iþ1

�1

bðiþ1Þn ¼ 0

ð25eÞ

Eqs. (25a), (25b), (25d) and (25e) should be solved to determinethe coefficients aðiÞn and bðiÞn . The coefficients of this set of equationsform a five diagonal matrix. In this study, Thomas algorithm isused to find these coefficients analytically. According to this algo-rithm, the reciprocity relations for calculating aðiÞn and bðiÞn are givenas follows:

bðnlÞv ¼ c2nl

aðiÞv ¼ c2i�1 � bðiÞv bi

bði�1Þv ¼ c2i�2 � aðiÞv ai

8<: i ¼ nl;nl � 1; . . . ;2

að1Þv ¼ c1 � bð1Þv b1

ð26Þ

The index v could be 0 or n to cover all unknown coefficients. Therelations related to a, b and c in each value of v are available inappendix.

5. Results and discussion

In this section, the ability of the presented analytical solution isexamined by applying it to solving two industrial applications: amultilayer spherical composite vessel under varying sun heat fluxand a multi-layer composite spherical shell with varying inside

Table 1Composite polymer properties [49].

Material number Filler Matrix k11 (W/mK) k22 (W/mK) Density (g/cc) Wt. (%) filler

1 Thermal Graph DKD X Lexan HF 110-11 N 11.4 0.74 1.46 402 Thermal Graph DKD X Lexan HF 110-11 N 8 0.6 1.38 303 Thermocarb CF300 Zytel 110 NC010 1.1 0.4 1.17 5

Fig. 4. Geometry and boundary conditions of composite spherical vessel.

Table 2Geometry and boundary conditions of composite spherical vessel.

Inner diameter (cm) 100Outer diameter (cm) 130Thickness of each layer (cm) 5Ambient temperature (K) 310Internal temperature (K) 300Convective coefficient (W/m2 K) 100

292 M. Norouzi et al. / Composite Structures 106 (2013) 288–295

temperature. Thermal properties of composite materials which areused in this study are presented in Table 1. In order to investigatethe effects of fibers’ angle on heat conduction and temperature dis-tribution of laminates, four common arrangements of fibers in lam-inate have been considered:

� Isotropic: Fibers in whole of laminate are in / direction. (Fibers’angles in each laminas are equal to 90�). The composite lami-nate is in form of an isotropic spherical laminate with conduc-tive coefficients krr = khh = k22.� Orthotropic: Fibers in whole of laminate are in h direction.

(Fibers’ angles in each laminas are equal to zero). The compositelaminate is such as a block orthotropic material with conductivecoefficients khh = k11 and krr = k22.� Cross-ply: Fibers’ angle is [0�, 90�, 0�, 90�, . . .].� [0�, 45�, 90�, 135�, . . .]: An intermediate arrangement that the

fibers’ angle change 45� in each layer.

An analytical solution for two-dimension steady heat conduc-tion in a single layer isotropic spherical laminate has been pre-sented by Arpaci [47]. Arpaci solution is simple part of currentresearch that the angle of fibers’ in a single-layer laminate is equalto. The result of Arpaci solution is used to investigate the validationof present analytical solution. As shown in Fig. 3, the result for thecurrent solution agrees completely with the analytical solution ofArpaci [47]. Here, because of the strong convergence of the temper-ature Fourier series, calculating the first ten terms of them issufficient.

� Case 1: Multi-layer spherical composite vessel

Heat conduction in a three-layer vessel with varying heat flux atoutside has been considered. It is considered that sun radiationheat flux varied as q00 = q�(1 + cos(h/2)), where q� is average ofsun heat flux on earth and is considered equal to 1357 W/m2

[50]. It is assumed that the vessel is cooled with air. Fig. 4 showsthe geometry and boundary conditions of this vessel. Table 2 pre-sents the properties of this vessel. The inner surface temperature isassumed to be constant.

Fig. 3. Temperature distribution of an isotropic sphere in radial direction(Tout = 500 K, rout = 1 m, h = 45�).

Fig. 5. Geometry and boundary conditions of composite spherical shell.

Table 3Geometry and boundary conditions of composite spherical shell.

Inner diameter (cm) 50Outer diameter (cm) 62Thickness of each layer (cm) 1Ambient temperature (K) 283Convective coefficient (W/m2 K) 500

� Case 2: Multi-layer spherical composite shell

Heat conduction through a composite spherical shell is consid-ered as the second example. This laminate is used for storing

M. Norouzi et al. / Composite Structures 106 (2013) 288–295 293

radioactive wastes in oceanic waters [51]. It is supposed that thisspherical shell is made of a five-layer composite with a stainlesssteel layer outside of sphere. The inner temperature of sphere isconsidered to vary in the form of T(h) = 300(1 + cos(h)) for morecomplexity of boundary conditions. Convective heat transfer hasbeen applied as the second radial boundary condition. The geome-try and boundary conditions of tank are presented in Fig. 5 andTable 3.

Fig. 6. Mean temperature of laminates for composite spherical vessel and shellcases.

Fig. 7. Contours of temperature distribution in r and h directions at differentarrangements of fibers for the composite spherical vessel case.

Fig. 6 shows the variation of mean temperature of laminate ver-sus fibers’ angle for two mentioned cases. It is supposed that thefibers’ angle in all laminates is similar and vary with each other.For the first case, mean temperature of laminate has beenincreased with growth of cone angle from 0� to 90�. Unlike the

Fig. 8. Contours of temperature distribution in r and h directions at differentarrangements of fibers for the composite spherical shell case.

Fig. 9. Temperature distribution of laminates in r direction under different coneangle for composite spherical vessel and shell cases.

Fig. 10. Contours of temperature distribution in r and h directions at different arrangements of composite materials for the composite spherical vessel case.

294 M. Norouzi et al. / Composite Structures 106 (2013) 288–295

second case, increasing cone angle results decreasing of meantemperature; this contrast is because of different outside/insidethermal boundary conditions.

Contours 7 and 8 depicted the temperature distribution indifferent fiber arrangements of spherical composite laminates forcase 1 and case 2, respectively. According to the figures, fiberarrangement has a significant effect on temperature distributionpattern in both cases. Respect to application and thermal condi-tions of design, the best arrangement of fibers in laminate shouldbe selected (see Fig. 7 and 8).

Fig. 9 shows the variation of temperature in radial direction in aspecific cone angle for case 1 and case 2, respectively. Here, it isassumed that the fibers’ angle is equal in whole of laminate.According to the figure, when the cone angle, W, is 90�, the temper-ature’s gradient is as an isotropic material. On the other hand,when cone angle far from 90� and near to 0�, the thermal behaviorof composite changed more and more until orthotropic behaviorappears in W = 0�.

Fig. 10 shows the effect of layer material arrangements ontemperature distribution for case 1. According to the figure, thesequence of composite materials in laminate could change thetemperature distribution pattern in laminate greatly and shouldbe considered as an important factor in multilayer compositematerials.

6. Conclusions

In present paper, an exact analytical solution for steady conduc-tive heat transfer in spherical laminates is presented as the firsttime. The solution is obtained under general linear boundary con-ditions so it could be applied simplicity for various conditionsincluding combinations of conduction, convection, and radiationboth inside and outside of the sphere. The results of present study

is useful for analyzing of thermal fracture, controlling directionalheat transfer through laminates and fiber placement in productionprocesses of spherical vessels. Here, we examined the capability ofthe present solution by applying it on two industrial applicationsfor different fiber arrangement of multilayer spherical laminates.The analytical solution indicated that the temperature distributionfor any arbitrary fiber arrangement will be intermediate betweenthose in single-layer laminates with fiber angles of 0% and 90%.The authors suggest that the future works could be focused on un-steady and non-Fourier heat conduction in spherical laminates.

Acknowledgments

This paper is presented based on a researching project which isGranted by Shahrood University of Technology. Therefore, theauthors appreciate for their financial support.

Appendix A. The coefficients of temperature distribution

In this section, the relations of coefficients of temperature dis-tribution of any lamina are presented:

b1 ¼B0v

A0v

c1 ¼ F0v � b1

8<: ðA1Þ

aiþ1 ¼�Aiþ1

v þBiþ1v �ðniÞ

ðBivþBiþ1

v �ðsiÞ�Aiv�biÞ

c2i ¼ �c2i�1 � Aiv � aiþ1

8><>: i ¼ 1;2;3; . . . ;nl � 1 ðA2Þ

biþ1 ¼ 1ðniÞ�aiþ1�ðsiÞ

c2iþ1 ¼ �c2i � B0iv � biþ1

(i ¼ 1;2;3; . . . ;nl � 1 ðA3Þ

M. Norouzi et al. / Composite Structures 106 (2013) 288–295 295

Coefficients ni and si are defined for simplicity:

ni ¼A0iv�Aiþ1

v �Aiv�A0iþ1

v

A0iv�Biþ1v �Ai

v�B0iþ1v

si ¼�A0iv�Bi

vþAiv�B0iv

A0iv�Biþ1v �Ai

v�B0iþ1v

8>>><>>>:

i ¼ 1;2;3; . . . ;nl � 1 ðA4Þ

The values of coefficients Av and Bv related to each pair of coeffi-cients are as follows (the coefficients A0v and B0v are the derivativeof Av and Bv, respectively):

A00 ¼ a1 ln r0

rnl

� �þb1

1r0

� �;A0

n ¼ a1rn

l2m

0 þb1nl2

0r

nl2

0

�1

0

B00 ¼ a1;B

01n ¼ a2r

�ðnþ1Þl2

00 þb2

�ðnþ1Þl2

0r�ðnþ1Þ

l20

�1

0

Ai0 ¼ ln ri

rnl

� �;Ai

1n ¼rirnl

� � nl2

i

Bi0 ¼ 1;Bi

1n ¼rirnl

� ��ðnþ1Þl2

i

Anl0 ¼

b2rnl;Anl

n ¼ a2rn

l2nl

nl þb2n

l2nl

rn

l2nl

�1

nl

Bnl0 ¼ a2;B

nln ¼ a2r

�ðnþ1Þl2

nlnl þb2

�ðnþ1Þl2

nl

r

�ðnþ1Þl2

nl

�1

nl

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

i¼ 1;2;3; . . . ;nl�1

ðA5Þ

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