Transient thermal analysis in 2D orthotropic FG hollow cylinder with heat source

International Journal of Heat and Mass Transfer 89 (2015) 977–984

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Transient thermal analysis in 2D orthotropic FG hollow cylinder with heat source K. Daneshjou a, M. Bakhtiari a,⇑, R. Alibakhshi a, M. Fakoor b a b

Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Department of Aerospace Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran 4399-55941, Iran

a r t i c l e

i n f o

Article history: Received 6 February 2015 Received in revised form 18 May 2015 Accepted 26 May 2015

Keywords: Transient heat conduction Orthotropic functionally graded material Laminate approximation theory Augmented state space method Time-dependent heat source

a b s t r a c t A straightforward implementation is applied to solve heat conduction equation for a 2D hollow cylinder made of orthotropic functionally graded material (FGM) in the presence of time-dependent heat source. All material properties are considered to vary continuously within the cylinder along the radial direction with arbitrary law. The transient solution can be obtained by augmented state space method, which leads to carry out the results easily, based on laminate approximation theory in the Laplace domain, and then the results obtained are converted into the time domain by applying the numerical Laplace transform inversion. By this method, the solution of heat conduction problem is obtained for general boundary conditions which can be included various combinations of arbitrary temperature, flux, or convection. Comparison of obtained results with special cases in the literatures shows the capability of the new presented method. Finally, in the presence of time-dependent heat source, the effects of circumferential to the radial thermal conductivity coefficient ratio and heat source size on temperature field are graphically shown. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) are extensively used in numerous engineering applications such as aerospace, engineering design and manufacture because of their desirable properties for specific applications under thermal loading in the heating area. They are a novel class of composite structures whose mechanical properties vary continuously along the specified directions. For example, in hollow cylindrical objects in which FGMs are used along the thickness, the mechanical properties vary from one side of cylinder to other side. This continuous spatial variation makes them sustainable in an extremely high temperature environment [1]. Considering that the thermal analysis is very considerable topic in manufacturing process, a large number of analytical investigations have been studied for heat conduction problem. Some presented analytical solutions of heat conduction problems in solid cylinder for unsteady conditions have been proposed in recent years [2–4]. Based on the Fourier heat conduction theory, heat flux is directly related to the temperature gradient using thermal

⇑ Corresponding author. Tel.: +98 2177240540; fax: +98 2177240488. E-mail address: [email protected] (M. Bakhtiari). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.104 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

conductivity [5]. In the study of heat conduction, Fourier law has provided successful and accurate results for many analyzed practical conditions (see for example [6]). Because the Laplace transform method may be the best to simplify the solving procedure of transient problems, this method is usually applied in the solving technique [7–11]. Additionally, in multi-dimensional problems, some analytical methods are available in which the combination of Laplace transform and other methods such as separation of variables (SOV), finite integral transform (FIT) and Green’s function have been considered [12–14]. As an example for this type of application, heat conduction problem has been solved using the combination of a Laplace transform approach and the separation of variables method for a composite cylinder with time-dependent boundary conditions [2]. Also, in a similar study of this application, the transient thermoelastic response of multilayered hollow cylinder with interface pressure has been investigated by employing the Laplace transform and finite difference methods [15]. In circular cylinders, the end effects of heat conduction of functionally graded materials and laminated composites have been studied for steady state problem [16]. The effects of material composition profile and two-dimensional distribution of temperature on the transient heat problem have been obtained for functionally graded hollow cylinder [17]. Recently, an easy-to-handle analytical approach has been proposed to solve heat conduction problem for

978

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

the non-uniform convective boundary conditions in a hollow cylinder [18]. An effective approach, which is valid for both constant and time-dependant boundary conditions, has been developed for transient thermal problem in a one-dimensional FG hollow cylinder by means of the state space method [19] and the transient thermal problem has been easily analyzed in FG cylindrical structures with arbitrary grading pattern and arbitrary thickness. While there are several reports on Fourier heat conduction analysis for hollow cylinders, very few reports can be found with the numerical investigation of time-dependant heat source and arbitrary boundary conditions in the literature. For example, an analytical solution has been proposed for the asymmetric transient heat conduction in a multilayer annulus with the presence of heat source and time-dependent boundary conditions [20]. In their numerical investigation, it is assumed that the heat source term is equal to zero in each layer. The proposed solution is also valid for any combination of time-dependant, inhomogeneous boundary conditions in the radial direction at inner and outer radii of the domain. Another special case has been numerically studied for steady heat conduction in polar coordinates for multi-layer domain with uniform and time-independent volumetric heat sources [21]. Separation of variables method has been employed to obtain the transient temperature distribution. This method is only valid for homogenous boundary conditions in the angular direction. Since the problems that can be solved using SOV method are relatively limited, it cannot be easily applied to include the influences of time-dependent boundary conditions and heat source terms. Above-mentioned limitations in solution procedure make researchers interested develop modified solving methods which can improve efficiency of computational codes. This paper addresses a new approach for transient thermal analysis in 2D orthotropic functionally graded hollow cylinder, where material properties of the FG cylinder are varied by arbitrary grading pattern in the radial direction, with the presence of time-dependant heat source and general boundary condition based on laminate approximation theory which can approach the solution to the exact one when the number of layers increases [23]. The main innovation of the current study is the introduction of augmented state space method, which reduces the mathematical complexity and also improves efficiency of computer code, for solving transient heat conduction problem. Compared to other numerical methods such as finite difference method, finite volume method and finite element method, augmented state space method can eliminate the tedious matrix inversion procedure. Furthermore, many general analytical solutions with complicated boundary conditions and time-dependant heat source can be derived by means of the presented technique. To guarantee the numerical accuracy of the current investigation, the presented results for the special case of the inhomogeneous cylinder are compared with those available published papers. At last, in order to prove the efficiency of presented method, the effects of orthotropy and different sizes of time-dependent heat source on transient temperature will be studied.

Fig. 1. Model of layered functionally graded hollow cylinder.

hN ¼ h=N; where h ¼ b  a; and consequently, inner radius for i-th layer can be defined as a½i ¼ a þ ði  1ÞhN for (i = 1, 2, . . . , N). All neighbouring layers are perfectly bonded at their interfaces and lined up in such away that their axes of symmetry coincide with each other [24]. Lamination approximation theory is suitable to solve problems with material inhomogeneity [19]. All the layers are assumed to be orthotropic in thermal properties and are in perfect thermal contact. For orthotropic materials, in the presence of heat source, the governing heat conduction equation for the temperature in the ith layer, T i ðr; h; tÞ, can be written as follows:

! ! ðiÞ ðiÞ 1 @ 1 @ ðiÞ @T ðr; h; tÞ ðiÞ @T ðr; h; tÞ þ 2 þ g ðiÞ ðr; h; tÞ rkr kh r @r @r r @h @h ¼ qðiÞ cðiÞ p

@T ðiÞ ðr; h; tÞ ; @t

ðiÞ

ð1Þ

ðiÞ

where kðÞ , T ðiÞ , g ðiÞ , qðiÞ and cp are, the conductive heat transfer coefficient in  direction, absolute temperature, volumetric heat source distribution, density and specific heat in the ith layer, respectively. Fourier equation for an orthotropic material in a cylindrical system is given as:

8 9 2 kr > < qr > = 6 qh ¼ 4 0 > : > ; qz 0

0 kh 0

9 38 0 > < @T=@r > = 7 0 5 @T=r@h > > : ; @T=@z kz

ð2Þ

where qr , qh and qz are the radial, circumferential and axial component of heat flux, respectively. In orthotropic materials, it is obvious that the tensor of the conductive heat transfer coefficients is diagonal [22]. In general, in the radial direction, from a mathematical point of view, the heat flux in i-th layer can be written as: ðiÞ

qrðiÞ ðr; h; tÞ ¼ kr 2. Mathematical formulation Consider a thick-walled and infinite functionally graded orthotropic hollow cylinder with inner radius a and outer radius b, as shown schematically in Fig. 1. Here, a cylindrical coordinate system ðr; hÞ is used to label the material point of the cylinder. For t > 0, both the inner and the outer surfaces of the annulus may be included all combination of arbitrary temperature, flux or convection boundary conditions. By employing laminate approximation theory, the orthotropic FG cylinder may be divided into N layers of equal thickness,

ai  r  bi

@T ðiÞ ðr; h; tÞ @r

ð3Þ

By substituting Eq. (3) into Eq. (1), a new form of Eq. (1), which is suitable to prepare the state space equation, can be obtained as:

! ðiÞ  1 @ 1 @  ðiÞ @T ðr; h; tÞ ðiÞ rqr ðr; h; tÞ þ 2 k þ g ðiÞ ðr; h; tÞ r @r r @h h @h ¼ qðiÞ cðiÞ p

@T ðiÞ ðr; h; tÞ @t

ð4Þ

The state space equations can be derived using Eqs. (3) and (4) as follows:

979

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

Fig. 2. Performance of augmented state space method in the ith layer.

(

@ @r

ðiÞ

qr ðr; h; tÞ

2

)

ðiÞ

T ðr; h; tÞ

¼4

 1r

ðiÞ @ @t

qðiÞ cp

 1ðiÞ kr

ðiÞ

þ

kh @ 2 r2 @h2

0

3(

ðiÞ

5 qr ðr; h; tÞ T ðiÞ ðr; h; tÞ

)

( þ

g ðiÞ ðr; h; tÞ 0

)

ð5Þ

Now, by considering the appropriate normal-mode expansions, the state vector in ith layer is expanded in terms of unknown modal functions as follows:

(

ðiÞ

qr ðr; h; tÞ

) ¼

T ðiÞ ðr; h; tÞ

þ1 X

(

n¼1

ðiÞ

qr;n ðr; tÞ

)

ðiÞ ðr; tÞ T ;n

einh

ð6Þ

where the subscript ‘‘; n ’’ represents the modal expansion of qr and T on h. Additionally, heat source term should be considered in the P ðiÞ inh form of g ðiÞ ðr; h; tÞ ¼ þ1 similar to above two states. n¼1 g ;n ðr; tÞ e By direct substituting the Eq. (6) into Eq. (5) and taking the Laplace transformation, the state equation for ith layer can be derived as:

@ @r

(

ðiÞ ~r;n q ðr;sÞ ðiÞ T~ ;n ðr; sÞ

2

)

¼4

ðiÞ

k

ðiÞ

 1r qðiÞ cp s  rh2 n2 

1 ðiÞ kr

0

3(

~ðiÞ ðr; sÞ q 5 r;n ðiÞ T~ ;n ðr;sÞ

)

( þ

ðiÞ g~;n ðr;sÞ

)

0 ð7Þ

where n and s are temperature mode and Laplace transform parameter, respectively. Here, it is assumed that the FG cylinder has initial condition equal to zero (i.e., T ;n ðr; 0Þ ¼ 0). Eq. (7) can be written in a simple form as follows: ðiÞ ðiÞ ðiÞ @Y ðiÞ n =@r ¼ An Y n þ Bn

ð8Þ

From Eq. (8), it can be seen immediately that the presence of heat source term causes the complexity of solution and consequently, the homogenous state space equation cannot be readily derived. Here, due to the difficulty of solving the corresponding problem, we introduce a new technique which can convert Eq. (8) into a simple and standard form of state equation as presented in [23]. In order to convert an inhomogeneous equation (see Eq. (7) or Eq. (8)) into homogeneous one, we can augment an obvious equation to Eq. (8). It is to be noted that the state variables used in Eq. (8) are in the modal form and in the Laplace domain. Consequently, this obvious equation should be augmented to Eq. (8) when it is in its modal form in the Laplace domain. For better description, similar to Eq. (6), let’s consider the following appropriate state variable as:

n;n ðr; tÞeinh

@ ~ ðn;n ðr; sÞÞ ¼ @r

! @ ~n;n ðr; sÞ ~ n;n ðr; sÞ ~n;n ðr; sÞ @r 1

ð10Þ

Eq. (10) can be easily augmented to Eq. (8) in order to convert this inhomogeneous equation to homogeneous one. Accordingly, the proposed technique used to reduce a difficult problem to a simpler one by converting a 2-by-2 matrix into a 3-by-3 matrix. Now, by applying this appropriate state variable, a new set of state variables can be observed. The advantage of this method is that the Eq. (8) can conveniently be obtained in the standard form as follows: ðiÞ

dV n ¼ DnðiÞ V nðiÞ dr

ð11Þ

h iT ~ðiÞ ~ ðiÞ ðiÞ is the modal state vector in the ith where V ðiÞ n ¼ qr;n ; T ;n ; n;n

2

3. Augmented state space method

þ1 X

Above conditions in selecting an appropriate variable allow to have a large number of arbitrary functions. In order to obtain the solutions in the short time domain, this variable can be considered as a linear function with respect to r. Now, according to the standard form of state equation, the suitable form of obvious equation can be written as follows:

layer, and Dn is a 3-by-3 modal coefficient matrix whose elements are obtained as:

h iT ðiÞ ~ ðiÞ ~r;n where Y nðiÞ ¼ q ; T ;n .

nðr; h; tÞ ¼

n;n ðr; sÞ) is dependent – Since this form of considered variable (i.e., ~ on the radial direction, it should be continuous (i.e., without any singular points) and differentiable (i.e., definable in the form of state equation) along the radial direction. – This appropriate variable is also considered to be continuous across the interface between two neighbouring layers as well as temperature and heat flux in radial direction.

 1r

6 6 1 6  ðiÞ DðiÞ ¼ n 6 kr 4 0

ðiÞ

qðiÞ cp s 

ðiÞ

kh r2

n2

ðiÞ g~;n ðr;sÞ n~;n ðr;sÞ

0

0

0

@ ~n;n ðr;sÞ 1 ~n;n ðr;sÞ @r

3 7 7 7 7 5

ð12Þ

ð9Þ

n¼1

Modal form of this new variable in the Laplace domain, which is ~ able to augment to state vector Y ðiÞ n , is denoted by n;n ðr; sÞ. An appro~;n ðr; sÞ) is selected so that the following condipriate variable (i.e., n tions are obviously satisfied.

Fig. 3. The distributions of non-dimensional temperature along the radial direction ða=b ¼ 0:2Þ.

980

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

The performance of augmented state space method is shown in Fig. 2. Two new boundary conditions, imposed on the two cylindrical surfaces, are added to the previous boundary conditions by ~;n ðr; sÞ). The values of n ~;n ðr; sÞ selecting the new state variable (i.e., n are easily obtained using a linear function considered above. It n;n ðr; sÞ at the inner and the outer surfaces means that the value of ~ n;n ða½1 ; sÞ and ~ n;n ðb½N ; sÞ; respectively. Since Eq. (10) augmented are ~ to Eq. (8) is completely obvious, two new added boundary conditions cannot affect the solving procedure of the problem. In general, adding an obvious equation to a set of equations does not influence on the solving procedure. Based on the augmented state space method, the heat conduction equation can be easily solved for the problems with inhomogeneous terms such as heat source and initial condition of temperature. Because the coefficient matrix DðiÞ n is not constant, it is difficult to get a direct solution to Eq. (11). To overcome the difficulty, by considering equal sufficiently small thickness of each layer (refer to laminate approximation theory as described in Section 2), the coefficient matrix Dn can be assumed constant within each layer [24]. The result of this approximation technique will gradually approach to exact solution when the number of layers increases [23]. In the following, the coefficient matrix is denoted as DðiÞ n ða½i Þ for the ith layer. Now, within the layer, the solution to Eq. (11) can be expressed as:

Fig. 5. Mechanical model of functionally graded hollow cylinder with heat source.

h i ðiÞ V nðiÞ ðrÞ ¼ V ðiÞ n ða½i Þ exp ðr  a½i ÞDn ða½i Þ for a½i 6 r 6 b½i ;

4. Numerical Laplace inversion

i ¼ 1; 2; . . . ; N

ð13Þ

Rewriting Eq. (13) at the outer surface of the ith layer, leads to useful formula as:

h i ðiÞ V nðiÞ ðb½i Þ ¼ V ðiÞ n ða½i Þ Exp ðb½i  a½i ÞDn ða½i Þ

ð14Þ

In a simple word, Eq. (14) relates the state variables at the outer surface of the ith layer to those at the inner surface. The continuity conditions between all interface layers leads to obtain final relation between state variables at the outer radius of cylinder (i.e., at b½N ) and those at the inner radius (i.e., at r ¼ a½1 ) via a 3-by-3 modal transfer matrix Mn ; as:

V n ðb½N Þ ¼ M n V n ða½1 Þ In which Mn ¼

Qq

i¼1 Exp

ð15Þ h

i

ðb½i  a½i Þ DðiÞ n ða½i Þ .

The unknown states in boundaries (V n ðb½N Þ and V n ða½1 Þ) can be complemented by use of Eq. (15) and arbitrary known boundary conditions at the inner and outer surfaces. Then, by applying Eq. (14), we can obtain the state vectors in all interface layers. Finally, in order to solve the problem in time domain, numerical inverse Laplace transform approach will be described.

To obtain the values of a function f in real time domain from its Laplace transform ~f , it is essential to apply a numerical inverse Laplace transform. It is worth noting that using analytical inverse Laplace transform can be complicated to achieve solutions in many situations. An integral formula for the inverse Laplace transform is given by [25]:

f ðtÞ ¼ L1 f ~f ðsÞ g ¼

1 lim 2pi T!1

Z

cþiT

est ~f ðsÞ ds

ð16Þ

ciT

where i is the complex number, and c is a real number greater than the real part of all singularities of ~f ðsÞ. In this study, the Durbin’s method [26] for inversion of the Laplace transforms is adopted to approximate the time function f ðtÞ by a periodic function of period T. To obtain smooth convergence and satisfactory solutions, Cesaro type of summation of the series is employed to effectively reduce the non-physical oscillations (Gibb’s phenomenon). In the Cesaro’s manner, a sufficiently large number of terms are required in order to approach to the true summation [11,27]. Using the first M terms of series, after some manipulations, the inverse Laplace transform with Cesaro’s summation can be written as:

( M ect ~f ðcÞ 1 X þ ðM þ 1  kÞ 2 M þ 1 k¼0 T            ipk ipk ipk ipk cos sin  Re ~f c þ t  Im ~f c þ t T T T T

f ðtÞ 

ð17Þ

Fig. 4. Transient temperature distribution in radial direction at h ¼ p=2.

where t is in the interval ½0 2T and cT is between 4 and 5 for sufficiently accurate solution [11]. In the current study, cT and the parameter M are taken to be 4.5 and 301, respectively. As k increases, the essence of the above summation is to gradually reduce the contribution of each term. It is evident from Eq. (17) that if M is small the early k terms could be prematurely over reduced and finally, Cesaro’s summation will not coverage uniformly to the true solution. In the following, some numerical examples will be investigated.

981

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

(a)

(b)

2.5 7.5

0 5

(c)

2.5 5 7.5

0 0

t=200 (sec)

0

0

0

0 2.5

0 2.5 7.5

12.5 17.5 20

10 15 20

0 10

t=600 (sec)

2.5

7.5

5

0

5

15

5

2.5 0 7.5 10 12.5 15

0

0 2.5

10

15 20

0

0

0

0

0

0

0

12.5

0

2.5

10

5

15

0

10

0

17.5

0

5

7.5

20

5

15 20

2.5

5

15

10

0

0 17.5

5

17.5

15

10

2.5 0

2.5

0 7.5

10 20

10

0

0

15

5

10

10

0

0

20

0

0 5

10 20 15 17.5

20

15

10 20

0

10

0 20

0

22.5

15

15

20

20

15

20

25

25

0 2.5

0

17.5

5

5 0

12.5 15

20

15

2.5

5

5

5 10 2.5 5 0

5

0

0

15

20

15

20

0 5

20

20

0 10

15

0

0

30

15 20

25

0

0 7.5

5

10 15

5

5 20

t=1400 (sec)

0

10

20

10

0

0

5

10

10 20

20

10

20

15 10

0 15

0

30

0

25

20

25

25

0

15

5

0 5

0

0

0

0

5

0

20

20

10

0

15

0

510 20

20

20

15 0

0

15

10 20

20 25

30

30

30

25

0

0

10

15

20

5

5

10

10 15

t=1800 (sec)

2.5 7.5

0 5

0

17.5

t=1000 (sec)

0

0

0

5

10

20

0

15

15

20

20

0

0 15

20

25

25

10 5

5 0

0

10

5

0 10

5

0

0

25

20

105 20 35

0 10

15

20

0 15

0

10

15

Steady state

(d)

0

0

0

0 15 20

30

30

10

10

0 15

0

20

20

0

0

20

15

20

15

30

35

30

25

25

25

20

5

10

5

5 0

0 15 10 20

0

0 15

10

0

0

10

0

5

0

15 15

20

20

Fig. 6. Contours of temperature distribution in r and h directions for orthotropic FG hollow cylinder: (a) g ¼ 0:5, (b) g ¼ 1, (c) g ¼ 2, (d) g ¼ 4.

5. Numerical scheme and discussion Some comparison studies should be accomplished in order to demonstrate the validity and efficiency of the present method.

For this reason, three different examples are presented. Due to lack of any data on the transient thermal analysis in 2D orthotropic FG cylinder with the presence of heat source, comparison is carried out with the available results for special cases presented in the

982

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

5.2. Verification with time-independent heat source

Fig. 7. Temperature variation with respect to time at r = 0.6 for different values of h and g.

At the second case, the three-layer semi-circular annular region is considered for an illustrative example. In this section, all computational parameters are assumed to be the same as those presented in [21]. It should be noted that spatially non-uniform, but time-independent volumetric heat sources can be assumed in all layers. For special case, when uniformly distributed heat source of constant magnitude is turned on at t = 0 in the first layer, transient temperature distribution in radial direction at h ¼ p=2 is shown in Fig. 4. The solution procedure seems to be accurate and the results are in excellent agreement with those obtained as reported in [21]. It is to be noted that the presented method has satisfactory convergence when the FG cylinder is fictitiously divided into less than 100 sub-layers with equal thickness, which causes a considerable decrease in computational time. 5.3. Effect of heat source on transient analysis

literature. The first example is concerned with a brief validation for transient analysis with no internal heat source. The second example follows the comparative results for transient analysis with time-independent heat source. The reader may refer to [19,21] for detail of material properties and non-dimensional quantities required in the first two examples. The last example is associated with the transient thermal analysis in 2D orthotropic FG cylinder containing time-dependant heat source. The material properties of the FG cylinder are assumed to be varied in the radial direction. Let us initially consider the following two special cases.

In this section, we consider the infinite orthotropic FG hollow cylinder with inner radius a = 0.5 m and outer radius b = 1 m, containing a time-dependant heat source with r ih ¼ 0:65 and r oh ¼ 0:9; whose inner surface is adiabatic and outer surface is subjected to prescribed time-dependent temperature boundary condition, as shown in Fig. 5. For the boundary conditions corresponded to this problem, after being expanded in Fourier series, the distribution of the heat source and the prescribed temperature can be written as: þ1 X

gðr; h; b; tÞ ¼ g 0n ðr; tÞ 5.1. Verification without heat source

for In the first case, the FG hollow cylinder is subjected to the time-dependant temperature at the inner surface while the outer surface is adiabatic. Furthermore, the inner and outer surfaces are considered to be made of carbon nanotubes and Aluminium, respectively. In this section, for convenience of comparison and discussion, all computational parameters are the same as those considered in [19]. Fig. 3 shows the effect of material inhomogeneity l1 and l2 on non-dimensional temperature along the radial direction at the non-dimensional times s ¼ 0:05 and 0.5. As can be seen from Fig. 3, there is a very good agreement between the numerical results and those obtained as reported in [19]. The validity of the presented method is obviously verified.

n

ðiÞ ðsinðnbÞ=npÞ einh

n¼1

p=2  b 6 h 6 p=2 þ b

Tðh; a; tÞ ¼ T 0 ðtÞ

þ1 X

ð18Þ

n

ðiÞ ð1  ðsinðnðp  aÞÞ=npÞÞ einh

n¼1

for 3p=2  a 6 h 6 3p=2 þ a where g 0n ðr; tÞ and T 0 ðtÞ can be expressed as:

g 0n ðr; tÞ ¼ S: ½Hðr  r in Þ  Hðr  roh Þ : HðtÞ T 0 ðtÞ ¼ T 0 : HðtÞ

ð19Þ

In which S is the magnitude of heat source and H denotes the Heaviside’s unit step function. It is noted that the temperature distribution is taken to be zero for all other angles.

Fig. 8. Temperature variation along the radial direction: (a) h ¼ p=2, (b) h ¼ 7p=6, (c) h ¼ 4p=3.

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

983

depicted at angular locations h ¼ p=3; p and 4p=3 for three different values of g. The radius is also fixed at r = 0.6. Also, the material inhomogeneity parameters l1 and l2 are fixed at l1 ¼ 0:5 and l2 ¼ 0:5. As can be seen, the temperature variation increases rapidly in the region near the heat source. In this figure, by decreasing the values of g, a monotonic variation of temperature at angular location h ¼ p can be concluded. This is because the temperature has slow distribution rate at h ¼ p. It can be also seen that the temperature variation increases with the increasing values of g. At three different angular locations, temperature variation in the radial direction for different values of t is plotted in Fig. 8. The parameter g and T 0 are fixed at g ¼ 2 and T 0 ¼ 20  C: As can be inferred from Fig. 8(a), which is related to a point with h ¼ p=2, due to the existence of heat source within the considered interval, there is an abrupt change in temperature field in the radial direction compared to Fig. 8(b) and (c) which are not directly subjected to the heat source. The effect of different sizes of heat source (i.e., parameter b) in temperature filed at angular locations h ¼ 2p=3; p and h ¼ 4p=3 is depicted in Fig. 9. Additionally, the radius is fixed at r = 0.55. In this calculation, in order to clearly show the influence of parameters involved, the outer surface is at zero uniform temperature (i.e., a ¼ 0) while the inner surface is adiabatic. As expected, increasing the angle of b, leads to increase the temperature variation for all the cases. In fact, it can be observed a distinct rise in temperature for points located at the regions near the heat source.

6. Concluding remarks

Fig. 9. Temperature variation with respect to time for different sizes of heat source at r = 0.55: (a) h ¼ 2p=3, (b) h ¼ p, (c) h ¼ 4p=3.

The FG material constants are taken as qa ¼ 1350 kg=m3 , ca ¼ 450 J=kg K, kra ¼ 35 W=m K, kh ¼ gkra for inner surface and qb ¼ 2707 kg=m3 , cb ¼ 896 J=kg K, krb ¼ 204 W=m K, kh ¼ gkrb for outer one. The parameter g is defined as conductive heat transfer coefficient in the circumferential to the radial direction. In order to demonstrate the mathematical robustness of the presented method, the grading pattern of the FG material is considered as:

GðrÞ ¼ Ga þ ðGb  Ga Þ

r  a l1 ba

þ l2 Ga sin

r  a  ba

p

ð20Þ

In which l1 and l2 are material inhomogeneity parameters. Fig. 6 deals with the temperature distribution of FG hollow cylinder in two directions with time-dependent heat source of magnitude S ¼ 150kw=m3 switched on at t = 0 and located in the domain b ¼ p=3; as shown in Fig. 5, for different values of conductive heat transfer coefficient ratio g ¼ kh =kr . Additionally, the prescribed temperature is assumed to be subtended the angle of a ¼ p=6: The material inhomogeneity parameters l1 and l2 are fixed at l1 ¼ 0:5 and l2 ¼ 0: According to Fig. 6, variation of the conductive heat transfer coefficient ratio g leads to different distribution rate of temperature. Since the conductive heat transfer coefficient in the circumferential to the radial direction is increasing, it is evident that the distribution of temperature should be rapidly grown in circumferential direction. It is important to mention that the thermal energy is thus retained for longer at region near the heat source, when ratio g decreases, allowing the temperature in this region to reach considerable values. The steady-state solution is also depicted for some different values of g: In the current computation, the FG cylinder is fictitiously divided into 100 sub-layers with equal thickness. For better description of the transient temperature analysis of FG hollow cylinder with time-dependent heat source, Fig. 7 is

An effective and straightforward technique, named augmented state space method, was introduced to solve the inhomogeneous heat conduction equation for a 2D hollow cylinder, which is made of orthotropic functionally graded material (FGM), containing a time-dependent heat source in this investigation. An attractive advantage of this new technique is that in comparison with other numerical solution methods, the proposed technique indeed can be applied to a large number of heat conduction problems including inhomogeneous term such as heat source. Proposed solution is valid for any combination of homogenous and inhomogeneous boundary conditions at the radial and circumferential directions. The main objective of this paper is that the transient solution can be conveniently obtained by the augmented state space method based on laminate approximation theory. In order to convert the results obtained from the Laplace domain into the time domain, a numerical inverse Laplace transformation has been applied. The presented results were found to be in very good agreement with well known published papers. To demonstrate the mathematical robustness of the presented method, the influences of orthotropy and different sizes of time-dependent heat source in transient temperature were also studied. The authors are hopeful that the proposed method may be considered as a robust tool in solving complicated inhomogeneous heat conduction problems.

Conflict of interest Authors confirm that there is not any conflict of interest.

References [1] H. Santos, C.M.M. Soares, C.A.M. Soares, J.N. Reddy, A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock, Compos. Struct. 86 (1–3) (2008) 10–21. [2] X. Lu, P. Tervola, M. Viljanen, Transient analytical solution to heat conduction in composite circular cylinder, Int. J. Heat Mass Transfer 49 (1–2) (2006) 341– 348.

984

K. Daneshjou et al. / International Journal of Heat and Mass Transfer 89 (2015) 977–984

[3] X. Lu, P. Tervola, M. Viljanen, Transient analytical solution to heat conduction in multi-dimensional composite cylinder slab, Int. J. Heat Mass Transfer 49 (5– 6) (2006) 1107–1114. [4] G.E. Cossali, Periodic heat conduction in a solid homogeneous finite cylinder, Int. J. Therm. Sci. 48 (4) (2009) 722–732. [5] David W. Hahn, M. Necati Özisßik, Heat Conduction, third ed., Wiley, 2012. [6] A. Zukauskas, J. Ziugzda, Heat transfer of a cylinder in crossflow, Academy of Sciences of the Lithuanian SSR, 1985. [7] A. Saleha, M. Al-Nimrb, Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique, Int. Commun. Heat Mass Transfer 35 (2) (2008) 204–214. [8] A.A. Delouei, M.H. Kayhani, M. Norouzi, Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates, Int. J. Heat Mass Transfer 55 (15–16) (2012) 4427–4436. [9] I. Keles, C. Conker, Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties, Eur. J. Mech. A/ Solids 30 (3) (2011) 449–455. [10] K. Ramadan, M.A. Al-Nimr, Analysis of transient heat transfer in multilayer thin films with nonlinear thermal boundary resistance, Int. J. Therm. Sci. 48 (9) (2009) 1718–1727. [11] Seyyed M. Hasheminejad, Ako Bahari, Saeed Abbasion, Modelling and simulation of acoustic pulse interaction with a fluid-filled hollow elastic sphere through numerical Laplace inversion, Appl. Math. Modell. 35 (1) (2011) 22–49. [12] X. Lu, P. Tervola, M. Viljanen, A new analytical method to solve heat equation for multi-dimensional composite slab, J. Phys. A: Math. Gen. 38 (13) (2005) 2873–2890. [13] Tzer-Ming Chen, A hybrid transform technique for the hyperbolic heat conduction problems, Int. J. Heat Mass Transfer 65 (2013) 274–279. [14] Tzer-Ming Chen, A hybrid Green’s function method for the hyperbolic heat conduction problems, Int. J. Heat Mass Transfer 52 (19–20) (2009) 4273–4278. [15] C.I. Hung, C.K. Chen, Z.Y. Lee, Thermoelastic transient response of multilayered hollow cylinder with interface pressure, J. Therm. Stresses 24 (10) (2001) 987– 1006.

[16] J.Q. Tarn, Y.M. Wang, End effects of heat conduction in circular cylinders of functionally graded materials and laminated composites, Int. J. Heat Mass Transfer 47 (26) (2004) 5741–5747. [17] M. Asgari, M. Akhlaghi, Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length, Heat Mass Transfer 45 (11) (2009) 1383–1392. [18] S. Tonini, G.E. Cossali, A novel analytical solution of the non-uniform convective boundary conditions problem for heat conduction in cylinders, Int. Commun. Heat Mass Transfer 39 (8) (2012) 1059–1065. [19] H.M. Wang, An effective approach for transient thermal analysis in a functionally graded hollow cylinder, Int. J. Heat Mass Transfer 67 (2013) 499–505. [20] Suneet Singh, Prashant K. Jain, Rizwan-uddin, Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions, J. Nucl. Eng. Des. 241 (1) (2011) 144– 154. [21] Suneet Singh, Prashant K. Jain, Rizwan-uddin, Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction, Int. J. Therm. Sci. 47 (3) (2008) 261–273. [22] J.C. Halpin, Primer on Composite Materials Analysis, CRC Press, Boca Raton, 1992. [23] W.Q. Chen, Z.G. Bian, C.F. Lv, H.J. Ding, 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid, Int. J. Solids Struct. 41 (3–4) (2004) 947–964. [24] Seyyed M. Hasheminejad, M. Rajabi, Acoustic scattering characteristics of a thick-walled orthotropic cylindrical shell at oblique incidence, Ultrasonics 47 (1–4) (2007) 32–48. [25] F.B. Hildebrand, Advanced Calculus for Applications, second ed., Prentice Hall, New Jersey, 1976. [26] F. Durbin, Numerical inversion of Laplace transforms: an effective improvement of Dubner and Abate’s method, Comput. J. 17 (4) (1973) 371– 376. [27] T.M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Massachusetts, 1957.