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On axisymmetric heat conduction problem for FGM layer on homogeneous substrate
International Communications in Heat and Mass Transfer 57 (2014) 157–162
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
On axisymmetric heat conduction problem for FGM layer on homogeneous substrate☆ Dariusz M. Perkowski Faculty of Mechanical Engineering, Białystok University of Technology, ul. Wiejska 45 C, 15-351 Białystok, Poland
a r t i c l e
i n f o
Available online 4 August 2014 Keywords: Temperature Heat flux FGM Layer Hankel transform
a b s t r a c t The paper deals with the problem of heat conduction in the FGM layer with heat conductivity coefficient dependent on depth from boundary surface. The non-homogeneous layer is ideally bounded with homogeneous half-space with constant heat conductivity coefficient. The boundary plane is heated by: a) given temperature as a function of radius r, or b) given heat flux as a function of radius r. The Hankel transform method is applied to obtain a solution of formulated problem. The influence of thermal and geometric properties of FGM layer on temperature distributions in the considered bodies was investigated. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Modern composite materials with functional graded properties (FGM) have a complex internal structure, due to the fact that they are composed of several different materials. The thermo-mechanical properties of the composite depend on the components included in such materials. There are two common types of FGM materials: with the continuous or discrete changing of the material properties. An example of FGM materials with the continuous changing of the material properties is the so-called FGM gradient zone created during the manufacturing process between the coating and the substrate component obtained under the action of temperature. Discrete changes in the properties are achieved by multilayer composites, which are very popular in technical applications. In the case of coatings with discrete changing of material properties the literature presented some models allowing the description ofa body by averaged technique [1–6]. However, this leads to a partial differential equations with variable coefficients and the solutions are possible in specific cases. For the case of the so-called gradient zone resulting from the heat treatment process there is a continuous changing of mechanical properties of the material with the depth. In the literature it is known as an approach to such bodies, namely core material whichis treated as a homogeneous medium, and the transition zone from its surface is described by non-homogeneous materials with functional graded properties. In many papers the researchers are concerned with the thermal and residual stress analysis. The FGM material has many applications in engineering construction and the heat conduction characteristics play a very important role in thermal and residual stress considerations
☆ Communicated by: W.J. Minkowycz
http://dx.doi.org/10.1016/j.icheatmasstransfer.2014.07.021 0735-1933/© 2014 Elsevier Ltd. All rights reserved.
[7,8]. These papers deal with exact solution of axisymmetric stationary conduction problem. Contact problems of thermo-elasticity taking into account heat generation for graded half-space were considered in [9,10] (threedimensional and axisymmetric problem) and [11] (two-dimensional problem). Thermal stresses in the FGM layer analyzed in the framework of the theory of plane state strain are studied by [12] and [13]. In all papers the thermo-mechanical properties are described by exponential function of distance to the surface of the body. A special type gradient coating used in practice is the one with a periodic multilayer structure ([14,15]). This coating is composed of two homogeneous layers repeated periodically. The paper [16] presents the newly proposed method using the proper transformation of variable, the Laplace transformation and the perturbation method of one-dimensional heat conduction problem in an FGM plate. The next paper [17] is devoted to analysis of the onedimensional temperature distribution in an FGM strip using multilayered approaches. Some analytical solution of the problem of one-dimensional transient heat conduction in a material where the thermal conductivity coefficient is described by a linear function but the thermal diffusivity is treated as a constant is presented in [18]. The matrix method for heat conduction in circular cylinders of functionally graded materials and laminated composites was applied in paper [15]. Some numerical methods are used to describe the temperature distributions in materials with functional gradation properties (for example, [19, 20] and [21]). The three-dimensional axisymmetric problem of thermal conduction of functionally graded circular plate, in which upper and lower boundary surfaces were kept in constant difference temperature was considered in [22]. Material properties were taken to be arbitrary distribution functions of the thickness. In this paper ordinary differential equations with variable coefficients ODE due to arbitrary distribution of
158
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
Nomenclature — thickness of layer [m]; — radius of heating zone [m]; — coefficient of heat conductivities of the FGM layer [W/(mK)]; — coefficient of heat conductivities of the homogeneous half-space [W/(mK)]; — the temperature in the half-space [K]; — the temperature in the functionally graded layer (FGM); — heat flux vector [W].
h a λ1(z) λ0 T0 T1 q
material properties along thickness coordinate was solved by the Peano–Baker series. The exact solution for three-dimensional axisymmetric problem for functionally graded circular plate obtained by variable separation method is presented in [23]. Namely, the axisymmetric problem of heat conduction in the nonhomogeneous body composed of the FGM layer and homogeneous half-space is considered. It is assumed that the upper surfaces are heated by: 1) given temperature dependent on the radius, or 2) given heat flux dependent on the radius. Moreover the ideal thermal contact of the FGM layer and the homogeneous substrate is taken into account. The heat conductivity coefficients are assumed to be a power function of depth from the boundary surface. The considered problem is axisymmetric and independent of time. The Hankel transformmethod is applied to solve the considered problem. The presented paper is one of continuity of research presented in [22,23] and obtained by Hankel transform method and two cases of heating on the upper boundary surface are considered: 1) given temperature dependent on the radius; and 2) given heat flux dependent on the radius.
The axisymmetric problem of heat conduction for FGM layer resting on the homogeneous space is considered. The analyzed problem is solved by using the cylindrical coordinate system (r, φ, z), where the z-axis is perpendicular to the lower boundary plane of the layer (see Fig. 1). It was assumed that T1(r, z) is the temperature in the functionally graded layer (FGM), and T0(r, z) is the temperature in the halfspace. Half-space is covered by a material with functional graded properties, and the coefficient of heat conductivity varies along with the depth
α
where λ*, c, α are given constants.
• Given temperature T 1 ðr; z ¼ hÞ ¼ T g ðrÞ ;
ð1Þ
ð2Þ
• Given heat flux ð1Þ
qz ðr; z ¼ hÞ ¼ qg ðr Þ
ð3Þ
where Tg(⋅), qg(⋅) are given functions. The thermal properties of half-space are described by a heat conductivity coefficient λ0 = const. In addition, the assumed ideal thermal contact between the layer and the half-space is assumed, which can be written as follows: lim T 1 ðr; zÞ ¼ lim− T 0 ðr; zÞ;
z→0þ
z→0
ð4Þ
∂T ðr; zÞ ∂T ðr; zÞ ¼ lim− λ0 0 : lim λ1 ðzÞ 1 z→0 ∂z ∂z z→0þ Moreover, radiation conditions at infinityare taken into account. Equation of heat conduction takes the following form • for homogeneous half-space ∂2 T 0 1 ∂T 0 ∂2 T 0 þ ¼ 0; z b 0; r N 0; þ r ∂r ∂r 2 ∂z2
ð5Þ
• FGM layer ∂2 T 1 1 ∂T 1 ∂2 T 1 1 ∂λ1 ðzÞ ∂T 1 þ þ þ ¼ 0; 0 b z b h; r N 0: r ∂r λ1 ðzÞ ∂z ∂z ∂r 2 ∂z2
2. The formulation and solution of the problem
λ1 ðzÞ ¼ λ ðc þ zÞ ;
On the upper boundary surface two cases of heating conditions are assumed, namely a temperature, or a known heat flux dependent on the radius in the form:
ð6Þ
Components of the heat flux vector: • in the homogeneous half-space ! q 0 ðr;
∂T ∂T ð0Þ ð0Þ zÞ ≡ qr ; 0; qz ¼ λ0 0 ; 0; λ0 0 ; ∂r ∂z
ð7Þ
• in the FGM layer ! q 1 ðr;
∂T ∂T ð1Þ ð1Þ zÞ ≡ qr ; 0; qz ¼ λ1 ðzÞ 1 ; 0; λ1 ðzÞ 1 : ∂r ∂z
ð8Þ
The solution of the above formulated problem are calculated by using the Hankel transform method, which is denoted by
ef ðs; zÞ ¼
Z∞ f ðr; zÞr J 0 ðsr Þdr:
ð9Þ
0
Using Eqs. (9) to (5) and (6) we obtain the ordinary differential equations; • for the homogeneous half-space
Fig. 1. The scheme of considered problem.
d2 Te0 2 −s Te0 ¼ 0; dz2
ð10Þ
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
• for the FGM layer d2 Te1 1 ∂λ1 ðzÞ dTe1 2 −s Te1 ¼ 0; þ λ1 ðzÞ ∂z dz dz2
ð11Þ
where Te0 and Te1 are temperature transformations. The solution for both considered problem can be written in the form
159
The Hankel transforms of temperature (Eq. (12)) are expressed by ^ _ six unknown functions C j ðsÞ ; C j ðsÞ ; j ¼ 0; 1; 2 for both considered boundary value problem, which should be determined from boundary conditions (2) and (3). From Eqs. (12) and (13) and boundary conditions (2) and (3) it follows that
• for the given temperature • for the homogeneous half-space which satisfies the radiation conditions at infinity 8 9 ^ ϑ Z∞ > < Teg ðsÞC 0 > = T 0 ðr; zÞ ¼ ð12Þ expðszÞsJ o ðsr Þds; e q _ ð s Þ q g > T 0 ðr; zÞ : ; C 0> 0 s
^
^
p
^
p
C 0 −C 1 c K p ðscÞ−C 2 c Ip ðscÞ ¼ 0 ^ p−1 ^ p−1 λ0 ^ C þ C1c K p−1 ðscÞ−C 2 c Ip−1 ðscÞ ¼ 0 λ1 ð0Þ 0 ^
p p C 1 ðc þ hÞ K p ðsðc þ hÞÞ þ C 2 ðc þ hÞ Ip ðsðc þ hÞÞ ¼ Teg ðsÞ
• for the FGM layer
• for the given heat flux
_
ϑ T 1 ðr; zÞ q T 1 ðr; zÞ 8 ^ 9 ^ Z∞ > = < Teg ðsÞ C 1 K p ðsðc þ zÞÞ þ C 2 Ip ðsðc þ zÞÞ > p ¼ ðc þ zÞ sJ o ðsr Þds e _ qg ðsÞ _ > > ; : C K ð s ð c þ z Þ Þ þ C I ð s ð c þ z Þ Þ 1 p 2 p 0 s ð13Þ
where Ip , Kp denotes the modified Bessel functions and 2p + α = 1.
z*
ϑ
T1 / ϑ0
T0ϑ / ϑ0
α = 1, β = 1, h* = 0.5 z
*
_
p
_
p
C 0 −C 1 c K p ðscÞ−C 2 c Ip ðscÞ ¼ 0 _ p−1 _ p−1 λ0 _ C þ C1c K p−1 ðscÞ−C 2 c Ip−1 ðscÞ ¼ 0 λ1 ð0Þ 0
e qg ðsÞ p p −C 1 ðc þ hÞ K p−1 ðsðc þ hÞÞ þ C 2 ðc þ hÞ Ip−1 ðsðc þ hÞÞ ¼ sλ1 ðhÞ _
ð15Þ
_
qg ðsÞ are the transforms of the boundary temperature where Teg ðsÞ and e and boundary heat flux given by Eqs. (2) and (3). The solution is valid for any positive value of parameter α.
z*
T1ϑ / ϑ0
α = 1, β = 4, h* = 0.5
r*
T0ϑ / ϑ0
r* T1ϑ / ϑ0
α = 1, β = 8, h* = 0.5
ð14Þ
^
T0ϑ / ϑ0
r* Fig. 2. The dimensionless temperature distribution Tϑ 1 /ϑ0 as a function of β = 1; 4; 8.
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D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
then (
Teg ðsÞ e qg ðsÞ
)
¼
ϑ0 q0
Za pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϑ0 −as cosðasÞ þ sinðasÞ a2 −r 2 r J 0 ðsrÞdr ¼ : q0 s3 0
ð18Þ 4. Numerical analysis The first problem of heating with given temperature on the boundary will be analyzed. Let us denote the dimensionless coordinate system (r*, z*) related to radius a of heating area:
r ¼ r=a; z ¼ z=a:
Fig. 3. The dimensionless temperature distribution Tϑ 1 /ϑ0 as a function of depth z* for β = 1; 4; 8.
3. Special case For numerical calculation and presentation of results we assume that the parameter α = 1 and λ1(0) = λ0. Eq. (13) takes the form:
ϑ T 1 ðr; zÞ q T 1 ðr; zÞ
8 ^ 9 ^ Z∞ > = < Teg ðsÞ C 1 K 0 ðsðc þ zÞÞ þ C 2 I 0 ðsðc þ zÞÞ > ¼ _ sJ o ðsr Þds: e _ ð s Þ q > > : g C 1 K 0 ðsðc þ zÞÞ þ C 2 I0 ðsðc þ zÞÞ ; 0 s ð16Þ
The boundary temperature and boundary heat flux taken are as follows
T g ðr Þ ¼ ϑ0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 −r 2 H ða−r Þ; qg ðr Þ ¼ q0 a2 −r 2 H ða−rÞ
ð17Þ
ð19Þ
Fig. 2shows the dimensionless distribution of temperature as a function of ratio of β = λ1(h)/λ0, and β = 1, 4, 8, then c = h/(β− α − 1). For β = 1 (see Fig. 2a) we obtained theresults for homogeneous half-space and it is some verification of the obtained solution. The changing of the temperature on the interface is presented in Fig. 2 (it's a line for z* = 0). We can observe that the maximal value of temperature on interface is connected with parameter β. The same situation is shown in Fig. 3. In this figure the distribution of temperature is shown as a function of depth from the boundary surface in the center of heating for three cases of parametersβ = 1; 4; 8. The temperature on the interface grows with parameter β and the temperature value in the case when half-space is covered by FGM coating is higher than in the homogeneous case, because it was assumed that the ratio of coefficient is β ≥ 1. In Fig. 4a and b the dimensionless distribution of the heat flux vector (1) component q(1) r /(ϑ0λ0) and qz /(ϑ0λ0) is shown. The first figure Fig. 4a (1) shows the distribution of qr /(ϑ0λ0) as a function of radius r* for three cases of parameter β. It can be observed that with the growth of parameter β, when the coated layer will be a better conductor of heat the max(1) imal value of q(1) r /(ϑ0λ0) is increased. Component qz /(ϑ0λ0) of the heat flux vector is shown inFig. 3b as a function of dimensionless thickness h*. Fig. 4 shows the dimensionless distribution of heat flux as a function (1) of parameter β = 1; 4; 8 on the interface: a) q(1) r /(ϑ0λ0); b) qz /(ϑ0λ0). In Fig. 4b we can observe that the difference between solutions for FGM coating and homogeneous case decreases with growth of thickness h*. Next, we consider another boundary condition of heating in the form of given heat flux. Fig. 5 shows the dimensionless distribution of
(1) Fig. 4. The dimensionless distribution of heat flux as a function of parameter β = 1; 4; 8 on interface: a) q(1) r /(ϑ0λ0) and b) qz /(ϑ0λ0).
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
z*
T1q a / ( q0 λ0 )
T0q a / ( q0 λ0 )
α = 1, β = 1, h* = 0.5 z*
z*
161
T1q a / ( q0 λ0 )
α = 1, β = 4, h* = 0.5
r*
T0q a / ( q0λ0 )
r* T1q a / ( q0 λ0 )
α = 1, β = 8, h* = 0.5
T0q a / ( q0 λ0 )
r* Fig. 5. The dimensionless temperature distribution Tqa/(q0λ0) as a function of β = 1; 4; 8.
temperature as a function of parameter β for three cases β = 1; 4; 8. In this case the maximal temperature on the center of heating area decreases with rise of parameter β. Next, Fig. 6 shows the dimensionless distribution of temperature as a function of depth from the boundary surface for three cases of parameters β = 1; 4; 8. We can formulate the conclusion that when the coating layer will be the better conductor the maximal value of temperature on the interface decreases. Fig. 7a and b shows the dimensionless distribution of components (1) q(1) r /q0 and qz /q0 of the heat flux vector for three cases of parameters β = 1; 4; 8. In both figures it can be observed that the value of the heat flux is smaller than the values in the case of homogeneous case body. 5. Conclusion
Fig. 6. The dimensionless temperature distribution Tqa/(q0λ0) as a function of depth z* for β = 1; 4; 8.
In this paper the exact solution in the form of Hankel's integrals of axisymmetric stationary heat conduction problem for FGM coating on homogeneous half-space is obtained. Eqs. (13) and (14) together with Eqs.(15) and (16) are adequate solutions for any positive value of parameter α. The influence of thermal and geometrical parameters on the temperature and heat flux distributions was investigated. The impact of non-homogeneity of the FGM layer represented by parameter β on the distribution of temperature on the interfaces was also examined. In the case of the temperature boundary condition with growing
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D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
(1) Fig. 7. The dimensionless distribution of heat flux as a function of parameter β = 1; 4; 8 on interface: a) q(1) r /q0 and b) qz /q0.
parameter β the temperature on the interface is greater than in the case of a homogeneous body (β = 1). The case is different when the boundary condition is in the form of heat flux the temperature decreases on the interface and it is lower than the homogeneity bodies. The obtained solution can be useful in the design of some FGM coating and further analysis of thermal stress for local axisymmetric heating is needed.
References [1] J. Aboudia, M.-J. Pinderab, S.M. Arnoldc, Higher-order theory for functionally graded materials, Compos. Part B 30 (1999) 777–832. [2] Woźniak Cz, E. Wierzbicki, Averaging Techniques in Thermomechanics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej, Poland, 2000. [3] A. Yevtushenko, S.J. Matysiak, E.G. Ivanyk, Influence of periodically layered material structure on the frictional temperature during braking, Int. J. Heat Mass Transfer 40 (1997) 2115–2122. [4] A. Kaczyński, S.J. Matysiak, On the complex potentials of the linear thermoelasticity with microlocal parameters, Acta Mech. 72 (1988) 245–259. [5] Stanisław J. Matysiak, Dariusz M. Perkowski, Axially symmetric problems of heat conduction in a periodically laminated layer with vertical cylindrical hole, Int. Commun. Heat Mass Transfer 38 (2011) 410–417. [6] R. Kulchytsky-Zhyhailo, S.J. Matysiak, D.M. Perkowski, Plane contact problems with frictional heating for a vertically layered half-space, Int. J. Heat Mass Transf. 54 (2011) 1805–1813. [7] A. Masoud, A. Mehdi, M.H. Seyed, Dynamic analysis of two-dimensional functionally graded thick hollow cylinder with finite length under impact loading [J], Acta Mech. 208 (3/4) (2009) 163–180. [8] A. Khosravifard, M.R. Hematiyan, L. Marin, Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method [J], Appl. Math. Model. 35 (9) (2011) 4157–4174. [9] S. Matysiak, A. Jevtuszenko, R. Kulczickij-Żyhailo, Kontaktnyje zadaczi termouprugosti dla poluprostranstva iz funkcionalno-gradijentnogo materiala, Mat. metody i fiz.mech. pola (1998) 45–57 (T. 41, No. 2).
[10] R. Kulczycki, Przestrzenne zagadnienia kontaktowe termosprężystości (Threedimensional Contact Problem of Thermo-elasticity), Wydawnictwo Politechniki Białostockiej, Poland, 2002. [11] S.P. Barik, M. Kanoria, P.K. Chaudhuri, Steady state thermoelastic contact problem in a functionally graded material, Int. J. Eng. Sci. 46 (2008) 775–789. [12] H.M. Shodja, A. Ghahremaninejad, An FGM coated elastic solid under thermomechanical loading: a two dimensional linear elastic approach, Surf. Coat. Technol. 200 (12–13) (2006) 4050–4064. [13] H.J. Choi, G.H. Paulino, Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation, J. Mech. Phys. Solids 56 (4) (2008) 1673–1692 (Elsevier). [14] Z.N. Farhat, Y. Ding, D.O. Northwood, A.T. Aplas, Nanoindentation and friction studies on Ti-based nanolaminated films, Surf. Coat. Technol. 89 (1997) 24–30. [15] A.A. Voevodin, E.V. Iarve, W. Ragland, J.S. Zabinski, S. Donaldson, Stress analyses and in-situ fracture observation of wear protective multilayer coatings in contact loading, Surf. Coat. Technol. 148 (2001) 38–45. [16] Y. Obata, N. Noda, Unsteady thermal stresses in a functionally gradient material plate, Trans. Jpn. Soc. Mech. Eng. Ser. A 59 (1993) 1090–1103. [17] T. Ishiguro, A. Makino, N. Araki, N. Noda, Transient temperature response in functionally gradient materials, Int. J. Thermophys. 14 (1993) 101–121. [18] Y. Tanigawa, T. Akai, R. Kawamura, N. Oka, Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties, J. Therm. Stress. 19 (1996) 77–102. [19] L. Byrd, V. Birman, An investigation of numerical modeling of transient heat conduction in a one-dimensional functionally graded material [J], Heat Transfer Eng. 31 (3) (2010) 212–221. [20] A. Sutradhar, G.H. Paulino, The simple boundary element method for transient heat conduction in functionally graded materials [J], Comput. Methods Appl. Mech. Eng. 193 (42/44) (2004) 4511–4539. [21] A. Sutradhar, G.H. Paulino, L.J. Gray, Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method [J], Eng. Anal. Boundary Elem. 26 (2) (2002) 119–132. [22] L.I.U. Wu-xiang, Analysis of steady heat conduction for 3D axisymmetric functionally graded circular plate, J. Cent. South Univ. 20 (2013) 1616–1622. [23] L.I.U. Wu-xiang, Exact solution of steady-state heat conduction for axisymmetric functionally graded circular plate [J], J. Tongji Univ. Nat. Sci. 38 (5) (2010) 716–719 (in Chinese).
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
On axisymmetric heat conduction problem for FGM layer on homogeneous substrate☆ Dariusz M. Perkowski Faculty of Mechanical Engineering, Białystok University of Technology, ul. Wiejska 45 C, 15-351 Białystok, Poland
a r t i c l e
i n f o
Available online 4 August 2014 Keywords: Temperature Heat flux FGM Layer Hankel transform
a b s t r a c t The paper deals with the problem of heat conduction in the FGM layer with heat conductivity coefficient dependent on depth from boundary surface. The non-homogeneous layer is ideally bounded with homogeneous half-space with constant heat conductivity coefficient. The boundary plane is heated by: a) given temperature as a function of radius r, or b) given heat flux as a function of radius r. The Hankel transform method is applied to obtain a solution of formulated problem. The influence of thermal and geometric properties of FGM layer on temperature distributions in the considered bodies was investigated. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Modern composite materials with functional graded properties (FGM) have a complex internal structure, due to the fact that they are composed of several different materials. The thermo-mechanical properties of the composite depend on the components included in such materials. There are two common types of FGM materials: with the continuous or discrete changing of the material properties. An example of FGM materials with the continuous changing of the material properties is the so-called FGM gradient zone created during the manufacturing process between the coating and the substrate component obtained under the action of temperature. Discrete changes in the properties are achieved by multilayer composites, which are very popular in technical applications. In the case of coatings with discrete changing of material properties the literature presented some models allowing the description ofa body by averaged technique [1–6]. However, this leads to a partial differential equations with variable coefficients and the solutions are possible in specific cases. For the case of the so-called gradient zone resulting from the heat treatment process there is a continuous changing of mechanical properties of the material with the depth. In the literature it is known as an approach to such bodies, namely core material whichis treated as a homogeneous medium, and the transition zone from its surface is described by non-homogeneous materials with functional graded properties. In many papers the researchers are concerned with the thermal and residual stress analysis. The FGM material has many applications in engineering construction and the heat conduction characteristics play a very important role in thermal and residual stress considerations
☆ Communicated by: W.J. Minkowycz
http://dx.doi.org/10.1016/j.icheatmasstransfer.2014.07.021 0735-1933/© 2014 Elsevier Ltd. All rights reserved.
[7,8]. These papers deal with exact solution of axisymmetric stationary conduction problem. Contact problems of thermo-elasticity taking into account heat generation for graded half-space were considered in [9,10] (threedimensional and axisymmetric problem) and [11] (two-dimensional problem). Thermal stresses in the FGM layer analyzed in the framework of the theory of plane state strain are studied by [12] and [13]. In all papers the thermo-mechanical properties are described by exponential function of distance to the surface of the body. A special type gradient coating used in practice is the one with a periodic multilayer structure ([14,15]). This coating is composed of two homogeneous layers repeated periodically. The paper [16] presents the newly proposed method using the proper transformation of variable, the Laplace transformation and the perturbation method of one-dimensional heat conduction problem in an FGM plate. The next paper [17] is devoted to analysis of the onedimensional temperature distribution in an FGM strip using multilayered approaches. Some analytical solution of the problem of one-dimensional transient heat conduction in a material where the thermal conductivity coefficient is described by a linear function but the thermal diffusivity is treated as a constant is presented in [18]. The matrix method for heat conduction in circular cylinders of functionally graded materials and laminated composites was applied in paper [15]. Some numerical methods are used to describe the temperature distributions in materials with functional gradation properties (for example, [19, 20] and [21]). The three-dimensional axisymmetric problem of thermal conduction of functionally graded circular plate, in which upper and lower boundary surfaces were kept in constant difference temperature was considered in [22]. Material properties were taken to be arbitrary distribution functions of the thickness. In this paper ordinary differential equations with variable coefficients ODE due to arbitrary distribution of
158
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
Nomenclature — thickness of layer [m]; — radius of heating zone [m]; — coefficient of heat conductivities of the FGM layer [W/(mK)]; — coefficient of heat conductivities of the homogeneous half-space [W/(mK)]; — the temperature in the half-space [K]; — the temperature in the functionally graded layer (FGM); — heat flux vector [W].
h a λ1(z) λ0 T0 T1 q
material properties along thickness coordinate was solved by the Peano–Baker series. The exact solution for three-dimensional axisymmetric problem for functionally graded circular plate obtained by variable separation method is presented in [23]. Namely, the axisymmetric problem of heat conduction in the nonhomogeneous body composed of the FGM layer and homogeneous half-space is considered. It is assumed that the upper surfaces are heated by: 1) given temperature dependent on the radius, or 2) given heat flux dependent on the radius. Moreover the ideal thermal contact of the FGM layer and the homogeneous substrate is taken into account. The heat conductivity coefficients are assumed to be a power function of depth from the boundary surface. The considered problem is axisymmetric and independent of time. The Hankel transformmethod is applied to solve the considered problem. The presented paper is one of continuity of research presented in [22,23] and obtained by Hankel transform method and two cases of heating on the upper boundary surface are considered: 1) given temperature dependent on the radius; and 2) given heat flux dependent on the radius.
The axisymmetric problem of heat conduction for FGM layer resting on the homogeneous space is considered. The analyzed problem is solved by using the cylindrical coordinate system (r, φ, z), where the z-axis is perpendicular to the lower boundary plane of the layer (see Fig. 1). It was assumed that T1(r, z) is the temperature in the functionally graded layer (FGM), and T0(r, z) is the temperature in the halfspace. Half-space is covered by a material with functional graded properties, and the coefficient of heat conductivity varies along with the depth
α
where λ*, c, α are given constants.
• Given temperature T 1 ðr; z ¼ hÞ ¼ T g ðrÞ ;
ð1Þ
ð2Þ
• Given heat flux ð1Þ
qz ðr; z ¼ hÞ ¼ qg ðr Þ
ð3Þ
where Tg(⋅), qg(⋅) are given functions. The thermal properties of half-space are described by a heat conductivity coefficient λ0 = const. In addition, the assumed ideal thermal contact between the layer and the half-space is assumed, which can be written as follows: lim T 1 ðr; zÞ ¼ lim− T 0 ðr; zÞ;
z→0þ
z→0
ð4Þ
∂T ðr; zÞ ∂T ðr; zÞ ¼ lim− λ0 0 : lim λ1 ðzÞ 1 z→0 ∂z ∂z z→0þ Moreover, radiation conditions at infinityare taken into account. Equation of heat conduction takes the following form • for homogeneous half-space ∂2 T 0 1 ∂T 0 ∂2 T 0 þ ¼ 0; z b 0; r N 0; þ r ∂r ∂r 2 ∂z2
ð5Þ
• FGM layer ∂2 T 1 1 ∂T 1 ∂2 T 1 1 ∂λ1 ðzÞ ∂T 1 þ þ þ ¼ 0; 0 b z b h; r N 0: r ∂r λ1 ðzÞ ∂z ∂z ∂r 2 ∂z2
2. The formulation and solution of the problem
λ1 ðzÞ ¼ λ ðc þ zÞ ;
On the upper boundary surface two cases of heating conditions are assumed, namely a temperature, or a known heat flux dependent on the radius in the form:
ð6Þ
Components of the heat flux vector: • in the homogeneous half-space ! q 0 ðr;
∂T ∂T ð0Þ ð0Þ zÞ ≡ qr ; 0; qz ¼ λ0 0 ; 0; λ0 0 ; ∂r ∂z
ð7Þ
• in the FGM layer ! q 1 ðr;
∂T ∂T ð1Þ ð1Þ zÞ ≡ qr ; 0; qz ¼ λ1 ðzÞ 1 ; 0; λ1 ðzÞ 1 : ∂r ∂z
ð8Þ
The solution of the above formulated problem are calculated by using the Hankel transform method, which is denoted by
ef ðs; zÞ ¼
Z∞ f ðr; zÞr J 0 ðsr Þdr:
ð9Þ
0
Using Eqs. (9) to (5) and (6) we obtain the ordinary differential equations; • for the homogeneous half-space
Fig. 1. The scheme of considered problem.
d2 Te0 2 −s Te0 ¼ 0; dz2
ð10Þ
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
• for the FGM layer d2 Te1 1 ∂λ1 ðzÞ dTe1 2 −s Te1 ¼ 0; þ λ1 ðzÞ ∂z dz dz2
ð11Þ
where Te0 and Te1 are temperature transformations. The solution for both considered problem can be written in the form
159
The Hankel transforms of temperature (Eq. (12)) are expressed by ^ _ six unknown functions C j ðsÞ ; C j ðsÞ ; j ¼ 0; 1; 2 for both considered boundary value problem, which should be determined from boundary conditions (2) and (3). From Eqs. (12) and (13) and boundary conditions (2) and (3) it follows that
• for the given temperature • for the homogeneous half-space which satisfies the radiation conditions at infinity 8 9 ^ ϑ Z∞ > < Teg ðsÞC 0 > = T 0 ðr; zÞ ¼ ð12Þ expðszÞsJ o ðsr Þds; e q _ ð s Þ q g > T 0 ðr; zÞ : ; C 0> 0 s
^
^
p
^
p
C 0 −C 1 c K p ðscÞ−C 2 c Ip ðscÞ ¼ 0 ^ p−1 ^ p−1 λ0 ^ C þ C1c K p−1 ðscÞ−C 2 c Ip−1 ðscÞ ¼ 0 λ1 ð0Þ 0 ^
p p C 1 ðc þ hÞ K p ðsðc þ hÞÞ þ C 2 ðc þ hÞ Ip ðsðc þ hÞÞ ¼ Teg ðsÞ
• for the FGM layer
• for the given heat flux
_
ϑ T 1 ðr; zÞ q T 1 ðr; zÞ 8 ^ 9 ^ Z∞ > = < Teg ðsÞ C 1 K p ðsðc þ zÞÞ þ C 2 Ip ðsðc þ zÞÞ > p ¼ ðc þ zÞ sJ o ðsr Þds e _ qg ðsÞ _ > > ; : C K ð s ð c þ z Þ Þ þ C I ð s ð c þ z Þ Þ 1 p 2 p 0 s ð13Þ
where Ip , Kp denotes the modified Bessel functions and 2p + α = 1.
z*
ϑ
T1 / ϑ0
T0ϑ / ϑ0
α = 1, β = 1, h* = 0.5 z
*
_
p
_
p
C 0 −C 1 c K p ðscÞ−C 2 c Ip ðscÞ ¼ 0 _ p−1 _ p−1 λ0 _ C þ C1c K p−1 ðscÞ−C 2 c Ip−1 ðscÞ ¼ 0 λ1 ð0Þ 0
e qg ðsÞ p p −C 1 ðc þ hÞ K p−1 ðsðc þ hÞÞ þ C 2 ðc þ hÞ Ip−1 ðsðc þ hÞÞ ¼ sλ1 ðhÞ _
ð15Þ
_
qg ðsÞ are the transforms of the boundary temperature where Teg ðsÞ and e and boundary heat flux given by Eqs. (2) and (3). The solution is valid for any positive value of parameter α.
z*
T1ϑ / ϑ0
α = 1, β = 4, h* = 0.5
r*
T0ϑ / ϑ0
r* T1ϑ / ϑ0
α = 1, β = 8, h* = 0.5
ð14Þ
^
T0ϑ / ϑ0
r* Fig. 2. The dimensionless temperature distribution Tϑ 1 /ϑ0 as a function of β = 1; 4; 8.
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then (
Teg ðsÞ e qg ðsÞ
)
¼
ϑ0 q0
Za pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϑ0 −as cosðasÞ þ sinðasÞ a2 −r 2 r J 0 ðsrÞdr ¼ : q0 s3 0
ð18Þ 4. Numerical analysis The first problem of heating with given temperature on the boundary will be analyzed. Let us denote the dimensionless coordinate system (r*, z*) related to radius a of heating area:
r ¼ r=a; z ¼ z=a:
Fig. 3. The dimensionless temperature distribution Tϑ 1 /ϑ0 as a function of depth z* for β = 1; 4; 8.
3. Special case For numerical calculation and presentation of results we assume that the parameter α = 1 and λ1(0) = λ0. Eq. (13) takes the form:
ϑ T 1 ðr; zÞ q T 1 ðr; zÞ
8 ^ 9 ^ Z∞ > = < Teg ðsÞ C 1 K 0 ðsðc þ zÞÞ þ C 2 I 0 ðsðc þ zÞÞ > ¼ _ sJ o ðsr Þds: e _ ð s Þ q > > : g C 1 K 0 ðsðc þ zÞÞ þ C 2 I0 ðsðc þ zÞÞ ; 0 s ð16Þ
The boundary temperature and boundary heat flux taken are as follows
T g ðr Þ ¼ ϑ0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 −r 2 H ða−r Þ; qg ðr Þ ¼ q0 a2 −r 2 H ða−rÞ
ð17Þ
ð19Þ
Fig. 2shows the dimensionless distribution of temperature as a function of ratio of β = λ1(h)/λ0, and β = 1, 4, 8, then c = h/(β− α − 1). For β = 1 (see Fig. 2a) we obtained theresults for homogeneous half-space and it is some verification of the obtained solution. The changing of the temperature on the interface is presented in Fig. 2 (it's a line for z* = 0). We can observe that the maximal value of temperature on interface is connected with parameter β. The same situation is shown in Fig. 3. In this figure the distribution of temperature is shown as a function of depth from the boundary surface in the center of heating for three cases of parametersβ = 1; 4; 8. The temperature on the interface grows with parameter β and the temperature value in the case when half-space is covered by FGM coating is higher than in the homogeneous case, because it was assumed that the ratio of coefficient is β ≥ 1. In Fig. 4a and b the dimensionless distribution of the heat flux vector (1) component q(1) r /(ϑ0λ0) and qz /(ϑ0λ0) is shown. The first figure Fig. 4a (1) shows the distribution of qr /(ϑ0λ0) as a function of radius r* for three cases of parameter β. It can be observed that with the growth of parameter β, when the coated layer will be a better conductor of heat the max(1) imal value of q(1) r /(ϑ0λ0) is increased. Component qz /(ϑ0λ0) of the heat flux vector is shown inFig. 3b as a function of dimensionless thickness h*. Fig. 4 shows the dimensionless distribution of heat flux as a function (1) of parameter β = 1; 4; 8 on the interface: a) q(1) r /(ϑ0λ0); b) qz /(ϑ0λ0). In Fig. 4b we can observe that the difference between solutions for FGM coating and homogeneous case decreases with growth of thickness h*. Next, we consider another boundary condition of heating in the form of given heat flux. Fig. 5 shows the dimensionless distribution of
(1) Fig. 4. The dimensionless distribution of heat flux as a function of parameter β = 1; 4; 8 on interface: a) q(1) r /(ϑ0λ0) and b) qz /(ϑ0λ0).
D.M. Perkowski / International Communications in Heat and Mass Transfer 57 (2014) 157–162
z*
T1q a / ( q0 λ0 )
T0q a / ( q0 λ0 )
α = 1, β = 1, h* = 0.5 z*
z*
161
T1q a / ( q0 λ0 )
α = 1, β = 4, h* = 0.5
r*
T0q a / ( q0λ0 )
r* T1q a / ( q0 λ0 )
α = 1, β = 8, h* = 0.5
T0q a / ( q0 λ0 )
r* Fig. 5. The dimensionless temperature distribution Tqa/(q0λ0) as a function of β = 1; 4; 8.
temperature as a function of parameter β for three cases β = 1; 4; 8. In this case the maximal temperature on the center of heating area decreases with rise of parameter β. Next, Fig. 6 shows the dimensionless distribution of temperature as a function of depth from the boundary surface for three cases of parameters β = 1; 4; 8. We can formulate the conclusion that when the coating layer will be the better conductor the maximal value of temperature on the interface decreases. Fig. 7a and b shows the dimensionless distribution of components (1) q(1) r /q0 and qz /q0 of the heat flux vector for three cases of parameters β = 1; 4; 8. In both figures it can be observed that the value of the heat flux is smaller than the values in the case of homogeneous case body. 5. Conclusion
Fig. 6. The dimensionless temperature distribution Tqa/(q0λ0) as a function of depth z* for β = 1; 4; 8.
In this paper the exact solution in the form of Hankel's integrals of axisymmetric stationary heat conduction problem for FGM coating on homogeneous half-space is obtained. Eqs. (13) and (14) together with Eqs.(15) and (16) are adequate solutions for any positive value of parameter α. The influence of thermal and geometrical parameters on the temperature and heat flux distributions was investigated. The impact of non-homogeneity of the FGM layer represented by parameter β on the distribution of temperature on the interfaces was also examined. In the case of the temperature boundary condition with growing
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(1) Fig. 7. The dimensionless distribution of heat flux as a function of parameter β = 1; 4; 8 on interface: a) q(1) r /q0 and b) qz /q0.
parameter β the temperature on the interface is greater than in the case of a homogeneous body (β = 1). The case is different when the boundary condition is in the form of heat flux the temperature decreases on the interface and it is lower than the homogeneity bodies. The obtained solution can be useful in the design of some FGM coating and further analysis of thermal stress for local axisymmetric heating is needed.
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