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A generalized self-consistent method for calculation of effective thermal conductivity of composites with interfacial contact conductance
International Communications in Heat and Mass Transfer 33 (2006) 142 – 150 www.elsevier.com/locate/ichmt
A generalized self-consistent method for calculation of effective thermal conductivity of composites with interfacial contact conductance☆ Yung-Ming Lee ⁎, Ruey-Bin Yang, Song-Sheun Gau Department of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan 40724, R.O.C. Available online 28 November 2005
Abstract The effective thermal conductivity of composites can be obtained by the generalized self-consistent method (GSCM). The effect of contact conductance that exists at the interface between the matrix and the inclusion, on the effective thermal conductivity is investigated. The particulate composite, transversely isotropic fiber composite, and multi-layered composite are considered in this study. Based on the energy balance, a simple criterion Beff = 0, yet new and analytical, for determining the effective thermal conductivity of the composite is rigorously derived. The temperature distribution of the matrix and the inclusion can also be obtained by GSCM. This is the by-product of the method, which many other methods do not provide. © 2005 Elsevier Ltd. All rights reserved. Keywords: Effective thermal conductivity; Contact conductance; GSCM; Composite
1. Introduction Composite materials are being used increasingly in a variety of modern engineering applications and this trend is likely to continue due to the fact that these materials possess a number of highly desirable engineering properties that can be exploited to design structures with high demand on their performance. Thermal conductivity is an important property of composites' application for electronic packaging, thermal insulation, heat spreader, etc. [1]. As a consequence, determination of the effective thermal properties of composites is essential for a successful design and manufacturing of such composite materials. Theoretical studies have shown that the effective thermal conductivity of composites is strongly dependent on the volume fraction, the distribution, inclusion size, the thermal conductivities of constituents and the interface conductance between inclusion and matrix [2–4]. Many theoretical studies have used micro-mechanics models such as dilute, self-consistent method, composite spheres (cylinders) model, differential method, and Mori-Tanaka method to predict the effective properties of the composite material [5,6].
☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-M. Lee).
0735-1933/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.10.004
Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
143
Christensen [7] evaluated those theoretical models for the calculation of the overall static elastic constants and concluded that only the Generalized Self-Consistent Model (GSCM) gives physical reasonable results at high concentrations. The generalized self-consistent method seems to be a more appropriate tool for homogenization process. It takes into account the complex interaction between matrix and inclusions by considering a representative unit cell inclusion, i.e., an inclusion and a surrounding ring matrix, which is itself embedded in the infinite effective medium. The Generalized Self-Consistent Model provides a physically realistic model of inclusion to inclusion interaction for two-phase system covering the full range of the volume fraction (0 ≦ ve ≦ 1). The idea of the Generalized Self-Consistent Model (originally termed as self-consistent scheme) was developed by Hashin [8] to determine the effective conductivity of the two-phase materials. The self-consistent field concept is extended to determine the composite reinforced with coated spheres [9]. The effective thermal conductivity is evaluated as the ratio of a volume average of the heat flux to temperature gradient in the model [8,9]. Although the final results agreed with that of Hasselman and Jonson [3], the derivation of the volume integral is not simple and the method may be difficult to treat multiple-coated cylinders or spheres. 2. General analysis Consider a large two-phase body of volume V and surface S. It is assumed that the material is isotropic. The intensity and heat flux are defined by t
H ¼ −jT Y
t
q ¼ kH
ð1Þ ð2Þ
In Eq. (1), T is the temperature solution. For a two-phase body, k in Eq. (2) represents the thermal conductivity of either one of the phases. The homogeneous boundary conditions of the first and second kinds on surface S are T ðSÞ ¼ −Hi0 xi
ð3aÞ
qn ðSÞ ¼ q0i ni
ð3bÞ
where Hi0 and qi0 are arbitrarily constant temperature and heat flux components, respectively, xi are Cartesian coordinates, the subscripts i are ranging from 1 to 3, and ni are the components of the outer normal vector on surface S. According to the tensor notation, a repeated subscript on i represents summation. The effective thermal conductivity of a two-phase body can be obtained with the aid of energy integrals [8]. The integral can be defined by Z 1 qi ð xÞHi ð xÞdV ð4Þ U¼ V V It was shown [8] that for boundary condition (3a) U ¼ keff Hi0 Hi0
ð5aÞ
and for boundary condition (3b) U¼
1 0 0 q q keff i i
ð5bÞ
In Eqs. (5a) and (5b), keff stands for the effective thermal conductivity of the two-phase composite. 3. Generalized self consistent method (GSCM) for spherical inclusions To account for the strong interactions between the matrix and inclusion at nondilute case, the Generalized SelfConsistent Model is shown in Fig. 1 where a spherical coordinate system (r,θ,ϕ) is used. The spherical inclusion of radius a is embedded in a concentric sphere matrix material of radius b, which is embedded in an infinite effective
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Y
r
X b
a Inclusion
Matrix
Fig. 1. The generalized self-consistent model.
medium possessing the unknown effective thermal conductivity. The ratio of the radii, a/b, is related the volume fraction of inclusions by ve = (a/b)3. The steady-state conduction equation of a particulate composite can be written as j2 Te ¼ 0
0VrVa
ð6Þ
j2 Tm ¼ 0
aVrVb
ð7Þ
j2 Teff ¼ 0
bVrVl
ð8Þ
The continuity of heat flux and the temperature jump condition should be satisfied between the inclusion and the matrix according to the conservation law of energy. However, the temperature solution is continuous between the matrix and the effective medium. These boundary conditions at the interfaces can be written as km ATm at r ¼ a ð9Þ Te ¼ Tm − C Ar r¼a km
ATm Ar
¼ ke r¼a
Teff ¼ Tm km
ATm Ar
ATe Ar
at r ¼ a
at r ¼ b
s ¼ keff r¼b
ð10Þ
r¼a
ATeff Ar
ð11Þ at r ¼ b
ð12Þ
r¼b
where the subscripts e, m and eff stand for the inclusion, the matrix and the effective medium, respectively, and C denotes thermal contact conductance, the reciprocal of thermal resistance, 1/Rt, between the matrix and the inclusion. For the imperfect contact case, C is finite; otherwise, C is infinite in case of perfect contact at the interface. The
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145
superscript s in this section represents the spherical inclusion model. The boundary condition far away from the inclusion is subjected to Eq. (2a) and is of the form Teff ¼ −bx3 ¼ −brcosh
ð13Þ
as rYl
The temperature solutions are taken in the forms [10] Te ¼ Ase rcosh
Tm ¼
ð14Þ
0VrVa
Bs Asm r þ 2m cosh r
aVrVb
ð15Þ
Bs cosh br þ eff r2
bVrVl
ð16Þ
Teff ¼
Substitution of Eqs. (14), (15), and (16) into Eqs. (9), (10), (11), and (12) becomes km s km s a3 Ase −a3 1− Am − 1 þ 2 B ¼0 Ca Ca m
ð17aÞ
a3 ke Ase −a3 km Asm þ 2km Bsm ¼ 0
ð17bÞ
b3 Asm þ Bsm −Bseff ¼ bb3
ð17cÞ
s s Bseff ¼ bb3 keff b3 km Asm −2km Bsm þ 2keff
ð17dÞ
s s s The unknown coefficients, Aes, Am , Bm , and Beff , can be determined from Eq. (17a) to (17d) if the effective thermal s s conductivity keff is known. The criteria for determining keff require the equality of the thermal energy in heterogeneous medium and in the equivalent homogeneous medium. The idea of mathematical derivation in the following comes from Beck et al. [11]. According to Eq. (9), the temperature solution experiences a step change at the contact interface and the derivatives of the temperature across the contact interface are singulars. Hence, substitution of Eq. (1) into the integral given in Eq. (4) is modified as
1 U¼ V
Z
Z Vm −Ve
kjT d jT dV þ
Ve þVe Ve −Ve
Z kjT d jT dV þ
Ve −Ve
kjT d jT dV
ð18Þ
where V is the summation of the volume of the inclusion Ve and the volume of the matrix Vm; while the contact zone is a very thin layer including the total volume of 2Vε. The following mathematical identity can be further used to simplify the integration. That is jd ½T ðkjT Þ ¼ Tkj2 T þ kjT d jT
ð19Þ
It is assumed that the composite material is homogeneous and isotropic inside either the matrix or the inclusion. Therefore, the thermal conductivity k can be treated as a constant. The first term on the right-hand side vanishes because the steady-state energy equation is nothing but the Laplace equation. Hence, Eq. (19) becomes kjT d jT ¼ jd ½T ðkjT Þ
ð20Þ
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After substituting Eq. (20) into Eq. (18), and then applying the divergence theorem to the resulting equation yields 2 Z Z 16 t t Tm km jTm d n dS þ Tm km jTm d ð−n ÞdS U¼ 4 V Smo Smi 0 B þ@
Z
Tm km jTm d t n dS þ Seo
1
Z Sei
C t Te ke jTe d ð−n ÞdS A þ
3
Z Sei
7 Te ke jTe d t n dS 5
ð21Þ
It is noted that the inner surface of the matrix equals to the outer surface of the contact zone and, similarly, the inner surface of the contact zone is the same as the outer surface of the inclusion. The unit normal vector t n is pointed outward. The directional derivative of the surface in the direction of n can be expressed as AT ¼ jT d t n An
ð22Þ
Substituting Eq. (22) to Eq. (21) becomes 2 ! ! Z Z Z Z 16 ATm ATm ATm ATe dS þ U¼ 4 dS þ dS þ dS Tm km T m km − Tm km T e ke − An An An An V Smo Smi Seo Sei 3
Z þ
Te ke Sei
ATe 7 dS 5 An
ð23Þ
Assuming the thickness of the contact zone is extremely small, one can easily find that the second term and the third term on the right side of Eq. (23) are cancelled out with each other. Also, the fourth term and the fifth term are cancelled out with each other, Eq. (23) is then of the form 1 U¼ V
Z Tm km Smo
ATm dS An
ð24Þ
Substitution of Eqs. (12) and (15) into Eq. (24) yields ks U ¼ eff b
2
2bBs bBs 2Bs bb2 − 3eff þ 2eff − 5eff b b b
! ð25Þ
From Eq. (5a), one has U¼
s keff H 0H 0
¼
s keff b2
ks ¼ eff b
2
2bBs bBs 2Bs bb − 3eff þ 2eff − 5eff b b b 2
! ð26Þ
After simple algebraic calculation, one can find Bseff ¼ 0:
ð27Þ
To the best knowledge of the authors, the rigorous and detailed derivation of the coefficient Bseff ¼ 0 is new in the s of composites. related literatures, and it can be used to determine the effective thermal conductivity keff
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147
Now, the effective thermal conductivity of the particulate composite can be deduced from solving for Beff in Eqs. (17a), (17b), (17c), and (17d) and setting it to zero according to Eq. (27). This will give the following result 2 3 km ð3ve Þke 3ve − 6 7 Ca ke −km s 7 keff ¼ km 6 ð28Þ 41 þ 3km km ð2 þ ve Þke 5 vm þ þ ke −km Ca ke −km Eq. (28) has the same form obtained in [3]. For the perfect contact case, Eq. (28) will reduce to the form obtained by Hashin [8]. 4. GSCM for transversely isotropic fiber composites The GSCM can be equally applied to compute the effective thermal conductivity of a fiber-reinforced composite. The derivation procedure is straightforward and is not presented in this section. The final result can be easily obtained by the GSCM and is given by 2 3 km ve ke ve − 6 7 Ca ke −km c 7 ¼ km 6 keff ð29Þ 4 1 þ vm 1 km ð1 þ ve Þke 5 þ þ 2km ke −km Ca 2km ðke −km Þ Eq. (29) has the same form obtained in Hasselman and Jonson [3]. 5. GSCM with multi-layer inclusions In a similar fashion, the method presented in this paper can be easily extended to find the effective thermal conductivity of either particulate composite or fibrous composite with multi-coated cylinders or spheres. The purpose of the coating may be to protect fibers or spheres from the formation of brittle reaction layers at their interfaces or to improve the heat conduction performance of the composite by applying a high thermal conductivity coating layers to the thermally poor inclusion. In this section, the effective thermal conductivity of a particulate composite with n concentric layers will be reported. The governing equations of this n-layer inclusion model are of the forms j2 T1 ¼ 0
0VrVR1
j2 Ti ¼ 0
Ri−1 VrVRi
j2 Teff ¼ 0
ð30Þ i ¼ 2;3; N ;n þ 1
ð31Þ
Rnþ1 Vrbl
ð32Þ
Again, the continuity of heat flux and the temperature jump condition should be satisfied between two adjacent layers according to the conservation law of energy. However, the temperature solution is continuous between the matrix and the effective medium. This composite material is subjected to a linear temperature gradient, β, far away from the inclusion. The temperature solution for composites reinforced with multi-coated spheres can be obtained from Carslaw and Jaeger [10]. Substitution of the temperature solutions into the boundary conditions becomes k2 k2 As2 − 1 þ 2 Bs ¼ 0 ð33aÞ R31 As1 −R31 1− C1 R1 C1 R1 2 R31 k1 As1 −R31 k2 As2 þ 2k2 Bs2 ¼ 0 kiþ1 s kiþ1 s s s s s A − 1þ2 B ¼ 0; Ri Ai þ Bi −Ri 1− Ci Ri iþ1 Ci Ri iþ1 R3i ki Asi −2ki Bsi −R3i kiþ1 Asiþ1 þ 2kiþ1 Bsiþ1 ¼ 0;
ð33bÞ i ¼ 2;3; N ;n
i ¼ 2;3; N ;n
ð33cÞ ð33dÞ
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Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
R3nþ1 Asnþ1 þ Bsnþ1 −Bseff ¼ bR3nþ1
ð33eÞ
s s R3nþ1 knþ1 Asnþ1 −2knþ1 Bsnþ1 þ 2keff Bseff ¼ bR3nþ1 keff :
ð33f Þ
Now, the effective thermal conductivity of a particulate composite with n concentric layers can be deduced from s in Eqs. (33a), (33b), (33c), (33d), (33e), and (33f) and setting it to zero. Furthermore, the effective solving for Beff thermal conductivity of a transversely isotropic fibrous composite with multi-coated layer can also be obtained by the similar procedures reported in this section. 6. Numerical example In order to calculate the temperature distribution at and along the interface between the matrix and inclusion, a sample composite having the same properties as [4] is used. The value for the radius of inclusion, a, is 1.25 cm and the temperature gradient, β, is 22.2 °C/cm. The thermal conductivity for the matrix and the inclusion are km = 0.72 W/mK, ke = 28.8 W/mK, respectively. Fig. 2 compares the temperature differentials of the interface at θ = π/2 in the particulate composite as a function of volume fractions for various interfacial contact conductance. It is shown that the results agree with those of Hasselman and Jonson [3]. In addition, for ve not equal to zero, the thermal interaction among inclusions becomes more significant as the volume fraction of the inclusions increases. The variation trend of the temperature differentials depends on the value of the interfacial thermal contact conductance. In Fig. 2, C = ∞ represents a perfect contact between the matrix and the inclusion. For small C, the temperature differential decreases as the volume fraction increases. On the contrary, the temperature differential increases as the volume fraction decreases for large interfacial contact conductance. For a wide range of the volume fraction, the temperature differential decreases as the interfacial contact conductance increases. It is noted that not only the effective thermal conductivity with interfacial contact conductance of many composites presented in this paper can be calculated but also the temperature distribution in the neighborhood of the interface can be obtained. The temperature solution can be further applied to compute the thermal stress field near the inclusion. Hence, thermal mechanism among inclusions can be examined by the generalized self-consistent method. 7. Final remarks The generalized self-consistent method has been successfully applied to calculate the effective thermal conductivities of particulate composite, transversely isotropic fiber composite, and multi-layered composite. The
Fig. 2. The temperature differentials among the matrix and the inclusions as a function of volume fractions for different values of interfacial thermal conductance.
Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
149
effect of thermal contact conductance and volume fractions of the inclusion on the effective thermal conductivity of composites is also presented in this study. A rigorous and detailed derivation of the coefficient Beff = 0 is new in the related literatures, and it can be used to determine the effective thermal conductivity, keff, of the above-mentioned composites. It is shown that the effective thermal conductivities, keff, of particulate composite and transversely isotropic fiber composite are exactly the same as those obtained in Hasselman and Jonson [3]. In addition, the effective thermal conductivity of a multi-layered particulate composite is also obtained. The GSCM can be easily extended to compute the effective thermal conductivities of a multi-layered fibrous composite. Also, it is believed that this method can be further applied to compute keff of a fibrous composite with isotropic matrix and cylindrically orthotropic fiber. Numerical example shows that contact conductance existing at the interface and the volume fraction of the inclusion have significant impact on the effective thermal conductivity. Furthermore, the temperature distribution of the matrix and the inclusion can also be obtained by GSCM. This is the by-product of the method, which many other methods do not provide. Therefore, the solutions can be used to investigate thermal interaction or thermal mechanism among the inclusions. Nomenclature A Coefficient in the temperature solution for the GSCM a Radius of the inclusion B Coefficient in the temperature solution for the GSCM b Radius of the matrix C Thermal contact conductance t Intensity vector H k Thermal conductivity Y Unit normal vector n t q Heat flux vector R Radius of the multi-layer body S Surface of a heterogeneous body T Temperature U Energy integral V Volume of a heterogeneous body ν Volume fraction β Temperature gradient in the far field Subscripts e Inclusion eff Effective medium i Index m Matrix Superscripts c Cylindrical model s Spherical model References [1] Y. Xu, K. Yagi, Automatic FEM model generation for evaluating thermal conductivity of composite with random materials arrangement, Comput. Mater. Sci. 30 (2004) 242–252. [2] Y.M. Lee, A. Haji-Sheikh, L.S. Fletcher, G.P. Peterson, Effective thermal conductivity in multidimensional bodies, J. Heat Transfer 116 (1994) 17–27. [3] D.P.H. Hasselman, L.F. Jonson, Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater. 21 (1987) 508–515. [4] R. Osiroff, D.P.H. Hasselman, Effect of interfacial thermal barrier on the thermal stresses near spherical inclusion in matrix subjected to linear heat flow, J. Compos. Mater. 25 (1991) 1588–1598.
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[5] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability materials, J. Appl. Phys. 33 (1962) 1514–1517. [6] R.M. Christensen, K.H. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids 27 (1979) 315–330. [7] R.M. Christensen, Two theoretical elasticity micromechanics, J. Elast. 50 (1998) 15–25. [8] Z. Hashin, Assessment of the self consistent scheme approximation: conductivity of particulate composites, J. Compos. Mater. 2 (3) (1968) 284–300. [9] J.D. Felske, Effective thermal conductivity of composite spheres in a continuous medium with contact resistance, Int. Heat Mass Transfer 47 (2004) 3453–3461. [10] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959, pp. 425–428. [11] J.V. Beck, K.D. Cole, A. Haji-Sheikh, B. Litkouhi, Heat Conduction Using Green's Function, Hemisphere Publishing Corporation, London, 1992, pp. 346–351.
A generalized self-consistent method for calculation of effective thermal conductivity of composites with interfacial contact conductance☆ Yung-Ming Lee ⁎, Ruey-Bin Yang, Song-Sheun Gau Department of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan 40724, R.O.C. Available online 28 November 2005
Abstract The effective thermal conductivity of composites can be obtained by the generalized self-consistent method (GSCM). The effect of contact conductance that exists at the interface between the matrix and the inclusion, on the effective thermal conductivity is investigated. The particulate composite, transversely isotropic fiber composite, and multi-layered composite are considered in this study. Based on the energy balance, a simple criterion Beff = 0, yet new and analytical, for determining the effective thermal conductivity of the composite is rigorously derived. The temperature distribution of the matrix and the inclusion can also be obtained by GSCM. This is the by-product of the method, which many other methods do not provide. © 2005 Elsevier Ltd. All rights reserved. Keywords: Effective thermal conductivity; Contact conductance; GSCM; Composite
1. Introduction Composite materials are being used increasingly in a variety of modern engineering applications and this trend is likely to continue due to the fact that these materials possess a number of highly desirable engineering properties that can be exploited to design structures with high demand on their performance. Thermal conductivity is an important property of composites' application for electronic packaging, thermal insulation, heat spreader, etc. [1]. As a consequence, determination of the effective thermal properties of composites is essential for a successful design and manufacturing of such composite materials. Theoretical studies have shown that the effective thermal conductivity of composites is strongly dependent on the volume fraction, the distribution, inclusion size, the thermal conductivities of constituents and the interface conductance between inclusion and matrix [2–4]. Many theoretical studies have used micro-mechanics models such as dilute, self-consistent method, composite spheres (cylinders) model, differential method, and Mori-Tanaka method to predict the effective properties of the composite material [5,6].
☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-M. Lee).
0735-1933/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.10.004
Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
143
Christensen [7] evaluated those theoretical models for the calculation of the overall static elastic constants and concluded that only the Generalized Self-Consistent Model (GSCM) gives physical reasonable results at high concentrations. The generalized self-consistent method seems to be a more appropriate tool for homogenization process. It takes into account the complex interaction between matrix and inclusions by considering a representative unit cell inclusion, i.e., an inclusion and a surrounding ring matrix, which is itself embedded in the infinite effective medium. The Generalized Self-Consistent Model provides a physically realistic model of inclusion to inclusion interaction for two-phase system covering the full range of the volume fraction (0 ≦ ve ≦ 1). The idea of the Generalized Self-Consistent Model (originally termed as self-consistent scheme) was developed by Hashin [8] to determine the effective conductivity of the two-phase materials. The self-consistent field concept is extended to determine the composite reinforced with coated spheres [9]. The effective thermal conductivity is evaluated as the ratio of a volume average of the heat flux to temperature gradient in the model [8,9]. Although the final results agreed with that of Hasselman and Jonson [3], the derivation of the volume integral is not simple and the method may be difficult to treat multiple-coated cylinders or spheres. 2. General analysis Consider a large two-phase body of volume V and surface S. It is assumed that the material is isotropic. The intensity and heat flux are defined by t
H ¼ −jT Y
t
q ¼ kH
ð1Þ ð2Þ
In Eq. (1), T is the temperature solution. For a two-phase body, k in Eq. (2) represents the thermal conductivity of either one of the phases. The homogeneous boundary conditions of the first and second kinds on surface S are T ðSÞ ¼ −Hi0 xi
ð3aÞ
qn ðSÞ ¼ q0i ni
ð3bÞ
where Hi0 and qi0 are arbitrarily constant temperature and heat flux components, respectively, xi are Cartesian coordinates, the subscripts i are ranging from 1 to 3, and ni are the components of the outer normal vector on surface S. According to the tensor notation, a repeated subscript on i represents summation. The effective thermal conductivity of a two-phase body can be obtained with the aid of energy integrals [8]. The integral can be defined by Z 1 qi ð xÞHi ð xÞdV ð4Þ U¼ V V It was shown [8] that for boundary condition (3a) U ¼ keff Hi0 Hi0
ð5aÞ
and for boundary condition (3b) U¼
1 0 0 q q keff i i
ð5bÞ
In Eqs. (5a) and (5b), keff stands for the effective thermal conductivity of the two-phase composite. 3. Generalized self consistent method (GSCM) for spherical inclusions To account for the strong interactions between the matrix and inclusion at nondilute case, the Generalized SelfConsistent Model is shown in Fig. 1 where a spherical coordinate system (r,θ,ϕ) is used. The spherical inclusion of radius a is embedded in a concentric sphere matrix material of radius b, which is embedded in an infinite effective
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Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
Y
r
X b
a Inclusion
Matrix
Fig. 1. The generalized self-consistent model.
medium possessing the unknown effective thermal conductivity. The ratio of the radii, a/b, is related the volume fraction of inclusions by ve = (a/b)3. The steady-state conduction equation of a particulate composite can be written as j2 Te ¼ 0
0VrVa
ð6Þ
j2 Tm ¼ 0
aVrVb
ð7Þ
j2 Teff ¼ 0
bVrVl
ð8Þ
The continuity of heat flux and the temperature jump condition should be satisfied between the inclusion and the matrix according to the conservation law of energy. However, the temperature solution is continuous between the matrix and the effective medium. These boundary conditions at the interfaces can be written as km ATm at r ¼ a ð9Þ Te ¼ Tm − C Ar r¼a km
ATm Ar
¼ ke r¼a
Teff ¼ Tm km
ATm Ar
ATe Ar
at r ¼ a
at r ¼ b
s ¼ keff r¼b
ð10Þ
r¼a
ATeff Ar
ð11Þ at r ¼ b
ð12Þ
r¼b
where the subscripts e, m and eff stand for the inclusion, the matrix and the effective medium, respectively, and C denotes thermal contact conductance, the reciprocal of thermal resistance, 1/Rt, between the matrix and the inclusion. For the imperfect contact case, C is finite; otherwise, C is infinite in case of perfect contact at the interface. The
Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
145
superscript s in this section represents the spherical inclusion model. The boundary condition far away from the inclusion is subjected to Eq. (2a) and is of the form Teff ¼ −bx3 ¼ −brcosh
ð13Þ
as rYl
The temperature solutions are taken in the forms [10] Te ¼ Ase rcosh
Tm ¼
ð14Þ
0VrVa
Bs Asm r þ 2m cosh r
aVrVb
ð15Þ
Bs cosh br þ eff r2
bVrVl
ð16Þ
Teff ¼
Substitution of Eqs. (14), (15), and (16) into Eqs. (9), (10), (11), and (12) becomes km s km s a3 Ase −a3 1− Am − 1 þ 2 B ¼0 Ca Ca m
ð17aÞ
a3 ke Ase −a3 km Asm þ 2km Bsm ¼ 0
ð17bÞ
b3 Asm þ Bsm −Bseff ¼ bb3
ð17cÞ
s s Bseff ¼ bb3 keff b3 km Asm −2km Bsm þ 2keff
ð17dÞ
s s s The unknown coefficients, Aes, Am , Bm , and Beff , can be determined from Eq. (17a) to (17d) if the effective thermal s s conductivity keff is known. The criteria for determining keff require the equality of the thermal energy in heterogeneous medium and in the equivalent homogeneous medium. The idea of mathematical derivation in the following comes from Beck et al. [11]. According to Eq. (9), the temperature solution experiences a step change at the contact interface and the derivatives of the temperature across the contact interface are singulars. Hence, substitution of Eq. (1) into the integral given in Eq. (4) is modified as
1 U¼ V
Z
Z Vm −Ve
kjT d jT dV þ
Ve þVe Ve −Ve
Z kjT d jT dV þ
Ve −Ve
kjT d jT dV
ð18Þ
where V is the summation of the volume of the inclusion Ve and the volume of the matrix Vm; while the contact zone is a very thin layer including the total volume of 2Vε. The following mathematical identity can be further used to simplify the integration. That is jd ½T ðkjT Þ ¼ Tkj2 T þ kjT d jT
ð19Þ
It is assumed that the composite material is homogeneous and isotropic inside either the matrix or the inclusion. Therefore, the thermal conductivity k can be treated as a constant. The first term on the right-hand side vanishes because the steady-state energy equation is nothing but the Laplace equation. Hence, Eq. (19) becomes kjT d jT ¼ jd ½T ðkjT Þ
ð20Þ
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Y.-M. Lee et al. / International Communications in Heat and Mass Transfer 33 (2006) 142–150
After substituting Eq. (20) into Eq. (18), and then applying the divergence theorem to the resulting equation yields 2 Z Z 16 t t Tm km jTm d n dS þ Tm km jTm d ð−n ÞdS U¼ 4 V Smo Smi 0 B þ@
Z
Tm km jTm d t n dS þ Seo
1
Z Sei
C t Te ke jTe d ð−n ÞdS A þ
3
Z Sei
7 Te ke jTe d t n dS 5
ð21Þ
It is noted that the inner surface of the matrix equals to the outer surface of the contact zone and, similarly, the inner surface of the contact zone is the same as the outer surface of the inclusion. The unit normal vector t n is pointed outward. The directional derivative of the surface in the direction of n can be expressed as AT ¼ jT d t n An
ð22Þ
Substituting Eq. (22) to Eq. (21) becomes 2 ! ! Z Z Z Z 16 ATm ATm ATm ATe dS þ U¼ 4 dS þ dS þ dS Tm km T m km − Tm km T e ke − An An An An V Smo Smi Seo Sei 3
Z þ
Te ke Sei
ATe 7 dS 5 An
ð23Þ
Assuming the thickness of the contact zone is extremely small, one can easily find that the second term and the third term on the right side of Eq. (23) are cancelled out with each other. Also, the fourth term and the fifth term are cancelled out with each other, Eq. (23) is then of the form 1 U¼ V
Z Tm km Smo
ATm dS An
ð24Þ
Substitution of Eqs. (12) and (15) into Eq. (24) yields ks U ¼ eff b
2
2bBs bBs 2Bs bb2 − 3eff þ 2eff − 5eff b b b
! ð25Þ
From Eq. (5a), one has U¼
s keff H 0H 0
¼
s keff b2
ks ¼ eff b
2
2bBs bBs 2Bs bb − 3eff þ 2eff − 5eff b b b 2
! ð26Þ
After simple algebraic calculation, one can find Bseff ¼ 0:
ð27Þ
To the best knowledge of the authors, the rigorous and detailed derivation of the coefficient Bseff ¼ 0 is new in the s of composites. related literatures, and it can be used to determine the effective thermal conductivity keff
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Now, the effective thermal conductivity of the particulate composite can be deduced from solving for Beff in Eqs. (17a), (17b), (17c), and (17d) and setting it to zero according to Eq. (27). This will give the following result 2 3 km ð3ve Þke 3ve − 6 7 Ca ke −km s 7 keff ¼ km 6 ð28Þ 41 þ 3km km ð2 þ ve Þke 5 vm þ þ ke −km Ca ke −km Eq. (28) has the same form obtained in [3]. For the perfect contact case, Eq. (28) will reduce to the form obtained by Hashin [8]. 4. GSCM for transversely isotropic fiber composites The GSCM can be equally applied to compute the effective thermal conductivity of a fiber-reinforced composite. The derivation procedure is straightforward and is not presented in this section. The final result can be easily obtained by the GSCM and is given by 2 3 km ve ke ve − 6 7 Ca ke −km c 7 ¼ km 6 keff ð29Þ 4 1 þ vm 1 km ð1 þ ve Þke 5 þ þ 2km ke −km Ca 2km ðke −km Þ Eq. (29) has the same form obtained in Hasselman and Jonson [3]. 5. GSCM with multi-layer inclusions In a similar fashion, the method presented in this paper can be easily extended to find the effective thermal conductivity of either particulate composite or fibrous composite with multi-coated cylinders or spheres. The purpose of the coating may be to protect fibers or spheres from the formation of brittle reaction layers at their interfaces or to improve the heat conduction performance of the composite by applying a high thermal conductivity coating layers to the thermally poor inclusion. In this section, the effective thermal conductivity of a particulate composite with n concentric layers will be reported. The governing equations of this n-layer inclusion model are of the forms j2 T1 ¼ 0
0VrVR1
j2 Ti ¼ 0
Ri−1 VrVRi
j2 Teff ¼ 0
ð30Þ i ¼ 2;3; N ;n þ 1
ð31Þ
Rnþ1 Vrbl
ð32Þ
Again, the continuity of heat flux and the temperature jump condition should be satisfied between two adjacent layers according to the conservation law of energy. However, the temperature solution is continuous between the matrix and the effective medium. This composite material is subjected to a linear temperature gradient, β, far away from the inclusion. The temperature solution for composites reinforced with multi-coated spheres can be obtained from Carslaw and Jaeger [10]. Substitution of the temperature solutions into the boundary conditions becomes k2 k2 As2 − 1 þ 2 Bs ¼ 0 ð33aÞ R31 As1 −R31 1− C1 R1 C1 R1 2 R31 k1 As1 −R31 k2 As2 þ 2k2 Bs2 ¼ 0 kiþ1 s kiþ1 s s s s s A − 1þ2 B ¼ 0; Ri Ai þ Bi −Ri 1− Ci Ri iþ1 Ci Ri iþ1 R3i ki Asi −2ki Bsi −R3i kiþ1 Asiþ1 þ 2kiþ1 Bsiþ1 ¼ 0;
ð33bÞ i ¼ 2;3; N ;n
i ¼ 2;3; N ;n
ð33cÞ ð33dÞ
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R3nþ1 Asnþ1 þ Bsnþ1 −Bseff ¼ bR3nþ1
ð33eÞ
s s R3nþ1 knþ1 Asnþ1 −2knþ1 Bsnþ1 þ 2keff Bseff ¼ bR3nþ1 keff :
ð33f Þ
Now, the effective thermal conductivity of a particulate composite with n concentric layers can be deduced from s in Eqs. (33a), (33b), (33c), (33d), (33e), and (33f) and setting it to zero. Furthermore, the effective solving for Beff thermal conductivity of a transversely isotropic fibrous composite with multi-coated layer can also be obtained by the similar procedures reported in this section. 6. Numerical example In order to calculate the temperature distribution at and along the interface between the matrix and inclusion, a sample composite having the same properties as [4] is used. The value for the radius of inclusion, a, is 1.25 cm and the temperature gradient, β, is 22.2 °C/cm. The thermal conductivity for the matrix and the inclusion are km = 0.72 W/mK, ke = 28.8 W/mK, respectively. Fig. 2 compares the temperature differentials of the interface at θ = π/2 in the particulate composite as a function of volume fractions for various interfacial contact conductance. It is shown that the results agree with those of Hasselman and Jonson [3]. In addition, for ve not equal to zero, the thermal interaction among inclusions becomes more significant as the volume fraction of the inclusions increases. The variation trend of the temperature differentials depends on the value of the interfacial thermal contact conductance. In Fig. 2, C = ∞ represents a perfect contact between the matrix and the inclusion. For small C, the temperature differential decreases as the volume fraction increases. On the contrary, the temperature differential increases as the volume fraction decreases for large interfacial contact conductance. For a wide range of the volume fraction, the temperature differential decreases as the interfacial contact conductance increases. It is noted that not only the effective thermal conductivity with interfacial contact conductance of many composites presented in this paper can be calculated but also the temperature distribution in the neighborhood of the interface can be obtained. The temperature solution can be further applied to compute the thermal stress field near the inclusion. Hence, thermal mechanism among inclusions can be examined by the generalized self-consistent method. 7. Final remarks The generalized self-consistent method has been successfully applied to calculate the effective thermal conductivities of particulate composite, transversely isotropic fiber composite, and multi-layered composite. The
Fig. 2. The temperature differentials among the matrix and the inclusions as a function of volume fractions for different values of interfacial thermal conductance.
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effect of thermal contact conductance and volume fractions of the inclusion on the effective thermal conductivity of composites is also presented in this study. A rigorous and detailed derivation of the coefficient Beff = 0 is new in the related literatures, and it can be used to determine the effective thermal conductivity, keff, of the above-mentioned composites. It is shown that the effective thermal conductivities, keff, of particulate composite and transversely isotropic fiber composite are exactly the same as those obtained in Hasselman and Jonson [3]. In addition, the effective thermal conductivity of a multi-layered particulate composite is also obtained. The GSCM can be easily extended to compute the effective thermal conductivities of a multi-layered fibrous composite. Also, it is believed that this method can be further applied to compute keff of a fibrous composite with isotropic matrix and cylindrically orthotropic fiber. Numerical example shows that contact conductance existing at the interface and the volume fraction of the inclusion have significant impact on the effective thermal conductivity. Furthermore, the temperature distribution of the matrix and the inclusion can also be obtained by GSCM. This is the by-product of the method, which many other methods do not provide. Therefore, the solutions can be used to investigate thermal interaction or thermal mechanism among the inclusions. Nomenclature A Coefficient in the temperature solution for the GSCM a Radius of the inclusion B Coefficient in the temperature solution for the GSCM b Radius of the matrix C Thermal contact conductance t Intensity vector H k Thermal conductivity Y Unit normal vector n t q Heat flux vector R Radius of the multi-layer body S Surface of a heterogeneous body T Temperature U Energy integral V Volume of a heterogeneous body ν Volume fraction β Temperature gradient in the far field Subscripts e Inclusion eff Effective medium i Index m Matrix Superscripts c Cylindrical model s Spherical model References [1] Y. Xu, K. Yagi, Automatic FEM model generation for evaluating thermal conductivity of composite with random materials arrangement, Comput. Mater. Sci. 30 (2004) 242–252. [2] Y.M. Lee, A. Haji-Sheikh, L.S. Fletcher, G.P. Peterson, Effective thermal conductivity in multidimensional bodies, J. Heat Transfer 116 (1994) 17–27. [3] D.P.H. Hasselman, L.F. Jonson, Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater. 21 (1987) 508–515. [4] R. Osiroff, D.P.H. Hasselman, Effect of interfacial thermal barrier on the thermal stresses near spherical inclusion in matrix subjected to linear heat flow, J. Compos. Mater. 25 (1991) 1588–1598.
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[5] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability materials, J. Appl. Phys. 33 (1962) 1514–1517. [6] R.M. Christensen, K.H. Lo, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids 27 (1979) 315–330. [7] R.M. Christensen, Two theoretical elasticity micromechanics, J. Elast. 50 (1998) 15–25. [8] Z. Hashin, Assessment of the self consistent scheme approximation: conductivity of particulate composites, J. Compos. Mater. 2 (3) (1968) 284–300. [9] J.D. Felske, Effective thermal conductivity of composite spheres in a continuous medium with contact resistance, Int. Heat Mass Transfer 47 (2004) 3453–3461. [10] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959, pp. 425–428. [11] J.V. Beck, K.D. Cole, A. Haji-Sheikh, B. Litkouhi, Heat Conduction Using Green's Function, Hemisphere Publishing Corporation, London, 1992, pp. 346–351.