Scattering of thermal waves and non-steady effective thermal conductivity of composites with coated particles

Applied Thermal Engineering 29 (2009) 925–931

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Scattering of thermal waves and non-steady effective thermal conductivity of composites with coated particles Xue-Qian Fang * Department of Engineering Mechanics, Shijiazhuang Railway Institute, Shijiazhuang 050043, PR China

a r t i c l e

i n f o

Article history: Received 18 January 2008 Accepted 27 April 2008 Available online 3 May 2008 Keywords: Particle-reinforced composite materials Scattering of thermal waves Coated particles Non-steady effective thermal conductivity

a b s t r a c t In this study, thermal wave method is proposed to predict the non-steady effective thermal conductivity of composites with coated particles, and the analytical solution of this problem is obtained. The Fourier heat conduction law is introduced to analyze the propagation of thermal waves in the particular composite. The scattering and refraction of thermal waves by a coated particle in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material zones are expanded by using the spherical wave functions and Legendre polynomial, and the expanded mode coefficients are determined by satisfying the boundary conditions of the coating layer. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of the material properties of the particles, coating and matrix on the effective thermal conductivity of composites under different wave frequencies are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the effective thermal conductivity under lower frequencies. In the region of lower frequency, the effect of the properties of the coating on the effective thermal conductivity is greater. Comparisons with the steady effective thermal conductivity obtained from other methods are also presented. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The subject of the effective thermal conductivity of composites is one of the classical problems in heterogeneous media. Recently, it has received considerable attention due to the increasing importance of high temperature systems, e.g., car manufacturing, dedicated space structures, etc. These composites usually undergo a complex thermal history. The design of composite materials for such applications requires a thorough understanding of heat conduction in them. The foundation of this understanding lies in the development of micromechanics models for accurately predicting the effective thermal conductivity of multiphase composites [1]. The methods used to measure the effective thermal conductivity are divided into two groups: the steady state method and the non-steady state method. In the first one, the sample is subjected to a constant heat flow. In the second group, a periodic or transient heat flow is established in the sample [2]. In the past, much attention has been focused on the problems of steady state. The earliest models predicting the thermal behavior of composites assumed that the two components were both homogeneous, and were perfectly bonded across a sharp and distinct interface. The Maxwell solution [3] was the starting point of finding the * Tel.: +86 451 86410268. E-mail address: [email protected] 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.04.020

effective conductivity of two-phase material systems, but it was valid only for very low concentration of the dispersed phase. Subsequently, many structural models, e.g., Parallel, Maxwell-Eucken [4], and Effective Medium Theory models [5], were proposed. Recently, Samantray et al. [6] applied the unit-cell approach to study the effective thermal conductivity of two-phase materials. The idea of the Generalized Self-Consistent Model was also developed by Hashin [7] to determine the effective thermal conductivity of the two-phase materials. Recently, the coating inclusions have been introduced in the design of composites to enhance the thermal properties. In the modeling, the coating was also introduced for other reasons: first, during the manufacturing process, a chemical reaction between the inclusion and matrix can create a third phase: the coating. Second, due to a mismatch between the two phases, the perfect interface assumption is not valid. Thus, the coating contributes to the character of the non-perfect interface. The dramatic effects of interfacial characteristics on the thermal conductivities and thermal diffusivities in particle [8] and fiber [9] reinforced composites have been experimentally demonstrated by Hasselman and his coworkers. Based on an equivalent inclusion concept, Hasselman and Johnson extended Maxwell’s theory to the systems of spherical inclusions with contact resistance [10]. Benveniste and his coworkers have proposed several analytical models to predict the effective thermal conductivity of composite materials which

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Nomenclature k D c

q kp Dp cp

qp kc Dc cc

qc a0 h Cp Cm T T0

x k # #sr #0

thermal conductivity of the matrix thermal diffusivity of the matrix specific heat capacity of the matrix density of the matrix thermal conductivity of the particle thermal diffusivity of the particle specific heat capacity of the particle density of the particles thermal conductivity of the coating thermal diffusivity of the particle specific heat capacity of the coating density of the coating radius of the particle thickness of the coating boundary between the particle and the coating boundary between the coating and the matrix temperature in composite materials average temperature incident frequency of thermal waves. incident wave number wave field far-field scattered wave the temperature amplitude of incident thermal waves

include the important effects of a thermal contact resistance between the fillers and matrix [11], and the coated cylindrically orthotropic fibers with a prescribed orientation distribution [12]. Lu and Song [13,14] investigated the coated or debonded inclusion, and developed a more general model to predict the effective thermal conductivity of composites. Due to the complexity of non-steady loading, there are few calculations on the effective thermal conductivity of composites under modulated conditions. Recently, Monde and Mitsutake [15] proposed a method to determine the thermal conductivity of solids by using an analytical inverse solution for unsteady heat conduction. By using modulated photothermal techniques, Salazar et al. [2] studied the effective thermal conductivity of composites made of a matrix filled with aligned circular cylinders of a different material. Recently, Fang and Hu investigated the distribution of dynamic effective thermal properties along the gradation direction of functionally graded materials by using Fourier heat conduction law [16] and non-Fourier heat conduction law [17]. Nevertheless, no attention has been paid to the non-steady effective thermal conductivity of composites with coated inclusions. With the wide application of materials in aerospace, automotive industries, and other high temperature situations, the study on the non-steady effective thermal conductivity of composites with coated inclusions plays very important role in the designing and manufacture of these materials. Thermal wave exists in high temperature situations, and is often applied with Fourier conduction law. Fourier’ law underlies ‘‘parabolic thermal wave” associated with a non-linear dependence of thermal conductivity on the temperature, and ‘‘thermal wave method” can measure the thermal properties of composites including the non-steady effective thermal conductivity of composites. Most recently, Fang et al. [18] have employed this method to predict the non-steady effective thermal conductivity of composites with coated fibers. The main objective of this paper is to extend the work of Fang et al. [18] to the three-dimensional case. The scattering of thermal waves from the coated particles and the effects of coating on the non-steady effective thermal conductivity of composites are investigated. The composite medium contains a random distribution of

spherical Bessel function of the first kind jn() ð1Þ spherical Hankel function of the first kind hn ðÞ ð2Þ Hankel function of the second kind hn Legendre polynomial Pn() An, Bn, En, Fn mode coefficients heat flow density in the radial direction qr far-field scattering amplitudes for the scattered thermal f(j, h) waves non-steady effective thermal conductivity keff qeff effective mass density of composites. effective heat capacity of composites. ceff K propagating wave number in the effective medium N number of the particles per unit volume volume fraction of particles Vp Superscripts (i) incident waves (s) scattered waves r refracted waves, m matrix p particle c coating

particular inclusions of same size with coating of the same thickness. Fourier heat conduction law is applied to analyze the heat transfer in composites. The temperature fields in different regions of the material are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions at the coating layer. The theory of Waterman and Truell [19] is applied to obtain the nonsteady effective thermal conductivity of composites. The non-steady effective thermal conductivity under different parameters is graphically illustrated and discussed. 2. Heat conduction equation in composites and its solution Consider a composite material containing randomly distributed coated particles embedded in an infinite matrix [20]. The particles of radius a0 have identical properties. Let k, c, q be the thermal conductivity, specific heat capacity and mass density of the matrix, and kp, cp, qp those of the particles. It is assumed that the thickness of the coating is h with material properties kc, cc, qc. In order to study the multiple scattering of thermal waves in composite materials with coated particles, we first consider the scattered field resulting from a single particle with coating layer. The inner radius of the particles is a0, and the outer radius is a1. The geometry is depicted in Fig. 1, where (x, y, z) is the Cartesian coordinate system with its origin being at the center of the particle, and (r, h, /) is the corresponding spherical coordinate system. Let the boundary between the particle and the coating be denoted by Cp, and that between the coating and the matrix by Cm. Based on the Fourier heat conduction law, the heat conduction equation in the composite material, in the absence of heat sources, is described as

r2 Tðr; tÞ ¼

1 oT ; D ot

ð1Þ

where r2 = o2/ox2 + o2/o y2 + o2/oz2 represents the threedimensional Laplace operator, T is the temperature in composite materials, and D is the thermal diffusivity with

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The refracted waves inside the particle are standing waves, and can be expressed as

#r ¼

1 X

Bn jn ðjp rÞP n ðcos hÞ;

ð9Þ

n¼0

where the superscript r stands for the refracted waves, and Bn are the mode coefficients of refracted waves. The temperature in the coating #c may be described by the sum of the two components (outgoing and ingoing waves) and is expressed in the following form [21]

#c ¼

1 X

ð1Þ

En hn ðjc rÞP n ðcos hÞ þ

n¼0

1 X

ð2Þ

F n hn ðjc rÞPn ðcos hÞ;

ð10Þ

n¼0 ð2Þ

Fig. 1. Coated spherical particle and the propagation of thermal waves in composites.

D ¼ k=ðqcÞ;

ð2Þ

where hn are the nth Hankel functions of the second kind, and denote the ingoing waves, and En and Fn are the mode coefficients of the waves in the coating. The wave numbers jc in the coating and jp in the spherical particle are given by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jc ¼ ð1 þ iÞ x=ð2Dc Þ;

ð11Þ

in which k, c and q are the thermal conductivity, specific heat at constant pressure and density of the matrix, respectively. The solution of periodic steady state is investigated. Suppose that

jp

ð12Þ

T ¼ T 0 þ Re½# expðixtÞ;

3. Boundary conditions and solution of the coefficients

ð3Þ

where T0 is the average temperature, and x is the incident frequency of thermal waves. Substituting Eq. (3) into Eq. (1), the following equation can be obtained:

r2 # þ j2 # ¼ 0;

ð4Þ

where j is the wave number of complex variables, and

j ¼ ð1 þ iÞk;

ð5Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi with k ¼ x=2D being the incident wave number. By using wave function expansion method, the incident thermal waves are expressed as

#ðiÞ ¼ #0 eiðjxxtÞ

1 X n ¼ #0 ð2n þ 1Þi jn ðjrÞPn ðcos hÞeixt ;

ð6Þ

The boundary conditions on Cm and Cp are given by c

# ¼ #m ; r

c

# ¼# ;

qcr ¼ qm r qrr

¼

qcr

for r ¼ a1 ;

ð13Þ

for r ¼ a0 ;

ð14Þ

where qr is the heat flow density in the radial direction, and . qr ¼ k o# or The continuous boundary condition of temperature on Cm gives

X1

ð1Þ

ð2Þ

½En hn ðjc a1 ÞPn ðcos hÞ þ F n hn ðjc a1 ÞPn ðcos hÞ 1 1 X X ð1Þ n i ð2n þ 1Þjn ðja1 ÞPn ðcos hÞ þ An hn ðja1 ÞPn ðcos hÞ; ¼ #0 n¼0

n¼0

ð15Þ

n¼0

By making use of the orthogonality relation of Legendre polynomials, one can obtain, for each n P 0, the following equation: ð1Þ

where the superscript (i) stands for the incident waves, #0 is the temperature amplitude of incident thermal waves, jn() are the spherical Bessel functions of the first kind, and Pn() is the Legendre polynomial. It should be noted that all wave fields have the same time variation eixt, which is omitted in all subsequent representations for notational convenience. When the thermal waves propagate in the particular composite material, the waves are scattered by the particles, and the scattered thermal waves are expanded in a series of outgoing spherical Hankel functions. The scattered field in the matrix is expressed in the following form:

#ðsÞ ¼

where Dc = kc/(qccc) and Dp = kp/(qpcp).

ð2Þ

n

En hn ðjc a1 Þ þ F n hn ðjc a1 Þ ¼ #0 i ð2n þ 1Þjn ðja1 Þ

n¼0

1 X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 þ iÞ x=ð2Dp Þ;

ð1Þ

An hn ðjrÞP n ðcos hÞ;

ð7Þ

n¼0

ð1Þ

þ An hn ðja1 Þ;

ð16Þ

The continuous boundary condition of temperature on Cp gives ð1Þ

ð2Þ

Bn jn ðjp a0 Þ ¼ En hn ðjc a0 Þ þ F n hn ðjc a0 Þ:

ð17Þ

According to the continuous boundary conditions of heat flux density on Cm and Cp, one can obtain

  o ð1Þ o ð2Þ kc En h ðjc a1 Þ þ F n h ðjc a1 Þ oa1 n oa1 s   o ð1Þ o n ¼ k An hn ðjan Þ þ #0 i ð2n þ 1Þ jn ðja1 Þ ; oa1 oa1     o o ð1Þ o ð2Þ kp Bn jn ðjp a0 Þ ¼ kc En hn ðjc a0 Þ þ F n hn ðjc a0 Þ ; oa0 oa0 oa0

ð18Þ ð19Þ

ð1Þ

where the superscript (s) stands for the scattered waves, hn ðÞ are the nth spherical Hankel functions of the first kind, and An are the mode coefficients that account for the distortion of the scattered spherical waves by the particles. The total temperature in the matrix should be produced by the superposition of the incident field and the scattered field, i.e.,

#m ¼ #ðiÞ þ #ðsÞ :

ð8Þ

After some manipulations, Eqs. (16)–(19) can be arranged as

½PfXg ¼ fQ g;

ð20Þ

where X = An, Bn, En, Fn, P is a coefficient matrix of 4  4, and f is a vector of 4 ranks, whose elements are shown in Appendix. After solving the linear equation system (20), the mode coefficients An, Bn, En, Fn(n = 0, 1, 2, . . .) can be obtained.

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X.-Q. Fang / Applied Thermal Engineering 29 (2009) 925–931

4. Effective propagating wave number of thermal waves We now consider a composite material with N particles randomly distributed in the matrix. The positions of these particles are denoted by the random variables (r1, r2, . . ., rN). The total temperature field at any point outside all particles can be expressed, in the multiple scattering form, as

#ðr; r 1 ; r 2 ; . . . ; r N Þ ¼ #i ðrÞ þ

N X

T s #i ðr m Þ

N X

N X

T s ðrm Þ

m¼1

T s ðrk Þ#i ðrm Þ þ    ;

ð21Þ keff ¼

k¼1;k–m

where the single summation denotes the primary scattered terms, and the double summation denotes the secondary scattered terms, and so on. The primary scattering is due to the incident waves alone, and the second scattering represents the rescattering of the primary scattered waves, etc. The multiple scattering theory of thermal waves takes into account the interaction among the distributed particles accurately. However, it is difficulty to deal with it, and the effective properties of composites can not be easily obtained. Here, we apply the effective field approximation to describe approximately the interaction among the distributed particles. Following the work of Waterman and Truell [19], the effective propagating wave number can be obtained from the scattered far-field. Once the scattered field due to a single particle is known, the phase velocities and attenuations of the coherent waves in the composite can be easily calculated by the double plane wave theory of Waterman and Truell [19]. The scattered fields for the incident thermal waves at a large distance from the particle can be obtained from Eq. (7) by letting r tend to 1. After applying the ð1Þ asymptotic expression of the radial function hn ðjrÞ, i.e.

  1 1 1 ð1Þ ; hn ðjrÞ  ei½jr2ðnþ1Þp þ o jr r

ð22Þ

the far-field scattered wave can be expressed asymptotically as

    1 X 1 1 f ðj; hÞ ijr 1 i ¼ e þo ; #sr  eijr iAn e2ðnþ1Þp Pn ðcos hÞ þ o r r r r n¼0 ð23Þ where

f ðj; hÞ ¼

1 X ðiÞn An P n ðcos hÞ:

ð24Þ

n¼0

The function f(j, h) is the far-field scattering amplitude of the scattered thermal waves. It is noted that the far-field scattered amplitudes are dependent on the angle h. The far-field scattered amplitudes at two specific angles, h = 0 and h = p, are of special interest, and are called the forward and backward scattering amplitudes, respectively. According to the theory of Waterman and Truell [19], in the case of three-dimensional scatterers, the effective propagating wave number is expressed as

 2  2  2 K 2pN 2pN ¼ 1 þ 2 f ðj; 0Þ  f ð j ; p Þ ; 2 k k k

ð25Þ

where K is the propagating wave number in the effective medium, and N is the number of the particles per unit volume with

N ¼ 3V p =ð4pa30 Þ;

5. Non-steady effective properties of the particle-reinforced composites According to Eq. (5), the non-steady effective thermal conductivity keff can be easily obtained from the effective propagating wave number as follows:

k¼1

þ

of a single scatterer. According to the theory of Waterman and Truell [19], the correlations between the particles are neglected. Thus, the validity of Eq. (25) is limited to the low volume concentration of particles.

ð26Þ

in which Vp is the volume fraction of the randomly distributed spherical particles in the matrix. It is noted that f(j, 0) is the forward scattering amplitude of a single scatterer, and f(k, p) is the backward scattering amplitude

qeff ceff k ½Reðk=KÞ2 ; qc

ð27Þ

where Re(z) denotes the real part of z, and qeff and ceff are the effective mass density and effective heat capacity of composites. From Ref.[2], it is known that they always follows the mixture rule, and qeffceff is given by

(

eff eff

q c

 2 )   h hV p h : þ qp c p V p þ ¼ qc 1  V p 1 þ qc c c 2 þ a0 a0 a0 ð28Þ

When the thermal properties of the particles and the coating approach to those of the matrix, the materials become homogeneous. In this case, the scattering of thermal waves do not exit, and the propagating wave number has no variation. From Eqs. (3) and (27), it is clear that the temperature field and effective thermal conductivity reduce to those determined for the pure matrix. 6. Numerical examples and discussion In the following analysis, it is convenient to make the variables dimensionless. To accomplish this step, a representative length scale a0, where a0 is the inner radius of particles, is introduced. The following dimensionless variables and quantities have been  chosen for computation: the incident wave number k ¼ ka0 ¼   0:1  2:0; h ¼ h=a0 ¼ 0:05  0:20; kp ¼ kp =k ¼ 2:0  8:0; cp ¼ cp =c ¼ 1:0  4:0, and qp ¼ qp =q ¼ 1:0  4:0; kc ¼ kc =k ¼ 0:5  4:0; cc ¼ cc =c ¼ 1:0  4:0, and qc ¼ qc =q ¼ 1:0  4:0, The dimensionless non-steady effective thermal conductivity is k* = keff/k. Through computation, it is found that it is numerically sufficient to truncate n at 10 for any desired incident frequency. The non-steady effective thermal conductivity of composites as a function of volume fraction of particles with parameters:  k ¼ 1:0; kp ¼ 4:0; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5 is presented in Fig. 2. It can be seen that the non-steady effective thermal conductivity increases with the increase of the coating thickness. Because the thermal conductivity of the particles is greater than that of the matrix, the non-steady effective thermal conductivity increases with the volume fraction of particles. The effect of the thickness of coating on the effective thermal conductivity also increases with the volume fraction of particles. Fig. 3 illustrates the non-steady effective thermal conductivity of composites as a function of volume fraction of particles with   parameters: k ¼ 1:0; h ¼ 0:1; kp ¼ 4:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5. It can be seen that the non-steady effective thermal conductivity increases with the increase of the values of cp and qp . With the increase of the volume fraction of particles, the effects of the values of cp and qp on the effective thermal conductivity also increase. Fig. 4 shows the non-steady effective thermal conductivity of composites as a function of volume fraction of particles with   parameters: k ¼ 1:0; h ¼ 0:1; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5. As expected, in the region of low frequency the non-stea-

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X.-Q. Fang / Applied Thermal Engineering 29 (2009) 925–931 2.6

1.9

3

2 h* = 0.10 1.7

1 λ*p = 2.0

2.4

Effective thermal conductivity λ*

Effective thermal conductivity λ*

1.8

1 h* = 0.05

2

3 h* = 0.20

1.6 1.5

1

1.4 1.3 1.2 1.1

2 λ*p = 4.0

2.2

3 λ*p = 8.0

2.0 3

1.8 1.6

2

1.4 1.2 1

1.0 0.8

1 0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.6

0.2

0

0.2

0.4

Fig. 2. Non-steady effective thermal conductivity as a function of volume fraction  of fibers ðk ¼ 1:0; kp ¼ 4:0; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5Þ.

1.0

1.2

1.4

1.6

1.8

2.0

3.0

1 c*p = ρ *p = 1.5

1 c*p = ρ *p = 1.5

2 c*p = ρ *p = 2.0 3

c*p =

ρ *p

Effective thermal conductivity λ*

Effective thermal conductivity λ*

0.8

Fig. 4. Non-steady effective thermal conductivity as a function of dimensionless  wave number ðh ¼ 0:1; V p ¼ 0:1; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5Þ.

2.5

= 3.0

2.0

3

2 1.5 1

1 0

0.6

Dimensionless wave number k*

Volume fraction Vp

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Volume fraction Vp

2 c*p = ρ *p = 2.0

2.5

3 c*p = ρ *p = 3.0

3

2.0

1.5

2

1

1.0

0.5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Dimensionless wave number k*

Fig. 3. Non-steady effective thermal conductivity as a function of volume fraction   of fibers ðk ¼ 1:0; h ¼ 0:1; kp ¼ 4:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5Þ.

Fig. 5. Non-steady effective thermal conductivity as a function of dimensionless  wave number ðh ¼ 0:1; V p ¼ 0:1; kp ¼ 4:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5Þ.

dy effective thermal conductivity increases with the increase of the value of h*. However, in the region of higher frequency, the nonsteady effective thermal conductivity decreases with the increase of the value of h*. So, it is very important to take the thermal loading into consideration when predicting the effective thermal conductivity of composites. Fig. 5 illustrates the non-steady effective thermal conductivity of composites as a function of the incident wave number with  parameters: h ¼ 0:1; V p ¼ 0:1; kp ¼ 4:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5. It can be seen that in the region of lower frequency, the variations the specific heat and density of the two phases nearly express no effect on the effective thermal conductivity, which is well known in the steady case. However, in the case of non-steady thermal loading, the effects of the specific heat and density of the two phases on the effective thermal conductivity of composites are quite different. With the increase of the incident wave number, the variations of the specific heat and density of the two phases begin to show great effect on the non-steady effective thermal conductivity of composites. The non-steady effective thermal conductivity increases with the increase of the values of cp and

qp , and the effects of the specific heat and density of the two phases on the non-steady effective thermal conductivity increase with the increase of dimensionless wave number. Fig. 6 illustrates the non-steady effective thermal conductivity of composites as a function of dimensionless wave number with parameters: V p ¼ 0:1; kp ¼ 4:0; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5. It can be seen that in the region of low frequency, the effective thermal conductivity increases with the increase of the value of h*. However, in the region of high frequency, it is clear that the non-steady effective thermal conductivity decreases with the increase of the value of h*. The greater the dimensionless wave number, the greater the effect of the value of h* on the non-steady effective thermal conductivity. Fig. 7 shows the non-steady effective thermal conductivity of composites as a function of dimensionless wave number with parameters:  V p ¼ 0:1; h ¼ 0:1; kp ¼ 4:0; cp ¼ qp ¼ 2:0; cc ¼ qc ¼ 1:5. It can be seen that in the region of low frequency, the non-steady effective thermal conductivity increases with the increase of the value of

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X.-Q. Fang / Applied Thermal Engineering 29 (2009) 925–931 1.6

1.7 EMT model

3 2

Steady effective thermal conductivity

Effective thermal conductivity λ*

1.5 1 1.4 1.3 1.2 1.1

1 h* = 0.05

1.0

2 h* = 0.10

0.9

Present model

1.6

Hasselman and Johnson

1.5

1.4

1.3

1.2

1.1

3 h* = 0.20 0.8

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1

0

0.02

Dimensionless wave number k*

Effective thermal conductivity λ*

2 3

= 0.5 = 2.5 = 5.0

2.0 3

2

1

0

0.2

0.10

0.12

0.14

0.16

0.18

0.2

Fig. 8. Comparison of the steady effective thermal conductivity with EMT model   and Hasselman and Johnson (Ref. [10]) ðkp ¼ 4:0; cp ¼ qp ¼ 2:0; h ¼ 0; k ¼ 0Þ.

7. Conclusions

1.5

1

0.08

mental data from Ref. [2]. In Ref.[2], the matrix is a polymer with bad conductor (k = 0.3 Wm1 K1), and the fibers have good thermal conductivity (kf = 15 Wm1 K1). Measurements were performed by a photopyroelectric technique. Through comparison, good agreement is found. In the case of this paper, no experimental data is available. So, only the comparison with the steady effective thermal conductivity is given.

2.5

1

0.06

Volume fraction Vp

Fig. 6. Non-steady effective thermal conductivity as a function of dimensionless wave number ðV p ¼ 0:1; kp ¼ 4:0; cp ¼ qp ¼ 2:0; kc ¼ 2:5; cc ¼ qc ¼ 1:5Þ.

λ*c λ*c λ*c

0.04

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Dimensionless wave number k* Fig. 7. Non-steady effective thermal conductivity as a function of dimensionless  wave number ðV p ¼ 0:1; h ¼ 0:1; kp ¼ 4:0; cp ¼ qp ¼ 2:0; cc ¼ qc ¼ 1:5Þ.

kc . However, in the region of high frequency, the non-steady effective thermal conductivity nearly expresses no variation with the value of kc . Finally, to demonstrate the validity of this dynamical thermal model, the steady effective thermal conductivity of two-phase composites without interface is given. As ka ? 0, the non-steady effective thermal conductivity tends to the steady solutions. In Fig. 8, the results obtained from the present model, Effective Medium Theory [4], and Hasselman and Johnson [10] are plotted. Close agreement is seen to exist between the models at low volume fractions; however, the present model predicts a lower value of effective thermal conductivity than the Effective Medium Theory when the volume fraction of particles is high. This is consistent with regards to criticism of the conventional Effective Medium Theory for overestimating the effective thermal conductivity of two-phase composites when kp > k. This is attributed to the assumption that the particles are regarded as the effective medium even at close range. It is noted that in the two-dimensional case (composites with coated fibers) [18], we can compare our results with the experi-

The scattering of thermal waves in composites with coated particles is investigated theoretically by employing wave functions expansion method. The analytical solution of the non-steady effective thermal conductivity of the composite is presented. The theory of Waterman and Truell is applied to obtain the non-steady effective thermal conductivity of composites. Comparison with the steady effective thermal conductivity demonstrates the validity of the dynamical thermal model. It has been found that the non-steady effective thermal conductivity of composites is dependent on the incident wave number, the material properties of the particles, coating and matrix, and the thickness of the coating. In different region of frequency, the effects of the properties of the particle and coating show great difference. In the region of low frequencies, the non-steady effective thermal conductivity of composites increases with an increase of the thickness of the coating, the thermal conductivity ratio of the coating and matrix, and the thermal conductivity ratio of the particles and matrix. The variations of the specific heat and density of the two phases nearly expresses no effect on the effective thermal conductivity. However, when the dimensionless wave number is greater than a certain number, the non-steady effective thermal conductivity of composites begins to decrease with the increases of the thickness of the coating and the thermal conductivity ratio of the particles and matrix. The thermal conductivity ratio of the coating and matrix nearly expresses no effect on the effective thermal conductivity. The effects of the specific heat and density of the two phases on the non-steady effective thermal conductivity increase greatly with the increase of dimensionless wave number. Therefore, to gain a higher effective thermal conductivity of composites, when the frequency of thermal loading is low, the greater thickness and thermal conductivity of coating and the greater thermal conductivity ratio of the particles and matrix should be chosen. However, in the region of higher frequencies

X.-Q. Fang / Applied Thermal Engineering 29 (2009) 925–931

(k* > 1.0), the smaller thickness of coating and the thermal conductivity of particles are preferable. The results of this paper can provide guidelines for the design of particle-reinforced composites in the presence of coating and would be helpful in understanding the thermal behavior of composites. Appendix The expressions of P and Q are given by ð1Þ

Pð1; 1Þ ¼ hn ðja1 Þ;

ð29Þ

Pð1; 2Þ ¼ 0;

ð30Þ

ð1Þ

ð31Þ

ð2Þ

ð32Þ

Pð1; 3Þ ¼ hn ðjc a1 Þ; Pð1; 4Þ ¼ hn ðjc a1 Þ; Pð2; 1Þ ¼ 0;

ð33Þ

Pð2; 2Þ ¼ jn ðjp a0 Þ;

ð34Þ

ð1Þ

Pð2; 3Þ ¼ hn ðjc a0 Þ;

ð35Þ

ð2Þ

Pð2; 4Þ ¼ hn ðjc a0 Þ; Pð3; 1Þ ¼ k½ðn 

ð36Þ

ð1Þ 1Þhn ð

ð1Þ a1 hnþ1 ð

ja1 Þ  j

ja1 Þ;

Pð3; 2Þ ¼ 0;

ð37Þ ð38Þ

ð1Þ

ð1Þ

ð39Þ

ð2Þ

ð2Þ

ð40Þ

Pð3; 3Þ ¼ kc ½ðn  1Þhn ðjc a1 Þ  jc a1 hnþ1 ðjc a1 Þ; Pð3; 4Þ ¼ kc ½ðn  1Þhn ðjc a1 Þ  jc a1 hnþ1 ðjc a1 Þ; Pð4; 1Þ ¼ 0;

ð41Þ

Pð4; 2Þ ¼ kp ½ðn  1Þjn ðjp a0 Þ  jp a0 jnþ1 ðjp a0 Þ;

ð42Þ

ð1Þ

ð1Þ

Pð4; 3Þ ¼ kc ½ðn  1Þhn ðjc a0 Þ  jc a0 hnþ1 ðjc a0 Þ;

ð43Þ

ð2Þ j  jc ap hnþ1 ðjc ap Þ; n Q ð1Þ ¼ #0 i ð2n þ 1Þjn ðja1 Þ;

ð44Þ

Q ð2Þ ¼ 0;

ð46Þ

Pð4; 4Þ ¼ kc ½ðn 

n

ð2Þ 1Þhn ð c ap Þ

ð45Þ

Q ð3Þ ¼ #0 i ð2n þ 1Þ½ðn  1Þjn ðja1 Þ  ja1 jnþ1 ðja1 Þ;

ð47Þ

Q ð4Þ ¼ 0;

ð48Þ

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