THE STATISTICAL SECOND-ORDER TWO-SCALE METHOD FOR HEAT TRANSFER PERFORMANCES OF RANDOM POROUS MATERIALS WITH INTERIOR SURFACE RADIATION

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 4, Number 2, Pages c 2013 Institute for Scientific Computing and Information THE STATISTICAL SECOND-ORDER TWO-SCALE METHOD FOR HEAT TRANSFER PERFORMANCES OF RANDOM POROUS MATERIALS WITH INTERIOR SURFACE RADIATION ZHIQIANG YANG, JUNZHI CUI, AND YIQIANG LI Abstract. In this paper, a statistical second-order two-scale SSOTS method is presented in a constructive way for predicting heat transfer performances of random porous materials with interior surface radiation. Firstly, the probability distribution model of porous materials with random distribution of a great number of cavities is described. Secondly, the SSOTS formulations for predicting effective heat conduction parameters and the temperature field are given. Then, a statistical prediction algorithm for maximum heat flux density is brought forward. Finally, some numerical results for porous materials with different random distribution models are calculated, and compared with the data by theoretical methods. The results demonstrate that the SSOTS method is valid to predict the heat transfer performances of random porous materials. Key words. Statistical second-order two-scale method, Interior surface radiation, Random porous materials, Maximum heat flux density. 1. Introduction Porous materials have been widely used in a variety of engineering and industrial products. Especially, with rapid development of space aircraft, people pay much more attention to porous materials. Inevitably, our attention is focused on the thermal properties of porous materials. So far, some methods to predict physical and mechanical properties of composite materials have been developed, such as the Maxwell-Eucken model [1], the Hashin-Shtrikman bounds [2], effective medium theory [3, 4], the self-consistent method [5] and so on. Although these methods effectively promoted the development of computational material science, the microstructure of real materials was greatly simplified to reduce the theoretical complexity. Furthermore, they are usually used to predict macroscopic heat conductivity parameters without considering the effect of radiation. In fact, radiative heat transfer plays a significant rule in modern technology. Especially, it is typically the major mode of heat transfer in high-porosity insulations at high temperature environment. In recent years, some worthwhile contributions in predicting thermal radiation properties of periodical porous materials have been achieved. Liu et al.[6] predicted the effective macroscopic properties of heat conduction-radiation problem by homogenization methods. Bakhvalov [7] obtained the formal expansions for heat conduction problem with radiation boundary conditions. Later, Allaire et al.[8] dealt with a linear heat equation with non-linear boundary conditions by two-scale asymptotic expansions method. Meanwhile, Yang et al.[9] presented a second-order two-scale method to solve the heat transfer performances of periodic porous materials with interior surface radiation, and gave the error estimation for the original solution and the asymptotic solution. In theory, [10-13] proved the existence and uniqueness of the heat conduction equation with non-linear radiation boundary conditions. Received by the editors December 20, Mathematics Subject Classification. 35R35, 49J40, 60G

2 152 Z. YANG, J. CUI, AND Y. LI Nevertheless, the heat transfer problem of random porous materials with interior surface radiation is not considered so far. Actually, composite materials with random distribution have been widely used in engineering. Such as metal-matrix composites, foamed plastics and polymer blends. For the composites with random distributions, Cui et al.established a statistical second-order two-scale analysis method by introducing a random sample model to predict the physical and mechanical properties of the composite structure [14-17]. Meanwhile, for the physics field problems of composite materials with stationary random distribution, Jikov et al. [18] proved the existences of the homogenization coefficients and the homogenization solution. However, the previous two-scale asymptotic expansion cannot be employed to the heat transfer problem with interior surface radiation. So, in this paper porous materials with random distribution will be investigated, and a new SSOTS method is developed by a constructive way to predict heat transfer properties, and calculate temperature and heat flux fields in meso-scopic level. This paper is organized as follows. In the following section, the meso-scopic configurations for porous materials with random distribution are represented. Section 3 is devoted to the formulation of the SSTOS method and the algorithm procedure for the maximum heat flux density. In Section 4 the numerical results for the heat transfer performances of random porous materials are shown. Finally, the conclusions are given. 2. Representation of meso-scopic configurations of porous materials with random distribution Suppose that the investigated porous materials are made from matrix and random cavities. Refer to Ref. [14, 20]. All the cavities are considered as ellipsoids or the polyhedrons inscribed inside the ellipsoids, which are randomly distributed in the matrix. In this paper all of the ellipsoid cavities are also considered as same scale, which means all of their long axes satisfy r 1 < a < r 2 where r 1 and r 2 are given upper and lower bounds. Then the porous materials with random distribution can be represented as follows: 1 There exists a constant ε satisfying 0 < ε << L, where L denotes the macro scale of the investigated structure Ω ε. Thus, the structure can be regarded as a set of cells with the ε-size, as shown in Fig.1a. 2 In each cell, the probability distribution of the cavities is identical. Then the investigated structure has periodically random distribution of cavities, and then can be represented by a probability distribution model of the cavities inside a typical cell. 3 Each ellipsoid can be defined by 9 random parameters, including the shape, size, orientation and spatial distribution of ellipsoid cavities: a 1, a 2, a 3, α 1, α 2, α 3, x 01, x 02, x 03, where a 1, a 2 and a 3 denote length of three axes; three Euler angles α 1, α 2, α 3 of the rotations; x 01, x 02 and x 03 the coordinates of the center. Let the random vector ζ = a 1, a 2, a 3, α 1, α 2, α 3, x 01, x 02, x 03. Their probability density functions are denoted by f a1 x, f a2 x, f a3 x,f α1 x, f α2 x, f α3 x,f x01 x, f x02 x, f x03 x, respectively. 4 Suppose that there are K ellipsoids inside a cell εy s, Y s represents a normalized cell, then its random sample is defined as ω s. s=1, 2, 3... denotes the index of samples, then we can define a sample of ellipsoids distribution as follows ω s = ζ s 1, ζ s 2, ζ s 3, ζ s K 1, ζ s K

3 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM 153 Therefore, the investigated structure Ω ε is logically composed of ε-size cells subjected to identical probability distribution model P. In this paper, the following material parameters for the porous materials with random distribution are considered { k ε ij x, ω s } K = kij 1 if x εy s where e i denotes the ith ellipsoid inside εy s, and kij 1 are the constants and satisfy that max { k 1 ij < M, M > 0. Thus, the k ε ij x, ω s } is a bounded and measurable tensor of random variables ω s. 3. Statistical second-order two-scale method 3.1. Statistical second-order two-scale formulation. In this section, a new SSOTS formulation is derived for calculating thermal properties of the heat transfer problem with interior surface radiation. Let Y ={y : 0 y j 1, j = 1...3} and ϖ be an unbounded domain of R 3 which satisfies the following conditions: B1 ϖ is a smooth unbounded domain of R 3 with a 1-periodic structure. B2 The cell of periodicity Y = ϖ Y is a domain with a Lipschitz boundary. The boundary of Y is Y = Y Γ, where Γ is the surfaces of the cavities, and Y is the reference cell, shown in Fig.1 b. B3 The set Y \ ϖ and the intersection of Y \ ϖ with the δ 0 neighborhood of Y consist of a finite number of Lipschitz domains separated from each other and from the edges of the cube Y by a positive distance. B4 The cavities of the domain Ω ε are convex. Then, the domain Ω ε as shown in Fig.1 a has the form: Ω ε = Ω εϖ, where Ω is a bounded Lipschitz convex domain of R 3 without cavities, and the cavities do not intersect with Ω. Since the gas is assumed to be transparent neither heat conduction nor absorption of radiation, this heat radiative process is modeled by an integral equation on the surfaces of cavities. Moreover, we suppose that the radiative surfaces are diffuse and grey, that is, the emissivity e of the surfaces does not depend on the wavelength of the radiation, and surface emits, absorb and reflect radiation in the same manner in all directions. For the structure Ω ε, the mixed boundary value problem of the heat transfer behavior with interior surface radiation can be expressed as follows k x ijx, ε ω T εx, ω = fx x Ω ε i x j T ε x, ω = T x x Γ 1 1 ν i kijx, ε ω T εx, ω = qx x Γ 2 x j i=1 ν i k ε ijx, ω T εx, ω x j = G ε σt 4 ε x, ω x Γ c ε where T ε x, ω denotes the temperature, k ε ij x, ω is the coefficients of the heat conductivity, and fx the internal heat source. σ is the Stefan-Boltzmann constant. And ω = {ω s, x εy s Ω ε }. Γ 1 and Γ 2 denote boundary portions where temperature and heat flux are prescribed, respectively, such that Γ 1 Γ 2 = Ω, Γ 1 Γ 2 = ; Γ c ε is the interior boundary that is composed of the surfaces of every e i

4 154 Z. YANG, J. CUI, AND Y. LI a b Figure 1. Porous materials with random distribution of cavities cavity Γcε,i, Γcε = m S i=1 Γcε,i, m is the numbers of cavities in Ωε. Gε is the operator defined as follows see [7, 8] 2 Gε σtε4 x, ω = eσtε4 x, ω e Z Rε x, ωf x, zdz Γcε,i Rε is the intensity of emitted radiation, and has the following relationship Z 4 3 Rε x, ω = eσtε x, ω + 1 e Rε z, ωf x, zdz Γcε,i For any fixed given sample ω, from [12, 13], we know that 3 has the unique solution Rε x, ω. F x, z is the view factor between two different points x and z on Γcε,i, and is defined on 3-D for a convex cavity is see [8, 12, 13] F x, z = nz x znx z x π z x 4 where nz denotes the unit normal at the point z, and for any x, z Γcε,i 2 for a closed surface, it satisfies the following properties Z F x, z 0, F x, z = F z, x, F x, zdz = 1 Γcε,i Then, for any fixed given sample ω P, [12, 13] proved the existence and uniqueness of Eq.1. In order to avoid the arguments on the mathematical properties of investigated functions below, we suppose that {k ω} is a bounded random variable, then there exists an expectation value. It is well known that the temperature increments of porous materials with random distribution do not only depend on its global behaviors, but also on random meso-scopic configurations, it can be expressed as Tε x, ω = T x, y, ω, where y = xε, x denotes the macroscopic coordinate and y is the local one.

5 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM 155 In order to obtain the two-scale expression of the temperature field, it is assumed that T ε x, ω can be expanded into the series of the following form: 4 T ε x, ω = T 0 x + εn α1 y, ω s T 0 + ε 2 N α1α x 2 y, ω s 2 T 0 α1 x α2 + C α1 y, ω s T0 3 T 0 + ε 3 P 1 x, y, ω s, x εy s Ω ε, y Y s where T 0 x only reflects the macroscopic behaviors of the structure, and is called as the homogenization solution. N α1 y, ω s, N α1α 2 y, ω s and C α1 y, ω s are called local solutions. They will be determined below. In addition, it is assumed that R ε x, ω have the following expansion 5 R ε x, ω = R 0 x + εh α1 y, ω s T 3 0 x T 0 + ε 2 P 2 x, y, ω s x εy s Ω ε, y Y s where R 0 x only depends on the macroscopic behavior of the structure, H α1 y, ω s is local solution, dependent on y, ω s. let l = z ε, where z is macroscopic coordinate of the structure, and l the local coordinate of 1-normalized cell. 4 and 5 are substituted into 3, which leads to the following identity 6 R 0 x + εh α1 y, ω s T0 3 x T 0 = eσt0 4 x + 1 e R 0 xf y, ldl Γ c +ε4eσt0 3 xt 1 x, y + 1 e H α1 l, ω s T0 3 x T 0 F y, ldl + Oε 2 Γ x c α1 Then, from 6, one obtains that 7 R 0 x = σt 4 0 x 8 H α1 y, ω s = 4eσN α1 y, ω s + 1 e H α1 l, ω s F y, ldl Γ c Similar to 3, we can prove that 8 has one unique solution H α1 y, ω s. Taking into account 9 x x + 1 ε y

6 156 Z. YANG, J. CUI, AND Y. LI Substituting 4 and 5 into 1 yields the equalities k x ijx, ε ω s T εx j x j 10 = ε 1 k ij y, ω s ε 0 k ij y, ω s 2 T 0 T0 + N α 1 y, ω s T 0 x j + k x i x ij y, ω s N α 1 y, ω s 2 T 0 j x i + kij y, ω s N α1 y, ω s 2 T 0 x j + k j y, ω s N α 1α 2 y, ω s 2 T 0 i x α2 + k j y, ω s C α 1 y, ω s T 3 T Oε i = f 11 G ε σtε 4 = eσt0 4 x e R 0 xf y, ldl Γ c +ε4eσt0 3 xn α1 y, ω s T 0 et0 3 xh α1 l, ω s T 0 F y, ldl Γ x c α1 +Oε 2 First, considering the coefficients of ε 1 one obtains that k iα1 y, ω s + k ij y, ω s N α 1 y, ω s T0 = 0 in Y s 12 ν i k ij y, ω s N α 1 y, ω s + k iα1 y, ω s T0 = eσt0 4 x e R 0 xf y, ldl y Γ Γ c where Γ is the surfaces of the cavities contained in Y s, Y s is the solid part of Y s. Taking the property F y, ldl = 1, it is easy to verify that Γ c 13 eσt0 4 x e R 0 xf y, ldl = 0 Γ c Since T0 to zero, by virtue of 13, 12 can be rewritten as 14 arises from the macroscopic behavior of the structure, it is not identical k ij y, ω s N α 1 y, ω s ν i k ij y, ω s N α 1 y, ω s + k iα1 y, ω s To attach the following boundary condition on Y s = k iα 1 y, ω s in Y s = 0 y Γ N α1 y, ω s = 0 on Y s

7 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM 157 So for any sample ω s, N α1 y, ω s is the solution of the following elliptic partial differential equation 15 k ij y, ω s N α 1 y, ω s = k iα 1 y, ω s ν i k ij y, ω s N α 1 y, ω s + k iα1 y, ω s N α1 y, ω s = 0 in Y s = 0 y Γ y Y s From Lax-Milgram theorem and Poincare s inequality, Eq.15 has one unique solution for any specified sample. Refer to [14, 15, 17], for any sample ω s the homogeneous parameters are defined as 16 ˆkij ω s = 1 k ip y, ω s N jy, ω s + k ij y, ω s dy Y Y y p Theorem 1 ˆk ij ω s M 1, M 1 is positive constants independent of y and ω s. Proof. See [14] Therefore, from supposition there exist one unique expectation value Eˆk ij x, ω s. Then, from theorem 1, ˆk ij ω s is bounded random function. Further the expectation value of ˆk ij ω s exists uniquely, by Kolmogorov s classical strong law of large number we can evaluate the expected homogenization coefficient, which is equivalent to the heat transfer coefficient ˆk ij, as follows: M 17 ˆkij = lim ˆk s=1 ij ω s M M From [19] we know that ˆk ij is symmetrical and positive definite. Then we can define the homogenized equation associated with Eq.17 in the following 18 T 0 ˆk ij = Y x i x j Y f in Ω T 0 = T x on Γ 1 ν iˆkij T 0 x j = qx on Γ 2 where T 0 x is called the expected homogenization solution on Ω. By Lax-Milgram theorem, Poincare s inequality, the homogenization problem 18 has a unique solution. Next, compare the coefficient of ε 0 on both sides of Eq.10. It can be noted that the right of the equation is independent of ε, so the coefficient of ε 0 on the left

8 158 Z. YANG, J. CUI, AND Y. LI side of the equation is equal to the right side, that is 19 k ij y, ω s 2 T 0 k ij y, ω s N α 1 y, ω s 2 T 0 x i x j x i k ij y, ω s N α1 y, ω s 2 T 0 x j k ij y, ω s N α 1α 2 y, ω s 2 T 0 x α2 k ij y, ω s C α 1 y, ω s T 3 T 0 0 = f Besides, from Eq.11, we have 20 ν i k ij y, ω s N α 1α 2 y, ω s ν i k ij y, ω s C α 1 y, ω s T 3 0 x T 0 = 4eσT0 3 xn α1 y, ω s T 0 e + k iα2 y, ω s N α1 y, ω s T 0 x α2 Γ c H α1 l, ω s T 3 0 x T 0 F y, ldl Making use of the symmetry of k ij y, ω s and the homogenized equation 18, Eq 19 can be written as: 21 k α1α 2 y, ω s Y Y ˆk α1α 2 + k iα2 y, ω s N α1 y, ω s +k α1α 2 y, ω s + k α2jy, ω s N α 1α 2 y, ω s + k ij y, ω s N α 1α 2 y, ω s 2 T 0 x α2 + k ij y, ω s C α 1 y, ω s T 3 T 0 0 = 0 Since 2 T 0 x α2 and T0 3 T0 are not identical to zero, by the constructing way analogous to determining N α1 y, ω s, for a given sample ω s, in each Y s one can define as follows: N α1α 2 y, ω s is the solution of the following problem: 22 k ij y, ω s Nα 1 α 2 y,ωs = Y ˆk Y α1α 2 y kiα2 i y, ω s N α1 y, ω s k α1α 2 y, ω s k y, α2j ωs Nα 1 y,ωs in Y s ν i k ij y, ω s Nα 1 α 2 y,ωs + k iα2 y, ω s N α1 y, ω s = 0 y Γ N α1α 2 y, ω s = 0 y Y s

9 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM 159 C α1 y, ω s is the solution of the following problem: k ij y, ω s C α 1 y, ω s = 0 in Y s ν i k ij y, ω s C α 1 y, ω s 23 = 4eσN α1 y, ω s e H α1 l, ω s F y, ldl y Γ Γ c C α1 y, ω s = 0 y Y s By Lax-Milgram theorem and Poincare s inequality, N α1α 2 y, ω s and C α1 y, ω s are determined uniquely. Summing up, one obtains the following theorem. Theorem 2 Temperature field for heat transfer problem with interior surface radiation 1 for porous materials with random distribution formally has a SSOTS asymptotic expansion as follows 24 T ε x, ω s = T 0 x + εn α1 y, ω s T 0 +ε N 2 α1α 2 y, ω s 2 T 0 + C α1 y, ω s T0 3 x α2 +ε 3 P 1 x, y, ω s x εy s Ω ε, y Y s T 0 where T 0 x is the solution of the homogenized Eq.18 with the parameters 17. N α1 y, ω s, N α1α 2 y, ω s and C α1 y, ω s are the local functions satisfying 15, 22 and 23, respectively. In practical computation of engineering only the sum for three terms in 24 is evaluated. 25 T ε x, ω = T 0 x + εn α1 y, ω s T 0 +ε N 2 α1α 2 y, ω s 2 T 0 + C α1 y, ω s T0 3 x α2 T 0 Furthermore, from 25 the expansion of the temperature gradients and heat flux density are given in the following. 26 T ε x, ω = T 0x + N α 1 y, ω s T 0 x x i x i +εn α1 y, ω s 2 T 0 x + ε N α 1α 2 y, ω s 2 T 0 x x i x α2 +ε 2 N α1α 2 y, ω s 3 T 0 x + ε C α 1 y, ω s T 3 T 0 x 0 x α2 x i +ε 2 C α1 y, ω s T0 3 2 T 0 x + 3T 2 T 0 x T 0 x 0 x i x i 27 q i x, ω s = k ij x, ω s T ε x, ω x j

10 160 Z. YANG, J. CUI, AND Y. LI 3.2. Algorithm procedure. The algorithm procedure of statistical second-order two-scale method for predicting thermal properties of the heat transfer problem with interior surface radiation is stated as follows 1. Generate a sample ω s for a unit cell Y s according to the probability distribution models P and the volume fraction V. Further, partition Y s into FE meshes. 2. Solve N α1 y, ω s according to the problem 15 by FE method. Furthermore, the homogenization coefficients ˆk ij ω s is evaluated by using the formula Repeat the steps 1-2 for M samples ω s s = 1,, M, where M is the number of samples. Then the expected thermal conduction ˆk ij is given by the formula The homogenization solution T 0 x is obtained by solving problem 18 in Ω. 5. According to obtained homogenization solution to determine the sub-domain, in which maximum heat flux density may occur. 6. Select the cell c s in sub-domain and generate a sample ω s. With the same meshes to 2, we evaluate N α1α 2 y, ω s and C a1 y, ω s corresponding to a sample ω s by solving the cell problems 22 and From 26 and 27, the temperatures and heat flux densities corresponding to the sample ω s are evaluated. 8. Suppose that the value of the heat flux density at x c s is q x, ω s. One obtains the following maximum heat flux density for a cell c s corresponding to the sample ω s 28 q extr ω s = max x c s qx, ωs 9. Repeating the steps 7-8 for M samples ω s s = 1,, M, one obtains M the maximum heat flux densities q extr ω s. By analogizing the expected homogenization parameters, the expected maximum heat flux density is given M q extr ω s s=1 29 q extr = M 4. Thermal properties of randomly distributed porous materials In order to investigate thermal properties of random porous materials with interior surface radiation, we consider three different types of microscopic distribution, and their effective computer generation algorithm has been developed by authors [20] based on the probability distribution model of cavities: Spherical cavities subject to uniformly stochastic distribution in a ε-cell; Spherical cavities subject to normal distribution around the centric point of ε-cell; Orientations of prolate ellipsoidal cavities, whose long axes is about two times of the middle axes and short axes, subject to normal distribution along x 3 -axis, and subject to uniformly stochastic distribution in ε-cell, shown in Fig.2. Due to cavities random dispersion, the numerical results of the different samples will vary even for the same distribution models of pores. So, to obtain more accurate prediction values a numbers of samples are required. It should be noted that the following computations are averaged for 50 samples Influence of micro-structures on the thermal conductivity parameters and maximum heat flux density. The thermal conductivity of ceramics is 4.41 W/mK in [21]. The internal heat source fx and the heat flux qx on lateral surfaces are taken as zero and the boundary temperatures in the z-direction are set as T 1 x =100K, T2 x =1000K. The emissivities of the cavities surfaces

11 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM a b 161 c Figure 2. a uniform distribution b location-normal distribution c orientation-normal distribution 2 are equal to 0.8. σ = W m K 4 1. The radii are taken as 0.04 for the spherical cavities subjected to uniform distribution and spherical cavities subjected to normal distribution. The orientations of spherical cavities are normal distribution. The size of their long axes is 0.1, their middle axes and short axes are both In order to show the thermal properties of porous materials, the effective thermal conductivity coefficients are obtained, and compared to Hashin-shtrikman bounds. The results are listed in Table 1 for different volume fractions, and all of spherical cavities are uniform random distribution. Table 2 shows the effective thermal conductivity for different volume fractions, and the spherical cavities are normal distribution around the centric point of ε-cell. Table 3 shows that the orientations of spherical cavities are normal distribution along x3 -axis. From the Tables 1, 2 and 3 the evaluated results satisfy the Hashin-shtrikman bounds, and the results show that SSOTS method is valid for predicting effective thermal conductivity of porous materials with random distribution. Table 1. The effective thermal conductivity with different volume fractions and the spherical cavities are uniform distribution Volume SSOTS W/mk HS-upper boundw/mk HS-lower boundw/mk Table 2. The effective thermal conductivity with different volume fractions and the spherical cavities are normal distribution Volume SSOTS W/mk HS-upper boundw/mk HS-lower boundw/mk

12 162 Z. YANG, J. CUI, AND Y. LI Table 3. The effective thermal conductivity with different volume fractions and orientations of spherical cavities are normal distribution Volume x1 x2 x HS-upper bound HS-lower bound Figs.3-5 illustrate the heat flux density for different cavities distribution. a b Figure 3. Heat flux density in a local cells with different volume fraction a 0.06 b a b Figure 4. Heat flux density in a local cells with different volume fraction and cavities subjected to normal distribution a 0.06 b Figs.3-4 show heat flux densities distribution for different random distributions with 0.06 and 0.22 of all cavities. One can see that heat flux density in the two local cells is different for different distribution, and local heat flux density with high volume fraction is relatively high.

13 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM 163 Figure 5. Statistically maximum heat flux densities as the functions of volume fractions for different distributions of cavities locations Fig.5 shows that the statistically maximum heat flux density increase with the increment of volume fractions of cavities, and the q extr of location-normal distribution is larger than the q extr of uniform distribution. So the curves of Fig.5 indicate that the statistically maximum heat flux density of the porous materials is concurrently affected by micro-structure The effects of radiation on maximum heat flux density. In order to consider the effect of radiation on maximum heat flux density of porous materials with different random distributions, we choose different emissivities of cavities surfaces. The coefficient of thermal conductivity and the boundary conditions are the same with that in section 4.1, and we also have the same random distribution as examples in section 4.1. Then some results are obtained in Figs.6-8 for different random distributions. Figure 6. Statistically maximum heat flux density as the functions of volume fractions and cavities subjected to uniformly distribution

14 164 Z. YANG, J. CUI, AND Y. LI Figure 7. Statistically maximum heat flux density as the functions of volume fractions and cavities subjected to normal distribution Figure 8. Statistically maximum heat flux density as the functions of volume fractions and orientation of cavities are normal distribution From Figs.6-8 we know that maximum heat flux density increase as the increment of emissivity in the same distribution. What is more, as the porosity of porous material increases, the maximum heat flux density changes acutely for different random distribution. From the above numerical results, it follows that the heat transfer problem with interior surface radiation for porous materials with random distribution can be effectively predicted by the SSOTS method, and the effect of radiation on thermal properties of random porous materials is significant, especially at a high temperature or high emissivity. Meanwhile, this two-scale model have been investigated in [9] for periodic porous materials, which shows that this method is effective to numerically solve the heat transfer problem with interior surface radiation.

15 THE SSOTS METHOD FOR HEAT TRANSFER PROBLEM Conclusions In this paper, a new statistical second-order two-scale methods for predicting the heat transfer properties of random porous materials is presented, including the second-order two-scale asymptotic expression on the temperature fields, and the formulations of the expected homogenized parameters, maximum heat flux density. The macroscopic thermal properties for the structures with varying probability distribution models, including volume fraction and location distributions of cavities, are shown. The numerical results show that the statistically maximum heat flux density is influenced not only by microscopic structure of random distribution of cavities, but also by the effect of radiation. The higher the emissivity is, the higher the maximum heat flux density is. It is noted that the SSOTS method in this paper can be employed to predict thermal properties of the random porous materials with interior surface radiation. Acknowledgments This work is supported by the Special Funds for National Basic Research Program of China 973 Program 2010CB832702, the National Natural Science Foundation of China , and also supported by the State Key Laboratory of Science and Engineering Computing, and Center for High Performance Computing of Northwestern Polytechnical University. References [1] Z. Hashin and S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys., [2] Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, [3] R. Landauer, The electrical resistance of binary metallic mixtures, J. Appl. Phys., [4] S. Kirkpatrick, Percolation and conduction, Rev. Mod. Phys., [5] S. Torquato, Random heterogeneous materials: Microstructure and macroscopic properties, New York, Springer, [6] S. T. Liu and Y. C. Zhang, Multi-scale analysis method for thermal conductivity of composite material with radiation, Computational Mechanics WCCM VI in conjunction with APCOM 04, sept.5-10, 2004, Beijing, China. [7] N. S. Bakhvalov, Averaging of the heat-transfer process in periodic media with radiative, Differ. Uraun., [8] G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model. Sim., [9] Z. Q. Yang, J. Z. Cui and Q. Ma, The second-order two-scale method for heat transfer performances of periodic porous materials with interior surface radiation, CMES: Comp. Model. Eng., [10] A. A. Amosov, Stationary nonlinear no local problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on radiation frequency, J. Math. Sci., [11] A. A. Amosov, Nonstationary radiative-conductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., [12] T. Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces, Math. Method. Appl. Sci., [13] N. Qatanani, A. Barham and Q. Heeh, Existence and uniqueness of the solution of the coupled-conduction-radiation energy transfer on diffuse-gray surfaces, Sur. Math. Appl., [14] Y. Y. Li and J. Z. Cui, Two-scale analysis method for predicting heat transfer performance of composite materials with random grain distribution, Sci. China Ser. A,

16 166 Z. YANG, J. CUI, AND Y. LI [15] Y. Yu, J. Z. Cui, F. Han and Y. Chen, The two-order and two-scale method for heat conduction properties of composite materials with random distribution of grains, Comput. Exp. Simul. Eng. Sci., [16] Y. Yu, J. Z. Cui and F. Han, The statistical second-order two-scale analysis method for heat conduction performances of the composite structure with inconsistent random distribution, Comp. Mater. Sci., [17] F. Han, J. Z. Cui and Y. Yu, The statistical second-order two-scale method for Mechanical properties of statistically inhomogeneous materials, Int. J. Numer. Meth. Eng., [18] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer: Berlin, [19] O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, [20] Y. Yu, J. Z. Cui and F. Han, An effective computer generation method for the composites with random distribution of large numbers of heterogeneous grains, Compos. Sci. Technol., [21] C. H. Yan, Thermal insulating mechanism and research on thermal insulating efficiency of thermal insulations of metallic thermal protection system, PhD Thesis, Harbin Institute of Technology, Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi an, China yzqyzq1984@126.com LSEC, ICMSEC, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China cjz@lsec.cc.ac.cn Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi an, China liyiqiang@mail.nwpu.edu.cn