Porous materials are of significance in engineering and industry on account of their high temperature resistance, corrosion resistance, fatigue resistance, and even in bio-medical engineering. In particular, as the rapid development of the aerospace industry, porous materials have drawn tremendous attention and wide research interest from scientists and engineers.^[1–7] Inevitably, our attention is focused on predicting the effective macroscopic properties of porous materials.

Convection takes place by flow of fluids and can be ignored at low pressure or in porous materials with a closed cell structure.^[1,2] Radiation is a way of heat transmission and plays a significant role in modern technology, especially as the temperature on a visible surface of the system is high enough. Some interesting work has been proposed to consider the surface radiation of porous materials in the past years. Liu and Zhang^[1] gave the effective macroscopic thermal properties of the coupled conduction and radiation problem with a small parameter ɛ. Bakhvalov^[3] introduced the asymptotic homogenization method for the solutions of those problems. Later, Cui et al.^[4,5] investigated the coupled heat transfer problem with rapidly oscillating coefficients, and gave the theoretical verification. Allaire and Ganaoui^[6] studied the coupled problem with non-linear interior surface radiation of porous materials by a two-scale homogenization method, and obtained the error estimates. Meanwhile, Ma and Cui^[7] discussed a second-order multiscale analysis and computation for the heat transfer problem, and justified the convergence results with an explicit rate ɛ^1/2. In theory, some authors^[8–11] gave detailed proofs of the existence and uniqueness of the heat conduction equation considering non-linear radiation boundary conditions.

Solving the heat conduction problem with a radiation boundary condition which arises in porous materials will be discussed in this study. Under such conditions, the direct finite element simulation in a very fine spatial mesh is necessary for porous materials to acquire the microscale effect, and thus a prohibitive amount of computation time and computer memory exceed the ability of a general computer. In this context, a homogenized method and the associated multiscale method, which are described in sources^[12–16] and other references, are introduced. The basic idea of the homogenized method is to describe the macroscopic behavior of the porous materials by absorbing the local oscillation due to the inhomogeneity, which cannot only save the computational effort but also insure the accuracy. On the basis of the homogenization method, He and Cui et al. established an SSOTS analysis method to predict the physical and mechanical performance of the composite structure through introducing a random sample model.^[17–21] However, the theoretical verification is not enough for the random composites.^[20] Meanwhile, Jikov et al.^[16] proved the existences of the homogenized coefficients and the homogenized solution for the randomly distributed composite. Yu et al.^[19] developed a novel algorithm for generating the random unit cell, which is based on a probability distribution model of the grains (pores). Compared with other micromechanics methods, such as the Mori–Tanaka method,^[22] the self-consistent method,^[23] the generalized self-consistent method,^[24] and the Maxwell–Eucken model,^[25] the homogenized method has a rigorous mathematical background and it contains almost all the microcosmic information of the composites.

The main new results obtained in this paper are an approximate error analysis in pointwise sense for the SSOTS method and to develop an efficient prediction algorithm for solving the nonlinear heat transfer problem.

This rest of this paper is organized as follows. In Section 2, the microscopic configurations for porous materials with random distributions are represented. Section 3 is devoted to the SSOTS formulae for prediction of the thermal properties of the porous materials. In Section 4, numerical results for the heat transfer performance of porous materials are obtained from the proposed two-scale analysis. Finally, Section 5 is devoted to the conclusions.

The investigated porous materials are composed of a matrix and random pores. All the pores are considered as ellipsoids, which are randomly distributed in the matrix. Similarly to Refs. [18] and [19], the microstructure of porous materials with random distribution can be described in detail as follows.

Fig. 1.

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	Fig. 1. Porous materials with random distribution of pores. (a) The whole structure Ωɛ and (b) unit cell Y*s.

In the following, the thermal conductivity parameters of the porous materials can be considered by

In this section, the SSOTS formulae are given for predicting effective properties of the heat transfer problem in detail, including expected homogenized parameters and the homogenized solution.

Let Y = {y : 0 ≤ y_j ≤ 1, j = 1,2,3} and be an unbounded domain of R³ which satisfies the conditions as follows.

Then, the investigated porous domain Ω^ɛ as shown in Fig. 1(a) has the form: Ω^ɛ = Ω∩ɛϖ and , where Ω is a bounded convex domain of R³ without cavities, and the cavities do not intersect. The is the domain made from interior surfaces of , such that , and m(ɛ) is the number of cavities contained in the porous materials.

The nonlinear heat transfer problem is firstly investigated by Bakhvalov,^[3] and later by Liu and Zhang^[1] and Cui et al.,^[4,5,26] in which the radiation in an interior surface of a closed cavity is defined by

Thus, under the aforementioned assumptions, the mixed boundary value problem of a heat conduction problem with radiation boundary condition can be expressed as follows:

In the following, we suppose that (i)

If {k (ω)} is a bounded random variable, and there exists an expectation value.

Remark 1 By suppositions (I)–(IV), (i) and (ii), for any fixed sample ω ∈ P, the boundary value problem (1) has the unique weak solution T_ɛ(x,ω).^[10,11]

It is well known that the temperature increments of porous materials with random distribution not only depend on the macroscopic behaviors, but also on microscopic configurations; hence it can be expressed as T_ɛ(x,ω) = T(x,y,ω), where y = x/ɛ, x denotes the macroscopic coordinate and y is the local one. In the following, we suppose that e = 1 to simplify the exposition. However, it is easy to extend the case 0 < e < 1, see Ref. [5] for details.

Enlightened by the work done by Yang et al.,^[4,5] T_ɛ (x,ω) can be obtained into the series of the following form:

For any fixed sample ω^s (s = 1,2,3,…), N_α1(y,ω^s) (α₁ = 1,2,3) is the solution of the following boundary problem:

Referring to Refs. [5], [18], and [19], for any fixed sample ω^s (s = 1,2,3,…), the homogeneous parameters are defined as

After the expected homogenized coefficients obtained from the unit cell Y*^s, the homogenized equations can be obtained as follows:

By the constructing way analogous to determining N_α1 (y,ω^s), N_{α1 α2}(y,ω^s) (α₁, α₂ = 1,2,3) is the solution of the boundary value problem as follows:

Remark 2 Similarly to the proof in Refs. [4] and [5] under suppositions (I)–(IV) and (ii), the existence and uniqueness of Eqs. (8) and (9) can be easily established.

Now, we can define the two-scale approximate solutions of problem (1) given by

For any fixed sample ω^s, to compare with the original solution (1), we substitute into Eq. (1), and have

In addition, by taking into Eq. (1) and considering the homogenized equation (7), the following results can be obtained:

Summing up, one obtains the following theorem.

Theorem 1 Ω^ɛ ⊂ R³ is a bounded Lipschitz convex domain. For any sample ω^s ∈ P (s = 1,2,3,…), the heat conduction problem with radiation boundary condition (1) for porous materials with random distribution formally has an SSOTS asymptotic expansion as follows:

In this section, some heat transfer examples are given to verify the validity and feasibility of the statistical two-scale method developed in this work.

4.1. Algorithm validation

A macrostructure Ω^ɛ, which is a union of entire periodic cells as illustrated in Fig. 2(a), is chosen, and the unit cell is shown in Fig. 2(b). The boundary temperatures in the x₃ direction are set as T̄₁(x) = 0 K, T̄₂ (x) = 0 K, respectively. σ = 5.669996×10⁻⁸ W·m⁻²·K⁻⁴, and the radius of the cavity in Y* is 0.25. k_ij = 10δ_ij W·m⁻¹·K⁻¹, f(x) = 10⁵ J·m⁻³·s⁻¹, where δ_ij (i,j = 1,2,3) is the Kronecker delta.

Fig. 2.

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	Fig. 2. (a) Domain Ωɛ = [0,0.25]3; (b) unit cell Y* = [0,1]3.

As the theoretical solution of the coupled heat transfer problem is difficult to find, we have to replace T_ɛ(x,t) by the FE solution T_FE with a very fine mesh. We employ the linear tetrahedral elements to compute the coupled problem of Eq. (1) by fine meshes, and solve the corresponding homogenized equation by a coarse mesh. The numbers of tetrahedrons and nodes are shown in Table 1.

Table 1.

Table 1.

Table 1. Computational cost for comparison. . Elements Original equation Unit cell Homogenized equation 736815 5817 93750 Nodes 137431 1351 17576

Table 1. Computational cost for comparison. .

It should be emphasized that T₀(x,ω) is the approximate solution of the homogenized equation (7), and are the first-order and the second-order two-scale approximate solutions based on Eq. (10).

Figures 3(a)–3(d) illustrate the computational results for T₀(x,ω), , , and T_FE at the intersection x₃ = 0.15.

Fig. 3.

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	Fig. 3. The temperature fields in the cross section x3 = 0.15; (a) T0 (x,ω); (b) ; (c) ; (d) T_FE.

Figure 4(a) and 4(b) show the computational results for temperature gradient ∂T_ɛ(x,ω)/∂x₁, ∂T_ɛ (x,ω)/∂x₂ along the lines of x₁ = x₂, x₃ = 0.15.

Fig. 4.

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	Fig. 4. (a) ∂Tɛ(x,ω)/∂x1; (b) ∂Tɛ(x,ω)/∂x2 along the lines of x1 = x2, x3 = 0.15.

From the computational results given in Figs. 3 and 4, it follows that the homogenized solution and the first-order solution are not effective to catch the local behavior of the solution for the coupled problems. However, the computational results displayed clearly verify that the second-order two-scale solution gives a more accurate numerical solution. Obviously, it can be concluded that the two-scale model proposed in this paper is sufficiently correct for computing the heat conduction problem with radiation boundary condition.

In addition, from Table 1, it can be seen that the mesh numbers of the second-order two-scale approximate solution previously are much less than that of the FE solution with fine meshes, and the computational time is also much less than that of the FE solution, especially for small ɛ. Consequently, the second-order two-scale method is of great importance in engineering computations owing to the importance of saving computer memory and CPU time. Actually, both the two-scale method and the direct finite element method are performed on the same computer (which has memory of 512 GB and 16 processors with CPU = 2.67 GHz). Considering the two-scale method previously, which is very cheap to solve the simulation (it takes about 1 s to finish solving the cell problems, and homogenized problem about 8 s for one iteration). On the other hand, the direct finite element simulation needs 620 s to compute the heat transfer problem for one iteration in the same examples since it requires very fine meshes.

4.2. Thermal properties of randomly distributed porous materials

In order to investigate the thermal performance of the porous materials, we consider three different types of microscopic distributions: spherical pores subject to a uniformly stochastic distribution in a ɛ-cell; spherical pores subject to a normal distribution around the centric point of ɛ-cell; orientations of ellipsoidal pores, whose long axes are two times that of the middle axes and short axes, which are subjected to normal distribution along the x₁ axis, and subjected to uniformly stochastic distribution in ɛ-cell. The distributions of orientations, shapes, and locations of pores on macroscopic properties of the materials are investigated by the SSOTS method, and due to pores random dispersion, the numerical results of different samples will vary even for the same distribution models of pores. Therefore, to obtain more accurate prediction values, a number of samples are required.

Figure 5 depicts the geometry structure of the plate studied with the length of 10 mm, width of 10 mm, and thickness of 5 mm, respectively. The temperatures in the x₃ direction are set as T̄₁(x), T̄₂(x) on the Γ₁. The radii are both taken as 0.04 for the pores subjected to a uniform distribution and pores subjected to a normal distribution. As the orientations of spherical pores are in a normal distribution, the sizes of their long axes are taken as 0.1, while middle axes and short axes are both 0.05.

Fig. 5.

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	Fig. 5. Schematic of porous material plate.

4.2.1. Microstructure effect of porous materials on the effect of thermal conductivity parameters

The thermal conductivity of ceramics is^[5,28] 4.41 W·m⁻¹·K⁻¹. The internal heat source f(x) is taken as zero and T̄₁(x) = 100 K, T̄₂(x) = 1000 K. At this level, the homogenized parameters for the effective thermal conductivity are computed for different samples and compared to corresponding measurements obtained from different numerical models.

The homogenized thermal conductivity coefficients in a unit statistic screen for some samples are displayed in Table 2 as a volume fraction of 20% with a uniform distribution of pores. Then, the expected homogenized parameters are computed by the procedure of the SSOTS algorithms previously.

Table 2.

Table 2.

Table 2. The homogenized coefficients for different samples as volume fractions for 20% with a uniform distribution of pores. . No. 5 3.3434216 0.00060946119 0.00010964318 3.3346926 0.0021270333 3.330518 10 3.3441258 0.0008014546 0.0012766311 3.3355239 0.0025972413 3.3301634 20 3.3429517 0.0016516943 0.0018878259 3.3356398 0.0010416966 3.3374127 30 3.342679 0.00094525486 0.0013600499 3.3355809 0.00056567049 3.3334383 40 3.3428481 −7.4151472×−5 0.0013087926 3.3357914 0.0016423277 3.3337254 50 3.3426518 −4.4637085×10−5 0.00086218149 3.3355769 0.00095091942 3.3333437

Table 2. The homogenized coefficients for different samples as volume fractions for 20% with a uniform distribution of pores. .

Figure 6 with the expected homogenized thermal conductivity in different samples illustrates the convergence of the expected homogenized parameters for volume fractions of 20% with a uniform distribution of pores. Statistically, depicted in Fig. 6, the different samples should have different prediction results. However, as the increasing number of samples with the same microstructure, the mathematical expectation of the computational results should converge. As a result, the scatter of data decreases with the increase of the number of samples, shown in Fig. 6. Thus, 50 samples are chosen in this work to avoid an unacceptable scatter of the computational results.

Fig. 6.

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	Fig. 6. Effective thermal conductivity k11 (a) and k22 (b) for different sample numbers.

Figure 7 displays the obtained values of the material coefficients from the statistical two-scale method as well as other computed values. The expected thermal conductivity results obtained from the proposed two-scale model agree well with the generalized self-consistent method^[24] and the Maxwell–Eucken model^[25] for a uniform distribution of pores. However, the empirical method cannot give satisfactory results for an orientation-normal distribution of spherical pores. In addition, the computational results based on the statistical two-scale model give better approximations of the empirical results at lower porosity, as the volume fractions increase, the deviations of the results increase. Obviously, it can be found that the SSOTS model developed in this paper is sufficiently accurate for predicting the effective properties of the porous materials.

Fig. 7.

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	Fig. 7. Comparison of the thermal conductivity coefficients obtained from different numerical models (a) uniform distribution of pores; (b) orientation-normal distribution of spherical pores.

4.2.2. Radiation effect on macroscopic effective performance with different random distributions

Heat transfer properties of the porous materials not only depend on the material properties of the matrix phases but also on the volume fractions, location, orientation, spatial distribution, and radiation effect of the pores. Therefore, it is absolutely essential to investigate their influence on the macroscopic effective performance of the porous materials. The thermal conductivity of porous materials is 4.41 W·m⁻¹·K⁻¹, and internal heat source f (x) is set to 5000 J·m⁻³·s⁻¹. The volume fractions of the materials are limited to 20%, and pores are subjected to a uniformly stochastic distribution in a ɛ-cell.

Figure 8(a) illustrates the computational results of the temperature field for boundary temperature T̄₁(x) = 100 K and T̄₂(x) = 1000 K along the lines of x₂ = 0.600167969, x₃ = 0.300844439, whereas figure 8(b) displays the computational results of the temperature field for boundary temperature T̄₁(x) = 500 K and T̄₂(x) = 1500 K.

Fig. 8.

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	Fig. 8. The temperature field for boundary temperature (a) T̄1(x) = 100 K, T̄2(x) = 1000 K; (b) T̄1(x) = 500 K, T̄2(x) = 1500 K along the lines of x2 = 0.600167969, x3 = 0.300844439.

It can be clearly seen from the figures that the SSOTS method developed in this work is effective to catch the local microscopic behavior of the porous materials. In particular, the detailed information of the radiation effect on the surfaces of the cavities can be captured accurately by the developed two-scale model.

This paper discusses the SSOTS analysis and computation for the heat conduction problem with radiation boundary condition in porous materials. The second-order two-scale analysis formulae for the heat transfer problem are derived in detail. Furthermore, the correctness of this two-scale model and the effectiveness of the SSOTS method proposed in this paper are verified by comparing with FE methods with fine meshes. During the process of the numerical verification, it is well shown that the SSOTS method is accurate to numerically solve the heat conduction equation with radiation boundary condition.

The macroscopic thermal performance for the porous structures with varying probability distribution models, including volume fraction, orientation, and location distributions of pores, is displayed. By comparing the obtained results with empirical formulas in the references, the proposed statistical two-scale model is validated. Further, in order to study the influence of the volume fractions and the radiation effect of pores on the macroscopic properties of porous materials, different microstructure models of these materials are chosen for a comparison. Numerical results clearly show that the microscopic structure has a significant impact on the macroscopic thermal properties. Especially, these properties vary with the probability model of random distributions of pores. Therefore, the SSOTS method and computation developed in this study can be used to predict the macroscopic thermal properties of the heat transfer problem and effectively catch the local behaviors caused by the microstructure inside the materials.