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Project supported by the China Postdoctoral Science Foundation (Grant Nos. 2015M580256 and 2016T90276).
This paper discusses a statistical second-order two-scale (SSOTS) analysis and computation for a heat conduction problem with a radiation boundary condition in random porous materials. Firstly, the microscopic configuration for the structure with random distribution is briefly characterized. Secondly, the SSOTS formulae for computing the heat transfer problem are derived successively by means of the construction way for each cell. Then, the statistical prediction algorithm based on the proposed two-scale model is described in detail. Finally, some numerical experiments are proposed, which show that the SSOTS method developed in this paper is effective for predicting the heat transfer performance of porous materials and demonstrating its significant applications in actual engineering computation.
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 ɛ1/2. 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. 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
![](cpb_25_9_090202/cpb160346ieqn1.gif)
![](cpb_25_9_090202/cpb160346ieqn2.gif)
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 ≤ yj ≤ 1, j = 1,2,3} and
Then, the investigated porous domain Ωɛ as shown in Fig.
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:
![](cpb_25_9_090202/cpb160346ieqn7.gif)
![](cpb_25_9_090202/cpb160346ieqn26.gif)
![](cpb_25_9_090202/cpb160346ieqn8.gif)
In the following, we suppose that
If {k (ω)} is a bounded random variable, and there exists an expectation value. {kij(x/ɛ,ω)} is symmetric and bounded, that is, there exist two positive constants c1 and c2 such that
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 Tɛ(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] Tɛ (x,ω) can be obtained into the series of the following form:
For any fixed sample ωs (s = 1,2,3,…), Nα1(y,ωs) (α1 = 1,2,3) is the solution of the following boundary problem:
![](cpb_25_9_090202/cpb160346ieqn9.gif)
Referring to Refs. [5], [18], and [19], for any fixed sample ωs (s = 1,2,3,…), the homogeneous parameters are defined as
![](cpb_25_9_090202/cpb160346ieqn10.gif)
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 Nα1 (y,ωs), Nα1 α2(y,ωs) (α1, α2 = 1,2,3) is the solution of the boundary value problem as follows:
Now, we can define the two-scale approximate solutions of problem (
![](cpb_25_9_090202/cpb160346ieqn11.gif)
![](cpb_25_9_090202/cpb160346ieqn12.gif)
For any fixed sample ωs, to compare
In addition, by taking
Summing up, one obtains the following theorem.
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.
A macrostructure Ωɛ, which is a union of entire periodic cells as illustrated in Fig.
As the theoretical solution of the coupled heat transfer problem is difficult to find, we have to replace Tɛ(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. (
![]() | Table 1. Computational cost for comparison. . |
It should be emphasized that T0(x,ω) is the approximate solution of the homogenized equation (
Figures
Figure
From the computational results given in Figs.
In addition, from Table
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
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
The homogenized thermal conductivity coefficients
![]() | Table 2. The homogenized coefficients |
Figure
Figure
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
![]() | Fig. 8. The temperature field ![]() |
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.
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