Statistical Multiscale Analysis of Transient Conduction and Radiation Heat Transfer Problem in Random Inhomogeneous Porous Materials

Yiqiang Li, Liang Ma, Zhiqiang Yang, Xiaofei Guan, Yufeng Nie and Zihao Yang, ,

1 Introduction

With rapid developments of space aircraft in recent years, porous materials have been widely used as insulation for thermal protection system owing to their various elegant qualities, such as low relative density, high heat resistance, low thermal conductivity and high radiation attenuation coefficient compared to the traditional materials [Grujicic,Pandurangan, Zhao et al. (2006); Ishizaki, Komarneni and Nanko (2013)]. And it is important to understand the thermal responses of porous materials and structures in many engineering applications. As we know, heat transfer in porous materials contains conduction, convection and radiation. The convective heat transfer can be neglected in closed-cell porous materials and radiation is typically the major mode of heat transfer in high-porosity insulations at high temperature environment [Modest (2013)].

Porous materials usually have multiple length scales [Park and Hou (2004); Zhang, Yang,Zhang et al. (2013); Wu and Xiao (2017)] and random microscopic configurations [Sundén and Yuan (2013); Wang, Wang, Pan et al. (2007)]. According to the arrangement characteristics of perforations, random materials can be classified into materials with consistent random distribution and those with inconsistent random distribution [Li and Cui(2005); Han, Cui and Yu (2010)]. The consistent random distribution means that the volume fraction and probability distribution model of perforations are the same everywhere in porous materials (Fig. 1), which will be considered in this paper. Considering interior surface radiation effect in the closed-cells, heat transfer response of random porous materials would become much complicated. Therefore, it is necessary to study the heat conduction-radiation problem of random porous materials.

Up to now, some theoretical models have been proposed and employed to calculate the effective thermal conductivity properties of composite materials, such as Maxwell-Eucken model [Hashin and Shtrikman (1962)], generalized self-consistent method [Chou, Nomura and Taya (1980)], Cheng-Vachon model [Cheng and Vachon (1969)], self-consistent method [Hill (1965)]. However, these methods simplified the microstructure of real composite materials by the homogenization schemes in order to reduce the computational complexity. Moreover, by using these theoretical methods the local temperature fields cannot be calculated. Some works have been performed on heat conduction-radiation problem of porous materials. Liu et al. [Liu and Zhang (2006)] calculated the effective macroscopic properties of static heat conduction-radiation problem. Allaire et al. [Allaire and El Ganaoui (2009)] gave the homogenization process of heat conduction problem with non-classical radiation boundary conditions by two-scale asymptotic expansion. It should be pointed out that the non-classical model cannot be used for the porous materials with low porosity, because it over-estimates the radiation behavior on the interior surface of cavities. The homogenization solutions of the heat conduction problem with radiation boundary condition were obtained in Bakhvalov [Bakhvalov (1981)]. After that, Yang et al. [Yang, Cui, Nie et al. (2012)] presented a second-order two-scale analysis method for the heat transfer problem of periodic porous materials with interior surface radiation.Besides, Amosov [Amosov (2010)] and Tiihonen [Tiihonen (1997)] proved the existence and uniqueness of the heat conduction equation with nonlinear radiation boundary conditions.

To our knowledge, we have not seen the study of the transient heat transfer problem of random porous materials with interior surface radiation in the existing literature. Actually,random porous materials have been widely used in engineering, such as metal-matrix composites and polymer blends. The transient heat conduction-radiation model is nonlinear,and it is difficult to find the analytical solutions. As for numerical solutions, due to the random structures and characteristic coefficients oscillating rapidly in small cells, in order to effectively capture the local fluctuation behaviors of temperature field and its derivatives,the mesh size must be very small while employing the traditional numerical methods,which will lead to tremendous amount of computer memory and CPU time. Therefore, it is necessary to develop highly efficient numerical method to solve the nonlinear transient heat conduction-radiation problem of random porous materials.

The homogenization method is developed to describe the overall behaviors of composite materials by incorporating the fluctuations due to the heterogeneities of the composites materials [Bensoussan, Lions and Papanicolaou (2011)]. However, numerous numerical results have demonstrated that the numerical accuracy of the standard homogenization method may not be satisfactory [Efendiev and Hou (2009); Yang, Zhang, Dong et al.(2017)]. And then, based on the homogenization method, various multi-scale methods have been proposed [Efendiev and Hou (2009); Juanes (2005); Engquist, Li, Ren et al. (2007)].However, they only obtained first-order asymptotic expansions, which are not enough to capture the local fluctuations of solutions in many physical and mechanical problems.Hence, it is necessary to develop more effective methods, which is the motivation for seeking higher-order multiscale methods. The existences of homogenization coefficients and homogenization solutions for the composite materials with consistent random distribution were proved in Jikov et al. [Jikov, Kozlov and Oleinik (2012)]. Cui et al. [Cui,Shih and Wang (1999); Yang, Ma and Ma (2017)] introduced the second-order multiscale method to predict different physical and mechanical behaviors of composite structures. By adding the second-order correctors, the microscopic fluctuation behaviors inside the composite materials can be captured more accurately. For the random composite material,Cui et al. [Li and Cui (2005); Guan, Liu, Jia et al. (2015); Han, Cui and Yu (2010)]established a statistical multiscale method by introducing a random sample cell. However,previous multiscale method cannot be directly applied to the transient conduction and radiation heat transfer problem because of the nonlinearity, randomness and time variation.Up to now, there is no effective numerical method found for the transient and nonlinear heat transfer problem of random porous materials.

The aim of this paper is to develop a novel statistical multiscale analysis method for the transient heat conduction problem in random porous materials with a nonlinear radiation boundary condition. The remainder of this paper is outlined as follows. In Section 2, the microscopic representation of random porous materials and the transient heat conductionradiation model are given. The formulation of multiscale asymptotic expansion is derived and the finite element algorithms based on the statistical multiscale method are given in details in Section 3. In Section 4, numerical results obtained by the statistical multiscale method are given to demonstrate the effectiveness of the present method. Finally, some conclusions are presented in Section 5. For convenience, we use the Einstein summation convention on repeated indices in this paper.

2 Transient conduction and radiation heat transfer problem

In this section, the microscopic representation of random porous materials is described.And then the transient heat conduction equation with a nonlinear radiation boundary condition is presented.

Figure 1: Porous materials with random distribution of pores (a) The whole structure ;(b) Unit cell

2.1 Microscopic representation of random porous materials

Suppose that the investigated porous material structureis made from matrix and pores.All the pores are considered as ellipsoids, which are randomly distributed in the matrix. In this paper, all the ellipsoid poresare considered as “one-scale”, which means all of their long axes satisfyandare given lower and upper bounds. And porous materials with random distribution of pores can be represented as follows

1) There exists a constantsatisfyingdenotes the size of.Thuscan be considered as a set of cells with size, as shown in Fig. 1(a).

2) Each ellipsoid in three-dimensional apace is defined by 9 random parameters, i.e. the coordinates of the central point, the sizes of the long, middle and short axisand 3 orientation parameters of the long and middle axis.Let, which contains all the information of an ellipsoid. If there areellipsoids in a cell,represents a normalized cell, then a random samplecan be defined as

The distribution of the pores in each cell satisfies the same probability distribution function.Then the investigated structure has periodically random distribution of pores, and then can be represented by the probability distribution of pores inside a typical cell. Therefore, the porous structureis logically composed ofsize cells subjected to the identical probability distribution modeland

wherehe index of samples andis the integer set. For the whole, define

Thus the thermal conductivity parameters of random porous materials can be considered by

whereekdenotes thekth pore insideand the material coefficientsatisfies thawhereC is a positive constant.

2.2 Heat conduction-radiation equation

Letandbe a smooth unbounded domain inwith random structures satisfying following conditions [Oleinik, Shamaev and Yosifian (1992)]:

1) The unit cellmain with a Lipschitz boundary.is the surfaces of cavities andis the solid part of, shown in Fig. 1(b).

2) The set(stands for the closure inof the domain) and the intersectionofwithneighborhood of consist of a finite number of Lipschitz domains separated from each other and from the edges ofby a positive distance.

The cavities of the domain are convex. Then, the investigated porous domainas shown in Fig. 1(a) can be expressed asis a bounded convex domain inwithout cavities, and these cavities do not intersect with.

Suppose that the radiative surfaces of cavities are diffuse and grey, that is, the emissivityof the surfaces does not depend on the wavelength of radiation. And based on the modeling of heat radiative process by an integral equation on the surfaces of cavities in periodic structures [Liu and Zhang (2006)] and microscopic representation of random structures, the nonlinear radiation boundary condition on interior surfaces of closed cavities in random porous materials can be defined as

wheredenotes the temperature,the Stefan-Boltzmann constant,unit outward normal onoundary composed of interior surfaces of cavities, andnumber of cavities inporous materials; the emissivitysatisfiesis the intensity of emitted radiation given by

whereis the view factor between two different pointsandon, and is defined on three dimensional form for a convex cavity as follows [Allaire and El Ganaoui (2009)]

whereanddenote the unit normal at different pointsz andx on, respectively.And for anyit satisfies the following properties

Under aforementioned assumptions, the transient heat transfer problem in random porous structures can be expressed as follows

The corresponding initial and thermal boundary conditions are

where1,2,3;,andare the mass density, specific heat and thermal conductivity, respectively;denotes the internal heat source,the prescribed boundary temperature,andthe parts of the boundary of this region, where temperature and heat flux are specified such thatThe timeplays the role of a parameter anddenotes the upper bound of time.In order to avoid the arguments on the mathematical properties of investigated functions below, we suppose that the random material parametersare bounded measurable functions of the random variable, and their mathematical expectations exist.Besides,is supposed to be symmetric.

3 Statistical multiscale method

In this section, the statistical multiscale formulation is derived by using the constructive way for calculating the nonlinear transient heat conduction behaviors of random porous materials. Moreover, the corresponding numerical algorithm is presented in details.

3.1 Statistical multiscale asymptotic expansions

Introducing the variablewhich denotes the local coordinates defined on 1-normalized cell, and then the material coefficients of microstructure incan be expressed asSince the temperature solutionof the heat conduction problem (1) depends on both global behaviors of the structureand local configuration in, and then it can be expressed asIn this paper, it is supposed thate=1to simplify the exposition.However, it is easy to extend the case0＜e＜1, see Park et al. [Park and Hou (2004)] for details. Further, it is assumed thatcan be expanded into the following form in two-scale variablesx andy

whereis the homogenization solution, which only reflects macroscopic behaviors,andare called as local solutions. Due toDifferentiation with respect to x is defined as

We now substitute (2) into (1) and match terms of the same order of ε, then the following equations can be obtained

A series of equations are obtained if the coefficients offrom both sides of above equations are required to equal to each other.

From the coefficients ofε-1the following local problem defined onare constructed

whereis the interior surfaces of the cavities contained inas shown in Fig. 2(b), such thatandN the number of cavities contained in the unit cellConsidering

it is easy to get that

Sincearises from the macroscopic behavior of the structure, it is not identical to zero. By virtue of (7), (6) can be rewritten as

To attach the following boundary condition on

Therefore, for any sampleis the solution of the following problem

Inspired by Li et al. [Li and Cui (2005)] and fromfor any fixed samplethe homogenization coefficients can be defined as

wheredenotes the Lebesgue measure ofFrom the symmetry and boundness ofand Lemma 3.2 in [Li and Cui (2005)], it follows thatandare bounded functions of random variable and their mathematical expectations exist. Thus, by takingL samplesL homogenization coefficients are obtained, and then according to the Kolmogorov’s classical strong law of large number,one can calculate the expected homogenization coefficients in formulas

Further, one can define the homogenized problem associated with (1) as follows

Similarly to Li et al. [Li and Cui (2005)], it is easy to prove thatis symmetric, positive and definite. By Lax-Milgram theorem and Poincare’s inequality, the homogenized problem (13) has a unique solution.

Next, comparing the coefficient ofon both sides of (4) and taking into account the homogenized Eq. (13), it can be obtained that

Sinceandare not identical to zero and considering the radiation asymptotic expansion (5), the auxiliary functionsandcan be defined by the constructing way analogous tonamely

whereis the macroscopic coordinate of the structure, and l is the local coordinate of 1-normalized cell.

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Remark 3.1 In order to ensure thatexpressed by (2) satisfy the boundary condition onandin the original problem (1), it is necessary to impose the boundary conditiononsincedo not generally equal to zero on the boundary. Besides, since the distribution of pores in all cells is random, the pore distributions in two adjacent cells are different. Therefore, the configuration of the cells cannot be extended periodically, and the periodicity conditions cannot be used here to construct cell problems (10) and (15)-(17). In order to ensure the existence, uniqueness and whole continuity of solutions, zero-boundary conditions are imposed on the cell problems(10) and (15)-(17).

Remark 3.2 In terms of Lax-Milgram theorem and Poincare’s inequality, it is easy to prove that above problems (10) and (15)-(17) have unique solution for any fixed

Now, we can define the multiscale approximate solutions of the problem (1) as follows

whereandrepresent the first-order and the second-order multiscale approximate solutions, respectively. Let

For any fixed sampleto comparewith the original solutionsubstitutinginto (1), we have

It can found that above residual is of orderO(1). However, in the practical engineering computation, εis a fixed smaller constant rather than tending to zero. So the first-order solution is not accepted by the engineers since it is not accurate enough to capture the micro-scale fluctuations inside materials.

In addition, takinginto (1) and considering homogenized Eq. (13),it yields that

where

By the definitions ofandgiven by (10) and(15)-(17), the first four terms on the right side of (19) are equal to zero. So the residual of(19) is of orderO()ε. It means that the second-order multiscale solutions are equivalent to the solutions of original problem (1) with orderO()εin nearly pointwise sense. This is the reason for seeking second-order two-scale expansions.

Summing up, one obtains following results

Theorem 3.1 The transient conduction and radiation heat transfer problem of random porous materials has statistical multiscale approximate solutions as follows

whereis the solution of the homogenized problem (13);are auxiliary functions defined by (10), and (15)-(17) on the normalized cellrespectively.

From the formula of temperature (21), the heat flux density is evaluated approximately by

3.2 Numerical implementation

3.2.1 Finite element (FE) formulations

The computer simulation algorithm developed in Yu et al. [Yu, Cui and Han (2008)] is used to generate a sample with random distribution of pores, and the FE meshes of sample cells applied in this study are those of Han et al. [Han, Cui and Yu (2010)], shown in Fig.2(a). It can be observed that cell problems (10) and (15)-(17) are all elliptic boundary value problems, and they can be solved by the standard finite element method to get the FE solutions of the first-order and second-order auxiliary functions. And then the homogenized coefficients are evaluated by (11-12) based on the FE solutions of first-order auxiliary functions. Further, the spatial region Ωis divided by using the FE mesh first,shown in Fig. 2(b), and then the temporal domain(0,t)*is divided by using the backward Euler full discrete format to solve the homogenized problem (13). According to (21) and the FE solutions of auxiliary problems, homogenized coefficients and homogenized problem, the statistical multiscale FE solution of temperature can be evaluated.

3.2.2 Algorithms for statistical multiscale method

The algorithm procedure of statistical multiscale method based on the FE computations for predicting the transient heat conduction-radiation performance is stated as follows

1) Generate a samplefor a unit cellaccording to the given probability distribution modelP , and determine the material coefficients in (1). Further, partitioninto FE meshes.

2) Solve the problem (10) by FE method to get the solution ofFurthermore,the homogenized coefficientsare calculated through formula (11).

3) Repeat steps 1-2 forL samples. And the expected homogenized coefficients and cane be evaluated by the formula (12).

4) With the expected homogenized coefficientsandobtained by step 3, the homogenization solutioncan be obtained through solving problem (13) inby using FE method and finite difference method.

5) Solve the cell problems (13)-(15) forby using the same meshes in step 2 to get the FE solutions ofand

6) From (21) and (22), the temperature and heat flux density distributions corresponding to the sampleare evaluated.

Figure 2: (a) The meshes of unit cell; (b) the meshes of the homogenized domainΩ

4 Numerical examples

In this section, in order to verify the validity and feasibility of the statistical multiscale analysis method for studying the transient heat conduction problem of random porous materials with nonlinear radiation boundary condition, some numerical examples are given here.

4.1 Validation of the statistical multiscale method

A macrostructure, which is a union of entire periodic cells as shown in Fig. 3(a), is chosen, and the unit cell is depicted in Fig. 3(b).=0K and=0K are boundary temperatures in-direction. The time step is=0.02, the radius of the cavity inis 0.25 and

Figure 3: (a) Domain; (b) Unit cell

Since it is difficult to find the analytical solution of above problem, we have to replacewith more precise FE solutionin a very fine mesh for comparison. The tetrahedron partition is implemented and the information of the FE meshes is listed in Table 1. Without confusion, it should be noted thatdenotes the numerical solution of the homogenized Eq. (13),andthe first-order and the second-order multiscale numerical solutions based on (21). We consider two cases

Case1：kij=15 W/m K, f( x)=10000 J/m3s

Case2：kij=15 W/m K, f( x)=100000 J/m3s

whereis the Kronecker delta.

Table 1: Comparison of computational cost

Fig. 4 and Fig. 5 show the numerical results forandat the intersection0.15and at time0.2of different cases. From Fig. 4 and Fig. 5, it can be found that the homogenized solutions, first-order and second-order approximate solutions are in accordance with the FE solutions in the very fine mesh. But Fig. 4 and Fig.5 demonstrate that the second-order approximate solutions are much better than the homogenized and first-order approximation solutions for temperature. It means that the homogenization solution and the first-order solution are insufficient to describe local fluctuation of the solution. But the second-order multiscale method gives more accurate numerical solutions.

In addition, from Tab. 1, it can be found that the mesh partition numbers of the newly statistical second-order multiscale approximate solution are much less than that of the refined FE solution. It means that the statistical multiscale method can greatly save computer memory and CPU time without losing precision, especially for smallε, which is very important in actual engineering computation. Both the multiscale and direct FE simulations are performed on the same computer with memory of 512 GB and 16 processors of CPU=2.67 GHz. Actually, it is very cheap by using the statistical multiscale method, taking about 0.8 seconds to finish solving the cell problem and homogenized problem about 6 seconds for one iteration, which takes the majority of the computational efforts. On the other hand, the direct FE simulation takes about 660 seconds because it requires fine meshes.

Figure 4: The temperature (K) in the cross section and at time0.2, case 1: (a)

Figure 5: The temperature (K) in the cross section 0.15 and at time0.2, case 2: (a)

4.2 Influence of micro-structures on the thermal properties and performance

The influence of micro-structures of random porous materials on thermal properties performance is investigated here. Fig. 6 depicts the geometry structure of a plate whose length is 10 mm, width is 10 mm and thickness is 5 mm. The internal heat sourcef( x, t)is taken as zero, the temperatures in-direction are set as100K and=1000K on, and the thermal conductivity of ceramics is 21.16 W/mK.

Porous material structures with three different random distributions of pores are considered:uniformly stochastic distribution of spherical pores in a cell; normally stochastic distribution of spherical pores around the centric point of a cell; uniformly stochastic distribution of ellipsoidal pores whose long axes is about two times of the middle axes and short axes, and which are subjected to normal distribution along x1-axis, in a cell. Fig. 7 depicts these three samples corresponding to those, which are generated based on Yu et al.[Yu, Cui and Han (2008)]. The radii are both taken as 0.0375 for the pores subjected to uniform distribution and pores subjected to normal distribution. As for the orientations of spherical pores are normal distribution, the sizes of their long axes are taken as 0.1, middle and short axes are both 0.05. The effect of locations, orientations and shapes of pores on thermal properties and performance of the plate are investigated by using the statistical multiscale method. However, due to the pores random dispersion, different samples with the same probability distribution may lead to different numerical results. Therefore, to obtain more accurate prediction values, a number of samples are required.

Figure 6: Schematic of porous materials plate

Figure 7: (a) Uniform distribution; (b) Location-normal distribution; (c) Orientationnormal distribution

Table 2: Orientation-normal distribution with a volume fraction of 20% for different samples

The expected homogenized thermal conductivity coefficients of porous material with orientation-normal distribution of pores and a volume fraction of 20% for different samples are displayed in Tab. 2. Fig. 8 shows the convergence of expected homogenized parameters with the number of samples. Statistically, different samples should have different numerical results, as shown in Fig. 8. However, as the increasing number of samples with the same random distribution, the mathematical expectation of the computation results should converge. Obviously, as shown in Fig. 8, the scatter of data decreases with increasing number of samples. As can be seen, 50 samples are chosen in this work to avoid an unacceptable scatter of the computation results.

Figure 8: Effective thermal conductivity with different number of samples for two different volume fractions (a) 15%; (b) 25%

Fig. 9 depicts the statistical multiscale results of material parameters and those computed by different analytical methods, i.e. Maxwell-Eucken model [Hashin and Shtrikman(1962)], generalized self-consistent method [Chou, Nomura and Taya (1980)], Cheng-Vachon model [Cheng and Vachon (1969)] and self-consistent method [Hill (1965)]. It can be found that the expected thermal conductivity parameter of porous material with uniform distribution of pores calculated by statistical multiscale method agrees well with the generalized self-consistent method and Maxwell-Eucken model. However, analytical methods cannot obtain satisfactory results for the orientation-normal distribution of spherical pores. Besides, the statistical multiscale results give better approximations than analytical results at lower porosity, as the volume fractions increasing, the deviations of the results increasing. Obviously, it can be concluded that the statistical multiscale method proposed in this paper is sufficiently accurate for predicting the effective properties of the random porous materials. Fig. 10 and Fig. 11 show the local heat flux density distributions in the cells of uniform and orientation-normal distributions with 5.0% and 15.0% of poresat time It can be found that local heat flux densities in different cells have a marked fluctuation. And local heat flux density distributions with a high volume fraction are relatively high. As a result, it can be concluded that the heat flux densities of random porous materials is concurrently affected not only by macroscopic properties, but also by the microscopic structure of random distribution of pores.

Figure 9: Comparison of the thermal conductivity coefficients obtained from different numerical models: Uniform distribution of pores and location-normal distribution of pores

Figure 11: Heat flux density in local cells with different volume fraction and pores subjected to orientation-normal distribution (a) 5%; (b) 15%

5 Conclusion

This paper presents a statistical multiscale analysis method for a transient heat conduction problem with nonlinear radiation boundary condition in random porous materials. The homogenization and the statistical multiscale formulation are derived in details. A linear homogenized problem independent of radiation effect is obtained, and the second-order correctors featuring nonlinear radiative heat transfer in perforations is introduced, which is absolutely necessary in multiscale simulation of this problem. Error analysis indicates that the second-order multiscale solution has a much better approximation to the solution of the original problem. The validity and effectiveness of the statistical multiscale method and numerical algorithm have been verified by two examples. It was concluded that the statistical multiscale method is accurate to numerically solve the transient conduction and radiation heat transfer problem in random porous materials. Moreover, numerical results also demonstrate that the thermal properties depend greatly on the micro-structures, like volume fraction, location, orientation and spatial distribution of perforations. And local fluctuations of temperature and its gradient can be captured more precisely by adding second-order correctors. As a result, the statistical multiscale analysis method and related numerical approximations techniques proposed can be practically employed to predict the transient heat transfer problem in random porous materials.

AcknowledgementThis work was financially supported by the National Natural Science Foundation of China (11501449), the Fundamental Research Funds for the Central Universities (3102017zy043), and the fund of the State Key Laboratory of Solidification Processing in NWPU (SKLSP201628) and the National Key Research and Development Program of China (2016YFB1100602).

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