Size- and shape-dependent effective conductivity of porous media with spheroidal gas-filled inclusions

Abstract

Porous media containing gas-filled inclusions embedded in a solid phase constitute an important class of natural or artificial materials of both theoretical and practical interest. In these materials, thermal conductivity is one of the most important properties. In a variety of situations of practical interest, when the characteristic size of gas-filled inclusions is comparable with the mean free path of gas molecules and when the slip flow regime is considered, the behavior of gas near solid surfaces cannot be described by classical thermal conductivity equations. In fact, the boundary conditions at the solid surfaces must be modified by considering that the temperature and normal heat flux simultaneously suffer a discontinuity. The first purpose of the present work is to develop an efficient and accurate micromechanical model capable of estimating the effective conductivity of porous materials while taking into account the discontinuities of the temperature and normal heat flux across solid surfaces and the non-spherical form of gas-filled inclusions. The second purpose of the present work is to study the dependencies of the effective conductivity on the size and shape of gas-filled inclusions. By applying the micromechanical model based on the differential scheme and by using the solution results obtained for auxiliary dilute problem accounting for modified boundary conditions on surface solids, the closed-form expression for the effective conductivity is obtained. Numerical results are provided to illustrate the dependence of the effective conductivity on the size and shape of gas-filled inclusions in the case of randomly oriented inclusions.

Appendices

Appendix 1: Algebraic calculations in orthogonal curvilinear coordinates

Consider a surface $\varGamma ^{(i)}$ situated in a three-dimensional Euclidean space. In order to obtain a mathematical characterization of $\varGamma ^{(i)}$ , a system of orthogonal curvilinear coordinates $\{y_{1},y_{2},y_{3}\}$ is defined such that the position vector ${\mathbf {x}}=(x_{1},x_{2},x_{3})$ of any point in space is expressed as

$$\begin{aligned} {\mathbf {x}}={\mathbf {x}} (y_{1},y_{2},y_{3})=[x_{1}(y_{1},y_{2},y_{3}), x_{2}(y_{1},y_{2},y_{3}),x_{3}(y_{1},y_{2},y_{3})]. \end{aligned}$$

(84)

The vector tangent to the $y_{i}$-coordinate curve is defined by

$$\begin{aligned} {\mathbf {t}}_{i}=\frac{\partial {\mathbf {x}}}{\partial y_{i}}=h_{i}{\mathbf {f}}_{i} \quad \hbox {with}\quad h_{i}=\left\| \frac{\partial {\mathbf {x}}}{\partial y_{i}}\right\| \end{aligned}$$

(85)

where the summation convention does not apply, $h_{i}$ is a metric coefficient and ${\mathbf {f}}_{i}$ is the unit vector tangent to the $y_{i}$ -coordinate curve. Since the curvilinear coordinates $y_{1}$, $y_{2}$ and $y_{3}$ are orthogonal, ${\mathbf {f}}_{1}$, ${\mathbf {f}}_{2}$ and ${\mathbf {f}}_{3}$ are orthonomal so that ${\mathbf {f}}_{i}\cdot {\mathbf {f}}_{j}=\delta _{ij}$. The surface $\varGamma ^{(i)}$ can be now defined by

$$\begin{aligned} \varGamma ^{(i)}=\left\{ {\mathbf {x}}={\mathbf {x}}(y_{1},y_{2},y_{3})\in {\mathbf {R}} ^{3}\mid y_{1}=\gamma _{0}\right\} \end{aligned}$$

(86)

where $\gamma _{0}$ is a constant scalar value.

Let us now introduce the temperature field $\varphi ({\mathbf {y}})$. The temperature gradient $\nabla \varphi$ is defined in the orthonormal curvilinear basis $\{{\mathbf {f}}_{1},{\mathbf {f}}_{2},{\mathbf {f}}_{3}\}$ by

$$\begin{aligned} \nabla \varphi ({\mathbf {y}})=\frac{1}{h_{1}}\frac{\partial \varphi }{\partial y_{1}}{\mathbf {f}}_{1}+\frac{1}{h_{2}}\frac{\partial \varphi }{\partial y_{2}} {\mathbf {f}}_{2}+\frac{1}{h_{3}}\frac{\partial \varphi }{\partial y_{3}} {\mathbf {f}}_{3}. \end{aligned}$$

(87)

Correspondingly, the resulting intensity field ${\mathbf {e}}({\mathbf {y}})$ is determined by

$$\begin{aligned} {\mathbf {e}}({\mathbf {y}})=-\frac{1}{h_{1}}\frac{\partial \varphi }{\partial y_{1}}{\mathbf {f}}_{1}-\frac{1}{h_{2}}\frac{\partial \varphi }{\partial y_{2}} {\mathbf {f}}_{2}-\frac{1}{h_{3}}\frac{\partial \varphi }{\partial y_{3}} {\mathbf {f}}_{3}. \end{aligned}$$

(88)

The heat flux divergence $\nabla \cdot {\mathbf {q}}$ is expressed in the system of orthonormal curvilinear coordinates $\{y_{1},y_{2},y_{3}\}$ by

$$\begin{aligned} \nabla \cdot {\mathbf {q}}=\frac{1}{h_{1}h_{2}h_{3}}\left[ \frac{\partial \left( h_{2}h_{3}q_{1}\right) }{\partial y_{1}}+\frac{\partial \left( h_{1}h_{3}q_{2}\right) }{\partial y_{2}}+\frac{\partial \left( h_{1}h_{2}q_{3}\right) }{\partial y_{3}}\right] . \end{aligned}$$

(89)

The surface Laplacian of the temperature field $\varDelta _{s}\varphi ({\mathbf {y }})$ is defined by

$$\begin{aligned} \varDelta _{s}\varphi =\frac{1}{h_{2}h_{3}}\left[ \frac{\partial }{\partial y_{2}}\left( \frac{h_{3}}{h_{2}}\frac{\partial \varphi }{\partial y_{2}} \right) +\frac{\partial }{\partial y_{3}}\left( \frac{h_{2}}{h_{3}}\frac{ \partial \varphi }{\partial y_{3}}\right) \right] . \end{aligned}$$

(90)

Now, we consider several important cases where the surface $\varGamma ^{(i)}$ is spherical or spheroidal. The previous definitions are now specified in detail.

• Spherical coordinate system

  First, when the surface $\varGamma ^{(i)}$ has a spherical form, the orthogonal curvilinear coordinate system described above reduces to the system of spherical coordinates $(r,\theta ,\phi )$, where $y_{1}\equiv r\in [0,+\,\infty ]$ is the radial coordinate, $y_{3}\equiv \phi \in [0,2\pi ]$ is the azimuthal angle and $y_{2}\equiv \theta \in [0,\pi ]$ is the elevation angle with

  $$\begin{aligned} r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}},\quad \tan {\theta }=\sqrt{ x_{1}^{2}+x_{2}^{2}}/x_{3},\quad \tan {\phi }=x_{2}/x_{1} \end{aligned}$$

  (91)

  or inversely,

  $$\begin{aligned} x_{1}=r\cos \phi \sin \theta ,\quad x_{2}=r\sin \phi \sin \theta ,\quad x_{3}=r\cos \theta . \end{aligned}$$

  (92)

  This spherical coordinate system $(r,\theta ,\phi )$ is associated to the corresponding spherical orthonormal basis $\{{\mathbf {f}}_{r},{\mathbf {f}}_{\theta },{\mathbf {f}}_{\phi }\}$ defined by

  $$\begin{aligned} {\mathbf {f}}_{1}\equiv \,& {} {\mathbf {f}}_{r}= \sin \theta (\cos \phi {\mathbf {j}} _{1}+\sin \phi {\mathbf {j}}_{2})+\cos \theta {\mathbf {j}}_{3},\nonumber \\ {\mathbf {f}}_{2}\equiv \,& {} {\mathbf {f}}_{\theta }=\cos \theta (\cos \phi {\mathbf {j}} _{1}+\sin \phi {\mathbf {j}}_{2})-\sin \theta {\mathbf {j}}_{3},\nonumber \\ {\mathbf {f}}_{3}\equiv \,& {} {\mathbf {f}}_{\phi }=-\sin \phi {\mathbf {j}}_{1}+\cos \phi {\mathbf {j}}_{2}. \end{aligned}$$

  (93)

  The metric coefficients of the spherical coordinate system $(r,\phi ,\theta )$ are given by

  $$\begin{aligned} h_{1}\equiv h_{r}=1,\quad h_{2}\equiv h_{\theta }=r, \quad h_{3}\equiv h_{\phi }=r\sin \theta . \end{aligned}$$

  (94)

• Prolate spheroidal coordinate system

  Second, when the surface $\varGamma ^{(i)}$ has a prolate spheroidal form, the orthogonal curvilinear coordinate system described above corresponds to the the system of prolate spheroidal coordinates $(\alpha ,\beta ,\gamma )$ with $y_{1}\equiv \alpha \in [0,+\infty ]$, $y_{3}\equiv \gamma \in [0,2\pi ]$ and $y_{2}\equiv \beta \in [0,\pi ]$. By introducing some additional notations as follows:

  $$\begin{aligned} p=\cos \beta ,\;\bar{p}=\sqrt{1-p^{2}}=\sin \beta ,\;\mu =\cosh \alpha ,\; \bar{\mu }=\sqrt{\mu ^{2}-1}=\sinh \alpha \end{aligned}$$

  (95)

  where b and a ($a>b$, or equivalently $w=a/b>1$) are the equatorial radius and the distance from center to pole along the symmetry axis of the spheroidal surface, respectively. The connection between the prolate spheroidal coordinates $(\alpha ,\beta ,\gamma )$ and the Cartesian coordinates $(x_{1},x_{2},x_{3})$ is specified by

  $$\begin{aligned} x_{1}=c\bar{p}\bar{\mu }\cos \gamma ,\quad x_{2}=c\bar{p}\bar{\mu }\sin \gamma ,\quad x_{3}=cp\mu , \end{aligned}$$

  (96)

  where $c=a/\mu$ is the distance from the focal point to the center of spheroidal surface. The prolate spheroidal coordinate system $(\alpha ,\beta ,\gamma )$ is relative to the corresponding prolate spheroidal orthonormal basis $\{{\mathbf {f}}_{\alpha },{\mathbf {f}}_{\beta },{\mathbf {f}}_{\gamma }\}$ given by

  $$\begin{aligned} {\mathbf {f}}_{1}\,\equiv\,& {} {\mathbf {f}}_{\alpha } = \frac{1}{\sqrt{\bar{p}^{2}+\bar{\mu }^{2}}}(\mu \bar{p}\cos \gamma {\mathbf {j}}_{1}+\mu \bar{p}\sin \gamma {\mathbf { j}}_{2}+\bar{\mu }p{\mathbf {j}}_{3}), \nonumber \\ {\mathbf {f}}_{2}\,\equiv\,& {} {\mathbf {f}}_{\beta } = \frac{1}{\sqrt{\bar{p}^{2}+\bar{\mu } ^{2}}}(\bar{\mu }p\cos \gamma {\mathbf {j}}_{1}+\bar{\mu }p\sin \gamma {\mathbf {j}} _{2}-\mu \bar{p}{\mathbf {j}}_{3}),\nonumber \\ {\mathbf {f}}_{3}\,\equiv\,& {} {\mathbf {f}}_{\gamma } = -\sin \gamma {\mathbf {j}}_{1}+\cos \gamma {\mathbf {j}}_{2}. \end{aligned}$$

  (97)

  The metric coefficients of the prolate spheroidal coordinate system $(\alpha ,\beta ,\gamma )$ are expressed as

  $$\begin{aligned} h_{1}\equiv h_{\alpha }= & {} c\sqrt{\mu ^{2}-p^{2}}\equiv 1/h,\quad h_{2}\equiv h_{\beta }=c\sqrt{\mu ^{2}-p^{2}}\equiv 1/h, \nonumber \\ h_{3}\equiv h_{\gamma }= & {} c\bar{\mu }\bar{p}\equiv 1/\bar{h}. \end{aligned}$$

  (98)

• Oblate spheroidal coordinate system

  Third, when the surface $\varGamma ^{(i)}$ exhibits an oblate spheroidal form, the appropriate orthogonal curvilinear coordinate system is naturally the oblate spheroidal coordinate one $(\alpha ,\beta ,\gamma )$ with $y_{1}\equiv \alpha \in [0,+\infty ]$, $y_{3}\equiv \gamma \in [0,2\pi ]$ and $y_{2}\equiv \beta \in [0,\pi ]$. As before, by defining the following additional notations:

  $$\begin{aligned} p= & {} \cos \beta ,\;\bar{p}=\sqrt{1-p^{2}}=\sin \beta ,\;\mu =\sinh \alpha ,\; \bar{\mu }=\sqrt{\mu ^{2}+1}=\cosh \alpha , \nonumber \\ \eta= & {} \iota \sinh \alpha =\iota \mu ,\;\bar{\eta }=\sqrt{\eta ^{2}-1} =\iota \cosh \alpha =\iota \bar{\mu },\;w=a/b, \end{aligned}$$

  (99)

  where $\iota =\sqrt{-1}$; b and a ($b>a$, or equivalently $w=a/b<1$) are the equatorial radius and the distance from center to pole along the symmetry axis of the spheroidal surface, respectively, the connection between the oblate spheroidal coordinates $(\alpha ,\beta ,\gamma )$ and the Cartesian coordinates $(x_{1},x_{2},x_{3})$ is specified as follows:

  $$\begin{aligned} x_{1}=c\bar{p}\bar{\mu }\cos \gamma ,\quad x_{2}=c\bar{p}\bar{\mu }\sin \gamma ,\quad x_{3}=cp\mu , \end{aligned}$$

  (100)

  where $c=a/\mu$ is the distance from the focal point to the center of spheroidal surface. The oblate spheroidal coordinate system $(\alpha ,\beta ,\gamma )$ is relative to the oblate spheroidal orthonormal basis $\{{\mathbf {f}} _{\alpha },{\mathbf {f}}_{\beta },{\mathbf {f}}_{\gamma }\}$ given by

  $$\begin{aligned} {\mathbf {f}}_{1}\,\equiv\, & {} {\mathbf {f}}_{\alpha }=\frac{1}{\sqrt{{p}^{2}+{\mu }^{2}}} (\mu \bar{p}\cos \gamma {\mathbf {j}}_{1}+\mu \bar{p}\sin \gamma {\mathbf {j}}_{2}+ \bar{\mu }p{\mathbf {j}}_{3}), \nonumber \\ {\mathbf {f}}_{2}\,\equiv\, & {} {\mathbf {f}}_{\beta }=\frac{1}{\sqrt{{p}^{2}+{\mu }^{2}}} (\bar{\mu }p\cos \gamma {\mathbf {j}}_{1}+\bar{\mu }p\sin \gamma {\mathbf {j}} _{2}-\mu \bar{p}{\mathbf {j}}_{3}),\nonumber \\ {\mathbf {f}}_{3}\,\equiv\, & {} {\mathbf {f}}_{\gamma }=-\sin \gamma {\mathbf {j}}_{1}+\cos \gamma {\mathbf {j}}_{2}. \end{aligned}$$

  (101)

  The metric coefficients of the oblate spheroidal coordinate system $(\alpha ,\beta ,\gamma )$ are determined by

  $$\begin{aligned} h_{1}\equiv h_{\alpha }= & {} c\sqrt{p^{2}-\eta ^{2}}=1/h,\;h_{2}\equiv h_{\beta }=c\sqrt{p^{2}-\eta ^{2}}=1/h, \nonumber \\ h_{3}\equiv h_{\gamma }= & {} -\iota c\bar{\eta }\bar{p}=1/\bar{h}. \end{aligned}$$

  (102)

Appendix 2: The expressions of $F_{nm}(\mu _{0})$, $L_{nm}(\mu _{0})$, $J_{nm}(\mu _{0})$ and $W_{nm}(\mu _{0})$ in (40)–(45) and (32)–(37) as well as the ones of $F_{nm}(\eta _{0})$, $L_{nm}( \eta _{0})$, $J_{nm}(\eta _{0})$ and $W_{nm}(\eta _{0})$ in (64)–(69) and (56 )–(61)

• Case of prolate spheroidal inclusion

  $$\begin{aligned} F_{nm}(\mu _{0})= & {} -\frac{2n+1}{2}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}p\bar{p }^{2}}{c\bar{\mu }_{0}\sqrt{(\mu _{0}^{2}-p^{2})^{3}}} \times P_{m}(\mu _{0})P_{m}^{ \prime }(p)P_{n}(p)\hbox {d}p \nonumber \\&+\frac{2n+1}{2}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}m(m+1)}{c\bar{\mu }_{0} \sqrt{\mu _{0}^{2}-p^{2}}}P_{m}(\mu _{0})P_{m}(p)P_{n}(p)\hbox {d}p, \end{aligned}$$

  (103)
  $$\begin{aligned} L_{nm}(\mu _{0})= & {} -\frac{2n+1}{2n(n+1)}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}p \bar{p}^{2}}{c\bar{\mu }_{0}\sqrt{(\mu _{0}^{2}-p^{2})^{3}}} P^{1}_{m}(\mu _{0})P_{m}^{1\prime }(p)P^{1}_{n}(p)\hbox {d}p \nonumber \\&+\frac{2n+1}{2n(n+1)}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}\sqrt{ \mu _{0}^{2}-p^{2}}}{c\bar{\mu }^{3}_{0}\bar{p}^{2}}P^{1}_{m}( \mu _{0})P^{1}_{m}(p)P^{1}_{n}(p)\hbox {d}p \nonumber \\&-\frac{2n+1}{2n(n+1)}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}}{c\bar{\mu }_{0} \sqrt{\mu _{0}^{2}-p^{2}}}P^{1}_{m}(\mu _{0})P^{1}_{n}(p)[-mP_{m+1}^{1 \prime }(p) \nonumber \\&+(m+1)P_{m}^{1}(p)+(m+1)pP_{m}^{1\prime }(p)]\hbox {d}p. \end{aligned}$$

  (104)
  $$\begin{aligned} J_{nm}(\mu _{0})= & {} -\frac{2n+1}{2}\int _{-1}^{1}\frac{\lambda C_{t}\bar{\mu } _{0}}{c\sqrt{\mu _{0}^{2}-p^{2}}}P^{\prime }_{m} (\mu _{0})P_{m}(p)P_{n}(p)\hbox { d}p, \end{aligned}$$

  (105)
  $$\begin{aligned} W_{nm}(\mu _{0})= & {} -\frac{2n+1}{2n(n+1)}\int _{-1}^{1}\frac{\lambda C_{t}\bar{ \mu }_{0}}{c\sqrt{\mu _{0}^{2}-p^{2}}} \times P^{1\prime }_{m}( \mu _{0})P^{1}_{m}(p)P^{1}_{n}(p)\hbox {d}p. \end{aligned}$$

  (106)

• Case of oblate spheroidal inclusion

  $$\begin{aligned} F_{nm}(\eta _{0})= & {} \frac{2n+1}{2}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}p\bar{p}^{2}}{c\bar{\eta }_{0}\sqrt{(p^{2} -\eta _{0}^{2})^{3}}}P_{m}(\eta _{0})P_{m}^{ \prime }(p)P_{n}(p)\hbox {d}p \nonumber \\&+\frac{2n+1}{2}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}m(m+1)}{c\bar{\eta }_{0} \sqrt{p^{2}-\eta _{0}^{2}}}P_{m}(\eta _{0})P_{m}(p)P_{n}(p)\hbox {d}p, \end{aligned}$$

  (107)
  $$\begin{aligned} L_{nm}(\eta _{0})= & {} \frac{2n+1}{2n(n+1)} \int _{-1}^{1}\frac{\lambda C_{q}k_{i}p \bar{p}^{2}}{c\bar{\eta }_{0}\sqrt{(p^{2}-\eta _{0}^{2})^{3}}} P^{1}_{m}(\eta _{0})P_{m}^{1\prime }(p)P^{1}_{n}(p)\hbox {d}p \nonumber \\&-\frac{2n+1}{2n(n+1)}\int _{-1}^{1}\frac{\lambda C_{q}k_{i}\sqrt{ p^{2}-\eta _{0}^{2}}}{c\bar{\eta }^{3}_{0}\bar{p}^{2}}P^{1}_{m}( \eta _{0})P^{1}_{m}(p)P^{1}_{n}(p)\hbox {d}p \nonumber \\&-\frac{2n+1}{2n(n+1)}\int _{-1}^{1} \frac{\lambda C_{q}k_{i}}{c\bar{\eta }_{0} \sqrt{p^{2}-\eta _{0}^{2}}}P^{1}_{m}(\eta _{0})P^{1}_{n}(p)[-mP_{m+1}^{1 \prime }(p) \nonumber \\&+(m+1)P_{m}^{1}(p)+(m+1)pP_{m}^{1\prime }(p)]\hbox {d}p. \end{aligned}$$

  (108)
  $$\begin{aligned} J_{nm}(\eta _{0})= & {} -\frac{2n+1}{2}\int _{-1}^{1} \frac{\lambda C_{t}\bar{\eta } _{0}}{c\sqrt{p^{2}-\eta _{0}^{2}}}P^{\prime }_{m}(\eta _{0})P_{m}(p)P_{n}(p) \hbox {d}p, \end{aligned}$$

  (109)
  $$\begin{aligned} W_{nm}(\eta _{0})= & {} -\frac{2n+1}{2n(n+1)} \int _{-1}^{1}\frac{\lambda C_{t}\bar{ \eta }_{0}}{c\sqrt{p^{2}-\eta _{0}^{2}}}P^{1\prime }_{m}( \eta _{0})P^{1}_{m}(p)P^{1}_{n}(p)\hbox {d}p. \end{aligned}$$

  (110)

Appendix 3: Expressions of the matrix $\left[ {\mathbf {Y}}_{1}\right]$ and $\left[ {\mathbf {Z}}_{1}\right]$ in (46) and $\left[ {\mathbf {Y}} _{2}\right]$ and $\left[ {\mathbf {Z}}_{2}\right]$ in (70)

$$\begin{aligned} \left[ {\mathbf {Y}}_{1}\right]= & {} \left[ \begin{array}{ccc} \left[ {\mathbf {Y}}_{1}^{(11)}\right] &{} \left[ {\mathbf {0}}\right] &{} \left[ {\mathbf {0}}\right] \\ \left[ {\mathbf {0}}\right] &{} \left[ {\mathbf {Y}}_{1}^{(22)}\right] &{} \left[ {\mathbf {0}}\right] \\ \left[ {\mathbf {0}}\right] &{} \left[ {\mathbf {0}}\right] &{} \left[ {\mathbf {Y}}_{1}^{(33)}\right] \\ &{} &{}\end{array}\right] ,\\ \left[ {\mathbf {Z}}_{1} \right]= & {} -[E_{3}^{0}cP_{1}(\mu _{0}),0, \ldots , 0, k_{0}E_{3}^{0}cP_{1}^{\prime }(\mu _{0}),0, \ldots , 0, E_{1}^{0}cP_{1}^{1}(\mu _{0}),0, \ldots , 0, k_{0}E_{1}^{0}cP_{1}^{1\prime }(\mu _{0}),0, \ldots , 0,E_{2}^{0}cP_{1}^{1}(\mu _{0}),0, \ldots , 0,k_{0}E_{2}^{0}cP_{1}^{1\prime }(\mu _{0}),0, \ldots , 0]^{T} \end{aligned}$$

where

$$\begin{aligned} \left[ {\mathbf {Y}}_{1}^{(11)} \right]= & {} \left[ \begin{array}{cccccccc} P_{1}(\mu _{0})-J_{11} &{} -J_{12} &{} \cdots &{} -J_{1N} &{} -Q_{1}(\mu _{0}) &{} 0 &{} \cdots &{} 0 \\ -J_{21} &{} P_{2}(\mu _{0})-J_{22} &{} \cdots &{} -J_{2N} &{} 0 &{}-Q_{2}(\mu _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -J_{N1} &{} -J_{N2} &{} \cdots &{} P_{N}(\mu _{0}) -J_{NN} &{} 0 &{}0 &{} \cdots &{} -Q_{N}(\mu _{0}) \\ k_{i}P_{1}^{\prime }(\mu _{0})-F_{11} &{} -F_{12} &{} \cdots &{} -F_{1N} &{} -k_{0}Q_{1}^{\prime }(\mu _{0}) &{} 0 &{} \cdots &{} 0 \\ -F_{21} &{} k_{i}P_{2}^{\prime }(\mu _{0})-F_{22} &{} \cdots &{} -F_{2N} &{} 0 &{}-k_{0}Q_{2}^{\prime }(\mu _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -F_{N1} &{} -F_{N2} &{} \cdots &{} k_{i}P_{N}^{\prime }(\mu _{0})-F_{NN} &{} 0 &{}0 &{} \cdots &{} -k_{0}Q_{N}^{\prime }(\mu _{0}) \end{array} \right] , \\ \left[ {\mathbf {Y}}_{1}^{(22)} \right]= & {} \left[ {\mathbf {Y}}_{1}^{(33)} \right] = \left[ \begin{array}{cccccccc} P_{1}^{1}(\mu _{0}) - W_{11} &{} -W_{12} &{} \cdots &{} -W_{1N} &{} -Q_{1}^{1}(\mu _{0}) &{} 0 &{} \cdots &{} 0 \\ -W_{21} &{} P_{2}^{1}(\mu _{0}) - W_{22} &{} \cdots &{} -W_{2N} &{} 0 &{}-Q_{2}^{1}(\mu _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -W_{N1} &{} -W_{N2} &{} \cdots &{} P_{N}^{1}(\mu _{0})-W_{NN} &{} 0 &{}0 &{} \cdots &{} -Q_{N}^{1}(\mu _{0}) \\ k_{i}P_{1}^{1\prime }(\mu _{0})-L_{11} &{} -L_{12} &{} \cdots &{} -L_{1N} &{} -k_{0}Q_{1}^{1\prime }(\mu _{0}) &{} 0 &{} \cdots &{} 0 \\ -L_{21} &{} k_{i}P_{2}^{1\prime }(\mu _{0})-L_{22} &{} \cdots &{} -L_{2N} &{} 0 &{}-k_{0}Q_{2}^{1\prime }(\mu _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -L_{N1} &{} -L_{N2} &{} \cdots &{} k_{i}P_{N}^{1\prime }(\mu _{0})-L_{NN} &{} 0 &{}0 &{} \cdots &{} -k_{0}Q_{N}^{1\prime }(\mu _{0}) \end{array} \right] , \end{aligned}$$

and $\left[ {\mathbf {0}}\right]$ is a null matrix. Note that $F_{nm}$, $L_{nm}$, $J_{nm}$ and $W_{nm}$ with $1\le n,m\le N$ are specified by Eqs. (103)–(106).

$$\begin{aligned} \left[ {\mathbf {Y}}_{2} \right]= & {} \left[ \begin{array}{ccc} \left[ {\mathbf {Y}}_{2}^{(11)} \right] &{} \left[ {\mathbf {0}} \right] &{} \left[ {\mathbf {0}} \right] \\ \left[ {\mathbf {0}} \right] &{} \left[ {\mathbf {Y}}_{2}^{(22)} \right] &{} \left[ {\mathbf {0}} \right] \\ \left[ {\mathbf {0}} \right] &{} \left[ {\mathbf {0}} \right] &{} \left[ {\mathbf {Y}}_{2}^{(33)} \right] \\ &{} &{}\end{array}\right] ,\\ \left[ {\mathbf {Z}}_{2} \right]= & {} \iota [E_{3}^{0}cP_{1}(\eta _{0}),0, \ldots , 0, k_{0}E_{3}^{0}cP_{1}^{\prime }(\eta _{0}),0, \ldots , 0, E_{1}^{0}cP_{1}^{1}(\eta _{0}),0, \ldots , 0, k_{0}E_{1}^{0}cP_{1}^{1\prime }(\eta _{0}),0, \ldots , 0,E_{2}^{0}cP_{1}^{1}(\eta _{0}),0, \ldots , 0,k_{0}E_{2}^{0}cP_{1}^{1\prime }(\eta _{0}),0, \ldots , 0]^{T} \end{aligned}$$

where

$$\begin{aligned} \left[ {\mathbf {Y}}_{2}^{(11)} \right]= & {} \left[ \begin{array}{cccccccc} P_{1}(\eta _{0}) - J_{11} &{} -J_{12} &{} \cdots &{} -J_{1N} &{} -Q_{1}(\eta _{0}) &{} 0 &{} \cdots &{} 0 \\ -J_{21} &{} P_{2}(\eta _{0}) -J_{22} &{} \cdots &{} -J_{2N} &{} 0 &{}-Q_{2}(\eta _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -J_{N1} &{} -J_{N2} &{} \cdots &{} P_{N}(\eta _{0})-J_{NN} &{} 0 &{}0 &{} \cdots &{} -Q_{N}(\eta _{0}) \\ k_{i}P_{1}^{\prime }(\eta _{0})-F_{11} &{} -F_{12} &{} \cdots &{} -F_{1N} &{} -k_{0}Q_{1}^{\prime }(\eta _{0}) &{} 0 &{} \cdots &{} 0 \\ -F_{21} &{} k_{i}P_{2}^{\prime }(\eta _{0})-F_{22} &{} \cdots &{} -F_{2N} &{} 0 &{}-k_{0}Q_{2}^{\prime }(\eta _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -F_{N1} &{} -F_{N2} &{} \cdots &{} k_{i}P_{N}^{\prime }(\eta _{0})-F_{NN} &{} 0 &{}0 &{} \cdots &{} -k_{0}Q_{N}^{\prime }(\eta _{0}) \end{array} \right] , \\ \left[ {\mathbf {Y}}_{2}^{(22)} \right]= & {} \left[ {\mathbf {Y}}_{2}^{(33)} \right] = \left[ \begin{array}{cccccccc} P_{1}^{1}(\eta _{0})-W_{11} &{} -W_{12} &{} \cdots &{} -W_{1N} &{} -Q_{1}^{1}(\eta _{0}) &{} 0 &{} \cdots &{} 0 \\ -W_{21} &{} P_{2}^{1}(\eta _{0})-W_{22} &{} \cdots &{} -W_{2N} &{} 0 &{}-Q_{2}^{1}(\eta _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -W_{N1} &{} -W_{N2} &{} \cdots &{} P_{N}^{1}(\eta _{0})-W_{NN} &{} 0 &{}0 &{} \cdots &{} -Q_{N}^{1}(\eta _{0}) \\ k_{i}P_{1}^{1\prime }(\eta _{0})-L_{11} &{} -L_{12} &{} \cdots &{} -L_{1N} &{} -k_{0}Q_{1}^{1\prime }(\eta _{0}) &{} 0 &{} \cdots &{} 0 \\ -L_{21} &{} k_{i}P_{2}^{1\prime }(\eta _{0})-L_{22} &{} \cdots &{} -L_{2N} &{} 0 &{}-k_{0}Q_{2}^{1\prime }(\eta _{0}) &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{}\vdots &{} \ddots &{} \vdots \\ -L_{N1} &{} -L_{N2} &{} \cdots &{} k_{i}P_{N}^{1\prime }(\eta _{0})-L_{NN} &{} 0 &{}0 &{} \cdots &{} -k_{0}Q_{N}^{1\prime }(\eta _{0}) \end{array} \right] , \end{aligned}$$

with $F_{nm}$, $L_{nm}$, $J_{nm}$ and $W_{nm}$ with $1\le n,m\le N$ provided by Eqs. (107)–(110).

Appendix 4: Derivation of Eqs. (73) and (74)

Due to the assumption that, in each step of the replace process, spheroidal gas-filled inclusions are randomly distributed and orientated into an isotropic porous material given from the previous step, the new porous material obtained will be also isotropic. By applying Eq. (15) together with Eq. (16), we obtain

$$\begin{aligned} k(t+\varDelta t){\mathbf {I}}=\,& {} k(t){\mathbf {I}} + {\mathcal {D}}k(t){\mathbf {I}} \nonumber \\= & {} k(t){\mathbf {I}} +\frac{c_{i}\varDelta t}{1+c_{i}\varDelta t}[k_i-k(t)]\oint [{\mathbf {R}}\cdot {\mathbf {F}} (\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))\cdot {\mathbf {R}}^{T}]\hbox {d}{\mathbf {n}} \end{aligned}$$

(111)

where $k(t+\varDelta t)$ is the effective thermal conductivity of the new isotropic porous material while k(t) is the effective thermal conductivity of the already constructed isotropic porous material of the previous step and $\oint \bullet \hbox {d}{\mathbf {n}}$ denotes the average over all randomly spatial orientations of spheroidal gas-filled inclusions which is defined by

$$\begin{aligned} \oint \bullet \hbox {d}{\mathbf {n}} = \frac{1}{8\pi ^{2}}\int _{\phi =0}^{2\pi }\int _{\theta =0}^{\pi } \int _{\gamma =0}^{2\pi }\bullet \sin {\theta }\hbox {d} \phi \hbox {d}\theta \hbox {d}\gamma . \end{aligned}$$

(112)

In addition, ${\mathbf {R}}$ in Eq. (111) is the rotation matrix between the global coordinate basis $\{{\mathbf {j}}_{1},{\mathbf {j}}_{2}, {\mathbf {j}}_{3}\}$ and the local coordinate basis $\{{\mathbf {j}}^{\prime }_{1},{\mathbf {j}}^{\prime }_{2}, {\mathbf {j}}^{\prime }_{3}\}$ associated to an arbitrary orientated inclusion and takes the following form

$$\begin{aligned} {\mathbf {R}}(\phi ,\theta ,\gamma ) = \begin{bmatrix} \cos \phi \sin \theta&\quad \sin \phi \sin \theta&\quad \cos \theta \\ \cos \phi \cos \theta \cos \gamma - \sin \phi \sin \gamma&\quad \sin \phi \cos \theta \cos \gamma +\sin \gamma \cos \phi&\quad -\sin \theta \cos \gamma \\ -\sin \gamma \cos \phi \cos \theta -\cos \gamma \sin \phi&\quad -\sin \phi \sin \gamma \cos \theta +\cos \gamma \cos \phi&\quad \sin \gamma \sin \theta \end{bmatrix} \end{aligned}$$

(113)

and ${\mathbf {R}}^{T}$ designates the transpose of the matrix ${\mathbf {R}}$; ${\mathbf {R}}\cdot {\mathbf {F}}(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))\cdot {\mathbf {R}}^{T}$ is the transformation of the generalized localization tensor ${\mathbf {F}}(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))$ after changing from the local coordinate system to the global coordinate one. It is clear that Eq. (111) is equivalent to

$$\begin{aligned} k(t+\varDelta t)= & {} k(t)+ {\mathcal {D}}k(t) = k(t) +\frac{c_{i}\varDelta t}{1+c_{i}\varDelta t}[k_i-k(t)]f(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t)) \end{aligned}$$

(114)

where $f(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))$ is given by

$$\begin{aligned} f(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t)) = \frac{1}{3}\oint \hbox {Tr}[{\mathbf {R}}\cdot {\mathbf {F}}(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))\cdot {\mathbf {R}}^{T}]\hbox {d}{\mathbf {n}}. \end{aligned}$$

(115)

Since $\hbox {Tr}[{\mathbf {R}}\cdot {\mathbf {F}}\cdot {\mathbf {R}}^{T}] = \hbox {Tr}[{\mathbf {F}}]$, the expression (115) of $f(\hbox {Kn},\alpha , \alpha _E, \xi ,k_i/k(t))$ reduces so that to (74).