Table 1 The summary of the equations for the homogenization of conduction and radiative transfer equations

T^ε

\(T^{\varepsilon } \left( {\mathbf{x}} \right) = T_{0} \left( {{\mathbf{x}},{\mathbf{y}}} \right) + \varepsilon T_{1} \left( {{\mathbf{x}},{\mathbf{y}},T_{0} } \right) + \varepsilon^{2} T_{2} \left( {{\mathbf{x}},{\mathbf{y}},T_{0} } \right)\)  

T₀

\(\frac{\partial }{{\partial x_{i} }}\left[ {K_{ij} \left( {T_{0} } \right)\frac{{\partial T_{0} }}{{\partial x_{j} }}} \right] - 4\overline{{\alpha \left( {T_{0} } \right)}} \sigma_{{\text{B}}} T_{0}^{4} + \overline{{\alpha \left( {T_{0} } \right)}} \int_{4\pi } {I_{0} {\text{d}}\Omega } = 0,\)

\(K_{ij}=\frac1{\left|Y\right|}\int\left[k_{ij}^\varepsilon\left(\mathbf y,T_0\right)+k_{i\alpha}^\varepsilon\left(\mathbf y,T_0\right)\frac{\partial N_j\left(\mathbf y,T_0\right)}{\partial y_\alpha}\right]\text{d}\mathbf y\) 

T₁

\(T_{1} \left( {{\mathbf{x}},{\mathbf{y}},T_{0} } \right) = N_{i} \left( {{\mathbf{y}},T_{0} } \right)\frac{{\partial T_{0} \left( {\mathbf{x}} \right)}}{{\partial x_{i} }}\)

N_α

\(\frac{\partial }{{\partial y_{i} }}\left( {k_{ij}^{\varepsilon } \left( {{\mathbf{y}},T_{0} } \right)\frac{{\partial N_{\alpha } \left( {{\mathbf{y}},T_{0} } \right)}}{{\partial y_{j} }}} \right) = - \frac{{\partial k_{i\alpha }^{\varepsilon } \left( {{\mathbf{y}},T_{0} } \right)}}{{\partial y_{i} }}\)

T₂

\(T_{2} = M_{\alpha \beta } \left( {{\mathbf{y}},T_{0} } \right)\frac{{\partial^{2} T_{0} }}{{\partial x_{\alpha } \partial x_{\beta } }} + P_{\alpha \beta } \left( {{\mathbf{y}},T_{0} } \right)\frac{{\partial T_{0} }}{{\partial x_{\alpha } }}\frac{{\partial T_{0} }}{{\partial x_{\beta } }} + C\left( {{\mathbf{y}},T_{0} } \right)\left( {4\sigma_{{\text{B}}} T_{0}^{4} - \int_{4\pi } {I_{0} {\text{d}}\Omega } } \right)\)

M_αβ

\(\frac{\partial }{{\partial y_{i} }}\left( {k_{ij}^{\varepsilon } \frac{{\partial M_{\alpha \beta } }}{{\partial y_{j} }}} \right) = - \left[ {k_{\alpha \beta }^{\varepsilon } - K_{\alpha \beta } + k_{\alpha i}^{\varepsilon } \frac{{\partial N_{\beta } }}{{\partial y_{i} }} + \frac{{\partial \left( {k_{\alpha i}^{\varepsilon } N_{\beta } } \right)}}{{\partial y_{i} }}} \right]\)

P_αβ

\(\frac{\partial }{{\partial y_{i} }}\left( {k_{ij}^{\varepsilon } \frac{{\partial P_{\alpha \beta } }}{{\partial y_{j} }}} \right) = - \left[ {\frac{{\partial k_{\alpha \beta }^{\varepsilon } }}{\partial T} + \frac{\partial }{\partial T}\left( {k_{\alpha i}^{\varepsilon } \frac{{\partial N_{\beta } }}{{\partial y_{i} }}} \right) - \frac{{\partial K_{\alpha \beta } }}{\partial T}} \right] - \frac{\partial }{{\partial y_{i} }}\left[ {\frac{{\partial \left( {k_{\alpha i}^{\varepsilon } N_{\beta } } \right)}}{\partial T} + N_{\beta } \frac{{\partial k_{ij}^{\varepsilon } }}{\partial T}\frac{{\partial N_{\alpha } }}{{\partial y_{j} }}} \right]\)

C

\(\frac{\partial }{{\partial y_{i} }}\left( {k_{ij}^{\varepsilon } \frac{\partial C}{{\partial y_{j} }}} \right) = \alpha^{\varepsilon } - \overline{\alpha }\)

I^ε

\(I^{\varepsilon } \left( {{\mathbf{x}},\Omega } \right) = I_{0} \left( {{\mathbf{x}},{\mathbf{y}},\Omega } \right) + \varepsilon I_{1} \left( {{\mathbf{x}},{\mathbf{y}},\Omega } \right)\)

I₀

\(\Omega_{i} \frac{{\partial I_{0} }}{{\partial x_{i} }} = - \overline{{\beta \left( {T_{0} } \right)}} I_{0} + \frac{{\overline{{\alpha \left( {T_{0} } \right)}} \sigma_{{\text{B}}} }}{\pi }T_{0}^{4} + \frac{1}{4\pi }\int_{4\pi } {I_{0} \left( {{\mathbf{x}},\Omega^{\prime}} \right)\overline{{\sigma \left( {T_{0} } \right)\Phi }} \left( {\Omega^{\prime},\Omega } \right){\text{d}}\Omega^{\prime}},\)

\(\overline{\varphi\left(T_0\right)}=\frac1{\left|Y\right|}\int\varphi\left(\mathbf y,T_0\right)\text{d}\mathrm y,\;\varphi\;\mathrm{is}\;\alpha,\;\beta\;\mathrm{or}\;\sigma\mathrm\Phi\)

I₁

\(\Omega_{i} \frac{{\partial I_{1} }}{{\partial y_{i} }} = - \left( {\beta^{\varepsilon } \left( {T_{0} } \right) - \overline{{\beta \left( {T_{0} } \right)}} } \right)I_{0} + \frac{{\left( {\alpha^{\varepsilon } \left( {T_{0} } \right) - \overline{{\alpha \left( {T_{0} } \right)}} } \right)\sigma_{{\text{B}}} }}{\pi }T_{0}^{4} + \frac{1}{4\pi }\int_{4\pi } {I_{0} \left( {{\mathbf{x}},\Omega^{\prime}} \right)\left( {\sigma^{\varepsilon } \left( {T_{0} } \right)\Phi^{\varepsilon } \left( {T_{0} } \right) - \overline{{\sigma \left( {T_{0} } \right)\Phi }} } \right){\text{d}}\Omega^{\prime}}\)