Laminate conductivity ($a^{\ast }=a_{H}n\otimes n+a_A(I-n\otimes n)$)

For isotropic phases with planar interfaces normal to $n$, normal conductivity is the weighted harmonic mean $a_H=(\sum p_i/a_i{)}^{-1}$ and tangential conductivity is the weighted arithmetic mean $a_A=\sum p_i{a}_i$. Continuity of normal flux and tangential temperature gradient gives the two averages directly.