# Perfect thermal contact

Perfect thermal contact of the surface of a solid with the environment (convective heat transfer) or another solid occurs when the temperatures of the mating surfaces are equal.

## Perfect thermal contact conditions

Perfect thermal contact supposes that on the boundary surface $A$ there holds an equality of the temperatures

$T{\big |}_{A}=T_{e}{\big |}_{A}\,$ and an equality of heat fluxes

$-k{\frac {\partial T}{\partial n}}{\bigg |}_{A}=-k_{e}{\frac {\partial T_{e}}{\partial n}}{\bigg |}_{A}\,$ where $T,~T_{e}$ are temperatures of the solid and environment (or mating solid), respectively; $k,~k_{e}$ are thermal conductivity coefficients of the solid and mating laminar layer (or solid), respectively; $n$ is normal to the surface $A$ .

If there is a heat source on the boundary surface $A$ , e.g. caused by sliding friction, the latter equality transforms in the following manner

$-k{\frac {\partial T}{\partial n}}{\bigg |}_{A}+k_{e}{\frac {\partial T_{e}}{\partial n}}{\bigg |}_{A}=q\,$ where $q$ is heat-generation rate per unit area.