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.