# 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

$T\big|_{}=T_e\big|_A \,$

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.

## References

• H. S. Carslow, J. C. Jaeger (1959). Conduction of heat in solids. Oxford: Clarendon Press.
• M. Shillor, M. Sofonea, J. J. Telega (2004). Models and analysis of quasistatic contact. Variational methods. Berlin: Springer.