# Perfect thermal contact

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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 ${\displaystyle A}$ there holds an equality of the temperatures

${\displaystyle T{\big |}_{}=T_{e}{\big |}_{A}\,}$

and an equality of heat fluxes

${\displaystyle -k{\frac {\partial T}{\partial n}}{\bigg |}_{A}=-k_{e}{\frac {\partial T_{e}}{\partial n}}{\bigg |}_{A}\,}$

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

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

${\displaystyle -k{\frac {\partial T}{\partial n}}{\bigg |}_{A}+k_{e}{\frac {\partial T_{e}}{\partial n}}{\bigg |}_{A}=q\,}$

where ${\displaystyle q}$ 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.