The conjugate heat transfer process of cooling a horizontal plate at the leading edge, in steady state condition, was solved considering the fluid flowing in laminar condition and hydro dynamically developed before interacting with a heated plate. The fluid was considered deep enough to allow the growth of a thermal boundary layer with no restrictions. The conservation of mass, momentum and energy equations at the solid and fluid were converted into a non dimensional form. The heated body presents a constant heat flux at the bottom side, and convective heat transfer at the top side. The interface temperature was obtained using the Chebyshev polynomial approximation. In order to verify the results obtained using the Chebyshev polynomial approximation, the results obtained from the analytical solution for the solid, were compared with the results attained with commercial CFD software, FIDAP®. The solution considered the calculation of the local and average heat transfer coefficient, the local and average Nusselt number, the local and average Biot number, and different temperature distributions at the interface.