We present the full derivation of a free surface pipe or open channel model including friction, changes of section and changes of slope/topography. This 1D free surface model (also named \textbf{FS}-model) is obtained from the incompressible 3D Navier-Stokes equation under shallow water assumptions with well-suited boundary conditions. It was introduced in the general framework of unsteady mixed flows in closed water pipes and is largely used by the engineers community.