In this paper, the steady temperature distribution and the Nusselt numbers are analytically determined for a Newtonian incompressible fluid in a rectangular duct, in fully developed laminar flow with viscous dissipation, for any combination of heated and adiabatic sides of the duct, in H1 boundary condition, and neglecting the axial heat conduction in the fluid. The Navier-Stokes and the energy balance equations are solved using the technique of the finite integral transforms. For a duct with four uniformly heated sides (4 version), the temperature distribution and the Nusselt numbers are obtained as a function of the aspect ratio and of the Brinkman number and presented in graphs and tables. Finally it is proved that the temperature field in a fully developed T boundary condition can be obtained as a particular case of the H1 problem and that the corresponding Nusselt numbers do not depend on the Brinkman number.