In this study, an analysis of laminar forced convection in a plane duct for a power-law fluid with constant thermophysical properties is performed by taking the viscous dissipation into account. This part of the study, Part 1, examines both hydrodynamically and thermally fully developed flow case. Two different thermal boundary conditions are considered: the constant heat flux (HI boundary condition) and the constant wall temperature (T boundary condition). The combined and interactive influences of the power-law index and the Brinkman number on temperature distribution and the Nusselt numbers are analytically determined both for the wall heating and cooling cases at T and H1 boundary conditions. Singularities are observed in Nu-Br-n behaviors and these are discussed in terms of the energy balance.