The heat transfer characteristics of a Newtonian fluid in a two-dimensional, planar, right-angled tee branch are studied over a range of inlet Reynolds numbers and Grashof numbers. The flow and heat transfer equations, subject to the Boussinesq approximation, are solved using a finite-element discretization. The effects of the branch length and the grid size on the interior flow field are examined to assess the accuracy of the solutions. Results are presented for two types of experimentally realizable boundary conditions—equal exit pressure at the outlet of each branch and specified flow split between the branches. The thermal boundary condition of uniform wall temperature is examined. The effect of increasing Reynolds number is to increase the size and strength of the recirculation zones in both the main and side branches, while that of increasing Grashof number is to decrease such an effect. For the case of equal exit pressures there is a significant flow reversal in the side branch and the exit flow rate from the main branch increases linearly with increasing Gr/Re2. For the case of specified flow split, an increasing back pressure is required to be maintained at the exit of the main branch to regulate the flow split at the desired level.