The study of flow in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants, where the principle issue of interest is convective heat transfer. For fully developed laminar flows, the problem can be characterised in terms of two coupled partial differential equations. In the case of perfectly electrically insulating boundaries, there is a well known analytical solution due to Shercliff, which provides the velocity and induced magnetic field profiles. In this paper, we demonstrate analytical solutions to and heat transfer problems for the Shercliff case in rectangular ducts and obtain temperature profiles and corresponding Nusselt numbers as functions of aspect ratio and Hartmann number.