Non-Newtonian fluids flow are very common in industrial processes, so it is important to know both the effect of the process on the fluid and vice-versa. Non-Newtonian Fluids exhibit complex mechanical behavior not found in Newtonian fluids, such as shear-rate-dependent viscosity and non-zero Yield stress. Nowadays there is a lack of understanding in the literature of the interaction among non-Newtonian fluids and different flow geometries, particularly as far as heat transfer is concerned. Some geometries are found more frequently in industrial processes, being, accordingly, a more frequent subject of research. Among these are the tubes and annuli. Most of the published articles about this subject are analytical studies or numerical simulations, while those based on experimental investigations are rather scarce. This work is focused in the evaluation of the thermal behavior of oil wells during the flow of the drilling fluid. The effect of fluid rheology on heat transfer in annular spaces and circular tubes was investigated experimentally. The purpose was to determine the convective heat transfer coefficient (Nusselt number). The boundary conditions for the annuli were uniform heat flux at the inner wall and adiabatic outer wall, while, for the tube, the heat flux at the wall was kept constant and uniform. To mimic the drilling fluid mechanical behavior, the working fluids were viscoplastic liquids at different concentrations. For the annuli, different radius ratios were studied. The experimental results showed that, for laminar and fully developed flow in the annuli, the fluid rheology does not affect the Nusselt number, which is governed by the radius ratio only. These results are in agreement with recently published theoretical predictions, and the main contribution of this work is to confirm this surprising result, which renders simpler the projects involving non-Newtonian fluids flowing in annuli under the thermal boundary conditions investigated.