At slightly supercritical pressure and in the neighborhood of the pseudo-critical temperature (defined as the temperature corresponding to the peak in specific heat at the operating pressure), the heat transfer coefficient between fluid and tube wall is strongly dependent on the heat flux. For large heat fluxes, a marked deterioration takes place in the heat transfer coefficient in the region where the bulk fluid temperature is below and the wall temperature above the pseudo-critical temperature. An analysis has been developed, based on the integration of the transport equations, to predict the deterioration in heat transfer at high heat fluxes, and the results have been compared with the previously available experimental results for steam. Experiments have been performed with carbon dioxide for additional comparison. Limits of safe operation in terms of the allowable heat flux for a particular flow rate have been determined both theoretically and experimentally. Experiments with twisted tape inserted in the test section to generate swirl have shown that the heat transfer rates can be improved by this method. Qualitative visual observations have been made of the flow under varying conditions of heat flux and flow rate.