At slightly supercritical pressure and in the neighborhood of the pseudocritical temperature (which corresponds to the peak in the 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 temperature is below the pseudocritical temperature and the wall temperature above the pseudocritical temperature. Equations have been developed to predict the deterioration in heat transfer at high heat fluxes and the results compared with previously available results for steam. Experiments have been performed with carbon dioxide for additional comparison. Limits of safe operation for a supercritical pressure heat exchanger in terms of the allowable heat flux for a particular flow rate have been determined theoretically and experimentally.