Heat transfer in water at supercritical pressures has been investigated numerically using a one-dimensional modeling approach. A 1D plug flow model has been developed in order to make fast predictions of the bulk-fluid temperature in a tubular flow. The chosen geometry is a vertical tube with an inner diameter of 10 mm and a heated length of 4.0 m. The simulations concern a heated upward flow with an imposed wall temperature profile. Viscous effects, internal conduction and enthalpy changes due to a pressure gradient have been neglected after evaluation of the governing equations in dimensionless form. The resulting set of equations is closed using Nusselt correlations found in literature and solved using an explicit Euler scheme to simulate heat transfer in a supercritical water flow. The results for three different cases show that the model is able to accurately predict the bulk temperature based on heat transfer rates provided by a suitable Nusselt correlation. However, there is also reason to assume that these correlations are very specific for the flow conditions, since boiling effects occurring at certain conditions can highly influence the heat transfer rate. As a consequence, the model may be unable to describe supercritical heat transfer over a broad range of configurations when only using one correlation. The agreement of these results with the two-dimensional simulations will be investigated in a separate article. The description of the model is preceded by a mathematical description of supercritical water flows and by an overview of the supercritical heat transfer phenomena as observed in earlier studies. (C) 2012 Elsevier B.V. All rights reserved.