We investigate the constant-wall-temperature convective heat-transfer characteristics of a model gaseous flow in two-dimensional micro and nano channels under hydrodynamically and thermally fully developed conditions. Our investigation covers both the slip-flow regime 0 ≤ Kn ≤ 0.1, and most of the transition regime 0.1 < Kn < 10, where Kn, the Knudsen number, is defined as the ratio between the molecular mean free path and the channel height. We use slip-flow theory in the presence of axial heat conduction to calculate the Nusselt number in the range 0 ≤ Kn ≤ 0.2, and a stochastic molecular simulation technique known as the direct simulation Monte Carlo (DSMC) to calculate the Nusselt number in the range 0.02 < Kn < 2. Inclusion of the effects of axial heat conduction in the continuum model is necessary since small-scale internal flows are typically characterized by finite Peclet numbers. Our results show that the slip-flow prediction is in good agreement with the DSMC results for Kn ≤ 0.1, but also remains a good approximation beyond its expected range of applicability. We also show that the Nusselt number decreases monotonically with increasing Knudsen number in the fully accommodating case, and that axial heat conduction increases the Nusselt number. The effect of axial heat conduction is largest at Kn = 0 and is of the order of 10%.