More and more evidence has shown that, at low Rayleigh numbers, the natural convective heat transfer from vertical isothermal channel walls is higher than the widely accepted fully-developed asymptote. This paper attempts to analyze the reason for this discrepancy by numerically solving the full Navier-Stokes equations and comparing with the results obtained from the parabolic type of equations. Calculations are made for the air on a wide range of the Rayleigh number and the channel aspect ratio. It is found that the vertical diffusion contained in elliptic formulation becomes important at a low Rayleigh number and cannot be neglected. The fully-developed asymptote supported by the parabolic solution appears to underpredict the Nusselt number except for special conditions.