  Boiling is an extremely effective way to promote heat transfer from a hot
surface to a liquid due to several mechanisms many of which are not understood
in quantitative detail. An important component of the overall process is that
the buoyancy of the bubbles compounds with that of the liquid to give rise to a
much enhanced natural convection. In this paper we focus specifically on this
enhancement and present a numerical study of the resulting two-phase
Rayleigh-B\'enard convection process. We make no attempt to model other aspects
of the boiling process such as bubble nucleation and detachment. We consider a
cylindrical cell with a diameter equal to its height. The cell base and top are
held at temperatures above and below the boiling point of the liquid,
respectively. By keeping the temperature difference constant and changing the
liquid pressure we study the effect of the liquid superheat in a Rayleigh
number range that, in the absence of boiling, would be between $2\times10^6$
and $5\times10^9$. We find a considerable enhancement of the heat transfer and
study its dependence on the bubble number, the degree of superheat of the hot
cell bottom and the Rayleigh number. The increased buoyancy provided by the
bubbles leads to more energetic hot plumes detaching from the hot cell bottom
and, as a consequence, the strength of the circulation in the cell is
significantly increased. Our results are in general agreement with recent
experimental results of Zhong et al., Phys. Rev. Lett. {\bf 102}, 124501 (2009)
for boiling Rayleigh-B\'enard convection.
