Bi-abundance photoionization models of planetary nebulae: determining
  the amount of Oxygen in the metal rich component
  We study the hypothesis of high metallicity clumps being responsible for the
abundance discrepancy found in planetary nebulae between the values obtained
from recombination and collisionaly excited lines. We generate grids of
photoionization models combining cold metal-rich clumps emitting the heavy
element recombination lines, embedded in a normal metallicity region
responsible for the forbidden lines. The two running parameters of the grid are
the metallicity of the clumps and its volume fraction relative to the whole
nebula. We determine the density and temperatures (from the Balmer jump and the
[OIII] 5007/4363 A line ratio), and the ionic abundances from the collisional
and recombination lines, as an observer would do. The metallicity of the
near-to-solar region is recovered, while the metallicity of the clumps is
systematically underestimated, by up to 2 orders of magnitude. This is mainly
because most of the H$\beta$ emission is coming from the "normal" region, and
only the small contribution emitted by the metal-rich clumps should be used. We
find that a given ADF(O$^{++}$) can be reproduced by a small amount of rich
clumps, or a bigger amount of less rich clumps. Finally, comparing with the
observations of NGC 6153 we find 2 models that reproduce its ADF(O$^{++}$) and
the observed electron temperatures. We determine the fraction of oxygen
embedded in the metal-rich region (with a fraction of volume less than 1%) to
be roughly between 25% and 60% of the total amount of oxygen in the nebula (a
few 10$^{-3} M_\odot$).
