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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$β$ 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$^{-3} M_\odot$).