  Models of chemical evolution of galaxies including the dust are nowadays
required to decipher the high-z universe. In a series of three papers we have
tackled the problem and set a modern chemical evolution model. In the first
paper (Piovan et al., 2011a) we revised the condensation coefficients for the
elements that typically are present in the dust. In the second paper (Piovan et
al., 2011b) we have implemented the dust into the Padova chemical model and
tested it against the observational data for the Solar Neighbourhood. In this
paper we extend it to the whole Disk of the Milky Way (MW). The Disk is used as
a laboratory to analyze the spatial and temporal behaviour of (i) several dust
grain families with the aid of which we can describe the ISM, (ii) the
abundances in the gas, dust, and total ISM of the elements present in the dust
and (iii) the depletion of the same elements. The temporal evolution of the
dust and gas across the Disk is calculated under the effect of radial flows and
a central Bar. The gradients of the abundances of C, N, O, Mg, Si, S, Ca and Fe
in gas and dust across the Disk are derived as a function of time. The
theoretical gradients nicely reproduce those derived from Cepheids, OB stars,
Red Giants and HII regions. This provides the backbone for the companion
processes of dust formation and evolution across the Disk. We examine in detail
the contributions to dust by AGB stars, SNae and grain accretion in the ISM at
different galacto-centric distances. Furthermore, we examine the variation of
the ratio silicates/carbonaceous grains with time and position in the Disk.
Finally, some hints about the depletion of the elements in regions of high and
low SFR (inner and outer Disk) are presented. The results obtained make it
possible to extend the model to other astrophysical situations or different
theoretical models like the chemo-dynamical N-Body simulations.
