  We used the Density Functional formalism, in particular the Scaled Particle
Theory, applied to a length-polydisperse hard-rectangular fluid to study its
phase behavior as a function of the mean particle aspect ratio ($\kappa_0$) and
polydispersity ($\Delta_0$). The numerical solutions of the coexistence
equations were calculated by transforming the original problem with infinite
degrees of freedoms to a finite set of equations for the amplitudes of the
Fourier expansion of the moments of the density profiles. We divided the study
into two parts: The first one is devoted to the calculation of the phase
diagrams in the packing fraction ($\eta_0$)- $\kappa_0$ plane for a fixed
$\Delta_0$ and selecting parent distribution functions with exponential (the
Schulz distribution) or Gaussian decays. In the second part we study the phase
behavior in the $\eta_0$-$\Delta_0$ plane for fixed $\kappa_0$ while $\Delta_0$
is changed. We characterize in detail the orientational ordering of particles
and the fractionation of different species between the coexisting phases. Also
we study the character (second vs. first order) of the Isotropic-Nematic phase
transition as a function of polydispersity. We particularly focused on the
stability of the Tetratic phase as a function of $\kappa_0$ and $\Delta_0$. The
Isotropic-Nematic transition becomes strongly of first order when
polydispersity is increased: the coexisting gap widens and the location of the
tricritical point moves to higher values of $\kappa_0$ while the Tetratic phase
is slightly destabilized with respect to the Nematic one. The results obtained
here can be tested in experiments on shaken monolayers of granular rods.
