of a uniform UV/X-ray background radiation [@2001cghr.confE..64H]; the feedback by supernovae, as originally prescribed by [@2003MNRAS.339..289S]; the chemical model by [@2007MNRAS.382.1050T] to account for the metal enrichment (for further details see @2014MNRAS.438..195P). The wind velocity associated to the kinetic supernovae feedback is 350 km s$^{-1}$.

[$\bullet$  [[[[agn]{}]{}]{}]{}.]{} This last run is built on the [[[csf]{}]{}]{} simulations with the addition of the AGN feedback, modeled following [@steinborn.etal.2015] who improved the original model by [@springel05] by introducing a detailed and phenomenologically-driven description. The thermodynamical and metal properties and the dynamical state of the main clusters of this sample are presented also in [@2015ApJ...813L..17R], [@planelles.etal.2016], and [@biffi.etal.2016].

The cosmological setting is a $\Lambda CDM$ model with the matter density $\Omega_m=0.24$, the dark energy density $\Omega_{\Lambda}=0.76$, $n_s=0.96$, the amplitude of the density fluctuation power spectrum $\sigma_8=0.8$, and the Hubble parameter $H=72$ km s$^{-1}$ Mpc$^{-1}$. The observational data, presented in Section 4, are rescaled to the simulated cosmology.

The Plummer gravitational softening of the DM particles is equivalent to 3.75 $h^{-1}\ \rm{kpc}$ in physical units up to $z=2$ and in comoving units at higher redshifts. The gas softening is, instead, fixed to the same value in comoving coordinates at all times. The mass of the 