Abstract: The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations for droplets of various sizes. The purpose of the present study is two-fold. First, we extend the multi-fluid model in order to cope with droplet trajectory crossings by using the quadrature method of moments in velocity phase space conditioned by size. We identify the numerical difficulties and provide dedicated numerical schemes in order to preserve the velocity moment space. Second, we conduct a comparison study and demonstrate the capability of such an approach to capture the dynamics of an evaporating polydisperse spray in a two-dimensional free jet configuration. We evaluate the accuracy and computational cost of Eulerian models and related discretization schemes versus Lagrangian solvers. It shows that, even for finite Stokes number, the standard Eulerian multi-fluid model is accurate at reasonable cost.