In this paper, we tackle the modeling and numerical simulation of dilute gas-droplet and gas-particle flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particle experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for more inertial particles, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive size moment of order 0 to N, N>2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment method resolved with high order finite volume method: the transport algorithm does not preserve the moment space Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. To achieve still a robust scheme, one possibility is to use projections onto the moment space but this leads to a substantial loss of accuracy and performance. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. By coupling this approach to a recently designed algorithm for evaporation which also preserves the moment space, polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. We show that such an approach is very competitive compared to multi-fluid approches where the size phase space is discretized into several sections and low order moment methods are used in each section. Two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.