A first attempt at deriving a fully Eulerian model for polydisperse evaporating sprays was developed by Tambour et al with the so-called sectional approach. However, the complete derivation of the sectional ‘multi-fluid' conservation equations from the Boltzmann-type spray equation was never provided, neither was the set of underlying assumptions nor the comparison with the classical Lagrangian model: the sampling method. In this paper, we clarify the set of assumptions necessary in order to derive the multi-fluid sectional model from the spray equation at the ‘kinetic level' and provide the derivation of the whole set of conservation equations describing the dispersed liquid phase. Whereas the previous derivation is conducted in any space dimension, we restrict ourselves to one-dimensional stationary flows where the droplets do not turn back and derive a Eulerian sampling model which is equivalent in this context to the usual Lagrangian particle approach. We then identify some situations, even within this restrictive framework, where the sectional approach fails to reproduce the coupling of the vaporization and dynamics of the spray, the sampling method then being required. In the domain of applicability of the sectional approach, the two methods are then compared numerically in the configuration of counterflow spray diffusion flames. The two methods, if refined enough, give quite similar results, except for some small differences, the origin of which is identified. It is proved that the sampling method is more precise even if it generates oscillations due to the intrinsic representation of a continuous function by Dirac delta functions. We thus provide a comprehensive analysis of the sectional approach from both the modelling and numerical points of view.