In this paper, an Eulerian model for single-species aerosol production and transport is introduced, and solved using the Pressure Implicit with Splitting of Operators (PISO) algorithm. The aerosol droplets are described in terms of two moments of the droplet size distribution, i.e., the droplet number concentration and the liquid mass fraction. The compressible PISO algorithm for reacting flows is extended to incorporate the transport equations of these two moments. The scheme is applied to the simulation of vapor-to-droplet conversion in a Laminar Flow Diffusion Chamber (LFDC). In that setting, we show the numerical properties of the method for, first, carrier gas flow without the presence of vapor or droplets, and second, the production and evolution of aerosol droplets through nucleation and condensation. The method is shown to be second order in time and space. We adopt a TVD scheme the handle unphysical oscillations that may arise near sharp nucleation fronts. Good agreement is found with experimental data, in terms of the predicted temperature centerline profile (within 1%) and LFDC outlet droplet number concentration. The detailed validation and analysis of the model in combination with PISO may be used for more advanced aerosol modeling.