Theoretical analysis of a method to measure size distributions of solid particles in water by aerosolization

Reliable measurement of the size-resolved number concentration (size distribution) of solid particles dispersed in water or melted ice is of critical importance in many geoscientific observational studies. Because physical and chemical properties of particles can be measured more unambiguously and accurately in rarefied media (air) than in condensed media (water), particle measurement after aerosolization using a nebulizer is a significant method for the observation of solid particles dispersed in water. We propose a mathematical theory for estimating the original size distribution of solid particles in water from the measured size distribution of aerosolized particles. We assume that the size distribution of water droplets produced by a nebulizer is given. The size distribution of solid particles in water can be estimated by solving a system of nonlinear equations. The complexity in solving the equations increases rapidly with the computational resolution of particle size and the assumed maximum number of particles within each droplet. For such a system of equations, we found rigorous error bounds of a true solution using INTLAB, an interval arithmetic package. Our theoretical framework will be useful in many fields in geoscience as a fundamental scheme to quantify solid particles in water. In particular, an application of the proposed theoretical method is shown to be useful for the quantitative observations of the size distribution of black carbon particles in rainwater.