On the uncertainty evaluation of Moment-Ratio mean diameters for multimodal distribution

Abstract

The uncertainty in Moment-Ratio (M-R) mean diameters for multi-modal or multi-peak particle systems has not been fully addressed within a comprehensive theoretical framework. This study presents a generalized approach that utilizes Gaussian decomposition and error propagation principles to tackle this issue. For a particle size distribution (PSD) expressed as a linear combination of independent log-normal distributions (LNDs), an expression for the overall M-R mean diameter is derived, based on the distribution parameters and weight coefficients of each LND. Since the terms in this expression are independent, the uncertainty in the M-R mean diameter for a multi-modal distribution is quantified through error propagation. Incorporating Masuda’s classical theory, a theoretical formula is developed. Numerical simulations validate the proposed theory, demonstrating high accuracy in estimating uncertainties for $D_{3,2}$ and $D_{4,3}$, with some underestimation observed for $D_{1,0}$ and $D_{3,0}$.