Least-mean-squares algorithm to determine submicrometer particle diameter, volume fraction, and size distribution width by elastic light scattering

Abstract

A computationally fast method to determine values and their uncertainty for particulate system volume median diameter, volume fraction, and size distribution width is presented. These properties cannot be obtained for submicrometer particulate by diffraction-based methods. The technique relies on a least-mean-squares method applied over a prespecified size range and distribution width. Prespecifying the range significantly reduces the number of calculations required to determine the particulate parameters from experimental data, allowing the practical evaluation of large data sets. The solution method that was developed has significant advantages over ratio-style calculations that are more commonly performed, the primary of which is a simple method to determine errors in the measurement parameters. We evaluated the predicted performance for a specific experimental system for various levels of noise, with monodisperse and log-normal distributions, by analyzing synthetic data with the algorithm. Results were a quantitative statement of system accuracy. In addition, synthetic log-normal data evaluated with monodisperse models revealed significant and systematic errors in the predicted volume median diameter. These errors indicate that, in general, systems with a significant size distribution width must be analyzed with a model that includes this size distribution. Finally, calibrated polystyrene spheres were measured with an experimental system that used four simultaneous scattering measurements, and all diameters were within the reported uncertainty.

Elastic light scattering measurements on non-uniform particle flow fields: theoretical considerations

“…These errors deteriorate the sizing performance of the diagnostics system and must be estimated from both the ratios a mn / n∞ and the transfer coefficients w mn,j for all detectors, if a quantitative statement about the sizing quality is to be made. By including estimates of this error in the sizing procedure, the effect of flow field variability is propagated through to the sizing results in a similar way as uncertainties in other input parameters (for an example, see [21]). A particular annoyance is that the errors in a multi-angle system are typically different for each angle affecting not only the uncertainty of the magnitude parameter n∞ but also that of the shape parameters θ. Estimation of the errors due to nonuniformity is based on ( 5) and ( 6) (to compute the transfer coefficients) and evaluation of flow field images such as figures 1 and 2 and or the structure of the field-generating device (arrangement of diffusion elements on the burner surface) for the ratios a mn / n∞ .…”