Crystallization is a major separation technique used in numerous chemical processes, and its operation plays a major role in determining properties of crystalline products and the effectiveness of downstream processes. Real-time and in-line observation of the evolution of population density constitutes major asset in understanding crystallizer operation and the phenomena that influence product quality.

Focused-beam reflectance measurement (FBRM) is a technique based on backscattering of laser light that is useful in monitoring a representation of the crystalline population. Properly installed, the FBRM records the chord-length density (CLD), which is a function of crystal geometry and is related to the population density. While estimation of CLD from the population density is relatively straightforward, the inversion of this procedure is problematic because the problem may be ill-conditioned for non-spherical particles. The focus of the present work is on this inversion process.

The principal component analysis offers a straightforward, constraint-free and timely alternative to restore the population density accurately from chord-length density data. It also provides guidance on those aspects of the population density that can be restored accurately. Since the relationship between chord-length and population densities is a function of crystal geometry, this manuscript considers various non-spherical shapes, including highly challenging needles. We describe efficient methods to estimate the population density from the measured CLD, to analyze the inversion process, and to quantify its limits. Since noise is present in the actual measurements, the case of restoration of the population density from noisy CLDs also is demonstrated. It is thought that the presented methodology has great potential to be the basis for a control scheme that manipulates the population density produced from FBRM raw data.