Determining cement constituent proportions from XRF-based oxide data and LOI: a hybrid approximation method for Portland cements with limestone additions

A hybrid approximation method, combining a numerical method with an algebraic solution and probabilistic evaluation, is proposed to determine the mass fractions of clinker (CLK), limestone (LS) and gypsum (GPS) in Portland–limestone cements (PLCs) based on the characteristic oxide compositions of each constituent: sulfur trioxide for GPS, loss on ignition (LOI) for LS and GPS, and calcium oxide for GPS, LS and CLK. In this method, oxides are systematically sampled within statistically defined ranges to generate coefficient combinations for a 3 × 3 linear system. Each system is solved algebraically using Cramer’s rule, with determinants evaluated using Sarrus’ rule, while singular or near-singular matrices are discarded. Solutions are filtered for physical feasibility: all mass fractions are non-negative, LS may be zero and extreme disproportionate solutions are excluded. Probabilistic weighting of valid solutions accounts for oxide variability, enabling estimation of expected values and uncertainties. The method is grounded entirely in data obtained from X-ray fluorescence (XRF). It is particularly useful when X-ray diffraction data are unavailable or when the main cement phases are amorphous, as it relies solely on oxide compositions rather than crystalline phase identification. The approach not only reveals the constituent mass fractions but also indirectly provides the oxide composition analysis of the cement, allowing inferences about the origin and characteristics of each constituent. By integrating numerical exploration, an exact algebraic solution and statistical evaluation within a single framework, this hybrid approach provides a computationally efficient, numerically robust and statistically informed framework for approximating the mass fractions of PLC constituents.