Incorporating the covariance effect in modelling batch flotation kinetics

Abstract

In this paper, recovery covariances were used to estimate the kinetic parameters from batch flotation tests to account for the lack of statistical independence and homoscedasticity in the cumulative recoveries. Non-linear parameter estimations were compared by using unweighted least squares estimation (ULSE), weighted least squares estimation (WLSE) and non-linear generalized least squares estimation (NLGLSE). Three autocorrelated time-recovery curves were used as base case to simulate theoretical kinetic response. Single Rate Constant, Rectangular and Gamma models were employed to describe the kinetic response.The NLGLSE allowed for significant precision improvements in the parameter estimation with respect to ULSE and WLSE, under known and constant covariance estimates. The variability of kSRC (SRC), kmax (Rectangular) and kmean (Gamma) decreased by approximately 40% with respect to ULSE, and 28% in comparison to WLSE. For R∞, the dispersion decreased 33% in comparison to ULSE, and 17% regarding WLSE. The limitations of ULSE and WLSE were caused by a lack of validity of the assumptions of independence and homoscedasticity in the time-recovery curves. The advantages of NLGLSE were only observed with accurate estimators of the covariance matrix, which were obtained in the simulations and in a laboratory flotation test that involved 11 replicates.Incorporating the covariance matrix in the parameter estimation allowed for improvements in the kinetic characterization. Thus, uncertainties related to the ore potential and circuit sizing (obtained from R∞ and the rate constant estimates) might be decreased using accurate covariance estimators in the objective functions.