Variance propagation in toll smelting operations treating multiple concentrate stockpiles
Metallurgical accounting for toll treatment smelters pose a number of challenges. Multiple feed stockpiles have to be accounted for as part of the monthly recovery estimation. Moreover, multiple metallurgical intermediates make up part of the in-process inventory, which also influence the monthly recovery calculation. Errors propagate from the measurements of volumes, assays, moisture fractions and bulk densities in the various material types through multiple steps up to the point where the final recovery is calculated. These errors contribute non-linearly to the variance in the final recovery estimate. This paper develops the mathematical formulation for variance propagation in toll smelting operations, including the effect of in-process inventory, assuming measurement biases have been eliminated beforehand. Operational data from a Southern African copper smelter is used for a case study. The method of propagation of variance showed that uncertainties in stockpile assays were the main contributors to variance in the recovery estimate. Variance in the volume and bulk density uncertainties contributed a secondary, yet significant, proportion to the overall recovery variance. It was determined, for the given case study, that the recovery variance depended on the calculation method used and that variance propagation via the two-product formula was smaller than recovery variances calculated via the standard recovery formula. However, the probability that the two product formula will give inaccurate (versus imprecise) results is significantly more due to the practical difficulty of equiprobable sampling of tailings streams from smelter plants (which includes materials such as slags and flue dust). Recommendations are made on how to achieve a reduction in overall uncertainty for toll treatment smelters. © The South African Institute of Mining and Metallurgy, 2006.