Sampling, Metallurgical Accounting and Reduction of BalanceEstimation Variance

摘要

A metallurgical balance drawn around a mineral processing plant orrnsmelter must involve all process streams crossing the plant boundary.rnModern sampling theory, as formulated by Gy, and sound statisticalrnmethod can be used to determine the uncertainties in the observed assaysrnfor the set of balance analytes and in the mass flow rates of the processrnstreams for the balance period. These uncertainties can then be usedrntogether with the material balance equations to arrive at adjustedrncomponent mass flows that satisfy the material balances exactly. Torndetermine whether or not the data is consistent with the level ofrnuncertainty in the data and with the material balance equations, arngoodness of fit test is applied. This goodness of fit criterion is thernprincipal tool to quantify the quality of the material balances.rnThe mathematical structure of the statistical material balance problemrnposes a number of challenges. For example, while the variances of thernadjusted component mass flows do not vanish, the variances of the sumsrnof the adjusted component mass flows about each balance node dornvanish, simply because the balancing computation demands that thernbalances close exactly. Even though it can be shown that the variances ofrnthe adjusted component mass flows are always lower than the estimatesrnfor the corresponding unadjusted component flows, it is still not possiblernto construct a confidence interval on the component mass flow into (andrnout of) a balance node using the adjusted component flows. On the facernof it, such a confidence interval is exactly what is most desired from thernbalance calculations. The problem arises because the balance problem isrna data adjustment rather than a parameter estimation problem.rnIt is possible, however, to calculate the recovery of a component to arnproduct stream and determine a confidence interval for the recoveryrnvalue. It is also possible to calculate an uncertainty in the mass flow ofrnany balance species in any process stream, before and after balancing.rnBalance results have little meaning without the confidence intervalrnestimates. Having invested in the manpower and procedures that make arnstatistically valid balance on an operation possible, the management mayrnfind that the levels of uncertainty associated with the balances are ratherrnhigher than they would like. The obvious solution to the wide confidencernintervals is to reduce the uncertainty in the assays and mass flow rates onrnthe principal process streams. The cost of this uncertainty reduction mayrnbe quite significant, as it will involve reduction in analytical variances orrnreduction in sampling and preparation variances which may requirerninvestment in and development of new sampling hardware or routinernduplication of assays to reduce the assaying variance.rnA second strategy for uncertainty reduction can be proposed whichrninvolves the use of additional sampling points to provide material balancernnodes internal to the processing operation. The information required tornmake these internal balances may well exist within the operation, but itrnmay not be recognised as being of value or the standard of sampling andrnassaying may not be deemed to be sufficient to use the data forrnmetallurgical accounting purposes. The marginal cost of elevating thernexisting sampling systems to metallurgical accounting standards mayrnwell be smaller than that to improve the sampling and assaying on thernboundary streams.rnThis paper presents a statistical analysis of an uncomplicated form ofrnthe balance calculations that permits exploration of accuracy issues andrnthe advantages that come from additional internal balancing nodes for thernprocess flow sheet. The calculations are based on a relatively simple flowrnsheet of common structure, but are of general applicability. The objectivernis to reveal broad issues. While each process flow sheet and sampling andrnanalysis system requires individual analysis to reveal precisely where therngains can be made most economically, a general result can berndemonstrated which provides an estimate of the limit to the variancernreduction which can be achieved by adding internal balance nodes.