All metallurgical balance techniques, old or new are based on mathematically equating the inflow and outflow of specific components at a balance point. This equality is always valid with a steady state process, but is also valid for a fluctuating process if the sampling period is sufficiently long to make inventory changes at the balance point negligible with respect to flows.

The two and three product formulas are based on the simultaneous solution of several equations : one representing the total solids balance, the others representing the balances for one or two specific components in the solids, i.e., chemical specie, mineral component, size fraction component, etc. The component balance is obtained by multiplying the appropriate solids flowrate by a component analysis (wt. component/ wt. solids). Application of the two or three product formula presumes absolute data accuracy and essentially limits the metallurgist to the use of only the data necessarily required for the calculation, i.e. one component for one feed- two products, two components for one feed-three products. If the same calculation were repeated using data for different mineral components in the same samples, the resultant weight split would be more or less different than the first, depending upon the statistical vagaries of the data. Any attempt to put together the calculation of flows throughout a complex process with these formulas always results in metallurgist frustration and usually metallurgical confusion, with the resultant balance more appropriately termed a metallurgical imbalance.

The now metallurgical balance techniques have been developed to make use of a much greater proportion of the available information, be it measured flows of solids or multi-component analyses. The results not only provide improved estimates of solids flows and component recoveries in the process, but also they permit one to obtain a perfect and consistent balance of all individual components by making appropriate adjustments to the original data.

The solids flow estimates obtained from the new balance are better because more data sets are used to obtain them. This situation is analogous to the averaging of measurements made by several techniques for the same variable, i. e. estimating the capacity of a container by: (1) calculating from the container dimensions, (2) weighing the contained water and (3) measuring the volume of contained water. The adjusted process measurements are in turn obtained by recognizing that data error exists and then making the smallest possible adjustments to the data to achieve complete consistency with the estimated flows. This also results in the adjusted data being more accurate, since in effect the more reliable data are being used to upgrade the less reliable data.

There are a number of practical benefits which accrue to the metallurgist as a result of using the new metallurgical balance techniques:

**Credibility**– Whether entirely deserved or not the development of internally consistent balances from what has hitherto been considered conflicting data significantly improves the credibility of the metallurgist with company and plant management, since it virtually eliminates negative criticism regarding balance inconsistencies.**Accuracy**– The use of much more of the available data, i.e. measured flows, several mineral component assays and several size fraction measures, in the balance provides more reliable flow and mineral recovery estimates and more accurate adjusted data. This permits rapid detection of the effects of process changes and greatly reduces the wasting of expensive information.**Clarity**– The metallurgically consistent balancing of multiple components provides a much broader and clearer view of what is actually occurring in the process, rather than focusing on the behavior of only a single component.**Consistency**– The use of a standard, systematic and mechanized calculation procedure with multiple component balances will remove individual bias, eliminate calculation error, and usually nullify the potentially catastrophic effects of infrequent spurious measurements.**Efficiency**– More of the balancing task can be shifted to a technician, since the decisions on data use are no longer necessary, and routine balances become only a matter of updating data for a routine computer calculation.

The use of these newer and better balancing techniques does necessitate some adjustment on the part of the metallurgist. Although some single balance point calculations can be made by improved techniques with a calculator, the calculations required for complex plant balances require access to a small to medium sized computer. The types of data being collected for each stream throughout the process must also be standardized to provide appropriate data for the balances.

It is also important to recognize that use of these new techniques cannot achieve miracles bad data begets lousy calculation results regard¬less of how the calculations are made.

**Search for Weight Split**

One other approach can be used to calculate weight splits if equation (5) is not a reasonable assumption, i.e. copper feed, concentrate and tail of 0.5, 15, and 0.05% Cu would not have equal standard deviations.

If one assumes a value of the weight split W(2)/W(1), which is bounded by values of 0 and 1, and then uses equations (11), (12) and (13) to calculate the analyses adjustments, it is then possible to calculate the sum of squares of adjustments defined by the first part of equation (4). The assumed value of the weight split can then be increased or decreased by some increment and a new sum of squares calculated. If the new sum of squares is smaller, continue in that direction, if the new value is larger, reverse directions and reduce the size of the increment by some reasonable factor, i.e. 0.5. This search would be continued until the change in weight split is acceptably small. Equations (11), (12) and (13) would then be used to calculate adjustments. This type of one variable search could be handled by a large programmable calculator, i.e. comparable to a TI 59.