# Balance of the useful components

BALANCE OF THE USEFUL COMPONENTS (EN: balance of useful components; DE: Nutzgehalt des Erzes, Nutzbilanz des Erzes; FR: bilan des mineraux utiles; ES: balance de componentes utiles; RU: баланс полезных компонентов) characterizes the distribution of the useful component (for example, of the metal within the ore) between the products of the processing (concentrates) and wastes (tailings) for the specified period of time (shift, day, ten days, month, quarter, year).

The balance of the useful components is compiled as the result of the testing and analytical control of the technological process at the beneficiation plant or metallurgical plant, and determines their major indicators: the quality and yield of the concentrate, the losses and content of the useful component within the wastes, the degree of utilization of the raw materials (the extraction of the useful component). The equation for the balance of the useful components is:

where

a - the content of the useful component within the source raw material;

b_{i} - the same within the i-th concentrate;

t - the same within the tailings;

g_{i} = P_{i}•100/Q_{i} - the yield of the i-th concentrate, %;

P_{i} - the mass of the concentrate;

Q_{i} - the mass of the ore.

For the operational monitoring and control of the process, they weigh and test the source ore, and determine the content of the useful components within all the products, the yield of the concentrate, and the rate of extraction according to the equation for the balance of the useful components, which in this case is named the technological balance. There is more precise the marketable balance, which is compiled according to the data of the weight accounting of the ore and marketable concentrates (with the correction for moisture), the determination of the content of the useful component within them, and also within the surplus reserves of the production, which has not been completed, and within the wastes, with taking into account the mechanical losses. The marketable balance of the useful components is compiled for each component in the units of tonnes (for the precious metals in the units of kilograms). For the single-component ore, from which only one concentrate is obtained during the beneficiation, the yield is calculated according to the equation for the balance

and the rate of extraction is calculated according to the equation

The marketable rate of extraction is calculated as

where

M_{k} - the mass quantity of the useful component within the marketable concentrate;

M_{p} - the same quantity within the ore, which has been processed during the same time.

For the multi-component ores and complicated technological schemes with the obtainment of several concentrates, they calculate the yield and rate of extraction with the help of the system of equations, the quantity of which is equal to the quantity of the researched components. The calculations may be performed using the digital or analog electronic computing machines.

Because in practice the testing and determination of the content of the components are performed with certain error, the balance of the useful components is also not precise. The inaccuracy manifests itself as the divergence of the so-called marketable and technological balances. In the case of calculation of the left and right parts of the balance equation for all measured parameters, there emerges the disparity of the balance:

D = aQ - (b•P + t•W),

where the mass of the tailings W = 0 for the ores of the precious metals and certain rare metals, because the yield of the concentrate from them amounts to the fractions of %. During the simultaneous determination of the contents of several components within all the products, the quantity of the equations exceeds the quantity of the unknow variables (overdetermined system), and there emerges the redundant information, which may be used for the improvement of the precision of the balance of the useful components. There have been developed the special programs for the computers, which permit to improve the precision not only for the balance of the useful components, but also for the measured values of the contents of the useful components. There act as the source data for the calculation the matrices with the contents of the components within the products, and the matrix with the acceptable errors of the measurement of the contents. The algorithm for the calculation is based on the criterion of the maximal verisimilitude, that is of the minimization of the function:

where

d_{ij} - the standard deviation;

S_{ij} - the errors of the determination.

The minimum of the function is found by the method of the indefinite Lagrange multipliers, which permits to compute all the corrections d_{ij} and yields g_{i}.

The balance of the useful components is the major reporting document of the beneficiation and metallurgical enterprises.

A a | B b | C c | D d | E e | F f |

G g | H h | I i | J j | K k | L l |

M m | N n | O o | P p | Q q | R r |

S s | T t | U u | V v | W w | X x |

Y y | Z z |