Design Institute, Ural Federal University, Yekaterinburg, Russia:

I. F. Garifulin, Chief Specialist
M. G. Viduetsky, Deputy Director of Science, vid.magr@mail.ru
V. A. Maltsev, Director, Doctor of Engineering Sciences
A. P. Purgin, Production Engineering Department

Advanced multi-operation dressing circuits can be described using systems composed of a few tens of linear equations. It is difficult to calculate the errors of product yields in these systems. The complex mineral dressing circuits composed of a few operations of separation and mixing, as well as circulating flows are increasingly widely calculated using the matrix methods which includes construction of a system of equations of material balances for all operations in the circuit, and solution of the system with the help of direct and inverse matrixes of the involved coefficients. The matrix method enables quick and accurate calculation of complex process circuits, and allows mathematical modeling to predict and analyze complex relationships between the input and output parameters. This article presents the estimate algorithm of the matrix method accuracy in calculation of mineral dressing circuits using the condition number of the matrix. The examples for the simplest circuit (one operation) and more complex circuits are given. The condition number can be a convenient tool for estimating the matrix accuracy in calculation of complex mineral dressing circuits when determination of other criteria (for instance, dressing product yield error) is complicated or impossible. The condition number is applicable to many problems in engineering, including new processing equipment design. For example, the described procedure was used to select optimal algorithm of comparative sampling in introduction of multiple-purpose flotation machines at processing plants.

1. Chanturia V. A., Bocharov V. A. Modern state and basic ways of technology development for complex processing of non-ferrous mineral raw materials. Tsvetnye Metally. 2016. No. 11. pp. 11–18. DOI: 10.17580/tsm.2016.11.01
2. Garifullin I. F., Maltsev V. A., Viduetskiy M. G., Poryvay E. B., Poryvay N. E. On the problem of calculation of complicated concentration schemes for mineral raw materials. Gornyi Zhurnal. 2006. No. 12. pp. 48–50.
3. Kozin V. Z. Testing of mineral resources. Yekaterinburg : Izdatelstvo UGGU, 2011. 315 p.
4. Shishkin A. S., Shishkin S. F. Engineering problem solution in EXCEL : tutorial. Yekaterinburg : Izdatelstvo UrFU, 2012. 364 p.
5. Kozin V. Z. Control of technological processes of concentration : Textbook. Yekaterinburg : Izdatelstvo UGGU, 2010. 302 p.
6. Kozin V. Z. Calculation and optimization of testing using relative random errors. Theory and Practice of Ore and Waste Processing : XXV International Conference Proceedings. Yekaterinburg : Fort Dialog-Iset, 2020. pp. 3–5.
7. Kozin V. Z., Komlev A. S. Random sampling error experimental determination at processing plants. Obogashchenie Rud. 2017. No. 2. pp. 44–48. DOI: 10.17580/or.2017.02.08
8. Kozin V. Z., Komlev A. S., Vodovozov K. A. Sampling results probable systematic inclination manifestation and calculation at concentrating mills. Izvestiya vuzov. Gornyi zhurnal. 2018. No. 6. pp. 69–76.
9. Rubinshteyn Yu. B., Volkov L. A. Mathematical methods in the concentration of mineral resources. Moscow : Nedra, 1987. 295 p.
10. Kozin V. Z. Commodity balance of concentrating factories. Yekaterinburg: Publ. UGGU, 2014. 133 p.
11. Morozov Yu. P. Projecting of the concentration plants : Textbook. Yekaterinburg : Izdatelstvo UGGU, 2009. Vol. 1. Project contents and design sequence. 304 p.
12. Yianatos J., Vinnett L., Panire I., Alvarez-Silva M., Díaz F. Residence time distribution measurements and modelling in industrial flotation columns. Minerals Engineering. 2017. Vol. 110. pp. 139–144.
13. Harbort G., Clarke D. Fluctuations in the popularity and usage of flo tation columns – An overview. Minerals Engineering. 2017. Vol. 100. pp. 17–30.
14. Viduetsky M. G., Garifulin I. F., Maltsev V. A., Purgin A. P. Some aspects of moder nization of pneumatic flotation machines of a column type. Tsvetnye Metally. 2021. No. 5. pp. 14–22. DOI: 10.17580/tsm.2021.05.02
15. Navia D., Villegas D., Cornejo I., de Prada C. Real-time optimization for a laboratory-scale flotation column. Computers & Chemical Engineering. 2016. Vol. 86. pp. 62–74.
16. Newcombe B. Comparison of flash and column flotation performance in an industrial sulphide rougher application. Minerals Engineering. 2016. Vol. 96-97. pp. 203–214.
17. Lavrinenko A. A. State and trends of development of flotation machines for solid mineral concentration in Russia. Tsvetnye Metally. 2016. No. 11. pp. 19–26. DOI: 10.17580/tsm.2016.11.02
18. Viduetskiy M. G., Garifulin I. F., Maltsev V. A., Purgin A. P. Pneumatic column flotation machines, evolution. Yekaterinburg : AMK “Den RA”, 2020. 149 p.