Mining Institute of the North, Siberian Branch, Russian Academy of Sciences, Yakutsk, Russia

Lebedev I. F., Candidate of Technical Sciences, Senior Researcher, ivleb@mail.ru
Yakovlev B. V., Doctor of Physical and Mathematical Sciences, Leading Research Fellow, b-yakovlev@mail.ru
Matveev A. I., Doctor of Technical Sciences, Chief Researcher, andrei.mati@yandex.ru

This paper examines the mathematical modeling of particle motion of varying density and size in a centrifugal pneumatic separator. A mathematical model of particle motion within the working volume of a centrifugal pneumatic separator rig developed by the authors has been developed. The pneumatic separator rig consists of a closed cylindrical drum housing equipped with inlet ports for the feedstock and airflow at the top and outlet ports at the bottom. A rotating disk is mounted coaxially beneath the inlet port to uniformly distribute the material and provide a controlled velocity for its ejection from the disk edge. Three coaxially arranged annular collection chambers are installed in the lower section to collect particles separated by aerodynamic size. The rig is designed to study the separation of mineral particles of varying density and size based on their migration capacity in an aerodynamic flow. This paper presents an algorithm for calculating the probability of determining the position of a particle within the separator’s working separation zone. The Gibbs ensemble method and the Runge-Kutta method are used to solve a system of differential equations, determining the possible particle positions in a pneumatic separator. The distribution of these positions is proportional to the probability density distribution of the particle positions. An algorithm for calculating the number of particles distributed across the radial accumulation chambers of separation products is developed and presented. This algorithm will ultimately allow for the determination of separation performance based on device parameters (chamber size, particle feed mode, air flow) and particle properties (size, density). Consequently, the model allows for the determination of separation parameters (e.g., air flow velocity, working element rotation speed) and the optimization of the separator design to achieve maximum heavy particle recovery. A comparison of the obtained data from field studies of particle distribution in a centrifugal pneumatic separator model with the calculated data showed high convergence of the results, confirming the validity of the mathematical model.
The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation (theme No. FWRS-2026-0056, reg. No. 126021217276-7).

1. Zhuravlev K. A., Vasyuta A. D. Analysis of factors influencing losses of recious metals during alluvial deposit exploitation. Mining Informational and Analytical Bulletin. 2023. No. 12-2. pp. 53–63.
2. Burakov A. M. An umbrella approach to volume minimization in gold sand washing. Mining Informational and Analytical Bulletin. 2023. No. 2. pp. 127–138.
3. Kovalev A. A. Main problems and high-efficiency equipment for complex processing of gold-bearing sands. Mining Informational and Analytical Bulletin. 1997. No. 3. pp. 43–47.
4. Solomakha A. E., Shvab A. V. Simulation of aerodynamics of a swirling turbulent flow in a centrifugal air classifier. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika. 2024. No. 87. pp. 150–162.
5. Evseev N. S., Zhukov I. A., Belchikov I. A. A study of aerodynamics and fractional separation of fine particles in a separation chamber. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika. 2023. No. 83. pp. 74–85.
6. Shvab A. V., Solomakha A. E. Numerical simulation of aerodynamics of air-centrifugal classifier. Izvestiya vysshikh uchebnykh zavedenii. Fizika. 2021. Vol. 64, Iss. 2-2. pp. 155–161.
7. Burkov A. I., Glushkov A. L., Lazykin V. A., Mokiev V. Yu. Substantiation of the main design parameters of the separation chamber of the pneumatic separator using various methods for calculating particle trajectories in the pneumoseparating channel. Agricultural Science Euro-North-East. 2022. Vol. 23, Iss. 3. pp. 402–410.
8. Penkov P. M., Morozov Yu. P., Khamidulin I. Kh. Improvement of centrifugal separation on the basis of pneumatic turbulization of separator cone wall layer. Mining Informational and Analytical Bulletin. 2022. No. 12-1. pp. 120–133.
9. Palvanov B. Y., Jafarov S. A. K., Matlatipov G. R. Mathematical model and numerical experiment for the study of the problem of separation of bulk mixtures on a pneumatic separator. 2024 IEEE 25^th International conference of young professionals in electron devices and materials (EDM). 2024. pp. 2580–2586.
10. Paten RU No. 238653. IPC B07B7/08. No. 2025100425. Centrifugal pneumatic classifier – concentrator. Matveev A. I., Lebedev I. F., Tsoi I. M., Pavlov A. Yu. Appl. 09.01.2025. Publ. 06.11.2025. Bull. No. 31.
11. Gibbs J. V. Termodinamika. Statisticheskaya mekhanika. Moscow: Nauka, 1982. 584 p.
12. Krylatsova S. R., Matveev A. I., Lebedev I. F., Yakovlev B. V. Simulation of particle motion in a screw pneumatic separator by statistical methods. Matematicheskie zametki SVFU. 2018. Vol. 25, Iss. 1. pp. 90–97.
13. Lebedev I. F. Experimental studies of the mineral particle separation processes in a laboratory model of the centrifugal pneumatic separator. Mining Industry Journal. 2025. Iss. S4. pp. 36–39.