Aidarkhan KALTAEV, Academician: Mathematical models of physical processes

Al-Farabi Kazakh National University hosted a lecture by Academician Aidarkhan Zhussupbekovich Kaltayev, Doctor of Physical and Mathematical Sciences and Professor of the National Academy of Sciences of the Republic of Kazakhstan, as part of its series of meetings with leading scholars dedicated to Kazakhstan's Day of Science. The lecture was entitled “Modeling Reactive Flows at Low, Moderate, and High Reynolds and Mach Numbers.”

The event brought together young researchers, master's students, and doctoral candidates from the Faculty of Mechanics and Mathematics at the university's Scientific Library. Professor Kaltayev emphasized the importance of applied science, highlighting several innovative developments in recent years.

One of the major trends in modern mining, he noted, is the increasing application of chemical and biochemical leaching technologies for extracting metals directly from underground ore deposits or from heap leaching operations. Today, mathematical modeling of chemical and biochemical mineral leaching processes has become one of the key methods in the mining industry. In this context, the lecturer presented advances in in-situ uranium recovery (ISR), including the development of a physical and mathematical model for the underground biochemical leaching of pyrite and uranium dioxide. A research team consisting of M. Tungatarova, M. Kurmanseit, and N. Shayakhmetov developed and successfully implemented an industrial digital model for uranium extraction using the ISR method. According to Academician Kaltayev, these technologies enable real-time process monitoring, efficient production management, and improved uranium recovery rates.

“Today I would like to discuss models of reactive flows at low, moderate, and high Reynolds and Mach numbers. One of the first applications concerns in-situ leaching of ore deposits, particularly uranium. Kazakhstan produces approximately 40 percent of the world's uranium using this method. If a uranium deposit lies, for example, 260 meters underground, a network of injection and production wells is constructed. Sulfuric acid, used in Kazakhstan as the leaching agent, is injected into the ore body through injection wells, while the uranium-bearing solution is pumped to the surface through production wells. The solution then undergoes sorption, where uranium oxide is separated using ion-exchange resins before being processed into yellowcake—the industrial form of uranium oxide.”

Professor Kaltayev further explained the geological processes behind uranium formation and accumulation in Kazakhstan. According to geological studies, uranium-bearing rocks originated in the northern part of the Karatau Mountains, in the Betpak-Dala region, as well as in the southern areas of Zhanakorgan and Shieli. The movement of uranium through underground waters and its subsequent accumulation in these regions can be accurately described using filtration theory, from the injection of the leaching solution through uranium recovery and sorption.

He noted that uranium in Kazakhstan occurs in two oxidation states: tetravalent uranium (U⁴⁺) and hexavalent uranium (U⁶⁺). While hexavalent uranium readily dissolves in sulfuric acid, tetravalent uranium dissolves much more slowly. However, ferric ions (Fe³⁺) significantly accelerate this process. Iron-oxidizing bacteria convert ferrous ions (Fe²⁺) into ferric ions, thereby acting as biological catalysts that greatly enhance uranium dissolution.

The lecture also examined biochemical leaching of pyrite (FeS₂). Iron-oxidizing bacteria, microscopic single-celled organisms approximately 1.5 micrometers long, may either remain suspended in solution or attach themselves to the negatively charged pyrite surface through electrochemical interactions. Experimental studies have shown that both bacterial attachment and detachment rates vary linearly with bacterial concentration on the mineral surface.

Academician Kaltayev currently heads the Engineering Mechanics and Modeling Research Laboratory at Satbayev University. Throughout his distinguished scientific career, he has published more than 120 scientific papers and is recognized as one of Kazakhstan's leading experts in fluid and gas mechanics. He has received research fellowships from the German Federal Ministry of Education and Research, the German Academic Exchange Service (DAAD), and the Japan Society for the Promotion of Cultural Relations.

His scientific achievements include the development of:

• mathematical models of vortex ring dynamics;
• computational models of carbon-air combustion at low Mach numbers;
• physical and numerical models describing the transition from deflagration to detonation;
• models of hydrogen combustion in supersonic airflows; and
• over the past two decades, together with his research team, a comprehensive 3D simulation model and digital control system for uranium extraction by in-situ recovery, enabling forecasting, planning, monitoring, and optimization of mining operations.

He also developed a 3D model of uranium oxide sorption processes, investigated technologies for solar thermal energy storage and utilization, and successfully commercialized the software package "Simulator Version 3.0" for uranium extraction by ISR, which is currently used by mining enterprises across Kazakhstan. His current research focuses on bacterial metal leaching using ISR technology, centrifugal pump design, and the application of artificial intelligence to engineering problems.

During the lecture, Professor Kaltayev also presented practical results from his research group, including joint studies with doctoral researcher Zhomart Ualiyev on flame propagation in enclosed spaces with complex geometries. Particular attention was given to computational models describing the transition from conventional combustion to detonation, enabling detailed analysis of hazardous gas-dynamic phenomena.

Discussing gas dynamics, the academician explained that conventional governing equations become increasingly difficult to apply accurately at low Mach numbers, often producing significant numerical errors. According to Professor Kaltayev, these challenges can be overcome by adopting an intermediate model of incompressible dynamic gas flow for deflagration modeling.

“From a theoretical perspective, there are no major difficulties. However, numerical solutions present significant challenges. Since the governing equations are nonlinear and can only be solved numerically, even slight variations in input data may lead to substantial deviations in the results. This becomes a critical issue when numerical modeling is intended for practical engineering applications.”

He explained that flame propagation through methane-air mixtures inside pipes containing obstacles such as rough surfaces and diaphragms can accelerate by several hundred times compared with free flame propagation. These obstacles generate intense turbulence, increase the flame surface area, accelerate heat transfer, and produce rapid pressure increases that may eventually trigger the transition from laminar combustion to detonation. Such processes require sophisticated mathematical treatment, particularly when dealing with irregular computational domains. To address this challenge, researchers apply the fictitious domain method, which allows calculations to be performed on regular computational grids while preserving the physical boundary conditions. This mathematical model was developed by Zhomart Ualiyev.

The lecture concluded with an active discussion, during which young researchers asked questions about 3D modeling of turbulent flows and the application of GPU-based high-performance computing for solving large-scale engineering problems.

Kairzhan Turezhanov