This paper presents the mathematical modeling of the oxygen transfer process using partial differential equations (PDEs). This process is crucial in various environmental and engineering applications, such as wastewater treatment, aeration systems, and natural water bodies, in order to maintain water quality. The authors solved the typical PDE for describing the change in oxygen concentration over time and present the developed model of the differential equation with the term “source”, indicating that the model could be used to optimize oxygen transfer in various environmental and engineering applications, contributing to improved water quality and system efficiency.

This paper presents a detailed thermodynamic and mathematical modeling study of the oxygenation processes in stationary water bodies, focusing on improving oxygen transfer efficiency, an essential factor in sustaining aquatic ecosystem health. The study employed mathematical models implemented in MATLAB R2024a to simulate the influence of temperature, bubble size, and mass transfer parameters. Key parameters, such as dissolved oxygen concentration, volumetric mass transfer coefficient (ak_L), and water temperature, were evaluated under different operational scenarios. The oxygenation system was powered by solar energy and included rotating fine-bubble generators mounted on a floating platform. Mathematical modeling carried out in MATLAB validated the theoretical models, showing how environmental factors such as temperature and bubble size influence oxygen dissolution. Initial experimental data, including dissolved oxygen levels (C₀ = 3.12 mg/dm³), saturation concentrations at various temperatures (Cs = 8.3 mg/dm³ at 24 °C; Cs = 7.3 mg/dm³ at 30 °C), and a mass transfer coefficient of ak_L = 0.09 s⁻¹, were used to support the model accuracy. The results highlight the potential of digitally controlled energy-efficient aeration technologies for applications in lake restoration, aquaculture, and sustainable water management. This paper introduces a coupled approach to oxygen transfer and temperature evolution validated experimentally, which has rarely been detailed in the literature. The novelty of this study lies in the combined thermodynamic modeling and exergy–entropy analysis along with real-time tracking, showing the relevance of energy-optimized, digitally monitored oxygenation platforms powered by solar energy.

Cyclone separators are widely used for gas–solid separation due to their robustness and low operating cost. This study focuses on the experimental validation of a single cyclone configuration and the development of a MATLAB-based numerical framework. The model employs a Euler–Lagrange approach to capture centrifugal, drag, and gravitational forces acting on spherical polyethylene particles (D = 5 mm). Laboratory-scale measurements of airflow, pressure drop, and separation efficiency showed strong agreement with the numerical model (deviation < 6%), confirming its reliability for the single cyclone case. Beyond this validated framework, exploratory simulations were carried out for series and parallel cyclone configurations to provide predictive insights into possible design trade-offs. Unlike high-fidelity CFD–DEM models, which are computationally intensive and allow detailed turbulence and particle–particle interactions, the present MATLAB model is simplified but transparent and fast to implement. Its originality lies in demonstrating a low-cost, experimentally calibrated tool that can support preliminary design decisions. The multi-cyclone results should be interpreted as predictive trends, as no direct experimental validation was possible within the present setup. These findings offer preliminary guidance for balancing efficiency, energy demand, and throughput in applied mechanics of multiphase flow systems.