Thermodynamic Analysis of Oxygenation Methods for Stationary Water: Mathematical Modeling and Experimental Investigation

Abstract

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.

1. Introduction

The oxygenation process in water consumes 50% to 90% of the total energy required by secondary wastewater treatment plants [1]. In light of the 1997 energy crisis and the ongoing rise in energy costs, improving the efficiency of oxygenation systems in wastewater treatment plants has become a critical research focus. This focus has spurred the development of fine-bubble immersion oxygenation systems, which are more energy-efficient than many other types of oxygenation systems [2,3].

Studies have shown that fine-bubble oxygenation devices can save up to 20–50% of energy consumption compared to coarse-bubble diffusers [4]. These energy savings have driven the conversion of over 1300 municipal and industrial wastewater treatment plants in the United States and Canada from coarse-bubble to fine-bubble oxygenation systems.

Dissolved oxygen (DO) refers to the amount of oxygen present in water. Its presence is a positive indicator of water quality, while its absence signifies severe pollution. The ecological quality of water largely depends on its oxygen content: higher dissolved oxygen levels indicate better water quality. Concentrations of dissolved oxygen can range from 0 to 15 mg/dm³, and adequate oxygen levels are crucial for natural rivers and certain water treatment processes. Many biological and chemical processes rely on oxygen, thereby decreasing its concentration in water.

Several industrial and environmental processes involve oxygenating a liquid by introducing air bubbles into the water [5,6,7]. Recent research on interphase mass transfer with fine-bubble generation focuses on key topics such as porosity (free volume or void fraction) [8], bubble characteristics [9], flow regimes, and computational fluid dynamics (CFD), which show that bubble column design parameters influence fluid dynamics through boundary conditions and end effects [10]. Other studies have examined heat transfer during bubble growth on nanostructured surfaces [11] and the benefits of nanobubbles in mass transfer [12,13,14].

The impact of liquid properties on the hydrodynamics of gas–liquid systems has been studied, with many papers discussing the effects of liquid viscosity (ν) and surface tension (σ) in Newtonian fluids [15,16,17]. These studies highlight that viscosity plays a dual role: an increase in viscosity hinders bubble thinning and prevents coalescence, but excessive viscosity can reduce liquid-phase turbulence, favor the formation of larger bubbles, and increase their number at the expense of smaller ones.

Studies on gas diffusion in liquids and the hydrodynamics of bubble columns conclude that bubble size and void fraction are primarily influenced by the physicochemical properties of the liquid phase and air distribution methods [8].

The authors of [18] presented a thermodynamic analysis and performance assessment of a real municipal wastewater treatment plant using energy and exergy methods. The plant treats about 222,000 m³ of domestic wastewater daily through primary and secondary treatment stages. A new parameter called the net exergy ratio (NExT) was introduced to represent the renewable and sustainable aspects of the sludge treatment process, as the sludge’s specific exergy increases during treatment. The NExT values were found to be 2.81 for the primary treatment and 1.31 for the secondary treatment. The exergetic efficiencies of the plant subsystems were calculated as 57.4% for the primary treatment system and 29.7% for the secondary treatment system, considering sludge outputs as waste.

In [19], a comprehensive exergy analysis of municipal wastewater treatment plants (WWTPs) in southern Tehran focusing on energy destruction, exergy flows, and biogas recovery is presented. The results show that chemical exergy dominates over physical exergy and that about 30% of the inlet wastewater’s exergy is destroyed in the aerobic reactors, accounting for 93% of total exergy destruction. Additionally, around 3700 kW of exergy is removed with dewatered sludge. A comparative analysis of six other WWTPs revealed that larger plants tend to have lower proportions of useful exergy, and increasing biogas recovery can reduce exergy losses from sludge. The research emphasizes the potential for energy recovery and efficiency improvement through anaerobic digestion and system design optimization.

Paper [1] presents a thermodynamic analysis of a compact hydrogen generation system intended for mobile fuel cell applications. The system combines a miniature autothermal reformer (ATR) and a water–gas shift (WGS) reactor, using methane as a model fuel to supply hydrogen for a 1 kW PEM fuel cell. The study focuses on optimizing feed composition and operating conditions to maximize hydrogen yield and purity. Challenges related to product purity and start-up behavior are addressed, and feasible technical solutions are proposed. The system was found to be suitable for remote and mobile energy applications.

Calise et al. (2020) [20] developed a detailed and validated model of a wastewater treatment process using activated sludge model 1 (ASM1) and simulated it in the INSEL environment. The model incorporates key biochemical reactions such as nitrification and denitrification based on the Ludzack–Ettinger process and solves the mass and energy balances using the explicit Euler method. Validation against literature data showed high accuracy (deviation below 1%). The study highlights the influence of temperature on biological kinetics, showing that denitrification is more affected than oxygen consumption.

The study in [21] presents a thermodynamic analysis of an experimental solar-heat supply system designed for cold regions such as northern Kazakhstan, where solar radiation is limited. The work evaluates a dual-circuit glazed flat-plate solar collector, examining how glazing, coolant flow, and absorber thickness affect heat transfer and system efficiency. The results showed that the thermal efficiency ranged between 2.40 and 2.53, confirming the system’s reliability and effectiveness in low-temperature environments. The study emphasizes the role of absorber layer thickness and collector design in improving thermal performance.

Recent studies have reported exergy efficiencies for water aeration systems ranging from 15% to 40%, depending on bubble size, water depth, and oxygen transfer methods.

The study in [22] investigates how water depth and diffuser coverage influence oxygen transfer performance in subsurface aeration systems. Experimental results revealed that both parameters significantly affect oxygen transfer capacity (OC), efficiency (E), and oxygen absorption percentage (δ). OC varied from 18 to 170 gO₂/m³·h, while E ranged from 2 to 17 gO₂/m³ air. Oxygen absorption ranged between 0.45% and 5.4%. The authors also derived mathematical models—an exponential model for water depth and linear models for other parameters—to describe these effects accurately.

A recent study [23] presented a validated CFD-based methodology to simulate dissolved oxygen (DO) distribution in a thin-layer cascade (TLC) reactor for microalgae cultivation, accounting for both gas–liquid mass transfer and oxygen production from photosynthesis, including inhibition effects. Using Ansys Fluent 2021 R1 and user-defined functions (UDFs), the model showed good agreement with experimental data across an 80 m TLC reactor, offering a reliable tool for in silico DO optimization and control in microalgae systems.

As global water demand rises, organizations are increasingly turning to digital technologies, such as artificial intelligence, sensors, smart water management systems, and networks, for monitoring water resources. These technologies help build more efficient water treatment and oxygenation infrastructure. In the last decade, there has been a shift toward autonomous systems powered by renewable energy sources like solar and wind. Solar-powered mobile platforms offer an innovative, energy-efficient solution for continuous oxygenation throughout the day, even in varying climatic conditions. Mathematical modeling of these oxygen transfer processes is essential for optimizing the operation of such platforms.

Improving water quality through oxygenation is crucial for maintaining ecosystem balance and preventing eutrophication. Mobile platforms equipped with microbubble generators offer an innovative solution for oxygenating stationary water bodies, enhancing the efficiency of oxygen transfer from air to water.

To study water oxygenation processes from a thermodynamic perspective, Gibbs free energy analysis and Henry’s law are used [24]. Mathematical models based on mass transfer equations, diffusion principles, and thermodynamic principles are essential for understanding and simulating the evolution of dissolved oxygen concentrations in water [25]. These models are valuable for optimizing operational parameters and assessing the impact of oxygenation systems on aquatic ecosystems.

This study aimed to examine the thermodynamic analysis methods applied to oxygenation processes in still waters. The main objective was to develop and validate a thermodynamic model for analyzing oxygen transfer processes in stationary water bodies using fine-bubble aeration systems powered by solar energy. The novelty of this study consists in the design and analysis of a mobile floating platform for water oxygenation that combines solar-powered microbubble generation with advanced thermodynamic modeling and real-time digital monitoring. Unlike conventional fixed aeration systems, the proposed solution integrates rotating fine-bubble generators with energy-autonomous operation and remote supervision capabilities, aligning with Industry 4.0 standards. Furthermore, we applied detailed thermodynamic and exergy analysis methods to evaluate the efficiency of oxygen transfer, supported by numerical approximations. This innovative, sustainable, and modular approach offers a new direction in eco-friendly water treatment technologies.

Potential future extensions of this study include experimental validation of the proposed models and the integration of real-time sensor data for adaptive control of the oxygenation process. Additionally, the evaluation framework could be expanded to include other environmental variables such as water flow dynamics, solar radiation variability, and biological oxygen demand.

2. Materials and Methods

This study investigated the design and materials of fine-bubble generators (FBGs), which play a crucial role in efficient water oxygenation. FBGs can be designed in various shapes (e.g., circular, rectangular, tubular, or spherical) and constructed from materials like glass, plastics, rubber, or shape-memory alloys, each chosen to meet specific performance criteria for optimal oxygen transfer in water [26,27]. Recent studies have introduced innovative FBG technologies that enhance energy efficiency by addressing mass transfer issues related to dispersion and movement in oxygenation systems [28,29].

The oxygen transfer process is driven by the need to restore thermodynamic equilibrium between the chemical potential of oxygen in water and air. The reaction proceeds from regions of higher oxygen concentration towards equilibrium [30]:

$$O_2(air)\leftrightarrow O_2(water)$$

2.1. Thermodynamic Analysis Methods

The oxygenation process was analyzed using thermodynamic principles, as it involves mass transfer between air and water. Various laws and concepts are applied.

2.1.1. First Law of Thermodynamics

The first law of thermodynamics is used to model heat transfer during the oxygenation process. It incorporates internal energy, mechanical work, and heat to determine the change in water temperature during aeration, as described by a heat energy equation [31,32]:

$$dU=\delta Q-\delta W[J]$$

(1)

This defines the notion of internal energy of a closed thermodynamic system.

Temperature data collected from the sensors were used to calculate the heat transfer, while the energy associated with oxygen dissolution was incorporated using known enthalpy values for oxygen dissolution.

The rise in water temperature during the aeration process can be modeled by the heat energy equation [32]:

$$Q=mc\mathrm{\Delta }T[J]$$

(2)

where m is the mass of water (kg), c is the specific heat of water (J/kg·K), and ΔT is the change in water temperature (T_f − T_i).

The efficiency of all pneumatic aeration systems is reduced with increasing water temperature (the energy transferred in the form of heat), because the solubility of oxygen decreases at higher temperatures [33].

2.1.2. Second Law of Thermodynamics

Equation (3) expresses the entropy generation rate due to irreversibilities during the oxygen transfer process. This thermodynamic formulation allows for quantification of the efficiency of the oxygenation process and understanding the degradation of usable energy. A MATLAB simulation tracks how entropy evolves in time based on the variation of temperature and oxygen concentration gradients [34].

$$\mathrm{\Delta }{S}_{gen}={\displaystyle \int \frac{\delta Q}{T}}\left[{\displaystyle \frac{J}{K}}\right]$$

(3)

This equation quantifies the entropy generation based on the heat transferred during the process and allows for the calculation of thermodynamic efficiency in oxygenation systems. The efficiency is maximized by minimizing this entropy generation, which is simulated in MATLAB to study the system’s behavior and its approach to thermodynamic equilibrium.

2.1.3. Thermomechanical Exergy Analysis

Exergy, which quantifies the amount of usable energy in a system, is used to evaluate the maximum work a system can do [34]:

$$Ex=U+pV-T_0{S}_{gen}$$

(4)

where U is the internal energy, pV the contribution of the mechanical interaction of the system, and T₀S the entropic contribution.

The specific physical exergy of water is calculated using the standard formulation:

$$e_{ph}=\left(h-h_0\right)-T_0\left(s-s_0\right)$$

(5)

where e_ph is the specific physical exergy of water, h and s represent the specific enthalpy and entropy of water at the system temperature, and h₀, s₀, and T₀ refer to the properties of the environmental state. For this study, the reference temperature T₀ was set at 298.15 K (25 °C).

The exergy destruction rate is calculated based on entropy generation and the temperature difference between the system and its surroundings:

$$\dot{E}{x}_{destruction}=T_0\cdot {\dot{S}}_{gen}$$

(6)

where T₀ is the reference (ambient) temperature and ${\dot{S}}_{gen}$ is the rate of entropy generation (previously calculated).

A detailed entropy generation balance was used to evaluate irreversibilities within the oxygenation process.

The total exergy input and destruction were evaluated using a control volume analysis, and the exergy efficiency of the oxygenation process was calculated as:

$${\eta }_{ex}={\displaystyle \frac{E_{useful}}{E_{input}}}={\displaystyle \frac{\left(\dot{m}\cdot e_{ph}\right)}{E_{supply}}}$$

(7)

where E_supply accounts for the energy delivered by mechanical aeration or microbubble generation.

2.1.4. Henry’s Law (Solubility of Oxygen in Water)

This law describes the solubility of oxygen in water, where the concentration of dissolved oxygen is in equilibrium with its partial pressure in the air. The solubility is affected by temperature, and a MATLAB model is used to explore how variations in partial pressure and temperature impact oxygen dissolution [35,36]:

$$C_{O_2}^{\ast }=H\cdot p_{O_2}$$

(8)

where $p_{O_2}$ is the partial pressure of oxygen in the atmosphere and H is Henry’s constant.

Henry’s constant k_H decreases with increasing temperature. It can be expressed using a temperature-dependent equation [36]:

$$k_H\left(T\right)=k_H^0\cdot \exp \left[{\displaystyle \frac{-\mathrm{\Delta }H}{R_M}}\cdot \left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_0}}\right)\right]$$

(9)

where $k_H^0$ is Henry’s law constant at a reference temperature T₀, ΔH is the enthalpy of dissolution for oxygen in water, R_M is the universal gas constant (R_M = 8314  J/kmolK), and T is the temperature in Kelvin.

2.1.5. Mass and Heat Transfer Analysis

Fick’s laws of diffusion are applied to describe the oxygen diffusion process in water. The concentration gradient over time and space is calculated, and numerical approximation in MATLAB help solve these diffusion equations.

The law connects the variation of concentration in time and space [37]:

$${\displaystyle \frac{\mathit{\partial }C}{\mathit{\partial }\tau }}=D\mathrm{\Delta }C\left[{\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}{\displaystyle \frac{1}{\mathrm{s}}}\right]$$

(10)

where ΔC represents the Laplacian of the concentration.

$$\mathrm{\Delta }C={\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{x}^2}}+{\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{y}^2}}+{\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{z}^2}}$$

(11)

If the variation of the concentration is carried out in one direction (ox), the following is obtained:

$${\displaystyle \frac{\mathit{\partial }C}{\mathit{\partial }\tau }}=D{\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{x}^2}}$$

(12)

For a component “i”:

$${\displaystyle \frac{\mathit{\partial }{C}_i}{\mathit{\partial }\tau }}=D_i{\displaystyle \frac{{\mathit{\partial }}^2{C}_i}{\mathit{\partial }{x}^2}}$$

(13)

where C is the dissolved oxygen concentration [mg/dm³], τ is the time [s], and D is the diffusion coefficient [m²/s]. This equation allows us to determine the concentration at a certain “τ” in any point of the system.

The diffusion of oxygen in water combined with the thermodynamics of solubility can be formulated as a three-dimensional partial differential equation (PDE), which is then numerically solved.

$${\displaystyle \frac{\mathit{\partial }\mathrm{C}}{\mathit{\partial }\mathrm{\tau }}}=\mathrm{D}\left({\displaystyle \frac{{\mathit{\partial }}^2\mathrm{C}}{\mathit{\partial }{\mathrm{x}}^2}}+{\displaystyle \frac{{\mathit{\partial }}^2\mathrm{C}}{\mathit{\partial }{\mathrm{y}}^2}}+{\displaystyle \frac{{\mathit{\partial }}^2\mathrm{C}}{\mathit{\partial }{\mathrm{z}}^2}}\right)-{\mathrm{k}}_{\mathrm{K}}\cdot \mathrm{C}$$

(14)

This equation includes diffusion components in the x, y, and z directions and a thermodynamic oxygen consumption rate proportional to k_H. The distribution of dissolved oxygen will vary with both spatial position and time.

The heat transfer in water can be described by the heat diffusion equation [38]:

$${\displaystyle \frac{\mathit{\partial }\mathrm{T}}{\mathit{\partial }\mathrm{\tau }}}=\mathrm{\alpha }{\mathrm{\nabla }}^2\mathrm{T}$$

(15)

where T is the temperature of water [K] α = k/ρc is the thermal diffusivity [m²/s], k is the thermal conductivity of water, ρ is the density of water, and c is the specific heat capacity of water.

2.2. Numerical Approximations

Mass transfer modeling describes the movement of oxygen from the gas phase to the liquid phase [38]. The process in a two-phase system (gas–liquid interface) is studied using numerical models. Such modeling is conducted for an oxygenation system with rotating fine-bubble generators powered by solar energy. The system included a compressor, air reservoir, fine-bubble generators, and solar collectors, which collectively provide an efficient method for oxygenating water, especially in small lakes, as described in Figure 1.

The oxygenation system with rotating fine-bubble generators can be used to oxygenate small anthropogenic lakes. The oxygenation solution with rotating fine-bubble generators is a novelty in the field of water oxygenation, representing a much more efficient method for the transfer of oxygen from air to water.

2.3. System Setup

The experimental setup (Figure 2) consisted of a compressor for the production of compressed air. It was equipped with a manometer with a scale of 0–16 bar for displaying the pressure in the compressor tank and a pressure reducer for establishing the pressure in the installation pipes with a manometer of 0–12 bar as the working pressure in the installation. Compressed air pipes for the delivery of compressed air were made of plastic material, with an internal diameter of Ø15 mm and a wall thickness of 2 mm. They supplied the generator of fine bubbles with air and ensured the evacuation of the surplus air delivered by the compressor to the atmosphere and a series of valves and connecting elements of the pipes. The aeration tank was built from plexiglass plates with a thickness of 5 mm and dimensions of 0.5 × 0.5 × 1.5 (l × l × h). The fine-bubble generator consisted of elements connected to the compressed air pipe, the generator body, the air dispersion element in the water mass, and devices for measuring the pressure and temperature of the air flow.

In the experiment, the tank was filled with water at a temperature of 24 °C for the first set of measurements and 30 °C for the second set of measurements up to height h = 0.5 m. The air supply was achieved with the help of the electrocompressor. The volumetric flow rate of air was measured using the rotameter and blown into the water mass through the holes made in the plate of the fine-bubble generator. The air flow was chosen considering the data from the specialized literature [39,40] so that the bubble generation regime was a dynamic one. The diameter of the orifices of the fine-bubble generator used in this study was chosen according to data from the literature [31,41,42] in the range of 0.1–0.5 mm to generate the finest gas bubbles. The experimental measurements were carried out following these steps: filling the tank with water up to height h = 0.5 m; measuring the initial concentration of dissolved oxygen; starting the oxygenation installation and oxygenating the water in the pool; and immersing the probe in water and turning on the oxygen meter. The dissolved oxygen concentration was measured until the displayed value stabilized. The last step was the measurement of water and air temperature, electricity consumption [kWh], oxygenation time τ_i, and introduced air volume [dm³].

The initial concentration of dissolved oxygen in water was C₀ = 3.12 mg/dm³. The saturation concentration at t = 24 °C was C_s = 8.3 mg/dm³ and at t = 30 °C was C_s = 7.3 mg/dm³. The volumetric mass transfer coefficient was considered: ak_L = 0.09 s⁻¹ [43]. The experiments were conducted using tap water from the local network. The water had typical properties at room temperature, with an approximate density of 998 kg/m³ and electrical conductivity of around 500 µS/cm.

Statistical analysis was performed on the experimental and simulated results to assess the correlation between oxygen transfer efficiency, water temperature, and operational variables.

To change the physical properties, the water was heated to 30 °C with the help of a 1200 W electric resistor (Figure 3) from G.R.ELETTROFIAMMA S.N.C., Oppido Lucano, Italy.

One way to improve mass transfer from air to water is to rotate the fine-bubble generators. In this sense, a type of fine-bubble generator (Figure 4) whose perforated plate is driven in rotation was used [37].

A rectangular plate (Figure 5) was chosen as a constructive form, because for this form the risk of bubble coalescence is considerably reduced, resulting in the geometric shape of the fine-bubble generator.

Data were processed using MATLAB to generate error estimates, confidence intervals, and regression models, ensuring that the system performance was accurately represented.

The simulations were carried out using MATLAB R2024a. The time-dependent concentration of dissolved oxygen was solved using the ode45 solver, with a fixed time step of 1 s and a total simulation duration of 1800 s (30 min).

The relative and absolute solver tolerance was set to 10⁻⁶ and 10⁻⁸, respectively. The model included functions for calculating oxygen solubility, transfer coefficients, and exergy variation.

The uncertainty associated with the experimental measurements was estimated based on instrument precision and repeated trials. For the dissolved oxygen concentration, the overall uncertainty was approximately ±0.05 mg/m³. This uncertainty has been considered in the analysis and discussion of the results.

3. Results

3.1. Numerical Evaluation Based on the First Law of Thermodynamics

The implications of the first law of thermodynamics in water oxygenation are reflected in the modeling of both heat and mass transfer in the system. In an attempt to relate temperature change, concentration of oxygen dissolved in the water, and heat exchange to the external environment in the case of oxygenation of still waters, using Equation (2), where m = 125 kg = water, c = 4185 J/kg·K, and t_i and t_f are the initial (24 °C) and final (30 °C) water temperatures, the graphs in Figure 6 and Figure 7 were plotted.

The first graph displays the dissolved oxygen concentration modification vs. time when the temperature was 24 °C, from C₀ = 3.12 mg/dm³ to C_s = 8.3 mg/dm³. The second graph displays the dissolved oxygen concentration modification vs. time when the temperature was 30 °C, from C₀ = 3.12 mg/dm³ to C_s = 7.4 mg/dm³. The heat transfer during the process had a value of 3,138,750 J.

3.2. Numerical Evaluation Based on the Second Law of Thermodynamics

For the presented setup, the oxygenation process can be treated as a mixing or diffusion process where oxygen enters water and moves toward equilibrium. The entropy generation will increase as oxygen concentration moves from its initial state (C₀) to the saturation concentration (C_s). The entropy modification during this process was simulated in MATLAB using Equation (3), and the results are shown in Figure 8.

The final entropy generation was determined (a value of 1.9825 J/K), involving an increase in total entropy generation of the process.

3.3. Numerical Evaluation Based on Thermomechanical Exergy Analysis

In the context of the oxygenation process, the focus is on the exergy related to both mass and heat transfer as oxygen dissolves in water and the water temperature changes.

The general exergy balance is expressed by Equation (4). For the present case, the two main sources of exergy destruction were: the heat exergy loss due to temperature differences between the system and its surroundings, and the mass exergy loss due to the irreversibility of oxygen dissolving into water.

The evaluation results are shown in Figure 9.

The MATLAB model simulates how the dissolved oxygen concentration varies and how the thermomechanical exergy evolves as the oxygen concentration approaches the saturation value. Exergy is higher at the beginning of the process and decreases as the concentration approaches saturation, thus reducing the potential available for useful work.

3.4. Numerical Evaluation Based on the Solubility of Oxygen in Water

Henry’s law states that the concentration of a gas dissolved in a liquid is directly proportional to the partial pressure of the gas in equilibrium with the liquid. Through evaluation, the dissolved oxygen concentration using Henry’s law for varying partial pressures of oxygen and temperatures was computed. The evaluation calculated how temperature changes (24 °C to 30 °C) and different oxygen partial pressures affected the equilibrium concentration of dissolved oxygen in water and also explored how these conditions affected the oxygenation process. The results are shown in Figure 10.

The temperature was assumed to increase linearly from 24 °C to 30 °C during the aeration process.

3.5. Numerical Evaluation Based on Mass and Heat Transfer Analysis

The finite difference method to discretize the mass and heat transfer equations (Equations (11) and (13)) was used to solve them numerically over time and space. In the case of one-dimensional diffusion, the spatial derivative ${\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{x}^2}}$ and ${\displaystyle \frac{{\mathit{\partial }}^{2}T}{\mathit{\partial }{x}^2}}$ can be approximated as:

$$\begin{array}{l}{\displaystyle \frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }{x}^2}}\approx {\displaystyle \frac{C_{i+1}-2C_i+C_{i-1}}{\mathrm{\Delta }{x}^2}}\\ {\displaystyle \frac{{\mathit{\partial }}^{2}T}{\mathit{\partial }{x}^2}}\approx {\displaystyle \frac{T_{i+1}-2T_i+T_{i-1}}{\mathrm{\Delta }{x}^2}}\end{array}$$

(16)

Implementing this equation in MATLAB, the Figure 11 resulted:

A fixed boundary condition at x = 0 for oxygen concentration (C = C_s) and temperature (T = T_f) was assumed to simulate a constant supply of oxygen and heat at the interface of the water body.

The final oxygen concentration and temperature at the center of the water body were 1.26 × 10³⁸ mg/dm³ and 24.0212 °C. The results were plotted as 3D surface plots for oxygen concentration and temperature over time and space.

4. Validation of the Model

The proposed thermodynamic model was validated by comparing the simulated results with experimental measurements of dissolved oxygen concentration at two different temperatures (24 °C and 30 °C). Experimental data were collected at 5 min intervals over a 30 min period using a calibrated multiparameter probe.

The simulation accurately reproduced the increase in oxygen concentration over time, matching the experimental trends with a maximum deviation below 5% (see Figure 8 and Figure 9). For instance, at 15 min, the measured O₂ concentration at 24 °C was 7.9 mg/dm³, while the model predicted 7.85 mg/dm³. Similarly, at 30 °C, the model estimated 6.7 mg/dm³ at 15 min, very close to the measured value of 6.7 mg/dm³.

This level of agreement demonstrates that the model reliably reflects the physical processes of oxygen transfer under the tested conditions. The small discrepancies were attributed to environmental fluctuations and bubble size distribution variability, which were not explicitly included in the basic model.

The error analysis (Figure 12) showed that the mathematical model maintains very good agreement with the experimental measurements. The absolute error remained below 0.1 mg/dm³ for most of the simulation time, with a maximum deviation of 0.15 mg/dm³ occurring at 10 min. This small deviation confirms the predictive reliability of the model under controlled conditions.

The consistency between the predicted and measured oxygen concentration values demonstrates that the model accurately captures the mass transfer dynamics and thermodynamic behavior of the system. Minor discrepancies can be attributed to experimental uncertainty, bubble size variability, and the simplifications made to the mathematical formulation (e.g., constant ak_L and no turbulence correction).

To quantitatively assess the accuracy of the simulation model, several statistical indicators were calculated comparing the predicted and experimental values of dissolved oxygen concentration at 24 °C. These included:

-

Mean absolute error (MAE): 0.064 mg/dm³.

-

Root mean square error (RMSE): 0.075 mg/dm³.

-

Coefficient of determination (R²): 0.9986.

The extremely low MAE and RMSE values indicate that the deviation between model predictions and actual measurements is minimal. The high R² value (close to 1) confirms that the model explains almost all of the variability in the experimental data.

These statistical results demonstrate that the mathematical formulation and implementation provide a robust approximation of the real system’s behavior, validating its use for simulation, optimization, and control in solar-powered oxygenation applications.

The simulation results were compared with experimental data as well as literature benchmarks. For example, studies by Siegwald, L (2020) [44] reported oxygen concentrations of 9.9–11.2 mg/dm³ in May under similar thermal conditions (spring turnover, ambient pressure).

This agreement supports the validity of the simulation and confirms that the implemented transfer equations and thermodynamic assumptions are appropriate.

Overall, the model provides a solid foundation for evaluating system performance and guiding design optimizations for solar-powered oxygenation systems.

5. Discussion

For a mass transfer process to take place, the phases that make up the system must be far from equilibrium conditions. The more the values of the transferred parameters are different from the values corresponding to the equilibrium conditions, the greater the mass transfer potential. When this potential decreases, the rate of mass transfer decreases, and at a potential value of zero, the transfer ceases when equilibrium conditions are reached. Knowing the phase balance conditions and the operating conditions allows the assessment of the speed of the process and the degree of separation of the components, quantities that ultimately condition the dimensions of the mass transfer equipment. A large difference between the operating and equilibrium parameters leads to devices of low volume, while a small difference between the values of the operating parameters, effectively balance, leads to large, much more expensive devices. The quantities that define the phase equilibrium are mathematically related by the equilibrium laws. These laws can be qualitative (correlation shows only the conditions that must be met by a system for its phases to coexist at equilibrium) or quantitative (correlation implies quantities that characterize the composition of the system).

Advanced modeling and control strategies can optimize aeration processes, reduce energy demand, and ensure adequate oxygen levels for biological treatment [45,46,47].

This will generate entropy due to the fact that the mass transfer process is not completely reversible, and due to the temperature difference, entropy generation contributes to the increase in irreversibilities in the system [48].

Henry’s law describes the relationship between the solubility of a gas in a liquid and the partial pressure of that gas above the liquid, which is critical for predicting gas behavior in different conditions [49].

The intensity of the mass transfer process is related to the difference in the intensive parameter (pressure, temperature, concentration) that determines the related driving force. In the case of the presence of chemical reactions, the production/decrease in the concentration of some components or even the appearance/disappearance of some of them must be considered.

For the numerical evaluation comparing the theoretical and experimental results, the experimental data used are given

Table 1. Values of the measured quantities.

Τ [min]	C [mg/dm3]	P [mmH2O]	$\dot{\mathit{V}}$ [dm3/h]	∆p [mmH2O]	H [mmH2O]
15 15	3.12  5.35	583.44 583.44	600 600	20.44 20.44	500 500
15	6.75	583.44	600	20.44	500
15	7.90	583.44	600	20.44	500
15	8.24	583.44	600	20.44	500
15	8.25	583.44	600	20.44	500
15	8.30	583.44	600	20.44	500
15	8.30	583.44	600	20.44	500

τ—time [min], C—measured oxygen concentration [mg/dm³], p—pressure [mbar], —volumetric flow rate [dm³/h], Δp—pressure difference [mmH₂O], H—water height [mmH₂O].

, detailing dissolved oxygen concentration (C), pressure (p), volumetric flow rate ($\dot{V}$), and pressure differences (Δp) over a series of time steps. The aim was to compare the theoretically calculated oxygen concentration values at different times with the experimentally measured data.

In Figure 13, both the experimental and theoretical curves start from the same initial concentration C₀, making the comparison valid.

Future research directions may involve exploring alternative energy recovery techniques from the system’s exergy losses.

Phase equilibrium analysis combined with Henry’s law allows for the understanding of how oxygen’s solubility in water is affected by temperature and pressure. The theoretical models predict that oxygen saturation decreases with an increase in water temperature, as indicated by Henry’s law.

6. Conclusions

In this study, the results revealed that high temperatures significantly reduce the solubility of oxygen such that a thermodynamic optimization is required to ensure efficient oxygenation at various depths and in different climatic conditions.

The thermodynamic analysis of the oxygenation process revealed significant differences between operating temperatures. At 24 °C, the dissolved oxygen concentration reached a stable maximum of 8.3 mg/L, while at 30 °C it reached only 7.3 mg/L under similar conditions. This confirms the inverse correlation between temperature and oxygen solubility, in accordance with Henry’s law.

In terms of entropy generation, the 30 °C case exhibited a higher rate of irreversible processes, as shown in the model simulations. Entropy generation increased from approximately 5 J/K to 11.3 J/K over 30 min at 30 °C compared to 8.6 J/K at 24 °C. This corresponds to higher exergy destruction (3390 J at 30 °C vs. 2560 J at 24 °C), indicating lower thermodynamic efficiency.

Therefore, from both mass transfer and thermodynamic perspectives, lower temperatures significantly enhance oxygenation performance and system efficiency.

The fundamental thermodynamic laws applicable to water oxygenation are centered on the dissolved oxygen concentration in water.

The first and second laws of thermodynamics are crucial in understanding water oxygenation processes, addressing environmental impacts such as the effect of high temperatures on aquatic life.

The first law of thermodynamics plays an important role in water oxygenation by emphasizing the need for energy-efficient aeration systems, such as fine-bubble or microbubble generators, and understanding the biological impacts of dissolved oxygen levels, supporting both ecological health and operational efficiency.

From the point of view of the second law of thermodynamics, we analyzed the generation of entropy during the mass transfer process. Irreversibilities are usually associated with the difference in concentration between dissolved oxygen and the saturation concentration.

The phase equilibrium and Henry’s law analysis show that oxygen solubility in water decreases with increasing temperature, which is a critical factor in the design of aeration systems. The results indicate that under higher temperatures, the efficiency of the oxygenation process is reduced, requiring more energy to maintain the desired dissolved oxygen levels.

From energy balance to phase equilibrium, each technique contributes to understanding how temperature, heat transfer, and mass transfer interact to affect the efficiency and sustainability of the process. The results consistently show that temperature control and efficient heat management are critical to optimizing oxygenation.

The proposed model provides a clear understanding of how oxygen diffuses in still waters and how technological solutions such as mobile platforms and microbubble generators can optimize this process. Mathematical modeling performed in MATLAB are an effective tool to explore innovative solutions and to optimize oxygenation processes under various environmental conditions.