Gas–Liquid Mixability Study in a Jet-Stirred Tank for Mineral Flotation

Abstract

Micro- and nano-bubble jet stirring, as an emerging technology, shows great potential in complex mineral sorting. Flow field characteristics and structural parameters of the gas–liquid two-phase system can lead to uneven bubble distribution inside the reaction vessel. Gas–liquid mixing uniformity is crucial for evaluating stirring effects, as increasing the contact area enhances reaction efficiency. To improve flotation process efficiency and resource recovery, further investigation into flow field characteristics and structural optimization is necessary. The internal flow field of the jet-stirred tank was analyzed using computational fluid dynamics (CFDs) with the Eulerian multiphase flow model and the Renormalization Group (RNG) k − ε turbulence model. Various operating (feeding and aerating volumes) and structural parameters (nozzle direction, height, inner diameter, and radius ratio) were simulated. Dimensionless variance is a statistical metric used to assess gas–liquid mixing uniformity. The results indicated bubbles accumulated in the middle of the vessel, leading to uneven mixing. Lower velocities resulted in low gas volume fractions, while excessively high velocities increased differences between the center and near-wall regions. Optimal mixing uniformity was achieved with a circumferential nozzle direction, 80 mm height, 5.0 mm inner diameter, and 0.55 radius ratio.

1. Introduction

The significance of mineral resources to economic and social development is indisputable. However, many mineral resources cannot be utilized directly after mining; they must undergo a series of sorting and processing procedures [1]. Among these procedures, flotation stands out as one of the most crucial and widely applied mineral sorting technologies [2]. The principle of flotation is based on the selective attachment between mineral particles and bubbles. During flotation, the slurry (a mixture of minerals and water) is introduced into a flotation cell, and air is injected through a gas–liquid two-phase mixing system, generating a large number of fine bubbles. Hydrophobic mineral particles preferentially attach to the surface of the bubbles and rise with them to the surface of the slurry, forming mineralized froth, thus separating from the hydrophilic minerals. This process takes full advantage of the differences in the surface properties of minerals to achieve mineral separation [3].

In stirred tank flotation operations, traditional stirred tanks utilize mechanical stirring, leading to challenges like high energy consumption, control difficulties, and bubble crushing [4]. To tackle these issues, jet-stirring technology was introduced. This innovative approach leverages the kinetic effect of high-speed jets. By generating numerous micro- and nano-bubbles through high-pressure jets, the stirred tank efficiently mixes them with suspended matter and colloids in the slurry, ultimately boosting the recovery rate of concentrates [5].

Research on jet mixing began as early as the 1950s [6], and, by the end of the 20th century, jet-mixing technology had been further developed. Researchers subsequently began to systematically study the flow characteristics of jets [7]. In the past 20 years, with the development and application of computational fluid dynamics (CFDs) technology, analyses of different mixing situations have become increasingly comprehensive [8]. In 2002, Patwardhan [9] predicted the mixing time and concentration distribution in a jet-mixing system using jet velocity, nozzle angle, and geometry as variables. In 2009, Wu [10,11,12] simulated and analyzed gas mixing and stirring, proposing a design of a stirring device from a multifaceted perspective. Parvareh et al. [13] experimentally investigated the effect of jet position on the neutralization performance of alkali streams. Experiments on the mixing of the neutralization reaction were conducted at optimal and worst positions, revealing that the jet position significantly affected neutralization performance. In 2017, Wang et al. [14] proposed a rotary jet-mixing device for the mixing of multiphase streams containing oxygenated air, lime slurry, and limestone particles to optimize the removal of sulfur dioxide. In 2019, Esmaeelzade’s team [15] calculated residence time and mixing within a jet-stirred reactor at different flow rates using a large eddy simulation, focusing on non-reacting mixtures under ambient conditions. In 2023, Zhang et al. [4] explored fluid flow and mass transfer patterns in a loop jet reactor, correlating these properties to guide reactor design.

Current research on jet mixing primarily examines the influence of operating parameters on performance, with a limited focus on structural parameters. Gas–liquid mixing uniformity is a critical parameter for evaluating the effectiveness of jet-stirred tanks, as improving this uniformity enhances production efficiency and reduces energy consumption, which is vital for effective jet stirring [16,17]. Many studies have explored this area, particularly in traditional mixing tanks where impeller design significantly affects hydrodynamic behavior. Yang et al. [18] designed a new type of impeller with a gridded structure, which demonstrated superior gas dispersion performance and lower energy consumption compared to conventional impellers. Zheng et al. [19] introduced a fan impeller into the gas–liquid mixing process and found it enhanced oxygen delivery efficiency. Li and Chen [20] investigated various impeller types and concluded that impellers with 45° and 60° slopes achieve more uniform gas–liquid mixing. Despite these advancements, research specifically addressing the uniformity of gas–liquid mixing in jet stirring remains limited. In the jet-stirred tank examined here, factors influencing gas–liquid mixing include jet velocity, nozzle orientation, nozzle height, nozzle inner diameter, and nozzle radius ratio. These parameters affect the dispersion and stability of bubbles in the liquid phase, which in turn impacts flotation effectiveness and resource recovery.

Therefore, simulations were conducted with different feed and aeration volumes to study their effects on the flow field distribution within the jet-stirred tank. Structural parameters such as nozzle direction, height, inner diameter, and radius ratio were analyzed to determine their impact on gas–liquid mixing. The originality of our work lies in the introduction of dimensionless variance as a statistical measure to evaluate gas–liquid mixing uniformity. This method allows for a more precise quantification of the effects of different parameter combinations on mixing performance by normalizing variance with the dimensionless Reynolds number. This dimensionless approach facilitates effective comparison and validation of results across various scales and conditions, ensuring that our research conclusions have broad applicability.

Unlike previous studies that mainly focused on operational parameters, our method provides a more accurate and generalizable tool for quantifying mixing effects. This innovation not only addresses gaps in current research on mixing uniformity but also offers a more scientific basis and practical guidance for the design and application of jet-stirring technology. By enhancing mixing uniformity, this study significantly improves flotation efficiency, which is its primary practical significance. Greater mixing uniformity leads to higher flotation efficiency, thus improving the recovery rate of mineral resources and the efficiency of resource utilization. These findings not only have theoretical value but also provide practical design guidance for industrial applications, advancing the practical use and optimization of jet-stirring technology in mineral flotation.

2. Methodology

2.1. Geometric Models

The 3D model of the jet-mixing tank was constructed with SpaceClaim 18.0 software. The structural dimensions of the jet-stirred tank are presented in

Table 1. Structural dimensions of jet-stirred tank.

Structures	Value
Inlet diameter/mm	10
Outlet diameter/mm	80
Cylinder diameter/mm	500
Cylinder height/mm	680
Distance of inlet from the bottom surface of the cylinder/mm	80
Nozzles/pcs (pieces)	4

. Figure 1 shows the computational domain model of the mixing tank, excluding wall thickness.

2.2. Multiphase and Turbulence Modelling

The transient three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations are analyzed in the context of turbulent fluid flow, focusing primarily on the flow field averaging effects rather than transient details. Each phase is treated as a continuous medium permeable to one another, with the slurry considered a continuous, incompressible fluid and the bubbles as proposed continuous phases. By treating bubbles as continuous phases, the RANS model can compute the average behavior of the flow field by integrating the overall impact of bubbles on the flow field. The jet-stirred tank operates under isothermal conditions, thus excluding considerations for heat transfer processes.

Guo et al. [21] investigated gas–liquid mass transfer in an aeration tank using the Euler–Euler model and mixing model, respectively, and compared these with experimental results. Their findings indicated that the Euler model is more suitable for simulating gas–liquid two-phase flow. Therefore, the interaction between air and liquid phases in the flow field inside the mixing tank is simulated using the Euler–Euler multiphase flow model in this study.

With the mixing tank as the control body, the continuity equation is:

$${\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\mathsf{\rho }\right)+\nabla \cdot \left(\mathsf{\rho }\mathsf{\nu }\right)=0$$

(1)

where $\mathsf{\nu }$ is the mass-average velocity of the mixture; $\mathsf{\nu }={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\alpha }}_{\mathrm{i}}{\mathsf{\rho }}_{\mathrm{i}}{\mathrm{u}}_{\mathrm{i}}}{\mathsf{\rho }}}$; ${\mathsf{\alpha }}_{\mathrm{i}}$, ${\mathsf{\rho }}_{\mathrm{i}}$, ${\mathrm{u}}_{\mathrm{i}}$ are the volume fraction, density, and average flow rate of phase i, respectively; $\mathsf{\rho }$ is the average density of the multiphase flow, $\mathsf{\rho }=\sum _{\mathrm{i}=1}^{\mathrm{n}} {\mathsf{\alpha }}_{\mathrm{i}}{\mathsf{\rho }}_{\mathrm{i}}$, and n is the number of phases.

The momentum equation can be expressed as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}\right)}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}\right)}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}\right)}{\partial \mathrm{z}}}=\\ -{\mathsf{\alpha }}_{\mathrm{q}}{\displaystyle \frac{\partial {\mathrm{P}}_{\mathrm{q}}}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial }{\partial \mathrm{x}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{xx}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{y}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{xy}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{z}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{xz}}\right)+{\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{g}}_{\mathrm{x}}+{\mathrm{M}}_{\mathrm{pq},\mathrm{x}}\\ \\ {\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}\right)}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}\right)}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}\right)}{\partial \mathrm{z}}}=\\ -{\mathsf{\alpha }}_{\mathrm{q}}{\displaystyle \frac{\partial {\mathrm{P}}_{\mathrm{q}}}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial }{\partial \mathrm{x}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{yx}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{y}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{yy}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{z}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{yz}}\right)+{\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{g}}_{\mathrm{y}}+{\mathrm{M}}_{\mathrm{pq},\mathrm{y}}\\ \\ {\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}{\mathrm{u}}_{\mathrm{q},\mathrm{x}}\right)}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}{\mathrm{u}}_{\mathrm{q},\mathrm{y}}\right)}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}{\mathrm{u}}_{\mathrm{q},\mathrm{z}}\right)}{\partial \mathrm{z}}}=\\ -{\mathsf{\alpha }}_{\mathrm{q}}{\displaystyle \frac{\partial {\mathrm{P}}_{\mathrm{q}}}{\partial \mathrm{z}}}+{\displaystyle \frac{\partial }{\partial \mathrm{x}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{zx}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{y}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{zy}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{z}}}\left({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\tau }}_{\mathrm{zz}}\right)+{\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{g}}_{\mathrm{z}}+{\mathrm{M}}_{\mathrm{pq},\mathrm{z}}\end{array}\right.$$

(2)

The equations describe the momentum conservation for each phase in the respective directions. The terms on the left side of the equations represent the changes in mass and momentum for each phase. On the right side, the terms include the pressure gradient, viscous stresses (including Reynolds stresses due to turbulence effects), volumetric forces (such as gravity), and interphase momentum coupling terms. ${\mathsf{\alpha }}_{\mathrm{q}}$ represents the volume fraction of phase $\mathrm{q}$; ${\mathsf{\rho }}_{\mathrm{q}}$ denotes the density of phase $\mathrm{q}$; ${\mathrm{u}}_{\mathrm{q},\mathrm{x}}$, ${\mathrm{u}}_{\mathrm{q},\mathrm{y}}$, ${\mathrm{u}}_{\mathrm{q},\mathrm{z}}$ are the velocity components of phase $\mathrm{q}$ in the three directions of x, y, z, respectively; ${\mathrm{P}}_{\mathrm{q}}$ represents the pressure of phase $\mathrm{q}$; ${\mathsf{\tau }}_{\mathrm{xx}}$, ${\mathsf{\tau }}_{\mathrm{yy}}$, ${\mathsf{\tau }}_{\mathrm{zz}}$ are the normal viscous stress components in the x, y and z directions, respectively; ${\mathsf{\tau }}_{\mathrm{xy}}$, ${\mathsf{\tau }}_{\mathrm{xz}}$, and ${\mathsf{\tau }}_{\mathrm{yz}}$ denote the tangential viscous stress components on the x-y, x-z, y-z planes, respectively; ${\mathrm{g}}_{\mathrm{x}}$, ${\mathrm{g}}_{\mathrm{y}}$, and ${\mathrm{g}}_{\mathrm{z}}$ represent the components of gravitational acceleration in the directions of x, y, z; in this study, ${\mathrm{g}}_{\mathrm{x}}$, ${\mathrm{g}}_{\mathrm{y}}$ are zero; ${\mathrm{g}}_{\mathrm{z}}$ represents the gravitational acceleration in the z direction; ${\mathrm{M}}_{\mathrm{pq},\mathrm{x}}$, ${\mathrm{M}}_{\mathrm{pq},\mathrm{y}}$, ${\mathrm{M}}_{\mathrm{pq},\mathrm{z}}$ are the interphase momentum coupling terms between phases $\mathrm{q}$ and $\mathrm{p}$ in the three directions of x, y, z, respectively.

In large-scale simulations, the k − ε turbulence model strikes a good balance between computational cost and accuracy when predicting shear flows. While other models, such as the Shear Stress Transport (SST) model, the Reynolds Stress Model (RSM), and Direct Numerical Simulation (DNS), offer higher accuracy, they require significantly more computational resources. Consequently, their practicality is limited given the scale and complexity of the present study. The flow field inside the mixing tank exhibits complex turbulent characteristics, with the standard k − ε turbulence model commonly used in current studies [22,23,24,25,26]. However, this model has limitations, particularly in accurately predicting shear flow in jet-mixing applications. Rahimi and Parvareh [27] conducted mixing experiments in a semi-industrial-scale tank (depicted in Figure 2a) using a side-incidence injector. Mixing profiles were determined by monitoring the homogenization process of a dark blue Nigrosine solution in the tank. The experiments involved varying jet angles of 0°, 22.5°, and 45°, with data recorded using an in-line detector photometer. Comparing experimental data with numerical simulations using various turbulence models, the mixing curves in Figure 2b demonstrate that the RNG model predicts peaks similar to experimental observations with a reasonable time delay. To further quantify the accuracy of the RNG model, the root mean square error (RMSE) was calculated and presented in

Table 2. Root-mean-square error for Standard, Realizable and RNG turbulence models.

	Standard	Realizable	RNG
RMSE	0.035	0.036	0.016

, showing that the RNG model yields results closer to experimental measurements compared to other models tested. The root mean square error (RMSE) is a commonly used metric to evaluate the discrepancy between model predictions and actual observed data. It reflects the magnitude of the difference between predicted and actual values [28]. The smaller the RMSE value, the closer the model’s predictions are to the actual observations, indicating higher model accuracy.

To enhance the validation of the accuracy and reliability of the RNG turbulence model in this study, we drew upon the work of Leszek Furman and Zdzislaw Stegowski [29], who combined tracer experiments with computational fluid dynamics (CFDs) simulation techniques to conduct an in-depth investigation into the complex fluid flow and mixing mechanisms within a jet mixer. By employing Residence Time Distribution (RTD), a statistical method for measuring the distribution of fluid or particle residence times within a chemical reactor, they systematically compared the experimentally measured RTD data with the RTD data predicted by CFDs simulations, as shown in Figure 3. We write the root mean square errors for the three different turbulence models in

Table 3. Root-mean-square error of the Standard, RSM and RNG turbulence models.

	Standard	RSM	RNG
RMSE	1.84 × 10−4	1.08 × 10−3	2.27 × 10−5

. The comparison results showed that the RTD curve predicted by the RNG k − ε model closely matched the experimental observations. This finding not only highlights the superiority of the RNG k − ε model in handling such complex flow problems but also provides strong external validation and support for the effectiveness and accuracy of using this model in this study.

The k − ε model based on the Reynolds-averaged Navier–Stokes (RANS) approach is widely applicable for the prediction of turbulent flow fields, in which the turbulent viscosity μt is obtained by two additional transport equations for the turbulent kinetic energy k and the turbulent dissipation rate ε [30]. The RNG k − ε model is based on the standard k − ε model and modified by a mathematical technique called the renormalization group (RNG) method, which can achieve a better response to rapidly strained and low Reynolds number flows [31]. The equations for turbulent kinetic energy k and turbulent dissipation rate ε are as follows:

$${\displaystyle \frac{\partial \left(\mathsf{\rho }\mathrm{k}\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }\mathrm{k}{\mathrm{u}}_{\mathrm{i}}\right)}{\partial {\mathrm{x}}_{\mathrm{i}}}}={\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left[{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathrm{k}}}}{\displaystyle \frac{\partial \mathrm{k}}{\partial {\mathrm{x}}_{\mathrm{j}}}}\right]+2{\mathsf{\mu }}_{\mathrm{t}}{\mathrm{E}}_{\mathrm{ij}}{\mathrm{E}}_{\mathrm{ij}} - \mathsf{\rho }\mathsf{\epsilon }$$

(3)

$${\displaystyle \frac{\partial \left(\mathsf{\rho }\mathsf{\epsilon }\right)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \left(\mathsf{\rho }\mathsf{\epsilon }{\mathrm{u}}_{\mathrm{i}}\right)}{\partial {\mathrm{x}}_{\mathrm{i}}}}={\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left[{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathsf{\epsilon }}}}{\displaystyle \frac{\partial \mathsf{\epsilon }}{\partial {\mathrm{x}}_{\mathrm{j}}}}\right]+{\mathrm{C}}_{1\mathsf{\epsilon }}{\displaystyle \frac{\mathsf{\epsilon }}{\mathrm{k}}}2{\mathsf{\mu }}_{\mathrm{t}}{\mathrm{E}}_{\mathrm{ij}}{\mathrm{E}}_{\mathrm{ij}} - {\mathrm{C}}_{2\mathsf{\epsilon }}\mathsf{\rho }{\displaystyle \frac{{\mathsf{\epsilon }}^2}{\mathrm{k}}}$$

(4)

where ${\mathrm{u}}_{\mathrm{i}}$ denotes the velocity component in each direction, and ${\mathrm{E}}_{\mathrm{ij}}$ denotes the component of the deformation rate with the constants ${\mathrm{C}}_{\mathsf{\mu }}$= 0.09, ${\mathsf{\sigma }}_{\mathrm{k}}$= 1.00, ${\mathsf{\sigma }}_{\mathsf{\epsilon }}$ = 1.30, ${\mathrm{C}}_{1\mathsf{\epsilon }}$= 1.44, ${\mathrm{C}}_{2\mathsf{\epsilon }}$= 1.92.

2.3. Boundary Conditions

The flow medium consists of a slurry and bubbles, with the slurry density set to 1300 kg/m³ and viscosity set to 0.001 kg/(m·s). Bubble density is set at 1.225 kg/m³ with a bubble size of 0.001 m. Gravitational acceleration is assumed to be 9.81 m/s².

In the simulation of the flow field within the jet-mixing tank, the boundary conditions are specified as follows:

(1)

Inlet boundary conditions: The jet inlet is set as a velocity inlet. For a total feed volume of 5.64 m³/h, each single nozzle receives 1.41 m³/h, corresponding to a velocity of 5 m/s. The volume fraction of the gas phase is fixed at 0.25, meaning that in the total flow rate of 5.64 m³/h, 25% is gas and the remaining 75% is liquid. Therefore, the gas flow rate is 1.41 m³/h (5.64 m³/h × 0.25), and the liquid flow rate is 4.23 m³/h (5.64 m³/h × 0.75).

(2)

Outlet boundary condition: The outlet boundary adopts a pressure outlet with a pressure value of 0 Pa. The outlet boundary backflow direction specification method is set to normal to boundary.

(3)

Wall boundary condition: A no-slip boundary condition was applied at the wall. Standard wall functions were employed in the turbulence model, with the distance from the wall to the nearest computational node (δ) set to 0.6 mm to ensure that the first computational node is located within the logarithmic region of the boundary layer.

2.4. Solver Settings

Simulations were performed using ANSYS Fluent 18.0. The system of equations is discretized using the finite volume method. Pressure–velocity coupling is achieved using the Phase Coupled SIMPLE algorithm with PRESTO! pressure interpolation. In traditional finite volume methods, velocity components and pressure are often discretized at different grid points. This can lead to inaccuracies in the coupling between the pressure and velocity fields. The PRESTO method reduces such discretization errors by calculating the pressure variable at different grid locations. The equations for momentum, volume fraction, turbulent kinetic energy, and turbulent dissipation rate are solved using the first-order upwind scheme. The second-order upwind scheme generally has an advantage over the first-order scheme in reducing numerical diffusion. However, due to the large grid size and the complexity of the flow characteristics in this study, using the second-order scheme directly would significantly increase computational costs. Additionally, considering the solution speed and resource limitations in large-scale industrial simulations, the first-order upwind scheme offers a reasonable balance between computational cost and accuracy. Therefore, the first-order upwind scheme was adopted in this study. The time step size is 0.2 s. The convergence criterion for the equations is set to 10⁻³.

Table 4. Under-relaxation factors.

Simulation Parameters	Value
Pressure	0.3
Density	0.6
Body Forces	0.6
Momentum	0.7
Volume Fraction	0.5
Turbulent Kinetic Energy	0.5
Turbulent Dissipation Rate	0.5
Turbulent Viscosity	0.8

shows the under-relaxation factors.

2.5. Verification of Grid Independence

The computational domain for the response was structured and meshed using Fluent Meshing 18.0 software, as illustrated in Figure 4. The Body of Influence method was employed for local mesh refinement. A total of five different meshes were generated, comprising 480,000, 1,150,000, 1,700,000, 2,010,000, and 2,360,000 ortho-hexahedral cells, respectively. A grid-independent analysis was conducted to compare these mesh configurations, and the results are presented in Figure 5.

For the vertical lines’ velocity magnitude, the average velocity deviations between the 1.7 million grid and the coarser grids (480,000 and 1.15 million) are 3.05% and 7.71%, respectively. In comparison, the deviations are only 0.53% and 0.63% when compared to the finer grids (2.01 million and 2.36 million). Regarding the horizontal lines’ velocity magnitude, the average deviations between the 1.7 million grid and the coarser grids (480,000 and 1.15 million) are 8.43% and 11.50%, respectively, while they are only 0.66% and 0.75% when compared to the finer grids (2.01 million and 2.36 million). Based on these comparisons, the structured grid with 1.7 million cells was chosen for the calculations, balancing accuracy and computational efficiency.

2.6. Verification of Reaction Time Independence

In CFDs simulations, assessing transient stability is crucial. Four monitoring points are positioned within the reactor in this study: point 1 at coordinates (0, 0.23, 0.1), point 2 at (0, 0.23, 0.2), point 3 at (0, 0.23, 0.3), and point 4 at (0, 0.23, 0.4). Transient flow is considered to have reached a steady state when the gas-phase velocity at these monitoring points stabilizes over time. The monitoring results are depicted in Figure 6.

Upon observation, it was noted that by the time the flow had reached 100 s, the relationship between velocity and time at each point gradually stabilized, indicating optimal mixing effects of the mixture. Therefore, for subsequent simulation calculations, 100 s was selected as the time at which flow stability was achieved.

3. Results and Discussion

In industrial settings, the efficiency of liquid–gas mixing in jet-stirred reactors is often influenced by multiple factors. The effects of operational conditions, specifically slurry feed rate and aeration rate, on the flow field characteristics of jet-stirred reactors are examined in this study. Additionally, the impact of various structural parameters on the gas volume fraction within the reactor is analyzed, offering insights for optimizing reactor design.

3.1. Influence of Feed Volume on Flow Field

Five specific operating conditions are investigated in this study, with total feed capacities set at 1.13 m³/h, 3.39 m³/h, 5.64 m³/h, 7.91 m³/h, and 10.17 m³/h, respectively. Adhering to the principle of material conservation, the inlet boundary conditions are detailed in

Table 5. Setting of velocity inlet boundary conditions.

Total Feed Volume	Amount of Feed per Nozzle	Individual Jet Velocity
1.13 m3/h	0.28 m3/h	1 m/s
3.39 m3/h	0.85 m3/h	3 m/s
5.64 m3/h	1.41 m3/h	5 m/s
7.91 m3/h	1.98 m3/h	7 m/s
10.17 m3/h	2.54 m3/h	9 m/s

.

3.1.1. Influence of Feeding Amount on Velocity Field in Stirred Tank

The instantaneous liquid-phase velocities of XY sections taken at different heights in the stirred tank are depicted in Figure 7. It is observed that despite varying inlet velocities across the five working conditions, the liquid-phase velocity follows a consistent distribution trend with spatial height: as height increases, the liquid-phase velocity decreases. This phenomenon results from gravity exerting a downward force on the liquid, countering the upward driving force of the jet and gradually reducing velocity. Across all planes, the liquid-phase velocity gradually increases from the center of the vessel body towards the vessel wall.

3.1.2. Influence of Feeding Amount on Gas Volume Fraction Distribution

Figure 8 illustrates the instantaneous distribution of the gas volume fraction in the central ZY plane at various injection velocities. It is evident from the figure that different injection velocities significantly influence the distribution of the gas volume fraction. At lower injection velocity (V = 1 m/s), the overall gas volume fraction remains low, with a relatively uniform distribution and no noticeable localized areas of high gas volume fraction. As the injection velocity increases to V = 5 m/s, the gas volume fraction distribution begins to change slightly, with gradual increases in certain regions, yet maintaining a generally uniform trend across the plane.

As the injection speed further increases, the rise in the gas volume fraction becomes notably pronounced, and the gas distribution begins shifting towards the center of the reactor body. This phenomenon indicates that higher feed volumes, corresponding to increased injection speeds, not only enhance the generation and dispersion of bubbles but also facilitate their more effective diffusion towards the central region of the tank body.

Figure 9 depicts the variation in the gas volume fraction with height under five different inlet velocity conditions, where Figure 8b zooms into the gray area shown in Figure 8a. Each data point in these curves represents the area-weighted average gas volume fraction at the corresponding height level. Observing these curves reveals that increasing the inlet velocity enhances the overall gas volume fraction in the reactor. The slope of each curve indicates the spatial uniformity of the gas volume fraction distribution. At higher inlet velocities, the gap in the gas volume fraction between points within the reactor increases, leading to reduced uniformity in gas volume fraction distribution.

To assess the mixing uniformity of the gas volume fraction in the stirred tank, 16 points were strategically selected on the ZY plane at the tank’s midpoint, and the gas volume fraction at each point was measured. Variance, a statistical measure, was utilized to evaluate gas–liquid mixing homogeneity, where a larger variance value indicates poorer homogeneity, and a smaller value signifies better mixing. The uniformity of gas volume fraction distribution across this plane was assessed by calculating the variance of the gas volume fraction values at these selected points. The specific locations of these points are illustrated in Figure 10, with coordinates detailed in

Table 6. Coordinate of a point.

Point	(X,Y,Z)
1	(0,−0.2,0.1)
2	(0,−0.1,0.1)
3	(0,0.1,0.1)
4	(0,0.2,0.1)
5	(0,−0.2,0.2)
6	(0,−0.1,0.2)
7	(0,0.1,0.2)
8	(0,0.2,0.2)
9	(0,−0.2,0.3)
10	(0,−0.1,0.3)
11	(0,0.1,0.3)
12	(0,0.2,0.3)
13	(0,−0.2,0.4)
14	(0,−0.1,0.4)
15	(0,0.1,0.4)
16	(0,0.2,0.4)

. In addition to using variance, to enhance the scientific validity and applicability of the study, the variance was also nondimensionalized by combining it with the Reynolds number, a dimensionless parameter. This nondimensionalization facilitates the comparison and verification of results across different scales and conditions, thereby ensuring the broad applicability of the research conclusions.

The Reynolds number equation is:

$$\mathrm{Re}={\displaystyle \frac{\mathsf{\rho }\mathrm{UD}}{\mathsf{\mu }}}$$

(5)

where $\mathsf{\rho }$ is the mixture density, $\mathrm{U}$ is the characteristic velocity (e.g., jet velocity), $\mathrm{D}$ is the characteristic length (e.g., nozzle diameter), and $\mathsf{\mu }$ is the mixture viscosity.

The dimensionless variance (DV) formula is:

$$Dimensionless Variance={\displaystyle \frac{Variance}{{\mathrm{Re}}^2}}$$

(6)

Table 7. Average and variance of different feeding quantities.

	1 m/s	3 m/s	5 m/s	7 m/s	9 m/s
Average	3.71 × 10−3	1.11 × 10−2	1.82 × 10−2	2.89 × 10−2	4.35 × 10−2
Variance	1.79 × 10−5	4.84 × 10−5	4.08 × 10−5	1.26 × 10−4	3.11 × 10−4
DV	1.68 × 10−16	5.09 × 10−16	1.54 × 10−16	2.41 × 10−16	3.59 × 10−16

presents the mean, variance, and dimensionless variance of the gas volume fraction. A transient simulation was conducted, and the mean gas volume fraction on the ZY plane at the moment when the flow reached a steady state was used. From the data in the table, it can be observed that, as the velocity increases, the average gas volume fraction also increases. Notably, at a velocity of 5 m/s, the mean gas volume fraction reaches a relatively high level, indicating that the generation and distribution of bubbles are optimal at this velocity. Furthermore, the dimensionless variance is at its lowest at 5 m/s, indicating that the distribution of the gas volume fraction has reasonable stability. Considering the mean, variance, and dimensionless variance, we selected 5 m/s as the parameter for subsequent experiments. This choice is significant because operating at 5 m/s can improve the uniformity of gas–liquid mixing, thereby enhancing flotation efficiency. At this velocity, the improved mixing uniformity allows for more effective mineral separation and increased recovery rates, thus optimizing the overall performance of the flotation process.

3.2. Influence of Aeration Volume on Flow Field

With the slurry feeding rate maintained at 5.64 m³/h (inlet velocity of 5 m/s), the impact of the flow field was thoroughly examined by varying the aeration rates (air fraction, AF = 0.1, 0.15, 0.20, 0.25, and 0.30).

3.2.1. Influence of Aeration on Velocity Field in Stirred Tank

Figure 11 depicts the instantaneous liquid-phase velocity contours. It is observed that at the 0.15 m position, the velocity contours under each condition exhibit a relatively similar distribution trend, indicating that at this height, changes in filling volume do not significantly affect the distribution of liquid-phase velocity. However, as height increases to 0.35 m, an increase in filling volume causes a decrease in liquid-phase velocity from 0.49 m/s to 0.42 m/s. This suggests that at higher positions (e.g., 0.35 m), increased filling volume may result in wider distribution of gas within the stirred tank, consequently affecting the distribution and velocity of the liquid phase.

3.2.2. Influence of Charging Volume on Gas Volume Fraction Distribution

Figure 12 illustrates the variation in the gas volume fraction with height under different aeration rates, while

Table 8. Average and variance of different aeration rates.

	AF = 0.10	AF = 0.15	AF = 0.20	AF = 0.25	AF = 0.30
Average	7.19 × 10−3	1.11 × 10−2	1.45 × 10−2	1.82 × 10−2	2.26 × 10−2
Variance	5.57 × 10−6	1.48 × 10−5	2.36 × 10−5	4.08 × 10−5	6.82 × 10−5
DV	2.08 × 10−17	5.58 × 10−17	8.90 × 10−17	1.54 × 10−16	2.58 × 10−16

provides the mean values, variances, and dimensionless variances of the gas volume fraction. Analysis of the table reveals that as the aeration rate increases, the mean gas volume fraction, variance, and dimensionless variance all exhibit a linear upward trend. This phenomenon does not clearly indicate the optimal aeration rate, but higher aeration rates do improve gas–liquid mixing efficiency. However, excessively high aeration rates result in significant disparities in the gas volume fraction among different points within the reactor, thereby reducing reaction uniformity to some extent. Therefore, considering all parameters, an aeration rate of 0.25 was selected for subsequent experiments.

3.3. Influence of Nozzle Direction on Mixing Uniformity

The influence of four structural parameters—nozzle direction, nozzle height, nozzle radius, and nozzle radius ratio—on the gas volume fraction of the jet-stirred tank is analyzed in this study. The geometrical model of these structural parameters is shown in Figure 13. Three main nozzle orientations are used in the jet-stirred tank: upward injection, downward injection, and circumferential injection. The nozzle radius for these three models is 5 mm, and the nozzles for the upward and downward jets are angled at 150° relative to the horizontal plane.

Figure 14 illustrates the instantaneous distribution of the gas volume fraction in the jet-stirred tank. The figure reveals that gas does not achieve uniform mixing inside the tank upon entering from the nozzle, whether injected upward or downward. Instead, it rapidly ascends along the walls and exits through the outlet. This flow pattern leads to a non-uniform distribution of gas within the tank, predominantly concentrated near the walls, resulting in significant variations in the gas volume fraction across different tank regions. Conversely, circumferential injection achieves a more uniform distribution of the gas volume fraction.

Figure 15 depicts gas volume fraction versus height curves for three different nozzle orientations. Upon observation, it is evident that gas volume fraction levels are generally higher across all points when using circumferential jetting compared to the other two orientations. Given the significant impact of the gas volume fraction on reaction efficiency and mixing uniformity in practical applications, the circumferential injection was selected as the preferred nozzle direction for subsequent experiments in the jet-stirred tank.

3.4. Influence of Nozzle Height on Mixing Uniformity

To investigate the effect of nozzle height on the gas–liquid mixing homogeneity of the jet-stirred tank, five different nozzle heights are explored: 40 mm, 60 mm, 80 mm, 100 mm, and 120 mm from the bottom surface of the vessel body. The nozzles in these models are oriented circumferentially, each with a nozzle radius of 5 mm. Figure 16 shows the contours of gas volume fraction distribution for different nozzle heights.

Additionally, to illustrate the variation in the gas volume fraction under different operational conditions, the gas volume fraction profile concerning height is depicted in Figure 17. From this plot, it is evident that the gradient of the gas volume fraction is less pronounced at nozzle heights of 80 mm and 100 mm from the tank bottom. This observation suggests a relatively minor variation in the gas volume fraction under these specific conditions.

Table 9. Average and variance of different nozzle heights.

	40 mm	60 mm	80 mm	100 mm	120 mm
Average	2.41 × 10−2	1.97 × 10−2	1.82 × 10−2	1.76 × 10−2	1.73 × 10−2
Variance	1.65 × 10−4	9.01 × 10−5	4.08 × 10−5	1.07 × 10−4	1.33 × 10−4
DV	6.22 × 10−16	3.74 × 10−16	1.54 × 10−16	4.03 × 10−16	5.02 × 10−16

presents the mean, variance, and dimensionless variance of the gas volume fraction in the ZY plane at various nozzle heights. It is evident from the table that at a nozzle height (H) of 80 mm, both the variance and dimensionless variance are minimized, indicating the most uniform distribution of the gas volume fraction. Considering all factors, the condition with H at 80 mm appears to be the most favorable. This is significant because achieving the most uniform gas distribution at this nozzle height enhances the efficiency of the flotation process. The improved gas–liquid mixing uniformity at 80 mm contributes to more effective mineral separation, leading to higher recovery rates and better overall flotation performance.

3.5. Influence of Nozzle Inner Diameter on Mixing Uniformity

In view of investigating the effect of nozzle inner diameter on the gas–liquid mixing uniformity in the jet-stirred tank, five different working conditions were analyzed, i.e., the ratios of the nozzle radius to the radius of the stirred tank were 1:20, 1:30, 1:40, 1:50, and 1:60, respectively. These ratios corresponded to a nozzle radius of 12.5 mm, 8.4 mm, 6.3 mm, 5.0 mm, and 4.2 mm, respectively. The models of these five nozzles were orientated in the circumferential direction, and the nozzle height was 80 mm from the bottom surface.

Table 10. Nozzle inner-diameter condition setting.

The Ratio of Nozzle Radius to Stirred Tank Radius	Nozzle Radius
1:20	12.5 mm
1:30	8.4 mm
1:40	6.3 mm
1:50	5.0 mm
1:60	4.2 mm

shows the nozzle inner-diameter condition setting. Figure 18 is the contour of gas volume fraction distribution for different nozzle diameters.

From the analysis of Figure 19, it is evident that gas volume fraction distribution trends are generally consistent among the cases with nozzle radii (R) of 6.3 mm, 5.0 mm, and 4.2 mm. Notably, between heights Z = 0.18 m and Z = 0.30 m, the gas volume fraction in the R = 4.2 mm case is 7.3% higher compared to the R = 5.0 mm and R = 6.3 mm cases. Below Z = 0.22 m, the R = 5.0 mm case predominates in the gas volume fraction, while, above this height, the R = 6.3 mm case marginally surpasses the gas volume fraction levels. Overall, the gas volume fraction distribution curve for the R = 5.0 mm case exhibits the shallowest slope, indicating a more uniform gas volume fraction distribution along the axial height.

Table 11. Average and variance of different nozzle inner diameters.

	12.5 mm	8.4 mm	6.3 mm	5.0 mm	4.2 mm
Average	1.31 × 10−2	1.48 × 10−2	1.81 × 10−2	1.82 × 10−2	1.93 × 10−2
Variance	1.49 × 10−4	8.85 × 10−5	5.22 × 10−5	4.08 × 10−5	6.13 × 10−5
DV	5.76 × 10−15	7.57 × 10−16	7.93 × 10−16	1.54 × 10−16	2.10 × 10−16

presents the mean, variance, and dimensionless variance of the gas volume fraction for different nozzle inner diameters. Analysis of the gas volume fraction distribution curves and data identifies a nozzle inner diameter of R = 5.0 mm as the optimal condition for subsequent experiments. This selection highlights the impact of nozzle diameter on gas volume fraction distribution and provides a basis for optimizing reactor performance.

3.6. Influence of Nozzle Radius Ratio on Mixing Uniformity

For the purpose of investigating the influence of the positional relationship between the nozzle and vessel body on the mixing homogeneity, the key parameter of nozzle radius ratio, the ratio of the position of the nozzle, r, to the radius of the vessel body, R, was introduced. Five representative working conditions are selected in this study: r: R = 0.45, r: R = 0.50, r: R = 0.55, r: R = 0.60, and r: R = 0.65. r: R = 0.65 represents the maximum ratio ensuring that the nozzle does not contact the wall. The nozzle direction for all five models is circumferential, with a nozzle radius of 5 mm and a nozzle height of 80 mm from the bottom surface. Figure 20 is the contour of gas volume fraction distribution for different radius ratios.

Figure 21 depicts the gas volume fraction profile as a function of height, elucidating the relationship between the gas volume fraction and the radius ratio. Below the height z = 0.19 m, the gas volume fraction exhibits a decreasing trend with increasing radius ratio at identical spatial heights. However, beyond this threshold, a reversal in the trend is observed, where the gas volume fraction gradually increases with higher radius ratios. Upon comprehensive analysis of the entire curve, it becomes evident that the slope is minimized at r: R = 0.55, indicating a smoother variation in the gas volume fraction with height and thereby enhancing the homogeneity of the reaction process under these conditions.

Table 12. Average and variance of different nozzle radius ratios.

	0.45	0.50	0.55	0.60	0.65
Average	1.82 × 10−2	1.72 × 10−2	1.62 × 10−2	1.56 × 10−2	1.53 × 10−2
Variance	4.08 × 10−5	3.86 × 10−5	3.66 × 10−5	3.89 × 10−5	4.31 × 10−5
DV	1.54 × 10−16	1.45 × 10−16	1.38 × 10−16	1.47 × 10−16	1.62 × 10−16

summarizes the mean, variance, and dimensionless variance for different nozzle radius ratios. Based on the distribution curves and data analysis, a nozzle radius ratio of r: R = 0.55 was determined to be the optimal operational condition. This finding underscores the importance of nozzle radius ratio in achieving effective gas–liquid mixing and uniform distribution.

4. Conclusions

The flow dynamics and gas volume fraction distribution in a jet-stirred tank are investigated in this study through numerical simulation. The analysis involves computing and evaluating the flow field of the stirred vessel under varying slurry feeding rates and aeration volumes to assess the impact of operational parameters. Furthermore, optimizations and adjustments are conducted on the structural parameters of the jet-stirred tank to enhance its performance. Despite the focus on a specific case, the results of this study provide valuable insights into the performance of jet-stirring technology. However, it is important to consider the generalizability of these findings. The simulation of a single case may not fully capture the variability of different operational conditions and structural parameters in industrial applications. To address this, we used dimensionless numbers, such as the Reynolds number, to normalize our results. This approach enhances the applicability of our findings by allowing for comparisons across different scales and conditions. The principal findings are outlined as follows:

(1)

In the jet-stirred tank, fluid flow rates decrease with increasing vessel height, leading to a notable velocity gradient and the formation of vortex phenomena. These vortices influence the aggregation of gas bubbles within the tank, resulting in a significantly higher internal gas volume fraction compared to the gas volume fraction near the tank walls.

(2)

When the inlet velocity is lower, the overall gas volume fraction remains low. Conversely, excessively high velocities exacerbate vortex effects, widening the disparity between the gas volume fraction within the vessel and that near the wall surface. Across varying filling volumes, gas movement ranges showed no significant differences, while the upward velocity of the liquid phase near the wall surface notably decreased from 0.49 m/s to 0.42 m/s, coinciding with an increased gas volume fraction.

(3)

During the optimization of the structural parameters, it was determined that the jet-stirred tank achieved optimal mixing homogeneity when employing circumferential jetting for the nozzles, with a nozzle height of 80 mm, a nozzle inner diameter of 5.0 mm, and a nozzle radius ratio of 0.55.