Improving Hydrodynamics and Energy Efficiency of Bioreactor by Developed Dimpled Turbine Blade Geometry

Abstract

The hydrodynamic efficiency of bioreactors is contingent upon the design of the impeller, particularly the blade geometry, which influences flow symmetry. This study evaluates the impact of dimpled surfaces on the blades of a turbine impeller on mixing processes. Investigations were conducted using simulations in ANSYS (2021R2) with the k-ε turbulence model and experiments measuring vortex funnel depth and power consumption at 247 rpm in an 11-L cylindrical vessel. Results indicate that dimples disrupt the rotational symmetry of the blades, increasing the volume-averaged flow velocity from 0.312 m/s to 0.321 m/s (a 2.9% increase); the maximum shear strain rate from 161 s⁻¹ to 1442 s⁻¹; and the turbulent vortex frequency from 183 s⁻¹ to 290 s⁻¹ (a 58% increase). The volume-averaged shear strain rate rose from 44 s⁻¹ to 63 s⁻¹ (a 43% improvement), and the vortex funnel depth increased from 44 mm to 50 mm (a 14% increase), indicating enhanced homogenization. This facilitates efficient processing of sensitive biological organisms, such as mycoplasmas, and more robust structures, including fungi and mycelium. However, power consumption increased by 4.5% (from 4.9 W to 5.1 W). Thus, disrupting symmetry with dimples intensifies hydrodynamic processes, enhancing mixing efficiency, but requires optimization to reduce energy costs, offering prospects for advancing biotechnological systems.

1. Introduction

Mixing processes are fundamental to biotechnological applications, particularly in bioreactors, where precise control of hydrodynamics is critical for optimizing bioprocess efficiency. Effective mixing ensures uniform distribution of nutrients, metabolites, dissolved oxygen, and temperature, which are essential for maintaining cell viability and maximizing product yield during submerged cultivation of cell cultures. However, the efficiency of mixing is intricately linked to the hydrodynamic environment, particularly the shear stress induced by impeller designs, which can significantly impact the viability of shear-sensitive biological organisms such as mycoplasmas, protoplasts, and mesenchymal stem cells. In bioprocessing, the challenge lies in achieving robust homogenization, dispersion, and suspension while minimizing cellular damage, especially for delicate systems used in biopharmaceuticals, regenerative medicine, and vaccine production. This necessitates innovative impeller designs that balance mixing intensity with energy efficiency and shear stress control, addressing a critical need in modern bioprocessing.

The design of turbine mixer blades plays a pivotal role in determining hydrodynamic performance, influencing flow symmetry, turbulence, and shear strain rates within the bioreactor. Conventional turbine mixers, characterized by rotational symmetry, provide stable but often suboptimal mixing for complex bioprocesses, particularly those involving shear-sensitive cultures. Recent advancements have focused on modifying blade geometry to enhance mixing efficiency, with surface modifications such as dimples—inspired by aerodynamic principles like those observed in golf balls—showing promise in generating micro-vortices that intensify turbulence and improve mass transfer. These modifications, however, often introduce trade-offs, such as increased power consumption, which must be carefully evaluated to ensure practical applicability in industrial settings.

This study investigates the impact of dimpled turbine mixer blades on bioreactor hydrodynamics, aiming to enhance mixing efficiency while assessing the associated shear stress and energy costs. By disrupting the rotational symmetry of standard impeller designs, dimples are hypothesized to improve homogenization, dispersion, and suspension processes, making them suitable for a wide range of biological organisms, from fragile mycoplasmas to more resilient fungi and mycelium. The research employs computational fluid dynamics (CFD) simulations using ANSYS (2021R2) and experimental validation to quantify the effects of dimpled surfaces on flow velocity, shear strain rate, turbulence eddy frequency, and power consumption. Furthermore, the study seeks to contextualize these findings within the broader framework of bioprocessing challenges, emphasizing the interplay between mixing efficiency, shear stress, and cell viability. By addressing these factors, the research aims to contribute to the development of optimized mixer designs that support diverse biotechnological applications, offering practical solutions for improving process scalability and productivity [1,2,3].

Ammar Al-Khalidi, in his article, explores the possibilities of improving the standard efficiency of oxygen transfer in aeration tanks using a turbine blade that generates energy to disperse and oxygenate the liquid [4]. The integration of specially designed turbine blades increases efficiency by 25%, and the electric generator recovers excess energy at high flows. The system has demonstrated that it is possible to recover 11% of the energy used, making the process more efficient without additional energy expenditure.

In their study, Mateusz Bartczak and Maciej Pilarek studied the colorimetric method for measuring mixing time in reusable and single-use bioreactors [5]. They drew attention to the importance of accurate measurement of mixing time for the efficient development of bioprocesses, which is critical in biopharmaceuticals, food processing, etc. The main advantage of this method is its non-invasiveness, ease of use, and the ability to obtain detailed information about fluid flows. The authors provided recommendations for setting up experiments, selecting reagents, and processing the obtained images for the most accurate results.

Anca-Irina Galaktion analyses the effect of biocatalyst size and mixer type on mixing efficiency in a bioreactor with immobilized S. Cerevisiae cells in alginate [6]. The study found that increasing the size of the biocatalysts improves mixing due to more intensive circulation at the top of the bioreactor with a pumped mixer and a turbine with curved blades. It is noted that the pump mixer and the Rushton turbine are the most efficient for this type of bioreactor.

Igor Korobiichuk uses a new approach to computer simulation of bioreactor hydrodynamics using ANSYS (2021R2) software, comparing different designs of turbine mixers [7]. In particular, the introduction of a new double-disc turbine with blades offset at different angles significantly reduces the risk of vortices during mixing. According to the results of simulations and experimental studies, the new design showed a low probability of vortex formation at different rotational speeds.

In their study, researchers Emily Liu, Kumar Perumal, and Yudi Samyoudi ilooked at the use of computer fluid dynamics (CFD) to study the mixing process in aerated bioreactors [8]. They focused on the role of aeration speed and mixing speed, analysing their impact on fermentation performance and yield. The use of CFDs has made it possible to simulate and predict processes in bioreactors, demonstrating that lack of proper mixing can significantly reduce fermentation efficiency. The accuracy of the developed model was confirmed by experimental data, which helped to describe the complex mixing mechanism.

Authors Mario Mand, Olga Hahn, and Juliane Meyer, in their study, examined the effects of high shear load on mesenchymal stem cells (MSCs) using a rotary rheometer in a cone-lamellar configuration [9]. They found that such cells, especially those isolated from adipose tissue, show resistance to loads of up to 18.38 Pa for 5 min, although longer exposure or higher levels of stress lead to cell damage. This research is important for the development of enzyme-free cell isolation methods that can minimize damage to MSCs while maintaining their therapeutic potential for regenerative medicine.

In his study, Xiaonuo Teng studied the scaling of BHK-21 cell culture based on similar hydrodynamic conditions in bioreactors [10]. The main goal was to optimize the process of growing cells for vaccine production by correlating hydrodynamic characteristics with mixing rates. Using CFD simulations, the researchers determined the optimal conditions for scaling from a laboratory bioreactor (5 L) to industrial bioreactors (42, 350, and 1000 L). They achieved similar results in terms of cell density and virus production, which confirmed the effectiveness of the proposed scaling strategy.

In their study, Julian Thiecks and Niels Tippkotter studied the increase in succinic acid production using the bacterium Actinobacillus succinogenes in an electrobioreactor [11]. The main goal of the work was to optimize electrochemical conditions to enhance the fermentation process by improving electron transfer, specifically using carbon nanotubes (CNTs). The authors showed that a 33% increase in mixing rate increased fermentation productivity, and CNTs significantly increased succinic acid yield due to better mixing.

Researcher Zaburko and his colleagues, in their study, focused on modelling and optimizing mixing and aeration processes in activated sludge bioreactors. The main goal of the work was to increase the efficiency of aeration and homogenization, which directly affects the efficiency of wastewater treatment. They ran numerical simulations in ANSYS (2021R2) Fluent to investigate the effects of different aeration parameters on turbulence and mixing rates in reactors. The obtained results made it possible to predict optimal conditions for providing the required amount of oxygen and uniform distribution of activated sludge [12].

Authors Seyed Ali Abtahi Mehrjardi and Karim Mazaheri, in their study, explored the enhancement of internal cooling performance in advanced turbine blades through the use of dimpled surfaces. They developed correlations for heat transfer coefficients in three dimpled tube designs (spherical, elliptical, and teardrop) and implemented these in a reduced conjugate heat transfer (RCHT) scheme to reduce computational costs. Using numerical simulations with the k-ω SST turbulence model, they validated their approach against experimental data for smooth tubes and applied it to the C3X turbine blade. Their findings show that dimples, particularly teardrop-shaped ones, reduce blade surface temperatures by up to 130 K and increase heat flux by up to 50%, with teardrop dimples lowering the midspan temperature by 19.8%. This research highlights the potential of dimpled surfaces to improve turbine blade cooling efficiency, offering a computationally efficient method for optimizing thermal performance in gas turbine engines [13].

Authors Leszek Ułanowicz, Paweł Szczepaniak, and Grzegorz Jastrzębski, in their study, explored the application of reverse engineering to model the geometry of turbine blade ring palisades for jet engines used in unmanned aerial vehicles (UAVs). They utilized coordinate-measuring, optical, and tomographic techniques to digitize and reconstruct the complex geometry of rotor and nozzle blades, employing Siemens NX 10 2014 and AutoCAD 2021 software for CAD modelling. Their findings demonstrate that reverse engineering enables accurate reproduction of turbine components at both macro and microscales, facilitating miniaturization and supporting further analyses, such as finite element method (FEM) simulations. This research is significant for designing lightweight, high-performance turbine jet engines for UAVs, enhancing their durability and operational reliability under extreme conditions [14].

Authors Heng Hu, Narmin Hushmandi, and Magnus Genrup, in their study, conducted a numerical investigation on the performance of gas turbine blades, focusing on the effects of simulation models and blade geometry. Using computational fluid dynamics (CFD) in ANSYS Fluent, they evaluated two turbulence models (Realizable k-ε and k-ω SST) with pressure- and density-based solvers to identify the most suitable model for 2D turbine blade analysis. The study compared two blade geometries with different exit blade angles (β2 = 79.5° for Geometry 1 and β2 = 70° for Geometry 2) and analysed the impact of varying inflow incidence angles (−48.8° to 10°). Their findings indicate that the pressure-based k-ω SST model provides accurate predictions with lower computational cost, and Geometry 2 with an incidence angle of 10° yields superior performance in terms of turbine work output and reduced losses. This research contributes to optimizing turbine blade design for enhanced aerodynamic efficiency in gas turbine systems [15].

Authors Lotfi Ben Said, Sarhan Karray, Wissem Zghal, Hamdi Hentati, Badreddine Ayadi, Alaa Chabir, and Muapper Alhadri investigated the mechanical strength and energy efficiency of 3D-printed wind turbine blades made from short carbon fiber-reinforced polylactic acid (SCFR-PLA) composites. Their study utilized fused filament fabrication (FFF) to produce blades with a novel HAWTSav design, inspired by the Savonius rotor, and compared them to conventional NACA-profile blades made from ABS polymer. Through experimental tensile tests and numerical simulations using ANSYS Fluent, they optimized FFF parameters (15% carbon content, 50 mm/s printing speed, 0° raster angle) to enhance the mechanical properties of SCFR-PLA. The HAWTSav blades demonstrated superior mechanical performance, with 68.5% lower strain than NACA blades, and improved energy efficiency, evidenced by higher power coefficients across various tip speed ratios. This research highlights the potential of SCFR-PLA composites for producing durable, efficient, and lightweight wind turbine blades, particularly for small-scale renewable energy applications [16].

Authors Ramesh S. Kempepatil, Mawaheb Al-Dossari, Ayyappa G. Hiremath, Jagadish Patil, A. Alqahtani, Jagadish V. Tawade, Shaxnoza Saydaxmetova, and M. Ijaz Khan, in their study, investigated the influence of surface roughness patterns and slip velocity on the lubrication performance of magnetohydrodynamic (MHD)-affected secant curved annular plates using couple stress lubricants. They derived a modified Reynolds equation based on Christensen’s stochastic theory to account for radial and azimuthal roughness patterns. Their findings show that azimuthal roughness significantly enhances squeeze film pressure, load-carrying capacity, and response time compared to smooth surfaces, while radial roughness has a detrimental effect. The presence of an applied magnetic field, slip velocity, and couple stress lubricants further improves bearing performance. This research is significant for optimizing bearing designs in industrial applications, particularly for enhancing durability and efficiency in high-frequency transducers and other large-scale systems [17].

Various studies reviewed have explored different mixer designs and blade geometries to enhance agitation, including the use of new configurations and surface modifications. In our research, we propose an improvement to the well-known design of the turbine mixer by modifying the surface of the blades. Inspired in idea by the dimpled surface of a golf ball, which reduces drag and enhances turbulence through the formation of micro-vortices, we applied indentations to the blade surface to achieve similar hydrodynamic benefits.

For example, in the study by Hélène Lange et al., shear stress values are given for two types of strains of micro-organisms E. coli and S. cerevisiae [18]. In another study by Biagini F. et al., shear stress values for strains of micro-organisms E. faecalis and Human gut microbiota are given [19]. A significant part of the research on the disintegration of micro-organisms was conducted by Fichte B. A., in whose works the limiting level of shear flow velocity for various micro-organisms is indicated, which makes it possible to assess and compare the resistance of bacteria, fungi, and animal and plant cells to shear stresses [20].

The analysis of the review of literature sources is summarized by us and presented in the form of a diagram (Figure 1), in which—depending on the type of micro-organism—the range of permissible shear stresses (flow shear rate) is indicated. If the shear velocity of the flow is multiplied by the dynamic viscosity of the culture fluid in which a certain culture of the producer is located, then we get the shear stress in Pa.

The values of the hydrodynamic gradients range from 5 to 200,000,000 s⁻¹. This indicates a wide range of conditions in which different types of micro-organisms can be present (Figure 1).

Mycoplasmas and cellular conglomerates exhibit relatively high resistance to landslides, with maximum gradients of 800 and 1000 s⁻¹, respectively.

Fragmentation of organelles, membranes, flagella, and DNA exhibits significant vulnerability, with a gradient of 1,000,000 s⁻¹, suggesting that such structures can be easily destroyed under high shear stresses.

Difference between Gram-positive and Gram-negative bacteria: Gram-negative bacteria require higher gradients to inactivate, which may be due to their greater resistance due to the presence of an outer membrane. At the same time, Gram-positive bacteria require lower gradients (up to 20,000 s⁻¹) to achieve inactivation.

Highly vulnerable micro-organisms—Actinomycetes, fungi, mycelium, and bacterial spores—show great sensitivity to shear stresses. For example, bacterial spores have a gradient of up to 10,000,000 s⁻¹, indicating their high resistance.

Generally, the higher the hydrodynamic gradient, the greater the effort required to kill or fragment micro-organisms. However, as can be seen from the data, even the most resistant micro-organisms (bacterial spores) require enormous gradients to achieve complete inactivation.

Symmetry in the geometry of turbine mixers and their associated flow fields plays a pivotal role in determining hydrodynamic efficiency within bioreactors. Conventional designs, such as the standard turbine mixer, rely on rotational symmetry to achieve uniform mixing of the nutrient medium. However, surface modifications, such as the introduction of dimples, may disrupt this symmetry, potentially enhancing turbulence and improving homogenization, dispersion, and suspension processes at the expense of increased energy consumption. In this context, the study of symmetry and its breaking is critical for optimizing bioprocess performance, rendering the analysis of such phenomena highly relevant to the advancement of biotechnological systems.

This study aims to investigate the turbine mixer in a bioreactor, focusing on modifying the blade surface with indentations to improve mixing efficiency and assess its hydromechanical characteristics. The study emphasizes determining the shear stress generated by the mixing device as a critical factor that may influence the yield of viable cells during the cultivation of producer cultures. Our goal is to intensify hydrodynamic processes during homogenization, dispersion, and suspension of liquid media. This approach needs further research through computer simulations and physical experiments, which are expected to enhance mixing performance. Additionally, based on a review of the current literature, we recognize the need to develop a generalized classification of stresses affecting micro-organisms, which will be addressed in future work.

It is expected that the use of indentations on the blade surface, with a diameter of a few millimetres, will influence the efficiency of homogenization, dispersion, suspension, and energy consumption of the mixer.

This study uses computational models using ANSYS (2021R2) software to comprehensively analyse the effect of recesses on fluid movement in the bioreactor volume. This study was conducted using real faucets printed on a 3D printer.

On the basis of the formed goal, the following research tasks are proposed:

• Verification of the influence of changes in the geometry of the surface of the turbine mixer blades on the intensification of hydrodynamic processes during homogenization, dispersion, and suspension of liquid.
• Study of the effect of depressions on the surface of the blades on the efficiency of the processes of homogenization, dispersion, suspension, and energy consumption of the mixer, through computer simulation and real experiment.
• To classify shear stresses for micro-organisms based on a review of modern scientific sources, with further comparison of their results with the results of our studies.
• Computer simulation of the influence of blade geometry on hydrodynamics in the bioreactor volume.
• The implementation of these tasks will be carried out in the following way.
• Computer simulation—using ANSYS (2021R2) software to analyse hydrodynamic processes under fluid mixing conditions, taking into account the geometry of the blades. This will help to obtain more accurate data on the impact of various parameters on the efficiency of the process.
• Experimental studies with real mixers made on a 3D printer to verify the results of computer simulations and confirm their authenticity.
• Evaluation of the impact of micro-recesses on the surface of the blades is a study of the efficiency of mixing processes depending on the microstructure of the surface, which will optimize energy consumption and increase efficiency.
• Analysis of scientific literature is the collection and generalization of information on different types of stresses for micro-organisms in bioreactors, which will allow research to classify and ensure the possibility of using this type of turbine mixer for the cultivation of different types of producers.

These tasks and methods will help to achieve the goal of the study and provide a comprehensive analysis of the mixing processes in the bioreactor.

2. Materials and Methods

The methodology of this article, which is shown in Figure 2, is designed to study the influence of dimpled geometry on the blades of a turbine mixer on hydrodynamic processes within a bioreactor, inspired by aerodynamic principles akin to the effect of dimples on a golf ball surface, which enhance turbulence through the formation of micro-vortices. The objective is to improve mixing efficiency in bioreactors by modifying the blade surface. The investigation encompasses computational modelling, laboratory experiments, and analysis of the obtained results.

The initial stage involves numerical simulation of hydrodynamic processes using specialized flow analysis software (ANSYS 2021R2). Two turbine mixer models are developed: a standard model with smooth blades and a modified model with dimpled surfaces. Both models share identical primary geometric parameters (diameter, blade height, disk diameter, blade length) but differ in blade thickness to maintain comparable material volumes. The dimples on the modified model are arranged with a specific fill angle and spacing.

The bioreactor model is represented as a cylindrical vessel with defined dimensions, partially filled with liquid, and equipped with a mixer positioned at a specified distance from the bottom. A computational mesh of tetrahedral elements is generated, with refinement near the blades to accurately capture flow dynamics. The liquid model corresponds to water (with specified density, viscosity, and temperature), and a turbulence model based on the Navier–Stokes equations is employed. Parameters such as mixer rotation speed, pressure, and boundary conditions accounting for rotation and reduced computational volume are defined. The analysis focuses on the distribution of flow velocity, shear strain rate, and turbulence vortex frequency.

To validate the simulation results, a laboratory experiment is conducted. Two mixer prototypes (standard and dimpled) are fabricated using 3D printing with materials capable of withstanding the required rotational speeds. The experimental setup includes a mixing device with adjustable power, a cylindrical vessel, and instruments for measuring power consumption, rotational speed, and flow geometry parameters.

The vessel is filled with liquid to a predetermined level, with the mixer positioned at a specified distance from the bottom, and a consistent liquid temperature is maintained. Both mixers are tested sequentially at varying rotational speeds. Power consumption is measured by comparing mixing power to idle operation. The depth of the vortex funnel is determined after flow stabilization. All measurements are repeated to ensure data reliability.

The final stage involves the systematization and analysis of the collected data. Experimental metrics (power consumption, vortex funnel depth) are averaged, tabulated, and visualized graphically. Hydrodynamic characteristics, including the Reynolds number, are calculated to confirm the turbulent flow regime. Mixing power is evaluated using an appropriate formula, with the coefficient for the standard mixer derived from the literature and that for the dimpled mixer determined experimentally. Relationships between hydrodynamic parameters are established.

Simulation results (distributions of velocity, shear strain rate, and vortex frequency) are compared with experimental data. Flow symmetry is analysed: the standard mixer generates an axial flow, while the dimpled mixer produces asymmetric circular flows with micro-vortices. A classification of shear strain rates relevant to biological objects is developed and compared with the values obtained for both mixers.

2.1. Materials and Methods for Computer Simulation

The computational analysis of hydrodynamic processes within the bioreactor was conducted using ANSYS (2021R2) CFX, selected for its robust capabilities in modelling turbulent flows. Two turbine mixer models were developed: a standard model with smooth blades and a modified model with dimpled surfaces, maintaining equivalent material volumes, as illustrated in Figure 3.

The dimensions of the turbine mixers are detailed in Figure 4 and Figure 5. The standard mixer has a diameter of 80 mm, blade height of 16 mm, disc diameter of 60 mm, blade length of 20 mm, and blade thickness of 1.6 mm. The dimpled mixer has a blade thickness of 2.136 mm, dimple radius of 2.67 mm, dimple spacing of 3.2 mm, and a dimple fill angle of 60°. These dimensions ensure equivalent material volumes for both mixers, as presented in

Table 1. Geometry characteristics of turbine mixers blades.

	Standard Turbine Mixer	Dimple Turbine Mixer
$\mathrm{Volume} ({\mathrm{m}\mathrm{m}}^3$)	27,000	27,692
$\mathrm{Surface} \mathrm{Area} ({\mathrm{m}\mathrm{m}}^2$)	10,620	11,468

.

The bioreactor was modelled as a cylindrical vessel (diameter 242 mm, height 246 mm), filled to 75% with water (density ρ = 998.23 kg/m³, viscosity μ = 0.001002 Pa·s) at 20 ± 2 °C, with the mixer positioned 50 mm from the bottom, as shown in Figure 6.

A computational mesh was generated using quadratic tetrahedral elements to accurately capture curved surfaces and turbulent flow features, as depicted in Figure 7. The mesh comprised 490,284 nodes and 326,028 elements, with adaptive refinement near the mixer blades to resolve high velocity and shear strain gradients. The element size near the blades was set to 2 × 10⁻³ m, determined through a mesh independence study. Three mesh density levels (coarse: 150,000 elements; medium: 326,028 elements; fine: 600,000 elements) were tested, with the medium mesh selected for its balance of accuracy (velocity and shear strain rate deviations < 2% compared to the fine mesh) and computational efficiency. The “Resolution” setting was set to 7, and “Mesh Defeaturing” was enabled to eliminate excessively small elements. The “Span Angle Center” was set to “Fine” to ensure precise modelling of blade curvature.

The standard k-ε turbulence model, based on Reynolds-averaged Navier–Stokes (RANS) equations, was employed to model turbulent flow. This model was chosen for its computational efficiency and validated accuracy in simulating turbulent flows in rotating machinery, as supported by prior studies [3,21]. The k-ε model calculates turbulent kinetic energy (k) and its dissipation rate (ε), using standard constants (C_μ = 0.09, C_ε1 = 1.44, C_ε2 = 1.92, σ_k = 1.0, σ_ε = 1.3). A sensitivity analysis was conducted to evaluate the impact of varying C_μ (0.07–0.11) and C_ε1 (1.40–1.48) on key outputs (volume-averaged velocity and shear strain rate), with deviations < 3%, confirming model robustness.

The motion of a fluid is described by the Navier–Stokes equations for an incompressible fluid:

$$\mathsf{\rho }·\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{t}}}+\left(\mathrm{u}\cdot \nabla \right)\mathrm{u}\right)=-\nabla \mathrm{p}+\mathsf{\mu }{\nabla }^2\mathrm{u}+\mathrm{f}$$

(1)

where

• $\mathsf{\rho }$—density of the liquid;
• $\mathrm{u}$—velocity vector;
• $\mathrm{t}$—time;
• $\nabla$—operator nabla;
• $\mathrm{p}$—pressure;
• $\mathsf{\mu }$—dynamic viscosity;
• ${\nabla }^2$—Laplace’s cameraman;
• $\mathrm{f}$—external volumetric forces (for example, gravity).

These equations describe the balance of forces in a fluid, including inertia, pressure, and viscous friction [22]. For turbulent flow, the k-ε model is used, which calculates the kinetic energy of turbulence ($\mathrm{k}$) and its dissipation ($\mathsf{\epsilon }$).

Transport equation for kinetic energy of turbulence k [22]:

$${\displaystyle \frac{Dk}{Dt}}={\displaystyle \frac{1}{\mathsf{\rho }}}·{\displaystyle \frac{\partial }{\partial x_k}}·\left[{\displaystyle \frac{{\mathsf{\mu }}_t}{{\mathsf{\sigma }}_k}}·{\displaystyle \frac{\partial k}{\partial x_k}}\right]+{\displaystyle \frac{{\mathsf{\mu }}_t}{\mathsf{\rho }}}·\left({\displaystyle \frac{\partial U_i}{\partial x_k}}+{\displaystyle \frac{\partial U_k}{\partial x_i}}\right)·{\displaystyle \frac{\partial U_i}{\partial x_k}}-\epsilon$$

(2)

where

• $k$ is the kinetic energy of turbulence per unit mass;
• ${\displaystyle \frac{Dk}{Dt}}={\displaystyle \frac{\partial k}{\partial t}}+U_j{\displaystyle \frac{\partial k}{\partial x_j}}$—a substantial derivative describing a change in time and space (convection), $k$;
• $\rho$—density of the liquid;
• ${\mu }_t$—turbulent (vortex) viscosity;
• ${\sigma }_k$ is the turbulent Prandtl number for the dimensionless constant, $k\left({\sigma }_k=1.0\right)$;
• ${\displaystyle \frac{\partial }{\partial x_k}}\left[{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}{\displaystyle \frac{\partial k}{\partial x_k}}\right]$—diffusion through turbulent transport, $k$;
• $U_i,U_k$—components of the average flow rate in the directions, $(x_i,x_k)$;
• ${\displaystyle \frac{\partial U_i}{\partial x_k}}$—velocity gradient;
• $\left({\displaystyle \frac{\partial U_i}{\partial x_k}}+{\displaystyle \frac{\partial U_k}{\partial x_i}}\right){\displaystyle \frac{\partial U_i}{\partial x_k}}$—generation through flow deformation (production period);
• $\epsilon$—the rate of dissipation of turbulent energy.

Transport equation for turbulence energy dissipation $\epsilon$ [22]:

$${\displaystyle \frac{D\mathsf{\epsilon }}{Dt}}={\displaystyle \frac{1}{\mathsf{\rho }}}{\displaystyle \frac{\partial }{\partial x_k}}\left[{\displaystyle \frac{{\mathsf{\mu }}_t}{{\mathsf{\sigma }}_{\mathsf{\epsilon }}}}{\displaystyle \frac{\partial \mathsf{\epsilon }}{\partial x_k}}\right]+{\displaystyle \frac{C_1{\mathsf{\mu }}_t}{\mathsf{\rho }}}{\displaystyle \frac{\mathsf{\epsilon }}{k}}\left({\displaystyle \frac{\partial U_i}{\partial x_k}}+{\displaystyle \frac{\partial U_k}{\partial x_i}}\right){\displaystyle \frac{\partial U_i}{\partial x_k}}-C_2{\displaystyle \frac{{\mathsf{\epsilon }}^2}{k}}$$

(3)

where

$\mathsf{\epsilon }$—the rate of dissipation of turbulent energy;

${\displaystyle \frac{D\mathsf{\epsilon }}{Dt}}={\displaystyle \frac{\partial \mathsf{\epsilon }}{\partial t}}+U_j{\displaystyle \frac{\partial \mathsf{\epsilon }}{\partial x_j}}$—is a substantial derivative for, $\mathsf{\epsilon }$;

${\mathsf{\sigma }}_{\mathsf{\epsilon }}$—is the turbulent Prandtl number for, a dimensionless constant, $\mathsf{\epsilon }\left({\mathsf{\sigma }}_{\mathsf{\epsilon }}=1.3\right)$;

${\displaystyle \frac{\partial }{\partial x_k}}\left[{\displaystyle \frac{{\mathsf{\mu }}_t}{{\mathsf{\sigma }}_{\mathsf{\epsilon }}}}{\displaystyle \frac{\partial \mathsf{\epsilon }}{\partial x_k}}\right]$—diffusion through turbulent transport, $\mathsf{\epsilon }$;

$C_1$—is the empirical constant that regulates the generation, $\epsilon \left(C_1=1.44\right)$;

${\displaystyle \frac{\mathsf{\epsilon }}{k}}$—is a relation that scales generation, $\mathsf{\epsilon }$;

$C_2$—is the empirical constant that regulates dissipation, $\mathsf{\epsilon }\left(C_2=1.92\right)$;

${\displaystyle \frac{{\mathsf{\epsilon }}^2}{k}}$—dissipation period, $\mathsf{\epsilon }$.

Alternative turbulence models, such as Large Eddy Simulation (LES), were considered but deemed computationally prohibitive due to the high mesh resolution and transient simulations required, increasing computational time by a factor of 10–20. The k-ω SST model was also evaluated but offered marginal improvements in near-wall accuracy at the cost of increased computational demand, which was unjustified for the study’s focus on volume-averaged hydrodynamic parameters. The k-ε model’s adequacy was further supported by its alignment with experimental results (e.g., vortex funnel depth and power consumption within 5% error).

The simulation utilized the “Frozen Rotor” approach to model steady-state rotation at 247 rpm, with the “Rotational Periodicity” condition applied to reduce the computational domain to one-sixth of the vessel, as illustrated in Figure 8. Boundary conditions included a no-slip wall for the tank and blades and atmospheric pressure (1 atm) at the liquid surface, as detailed in

Table 2. Boundary conditions.

Name of the Surface	Type of Boundary Condition
r_f_r_1, r_f_r_2, r_f_r_3, s_f_r_1, s_f_r_2, s_f_r_3	Frozen Rotor
r_r_p_1, r_r_p_2, s_r_p_1, s_r_p_2	Rotational Periodicity
wall, mixer	Wall

. The Navier–Stokes equations were solved iteratively using a conjugate gradient solver, with convergence criteria set to a root-mean-square residual of 10⁻⁵ for momentum, continuity, and turbulence equations. Convergence was achieved within 500–700 iterations, with residuals monitored for stability. A time step sensitivity analysis confirmed the validity of the steady-state assumption.

Post-processing involved analysing distributions of velocity, shear strain rate, and turbulence eddy frequency. Results were validated against experimental measurements of vortex funnel depth (44 mm vs. 50 mm for standard and dimpled mixers, respectively) and power consumption (4.9 W vs. 5.1 W), with discrepancies < 5%. The Reynolds number (26,368) confirmed turbulent flow, aligning with the k-ε model’s applicability.

This methodology ensures reproducibility through detailed documentation of mesh settings, turbulence model parameters, and convergence criteria, providing a robust framework for analysing the hydrodynamic impact of dimpled turbine blades.

2.2. Fractal Analysis of Dimpled Blade Geometry

The description of the design of a potentially possible fractal is demonstrated in Figure 9. Three large circles of the same diameter are placed in such a way that, when in contact, they form a curved triangle in arcs. The side of this triangle can be defined as the length of the arc between the points of intersection of two large circles equal to the length of the great circle. The process is repeated recursively, thus forming a fractal ${\displaystyle \frac{1}{4}}\mathrm{s}={\displaystyle \frac{1}{10}}$.

Let the radius of great circles be the length of one great circle. The length of the arc forming the side of the curvilinear triangle: The length of the arc $\mathrm{R}\mathrm{L}=2\mathsf{\pi }\mathrm{R}{\displaystyle \frac{1}{4}}·2\mathsf{\pi }\mathrm{R}={\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}$. Given the scale factor between the levels $\mathrm{s}$, it can be argued that the radius of the circles at the next level will be ${\mathrm{R}}_1=\mathrm{s}\mathrm{R}={\displaystyle \frac{\mathrm{R}}{10}}$.

The fractal iteration will be based on each level; 3 smaller circles are added to the curved triangle, respectively; at the nth level, the radius of the circles ${\mathrm{R}}_{\mathrm{n}}={\mathrm{s}}^{\mathrm{n}}·\mathrm{R}=\mathrm{R}·{{\displaystyle \frac{1}{10}}}^{\mathrm{n}}$, with each level forming a new curved triangle and the process repeated.

The Hausdorff dimension for a fractal is determined by the similarity equation by [23]:

$$\mathrm{N}·{\mathrm{s}}^{\mathrm{D}}=1$$

(4)

$\mathrm{N}$—the number of copies of the fractal at each level (in our case, because we add 3 smaller circles), $\mathrm{N}=3$;

$\mathrm{s}$—scale coefficient;

$\mathrm{D}$ is the Hausdorff dimension.

Let us convert the equation to establish the Hausdorff dimension of the fractal:

$$3·{\left({\displaystyle \frac{1}{10}}\right)}^{\mathrm{D}}=1$$

$${\left({\displaystyle \frac{1}{10}}\right)}^{\mathrm{D}}={\displaystyle \frac{1}{3}}$$

$${10}^{\mathrm{D}}=3$$

$$\mathrm{D}·\log 10=\log 3$$

$$\mathrm{D}={\displaystyle \frac{\log 3}{\log 10}}\approx 0.4771$$

This value is less than 1, which is logical, because the fractal is made up of curves (one-dimensional objects) that scale with a very small factor, and its “density” in two-dimensional space is small.

To determine the length of the fractal boundary, an analogy with Sierpinski’s triangle and Koch’s snowflake [24] will be used:

In the Sierpinski triangle, the parts are removed, and the Hausdorff dimension is defined by [23]:

$$\mathrm{D}={\displaystyle \frac{\log 3}{\log 2}}\approx 1.585$$

(5)

In Koch’s snowflake, the length of the boundary increases to infinity, because new segments are added at each level [24]

$$\mathrm{D}={\displaystyle \frac{\log 4}{\log 3}}\approx 1.262$$

(6)

The length of the boundary at the nth level increases as ${\left({\displaystyle \frac{4}{3}}\right)}^{\mathrm{n}}$.

In our case, we add new arcs (circle boundaries), and their total length increases with each iteration.

Length at the first level (initial length of the boundary) is the total length of three arcs of a curved triangle formed by large circles.

The length of one arc, therefore, the initial length of the boundary (3 arcs): ${\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}$

$${\mathrm{L}}_0=3·{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}$$

(7)

Length at the following levels:

In the first iteration, add 3 smaller circles with a radius ${\mathrm{R}}_1={\displaystyle \frac{\mathrm{R}}{10}}$.

The circumference length of one small circle is: $2\mathsf{\pi }·{\displaystyle \frac{\mathrm{R}}{10}}={\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{5}}$.

Suppose that each small circle adds to the boundary an arc proportional to the initial arc, i.e., its circumference (by analogy with the initial condition): ${\displaystyle \frac{1}{4}}$

$${\displaystyle \frac{1}{4}}·{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{5}}={\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{20}}$$

The total length added by the three small circles is

$$3·{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{20}}={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{20}}$$

(8)

At the nth level, we add a circle (since each curved triangle gives rise to 3 new circles, and the number of triangles at the nth level is $3^{\mathrm{n}}$).

The radius of the circles at the nth level:

$${\mathrm{R}}_{\mathrm{n}}=\mathrm{R}·{\left({\displaystyle \frac{1}{10}}\right)}^{\mathrm{n}}$$

(9)

The length of the arc added by one circle at the nth level is given by the following equation:

$${\displaystyle \frac{1}{4}}·2\mathsf{\pi }{\mathrm{R}}_{\mathrm{n}}={\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·{\left({\displaystyle \frac{1}{10}}\right)}^{\mathrm{n}}$$

(10)

Total length added at the nth level is as follows:

$$3^{\mathrm{n}}·{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·{\left({\displaystyle \frac{1}{10}}\right)}^{\mathrm{n}}={\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{n}}$$

(11)

The total length of the boundary after n iterations is the sum of the initial length and all added arcs:

$${\mathrm{L}}_{\mathrm{n}}={\mathrm{L}}_0+\sum _{\mathrm{k}=1}^{\mathrm{n}}\left({\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{k}}\right)$$

(12)

Let us substitute the following: ${\mathrm{L}}_0={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}$

$${\mathrm{L}}_{\mathrm{n}}={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}+{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·\sum _{\mathrm{k}=1}^{\mathrm{n}}{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{k}}$$

(13)

Let us calculate the sum of the geometric progression:

$$\sum _{\mathrm{k}=1}^{\mathrm{n}}{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{k}}={\displaystyle \frac{\frac{3}{10}\left(1-{\left(\frac{3}{10}\right)}^{\mathrm{n}}\right)}{1-\frac{3}{10}}}={\displaystyle \frac{3}{7}}·\left(1-{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{n}}\right)$$

$${\mathrm{L}}_{\mathrm{n}}={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}+{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·\sum _{\mathrm{k}=1}^{\mathrm{n}}{\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{k}}$$

(14)

When limit n→∞, therefore ${\left({\displaystyle \frac{3}{10}}\right)}^{\mathrm{n}}\to 0$

$${\mathrm{L}}_{\mathrm{\infty }}={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}+{\displaystyle \frac{\mathsf{\pi }\mathrm{R}}{2}}·{\displaystyle \frac{3}{7}}·1={\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{2}}+{\displaystyle \frac{3\mathsf{\pi }\mathrm{R}}{14}}=\mathsf{\pi }\mathrm{R}·{\displaystyle \frac{12}{7}}$$

(15)

Therefore, the length of the fractal boundary at n→∞:

$${\mathrm{L}}_{\mathrm{\infty }}=\mathsf{\pi }\mathrm{R}·{\displaystyle \frac{12}{7}}$$

(16)

Unlike Koch’s snowflake, where the length of the boundary goes to infinity, in our fractal the length of the boundary has a finite limit, since the scale factor and the contribution of new arcs decreases exponentially.

The fractal design of a turbine impeller blade incorporates first-generation dimples, into which second-generation dimples—ten times smaller in size—are inscribed, and so forth for subsequent generations, with each successive generation scaled down by a factor of $\mathrm{s}={\displaystyle \frac{1}{10}}$. This creates a hierarchy of dimples, where the sizes decrease exponentially; for instance, third-generation dimples are 100 times smaller than the first. Such a design theoretically impacts the hydrodynamics, particularly shear stress, friction, and turbulence, by modifying the boundary layer. However, the second and subsequent generations of dimples are so small that their influence on macroscopic fluid volumes becomes negligible. For example, while first-generation dimples can alter pressure drag by delaying flow separation, second-generation dimples contribute an order of magnitude less to this effect, and the third generation even less, due to the exponential reduction in scale. Turbulence in macroscopic fluid volumes primarily depends on large-scale vortical structures generated by first-generation dimples, whereas smaller dimples only affect microscales, exerting minimal impact on overall flow dynamics. Shear stress and viscous friction are also minimally affected by second-generation dimples, as their contribution to the velocity gradient $\left({\displaystyle \frac{\partial u}{\partial y}}\right)$ is insignificant. Consequently, for macroscopic fluid volumes, the second and subsequent generations of dimples can be disregarded, as their influence constitutes less than 1% of the total effect. This allows for a simplified blade design, limited to first-generation dimples, significantly facilitating manufacturing without compromising efficiency. Focusing on first-generation dimples ensures the primary effect of drag reduction and turbulence control, while neglecting subsequent generations does not impact the hydrodynamic characteristics at a macroscopic scale. Therefore, optimization involving second and subsequent generations of dimples is not practical due to their negligible influence.

2.3. Materials and Methods for Experimental Research

To verify the accuracy of the computer simulation results, we performed a laboratory experiment. A 50 W mixing stand, designed to drive the turbine mixer, was used to study homogenization processes in the tank shown in Figure 6. The laboratory setup replicated the exact parameters of the computer simulation, ensuring that all data input into ANSYS (2021R2) corresponded fully to the conditions observed during the physical tests.

Figure 10 shows a diagram of a mixing bench that was used for laboratory studies of the efficiency of mixing using various turbine mixers. The scheme includes the following elements: 1—Turbine mixer (available in standard and dimple variants), 2—locking screw, 3—shaft, 4—container, 5—mixing device, 6—tachometer, 7—wattmeter.

The mixing stand setup allows for comparing the efficiency of two types of turbine mixers under real conditions similar to those in computer simulations. This confirmed the simulation results and helped analyse how the geometry of turbine mixer blades affects homogenization and energy efficiency.

2.4. Methodology of the Laboratory Experiment

To investigate the influence of turbine mixer blade surface geometry on mixing efficiency and power consumption, a laboratory experiment was conducted using a 50 W test stand.

The following materials and equipment were used:

• Mixing device with a shaft.
• Turbine mixers (3D-printed): standard (smooth blades) and modified (dimpled blades).
• Cylindrical container (D = 242 mm, H = 246 mm), filled with water to 75% of its volume.
• Wattmeter (±2%), tachometer (±1 rpm), ruler (±1 mm).

The following experimental procedure was followed:

• Preparation: The standard and modified turbine mixers were sequentially mounted onto the shaft of the mixing device. The container was filled with water to a height of 184.5 mm, and the mixer was positioned 50 mm above the container bottom.
• Measurement: The rotational speed was adjusted (0–534 rpm: 0, 14, 45, 135, 247, 377, 534 rpm) and monitored using a tachometer. Power consumption was recorded with a wattmeter over a 1 min period, with mixing power determined as the difference from the idle state.
• Funnel Depth: The depth of the vortex was measured using a ruler after flow stabilization (30 s) at each speed.
• Data Processing: Averaged values (three replicates) of power consumption and funnel depth were analysed, systematized in tables, and presented graphically.
• The following conditions were applied:
• Water temperature: 20 ± 2 °C (density ρ = 998.23 kg/m³, viscosity μ = 0.001002 Pa·s).

3. Results and Discussion

3.1. Result of the Analytical Calculation of Power

The equation for calculating the power of a turbine agitator in a reactor depends on several parameters, including the type of liquid, the design of the agitator, the rotational speed, and the diameter of the agitator. The basic formula is expressed in terms of the power number (P) and is as follows [25]:

$$P=K_N·\rho ·n^3·d^5$$

(17)

where $\rho =1000\left[{\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}\right]$—density of the liquid; $n=4.12\left[{\mathrm{s}}^{-1}\right]$—rotational speed; $d=0.08\left[\mathrm{m}\right]$—Turbine mixer diameter; $\mu =0.001\left[\mathrm{P}\mathrm{a}·\mathrm{s}\right]$—dynamic viscosity of water

$$Re={\displaystyle \frac{\rho ·n·d^2}{\mu }}={\displaystyle \frac{1000·4.12·0.0064}{0.001}}=\mathrm{26,368}$$

The Reynolds number is 26,368, which corresponds to turbulence $\left(Re>{10}^4\right)$, $K_N=1.7$—power number, a dimensionless coefficient depending on the type of agitator and the flow mode (laminar or turbulent) [25].

$$P=K_N·\rho ·n^3·d^5=1.7·1000·69.93·0.000032768=3.89\left[\mathrm{W}\right]$$

3.2. Symmetry Analysis of Turbine Mixer Designs

The standard turbine mixer with smooth blades exhibits a high degree of rotational symmetry, reflected in a symmetric flow velocity distribution, predominantly in the axial direction (Figure 11a, maximum velocity 0.88 m/s). This symmetry yields a stable but less turbulent flow, as evidenced by a maximum shear strain rate of 161 s⁻¹ (Figure 12a). In contrast, the dimpled mixer, with its irregular blade surface, disrupts radial symmetry, leading to the formation of asymmetric circular flows (Figure 11b) and significantly higher localized turbulence zones (turbulence eddy frequency of 290 s⁻¹ compared to 183 s⁻¹ for the standard design, Figure 13). This symmetry-breaking effect is corroborated by the shear strain rate distribution, where a maximum value of 1442 s⁻¹ is observed near the dimples (Figure 12b), compared to a more uniform distribution in the standard mixer.

3.3. Computer Simulation Results

In Figure 11c (top view), there are practically no significant differences between the two designs, except for small local zones near the edges of the turbine mixer blades. Below are the results of a comparison between the two proposed turbines: an open turbine and an open dimpled turbine. Figure 11a–c presents velocity diagrams in different planes, and analysis of these diagrams indicates that the maximum velocities do not exceed 0.88 m/s. As shown in Figure 11a, there is a significant local difference in the area beneath the turbine mixer blade, where the maximum velocities for a smooth turbine mixer are noticeably higher than those for a dimpled turbine mixer.

In turn, in Figure 11b, where the lateral view of the blade is depicted, the situation is reversed; for a dimple turbine mixer near the surface of the blade, a zone with a higher velocity is observed than in the case of a smooth blade. This may indicate the creation of different flows, namely smooth turbine mixer–axial flows, dimple turbine mixer–circular flows.

Shear strain rate is a value that characterizes the rate of change of deformation in a liquid or material under the action of shear stress. It determines how quickly the shape or volume of a material changes when a shear force is applied to it.

Figure 12a–c presents diagrams of shear deformation velocities in different projections; the analysis of the obtained diagrams allows us to establish that the maximum values of shear deformation velocities do not exceed 594 s⁻¹. However, in turn, the pattern of the distribution of shear deformation velocities has slight differences.

As can be seen in Figure 12a, there is a significant local difference in the area under the mixer blade; for a turbine mixer with smooth blades the value of the shear deformation rate is noticeably greater than for a dimple turbine mixer. It is also possible to mark the central zone of the diagram, where it is possible to mark a more developed zone of shear stresses when using a dimpled turbine mixer.

In turn, in Figure 12b, where the lateral view of the blade is depicted, the following situation is observed: for a dimple turbine mixer near the surface of the blade, there are zones with a greater level of shear stresses than in the case of a smooth blade. This may indicate the creation of different flows, namely smooth turbine mixer–axial flows, dimple turbine mixer–circular flows.

In Figure 12c, i.e., the top view, we can observe a high level of difference, namely a large number of shear stresses in the surface area of the turbine mixer blades when using a dimple turbine mixer.

The frequency of turbulence vortices (eddy frequency) is a measure of how often vortex structures form and collapse in a turbulent flow. In the turbulent flow of a liquid or gas, there are many vortex motions of different sizes and speeds, and the frequency of the vortices describes the speed at which these vortices appear and disappear. The frequency of turbulence vortices (eddy frequency) is measured in the inverse of time, i.e., in hertz (Hz).

Figure 13 shows a diagram of the frequency of turbulence vortices; the analysis of the obtained diagrams allows us to establish that the maximum values of the frequency of turbulence vortices do not exceed 23,906 s⁻¹.

The pattern of frequency distribution of turbulence vortices has slight differences, namely, when using a turbine mixer with dimples, we get a higher frequency of turbulence vortices in the area of the turbine mixer surface. This may indicate the creation of more turbulent vortices when using dimples as the surface of the turbine mixer.

4. Experimental Results

In order to verify the results of the computer simulation, we conducted a study in laboratory conditions. We were able to obtain a wide range of data, which allowed us to draw certain conclusions.

As can be observed, Figure 14a,b clearly shows the difference between the two variants studied. The dimple turbine mixer forms a larger funnel, with a depth of 50 mm, than a standard turbine mixer, with a depth of 44 mm, which indicates more intense homogenization on the dimple side of the mixer, which in turn confirms our studies in computer simulations.

Our theoretical calculations show a high level of agreement with the results presented in

Table 3. Dependence of average power on rotational speed for standard and dimple turbine mixers (water).

Standard Turbine Mixer (Water)
Mixing Stand Speed	No Speed	0	1	2	3	4	5
Speed (rpm)	0	14	45	135	247	377	534
Total power (W) average	3.1	3.2	3.3	3.9	4.9	6.85	9.65
Power consumption (W) average	0	0.1	0.2	0.8	1.8	3.75	6.55
Dimple Turbine Mixer (Water)
Mixing Stand Speed	No Speed	0	1	2	3	4	5
Speed (rpm)	0	14	45	135	247	377	534
Total power (W) average	3.1	3.2	3.4	4	5.1	7.15	10.05
Power consumption (W) average	0	0.1	0.3	0.9	2.0	4.05	6.95

: the calculated value of 3.89 W is close to the practical value of 4.9 W. The reported values are averaged, with errors within ±2% for power consumption and ±1 mm for funnel depth, reflecting the accuracy of our setup. Standard deviations (±1.2 mm and ±1.5 mm for funnel depth; ±0.1 W for power) were calculated but omitted for brevity.

As mentioned in the section “Result of the Analytical Calculation of Power”, in order to determine the power consumed by the mixing process, it is necessary to know the value of the density of the liquid, the speed of revolutions of the agitator, its diameter, and the power factor, which depends on the type of agitator design. For typical designs, this value can be found in the reference scientific literature, usually from experimental monograms depending on the Reynolds number. For a standard open turbine agitator, the following values established by the researchers lie in the range from at Reynolds less than 10, to at Reynolds greater than [26]:

$$40{\le K}_N\le 0.8·{10}^5$$

For the designed dimple turbine agitator, based on the results of experimental data on the power consumed for agitation and various values of the modified Reynolds number, an array of values of power factors was obtained, which are given in

Table 4. Data array for calculating the power factor of agitators.

№	n—Rotation Velocity, (s−1)	d—Diameter of the Mixer, (m)	ρ—Density of Water at 20 °C (kg/m3)	μ—Dynamic Viscosity of Water at 20 °C, (Pa∙s)	N1—Power of the Standard Homogenizer, (W)	N2—Power Consumption of a Standard Homogenizer, (W)	Re	Kn1—Standard Mixer	Kn2—Modified Mixer
1	0	0.08	998.23	0.001002	0	0	0	0	0
2	0.23	0.08	998.23	0.001002	0.1	0.1	1487	2406	2406
3	0.75	0.08	998.23	0.001002	0.2	0.3	4781	144	217
4	2.25	0.08	998.23	0.001002	0.8	0.9	14,345	21.4	24.1
5	4.11	0.08	998.23	0.001002	1.8	2.0	26,247	7.88	8.76
6	6.28	0.08	998.23	0.001002	3.75	4.05	40,062	4.62	4.99
7	8.9	0.08	998.23	0.001002	6.55	6.95	56,745	2.84	3.01

.

Based on the data presented in Table 4, a nomogram was constructed (Figure 15), enabling the determination of precise power factor values for both a standard turbine mixer and a dimpled mixer. This nomogram facilitates approximate theoretical calculations for turbine mixers. The results also indicate that the power factor for both types of agitators depends on the Reynolds number and, consequently, on the flow regime. It is well established in the literature that in laminar flow, the power factor is two orders of magnitude higher than in turbulent flow [26]. Moreover, the curve demonstrates good agreement with published data, particularly in the transitional and turbulent flow regimes [25].

One of the studied parameters was the power of the turbine mixer. We expected the new geometry to improve energy efficiency. However, as seen in Figure 16, the laboratory results show that the new turbine mixer geometry actually increases mixing power. This further confirms the reliability of computer simulations.

Paying attention to Figure 16, we can see a certain progression in the increase in energy consumption of the mixing stand when using different types of turbine mixers; the higher the rotational speed, the more energy the dimple turbine mixer consumes compared to a standard turbine mixer; therefore, our initial assumption about the energy efficiency of the dimple geometry of the blade surface is incorrect. Instead, we can confidently say that the use of modified geometry increases both power consumption and mixing efficiency. The margin of error did not exceed 2%.

Analysis of the diagram (Figure 17)—which depicts the ranges of shear strain rates for various biological organisms compared to the average and maximum values for a standard turbine mixer (44 s⁻¹ and 161 s⁻¹) and a dimple turbine mixer (63 s⁻¹ and 1442 s⁻¹) at a rotational speed of 247 rpm—allows for several conclusions. These shear strain rate values are specifically determined by this rotational frequency, which explains their positioning within a distinct zone of the diagram. Both mixers generate average shear rates (44 s⁻¹ and 63 s⁻¹) that intersect the ranges of most sensitive biological organisms, such as cellular conglomerates, mycoplasmas, organelles, and protoplasts, indicating their effectiveness in impacting these structures at 247 rpm.

The maximum shear rate of the standard turbine mixer (161 s⁻¹) encompasses a broader spectrum, including Gram-negative bacteria, while the maximum shear rate of the dimple turbine mixer (1442 s⁻¹) extends further to affect more resilient organisms such as fungi, mycelium, and Chlorella, owing to its higher intensity at the same rotational speed. The logarithmic scale highlights the substantial disparity between the minimum and maximum values for organisms like bacterial spores (ranging from 2 × 10⁸ to 10¹⁰ s⁻¹), which neither mixer can effectively process at 247 rpm due to insufficient maximum shear rates. Thus, at this rotational speed, the standard turbine mixer is optimal for soft to moderately resistant biological organisms, whereas the dimple turbine mixer is better suited for more resistant structures. However, neither mixer reaches the threshold required for the most resilient organisms, a limitation attributable to the shear rate constraints at 247 rpm.

The full results of CFD studies of the average volume parameters of hydrodynamics for two types of agitators are given in

Table 5. Volume-averaged hydrodynamic parameters in the mixer and dimple mixer zone.

	Default Mixer	Dimple Mixer
Velocity Volume Average $[\mathrm{m}·{\mathrm{s}}^{-1}]$	0.312	0.321
Water Shear Strain Rate Volume Average ${[\mathrm{s}}^{-1}]$	44	63
Water Shear Strain Rate Volume Max ${[\mathrm{s}}^{-1}]$	161	1442
Turbulence Eddy Frequency Volume Average ${[\mathrm{s}}^{-1}]$	183	290
Turbulence Eddy Frequency Volume Max ${[\mathrm{s}}^{-1}]$	1239	63,170

.

This analysis underscores the critical importance of considering rotational speed when selecting a mixer type based on the target biological material and the desired level of mechanical impact. The use of dimples on the surface of the turbine mixer blades led to more intensive mixing, which is confirmed by both computer simulations and experimental studies. The dimples create a more turbulent flow, which helps improve homogenization and dispersion processes.

The distribution of velocities and shear rates has shown that the dimple turbine mixer creates more uniform and intense fluid flows compared to a standard turbine mixer. This ensures that the medium in the bioreactor is mixed more efficiently.

The frequency of turbulence vortices in the surface area of the dimple turbine mixer was higher, indicating the creation of more turbulent vortices. This can have a positive effect on the processes of mass transfer and dissolution of oxygen in the liquid.

The dimple turbine mixer consumed more energy compared to a standard turbine mixer. This suggests that the use of modified blade surface geometry increases mixing efficiency, but is not an energy-efficient solution. Experimental data confirmed the results of computer simulations.

In further research, it is advisable to consider the possibilities of optimizing the geometry of the dimples to achieve a balance between mixing efficiency and energy consumption. Additionally, it is worth conducting a study of the influence of other geometric parameters on the hydrodynamic properties of the turbine mixer.

Thus, the results of this study confirm that modifying the geometry of turbine mixer blades by adding dimples is a promising area for improving mixing processes in bioreactors, but needs to be further optimized in terms of energy efficiency.

5. Discussion

The findings of this study provide valuable insights into the influence of turbine mixer blade surface geometry on mixing efficiency, particularly through the introduction of dimples, and align with broader trends observed in the literature. Our results demonstrate that the dimple turbine mixer enhances mixing intensity by generating a more developed turbulent flow, as evidenced by the increased maximum shear strain rate (1442 s⁻¹ compared to 161 s⁻¹ for the standard mixer) and a higher turbulence eddy frequency (290 s⁻¹ volume average versus 183 s⁻¹) at 247 rpm. This aligns with the experimental observation of a deeper vortex funnel (50 mm versus 44 mm) and corroborates the simulation data, indicating improved homogenization, dispersion, and suspension processes. However, this enhancement comes at the cost of a 4.5% increase in power consumption (5.1 W versus 4.9 W at 247 rpm), challenging our initial hypothesis that dimple geometry might improve energy efficiency alongside mixing performance.

A critical evaluation of these results reveals both strengths and limitations. The intensification of hydrodynamic processes is consistent with the underlying principle of dimple-induced micro-vortices, inspired by golf ball aerodynamics, which reduce drag and enhance turbulence. This observation is supported by similar findings in the literature, such as the work of Karki, P.J. et al. (2021) [27] in their study “Comparative numerical and experimental study of golden angle and conventional agitator impellers”. Kumar et al. reported that non-conventional impeller designs, such as those with angular modifications, increased turbulence and mixing efficiency, albeit with elevated energy demands, mirroring our findings with the dimple turbine mixer. Their study showed a comparable trade-off, where a golden angle impeller improved flow patterns but required higher power input compared to a standard Rushton turbine, reinforcing the notion that surface modifications often enhance mixing at the expense of energy efficiency.

However, our results also highlight a need for caution in interpreting the practical implications. While the dimple mixer excels in processing more resilient biological organisms like fungi and mycelium (shear strain rates up to 1442 s⁻¹), it falls short of the thresholds required for highly resistant organisms such as bacterial spores (2 × 10⁸ to 10¹⁰ s⁻¹), a limitation also noted in the literature for conventional turbine designs. This suggests that while our modification is a step forward, it does not fully address the spectrum of biotechnological needs, particularly for applications requiring extreme shear forces. Furthermore, the increased hydraulic resistance observed in the dimple mixer, as indicated by higher power consumption, points to a design trade-off that may not be universally beneficial. In contrast, studies like that of Korobiichuk et al. (2022) [7] achieved reduced vortex formation with a double-disc turbine, suggesting that alternative geometries might offer energy savings that our dimple design does not.

The consistency between our computational fluid dynamics (CFD) simulations using ANSYS (2021R2) and experimental outcomes strengthens the reliability of our methodology. For instance, the simulated shear strain rate distribution (Figure 12) and velocity profiles (Figure 11) accurately predicted the experimental increase in vortex depth and power consumption, validating the use of the k-ε turbulence model and the Frozen Rotor approach. Yet, a critical reflection reveals that our study could benefit from exploring a wider range of rotational speeds beyond 247 rpm to better map the operational envelope of both mixers. This limitation is evident when compared to Teng et al. (2021) [10], who scaled hydrodynamic conditions across multiple bioreactor sizes, suggesting that our conclusions might be specific to the tested conditions.

The introduction of dimples on the blade surface induces a symmetry-breaking phenomenon that significantly influences hydrodynamic characteristics. In the standard mixer, symmetric geometry promotes predominantly axial flows with lower turbulence (turbulence eddy frequency of 183 s⁻¹), whereas dimples disrupt this symmetry, generating micro-vortices that elevate the turbulence eddy frequency to 290 s⁻¹ and the maximum shear strain rate to 1442 s⁻¹. This symmetry-breaking enhances homogenization and dispersion, as evidenced by a deeper vortex funnel (50 mm versus 44 mm), but also increases hydraulic resistance, reflected in a 4.5% rise in power consumption (5.1 W versus 4.9 W). Thus, symmetry-breaking is a critical factor in improving mixing efficiency, though it necessitates optimization to mitigate energy costs.

In comparison to global research, our findings echo the broader trend of optimizing mixer geometry for specific bioprocess outcomes, as seen in Galaction et al. (2010) [6] and Liew et al. (2010) [8], where tailored impeller designs improved mixing for immobilized cells and aerated systems, respectively. However, unlike Alkhalidi et al. (2016) [4], who achieved energy recovery with a self-powered mixer, our dimple design does not offer an energy-efficient solution, highlighting a gap in our approach that warrants further investigation. The increased turbulence and shear rates are advantageous for shear-sensitive cultures like mycoplasmas and protoplasts, yet the energy penalty may limit scalability in industrial settings, a concern also raised by Karki, P.J. et al. (2021) [27] regarding non-conventional impellers.

Moving forward, the dimple geometry presents a promising avenue for enhancing mixing efficiency, but its practical application requires optimization to mitigate energy costs. Future research should focus on varying dimple size, shape, and distribution to strike a balance between turbulence generation and power consumption, potentially drawing inspiration from Kumar et al.’s angular modifications. Additionally, integrating our shear stress classification (Figure 17) with real-time bioprocess monitoring could refine its utility for specific producer cultures. In conclusion, while our study confirms the hydrodynamic benefits of dimple-modified blades, aligning with trends in the field, it also underscores the need for a nuanced approach to design optimization to fully realize its biotechnological potential.

The results of the study conducted in a laboratory bioreactor demonstrate a significant impact on the hydrodynamics of flows generated by a dimpled turbine impeller. The obtained data, based on the theory of similarity of physical phenomena and processes, confirm the potential for their scalability in the design of industrial bioreactors of various sizes and capacities, ensuring effective homogenization and adaptation of mixing device configurations for a wide range of biotechnological applications, from laboratory to industrial scales.

6. Conclusions

This investigation elucidates the impact of dimpled turbine mixer blades on bioreactor hydrodynamics, validated through ANSYS (2021R2) simulations and experimental studies. The dimpled design disrupts radial symmetry, yielding a 2.9% increase in volume-averaged flow velocity (0.321 m/s vs. 0.312 m/s), maximum shear strain rate (1442 s⁻¹ vs. 161 s⁻¹), and a 58% enhancement in turbulence eddy frequency (290 s⁻¹ vs. 183 s⁻¹) at 247 rpm. These improvements, driven by symmetry-breaking micro-vortices, result in a 14% deeper vortex funnel (50 mm vs. 44 mm), enhancing homogenization, dispersion, and suspension processes. The dimpled mixer proves effective for resilient biological organisms like fungi and mycelium, though inadequate for highly resistant organisms such as bacterial spores.

However, the enhanced mixing efficiency incurs a 4.5% increase in power consumption (5.1 W vs. 4.9 W), challenging initial assumptions of energy efficiency. A shear strain rate classification (Figure 17), aligned with the literature, supports tailored mixer selection for specific bioprocesses. While the dimpled design advances bioreactor mixing performance, optimizing dimple parameters (size, shape, distribution) is critical to balancing efficiency and energy costs. These findings contribute to biotechnological advancements, with future research needed to refine energy-efficient mixer geometries.