Study of Turbulent Kinetic Energy and Dissipation Based on Fractal Impeller

Abstract

Turbulent kinetic energy and turbulent dissipation are important aspects of the flow field characteristics, which can affect the wear and energy loss in mixing equipment. In order to increase equipment wear and energy loss in the mixing process, a series of fractal impellers were designed based on the fractal iteration method, and the effects of the fractal dimension and the number of iterations on the flow field characteristics were investigated. Firstly, the distribution characteristics of turbulent kinetic energy and its uniformity were studied. Then, the distribution characteristics of the turbulent dissipation rate were studied and interpreted using vortex analysis, and the mixing power of the device was further investigated. The results showed that: for the turbulent kinetic energy of the flow field, an increase in the fractal dimension and the number of iterations makes the turbulent kinetic energy intensity of the flow field decrease and the distribution more uniform, compared to the non-iterative impeller, specifically the rectangular secondary iterative impeller caused a 30% reduction in the turbulent kinetic energy intensity and a 50% increase in the uniformity; for the turbulent dissipation of the flow field, in general an increase in the fractal dimension reduces the turbulent dissipation in the flow field, and an increase in the number of iterations increases it slightly, this influence law is due to a change in the trailing vortex caused by the blade structure; and a change in the law of turbulent dissipation also causes a corresponding change in the stirring power, from the non-iterative impeller to a rectangular one iteration impeller, the power decreases by 20% while the average speed decreases by only 5%. In conclusion, the special boundary of the fractal iterative impeller can reduce the turbulent kinetic energy and turbulent dissipation of the flow field to a large extent, and its two characteristics, the fractal dimension and the number of iterations, affect the reduction effect. The results of the study can be used as a reference for the design of mixing equipment to reduce turbulent kinetic energy and turbulent dissipation.

1. Introduction

Mixing equipment is necessary equipment for the production process in the chemical industry and is widely used in a variety of operations, such as intensive dissolution, heat transfer, gas diffusion, extraction, and flocculation [1,2,3,4]. During the mixing process, the turbulent kinetic energy and turbulent dissipation affect lots of aspects of the stirring ability [5,6,7].

Turbulent kinetic energy is mainly derived from the mean kinetic energy of the flow field, which provides energy to the turbulence through Reynolds shear stress work, characterizing the pulsation strength of fluids [8,9]. Too much turbulent kinetic energy will lead to violent action between the fluid and the equipment, causing wear hazards. Many pieces of research to optimize the turbulent kinetic energy characteristics have been conducted. Liu et al. [10] demonstrated experimentally that the addition of an inlet guide vane can decrease the high turbulent kinetic energy zone of a centrifugal pump, therefore, the impact intensity can be reduced, and the impeller performance can be improved. Zhang et al. [11] demonstrated that the open-hole treatment of the vanes helps to reduce the turbulent kinetic energy intensity, making the internal flow field of the centrifugal pump more stable and reducing the wear. Zhao et al. [12] found through simulation experiments that the addition of small particles would weaken the turbulent kinetic energy in the centrifugal pump and make the flow in the pump more stable through simulation experiments.

While the turbulent kinetic energy provides energy to the turbulence, a part of it is converted to the kinetic energy of molecular thermal motion, which forms the turbulent dissipation. and the turbulent dissipation rate characterizes the influences on the energy loss in the flow field. In order to increase the energy efficiency of mixing equipment, it is necessary to reduce the turbulent dissipation rate of the flow field. Chen et al. [13] used PIV (particle image velocimetry) and POD (proper orthogonal decomposition) techniques to experimentally analyze the turbulent dissipation of channel flow and revealed the characteristics of the turbulent dissipation rate distribution in the channel flow. Based on the k-ε model, Li et al. [8] investigated the influence law of inlet conditions on the turbulent dissipation characteristics of shear flow, and the results showed that the smaller the inlet fluid strength and the greater the viscosity, the higher the intensity of turbulent dissipation. Trad et al. [14] designed a 3EE82/4RT56 impeller, which could obtain a flow pattern with a lower turbulent dissipation rate and power input, and the solid suspension and gas–liquid mass transfer in a baffle-free tank were simultaneously enhanced. Chen et al. [15] studied the turbulent energy loss characteristics in centrifugal pumps and proved that the eddy component P_θθ can suppress turbulent dissipation.

From the relevant articles, concerning the turbulent kinetic energy and turbulent dissipation of the flow field, scholars’ research mainly rests on the influence law of the equipment structure on its distribution and intensity. In order to reduce the turbulent energy and turbulent dissipation, through trying different treatment methods, scientists proved that the characteristics of the impeller are an important factor, and obtained a number of impeller structures with optimized effects. However, the effect of the specific characteristics on the turbulent kinetic energy and turbulent energy dissipation of an optimized impeller is not studied in detail.

Fractal geometry swept fluids tend to produce unique turbulence characteristics and the design of a fractal impeller with a fractal boundary has a good optimization effect in many cases. Nedi’c [16] studied the air sweeping process of fractal structured plates and found that the drag coefficient increases with the number of fractal iterations and the wake size decreases with the increase in the fractal dimension. Steiros [17] designed a class of fractal impellers and found that the fractal blades can reduce the mixing power and drag. Bas bug et al. [18] further investigated the reason for the ability of fractal impellers to reduce energy consumption and drag, and found that it comes from the tail vortex refinement due to the special boundary. Gu et al. [19,20] conducted numerical simulation experiments on the mixing process of a kind of fractal impellers and found that this type of fractal impellers can improve the mixing uniformity and reduce the mixing power.

From the above studies, the fractal geometry has unique turbulence characteristics, and the fractal impeller has great potential to optimize the flow field characteristics. However, relevant studies are still lacking and most researchers choose to evaluate the optimization effect of the fractal impeller relative to the normal impeller, and less detailed studies are conducted on the change law of the optimization effect, which is influenced by specific factors, such as the fractal dimension and number of iterations.

In the flow field, the distribution and intensity of turbulent kinetic energy and turbulent dissipation is significantly affected by the structure of the factual impeller. Different fractal boundaries can have different turbulence characteristics; if we can design a fractal blade and find a suitable type of impeller to make a flow field with low turbulent kinetic energy and turbulent energy dissipation intensity, and further study the specific characteristics to optimize the effect of the law, the results of the study can provide a new reference direction in the design of impeller blades where low wear and low energy losses are required, and scholars can build on the findings of this paper and continue to change the structure to achieve better optimization results, or to design in a new optimization direction.

Therefore, based on the turbine impeller and fractal geometry, we designed a new class of turbine impeller with fractal geometry (referred to as fractal impeller). The k-ε model of computational fluid dynamics was used to investigate the effects of the fractal dimension and number of iterations on the turbulent kinetic energy and turbulent dissipation rate. The results will provide a new design reference for the design of high-speed centrifugal stirrers.

In this paper, a series of fractal impellers were designed to reduce the turbulent kinetic energy and turbulent dissipation rate of the flow field as a way to optimize the performance of the mixing equipment in terms of energy consumption and wear-related performance. The CFD numerical simulations were carried out using the relevant mathematical models. In order to investigate the effect of the optimized turbulent kinetic energy and turbulent dissipation rate, firstly, the intensity characteristics and uniformity of the distribution of the turbulent kinetic energy were explored. Next, the distribution characteristics of the turbulent dissipation rate were studied and explained by tail vortex analysis. Lastly, the stirring power of the equipment was investigated to evaluate the optimization effect.

2. Physical Model

2.1. Fractal Impeller Model

The fractal structure has two main properties: the result of the observed object is independent of the observation scale, and its whole and parts are similar. In contrast to traditional geometry with integer dimensions, such as one-dimensional lines and two-dimensional surfaces, the dimensions of fractal geometry are generally between the integers [21,22,23].

Based on fractal geometry and the turbine blade, a series of fractal blades are formed. The no iteration blade is a 48 mm × 32 mm rectangle. On the basis of the no iteration blade, the boundary line is divided into four equal parts, and the middle two equal parts are each formed into a square triangle and a square with the same top and bottom, that is, the triangular and rectangular fractal iteration blade of one iteration. The blades after the first iteration are iterated again on the newly formed sides to form the respective second iteration blades. The area of each type of blade remains the same and only the shape differs. The fractal dimension of the non-iterative blade can be considered as D₀ = 1, the triangular iterative blade is D₃ = log6/log(1/4) = 1.29, and the rectangular iterative blade is D₄ = log8/log(1/4) = 1.5 [24].

The fractal blades are shown in Figure 1, including one non-iterative blade and four fractal iterative blades.

The fractal impellers are installed in the mixing tank as shown in Figure 2. The origin of the mixing equipment is at the center, the Z-axis is facing downward, and the center cross-section of the impeller is at Z = 110 mm. The mixing tank is a flat-bottomed cylindrical tank with a central combination. The tank surface height is H = 510 mm, the tank inner diameter is T = 400 mm, the height of four uniformly distributed flat baffles is W = T/10 = 40 mm, the diameter of the stirring impeller is d = 160 mm, the height of the impeller from the bottom is C = T/3 = 133 mm, and the diameter of the disk inside the impeller is d₁ = 90 mm. The stirring fluid is water, with a density of ρ = 1000 kg/m³, and a viscosity of μ = 0.00103 pa·s.

3. Hydrodynamic System

3.1. Computational Fluid Dynamics Model

It is generally accepted that no matter how complex the turbulent motion is, the NS equation is still applicable for the transient motion of turbulent flows [25,26]. The vector form is as follows:

$$\frac{\partial V}{\partial t}+\left(V·\nabla \right)V=f-\frac{1}{\rho }\nabla p+\frac{\mu }{\rho }{\nabla }^{2}V$$

(1)

where ρ is the fluid density; t is the time; V is the velocity vector as the external force per unit volume of the fluid; μ is the dynamic viscosity; p is the pressure; f is the external force.

To consider the effect of fluid pulsation and to simplify the calculation to calculate the variables of the averaged motion, the Reynolds mean method was used to average the three-dimensional NS equations for the incompressible case to obtain the RANS control equations [27,28] as follows. The continuity equation is:

$$\begin{array}{c}\frac{\partial \left(\rho u_i\right)}{\partial x_i}=0\end{array}$$

(2)

The momentum equation is:

$$\begin{array}{c}\rho \left(\frac{\partial u_i}{\partial t}+\frac{\partial u_j{u}_j}{\partial x_j}\right)=-\frac{\partial p}{\partial x_i}+\frac{\partial \left(-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)}{\partial x_j}+\frac{\partial }{\partial x_j}[\mu \left(\frac{u_i}{u_j}+\frac{u_j}{u_i}\right)-\frac{2}{3}\mu {\delta }_{ij}\frac{u_k}{x_k}]+\rho f_i\end{array}$$

(3)

where u_i, u_j, u_k are the mean velocity components and p is the mean pressure; $-\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ denotes the Reynolds stress, where $u_i^{\prime }$, $u_j^{\prime },$ are the pulsation velocities of the mean velocity components; x_i, x_j, x_k, are the flow components; ${\delta }_{ij}$ is the Kronecker tensor component; f_i is the i directional mass force.

In order to calculate the turbulent kinetic energy and turbulent energy dissipation, the turbulence model used in this paper is the standard k-ε model with good applicability [29]. The satisfied transport equation is:

$$\begin{array}{c}\raisebox{1ex}{$\partial \left(\rho k\right)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.+\raisebox{1ex}{$\partial \left(\rho k{u}_i\right)$}\!\left/ \!\raisebox{-1ex}{$\partial x_i$}\right.=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+P_k-\rho \epsilon \end{array}$$

(4)

$$\begin{array}{c}\raisebox{1ex}{$\partial \left(\rho \epsilon \right)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.+\raisebox{1ex}{$\partial \left(\rho k{u}_i\right)$}\!\left/ \!\raisebox{-1ex}{$\partial x_i$}\right.=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\frac{C_{1\epsilon }\epsilon P_k}{k}-\frac{C_{2\epsilon }\rho {\epsilon }^2}{k}\end{array}$$

(5)

where k is the turbulent kinetic energy; ε is the turbulent dissipation rate; μ_t = ρC_μk²/ε is the vortex viscosity coefficient; the turbulent kinetic energy generated by the mean velocity gradient is P_k = μ_tS²; S is the mode of the mean strain rate tensor. C_μ = 0.09, C_1ε = 1.44, C_2ε = 1.92, σ_μ = 1, σ_ε = 1.3.

3.2. Simulation Experiments

The MRF method was used to deal with the dynamic–static relationship of the flow field modeled by solid works, and imported into ANSYS Mesh for meshing. The 4-sided unstructured mesh with strong adaptability was chosen, and the mesh encryption of the dynamic region was carried out, as shown in Figure 3.

In order to verify the irrelevance of the number of meshes on the simulation results, multiple meshes were divided for the simulation for the non-iterative impeller. The stirring power of the stirring equipment and the turbulent kinetic energy at one point in the flow field were selected as the judging criteria for the irrelevance, as shown in

Table 1. Agitation power of each grid and turbulent kinetic energy of the monitoring points.

Mesh Number	Stirring Power (w)	Monitoring Turbulent Kinetic Energy [m2/s2]
621,452	75.23	0.002
1,132,770	96.07	0.043
2,354,935	108.15	0.053
4,102,625	107.82	0.052

.

Considering the computational accuracy and speed, 2,354,935 was chosen as the mesh number in this experiment. With similar shape and mesh division settings, the mesh numbers of five fractal impellers were kept at about 2,350,000, which has no great influence on the results.

Considering the complex structural edges of the fractal impeller, the quality of the mesh divided by the flow field model may still be poor, despite the local refinement. The number of mesh for this simulation experiment is large and multiple sets of experiments are required. The adequate accuracy of the standard k-ε model in simulating the stirred flow field has been demonstrated in some papers. Considering the accuracy and economy, the k-ε model with better applicability and more stability was chosen as the turbulence model for this paper.

The turbulent kinetic energy and turbulent dissipation characteristics were studied using steady-state analysis, solved with double high accuracy, the chosen speed was 300 r/min, and the convergence standard was RMS = 1 × 10⁻⁴.

3.3. Verification of Simulation Models

In order to verify the accuracy of the adopted numerical simulation model, the experimental and simulation results (common turbine impellers) were compared. Simulations and experiments were carried out using the same no iteration impeller and mixing equipment, the size of which is as described in Section 2.1. The specific steps are as follows:

(1)

Four experiments at 300 r/min using the mixing equipment were performed. We used a PIV velocimeter to measure the fluid velocity at 25 observation points in a vertical line at the same position, as shown in Figure 4, and took the average value of each experiment as the velocity at this point;

(2)

Using the mesh and boundary settings above, three sets of simulations were performed for the same case as the experiments. The vertical line velocity at the same position was derived for each simulation, and the curve was fitted and compared with the experimental data in step (1), as shown in Figure 5, where the horizontal axis is the axial distance, and the vertical axis is the speed of the monitoring point;

(3)

The accuracy of the simulated experiments was measured using the average difference between the data of each group of simulated experiments and the average value of each point of the experimental data relative to the percentage average of the experimental data.

As shown in Figure 5, the difference between the velocity curves of the simulated experiment and the actual experiment at the same position is small, and the average percentage differences of each point are 10.2%, 5.5%, and 8.1%, respectively, indicating that the numerical simulation model has a certain degree of accuracy and can complete the simulation of the experimental situation.

4. Result and Discussion

4.1. Turbulent Kinetic Energy Characteristics

Turbulent kinetic energy (k) is usually generated in a large mesoscale vortex, and in time-averaged flows, where $k=\frac{1}{2}m(\overline{{u_i^{\prime }}^2})$, defined as 1/2 of the product of the mean turbulent velocity rise and fall variance $(\overline{{u_i^{\prime }}^2})$ and the fluid mass m [30]. The turbulent kinetic energy is often estimated using the turbulent intensity in situations where it is difficult to calculate directly, which is calculated as $k=\frac{3}{2}{\left(UI\right)}^2$. Where U is the mean velocity of the flow field, and I is the turbulent intensity.

The turbulent kinetic energy characterizes the pulsation intensity of the flow field. The smaller the turbulent intensity, the smaller the turbulent kinetic energy, and the more stable the flow field is for a certain velocity of the flow field.

In the following, the turbulent energy distribution and intensity characteristics of the fractal impellers’ flow field are investigated. The turbulent kinetic energy in the plane of the Z = 110 mm cross-section at the center of the fractal impellers is monitored, as shown in Figure 6.

As shown in Figure 6, for the non-iterative impeller of the flow field, the jet of the impeller and the vortex caused by the impact on the tank wall are the main locations for turbulent energy generation in the flow field, and the jet has the highest intensity near the impeller. Therefore, the turbulent kinetic energy intensity is highest in the zone between the six blades in the shape of a “petal”. As the jet spreads, it impacts the tank wall and also generates a vortex, which is relatively weak, therefore the turbulent kinetic energy at the tank wall also exists at a certain intensity. Weakly influenced by vortices from jets and impacts, the intensity in the middle zone between the tank wall and the impeller area is low. Apart from that, due to the simple fluid motion, the turbulent kinetic energy has the lowest intensity at the backflow side of the impeller.

For the triangular iterative impeller, the turbulent kinetic energy is also mainly concentrated in the “petal” near the impeller, but the intensity is significantly reduced, and further reduced as the number of iterations increases. It reduces the middle zone of the low turbulent kinetic energy intensity and expands the low intensity at the backflow side. The reason for this phenomenon may be that: the special blade boundary of triangular iterations reduces the intensity of the jet, and the reduction effect decreases with the number of iterations, causing the intensity of the vortex in each zone to decrease accordingly.

The turbulent kinetic energy distribution intensity characteristics of the rectangular iterative impeller are quite different. With much lower intensity than other impeller flow fields, the rectangular iterative impeller zone does not form a high-intensity “petal”, but a “windmill” low-intensity zone centered on the backflow side. The turbulent kinetic energy of the flow field is mainly concentrated in the tank wall zone and the middle zone. With the increase in the number of iterations, the intensity of the “windmill” on the backflow side decreases and gradually expands to a larger area, which shows that the number of rectangular iterations is more effective in reducing the jet compared to the number of triangular iterations.

The turbulent kinetic energy distribution characteristics of the impeller zone show different characteristics with the changes in the fractal dimension and the number of iterations, but the distribution characteristics near the tank wall zone and the middle zone are not obvious. In order to further study the intensity characteristics, an observation line was selected every 10 mm from the edge of the crossing impeller region, where 2R/T = 0.5 occurred to the edge of the mixing tank, 2R/T = 1 occurred in the middle 45° plane, and an additional observation line of 2R/T = 0.95 was added. The turbulent kinetic energy variation of the fractal impellers with the off-impeller radius is shown in Figure 7, where the horizontal Z-axis is the axial distance, and the vertical axis is the turbulent kinetic energy of the monitoring point.

The intensity of the turbulent kinetic energy represents the pulsating intensity of the flow field and affects the movement and collision of solid particles. As mentioned above, the jet of the stirring impeller is the main location for turbulent kinetic energy generation. Therefore, the turbulent kinetic energy of the non-iterative impeller flow field mainly exists in the impeller zone (2R/T = 0.5 to 2R/T = 0.7). The higher intensity jet exists only within the zone near the impeller, and the jet has a strong influence effect on the radial direction; relatively, the axial influence effect is poor, which causes the similar characteristics of the area. The zone near the tank wall (2R/T = 0.95 to 2R/T = 1) has a medium intensity. The turbulent kinetic energy of this zone is generated from an up and down vortex, which is formed by the impact of the jet at the tank wall. This zone exists at around 2R/T = 0.95, which has a large effect on the axial direction and a relatively poor effect on the radial direction, which causes the similar characteristics of the zone. The middle region (2R/T = 0.8 to 2R/T = 0.9) has a lower intensity due to less influence by the jet and impact vortex, but the distribution range characteristics of the other two zones make this region have a large axial and longitudinal range. Therefore, the impeller and the tank wall are the parts that are most severely worn during the mixing process.

Compared with the non-iterative impeller, the fractal iterative impeller has a lower intensity for turbulent kinetic energy distribution. With the increase in the fractal dimension and iteration numbers, the intensity of the “petal” high turbulent kinetic energy zone in the impeller area decreases, the “windmill” low intensity area is formed on the backflow surface of the blade, the turbulent kinetic energy intensity of the zone near the tank wall with a large range decreases, and the range of the low-intensity middle zone is reduced and expanded. The turbulence of the flow field is mainly concentrated in the impeller zone and the tank wall zone, for these areas turbulent energy intensity reduction is achieved through the fractal blade, and the wear of impeller and tank wall may be reduced.

4.2. Turbulent Kinetic Energy Uniformity Analysis

The uniformity of turbulent kinetic energy characterizes the uniformity of the pulsation in the flow field and the wear in the equipment. In order to study the turbulent kinetic energy distribution interval of the flow field of the fractal impeller, the turbulent kinetic energy of the flow field is divided into n = 50 intervals. The volume share of each interval can be calculated, and the value is represented by the median turbulent energy of the interval.

The distribution of the turbulent energy interval and the average turbulent kinetic energy X of the flow field for each impeller is shown in Figure 8, the red line represents the average turbulent kinetic energy of each fractal impeller, and the blue point represents the proportion of turbulent kinetic energy in each section.

As shown in Figure 8, the mean turbulent kinetic energy of the flow field decreases with the increase in the fractal dimension and the number of iterations. The turbulent kinetic energy distribution intervals of each fractal impeller are similarly characterized by two peaks. The first peak is located at a smaller value near 0, the turbulent kinetic energy of this peak comes from the middle zone, which is less intense but more extensive. The second peak is around 0.02, the turbulent kinetic energy of this peak comes from the tank wall zone with medium intensity and medium range, this peak tends to decrease as the fractal dimension and the number of iterations increase. The percentage of the interval decreases as the turbulent kinetic energy increases beyond the second peak. The turbulent kinetic energy in this part comes from the impeller zone, which is more intense but less extensive. Besides, the peak of the rectangular iterative impeller is significantly flatter, and the value of the interval with more volume is closer to the average value of the turbulent kinetic energy, which may improve the uniformity of the turbulent kinetic energy.

To compare the uniformity of turbulent kinetic energy distribution of each impeller, the standard deviation σ_k is calculated by considering the interval share as the probability of distribution:

$$\begin{array}{c}{\sigma }_k=\sqrt{{\displaystyle \sum _{i=1}^n}{{(k}_i-k_{avg})}^2\ast P_i}\end{array}$$

(6)

where k_i is the average turbulent kinetic energy for an interval; k_avg is the weighted average turbulence energy of the total interval; P_i is the share of a certain turbulent energy interval.

The standard deviation of the turbulent kinetic energy distribution for each impeller is obtained as follows:

No iteration: σ₀ = 0.3142;

Triangular one iteration: σ₃ = 0.2456;

Triangular secondary iteration: σ₃₍₂₎ = 0.2397;

Rectangular one iterative: σ₄ = 0.1794;

Rectangular secondary iteration: σ₄₍₂₎ = 0.1713.

The increase in the fractal dimension is able to reduce the σ_k, which is reduced by about 50% from the no iteration impeller to the rectangular iteration impeller. This change law means that the turbulent kinetic energy in the flow field is more evenly distributed, and the intensity of the turbulent energy in the impeller zone, middle zone, and tank wall zone are not only reduced, but are also more evenly distributed, indicating that the turbulent kinetic energy distribution can be relatively more uniform. Compared to the fractal dimension, the reduction effect of the number of iterations on the σ_k is weaker. It indicates that as the fractal dimension and the number of iterations increase, the wear of the mixing equipment is not only smaller but also more uniform. The rectangular second iteration impeller has the best wear uniformity characteristics. To a certain extent, it can be predicted that as the fractal dimension and the number of iterations increase, the wear of the mixing equipment may become smaller and more uniform.

From the above research, it is clear that the increase in the fractal dimension and the number of iterations reduces the turbulent kinetic energy intensity of the flow field and improves the uniformity, with a decrease of about 30% in the turbulent kinetic energy intensity and an increase of about 50% in the uniformity of the rectangular secondary iteration impeller compared to the no iteration impeller. When compared with previous optimization work on turbulent energy intensity reduction, for e.g., Zhang et al. [11] use of open 4 × 1 mm holes in the blades resulted in about a 5% reduction in turbulent kinetic energy, and Liu et al. [10] adding of inlet guide vanes resulted in a small reduction in the turbulent kinetic energy of centrifugal pumps, the class of fractal designed impellers in this paper has a large reduction effect, indicating that the reduction of turbulent kinetic energy in the flow field by using fractal blades is effective.

4.3. Turbulent Dissipation Characteristics

The energy dissipation in the flow field comes from two main sources, viscous dissipation and turbulent dissipation, the former from the energy dissipation due to the physical viscosity of the fluid and the latter from the dissipation of the turbulent kinetic energy converted from the mean flow kinetic energy [15]. Most of the energy dissipation in the flow field comes from turbulent dissipation [31].

The turbulent dissipation rate ε is the rate of conversion from the turbulent kinetic energy of large- and medium-scale vortices to the kinetic energy of molecular thermal motion in small-scale vortices under the effect of molecular viscosity. The formula for ε in time-averaged flow can be obtained from the incompressible NS equation [30,32], $\epsilon =\frac{\mu }{\rho }\overline{\frac{\partial u_i^{\prime }}{x_j}\frac{\partial u_i^{\prime }}{x_j}}$. The turbulent kinetic energy k and the turbulent intensity scale l were used to estimate it using $\epsilon ={C_{\mu }}^{3/4}\frac{k^{3/2}}{l}$. It can be seen from this formula that the intensity of the turbulent dissipation is proportional to the intensity of turbulent kinetic energy and inversely proportional to the turbulence scale.

In the following, the turbulent dissipation rate distribution and the intensity characteristics of fractal impellers are studied.

The turbulent dissipation rate distribution of the flow field with the central cross-section of the agitator at Z = 110 mm is shown in Figure 9.

As shown in the Figure 9, the distribution of the turbulent dissipation rate is similar. The turbulent dissipation of the equipment mainly originates from the tailing vortex and jet, with the tailing vortex as the main intensity. Due to the small-scale vortices generated by the jet, the turbulent dissipation in the flow field is mainly in the impeller zone with six medium-strength “petals”. Due to the high intensity of the tailing vortex near the surface of the impeller blade, the turbulent dissipation with high intensity is distributed on the surface of the impeller. Due to the impact of the jet on the tank wall, the medium intensity is distributed on the tank wall. Likewise, due to less influence by the jet and tailing vortex the intensity of the middle zone between the impeller and tank wall is low.

Similar to the variation pattern of the turbulent kinetic energy distribution, the intensity of both the jet and the impact vortex becomes lower because the fractal blade boundary reduces the jet intensity, and the effect of the number of iterations is smaller than that of the fractal dimension. Therefore, compared with the non-iterative impeller, the fractal iterative impeller significantly reduces the turbulent dissipation intensity of the flow field, the rectangular iterative method has a greater reduction effect than the triangular iterative method, and the turbulent dissipation intensity of the respective secondary iterative method is slightly lower than that of the one iterative method.

It is worth mentioning that the fractal dimension and iteration number have different effects near the surface of the impeller.

The characteristics of the turbulent dissipation in the impeller region of the flow field show different characteristics with the change in fractal dimension and iteration number, but do not change significantly in the flow field near the tank wall and in the middle region.

In order to further study the distribution characteristics of turbulent dissipation in the flow field, an observation line, similar to the analysis of turbulent kinetic energy, was built.

As shown in Figure 10, where the horizontal Z-axis is the axial distance, and the vertical axis is the turbulent dissipation rate of the monitoring point, the longitudinal turbulent dissipation rate distribution characteristics of each fractal impeller are similar. According to the radius length away from the Z-axis, it is mainly divided into three parts: from the edge of the impeller 2R/T = 0.5 to 2R/T = 0.8 is the single-peak and medium-intensity zone, the double-peak and low-intensity zone at 2R/T = 0.9 and 2R/T = 0.95, and the single-peak and high-intensity zone at 2R/T = 1.

The single-peak and medium-intensity zone exists near the impeller, with a maximum value of near Z = 0.11 m. This zone comes from the influence of the jet near the impeller, the effect of radial influence is high, but axially is poor, therefore this zone has a large radial coverage, but the axial coverage is narrow, and as the radius increases, the jet strength decreases, so the intensity of the turbulent dissipation decreases with the increase in the radius. Then, it gradually transits to the low-intensity area with double-peak values. In this zone, the turbulent energy dissipation from jet and impact vortex, the intensity increased with the increase in the radius, and due to the increasing influence of the impact vortex, a more obvious peak gradually appeared near Z = 0.05 m. This zone has low strength, small radial range, but large axial coverage. Finally, it transits to a single-peak and high-intensity zone, which is located at the tank wall of the mixing tank. In this location, the jet strikes the tank wall directly, and this behavior generates a large number of small-scale vortices, resulting in relatively more turbulent dissipation, this zone has the highest dissipation strength and the largest Z-direction coverage, but the radial range is small.

From the edge of the impeller to the tank wall, the intensity changes from large to small and then to large. In the longitudinal direction, its coverage is an increasing sector, and there is maximum range and intensity at the mixing tank wall.

To sum up the abovementioned findings, the turbulent dissipation of the flow field is mainly concentrated in the impeller area, which is divided into two parts. One part is distributed around the blade in the shape of “petals” with medium intensity, this part is derived from the influence of the jet, and the other part is distributed near the blade surface with the highest intensity, this part comes from the influence of the tail vortex. In addition, the turbulent dissipation of higher intensity is also distributed at the tank wall. The direct strike of the jet in this zone on the tank wall produces a large amount of turbulent energy dissipation. In general, turbulent energy dissipation, from the blade surface to the tank wall, does so with a gradually expanding range. The increase in the fractal dimension and iteration number will reduce the intensity of the turbulent dissipation in each part of the flow field. However, the turbulent dissipation near the blade surface does not change with the same intensity as the flow field distribution.

To further reveal the law of turbulent dissipation on the surface of fractal impellers, the surface weighted average turbulent dissipation rate of each side of the impeller was used to characterize the dissipation intensity, as shown in Figure 11.

As shown in Figure 11, the turbulent dissipation intensity of each side of the fractal iterative impeller is reduced compared with the non-iterative impeller. The distribution of the one iteration impeller and no iteration impeller is similar, with the highest dissipation intensity on the upper surface and the lowest on the backflow surface. The increase in the fractal dimension decreases the average turbulent energy dissipation rate of each surface, and the increase in the number of iterations can increase it. The turbulent dissipation on the impeller surface is mainly influenced by the trailing vortex, and such a pattern may be related to the variation of the trailing vortex intensity caused by the special boundary of the fractal blade.

When combining the turbulent dissipation variation law of the flow field and the impeller surface we can surmise the following: the increase in the fractal dimension will reduce the turbulent dissipation intensity of each part, and the increase in the iteration number will reduce the flow field intensity but increase the blade surface turbulent dissipation intensity.

Since the turbulent energy dissipation in the flow field mainly exists in the trailing vortex, in general an increase in the number of iterations increases the turbulent energy dissipation in the flow field.

4.4. Trailing Vortex Analysis

It can be seen that different fractal impellers have different turbulent energy dissipation near the blade, which may be related to the blade trailing vortex [33,34]. To verify the similar influence mechanism, the characteristics of the tailing vortex variation due to the fractal blade are investigated.

In this paper, the λ2 criterion was used to identify the vortex, λ2 is the second (intermediate) eigenvalue of the symmetric tensor S² + Ω² (λ3 ≥ λ2 ≥ λ1), where S and Ω are the symmetric and antisymmetric matrices of the velocity gradient tensor, respectively [18]. λ2 = −7000 as the iso-surface is selected, as shown in Figure 12, the blue part represents the trailing vortex area.

As shown in Figure 12, the wake of the turbine impeller is mainly composed of two streamlined large trailing vortices. Compared with the non-iterative impeller, the strength of the trailing vortex of the fractal impeller is significantly lower. The intensity of the tailing vortex decreases with the increase in the fractal dimension, is changed into small fragments by the fractal structure, and increases slightly with the increase in iterations. Turbulent energy dissipation mainly exists in the tailing vortex of the impeller, which occupies the main intensity of the vortex in the flow field, so the turbulent energy dissipation in the flow field is mainly concentrated near the impeller. The change to the tailing vortex intensity with the fractal dimension and the number of iterations also leads to a change in the intensity of the turbulent dissipation near the impeller, with the same trend.

4.5. Analysis of Mixing Power

The input power represents the energy consumption intensity of the mixing equipment. In a stable flow field system, the input energy of the flow field should be equal to the fluid kinetic energy, the turbulent kinetic energy, and the heat energy generated by the turbulent energy dissipation and the direct viscous dissipation. The heat energy generated by the direct viscous dissipation can almost be ignored.

The torque M of each impeller is obtained through the post-processing of CFX.

The power expression of six straight blade turbine impellers [1] is

$$\begin{array}{c}P=M\omega =\frac{\pi MN}{30}\end{array}$$

(7)

where M is the torque; ω is the angular velocity; N is the speed.

The stirring power P, and the average velocity V, of the flow field of each impeller are derived as shown in

Table 2. The stirring power and the average velocity of the flow field of each impeller.

	Non-Iterative	Triangular One	Triangular Secondary	Rectangular One	Rectangular Secondary
Average speed (m/s)	0.649	0.636	0.628	0.620	0.613
Mixing power (W)	108.1	94.6	96.1	79.78	80.59

.

As shown in the table, the increase in the fractal dimension makes the stirring power decrease, from no iteration impeller to rectangular iteration impeller, the stirring power decreases about 20%, but the average speed only decreases by 5%, which proves that the fractal impeller has good energy dissipation characteristics.

The findings from some previous related studies: Steiros [17] designed a class of fractal and perforated impellers with about a 10–18% reduction in stirring power, and Gu [19] designed a class of fractal impellers with only a slight reduction in stirring power. This paper designed a class of fractal impeller that can achieve a 20% reduction in mixing power effect. It shows that the fractal impeller designed in this paper has a better energy loss effect and can effectively reduce the stirring power.

5. Conclusions

In order to investigate the good turbulence performance of the fractal iterative impellers on energy consumption and relative wear, we firstly investigated the distribution intensity characteristics and uniformity of the turbulent kinetic energy. Then the distribution characteristics of the turbulent dissipation were investigated and then interpreted using tail vortex analysis. Lastly, the mixing power of the device was further investigated in order to evaluate the optimization effect. The following conclusions were obtained:

(1)

For the non-iterative impeller, the turbulent kinetic energy is mainly distributed in the impeller region, in addition to the medium-intensity of the tank wall region and the low-intensity of the intermediate region. The designed fractal impeller reduces the intensity of the turbulent energy in each region of the flow field to a large extent, and it is found that the increase in the fractal dimension of the impeller boundary and the number of iterations leads to a further reduction in the intensity and a further increase in the uniformity of its distribution. Compared with the no iteration impeller, the turbulent kinetic energy intensity of the rectangular secondary iteration impeller decreases by 10% and the uniformity increases by 50%, which may generate the greatest wear characteristics of the mixing device;

(2)

The turbulent dissipation of the non-iterative impeller is mainly concentrated in the impeller region. The designed fractal impeller reduces the turbulent dissipation intensity of the flow field to a greater extent, and therefore reduces the energy loss of the mixing equipment. It is also found that the increase in the fractal dimension and the number of iterations reduces the turbulent dissipation intensity of the flow field except at the impeller surface. At impeller surface, the increase in the fractal dimension reduces the turbulent dissipation, while the increase in the number of iterations improves it, this influence law originates from the changes in the trailing vortex due to the fractal boundary. Compared to the no iteration impeller, the stirring power of the rectangular one iteration impeller decreases by 20% and the average velocity decreases by only 5%; it can be predicted that the mixing equipment with the rectangular one iteration impeller may generate the greatest energy loss characteristics.

According to the conclusion from this study, the fractal iterative impeller has lower turbulent kinetic energy and turbulent dissipation intensity compared to the non-iterative impeller (ordinary turbine impeller). The two influencing factors, the fractal dimension and the number of iterations, affect the reduction influence. In contrast to the study on the optimization characteristics of the designed fractal impellers compared with the ordinary impellers, this paper focuses on the comparison of the flow field characteristics between fractal iteration impellers, and investigates the influence of fractal characteristics and the number of iterations on them.

However, there are still limitations to this paper, and we only study the characteristics of the designed class of fractal impellers. By comparing the fractal impellers designed by other scholars, we find that the impeller type, blade inclination, blade boundary, and installation method affect the flow field characteristics of the fractal impeller to a large extent, but there is still a lack of research on how these factors affect the fractal dimension and the number of iterations on the flow field characteristics.

Further research should focus on the design of various types of fractal impeller and summarize their special optimization effects. In the future, if scholars are able to conduct a lot of research, they will be able to develop a good understanding of the influence of each feature of the fractal impeller on the flow field characteristics. The results of the research will facilitate the design of various unique features of the fractal impeller, which may result in a new type of impeller system, providing a choice of impeller for various types of mixing equipment.