Energy-Efficient Design Optimization of Two-Bladed Agitators in Cylindrical Tanks

Mechanical agitation plays a pivotal role in industries such as chemicals, pharmaceuticals, food, paint, and wastewater treatment. A critical challenge faced in these sectors involves reducing energy consumption and optimizing mixing time for efficient operations, a problem partially addressed in previous research [1, 2].

In industrial settings, stirred tanks of various shapes, equipped with simple and complex agitators, are commonplace. The design of agitator blades emerges as a crucial parameter in mixing systems, a subject extensively studied in numerous research laboratories globally. Ameur et al. [3] developed a novel blade design aimed at minimizing energy consumption while enhancing mixing quality, factors that directly influence the cost of mixing operations. Their research analyzed the effects of blade curvature, diameter, and number. The findings indicated that curved blades are more effective in reducing energy consumption compared to straight blades. Bouzit et al. [4] conducted a numerical simulation in a tank filled with a Newtonian fluid, stirred by a two-bladed agitator. This study confirmed that such geometries predominantly generate tangential flow, with the second agitator behaving akin to a turbine, generating higher axial and radial speeds. The influence of the agitator's position within the tank on axial velocity was also examined. Laidoudi [5] numerically simulated the flow in a tank stirred by a two-bladed agitator with holes of varying diameters. Results showed that blades with a hole diameter ratio of 0.133 enhance mixing speed and decrease the power number. Abid et al. [6] simulated the flow of viscous fluids in a vessel stirred by two blades of different heights, observing that agitator size significantly impacts flow structure. Foukrach et al. [7] compared the effects of a three-blade Rushton turbine shape on energy consumption and vortex size, revealing that turbines with converging triangular blades increase power consumption, while those with diverging triangular blades reduce vortex size. Devi and Kumar [8] analyzed flow characteristics in a stirred tank using a curved blade impeller, identified as the most efficient type. Luan et al. [9] employed a novel curved six-blade turbine to improve mixing time and energy consumption in a resolubilizing fluid, noting that mixing rate and efficiency are influenced by turbine eccentricity and clearance from the tank bottom. Youcefi et al. [10] observed a significant reduction in energy consumption and vortex size in a slotted stirred tank with a Rushton turbine. Hassouni et al. [11] demonstrated that thermal effects enhance mixing quality in a four-bladed turbine-agitated tank, with no discernible influence of the Richardson number on the power number. Finally, Kamla et al. [12] conducted a comparative study of three modified anchor-type agitator geometries, aiming to improve fluid mixing in tanks and reduce energy consumption, especially in the lower part of the tank.

In the realm of agitation systems, the power number stands as a critical global parameter, intricately linked to the flow structures determined significantly by the agitation system design and fluid rheology. The pursuit of reducing energy consumption has captivated many researchers in this field. Youcefi and Youcefi [13] conducted measurements of the power number and mixing time in a tank stirred by a two-blade mobile in viscoelastic fluids, revealing a strong dependence of mixing time on fluid elasticity. Ameur and Bouzit [14] embarked on a numerical study to investigate energy consumption and velocity profiles in a flat-bottomed cylindrical tank agitated by a two-blade mobile, focusing on both Newtonian and non-Newtonian rheofluidifying fluids. They established a correlation for estimating energy consumption, taking into account variations in Reynolds number, fluid flow index, and blade dimensions. Yang et al. [15] demonstrated that, for equivalent power consumption, the velocity components in a stirred tank with RT-G are enhanced in comparison to those in RT. Basavarajappa et al. [16] observed the highest power number values with larger agitator sizes, noting a substantial decrease of nearly 70% in power number when the bottom clearance was reduced from 60 to 100 mm for the second turbine. Ge et al. [17] through simulation studies, found that alterations in blade shape significantly affect velocity distribution and energy consumption. Kato et al. [18, 19] correlated the energy consumption of various wide Maxblend type impellers using established expressions from the literature. Haitsuka et al. [20] investigated energy consumption and the volumetric mass transfer coefficient in tanks fitted with diverse impellers, concluding that larger impellers do not necessarily reduce aerated energy consumption. Lane and Koh [21] employed numerical methods to calculate the flow and power dissipation in a tank baffled by a Rushton turbine, contributing further to the understanding of energy dynamics in agitation systems.

In the field of agitation, researchers have increasingly focused on utilizing both numerical and experimental simulations to enhance hydrodynamic performance in tanks containing various types of fluids, including complex fluids, suspensions, and nanofluids. Youcefi [22] conducted an experimental investigation into the power number, mixing time, and velocity profiles for viscoelastic fluids in a cylindrical tank stirred by a two-blade mobile. His findings indicated a decrease in power number at a Reynolds number Re*=7. Bertrand [23], in his study on the agitation of Newtonian and non-Newtonian fluids using paddle, anchor, and barrier mobiles, applied both numerical and experimental techniques. Ameur [24, 25] explored the impact of blade inclination on the flow of non-Newtonian fluids in a stirred tank. The results demonstrated an increase in both the power number and vortex size with shear thinning and the angle of attack of the blades. Energy consumption was observed to increase significantly when the blade height ratio was adjusted from 0.1 to 0.95 and the blade diameter ratio from 0.2 to 0.8. Mokhefi et al. [26, 27] studied the thermal and hydrodynamic effects of a non-isothermal laminar flow of an Al₂O₃-water nanofluid in both standard and wavy tanks equipped with an anchor. They concluded that optimal heat transfer was achieved with a high-volume fraction nanofluid and a low behavior index, and corrugated walls contributed to high thermal efficiency. Montante et al. [28] compared mixing times determined through Computational Fluid Dynamics (CFD) with experimental data obtained using a conductimetric method for rheofluidifying fluids. They also examined the influence of tracer injection position on mixing time. Wang et al. [29] presented findings on the energy required for suspension in a mechanically agitated tank, suggesting that solid suspensions could be enhanced by removing baffles. Pakzad et al. [30] utilized CFD data to study the effects of turbine power, fluid rheology, and turbine size on the mixing time of rheofluidifying fluids.

Upon conducting a thorough review of existing literature in the field of mixing, it becomes clear that the design of the agitator is crucial in optimizing mechanical agitation performance. Despite this, there is a notable gap in the literature regarding two-blade agitators, particularly in terms of enhancing hydrodynamic mixing processes and reducing energy consumption. These agitators are especially relevant in the context of highly viscous fluids, where energy efficiency is a significant concern, and axial movements are typically minimal. However, the introduction of some axial motion is necessary to disrupt stagnation at the bottom of the tanks. Addressing this gap, the current study aims to develop novel agitator configurations that not only enhance axial mobility but also preserve the predominance of tangential motion. This geometric approach to improving flow behavior and energy efficiency represents a novel area of exploration, absent in the existing body of research. This gap thus forms the primary motivation for this investigation, promising to contribute significantly to the field of mechanical agitation.

A theoretical analysis of the hydrodynamic and energy performance of agitated systems, particularly the current agitation system, relies on a thorough understanding of fluid flow velocity (u, v, w) and pressure p at every point. In experimental analysis, these two parameters are determined using measuring instruments. However, in theoretical analysis, they are established by solving a set of differential mathematical equations known as the Navier-Stokes equations. The flow inside the agitated tank equipped with the new two-blade impeller design is hence governed by the fundamental equations of Computational Fluid Dynamics of Navier-Stokes, including mass and momentum equations. Adopting Cartesian coordinates and laminar stationary regime, the governing equations are given as follow:

Mass equation

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$               (3)

Momentum equations

$\begin{align}

  & \rho \left( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z} \right)=-\frac{\partial p}{\partial x} \\ 

 & \text{                     }+\mu \left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{z}^{2}}} \right) \\ 

\end{align}$               (4)

$\begin{align}

  & \rho \left( u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z} \right)=-\frac{\partial p}{\partial y} \\ 

 & \text{                     }+\mu \left( \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{z}^{2}}} \right) \\ 

\end{align}$               (5)

$\begin{align}

  & \rho \left( u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z} \right)=-\frac{\partial p}{\partial z} \\ 

 & \text{                   }+\mu \left( \frac{{{\partial }^{2}}w}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}w}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}w}{\partial {{z}^{2}}} \right)-\rho g \\ 

\end{align}$               (6)

In the present study, the rotational speed of the impeller is commanded by the Reynolds number which is calculated for the modified impeller using the characteristic dimension d (diameter of the largest part two-blade):

$\operatorname{Re}=\frac{\rho N{{d}^{2}}}{\mu }$               (7)

The power number Np is a dimensionless parameter used in fluid mechanics and chemical engineering to characterize the power consumption of an agitator or impeller in a mixing process. It represents the ratio of the power input to the fluid by the agitator to the power input required for ideal, uniform mixing. The power number is essential for understanding the efficiency and effectiveness of mixing equipment and is often used to compare different agitator designs and their performance in various mixing applications. In the case of the current modified two-blade agitator, the power number using the following formula:

$Np=\frac{P}{\rho {{N}^{3}}{{D}^{5}}}$               (8)

where, P denotes the power consumed by the stirred system. It can be calculated using the viscous dissipation function.

$P=\mu \iiint\limits_{\text{Vessel volume}}{{{Q}_{v}}\text{d}x\text{ d}y\text{ d}z}$               (9)

where, Q_v presents the viscous dissipation function which is calculated based on the fluid flow velocities (u, v, w) within the agitated vessel, obtained from Eqs. (3)-(6), using the following formula:

$\begin{align}

  & {{Q}_{v}}=2{{\left( \frac{\partial u}{\partial x} \right)}^{2}}+2{{\left( \frac{\partial v}{\partial y} \right)}^{2}}+2{{\left( \frac{\partial w}{\partial z} \right)}^{2}}+{{\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)}^{2}} \\ 

 & \text{                 }+{{\left( \frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \right)}^{2}}+{{\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right)}^{2}} \\ 

\end{align}$               (10)

The tangential, radial, and axial velocities play an important role in the study of fluid agitation in a mixing system. They allow to characterize the fluid movements induced by the agitator, which is essential for understanding the performance and efficiency of the mixing process. The tangential and the radial velocities are calculated according to the Cartesian velocities u and v as follow:

${{V}_{\theta }}=\frac{-uy+vx}{\sqrt{{{x}^{2}}+{{y}^{2}}}},\text{ }Vr=\frac{ux+vy}{\sqrt{{{x}^{2}}+{{y}^{2}}}}$               (11)

It should be noted here that the axial velocity is equal to w in the Cartesian referential. On the other hand, it is worth noting that the results will be presented with dimensionless quantities allowing to normalize physical quantities and generalize the study. Hence, certain dimensionless quantities, namely the tangential, radial, and axial velocities are defined as follows:

$V_{\theta }^{*}=\frac{{{V}_{\theta }}}{\pi ND},\text{ }V_{r}^{*}=\frac{{{V}_{r}}}{\pi ND},\text{ }{{W}^{*}}=\frac{w}{\pi ND}$               (12)

The hydrodynamic results of the new two-blades configuration shown in Figure 4 in the agitated tank have been analyzed in different planes and lines, as shown respectively in Figures 5 and 6. The influence of the parameters, namely the Reynolds number and the height of the enlarged part of the two-blades, on the hydrodynamic and energetic behavior in the agitated tank has been highlighted. The results are presented in the form of axial and radial tangential velocity contours, streamlines, and power number curves. For a purely laminar regime, a Reynolds number range between 0.1 and 100 has been adopted in this study. Moreover, following the design conditions, the shape ratio, defined as the height of the enlarged part of the two-blades relative to the height of the standard one, has been varied between 0 and 1.

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Figure 4. Different geometries of the suggested two-blades

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Figure 5. Different measurement lines adopted in the present analyze

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Figure 6. Different measurement planes adopted in the present analyze

5.1 Hydrodynamic behavior

In this subsection, we focus on analyzing the hydrodynamic behavior inside the agitated tank, taking into account the influence of the Reynolds number, which reflects in fact the rotational speed of the two-blade, as well as the height of its enlarged part of this impeller. The objective is to identify the different behaviors of the three velocity components namely tangential, radial and axial velocities, which are important for assessing the mixing quality. We aim to gain a better understanding of how these parameters affect the fluid flow in the tank and its ability to achieve efficient mixing. It is important to remember that the variation in height of the enlarged part of the two-blade is accompanied by the conservation of its total volume, and therefore, the same volume of fluid swept.

5.1.1 Global hydrodynamic behavior

For a more in-depth understanding, the global velocity con-tours and streamlines are presented in different planes, as shown in Figure 6.

Figure 7 illustrates the distribution of the velocity magnitude and the streamlines’ behavior at the agitator plane, extending towards the tank wall for different heights of the enlarged part of the two-bale under two Reynolds number values: Re=10 and Re=30.

With varying the targeted height from h_m/h=1 to 0, regardless of the Reynolds number, a noticeable decrease in the overall kinetic intensity is observed in this plane as the height of the widened part decreases (where the largest height corresponds to the standard configuration).

pi_zhu_2023-12-30_180719.png

Figure 7. Velocity contours and streamlines in the plane of the agitator for two Reynolds number

pi_zhu_2023-12-30_181141.png

Figure 8. Velocity contours and streamlines on the median plan for two Reynolds number

pi_zhu_2023-12-30_181216.png

Figure 9. Velocity contours and streamlines on the horizontal planes for two different Reynolds numbers

pi_zhu_2023-12-30_181331.png

Figure 10. Tree-dimensional streamlines for different configurations of the two-blade

pi_zhu_2023-12-30_181405.png

Figure 11. Tangential velocity for different geometries at different positions

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Figure 12. Radial velocity for different geometries at different positions

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Figure 13. Axial velocity for different geometries at different positions

5.2 Power consumption

Agitation power, also known as mixing power, is an essential measure of the energy required to stir or mix a liquid in an agitated tank. It represents the amount of mechanical energy provided by the agitator to overcome shear forces and flow resistance of the fluid. The agitation power depends on several factors, such as the agitator's rotational speed, the type of agitator, the diameter and height of the agitator, the fluid's density and viscosity, the Reynolds number, and the tank's geometry. A lower power consumption is a sought-after advantage in the design of an agitator. A design that allows for the reduction of agitation power is beneficial in several ways. Therefore, it is essential to assess the power consumption rate of our agitation system with the new two-blade design featuring two expanded and narrowed parts in order to optimize the operating conditions. The power number is thus an indicator of consumed power.

Figure 14 depicts the power number of agitation as a function of the Reynolds number for different two-blade configurations, including the two standard geometries.

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Figure 14. Power number versus Re for different geometries

The power number demonstrates a distinct decreasing trend as the Reynolds number increases, and it significantly diminishes with the implementation of the new design, especially for decreasing h_m values. This aspect represents a significant advantage of our research, as it indicates that we have achieved a more efficient and optimal energy consumption while simultaneously recovering the deficit in tangential movement. Importantly, this achievement has been accomplished while maintaining the same volume as the standard two-blade configurations, underscoring the important role of hydrodynamic modifications in reducing the power number.

In the case of the standard two-blade configuration with h_m/h=0, it is noteworthy that its power consumption is lower than that of the primary standard agitator but higher than the agitator with h_m/h=1/2. This discrepancy can be attributed to the variations in hydrodynamic behavior between the configurations, despite the volume being conserved.

Indeed, numerical data in tables can provide a more detailed view and allow us to identify the percentage of improvement in energy consumption compared to the primary standard case. Table 4 presents the power numbers and their corresponding percentage of improvement relative to the standard two-blade for various configurations and Reynolds numbers. At Re=10, a remarkable percentage improvement of 93.40% was achieved with a configuration having h_m/h=1/4. This specific design offers the most energy-efficient performance regardless of the Reynolds number. Additionally, the second standard geometry shows a substantial improvement percentage of 78.95% compared to the primary standard and ranks as the second-best geometry in terms of energy consumption.

Table 4. Power number for various modified height and Reynolds number

This knowledge can greatly benefit industrial applications seeking to enhance their energy efficiency and reduce operational costs.

It's worth noting that the approach taken in the current two-blade design yields better energy savings compared to the hole technique applied to two-blades as presented by Laidoudi [5], owing to several determinative factors. Furthermore, our technique, which combines axial mobility and tangential predominance, allows to minimize power consumption while sweeping the same volume of fluid without altering the blade's construction material. Consequently, the fluid's hydrodynamics is the only driver of this significant reduction. In contrast, in the hole technique, the analysis reveals an overall decrease primarily attributable to the reduction in the blade's construction material, which is not as straightforward.