Abstract
The design parameters of a mixing system have a major impact on the quality of the final product. Therefore, identifying the optimum parameters of mixing systems is highly relevant to various industrial processes dealing with particulate flows. However, the studies on the influences of the mixer's design features are still insufficient. In this study, the Discrete Element Method (DEM) is used to examine the impact of paddle angle, width, and gap on the mixing performance of a twin paddle blender. The mixing performance and particle flow are assessed using the relative standard deviation (RSD) mixing index, velocity field, diffusivity coefficient, granular temperature, the force acting on particles, and the mixer's power consumption. The mixing performance is highest for a paddle angle of 0° at the cost of the highest forces acting on particles. The paddle width is indicated as a critical factor for achieving better mixing quality. In contrast, the powder mixing efficiency and the mixer's power consumption are not significantly affected by the paddle gap. The results regarding the power consumption denote that the mixer using the paddle angle of 60° has the minimum power consumption. Moreover, increasing the paddle width results in the enhancement of the mixer's power consumption.

 

1. Introduction
Mixing and particulate processing are carried out in various manufacturing industries, including chemical, cosmetics, food, and pharmaceuticals. Optimizing the performance of a mixer is a critical task for industries dealing with particle processing. Achieving higher product homogeneity in a shorter time is usually the main goal of mixer performance optimization. Insufficient mixing can cause the final product to be rejected due to poor quality. It is generally possible to enhance a mixer's performance by altering its design or adjusting operational parameters.

It is vital to closely monitor the changes in power consumption and stress experienced by particles at each condition when manipulating the operational parameters or design variables of a mixing system to identify the optimum conditions. For instance, increasing the mixer's rotational speed in most cases leads to shortening the mixing time, but simultaneously, it could also increase the stress acting on particles, which may lead to particle breakage for brittle materials. Therefore, a detailed analysis of mixing performance (usually quantified by the RSD mixing index), power consumption, and stress analysis need to be conducted when trying to optimize the mixing system performance for a specific process.

Various manufacturing processes employ mixers equipped with mechanical impellers (agitators). This group of mixers (convective mixers) consists of single/double stationary vessels and a rotating blade(s) or impeller(s). The impeller moves through the bed of particles and mixes them. Convective mixers can be classified according to the shape of their impellers, including ribbon, plow, paddle, and screw blenders. There are several studies in the literature analyzing the efficiency of these mixers using experiments and numerical simulations, both qualitatively and quantitatively.

The Discrete Element Method (DEM) approach, initially introduced by Cundall and Strack, is the most common numerical technique used to analyze the behavior of granular materials in mixing systems. The method provides particle-level information that is currently impossible or very difficult to measure experimentally (e.gs., the forces acting on each particle individually as well as particles’ positions and velocities).

In several research studies on various agitated blenders, the effect of operating parameters (e.g., RPM, vessel fill level, loading arrangement) on the mixing performance was assessed using both experiments and DEM simulations. Impeller configuration/design is another decisive parameter that affects the mixing quality in a convective mixer. The amount of momentum imparted to the particle bed is generally determined by the agitator design, which directly affects the particle flow pattern and the mixing efficiency of powder blenders. Furthermore, the impeller design can also enhance or reduce segregation in a mixing system containing a polydisperse particle mixture. Therefore, the design and development of an optimal agitated mixer greatly depend on the understanding of the effect of impeller configuration on mixing. Despite the significant role of the impeller design on mixing performance, only a few studies in the literature have reported its impact on the mixing process in an agitated mixer. For example, Chandratilleke et al. and Siraj et al. used DEM to study the effects of the blade's shape and angle on particulate flow behavior in a system equipped with a single blade containing a bed of binary mixture. They found that both forces that are acting on the blade and interparticle forces decreased by increasing the blade angle. Daraio et al. explored the effect of impeller arm length on particulate flow quality in a vertically stirred mill consisting of spherical particles. Calculating velocity, collision frequency, energy dissipation, and hydrostatic pressure from their DEM simulations, the authors concluded that the long-arm impeller design was the most effective in shortening the mixing time and enhancing the particle bed agitation. In order to assess the ribbon design effect on mixer performance, Jin et al. analyzed the mixing rate, particles' path line, velocity distribution, and forces acting on particles for three different impeller designs using both DEM and experimental investigation. It was found that the impeller (ribbon) design significantly affected the mixing performance. In another research, Tsugeno et al. studied the impacts of blade width and blade pitch on the particle mixing quality in a ribbon blender. The results revealed that the mixing efficiency, quantified by mixing progress per one paddle rotation, improved noticeably when the blade width was increased. On the other hand, the blade pitch did not affect the mixing efficiency significantly. Therefore, the blade width was identified as a critical design parameter to enhance ribbon mixer performance.

Ebrahimi et al. used DEM to examine the mixing efficiency of a single paddle mixer as a function of impeller design. Five different impeller configurations were examined, and by calculating RSD, the force acting on particles, diffusion coefficient, Peclet number, and granular temperature, the authors demonstrated that the impeller configurations significantly affected mixing performance and granular behavior. Chandratilleke et al. used DEM to explore the influences of blade rake angle and gap on powder mixing in a cylindrical bladed blender. The simulation results demonstrated that the blade with a 90° rake angle achieved the maximum mixing rate with the cost of the highest inter-particle forces. In contrast, a blade with a rake angle of 135° produced the slowest mixing rate when the inter-particle stresses were minimum.

Boonokanokwong et al. studied the influence of the number of impeller blades on mixing kinetics and particulate flow in a bladed mixer. The RSD and Lacey mixing index values were calculated for various mixer designs. The results revealed that mixing systems equipped with two or three impeller blades had superior mixing efficiency compared to mixing systems with one or four blades. Particle diffusivity and granular temperature were also higher in the two- and three-bladed mixers than in the one- and four-bladed mixers.

It is noticeable that in all the studies summarized above, the influence of the impeller design on the mixing performance of laboratory-scale single-vessel mixers has been analyzed. In this study, however, we use the DEM method to study the effect of the impeller configuration (blade's gap, width, and angle) on the mixing quality of a large-scale double paddle blender, which has not been reported in the literature previously. In order to achieve the research's objectives, particle velocity, mixing kinetics, granular temperature, particle diffusivity, forces acting on particles, power consumption, and impeller torque were calculated using the DEM simulations. The RSD mixing index was also calculated to evaluate the mixing degree. In this study, all the simulations were carried out with a constant impeller rotational speed (40 RPM), fill level (40%), and top-bottom loading arrangement. These operational conditions were selected based on our previous studies on this double-paddle blender.

Following is the outline of the present paper:

Section 2 discusses the DEM model, the geometry configurations, and the characterization methods used in this study.

Section 3 discusses the results, including the effects of the impeller configurations on the mixing performance, mixing kinetics, particle diffusivity, forces acting on particles, and the mixer power consumption.

Finally, the last section (Section 4) concludes the paper.

 

2. Modelling and Simulations
In order to determine the position, velocity, and acceleration of each particle in the mixer, the Discrete Element Method (DEM) approach is employed in our study. In this method, the collision dynamics of particles are modeled during the mixing process. The DEM approach and the simulation conditions are presented in this section, followed by the descriptions of mixing characterization methods used in the current research.

2.1 Discrete Element Method (DEM)
Newton's equations of motion are solved to calculate the velocity of each particle over time. For small time steps, it can be assumed that the gravitational force and direct collisions with other particles are the only factors affecting the dynamics of the particles. Using The DEM method, particles' positions, velocities as well as forces acting on particles can be obtained. In the present study, in order to calculate the normal and tangential forces, the Hertz–Mindlin contact model is employed. The torque and contact force expressions can be found in our previous publications and literature, and for brevity, those equations are not presented in this study. LIGGGHTS (DCS, Linz, Austria), an open-source DEM package, is utilized for all the simulations reported in the current study.

It has been widely verified that the DEM is reliable in many numerical simulations of particle mixing in the paddle mixers. However, prior to starting the simulation runs, the DEM model needs to be validated. It should be mentioned that the DEM model utilized in this paper has been validated in our previous study against experimental data obtained from a rotary drum. Consequently, the physical properties employed in the simulations can be found in our previous study, and for concision, they are not reiterated in this study.

2.2 Simulation Conditions
2.2.1 Geometry of the Mixer
The mixer investigated in this study is the same as the mixer analyzed in our previous studies. For brevity, the dimensions of the mixer used in this study are not provided in this paper. However, a detailed description of the mixer's specifications can be found in our previous publication. Briefly, the double paddle blender consists of two cylindrical vessels with two identical impellers arranged alternately at 90°. In this article, the coordinate system's origin is placed at the center of the intersection of two cylindrical-shaped vessels. Figure 1 and Figure 2 illustrate the vessel geometry and impeller geometry (used in our base case simulation).

Figure 1. Mixer's geometry used in the base case simulation; (a) Front view, and (b) Isometric view.

 

Figure 2. Impeller's geometry used in the base case simulation.

2.2.2 Mixer's Design Parameters
In this section, all the investigated mixer configurations are mentioned in detail. Three different mixer configurations’ parameters are studied, including gap (distance between the blades and mixer's wall), width, and angle of blades. The length of the impellers (B in Figure 2) is altered to adjust the gap between the blades and the mixer's wall. Six blade gaps are investigated: 0.25 d_p, 0.5 d_p, 0.75 d_p, 1 d_p, 1.5d_p, and 2 d_p where d_p stands for particle diameter, which is 5 mm in the simulations. Moreover, the effect of paddle width (A in Figure 2) on the mixing quality is investigated. Like the investigated mixer in our previous study, the paddle width is selected to be 0.053 m for the base case in the current study. Different multiplications of the base paddle width are selected to examine the impact of the paddle width on the particle mixing process, including 0.25, 0.5, 0.75, 1.0, and 1.25 times the base paddle width (A). Moreover, to examine the influence of the paddle angle on the mixing quality, the mixing process is simulated using various paddle angles. In this study, the angle between the blade surface and the horizontal z-direction is referred as the paddle angle (α). Four different paddle angles are selected: 0°, 30°, 45° and 60°. Figure 3 displays a schematic of the impeller blade’s angle configurations used in our simulations (0°, 30°,45°, and 60°).

Figure 3. Schematic of various configurations for paddle angle (a) α = 0°, (b) α = 30°, (c) α = 45° and (d) α = 60°.

2.2.3 Particles' Properties
Spherical glass beads with a density of 2500 and a diameter of 5 mm are employed in this study. A total of 40% of the mixer's volume is loaded with particles (in total 264,600 particles). First, 132,300 blue particles with a diameter of 5 mm are added to the mixing system. The particles are allowed to settle under the gravitational force, and when the average particle kinetic energy (K_e) of all particles is less than 1.0 × 10⁻⁷ J, then 132,300 red particles with the same diameter are added on top of the blue particles and allowed to settle under the gravity force. In this study, this loading configuration is known as the top-bottom loading arrangement. In order to analyze the mixing process, the particles are colored red and blue, but their material properties are the same. The impellers remain stationary as the particles are loaded into the mixer during the simulation. When the particle loading step is performed, and the average kinetic energy of all particles is less than 1.0 × 10⁻⁷ J, the impellers are allowed to rotate at 40 RPM for 30 s of simulation time. Figure 4 illustrates the initial state of the mixing system after the generation of all the particles from various views.

Figure 4. The initial state of the mixing system after loading all the particles; (a) Side view, (b) Front view.

2.3 Mixing Characterization Analysis
2.3.1 Mixing Performance
The relative standard deviation (RSD) mixing index is employed to quantify the performance of the mixing system under examination:

where C_avg denotes the average concentration of all the samples and σ stands for standard deviation, which is calculated as follows:

where x_i, x_m, and N are the number fraction of each type of particle presented in grid i, the average number fraction of particles presented in all grids, and the total number of grids, respectively.

2.3.2 Solid Dispersion
In our previous study, it was found that the dominant mixing mechanism in the mixing system is diffusion. Thus, the diffusion intensity as another microscopic property of the system is calculated, which helps to characterize the mixing system under investigation. The diffusion intensity of the particulate flow is quantified by calculating the diffusion coefficient. Several studies have shown that mixer's geometrical structure greatly influences diffusion coefficients. The diffusion coefficient is defined as:

where D_ij denotes the diffusion coefficient in direction i due to a concentration gradient in the direction j. Δxi and  represent the particle displacement and mean displacement during the time interval (Δt), respectively. Angle bracket ⟨ ⟩ denotes the averaging over all the particles for the contents within those brackets.

2.3.3 Granular Temperature
The degree of random movement of particles is measured by the granular temperature. In addition, it can be used to evaluate how particles diffuse and segregate. The diffusive mechanism plays a crucial role in particle mixing in systems with higher granular temperatures. The granular temperature is defined as follows:

where u′ and di denote each particle's velocity fluctuation at a given time in a control volume and the spatial dimension, sequentially. Moreover, ⟨ ⟩ averages within the control volume. This study uses the same control volume and time interval as our previous study (Δt=0.1 revolution time and control volume size = 4 d_p) .

Temperature profiles at granular scales can also assist with the development of continuous models to describe the mixing system. In this study, granular temperature values are calculated on a plane along the shaft (contour plot) as well as in series of control volumes (highlighted in blue in Figure 5) and are reported in the results section to examine the effect of impeller design on the granular temperature distribution in the mixer. The plane is selected in the location where it is fully submerged in the particle mixture during the mixing process, and the control volumes are selected at the tip of the impeller's blades. Figure 5 illustrates the plane and the control volumes used in the calculation of the granular temperature.

Figure 5. Plane and control volumes used for calculation of granular temperature profile from (a) Isometric, (b) Side, and (c) Front view.

2.3.4 Force Acting on Particles and Power Consumption
As mentioned in Section 2.1, contact and gravitational forces are only considered forces acting on particles in each timestep. During the mixing process, both contact and gravitational forces play a vital role in determining the hydrodynamics of the particulate flow. Therefore, in this study, to analyze the granular flow behavior, mean and time-averaged forces acting on particles are calculated. The mean force (F_m) is defined here as the average forces acting on all particles at a specific time. The time-averaged force (F_t) is determined by averaging the mean forces over entirety of the mixing time (30 s). In addition, to show the system's force profile, the time-averaged force acting on all the particles on a plane along shaft (depicted in Figure 5) is calculated and reported in each simulation case for comparison. It is important to also investigate how the power and torque values change when modifying the impeller configuration. In general, power consumption is also a main parameter in evaluating the performance of a mixing system. Thus, to evaluate the performance of the mixer, the power consumption (P) for each impeller is calculated as follows:

where s and M are impeller rotational speed (RPM) and torque acting on the impeller (N.m), respectively. In this study, since the mixer contains two parallel impellers, the system's total power consumption can be calculated from the summation of the power consumption of each impeller.

 

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