A comprehensive review of the application of DEM in the investigation of batch solid mixers

Behrooz Jadidi; Mohammadreza Ebrahimi; Farhad Ein-Mozaffari; Ali Lohi

doi:10.1515/revce-2021-0049

Publicly Available Published by De Gruyter March 28, 2022

A comprehensive review of the application of DEM in the investigation of batch solid mixers

Behrooz Jadidi , Mohammadreza Ebrahimi , Farhad Ein-Mozaffari and Ali Lohi

From the journal Reviews in Chemical Engineering

https://doi.org/10.1515/revce-2021-0049

Abstract

Powder mixing is a vital operation in a wide range of industries, such as food, pharmaceutical, and cosmetics. Despite the common use of mixing systems in various industries, often due to the complex nature of mixing systems, the effects of operating and design parameters on the mixers’ performance and final blend are not fully known, and therefore optimal parameters are selected through experience or trial and error. Experimental and numerical techniques have been widely used to analyze mixing systems and to gain a detailed understanding of mixing processes. The limitations associated with experimental techniques, however, have made discrete element method (DEM) a valuable complementary tool to obtain comprehensive particle level information about mixing systems. In the present study, the fundamentals of solid-solid mixing, segregation, and characteristics of different types of batch solid mixers are briefly reviewed. Previously published papers related to the application of DEM in studying mixing quality and assessing the influence of operating and design parameters on the mixing performance of various batch mixing systems are summarized in detail. The challenges with regards to the DEM simulation of mixing systems, the available solutions to address those challenges and our recommendations for future simulations of solid mixing are also presented and discussed.

Keywords: batch mixing; discrete element method (DEM); mixing indices; mixing kinetics and mechanism; solid blending

1 Introduction

Mixing or blending is a process which is utilized to reduce the extent of inhomogeneity and to reach a desirable blend of components (Paul et al. 2003; Vasudeo Rane et al. 2017). Inhomogeneity may be caused by differences in concentration, phase, or temperature of the mixture (Paul et al. 2003). There are many examples of mixing in our daily life (Vasudeo Rane et al. 2017). Mixing operations are classified into six categories: liquid–liquid (miscible), liquid–liquid (immiscible), solid–liquid, gas–liquid, gas–liquid–solid, and solid–solid (Lindley 1991). Solid–solid mixing is one of the most widely used types of mixing in people’s daily lives. For example, different types of solid–solid blenders are utilized in the production of breakfast cereals, animal foods, and medications used by patients. Solid–solid mixing is a vital process in many industries such as chemical, food, cosmetics, pharmaceutical, paper, metallurgy, coal, plastics, and ceramics (Alian et al. 2015a; Bridgwater 2003; Harnby et al. 1985). In the pharmaceutical industry, for instance, the homogeneous mixing of active pharmaceutical ingredients and expedients is critical in order to avoid producing sub or super-potent capsules and tablets leading to the rejection of the final products (Radl et al. 2010). The knowledge of the underlying phenomena happening in particle mixers can help industries select the most efficient design and operating conditions of particle mixers (Remy et al. 2009). This will ultimately lead to a final product being delivered by industries to their end-users that is more affordable and of a higher quality. In the food and pharmaceutical industries, this can result in saving more lives and improving food security around the globe.

Despite the common use of mixing systems in various industries, often due to the complex nature of mixing systems, the effects of operating and design parameters on the mixers’ performance and final blend are not fully known, and optimal parameters are selected through experience or trial and error (Chandratilleke et al. 2009; Radl et al. 2010). Both experimental and numerical techniques have been widely used to analyze mixing systems and to gain a detailed understanding of mixing processes. Due to the limitations associated with experimental techniques, the discrete element method (DEM) has recently emerged as a powerful numerical technique to simulate and analyze processes, including granular materials, complementary to the experimental measurements (Hassanpour and Pasha 2014). This approach can reveal detailed information about the processes/systems which can be extremely challenging if not impossible to obtain with the current development of experimental techniques.

Various types of mixers are used in industry depending on the application, material properties (physical and chemical), and cost. Solid mixers are generally divided into batch and continuous mixers. In a batch mixer, the mixture components are placed in the mixer, mixed, and discharged for one batch before another batch is introduced. Today, these mixers are the most popular type of mixers in the food and pharmaceutical industries (Yaraghi 2018). Batch mixers, based on the dominant mixing mechanism in the system, can be categorized into two main groups: tumbling mixers and convective mixers (Ortega-Rivas 2012). In tumbling mixers, the diffusive mechanism is dominant. In convective mixers, however, usually the convection mechanism dominates the diffusion mechanism (Ortega-Rivas 2012). In a continuous mixer, the mixture ingredients are introduced continuously to the mixing vessel and get mixed as they pass through the mixer and get discharged continuously. Continuous mixers are not the focus of this review, and interested readers are referred to other studies in literature (Bhalode and Ierapetritou 2020; Gao et al. 2012; Kehlenbeck 2011; Kingston and Heindel 2014; Koller et al. 2011; Langston et al. 1994; Palmer et al. 2020; Portillo et al. 2010, 2009; Vanarase and Muzzio 2011).

1.1 Tumbling mixers

Tumbling mixers contain a closed vessel that is partially filled and rotates around its axis (Saberian et al. 2002). The vessel can rotate to a speed of up to 100 revolutions per minute with a working volume capacity of 50–60% of the total vessel volume (Niranjan et al. 1994). Tumbling mixers operate by repeatedly lifting, tossing, and rolling the powder materials within the vessel. The rate of mixing is rather slow when compared to other blenders. However, a homogenous mix can be anticipated given sufficient time for mixing (Jones and Bridgwater 1998). Tumbling mixers are generally classified into four types, namely the twin-shell mixers (V-mixers), the tote mixers, the double-cone mixers, and the horizontal rotary mixers. The twin-shell mixers are the most widely used types of tumbling mixers (Alexander et al. 2004a; Chang et al. 1992; Cullen et al. 2015; Perrault et al. 2010). Generally, tumbling blenders are not suitable for use in the mixing of agglomeration mixtures; however, weak particle agglomeration can be broken by changing the angle or rotational direction of the shell (Alexander et al. 2004b; Paul et al. 2003). One of the important advantages of tumbling blenders is that cleaning and maintenance of tumbling mixers is rather easy, and it only takes short time to clean the vessel and make it ready for the next batch (Niranjan et al. 1994). These types of mixers also generally require less power when compared to convective mixers. A disadvantage of this type of mixer is the possibility of de-mixing while discharging the contents (Paul et al. 2003). The general advantages and disadvantages of tumbling mixers are tabulated in Table 1.

Table 1:

Advantages and disadvantages of tumbling blenders.

Advantages	Disadvantages
Low-cost service and maintenance (cleaning and emptying)	Only batch mixing
Ease of sampling	De-mixing can occur during discharging
Complete release of the final mixture	High chance of segregation of the solid particles
	Not suitable for cohesive materials

1.2 Convective mixers

A convective solid mixer consists of a stationary vessel (positioned vertically or horizontally) and an impeller mounted on a shaft. Unlike tumbling mixers, the shell stays stationary while the contents within are mechanically mixed using paddles, ribbons, or ploughs. Based on the vessel’s orientation, convective blenders can be categorized into horizontal and vertical blenders. Common types of horizontal convective blenders are ribbon blender, ploughshare blender, paddle blender, and sigma blade mixer. Moreover, high-shear mixers, screw blenders, and bladed mixers are some common types of vertical convective blenders. Convective mixers can be designed as a single or twin shaft configuration (Cullen et al. 2015). In convective mixers, the impeller rotates inside a static shell, and particle groups are transported in the bulk of a mixture from one position to another (Harnby et al. 1985). Transportation of the groups of particles would lead to mixing in the system due to the convective and shear mechanisms. However, there are some studies in literature reporting that the diffusion mechanism can be dominant in convective blenders (Ebrahimi et al. 2018; Yaraghi et al. 2018). Convective mixers are used for a wide variety of powders ranging from free-flowing to cohesive powders. In addition, they can operate in a wide range of mixing capacity (Harnby et al. 1985). The general advantages and disadvantages of the convective mixers are summarized in Table 2.

Table 2:

Advantages and disadvantages of convective blenders.

Advantages	Disadvantages
Suitable for both batch and continuous operation	High utility and maintenance cost
Applicable for both free-flowing and cohesive materials	Complicated mixing quality analysis due to sampling difficulties
Wide range of capacities	Hard to clean

The final goals of this review paper are to present the current status of the use of the DEM technique in studying mixing systems, discuss the challenges faced when using the DEM approach and available solutions to address those challenges and recommend future possibilities to use the DEM technique in studying mixing systems. The current study is organized as follows: initially, the mixing and segregation mechanisms are briefly reviewed. The characterization of solid-solid mixing systems through experiments and the DEM techniques is also covered. The application of DEM to investigate the effect of design parameters and operating conditions on the mixing performance is thoroughly reviewed and summarized in Section 4. The possible challenges in implementing DEM in analyzing mixing systems are discussed and the possible solutions to address those challenges are presented in Section 5. In the final section, our recommendations for the future use of DEM to investigate mixing systems are presented.

2 Mixing mechanisms and segregation

2.1 Mixing mechanisms

The main mixing mechanisms in solid–solid blenders can be classified into three categories: convective mixing, diffusive mixing, and shear mixing (Bridgwater 1976; Lacey 1954; Paul et al. 2003). All three mixing mechanisms are likely to occur in a mixing operation to various extents simultaneously (Bridgwater 1976; Lacey 1954). However, depending on the mixer type, mixing operating conditions (e.g., initial loading pattern and rotational speed), and the flowability of the solid particles, often one of them is the dominant mechanism (Bridgwater 1976; Masuda et al. 2006).

The convective mechanism can be defined as the motion of a group of adjacent particles from one location to another location in the mixture (Lacey 1954; Paul et al. 2003). For instance, the convective mixing mechanism can occur when an agitator or impeller moves a relatively large mass of particulate material from one part of the mixer to another part (Bridgwater 1976; Kingston 2013). Figure 1(a) shows how particles move in groups during the convective mixing mechanism. The convective mixing mechanism is a fast mechanism when compared to other mechanisms (diffusive and shear), and it can distribute particles in the entire mixing vessel (Alizadeh 2013).

Figure 1:

Schematic of movement of particles during each mixing mechanism: (a) convective mechanism, (b) diffusive mechanism, (c) shear mechanism.

Diffusive mixing occurs due to the micro-scale random motion of particles inside the mixture, which leads to a change in their positions relative to each other (Lacey 1954; Paul et al. 2003). The diffusive mixing mechanism is essential to reach microscopic homogenization (Masuda et al. 2006). The schematic of the random path of particles during the diffusive mixing mechanism is shown in Figure 1(b). Since the diffusive mixing involves the motion of individual particles, a better randomly mixed mixture could be yielded by this mechanism in comparison with other mechanisms (Masuda et al. 2006). Also, compared to convective and shear mixing, this mechanism happens at lower mixing rates. Thus, when diffusion is the dominant mixing mechanism, the mixing process would be slow and take a long time to achieve the required degree of homogeneity (Masuda et al. 2006). The shear mixing mechanism results from the momentum interchange between particle layers that are moving with different velocities. This causes some particles to move from one layer to another (Masuda et al. 2006). In other words, the shear mixing mechanism occurs when the motion of a layer of particles, which is adjacent to the other layers, causing some of the particles to be transported to another layer (Alizadeh et al. 2014a). Figure 1(c) illustrates the momentum exchange during the motion of particle layers in a mixture. A wide distribution of particle velocities happens in the regions close to the impellers in a mixing system. Thus, the shear mixing mechanism may happen in those regions caused by the motion of a large mass of particles via the convective mechanism (Kingston 2013). Furthermore, due to the compression and dilation of the particles, this mixing mechanism may be developed in the vicinity of the mixer’s walls (Masuda et al. 2006). It is worth mentioning that the shear mechanism plays an essential role in breaking the agglomerates of particles. Thus, when the mixing of cohesive powders is being conducted, the mixers in which the shear mixing mechanism is dominant are favorable (Hogg 2009).

2.2 Segregation mechanisms

In various processing industries, achieving a homogenous mixture of particulate solids with different sizes and densities is essential. However, when particles have different physical or mechanical characteristics such as density, size, roughness, elasticity, and shape, they tend to exhibit some degree of segregation (Alizadeh 2013; Hogg 2009; Mosby et al. 1996). Although all aforementioned differences in particle properties can cause segregation, the difference in particle sizes in a mixture is the most influential parameter for segregation among others (Alizadeh 2013; Alizadeh et al. 2013a, 2013b). Particle shape-induced segregation in mixing systems also has been the subject of some studies in the literature (Govender et al. 2018; Sinnott and Cleary 2016). He et al. (2019, 2021 demonstrated that particle shape had a significant impact on axial and radial segregation. Segregation is one of the most influential phenomena which adversely impacts the uniformity of bulk materials and makes particle handling and processing such as mixing even more challenging (Alizadeh et al. 2013a; Marucci et al. 2018).

Segregation mechanisms based on particle size can be classified into four categories: trajectory, elutriation, percolation (sifting), and push-away (Rhodes 2008). Figure 2 illustrates various types of segregation. The trajectory and elutriation segregation mechanisms become important in two-phase systems when the fluid–particle interaction force has a significant influence on the particulate phase motion. The percolation mechanism happens due to the particle size difference. The gap between larger particles allows the smaller particles to slide downward in the system and consequently the particles in the system are segregated due to their particle size. The main reason for the push-away segregation mechanism is the difference between particles’ density. When a high-density particle comes into contact with two lighter particles, it can push them away. As a result, heavier particles could deposit in the center of a system and the lighter ones may be pushed toward the walls (Silva et al. 2000). As stated by Tang and Puri (2004) and Rhodes (2008), the segregation can be diminished by changing the particles’ properties and mixing operating conditions.

Figure 2:

Segregation mechanisms: (a) trajectory, (b) elutriation, (c) percolation (sifting), and (d) push-away.

3 Characterization of the solid–solid mixing

3.1 Experimental methods

In recent decades the rapid advancement in experimental techniques has facilitated the investigation of mixing systems in more detail (Asachi et al. 2018; Kingston 2013; Radl et al. 2010; Remy et al. 2009, 2010b, 2011; Zheng et al. 2012; Zhu et al. 2007). There are several experimental techniques that can be used to assess the mixture homogeneity and they differ in accuracy, fundamental basis, cost, and operating conditions (Asachi et al. 2018; Kingston 2013; Mendez et al. 2010). In general, the experimental techniques used to assess the mixing quality can be classified into two categories: (1) invasive methods, and (2) non-invasive methods (Kingston 2013). The invasive method usually referred to as the sampling method is a very common approach for assessing the mixing quality in an industrial mixer (Paul et al. 2003; Yaraghi 2018; Yaraghi et al. 2018). Samples are taken from different parts of a mixer and are considered representative of the entirety of the mixing system. Since samplers usually can not reach the mixer’s wall, mixing quality near the wall cannot be easily investigated by the sampling method. It is also worth mentioning that the sample size, number of samples taken from a mixing system, and the location of sampling points can affect the accuracy of the mixing assessment (Poux et al. 1991). In the sampling method, a Thief sampler is commonly used in order to take samples from a mixer (Alizadeh 2013; Chen and Yu 2004; Ortega-Rivas 2012; Paul et al. 2003). Thief sampling can be categorized into three groups: (1) side sampling, (2) end sampling, and (3) core sampling (Alian et al. 2015a; Paul et al. 2003). Muzzio et al. (1997, 2003 reviewed different sampling methods and discussed both sampling technology and sampling procedures.

Recently, noninvasive experimental techniques have gained some attention in analyzing mixing systems because they can address the issues associated with invasive methods such as disturbance of the bed of material due to the sampler insertion, and disturbance of the granular flow path (Nadeem and Heindel 2018). The noninvasive methods can provide experimental data during the mixing process without the need to stop the mixing system to take samples (Nadeem and Heindel 2018). In recent studies, Asachi et al. (2018) and Nadeem and Heindel (2018) reviewed in detail various non-invasive experimental methods and summarized the advantages and disadvantages of them. The reader is referred to the above-mentioned references for detailed information on noninvasive experimental methods. Image analysis techniques, spectroscopic methods, and tomographic methods are the most common noninvasive methods using for qualification of the mixing process (Asachi et al. 2018; Nadeem and Heindel 2018). A summary of advantages and disadvantages of both invasive and noninvasive experimental techniques is presented in Table 3. Based on the limitations associated with the sampling methods summarized in Muzzio et al. (2003)’s work and the disadvantages of noninvasive experimental methods presented in Table 3, one can conclude that detailed analysis of solid mixing processes only through experimental studies is not feasible. This underlines the necessity of particulate simulations in investigating solid mixing (Swarbrick 2013).

Table 3:

Advantages and disadvantages of invasive and noninvasive experimental techniques.

	Advantages	Disadvantages
Invasive	Easy to implement Cost-effective	Destruction of the bed of material during sampler insertion Disturbance of the granular flow path Lower accuracy compared to noninvasive methods
Non-invasive	In-line measurements Implementation for various mixing systems	Expensive Cumbersome to implement

3.2 Numerical method

Today with the great advancement in the computational facilities, numerical simulation of mixing systems has become a powerful tool to investigate mixing systems. The simulations can be used as a predictive tool for designing a new mixing system or can be used in order to optimize the mixing system performance (Hassanpour and Pasha 2014). Among all the methods for simulation of particulate flows, DEM is in fact the most widely used technique and over the past two decades, has been used extensively to obtain detailed particle-level information about particulate flows (Alian et al. 2015b). DEM is a Lagrangian approach and is capable of tracking the motion of each individual particle in the system. The principles of discrete modeling were pioneered by Cundall and Strack in the late 1970s (Cundall and Strack 1979).

3.2.1 Fundamentals of discrete element method (DEM)

In the DEM approach, Newton’s second law of motion is solved to estimate the particle motion, and contact models are applied for the description of particle interactions with the neighboring particles and the geometry surfaces (Abbaspour-Fard 2000; Knowlton and Pearce 2013; Norouzi et al. 2016). Using time integration of equations of motion for each particle in the system, the DEM technique enables the evaluation of the dynamic behaviors of the entire system (Norouzi et al. 2016). Figure 3 illustrates the application of Newton’s second law and contact model in DEM.

Figure 3:

Application of Newton’s second law and contact models in DEM.

In a granular flow, each particle can have two types of movement: translational and rotational (Hassanpour and Pasha 2014). Newton’s second law of motion estimates the translational motion, and the rotational motion is evaluated by Euler’s second law (Di Renzo and Di Maio 2004; Lindley 1991). The velocities and locations of particles are adjusted by integrating accelerations over a time period (Hassanpour and Pasha 2014). The governing equations for translational and rotational motion in the system are as follows (Hassanpour and Pasha 2014):

(1) m i d v → i d t = ∑ j N c ( F → c , n i j + F → c , t i j ) + ∑ k N n c F → nc i k + F → f − p i + F → ext i

(2) I i d ω → i d t = ∑ j N c ( M → i j + M → r i j )

where m_i, v → i , ω → i are mass, translational velocity, and angular velocity of the individual particle i, respectively. F → c , n i j and F → c , t i j are normal and tangential contact forces between particle i and j and F → nc i k represents noncontact forces between particle i and k, acting on particle i. Correspondingly, I_i, M → i j , and M → r i j are the moment of inertia, rotational torque, and rolling resistance torque of the particle i. F → f − p i represents the fluid–particle interaction forces acting on particle i such as drag and lift forces and F → ext i describes all external forces (both uniform and nonuniform) acting on particle i such as gravity force and electromagnetic field. It should be noted that this force often is ignored in the simulation of solid–solid mixing systems as fluid effects are negligible on the trajectory of particles. By calculating the ratios between the gravity force, drag force, and contact forces, one can determine to whether include or exclude the influence of the surrounding fluids in the mixing simulations. It should however be noted that including the effect of air in the simulation of the mixing process, for example by employing the CFD-DEM approach, increases the computational time noticeably. For spherical particles, M → i j acting on a particle is generated only due to the tangential component of contact force. However, when simulating non-spherical particles M → i j can be generated as a result of both normal and tangential components of contact force (Favier et al. 1999).

Modeling interactions between particles and particle-geometry may be the most critical aspect of DEM simulation (Yeom et al. 2019). Usually, simplified models or equations are utilized to determine the forces between any two particles (Zhu et al. 2007). Extensive investigations have been done to develop accurate force-displacement models based on theories of contact mechanics that can be used in predicting particulate flow behavior in DEM simulations (Di Renzo and Di Maio 2004; Elperin and Golshtein 1997; Norouzi et al. 2016; Stevens and Hrenya 2005; Zhu et al. 2007). Numerous force-displacement laws for estimation of interaction forces have been proposed in literature (Cooke et al. 1976; Cundall and Hart 1992; Di Renzo and Di Maio 2004; Kruggel-Emden et al. 2007; Stevens and Hrenya 2005). Contact force models can be defined as a function of particle overlap and the relative particle velocities. The normal contact force models can be categorized into four main groups (Alizadeh 2013): the continuous potential models, the linear viscoelastic models, the nonlinear viscoelastic models, and the hysteretic models. A summary of the most common normal contact forces used in DEM is illustrated in Table 4. The most common and widely applied tangential contact force model is the one suggested by Mindlin and Deresiewicz (Mindlin and Deresiewicz 1953). The contact force models were reviewed by Norouzi et al. (2016), Kruggel-Emden et al. (2007), Zhu et al. (2007), and Stevens and Hrenya (2005) in detail. In this study, however, for the sake of brevity, contact models and their equations are not reported. Thus, the interested reader can refer to the previously mentioned literature.

Table 4:

Summary of the common normal forces used in DEM.

	Elastic	Inelastic
Free flowing	Hertzian model (Norouzi et al. 2016) Linear spring model (Norouzi et al. 2016)	Hysteretic spring model (Walton and Braun 1986a, 1986b)
Cohesive	JKR (Barthel 2008) DMT (Prokopovich and Perni 2011)	Elasto-plastic adhesive models (Hare et al. 2015; Luding 2008; Pasha et al. 2014; Thakur et al. 2014)

F → nc i k can include van der Waals force, electrostatic force, or liquid bridge force. These noncontact forces become important when dealing with fine particles or when moisture affects the flow behavior of particles. Therefore, depending on the particle type, size, and operational conditions, these forces may be included or excluded from DEM simulations. The fluid–particle interaction force (F_p−f) needs to be considered in DEM simulations when the surrounding fluid has a significant influence on the particle motion. It should be noted that this force often is ignored in the simulation of solid–solid mixing systems as fluid effects are negligible on the trajectory of particles. The rotational torque happens due to the particle contacts in the tangential direction and rolling resistance torque arises due to the uneven contact pressure in the contact area (Brilliantov and Pöschel 1998; Kondic 1999; Norouzi et al. 2016). The latter can be calculated through one of the three main models; constant torque model (Model A), viscous model (Model B), or elastic–plastic spring-dashpot model (Model C). Ai et al. (2011) reviewed the influence of a rolling resistance model on particle motion and concluded that Model C was the most suitable rolling resistance model in their DEM simulations.

3.2.2 Assessment of the mixing mechanisms using DEM

The diffusivity coefficient can be used to assess the significance of the diffusive mechanism in a mixing system (Yaraghi et al. 2018). The diffusivity coefficient determines the particle mass flux caused by their random movement. The diffusion coefficient, D_ij, for the diffusion occurring in the i direction due to a composition gradient in the j direction is given by (Yaraghi 2018):

(3) D i j = ( Δ x i − Δ x i ‾ ) ( Δ x j − Δ x ‾ j ) 2 Δ t

where Δx_i is the particle displacement in the i direction, and Δ x ‾ i is the mean displacement of all particles in the i direction during Δt. Hwang and Hogg (1980) investigated the mixing of dry powders over an inclined surface and proposed a simple linear relationship for the diffusion coefficient of the diffusion occurring in the i direction due to a composition gradient in the j direction:

(4) D i j = D 0 ( 1 + α ∂ v → i ∂ j )

where D₀ and α are constant, and v → i is the velocity of the particle in the i direction. Peclet number can be used to gauge the ratio between the diffusive and convective mechanisms and it is defined as follows (Yaraghi et al. 2018):

(5) P e i j = U i L c D i j

where U_i is the average velocity of the particles in the i direction, and L_c represents the characteristic length of the mixer. The diffusive mechanism is dominant if the Peclet number value is smaller than one. On the other hand, when the Peclet number is larger than one the convective mixing mechanism is the dominant mechanism (Yaraghi 2018).

The granular temperature which can be calculated from DEM simulation results is another way to gauge the contribution of diffusion to the mixing of particles. The granular temperature is calculated as follows (Boonkanokwong et al. 2016):

(6) T = 1 3 u ′ u ′

where u^′ is the fluctuation velocity of each particle, which is calculated by subtracting the mean velocity from the individual velocity of a group of particles in a control volume at a specific time, and the 〈〉 symbol represents the temporal averaging within the control volume (Boonkanokwong et al. 2016; Remy et al. 2009; Yaraghi et al. 2018). In general, higher granular temperature means a higher velocity fluctuation and less uniform particle flow. Higher granular temperature values in a mixing system indicate that the diffusive mechanism has a dominant contribution in mixing (Boonkanokwong et al. 2016). Since the diffusion and shear usually happen together, the intensity of the shear mechanism can also be gauged by granular temperature (Sacher and Khinast 2016). Thus, a region in mixing systems with a high granular temperature value represents a part of the mixer with a high shear rate.

There are a couple of formulations available in literature to assess the shear mechanism. In studying a bladed mixer, Radl et al. (2010) proposed that the granular bed can be described as a continuum system and therefore the normalized shear rate distribution in the x–y plane can be calculated as:

(7) γ ∗ ⋅ = 2 ∗ ( ∂ U ∗ ∂ x ∗ ) 2 + 2 ∗ ( ∂ V ∗ ∂ y ∗ ) 2 + ( ∂ U ∗ ∂ x ∗ + ∂ V ∗ ∂ y ∗ ) 2

where U^∗, V^* are the mean velocity of the particles in the x and y directions normalized with the impeller velocity, and x^*, y^* are coordinates in the x and y directions normalized with the radius of the mixer. The shear rate distribution enables the assessment of the shear rate at each part of the mixer (Radl et al. 2010). Thus, the contribution of shear mechanisms in the mixing process can be explored.

In order to assess the shear mechanism contribution during mixing operation, Remy et al. (2010b) calculated shear stress using Eq. (8) proposed by Campbell (2002). The collisional stresses in a control volume with the size of V_c were calculated as:

(8) τ i j = d p V c F i k j

where d_p is the particle diameter, F_i is contact forces acting on particles in i direction, k_j is the unit vector in the j direction. Both methods presented by Remy et al. (2009) and Radl et al. (2010) can help to find the contribution of the shear mechanism in the mixing process.

3.2.3 Assessment of mixing quality

Mixing indices are commonly utilized to evaluate mixing quality in a blender (Poux et al. 1991). Various mixing indices have been introduced and used in literature by different researchers (Chandratilleke et al. 2012; Danckwerts 1953; Halidan et al. 2014; Lacey 1954). Fan et al. (1970) reviewed some of these indices proposed from 1958 to 1969. Wen et al. (2015) summarized different mixing indices in detail and assessed the advantages and disadvantages of each one of them. In other research, Deen et al. (2010) compared four different mixing indices in order to evaluate mixing in pressurized fluidized beds. Recently, Bhalode and Ierapetritou (2020) critically reviewed the commonly used mixing indices. The authors categorized different mixing indices based on the method of calculation into three main groups: (1) variance-based, (2) distance-based, and (3) contact-based. Readers are referred to references (Bhalode and Ierapetritou 2020; Deen et al. 2010; Wen et al. 2015) for more information regarding various mixing and segregation indices available in literature. A summary of the most common mixing and segregation indices presented in literature for evaluating the mixing performance of various mixing systems is presented in Table 5. Based on Table 5, one can conclude that RSD and Lacey mixing indices are the most common mixing indices used in the analysis of mixing systems in literature.

Table 5:

The summary of mixing/segregation indices and their applications.

Mixing/segregation index	Index equation	Range (completely segregated- completely randomized)	Application of mixing/segregation index in the evaluation of mixing performance
Lacey index (Lacey 1954)	LI = σ 0 2 − σ 2 σ 0 2 − σ r 2	0–1	Bladed mixer (Chandratilleke et al. 2009, 2010, 2012, 2014), ribbon blender (Basinskas and Sakai 2016a; Chandratilleke et al. 2018; Gao et al. 2019), Slant cone mixer (Alian et al. 2015b), ploughshare mixer (Alian et al. 2015a; Hassanpour et al. 2011), rotary drum (Alchikh-Sulaiman et al. 2016; He et al. 2019; Ji et al. 2020; Liu et al. 2013), double-screw conical mixer (Cai et al. 2019)
Intensity of segregation	I s = σ 2 − σ r 2 σ 0 2 − σ r 2	1–0	Rotary drum (Chaudhuri et al. 2006; Qi et al. 2017; Yazdani and Hashemabadi 2019), cone blender (Moakher et al. 2000), bin blender (Portillo et al. 2008)
Relative standard deviation (RSD)	RSD = σ C avg	1–0	Paddle blender (Ebrahimi et al. 2020, 2018; Jadidi et al. 2022; Yaraghi et al. 2018), bin-blender (Alexander et al. 2004b; Arratia et al. 2006a, 2006b), V-blender (Lemieux et al. 2007), bladed mixer (Remy et al. 2009, 2010a, 2010b, 2011), rotary drum (Sebastian Escotet-Espinoza et al. 2018)
Particle scale mixing index (Chandratilleke et al. 2012)	PSMI = S 0 2 − S 2 S 0 2 − S r 2	0–1	Ribbon blender (Halidan et al. 2014), screw blender (Qi et al. 2017), bladed mixer (Qi et al. 2017)
Siria mixing index (Siiriä and Yliruusi 2009)	S = ∑ i = 1 N ∑ j = 1 N M i j N 2	0–1	Rotary drum (Wen et al. 2015), multiple-spouted bed (Chen et al. 2018; Wen et al. 2015)
Segregation index (Stambaugh et al. 2004)	SI = C A A C A A + C A B + C B B C B B + C A B	2–0	Rotating (Hlosta et al. 2020; Marigo et al. 2012), hoop mixer (Marigo et al. 2012), turbula mixer (Marigo et al. 2012)
Graphic segregation index (Dai et al. 2020)	GPSI = A m g A f g	1–0	Vibrated bed (Dai et al. 2020)
Subdomain-based mixing index (Cho et al. 2017)	SMI = 1 N ∑ i = 1 M [ SMI ( S i ) ∑ k = 1 Q n k i ]	0–1	Screw blender (Cho et al. 2017), ribbon blender (Harish et al. 2019)

4 Effects of design parameters and operating conditions on the mixing quality

As mentioned before, mixing systems are essential unit operations in various industries and it is vital to ensure that a blender operates in an optimal condition to guarantee the quality of the final product. In general, the key parameters which significantly influence the mixer performance and mixing quality can be classified into three categories (Table 6).

Operational parameters (e.g. fill level, initial loading pattern, impeller or vessel rotational speed, impeller or vessel rotational axis).
Mixer’s design parameters (e.g. type and dimension of impeller and vessel, vessel and impeller orientation, clearance between impeller and vessel, mixer’s sizes, baffle).
Particles’ properties (e.g. shape, size, density, type of particles (cohesive and noncohesive)).

Table 6:

Key parameters influencing the mixing process.

Parameters affecting the mixing
Operational conditions	Mixer’s design parameters	Particle’s properties
Initial loading pattern	Type and dimension of impeller and vessel	Size
Vessel fill level	Vessel orientation	Type of particles (cohesive and non-cohesive)
Impeller or vessel rotational speed	Clearance between impeller and vessel	Shape
Impeller or vessel rotational axis	Impeller orientation	Density
Impeller or vessel rotational axis	Baffle	Density

The influence of the operational parameters on the mixing performance has been the subject of several studies. For example, Sakai et al. (2015) and Basinskas and Sakai (2016b) reported that increasing the fill level improved the mixing efficiency. In contrast, Qi et al. (2017) found that decreasing the fill level enhanced the mixing quality for twin-screw blenders. There are also some studies which demonstrated that the fill level did not have a significant influence on the mixing performance when studying paddle mixers (Jadidi et al. 2022; Yaraghi et al. 2018). In the majority of studies, increasing the impeller or vessel rotational speed led to an enhancement of mixing quality (Basinskas and Sakai 2016b; Chandratilleke et al. 2010; He et al. 2019; Jadidi et al. 2022; Lemieux et al. 2007; Tahvildarian et al. 2013; Yaraghi et al. 2018; Zhou et al. 2003). However, as mentioned by Halidan et al. (2018), this improvement was achieved by increasing the particle-geometry contacts force, which may not be favourable, when mixing brittle particles. As reported in the various literature, the initial loading pattern can also have a decisive influence on the mixing quality in both convective and tumbling blenders (Alian et al. 2015a; Alizadeh et al. 2014a; Arratia et al. 2006a, 2006b; Basinskas and Sakai 2016a; Gao et al. 2019; Jadidi et al. 2022; Marigo et al. 2012). However, some studies demonstrate the negligible influence of this operational parameter on the mixing performance (Ebrahimi et al. 2020; Yaraghi et al. 2018). Some researchers have also explored the applicability of DEM in optimizing mixing systems design. Remy et al. (2010b), Ebrahimi et al. (2018), and Boonkanokwong et al. (2016) analyzed the influence of the impeller configuration on mixing quality and reported the vital role of this design parameter on the mixing performance of a blender. The importance of the blade rake angle and clearance on mixing performance of a vertical bladed mixer was reported by Chandratilleke et al. (2009).

It is worth considering that the suitability and performance of a mixing system for a specific process can vary greatly depending on the particle properties. For example, a mixing system which operates optimally for free-flowing particles may not operate suitably for the mixing of cohesive powders. Changing particle size distribution, density, and shape can make a mixing system appropriate or inappropriate for a specific process. For example, Halidan et al. (2018) reported that the two bladed ribbon mixer was not a suitable mixing system for mixing cohesive materials, while it could properly mix free-flowing particles. Chaudhuri et al. (2006) reported that cohesive particles mixed better compared to free-flowing particles in a rotary drum. On the other hand, Yazdani and Hashemabadi (2019) found that the rotary drum mixing performance was deteriorated for highly cohesive particles. Ji et al. (2020) demonstrated that the mixing rate of the rotary drum was better for nonspherical particles compared to spherical particles. On the other hand, for a bladed mixer, Govender et al. (2018) reported that the mixing system comprising spherical particles reached the final mixing index value faster compared to the system including nonspherical particles. Recently, Hlosta et al. (2020) thoroughly investigated the effect of particle properties (i.e. shape, size, and density) on the mixing performance of a rotary drum. The influence of particle properties on bladed mixing systems has been reported in studies by Boonkanokwong et al. (2018) and Halidan et al. (2014). In those studies, it was highlighted that the performance of a mixing system depends greatly on the particle properties.

Therefore, it is essential to thoroughly understand the effect of those parameters on the mixing performance in order to find an optimal design and operating parameters for a mixing system. The studies which have used the DEM techniques to analyze various mixing systems and to investigate the effect of operating and design parameters on the mixing quality are summarized in Tables 7 –13. The summary includes the analysis of both tumbling and convective mixing systems. The available information regarding the type of mixer, particle data, type of contact model, experimental technique for validation, mixing quantification methods, and the main findings/results are included in Tables 7 –13.

Table 7:

Summary of application of DEM to investigate mixing in the cone blenders.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Moakher et al. (2000)	Double-cone blender V-blender	Spherical Noncohesive Monodisperse Bidisperse	Walton and Braun (Walton and Braun 1986a, 1986b)	Image analysis (Brone et al. 1997; Brone and Muzzio 2000)	Mixer’s type	Intensity of segregation Mixing mechanism Velocity fields	Nonsymmetric mixers provided better mixing performance
Arratia et al. (2006a)	Bin blender	Spherical Noncohesive Monodisperse Bidisperse	Walton and Braun (Walton and Braun 1986a, 1986b)	NIR spectroscopy (Duong et al. 2003)	Initial loading pattern Fill level	Intensity of segregation RSD Mixing time Mixing mechanisms Segregation mechanisms	For monodisperse particles, mixing worsened with increasing fill level for top-bottom and side-side initial loadings Mixing of monodisperse particles was sensitive to the initial loading pattern (top-bottom yielded a better mixing quality) For bidisperse particles, small particles moved toward the blender walls while larger particles remained in the center, leading to considerable segregation The segregation tendency was sensitive to fill level, and the intensity of the segregation enhanced with increasing the fill level
Ren et al. (2013)	Tote blender	Spherical Noncohesive Monodisperse	Zhou et al. (1999)	Thief sampling (Lemieux et al. 2007)	Fill level Vessel rotational speed Vessel angle	RSD Energy consumption	A fill level of 50%–60% was found to provide high efficiency and productivity 30 RPM was found to be the optimum rotation rate Changing the angle of inclination of the blender was found to be an effective and simple way to speed-up mixing
Alian et al. (2015b)	Slant cone mixer (with T-shaped blades)	Spherical Noncohesive Monodisperse	Hertz-Mindlin	Image analysis and thief sampling	Initial loading pattern Vessel rotational speed Fill level Agitator speed	Lacey index Average velocity	The mixing rate was almost the same for side-side and top–bottom initial loadings, and was higher than back-front initial loading Better performance was observed at a 70% fill level compared to a 100% fill level when the agitator was stationary Increasing agitator speed improved mixing quality Better mixing performance was obtained when the agitator and vessel rotated in the same direction (co-rotating mode) compared to when the agitator and vessel rotated in the opposite direction (counter-rotating mode)
Alchikh-Sulaiman et al. (2015)	Slant cone mixer (with T-shaped blades)	Spherical Noncohesive Monodisperse Bidisperse Tridisperse Polydisperse	Hertz-Mindlin	Image analysis, thief sampling	Initial loading pattern Vessel rotational speed Agitator speed	Lacey index	For bidisperse, the best mixing was obtained when top–bottom initial loading was used For tri and polydisperse mixtures, mixing efficiency decreased by increasing the drum rotational velocity Well-mixed particles were not attained due to segregation (smaller particles were pushed around the drum wall and larger ones were collected in the center) for the bidisperse, tridisperse, and polydisperse particles The best mixing performances were achieved at 45 RPM and 55 RPM for bidisperse and polydisperse mixtures, respectively
Basinskas and Sakai, (2016b)	Batch mixer	Spherical Noncohesive Monodisperse	Linear spring-dashpot	PIV, visual observations	Fill level Rotational speed Secondary-axis speed and position	Lacey index Granular temperature Variances of the transportation velocity (VTV)	An increase in the fill level led to a decrease in granular temperature and consequently worsened the mixing performance The granular temperature was increased by increasing the rotational speed The VTV was independent of the fill level The mixing performance, granular temperature, and VTV did not depend on the secondary movement of the mixer

Table 8:

Summary of application of DEM to investigate mixing in the V-blenders.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Lemieux et al. (2007)	V-blender Bin blender	Spherical Noncohesive Monodisperse	Linear spring-dashpot	Thief sampling	Fill level Initial loading pattern Vessel rotational speed	RSD Particle mean velocity Granular temperature	The best performances of both mixers were achieved at the low fill level, high rotational speed, and top–bottom initial loading pattern
Tahvildarian et al. (2013)	V-blender	Spherical Noncohesive Monodisperse	Hertz-Mindlin	PEPT (Kuo et al. 2005)	Fill level Vessel rotational speed	Circulation intensity Particles axial dispersion coefficient	An increase in the vessel rotational speed led to enhancement of the circulation intensity The circulation intensity declined as the fill level was increased With an increase in the vessel rotational speed, the axial dispersion coefficient increased linearly, but it decreased when increasing the fill level
Alizadeh et al. (2014a)	Tetrapodal blender V-blender	Spherical Noncohesive Monodisperse Bidisperse	Zhou et al. (1999)		Fill level Initial loading pattern Vessel rotational speed	RSD The axial flux of granules Mean velocities Mixing mechanisms Segregation mechanisms	For monodisperse and bidisperse particles tetrapodal blender had a better mixing performance than the V-blender

Table 9:

Summary of application of DEM to investigate mixing in the rotary drums.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Chaudhuri et al. (2006)	Rotary drum	Spherical Cohesive Noncohesive Monodisperse Bidisperse	Walton and Braun (Walton and Braun 1986a, 1986b) + square-well potential	Image analysis	Particle cohesion Vessel rotational speed	Intensity of segregation	The validated DEM model predicted that increasing the cohesion led to an increase of the avalanche size and the dilation of the granular bed Enhanced mixing was observed for cohesive particles compared to the free-flowing particles (for a very mild cohesive case ( F cohesion / W p =0.1)) Increasing the vessel rotational speed resulted in better mixing due to the breaking up of the coherent particle bonds In the bidisperse case, better mixing was achieved by increasing the adhesion between different particles
Marigo et al. (2012)	Rotary drum Turbula mixer Hoop mixer	Spherical Noncohesive Monodisperse	Hertz-Mindlin	PEPT (Marigo et al. 2013)	Initial loading pattern Vessel rotational speed	Segregation index	Increasing the vessel rotational speed improved the mixing rate of the rotary drum and deteriorated the mixing performance of the turbula mixer In the hoop mixer, as the rotational speed increased the axial mixing increased, while the radial mixing slightly diminished
Alizadeh et al. (2014b)	Rotary drum	Spherical Noncohesive Polydisperse	Zhou et al. (1999)	RPT	Static friction coefficient Young’s modulus	Weak and strong sense mixing index Axial dispersion Velocity profiles	The static friction coefficient and Young’s modulus had significant impacts on the particle mixing dynamics
Alchikh-Sulaiman et al. (2016)	Rotary drum	Spherical Noncohesive Monodisperse Bidisperse Tridisperse Polydisperse	Hertz-Mindlin	Image analysis	Vessel rotational speed Particle size Initial loading pattern	Lacey index	The degree of mixing obtained from the polydisperse mixture was lower than that of monodisperse, due to segregation Final mixing indices were higher for polydisperse than tridisperse and bidisperse The best mixing performance for bidisperse and tridisperse was achieved from the TB initial loading pattern Segregation decreased as the degree of polydispersity increased (due to diffusive and convective particle motion)
Sebastian Escotet-Espinoza et al. (2018)	Rotary drum	Spherical Cohesive Bidisperse	Hertz-Mindlin + JKR	PEPT (Parker et al. 1997)	Particle surface energy Particle diameter Particle density	RSD Mixing time Axial dispersion coefficient	A better mixing rate was achieved by decreasing the particle surface energy and particle density, and increasing the particle diameter The axial dispersion coefficient was found to have an empirical relationship with particle properties
He et al. (2019)	Rotary drum	Spherical Nonspherical Noncohesive Bidisperse	Hertz-Mindlin		Particle shape Vessel rotational speed	Lacey index Mean velocity of particles	The ellipsoids particles remained in the center of the vessel while the spherical particles tended to accumulate close to the drum wall Increasing the vessel rotational speed decreased the segregation rate
Yazdani and Hashemabadi (2019)	Rotary drum	Spherical Noncohesive Cohesive Bidisperse	Hertz-Mindlin + SJKR		Vessel rotational speed Cohesion energy density (CED)	Intensity of segregation	Increasing the vessel rotational speed resulted in better mixing performance Segregation was observed in the case of mixing of cohesive and noncohesive particles For the cases when there was no interparticle cohesion between two different cohesive particles, the uniform mixture was not achieved Increasing CED for highly-cohesive particles (CED >80 kJ/m³) worsened the mixing performance (attributed to strong cohesive force between particles)
Ji et al. (2020)	Rotary drum	Spherical Nonspherical Noncohesive Monodisperse Bidisperse	Hertz-Mindlin	Image analysis (You and Zhao 2018)	Particle shape Vessel rotational speed	Lacey index	The mixing rate of nonspherical particles was higher than the mixing rate of spherical particles An increase in the vessel rotational speed led to an increase in the mixing rate
Pachón-Morales et al. (2020)	Rotary drum	Nonspherical Cohesive Monodisperse	Hertz-Mindlin + SJKR	Image analysis	Particle cohesion Particle shape	Centroid angle	Elongated particles showed a higher centroid angle when compared to spherical particles The centroid angle increased by increasing the particle cohesion
Hlosta et al. (2020)	Rotary drum	Spherical Nonspherical Noncohesive Monodisperse Bidisperse	Hertz-Mindlin	PIV	Shape, size, and density of particles Vessel fill level Vessel rotational speed Initial loading pattern	Segregation index	In general, worse mixing performance was obtained when mixing particles with different densities compared to cases when mixing particles with different shapes or sizes Increasing the vessel rotational speed led to an improvement of the mixing performance The optimum vessel fill level was different for spherical (40–50%) and sharp-edged particles (30–40%) The vessel fill level had the most significant influence on mixing homogeneity
He et al. (2020)	Rotary drum	Nonspherical Spherical Noncohesive Mono-disperse	Hertz-Mindlin	Image analysis (Ma and Zhao 2017)	Vessel rotational speed Aspect ratio of ellipsoids	Lacey index	For both spheres and ellipses, the mixing rate decreased substantially with increasing rotation speed In general, ellipsoids mixed faster than spheres in the cascade or rolling regime As the aspect ratio deviated from 1, particles rotated more violently, especially those against the drum wall Spheres had lower convective mixing and higher diffusive mixing than ellipsoids at 40 rpm
He et al. (2021)	Rotary drum	Mixture of spherical and nonspherical Noncohesive	Hertz-Mindlin	Image analysis (Ma and Zhao 2017)	Aspect ratio of ellipsoids	Lacey index	Axial segregation was observed when mixing particles with different shapes Spheres accumulated in the middle while ellipsoids collected at the periphery

Table 10:

Summary of application of DEM to investigate mixing in the bladed mixers.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Zhou et al. (2003)	Bladed mixer	Spherical Noncohesive Bidisperse	Zhou et al. (1999)	PEPT (Stewart et al. 2001b)	Impeller rotational speed Volume fraction Particle density and size	Lacey index	The small particles accumulated at the bottom of the mixer and the large particles gathered at the top of the mixer Decreasing the density or size differences between particles led to an enhancement in the mixing performance As the impeller rotational speed was increased, better mixing was achieved during the beginning stages of the mixing process. The impeller rotational speed, on the other hand, had no effect on the final mixing quality
Remy et al. (2009)	Bladed mixer	Spherical Noncohesive Monodisperse Bidisperse Polydisperse	Tsuji et al. (1992)	PIV	Blade orientation Blade configuration Fill level	RSD Granular temperature Particle diffusivities The particle bed pressure Shear stress Velocity profile	Mixing due to diffusion and granular temperature values decreased when using the acute blade orientation compared to when using obtuse blade orientation The ratio of time-averaged shear stress to bed pressure increased in a radial direction from the center of the mixer to the wall The best mixing performance was achieved when the fill level was to the top of the blades
Chandratilleke et al. (2009)	Bladed mixer	Spherical Noncohesive Monodisperse Bidisperse	Zhou et al. (1999)	PEPT (Stewart et al. 2001b)	Blade rake angle Blade clearance (the gap between blade and bottom of the vessel)	Lacey index Particle speed Interparticle forces and force network	When changing the rake angle to a range between 45° and 135°, the fastest mixing rate was achieved in the case of a 90° rake angle
Remy et al. (2010b)	Bladed mixer	Spherical Noncohesive Monodisperse Bidisperse	Tsuji et al. (1992)	PIV	Surface roughness (particles, wall, blade) Blade speed	RSD Granular temperature Particle diffusivities	Surface roughness and blade speed had a significant impact on mixing Velocity fluctuation amplitudes, bed dilation, granular temperature, and particle diffusivities increased with an increase in particle roughness Increasing the wall and particle roughness values had similar effects on mixing behavior
Chandratilleke et al. (2010)	Bladed mixer	Spherical Non-cohesive Monodisperse Bidisperse	Zhou et al. (1999)	PEPT (Stewart et al. 2001b)	Blade speed	Lacey index Flow fields Blade torque Interparticle forces Force network	An increase in the blade speed created a unidirectional flow close to the bed surface The mixing rate and interparticle forces (normal and tangential) increased by increasing the blade speed
Remy et al. (2011)	Bladed mixer	Spherical Noncohesive Monodisperse Bidisperse Polydisperse	Tsuji et al. (1992)	PIV	Polydispersity	Segregation patterns Mixing kinetics Normal and shear stress profile Granular temperature	The segregation caused by the sieving mechanism was minimized by increasing the degree of polydispersity By increasing the degree of polydispersity, lower stress values were reported The highest granular temperature values were found to be near the vessel wall and at the top of the particle bed surface in both the experiments and simulations Both particle diffusivity and convection increased by increasing the polydispersity of the particles in the mixer
Nakamura et al. (2013)	High shear mixer	Spherical Noncohesive Monodisperse	Hertz-Mindlin	PEPT (Stewart et al. 2001b)	Mixer’s size	Dimensionless shear rate Particle’s velocity Particle’s collision energy	A constant impeller tip speed can guarantee the similarity of internal particle shear flow magnitude and collision energy of particles at different vessel sizes With increasing the vessel size, particle collision energy per unit time decreased significantly
Chandratilleke et al. (2014)	Bladed mixer	Spherical Cohesive Monodisperse	Hertz-Mindlin + van der Waals	PEPT (Stewart et al. 2001b)	Interparticle cohesion Particle-wall cohesion Rake angle	Lacey index	For the highly cohesive material, the mixing performance was ineffective as the bed of particles was lifted, leading to limited interaction between particles and blade Decreasing the particle-wall cohesion led to an enhancement of the mixing performance A blade with a 90° rake angle yielded the best mixing rate
Halidan et al. (2014)	Bladed mixer	Spherical Noncohesive Bidisperse	Zhou et al. (1999)	PEPT(based on Stewart et al. (2001a))	Particle size ratio (r_s) Density ratio (r_d) Volume fraction (x_l)	PMSI Mixing mechanisms Velocity field Force analysis Mixing trends	At a given volume fraction, there was an optimum combination of particle density and particle size ratios which led to the best mixing performance An increase in r_s, r_d, and x_l values increased the mixing index value to a maximum point and after that an increase in r_s, r_d, and x_l values decreased the mixing index value
Boonkanokwong et al. (2016)	Bladed mixer	Spherical Monodisperse Noncohesive	Tsuji et al. (1992)	PEPT (Stewart et al. 2001b)	Number of impeller blades The ratio of mixer diameter to particle diameter	RSD Lacey index Granular temperature Particle diffusivities Velocity fields Bulk density, void fraction, and solid fraction Normal contact force network	Granular temperature and particle diffusivity values were higher when using two and three blade agitators compared to when using one and four blade agitators Mixing performance was better for two or three blade agitators than one or four blade agitators A higher number of impeller blades increased the blade-particle forces, leading to an increase in bulk density
Boonkanokwong et al. (2018)	Bladed mixer	Spherical Monodisperse Noncohesive	Tsuji et al. (1992)	PEPT (Stewart et al. 2001b)	Particle properties Impeller blade design Fill level	RSD Lacey index Impeller torque Power consumption	The position of the blade, the fill level, friction coefficient, and size of the particles had a significant impact on the torque and power consumption
Govender et al. (2018)	Four-bladed mixer	Spherical Nonspherical Noncohesive Monodisperse Polydisperse	Hertz-Mindlin		Particle shape	Lacey index RSD Interparticle forces	The mixing system comprising spherical particles reached the final mixing index value (i.e. 0.7) faster compared to other mixing systems including nonspherical particles The particle shape should be accurately taken into account in DEM when simulating the mixing process of non-spherical particles
Herman et al. (2021)	Bladed mixer	Spherical Noncohesive Monodisperse	Zhou et al. (1999)	PEPT (Stewart et al. 2001b)	Mixer’s scale-up ratio	Lacey index Force and torque on blades Particle’s velocity	In constant Froude numbers, scaling-up did not affect the quality of mixing. It however affected the mixing rates A longer mixing time was required to achieve a steady-state condition for a large mixing system mixing compared to a small mixing system As the mixer size increased, the average particle velocity also increased Based on a scaling-up ratio and Froude number (or speed of rotation), a series of correlations were derived to predict mixing rate, particle velocity, total forces, average blade torque, and contact forces

Table 11:

Summary of application of DEM to investigate mixing in the ribbon blenders.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Basinskas and Sakai, (2016a)	Ribbon blender	Spherical Noncohesive Monodisperse	Linear spring-dashpot	PIV, visual observations	Fill level Blade speed Initial loading pattern	Lacey index	An increase in the fill level and blade speed led to an enhancement in mixing performance The mixing quality obtained from the front-back initial loading pattern was worse than the mixing quality obtained from the side–side and top–bottom initial loading patterns
Halidan et al. (2018)	Ribbon blender	Spherical Noncohesive Cohesive Bidisperse	Zhou et al. (1999)	Core sampling (based on Halidan et al. (2016))	Impeller rotational speed Fill level Cohesiveness Blade design (two-bladed and four-bladed)	Lacey index Velocity and flow pattern Contact forces (particle–particle and particle–wall)	For both cohesive and noncohesive particles, the optimal impeller rotational speed was 100 RPM For noncohesive materials, contact forces (particle–particle and particle–wall) increased by increasing the impeller speed For both two-bladed and four-bladed mixers, the mixing quality was deteriorated by an increase in the fill level and cohesiveness A two-bladed ribbon mixer was not a suitable mixer to blend cohesive materials
Chandratilleke et al. (2018)	Ribbon blender	Spherical Noncohesive Cohesive Monodisperse	Zhou et al. (1999) + van der Waals	PEPT (Stewart et al. 2001b)	Blade supporting spoke number Particle cohesion Vessel fill level	Lacey index Contact forces Diffusion coefficient Axial velocity variation	Increasing the number of spokes increased the mixing rate for noncohesive particles For cohesive materials, the optimum number of spokes depended on the fill level. Particle dispersion caused by axial velocity variations had a dominant effect on the mixing performance compared to axial diffusion At the low fill levels, axial diffusion was the dominant mixing mechanism, but, at the high fill levels, the dominant mixing mechanism was particle dispersion
Gao et al. (2019)	Ribbon blender	Spherical Noncohesive Monodisperse Bidisperse	Hertz-Mindlin	–	Initial loading pattern Particle size Impeller rotational speed Inner blades Number of vessels	Lacey index Relative velocity components between particles after collision	The mixing efficiency was significantly influenced by the initial loading pattern and impeller RPM, while particle size and inner blades had negligible effects Except for the side–side initial loading pattern, the ribbon blender with a double U-shaped vessel demonstrated better mixing efficiency than the single vessel
Tsugeno et al. (2021)	Ribbon blender	Spherical Noncohesive Monodisperse	Linear spring-dashpot		Impeller rotational speed Impeller width Pitch of the impeller	Lacey index	A novel approach was proposed to identify the dominant mixing mechanism in the ribbon blender The impeller rotational speed and impeller pitch had a negligible influence on mixing performance Increasing the impeller width improved the mixing performance

Table 12:

Summary of application of DEM to investigate mixing in the ploughshare and paddle blenders.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Hassanpour et al. (2011)	Double paddle blender	Spherical Noncohesive Monodisperse Bidisperse Polydisperse	Hertz-Mindlin	PEPT		Time-averaged velocity	DEM could replicate the vertical and horizontal mixing patterns observed experimentally The time-averaged velocity values obtained from DEM and experiments were in good agreement
Alian et al. (2015a)	Ploughshare mixer	Spherical Non–cohesive Monodisperse	Hertz-Mindlin	PEPT (based on Laurent and Cleary (2012))	Initial loading pattern Particle size Fill level Blade rotational speed	Lacey index	Initially, a top–bottom loading pattern provided better mixing than a side–side loading pattern. The mixing indices however approached the same value for both patterns after a few seconds regardless of the initial loading pattern Mixing time reduced with increased impeller speed Fill level had a significant impact on mixing quality (lower fill level resulted in higher mixing quality) Better mixing was obtained for larger particles due to higher average velocity
Pantaleev et al. (2017)	Paddle blade blender	Dry and wet powder Cohesive Noncohesive Monodisperse	Visco-elasto-plastic adhesive (Thakur et al. 2014)	FT4	Moisture content Blade tip speed	Bulk density Total energy	The DEM model and experimental results were in good agreement qualitatively for both dry and wet powders Despite the close agreement between the DEM and FT4 measurements at the calibration stage, the mixing system DEM model could predict the mixing rate measured experimentally only qualitatively
Yaraghi et al. (2018)	Single paddle blender	Spherical Noncohesive Monodisperse	Hertz-Mindlin	Thief sampling	Fill level Initial loading pattern Impeller rotational speed	RSD Granular temperature Particle diffusivities	Generally, increasing the impeller rotational speed (10–40 RPM) resulted in a better degree of mixing regardless of selected initial loading pattern The mixing efficiency was improved by increasing the fill level from 40% to 60%. At 10 RPM, however, the mixing efficiency was independent of fill level The initial loading pattern did not have a significant effect on mixing quality
Ebrahimi et al. (2018)	Single paddle blender	Spherical Noncohesive Monodisperse	Hertz-Mindlin	Thief sampling	Impeller configuration	RSD Granular temperature Particle diffusivities particle–particle and particle-geometry normal contact forces	The mixing efficiency and granular behaviour was significantly influenced by impeller configuration Impellers angled at 30° and 45° provided better mixing than 0° and 60° Initially, the mixing efficiency of the 60° angled paddles was the worst Regardless of the impeller configuration, diffusion was the dominant mixing mechanism Angled paddles imposed more force to particles than a paddle with 0° angle
Ebrahimi et al. (2020)	Single paddle blender	Spherical Bidisperse Noncohesive	Hertz-Mindlin	Thief sampling	Initial loading pattern Fill level Impeller rotational speed Particle number ratio	RSD Particle diffusivities	The dominant mixing mechanism was diffusion The initial loading pattern did not have a noticeable impact on the mixing efficiency The particle number ratio had the greatest impact on the mixing performance Reducing the particle number ratio resulted in a deterioration of the mixing performance
Jadidi et al. (2022)	Double paddle blender	Spherical Monodisperse	Hertz-Mindlin	Dynamic angle of repose	Initial loading pattern Fill level Impeller rotational speed	RSD Granular temperature Particle diffusivities	The dominant mixing mechanism was diffusion Based on the operating parameters, a statistical model was developed for RSD Mixing performance was affected by impeller speed and loading pattern

Table 13:

Summary of application of DEM to investigate mixing in the screw blenders.

Authors	Mixer type	Particle data	DEM contact model	Experiments for validation	Variation of operating and design parameters	Parameters calculated to analyze mixing quality	Findings/results
Sakai et al. (2015)	Twin-screw blender	Spherical Noncohesive Monodisperse	Linear spring-dashpot	Image analysis	Fill level Impeller rotational speed	Lacey index Velocity distribution of particles	The mixing performance was improved by increasing the fill level The influence of the impeller rotational speed on the mixing efficiency was negligible compared to the influence of fill level on the mixing efficiency
Qi et al. (2017)	Double screw blender	Spherical Nonspherical Bidisperse Polydisperse	Hertz-Mindlin	Optical visualization (based on Kretz et al. (2016) and Kingston et al. (2015))	Screw rotational speed Screw pitch length Fill level	Lacey index Mixing time Visual observation	The mixing pattern of the binary system was similar to the mixing pattern of a polydisperse system The screw rotational speed had a minimal effect on the mixing performance. However, increasing the pitch length and decreasing the fill level led to a decline in the mixing quality
Golshan et al. (2017)	Screw conical blender (Nauta blender)	Spherical Noncohesive Monodisperse	Hertz-Mindlin	PEPT (based on Schutyser et al. (2003))	Sweeping speed Rotational speed Impeller diameter	Lacey index Granular temperature Velocity profile of particles	At a constant rotational speed, the mixing performance was worsened by increasing the sweeping speed The mixing performance was enhanced by increasing the rotational speed and the impeller diameter
Cai et al. (2019)	Double-screw conical blender	Spherical Noncohesive Bidisperse	Linear spring-dashpot	Image analysis	Sweeping speed Rotational speed Particle diameter ratio	Lacey index Device power consumption Collision energy distribution	The mixing quality decreased by increasing the sweeping speed An increase in rotational speed led to an enhancement in mixing performance The larger the particle diameter ratio chosen, the worse the mixing quality was observed (due to the size segregation)

If we have a closer look at the studies summarized in Tables 7 –13 and quantify them (Figure 4) based on inclusion or exclusion of 1- non-spherical particles, 2- cohesive powders and 3- mono, bi, and poly-dispersed particles in the DEM simulation of mixing systems. It is revealed that particle shape has been considered in only 20% of DEM studies when analyzing a mixing system. Mixing of cohesive powders in the literature has only been the subject of 18% of studies. Mono-disperse particles have been used in the majority (46%) of the papers while two or more particle types in industrial applications are usually mixed. These measures reflect that the focus of DEM studies of mixing systems in academia may not be aligned well with what industrial users expect to learn from DEM simulation of mixing systems. We believe including particles’ shape, cohesive powders, and poly-dispersity in DEM simulations of mixing systems can help to close this gap.

Figure 4:

Quantification of the studies listed in Tables 7 –13, based on inclusion or exclusion of 1- nonspherical particles (a), 2- cohesive powders (b) and 3- mono, bi, and poly-dispersed particles (c).

5 Possible challenges and solutions in using DEM to simulate mixing systems

Based on what has been presented thus far, one can conclude that DEM is a very common approach to study mixing processes. There are, however, some challenges associated with the use of this method, not only when it is applied for mixing system analyses, but in general for other unit operations comprised of particles and powders. These challenges are presented in detail in this section, along with the available solutions that have been proposed by researchers.

5.1 Challenges and difficulties in implementing DEM for mixing systems

5.1.1 Computational time

The DEM method in general is computationally intensive as every individual contact between particles and particle-geometry is resolved throughout the simulation time (Norouzi et al. 2016). This, therefore, limits the number of particles that can be simulated in a DEM simulation. Even using high-end GPU cards, the number of particles in most simulations does not exceed several million (based on our literature review, currently, the majority of DEM simulations are still performed using 16–24 CPU cores). This consequently limits the size of the unit operation which can be analyzed (Blais et al. 2019). The time step used in DEM simulations must also be made small enough that the surface waves generated due to the contact between two particles do not propagate more than the distance between the center of particles in contact within a few time steps (Blais et al. 2019). Commonly the time step which guarantees the stability of the explicit time integration scheme of the DEM method is chosen as a fraction (usually 20–30%) of the Rayleigh time (Blais et al. 2019; Norouzi et al. 2016). The result is a very small timestep, usually no more than a few microseconds. DEM’s small time step makes it a very computationally demanding method. Therefore, despite its computational modeling potentials, the DEM method is only able to simulate a maximum of a few minutes of the mixing process in a reasonable time frame, which in some cases may not be enough to thoroughly analyze the mixing process. The challenge related to the long computational time becomes even more burdensome when simulating fine submillimeter powders (as a smaller time step needs to be used). To avoid this challenge, the majority of simulation cases in literature (summarized in Tables 7 –13) have focused on the simulation of laboratory or pilot scale mixers comprising a maximum of several hundred thousand particles with a diameter in the millimeter range. In other words, Tables 7 –13 imply that the use of the DEM technique in order to investigate large scale industrial mixing systems including a large number of free-flowing particles or cohesive powders (in the range of several hundred million or billions) is not currently possible with the current computational hardware available.

Simulating the mixing process of nonspherical particles in the mixing systems, either by the multisphere approach (Favier et al. 1999), superquadrics (Lillie and Wriggers 2006; Williams and Pentland 1992) or by the polyhedral approach (Govender et al. 2020, 2018; Hopkins 2014; Nassauer et al. 2012), can also increase the computational time considerably (Lu et al. 2015). When simulating nonspherical particles, the contact detection step can be very challenging and time-consuming. For example, 80 percent of the total computational time can be easily taken up for detecting contacts between particles when simulating three-dimensional polyhedron particles. Therefore, the application of DEM to simulate nonspherical particles heavily depends on the approach of contact detection employed (Kodam et al. 2010; Nezami et al. 2004). Therefore, although most of the solid particles in nature and industry do not have a perfect sphere shape, to avoid long simulation time, the particle shape in the majority of studies summarized in Tables 7 –13 has been approximated with a sphere shape.

5.1.2 Calibration procedure

The calibration process is the key to achieving realistic simulation results (Coetzee 2017; Hoshishima et al. 2021). The calibration process or selection of input parameters (for contact models and particle properties) for a DEM simulation is however very challenging and one of the limiting aspects of this approach (Yan et al. 2015). In general, obtaining the DEM input parameters for a specific particle or powder through direct microscopic measurements is very difficult and costly, if not impossible (Benvenuti et al. 2016). Commonly, the calibration process is performed by trial and error. In the calibration process, the DEM interaction parameters such as coefficient of static and rolling frictions, coefficient of restitution, surface energy, plasticity ratio, and adhesion stiffness as well as particle properties such as shear modulus are systematically varied and their influence on the bulk response (i.e. macroscopic response obtained from standard characterization tests such as angle of repose or shear cell tests) is monitored. A set of DEM input parameters which provides a macroscopic response in a reasonable agreement with bulk experiments is selected as calibrated input parameters. There are however several particle properties and contact model input parameters which need to be selected/calibrated, and this means that generally DEM calibration is a time-consuming and resource-intensive process. In some cases, the calibration process takes more time and effort than simulating the process itself. The calibration process becomes even more challenging when simulating cohesive particles as there are more input parameters to be set by users for contact models and particle properties (Pantaleev et al. 2017; Safranyik et al. 2017). This issue, along with the extremely long computational time, can explain why there is a limited number of studies available in literature on the simulation of mixers including cohesive material.

5.1.3 A time-consuming approach for mixing systems optimization

As mentioned in Section 4, there is a large number of design and operating parameters that can have a decisive effect on the performance of a mixing system. To find the optimal design and operating conditions, one needs to carry out a large number of DEM simulations and analyze the combinations of all design and operating parameters on mixing performance. As mentioned before, DEM simulations can be very time-intensive and therefore running several simulations to find the optimum design and operating parameters can be a slow and time-consuming task. Consequently, this can hinder the application of the DEM technique as a complete optimization tool.

5.2 Available/potential solutions

From what has been summarized above, one can conclude that the major obstacle in using the DEM technique in studying mixing systems is mainly related to the high computational time involved and calibration process. To tackle these challenges, some solutions have been employed in the research literature.

5.2.1 GPU and high-performance computing (HPC)

To decrease the computational time, some parallelization approaches have been developed allowing execution of the DEM codes on clusters on a large number of CPUs (Berger et al. 2015; Maknickas et al. 2006). Shigeto and Sakai (2011) proposed multithread parallel computation of DEM, which effectively used the available memory and accelerated the DEM computation considerably. Berger et al. (2015) developed an MPI/OpenMP hybrid parallelization for LIGGGHTS and demonstrated a reduction in the DEM computational time for some large-scale simulations. OpenMP and hybrid MPI/OpenMP parallelization of MFIX DEM solver also have been reported in the literature (Amritkar et al. 2014; Liu et al. 2014). The use of modern computer architectures such as graphic processing units (GPU) has also recently become prevalent for scientific computations that can significantly reduce the computational time (Kurowski et al. 2011). Compared to CPUs which are composed of a few cores and can handle a few software threads at a time, GPUs are composed of hundreds of cores which can execute different sequences of instructions simultaneously (Washizawa and Nakahara 2013). As a result, GPU can process far more calculations per second than a typical CPU, ultimately reducing the computational time required for a DEM simulation. Researchers have demonstrated the capabilities of GPU to simulate particulate systems and large scale mixing systems in literature (Gan et al. 2016; Govender et al. 2018; Radeke et al. 2010). Gan et al. (2016) reported a speed up of 40–75 times in their DEM simulations when using a single GPU compared to when using a single CPU. They mentioned that the amount of speed up depended on the type of GPU accelerator card and DEM algorithms applied. Gan et al. (2016) also demonstrated that using 32 GPUs with a message passing interface (MPI) could reduce the computational time 18 times compared to when they used a single GPU. It is expected to see a high volume of research studies in the field of GPU-based DEM in near future. Moreover, the use of multi-GPU in DEM is an emerging tool that can potentially further reduce the computational time.

5.2.2 Scaling relationships and coarse-graining

The use of scaling relationships (scaling laws) and coarse-graining can also be considered in order to shorten the computational time (Ebrahimi et al. 2017; Nakamura et al. 2020; Sakai 2016). In the scaling relationship approach, some relevant dimensionless numbers are kept constant for both original and scaled-up systems (dimensionless analysis). In the scale-up simulations, both particles and mixing systems are scaled. The computational time required to simulate the scaled-up system is shorter than the computational time required to simulate the original system as larger time step can be selected in the scaled-up system. The similarity of the dynamic behavior of the scaled system and the original system is guaranteed through the dimensionless analysis (Ding et al. 2001). In the coarse-grain DEM approach, the coarse-grain particle has a diameter l times larger than the original particle diameter. This means l³ original particles are represented by one coarse-grain particle. The equations of motion and contact models are adjusted for coarse-grain particles in a way that the dynamic behavior of the coarse-grained particles and the average behaviour of original particle assembly is identical (Sakai 2016). Simulating a lower number of particles with larger diameter in a coarse-grained DEM model reduces the computational time considerably compared to the original model. The successful application of coarse-grained DEM in simulating fluidized bed and pneumatic conveying has been reported in literature and can also be employed for mixing systems (Sakai et al. 2012; Sakai and Koshizuka 2009).

5.2.3 Data-driven DEM and time-extrapolation approach

Extrapolation strategies have been proposed in literature in order to reduce the computational intensity of DEM simulation of mixing systems (Bednarek et al. 2019; Doucet et al. 2008). The main idea of the extrapolation approaches is to perform DEM simulation for a short period of time (in the range of a few seconds), identify the system characteristics and dynamic behaviour, and then extrapolate data for longer time frames. Moreover, recently the use of machine learning (ML) in various scientific disciplines has been explored (He and Tafti 2019; Zhao et al. 2020). The ML technique can be used in combination with the DEM technique in order to:

Decrease the number of simulation cases required to identify the optimum design and operating conditions and,
Forecast the behavior of the mixing system for a time frame of several minutes or hours.

ML codes can be trained through the use of data generated by DEM simulations and then can predict the key relevant quantities of interest (e.g. degree of homogeneity/uniformity) for other simulation cases with different operating conditions and designs and for a longer time frame of several minutes or hours (Kumar et al. 2020). Although the use of the hybrid DEM-ML approach sounds promising it should be noted that ML can only perform well in the design space used to train the code and a few seconds or minutes of DEM simulation may not provide enough data for training the ML code when simulating nonlinear and complex systems. The hybrid ML-DEM approach seems to have great potential and it is expected to ameliorate the issues associated with the high computational demand of the DEM technique. More research and studies however in this field need to be conducted to better evaluate the capabilities of the hybrid DEM-ML method.

5.2.4 Optimization based calibration

As previously stated, calibration of DEM input parameters is laborious and resource-intensive. As the application of DEM in analyzing particulate systems and mixing equipment is rapidly increasing there is a clear need for identifying the DEM input parameters through an efficient calibration method. One common approach to improve the calibration process is the use of design of experiment approaches (Johnstone 2010; Pantaleev et al. 2017; Rong et al. 2020). The disadvantage of this method is that a large number of DEM simulations still need to be performed to determine the optimal DEM input parameters (Richter et al. 2020; Zhu et al. 2022). Another approach to ease the calibration process that has gained great attention in scientific communities is the use of optimization methods (Orefice and Khinast 2020). Various optimization methods such as generic algorithms (Do et al. 2018), response surface models (Yoon 2007), Kriging (Rackl and Hanley 2017), artificial neural network (Benvenuti et al. 2016), and generalized surrogate modeling-based calibration (Richter et al. 2020) have been reported in literature to efficiently calibrate the DEM input parameters. We believe that the optimization based calibration process has demonstrated great potential to reduce the burdensome calibration effort. Addressing the challenge associated with the calibration process will consequently increase the use of the DEM technique in the simulation of particulate systems such as mixing.

6 Conclusions and recommended future studies

In the current study, initially different types of mixing systems and their applications, advantages, and disadvantages were reviewed. Then the mixing and segregation mechanisms were discussed. Both invasive and noninvasive experimental tools used to characterize the mixing processes were summarized. The DEM method as the most common approach to simulate solid mixers was also presented. Also, demonstrated was how DEM has been used in literature in order to investigate the effect of design and operating conditions on the mixing quality.

Based on the detailed review presented in this report, it was found that the majority of studies have used DEM to analyze small/laboratory scale mixing systems including free-flowing mono or bi-disperse spherical particles. This indicates that the way the DEM technique is used in studying mixing systems in academia and in a research environment differs greatly from practical applications and does not reflect industries’ needs. We, therefore, recommend conducting the following studies in order to reduce this gap and further investigate the application of DEM in analyzing mixing systems.

The mixing performance of mixing systems including nonspherical particles with definite, well-defined shapes is scarce in literature (Govender et al. 2018). The extremely long computational time can be the reason for this shortcoming. However, with access to proper computational facilities, it would be possible to run DEM simulations of mixing systems including tablets, capsule shape, and needle shape particles and compare those simulations with mixing systems including spherical particles. This can likely help to analyze the relationships between the mixing processes of spherical particles versus the mixing process of nonspherical particles. As the majority of computational time (almost 80%) is taken up on contact detection and the calculation of the contact region geometric details (Zhong et al. 2016), therefore developing and employing an efficient contact detection algorithm can also help to decrease the computational time when simulating particles with irregular shapes (Nezami et al. 2004). Available contact detection algorithms based on particle shape representation have recently been reviewed by Zhong et al. (2016) and Lu et al. (2015).
An understanding of the mixing process of fine cohesive powders is of great importance for the pharmaceutical and food industries. However, DEM simulations of mixing systems including cohesive fine powders have rarely been reported in literature. We believe that using the particle scaling approach suggested by Thakur et al. (2016) and Pantaleev et al. (2017), employing the optimization based calibration methods (mentioned in Section 5.2.4) and by benefiting from GPU capabilities, it should now be possible to simulate mixing systems including fine cohesive powders within a reasonable computational time.
Scale-up of solid mixers has been a longstanding issue and is of great importance in different industries such as the pharmaceutical and food industries (Herman et al. 2021). In these industries it is imperative that the final solid mixture quality obtained from a large scale mixer in the production line is identical to the final solid mixture quality yielded from a laboratory scale mixer at the product development stage. DEM can provide detailed microscale information which helps to analyze the dynamic and kinematic behaviour of particles during scale-up. The studies conducted by Herman et al. (2021) and Nakamura et al. (2013) have shown the potential capabilities of DEM to investigate the scale-up procedure of mixing systems. Using GPU and multi-GPU, one now should be able to simulate various mixing system sizes (i.e. from laboratory scale to pilot or even industrial scale including a high number of particles) in order to establish the scale-up methodologies for solid mixers.
In most industrial applications several particles with distinct properties (size, density, and shape) are blended to a specific degree of homogeneity. However, as the summary of the literature review in Tables 7 –13 demonstrated, the majority of studies have only focused on either mono-disperse or bi-disperse particles with the same density and shape. This again demonstrates that the research studies in the DEM field are not aligned well with the DEM simulation needs of industries. It is therefore recommended to perform DEM simulations to analyze the effect of polydispersity on mixing quality. It is also proposed to utilize DEM to simulate the mixing process of particles with different densities and different shapes. These simulations can definitely help industries overcome their hurdles with operating mixing systems.

Nomenclature

m _i: Mass of particle i (kg)
v → i: Velocity of particle i ( m s )
F → c , n i j: Normal contact forces (N)
F → c , t i j: Tangential contact forces (N)
F → nc i k: Noncontact forces between particle i and k (N)
F → f − p i: Fluid–particle interaction forces acting on particle i (N)
F → ext i: All external forces (both uniform and nonuniform) acting on particle i (N)
I _i: Moment of inertia of the particle i (kg m²)
ω → i: Angular velocity of particle i ( rad s )
M → i j: Rotational torque of particle i (N m)
M → r i j: Rolling resistance torque of particle i (N m)
D _ij: Diffusion coefficient for the diffusion occuring in the i direction due to a composition gradient in the j direction
Δx_i: Particle displacement in the i direction (m)
Δ x ‾ i: Mean displacement of all particles in the i direction (m)
Pe_ij: Peclet number for the ij direction
U _i: Average velocity of the particles in the i direction ( m s )
L _c: Characteristic length of the mixer (m)
T: Granular temperature ( m 2 s 2 )
u ^′: Fluctuation velocity of each particle ( m s )
γ ^∗: Normalized shear rate distribution in the x–y plane
U ^∗: Mean velocity of the particles in the x direction normalized with the impeller velocity
V ^*: Mean velocity of the particles in the y direction normalized with the impeller velocity
x*: Coordinate in the x direction normalized with the radius of the mixer
y*: Coordinate in the y direction normalized with the radius of the mixer
d _p: Particle diameter (m)
F _i: Contact forces acting on particles in i direction (N)
k _j: Unit vector in the j direction
τ _ij: Collisional stresses in a control volume with the size of V_c ( N m 2 )
LI: Lacey index
I _s: Intensity of segregation
RSD: Relative standard deviation
PSMI: Particle scale mixing index
S: Siria mixing index
SI: Segregation index
GPSI: Graphic segregation index
SMI: Subdomain-based mixing index

Corresponding author: Farhad Ein-Mozaffari, Department of Chemical Engineering, Ryerson University, 350 Victoria Street, Toronto M5B 2K3, Canada, E-mail: fmozaffa@ryerson.ca

Funding source: Natural Sciences and Engineering Research Council of Canada (NSERC)

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-06-06

Revised: 2022-02-21

Accepted: 2022-02-22

Published Online: 2022-03-28

Published in Print: 2023-07-26

A comprehensive review of the application of DEM in the investigation of batch solid mixers

Abstract

1 Introduction

1.1 Tumbling mixers

1.2 Convective mixers

2 Mixing mechanisms and segregation

2.1 Mixing mechanisms

2.2 Segregation mechanisms

3 Characterization of the solid–solid mixing

3.1 Experimental methods

3.2 Numerical method

3.2.1 Fundamentals of discrete element method (DEM)

3.2.2 Assessment of the mixing mechanisms using DEM

3.2.3 Assessment of mixing quality

4 Effects of design parameters and operating conditions on the mixing quality

5 Possible challenges and solutions in using DEM to simulate mixing systems

5.1 Challenges and difficulties in implementing DEM for mixing systems

5.1.1 Computational time

5.1.2 Calibration procedure

5.1.3 A time-consuming approach for mixing systems optimization

5.2 Available/potential solutions

5.2.1 GPU and high-performance computing (HPC)

5.2.2 Scaling relationships and coarse-graining

5.2.3 Data-driven DEM and time-extrapolation approach

5.2.4 Optimization based calibration

6 Conclusions and recommended future studies

Nomenclature

References

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