A Comprehensive Review of Discrete Element Method Studies of Granular Flow in Static Mixers

Abstract

The Discrete Element Method (DEM) has become a cornerstone for analysing granular flow and mixing phenomena in static mixers. This review provides a comprehensive synthesis that distinguishes it from previous studies by: (i) covering a broad range of static mixer geometries, including Kenics, SMX, and Sulzer designs; (ii) integrating experimental validation methods, such as particle tracking, high-speed imaging, Particle Image Velocimetry (PIV), and X-ray tomography, to assess DEM predictions; and (iii) systematically analyzing computational strategies, including advanced contact models, hybrid DEM-CFD/FEM frameworks, machine learning surrogates, and GPU-accelerated simulations. Recent advances in contact mechanics—such as improved cohesion, rolling resistance, and nonspherical particle modelling—have enhanced simulation realism, while adaptive time-stepping and coarse-graining improve computational efficiency. DEM studies have revealed several non-obvious relationships between mixer geometry and particle dynamics. Variations in blade pitch, helix angle, and element arrangement significantly affect local velocity fields, mixing uniformity, and energy dissipation. Alternating left–right element orientations promote cross-sectional particle exchange and reduce stagnant regions, whereas higher pitch angles enhance axial transport but can weaken radial mixing. Particle–wall friction and surface roughness strongly govern shear layer formation and segregation intensity, demonstrating the need for geometry-specific optimization. Comparative analyses elucidate how particle–wall interactions and channel structure influence segregation, residence time, and energy dissipation. The review also identifies current limitations, highlights validation and scale-up challenges, and outlines key directions for developing faster, more physically grounded DEM models, providing practical guidance for industrial mixer design and optimization.

1. Introduction

Granular materials are widely used in industries such as pharmaceuticals, food processing, mining, and chemical engineering [1], where efficient mixing is crucial for product uniformity and process performance [2]. Static mixers are increasingly studied because they lack moving parts, require little energy, and enable continuous mixing in confined geometries [3]. However, granular flow in static mixers is complex due to the non-linear, non-cohesive, and segregating nature of particles [4]. Experimental studies provide insights into flow regimes, particle distribution, and mixing quality across different geometries and conditions [5], but they are limited in spatial and temporal resolution. The Discrete Element Method (DEM) offers a powerful alternative [6], enabling detailed analysis of particle interactions, flow dynamics, and mixing mechanisms [7]. Integrating experimental data with DEM simulations has improved static mixer design and optimized processing conditions [8].

While many types of mixers exist, this review focuses on static mixers because they are widely employed across the chemical, pharmaceutical, and food industries for both batch and continuous processes. Their simple construction, low energy consumption, and well-defined flow patterns make them ideal candidates for DEM analysis, allowing detailed evaluation of particle-level interactions, residence times, and segregation phenomena. Concentrating on static mixers also enables the synthesis of practical guidance for industrial design, optimization, and scale-up. Static mixers are devices which can be used in pharmaceutical industry, agriculture, chemical industry, food production, and civil construction for processing granular materials [9]. Unlike dynamic mixers with mechanical agitation, they lack moving parts and rely on engineered geometries within a fixed conduit to induce mixing through controlled flow redirection and stratification [10]. This design reduces maintenance, lowers energy use, and improves reliability, making static mixers suitable for continuous processes and sensitive materials [11].

To clarify the discussion of flow and mixing phenomena in static mixers,

Table 1. Terminology note.

Term	Definition	Typical Cause/Context
Dead zones	Regions within the mixer where particle motion is minimal or stagnant, leading to poor mixing efficiency.	Low shear regions, poor geometry design, or excessive friction near walls.
Channelling	Preferential flow of particles along specific low-resistance paths, bypassing other regions of the mixer.	Uneven flow distribution, high feed rates, or geometry-induced flow asymmetry.
Clogging (arching)	Formation of stable particle bridges or arches that obstruct flow through the mixer channels.	High cohesion, irregular particle shape, or narrow flow passages.
Segregation	Spatial separation of particles differing in size, density, or shape, reducing mixture homogeneity.	Differences in particle properties, vibration, or gravitational settling during flow.

summarizes key terms commonly used to describe characteristic flow behaviours and challenges, providing a consistent basis for interpreting experimental and numerical results.

In granular systems, static mixers improve particle distribution, reduce segregation, and enhance downstream operations such as coating, dosing, and compaction [12]. Granular flow is complex, influenced by particle shape, size distribution, density, friction, and interparticle forces [13], which can cause channelling, dead zones, and radial segregation, limiting mixing efficiency [10]. Understanding and optimizing this flow is essential for process efficiency and product quality [14]. Experimental and numerical studies, particularly using DEM, allow analysis of particle-scale dynamics and the influence of mixer design and operating conditions, supporting more efficient static mixing technologies [8].

Granular materials exhibit non-Newtonian behaviours such as jamming, arching, segregation, and non-uniform velocity profiles [15], highly sensitive to particle properties like size, shape, cohesion, density, and surface roughness, complicating predictions of mixing performance [16].

Achieving efficient mixing while minimizing segregation is a key challenge [17]. Poorly designed static mixer geometries can cause flow channelling, dead zones, or stratification, reducing efficiency and product consistency [10]. Granular flow lacks fluid-like diffusion, so homogeneity requires mixer designs that promote repeated particle reorientation without excessive shear or attrition [18]. DEM studies of static mixers provide actionable insights that directly address industrial challenges. For example, by predicting segregation patterns and residence time distributions, DEM supports scale-up from laboratory to production scale, ensures uniform product quality, and informs design optimization of mixer geometry and operating conditions. These capabilities are particularly valuable in the pharmaceutical industry, where cohesive powders must be blended uniformly, in the food industry, for consistent ingredient mixing, and in the chemical industry, for optimizing energy use and minimizing process inefficiencies.

Previous reviews on DEM applications in mixing systems often focus on individual elements such as contact models, algorithmic developments, or specific mixer geometries. However, they rarely provide a systematic synthesis that integrates experimental validation, computational strategies, and practical implications for industrial-scale applications. This review addresses these gaps by combining insights from multiple static mixer geometries, validated simulation approaches, and computational innovations, offering a holistic perspective on challenges like segregation, channelling, and dead zones, and providing guidance for future research and industrial design.

Scalability is another difficulty, as lab- or pilot-scale designs may not translate directly to industrial systems due to scale-dependent flow regimes and boundary effects [10], and the absence of standardized design criteria limits systematic optimization [19]. Experimental methods often struggle to visualize particle movement, while analytical models cannot fully capture discrete interactions [20]. DEM provides a powerful computational approach for simulating particle dynamics, capturing individual motions and contact forces based on Newtonian mechanics [5,21,22].

In static mixers, DEM offers particle-scale insight into flow patterns, shear zones, and particle rearrangements, enabling analysis of local mixing, stagnant regions, and segregation phenomena that are difficult to capture experimentally [5,23]. DEM incorporates realistic particle properties—size, shape, friction, cohesion, and restitution—allowing high-fidelity simulations that reflect real materials [24]. It can also be coupled with methods like CFD to model multiphase systems and assess the impact of process conditions such as velocity, pressure drop, or fluidization [25].

Table 2 summarizes the principal features, advantages, and challenges associated with granular flow in static mixers. It highlights their industrial relevance, common experimental and simulation approaches, as well as current design limitations and potential future research directions.

Despite its computational demands, DEM is a valuable tool for static mixer design and optimization, enabling virtual prototyping and sensitivity analyses that are often impractical experimentally [26]. With increasing computational power and advanced modelling frameworks, DEM is expected to play a growing role in studying granular flow in static mixers and beyond [21].

Table 2. Key Aspects of Granular Flow and Static Mixers.

Aspect	Details	References
Industrial relevance	Granular materials are essential in pharmaceuticals, food, mining, chemical engineering, agriculture, and construction; mixing ensures product uniformity and process performance.	[1,2,9]
Static mixer advantages	No moving parts, low energy consumption, continuous operation in confined geometries, reduced maintenance, enhanced reliability.	[3,10,11]
Challenges of granular flow	Non-linear, segregating behaviour; influenced by particle shape, size, density, cohesion, friction; issues like channelling, dead zones, radial segregation.	[4,13,15,16]
Experimental approaches	Provide insights into flow regimes, particle distribution, mixing quality; limited by resolution and difficulty in visualizing opaque granular media.	[5,20]
DEM principle and interactions	DEM simulates discrete particles by solving Newton’s laws for translational/rotational dynamics with time-stepping; contact forces (elastic, damping, frictional) typically modelled with spring-dashpot or Hertz-Mindlin laws.	[27,28,29,30,31]
Particle-scale details	Captures particle size, shape, roughness, cohesion, adhesion, and explicitly defined boundaries (walls, blades, mixers).	[9,32,33]
Dynamic phenomena	Segregation, clogging, arching, compaction, mixing efficiency captured at particle scale.	[33]
Time integration & computation	Explicit schemes (velocity-Verlet, Euler-Cromer) require small time steps (10–30% of contact time); simulations are computationally demanding, often needing millions of steps.	[29,34,35,36,37]
Computational strategies	Efficiency improved by parallel computing (multi-core CPUs, GPUs, HPC), neighbour list optimization (linked-cell, Verlet), or simplified contact models.	[38,39,40,41,42]
Applications in static mixers	DEM predicts particle trajectories, residence time, segregation, flow regimes; supports geometry optimization, design calibration, and scale-up studies.	[5,6,7,21,22,24,43,44]
Contact models	Hertzian (nonlinear, accurate, heavy); linear spring-dashpot (simpler, widely used); tangential models include Coulomb friction, rolling/torsional resistance, and cohesion for powders.	[5,8,44,45,46,47,48,49]
DEM advantages	High fidelity; can couple with CFD for multiphase flows; enables virtual prototyping, sensitivity analyses, and scalability studies.	[23,25,26]
Design & optimization challenges	Achieving uniform mixing without segregation; scale-up difficulties; absence of standardized design rules; requires careful geometry tailoring.	[10,17,18,19]
Future directions	Enhanced experimental validation, multi-scale and hybrid modelling, integration with machine learning, improved contact models, better parameter calibration.	[21,26]

Table 2. Key Aspects of Granular Flow and Static Mixers.

Aspect	Details	References
Industrial relevance	Granular materials are essential in pharmaceuticals, food, mining, chemical engineering, agriculture, and construction; mixing ensures product uniformity and process performance.	[1,2,9]
Static mixer advantages	No moving parts, low energy consumption, continuous operation in confined geometries, reduced maintenance, enhanced reliability.	[3,10,11]
Challenges of granular flow	Non-linear, segregating behaviour; influenced by particle shape, size, density, cohesion, friction; issues like channelling, dead zones, radial segregation.	[4,13,15,16]
Experimental approaches	Provide insights into flow regimes, particle distribution, mixing quality; limited by resolution and difficulty in visualizing opaque granular media.	[5,20]
DEM principle and interactions	DEM simulates discrete particles by solving Newton’s laws for translational/rotational dynamics with time-stepping; contact forces (elastic, damping, frictional) typically modelled with spring-dashpot or Hertz-Mindlin laws.	[27,28,29,30,31]
Particle-scale details	Captures particle size, shape, roughness, cohesion, adhesion, and explicitly defined boundaries (walls, blades, mixers).	[9,32,33]
Dynamic phenomena	Segregation, clogging, arching, compaction, mixing efficiency captured at particle scale.	[33]
Time integration & computation	Explicit schemes (velocity-Verlet, Euler-Cromer) require small time steps (10–30% of contact time); simulations are computationally demanding, often needing millions of steps.	[29,34,35,36,37]
Computational strategies	Efficiency improved by parallel computing (multi-core CPUs, GPUs, HPC), neighbour list optimization (linked-cell, Verlet), or simplified contact models.	[38,39,40,41,42]
Applications in static mixers	DEM predicts particle trajectories, residence time, segregation, flow regimes; supports geometry optimization, design calibration, and scale-up studies.	[5,6,7,21,22,24,43,44]
Contact models	Hertzian (nonlinear, accurate, heavy); linear spring-dashpot (simpler, widely used); tangential models include Coulomb friction, rolling/torsional resistance, and cohesion for powders.	[5,8,44,45,46,47,48,49]
DEM advantages	High fidelity; can couple with CFD for multiphase flows; enables virtual prototyping, sensitivity analyses, and scalability studies.	[23,25,26]
Design & optimization challenges	Achieving uniform mixing without segregation; scale-up difficulties; absence of standardized design rules; requires careful geometry tailoring.	[10,17,18,19]
Future directions	Enhanced experimental validation, multi-scale and hybrid modelling, integration with machine learning, improved contact models, better parameter calibration.	[21,26]

This review synthesizes current knowledge on granular flow, focusing on experimental studies and DEM simulations. It covers mixing mechanisms, the effects of mixer geometry, operating conditions, and material properties, as well as experimental methods for assessing mixing efficiency and flow visualization. DEM modelling, including parameter calibration and validation, is discussed, along with comparisons of mixer designs and DEM-informed innovations. Finally, methodological gaps and future directions—such as enhanced measurements, multi-scale modelling, and machine learning integration—are highlighted, emphasizing the combined value of experimental and numerical approaches for improving static mixer performance.

To quantitatively map the intellectual structure and emerging trends in DEM research related to granular flow in static mixers, a bibliometric analysis was conducted using the VOSviewer software, ver. 1.6.20, https://www.vosviewer.com, accessed on 30 January 2025. Author and index keywords were extracted from the Scopus database to visualize the co-occurrence network and identify major research clusters. A total of 151 records were analysed based on their abstracts to provide an overview of the field. As illustrated in Figure 1, the keyword co-occurrence map revealed six distinct clusters, representing thematically related research directions within DEM studies of static mixers. The size of each node corresponds to the frequency of keyword occurrence, while different colours indicate separate clusters of closely associated terms. This visualization effectively highlights the primary topics and interconnections shaping current DEM research in granular flow analysis.

This review aims to (i) synthesize DEM applications for granular flow in static mixers, (ii) highlight experimental validation methods and computational strategies, and (iii) identify key challenges and future research directions, including scale-up, cohesive particle behaviour, and integration with hybrid and machine-learning approaches. The manuscript is organized to first provide an overview of DEM principles, followed by discussions of validation techniques, computational innovations, industrial applications, and emerging research directions.

2. Fundamentals of DEM for Granular Flow

Discrete Element Method simulations have become indispensable for analysing granular flow behaviour in static mixers, offering particle-scale insights that are often inaccessible through experimental observation. DEM is a numerical technique that simulates the mechanical behaviour of systems of discrete, interacting particles [27], Figure 2. It is well-suited for granular materials, whose macroscopic behaviour arises from complex interparticle interactions. DEM solves Newton’s second law for each particle, updating positions, velocities, and angular velocities over discrete time steps [28,29]. The reliability and interpretability of DEM predictions strongly depend on the modelling choices, calibration strategies, and the physical assumptions embedded in the simulation framework.

A central challenge in applying DEM to static mixers arises from the inherently complex flow regimes characterized by non-uniform shear, localized compaction, and intermittent particle contacts. Unlike simpler conveying systems, static mixers generate strong spatial gradients in velocity and porosity due to their geometrically induced recirculation zones. Accurately capturing these effects requires fine temporal and spatial resolution, often leading to prohibitive computational demands. Particle interactions are modelled via contact forces, typically combining elastic, damping, and frictional components [27]. Normal and tangential forces are often described by spring-dashpot models, such as linear spring or Hertz-Mindlin laws, capturing deformation, energy dissipation, and frictional sliding [30,31].

DEM excels at representing particle-scale features, including size, shape, surface roughness, and cohesive forces [32], while explicitly defining boundaries like walls or mixer elements [9]. It captures dynamic phenomena such as segregation, clogging, arching, and mixing efficiency, which are challenging for continuum models or experiments [33]. Time integration uses explicit schemes such as velocity-Verlet or Euler) [29], requiring small time steps and making simulations computationally intensive [34]. Scaling DEM simulations to industrial dimensions remains problematic, as particle size reduction for computational feasibility can alter flow behaviour and mixing efficiency. Advances in parallel computing and algorithms continue to expand DEM’s applicability to complex, industrial-scale systems [42].

Several critical modelling decisions influence the predictive accuracy of DEM. The selection of the contact model (such as linear vs. Hertz–Mindlin, inclusion of rolling or cohesive forces) must reflect the physical characteristics of the material under realistic operating conditions. The boundary conditions and geometry representation of the mixing elements also dictate flow behaviour, where simplified or smoothed surfaces can suppress key flow features such as dead zones and arching. In static mixers, DEM provides a particle-level view of how geometry, flow rate, and material properties affect flow and mixing quality [43]. It enables visualization and quantification of particle trajectories, residence times, and contact networks, making it essential for analysing and optimizing granular flow [5]. Particles are typically modelled as spheres for simplicity, though ellipsoids, polyhedral, or clumped-sphere models can capture non-spherical behaviour relevant to industrial mixers [50,51].

The motion of each particle is governed by Newton’s second law, with separate equations for translational and rotational dynamics. For translation, the equation is [5,28]:

$$m_i{\displaystyle \frac{d^2{\mathit{r}}_i}{d{t}^2}}=\sum ({\mathit{F}}_i^{contact}+{\mathit{F}}_i^{body})$$

(1)

where $m_i$ is the mass of the particle $i$, ${\mathit{r}}_i$ is its position vector, ${\mathit{F}}_i^{contact}$ denotes the total contact force resulting from interactions with neighbouring particles and walls, and ${\mathit{F}}_i^{body}$ describes body forces such as gravity or fluid drag.

Rotational motion is described by:

$${\mathit{I}}_i{\displaystyle \frac{d{\mathit{\omega }}_i}{dt}}=\sum {\mathit{T}}_i^{contact}$$

(2)

where ${\mathit{I}}_I$ is the moment of inertia of the non-spherical particle, ${\mathit{\omega }}_I$ is the angular velocity vector, and ${\mathit{T}}_i^{contact}$ is the torque arising from tangential contact forces and rolling resistance.

Figure 3 illustrates the fundamental interaction mechanisms governing particle motion within the Discrete Element Method framework. Each particle is subjected to forces arising from direct contact interactions with neighbouring particles, as well as long-range or non-contact forces.

Interparticle forces in DEM are resolved into normal and tangential components using contact laws like linear spring-dashpot or Hertz-Mindlin models [27,45]. Tangential forces account for sliding friction, while rolling and torsional resistances can model realistic behaviour in dense or cohesive flows [46]. Accurate representation of these forces is essential in static mixer simulations to capture particle trajectories, mixing dynamics, and flow regimes [43], with parameters such as stiffness, restitution, friction, and contact time carefully calibrated against experimental data [5].

Accurate particle–particle and particle–wall interactions are fundamental in DEM and governed by contact mechanics models [5,8,47]. These models capture normal and tangential forces during collisions, essential for simulating granular dynamics in static mixers [8].

To provide a clear and practical overview of the DEM contact models relevant to granular flow in static mixers,

Table 3. Summary table for DEM contact models.

Contact Model/Interaction	Governing Equation	Physical Interpretation	Practical Implications for Static Mixers	Computational Considerations	Calibration and Sensitivity	Typical Applications	Ref.
Hertzian (nonlinear)	$F_n=k_n{\delta }_n^{3/2}$	Nonlinear elastic contact; models particle deformation accurately	Captures realistic collision behaviour; important for deformable particles and high-fidelity simulations	High computational cost; small time steps needed for stability	Sensitive to stiffness; requires careful experimental or inverse calibration; sensitive to particle shape	Simulations requiring accurate deformation, energy dissipation, or contact stress analysis	[27,39]
Linear spring–dashpot	$F_n=k_n{\delta }_n+{\gamma }_n\dot{{\delta }_n}$	Linear elastic contact with energy dissipation	Efficient for large-scale simulations; suitable for rigid or nearly elastic particles; controls restitution	Computationally cheaper; may produce unrealistic overlaps if stiffness is too low	Easier calibration; damping strongly affects energy loss; less sensitive to particle shape	Industrial-scale mixers, fast screening of process parameters, coarse-grained simulations	[52]
Tangential (elastic-damped Coulomb friction)	$F_t=\mathrm{m}\mathrm{i}\mathrm{n}(k_t{\delta }_t+{\gamma }_t\dot{{\delta }_t},\mu F_n)$	Tangential resistance and friction; accounts for rolling/sliding	Influences energy dissipation, particle interlocking, and segregation; critical for predicting mixing efficiency	Moderate computational cost; stability depends on time step and stiffness selection	Parameters require calibration; sensitive to surface roughness and particle shape	Predicting particle segregation, flow uniformity, and residence time distributions	[53]
Rolling resistance/torque	$T_i={\mu }_r{F}_{n}R$	Models energy loss due to particle rotation and rolling	Reduces unrealistic rolling; affects segregation and mixing patterns	Adds modest computational cost	Rolling friction coefficient must be calibrated; depends on particle shape	Systems with non-spherical particles or elongated shapes; controlling axial mixing	[54]
Cohesive interactions (van der Waals/liquid bridges)	$F_c=f(\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y},$ $\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e},$ $\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})$	Models attractive forces between particles	Critical for high-moisture or fine powders; affects agglomeration, arching, and flow blockage	Increases computational complexity; smaller time steps may be needed	Difficult to measure directly; often fitted to experimental bulk behaviour	Wet or fine granular systems; silage, powders, high-moisture feedstock’s	[55]
Boundary/wall interactions	$F_t=f(contactmodel,{\mu }_{\omega })$	Interaction between particles and mixer walls	Influences dead zones, particle segregation, and flow uniformity	Computational cost depends on wall discretization and contact model	Wall friction calibration critical for accurate flow	All static mixer designs; particularly important for narrow or complex geometries	[56]

summarizes the commonly used normal, tangential, rolling, cohesive, and wall interaction models. Beyond the governing equations, the table highlights the physical interpretation of each model, their practical implications for flow behaviour, and the computational trade-offs associated with their implementation. Additionally, it outlines the calibration challenges and sensitivity to particle properties, as well as typical applications in static mixer simulations.

$k_n$ is a material- and geometry-dependent stiffness coefficient. ${\gamma }_n$ is the normal damping coefficient, ${\delta }_n$ is the particle overlap, $\dot{{\delta }_n}$ is the relative normal velocity. $k_t$ and ${\gamma }_t$ representing tangential stiffness and damping coefficients, ${\delta }_t$ the accumulated tangential displacement, $\dot{{\delta }_t}$ the relative tangential velocity, and μ the static friction coefficient. The particle shape approximation, often simplified as spheres for computational tractability introduces additional uncertainty, particularly in systems dominated by interlocking or irregular particle geometries. The frictional and restitution coefficients typically derived from laboratory tests or inverse fitting, can further affect flow predictions, underscoring the importance of rigorous model calibration and sensitivity analysis.

For systems with significant particle shape or roughness, rolling resistance and torsional friction models add torques based on angular displacement and contact forces. Cohesive or fine powders may include capillary, van der Waals, or electrostatic forces to simulate adhesion [49]. In static mixers, selecting and calibrating the contact model is essential for accurate prediction of particle trajectories, mixing dynamics, residence times, and segregation [43,44]. Choosing between linear and nonlinear laws balances computational efficiency and physical fidelity, depending on the application and simulation scale.

In DEM, particle motion is tracked by integrating Newton’s equations of motion, with accuracy and stability critical for realistic trajectories and collective behaviour in systems like static mixers [5,34]. Time integration schemes, such as explicit velocity-Verlet or Euler-Cromer methods, update positions and velocities sequentially using forces at each step, providing computational simplicity [29]. The Velocity-Verlet algorithm is the dominant integration scheme in DEM because it achieves a favourable balance between numerical accuracy, stability, and computational cost. It is simplistic and time-reversible, ensuring good energy conservation even with small numerical errors over long simulation periods. This algorithm requires only a single force evaluation per time step, which substantially reduces computational cost compared with higher-order schemes. Its second-order accuracy allows for stable integration at relatively small time steps, typically governed by the Rayleigh wave propagation limit [37], thereby maintaining realistic particle–contact dynamics without compromising efficiency.

The velocity-Verlet algorithm, for example, integrates positions and velocities as [29]:

$$\mathit{r}\left(t+\Delta t\right)=\mathit{r}\left(t\right)+\mathit{v}\left(t\right)\Delta t+{\displaystyle \frac{1}{2m}}\mathit{F}\left(t\right)\Delta t^2$$

(3)

$$\mathit{v}\left(t+\Delta t\right)=\mathit{v}\left(t\right)+{\displaystyle \frac{1}{2m}}⌊\mathit{F}\left(t\right)+\mathit{F}(t+\Delta t)⌋\Delta t$$

(4)

where $\mathit{r}\left(t\right)$ and $\mathit{v}\left(t\right)$ are the particle’s position and velocity at time $t$, $\Delta t$ is the time step, and $\mathit{F}\left(t\right)$ is the total force acting on the particle. Angular motion is updated similarly using torques and moments of inertia.

Explicit methods in DEM are computationally efficient but require small time steps for stability, particularly when modelling stiff contacts or high particle stiffness [5,29]. Excessively large time steps can lead to instability or unrealistic particle overlaps, whereas very small steps increase computational cost [7]. Implicit schemes are rarely used due to their complexity. In static mixers, frequent collisions and shearing make the choice of time step critical for accurate flow prediction [14], and adaptive time-stepping can improve efficiency without compromising stability [35]. Overall, time integration schemes and computational strategies directly influence the accuracy and stability of DEM simulations in static mixers, affecting predictions of segregation, dead zones, and residence time distributions. Efficient approaches, including adaptive time-stepping and parallel computing, enable realistic simulation of industrial-scale mixers without prohibitive computational cost.

DEM simulations of granular flow in static mixers are computationally demanding due to resolving individual particle interactions over many time steps [5,36]. Key considerations include time step selection, scalability, and parallel computing [38]. Time steps must remain below the characteristic contact time to prevent particle overlap and instability [37], which depends on particle mass and stiffness [27]. High stiffness or fine particles may require millions of steps, with typical time steps set to 10–30% of the Rayleigh time step.

Due to the high computational cost of DEM, especially for industrial-scale simulations, parallel computing is essential [57]. Modern DEM software utilizes multi-core CPUs and GPUs to calculate contact forces, particle motions, and neighbour lists simultaneously [42]. Efficiency depends on domain decomposition, load balancing, and communication overhead [39]. For large systems, HPC clusters or cloud computing significantly reduce runtime. Additional strategies include neighbour list optimization, adaptive time-stepping, and simplified contact models, such as linear spring-dashpot, to lower computational load while maintaining adequate accuracy [40,41].

In static mixers, resolving flow patterns, mixing quality, and residence times in confined, dynamic conditions requires balancing simulation resolution and runtime [58]. Various DEM software packages are available for specific applications [59]. EDEM (Altair) supports GPU acceleration and CFD/FEA integration, Rocky DEM (ESSS) handles non-spherical particles and coupled DEM-CFD-FEM simulations, PFC (Itasca), Minneapolis, MI, US, is used in geomechanics, and LIGGGHTS (CFDEM) excels in large-scale DEM-CFD coupling. Simcenter STAR-CCM+ (Siemens) integrates DEM and CFD, MFiX-DEM, (NETL) specializes in gas-solid flows, XPS Fraunhofer) targets granular processes, and MercuryDPM supports multiphase flow. For bulk material handling and mining, EDEM, Rocky DEM, and LIGGGHTS are common, while Rocky DEM, LIGGGHTS, and STAR-CCM+ suit large-scale simulations.

Successful DEM simulations of granular flow in static mixers rely on accurate contact models, appropriate time step selection, and optimized computational strategies. These factors are essential for achieving precise, efficient, and scalable simulations that capture detailed particle-scale dynamics and support large-scale studies of granular mixing.

3. DEM Implementation in Static Mixers

DEM provides a powerful tool for investigating particle-scale phenomena governing granular flow and mixing in static mixers, capturing transport, shear, dispersion, and segregation within fixed geometries [44]. Static mixers are passive devices that promote particle reorientation and homogenization without moving parts [60]. Various geometries offer advantages depending on material properties and flow conditions [21]: Kenics mixers use alternating helical elements for distributive mixing but may clog with cohesive materials; Sulzer mixers (SMX, SMXL) create complex axial and radial flows with high efficiency at higher pressure drop [61]; helical mixers reduce segregation under gravity-driven flow; split-and-recombine (SAR) mixers divide and merge streams to achieve high mixing in short residence times [62]. DEM-informed novel geometries, including twisted baffles and multi-inlet designs, allow tailoring to particle properties, flow rate, wall friction, and cohesion to optimize mixing, minimize dead zones, and reduce blockages.

A DEM model for a static mixer starts with an accurate representation of the mixer geometry, such as Kenics, Sulzer SMX, or custom baffles, implemented as rigid boundaries [63]. Geometric detail depends on resolution and computational resources, with mixer elements typically assumed stationary and impermeable, using the same contact mechanics as particle–particle interactions. Particles are introduced with defined initial conditions (velocity, position, mass flow rate) [63] and simulations may model batch, continuous, or quasi-steady flow [64]. Material properties, including size distribution, density, friction, and cohesion, are calibrated to experimental conditions, and particle trajectories are computed under forces like gravity or inlet velocities.

Figure 4 illustrates the mixing mechanism within a Kenics-type static mixer, adapted for granular materials such as powders or pellets. Two distinct particle streams enter the mixer and pass through a series of fixed helical or curved elements that repeatedly split, rotate, and recombine the flow. The movement of particles through the mixer is driven primarily by gravity, while the static elements induce transverse motion and shear, causing the granular streams to interlace and progressively improve their spatial distribution. As the material advances through successive elements, the initially segregated particles become more uniformly distributed, resulting in enhanced homogeneity at the outlet. This gravity-driven process, achieved without moving parts, efficiently promotes both macro- and micro-scale mixing of granular systems while maintaining low energy consumption and mechanical simplicity.

Several key outputs are derived from DEM simulations of static mixers, including residence time distribution (RTD), mixing indices (Lacey’s or Danckwerts’ indices), velocity profiles, and particle dispersion metrics. These outputs enable quantification of mixing performance and identification of inefficiencies such as dead zones, channelling, or segregation.

Advanced modelling approaches may also include:

• Multiphase coupling, where DEM is integrated with CFD to model gas–solid or liquid–solid flows in static mixers used for fluidized or wetted granular systems.
• Non-spherical particle modelling enhances the realism of simulations for elongated, angular, or irregularly shaped particles commonly found in agricultural and pharmaceutical applications.
• Thermal and reactive modelling, where temperature profiles or chemical transformations are coupled to the mechanical mixing process.

DEM has been used for parametric studies, varying mixer geometry, particle properties, flow rates, and fill levels to evaluate mixing performance, enabling virtual prototyping and design optimization [5]. However, DEM remains computationally demanding, especially for industrial-scale systems or long residence times, prompting hybrid approaches that combine experiments with reduced-order or surrogate models [65]. Accurate representation of static mixer geometries is crucial, as repeated flow redirection through stationary elements strongly affects particle trajectories, shear zones, and mixing efficiency [66]. These geometries are typically modelled as rigid, non-deformable boundaries interacting with particles via contact forces [45]. The geometry can be represented using either: analytical primitives (such as cylinders, planes, and boxes, for simplified or idealized mixer designs); or triangulated surface meshes (derived from computer-aided design models for complex or industrially realistic geometries).

Common static mixer designs—such as Kenics^®, Sulzer SMX^®, Ross LPD^®, and custom baffles—are imported into DEM as surface meshes, discretized as internal obstacles that redirect particle flow [67]. Particle–mixer interactions use the same contact mechanics as particle–particle forces, including normal and tangential components with friction, restitution, and stiffness parameters [66]. Surface roughness, represented by higher friction or micro features, affects flow resistance and particle adhesion [68]. Geometry resolution influences computational performance: simplified meshes allow faster simulations, while high-resolution geometries improve realism at increased cost, with adaptive meshing reducing element count without major loss of accuracy [69].

The analysis revealed that variations in blade pitch, helix angle, and element arrangement exert a pronounced influence on the local velocity fields, mixing uniformity, and energy dissipation within static mixers. DEM simulations showed that changes in pitch and helix geometry directly modify the shear and circulation patterns governing particle transport [14]. A lower helix angle typically enhances radial motion and interlayer exchange, promoting uniform mixing but increasing localized shear stresses and energy dissipation. Conversely, higher pitch angles facilitate faster axial transport and reduced residence time, though often at the expense of radial mixing efficiency and homogeneity of concentration fields [36].

Alternating left–right element orientations were found to play a crucial role in promoting cross-sectional particle exchange [59]. This configuration disrupts coherent flow paths and suppresses the formation of persistent velocity gradients, thereby reducing stagnant regions and dead zones frequently observed in unidirectional geometries. Such alternation creates a continuous redistribution of particles between peripheral and central regions, resulting in improved mixing efficiency and more uniform residence time distributions. Particle–wall interactions emerged as another critical determinant of granular flow behaviour. Increased wall friction and surface roughness intensify the formation of shear layers adjacent to the channel boundary, leading to more pronounced energy dissipation and occasionally enhanced segregation effects, depending on particle size and density. However, moderate wall roughness can prevent slip and promote momentum exchange between flow layers, contributing to a more effective mixing process.

Periodic DEM simulations model an infinitely repeating mixer section using a representative unit cell with periodic boundary conditions, reducing computational effort [5]. Accurate material properties are essential for realistic particle behaviour, as flow depends on size, shape, density, friction, and cohesion [43]. Particle size distribution (PSD) strongly influences packing, flowability, and mixing. DEM can incorporate mono-sized, bi-modal, or polydisperse distributions to capture segregation, contact networks, and dynamic responses, typically based on experimental data or theoretical fits (such as log-normal or Gaussian) [70,71].

Particle shape strongly affects flow in static mixers, as anisotropic shapes enhance interlocking, shear resistance, and orientation-dependent behaviour [72]. While early DEM models used spheres, modern simulations employ ellipsoids, superquadric, clumped spheres, or polyhedral, improving realism but increasing computational cost [73,74]. Frictional properties, including static and dynamic coefficients, govern sliding resistance, shear strength, energy dissipation, and flow regimes, and are obtained experimentally or via calibration [75].

Cohesion is critical in systems with fine powders, moisture, or interparticle forces like van der Waals or capillary effects, leading to agglomeration, arching, and flow blockage [76,77]. In DEM, cohesion is modelled via attractive force potentials, capillary bridges, or bonded-particle frameworks [78]. Particle density, elastic modulus, and restitution coefficient also affect collisions, energy loss, and force transmission, often requiring experimental calibration [79].

Proper boundary conditions and flow initialization are essential for realistic DEM simulations [5]. Boundaries, including walls, inlets, outlets, and internal elements, are treated as rigid, impenetrable surfaces interacting via contact forces with friction and restitution [80,81]. Surface roughness and curvature influence flow, especially in confined regions [82].

Periodic boundary conditions can be applied axially or laterally to simulate an infinite or repeating section of a static mixer, reducing computational cost while preserving representative mixing dynamics [83]. Flow initialization defines how particles are introduced and positioned at the start of a simulation [66]. Simulations may be initialized as:

• Batch systems involve placing a fixed number of particles into the domain and allowing them to settle under gravity before initiating the flow.
• Continuous systems, where particles are continuously inserted at the inlet with specified velocities or mass flow rates and exit the system through an outlet.
• Quasi-steady-state setups, where a small representative volume is used with periodic inlets and outlets to maintain a constant particle population.

To avoid artificial effects from abrupt particle injection, a ramp-up period gradually introduces particles and external forces, stabilizing the system and preventing unrealistic overlaps or compaction [84]. Inlet velocity profiles, feed funnels, or particle generation planes are used to mimic real feeding, while outlet boundaries may absorb particles to assess residence times and steady-state mixing [85]. External forces such as gravity, vibration, or pressure gradients are applied depending on process conditions [86]. For multiphase flows involving particles and fluids, DEM is often coupled with CFD to capture interphase momentum exchange, particle dispersion, and flow instabilities [21].

DEM-CFD coupling integrates the discrete treatment of particles with the continuum modelling of the fluid phase, enabling simultaneous resolution of particle trajectories and fluid flow fields. The fluid dynamics are typically governed by the Navier–Stokes equations, solved over an Eulerian grid, while particles are tracked in a Lagrangian framework using DEM. The two phases exchange momentum through interphase drag, lift, and added mass forces, allowing for bidirectional coupling.

In the context of static mixers, DEM-CFD coupling is particularly relevant in applications involving:

• Wetted or paste-like granular flows, where the presence of a liquid modifies cohesion and flowability.
• Gas–solid systems, such as pneumatic conveying or aerated mixing.
• Liquid–solid mixing, where high-viscosity fluids or slurries are processed in static mixer geometries.
• Heat and mass transfer analysis, where fluids serve as heat carriers or solute dispersants interacting with solid particulates.

Several coupling strategies exist, including:

• One-way coupling, where the fluid affects particle motion (e.g., drag forces), but particles do not influence the fluid flow-suitable for dilute systems.
• Two-way coupling, where particle motion also alters the local fluid velocity and pressure fields-appropriate for denser flows where particle–fluid interactions are strong.
• Four-way coupling, which additionally includes inter-particle collisions and turbulence–particle interactions, is often used in highly dynamic multiphase environments.

DEM-CFD simulations face challenges such as the need for acceptable spatial and temporal resolution near complex boundaries and the high computational cost of fluid–particle interactions over many time steps, making advanced solvers and parallel computing essential. Realism can be improved by incorporating capillary forces, liquid bridging, fluid-induced cohesion, and turbulent dispersion [86]. Experimental validation using techniques like Particle Image Velocimetry (PIV) and X-ray tomography is crucial for comparing simulated and observed flow behaviour [87,88].

DEM offers a powerful and adaptable framework for modelling granular flow in static mixers, supporting the analysis and optimization of mixing performance. Accurate implementation of mixer geometry, material properties, boundary conditions, and flow initialization is crucial for realistic simulations. Coupling DEM with CFD further expands its capability to model complex multiphase systems. As computational tools advance, DEM is expected to play an increasingly vital role in both research and industrial applications involving granular mixing.

4. Performance Evaluation and Validation

DEM enables detailed analysis of static mixer performance by providing particle-scale data often inaccessible experimentally [89]. Key metrics include mixing quality, assessed via indices such as the Lacey index, segregation intensity, or entropy-based measures [90], and residence time distribution (RTD), determined by tracking particle exit times to reveal flow patterns, channelling, or dead zones [91]. DEM also evaluates local shear and stress distributions, allowing identification of high-stress regions and prediction of particle breakage or degradation [92,93].

DEM simulations reveal segregation due to particle size, density, or shape, enabling analysis of how mixer geometry and flow conditions affect demixing and guiding design improvements [5]. Visualization of particle flow, contact maps, and concentration profiles, along with time-averaged data and spatial binning, helps assess the impact of geometric features on mixing efficiency. DEM also allows parametric and sensitivity studies of feed rate, particle properties, or mixer configurations, facilitating virtual design screening without extensive experiments [94].

Assessing the efficiency of granular mixing in static mixers requires robust, quantifiable metrics that capture the spatial and temporal distribution of particles. In the context of DEM simulations, a range of statistical and dynamic indicators can be used to evaluate mixing performance [95]. These metrics offer insights into the homogeneity, predictability, and functional quality of the mixing process and support comparative analysis across mixer geometries and operating conditions.

One of the most widely used quantitative tools is the mixing index, which evaluates the degree of spatial uniformity of different particle types or tracers. Among the common indices are:

• Lacey’s mixing index, which compares the variance of a sample’s composition to the variance in a fully segregated and ideally mixed state.
• Relative Standard Deviation (RSD), which expresses the variability of sample composition as the standard deviation relative to the mean concentration.
• Intensity of segregation (I), defined as the normalized variance of component concentration over time or space.
• Entropy-based indices, which interpret mixing as a measure of disorder in the system.

Mixing indices are computed by dividing the mixer into sampling bins and tracking particle identities or concentrations, with values near zero (segregation) or one (mixing) indicating ideal homogeneity. RTD, obtained by tracking tagged particles, characterizes the temporal spread through the mixer and identifies flow regimes such as plug flow, back-mixing, or dead zones [96,97]. Coefficient of variation (CoV) of particle concentration, velocities, or shear rates provides a statistical measure of uniformity, with low CoV indicating consistent flow and high CoV indicating segregation [98].

Shear exposure metrics, derived from local velocity gradients or contact forces, quantify stresses on particles, affecting breakage, agglomeration, or coating uniformity. Particle trajectory and dispersion analyses, including mean squared displacement (MSD) and relative displacement, provide dynamic insights into particle redistribution within the mixer [44]. Evaluated over time, position, or processing conditions, these metrics enable detailed comparisons and optimization. DEM offers a particle-resolved view of flow patterns, aiding the understanding of transport, dispersion, and segregation mechanisms critical for mixing efficiency and homogeneity [43,87].

Velocity profiles, showing how particle velocities vary within the mixer, are a primary descriptor of flow behaviour [66]. DEM extracts instantaneous and time-averaged velocity fields by tracking particle motion, highlighting regions of high shear, stagnation, and preferential flow. Well-designed mixers promote redistribution and reorientation of particles, avoiding channelling or dead zones. Segregation patterns, influenced by particle size, shape, or density, can also be quantified using DEM, revealing phenomena such as percolation of fine particles, accumulation of heavy particles, or radial separation [99]. Predicting segregation is essential for maintaining mixing quality and process repeatability.

Streamline and trajectory analyses visualize particle motion, revealing dominant flow paths, recirculation zones, and axial or radial dispersion [100]. Combined with residence time data, they provide a detailed view of particle transport and mixing dynamics. Shear rate distributions indicate mixing intensity and potential particle attrition, with high-shear zones near walls or baffles enhancing mixing but risking breakage in sensitive materials [44]. DEM flow patterns can be validated against experimental data from PIV, MRI, or X-ray tomography, supporting parameter calibration and confidence in predictions [101].

While high-resolution methods such as MRI and X-ray tomography provide unparalleled insight into local flow and mixing dynamics, their application remains largely restricted to laboratory-scale systems due to size constraints, imaging time, and equipment cost. For industrial-scale static mixers, non-invasive optical and tracer-based techniques—such as Particle Image Velocimetry (PIV), Positron Emission Particle Tracking (PEPT), or advanced tracer residence time distribution (RTD) analyses—offer more practical validation options, albeit with lower spatial resolution. Significant validation gaps persist for large, opaque, or high-load systems, where direct visualization is limited and scaling effects are not fully captured. Developing hybrid validation strategies that combine coarse-scale tracer data with DEM or CFD predictions represents a promising direction for improving model reliability under real operating conditions.

Analysis of particle–particle and particle–wall interactions is key to understanding granular flow in static mixers [14]. These interactions govern momentum transfer, energy dissipation, and particle rearrangement, affecting overall mixing performance. DEM explicitly resolves contacts, allowing detailed investigation of their effects. Particle-particle interactions are defined by normal and tangential forces, influenced by stiffness, restitution, friction, and cohesion [102]. Normal forces control deformation and penetration resistance, while tangential forces govern sliding, rolling, and interlocking. Accurate modelling captures force chain formation, clustering, and dilation, with DEM quantifying contact force distributions, durations, and coordination numbers to reveal the evolving internal granular structure.

In static mixers, frequent particle–particle collisions arise from complex geometry and induced shear [44]. Collision frequency and energy dissipation influence mixing efficiency and may cause particle breakage or attrition, particularly for fragile materials. DEM collision statistics help evaluate mechanical stresses and optimize process parameters. Particle-wall interactions also shape flow patterns and mixing, often exhibiting higher friction or adhesion depending on surface properties [100]. DEM incorporates these effects through contact parameters, simulating slip, stick-slip transitions, and localized shear near walls.

The geometry and surface properties of static mixer elements affect particle–wall collision angles, residence times near boundaries, and shear layer development [47]. DEM identifies zones prone to buildup, clogging, or wear, guiding design improvements. It also captures phenomena such as rolling resistance, particle shape effects, and cohesive bonding in wet or agglomerated systems [76], enhancing predictive accuracy. Optimizing element shape, size, and arrangement is crucial for mixing efficiency, pressure drop, and process reliability, and DEM provides a powerful tool for systematic design optimization [5].

Element shape strongly affects flow diversion, shear generation, and particle reorientation in static mixers [103]. Common designs—helical, twisted, or baffle-like (e.g., Kenics, SMX, custom geometries)—induce distinct flow dynamics. DEM enables analysis of subtle geometric variations, such as pitch, twist angle, or cross-section, on particle trajectories, shear rates, and mixing indices, supporting virtual prototyping to enhance mixing while reducing dead zones.

Static mixer geometry strongly influences granular flow patterns, as shown in Figure 5. DEM simulations reveal that Kenics KMX helical elements induce repeated splitting and swirling of particle streams, promoting radial and axial mixing and reducing dead zones. The Ross ISG design, with its helical blades, generates multilayer particle division and recombination, enhancing particle randomization and mitigating axial segregation. Sulzer SMV V-shaped elements produce layered splitting and convergence, increasing particle–particle interactions and contact frequency, which improves homogenization. In contrast, the Sulzer SMX grid-like crisscross structure directs particles through multiple redirection points, creating high turbulence, enhanced mixing uniformity, and improved transport of dense or cohesive particles. These DEM-based insights illustrate how internal geometry governs particle trajectories, swirling, splitting, and randomization, providing a direct link between design features and granular flow performance.

The accuracy of DEM predictions can vary depending on mixer geometry. Complex geometries such as Kenics KMX or Sulzer SMX may require finer particle resolution and smaller time steps to capture local collisions, shear layers, and recirculation zones, while simpler designs like Ross ISG or Sulzer SMV allow reliable predictions with coarser discretization, reducing computational cost. Validation studies using particle tracking, tracer tests, and residence time measurements confirm that DEM can accurately reproduce mixing efficiency and segregation trends across these geometries. For Kenics and Sulzer mixers, multiple experimental studies demonstrate close agreement between DEM predictions and observed flow behaviour. Nevertheless, scaling from laboratory to industrial scale introduces challenges, such as increased particle numbers, more complex flow interactions, and higher computational demands, which may limit predictive accuracy. To address these limitations, hybrid approaches, coarse-graining techniques, and careful calibration of material and contact properties are recommended, ensuring that DEM remains a reliable tool for industrial mixer design and optimization. Element arrangement—spacing, orientation, and sequence—also critically influences performance [104]. Alternating or staggered configurations improve dispersion through repeated bifurcation and recombination, while insufficient spacing may cause jamming or attrition. DEM allows systematic optimization to balance mixing quality with operational constraints.

Scale-up considerations are integrated into design optimization [105]. DEM helps predict performance from lab to industrial scale, accounting for changes in particle load, velocity, and residence time, preventing scale-up issues like segregation or poor mixing.

Optimization often involves multi-objective criteria, balancing mixing uniformity, energy use, throughput, and equipment durability [106]. Coupling DEM with algorithms such as genetic algorithms, response surface methods, or machine learning enables efficient exploration of design spaces and identification of Pareto-optimal solutions.

Recent DEM-based optimizations also incorporate material-specific properties like particle shape, cohesion, and moisture, allowing tailored improvements for challenging granular systems. Reliable DEM predictions depend on rigorous validation against experimental data [5], ensuring contact models, material parameters, and boundary conditions accurately represent real-world behaviour for design and scale-up.

DEM has been widely validated against experimental measurements of granular flow in static mixers, showing good agreement for particle trajectories, residence time distributions, axial and radial mixing, and segregation patterns. Studies indicate that, for dry, non-cohesive particles, DEM can reproduce particle motion and mixing indices within 10–20% of experimental values, depending on particle size distribution and mixer geometry. DEM excels at capturing microscale phenomena, including collision dynamics, particle–particle and particle–wall interactions, and the influence of mixer internal geometry on flow behaviour. However, DEM has limitations. Its predictive accuracy decreases for highly cohesive, wet, or fine powders, where agglomeration and complex interstitial forces dominate. Scale-up effects may also be poorly captured if computational domain sizes are limited. In such cases, DEM may overestimate particle mobility or fail to reproduce blockage and arching phenomena accurately. Additionally, DEM simulations are computationally intensive, especially for large systems with millions of particles, making it less suitable for rapid industrial-scale assessments unless coarse-graining or hybrid modelling approaches are applied.

To illustrate the range of recent developments and applications, representative case studies on static mixers involving granular and multiphase materials are summarized in

Table 4. Case Studies on Static Mixers for Granular Materials.

No.	Mixer Type/Geometry	Material or System Studied	Methodology	Key Findings	Reference
1	Laminar static mixer with baffles and vortex generators	Liquid–liquid mixing under laminar conditions	CFD (3D RANS)	Demonstrated that optimized vortex-generator geometries significantly enhance mixing efficiency while reducing pressure drop.	[3].
2	Static mixer (cross-blade type)	Granular solids	DEM simulation	Quantified granular mixing patterns and residence time distributions; validated DEM predictions against experiments.	[8]
3	Various static mixer designs	Liquids and suspensions	Review (CFD, experimental)	Comprehensive review of mixing mechanisms, design correlations, and performance metrics in static mixers.	[10]
4	Kenics static mixer	Single- and two-phase flows	CFD and experimental validation	Quantified the effect of pitch angle and element number on pressure drop and mixing uniformity.	[11]
5	Revolving static mixer (FixMix type)	Granular materials	DEM–CFD hybrid model	Demonstrated that rotation combined with static elements improves particle homogeneity and reduces segregation.	[14]
6	Helical static mixer (Komax type)	Granular and particulate systems	DEM–CFD coupling	Provided quantitative insight into local shear and particle dispersion mechanisms within static mixing zones.	[43]
7	SMX-type static mixer	Active pharmaceutical ingredients (API)	CFD modeling	Developed a validated CFD model for static mixer flow; used for process scale-up in continuous pharmaceutical manufacturing.	[58]
8	Milli-scale split-and-recombine mixer	Microreactor applications	CFD	Evaluated micromixing and residence time uniformity; identified optimized geometrical ratios for efficient micro-scale mixing.	[62]
9	Helical static mixer	Turbulent liquid flow	CFD (k–ε model)	Showed flow enhancement and reduced axial dispersion through helical insert geometries.	[81]
10	Industrial helical static mixer	Single-phase turbulent flow	CFD (3D Navier–Stokes)	Predicted pressure loss and flow development; validated simulation with industrial-scale data.	[83]
11	Pipeline static mixers (Kenics, Sulzer SMX, etc.)	Various industrial fluids	Experimental and industrial review	Provided design guidelines and performance maps for industrial static mixing applications across scales.	[27]

. These studies encompass both experimental and numerical approaches, highlighting the influence of mixer geometry, scale, and flow regime on mixing performance. Case studies include laminar, turbulent, and particulate systems modeled through Computational Fluid Dynamics (CFD), Discrete Element Method (DEM), and hybrid DEM–CFD frameworks.

Particularly, DEM-based investigations [8,14,43] provide insights into granular flow behaviour, segregation, and residence time distributions, while CFD studies [3,11,81] focus on fluid mixing enhancement and energy optimization. Hybrid and micro-scale designs [58,62] extend the applicability of static mixers to continuous pharmaceutical and milli-reactor systems, confirming their versatility across scales. Collectively, these case studies demonstrate how static mixer configurations can be optimized to achieve improved homogeneity, process intensification, and scale-up reliability.

Recent advances in DEM modelling have increasingly been applied to industrial-scale mixing processes, demonstrating the method’s capacity to capture realistic particle behaviour under complex operating conditions. Applications in static mixer design now extend beyond laboratory validation to process optimization in pharmaceutical, food, and chemical industries, where DEM simulations support geometry refinement, scale-up, and energy-efficiency assessments. Case studies integrating DEM with experimental monitoring or CFD coupling have provided actionable insights into flow uniformity, residence time distribution, and fouling tendencies. In the pharmaceutical sector, DEM simulations have guided the design of static mixers used for continuous granulation and blending of excipients, where controlling residence time distribution and minimizing shear-induced degradation are critical. In the chemical industry, DEM-assisted analyses of Sulzer and Kenics mixer geometries have improved the understanding of particle–wall interactions, fouling behaviour, and mixing uniformity in polymer compounding and catalyst coating operations. These studies demonstrate that DEM is increasingly used as a predictive tool to refine static mixer geometries, reduce experimental prototyping, and support efficient scale-up from laboratory to industrial systems. These examples underscore the practical relevance of DEM as a predictive tool, bridging the gap between theoretical model development and industrial implementation.

Experimental validation of DEM often targets key macroscopic and microscopic flow characteristics, such as RTD, mixing indices, velocity fields, and segregation patterns. Particle tracking using PIV, MRI, or X-ray tomography provides detailed velocity and concentration data for comparison with DEM predictions [107], while tracer or dye experiments benchmark residence times and mixing dynamics.

Critical material properties—friction, restitution, and cohesion—are calibrated through independent tests (such as shear cell, angle of repose, or direct contact measurements) to improve model fidelity [108], and sensitivity analyses identify influential parameters. Among the mentioned experimental approaches, shear cell calibration is generally considered the most dependable for characterizing industrial powders. Unlike particle impact or angle of repose tests, which primarily capture bulk or surface flow tendencies under specific conditions, shear cell testing provides quantitative and reproducible parameters such as the cohesion, internal friction angle, and flow function. These parameters are directly relevant to powder flowability in industrial-scale handling and can be incorporated into constitutive models and numerical simulations (DEM or continuum-based frameworks). Geometry and boundary conditions are validated by comparing DEM flow patterns and pressure drops with experimental data from scaled or full-scale mixers [109].

DEM validation includes qualitative and quantitative aspects [110]. Qualitative validation compares flow patterns, mixing uniformity, and segregation visually, while quantitative validation uses metrics like RMSE, R², or deviations from experimental mixing indices.

Validation is challenging due to granular flow complexity and experimental limitations, such as restricted spatial resolution or difficulty measuring internal variables [102]. Hybrid approaches combining DEM with reduced-order models or surrogate experiments help bridge these gaps. Direct comparison with experimental data is essential for confirming DEM accuracy and guiding model refinement, using methods from bulk mixing tests to advanced particle tracking [85].

Physical mixing experiments provide foundational benchmarks for mixer performance [111], blending granular materials distinguished by colour, size, or density and assessing mixing via indices, concentration variance, and segregation along the mixer. These macroscopic measures validate DEM predictions of key flow and mixing phenomena.

Particle tracking techniques enable detailed validation at the particle scale. Methods such as PIV, MRI, PEPT, and X-ray tomography offer spatially and temporally resolved data on velocities, trajectories, and distributions [112]. PIV captures near-surface velocities in transparent systems, PEPT tracks tracer particles in 3D, and MRI or X-ray tomography visualize internal particle arrangements in opaque or dense systems.

While high-resolution methods such as MRI and X-ray tomography provide unparalleled insight into local flow and mixing dynamics, their application remains largely restricted to laboratory-scale systems due to size constraints, imaging time, and equipment cost. For industrial-scale static mixers, non-invasive optical and tracer-based techniques—such as PIV, PEPT, or advanced tracer residence time distribution (RTD) analyses—offer more practical validation options, albeit with lower spatial resolution. Significant validation gaps persist for large, opaque, or high-load systems, where direct visualization is limited and scaling effects are not fully captured. Developing hybrid validation strategies that combine coarse-scale tracer data with DEM or CFD predictions represents a promising direction for improving model reliability under real operating conditions.

DEM simulations reproduce experimental observations by tracking particle positions, velocities, and species concentrations over time [113]. Quantitative validation uses metrics such as velocity histograms, residence time distributions, and spatial concentration profiles, with good agreement indicating accurate modelling of particle interactions, boundaries, and flow dynamics.

Challenges remain due to scale differences, measurement uncertainties, and simplifications in both simulations and experiments [114]. DEM often employs idealized particle shapes or simplified contact laws, while experimental data may include noise or sampling bias, requiring iterative calibration and sensitivity analyses.

DEM provides detailed particle-scale insights into granular flow, but comparisons with other numerical methods, such as CFD and FEM, highlight their strengths, limitations, and appropriate applications [21].

CFD models granular materials as a continuum, often using Eulerian–Eulerian or Eulerian–Lagrangian frameworks [115]. It excels at simulating fluid–solid interactions in slurries or pneumatic conveying and requires less computational effort than DEM for large-scale systems [116], but lacks particle-level detail. Coupled DEM-CFD approaches combine CFD for fluid flow with DEM for discrete particle tracking [117].

FEM is used for deformation and stress analysis in solids and fluids but is less suited for discrete granular flow due to continuum assumptions [118]. It is useful for simulating mixer component deformation, powder compaction, or cohesive granular masses, but cannot capture individual particle interactions.

DEM provides superior resolution of micro-scale phenomena—collisions, friction, and segregation—essential for static mixers handling dry or heterogeneous granular materials [119]. Its high computational cost limits applications to smaller or simplified systems.

Hybrid frameworks combining DEM with CFD or FEM address DEM’s limitations [120]. DEM-FEM simulates particle flow with flexible mixer components, while DEM-CFD captures multiphase fluid–particle interactions, enhancing realism and applicability to industrial scenarios.

Sensitivity analysis and parameter calibration are essential for robust DEM simulations of granular flow in static mixers [121]. Key input parameters—particle stiffness, friction (particle–particle and particle–wall), restitution, density, shape, and cohesion—are systematically evaluated. Calibration uses experimental benchmarks such as shear cell tests, angle of repose measurements, and single-particle collisions to align simulations with observed behaviour and reduce uncertainty [102].

Sensitivity analysis complements calibration by quantifying how input variations affect outputs such as mixing indices, RTD, velocity profiles, and segregation [5]. Methods range from one-factor-at-a-time (OFAT) to global approaches, including variance-based techniques and DOE. Identifying influential parameters guides calibration and highlights sources of model uncertainty.

In static mixers, particle friction and restitution strongly affect flow and mixing, while stiffness mainly impacts computational stability. Cohesion and particle shape are critical for fine or moist powders, where interparticle forces alter flow behaviour. Iterative calibration and sensitivity cycles refine parameter ranges and validate DEM against experimental data, enhancing predictive confidence for design, scale-up, and optimization [122]. Emerging methods use optimization algorithms and machine learning to accelerate calibration and improve efficiency.

DEM is a powerful tool for analysing and optimizing static mixer performance, offering particle-level resolution and quantitative metrics to assess mixing efficiency, flow patterns, and segregation. Accurate modelling of interactions, geometry, and material properties, along with sensitivity analysis and validation against experiments-is critical for predictive capability. Coupling DEM with experimental techniques and complementary methods like CFD or FEM enhances model robustness and applicability. This integrated approach enables informed design, optimization, and scale-up of efficient static mixing systems for industrial granular processes.

Table 5. DEM Analysis of Static Mixer Performance.

Section	Category	Details	References
Granular Flow	Industrial relevance	Granular materials are essential in pharmaceuticals, food, mining, chemical engineering, agriculture, and construction; mixing ensures product uniformity and process performance.	[1,2,9]
	Challenges	Non-linear, segregating behaviour; influenced by particle shape, size, density, cohesion, friction; issues like channeling, dead zones, radial segregation.	[4,13,15,16]
	Experimental approaches	Insights into flow regimes, particle distribution, and mixing quality; limited by resolution and difficulty in visualizing opaque granular media.	[5,20]
Static Mixers	Advantages	No moving parts, low energy consumption, continuous operation, reduced maintenance, enhanced reliability.	[3,10,11]
	Types and geometry	Kenics®, Sulzer SMX/SMXL®, helical, SAR, and novel AM designs; geometry represented by analytical primitives or CAD meshes; adaptive meshing and periodic geometries balance accuracy and computational cost.	[21,45,61,62,66,67,69]
	Design and optimization challenges	Achieving uniform mixing without segregation; scale-up difficulties; absence of standardized design rules; requires careful geometry tailoring.	[10,17,18,19]
DEM Modelling	Purpose and applications	Investigates particle-scale mechanisms (transport, shear, dispersion, segregation); supports design optimization, virtual prototyping, predictive analysis, and scale-up.	[5,27,28,29,30,31,44,45,60,89,94,95]
	Material and boundary properties	Particle size, shape, friction, cohesion, density, elastic modulus, restitution coefficient; rigid walls; batch, continuous, or quasi-steady simulations; inlet/outlet conditions; ramp-up injection avoids artefacts.	[5,43,66,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86]
	Outputs and flow phenomena	RTD, mixing quality indices (Lacey’s, Danckwerts, entropy-based), velocity fields, dispersion, dead zones, segregation, channeling, particle–particle/wall interactions, shear/stress fields, flow visualization.	[14,44,63,64,66,87,88,89,90,91,95,96,97,98,99,100]
	Contact models	Hertzian (nonlinear, accurate, heavy) and linear spring-dashpot (simpler, widely used); tangential models include Coulomb friction, rolling/torsional resistance, cohesion.	[5,8,44,46,47,48,49]
	Advanced modelling and comparison	Multiphase coupling (DEM–CFD, reactive/thermal systems); one-/two-/four-way strategies; drag, lift, turbulence, capillarity; DEM vs. CFD/FEM; hybrid models combine strengths.	[21,86,87,88,115,116,117,118,119,120]
	Sensitivity, calibration and validation	Input parameters tuned via experiments (shear cell, angle of repose); validated against tracer studies, PIV, MRI, PEPT, X-ray tomography.	[5,85,101,102,107,108,109,110,111,112,113,114,121,122]
	Strengths and limitations	High particle-level realism; supports optimization and scale-up. Limitations: high computational cost; industrial-scale DEM often requires hybrid or reduced-order approaches.	[5,65]
	Future directions	Enhanced measurement techniques, multi-scale and hybrid modelling, machine learning integration, improved contact models, better parameter calibration.	[21,26]

summarizes the key aspects of using DEM models applied to evaluate static mixer performance. It includes mixing quality indices, residence time distribution, shear and stress analysis, segregation assessment, and flow visualization. The particle–particle and particle–wall interactions, design optimization strategies, and validation approaches are outlined. The comparison with CFD/FEM methods, parameter calibration, and industrial applications further highlight the scope and significance of DEM in advancing static mixer research and design.

One-way coupling is most suitable for dilute particulate systems or tracer studies, where particles are influenced by the carrier fluid but their feedback on the flow field is negligible, for example, monitoring dispersion efficiency of seeds or microcapsules in low-solid suspensions. Two-way coupling becomes relevant in dense suspensions or slurries, such as mixing starch or protein suspensions in food processing, where the momentum exchange between the particles and the fluid affects both phases. Four-way coupling is necessary in highly concentrated systems involving frequent particle–particle and particle–wall collisions, as encountered in granular or paste-like materials processed in the chemical and pharmaceutical industries (powder blending, granulation, or pigment mixing).

5. Strengths, Limitations and Challenges of DEM

DEM is a powerful tool for simulating granular flow in static mixers, providing particle-scale insights often inaccessible to experiments or continuum models [109]. Its strengths include detailed resolution of particle motions, collisions, and interactions, enabling analysis of force chains, contact dynamics, and segregation [123]. DEM allows precise evaluation of mixing patterns, RTD, shear environments, and the effects of particle properties (size, shape, friction, cohesion) on flow, while flexibly handling complex geometries and boundary conditions.

DEM enables parametric studies and virtual prototyping, allowing efficient exploration of design, operational, and material variations without costly experiments [124]. Coupled with advanced visualization, DEM aids in optimizing mixer performance, scale-up, and product quality. However, DEM faces challenges from high computational costs due to simulating millions of particles over many time steps, often requiring simplifications that may affect accuracy [74]. It also depends on empirical contact models and parameters—such as stiffness, friction, and restitution—that are difficult to measure precisely, making calibration and validation essential [27]. Modelling cohesive, adhesive, or irregularly shaped particles remains a continuing challenge.

DEM primarily focuses on particulate phases and is less suitable for fluid dynamics unless coupled with CFD or continuum methods, which increases complexity and computational cost [21]. Nevertheless, DEM provides detailed particle-level insights [125] that are difficult to obtain experimentally or via continuum models, enabling a deeper understanding of mechanisms driving mixing performance. It simulates individual particle trajectories, rotations, and interactions, capturing microscale phenomena such as collisions, friction, rolling, sliding, and force chain formation [24]. This allows identification of high-stress regions, clustering, and localized flow patterns affecting macroscopic outcomes. DEM data also support computation of metrics like residence time distributions, mixing indices, velocity, and shear rate profiles, facilitating robust statistical analysis and visualization [126].

DEM captures particle-scale effects of material heterogeneity, including density, shape, and surface roughness [127], which are often neglected in continuum models. This is crucial for complex or fragile granular systems, where individual particle behaviour affects overall performance. DEM outputs also reveal flow inefficiencies such as dead zones, preferential pathways, or segregation, guiding targeted modifications to mixer geometry or operating conditions. Particle-level modelling enables the study of dynamic processes like breakage, attrition, agglomeration, and coating [128,129]. Accurate parameter calibration remains critical, as DEM’s predictive reliability depends on correctly specifying input parameters, which are often difficult to determine due to the complex nature of granular materials [14].

Applying DEM to static mixers involves several critical challenges and modelling choices that determine the reliability of simulation outcomes [5,6]. The foremost difficulty lies in accurately representing particle properties, such as size, shape, surface roughness, and mechanical characteristics, which strongly influence contact forces, frictional behavior, and cohesion. Selecting appropriate contact models and parameter values is therefore essential, yet often limited by experimental uncertainty and scale dependence [14]. Calibration of friction, restitution, and stiffness parameters remains nontrivial, as measurements obtained from shear cells, tribometers, or collision tests may not directly translate to DEM conditions. Additionally, scaling from laboratory to industrial geometries introduces further complexity, requiring validation strategies and potential parameter adjustments to maintain physical consistency [57]. Together, these issues highlight the need for systematic parameter calibration, geometry-specific optimization, and multi-scale validation approaches in DEM applications to static mixers.

The absence of standardized calibration protocols in DEM adds variability and uncertainty, as differences in methodologies, experimental setups, and interpretation hinder direct comparison and the establishment of universal parameter sets [130]. High-fidelity calibration is further limited by DEM’s computational cost, restricting the number of parameter combinations that can be explored [131]. A major challenge of DEM for static mixers remains its high computational demand, as tracking individual particle motions and interactions requires substantial processing power, memory, and simulation time [92].

The computational expense of DEM simulations grows rapidly with the number of particles, as the method involves continuous calculation of contact forces and motion integration for every particle and its neighbours at each time step. For industrial-scale static mixers, which may contain millions of particles, this can lead to prohibitive simulation times, often spanning days or weeks on conventional computing resources.

DEM’s computational cost increases rapidly with particle number, as contact forces and motions must be calculated for each particle and its neighbours at every time step. Industrial-scale static mixers with millions of particles can require days or weeks of simulation. Small integration time steps, needed to accurately resolve collisions, further multiply the number of steps [21]. Memory demands are also high, storing positions, velocities, forces, contact histories, and material properties. Strategies to improve DEM scalability have been developed to mitigate these challenges [132]:

• Parallel computing on multi-core CPUs, GPUs, or HPC clusters distributes workload, reducing simulation time.
• Domain decomposition partitions the simulation space for concurrent processing, enabling larger systems.
• Model simplifications—spherical particles, fewer particles, or coarse-graining—lower computational load but may reduce fidelity.
• Adaptive time stepping and event-driven algorithms refine resolution during collisions, improving efficiency without losing accuracy.

Despite advances, trade-offs between model detail, system size, and simulation time remain critical in DEM [133]. Modellers must balance physical realism with computational resources, often using hybrid approaches or representative subsystems. Accurately representing particle shape and cohesion is essential for realistic granular flow in static mixers but remains challenging [5].

Most DEM models simplify particles as spheres or spheroids to reduce computational cost and simplify contact calculations [134]. Yet irregular, angular, elongated, or flat particles influence packing, flow, and mixing, affecting interlocking, rolling resistance, and orientation-dependent friction.

Advanced approaches include multi-sphere or clumped-sphere models, representing irregular particles as rigid assemblies of overlapping spheres [135], and polyhedral or superquadric models, capturing angular and non-convex geometries at higher computational cost [136]. The choice between clumped-sphere, polyhedral, and voxel-based models primarily depends on the trade-off between computational cost and geometric fidelity required for the application. For large-scale industrial simulations, where the focus is on bulk flow behaviour rather than detailed contact mechanics, clumped-sphere models are often preferred due to their computational efficiency and ease of implementation. In contrast, polyhedral models offer a good balance between shape accuracy and computational expense, making them suitable for systems where particle interlocking, angularity, or anisotropic packing play an important role. Voxel-based models provide the highest geometric fidelity and are most appropriate for irregular or highly non-convex particle geometries, but they demand significantly greater computational resources and are typically reserved for small-scale or high-precision studies.

Voxel-based and mesh-based shape representations offer detailed particle geometry but require significant processing resources, limiting system size or simulation duration.

Alongside shape complexity, cohesive forces-arising from van der Waals attractions, liquid bridges, electrostatic interactions, or chemical bonding-play a pivotal role in many granular systems, especially powders, moist materials, and fine particles. Cohesion affects flowability, agglomeration, segregation, and mixing homogeneity, often leading to phenomena such as arching, clogging, or reduced shearability in static mixers.

Incorporating cohesion in DEM adds force models accounting for short-range attractions influenced by particle separation, contact area, and environmental conditions (e.g., humidity) [137]. Common models include JKR theory, liquid bridges, and simplified adhesive laws, which enhance realism but increase computational and calibration demands.

Modelling complex shapes with cohesion is especially challenging, as shape affects contact area and force distribution, influencing cohesion magnitude and direction [138]. Accurate simulations require careful integration of shape and cohesion models, validated experimentally.

Advances in computing and algorithms are enabling broader use of sophisticated shape and cohesion modelling in DEM, improving predictive accuracy and supporting static mixer design and optimization across diverse materials and conditions [24].

DEM provides unmatched particle-scale insight into granular flow and mixing in static mixers, enabling accurate, predictive modeming of complex behaviours. However, its broader application is limited by computational demands, parameter calibration challenges, and modeming assumptions. Overcoming these limitations requires advanced calibration techniques, improved algorithms, hybrid modeming approaches, and experimental validation. Progress in modelling particle shape, cohesion, and scalability will further enhance DEM’s effectiveness in both research and industrial applications.

Table 6. Capabilities, Advantages, and Limitations of DEM in Static Mixer Simulations.

Aspect	Description	References
Particle-scale resolution	Tracks individual particle motions, collisions, rotations, and interactions; resolves microscale phenomena (force chains, contact dynamics, segregation).	[24,123,125]
Mixing behaviour analysis	Quantifies residence time distributions, velocity fields, shear rates, mixing indices; identifies dead zones, preferential pathways, clustering.	[109,126]
Material property influence	Captures effects of particle size, shape, density, friction, cohesion, surface roughness on flow and mixing behaviour.	[123,127]
Geometry and boundary modelling	Flexibility to represent irregular geometries, mixer internals, and complex boundary conditions.	[109]
Virtual prototyping and parametric studies	Enables systematic evaluation of design variations, operating conditions, and material properties.	[124]
Process visualization	Provides advanced visualization of flow fields, stress maps, and particle trajectories for performance diagnosis and optimization.	[125]
Expanded applications	Supports modelling of breakage, attrition, agglomeration, and coating processes beyond simple mixing.	[128]
Computational cost	DEM requires tracking millions of particles with small time steps → high memory and CPU demand; industrial-scale simulations remain challenging.	[21,74,92,132,133]
Simplifications and scaling	Often relies on spherical particles, reduced counts, or scaled-down mixers → may reduce accuracy and generalizability.	[5,134]
Contact model limitations	Uses empirical contact laws (friction, restitution, stiffness, cohesion) that are difficult to measure and calibrate accurately.	[14,27,94]
Parameter calibration	Requires iterative, multi-parameter tuning; influenced by particle variability, environmental conditions, and lack of standard protocols.	[94,129,130,131]
Fluid-phase coupling	DEM alone cannot capture fluid dynamics; DEM–CFD or DEM–FEM coupling needed for multiphase systems, but increases cost/complexity.	[21]
Shape modelling	Advanced models (clumped spheres, polyhedral, superquadric, voxel/mesh) improve realism but raise computational demands.	[24,134,136]
Cohesive forces	Cohesion models (JKR, liquid bridge, adhesive laws) capture van der Waals, capillary, or electrostatic effects but add complexity and calibration challenges.	[137,138]

outlines the principal strengths and constraints of applying the DEM modelling to static mixer simulations. It highlights DEM’s ability to resolve particle-scale dynamics, analyse mixing behaviour, and capture the influence of material properties and mixer geometries. Applications such as virtual prototyping, process visualization, and modelling of complex particle phenomena are emphasized alongside key limitations, including computational costs, simplified assumptions, contact model uncertainties, and challenges in parameter calibration. The role of coupling with fluid-phase models and advanced shape or cohesion representations is also considered, underscoring both the versatility and ongoing challenges of DEM in industrial-scale mixing studies.

6. Future Directions

Despite advances in DEM and experiments on granular flow in static mixers, several research areas remain.

Improving computational efficiency and scalability is a priority. Enhanced algorithms, including advanced parallelization, GPU acceleration, and adaptive resolution techniques, will allow simulation of larger, industrially relevant systems over longer timescales without losing accuracy [139].

Refinement of particle shape and material models is also essential. Future work should incorporate realistic non-spherical geometries, surface roughness, deformability, and improved cohesive force models capturing complex interparticle interactions under varying conditions [140].

Finally, multi-physics coupling offers opportunities to expand DEM’s scope. Integration with CFD, thermal or chemical reaction models will enable simulation of multiphase, reactive, or temperature-dependent granular processes, supporting comprehensive analysis of industrial operations [141].

Machine learning and data-driven approaches can accelerate parameter calibration, sensitivity analysis, and surrogate modelling by leveraging large DEM and experimental datasets to optimize inputs and predict mixing outcomes with lower computational cost [142]. Surrogate models leveraging machine learning are increasingly used to reduce DEM simulation costs while maintaining predictive accuracy. Artificial neural networks (ANNs) can learn the nonlinear mapping between input parameters (particle size, mixer geometry, rotational speed) and DEM outputs such as residence time distribution, mixing index, or energy dissipation. Gaussian process regression (GPR) provides a probabilistic framework for predicting DEM results with associated confidence intervals, enabling efficient exploration of the input space with fewer full DEM runs. Other techniques, including random forests and support vector regression, have been applied in sensitivity analyses and parametric studies, but ANNs and GPR remain the most effective for high-dimensional systems where computational savings are critical. These surrogate approaches allow practitioners to conduct rapid “what-if” analyses, optimize mixer designs, and perform uncertainty quantification with significantly lower computational cost compared to full DEM simulations.

Improved experimental techniques for in situ characterization—such as advanced imaging, tomography, and tracer methods—will provide richer validation datasets, especially under industrial conditions [21].

Addressing scale-up challenges is also crucial. Linking laboratory-scale findings to full-scale mixer performance using validated DEM models will support reliable process design and optimization [143]. While DEM has matured into a reliable tool for research and validation, particularly for laboratory- and pilot-scale systems, its direct use for industrial-scale design remains limited. The main challenges include the computational cost of simulating millions of particles, uncertainties in material property calibration, and complexities associated with multiphase interactions under real process conditions. Nonetheless, DEM increasingly contributes to hybrid modelling and digital twin frameworks, providing validated local-scale insights that support design optimization, process troubleshooting, and scale-up decisions, thereby enhancing confidence in industrial applications even if it is not yet fully predictive on its own.

To enable wider industrial use of DEM, improvements are needed in computational efficiency (GPU acceleration, parallelization, adaptive resolution), robust validation protocols linking lab- and plant-scale mixers, and integration with hybrid or surrogate models that reduce simulation time while maintaining accuracy. Such advances would allow DEM to inform practical decisions in pharmaceutical blending, food ingredient mixing, and chemical process optimization, providing designers with reliable, actionable insights for scale-up, quality control, and process efficiency.

From an industrial perspective, future developments in DEM are expected to significantly shorten static mixer design cycles by enabling rapid virtual prototyping and optimization before fabrication. Through detailed predictions of mixing efficiency, segregation patterns, and energy consumption, DEM can support informed design decisions and minimize costly trial-and-error testing. Furthermore, machine-learning-assisted parameter calibration offers a promising route to reduce model setup time and improve generalization across different materials and operating conditions. Such integration of data-driven calibration and process simulation could accelerate the adoption of DEM as a routine design and scale-up tool in the pharmaceutical and food industries, where reproducibility, efficiency, and regulatory compliance are of key importance. The computational intensity of DEM, particularly for industrial-scale systems with millions of particles, remains a major bottleneck [5]. Algorithmic improvements targeting contact detection and force calculations—using hierarchical spatial partitioning (octrees, k-d trees), linked-cell methods, neighbour lists, and adaptive techniques—can reduce complexity and enhance efficiency [144].

Parallel computing is essential for scaling DEM simulations, with multi-core CPUs, GPUs, and distributed clusters enabling concurrent particle calculations and substantial speed-ups, often by one to two orders of magnitude [34].

Hybrid modelling, combining DEM with continuum or coarse-grained approaches, reduces the number of explicitly simulated particles while preserving key flow features, applying DEM only where detailed resolution is needed [145].

Adaptive time-stepping and event-driven algorithms further enhance efficiency by refining time resolution during frequent or localized collisions [146].

Machine learning for surrogate modelling offers a promising approach to accelerate DEM simulations [147], approximating particle behaviour and mixing outcomes from DEM-generated data at reduced computational cost.

Accurate particle interaction modelling remains central to DEM’s predictive power in static mixers [43]. Traditional Hertzian contact models with spring-dashpot elements work for spherical, non-cohesive particles but struggle with irregular shapes, cohesion, plasticity, or time-dependent effects.

Future contact models should incorporate particle shape complexity, adhesive/cohesive forces, plastic or viscoelastic deformation, and time-dependent phenomena like creep and fatigue. Experimental micromechanical data are crucial for parameter calibration and validation, enhancing DEM’s ability to predict flow resistance, shear behaviour, and agglomeration, thereby broadening its industrial applicability.

Integrating DEM with complementary computational techniques advances the simulation of granular flow in static mixers [5]. While DEM captures particle-scale interactions, many practical flows involve coupled phenomena—fluid flow, heat transfer, chemical reactions, or structural deformation—that DEM alone cannot fully address. Hybrid frameworks combining DEM with other methods enable multiphysics analyses, improving accuracy and applicability.

DEM-CFD coupling simulates fluid–particle interactions in gas-solid or liquid-solid systems, with DEM tracking particle motion and collisions and CFD modelling the fluid’s drag, lift, and pressure effects [148]. This approach predicts particle trajectories, dispersion, and mixing efficiency under realistic multiphase conditions, aiding design and scale-up of mixers handling wet or fluidized granular materials.

DEM-FEM coupling captures elastic or plastic deformation of particles and mixer components, stress distributions, wear, and potential damage, supporting equipment durability and flow-structure feedback analysis [149]. Other methods, such as Lattice Boltzmann Method (LBM) [150] for fluid flow or Population Balance Models (PBM) for particle size evolution [151], can be combined with DEM to address multiphase or multiphysics challenges [152]. For example, DEM-PBM models particle breakage, agglomeration, and growth, linking microscale dynamics to particle size distributions that affect mixing quality.

LBM model has emerged as a powerful numerical approach for simulating fluid–particle interactions in granular systems and offers a complementary perspective to conventional DEM simulations. Unlike traditional Navier–Stokes solvers, LBM models fluid flow at a mesoscopic scale by tracking particle distribution functions on a lattice grid, which enables efficient handling of complex geometries, multiphase flows, and moving boundaries, Figure 6.

When coupled with DEM, LBM provides a fully resolved description of particle–fluid interactions, capturing hydrodynamic forces acting on individual particles. In such simulations, DEM tracks the discrete motion of granular particles (translational and rotational dynamics), while LBM resolves the surrounding fluid flow, including local pressure and velocity fields. This approach has been increasingly applied to study granular flows in static mixers under wet or high-moisture conditions, sediment transport, and slurry mixing.

The coupling of Lattice Boltzmann Method (LBM) and DEM offers several advantages, including accurate representation of particle–fluid interactions such as drag, lift, and added-mass effects, efficient handling of complex mixer geometries and dynamic boundaries, and the capability to simulate multiphase flows and interactions with fine particles or high-viscosity fluids. This approach also presents limitations and challenges: computational costs are substantially higher than DEM-only simulations, particularly for dense particle suspensions; the parameterization of fluid–particle interaction forces, such as drag correlations, can significantly influence accuracy; and coupled simulations require careful time-step synchronization between DEM and LBM solvers to ensure numerical stability. Recent studies have demonstrated that LBM–DEM coupling provides improved predictive capability for particle distribution, mixing efficiency, and segregation patterns in static mixers compared with DEM alone, particularly when fluid effects are non-negligible. Nevertheless, for dry, non-cohesive granular flows where fluid effects are minimal, DEM-only simulations remain computationally more efficient and sufficiently accurate.

Hybrid frameworks enable multiscale simulation, applying DEM in critical regions and continuum methods elsewhere, reducing computational cost while maintaining local accuracy [153]. Coupled methods pose challenges in numerical stability, data exchange, and computational overhead, motivating research on efficient algorithms and parallel computing.

DEM is invaluable for developing and optimizing static mixer designs and analysing complex industrial granular flows [154], providing particle-level insights into how geometry and operating conditions influence mixing efficiency, segregation, wear, and throughput beyond what experiments or continuum models can achieve.

Table 7. Comparison of DEM with CFD, FEM, and Hybrid Modelling Approaches for Static Mixers.

Research Area	DEM Advantages/Focus	Comparison/Appropriate Scenarios	References
Computational efficiency and scalability	Advanced algorithms (parallelization, GPU acceleration, adaptive resolution) enable larger particle simulations	Hybrid methods can reduce cost in less critical regions	[5,34,92,139,144,145,146]
Particle shape and material modelling	Realistic particle geometries, friction, cohesion, deformability	CFD/FEM treat particles as continuum; less detail	[5,24,99,134,140]
Multi-physics coupling	Integration with CFD, FEM, LBM, PBM, thermal/chemical models	Hybrid models capture coupled phenomena efficiently	[5,21,43,141,148,149,150,151,152,153]
Machine learning and data-driven methods	Parameter calibration, surrogate modelling, predictive acceleration	Applicable across all simulation types	[142,147]
Experimental validation and data acquisition	DEM provides detailed particle-scale insights to validate experiments	CFD/FEM rely more on macroscopic measurements	[21]
Scale-up strategies	Links lab-scale DEM results with full-scale mixer performance	Hybrid or continuum models used for large-scale predictions	[92,129,143,155]
Advanced contact models	Captures adhesion, cohesion, plastic/viscoelastic deformation	Continuum models cannot represent individual particle contacts	[43]
Hybrid and multiscale modelling	Combines DEM detail with continuum efficiency	Ideal for industrial-scale simulations	[145,151]
Industrial applications and design innovation	Virtual prototyping, optimization, troubleshooting, novel materials	Supports design decisions without extensive experiments	[92,99,154,155]
Digital twins and process monitoring	Real-time DEM coupling with sensor data for adaptive control	Enhances predictive maintenance and process optimization	[154]

highlights the emerging research areas that aim to advance the use of the DEM models in static mixer studies. Key directions include improving computational efficiency and scalability, enhancing particle shape and material modelling, and extending DEM through multi-physics and hybrid approaches. The integration of machine learning, improved experimental validation, and scale-up strategies is emphasized as critical for bridging laboratory and industrial applications. The advances in contact models, digital twin development, and process monitoring underline the growing role of DEM in enabling innovation and reliability in static mixer design and operation.

In the realm of novel static mixer designs, DEM facilitates virtual prototyping by allowing designers to test and refine complex internal element shapes, configurations, and arrangements prior to physical fabrication. This capability accelerates innovation by providing insights into how subtle changes in blade angle, element twist, baffle placement, or modular components affect flow patterns and particle trajectories. Such studies support the development of mixers tailored to specific granular materials or process objectives, including enhanced homogeneity, reduced energy consumption, or minimized segregation.

DEM is critical for scaling mixer designs from laboratory to industrial scale, simulating particle flow under varied sizes and operational parameters to identify scale-up challenges such as dead zones, arching, or uneven residence times [155]. It also supports optimization and troubleshooting in industrial processes like mixing, conveying, coating, or drying [92], predicting the effects of feed rate, moisture, temperature, or rotational speed on granular behaviour and mitigating issues such as clogging, particle breakage, or wear.

DEM enables assessment of complex granular materials with irregular shapes, polydispersity, or cohesion, guiding mixer customization for pharmaceuticals, food, chemicals, and additive manufacturing [99]. Integration with real-time sensor data and process monitoring facilitates digital twins and predictive control, enabling adaptive management and continuous process improvement.

Future research in DEM for static mixers will benefit from advances in computational power, improved contact models, and integration with other modelling techniques. These developments will enhance scalability, realism, and predictive accuracy, bridging the gap between academic studies and industrial applications. By combining DEM with experimental innovations and data science, researchers can accelerate mixer design, optimize processes, and enable robust, efficient granular handling across various industries.

7. Conclusions

This review has systematically examined the evolution, capabilities, and challenges of the Discrete Element Method (DEM) in modelling granular flow and mixing behaviour in static mixers. By integrating insights from theoretical modelling, experimental validation, and numerical simulation, DEM has demonstrated unique advantages in resolving micro-scale particle dynamics, capturing mixing heterogeneity, and predicting process behaviour under varying design and operating conditions.

Although DEM provides detailed mechanistic insights, its industrial adoption remains constrained by high computational costs, complex parameter calibration, and limited scalability to realistic process conditions. To overcome these barriers, future research should prioritize hybrid and multiphysics frameworks that combine DEM with complementary methods such as Computational Fluid Dynamics (CFD) and Finite Element Method (FEM). Such coupling enables simultaneous resolution of solid–fluid interactions, particle deformation, and stress distribution, thus improving the fidelity of simulations for viscoelastic, cohesive, and multiphase materials typical in pharmaceutical, food, and chemical processing.

Advancements in machine-learning-assisted DEM calibration and AI-based surrogate modelling are expected to substantially reduce computational demands and accelerate parameter optimization, ultimately shortening design and validation cycles. Data-driven hybrid models will also facilitate predictive process control and adaptive mixer design, linking simulation outputs directly to measurable process parameters.

Looking forward, DEM is poised to become a cornerstone methodology for process design and optimization, provided that future developments emphasize: rigorous model validation through standardized experimental datasets, algorithmic acceleration via GPU and cloud-based computing, integration into multiphysics simulation environments, and establishment of industrially relevant benchmarks for performance evaluation.

When these advances converge, DEM, supported by CFD, FEM, and AI-driven calibration—will evolve from a primarily research-oriented technique into a routine design and optimization tool, enabling digital-twin development and real-time process insight for static mixers in diverse industrial applications.