Coarse graining Euler-Lagrange simulations of cohesive particle fluidization
Graphical abstract
The left panel shows the determination of relative smoothing lengths for different parcel sizes by investigating the relative slip velocity of fluidized powders, while the right panel depicts the comparison of the original cohesive system with the coarse-grained system based on the identified smoothing length and stress-based scaling rules.
Introduction
Cohesive particulate systems, such as agitated fine powders and wet fluidized beds, arise in a variety of industrial processes. Examples include chemical processes such as cracking of hydrocarbons [1], and the use of cohesive (often wet) powders in the pharmaceutical industry [2]. Mathematical description of the gas-solid flow characteristics in these industrial processes is useful for equipment design and optimization. The occurrence of coherent, inhomogeneous flow structures (often called “mesoscale structures”) in these systems affects essential process characteristics. These structures greatly add to the complexity of modeling these flows even in dry systems [3,4]. Even more challenging are cohesive systems: the addition of liquid, or the dominance of van der Waals forces in fine powders, increases the complexity of the system behavior markedly. For example, the presence of a liquid film on the particles enables liquid bridge formation during particle–particle or particle–wall interactions. These liquid bridges may cause desired or unwanted agglomeration [5]. The flow structure is impeded by agglomeration, which leads to slumping or roughening of the fluidization process. Overall, inter-particle cohesion may affect the stresses developing in the granular phase, the rates of interphase (gas-particle) exchange of momentum, heat and mass, and the rates of chemical reactions. This impact on fluidization and process performance have been explored by several researchers in the past [[6], [7], [8], [9], [10], [11]]. In what follows, we focus on computational models that are used to describe cohesive gas-particle mixtures, as well as the associated challenges.
Gas-particle systems can be simulated by one of the following methods: (i) Direct Numerical Simulation (DNS), (ii) Euler–Euler (EE), and (iii) Euler–Lagrange (EL) methods [11]. In DNS, the Newton's equation of motion is solved for each particle and the interstitial fluid flow around the particles is completely resolved by solving the Navier–Stokes equation [12], which makes the approach computationally very demanding. As a result, DNS is limited to small systems (typically 103–105 particles). DNS is mainly used for the simulation of microscale flow features to derive closures for other simulation approaches, which is not the focus of our present study. The EE method, commonly referred to as the two-fluid model (TFM), treats both phases as continua. TFMs rely on solving locally averaged equations [13,14]. This concept fits well for large-scale flow characteristics, although constitutive models are needed for the interaction force between the two phases, and for the effective stress in each phase. TFMs are frequently used for noncohesive systems [13,15], however, rarely for cohesive systems since a rigorous set of constitutive models (most importantly for the granular stress tensor) is currently not available.
In the EL approach commonly referred to as the Computational FluidDynamics–Discrete Element Method (CFD–DEM) [[16], [17], [18]] one solves the locally averaged equations for the fluid phase and the Newton's equation of motion for the individual particles while resolving particle-particle contact through some version of the spring-dashpot model [2,19]. In addition to the specification of the characteristics of the spring-dashpot model, the CFD-DEM approach requires closure relations for fluid–particle interactions, e.g., the drag force. Results from DNS simulations have been used to construct drag closure relations in the literature; for example, see [12,20] The CFD-DEM approach offers opportunities to specify a wide range of inter-particle forces to model cohesion [11,[21], [22], [23]], as well as other particle-particle transfer processes, e.g. liquid redistribution upon collisions [[24], [25], [26]], making it attractive for the investigation of flows of cohesive powders. The present study is concerned with scaling issues related to coarse-grained CFD-DEM approach.
Nowadays, CFD-DEM simulations with several million particles can be performed with standard computer clusters. However, as industrial scale processes typically involve orders of magnitude more particles, some kind of coarse-graining is necessary if one wishes to apply this approach to probe industrial processes. Such a coarse-graining approach effectively reduces the number of computational entities that need to be tracked. The challenge associated with coarse-grained models is that they require modifications to the way in which particle-fluid and particle-particle interaction are calculated, which are not straightforward to identify. Broadly, there are two different strategies for reducing the complexity of the EL simulations: (i) in the Discrete Parcel Method (DPM), which is the focus of our present study, each “parcel” consists of a specified number of primary particles and is treated as a single large sphere and all parcel-parcel and parcel-wall interactions are resolved [27,28]; (ii) alternately, in the Multiphase Particle-In-Cell method (MP-PIC) [29], the parcel-parcel interactions are not resolved, but are treated through a postulated effective stress model. Clearly, DPM is computationally more intensive than MP-PIC as it requires resolution of contact interactions. However, as validated effective stress models are not available for cohesive systems at the present time, the MP-PIC approach is not suitable for probing the effect of inter-particle forces on flow characteristics, and hence it is not pursued in this study.
In DPM, DEM-inspired contact force models are used to quantify parcel-parcel and parcel-wall interactions, where one can also introduce inter-parcel forces as effective equivalents for the inter-particle forces between the primary particles (e.g., that due to liquid bridge between the particles). The present study examines closely the liquid bridge force model that one should employ in DPM to make it an effective equivalent for the liquid bridge forces between the primary particles in wet-particle fluidization simulations.
Another option to reduce the computational cost associated with CFD-DEM models is the coarsening of the fluid phase so that coarse grids can be used to solve the local-average fluid phase equations [30]. Such fluid coarsening would require corrections to the filtered drag model [31], especially in wet gas-particle systems [11], and in other cohesive systems [30]. Much work has already been done on development of methodologies for fluid coarsening and so we do not address it in the present study; instead, we focus on particle coarsening rules associated with the use of DPM and CFD-DPM.
Bierwisch et al. [28] studied cavity filling with powders, whose cohesive interaction was quantified through the well-known JKR model [32]. They examined two particle coarsening approaches, one based on matching stresses between the original and the coarse-grained systems and the other based on matching the energy densities. They concluded that the energy density-based scaling is suitable for dilute systems, while the stress-based scaling led to better results for dense systems.
Sakai et al. [33] performed CFD-DEM simulations of 2-D fluidized beds accounting for van der Waals interaction between the particles. They presented their so-called “direct force scaling”, based on quadratic scaling of the cohesion force and cubic scaling of the contact force, which led to very good agreement between the original (full) and coarsened simulations. Similarly, Thakur et al. [34] have found DPM simulations with quadratic scaling of the (van der Waals) adhesive force parameters to approximate the full DEM simulations of uniaxial compression test. Chana and Washinoa [35] coarse-grained their simulations of wet particle flow in an agitated mixer and found that quadratic scaling of the liquid bridge force between the particles captured the bulk flow field and the mixing behavior of the original system for a ratio of parcel- and particle radius up to four.
In summary, two ways of scaling particle-particle interaction forces, quadratic or cubic, have been presented in previous work. In the context of cohesive DPM reasonable agreements were found in their specified system. However, the focus of the presented coarse-graining approaches did not continue to a deeper analysis of various particle coarsening approaches for gas-particle flows, motivating our present contribution.
We asses these (effectively two) different coarse-graining rules using fluidization of particles in a periodic domain as a test problem. As fluid grid resolution is known to have a large effect on predictions [36], we use the same fluid grid cell in the original and coarse-grained systems so that we can focus exclusively on particle coarsening. The CFD-DPM approach requires smoothing of exchange fields (e.g., particle volume fraction, drag force) that are projected on the fluid grid, since parcel size could become comparable to or even larger than a fluid cell depending on the degree of particle coarsening employed (when one chooses not to coarsen the fluid grid). Consequently, we begin by identifying a relationship between the optimal smoothing length and the parcel size that would lead to the same domain-average slip velocity (defined as the difference in the domain-average velocities of the gas and particle phases) for different coarse-graining ratios (i.e. parcel radius/particle radius) in the case of noncohesive particles. We then use this exchange field coarsening in cohesive particle fluidization to assess the two sets of coarse graining rules referred to above. It will be shown that the stress-based coarsening of CFD-DEM to CFD-DPM yields superior results.
This paper is organized as follows: We present the computational method and the closures for the different forces in Section 2. A theoretical analysis of coarse graining strategies for cohesive systems is presented in Section 3. In Section 4, the merits of the coarse-grained models are assessed. The main findings are summarized in Section 5.
Section snippets
Computational method
The CFD part of the CFD–DEM approach is realized within the framework of OpenFOAM® [37], and the DEM part is solved using LIGGGHTS® [38]. The coupling between these two tools is performed with CFDEM® [38], which also implements the exchange field smoothing and the main flow solution algorithm.
The translational and rotational motions of the spherical particles considered in this study are described by:
Here, mi is the mass of particle
Theoretical analysis of coarse-graining strategies for cohesive systems
In the parcel approach, the diameter of the parcel is defined via the scaling ratio α according to dparcel = α ∙ dprim. In terms of radius, this is equivalent to Rparcel = α ∙ Rprim. Consequently, one parcel consists of α3 particles.
Proper scaling for the fluid-particle interaction force is fairly easy to identify. As the fluid-particle interaction force experienced by a parcel in a CFD-DPM simulation must be the same as that experienced by the α3 particles in it, the parcel drag force should
Simulation setup and post processing strategy
The parameters of the periodic box simulations are displayed in Table 1. The calculated terminal settling velocity vt for the primary particle is 0.8562 [m/s], and the reference time is tref = vt/g = 0.0873 [s]. These quantities are used to scale the results presented in the next sections.
Fig. 1 shows illustrative simulations of the original system and two coarse-grained systems with α = 4 and different smoothing lengths. One has to note that simulation snapshots can only compare qualitative
Summary
Previous studies seeking to evaluate rules for particle coarsening considered only DEM simulations of particulate flows [28,34,35], or simplified 2D CFD-DEM simulations of fluid-particle flows [33]. For fluid-particle systems, coarsening of the fluid grid and sequential fluid/particle coarsening have also been considered [11,27],. No prior study has examined particle coarsening in the presence of liquid transfer between particles. The present study focuses on coarse-graining rules for
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to acknowledge the use of HPC resources (dcluster) provided by the ZID of Graz University of Technology. Partial financial support for this work by Syncrude (Canada) Ltd. is gratefully acknowledged.
Nomenclature
Latin symbols
- ai
- Dimensionless filling rate [-]
- Bo
- Bond number [-]
- c
- Damping coefficient [kg/s]
- cwidth
- Width coefficient laplace filter [-]
- Ca
- Capillary number [-]
- C
- Constant [-]
- d
- Diameter [m]
- D
- Diffusion coefficient [m2/s]
- e
- Coefficient of restitution [-]
- Ekin
- Kinetic energy [J]
- f
- Function [-]
- F
- Force [N]
- g
- Gravity [m/s2]
- k
- Stiffness [N/m]
- hε
- Particle roughness [m]
- hij
- Surface-Surface distance [m]
- h0
- Initial liquid height [m]
- hrup
- Rupture distance [m]
- I
- Moment of inertia [kg m2]
- Ksl
- Momentum exchange coefficient [kg/(m3 s)]
- Lc
- Contact loss energy
References (53)
- et al.
Powder technology in the pharmaceutical industry: the need to catch up fast
Powder Technol.
(2002) - et al.
The effect of thin liquid layers on fluidisation characteristics
Powder Technol.
(1984) - et al.
A fracture mechanics approach to the breakage of particle agglomerates
Chem. Eng. Sci.
(1987) - et al.
Effect of cohesive forces on fluidized and spouted beds of wet particles
Powder Technol.
(2000) - et al.
Prediction of fluidized bed behaviour in the presence of liquid bridges
Powder Technol.
(2001) - et al.
Simulating wet gas-solid fluidized beds using coarse-grid CFD-DEM
Chem. Eng. Sci.
(2016) - et al.
Review of discrete particle modeling of fluidized beds
Chem. Eng. Sci.
(2007) - et al.
Discrete particle simulation of two-dimensional fluidized bed
Powder Technol.
(1993) - et al.
Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics
Chem. Eng. Sci.
(1997) - et al.
Micromechanical modeling and analysis of different flow regimes in gas fluidization
Chem. Eng. Sci.
(2012)