Challenges in the Simulation of Drying in Fluid Bed Granulation
- A focused review of drying-related models in mainly CFD, DEM, and PBM models for a pilot to large-scale fluid bed granulators, i.e., drying of agglomerates experiencing size change simultaneously;
- A summary of methodologies for the simulation and modeling of simultaneous agglomeration and drying;
- A summary of the challenges imposed upon the simulation of drying in fluid bed granulation systems.
2.1. Why Drying Influences the Agglomeration Rate
2.2. Mechanisms of Drying in Fluid Bed Granulation
- Constant-rate drying: The delivery of the water from the interior to the surface is sufficient to keep the surface completely wet; hence, the drying rate is constant. The pores are progressively depleted of water.
- Falling-rate drying: The surface layer of water starts to deplete inside the solid. Two mechanisms are expected based on the amount of liquid available inside the pores:
- First falling-rate drying: Initially, the liquid is dragged from the larger pores to the solid surface; the primary drying mechanism is the same as in the constant-rate mechanism. The only difference is that the wetted surface area reduces over time. The water inside the pore is the continuous phase, while the air is the dispersed one. It should be noted that the rate of drying in the first falling-rate period is typically linear (see Figure 5 for a visualization).
- Second falling-rate drying: Progressive water removal from the solid gives rise to the air volume fraction inside the pore. A continuous liquid film cannot be maintained inside the pores below a specific moisture level. Consequently, air will fill the pore, forming the continuous phase. Therefore, the remaining water is relegated to small, isolated pools in the corners and interstices of the pores, resulting in a sudden drop in the drying rate, as illustrated in Figure 5.
- Free surface moisture: The liquid covering the outer surface of the granule. Drying of this type of moisture follows the evaporation from free liquid (wet-bulb evaporation).
- Inter-particle moisture: The liquid bridging the primary particles inside the granule.
- Intra-particle moisture: The liquid trapped inside the pores of each primary particle. Drying of this type of moisture is governed by the internal drying resistance.
3. Key Influence Parameters
4. Simulation Approaches
- Micro-scale numerical approaches: in these approaches, the particles and droplets are resolved to a sub-particle/droplet level. For deterministic approaches, the balance equations for momentum, heat, and mass transfer are solved in the intra-particle or intra-droplet domain. One example of this approach is a direct numerical simulation (DNS). Metzger  considered a pore-network model as an example of this group of approaches. Another group of micro-scale methods is based on stochastics and probabilities. One example of stochastic micro-models is the Monte Carlo method: According to Terrazas-Velarde [19,20,21], in this approach, a limited number of particles are simulated to extract the agglomeration kernel. As described in Section 4.3, the advantage of this approach is that the deposited liquid’s distribution and thickness can be captured on the particle. This is advantageous in investigating the effect of liquid drying on the agglomeration rate. However, the computational cost is too high to study a whole process.
- Meso-scale numerical approaches: The balance equations are resolved down to the single particle level (as in the Discrete Element Method, CFD-DEM) or a continuum level (as in the Two-Fluid Model, TFM, and the Multi-Fluid Model, MFM). Newton’s second law is solved for each particle in the CFD-DEM approach. In contrast, in the TFM and MFM approach, solid particles are considered as one or several continua, respectively. In this manner, one needs to define the solids rheology, including solids viscosity, pressure, and granular temperature, for TFM or MFM approaches.
- Macro-scale modeling approaches: The FBs are divided into several well-mixed compartments. In each compartment, a specific phenomenon dominates. The exchange rate between different compartments needs to be defined a priori or determined using detailed simulations, e.g., mesoscale or micro-scale approaches. The population balance model (PBM) is typically used to model the granule growth.
4.1. Two-Fluid Models
4.1.1. No Agglomeration Models
4.1.2. Decoupled TFM-PBM
4.1.3. Loosely Coupled TFM-PBM
4.1.4. Strongly Coupled MFM-PBM
4.1.5. Modified Solids Rheology
|Phenomena Considered||Number of Papers||References|
|Granulation and drying||2||Li et al. , Li et al. |
|Breakage||1||Liu and Li |
|The success factor of collision||0||-|
|Accounting for surface coverage||1||Askarishahi et al. |
|Intra-particle layer thickness variability||0||-|
|Stefan diffusion effects in evaporation (Spalding mass transfer numbers)||0||-|
|Falling rate drying||2||Wang et al. , Tu et al. |
|Modified solids rheology||0||-|
4.2. CFD-DEM-Based Models
4.2.1. Drying in the Context of Granulation Research
- A particle located in the spray zone at a certain distance from the spray source accumulates “wet surface energy” (i.e., the ability to build cohesive forces) based on an exponential function of residence time in the spray zone.
- Drying of the deposited liquid initially increases and subsequently decreases particle–particle cohesive forces (i.e., the binder is assumed to become more viscous at the initial phase of drying and then solidifies)
- The liquid in a particle–particle bond is also dried based on an exponential function of the liquid bond age. The surface energy (i.e., the strength of the cohesive force) is increased based on a “dry-out factor” that quantifies the final (i.e., dry) strength of the bond. This considers the well-known fact that solidified liquid bridges can withstand substantial forces .
4.2.2. Drying in the Context of Coating
4.3. Stochastic Models
4.4. Compartment Models
5. Challenges in the Simulation of Drying in Granulation
- Droplet tracking and droplet–particle interactions;
- Particle–particle interactions, agglomerate growth, and breakage during spraying and drying;
- Multi-component liquid evaporation and binder effect on drying and consolidation (e.g., dependency of binder solution viscosity on the shear rate and temperature due to non-Newtonian behavior);
- Possible dissolution of solid powders in the liquid phase;
- Primary particles’ pore network (intra-particle voidage) and the changing granule pore structure (inter-particle voidage inside a granule) during the process;
- Redistribution of liquid among primary particles upon collision and agglomerate formation;
- Drying rates as affected by surface moisture and internal moisture;
- The contribution of various phenomena to drying, including:
- Hydrodynamics (e.g., surface tension force-driven drying in a pore, immigration of pore liquid to the surface due to capillary);
- Mass transfer (e.g., wet-bulb phenomenon and saturation of fluid, diffusion-driven drying in the pore);
- Heat transfer (e.g., temperature effect on fluid phase capacity to carry vapor).
- The high computational cost for detailed simulations due to the high number of particles and droplets involved.
5.1. Droplet Tracking and Droplet–Particle Interaction
5.1.1. Droplets as Lagrangian Points
- It is not clear whether evaporating droplets follow the gas flow (i.e., the limit of zero Stokes number is reached, i.e., small droplets) or have enough inertia to simply pierce the gas flow and follow predominantly straight trajectories (i.e., the limit of infinite Stokes number applies, i.e., large droplets);
- A wide droplet size distribution is present, such that the droplet Stokes number covers a wide range such that criterium (1) becomes relevant;
- Droplets change their properties (i.e., diameter, composition) significantly during their journey through the gas phase as a result of drying;
- The droplet impact speed on particles or walls is of central interest, e.g., to account for phenomena such as splashing.
- The early study of Goldschmidt [101,102] and co-workers considered droplets as discrete entities in addition to the particles. Drying was absent since they focused on a melt granulation process in a two-dimensional setup. Droplets were injected into a spray zone, allowing for randomized droplet velocities. The collision of particles and droplets leads to wetted particles (“coalescence”), and the collision of wet particles can result in the formation of granules (“agglomeration”). Even the “masking” of the wetted surface inside a granule was considered. Since they used a hard-sphere approach, their model did not allow consideration of multiple contacts.
- Barrasso and Ramachandran  performed 3D flow simulations in an unrealistically small domain (a “drum” with a diameter of 40 mm and a length of 60 mm) and huge primary particles (1 mm diameter). Liquid droplets were considered to have the same size and composition as the solid particles. A cylindrical region (diameter of 8 mm, full drum length) was considered as the liquid addition zone. No cohesive interactions or drying model were characteristics of this conceptual study to demonstrate how the coupling between the DEM and the PBM can be achieved.
- In the studies of Jiang et al. [62,77], solid-like droplets are considered to directly investigate the droplet deposition rate in the spray zone. In detail, they considered droplet–particle impacts and analyzed their outcome depending on the Weber and Reynolds number (“depositing” or “splashing”) . This study is one of the few articles in which intra-particle variations of the coating layer were considered.
- In the CFD-DEM study of Grohn et al.  for layering granulation, the droplets are generated as a second particulate phase in the DEM part of the code. A “loading coefficient” of the liquid α, is introduced to consider the solid concentration in a solution used for coating. This hence enabled advanced consideration of the coating process of individual particles. This study considered a fixed global drying rate. Moreover, droplet drying (in flight) has not been studied in detail.
5.1.2. Spray Zone Approach
Identification of the Spray Zone
- Spray (or wetting) zone: the region close to the nozzle where droplet formation, droplet/particle collision, and spreading droplets on the particle surface occur. This region features high humidity and low temperature.
- Drying zone: the region below the spray zone, identified by high fluctuation in temperature and humidity.
- Heat transfer zone: the region above the distributor where there is significant heat exchange between fluidization air and particles. This region features high temperatures and constant humidity.
- Non-active zone: the region between the heat transfer and drying zones, featuring constant temperature and humidity.
No Interaction (Droplet–Particle Interaction Lumped into the Agglomeration Kernel)
Uniformly Distributed Droplet Deposition on Particles
- Track the particle residence time in the spray zone. This was performed by tracking the time particles needed to pass the spray zone. In this manner, higher residence time means higher liquid content of the particle;
- Obtain the exchange rate of particles between different compartments (inter-compartmental particle transfer) and particle distribution in each compartment;
- Obtain particle collision dynamics, including the collision frequency and collision energy.
Calculated Rate of Deposition
5.1.3. Surface Energy Pick-Up Concept
- Wetting: “surface of particle becoming ‘active’ as a result of picking up surface energy in the spray zone”.
- Drying: “increase in surface energy or adhesion energy value with time”.
- Wet surface energy: “surface energy that can still form bonds and has not reached some terminal dried-out value”.
5.1.4. Continuum Spray Filtration Model
5.1.5. Droplets in Stochastic Models
5.1.6. Droplets in PBM Models
5.2. Surface Area Available for Drying
5.2.1. Continuous Film-Based Models
5.2.2. Correction Factor-Based Models
5.2.3. Surface Coverage-Based Models
5.2.4. Droplet Height-Based Models
5.2.5. Balance Equation-Based Model
5.2.6. Common Deficiencies of Previous Models
- Freely accessible liquid, which includes accessible surface and bridge liquid: here, no significant mass transfer limitation is present, and the entire “wet” (i.e., liquid) surface area can be considered (for bridge liquid, this is difficult to estimate, but in principle doable).
- Liquid with limited accessibility, which includes surface and bridge liquid that is not interfacing with the bulk gas phase, as well as accessible pore liquid. For this group, the evaporation rate will be limited but finite.
- Liquid with no or very minor accessibility includes inaccessible pore liquid. For this group, evaporation rates can be considered too small to have any relevance for industrial processes.
5.3. Mass Transfer Coefficient and Reference Concentration Difference Estimation
5.3.1. External Particle Surface-to-Bulk Transport
- Equimolar diffusion in the gas next to the external surface of the particle is assumed. Thus, it is assumed that there is no net flux of gas that exits the particle, and hence there exists no net radial gas velocity directly at the liquid-gas interface (note that this is unrelated to the fact that flow around the particle is considered in many publications). The resulting model equation is linear in the concentration difference, which leads to an underestimation of the evaporation rate in case of high vapor mole fractions, i.e., high temperatures. A fix to this shortcoming would be to use the Spalding mass transfer number, or recently published corrections ;
- An isolated wet particle is often considered, as well documented in many publications, e.g., that of Li et al. . This assumption dates back to the early work of Heinrich and Mörl , which was clearly motivated at that time by the lack of closures for dense suspensions. It should be noted that the work of Heinrich and Mörl  already considered the fraction of the wetted surface of a particle when estimating the area available for mass transfer. In the future, improved correlations for mass transfer coefficients should be used—e.g., as already carried out by Askarishahi et al.  by exploiting the correlation of Deen et al. for heat transfer  and an analogy between heat and mass transfer.
- The entire previous works on drying used the local average concentration (i.e., an ensemble-averaged quantity as performed in RANS-type of simulations, or spatially averaged quantities as in the case of LES-type simulations). Thus, a resolution-dependency is expected since gas–particle systems spontaneously cluster [127,128], and this will affect the evaporation rate: as shown in our illustration (see Figure 11), even in case there is a difference between the region-average vapor concentration and the vapor equilibrium concentration at the external particle surface, the net evaporation rate might be zero! The size of these clusters is in the order of the particle diameter, with details depending on the particle Froude number . Thus, it would be computationally very expensive to resolve the vapor concentration field in the gas phase since cluster size and the length scale of vapor concentration fluctuations will be of similar magnitude. Fortunately, an approach to overcome this challenge has been presented by Agrawal et al. ; however, in a different context: this early study revealed that a correction accounting for unresolved concentration fluctuations can be constructed and that this correction is similar to what is long known for the drag force. Most important, Agrawal et al.  showed that these unresolved structures lead to an extremely strong suppression of mass transfer—one or two orders of magnitude (!) lower mass transfer rates over a wide range of particle volume fractions were report. Physically, this means that the clustering phenomenon dominates the evaporation rate, i.e., mass transfer (drying) is so fast that the gas phase is locally saturated with vapor as long as a wet particle exists. In simple words, one can think of a “vapor concentration slip” that exists, i.e., a mismatch between the local average vapor concentration and the vapor concentration seen by individual particles. This message was reinforced by several recent studies, e.g., Guo and Capecelatro  or Rauchenzauner et al. [132,133].
- It must be clear that in models that are not able to track each individual particle, imperfect mixing (within a region of interest) will cause a similar effect: if wet particles with high vapor pressure (caused by high temperature) are in regions that are saturated with vapor, and dry particles in low vapor content regions, the net evaporation rate might be zero (see Figure 12). In contrast, if one only considers particle–average vapor pressures and liquid content, the predicted evaporation rate would be non-zero! While this scenario is certainly rare (since particle-phase mixing is typically fast), it should be mentioned that such a correction could not be found in the open literature.
- Another detail is the concentration that should be considered at the particle surface: most studies consider the vapor concentration in equilibrium with the liquid at the external particle surface. Aziz et al.  simulated the evaporation process based on the modification for the concentration difference as proposed by Putranto and Chen , i.e., the so-called “reaction engineering approach” (using an Arrhenius-type of function). This more advanced approach allows us to consider hygroscopic materials.
5.3.2. Intra-Particle Transport
5.3.3. Analogy to Heat Transfer
5.4. Drying Kinetics
5.4.1. Phenomenological Single-Particle Drying Models
5.4.2. Sessile Droplet-Based Drying Models
5.4.3. Pore Network Models
5.5. Competition between Granule Drying and Suspended Droplet Evaporation
5.6. Effect of Drying on the Agglomerate Size Distribution and Strength
- Drying can change the liquid binder strength due to the dependency of viscosity and binder concentration on temperature. As discussed in Section 2.1, this can be quantified through the Stokes number;
- Drying affects the number and distribution of wet spots on the granule and, consequently, the probability of successful agglomeration; this can influence the granule morphology, as discussed in Section 2.1;
- Drying can influence the consolidation rate through the evaporation of volatile components of the biner solution.
6. Identified Gaps and Way Forward
- Performing experiments that isolate evaporation and granulation phenomena as much as possible but use granules that are typical for a real-world granulation process. Such experiments would then have a very high value for the validation of individual simulation models. These experiments could use 3D-printed reference granules that are then wetted in a defined way, similar to what has been performed by Ge et al.  for the mechanical strength of granules.
- Modeling of the final strength of particle–particle bonds after evaporation of liquid bridges as a follow-up to the early work of Kafui and Thornton .
- Modeling of the effect of drying on the binder solution properties and its contribution to the strength of granule, as a follow-up to the recent work of Singh and Tsotsas . The effect of drying will be even more complicated if the binder forms a crust in the droplet deposited on the particle surface, as reported by Dernedde et al. .
- A differentiated description of the amount of liquid on the surface and inside a granule: this is necessary to correctly model the amount of liquid that is freely accessible (or not) for drying. Indeed, such a modeling work will rely to a large degree on experimental data, and hence dedicated experiments closely linked to computer models should be followed.
- The tendency to form liquid bridges is massively affected by the roughness of the particles or granules . Accounting for roughness effects when estimating the surface area available for evaporation would be a leap forward, same as a model to predict the effect of solid depositions on the roughness evolution. In addition, such a model refinement could improve the fidelity of cohesion models.
- Performing a rigorous simulation study that uses “best in class” models (i.e., Lagrangian droplet tracking, drying of droplets and particles considering surface covering, sophisticated drying kinetics model, corrections to the mass transfer rate due to clustering) to support or reject key assumptions made in the field. Based on such a reference study, regime maps could potentially be developed.
- Diez et al.  conducted a detailed experimental study on the effect of drying on the granule properties, including the morphological structure, particle moisture content, porosity, density, compression strength, and wetting behavior in a horizontal fluidized bed.
- Askarishahi et al. [9,10] conducted a numerical and experimental study on the agglomeration and drying of a placebo formulation in a fluid bed granulator. They provided a set of experimental data for a wide range of operating conditions (spray rate, binder concentration, and fluidization gas temperature). Their data can be used for validation of FBG performance in macro-scales. Muddu et al.  and Tamrakar and Ramachandran  conducted a similar study.
- Närvänen et al.  conducted an experimental study on the particle size distribution in an FBG for spraying and drying phases.
- Schmidt et al.  presented a set of experimental data on the effect of drying conditions on the particle size distribution for layering granulation.
- Bouffard et al.  reported a set of experimental data on the impact of binder solution atomization on the granule growth kinetics.
- Dadkhah et al.  presented a set of experimental data on the dependency of granule morphology on the process variables.
Data Availability Statement
Conflicts of Interest
|CFD||Computational Fluid Dynamics|
|CNMC||Constant-Number Monte Carlo|
|CVMC||Constant-Volume Monte Carlo|
|DEM||Discrete Element Method|
|DoW||Degree of Wetness|
|FBC||Fluid Bed Coater|
|FBD||Fluid Bed Dryer|
|FBG||Fluid Bed Granulator|
|LoD||Loss on Drying|
|PBE||Population Balance Equation|
|PBM||Population Balance Method|
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|Parameter/Variable||Influence on Fluid Bed Performance|
|number of nozzles|
|droplet size and its distribution|
|atomization air pressure|
|heat of evaporation|
|size and size distribution|
|Phenomena Considered||Number of Papers||References|
|Accounting for surface coverage||4||Askarishahi et al. , Kieckhefen et al. , Fries et al. , Madlmeir |
|Finite cohesion force after drying is complete||1||Kafui and Thornton |
|Intra-particle layer thickness variability||1||Jiang et al. |
|Stefan Diffusion effects in evaporation (Spalding mass transfer numbers)||1||Madlmeir and Radl |
|Systematic coarse-graining||1||Kieckhefen et al. , Madlmeir and Radl |
|Falling rate drying||0||-|
|Phenomenon Considered||Number of Papers||References|
|Granulation and drying||13||Dernedde et al. , Terrazas et al. [19,20,21], Marshall , Rieck et al. [70,71], Hussain et al. (Hussain et al., 2013b), Singh and Tsotsas [8,69,72], Das and Kumar , Du et al. |
|Pre-deposition droplet evaporation||1||Dernedde et al. |
|Imbibition of droplets into particles||1||Terrazas et al. |
|Breakage||10||Terrazas et al. [19,20,21], Marshall , Rieck et al. [70,71], Hussain et al. (Hussain et al., 2013b), Singh and Tsotsas [8,69,72], Das and Kumar , Du et al. |
|Accounting for surface coverage||13||Dernedde et al. , Terrazas et al. [19,20,21], Marshall , Rieck et al. [70,71], Hussain et al. , Singh and Tsotsas [8,69,72], Das and Kumar , Du et al. |
|Liquid binder viscosity change||8||Dernedde et al. , Terrazas et al. [19,21], Marshall , Rieck et al. , Singh and Tsotsas [8,69,72]|
|Falling rate drying (of intra-particle moisture or intra-granule moisture)||0||-|
|Droplet deposition on particle||13||Dernedde et al. , Terrazas et al. [19,20,21], Marshall , Rieck et al. [70,71], Hussain et al. (Hussain et al., 2013b), Singh and Tsotsas [8,69,72], Das and Kumar , Du et al. |
|Phenomenon Considered||Number of Papers||References|
|Granulation and drying||4||Hussain et al. , Peglow et al. , Askarishahi et al. [9,10],|
Das and Kumar 
|Breakage||1||Liu and Li |
|The success factor of collision||4||Hussain et al. , Das and Kumar , Askarishahi et al. [9,10]|
|Accounting for surface coverage||1||Hussain et al. |
|Intra-particle layer thickness variability||0||-|
|Falling rate drying||3||Arthur et al. , Peglow et al. , Askarishahi et al. [9,10]|
|Droplet deposition on particle||2||Heinrich et al. , Hussain et al. |
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Askarishahi, M.; Salehi, M.-S.; Radl, S. Challenges in the Simulation of Drying in Fluid Bed Granulation. Processes 2023, 11, 569. https://doi.org/10.3390/pr11020569
Askarishahi M, Salehi M-S, Radl S. Challenges in the Simulation of Drying in Fluid Bed Granulation. Processes. 2023; 11(2):569. https://doi.org/10.3390/pr11020569Chicago/Turabian Style
Askarishahi, Maryam, Mohammad-Sadegh Salehi, and Stefan Radl. 2023. "Challenges in the Simulation of Drying in Fluid Bed Granulation" Processes 11, no. 2: 569. https://doi.org/10.3390/pr11020569