Predicting Defluidization in Fluidized Bed Conversion: From Plastics Pyrolysis to Biomass Combustion via Surface Coating Models

Abstract

In fluidized bed conversion processes such as pyrolysis and combustion, defluidization mainly arises from particle agglomeration, which originates from the surface coating of primary bed materials (e.g., sand) by partially liquefied feedstock components, e.g., plastics or biomass. For reliable operation, the probability of occurrence of defluidization must be quantifiable. However, existing models are either computationally expensive or difficult to transfer across feedstocks with different rheological behaviors. Furthermore, such transferability challenges are particularly pronounced in technically relevant systems involving liquefied components, such as molten polymers and ash-derived silicate melts. In this study, we propose two new coating approaches: (i) a simplified full coating model, where a fraction of bed particles is directly assumed to be fully covered upon feed introduction, and (ii) a partial coating model, where only local surface regions of particles are coated. The proposed models are implemented within a Monte Carlo framework and validated against experimental data reported in the literature for polyethylene and polypropylene pyrolysis as well as for wheat straw combustion. Across all cases, the model predictions capture the experimentally observed defluidization behavior reported in reference studies (e.g., with coefficients of determination of $R^2=0.912$ for the polymer series and $R^2=0.917$ for the wheat straw series). Beyond model validation, several model-based analyses and discussions are further conducted based on the characteristics of the proposed framework. Overall, the developed methodology provides a generalized basis for analyzing coating-driven defluidization across polymers and biomass, with potential extensions to co-pyrolysis, co-gasification, and other thermochemical conversion processes.

1. Introduction

Thermochemical conversion of solid feedstocks such as waste plastics and biomass has attracted growing attention as a sustainable route for resource recovery and energy production. The pyrolysis of waste plastics, including polyolefins such as polyethylene (PE) and polypropylene (PP), enables the chemical recycling of end-of-life polymers into valuable hydrocarbon products [1,2,3,4], whereas biomass conversion provides a renewable and carbon-neutral source of energy [5,6,7,8], playing an essential role in the global transition towards low-carbon energy systems. Among the various thermochemical conversion technologies, fluidized-bed systems have been widely recognized for their excellent heat and mass transfer characteristics, uniform temperature distribution, and high operational flexibility toward diverse solid feedstocks. In particular, the bubbling fluidized bed (BFB) has been extensively employed in both fundamental investigations and industrial-scale processes owing to its stable hydrodynamics, efficient mixing behavior, and straightforward scalability [9,10,11,12]. Nevertheless, BFB systems may encounter potential operational issues when processing plastic or biomass feedstocks, such as bed defluidization, which often results from the agglomeration of bed materials.

A comprehensive overview of defluidization phenomena during polymer pyrolysis was provided in our previous work [13], where several key studies were discussed. Kaminsky, Arena, Mastellone, and co-workers [14,15,16,17,18,19,20,21,22,23,24,25] conducted a series of experimental studies to investigate the fluidization behavior of plastics during pyrolysis in BFBs. Their findings revealed the fundamental mechanisms governing particle agglomeration during pyrolysis, showing that molten polymers or by-products can adhere to inert bed materials and form agglomerates, which ultimately lead to defluidization.

Although similar agglomeration phenomena are frequently reported during biomass combustion and gasification in BFB reactors, the underlying mechanism differs fundamentally from that observed in polymer pyrolysis. Experimental investigations have reported bed agglomeration and defluidization for a wide variety of biomass fuels, including agricultural residues such as wheat straw [11,26,27], olive husk [28], oat hull and sunflower husk [27], herbaceous energy biomasses such as Cynara cardunculus L. [29] and Miscanthus (Miscanthus × giganteus) [11], and woody biomasses such as pine wood [27,30], white wood [11], birch wood, and grot (branches, roots, and tops) [30]. Across these studies, potassium-rich ashes consistently accelerated agglomeration and promoted defluidization when quartz sand was employed as the bed material. The underlying mechanisms have also been explored in several studies. Liu et al. [31] found that fine ash particles tend to form surface coatings on bed materials, whereas larger ash fragments are more likely to undergo direct melting, leading to two distinct agglomeration behaviors. Subsequent studies [27,32] revealed that, under practical operating conditions, two agglomeration mechanisms often coexist: coating-induced agglomeration, in which potassium compounds react with SiO₂ to form low-melting potassium silicates that wet and bind sand particles, and melting-induced agglomeration, where portions of the ash melt directly and act as viscous binders on the particle surface.

To mitigate bed agglomeration, various studies have explored alternative bed materials and additives. Serrano et al. [29] demonstrated that sepiolite, due to its lower silica content and the presence of magnesium oxide and calcium oxide, effectively suppresses the formation of potassium silicate. Morris et al. [11] reported that olivine delays bed collapse, as its magnesium–iron silicate composition exhibits low reactivity with alkali species and promotes the formation of a stable calcium silicate layer with a relatively higher melting temperature. In addition, Zhou et al. [30] demonstrated that the use of magnesite or dolomite as bed material markedly prevents agglomeration, since the resulting basic oxides (magnesium oxide and calcium oxide) capture potassium and inhibit the formation of potassium silicate compounds. Likewise, Anicic [27] found that kaolin, lime, and dolomite can effectively eliminate defluidization through a similar mechanism, despite being employed only as additives.

In addition to the selection of bed materials, operating parameters play a crucial role in governing the evolution of agglomeration behavior. Extensive experimental studies consistently indicate that the defluidization time decreases markedly with increasing bed temperature [26,28,31], whereas a reduction in bed material size [11,26,28] or an increase in superficial gas velocity [26,30,31,33] tend to enhance stable fluidization. Moreover, excessive fuel feeding has been shown to accelerate bed collapse [31], while the bed height exhibits an optimal range, beyond which defluidization occurs earlier [11]. Overall, these studies demonstrate that operating conditions have a significant impact on the stable operation of BFB systems.

Beyond the conventional operating conditions of BFB systems, several process modifications have also been proposed to reduce the tendency toward bed agglomeration and defluidization. For instance, Zhou et al. [30] demonstrated that a pressurized steam-oxygen BFB configuration enhances gas–solid mixing and temperature uniformity, thereby reducing local overheating and sticky ash formation. Similarly, Zhao et al. [34] applied the concept of a pulsating-gas-flow BFB and showed that controlled gas pulsation can effectively suppress agglomeration, with the pulsation duty cycle identified as the most influential parameter.

Lin et al. [26] developed one of the earliest analytical models to predict the complete defluidization time during straw combustion. The model was able to accurately reproduce the defluidization time under different temperatures, gas velocities, and bed particle sizes. Building upon this work, Chaivatamaset et al. [35] proposed two mathematical models for predicting the complete defluidization time: an empirical multivariable regression model based on a large experimental dataset, and a modified version of the force-balance model proposed by Lin et al. [26]. Both models identified the alkali content in ash, bed temperature, and gas velocity as the main controlling factors governing agglomeration and defluidization. Validated against laboratory- and pilot-scale experiments, these models provide an effective means for assessing bed stability under varying operating conditions. However, similar to most analytical and empirical approaches, their applicability remains limited to macroscopic prediction of defluidization time, without accounting for the agglomeration rate or the dynamic evolution of particle size distribution (PSD).

To gain a more detailed understanding of the mechanisms underlying bed agglomeration, researchers have increasingly employed simulations based on multiphase computational fluid dynamics (CFD). Anicic [27] applied an Eulerian–Eulerian two-fluid model to investigate the segregation and mixing behavior in a binary solid system consisting of agglomerates and primary sand particles. The simulations revealed that, due to differences in density and particle size, agglomerates rapidly segregate from sand, leading to a loss of local fluidization uniformity and possibly accelerating bed defluidization. Tasleem et al. [36] subsequently developed a coupled Eulerian–Eulerian two-fluid and population balance model (PBM) framework to simulate the formation and evolution of bed agglomerates during biomass combustion. The model, incorporating combustion reactions, ash release, and particle adhesion kinetics, successfully reproduced key experimental phenomena such as bed expansion, agglomerate growth, and defluidization. In a later study, Tasleem et al. [37] extended the model to a larger scale system and introduced an alkali-silicate adhesion mechanism, thereby improving the accuracy of agglomeration modeling. These CFD-PBM approaches have demonstrated the capability to capture the early onset of agglomeration and to predict defluidization behavior with reasonable accuracy, providing valuable insights into bed instability. Nevertheless, such computationally intensive CFD simulations typically require substantial computational effort, especially when apparatus geometry is complex and huge.

On the other hand, the Monte Carlo (MC) method, as a statistical modeling approach, offers an alternative pathway to capture the stochastic nature of particle interactions in BFBs. Daw and Halow [38] and Garcia-Gutierrez et al. [39] developed MC-based mixing models that successfully described particle dispersion, residence time, and circulation behavior in BFBs, providing a probabilistic framework for analyzing mesoscale transport phenomena. Beyond mixing, the formation of agglomerates in BFBs has also been successfully modeled using MC approaches in several studies on spray agglomeration [40,41,42,43,44,45], demonstrating the capability of this approach to capture dynamic particle growth. Moreover, our previous work [13] further showed the capability of the MC framework to simulate polymer pyrolysis in BFBs, enabling quantitative prediction of bed defluidization and PSD evolution caused by coating-induced agglomeration.

Despite the demonstrated potential of MC-based modeling for capturing agglomeration and defluidization phenomena in BFBs, a generalized and computationally efficient framework capable of consistently describing coating-induced agglomeration across different thermochemical conversion processes is still lacking. Existing MC-based studies have mainly focused on agglomeration in spray processes or during polymer pyrolysis. However, bed material agglomeration leading to defluidization is also frequently observed during biomass combustion. The underlying agglomeration mechanism in this case, namely, ash-driven coating of bed particles, differs fundamentally from those governing polymer melt-induced or spray agglomeration, and has not yet been incorporated into a unified modeling concept. As a consequence, the transferability of existing models across feedstock systems with fundamentally different coating formation mechanisms and resulting melt rheological properties remains limited. Motivated by these gaps, the present study aims to develop a simplified and generalized MC-based modeling approach to describe coating-induced agglomeration in BFBs and to predict bed defluidization behavior as well as PSD evolution across different feedstock systems.

Building upon the previously developed MC framework for polymer pyrolysis [13], the present work introduces a simplified and generalized stochastic modeling approach capable of predicting bed defluidization and PSD evolution during both polymer pyrolysis and biomass combustion. To achieve this, a key parameter, the coating percentage, is introduced to represent the degree of surface coverage and to simplify the modeling of coating-induced agglomeration. Based on this concept, a full coating model is first proposed to describe the defluidization behavior induced by molten polymer coating that fully covers bed particles within a localized region of the BFB. The framework is then extended to a partial coating model, which considers stochastic and partial surface reactions between potassium-rich ashes and bed materials during biomass combustion, leading to random coating formation on particle surfaces across the bed. The model predictions are validated against representative experimental datasets from the literature, and the effects of key operating parameters, such as gas velocity, bed temperature, bed mass, as well as bed material size, on the scaling of the coating percentage are introduced to clarify the concept. In addition, this study demonstrates the capability of the MC framework to capture particle-scale information and the dynamic evolution of the system, followed by a comparative discussion of the two newly developed models.

2. Model Development

2.1. Conceptual Background

In previous work [13], a two-period MC framework was developed to simulate particle agglomeration during polymer pyrolysis in a BFB. The model coupled a layering period (LP) and an agglomeration period (AP) to reproduce the overall process from polymer feeding to bed defluidization. During the LP, each large polymer particle injected into the bed was surrounded by numerous primary sand particles. Driven by random collisions and relative motion, these sand particles gradually adhered to the softened polymer surface and migrated inward, forming a dense polymer–sand aggregate. As the polymer continuously decomposed, its volume dynamically shrank while continuously capturing new sand particles, leading to a progressive structural evolution of the aggregate. When the volume fraction of sand within the polymer reached a critical threshold, the aggregate crumbled and broke apart into a number of individual sand particles, each fully coated with a layer of molten polymer. These fully coated sands were then introduced into the AP, replacing an equivalent number of uncoated primary particles. During the AP, random collisions occurred among coated particles, uncoated particles, and previously formed agglomerates, leading to the formation and growth of agglomerates, eventually resulting in defluidization.

In the present study, the overall structure of the MC framework during the AP remains unchanged. However, the LP has been fundamentally simplified. Specifically, the simulation of the LP is replaced by its final outcome, the generation of fully coated particles, represented by a new model parameter, the coating percentage ($\theta$). From a physical perspective, the coating percentage is introduced as a lumped parameter representing the fraction of particles or particle surface area that becomes sticky during a single feeding event, or that contributes to the further thickening of an existing coating layer on the surface. It should be emphasized that the coating percentage is not an arbitrarily chosen fitting parameter; rather, it reflects the combined influence of multiple factors, including feedstock properties, surface wetting and adhesion characteristics, as well as the contact and distribution probability between the feedstock and bed particles. These effects are further governed by the operating conditions of the process. Since these underlying micro-scale processes cannot be directly resolved or measured in practical BFBs, the coating percentage is determined through calibration combined with physically motivated scaling relations. This approach avoids the introduction of non-measurable microscopic parameters while retaining physical interpretability. Details of the calibration procedure are provided in Section 4.

This simplification not only improves computational efficiency but also, more importantly, enables the extension of the modeling framework from polymer pyrolysis systems to biomass combustion systems. Due to the fundamentally different coating mechanisms occurring on the surface of sand particles in these two processes, two corresponding approaches are proposed: the full coating model (FCM) and the partial coating model (PCM). The former considers the complete coverage of sand particles by molten polymer within a certain region of the bed, whereas the latter describes a coating process induced by reactions between dispersed ash particles and sand surfaces throughout the BFB. By introducing these two distinct coating mechanisms, the MC framework is further extended to consistently simulate coating-driven agglomeration phenomena in both systems within a unified modeling structure. The adapted flowchart of both models is illustrated in Figure 1. The detailed framework and the two proposed coating approaches are described in the following sections.

2.2. Full Coating Model

Similar to the previous work, the present study focuses on the pyrolysis of PE and PP. The polymer-induced coating of bed particles can be well represented by the FCM, since in the previous LP-AP coupled framework, the particles released from the LP into the AP were also fully coated by molten polymer. In the FCM, at each feeding step $t_{\mathrm{feed},j}$, a certain fraction of bed particles is assumed to become entirely covered by molten polymer. Based on the total number of particles currently present in the MC box, $n_{\mathrm{p},\mathrm{tot}}$, and the number of particles randomly selected for coating, $n_{\mathrm{p},\mathrm{fc}}$, the full-coating percentage is defined as:

$${\theta }_{\mathrm{fc}}={\displaystyle \frac{n_{\mathrm{p},\mathrm{fc}}}{n_{\mathrm{p},\mathrm{tot}}}}.$$

(1)

It quantifies the fraction of bed materials that are fully covered by molten polymer, including both individual sand particles and existing agglomerates, which, unlike the the previously developed LP-AP model, are not restricted to particles released from the layering stage. During each feeding event, a fraction of all particles is randomly selected according to this ratio and undergoes coating, representing the stochastic spatial distribution of molten polymer layers within the fluidized bed. Moreover, it is assumed that the amount of molten polymer distributed to each selected particle is identical. Consequently, larger particles with greater external surface area build up thinner coating layers. Both assumptions provide a more realistic representation of the coating behavior observed in actual fluidized beds.

It should be noted that, in real BFB operation, fluctuations in the raw material feeding rate and variations in particle collision frequency exist. In the present model, the feeding process is discretized into feeding events. Due to the short MC time step, short-term feeding fluctuations are averaged out, so that each feeding event corresponds to the introduction of one feedstock particle. Particle collision frequency is evaluated based on bed-averaged properties rather than at a local scale. Since the MC simulation box represents the overall behavior of the fluidized bed rather than a local region, randomness in the process is introduced by the MC framework itself through random particle selection, rather than by explicitly modeling fluctuations in feeding or collision frequency. Consequently, the assumption of equal coating material allocation could be understood as a statistically averaged distribution per selected particle or surface, allowing the model to capture coating and agglomeration behavior at the bed scale. The same statistical interpretation applies to both the FCM and the PCM, as both models are developed within the same MC framework and differ only in the underlying coating mechanism.

2.3. Partial Coating Model

Though the FCM effectively describes polymer-induced coating during certain types of plastic pyrolysis, the coating mechanism in biomass combustion systems is fundamentally different, as the coating originates from combustion-generated ashes that interact with the surfaces of bed materials. Therefore, in this work, a PCM is developed to extend the coating-based MC framework to biomass combustion systems. The Danish wheat straw combustion experiments focusing on the defluidization phenomenon, reported by Lin et al. [26], are adopted as the reference case in this study. During combustion, potassium compounds released from the fuel ash usually react with the surfaces of silica sand particles. More specifically, the reaction between K₂O and SiO₂ is considered, leading to the formation of various K₂O-SiO₂ compounds (potassium silicates). Because these compounds have melting temperatures within the typical range of wheat straw combustion, they exist as a molten phase after the reaction. The specific composition of the formed potassium silicates and their corresponding viscosities are discussed in Section 3.2.4. Although the direct melting or deposition of potassium-rich ash may also result in a melt-induced type of coating layer formation, this study focuses exclusively on the reaction-driven formation of potassium silicate on sand surfaces, which is regarded as the primary mechanism of coating-induced agglomeration. Moreover, because ash particles are extremely fine (typically smaller than 10 μm [31]) and are dispersed throughout the bed with particle motion, this reaction is considered to occur instantaneously and locally on limited areas of the sand surface. To capture this localized nature, the PCM assumes that only a portion of the particle surface is coated rather than the entire surface. Using the concept of positions (CoP, [40,41,42]), each primary sand particle surface is divided into six equally sized surface sectors. The choice of six sectors is based on the structural assumption that a spherical primary particle can be in contact with up to six neighboring ones. During each feeding event in the PCM, a fraction of all exposed sectors is randomly selected and assigned as reactive regions, regardless of whether the selected particle is single or agglomerated, representing the localized reaction between the silica sand surface and K₂O released from the fuel ash, forming molten potassium silicate. Overall, these model assumptions are consistent with the main coating mechanism, as observed in the reference experiments. The partial coating percentage is accordingly defined as:

$${\theta }_{\mathrm{pc}}={\displaystyle \frac{n_{\sec ,\mathrm{pc}}}{n_{\sec ,\mathrm{ext},\mathrm{tot}}}},$$

(2)

where $n_{\sec ,\mathrm{pc}}$ represents the number of selected surface sectors, and $n_{\sec ,\mathrm{ext},\mathrm{tot}}$ denotes the total number of external surface sectors within the MC box. Because the PCM operates on a sector-based selection, the amount of K₂O reacting within each sector is assumed to be constant. Consequently, the total amount of coating formed does not depend on whether the selected surface belongs to a single particle or an agglomerate, which distinguishes the PCM from the FCM in principle.

Once the reactive sectors are selected, the sand volume within each selected sector is reduced according to the amount of SiO₂ consumed by the surface reaction, while the corresponding amount of molten potassium silicate is added to the same sector. The volume balance for each feeding moment j is given by:

$$V_{\mathrm{s},\sec ,j}=V_{\mathrm{s},\sec ,j-1}-V_{\mathrm{s},\mathrm{cons},\sec ,j},$$

(3)

$$V_{\mathrm{p},\sec ,j}=V_{\mathrm{p},\sec ,j-1}+V_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,\sec ,j}.$$

(4)

Here, $V_{\mathrm{s},\sec ,j}$ and $V_{\mathrm{p},\sec ,j}$ denote, respectively, the unreacted sand volume and the total coated particle volume of the corresponding sector, whereas $V_{\mathrm{s},\mathrm{cons},\sec ,j}$ and $V_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,\sec ,j}$ represent the sand volume consumed and the molten coating formed during the corresponding feeding period. The detailed evaluation of these quantities for the wheat straw combustion case, including the reaction stoichiometry, ash composition, and elutriation correction, is presented in Section 3.2.2. The coating thickness of each selected sector is then calculated as:

$$h_{\sec ,j}={\left[{\displaystyle \frac{9V_{\mathrm{p},\sec ,j}}{2\pi }}\right]}^{1/3}-{\left[{\displaystyle \frac{9V_{\mathrm{s},\sec ,j}}{2\pi }}\right]}^{1/3},$$

(5)

where the updated thickness $h_{\sec ,j}$ is subsequently used to revise the particle geometry and state information, which in turn affects the particle properties (e.g., mass, density, and volume) as well as the agglomeration dynamics during subsequent events (e.g., collisions between particles). In addition, based on the updated particle volumes, the mean particle diameter and PSD are recalculated to track the dynamic evolution of the granulometric bed properties.

2.4. Common MC Framework

Apart from the updated feeding event, the dynamics of other events follow the same framework as that employed in the AP simulations of our previous work, using the same stochastic collision and agglomeration routines. Briefly, the MC agglomeration framework treats the BFB reactor as a statistically representative sample box that reproduces the evolution of the processed materials, particle collisions, and agglomeration behavior occurring within the real system. A direct-simulation MC approach [46,47] is employed to ensure a time-driven progression of events. At each time step, all particles within the MC box are randomly paired and undergo collisions. The time-step length is determined by the collision frequency, which depends on the mean particle velocity and the number concentration in the bed [43,48]. The surface labeling method for tracking particle states is consistent with our previous work, since the CoP allows for an effective representation of the surface as well as inter-particle conditions. When a collision involves coated or wet surfaces, agglomeration occurs if the viscous Stokes criterion [49,50] is satisfied; otherwise, particles rebound. In parallel, the reactions of the processed material are also incorporated as one discrete event that updates the coating state (e.g., polymer degradation or the formation of potassium silicate melt), thereby influencing subsequent collisions. Particle properties (e.g., mass, volume, and mean diameter) are updated after each time step, and their evolution subsequently affects the following collision processes and the hydrodynamic behavior of the fluidized bed.

The hydrodynamic parameters required in the MC model, including the particle velocity, bed porosity, and minimum fluidization condition, are evaluated using well-established correlations developed for BFBs [51,52]. These correlations relate the mean particle size and density to characteristic dimensionless numbers, such as the Reynolds and Archimedes numbers. By assuming a constant agglomerate porosity [40,44], the equivalent agglomerate diameter of particles in the MC box can be calculated. Accordingly, the volume-based PSD is reconstructed from the particle size classes, and its representative parameters (e.g., mean diameter) are evaluated using the method of moments [53,54]. Defluidization is considered to occur once the calculated minimum fluidization velocity, evaluated based on the current mean particle diameter, exceeds the superficial gas velocity, following the approach of Saleh et al. [55].

Overall, the MC box in this work still serves as a reduced representation of the real reactor, and the numerical implementation, including scaling and particle number regulation, follows the same procedures as in our previous work, with details summarized in Section 3.2.1. An updated flowchart of the model framework is presented in Figure 1.

3. Simulation Setup

3.1. Reference Experiments

It should be noted that no new experiments were conducted in the present study. All experimental data used for model validation were taken from previously published literature and are briefly summarized here for clarity. The experimental datasets used for model validation originate from two sets of semi-batch BFB experiments, both employing quartz sand as bed material and continuous feeding of solid waste fuels (PE/PP or wheat straw). The polymer pyrolysis dataset is identical to that used in our previous work [13], originally compiled from the experiments of Arena and Mastellone [21,22,24,56], which focused on the defluidization behavior during PE and PP pyrolysis. In these experiments, the reactor was operated at 450 °C under a nitrogen atmosphere, while PE or PP granules were continuously introduced into the preheated bed, and the defluidization time was measured under varying bed mass, superficial gas velocity, and polymer feed rate. The biomass combustion dataset was taken from the study by Lin et al. [26], which systematically investigated bed defluidization behavior during wheat straw combustion under varying operating conditions. In these experiments, the influence of bed temperature, gas flow rate, and bed particle size was systematically examined. For the present work, only the subset of the biomass experiments with complete and reproducible operating conditions was considered for simulation.

Table 1. Comparison of experimental systems and main operating ranges used for model validation.

Parameter	Polymer Pyrolysis	Biomass Combustion
Reactor type	Semi-batch BFB	Semi-batch BFB
Reactor diameter	0.055 $\mathrm{m}$	0.068 $\mathrm{m}$
Operating temperature	450 °C	796–820 °C
Bed material	Silica sand	Silica sand
Bed material density	2600 $\mathrm{kg}{\mathrm{m}}^{-3}$	2600 $\mathrm{kg}{\mathrm{m}}^{-3}$
Bed material diameter	0.350 $\mathrm{mm}$	0.275–0.460 $\mathrm{mm}$
Bed mass	0.240–0.480 $\mathrm{kg}$	0.635 $\mathrm{kg}$
Gas atmosphere	N2	Air + N2 ($\lambda =1.2$)
Superficial gas velocity	0.22–0.40 $\mathrm{m}{\mathrm{s}}^{-1}$	14.0–24.5 $\mathrm{NL}{\min }^{-1}$
Feed material	PE/PP pellets	wheat straw pellets
Feed rate	0.020–0.073 $\mathrm{g}{\mathrm{s}}^{-1}$	0.047 $\mathrm{g}{\mathrm{s}}^{-1}$

summarizes the main characteristics of the two experimental systems, including the reactor configurations, material properties, and ranges of the core operating variables.

In addition, for the wheat straw series, the solid feed rate used in the simulations was back-calculated from the reference air flow rate (14.0 NL min⁻¹) and stoichiometric factor ($\lambda =1.2$) reported by Lin et al. [26], based on complete-combustion stoichiometry and the ultimate analysis of wheat straw (wt.% DAF). Although Lin et al. [26] also examined the effect of the stoichiometric factor on defluidization behavior, the influence was found to be minor; therefore, all simulations in this study were performed under the reference condition. The density of the process gas mixture was calculated assuming ideal behavior. The temperature dependence of the pure-component viscosity was determined using Sutherland’s law [57], and the viscosity of the gas mixture was obtained from the Herning–Zipperer mixing rule [58]. Since these properties were derived from standard correlations rather than directly measured, they are not included in Table 1.

3.2. Simulation Parameters and Setup

In the following, the numerical setup and implementation details of the MC simulations are described in more detail. All MC simulations were implemented in MATLAB (R2025b, MathWorks) and performed on a standard commercial workstation equipped with an Intel Core i9-10900K CPU (10 physical cores) and 64 GB RAM.

3.2.1. MC Box Configuration and Scaling

In this study, the MC simulation box is regarded as a representative sample of the fluidized bed, used to reproduce the particle fluidization, agglomeration, and eventual defluidization processes. This concept has already been applied and validated in our previous work [13], and has proven effective in mapping the macroscopic behavior of the real bed onto a finite simulation domain. The underlying assumption is that, within a properly scaled sample box, the coating and agglomeration phenomena can represent the behavior of the entire fluidized bed system. For the polymer pyrolysis simulations, the parameter settings are consistent with those in the last work. In particular, the same MC box size is employed, with an initial number of primary particles of $n_{s,MC,0}$ = 20,000, and a maximum particle number of $n_{p,MC,max}$ = 20,000. However, for the biomass combustion simulation, the situation is different. Due to the distinct coating mechanisms in the PCM, as compared to the FCM and the LP, the MC box size cannot simply be taken over, and further testing was required. The convergence validation and the comparison between simulation and experimental results are presented in detail in Section 4.2. It should be noted that, regardless of whether FCM or PCM is applied, the corresponding parameter scaling methods (e.g., feed rate of polymer or biomass) and particle number regulation strategies (i.e., duplication of all particles once the number falls below half of the initial value) are also identical to those in our former study, and are only briefly mentioned here.

3.2.2. Coating Formation During Wheat Straw Combustion

The following subsection describes the procedure used to estimate the formation of molten potassium silicate on selected surface sectors of bed particles in the wheat straw combustion simulations. This step is specific to biomass combustion, since such a mechanism is absent in polymer pyrolysis, where the coating layer originates from molten polymer rather than from ash-bed reactions. In wheat straw combustion, potassium released from the fuel ash reacts with the SiO₂ surface of the bed material, resulting in localized wetting by molten potassium silicate. For simulation purposes, the biomass feed rate alone is insufficient to quantify this coating process. Therefore, taking wheat straw combustion as an example, the formation of K₂O and its subsequent reaction with the silica sand surface are analyzed in detail below.

To begin with, the ash content in the biomass is obtained from the proximate analysis of the fuel, whereas the K₂O fraction in the ash is determined from the chemical analysis of the ash produced from combustion. Accordingly, the generation rate of K₂O can be calculated from the biomass feed rate (${\dot{m}}_{\mathrm{feed}}$), the ash content of the fuel ($x_{\mathrm{ash}}$), and the K₂O fraction in the ash ($x_{{\mathrm{K}}_2\mathrm{O}}$) as follows:

$${\dot{m}}_{{\mathrm{K}}_2\mathrm{O}}={\dot{m}}_{\mathrm{feed}}·x_{\mathrm{ash}}·x_{{\mathrm{K}}_2\mathrm{O}}.$$

(6)

In the case of wheat straw, the ash content was 7.3 wt%, and the K₂O fraction in the ash was 29.0 wt% [26].

However, a portion of the fine ash particles is entrained by the gas flow and leaves the fluidized bed, meaning that not all of the K₂O generated remains available for reaction within the reactor. Considering the elutriation factor (${\zeta }_{\mathrm{elu}}$), which represents the fraction of K₂O lost from the bed due to ash entrainment, the actual K₂O mass rate retained in the bed can be expressed as:

$${\dot{m}}_{{\mathrm{K}}_2\mathrm{O},\mathrm{ret}}=(1-{\zeta }_{\mathrm{elu}})·{\dot{m}}_{{\mathrm{K}}_2\mathrm{O}}.$$

(7)

The value of ${\zeta }_{\mathrm{elu}}$ can be obtained by fitting the experimental data, which will be described in Section 3.2.3.

In the MC simulations, since the time step is relatively small, a discrete feeding approach similar to that used in the polymer pyrolysis simulation is adopted. That is, a wheat straw particle is introduced into the system at regular time intervals, and the total amount of K₂O added to the simulation domain during each feeding event is given by:

$$m_{\mathrm{feed},{\mathrm{K}}_2\mathrm{O},j}={\displaystyle \frac{{\dot{m}}_{{\mathrm{K}}_2\mathrm{O}}·\left(t_{\mathrm{feed},j}-t_{\mathrm{feed},j-1}\right)}{S}}\mathrm{for}j=1,2,3,\dots ,N,$$

(8)

where $t_{\mathrm{feed},j}$ denotes the time of the j-th feeding moment, which is derived in the same manner as in the polymer MC simulations. S is the scaling factor, defined as the ratio between the total number of primary particles in the bed and the number of primary sands represented in the MC box.

Since the total amount of K₂O participating in the reaction is known, the corresponding amount of SiO₂ involved can be determined from the stoichiometric relationship. The total mass of potassium silicate formed during each feeding step can be expressed as:

$$m_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,j}={\displaystyle \frac{m_{\mathrm{feed},{\mathrm{K}}_2\mathrm{O},j}}{x_{{\mathrm{K}}_2\mathrm{O},\mathrm{melt}}}},$$

(9)

where $x_{{\mathrm{K}}_2\mathrm{O},\mathrm{melt}}$ is the mass fraction of K₂O in the corresponding molten potassium silicate. The detailed stoichiometric coefficients adopted will be described in Section 3.2.4. The total volume of sand consumed from all selected particles can then be calculated as:

$$V_{\mathrm{s},\mathrm{cons},j}=m_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,j}·(1-x_{{\mathrm{K}}_2\mathrm{O},\mathrm{melt}})·{\displaystyle \frac{1}{{\rho }_{\mathrm{s}}}},$$

(10)

where ${\rho }_{\mathrm{s}}$ is the density of sand.

The molten potassium silicate generated during each feeding step is evenly distributed over $n_{\sec ,\mathrm{pc},j}$ randomly selected external surface sectors of the particles. As described in Section 2.3, the PCM assumes that each selected sector receives the same amount of coating material. Accordingly, the molten volume allocated to each sector can be expressed as:

$$V_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,\sec ,j}={\displaystyle \frac{m_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,j}}{n_{\sec ,\mathrm{pc},j}·{\rho }_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2}}},$$

(11)

where ${\rho }_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2}$ is the density of the molten potassium silicate. On the other hand, since each surface sector receives the same amount of K₂O, the corresponding consumption of SiO₂ is also identical among these sectors. Thus, the consumed sand volume per sector can be written as:

$$V_{\mathrm{s},\mathrm{cons},\sec ,j}={\displaystyle \frac{V_{\mathrm{s},\mathrm{cons},j}}{n_{\sec ,\mathrm{pc},j}}}.$$

(12)

The calculated values of $V_{{\mathrm{K}}_2\mathrm{O}-{\mathrm{SiO}}_2,\sec ,j}$ and $V_{\mathrm{s},\mathrm{cons},\sec ,j}$ are subsequently used in the volume-balance relations introduced in Section 2.3 (Equations (3)–(5)) to update the coating thickness on each reacted sector ($h_{\sec ,j}$). This procedure establishes a consistent link between the chemical formation of molten potassium silicate and the geometric evolution of coated particles within the PCM framework.

3.2.3. Elutriation Factor Estimation

As discussed in the previous subsection, part of the potassium species originating from the fuel ash may be removed from the bed due to ash elutriation, meaning that not all K₂O generated during combustion reacts with the sand surface. Lin et al. [26] experimentally measured the accumulation of potassium in the bed at various temperatures and identified the occurrence of elutriation of K-containing fine ash particles. To quantitatively evaluate this effect, their data were re-plotted and analyzed, as shown in Figure 2.

The dashed line represents the theoretical accumulation of potassium in the bed assuming complete retention, while the solid line corresponds to the global linear fit of the measured data. As can be seen, elutriation is negligible during the early stage of combustion but becomes progressively more pronounced with time. Based on the linear fits, the slope of the theoretical line is 0.0455, whereas the average slope of the experimental data is 0.0354. The ratio between these two slopes yields a retention fraction of approximately 0.778, indicating that about 77.8% of potassium was retained in the bed, while roughly 22.2% was entrained with fine ash. In the subsequent modeling, the corresponding elutriation factor (${\zeta }_{\mathrm{elu}}=0.222$) is assumed to remain constant throughout the process and is applied to correct the actual K₂O supply rate in the simulation.

3.2.4. Viscosity of Molten Potassium Silicate

As summarized in Table 1, the combustion temperature of wheat straw in the present simulations ranges from 796 to 869 °C. This implies that, when the bed temperature exceeds the lowest value (796 °C), the potassium silicate formed on the sand surface is expected to enter the molten state. Such a melting transition is particularly important, as it serves as a key factor leading to particle agglomeration. Moreover, the viscosity of molten potassium silicate varies strongly with temperature, which directly affects both the probability and rate of agglomeration. Bockris et al. [59] systematically measured the viscosity of K₂O-SiO₂ binary melts with different compositions at high temperatures, providing fundamental data for establishing the quantitative relationship between melt viscosity and temperature. Figure 3 shows the experimental viscosity data and the corresponding fitted curves for two representative compositions (Case A: 31.1 wt% K₂O; Case B: 35.2 wt% K₂O).

As can be observed, the respective liquidus temperatures ($T_{\mathrm{liquidus}}$) of these two compositions are 780 and 820 °C, which can be regarded as the representative melting points of potassium silicate. In addition, the viscosity decreases exponentially with increasing temperature, indicating a typical Arrhenius expression. Considering that Case A matches well with the simulated temperature range (796–869 °C), this composition was selected to represent the melt in this study. The viscosity-temperature relationship was therefore fitted and expressed as follows:

$$\mu \left(T\right)=3.11\times {10}^{-5}exp\left({\displaystyle \frac{1.88\times {10}^5}{R\left(T+273.15\right)}}\right),$$

(13)

which allows the estimation of the potassium silicate melt viscosity at different temperatures during the simulations. Meanwhile, the corresponding chemical composition (22.3 mol% K₂O and 77.7 mol% SiO₂) was adopted to represent the reaction stoichiometry between K₂O and SiO₂ on the sand surface.

4. Results and Discussion

4.1. Polymer Pyrolysis

4.1.1. Preliminary Simulations

As mentioned above, since the polymer pyrolysis simulations in this work follow the same configuration as the previous study [13], the MC box size was directly adopted from that work (20,000 particles). In the FCM, each selected particle is assumed to be entirely coated by a certain polymer coating layer. However, the fraction of bed particles that gets fully coated during the pyrolysis process cannot be experimentally measured, as the coating process occurs dynamically within the fluidized bed. Therefore, the coating percentage (${\theta }_{\mathrm{fc}}$) must be determined through numerical calibration based on one existing experimental result. Here, similar to our previous work, Experiment No. 7 was chosen as the benchmark case for the calibration of polymer pyrolysis simulation.

To establish the quantitative relationship between coating percentage and defluidization behavior, a series of simulations were performed under the same operating conditions as the benchmark experiment, but with different assumed coating percentages. For each case, the corresponding defluidization time was obtained from the simulation, and the results are plotted in Figure 4.

It can be seen that the simulated defluidization time decreases with increasing coating percentage, and the simulated data can be well correlated by the following power-law relationship:

$$t_{\mathrm{def}}=0.0606{\theta }_{\mathrm{fc}}^{-0.891}.$$

(14)

The coating percentage corresponding to the experimentally measured defluidization time ($127.0\mathrm{s}$) was obtained from the fitted correlation, as indicated by the intersection between the fitted curve and the benchmark line in Figure 4. This yields a calibrated coating percentage of

$${\theta }_{\mathrm{fc},\mathrm{cal}}=0.0187\%.$$

(15)

This value represents the fraction of sand particles that become fully coated after each polymer feeding moment under the benchmark condition, and it is adopted as the reference coating percentage for all subsequent PE and PP pyrolysis simulations in this study. It is worth noting that the use of such a single calibrated coating percentage for both PE and PP is physically reasonable under the present operating conditions. Within the FCM, the coating percentage implicitly reflects material-related properties such as polymer melting behavior and surface wetting characteristics. At the investigated process temperature of 450 °C, PE and PP exhibit similarly low melt viscosities and comparable rheological behavior, leading to similar coating and agglomeration characteristics. As a result, ${\theta }_{\mathrm{fc},\mathrm{cal}}$ can be regarded as effectively material-insensitive for these two polymers under the studied conditions, which supports the use of a single calibrated value in the FCM for both PE and PP cases.

In addition, Equation (14) enables a direct sensitivity assessment of the coating percentage. As defined earlier, the coating percentage is a non-measurable lumped parameter that integrates the combined effects of multiple underlying mechanisms, thereby representing unresolved micro-scale process uncertainties within the model. Accordingly, the sensitivity analysis is focused on ${\theta }_{\mathrm{fc}}$. By taking the logarithm of Equation (14) and performing a differential, the local relative sensitivity can be expressed as

$${\displaystyle \frac{\Delta t_{\mathrm{def}}}{t_{\mathrm{def}}}}\approx -0.891{\displaystyle \frac{\Delta {\theta }_{\mathrm{fc}}}{{\theta }_{\mathrm{fc}}}}.$$

(16)

This relation indicates that an uncertainty of $\pm 10\%$ in the coating percentage leads to an uncertainty of approximately $\mp 8.9\%$ in the predicted defluidization time for the benchmark case with fixed operating conditions. More specifically, changing the coating percentage to $0.9{\theta }_{\mathrm{fc},\mathrm{cal}}$ or $1.1{\theta }_{\mathrm{fc},\mathrm{cal}}$ results in predicted defluidization times of 139.7 s (+9.8%) and 116.8 s (−8.1%), respectively. This magnitude is consistent with the role of ${\theta }_{\mathrm{fc}}$ as a key effective parameter controlling coating-driven agglomeration within the FCM. The influence of operating parameters is not examined through individual perturbation-based sensitivity analyses. Instead, their effects are evaluated implicitly through the cross-condition model validation presented in the following sections, where ${\theta }_{\mathrm{fc}}$ is scaled according to these operating inputs. If the model is able to predict defluidization times consistently under different operating inputs, this is regarded as a more representative way to assess the influence of parameters at the engineering scale.

4.1.2. Model Validation

For operating conditions other than the benchmark experiment, the coating percentage was not recalibrated but scaled linearly with respect to the variations in bed mass and gas velocity. It is assumed that a smaller bed mass (at a fixed bed material size) leads to a larger number of particles being coated, while a higher gas velocity enhances particle-polymer collisions and promotes the spatial distribution of the molten polymer. Accordingly, the coating percentage for each operating condition was calculated using the following proportional relation:

$${\theta }_{\mathrm{fc}}={\displaystyle \frac{m_{\mathrm{b},\mathrm{ref}}}{m_{\mathrm{b}}}}·{\displaystyle \frac{u_{\mathrm{g}}}{u_{\mathrm{g},\mathrm{ref}}}}·{\theta }_{\mathrm{fc},\mathrm{cal}},$$

(17)

where $m_{\mathrm{b},\mathrm{ref}}$ and $u_{\mathrm{g},\mathrm{ref}}$ are the bed mass and the gas velocity of the benchmark experiment (No. 7), respectively. This simple linear scaling, which considers only the variations in bed mass and gas velocity, preserves the proportionality between coating intensity and the operating conditions of the polymer pyrolysis experiments.

Using the scaled coating percentages, simulations were carried out for all other experimental conditions listed in

Table 2. Comparison of experimental and simulated defluidization times under various conditions during the PE pyrolysis.

PE	${\mathit{m}}_{\mathbf{b}}$ [kg]	${\dot{\mathit{m}}}_{\mathbf{feed}}$ [$\mathbf{g}{\mathbf{s}}^{-1}$]	${\mathit{u}}_{\mathbf{g}}$ [$\mathbf{m}{\mathbf{s}}^{-1}$]	Exp. ${\mathit{t}}_{\mathbf{def}}$ [$\mathbf{s}$]	Sim. ${\mathit{t}}_{\mathbf{def}}$ ([13]) [$\mathbf{s}$]	Sim. ${\mathit{t}}_{\mathbf{def}}$ (FCM) [$\mathbf{s}$]
1	0.240	0.020	0.22	312.0	209.5	244.3
2	0.240	0.040	0.22	135.0	114.1	125.2
3	0.240	0.060	0.22	94.0	88.2	87.3
4	0.240	0.073	0.22	81.0	76.9	75.1
5	0.360	0.020	0.22	333.0	312.4	356.3
6	0.360	0.040	0.22	150.0	161.8	176.1
7	0.360	0.060	0.22	127.0	115.2	128.7
8	0.360	0.073	0.22	111.0	97.9	108.1
9	0.480	0.040	0.22	190.0	222.8	222.8
10	0.480	0.060	0.22	157.0	171.0	168.9
11	0.480	0.073	0.22	141.0	159.3	142.6
12	0.360	0.040	0.40	140.0	137.7	186.2
13	0.360	0.060	0.40	113.0	119.4	128.4

and

Table 3. Comparison of experimental and simulated defluidization times under various conditions during the PP pyrolysis.

PP	${\mathit{m}}_{\mathbf{b}}$ [kg]	${\dot{\mathit{m}}}_{\mathbf{feed}}$ [$\mathbf{g}{\mathbf{s}}^{-1}$]	${\mathit{u}}_{\mathbf{g}}$ [$\mathbf{m}{\mathbf{s}}^{-1}$]	Exp. ${\mathit{t}}_{\mathbf{def}}$ [$\mathbf{s}$]	Sim. ${\mathit{t}}_{\mathbf{def}}$ ([13]) [$\mathbf{s}$]	Sim. ${\mathit{t}}_{\mathbf{def}}$ (FCM) [$\mathbf{s}$]
14	0.240	0.020	0.22	177.0	169.7	218.6
15	0.240	0.040	0.22	135.0	99.4	117.6
16	0.240	0.060	0.22	72.0	71.8	80.0
17	0.240	0.073	0.22	68.0	63.1	67.5
18	0.360	0.020	0.22	384.0	267.8	369.2
19	0.360	0.040	0.22	166.0	140.9	167.3
20	0.360	0.060	0.22	135.0	117.2	117.5
21	0.360	0.073	0.22	88.0	98.6	89.9
22	0.480	0.040	0.22	264.0	205.7	226.8
23	0.480	0.060	0.22	155.0	149.6	156.8
24	0.480	0.073	0.22	126.0	133.4	127.3
25	0.240	0.020	0.40	168.0	181.6	196.5
26	0.240	0.040	0.40	115.0	109.8	114.1
27	0.240	0.060	0.40	106.0	83.4	83.6
28	0.240	0.073	0.40	65.0	70.5	64.0
29	0.360	0.040	0.40	147.0	144.2	162.3
30	0.360	0.060	0.40	128.0	105.8	115.6
31	0.360	0.073	0.40	94.0	96.0	91.3

, and the predicted defluidization times were compared with the corresponding experimental data. The experimental dataset employed here is identical to that used in our previous research, which was originally compiled from the works of Arena and Mastellone [21,22,24,56]. In total, 31 experiments were included, comprising 13 cases for PE (No. 1–13, Table 2) and 18 cases for PP (No. 14–31, Table 3) pyrolysis.

For comparison, the MC model proposed in our earlier work was reanalyzed using the same dataset. In addition to the coefficient of determination originally reported ($R^2=0.804$), the reanalysis also provided a correlation coefficient of 0.924, a root-mean-square error (RMSE) of 33.2 s, and a mean absolute percentage error (MAPE) of 11.1%. The results of the present FCM are shown in Figure 5. The FCM predictions exhibit an overall strong agreement with the experimental data, yielding a correlation coefficient of 0.956, an $R^2$ of 0.912, an RMSE of 22.2 s, and a MAPE of 9.4%. Most data points are located close to the parity line, demonstrating that the linearly scaled coating percentage successfully transfers the calibrated coating behavior to other operating conditions. The model consistently reproduces the defluidization behavior for both PE and PP series, confirming the robustness of the proposed coating-based framework. Although the calibration was conducted using a random representative PE case, the satisfactory prediction accuracy for the PP series suggests a similar coating mechanism.

A more detailed inspection of the results shows that the proposed FCM is able to consistently reproduce the influence of key operating parameters on bed defluidization behavior. Specifically, an increase in polymer feed rate leads to a systematic reduction in defluidization time, which can be attributed to the accelerated formation and growth of agglomerates driven by enhanced polymer coating. In addition, a reduction in bed mass results in earlier bed collapse, since the same amount of molten polymer is distributed over a smaller number of bed particles, leading to a more rapid increase in effective surface coverage. Regarding the effect of superficial gas velocity, the FCM predictions indicate that its influence on defluidization time is relatively weak. In particular, increasing gas velocity does not necessarily result in a pronounced delay of bed collapse. All of these trends are consistent with experimentally observed defluidization behavior [21,22,24,56] and with conclusions drawn in previous LP-AP coupled MC modeling studies [13].

Overall, the introduction of calibration and scaling of coating percentage, which replaces the layering period in the previous LP-AP coupled MC model, significantly improves the quantitative accuracy of the predictions. This improvement may be attributed to the fact that the previous AP did not consider the possibility of sand agglomerates being fully coated by the molten polymer. Nevertheless, the earlier LP-AP coupled model was still able to achieve satisfactory predictive performance without calibration, indicating that the newly developed FCM provides a mechanistic complement rather than a replacement to the former framework.

4.1.3. Discussion Based on PSD

Besides the estimation of the defluidization time, the MC simulations also provide the time evolution of the volume-weighted cumulative particle size distribution ($Q_3$). Figure 6 summarizes the evolution of $Q_3$ for all 31 PE and PP simulations. Since the final mean particle size and the corresponding PSDs differ with the superficial gas velocity, the data are grouped into two subfigures according to $u_{\mathrm{g}}$: (a) $u_{\mathrm{g}}=0.22\mathrm{m}{\mathrm{s}}^{-1}$ and (b) $u_{\mathrm{g}}=0.40\mathrm{m}{\mathrm{s}}^{-1}$. In each subfigure, four representative time points are displayed, i.e., $\tau =0.25$, $0.50$, $0.75$, and $1.00$, respectively, where $\tau =t/t_{\mathrm{def}}$ is the normalized time, defined as the ratio of the process time to the corresponding defluidization time. At these four time snapshots, to enable direct comparison across different operating conditions, the shaded regions denote the across-case envelopes (min-max ranges) of all simulated cases. In addition, one representative simulation was randomly selected for each gas velocity to visually illustrate the characteristic PSD profiles and their temporal evolution, namely example cases: (a) Sim. No. 7 and (b) Sim. No. 31.

As can be seen, both the shaded envelopes and the representative cases show that the PSDs at different times are well separated, with overlap occurring only at the coarse size classes. For all simulations, as the process proceeds, an increasing number of primary sand particles become coated with molten polymer, leading to a gradual decrease in the fraction of uncoated primary particles (reflected by the $Q_3$ value at $d_{\mathrm{p},0}$). Accordingly, the $Q_3$ curves shift progressively toward larger size classes and become broader with time, indicating a continuous increase in the mean particle size and an expansion of the distribution toward the coarse end. Even at $\tau =1.00$ (the defluidization moment), a portion of uncoated primary sand particles still remains in the bed. In addition, the PSDs of PE and PP show highly similar shapes and temporal evolution, which can be attributed to their comparable physical properties at the process temperature. These observations are consistent with the previous LP-AP model results and clearly reflect the influence of the scaled coating percentage.

It is noteworthy that these trends also directly reflect the influence of the scaled coating percentage. At identical gas velocity and process time, the occurrence of the envelopes originates not only from the stochastic nature of the MC simulation or from differences in feed rate, but also from the variations in coating percentage induced by the mass-based scaling. On the other hand, under higher gas velocities, the increased coating fraction resulting from the velocity-based scaling naturally leads to a broader PSD.

Overall, based on the capability of the MC model, the PSD evolution observed across different materials and operating conditions demonstrates the general applicability of the calibration and scaling of the coating percentage concept, which was further extended to the modeling of biomass combustion.

4.2. Biomass Combustion

4.2.1. Preliminary Simulations

Before performing the main simulations of wheat straw combustion, several preliminary tests were conducted. First, the size of the MC box was evaluated. The test was carried out for experimental case No. 36, which represents a mid-range condition in terms of defluidization time among all applied wheat straw combustion experiments (see

Table 4. Comparison of experimental and simulated defluidization times and corresponding PSD statistics at the defluidization moment during wheat straw (WS) combustion.

WS	${\mathit{d}}_{\mathbf{p},0}$ [$\mathbf{mm}$]	${\mathbf{\Phi }}_{\mathbf{g}}$ [$\mathbf{NL}{\mathbf{min}}^{-1}$]	${\mathit{T}}_{\mathbf{b}}$ [°C]	Exp. ${\mathit{t}}_{\mathbf{def}}$ [$\mathbf{s}$]	Sim. ${\mathit{t}}_{\mathbf{def}}$ [$\mathbf{s}$]	Sim. ${\overline{\mathit{d}}}_{\mathbf{p},\mathbf{def}}$ [$\mathbf{mm}$]	Sim. ${\mathit{\sigma }}_{\mathbf{p},\mathbf{def}}$ [$\mathbf{mm}$]
32	0.275	14.0	796	2526.0	2709.5	0.970	0.387
33	0.275	14.0	797	2712.0	2666.5	0.970	0.370
34	0.275	14.0	822	1560.0	1705.2	0.989	0.376
35	0.275	14.0	823	1584.0	1674.7	0.989	0.373
36	0.388	17.5	843	1158.0	1171.1	1.118	0.400
37	0.388	17.5	845	1152.0	1133.8	1.119	0.404
38	0.388	17.5	866	1098.0	796.9	1.138	0.423
39	0.388	17.5	869	1062.0	754.9	1.141	0.403
40	0.460	24.5	819	1560.0	1515.7	1.316	0.513
41	0.460	24.5	820	1590.0	1499.6	1.318	0.493

). Owing to its balanced agglomeration behavior, this condition serves as a representative and moderately sensitive case, where the influence of the MC box size on the simulation results can be reasonably evaluated. Once numerical convergence was verified under this condition, other operating cases (with slower agglomeration and later defluidization) are expected to be less sensitive to the box size.

Figure 7 shows the results of the convergence test for the MC box size. The predicted defluidization time was compared across different box sizes ranging from 700 to 20,000. As can be observed, noticeable fluctuations appear when the box size is smaller than 2500, whereas the results gradually level out for larger box sizes. In particular, when $n_{\mathrm{s},\mathrm{MC},0}>5000$, the predicted values become nearly stable, with an average deviation from the experimental defluidization time (1158.0 s) of less than one percent (0.98%). For a box size of 10,000, the relative deviation from the largest case (20,000) is only 0.32%, calculated with respect to the corresponding defluidization time. In contrast, the simulation runtime increases markedly from 119 to 305 min. Considering both numerical accuracy and computational cost, a box size of 10,000 was therefore adopted as the representative size for all wheat straw combustion simulations in this work.

Subsequently, the coating percentage (${\theta }_{\mathrm{pc}}$) involved in the wheat straw combustion process was also calibrated. The benchmark experiment selected for this calibration is identical to that used in the MC box size test (Experiment No. 36). The calibration procedure follows exactly the same approach as for the FCM model, and the resulting correlation is illustrated in Figure 8.

It can be seen that the correlation between coating percentage and defluidization time can again be well described by a power-law:

$$t_{\mathrm{def}}=0.234{\theta }_{\mathrm{pc}}^{-0.970}.$$

(18)

Moreover, compared to the calibration of the FCM, all simulation data points here show an even better agreement with the fitted curve. From the experimentally measured defluidization time, the corresponding calibrated coating percentage for the series of wheat straw combustion simulations was obtained from the fitted correlation:

$${\theta }_{\mathrm{pc},\mathrm{cal}}=0.0155\%.$$

(19)

This value represents the fraction of external surface sectors of sand particles that react with ${\mathrm{K}}_2\mathrm{O}$ after each feeding event under the benchmark condition. For all other PCM simulations, it is used as the reference value for the subsequent scaling of the coating percentage.

Similar to the analyse for the FCM, Equation (18) also allows a direct assessment of the sensitivity of the partial coating percentage. As ${\theta }_{\mathrm{pc}}$ is likewise defined as a non-measurable lumped parameter integrating multiple underlying mechanisms during biomass combustion, its sensitivity provides insight into the robustness of the PCM predictions. For the benchmark case, the fitted power-law exponent implies that a relative uncertainty of $\pm 10\%$ in ${\theta }_{\mathrm{pc}}$ leads to an uncertainty of approximately $\mp 9.7\%$ in the predicted defluidization time. Accordingly, adjusting the coating percentage to $0.9{\theta }_{\mathrm{pc},\mathrm{cal}}$ or $1.1{\theta }_{\mathrm{pc},\mathrm{cal}}$ yields predicted defluidization times of 1285.2 s (+10.8%) and 1057.9 s (−8.8%), respectively. These analyses indicate that ${\theta }_{\mathrm{pc}}$ plays an important role in governing coating-induced agglomeration within the PCM. The influence of operating parameters is assessed based on the ability of the PCM to consistently reproduce defluidization behavior under different operating conditions.

In the previous section, the necessity of scaling the coating percentage based on the calibrated reference value was discussed. For the present series of wheat straw combustion experiments, the main varying parameters are the volumetric gas flow rate (${\Phi }_{\mathrm{g}}$), the bed material diameter ($d_{\mathrm{s}}$), and the process temperature ($T_{\mathrm{b}}$). It should be noted that the volumetric gas flow rate can be directly converted to the superficial gas velocity ($u_{\mathrm{g}}$) since all experiments were conducted in the same reactor. Accordingly, the overall scaling of the partial coating percentage (${\theta }_{\mathrm{pc}}$) can be expressed as

$${\theta }_{\mathrm{pc}}=f\left(u_{\mathrm{g}}\right)·g\left(d_{\mathrm{s}}\right)·h\left(T_{\mathrm{b}}\right)·{\theta }_{\mathrm{pc},\mathrm{cal}},$$

(20)

where $f\left(u_{\mathrm{g}}\right)$, $g\left(d_{\mathrm{s}}\right)$, and $h\left(T_{\mathrm{b}}\right)$ are the three corresponding scaling factors.

Similar to the FCM model, a linear scaling approach is employed, in which the coating percentage varies linearly with the key operating parameters relative to their respective reference values. For the gas velocity, as has been validated in the FCM model, the linear dependence follows the same definition, and the corresponding scaling factor is given by

$$f\left(u_{\mathrm{g}}\right)={\displaystyle \frac{u_{\mathrm{g}}}{u_{\mathrm{g},\mathrm{ref}}}}.$$

(21)

However, for the bed material diameter and the bed temperature, the scaling relations cannot be directly expressed in terms of these variables, but must be formulated based on their physical influences. Assuming that the amount of ${\mathrm{K}}_2\mathrm{O}$ available for reaction is proportional to the total external surface area of the bed, the particle size related scaling factor $g\left(d_{\mathrm{s}}\right)$ can be derived from the ratio of total surface areas:

$$g\left(d_{\mathrm{s}}\right)={\displaystyle \frac{A_{\mathrm{s},\mathrm{tot}}}{A_{\mathrm{s},\mathrm{tot},\mathrm{ref}}}}=\left({\displaystyle \frac{m_{\mathrm{b}}}{{\rho }_{\mathrm{s}}(\pi d_{\mathrm{s}}^3/6)}}\pi d_{\mathrm{s}}^2\right){\displaystyle /}\left({\displaystyle \frac{m_{\mathrm{b}},\mathrm{ref}}{{\rho }_{\mathrm{s}}(\pi d_{\mathrm{s},\mathrm{ref}}^3/6)}}\pi d_{\mathrm{s},\mathrm{ref}}^2\right).$$

(22)

Since all experiments in this series were conducted with the same total bed mass, the total number of particles varies inversely with the particle volume, resulting in a corresponding change in the overall external surface area. Accordingly, Equation (22) can be simplified as:

$$g\left(d_{\mathrm{s}}\right)={\displaystyle \frac{d_{\mathrm{s},\mathrm{ref}}}{d_{\mathrm{s}}}}.$$

(23)

This implies that smaller sand particles exhibit a higher coating percentage due to their larger total external surface area. For the bed temperature, the scaling is mapped to the viscosity of the molten potassium silicate, following the interpretation proposed by Lin et al. [26]. They reported that an increase in bed temperature reduces the viscosity of the molten layer, which enhances the coating coverage on sand surfaces. Accordingly, the temperature-related scaling factor is expressed as

$$h\left(T_{\mathrm{b}}\right)={\displaystyle \frac{\mu (T_{\mathrm{b}},\mathrm{ref})}{\mu \left(T_{\mathrm{b}}\right)}}.$$

(24)

Finally, based on Equations (20)–(24), the expression used in the PCM to scale the coating percentage under different operating conditions is obtained as

$${\theta }_{\mathrm{pc}}={\displaystyle \frac{u_{\mathrm{g}}}{u_{\mathrm{g},\mathrm{ref}}}}·{\displaystyle \frac{d_{\mathrm{s},\mathrm{ref}}}{d_{\mathrm{s}}}}·{\displaystyle \frac{\mu (T_{\mathrm{b}},\mathrm{ref})}{\mu \left(T_{\mathrm{b}}\right)}}·{\theta }_{\mathrm{pc},\mathrm{cal}}.$$

(25)

In this context, ${\theta }_{\mathrm{pc}}$ serves as a lumped parameter quantifying the fraction of bed particle surface regions coated by reaction-generated molten potassium silicates during wheat straw combustion. From a physical perspective, this coating percentage integrates the combined effects of fuel ash composition, the extent of ash elutriation, and the probability of ash-bed material surface reactions leading to the formation of effective coated surface regions. Together with the operating parameters explicitly included in Equation (25), ${\theta }_{\mathrm{pc}}$ provides a physically meaningful link between raw biomass properties, operating conditions, and coating-induced agglomeration behavior within the PCM framework.

4.2.2. Model Validation

After the calibration based on the benchmark experiment and the implementation of the scaling of the coating percentage, model validation was conducted for all experimental cases of the wheat straw combustion series. The corresponding comparisons between the experimental and simulated defluidization times are summarized in Table 4. Figure 9 presents the overall correlation between the experimental and simulated defluidization times.

It can be seen that the PCM predictions exhibit a generally good agreement with the experimental data across the entire range of operating conditions. Most data points are closely distributed around the reference line, indicating that the applied calibration and scaling approach can also accurately predict the coating percentage under different operating conditions. Quantitatively, the correlation analysis yields a coefficient of determination of $R^2=0.917$, a correlation coefficient of 0.980, an RMSE of 158.5 s, and a MAPE of 8.8%, confirming the predictive capability of the PCM.

A closer examination of the listed defluidization times in Table 4 shows that, for all three gas velocities and the corresponding particle sizes, the simulation results show a systematic decrease in defluidization time with increasing temperature, which is consistent with the overall trend summarized by Lin et al. [26]. The largest discrepancies are observed in the highest temperature range (Sim. No. 38 and 39), where the simulated defluidization occurs earlier than the experimental results, corresponding to relative errors of −27.6% and −28.9%, respectively, whereas for all other cases the deviations remain within −0.8% to +8.5%. From the modeling perspective, this behavior may be attributed to temperature-induced changes in the composition of the molten potassium silicate, which could alter its viscosity-temperature relationship. For instance, as shown in Figure 3, increasing temperature allows a wider range of potassium silicate compositions to reach the liquidus state, resulting in more complex viscosity-temperature behavior of the melt. At high temperatures, the overall melt viscosity tends to decrease, while stronger surface reactions and the formation of additional ash-derived species may simultaneously occur, thereby altering the effective properties of sticky surface regions on particles, such as coating percentage and the potential loss of stickiness due to deposition of other ash species. In the present model, however, such temperature-dependent effects are simplified by a linear scaling of the coating percentage based on a fixed melt composition, which may lead to an overestimation of the onset and growth rates of agglomeration. In addition, the reaction between K₂O and sand particle surfaces is expected to proceed more rapidly at high temperatures. Even though this reaction is assumed to occur instantaneously in the current model, the resulting local changes in particle properties may further amplify deviations in the particle surface properties. Consequently, the viscosity-based linear scaling used to represent coating percentage may no longer be fully applicable under such high temperature conditions, leading to an earlier prediction of the defluidization moment.

Based on the features of the MC simulation, the PSD statistics at the defluidization moment are also listed in Table 4. Since the simulated results for the paired experiments conducted under nearly identical operating conditions show negligible differences, only one representative case from each pair (i.e., Sim. No. 32, 34, 36, 38, and 40) is selected for further analysis. The corresponding PSDs at the defluidization moment are presented in Figure 10.

It is evident that, as the gas velocity and initial sand particle size increase, the overall PSD tends to shift toward larger particle diameters and becomes broader. For particles with the same initial diameter and at a constant gas flow rate (i.e., Sim. No. 32 vs. 34 and Sim. No. 36 vs. 38), an increase in bed temperature enlarges the mean particle size, as reflected by the higher ${\overline{d}}_{\mathrm{p},\mathrm{def}}$ values. This behavior is mainly attributed to the temperature-dependent properties of the gas phase, such as its density and viscosity, which modify the fluidization hydrodynamics and thereby influence the mean particle size at the defluidization moment in the simulations. In contrast, although an increase in temperature generally leads to a higher coating percentage, the standard deviation (${\sigma }_{\mathrm{p},\mathrm{def}}$) does not necessarily follow the same trend. For instance, in Sim. No. 32 vs. 34, ${\sigma }_{\mathrm{p},\mathrm{def}}$ decreases with temperature, whereas in Sim. No. 36 vs. 38 it slightly increases. This indicates that the spread of the PSD is not directly determined by the coating percentage. A higher coating extent indeed promotes the participation of more particles in the agglomeration process. However, it may either lead to the formation of a few large agglomerates or a bigger number of smaller clusters. This variability originates from the stochastic nature of the MC model, in which coating and collision events are governed by probabilities. Overall, the PCM shows its capability to capture the defluidization moment during the wheat straw combustion, as well as to estimate the characteristics of the PSD under various operating conditions.

From a broader modeling perspective, it is instructive to position the proposed MC framework relative to CFD-PBM approaches mentioned in Section 1. Reactive CFD-PBM simulations of BFBs typically require the simultaneous resolution of the gas–solid flow field and the solution of population balance equations, leading to a large number of spatial degrees of freedom, very small time steps, and substantial computational effort. For instance, in the work of Tasleem et al. [37], the CFD-PBM framework with approximately 60,200 hexahedral mesh elements and a time step of $1 \times {10}^{-5}\mathrm{s}$, relying on a 40-core high-performance computer to simulate more than 20 min of physical process time and to predict the defluidization behavior of the bed with good agreement with experiments. Such simulations provide detailed information on local flow structures and PSDs but are associated with a considerable computational cost. In contrast, the MC framework proposed in this study focuses on bed-scale agglomeration and defluidization behavior and describes the system evolution using a statistically representative simulation box with a limited number of effective parameters, without explicitly resolving local instantaneous flow fields. All MC simulations in this work were performed on a standard workstation, as specified in Section 3. As illustrated in Figure 7, for the benchmark case, even with a relatively large MC box size of 20,000, the simulation of approximately 20 min of physical process time requires about 5 h of computation. While CFD-PBM models offer superior spatial resolution and access to local hydrodynamic quantities, their application to coating-induced defluidization is often system specific and computationally demanding. The MC approach adopted here can therefore be regarded as a complementary modeling strategy, particularly suited for rapid prediction, parametric studies, and extension to systems involving different coating mechanisms, such as polymer pyrolysis and biomass combustion. To further evaluate the performance of the present model and analyze the coating-related phenomena, a detailed analysis is presented in the following sections.

4.2.3. Model-Based Analyses

Coating Layer Thickness

Besides the PSD, the MC simulations also recorded the historical external coating thickness on each reacted sector during all successful collision events, as shown in Figure 11a.

In the present model, once a sector is selected, it is assumed to react uniformly with ${\mathrm{K}}_2\mathrm{O}$, forming a homogeneous potassium silicate layer on the particle surface. As shown, most coating layers exhibit thicknesses below 5 μm, while a few cases reach approximately 9 μm. These values are smaller than the average external coating thickness of 10–20 μm reported by Lin et al. [26], although their SEM images also indicated the presence of locally thinner surface layers as well as internal bridge structures.

To further analyze the influence of the surface reaction and coverage assumptions on the simulation results, an additional comparative test was conducted. In this new model assumption, the definition of the activated position was adapted from Du et al. [45]: when one external sector is selected, the fraction of its surface participating in the reaction is randomly assigned to $25\%$, $50\%$, $75\%$, or $100\%$ (partial sector coverage), instead of assuming full sector coverage (i.e., $100\%$) as in the original model. This modification aims to represent the non-uniform reactions on particle surfaces under realistic conditions and to assess the effect of spatial randomness in the formation of potassium silicate layers, which is also physically reasonable because the original assumption may overestimate the actual reaction area, particularly when the size of ash particles is small.

As shown in Figure 11a, the random partial sector activation scheme results in a higher mean thickness and a broader coating thickness distribution. According to the simulation statistics, the maximum coating thickness reaches approximately 15 μm. Although the overall values remain lower than the experimental reference, the new assumption presents a closer agreement with realistic coating behavior. It should be noted that the present study only considers a chemically driven, coating-induced agglomeration mechanism, in which particle aggregation occurs through the formed potassium silicate layers at the contact interfaces. Physical mechanisms such as melt-induced adhesion (e.g., the molten $\mathrm{KCl}$ on the particle surface discussed by Anicic [27]) are not included in the current model. These physical processes may also lead to an increase in coating thickness under real operating conditions, resulting in thicker surface layers observed in experiments.

Nevertheless, this modification does not alter the overall interpretation presented in the previous section. As plotted in Figure 11b, the obtained PSDs under both assumptions at the defluidization moment exhibit negligible differences. This is because the formed bridge necks are extremely thin compared with the particle or agglomerate sizes. Therefore, even though the historical external coating thickness distribution changes significantly, the predicted defluidization time and PSD evolution remain essentially unaffected.

Plastic Pyrolysis vs. Biomass Combustion

To compare the characteristics of the two modeling frameworks, one representative polymer pyrolysis case (FCM) and one biomass combustion case (PCM) were selected and plotted together for qualitative analysis, as shown in Figure 12.

Although the two simulated BFB systems differ in materials and operating conditions, their comparison provides valuable insight into how distinct coating mechanisms influence agglomeration behavior. Figure 12a shows that the PSD obtained from the PCM is smoother and more continuous, whereas the FCM curve displays a more step-like pattern. This difference originates from the distinct coating mechanisms adopted in the two models. In the FCM, the molten polymer uniformly and completely coats the selected particles, representing a situation in which the polymer is suddenly introduced into a local region of the fluidized bed and makes the particle surfaces in that region sticky. In contrast, in the PCM, the potassium silicate coating forms on randomly distributed surface sectors of the particles. Consequently, the FCM promotes size growth in limited particle size classes, whereas the PCM yields a broader and more gradual particle size transition across the bed. Figure 12b further supports this interpretation by presenting the primary particle number distributions (PPDs) at the defluidization moment, which can be directly accessed through the MC framework. It can be seen that the PPD in the FCM case is more discrete, meaning that most particles are concentrated in only a few classes. As an example, the first plateau in the FCM PSD arises from two major particle classes, unagglomerated primaries, and clusters of seven primaries, representing the dominant peaks in the corresponding PPD. Intermediate agglomerates such as dimers and trimers are relatively scarce, resulting in a large gap between these two main classes and the pronounced plateau observed in the PSD. Such behavior originates from the CoP and the full-coating assumption in the FCM, which allow each particle to be fully covered by neighboring particles, leading to a discontinuous population structure that explains the step-wise character of the corresponding PSD. In contrast, the PCM exhibits a more gradual transition across agglomerate size classes, with dimers, trimers, and medium-sized clusters occurring more frequently, resulting in a smoother overall PSD and indicating a more progressive agglomeration pathway. Overall, the comparison highlights how the stochastic detail resolved by the MC model reveals the intrinsic differences in coating-induced growth between the two mechanisms.

Figure 13 presents the evolution of the normalized mean particle diameter (${\overline{d}}_{\mathrm{p}}/d_{\mathrm{p},0}$) with the dimensionless time $\tau =t/t_{\mathrm{def}}$. Both cases exhibit an initial linear growth regime, indicating a similar early-stage agglomeration behavior. In the polymer pyrolysis case (FCM), the mean diameter remains nearly linear throughout the process, as defluidization occurs relatively early and the process terminates before the onset of the accelerated growth phase. In contrast, in the biomass combustion case (PCM), the process continues for a longer duration, clearly showing a transition from the initial linear regime to a nonlinear, exponential-like growth behavior before defluidization. Such accelerated growth can be attributed to the increasing fraction of coated and already agglomerated particles in the bed, which raises the probability of forming larger clusters and thereby promotes faster particle growth at later stages.

In general, both FCM and PCM effectively illustrate how two distinct coating formation mechanisms result in significantly different growth behaviors and agglomeration rates, even though both models are developed within a unified MC framework.

5. Conclusions

In this study, a generalized MC modeling framework was developed to predict bed defluidization and the temporal evolution of particle properties during thermochemical conversion processes in BFBs. Building upon a previously proposed LP-AP MC model for polymer pyrolysis, two surface-coating-based model variants were introduced. The FCM describes the complete coverage of bed particles by molten polymer within localized regions, whereas the PCM represents the stochastic mechanism by which potassium-rich ashes, fully dispersed within the BFB, randomly react with selected surface sectors of bed materials during biomass combustion. A key methodological contribution of the present work is the introduction of the coating percentage as a physically interpretable lumped parameter within the MC framework. Rather than explicitly resolving the complex and largely inaccessible micro-scale coating processes in BFBs, the coating percentage effectively integrates the combined influence of feedstock properties, surface interaction mechanisms, and operating conditions into a single model parameter. This concept enables a substantial simplification of the modeling framework while preserving the essential physics governing coating-induced agglomeration and defluidization.

Both models successfully reproduced the characteristic agglomeration trends induced by surface coating under varying operating conditions, showing good agreement with reference experimental datasets from the literature. This agreement further demonstrates the validity of the proposed linear scaling of the coating percentage, confirming that the introduced correlation between coating formation and key process parameters can quantitatively describe the influence of operating conditions on bed stability. More specifically, the FCM reproduces the experimentally observed decrease in defluidization time with increasing polymer feed rate and decreasing bed mass, while indicating that variations in superficial gas velocity do not significantly alter the onset of bed collapse within the investigated parameter range. On the other hand, the PCM simulations of biomass combustion successfully capture the reduction in defluidization time with increasing bed temperature. Together, these results demonstrate that the proposed coating-based MC framework is capable of capturing not only overall agglomeration behavior, but also the characteristic parameter dependencies governing defluidization across both polymer pyrolysis and biomass combustion systems.

Beyond predicting defluidization time, the developed models enable tracking the evolution of coating thickness distribution, PSD, and PPD, thereby providing valuable mesoscale insights that are difficult to obtain experimentally. Within the unified MC framework, the comparison between PE pyrolysis and wheat straw combustion demonstrates that distinct coating mechanisms, molten polymer coating versus potassium silicate deposition, govern markedly different particle growth rates and PSD characteristics. More specifically, the FCM results are characterized by a comparatively higher fraction of unagglomerated primary particles and a more discrete agglomerate population, leading to step-like features in the resulting PSDs. This behavior reflects the localized and fully coating nature of molten polymer deposition, which promotes the formation of dominant agglomerate classes while suppressing intermediate sizes. In contrast, the PCM yields smoother and more continuous PSDs, as partial and spatially distributed surface activation favors a more gradual and progressive agglomeration pathway involving a broader range of agglomerate sizes.

From a broader perspective, the proposed framework demonstrates that coating-induced agglomeration in BFBs can be described within a unified stochastic modeling concept, despite fundamentally different coating formation mechanisms. By representing different coating formation mechanisms through the coating percentage and embedding them into a common MC framework, the model achieves both conceptual generality and computational efficiency, which are essential for engineering-scale applications. Nevertheless, it should be emphasized that the FCM and PCM are not restricted to specific feedstock systems; that is, the FCM is not limited to polymer pyrolysis, and the PCM is not confined to biomass combustion. Rather, they represent two fundamental coating mechanisms that may coexist in practical fluidized bed processes. For example, during the pyrolysis of polyethylene terephthalate, viscous tar residues may partially cover the bed material surfaces, a phenomenon that can be effectively captured by the PCM concept.

In future work, incorporating the size information of ash particles may enable the model to describe stochastic surface coverage in a manner similar to spray agglomeration, thereby potentially eliminating the need for empirical calibration of the coating percentage in biomass combustion simulations. Moreover, owing to its simplicity and scalability, the proposed MC framework can be further extended to other multi-feedstock thermochemical conversion processes, such as co-pyrolysis and co-gasification.