Eularian–Eularian Model for Agglomeration Behavior of Combusted Iron Particles

Abstract

Direct reduction of iron (DRI) technology in fluidized beds has been identified as a promising approach due to its environmental benefits over other methods. Nevertheless, the process of iron particle sintering in the DRI approach poses a significant obstacle to its advancement. The present work investigated the phenomenon of agglomeration in fine iron particles across various temperatures and with multiple sintering force models of different intensities of solid bridge force. The study utilized a simple but comprehensive and cost-effective CFD model developed using the Eularian–Eularian two-fluid model. The model was explicitly incorporated with user-defined subroutines for the solid phase, while the gas phase was modeled with AVL Fire advance simulation software. The solid bridge force between solid particles was modeled as the inter-particle cohesive force. The model was validated with the experimental results and results from another CFD-DEM model for the same experiment. High temperatures with increased sintering forces were observed to have the most impact on the iron particle agglomeration, while the gas’s superficial velocity had a minimal effect on it. The predictions of this model closely align with the CFD-DEM model results, providing sufficient reliability to implement this model on a large scale.

1. Introduction

The current state of the global energy supply requires cleaner, renewable, and CO₂-neutral energy sources to replace fossil fuels and address the threats of climate change globally. In recent times, the DRI method has been identified as one of the core technologies for the decarbonizing iron process in favor of the road map of climate change mitigation targets [1]. Further, industry metal fuels were identified as a very encouraging and environmentally beneficial energy source due to their higher energy density and zero CO₂ emissions when in the DRI process [1,2]. Iron powders, particularly those in micron size, are initially prioritized because of their high volumetric energy density and widespread availability [2,3,4,5]. Iron powders undergo combustion to produce thermal energy. The essential step in achieving a carbon dioxide-free iron fuel cycle is the DRI, utilizing sustainable energy sources such as hydrogen (H₂) after the combustion process [6,7]. It is beneficial to use fluidized bed combustion plants for the DRI process due to their flexibility in using fine iron particles and their cost-effectiveness. It is estimated that a fluidized bed reduces the material cost by half compared with the MIDREX method using a shaft furnace of DRI technology [8].

Nevertheless, a significant issue arises in agglomeration caused by particle sintering; this often occurs when iron oxides are chemically reduced at high temperatures. Agglomeration/sintering has been detected in reactors that promote efficient mass/heat transmission and adequate solid mixing, such as fluidized beds [9]. As a result, when agglomeration is not avoided, fluidization is significantly reduced or even stopped completely, negatively impacting the operation’s efficiency. A detailed understanding of the agglomeration, as well as defluidization characteristics of iron/iron oxide fines at high temperatures, is necessary to create an operational design of fluidized bed DRI.

Agglomeration occurs in the fluidized bed combustion due to several types of cohesive forces [10,11,12,13]. Those inter-particle cohesive forces fall under several categories according to their natures. They include electrostatic forces, van der Walls forces, solid bridge, or liquid bridge forces. Any of these forces can become dominant according to the specific case. Out of these force categories, it was identified that the liquid bridge force is most influential at high temperatures of the DRI method [14].

Liquid bridges are more versatile than van der Waals forces in practical applications. They can be adjusted. Liquid bridges are more complex as they involve both dynamic and static forces, and they use up energy. The static liquid bridge force arises from the combination of surface tension and pressure deficit within the liquid bridge. This emphasizes how these factors work together to influence liquid bridges in industrial processes. When a particle’s surface is coated with a mobile liquid film, liquid bridges will develop at points of contact with other particles and surfaces. Moreover, if the partial pressure of a suitable vapor in the surrounding gas is high enough, condensation will take place at these contact points, leading to the formation of additional liquid bridges [14].

The geometrical/physical parameters of the iron/iron oxide particles and operating conditions play a major role in particle agglomeration behavior [15,16]. The physical parameters include the composition of the particle, size, shape and microstructure, while operational conditions encompass operating temperature, reduction of gas composition, and flow rate of the gas. Hence, it has become very difficult to model the solid bridge force accurately in agglomeration. Although experimental studies can be conducted to understand the agglomeration behavior, the operational costs of these experiments are very high. Therefore, agglomeration formation in the DRI method has become a challenge to be addressed because it mitigates the efficiencies of the process [17]. Therefore, it is crucial to understand the process of inter-particle cohesion by solid bridge force in the DRI method, which causes defluidization of fluidized bed combustion plants.

Thus, different approaches to numerical simulations, such as the two-fluid method and the discrete element method (DEM), provide tools to investigate the fluidization phenomena that take place in fluidized bed combustion. The numerical model simulation strategy developed by Tazleem et al. uses two fluid approaches based on the Eularian–Eulatian technique to predict agglomeration. The established modeling technique is verified using experimental data from the literature for a simplified two-dimensional geometry of a pilot-scale fluidized bed combustor [18]. Another numerical model based on the DEM approach was developed by Kuwagi et al. [12] to investigate metallic solid bridge forces in agglomeration. This model encompasses a surface diffusion mechanism that considers the surface roughness. Further, particle sintering in agglomeration is analyzed by Mansourpour et al. [19,20] with a CFD-DEM simulation. Their study incorporates the effect on the agglomerate sizes by temperature and gas velocity.

There are several studies that have utilized CFD-DEM simulations to investigate the sintering behavior of iron/iron oxide fines in fluidized beds. However, the high computing cost makes it challenging to simulate large-scale experiments using the CFD-DEM approach. The computer simulation cost of the DEM is significantly affected by the enormous number of particles in the micron-sized cohesive iron/iron oxide powder. To overcome this challenge and to expand the modeling capability to a large/industrial scale, the two-fluid modeling approaches can be applied. However, there is very little research work conducted using the two-fluid approach to simulate agglomeration in fluidized beds for the production of iron/iron oxide powder. Even the available literature does not provide proper agglomeration behavior with significant parameters, like solid pressure. They do not provide solid fraction variations with defluidization in the fluidized bed, which is crucial for understanding the agglomeration patterns in the fluidized bed. In this model, solid fraction variation aligned with the solid pressure variation is analyzed in detail and validated with the CFD-DEM study. Moreover, this study provides a cost-effective multiphase CFD model that can be scaled up for industrial-level reactors to predict agglomeration.

Therefore, in this study, a simplified yet comprehensive CFD 3D numerical model using the Eularian–Eularian method was developed using AVL Fire advanced simulation software (2024 R1) and user-defined FORTRAN subroutines. The gas phase of the model was modeled using the inbuilt AVL Fire algorithms, while the solid phase was developed explicitly using user-defined subroutines and coupled with the software AVL Fire.

In this work, defluidization behavior of iron particles were analyzed at various temperatures in a controlled adiabatic and reduced atmosphere. It is crucial to understand the process of sintering iron particles in DRI method. When in the DRI process, sintering can also occur at lower temperatures due to chemical reactions of converting iron oxide to metallic iron [6]. This model includes a variety of temperature settings, from low to high, to analyze the micron-sized iron particle sintering in the fluidized bed.

Further, the paper provides a detailed explanation modeling procedure and the application of solid bridge force into the model in Section 2. Section 3 of this paper includes a detailed description of the geometry of the reactor and other parameters. Section 4 covers the numerical results achieved with this model and validation of the results with CFD-DEM model results produced for the same experiment. Section 5 outlines the conclusions drawn from the previous section of results and discussion and the intended future work of this model.

2. Model Description

Within the framework of this model, the solid phase is explicitly simulated by means of user-defined subroutines, and it is coupled with the AVL fire gas phase simultaneously. Primarily, the model is developed to predict the solid agglomeration based on the sintering force estimations for four different temperatures and three different sintering force factors. Further, the modeling of solid phases is comprised of several other sub-models that have been developed by a number of authors and validated using experimental data [21,22].

2.1. Modeling of Bed Hydrodynamics

The equations of continuity and momentum are solved for both gas and solid phases with the Eulerian–Eulerian modeling approach. Equations (1) and (2) depict the continuity equations for gas and solid, illustrating mass flow from and into the controlled volume, as well as the net mass transfer in and out of the control volume. Equations (3) and (4) depict momentum transfer for gas and solid phases, encompassing all factors associated with momentum transfer into and out of a regulated volume.

$${\displaystyle \frac{\partial \left({\epsilon }_g{\rho }_g\right)}{\partial t}}+\nabla \cdot \left({\epsilon }_g{\rho }_g{u}_g\right)=S_{gs}$$

(1)

$${\displaystyle \frac{\partial \left({\epsilon }_s{\rho }_s\right)}{\partial t}}+\nabla \cdot \left({\epsilon }_s{\rho }_s{u}_s\right)=S_{sg}$$

(2)

$${\displaystyle \frac{\partial \left({\epsilon }_g{\rho }_g{u}_g\right)}{\partial t}}+\nabla \cdot \left({\epsilon }_g{\rho }_g{u}_g{u}_g\right)=-{\epsilon }_g\nabla P+\nabla \cdot \overline{\overline{{\tau }_g}}+S_{mom}\left(u_g-u_s\right)+{\epsilon }_g{\rho }_{g}g+S_{gs}$$

(3)

$${\displaystyle \frac{\partial \left({\epsilon }_s{\rho }_s{u}_s\right)}{\partial t}}+\nabla \cdot \left({\epsilon }_s{\rho }_s{u}_s{u}_s\right)=-{\epsilon }_s\nabla P+\nabla \cdot \overline{\overline{{\tau }_s}}-S_{mom}\left(u_g-u_s\right)+{\epsilon }_s{\rho }_{s}g+S_{sg}$$

(4)

Interphase Momentum Transfer

Drag force calculation is conducted using two approaches in this study. The calculation is based on Equations (5) and (8), utilizing the solid volume percentage in each cell of the domain. The flow resistance is calculated using the empirical pressure drop correlation proposed by Ergun [21] when the solid volume fraction exceeds 0.2 (ε > 0.2, dense regions). The Gidaspow model [21] is utilized to determine the pressure drop for regions with a solid volume fraction below 0.2 (ε < 0.2, dilute regions). The coefficients of permeability and inertial are denoted by η and γ, computed in Equations (6) and (7), respectively. They are utilized to determine the $S_{mom}$ for the dense phase in Equation (5). The gas phase flow resistance factor is added to the solid phase momentum equation with a sign that is opposite. Equations (9) and (10) determine the drag coefficient of the gas phase and the Reynolds number of the gas flow.

$$S_{mom}=-\left({\displaystyle \frac{{\mu }_g}{\eta }}\left|v_{\mathrm{\infty }}\right|+\gamma {\displaystyle \frac{1}{2}}{\rho }_g{v}^2\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left.{(\mathrm{f}\mathrm{o}\mathrm{r} \epsilon }_s>0.2\right)$$

(5)

$$\eta ={\displaystyle \frac{{\psi }^2{d}_{eq}^2}{150}}{\displaystyle \frac{{\left(1-{\epsilon }_s\right)}^3}{{\epsilon }_s^2}}$$

(6)

$$\gamma ={\displaystyle \frac{3.5}{\psi d_{eq}}}{\displaystyle \frac{{\epsilon }_s}{{\left(1-{\epsilon }_s\right)}^3}}$$

(7)

$$S_{mom}={\displaystyle \frac{3}{4}}{C}_d{\displaystyle \frac{{\epsilon }_g{\rho }_g\left|u_g-u_s\right|{\epsilon }_s}{d_s}}{\epsilon }_g^{-2.65}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{(\mathrm{f}\mathrm{o}\mathrm{r}\epsilon }_s\le 0.2)$$

(8)

$$C_d=\left\{\begin{array}{rr}{\displaystyle \frac{24\left(1+0.15R{e}^{0.687}\right)}{Re}},& Re\le 1000\\ 0.44,& Re\ge 1000\end{array}\right.$$

(9)

$$Re={\displaystyle \frac{{\rho }_g\left|{\mathit{u}}_g-{\mathit{u}}_s\right|{\epsilon }_g{d}_s}{{\mu }_g}}$$

(10)

2.2. Cohesive Force Model

The particle agglomeration model requires a model of the chosen interparticle cohesive force and adding it to the momentum equation in addition to the other main forces. The contact force can be an electrostatic force, a liquid bridge force, etc. In this model, micron-sized iron powder is combusted for different temperatures. Therefore, solid bridge force is recognized as the major contributor to the particle agglomeration in the DRI method in a fluidized bed. When two or more particles are bonded together at a temperature lower than that particle’s melting point, the solid bridge force emerges among those agglomerated particles. The DRI process is carried out at elevated temperatures of more than 500 °C. These temperatures cause the outer layer of the iron oxide particles to become adhesive and malleable. Hence, the environment favorable for the particle sintering is created in the DRI method when in higher temperatures. According to Knight et al., surface diffusion is driven by the atoms of the particles moving to the junction of the solid bridge [23]. Takafumi et al. explain the surface tension mechanism in detail based on the balance between the buoyancy force of the bubbles and the breaking force of the solid bridge [24]. Equation (11) represents the solid bridge force between two particles.

$$F_{\mathrm{s}\mathrm{b}}=\pi \sigma x^2,$$

(11)

where tensile strength and the neck radius are represented by σ and $x$, respectively. The neck radius is modeled by using Kuczynski’s surface diffusion model, which is explained in Equation (12) [25].

$$x=({{\displaystyle \frac{56\gamma {\delta }^4}{KT}}{D}_s{a}^{3}t)}^{1/7},$$

(12)

where the variables γ, δ, K, T, a, and t stand for surface tension, lattice constant, Boltzmann constant, temperature, curvature radius, and contact time, respectively. The surface diffusion coefficient ($D_s$) in Equation (13) is determined by the frequency factor ($D_{o,s}$) and activation energy ($E_s$).

$$D_s=D_{o,s}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{{-E}_s}{RT}}\right)$$

(13)

Moreover, a general equation for solid bridge force modeling can be derived using Equation (14) [12,26].

$$F_{\mathrm{s}\mathrm{b}}=\mathrm{C}\pi \sigma {({\displaystyle \frac{56\gamma {\delta }^4}{KT}} D_s{r}_p^{3}t)}^{2/7},$$

(14)

where $r_p$ is the minimum radius value of two particles in contact.

In this model, solid bridge force is modeled in three different impact models by varying the C value in Equation (14). The model for value C = 0.417 is called the three-micro contact-point model. Similarly, C = 1 represents the smooth surface model, and C = 1.251 refers to the nine-microcontact-point model. This classification for the model for different intensities of sintering force is similar to the CFD-DEM study conducted by Liu et al. [26].

Another key parameter in Equation (14) is the contact time, t. Initially, the contact time is zero. Then, if the consecutive cells have similar velocities within the (+/−5%) range, solid mass in those cells is said to be in contact, and the contact time is added to the stored contact time of the previous time step. If the velocities fall out of the boundary, the contact time remains the same.

3. Experimental Setup and Grid

3.1. Setup

Figure 1 represents the simulation setup used by Liu et al. for their CDF-DEM modeling [26]. In this model, a grid with the same dimensions is also applied as a cylindrical reactor with a 1.6 cm diameter and 4.8 cm height. The reactor has an inner diameter of 1.6 cm, while the height of the cylinder is maintained at 4.8 cm and 9.6 cm for the simulation. In the experiment, three distinct diameters of particles were used. However, Liu et al.’s coarse-grained method results for particle size, which is 58 μm, were used in this study. The total weight of the 0.01 kg iron particles was used to fill the bed [17].

Grid independence was analyzed for three different grids depending on the number of cells. The number of cells in grid 1, grid 2, and grid 3 are 12,200, 22,000, and 35,000, respectively. The average values of flow velocities taken from cross-sectional cuts at different reactor heights from the bottom of the reactor are compared and presented in Figure 2. The variation of these parameters in the three different meshes was found to be reasonably acceptable and within the expected range. Therefore, grid 2 was selected as the mesh in the simulations of this study.

3.2. Settings and Parameters

The simulations were carried out in a reduced environment by feeding inert gas (N₂) from the bottom of the gasifier uniformly into the bed. Cylinder walls were given zero gradient boundary conditions. Environmental conditions were set to ambient pressure settings. All the simulation settings associated with iron combustion were taken from the CFD-DEM model, which is listed in

Table 1. Physical properties and settings used in the simulations [26].

Particle Properties	Value	Source
Coefficient of restitution e	0.9
Coefficient of friction μ	0.5
Density ρp	5240 kg/m3
Lattice constant δ	1.38 × 10−9 (m)
Surface tension γ	1 N/m	[28]
$\mathrm{Frequency} \mathrm{factor} D_{o,s}$	2.8 × 10−13 (≤900 °C) 1.6 × 105 (>900 °C) m2/s	[29]
Activation energy	1.74 × 105 (≤900 °C) 5.79 × 105 (>900 °C) J/mol	[29]

[26]. Tensile strength was taken from an existing analysis performed by Hidaka et al. to determine tensile strength variation with differential temperatures [27]. Gas phase parameters were given according to the AVL Fire in-built algorithms that are specific to temperatures. The simulations were performed for a maximum of 20 s with a time step of 0.001 s. Simulation runs were conducted at four different temperatures, from 500 °C to 1000 °C. Benchmark simulations for zero agglomeration (C = 0) were conducted for all four temperatures for the purpose of comparison. Then, the three-micro contact-point model l (C = 0.417), smooth-surface model (C = 1), and nine-micro contact-point model (C = 1.251) were performed as agglomeration force models.

4. Results and Discussion

This section examines the findings of simulated cases that were carried out with varying degrees of sintering force and temperatures during the process. The primary focus of this investigation was on the downward trend in the pressure drop of the solid bed surface with the presence of a sintering force. Further, visual representations of the solid fraction and solid velocity are also presented. The features of (de)fluidization of combusted iron powder at increased gas velocities are presented and evaluated in this section.

4.1. Minimum Fluidized Velocity and Bed Conditions

To validate the CFD model for solid hydrodynamics of combustion iron powder, the minimum fluidization velocity (u_mf) must be estimated. Minimum fluidization velocity is the minimum velocity required to make the bed fluidized by keeping the upward drag force of the gas equal to the total solid bed weight. This is one of the most crucial elements in fluidized bed combustion. Numerical simulations were run to estimate the u_mf with an N₂ environment by varying the flow rate at a constant temperature of 20 °C.

Minimum fluidized velocity (u_mf) for this simulation setup was found to be 0.01 m/s, while 0.15 m/s was found to be excessive based on the range of simulations carried out from 0.001 m/s to 0.15 m/s. Solids climbed to the top of the gasifier at a velocity of 0.15 m/s, suggesting that the velocity was too high. On the other hand, the velocity of 0.001 m/s resulted in non-fluidized behavior because the velocity was not enough to fluidize the bed. This result for minimum fluidization velocity estimation aligns with DEM simulations conducted by Liu et al., indicating that 0.01 m/s is the minimal fluidization velocity for this arrangement, though there are some discrepancies with theoretical calculations derived using Beestra’s correlation [26]. The variation of the particle size distribution can be attributed to the slight variation in theoretical and numerical results [17,30].

Moreover, in the CFD-DEM study, 0.15 m/s is used to analyze the effect of fluidization velocity on particle agglomeration. However, 0.15 m/s was found to be excessive for this simulation setup. This distinction can be ascribed to the fact that these simulations were carried out using the Eularian–Eularian two-fluid approach, in which the solid is considered as a fluid, but the CFD-DEM method is based on the Lagrangian approach, which considers solid as discrete particles. However, in general, the simulation outcomes reasonably agree with the outcomes of the other referenced literature mentioned above, providing evidence for the applications of accurate hydrodynamic theories.

4.2. Solid Bed Hydrodynamics

In a proper bubbling fluidized bed, bubbles originate at the bottom of the solid bed and rise to the top, increasing in bubble size until their buoyancy force exceeds the solid drag force. The buoyancy force rises as the bubble diameter increases, causing the bubbles to climb to the surface of the solid substrate and burst. This situation recurs in the fluidized bed reactor to enhance fuel mixing capabilities for improved combustion. As the bubble ascends, solid particles move, resulting in the formation of zones with lower solid fraction. However, when agglomeration occurs, it affects the bubble growth in the fluidized bed, driving the bed toward defludization. This numerical model has the capability to analyze the strength of the solid bridge force for various temperatures and four different agglomeration force models.

Figure 3 illustrates the normalized bed pressure drop profiles over time for each sintering force model temperature varying from 500 °C to 1000 °C. The standard deviation (STD) of the pressure drop curves was calculated from 0.5 s to 15 s. At low temperatures (e.g., 500 °C), the cohesive force impact was observed to be negligible in any sintering force model. This can be identified in the snapshots of Figure 4, as both cohesive and non-cohesive cases have healthy bubbling behavior at 500 °C. Comparatively, the standard deviation was increased in the three-micro contact-point model (C = 0.417) at 650 °C compared with the 500 °C model cases despite there being no agglomeration visible in Figure 4b. This is because even with reduced sintering force, the bed may have difficulty maintaining bubbling at temperatures exceeding 650 °C. Additionally, when C = 0.417 (three-micro contact-point model) at 1000 °C, a consistent reduction in pressure drop is observed after 8 s. It is further demonstrated by the snapshots in Figure 4d. Images captured at 1000 °C with a C value of 0.417 display blue areas indicating minimal solid presence and channeling caused by particle sintering. The void fraction snapshots from the CFD-DEM model performed by Liu et al. provide similar evidence to this discussion. However, there are some distinctions in the three-micro contact-point model (C = 0.417) at 1000 °C in the CFD-DEM study, as it shows good bubbling hydrodynamics in the CFD-DEM study even with a higher STD value. In this study, agglomeration occurred after 8 s, which shows significant pressure reduction. To illustrate that behavior, 15 s of run time was maintained in this study [26].

In sintering force models at higher temperatures (i.e., 800 °C and 1000 °C), the fluctuations of the pressure drop increased gradually, showing higher standard deviation (STD) values. This notable variation proposes an increase in bubble sizes, resulting in more extreme bed dynamics. The sintering force models of a smooth surface (C = 1) and the nine-microcontact-point model (C = 1.251) had significant differences in terms of the magnitude (approximately 2.4 to 3 times) compared with the three-micro contact-point model (C = 0.417). Hence, the rate of particle clumping was high when in higher sintering force models (C ≥ 1), leading to inhomogeneity in the bed. This scenario can be well captured in a three-micro contact-point model sintering model at 1000 °C. Then, when the sintering force was further increased by applying the smooth surface model (C = 1) and the nine-microcontact-point model (C = 1.251) at the maximum temperature used in this study (i.e., 1000 °C), the pressure drop and the standard deviation (STD) decreases rapidly, clearly indicating particle agglomeration in Figure 3d and Figure 4d. Despite the similar trend observed in the CFD-DEM study, the sintering bonds remain permeable in the CFD-DEM cases, whereas in this model, some solid movements were still visible. This model demonstrates the occurrence of cohesive bonding, where both the formation and dissolution of bonds take place. However, it is important to note that healthy bubbling is entirely halted at 1000 °C in both the smooth surface and nine-micro contact-point model. These arguments are supported by Figure 5, as solid velocities decrease with the increase of sintering force impact and with the rise of temperature.

In Figure 3c, a decrease in bed height and flow inhomogeneity can be observed in sintering force models, especially when increasing the impact of sintering force at 800 °C. This is attributed to the impact of agglomeration, which is noticeable even without complete defluidization.

A reduction in the bed expansion can be seen Figure 4d from three-micro contact point model after 8 s. Further, a significant reduction in bed expansion is observed in Figure 4d for the smooth and nine-micro contact-point models. This was also clearly observed in the CFD-DEM study.

The creation of significantly larger agglomerates and the strong adhesion of particles to the bed wall within a few seconds is observed, as shown in Figure 4c,d, which depicts smooth and nine-micro contact-point model situations, respectively. In Figure 4c, in the smooth and nine-micro contact-point models, making and breaking agglomerates can be seen closer to the wall after 10 s. In Figure 4d, for sintering force cases, the formation of agglomerates can be observed. This behavior is indicated in the smooth and nine-micro contact-point models due to the high intensities of the sintering force. Agglomeration can be observed clearly in those images of smooth and nine-micro contact-point model situations at 1000 °C, which suggests that the solid bridge force that was forecasted with this model at this temperature is very high. This observation, similar to the CFD-DEM study results, suggests that this two-fluid model reasonably accurately predicts the particle agglomeration leading to defluidization in fluidized bed combustion.

In the CFD-DEM work, it is demonstrated that there is no movement in any of the particles in the smooth and nine-micro contact-point model situations when the temperature is 1000 °C, which indicates that the defluidization pressure drop remains constant. Nevertheless, this study demonstrates that even in smooth and nine-micro contact-point model cases at a temperature of 1000 °C, there is little motion in solid particles, which indicates that there are slight differences in pressure drops, and certain solid particles have low-velocity values. There is evidence of this in the pressure drop graphs shown in Figure 3d and the solid velocity snapshots shown in Figure 5b. A potential reason for this variance is that the different methods of calculating contact time that were discussed in the previous section can cause it.

The results of this model reasonably reflect the observation of the experiment and CFD-DEM findings. It is noticed that, in general, low-temperature runs (500 °C and 650 °C) could operate continuously for more than two hours without any defluidization appearance. Nonetheless, the experiments conducted at 950 °C were soon found to be non-fluidized behavior within several minutes [17].

Moreover, Figure 5b depicts that the solid toward the bottom of the bed has lower velocities than the top in agglomerated models, whereas when C = 0, the bottom of the bed has higher velocities than the top. According to the experimental findings, non-fluidized zones initially started from the bottom of the bed. Thus, it is fair to say that the smooth-surface model and the nine-micro contact-point model provide better predictions of solid sintering force, which is closely related to the experimental observations. The neck size of the solid bridge grows with the particle temperature and the contact time according to Equation (14). Particle cohesion begins from the lower part of the bed, particularly at the beds’ corners, because of the high density, as depicted in Figure 4c,d. Then, with the contact period of the particles rising, more particles tend to accumulate into the existing agglomerates, increasing the size of the agglomerates [17].

4.3. Effects of the Gas Superficial Velocity

Defluidization occurs due to the interaction between the sintering force (solid bridge force) and the breakage force in the reactor. Combining forces of drag force and the particle collision force establishes the breakage force needed to break down the agglomerated particles. Increasing the gas’s superficial velocities can provide the breakage force to separate particles, reducing the contact time. Thus, high fluidization velocities have the ability to break the particles that stick together. Another set of simulations was performed to study the effect of gas superficial velocity on particle agglomeration by increasing the velocity to 0.12 m/s. The simulations were carried out for a smooth surface (C = 1) and a nine-micro contact-point model (C = 1.251) at 1000 °C.

When the gas superficial velocity is increased to 0.12 m/s, the bed seems more dynamic for both the smooth and nine-micro contact-point cases model compared with the previously stated scenarios. After 13 s, the smooth surface model bed begins to lose its fluid-like properties, whereas the nine-micro contact-point model starts to do so after 9 s.

After 9 s of the nine-micro contact point model, the sintering region has expanded from the bottom of the bed over the whole bed, indicating almost defluidization. The results indicate that higher gas surface velocities can lead to an increased gas flow through the bed, resulting in the formation of big bubbles and higher collision frequencies between particles. Consequently, the decreased contact time between colliding particles weakens the solid bridge, potentially delaying defluidization. However, defluidization could still be observed in those simulations, as indicated in Figure 6, the pressure drop curve, and the snapshots of solid fraction in Figure 7. This phenomenon is further proved by the solid velocity snapshots depicted in Figure 8.

5. Conclusions

The agglomeration hydrodynamics of the iron powder particles in a fluidized bed were analyzed using a CFD numerical model. The model was developed with a simplified approach of a two-fluid method involving the implementation of the user-defined subroutines written by the authors. The conclusions of the study are as below:

(1)

An Eulerian–Eulerian multiphase model, where the solid phase is explicitly simulated, was implemented through user-defined subroutines and is successfully extended to cohesive dense gas-solid flows by including temperature-dependent solid bridge (sintering) force.

(2)

The iron fine powder agglomeration due to the sintering force developed in the model qualitatively agrees with the experiment and CFD-DEM studies. The solid temperature of the iron particles was found to be a significant parameter in estimating the sintering force effect. Higher gas superficial velocities can delay the defluidization in the fluidized bed combustion. However, the bed still leads toward the defluidization regardless of the gas’s superficial velocity.

(3)

It was observed that there is a significant impact on particle cohesion of the smooth surface/nine-micro contact-point model (C = 1/1.251) model (2.4/3 scaling factor) results in higher temperatures, which is closely aligned with the realistic phenomena. This behavior is similar to that of the CFD-DEM study of the same experiment. Therefore, the smooth surface/nine-micro contact-point model sintering force model is more suitable for representing the true phenomena observation in the study of combusted iron powder.

(4)

The normalized pressure drop curve follows a similar pattern for each temperature and model type similar to the experiment. With the temperature and sintering force factor increase, the higher pressure drop variation could be observed attributing to higher standard deviations. Nevertheless, the standard deviation of the pressure drop significantly drops in clear agglomeration cases at higher temperatures.

(5)

The study proposed a cost-effective, simplified CFD model for predicting particle agglomeration in fine iron particles. The results of the model reasonably agree with the other referenced CFD-DEM work along with the experiment observations. Future work can lead to the implementation of the model on a larger scale to minimize cost.