Hoffmann et al. (1993) carried out experimental investigations in binary mixtures to study mixing and segregation behavior. A new empirical correlation was developed for bubble wake angles for Geldart B particles. The rate of material interchange between the wake of a rising bubble and the surrounding bubbles is independent of minimum fluidization velocity. The authors modeled mixing/segregation behavior in binary systems.

Singh et al. (2019) carried out mixing and segregation studies for unary and binary beds using ECT measurements to understand the temporal particle velocity and bubble size distribution in binary beds. The authors investigated the effect of different parameters like the same and different size ratios, namely, 96 μm, 430 μm, 922 μm, and 3,500 μm, on the hydrodynamics of beds with binary mixtures and unary beds of particle sizes 96 μm and 922 μm. The gas superficial velocity was varied in the range of 0.006–0.684 m/s. The authors found extremely novel and interesting facts on the dynamics of beds with binary mixtures in transient situations and segregating beds. Some salient features revealed were the characterization of segregated regions in the bed large diameter ratios (96 μm and 922 μm), the effect of bubbling behavior on segregation, and the effects of gas velocity and mixture composition on transient segregation of binary beds. The authors also found the importance of a higher amount of smaller particles in the reduction of segregation in beds. According to the authors, the data provided by their study would be helpful in building robust Eulerian–Eulerian CFD models for predicting dynamics of segregation and mixing in unary and binary fluidized beds.

Kalo et al. (2019) studied the dynamics of unary and binary fluidized conical beds using RPT composed of particle sizes of 0.6 and 1 mm, respectively. The authors found interesting results in terms of gas–solid and particle–particle interactions in conical beds using time-averaged quantities (mean and rms velocities). One of the major findings was the ability of conical beds to provide better mixing even at lower superficial velocities compared to cylindrical beds. Furthermore, the authors observed that gas–solid interactions played a vital role at the bottom while particle–particle interactions played an important role at the top in the dynamics of binary conical beds with 50–50 wt.% composition.

Gupta and De (2021) performed experimental measurements for a dual fluidized bed under a fast fluidization regime. The authors found that poly-disperse binary mixtures have different pressure profiles than unary (uniformly sized) sand particles with narrow particle size distribution. Segregation is high in the bubbling fluidized bed riser and decreases with increases in superficial velocities. Furthermore, the authors highlighted the influence of pressure drop on poly-disperse mixtures and presented an analytical model to justify their results. The authors also found that the trends of pressure drop with an increase in superficial velocities were similar to those of bubbling fluidized beds.

Roy et al. (2021) carried out experimental investigations to study bed dynamics in a binary fluidized bed with different particle sizes (0.5 and 2 mm sized particles). The authors found that with an increase in the 2 mm fraction from 10 wt.% to 40 wt.%, the computed bubble-rise velocity decreased, which in turn caused a decrease in the velocity of particles. This is in contrast to the results for a bed of uniformly sized particles (also termed a unary bed), where the influence of superficial velocity on bubble diameter and rise velocity has an increasing trend (Penn et al., 2019). In unary beds, as the ratio of superficial velocity ( U g s ) to minimum superficial velocity ( U m f ) increased from 1 to 3, a corresponding increase in bubble-rise velocity was observed. For a particle diameter of 0.5 mm, the difference in particle velocity at the center increased, while when a coarser fraction was added to 0.5 mm particles, the bubble-rise velocities decreased. The authors also investigated the axial velocity profiles for both particles and gas. For lower superficial velocities, the axial velocities of the particles were fully developed while having an inversion near the walls. With the increase in the coarse fraction of particles, the authors observed an increase in particle velocities. The authors emphasize that the data for velocity distribution are not available for binary/polydisperse beds.

The following points have been deduced from the literature review: 1. few numerical investigations on velocity and volume fraction distributions of both phases in binary fluidized beds exist, while experimental data for binary systems for bubbling fluidized beds are available using RPT and ECT. 2. Researchers have found that more than a 50 wt.% presence of small particles in a binary mixture caused changes in the axial profiles of particle holdup and abrupt changes in the bed height at the final steady state. 3. Few correlations predict minimum fluidization velocity and pressure drop for binary mixtures and Geldart B particles.

This study envisages the following numerical simulations using six different binary mixtures: 1. Characterization of the bubbling fluidized bed in terms of time-averaged mean gas and particle velocities (of both small and large particle sizes) in the bottom, middle, and dilute zones. 2. Characterization of bed dynamics in terms of time-averaged solid holdup in the axial direction. 3. Investigation of the effect of operating parameters like superficial velocity and initial bed height on bed dynamics (minimum fluidization velocity and pressure drop). 4. Development of correlations for minimum fluidization velocity and pressure drop for binary systems.

Two-phase modeling has been performed using the Eulerian–Eulerian multiphase model coupled with an RNG k–ε model. The Syamlal et al. (1993) model was used for the modeling of granular viscosity, while the Syamlal and O’Brien (1987) model was used for modeling the drag of the system. The Schaeffer (1987) model (Schaeffer model) was used for simultaneous calculations of frictional pressure and viscosity. Granular bulk viscosity was modeled by the Lun et al. (1984) model; the Ahmadi and Ma (1990) model was used for simultaneous modeling of solids pressure and radial distribution. All the models and governing equations and various other parameters were kept the same in Ansys Fluent 18.1. A detailed description can be found in Ganguli and Bhatt (2023).

A 3D cylindrical geometry of height 1.4 m and diameter of 0.072 m was chosen for simulations. Figure 1A shows the 2D schematic of the geometry created in Ansys Fluent 18.1. Three different radial positions were used for collection of data, as shown in Figure 1A.

Air as a fluid phase with a combination of glass particles as a solid phase is used in the system. The density of air is 1.22 kgm⁻³, and the dynamic viscosity is 0.000017 kgm⁻¹s⁻¹. The glass particles have a varying diameter between 154 μm (fine particles) and 488 μm (large particles) (with a size ratio of 3.2) based on the percentage of large particles in the mixture. The density of glass particles is 2,485 kgm⁻³, and the dynamic viscosity is 0.00082 kgm⁻¹s⁻¹. The phase properties have also been used for simulation purposes by Jayarathna and Halvorsen (2011).

Figure 2 shows the radial gas velocity profile at Position 2 of Figure 1A for three different meshes. The mesh elements are as follows: Mesh 1 has 173,040 elements, Mesh 2 has 267,786 elements, and Mesh 3 has 497,568 elements. As the mesh number increases, the mesh refinement near the wall also increases. The error between Mesh 1 and Mesh 2 is 10%, while that between Mesh 2 and Mesh 3 is 2%. Therefore, Mesh 2 was used for further simulations. Figure 1A shows axial and radial views of Mesh 2. The mesh used is the same as the one used by Ganguli and Bhatt (2023).

The solution method for all the simulations is the same as our previous work (Ganguli and Bhatt, 2023). Only a brief description is provided. The convergence criterion was kept as 10⁻³ for the continuity equation and 10⁻⁴ for the other equations. A first-order implicit scheme was used for the transient formations. Pressure–velocity coupling was achieved using the phase-coupled SIMPLE scheme. For the calculations of volume fraction, momentum, turbulent dissipation rate, and turbulent kinetic energy, the first-order upwinding scheme was used. Table 1 gives the details of the simulations about the mixture types, particle size, initial bed heights, and superficial gas velocities.

The validated model used in our previous work has been used (Ganguli and Bhatt, 2023). Further validation of the model for pressure drop has been performed with additional data available for the mixtures of 40 wt.% large particles and 60 wt.% large particles, while for work to determine the mean velocity distribution of small and large particles has been performed with the experimental data of Roy et al. (2021). For pressure drop predictions, simulations were performed for a bed with an initial bed height of 0.235 m, and the superficial gas velocity was varied in a range of U g s = 0.08–0.225 m/s. Figure 3 shows the variations of pressure drop with variations in superficial gas velocity for 40 wt.% and 60 wt.% large particles mixtures. The average error between the model and the experimental value was 6%. Figures 4A–C shows the comparison of predicted results of mean velocity profiles with the results of published literature (Roy et al., 2021) for the case of a 10 wt.% binary mixture composition and superficial velocities in the range of 1.1–2.1 m/s. Deviations in the range of 10% have been observed for the compositions considered in the literature. This clarifies two aspects: the first being that validation with pressure drop measurements is not a sufficient criterion for model validation, and second, Eulerian–Eulerian models need to be tested with more experimental data. A further argument can also be made that the investigations of published literature have been made with a size ratio of 4 with larger sized particles in the size range (0.5–2 mm), while the present work has used the size range of (0.154–0.488 mm) and a size ratio of 3.2. The comparison for the other two compositions (30 wt.% and 40 wt.%) has been provided in the Supplementary Material. The deviations in these cases were less than 10%, suggesting that the Eulerian–Eulerian models with KTGF model predict well for binary mixtures.

The gas and particle dynamics for both small and large particles are quantified in the validated model by plotting each of the mean velocity profiles for all particle mixtures at three different superficial velocities for three different bed heights. For understanding, the dynamics of solid particle quantification have been further determined in the form of particle axial volume fraction profiles for the centerline. The axial locations at which the radial velocity profiles have been calculated are shown in Table 2.

The gas and particle velocity profiles at different axial locations for different superficial velocities and bed heights for the 20 wt.%, 40 wt.%, 60 wt.%, and 80 wt.% mixtures of large-sized particles are plotted in Figures 5–8, while those for the 0 wt.%, 80 wt.%, and 100 wt.% mixtures of large-sized particles are provided in the Supplementary Material because the main focus of the present work is understanding segregation and mixing in binary beds. Numbers have been used to denote the axial locations for the collection of axial velocity data in radial positions. The axial positions are chosen in the bottom, middle, and dilute zone based on the experimental data collection positions as per Jayarathna and Halvorsen (2011). Furthermore, the axial positions depend on the bed height considered. The axial positions and radial variations have been represented in dimensionless form for all figures. This is because the steady state bed height is different for different initial bed heights, and the axial positions are non-dimensionalized using the final steady state bed height. The mean gas and particle velocities have been time-averaged over 7 s before the results are presented. Similarly, the particle volume fractions for the vertical centerline of the bed have been presented after time-averaging. The size ratio has been considered constant as 3.2 for all simulations, and only compositions have been varied [binary mixtures having 0 wt.%, 20 wt.%, 40 wt.%, 60 wt.%, 80 wt.%, and 100 wt.% of large particles (488 μm), respectively].

The dimensionless distance range is taken as −1 < r/R < 1. During the profile analysis, the start position is the leftmost radial position (or left-hand wall), and the end position is the rightmost radial position (right-hand wall). The terminology for axial positions and superficial velocities describing velocity profiles for an initial bed height of 0.335 m (z/H = 0.24) and 0.635 m (z/H = 0.45) has been tabulated in Table 2.

In this section, solid phase volume fractions across the axial centerline of the final bed height have been presented for the three superficial velocities and two mixture compositions considered, namely, 20 wt.% and 40 wt.%. Both bed heights have been chosen for each of the compositions. The discussion has been restricted to only two compositions to demonstrate the segregation and mixing that can be analyzed using the solid volume fractions. The other compositions also have similar trends with slight differences due to the presence of higher amounts of large particles but have not been included.

Figure 9A, B shows solid fraction profiles for both large and small volume fractions for the 20 wt.% mixture case for both bed heights. Similar profiles have been reported by Lan et al. (2014). For all three superficial velocities and the lower bed height of 0.335 m (z/H = 0.24), segregation is observed in the bottom layer with a larger amount of large particles and a relatively lower volume fraction of smaller particles, and vice versa in the top layer. The middle layer showed some mixing between the large and small particles. Furthermore, the dispersion height was found to be higher for smaller particles for all three superficial velocities. For the lowest superficial velocity of 0.3 m/s, the dispersion height for large particles is 0.31 m (z/H = 0.22), while the dispersion height for small particles is 0.36 m. This also compliments the findings observed in the velocity distributions where the particle velocities are not observed for Position 3. However, volume fractions lower than 0.0005 are observed for the superficial velocity of 0.6 m/s, while corresponding volume fraction for 0.45 m/s is 0.007. This is due to the higher kinetic energy of the gas and the smaller particles, and only a very few large particles are carried over. Because very few particles reach the top or the dilute zone, the velocity distributions depict the larger particles in the profiles. Dispersion heights, however, are 0.5–0.7 m. Furthermore, the undulations or decrease in volume fractions indicate bubble-containing solids and can also be seen in volume fraction contours as depicted in our earlier work (Ganguli and Bhatt, 2023). These patterns confirm a bubbling regime and well mixed patterns. For an initial height of 0.635 m (z/H = 0.45), complete segregation is observed in all the layers with all small particles in the top layer and large particles in the bottom layer. Interestingly, here, two layers are formed, while three layers are formed in the previous case with a lower bed height. Some large particles, however, reach the top layer in all three cases, although the dispersion heights for all three heights are different.

Figure 9A, B shows solid fraction profiles for the 40 wt% mixture case for both bed heights. For the lower initial bed height (z/H = 0.24), segregation at the bottom layer is evident for all superficial velocities, and the highest segregation occurs at 0.3 m/s. However, from z/H = 0.15, some amount of mixing is observed until the dispersed bed height is reached. The volume fractions of both large and small particles are similar in the middle of the bed (starting at z/H = 0.12 and evident until z/H = 0.3 for all superficial velocities), with small particles becoming higher at the top of the bed (or the dilute zone), especially for superficial velocities of 0.3 and 0.45 m/s. The mixing is most evident for z/H = 0.13 to 0.4 for a superficial velocity of 0.6 m/s, where substantial volume fractions of large particles are observed compared to small particles, indicating good mixing. This may be attributed to three major factors: the influence of gas velocity and bubbles that increase the kinetic energy of large particles, higher GS interactions dominating the particle–particle interactions, and lower hindrance of small particles. For a bed height of 0.635 m (z/H = 0.45), the particle volume fraction profiles also depict segregation at the bottom (volume fractions of larger particles are higher by 10%–35%) and mixing at both the middle and top layers with small particles having higher volume fractions (15%–20% higher) compared to larger particles. The undulations at the middle and top portions depict the presence of bubbles. For U g s = 0.6 m/s, a dip in the volume fraction both of small and large particles is observed that denotes the formation of larger bubbles.

Figures 10A, B represent the final bed height for the fluidized bed after reaching a steady state for two initial bed heights, 0.335 m (z/H = 0.24) and 0.635 m (z/H = 0.45). The bed height considered does not indicate the bed height for two different particle sizes, but the highest bed height that can be observed at a steady state is taken, which mostly consists of small particles. In Figure 10A, it can be observed that for a mixture with 20 wt.% large particles, there is a linear increase in the bed expansion ratio for all superficial velocities. Furthermore, the bed expands 1.5 times the initial bed height for a superficial velocity of U g s = 0.3 m/s and goes up to 2.4 m for a superficial velocity of U g s = 0.6 m/s. This reiterates the significant role of bubble volume in occupying the expanded bed volume, clearly indicating a bubbling fluidized bed regime. For a 40 wt.% mixture, however, the bed expansion increases for superficial velocities up to U g s = 0.45 m/s (1.28 times for U g s = 0.3 m/s and 1.49 times for U g s = 0.45 m/s), and there is a decrease in bed expansion for higher superficial velocities. This indicates that for this binary mixture, a bubbly regime exists for initial superficial velocities, but a transition to a turbulent regime that breaks the bubbles, reduces the bubble volume, and promotes mixing takes place. This causes a lower increase in bed height than expected, as seen in the 20 wt.% mixture. For the 60 wt.% mixture, however, the increase in bed volume is lower (1.1 times the initial height) for a superficial velocity of U g s = 0.3 m/s, indicating a homogeneous regime. This is followed by 1.3 times initial height for U g s = 0.45 m/s, which indicates a transition to a bubbling regime, and then an increase of 1.73 times the initial bed height for U g s = 0.6 m/s, which indicates a bubbling regime with a considerable number of bubbles occupying the expanded bed volume. A similar trend of increase in expanded bed height is shown for an 80 wt.% mixture of large particles for the superficial velocities considered. For a 100 wt.% large particle mixture, a linear increase of height for the two superficial velocities is observed with a bubbling/slugging regime for a superficial velocity of U g s = 0.6 m/s.

Figure 10B shows the effect of superficial velocity on bed expansion for all the binary mixtures considered with an initial bed height of 0.635 m (z/H = 0.45). It can be observed that for the 0 wt.% large particles, the bed expands to approximately two times the initial bed height. However, there is no significant effect on the bed expansion for the superficial velocities considered. For 20 wt.% large particles in the binary mixture, the bed expansion is 1.4 times for the lowest superficial velocity and then tapers for a velocity of U g s = 0.6 m/s, indicating a transition from a bubbling to a turbulent regime. For 40 wt.% large particles in the binary mixture, there is a linear increase in the bed expansion ratio, indicating that for all superficial velocities, the bed follows a bubbling regime. For 60 wt.% and 80 wt.% large particles in the binary mixtures, a minor relative increase in bed expansion ratios is observed at lower superficial velocities, while a steep increase is observed for superficial velocities approximately U g s = 0.6 m/s. A linear increase in bed expansion ratio is also observed for 100 wt.% large particles.

Table 3 summarizes the correlations for minimum fluidization velocities and pressure drops available in the literature. Because a comprehensive analysis in terms of quantification of pressure drop and velocity distribution of both gas and particle velocities has been made in the present work, it was thought worthwhile to find a correlation between minimum fluidization velocity and pressure drop. To determine the pressure drop, simulations for different superficial velocities (0.02–0.75 m/s) starting from fixed bed to fluidized bed were carried out for all the binary mixtures and bed pressure drop versus superficial velocity were plotted. The minimum fluid velocity was determined from the intersection of the linear graph where the bed changes from fixed to fluidized bed. The corresponding gas volume fraction in the bed was found by taking the time-averaged volume fraction of the bed after 7 s for all the cases while the particles were considered to be spherical [as per experimental data of Jayarathna and Halvorsen (2011)] with sphericity equal to 1. Because the pressure drop was found to be a function of the initial bed height to column diameter ratio, it was also considered for the correlation of pressure drop, although only two bed heights have been considered.

A new correlation for minimum fluidization velocity is given by the following equation: R e m f = 0.143 A r s i 0.58 ε g ∅ s i 0.63 . (1)

The correlation for maximum pressure drop is given by E u m f = 261.45 * 10 3 A r s i − 0.33 ε g ∅ s i − 3.57 h D c o B − 0.99 . (2)

Figure 11A shows the parity plot of minimum fluidization velocity predicted by previous works with respect to the experimental data used from the literature. It can be observed that the predictions from the correlation developed fall within a ±15% deviation from experimental data. The correlations available in the literature predict with deviations of more than 80%. This is due to the fact that the superficial velocities considered by the authors are much lower than the ones considered in the present analysis. Figure 11B shows the parity plot of pressure drop with experimental data. The developed correlation predicts well within ±15% deviation. The predictions of Sau et al. (2008) also fall within a ±15% deviation from experimental data, while other correlations show very high deviations (>80%).