Solids Mixing Intensification in Tapered Fluidized Beds with an Inlet Jet: Experimental Validation and CFD Simulation

Abstract

In this article, the intensification of solids mixing in tapered fluidized beds equipped with an inlet jet and varying apex angles (2.86°, 5.71°, and 8.53°) has been investigated. In this regard, a particle segregation number (PSN) and multi-fluid modeling (MFM) approach were employed to analyze the mixing process. The study utilized solid mixtures composed of particles with a density of 2500 kg/m³ and diameters of 240 and 510 µm. Simulation results were validated against our experimental data obtained using a small tapered bed without an inlet jet and those obtained using a larger tapered bed with an inlet jet, as Huilin et al. (2003) reported. This validation demonstrates satisfactory agreement between the simulation results and experimental data. The solids mixing process in a columnar fluidized bed was found to resemble that in a tapered bed with an apex angle of 2.86°. Increasing the apex angle leads to a larger equilibrium mixing value. In addition, the influences of inlet jet velocity and nozzle diameter on the solids mixing process were investigated. The simulation results indicated that higher inlet jet velocities and larger nozzle diameters enhance the equilibrium mixing index value. Notably, inlet jet velocities of 0.7 and 0.8 m/s exhibited three distinct solids mixing stages: rapid, slow, and equilibrium, whereas higher jet velocities only involved rapid and equilibrium mixing stages. Moreover, this study further examined how the initial arrangement of solid particles affects the mixing index, providing valuable insights into optimizing the solids mixing process. Furthermore, the present work sheds light on the factors influencing the mixing of solids in tapered fluidized beds, offering valuable insights for further research and industrial applications.

Introduction

Solid mixtures are ubiquitous in process industries such as food processing, pharmaceuticals, chemical processing, and mineral processing. However, these mixtures rarely contain particles of uniform size or density. In reality, particle size distributions and variations in density are common in practice, significantly impacting the overall process. Achieving adequate mixing of such solid mixtures is crucial for various industrial applications.

Fluidized beds serve as widely used gas–solid contacting devices in these industries. This necessitates investigating the hydrodynamic behavior of solid mixtures within them, specifically focusing on the mixing and segregation of solid particles [1]. While fluidized beds offer advantages in enhancing solids mixing, it is essential to remember that perfect mixing of solid particles within these beds cannot be assumed. The minimum fluidization velocity (U_mf) and superficial gas velocity (U₀) are the primary variables influencing mixing and segregation phenomena in fluidized beds [2]. Various mixing (segregation) indices have been proposed and rigorously tested to study these phenomena. Several mixing indices have been proposed in the literature to quantify solids mixing or segregation in fluidized beds [3,4,5,6,7,8,9,10,11].

Previous studies using the discrete particle modeling (DPM) approach have shown that an increase in superficial gas velocity, solid density, and a decrease in particle diameter lead to a higher mixing index value [12]. Norouzi et al. [13] investigated solids circulation patterns in 2D bubbling fluidized beds and reported an increased axial dispersion coefficient with higher superficial gas velocities. Zhu et al. [14] employed Ashton’s mixing index to examine the circulation of dry and wet binary solid mixtures in 2D spouted beds. They found that the rate of solids mixing vigorously depends on both the mixing rate within the annulus region and the solids’ moisture content. Peng et al. [15] investigated combustion fluidized bed reactors and concluded that a decrease in the particle diameter ratio increases the Lacey mixing index value, indicating an improved state of solids mixing.

The multi-fluid modeling (MFM) approach has also been employed to investigate solids mixing in fluidized beds. Huilin et al. [16, 17] utilized this approach in 2D bubbling fluidized beds. They observed a decrease in the average particle diameter along the bed height, accompanied by an increase in the mass fraction of smaller particles.

Coroneo et al. [18] employed theoretical and experimental methods to investigate the influence of various parameters, namely computational grid size, time step, and residual errors, on the simulation of 2D fluidized beds for predicting the segregation of bidisperse solid mixtures. Zhong et al. [19] explored the effects of wall boundary conditions in 2D bubbling fluidized beds containing binary solid mixtures. They reported that the restitution coefficient for particle–wall impacts has a negligible impact on predicting solids segregation. In a separate study, Mostafazadeh et al. [20] investigated the influence of solid particle proportion on solids mixing in 2D bubbling fluidized beds under varying gas velocities.

Banaei et al. [21, 22] proposed a novel technique for studying the rate of solids mixing in dense gas–solid fluidized beds. Their approach involved coupling the two-fluid modeling (TFM) approach with particle tracking, enabling them to investigate the impacts of superficial gas velocity and the restitution coefficient on the mixing rate.

Hernández-Jiménez et al. [23] investigated the solids mixing process in a pseudo-2D fluidized bed using the TFM approach. They proposed a novel mixing index for qualitative evaluation of solids mixing. This index utilizes information from all three phases (gas, particles, and interstitial fluid) involved in the fluidized bed. Compared to the Lacey and meso-mixing indices, it requires less information while still providing accurate mixing process predictions beyond the experimental setup’s limitations.

Extensive research has been conducted on tapered fluidized beds, focusing on various aspects such as:

• 1. Expanded bed height and pressure drop: Depypere et al. [24], Maruyama and Sato [25], and Sau and Biswal [26] studied the influence of bed geometry on the expansion and pressure drop of solids.

• 2. Minimum partial fluidization velocity and maximum pressure drop: Agarwal and Roy [27], Gan et al. [28], Jean and Fan [29], Kaewklum and Kuprianov [30], Khani [31], Kmiec [32], Lucas et al. [33], and Singh et al. [34] investigated the minimum fluidization velocity and maximum pressure drop in tapered beds, considering the impact of various parameters.

Mardanloo et al. [35] employed computational fluid dynamics (CFD) techniques with a two-fluid model approach to investigate solids mixing in a two-dimensional tapered fluidized bed. The study utilized the Lacey mixing index for quantitative evaluation of solids mixing. They meticulously investigated the effects of various operating and design parameters on the mixing index, including superficial gas velocity, solids mass fraction, operating pressure, apex angle, and particle density.

The study of gas jet flow and jet penetration length in fluidized beds with an inlet jet in the gas distributor is crucial. The jet penetration length is the distance from the gas distributor at which the gas volume fraction reaches approximately 0.8 [19, 36]. Wang et al. [37] employed the MFM approach to investigate the mixing and segregation of binary solid mixtures in 3D spouted beds with two inlet jets. They reported that the mechanism of solid particle segregation can be identified by tracking the solid phase velocity.

Extensive research on solids mixing in tapered fluidized beds underscores the significant impact of bed geometry on the mixing process. Its vital role in various industries necessitates further investigations into the influence of other crucial factors to gain a comprehensive understanding of the phenomenon. This study aims to address this need through the following objectives:

• 1. Apply the MFM approach for simulating the intensification of solids mixing in binary solid mixtures within 2D tapered fluidized beds equipped with an inlet jet.

• 2. Investigate the effects of tapered fluidized bed geometry, inlet jet velocity, and its diameter on enhancing solids mixing.

• 3. Quantitatively evaluate the enhancement of solids mixing using the particle segregation number (PSN).

• 4. Explore the effects of various design and operating parameters on the time variations of PSN values.

The subsequent sections are organized as follows:

“Governing Equations”: Governing equations.

“Experimental Section”: A small-scale experimental setup without an inlet jet and experimental procedures.

“Simulation Procedure and Analysis”: Simulation procedures and analysis.

“Results and Discussion”: Experimental analysis, simulation results, and discussion.

“Limitations and Guide for Future Works”: Concluding remarks.

Governing Equations

This study employed the multi-fluid modeling (MFM) approach, integrating both the Eulerian–Eulerian framework and the kinetic theory of granular flow (KTGF), to investigate the hydrodynamic behavior of binary solid mixtures within tapered fluidized beds. The gas and individual solid phases were considered fully interpenetrating within this modeling framework. The approach utilizes continuity equations and conservation equations of linear momentum for both the gas and solid phases. In addition, the solid phase pressure, solid shear viscosity, and the solid bulk viscosity are expressed as functions of the granular temperature [38]. The governing equations encompass continuity and momentum equations for each phase, alongside granular temperature equations for the solid phases, as detailed by Benyahia et al. [39]:

Continuity Equations

The continuity equation for the gas phase can be expressed as

$$\frac{\partial ({\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}})}{\partial t}+\nabla \cdot \left({\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}}{{\varvec{u}}}_{{\text{g}}}\right)=0.$$

(1)

In addition, for systems consisting of M solid phases, the continuity equation for each solid phase (m = 1, …, M) can be expressed as follows:

$$\frac{\partial ({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}})}{\partial t}+\nabla \left({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{{\varvec{u}}}_{{\text{m}}}\right)=0,$$

(2)

where ρ_g and ρ_m represent the densities of the gas and the mth solid phases, respectively; in addition, u_g, u_m, ε_g, and ε_m correspond to the velocities of the gas and mth solid phase, as well as the volume fractions of the gas and the mth solid phases, respectively. It is important to note that the sum of all volume fractions should be equal to one.

$${\varepsilon }_{{\text{g}}}+\sum_{{\text{m}}=1}^{{\text{M}}} {\varepsilon }_{m}=1.0.$$

(3)

Equations of Momentum

The equation of momentum for the gas phase can be expressed as

$$\frac{\partial \left({\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}}{{\varvec{u}}}_{{\text{g}}}\right)}{\partial t}+\nabla \left({\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}}{{\varvec{u}}}_{{\text{g}}}{{\varvec{u}}}_{{\text{g}}}\right)=-{\varepsilon }_{{\text{g}}}\nabla {P}_{{\text{g}}}+\nabla {\tau }_{{\text{g}}}-\sum_{{\text{m}}=1}^{{\text{M}}} {I}_{{\text{gm}}}+{\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}}g,$$

(4)

where τ_g is the stress tensor of the gas phase, which can be given by

$${\tau }_{{\text{g}}}=-{\upvarepsilon }_{{\text{g}}}{\mu }_{{\text{g}}}\left[\nabla {{\varvec{u}}}_{{\text{g}}}+{\left(\nabla {{\varvec{u}}}_{{\text{g}}}\right)}^{{\text{T}}}\right]+{\varepsilon }_{{\text{g}}}\left({\lambda }_{{\text{g}}}-\frac{2}{3}{\mu }_{{\text{g}}}\right)\left(\nabla {{\varvec{u}}}_{{\text{g}}}\right)I.$$

(5)

In Eq. 4, P_g represents the gas pressure, I is the unit tensor, and μ_g and λ_g are, respectively, the shear and bulk viscosities of the gas phase, with λ_g = 0. Moreover, I_gm represents the momentum exchange between the gas phase and the m^th solid phase and can be defined as follows:

$${{\varvec{I}}}_{{\text{gm}}}={\beta }_{{\text{gm}}}\left({{\varvec{u}}}_{{\text{g}}}-{{\varvec{u}}}_{{\text{m}}}\right).$$

(6)

The interphase momentum exchange coefficient between the gas and the m^th solid phases is denoted by β_gm. Various models have been proposed to evaluate this parameter. For instance, the Syamlal and O’Brien [40] model can be used as follows:

$${\beta }_{{\text{gm}}}=\frac{3}{4}\frac{{\varepsilon }_{{\text{m}}}{\varepsilon }_{{\text{g}}}{\rho }_{{\text{g}}}}{{{\upsilon }_{{\text{r}},{\text{m}}}}^{2}{d}_{{\text{pm}}}}{C}_{{\text{D}}}\left(\frac{{Re}_{{\text{m}}}}{{\upsilon }_{{\text{r}},{\text{m}}}}\right)\left|{{\varvec{u}}}_{{\text{g}}}-{{\varvec{u}}}_{{\text{m}}}\right|,$$

(7)

$${C}_{{\text{D}}}={(0.63+\frac{4.8}{\sqrt{{Re}_{{\text{m}}}/{\upsilon }_{{\text{r}},{\text{m}}}}})}^{2},$$

(8)

$${{\text{Re}}}_{{\text{m}}}=\frac{{d}_{{\text{pm}}}\left|{{\varvec{u}}}_{{\text{g}}}-{{\varvec{u}}}_{{\text{m}}}\right|{\rho }_{{\text{g}}}}{{\mu }_{{\text{g}}}},$$

(9)

$${\upsilon }_{{\text{r}},{\text{m}}}=0.5\left[A-0.06{Re}_{{\text{m}}}+\sqrt{{\left(0.06{Re}_{{\text{m}}}\right)}^{2}+0.12{Re}_{{\text{m}}}\left(2B-A\right)+{A}^{2}}\right],$$

(10)

$$\left\{\begin{array}{ll}A={{\varepsilon }_{{\text{g}}}}^{4.14},B=0.8{{\varepsilon }_{{\text{g}}}}^{1.28}&\quad {\varepsilon }_{{\text{g}}}\le 0.85\\ A={{\varepsilon }_{{\text{g}}}}^{4.14},B={{\varepsilon }_{{\text{g}}}}^{2.65} & \quad {\varepsilon }_{{\text{g}}}>0.85\end{array}\right..$$

(11)

In addition, the equation of momentum for the mth solid phase can be expressed as follows:

$$\frac{\partial \left({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{{\varvec{u}}}_{{\text{m}}}\right)}{\partial {\text{t}}}+\nabla \left({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{{\varvec{u}}}_{{\text{m}}}{{\varvec{u}}}_{{\text{m}}}\right)=-{\varepsilon }_{{\text{m}}}\nabla {P}_{{\text{g}}}+\nabla {\tau }_{{\text{m}}}+{{\varvec{I}}}_{{\text{gm}}}-\sum_{\begin{subarray}{c}k=1\\ k\ne m\end{subarray}}^{{\text{M}}} {{\varvec{I}}}_{{\text{mk}}}+{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}g+{S}_{{\text{m}}}.$$

(12)

The stress tensor for the mth solid phase, denoted by τ_m, can be given by a Newtonian-type viscous approximation as follows:

$${\tau }_{{\text{m}}}=-\left({P}_{{\text{m}}}+\mathfrak{y}{\mu }_{{\text{b}}}\nabla {{\varvec{u}}}_{{\text{m}}}\right)I+{{\upvarepsilon }_{{\text{m}}}\mu }_{{\text{m}}}\left[\frac{1}{2}\left[\nabla {{\varvec{u}}}_{{\text{m}}}+{\left(\nabla {{\varvec{u}}}_{{\text{m}}}\right)}^{{\text{T}}}\right]+\frac{1}{3}\left(\nabla {{\varvec{u}}}_{{\text{g}}}\right)I\right],$$

(13)

$$\mathfrak{y}=\frac{1}{2}\left(1+e\right).$$

(14)

Moreover, the solid phase pressure, denoted by P_m, shear viscosity, denoted by μ_m, and the bulk viscosity, denoted by μ_b, are expressed as functions of the granular temperature. P_m can be evaluated using the following equation:

$${P}_{{\text{m}}}={\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{\theta }_{{\text{m}}}\left[1+4\mathfrak{y}\sum_{{\text{n}}=1}^{{\text{M}}}{\varepsilon }_{{\text{n}}}{g}_{0,{\text{mn}}}\right],$$

(15)

where g_0,mk represents the radial distribution function (RDF), which expresses the statistics of the spatial arrangement of the particles.

$${g}_{0,{\text{mk}}}=\frac{1}{{\varepsilon }_{{\text{g}}}}+\frac{3{d}_{{\text{m}}}{d}_{{\text{n}}}}{{{\varepsilon }_{{\text{g}}}}^{2}({d}_{{\text{m}}}{+d}_{{\text{n}}})}\sum_{{\text{k}}=1}^{{\text{M}}}\frac{{\varepsilon }_{{\text{k}}}}{{d}_{{\text{k}}}}.$$

(16)

Furthermore, I_mk represents the momentum exchange between solid–solid phases and can be defined as follows:

$${{\varvec{I}}}_{{\text{mk}}}={\beta }_{{\text{mk}}}\left({{\varvec{u}}}_{{\text{m}}}-{{\varvec{u}}}_{{\text{k}}}\right),$$

(17)

where β_mk is the rate of exchange of solid–solid momentum, which can be determined using the equation proposed by Syamlal et al. [40].

$${\beta }_{{\text{mk}}}=3\left(1+e\right)\left(\frac{\uppi }{2}+{C}_{{\text{f}}}\frac{{\pi }^{2}}{8}\right)\frac{{\left({d}_{{\text{m}}}+{d}_{{\text{k}}}\right)}^{2}}{2\uppi \left({\rho }_{{\text{m}}}{{d}_{{\text{m}}}}^{3}+{\rho }_{{\text{k}}}{{d}_{{\text{k}}}}^{3}\right)}{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{\varepsilon }_{{\text{k}}}{\rho }_{{\text{k}}}{g}_{0,{\text{mk}}}\left|{{\varvec{u}}}_{{\text{k}}}-{{\varvec{u}}}_{{\text{m}}}\right|.$$

(18)

In Eqs. 14 and 18, C_f represents the friction coefficient, and e is the restitution coefficient [41].

The shear viscosity of solid phases (μ_m) consists of collisional, kinetic, and frictional terms, which can be expressed as follows:

$${\mu }_{{\text{m}}}={\mu }_{{\text{m}},{\text{col}}}+{\mu }_{{\text{m}},{\text{kin}}}+{\mu }_{{\text{m}},{\text{fr}}}.$$

(19)

Note that the basic assumption of this model is that the stress during particle flow is not constant but fluctuates around an average value.

$${\mu }_{{\text{m}}}=\left(\frac{2+\alpha }{3}\right)\left[\frac{{\mu }_{m}^{*}}{{g}_{0,{\text{mm}}}\upeta \left(2-\eta \right)}\left(1+\frac{8}{5}\eta \sum _{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right)\left(1+\frac{8}{5}\eta \left(3\eta -2\right)\sum _{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right)\right]+\frac{{P}_{{\text{c}}}{\text{sin}}\left(\varnothing \right)}{2\sqrt{{I}_{2{\text{D}}}}}\frac{{\varepsilon }_{{\text{m}}}}{{\sum }_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}},$$

(20)

$${\mu }_{m}^{*}=\frac{{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{g}_{0,{\text{mm}}}{\theta }_{{\text{k}}}{\mu }_{{\text{s}}}}{{\rho }_{{\text{m}}}\left({\sum }_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right){\theta }_{{\text{m}}}+(\frac{2{\beta }_{{\text{gm}}}{\mu }_{{\text{s}}}}{{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}})},$$

(21)

$${\mu }_{{\text{s}}}=\frac{5}{96}{\rho }_{{\text{m}}}{d}_{{\text{m}}}\sqrt{\pi {\theta }_{{\text{m}}}},$$

(22)

$${\mu }_{{\text{b}}}=\frac{256}{5\pi }{\mu }_{{\text{s}}}{\varepsilon }_{{\text{m}}}\sum _{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}},$$

(23)

where α = 1.6 and I_2D is the second invariant of the deviatoric stress tensor, which can be calculated using the following equation:

$$\begin{aligned}{I}_{2{\text{D}}}&=\frac{1}{6}\left[{\left({D}_{11}-{D}_{22}\right)}^{2}+{\left({D}_{22}-{D}_{33}\right)}^{2}+{\left({D}_{33}-{D}_{11}\right)}^{2}\right]\\ & \quad +{{D}_{12}}^{2}+{{D}_{23}}^{2}+{{D}_{31}}^{2},\end{aligned}$$

(24)

$${D}_{{\text{ij}}}=\frac{1}{2}\left(\frac{\partial {u}_{{\text{m}},{\text{i}}}}{\partial {x}_{{\text{j}}}}+\frac{\partial {u}_{{\text{m}},{\text{j}}}}{\partial {x}_{{\text{i}}}}\right),$$

(25)

$${P}_{{\text{c}}}=\left\{\begin{array}{ll}{10}^{25} {({\varepsilon }^{*}-{\varepsilon }_{{\text{g}}})}^{10} &\quad {\varepsilon }_{{\text{g}}}<{\varepsilon }^{*}\\ 0& \quad {\varepsilon }_{{\text{g}}}\ge {\varepsilon }^{*}\end{array}\right..$$

(26)

Moreover, ε* = 1 – ${\varepsilon }_{{\text{mixture}}}^{{\text{max}}}$, where ${\varepsilon }_{{\text{mixture}}}^{{\text{max}}}$ represents the maximum solid packing in binary solid mixtures, which can be determined using empirical correlations. In this study, the value of the maximum packing of the solid phases in the fluidized beds consisting of binary solid mixtures could be calculated using the correlations proposed by Landel and Fedors [42], assuming that d₁ > d₂ and X₁ = $\frac{{\varepsilon }_{1}}{{\varepsilon }_{1}+{\varepsilon }_{2}}$, as follows:

$$\begin{aligned}{X}_{1}& \le {{X}_{1} }^{*}\\ &\quad {{\varepsilon }_{\text{mixture}}^{\text{max}}}=\Bigg[({\varepsilon }_{1,{\text{max}}}-{\varepsilon }_{2,{\text{max}}})+\left(1-\sqrt{\frac{{d}_{2}}{{d}_{1}}}\right)(1-{\varepsilon }_{1,{\text{max}}}){\varepsilon }_{2,{\text{max}}}\Bigg] \\ & \quad [{\varepsilon }_{1,{\text{max}}}+(1-{\varepsilon }_{1,{\text{max}}}){\varepsilon }_{2,{\text{max}}}]\frac{{X}_{1}}{{\varepsilon }_{1,{\text{max}}}}+{\varepsilon }_{2,{\text{max}}},\end{aligned}$$

(27)

$$\begin{aligned}&{X}_{1}>{{X}_{1} }^{*} \\ &\quad {\varepsilon }_{{\text{mixture}}}^{{\text{max}}}=\left(1-\sqrt{\frac{{d}_{2}}{{d}_{1}}}\right)\left[{\varepsilon }_{1,{\text{max}}}+\left(1-{\varepsilon }_{1,{\text{max}}}\right){\varepsilon }_{2,{\text{max}}}\right]\\ & \quad \left(1-{X}_{1}\right)+{\varepsilon }_{1,{\text{max}}},\end{aligned}$$

(28)

$${{X}_{1} }^{*}=\frac{{\varepsilon }_{1,{\text{max}}}}{{\varepsilon }_{1,{\text{max}}}+(1-{\varepsilon }_{1,{\text{max}}}){\varepsilon }_{2,{\text{max}}}}.$$

(29)

Modeling the hydrodynamic behavior of tapered fluidized beds involves incorporating a solid source term (S_m) into the momentum equation for the solid phase(s). Peng and Fan [43] employed a mechanical energy balance for solid particles within a liquid–solid tapered fluidized bed and derived a relationship for evaluating the pressure drop in such beds. Assuming an incompressible gas phase and considering frictional dissipation energy between solid particles and the gas phase, the equation of energy for the gas phase can be expressed as follows:

$$\frac{\Delta P}{{\rho }_{{\text{g}}}}+\frac{1}{2}\Delta {\left(\frac{u}{{\varepsilon }_{{\text{g}}}}\right)}^{2}+g\Delta h+{E}_{{\text{fr}}}=0,$$

(30)

where ∆P represents the pressure drop, ρ_g is the gas density, ∆h is the bed height, E_fr is the rate of frictional dissipation energy, u is the gas velocity, and ε_g is the gas volume fraction. Taking the net pressure as P_N = P + ρ_ggh, we obtain:

$$\left(-\Delta {P}_{{\text{N}}}\right)=\frac{1}{2}{\rho }_{{\text{g}}}\Delta {\left(\frac{u}{{\varepsilon }_{{\text{g}}}}\right)}^{2}+{{\rho }_{{\text{g}}}E}_{{\text{fr}}}=0,$$

(31)

where the net pressure drop (–∆P_N) is the sum of the frictional and kinetic terms (–∆P_fr and –∆P_kin).

$$\left(-\Delta {P}_{{\text{N}}}\right)=\left(-\Delta {P}_{{\text{fr}}}\right)+\left(-\Delta {P}_{{\text{kin}}}\right),$$

(32)

$$\left(-\Delta {P}_{{\text{fr}}}\right)={{\rho }_{{\text{g}}}E}_{{\text{fr}}},$$

(33)

$$\left(-\Delta {P}_{{\text{kin}}}\right)=\frac{1}{2}{\rho }_{{\text{g}}}\Delta {(\frac{u}{{\varepsilon }_{{\text{g}}}})}^{2}.$$

(34)

Peng and Fan [43] investigated a tapered fluidized bed with a bottom diameter of D₀ and a top diameter of D₁. They derived relationships for the frictional, kinetic, and net pressure drop in various flow regimes, including the fixed, partially fluidized, and completely fluidized. These relationships are presented as follows:

$$\left(-\Delta {P}_{{\text{kin}}}\right)=\frac{1}{2}{\left(\frac{{u}_{0}}{{\varepsilon }_{{\text{g}}}}\right)}^{2}\left[{\left(\frac{{D}_{0}}{{D}_{1}}\right)}^{4}-1\right]{\rho }_{{\text{g}}},$$

(35)

$$\left(-\Delta {P}_{{\text{fr}}}\right)=\left(1-{\varepsilon }_{{\text{g}}}\right)\left( {\rho }_{{\text{s}}}-{\rho }_{{\text{g}}}\right)g{H}_{{\text{exp}}},$$

(36)

where H_exp represents the expanded bed’s height, and u₀ represents the superficial gas velocity. Wang et al. [44] simulated the hydrodynamic behavior of spouted beds and categorized the bed space into two regions: diluted and dense. They further reported that the inclined walls of tapered beds can exert stress on the gas phase, leading to a higher pressure drop than columnar beds. Two coefficients relevant to the dense and diluted regions were defined to assess the pressure drop ratio as follows:

$${k}_{{\text{a}}}=\frac{\left(-\Delta {P}_{{\text{top}}}\right)}{\left(-\Delta {P}_{{\text{col}}}\right)},$$

(37)

$${k}_{{\text{c}}}=f\left({\varepsilon }_{{\text{g}}},{\rho }_{{\text{g}}},{\rho }_{{\text{s}}},{d}_{{\text{s}}},{\mu }_{{\text{g}}},{u}_{{\text{g}},{\text{z}}}\right),$$

(38)

where ∆P_top and ∆P_col represent the pressure drops in tapered and columnar beds, respectively, and u_g,z denotes the gas velocity component in the axial direction. Sau and Biswal [26] assumed a value of k_c = 1.0 and derived the following relationship for the coefficient k_a:

$${k}_{{\text{a}}}=\frac{\left(1-{\varepsilon }_{{\text{g}}}\right)\left({{\rho }_{{\text{s}}}-\rho }_{{\text{g}}}\right)g+\frac{1}{2{H}_{0}}{(\frac{{u}_{0}}{{\varepsilon }_{{\text{g}}}})}^{2}[({\frac{{D}_{0}}{{D}_{1}})}^{4}-1]{\rho }_{{\text{g}}}}{\left(1-{\varepsilon }_{{\text{g}}}\right)\left({{\rho }_{{\text{s}}}-\rho }_{{\text{g}}}\right)g},$$

(39)

where H₀ represents the static bed height of solid particles, this equation indicates that for a columnar bed with D₀ = D₁, the k_a value equals 1. For tapered beds with coefficients of k_a and k_c, Sau and Biswal [26] reported relationships to evaluate the solid source term for both the diluted and dense regions as follows:

$${S}_{{\text{m}}}=\left(k-1\right){\varepsilon }_{{\text{s}}}{\rho }_{{\text{s}}}g,$$

(40)

$$k=\left\{\begin{array}{ll}{k}_{{\text{a}}} &\quad {\varepsilon }_{{\text{g}}}\le 0.8, z\le {H}_{0}\\ {k}_{{\text{c}}} & \quad {\varepsilon }_{{\text{g}}}>0.8\end{array}\right..$$

(41)

Kinetic Theory of Granular Flow

Applying the kinetic theory of gasses to evaluate the kinetic energy due to the random fluctuating velocity of solid particles, the granular temperature can be expressed as the squared average of the fluctuations of solid particles (${\theta }_{s}=\frac{1}{3}\overline{{{({v}_{s}}^{\prime})}^{2}}$). The specifications of the solid phase in the multi-fluid model, such as pressure, viscosity and bulk viscosity, diffusion coefficient of granular energy, and stresses, are defined using the granular temperature. The equation of granular temperature for the mth solid phase can be expressed as follows:

$$\begin{aligned}\frac{3}{2}\left[\frac{\partial \left({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{\theta }_{{\text{m}}}\right)}{\partial {\text{t}}}+\nabla \left({\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{\theta }_{{\text{m}}}{{\varvec{u}}}_{{\text{m}}}\right)\right]&=-\nabla {\varvec{q}}-{(\tau }_{{\text{m}}}:\nabla {{\varvec{u}}}_{{\text{m}}})\\ & \quad +{\Gamma }_{{\text{slip}}}-{J}_{{\text{coll}}}-{J}_{{\text{vis}}},\end{aligned}$$

(42)

where q represents the pseudo-thermal energy flux vector, Γ_slip denotes the production of granular energy due to the slip between the phases, J_coll signifies the dissipation of granular energy via inelastic collisions, and J_vis represents the dissipation of granular energy via viscous dissipation of the gas phase. These parameters can be evaluated as follows:

$${\varvec{q}}=-\lambda \nabla {\theta }_{{\text{m}}},$$

(43)

$${\varvec{q}}=\frac{{{\kappa }_{{\text{m}}}}^{*}}{{g}_{0,{\text{mm}}}}\left\{\left[1+\frac{12}{5}\eta \sum_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right]\left[1+\frac{12}{5}{\eta }^{2}\left(4\eta -3\right)\sum_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right]+\left[\frac{64}{25\pi }\left(41-32\eta \right){\eta }^{2}{\left(\sum_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}\right)}^{2}\right]\right\}\nabla {\theta }_{{\text{m}}},$$

(44)

$${{\kappa }_{{\text{m}}}}^{*}=\frac{{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}{g}_{0,{\text{mm}}}{\theta }_{{\text{m}}}\upkappa }{{(\rho }_{{\text{m}}}{\theta }_{{\text{m}}}\sum_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}})+\frac{6{\beta }_{{\text{gm}}}\upkappa }{5{\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}}},$$

(45)

$$\upkappa =\frac{75{{\rho }_{{\text{m}}}d}_{{\text{m}}}\sqrt{\pi {\theta }_{{\text{m}}}}}{48\eta (41-33\eta )},$$

(46)

$${J}_{{\text{coll}}}= \frac{48}{\sqrt{\pi }}\eta \left(1-\upeta \right){\varepsilon }_{{\text{m}}}{\rho }_{{\text{m}}}\frac{\sum_{{\text{k}}=1}^{{\text{M}}}{\varepsilon }_{{\text{k}}}{g}_{0,{\text{mk}}}}{{d}_{{\text{m}}}}{{\theta }_{{\text{m}}}}^{3/2},$$

(47)

$${\Gamma }_{{\text{slip}}}=\frac{81{\upvarepsilon }_{{\text{m}}}{{\mu }_{{\text{g}}}}^{2}{\left(\left|{\mathbf{u}}_{{\text{g}}}-{\mathbf{u}}_{{\text{m}}}\right|\right)}^{2}}{{g}_{0,{\text{mm}}}{{d}_{{\text{m}}}}^{3}{\rho }_{{\text{m}}}\sqrt{\pi {\theta }_{{\text{m}}}}},$$

(48)

$${J}_{{\text{vis}}}=3{\beta }_{{\text{gm}}}{\theta }_{{\text{m}}}.$$

(49)

Experimental Section

Small-Scale Tapered Fluidized Bed Without an Inlet Jet

Figure 1 presents a schematic diagram of the experimental setup employed in this study. The tapered bed, made of Plexiglas, had a height of 0.7 m and an apex angle of 8.13°. A gas inlet or bed base with a diameter of 0.07 m was utilized. A stainless-steel sieve with a mesh count of 170 (equivalent to a mesh size of 88 µm) served as the gas distributor. A plenum chamber was incorporated to ensure well-developed fluid flow within the tapered fluidized bed. The pressure drop across the bed was measured using a manometer positioned in the upper region of the bed, just above the gas distributor. A pre-calibrated gas rotameter, located downstream of the compressor (model TK1500/100-5.5 hp) and the 550 L air vessel, regulated the inlet gas flow rate. The experimental runs were conducted at an operating temperature of 25 °C and atmospheric pressure. Air was employed as the fluidizing gas, with a density (ρ_g) of 1.22 kg/m³ and a dynamic viscosity (μ_g) of 1.85 × 10^–5 Pa.s. Spherical glass beads with a 500 µm diameter and 2500 kg/m³ density were used as the fluidized particles.

Fig. 1

Experimental apparatus: (1) air compressor and tank, (2) pressure regulator, (3) flowmeter, (4) lamps, (5) fluidized bed, (6) digital camera, and (7) manometer

Initial Layout of Solid Particles

Figure 2 illustrates the initial arrangement of solid particles. Jetsam particles were placed in the bottom layer of the bed, while flotsam particles were located in the layer above. The bed height was maintained at a constant 7 cm, although the final heights of both jetsam and flotsam layers could vary depending on the relative volume occupied by each particle type within the bed.

Fig. 2

The initial arrangement of particles in the tapered bed without an inlet jet

Camera Settings and Analyzing Images

During each experimental run, several images of the bed at 240 frames per second (fps) were captured using a Casio Exilim ZR1200 Digital camera. For optimal visibility, consistent and intense illumination was provided by two 25 W LED bulb lamps positioned on either side of the tapered bed. Moreover, a black opaque curtain strategically positioned over the camera prevented direct reflections from the surrounding environment onto the bed. The camera produced digital images with a resolution of 512 × 384 pixels, deemed sufficient for digital image analysis (DIA). The DIA technique was employed to process and examine the captured images. To ensure the robustness and reproducibility of the quantitative DIA methodology, each experimental run was repeated a minimum of three times. This iterative procedure validated the consistency of the experimental data, enhancing its reliability. Notably, the presence of non-perfectly spherical particles with a size distribution between 450 and 550 μm introduced potential sources of error.

All experimental runs were conducted in a pseudo-two-dimensional tapered bed configuration. While dedicated efforts were made to ensure a consistent gas flow throughout the bed, achieving absolute uniformity was only partially achievable. Minor discrepancies in individual measurements could arise due to varying environmental factors. Nevertheless, meticulous measures were taken to minimize error margins. These included minimizing environmental disturbances, calibrating the equipment, conducting multiple repetitions of experimental trials, capturing multiple parameter measurements, and carefully considering potential errors related to human operators during variable measurements and digital image analysis for mixing index evaluation. The uncertainties and errors associated with the gas flow rate are concisely summarized in Table 1.

Table 1 Gas flow rate error and uncertainty

Experimental Procedure

The solid particles were meticulously introduced into the bed to achieve uniform dispersion to start each experimental run. Subsequently, the gas inlet flow rate was carefully adjusted and introduced into the plenum chamber of the tapered bed. Images captured from the fluidized bed were rigorously examined using an image analysis methodology. This procedure was replicated a minimum of three times, and the resulting data were averaged and reported. Representative images captured during the experimental run are presented in Fig. 3. A detailed breakdown of the uncertainties and inaccuracies associated with the gas flow rate is provided in Table 1.

Fig. 3

Images taken from the tapered bed containing various jetsam and flotsam solid particles. a) X_j = 0.23, b) X_j = 0.47, and c) X_j = 0.73

Simulation Procedure and Analysis

Simulation Procedure

An in-house code developed by our research group was used for the numerical simulations. The simulations involved solving governing equations for three-phase systems, encompassing continuity equations, momentum equations for the gas and solid phases, and granular temperature equations for the solid phases. The simulations were conducted considering the 2D geometry of the tapered fluidized beds. The governing equations were discretized using the finite volume method and the pressure–velocity coupling algorithm to achieve this. The semi-implicit method for pressure-linked equations (SIMPLE) was implemented. Furthermore, the time-dependent and spatial terms of the governing equations were discretized using the first-order implicit scheme and the Superbee second-order scheme, respectively. All simulation runs were performed for 30 s with a time step of 0.0001 s. The maximum number of iterations was set to 800 to ensure convergence of the solved equations at each time step.

Specifications of Tapered Fluidized Beds with an Inlet Jet

Figure 4 presents a schematic diagram of the tapered fluidized bed with an inlet jet investigated in this study. The initial arrangement of the two solid phases was vertical, exhibiting complete segregation. In all simulation runs, the bottom diameter of the tapered bed was maintained at 10 cm, and the bed height was 50 cm. Notably, under the initial conditions, the solid particles experienced minimum fluidization, with the minimum fluidization velocity of binary solid mixtures determined using an experimental correlation proposed by Khani [36]. Throughout the simulations, a no-slip boundary condition was applied for the gas phase, while a partial slip boundary condition was applied for the solid phases. In addition, a specularity coefficient of 0.6 was used for the solid phase.

Fig. 4

Schematic diagram of simulated tapered fluidized bed with an inlet jet

The gas distributor was equipped with an inlet jet, and the gas velocity was maintained uniformly within the jet and the distributor. The gas distributor’s velocity was set to match the minimum fluidization velocity of the solid particles. The solid particle velocity was zero at the inlet boundary, and the gas phase volume fraction was set to one. The outlet pressure was fixed at 1 atm as the outlet boundary condition. Table 2 summarizes the geometric specifications used in the simulation runs. Glass bead particles with diameters of 240 and 510 µm (representing a solid mixture with the same density but different diameters) were employed to investigate the mixing process of binary solid mixtures. Table 3 summarizes the remaining physical properties of the solid particles and the gas phase used in this study.

Table 2 Specifications of fluidized beds

Table 3 Physical properties of gas and solid phases

Mixing Index

Keller [7] proposed the particle segregation number (PSN) as a segregation index for solid particles in a fluidized bed, calculated based on the normalized average heights of the solid phase as follows:

$${\text{PSN}}=2\left(\Delta {h}_{{\text{flotsam}}}-\Delta {h}_{{\text{jetsam}}}\right),$$

(50)

where PSN values of zero and one represent the completely random mixture and the complete segregation of particles, respectively, and ∆h is the average height of flotsam and jetsam, which can be calculated by the following equation:

$$\Delta h=\frac{{\sum }_{{\text{k}}}^{{\text{N}}}{\varepsilon }_{{\text{s}},{\text{k}}}{h}_{{\text{k}}}{V}_{{\text{k}}}}{H{\sum }_{{\text{k}}}^{{\text{N}}}{\varepsilon }_{{\text{s}},{\text{k}}}{V}_{{\text{k}}}}.$$

(51)

In this equation, h_k is the distance of the kth cell center from the gas distributor, V_k is the volume of the kth cell, ε_s,k is the volume fraction of the solid phase (flotsam or jetsam) in the kth cell, N is the number of grid cells, and H is the static bed height. In addition, the relationship between the mixing and segregation indices can be expressed as follows:

$${\text{Mixing index }} = { 1 }{-}{\text{segregation index }} = { 1 }{-}{\text{ PSN}}{.}$$

(52)

This study used the mixing index mentioned above to investigate the solids mixing in the tapered fluidized beds.

Results and Discussion

Grid Independence Study

Ensuring the independence of simulation results on grid size was a crucial aspect of this study. Selecting an appropriate grid size was essential to guarantee that the results remained independent of the grid resolution, as inaccurate choices could lead to unreliable findings. In the Eulerian–Eulerian approach, where the solid phase is treated as a continuum, the grid size should be chosen such that each cell contains a sufficient number of solid particles to satisfy the continuum assumption for the solid phase(s). Several criteria have been proposed in CFD simulations to determine an appropriate grid size. Guenther and Syamlal [45] suggested setting the grid size at ten times the solid particle diameter as one such criterion. This study investigated the effects of different grid sizes on the axial velocity of solid particles in a tapered fluidized bed to examine the independence of simulation results on grid size. The bed had an apex angle of 8.53°, an inlet jet velocity of 2.0 m/s, and an inlet jet diameter of 5 mm (equal to 5% of the bottom diameter of the bed).

This study explored various computational grid sizes, including 25 × 50, 35 × 60, 45 × 80, and 55 × 100, to ensure grid independence. Figure 5 exemplifies the axial velocity of solid particles with a diameter of 240 µm at a height of 13 cm as a function of grid size. This figure demonstrates minimal discrepancies between the results obtained for grid numbers 35 × 60, 45 × 80, and 55 × 100. This suggests that the obtained results are likely independent of the grid size within this range. Consequently, a grid number of 45 × 80 was chosen as an appropriate size for subsequent simulations. The Grid Convergence Index (GCI) was also employed to assess grid convergence further. This, widely used in CFD studies, provides a quantitative measure of the reduction in errors achieved through grid refinement, facilitating the evaluation of solution convergence trends. The approach advocated by Celik et al. [46] was adopted for GCI assessment. In the context of two-dimensional simulations, the mesh or grid size (h) was determined using the following equation:

$$h = \left[ {\frac{1}{N}\sum\limits_{i = 1}^{N} {(\Delta A_{i} )} } \right]^{1/2} ,$$

(53)

where N represents the total number of cells utilized for the computations, while ΔAi corresponds to the area of the ith specific cell. By taking into account the sequence h₁ < h₂ < h₃ < h₄ and defining r₂₁ = h₂/h₁, as well as r₃₂ = h₃/h₂, the apparent order (denoted as p) of the method was evaluated using the subsequent equation.

$$p_{21} = \frac{1}{{\ln (r_{21} )}}\left| {\ln \left| {\frac{{\varepsilon_{32} }}{{\varepsilon_{21} }}} \right| + q(p)} \right|,$$

(54)

$$q(p) = \ln \left( {\frac{{r_{21}^{{\text{p}}} - s}}{{r_{32}^{{\text{p}}} - s}}} \right),$$

(55)

$$s = 1 \cdot {\text{sign}}\left( {\frac{{\varepsilon_{32} }}{{\varepsilon_{21} }}} \right),$$

(56)

where ε₃₂ = V_f3 –V_f2, ε₂₁ = V_f2 –V_f1, V_fk are the flotsam axial velocity on the k^th grid.

Fig. 5

Sensitivity analysis and grid independence study for the particle diameter of 240 µm and at the height of 13 cm

The fine-grid convergence index (GCI_fine) computation entailed utilizing the following error estimates.

Approximate relative error:

$$e_{a}^{21} = \left| {\frac{{V_{{{\text{f}}1}} - V_{{{\text{f}}2}} }}{{V_{{{\text{f}}1}} }}} \right|.$$

(57)

The fine-grid convergence index:

$${\text{GCI}}_{{{\text{fine}}}}^{21} = \frac{{1.25e_{a}^{21} }}{{r_{21}^{p} - 1}}.$$

(58)

Similarly, GCI_fine³² was calculated. Table 4 summarizes the obtained GCI values, demonstrating that the discrepancies between the simulation results obtained with different grid sizes are not statistically significant. This confirms that the chosen grid size 45 × 80 is appropriate for the simulations.

Table 4 Calculations of discretization errors in different bed dimensionless lengths [–]

Validation of Simulation Results

Validation of Simulation Results Against Our Own Experimental Data Using Mixing Index

A two-step approach was implemented to validate the simulation results against our own experimental data. First, a slightly tapered bed without an inlet jet was used to compare the simulated Lacey mixing index with the experimentally determined value. This served as an initial validation of the simulation methodology. Subsequently, a larger tapered bed with an inlet jet was used to generate further findings. Figure 6 presents the mixing index values obtained from simulation and experimental measurements at various gas velocities. The root mean square error (RMSE) was calculated to quantify the agreement between simulation and experimental results. The computed RMSE values indicate that the deviations remain within an acceptable range, demonstrating the reliability of the simulation results and providing confidence in their applicability for further analysis and interpretation.

Fig. 6

Variations of mixing index value vs. time for various volume fractions of jetsam particles (X_j = 0.23 (a, b), 0.47 (c, d), and 0.73 (e, f))

$${\text{RMSE}}=100\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left(1-\frac{{x}_{{\text{i}}({\text{simulation}})}}{{x}_{{\text{i}}({\text{experimental}})}}\right)}^{2}}.$$

(59)

Validation of Simulation Results Against Experimental Data Reported by Huilin et al. [16]

Furthermore, the simulation results predicted by the developed model were validated against experimental data reported by Huilin et al. [16]. Specifically, the profile of the average diameter of the binary solid mixture within the bed was obtained for a gas velocity of 2.2 m/s and an initial mass fraction of small particles (x_s0) of 0.5. This profile was then compared with the experimental data for validation. The average diameter (d_av) of solid particles in a binary solid mixture can be calculated using the following equation:

$$\frac{1}{{d}_{{\text{av}}}}=\frac{{x}_{{\text{s}}}}{{d}_{{\text{s}}}}+\frac{{x}_{{\text{b}}}}{{d}_{{\text{b}}}}.$$

(60)

A comparison was made with experimental data reported by Huilin et al. [16] to validate further the simulation results predicted by the developed model. This validation involved obtaining the profile of the average diameter of a binary solid mixture within the bed for a gas velocity of 2.2 m/s and an initial mass fraction of small particles (x_s0) of 0.5. The comparison aimed to validate the accuracy of the simulation model. The solid mixtures used for this validation consisted of glass bead particles with diameters of 2.30 and 4.26 mm, and with a 2600 kg/m³ density. These particles were tested in a columnar bed with a diameter of 10.0 cm and a height of 1.2 m. The gas velocity used for the validation was 2.2 m/s, and the initial mass fraction of small particles was x_s0 = 0.5. Figure 7 compares the simulation results and the corresponding experimental data. The root mean square error (RMSE) was calculated to quantify the discrepancy between the simulated and measured values. The obtained RMSE value of 9% is acceptable, indicating a reasonable agreement between the simulation and experimental results. This figure clearly illustrates that the average diameter of the solid mixture decreases with increasing bed height. This trend reflects the settling behavior of the particles, with coarser particles accumulating in the bottom section of the bed and finer particles accumulating at the top [47, 48]. It is essential to acknowledge that the MFM-Eulerian approach used in the modeling may introduce some uncertainties due to the inherent continuum hypothesis for the solid phase(s) and certain non-physical assumptions within the approach. Moreover, the large particle diameter and small bed dimensions used in the simulation runs could lead to insufficient particles per grid cell, potentially violating the continuum assumption for the solid phase(s) and contributing to discrepancies between the simulation results and experimental data.

Fig. 7

Comparison of simulation result with the experimental data reported by Huilin et al. [16] for the gas velocity of 2.2 m/s and the initial mass fraction of fine particles of x_s0 = 0.5

Solids Mixing Process in Tapered Fluidized Beds

Effect of Bed Geometry

To investigate the effects of fluidized bed geometry on solids mixing, we examined one columnar fluidized bed and three tapered fluidized beds with apex angles of 2.86°, 5.71°, and 8.531°. The inlet jet diameter was set to 5 mm (5% of the bottom diameter), and the inlet gas velocity was maintained at 1.0 m/s. Figure 8 presents the contours of solid density (i.e., ε_s1ρ_s1) for particles with a diameter of 510 µm at t = 30 s. After 30 s, it is evident that particles with a diameter of 510 µm accumulate at the bottom of the columnar bed and the tapered bed with an apex angle of 2.86°, demonstrating poor mixing. However, the tapered beds with apex angles of 5.71° and 8.53° exhibit partial mixing after 30 s. This finding suggests that these two tapered beds mix better solids than the columnar and the tapered bed with the smallest apex angle.

Fig. 8

Solid density in different fluidized beds (d_p = 510 µm, U_J = 1.0 m/s, and t = 30 s), a columnar, b apex angle = 2.86°, c apex angle = 5.71°, and d apex angle = 8.53°

Figure 9 illustrates that the mixing index slope increases with an increasing apex angle, from 2.86° to 8.53°, indicating a faster mixing rate. Moreover, the mixing index in the tapered fluidized beds reaches equilibrium within a 5-s interval. In contrast, the mixing index in the columnar fluidized bed increases slowly. These results suggest that the tapered fluidized beds exhibit improved mixing compared to the columnar fluidized bed. The larger space available in the tapered beds with larger apex angles facilitates the formation of larger gas bubbles, enhancing the solids mixing process. As a result, the tapered beds with larger apex angles have a higher potential for achieving better and more efficient solids mixing.

Fig. 9

Effect of fluidized bed geometry on the mixing index value (U_J = 1.0 m/s)

Effect of Inlet Jet Velocity

The inlet jet velocity directly impacts the size and rising velocity of gas bubbles in fluidized beds, influencing the circulation and mixing of solid particles. To investigate these effects, we examined a tapered bed with an apex angle of 8.531° and an inlet jet diameter of 5 mm at different inlet jet velocities (U_J = 0.5, 0.6, 0.7, and 0.8 m/s). Figure 10a presents the temporal variations of the mixing index for these velocities. The simulation results show that a low inlet jet velocity of U_J = 0.5 m/s leads to the formation of small gas bubbles, resulting in a slow rate of solids mixing and minimal variation in the mixing index over time. As the inlet jet velocity increases from 0.5 to 0.8 m/s, the rate of solids mixing increases, and the time required to reach equilibrium mixing decreases. This observation aligns with the three stages of solids mixing dynamics identified by Zhu et al. [14] in spouted beds: rapid, slow, and equilibrium. These stages are discernible in Fig. 10a, particularly for U_J = 0.8 m/s. Further increases in the inlet jet velocity beyond 1 m/s lead to a dominant rapid mixing stage, rendering the slow mixing stage indistinguishable. Consequently, the mixing process only involves rapid mixing during the 0–5 s interval and equilibrium mixing from 5 to 30 s. Notably, the equilibrium mixing index value remains constant for higher inlet jet velocities, independent of the specific velocity.

Fig. 10

Effect of inlet jet velocity on the mixing index value

Effect of Initial Arrangement of Two Solid Phases

To investigate the effects of the initial arrangement of solid phases (lateral and vertical) on the solids mixing, two tapered beds with an apex angle of 8.531°, U_J = 1.0 m/s, and an inlet jet diameter of 5 mm were examined. Figure 11 demonstrates the variations of solid density of solid phase 1 (i.e., ε_s1 ρ_s1) for particles with a diameter of 510 µm in the tapered fluidized bed with an apex angle of 8.531°, U_J = 1.0 m/s, and an initial lateral arrangement of solids.

Fig. 11

Solid density of solid phase 1 in the tapered fluidized beds with the apex angle of 8.531°, d_p = 510 µm, and U_J = 1.0 m/s for the initial lateral arrangement of solids

The results obtained for solids mixing with the initial lateral arrangement of solids are presented in Fig. 12. This figure demonstrates that the temporal variations of the average height for both solid phases (small and large particles) evaluated using Eq. 51 are nearly identical. However, the mixing index (Eq. 52) needs to be revised to accurately predict the average height of the two solid phases due to the initial lateral arrangement of solids.

Fig. 12

Comparison of the average height of solid phases and the mixing index value for the initial lateral arrangements of solid phases in the tapered fluidized bed with the apex angle of 8.531°, d_p = 510 µm, and U_J = 1.0 m/s

In the present work, to correctly predict the mixing process in the case of lateral arrangements of solid phases, Eq. 51 was appropriately modified. In this regard, an average lateral distance between the two solid phases was defined (i.e., Eq. 55), and the mixing index was evaluated using this average lateral distance. Equation 56 represents the normalized distance of the lateral layers for the two solid phases.

$$\Delta x=\frac{\sum_{{\text{k}}}^{{\text{N}}}{\varepsilon }_{{\text{s}},{\text{k}}}{x}_{{\text{k}}}{V}_{{\text{k}}}}{X\sum_{{\text{k}}}^{{\text{N}}}{\varepsilon }_{{\text{s}},{\text{k}}}{V}_{{\text{k}}}}.$$

(61)

The expression for PSN is given by Eq. 57, where x_k is the lateral distance of the kth cell center from the central line of the tapered bed, V_k is the volume of the kth cell, ε_s,k is the volume fraction of the sth solid phase, N is the number of cells, and X is the distance of the central line of the tapered bed from the walls. Thus, PSN can be written as follows:

$$PSN=\left(\left|\Delta {x}_{{\text{flotsam}}}\right|-\left|\Delta {x}_{{\text{jetsam}}}\right|\right).$$

(62)

The combined effects of the initial arrangement (lateral and vertical) and inlet jet velocity (U_J = 1.0 and 2.0 m/s) on solids mixing in the tapered fluidized bed with an apex angle of 8.531° and an inlet jet diameter of 5 mm were investigated using a combination of Eqs. 52 and 62. Figure 13a presents the solids mixing process for both lateral (obtained using Eq. 56) and vertical (obtained using Eq. 51) arrangements of the two solid phases. This figure demonstrates that solid particles in a tapered fluidized bed with the initial lateral arrangement exhibit a lower tendency to reach a completely random mixture state. Consequently, the mixing index reaches its equilibrium value slowly. Conversely, the mixing index value rapidly increases for the initial vertical arrangement and reaches the equilibrium value 0.9 within 10 s. This behavior indicates that solid particles in fluidized beds with the initial vertical arrangement tend to mix significantly more than those with the initial lateral arrangement. Furthermore, increasing the inlet jet velocity to U_J = 2.0 m/s shows that both the lateral and vertical arrangements have a negligible influence on the mixing index, as shown in Fig. 13b. It should be added that the obtained results demonstrate good agreement with those reported by Luo et al. [49].

Fig. 13

Effect of initial arrangement of solid phases on the mixing index. a U_J = 1.0 m/s and b U_J = 2.0 m/s

Effect of Inlet Jet Diameter

Solids mixing in fluidized beds with an inlet jet can be influenced by several factors, including the initial diameter of gas bubbles, the inlet jet diameter, and the frequency of gas bubble formation [50, 51]. Tapered fluidized beds with an apex angle of 8.531° and various inlet jet diameters were examined to investigate the effect of the inlet jet diameter on solids mixing. The chosen diameters were 2.5, 5.0, and 10 mm, corresponding to 2.5%, 5%, and 10% of the bottom diameter of the bed, respectively. Figure 14a and b presents the temporal variations of the mixing index for inlet jet velocities of 1.0 and 2.0 m/s, respectively. At U_J = 1.0 m/s, the trend of the mixing index shows that increasing the inlet jet diameter leads to a higher gas flow rate and a greater frequency of large gas bubble formation. This results in a faster increase in the mixing index, which reaches its equilibrium value around 5 s. Conversely, reducing the inlet jet diameter to 5% of the bottom diameter yields a slower increase in the mixing index, reaching equilibrium at around 10 s with a lower gradient. Furthermore, Fig. 14b demonstrates that at an increased inlet jet velocity of 2.0 m/s, the diameter of the inlet jet has a minimal impact on solids mixing. The time required to attain the equilibrium mixing value becomes nearly identical for all tested inlet jet diameters.

Fig. 14

Effect of inlet jet diameter on the mixing index. a U_J = 1.0 m/s and b U_J = 2.0 m/s

Limitations and Guide for Future Works

This study made assumptions that introduced limitations and potential inaccuracies in the obtained results. One fundamental limitation was the assumption of a laminar gas flow field, which may not accurately represent fluidized beds’ actual turbulent flow behavior. In addition, the 2D bed geometry and the absence of a real gas distributor at the inlet simplified the model and could influence the mixing dynamics. To enhance the robustness of future studies in this area, we recommend addressing these limitations through the following measures:

• 3D bed geometry: Conducting simulation studies with 3D bed geometries will provide a more accurate representation of the actual fluidized bed and improve the predictions of mixing behavior.

• Gas distributor at the inlet: Introducing a gas distributor at the inlet can significantly impact the flow field and mixing patterns. Modeling this component will provide a more realistic representation of the gas injection process.

• Other computational fluid dynamics models: Exploring different CFD modeling approach such as CFD-DEM may be necessary to evaluate the solids mixing more accurately.

• Other conditions: Investigating the influence of additional parameters, such as bed vibration, gas type, operating temperature, drag coefficient, initial solid arrangement, and a more comprehensive range of solid particle sizes, is recommended for further research. These factors can affect the mixing index and provide a more comprehensive understanding of the solids mixing process in fluidized beds.

By addressing these limitations and incorporating additional complexities, future research can improve the accuracy of the model predictions and contribute to a deeper understanding of solids mixing in tapered fluidized beds.

Conclusion

This study investigated the intensification of solids mixing in binary mixtures within two-dimensional tapered fluidized beds with an inlet jet using the multi-fluid modeling (MFM) approach. The simulation results demonstrated good agreement with our own experimental data and those reported in the literature, validating the model’s accuracy. The study employed glass bead particles with 240 and 510 µm diameters and a 2500 kg/m³ density for the binary solid mixtures. Various factors influencing solids mixing intensification were examined, including bed geometries (columnar and tapered beds with apex angles of 2.862°, 5.711°, and 8.531°), inlet jet velocity, initial solid arrangement (vertical and lateral), and inlet jet diameter. The particle segregation number (PSN) was the primary metric for evaluating mixing performance.

The findings revealed that tapered fluidized beds with larger apex angles (5.711° and 8.531°) achieved significantly higher equilibrium mixing values than the columnar bed and the tapered bed with the smallest apex angle (2.862°). At a low inlet jet velocity (U_J = 0.5 m/s), solids mixing was insufficient, with an equilibrium mixing index value of approximately 0.1. Increasing the inlet jet velocity increased the mixing index rapidly, reaching the equilibrium value quickly. The observed solids mixing mechanism for inlet jet velocities of 0.7 and 0.8 m/s involved three distinct stages: rapid, slow, and equilibrium. Notably, only the rapid mixing and equilibrium stages were observed at higher inlet jet velocities.

Regarding the initial solid arrangement, the standard mixing index proved inadequate for predicting the rate of solids mixing for the lateral arrangement. Consequently, a modified mixing index was implemented. This revealed that the vertical arrangement facilitated faster and more effective solids mixing than the lateral arrangement, particularly at lower inlet jet velocities. However, at U_J = 2.0 m/s, both arrangements exhibited similar mixing behavior.

Finally, the study on the effect of the inlet jet diameter demonstrated that increasing the diameter for smaller inlet jet velocities resulted in the rapid achievement of equilibrium solids mixing. Conversely, the diameter had no significant influence on the mixing process at larger inlet jet velocities.