# Continuum Modeling of Slightly Wet Fluidization with Electrical Capacitance Tomograph Validation

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## Abstract

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## 1. Introduction

## 2. Motivation and Objectives of the Study

## 3. Theory

#### 3.1. Solid Stress Tensor

#### 3.2. Inter-Particle Gap and Liquid Bridge Rupture Distance

^{−2}), after which the variation becomes negligible.

#### 3.3. Energy Dissipation

## 4. Experiments

_{mf}= 0.17 m/s). The pressure drop across the distributor in the fluidization columns was designed to give around 40% of the dry bed pressure drop, which falls within the “rule of thumb” for distributor design [33]. In each experiment, the liquid loading was incrementally varied up to a maximum of δ = 0.15 × 10

^{−2}(mass of liquid to mass of dry bed). Additional details of the experiment operating conditions are given in Table 2.

## 5. Computational Model

#### 5.1. Meshing of the Simulation Domain and Solution Procedure

_{max}is specified by the CFL condition to fall within the range of ~1–5 [35], $\u2206\mathrm{t}$ is the solution time step and $\u2206\mathrm{x}$, $\u2206\mathrm{y}$ and $\u2206\mathrm{z}$ are the grid sizes in the x, y and z coordinates, respectively. ${\mathrm{u}}_{\mathrm{s},\mathrm{x}}$, ${\mathrm{u}}_{\mathrm{s},\mathrm{y}}$ and ${\mathrm{u}}_{\mathrm{s},\mathrm{z}}$ are the particle velocity in the x, y and z coordinates, respectively. In this study, two different time steps of $\u2206\mathrm{t}$ = 0.1 × 10

^{−3}and 0.3 × 10

^{−4}s were used for dry and wet conditions, respectively. The wet case was found to require a smaller time step to avoid the stiffness and instability in the solution of the solid momentum equation. The convection terms in the continuity, momentum and granular temperature equations were solved using first-order discretization schemes with a maximum number of iterations of 20 per time step and a convergence criterion of 10

^{−3}for each scaled residual component. For coupling the pressure–velocity fields, a modified SIMPLE algorithm was used [36]. The computer used in the simulation was HP Workstation (3.20 Ghz 4 Core processor with 24 GB RAM).

#### 5.2. Boundary Conditions and Simulation Parameter

## 6. Results and Discussion

#### 6.1. Boundary Conditions and Simulation Parameter

- At a small liquid presence (δ < 0.08 × 10
^{−2}) there is a decrease in the bubble size, as a result of bubbles splitting, and intensified wall-bubble movements. This may also change to gas channeling with most of the gas rising through a channel adjacent to the wall. In some way, this resembles the fluidization behavior of Geldart A dry particles [2]. - At an intermediate liquid presence (0.08 × 10
^{−2}< δ < 0.12 × 10^{−2}) slugging is observed (jointly rising bubbles occupying more than half of the column cross-section), typically causing the bed to rise and collapse as a piston. This is of great similarity to the fluidization behavior of highly cohesive powder such as Geldart C dry particles [2]. - At a considerably large liquid presence (δ > 0.12 × 10
^{−2}) the bed becomes de-fluidized with the gas mainly escaping through one or more channels.

#### 6.2. Comparison of the Experiment Results with the Model Predictions

^{−2}and δ = 0.055 × 10

^{−2}is lower than that of the dry case and appears to show a peak at the point of incipient fluidization velocity of 0.2 m/s. In this range of wetting conditions, the pressure drop peak is believed to be associated with the breakdown of interparticle cohesion force and particle rearrangement before fluidization. The peak is then followed by reduced pressure due to the channeling phenomena. At the high range of liquid presence of δ > 0.1 × 10

^{−2}, the pressure drop is consistently higher than that of the dry case, mainly due to the commencement of slugging and de-fluidization.

^{−2}, the experimental data shows the value of $FI$ to increase above unity due to severe slugging (piston-like behavior) and increased wall shearing and friction, eventually leading to de-fluidization. This is to a great extent similar to the behavior of a highly cohesive powder, where the $FI$ value is usually greater than 1.4 [38]. The model fails to provide a solution at a high liquid loading (δ > 0.1 × 10

^{−2}) because at this condition the bed is virtually static (de-fluidized) and hence, the solution diverges.

#### 6.3. Solid Shear Stress, Energy Dissipation, and Granular Temperature

^{−2}. Due to the anisotropic nature of the stresses, the reported values are for the root sum square given by $\sqrt{{\tau}_{xy}^{2}+{\tau}_{xz}^{2}+{\tau}_{yz}^{2}}$). The values of the kinetic, collisional, and frictional contribution to the overall shear stress are numerically close to the range reported in the literature [41]. The liquid-induced shear stress within the intermediate flow regime dominates the overall shear stress and approaches the frictional stress at close to maximum packing. It should be noted that the scattering of the various solid stress terms appearing in this figure, i.e., different stress values at the same volume fraction, is owing to the variations of the shear rate and granular temperature at the same solid concentration.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Notations | |

$C$ | Courant number (-) |

${C}_{D}$ | drag coefficient (-) |

${d}_{s}$ | particle diameter of solid phase (m) |

${e}_{dry}$ | dry particle-particle restitution coefficient (-) |

${e}_{ss}$ | particle-particle restitution coefficient (-) |

${e}_{{s}_{i},w}$ | particle-wall restitution coefficient (-) |

${e}_{wet}$ | wet particle-particle restitution coefficient (-) |

${\dot{F}}_{liquid}$ | dynamic liquid bridge force (kg m s^{−2}) |

$g$ | gravity (m s^{−2}) |

${g}_{0,ss}$ | Radial distribution function (-) |

$h$ | inter-particle gap (m) |

${h}_{a}$ | particle surface asperity (m) |

${h}_{critical}$ | critical separation distance (m) |

$\overline{\overline{I}}$ | Unit vector (-) |

${I}_{2D}$ | second invariant of the deviatoric stress tensor (s^{−2}) |

${M}_{l},{M}_{p}$ | liquid and solid mass (kg) |

${N}_{p}$ | number of particles per unit area (m^{−2}) |

$P$ | pressure (pa) |

$\overline{\overline{S}}$ | strain rate (s^{−1}) |

$St$ | dimensionless Stokes number (-) |

${St}^{*}$ | critical dimensionless Stokes number (-) |

${Re}_{s}$ | Reynolds number of solid phase (-) |

$t$ | Time (s) |

$\dot{u}$ | particle approach/collission velocity (m s^{−1}) |

${\stackrel{\u20d1}{u}}_{g},{\stackrel{\u20d1}{u}}_{s}$ | gas and solid velocity vector (m s^{−1}) |

${u}_{s,w}$ | particle velocity at wall (m s^{−1}) |

${v}_{r,s}$ | terminal velocity correlation (-) |

Greek symbols | |

${\alpha}_{g},{\alpha}_{s}$ | volume fraction of gas and solid phase $i$, respectively (-) |

${\alpha}_{s-min}$ | minimum solid volume fraction to trigger friction (=0.52) (-) |

${\alpha}_{s-max}$ | maximum solid volume fraction at packing (=0.63) (-) |

$\beta $ | momentum exchange coefficient (kg m^{−3} s^{−1}) |

${\mathsf{\gamma}}_{{\Theta}_{s}}$ | collisional energy dissipation (kg m^{−1} s^{−3}) |

$\delta $ | liquid to dry solid mass ratio (-) |

${\Theta}_{s}$ | granular temperature of solid phase $i$ (m^{2} s^{−2}) |

${\kappa}_{{\Theta}_{s}}$ | diffusion coefficient of granular energy (kg m^{−1} s^{−1}) |

${\mu}_{L}$ | liquid dynamic viscosity (kg m^{−1} s^{−1}) |

${\mu}_{s,col}$ | viscosity of solid phase due to collision (kg m^{−1} s^{−1}) |

${\mu}_{s,kin}$ | viscosity of solid phase due to kinetics (kg m^{−1} s^{−1}) |

${\mu}_{s,fr}$ | viscosity of solid phase due to friction (kg m^{−1} s^{−1}) |

${\mu}_{wet}$ | viscosity of solid phase due to liquid bridge (kg m^{−1} s^{−1}) |

${\rho}_{s},{\rho}_{g},{\rho}_{L}$ | Solid, gas, and liquid densities, respectively (kg m^{−3}) |

$\overline{\overline{\tau}}$ | shear stress tensor (kg m^{−1} s^{−2}) |

$\varphi $ | angle of Internal friction (Degree) |

${\varphi}^{\prime}$ | internal angle of lubrication (Degree) |

${\varphi}_{gs}$ | energy exchange between gas-solid phase (kg m^{−1} s^{−1}) |

$\gamma $ | liquid bridge angle (rad) |

$\phi $ | specularity coefficient (-) |

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**Figure 1.**Wet particle interaction: (

**a**) microscope image of wet particles (glass beads wetted with silicone oil) at static conditions demonstrating the formation of liquid bridges at the contact point; (

**b**) schematics of the proposed flow regime and the corresponding particle-particle contacts.

**Figure 2.**(

**a**) Inter-particle distance and (

**b**) the critical separation distance for a particle diameter of 350 µm.

**Figure 4.**Electric capacitance tomography (ECT) images of 7.6 cm above the fluidized bed gas distributor. Each column is produced by stacking a series of images recorded at the rate of 100 frames per second and representing the spatial average solid concentration over 3.8 cm height. The data was produced in a 15 cm diameter column with the bed material consisting of 3.5 kg glass beads fluidized at the gas velocity of 0.8 m/s.

**Figure 5.**Fast Fourier transform (FFT) analysis of solid fraction fluctuation obtained by ECT measurement in a bubbling fluidized bed (

**a**) dry bed (

**b**) wet bed at δ = 0.055 × 10

^{−2}. The data were produced at 0.8 m/s gas velocity.

**Figure 6.**Comparison of experimental and predicted fluidized bed material distribution at various wetting conditions. The experiments were carried out in a small fluidization column of 5 cm diameter at a fluidization velocity of 0.8 m/s.

**Figure 7.**Experimental fluidized bed pressure drop at various gas velocities and wetting conditions. The data was produced in a 15 cm diameter column with the bed material consisting of 3.5 kg glass beads. Each data point represents the average of three measurements (maximum deviation of $\mp $10%).

**Figure 8.**Comparison of the predicted and experimentally determined fluidization index (FI). The experimental data was produced in a 15 cm diameter column with the bed material consisting of 3.5 kg glass beads fluidized at the gas velocity of 0.8 m/s. Each data point represents the average of three measurements (maximum deviation of $\mp $10%).

**Figure 9.**Predicted solid shear stress in a slightly wet fluidized bed of 15 cm diameter at the gas velocity of 0.8 m/s and liquid presence of δ = 0.1 × 10

^{−2}wt% liquid.

**Figure 10.**Predicted (

**a**) energy dissipation rate and (

**b**) granular temperature as a function of the solid concentration in a dry and a slightly wet fluidized bed of 15 cm diameter at the gas velocity of 0.8 m/s.

Process | Problems |
---|---|

Coal/biomass gasification | Oil/tar leading to agglomeration and severe degradation in the low-temperature regions of fluidized bed reactors |

Catalytic cracking | Surface catalyst melting at high temperatures leads to dead zones and de-fluidization. |

Pneumatic conveying | Moisture leading to solid slugging, high-pressure drop, wear, and line blockage |

Fluidized bed coating/drying | Liquid presence leads to undesired agglomeration and particle segregation |

Particle | Glass beads, Density = 2500 [kg/m ^{3}] Diameter = 350 [µm] |

Fluidization velocity | Up to 0.8 [m/s] |

Fluidization medium | Air at ambient condition |

liquid used in wet condition | Silicon oil (Fluid 500, Dow Corning Ltd., UK) dynamic viscosity = 0.4945 [kg/m.s] surface tension = 0.0165 [N/m] density = 969 [kg/m ^{3}] |

Statics bed height | 13 [cm] (large column) 5 [cm] (small column) |

Liquid content in wet condition | $\delta $ = 0.027 − 0.138 × 10^{−2} [-] |

Solids pressure: ${P}_{s}={P}_{s,kin}+{P}_{s,col}+{P}_{s,fr}$ where |

${P}_{s,kin}={\alpha}_{s}{\rho}_{s}{\Theta}_{s}$ |

${P}_{s,col}=2{\rho}_{s}\left(1+{e}_{s,dry}\right){\alpha}_{s}^{2}{g}_{0,ss}{\Theta}_{s}$ |

${P}_{s,fr}=0.05\frac{{\left({\alpha}_{s}-{\alpha}_{s,min}\right)}^{2}}{{\left({\alpha}_{s,max}-{\alpha}_{s}\right)}^{3}}$ |

Radial distribution function: |

${g}_{0,ss}={\left[1-{\left(\frac{{\alpha}_{s}}{{\alpha}_{s,max}}\right)}^{1/3}\right]}^{-1}$ |

Gas-solid drag coefficient: |

${\beta}_{gs}=\frac{3{\rho}_{g}{\alpha}_{s}{\alpha}_{g}}{4{u}_{r,s}^{2}{d}_{s}}{C}_{D}\left(\frac{{Re}_{s}}{{v}_{r,s}}\right)\left|{\stackrel{\u20d1}{u}}_{g}{-\stackrel{\u20d1}{u}}_{s}\right|$ |

where ${v}_{r,s}=0.5\left(A-0.06{Re}_{s}+\sqrt{{\left(0.06{Re}_{s}\right)}^{2}+0.12{Re}_{s}\left(2B-A\right)+{A}^{2}}\right)$ |

$A={\alpha}_{g}^{4.14},\left\{\begin{array}{c}B=0.8{\alpha}_{g}^{1.28}\left({\alpha}_{g}\le 0.85\right)\\ B={\alpha}_{g}^{2.65}\left({\alpha}_{g}0.85\right)\end{array}\right.$ |

${C}_{D}={\left(0.63+\frac{4.8}{\sqrt{{Re}_{s}/{v}_{r,s}}}\right)}^{2}$ |

${Re}_{s}=\frac{{d}_{s}{\rho}_{g}\left|{\stackrel{\u20d1}{u}}_{g}{-\stackrel{\u20d1}{u}}_{s}\right|}{{\mu}_{g}}$ |

Diffusion coefficient of granular energy: ${\mathsf{\kappa}}_{{\Theta}_{s}}=\frac{150{\rho}_{s}{d}_{s}{\left({\mathsf{\pi}\Theta}_{s}\right)}^{\frac{1}{2}}}{384\left({e}_{ss}+1\right){g}_{0,ss}}{\left[1+\frac{6}{5}{\alpha}_{s}{g}_{0,ss}\left({e}_{s,dry}+1\right)\right]}^{2}+2{{\alpha}_{s}}^{2}{\rho}_{s}{d}_{s}{g}_{0,ss}\left({e}_{s,dry}+1\right){\left(\frac{{\Theta}_{s}}{\mathsf{\pi}}\right)}^{\frac{1}{2}}$ |

Kinetic viscosity: ${\mu}_{s,kin}=\frac{{\alpha}_{s}{\rho}_{s}{d}_{s}\sqrt{{\Theta}_{s}\mathsf{\pi}}}{6\left(3-{e}_{ss}\right)}\left[1+\frac{2}{5}\left({e}_{s,dry}+1\right)\left(3{e}_{s,dry}-1\right){\alpha}_{s}{g}_{0,ss}\right]$ |

Collisional viscosity: |

${\mu}_{s,col}=\frac{4}{5}{\alpha}_{s}{\rho}_{s}{d}_{s}{g}_{0,ss}\left({e}_{s,dry}+1\right){\left(\frac{{\Theta}_{s}}{\mathsf{\pi}}\right)}^{1/2}$ |

Frictional viscosity: ${\mu}_{s,fr}={P}_{s,fr}\mathrm{s}\mathrm{i}\mathrm{n}{\varphi}^{\prime}$ |

Strain rate ($i$ = gas or solid): $\stackrel{\u033f}{S}=\left(\begin{array}{ccc}\frac{\partial {U}_{i,x}}{\partial x}& \frac{1}{2}\left(\frac{\partial {U}_{i,y}}{\partial x}+\frac{\partial {U}_{i,x}}{\partial y}\right)& \frac{1}{2}\left(\frac{\partial {U}_{i,x}}{\partial z}+\frac{\partial {U}_{i,z}}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial {U}_{i,y}}{\partial x}+\frac{\partial {U}_{i,x}}{\partial y}\right)& \frac{\partial {U}_{i,y}}{\partial y}& \frac{1}{2}\left(\frac{\partial {U}_{i,z}}{\partial y}+\frac{\partial {U}_{i,y}}{\partial z}\right)\\ \frac{1}{2}\left(\frac{\partial {U}_{i,x}}{\partial z}+\frac{\partial {U}_{i,z}}{\partial x}\right)& \frac{1}{2}\left(\frac{\partial {U}_{i,z}}{\partial y}+\frac{\partial {U}_{i,y}}{\partial z}\right)& \frac{\partial {U}_{i,z}}{\partial z}\end{array}\right)$ |

Fluidization velocity, U | 0.8 [m/s] |

Gas outlet pressure, ${P}_{out}$ | 0 [Pa_{g}] |

Wall-particle restitution coefficient, ${e}_{s,w}$ | 0.8 [-] |

Dry particle-particle restitution coefficient, ${e}_{s,dry}$ | 0.9 [-] |

Maximum allowable solid concentration, ${\alpha}_{s}$ | 0.61 [-] |

Critical frictional solid concentration | 0.58 [-] |

Internal angle of friction, ${\varphi}^{\prime}$ | 30 [deg] |

liquid contact angle, $\gamma $ | 0.175 [rad] |

Specularity coefficient, $\phi $ | 0.5 [-] |

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**MDPI and ACS Style**

Makkawi, Y.; Yu, X.; Ocone, R.; Generalis, S.
Continuum Modeling of Slightly Wet Fluidization with Electrical Capacitance Tomograph Validation. *Energies* **2024**, *17*, 2656.
https://doi.org/10.3390/en17112656

**AMA Style**

Makkawi Y, Yu X, Ocone R, Generalis S.
Continuum Modeling of Slightly Wet Fluidization with Electrical Capacitance Tomograph Validation. *Energies*. 2024; 17(11):2656.
https://doi.org/10.3390/en17112656

**Chicago/Turabian Style**

Makkawi, Yassir, Xi Yu, Raffaella Ocone, and Sotos Generalis.
2024. "Continuum Modeling of Slightly Wet Fluidization with Electrical Capacitance Tomograph Validation" *Energies* 17, no. 11: 2656.
https://doi.org/10.3390/en17112656