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^{1}

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The difference between homogeneous and bubbling fluidization behaviors has been studied for the past 70 years, where several researchers have reported on the influence of interparticle forces in fluidization. Although interparticle forces such as van der Waals forces are evident in a real system, these forces are not the determinant in homogeneous fluidization, which can be simulated without any interparticle forces. In our previous study, the difference in fundamental mechanisms of the two fluidization states was analytically determined with a dimensionless gravity term, comprising the Reynolds number, Archimedes number, and density ratio. Nevertheless, some researchers insist that interparticle forces are dominant and a hydrodynamic force is not dominant. In this study, a dimensional analysis was applied to obtain a dominant parameter for distinguishing two fluidizations. Furthermore, some parameters were examined by comparing the experimental data in previous studies. The results indicated that hydrodynamic force is the dominant factor and the dimensionless gravity term is the dominant parameter in differentiating the two fluidized states.

Fluidization can be observed in several natural phenomena such as avalanches, sandstorms, pyroclastic flows, and ground liquefaction. This characteristic is utilized in industrial processes as fluidized beds, where a balance between the drag and gravitational forces acting on the particle bed fluidizes the particles when the fluid velocity from the bottom wall exceeds a certain limit. The velocity at this instant is termed the minimum fluidization velocity, u_{mf}.

There are two kinds of fluidization: homogenous or particulate fluidization, where the particle bed expands homogeneously and bubbling or aggregative fluidization, where voids (bubbles) occur in the particle bed as the injected fluid velocity increases. Geldart [_{mf}, and bubbles were observed at a certain fluid velocity, which is called the minimum bubbling fluidization velocity, u_{mb}. However, homogeneous fluidization did not occur in the powder beds of Groups B and D. Moreover, bubbling fluidization is generally not observed in liquid systems such as glass bead-water systems.

As summarized in

The second approach derives an equation to estimate the minimum bubbling fluidization velocity u_{mb}. Primarily, Geldart [

u m b = 100 d p . (1)

Subsequently, Abrahamsen and Geldart [

u m b = 2.07 d p ρ f 0.06 μ f 0.347 exp ( 0.716 F 45 ) , (2)

where F_{45} is the mass fraction of particles with diameters less than 45 μm. The equation derived by Abrahamsen and Geldart [

Criterion | Homogeneous | Bubbling | Reference |
---|---|---|---|

For estimation of fluidization state | |||

F r m f = u m f 2 g d p | Fr_{mf} < 0.13 | Fr_{mf} > 1.3 | Wilhelm and Kwauk [ |

R = F r m f R e m f ( ρ p − ρ f ρ f ) ( L m f D c ) | R < 100 | R > 100 | Romero and Johanson [ |

N t r = R e F r 0.5 ( ρ p − ρ f ρ f ) | N_{tr} < C_{1 } depending on ε and ε_{mf} | N_{tr} > C_{1} | Verloop and Heertjes [ |

N f = A r m ( ρ a ρ f ) 0.5 ρ_{a}: average density of particulate phase | N_{f} < C_{2 } m and C_{2} depend on range of operation | N_{f} > C_{2} | Doichev et al. [ |

U e = 1 F r t 0.5 ( ρ p − ρ f ρ p ) 0.5 U ε = 0.56 n ( 1 − ε b ) 0.5 ε b n − 1 F r t = u t g d p | U_{e} < U_{e} | U_{e} > U_{e} | Foscolo and Gibilaro [ |

D n = ( A r R e m f ) ( ρ p − ρ f ρ f ) | D_{n} < 10^{4} | D_{n} > 10^{6} | Liu et al. [^{ } |

F G * = A r R e 2 ρ * | F G * > 11.3 | F G * < 11.3 | Kogane et al. [ |

For estimation of minimum bubbling fluidization velocity | |||

u m b = 100 d p | u_{0} < u_{mb} | u_{0} > u_{mb} | Geldart [ |

u m b = 2.07 d p ρ f 0.06 μ f 0.347 exp ( 0.716 F 45 ) F_{45}: mass fraction of particles having a diameter less than 45 μm | u_{0} < u_{mb} | u_{0} > u_{mb} | Abrahamsen and Geldart [ |

R e m b = 0.263 ρ ∗ − 0.553 A r 0.612 | Re < Re_{mb} | Re > Re_{mb} | Kuwagi et al. [ |

R e m b = 0.297 ρ ∗ − 0.5 A r 0.5 | Re < Re_{mb} | Re > Re_{mb} | Kogane et al. [ |

on a three-dimensional (3-D) graph consisting of three dimensionless numbers—Re, Ar, and ρ*. Thus, the following equation was derived from the boundary plane between homogeneous and bubbling fluidization states:

R e m b = 0.263 ρ ∗ − 0.553 A r 0.612 . (3)

The derivation method is similar to that of Abrahamsen and Geldart [

R e m b = 0.297 ρ ∗ − 1 / 2 A r 1 / 2 . (4)

Certain explanations have been proposed to adequately clarify the basic mechanisms that differentiate homogeneous and bubbling fluidization. Researchers [

In this study, we used larger particles, i.e., Groups A, B, and D to determine the differences between the two fluidization states. The interparticle force includes a repulsion force resulting from the collision of particles, and the repulsion force can be calculated in a numerical simulation using softer springs to reduce computational time [_{n} = 7 or 3.5 N/m for calculating the repulsion force, whereas k_{n} = 800 N/m in the current study. In addition, Thornton et al. [

Furthermore, as mentioned before, dimensional analysis has not been effectively utilized in previous studies for derivation of the discriminant equation to determine the criterion for transition between homogeneous and bubbling fluidizations. When the Froude number or Reynolds number is defined using the minimum fluidization velocity as a reference velocity, the dimensionless number is treated as a physical property, such as the Prandtl number. On the other hand, the velocity of the injected fluid, i.e., the fluidizing velocity, can be used as a reference velocity for the dimensionless numbers. Valverde et al. [

The analogy between the dimensional and dimensionless simulations has already been presented [

The governing equations for the DEM-CFD coupling model [

Fluid phase:

∂ ε ∂ t + ∂ ∂ x i ( ε u i ) = 0 , (5)

ρ f ∂ ∂ t ( ε u i ) + ρ f ∂ ∂ x i ( ε u i u j ) = − ε ∂ p ∂ x i − ε f p f − ρ f ε g . (6)

Particle phase:

m p d v d t = − V p ∇ p + ( F p p + F p f ) − m p g , (7)

I d ω d t = | F t | r p . (8)

There are two definitions of pressure: pressure perturbation, p', and the summation of the pressure in the hydrostatic equilibrium and the pressure perturbation, p. Equations (6) and (7) are expressed with terms in relation to p.

However, Equations (6) and (7) can be expressed with terms in relation to p' as

ρ f ∂ ∂ t ( ε u i ) + ρ f ∂ ∂ x i ( ε u i u j ) = − ε ∂ p ′ ∂ x i − ε f p f , (9)

m p d v d t = − V p ∇ p ′ + ( F p p + F p f ) − ( m p − m f ) g . (10)

Thereafter, the following equations were derived by nondimensionalizing Equations (5)-(8).

Fluid phase:

∂ ε ∂ T + ∂ ∂ X i ( ε U i ) = 0 , (11)

∂ ∂ T ( ε U i ) + ∂ ∂ X i ( ε U i U j ) = − ε ∂ P ∂ X i − ε f p f * − ε F r 2 . (12)

Particle phase:

D V d T = − 1 ρ * ∇ P + ( F p p * + F p f * ) − G a R e 2 ⋅ g | g | , (13)

d ω * d T = | F t * | R (14)

Furthermore, Equations (9) and (10) can be rewritten as

∂ ∂ T ( ε U i ) + ∂ ∂ X i ( ε U i U j ) = − ε ∂ P ′ ∂ X i − ε f p f * , (15)

D V d T = − 1 ρ * ∇ P ′ + ( F p p * + F p f * ) − 1 ρ * ⋅ A r R e 2 ⋅ g | g | . (16)

In addition, the following reference qualities were used in the nondimensionalization process:

x 0 = d p , v 0 = u 0 , p 0 = ρ f u 0 2 , t 0 = d p u 0 , f 0 = ρ f u 0 2 d p , F 0 = π 6 ρ p d p 2 u 0 2 , ω 0 = 10 u 0 d p (17)

Further details on the related nondimensionalization procedure can be found in our previous study [

The fluidization behavior of homogeneous or bubbling fluidization was determined under each condition by performing a DEM-CFD simulation [_{p} in width, 700d_{p} in height, and d_{p} in thickness. This indicates that the present simulations are two-dimensional. Since the simulation mesh size was set to 5d_{p}, the grid numbers are 40 × 140 for the horizontal direction and vertical direction, respectively. The fluid was uniformly injected from the bottom wall at various velocities u_{0}.

The simulation conditions are listed in

Equations (3) and (4) were compared with Equations (1) and (2) to be rewritten with dimensional numbers (physical properties) as follows:

Particle diameter: d_{p} [μm] | 50, 60, 100, 200, 500, 1000 |
---|---|

Particle density: ρ_{p} [kg/m^{3}] | 265, 1000 (FCC), 2650, 5300 (glass), 11,340 (lead) |

Particle number | 45,000 |

Fluid | Density: ρ_{f} [kg/m^{3}] | Viscosity: μ_{f} [Pa∙s] |
---|---|---|

Air | 1.20 | 1.82 × 10^{−5} |

Air (1 MPa) | 12.0 | 1.82 × 10^{−5} |

Air (10 MPa) | 120 | 1.82 × 10^{−5} |

Air (500˚C) | 4.56 × 10^{−1} | 3.55 × 10^{−5} |

Water | 9.98 × 10^{2} | 1.00 × 10^{−}^{3} |

u m b = 1.06 d p 0.836 ρ f 0.165 μ f 0.224 × ( ρ p − ρ f ) 0.612 ρ p 0.553 (18)

u m b = 0.930 d p 0.5 ( 1 − ρ f ρ p ) 0.5 . (19)

The indices of d_{p}, ρ_{f}, and μ_{f} are listed in _{p} are 1 and 0.836, those of ρ_{f} are 0.06 and 0.165, and those of μ_{f} are 0.347 and 0.224 in Equations (2) and (18), respectively. Notably, Equation (2) is based on experimental data and Equation (18) is based on simulated data, but the derivation method is similar for fitting the data. When assuming all the fluid property values as constants, Equation (18) becomes a function of d p 0.836 , which is close to the index of 1.0 in Equation (1). Moreover, the absence of the fluid viscosity μ_{f} term in Equation (19) indicates that the fluid viscosity is not an essential parameter. Furthermore, the fluid density ρ_{f} was expressed only in the buoyancy term: (1 − ρ_{f}/ρ_{p}). Therefore, the form of Equation (19) is similar to Equation (1).

Certain equations from _{mf} based on u_{mf} [_{mf}= 1, bubbling fluidization occurs at Fr_{mf}> 0.1, as shown in

Fluid | d_{p} | ρ_{f} | μ_{f} |
---|---|---|---|

Geldart (1973) (Equation (1)) | 1 | - | - |

Abrahamsen and Geldart (1980) (Equation (2)) | 1 | 0.06 | 0.347 |

Kuwagi et al. (2014) (Equation (3)) | 0.836 | 0.165 | 0.224 |

Kogane et al. (2019) (Equation (4)) | 0.5 | - | - |

Fr_{mf} < 0.1. The experimental data obtained from the literature [^{−1} to 10^{−3} [_{mf} based on u_{mf}. Therefore, the conclusion that the Froude number criterion fails to estimate the difference [

The values of U_{e} − U_{ε} calculated using the criterion from Foscolo and Gibilaro [

Furthermore, _{n} proposed by Liu et al. [_{n} < 10^{4}, transitional region ranges from 10^{4} to 10^{6}, and bubbling fluidization occurs at D_{n} > 10^{6}. However, the boundary between the bubbling fluidization and transitional states evaluated from our simulation was at 10^{7}, as shown in ^{7}.

The values of the dimensionless gravity term F G * proposed by the present authors [_{mf} and D_{n} are based on u_{mf}, and therefore cannot be used as the discriminant in varying the fluidizing velocity or superficial velocity u_{0}. On the contrary, the dimensionless gravity term is a dimensionless parameter based on u_{0}; thus, it can be applied to alter the fluidization pattern by varying u_{0}. The homogeneous and bubbling fluidizations were well separated in the obtained results, and the boundary value was consistent with the experimental values of u_{mb} referred from the literature [

In this study, we considered a Froude number based on u_{0} instead of u_{mf} [

the homogeneous and bubbling fluidizations were well separated by a boundary value of 0.1. Moreover, this result is similar to that presented in

F r = R e 2 G a (20)

F G * = A r R e 2 ρ * . (21)

These equations correspond to the expressions of gravity in Equations (13) and (16). Therefore, both Fr and F G * were derived from the dimensionless governing equations discussed earlier, and the distinction is highlighted from the treatment of pressure, i.e., p or p', as shown in Equations (7) and (10). Notably, Fr comprises two dimensionless numbers, whereas F G * comprises three dimensionless numbers. The dimensional analysis requires three dimensionless parameters [

The estimation equation for u_{mb} was quantitatively verified by comparing the experimental values with that referred from the literature [

As shown in

The deviation from the experimental data is summarized in _{mb} increased; a large u_{mb} indicates a high Reynolds number. Moreover, the reason for the deviation is the same as that discussed in

Estimation equation | Deviation from exp. data [%] |
---|---|

Abrahamsen and Geldart [ | 27 |

Foscolo and Gibilaro [ | 39 |

Kuwagi et al. [ | 18 |

Kogane et al. [ | 33 |

In this study, the distinct fundamental mechanisms of the homogeneous and bubbling fluidization behaviors were determined using a dimensionless gravity term that was derived to distinguish the two fluidization states as per our previous report [

The estimation equations for dimensional analysis were derived from a 3-D flow regime map and the dimensionless gravity term was compared with the criteria reported in previous studies. A comparative analysis on the indices of physical properties confirmed that our derived equations were almost equivalent to the equations established in prior research. Thus, the dimensionless gravity term is signified as a dominant parameter for distinguishing between homogeneous and bubbling fluidization. In addition, the dimensionless gravity term was expressed in the dimensionless governing equations. In comparison to interparticle forces, the results of the current research indicated that hydrodynamic force is the most important and dominant factor in differentiating the two fluidized states. However, interparticle forces, such as cohesive and repulsion forces and the effect of particle inertia can be considered as additional factors to demonstrate more accurate simulations. Notably, the distinction between the two fluidization states can be simulated and observed without considering these additional factors as well.

Furthermore, the estimation equations were compared with existing discriminant equations. The minimum bubbling fluidization velocities estimated in the current study were consistent with the experimental data, and the estimation equations accurately distinguished the two fluidization behaviors.

The authors declare no conflicts of interest regarding the publication of this paper.

Kuwagi, K., Kogane, A., Sasaki, Y. and Hirano, H. (2021) Validation of Dimensionless Parameters for Distinguishing between Homogeneous and Bubbling Fluidizations. Open Journal of Fluid Dynamics, 11, 81-97. https://doi.org/10.4236/ojfd.2021.112006

Ar Archimedes number, –

d_{p} diameter of particle, m

f_{pf} particle-fluid interaction force per unit volume, N/m^{3}

f p f * dimensionless particle-fluid interaction force, –

F_{pp} particle-particle interaction force, N

F p p * dimensionless particle-particle interaction force, –

F_{pf} particle-fluid interaction force, N

F p f * dimensionless particle-fluid interaction force, –

Fr Froude number, –

F_{t }tangential soft sphere interaction at contact, N

F t * dimensionless tangential soft sphere interaction at contact, –

g gravitational acceleration, m/s^{2}

GaGalilei number, –

I inertial moment of particle, kg·m^{2}

m_{ }mass, kg

p_{ }pressure, Pa

p'_{ }pressure perturbation, Pa

P_{ }dimensionless pressure, –

P' dimensionless pressure perturbation, –

r_{p} radius of particle, m

R dimensionless radius of particle, –

Re particle Reynolds number, –

ttime, s

T dimensionless time, –

ufluid velocity, m/s

U dimensionless fluid velocity, –

u_{0}fluidizing velocity, m/s

u_{t}terminal velocity, m/s

vparticle velocity, m/s

V dimensionless particle velocity, –

V_{p}volume of particle, m^{3}

xcoordinate, m

X dimensionless coordinate, –

ε voidage, –

μ viscosity, Pa·s

ρ density, kg/m^{3 }

ρ^{*} density ratio (=ρ_{p}/ρ_{f}), –

ω angular velocity, rad/s

ω^{*} dimensionless angular velocity

f fluid

mf minimum fluidization

mb minimum bubbling fluidization

o arbitrary reference quality

p particle