MP-PIC investigation of the multi-scale gas–solid flow in ... · solid phase), and the strongly non-linearly coupling of these structures (e.g., the bubble–particle interaction,

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MP-PIC investigation of the multi-scale gas–solid flow in the bubbling fluidized bed

Zhanghao Wan, Shiliang Yang (), Hua Wang

State Key Laboratory of Complex Nonferrous Metal Resources Clean Utilization, Kunming University of Science and Technology, Kunming 650093, Yunnan, China

Abstract Numerical study of the dense gas–solid flow in a bubbling fluidized bed has been conducted via

the multiphase particle-in-cell approach. Based on the numerical results, the mesoscale bubble dynamics combined with the particle-scale information of gas–solid phase are explored to understand the multi-scale flow property of the gas–solid phase in this apparatus. Several tips

regarding the multi-scale gas–solid flow can be obtained as: (i) Bubble volume in the bed behaves with a log-normal distribution pattern and the largest bubble appears in the central part of the bed. (ii) The horizontal chord length of bubbles is the largest in the central region but

small near the wall. Large vertical chord length of bubbles exists near the bed wall. (iii) Elongation of the rising bubbles in the upper part of the bed enlarges the aspect ratio. Large aspect ratio and small pressure of the bubbles appear near the bed wall. Bubble pressure

decreases along the bed height. (iv) Large particle Reynolds number can be observed in the dome and tail of the bubbles while small value appears at both sides of bubbles. The operating parameters alter the bubble dynamics and also the particle-scale information of solid phase.

Keywords bubbling fluidized bed (BFB) mesoscale structure multi-scale hydrodynamics multiphase particle-in-cell (MP-PIC) numerical study

Article History Received: 14 February 2020

Revised: 1 April 2020

Accepted: 2 April 2020

Research Article © Tsinghua University Press 2020

1 Introduction

Dense gas–solid bubbling fluidized bed (BFB) has been widely adopted in many industrial applications involving physical and chemical operations for many years, such as the drying, biomass gasification, and waste treatment, due to its excellent heat and mass transfer rate, the good mixing, fast chemical reactions, and also the ability to process wide particle size distribution. The gas–solid flow in the BFB behaves with a wide range of solid concentration and varying velocity. The bubbling fluidizing regime of the dense fluidized bed is characteristics with the presence of the complicated multi-scale flow structure (e.g., the mesoscale bubble structure and the particle-scale solid phase), and the strongly non-linearly coupling of these structures (e.g., the bubble–particle interaction, the particle–particle collision, and particle–wall interaction), which makes the design, operation, and scale-up of the industrial-scale equipment a challenge. Thus, the technical improvement for the large-scale commercial units is still slow, giving rise to the essential role of deeply understanding the multi-scale flow structure of the BFB for

the practical operation. Due to the widespread application of this apparatus,

there exist a vast amount of the experimental reports for the multiphase flow hydrodynamics in the bubbling fluidized bed, such as the effect of immersed tube (Hadi et al., 2012), solid circulation pattern (Laverman et al., 2012), entrainment and electrostatic charges (Alsmari et al., 2016), and segregation (Chew and Hrenya, 2011). Valuable results regarding the multi-scale flow property of the gas–solid phase have been provided. However, the collection of the experimental data in this kind of dense fluidizing system is challenging and costly due to the opaque property of the internal particulate flow.

Besides the experimental approach, computer simulation becomes a powerful tool for exploring the gas–solid flow dynamics for several decades (Gu et al., 2019; Liu et al., 2020), due to its ability to obtain the local detailed information (e.g., porosity and velocity) of gas–solid phase, and the low cost for testing with different operating parameters (Yang et al., 2019a; Yan et al., 2020). Based on the scale choosing to resolve the solid motion, the available numerical approaches for simulating the dense gas–solid

Vol. 2, No. 4, 2020, 00–00Experimental and Computational Multiphase Flow https://doi.org/10.1007/s42757-020-0067-1

Z. Wan, S. Yang, H. Wang

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flow in the fluidizing apparatus can be generally divided as the two-fluid model (TFM) (Li et al., 2013; Lin et al., 2019; Zhu et al., 2019), combined computational fluid dynamics– discrete element method (CFD–DEM) (Tang et al., 2017; Gui et al., 2018; Wu et al., 2018; Hu et al., 2019), and multiphase particle-in-cell (MP-PIC) (Snider, 2001; Snider et al., 2011). The Eulerian–Eulerian TFM method treats both the gas and solid phases as the interpenetrating continua, and has been widely chosen for the multiphase flow simulation due to its low computational cost. Via TFM, plenty of the numerical simulations have been reported, such as Yang et al. (2017). However, it cannot provide the individual motion of solid phase. The CFD–DEM method tracks the solid phase at the particle-scale level and the particle collision event with the soft-sphere collision model initially proposed by Cundall and Strack (1979). Thus, it can provide a high-fidelity solution for the dense two-phase flow and has been widely adopted to simulate the bubbling fluidizing apparatus. Via CFD–DEM, Gui and Fan (2011) studied the particle mixing in a bubbling fluidized bed based on fractal and entropy analysis. He et al. (2012) numerically explored the granular temperature distribution in a bubbling fluidized bed. Li et al. (2016) reported a numerical investigation of the particle–wall interaction in a pseudo-2D fluidized bed with the CFD–DEM approach. Xu et al. (2017) used the CFD–DEM to simulate the fluidization behavior of Geldart B particles in a micro fluidized bed. Their results found that the column diameter and particle–wall friction contribute to the variation of the minimum fluidization velocity. Recently, Yang et al. (2019b, 2019c) extended the cold CFD–DEM with the consideration of heat and mass transfer, the chemical reactions to explore the biomass gasification process in a bubbling fluidized bed gasifier. However, the limitation of this approach lies in the requirement of huge computational resources and thus the abovementioned reports are mainly conducted in a small-scale system. With the introduction of the numerical parcel and the adoption of empirical closures for modeling the particle–particle collision, the MP-PIC approach can significantly decrease the computational cost, and thus has been chosen to explore the large-scale fluidizing apparatus. Using MP-PIC, Feng et al. (2018) numerically studied the bubbling fluidized bed with Geldart A particles and found that the drag model significantly impacts the numerical results. Wang et al. (2018) simulated the biomass gasification in a fluidized bed reactor with MP-PIC to explore the operating parameters on the gasification process. To examine the effect of baffles, Yang et al. (2018a) conducted a simulation in a bubbling fluidized bed with MP-PIC. Recently, Yang et al. (2019d) explored the segregation behavior of the solid phase in a bubbling fluidized bed gasifier with the numerical

simulation carried out with the MP-PIC approach. Bubble characteristics (e.g., bubble size, bubble shape,

bubble aspect ratio, rising velocity) have the dominant effect on the fluidizing quality and system performance of the processes involved in the apparatus as the mesoscale bubble phase controls the macroscopic behavior of solid phase. Bubbles in the bed promote the particle mixing but are negative to the gas–solid contact. The volume and rising velocity of bubble phase can be adopted to evaluate the bed expansion. Moreover, an in-depth understanding of the microscopic mechanism of solid phase at the particle-scale level is the key to obtain the detailed micro-dynamic information. Furthermore, the mutual interaction of the bubble phase and solid phase gives rise to the non-linear chaotic motion of the whole bed.

In view of the critical role of the inherent multi-scale flow of dense gas–solid phase, the current work conducts the numerical simulation of the dense gas–solid flow in the bubbling fluidized bed with the MP-PIC approach to explore the flow characteristics of mesoscale bubble structure and the particle-scale information of solid phase. To achieve this goal, we firstly describe the model details for the MP-PIC approach. Then, we compare the numerical results via the MP-PIC approach with the experimental data and also the numerical results obtained with TFM, followed by evaluating the general gassolid flow behavior in this apparatus. Subsequently, the dynamical characteristics (e.g., volume, chord length, aspect ratio, pressure) of the rising bubbles together with the effect of operating parameters are investigated. Finally, the particle-scale level information (e.g., the Reynolds number, dispersion coefficient) of the solid phase in the bed are discussed.

2 Numerical methods

2.1 Governing equations of gas phase

In the MP-PIC approach, the gas phase is treated as the continuous medium, which is governed by the volume- averaged Navier–Stokes equations. In the current work, large eddy simulation is chosen for solving the gas motion. In the dense gas–solid motion, the momentum exchange of gas phase with solid phase should be considered, which is achieved with the introduction of voidage and momentum source term. The governing equations for the mass and momentum of gas phase can be expressed as

g gg g g

( )( ) 0

ε ρε ρ

t¶

+⋅ =¶

u

(1)

g g gg g g g g

g g g g gs

( )( )

( )

ε ρε ρ p

tε τ ε ρ

¶+⋅ =-

¶+⋅ + -

uu u

g F (2)


3

where εg is the volume fraction of gas phase; ρg is the density of gas phase; ug is the gas velocity; and pg is the gas pressure. g represents the gravitational acceleration. The sinking term Fgs represents the momentum source term due to the presence of solid phase. The stress tensor of gas phase τg is given by

g , g g

23

ji kij ij

j i k

uu uτ μ μ δx x x

æ ö¶¶ ¶÷ç ÷= + -ç ÷ç ÷ç ¶ ¶ ¶è ø (3)

The gas shear viscosity gμ is evaluated as the summary of the laminar and turbulent viscosity. In our work, the turbulent viscosity is calculated through the Smagorinsky model (1963) described as

22

t s gji

j i

uuu C ρx x

æ ö¶¶ ÷ç ÷= D +ç ÷ç ÷ç ¶ ¶è ø (4)

( )1 3= x y zD D D D (5)

where Cs and Δ are the Smagorinsky constant and the filter scale, respectively.

2.2 Governing equations of solid phase

In the MP-PIC approach, the solid motion is tracked in the Lagrangian framework. To speed up the simulation, the concept of the numerical parcel is chosen. Specifically, a numerical parcel means that the particle tracked in this approach is a collection of several particles with the same properties (e.g., density, particle size, speed, volume, position). With this assumption, the particle number tracked in this work can be reduced, and thus the simulation speed can be significantly enhanced and the computational resources can be greatly reduced. To track the solid phase, the particle distribution function (PDF) fs is introduced to track the solid motion, and the governing equations can be formulated as (Snider, 2001; Snider and Banerjee, 2010):

s s s s s D s

s D

( ) ( Γ )f f f f ft t τ

¶ ¶ ¶ -+ + =

¶ ¶ ¶u

u (6)

where us is the particle velocity. fD and Dτ stand for respectively the PDF for the local mass-averaged particle velocity and collision damping time (Snider and Banerjee, 2010). sΓ stands for the particle acceleration, which is calculated as

ss s g s g s

s p s

s s

D

d 1 1Γ ( )d

2

D p τt ρ ρ ε

τ

= = - - -

-+ +

u u u

u ug (7)

where sρ is the particle density. Ds represents the drag function. In this study, we use the drag correlation proposed by Gidaspow (1994) to evaluate the drag force

coefficient gsβ , which can be expressed as

g g g g s 2.65D g g

sgs 2

g g g g g sg2

sg s

(1 )3 , 0.84

150(1 ) 1.75 (1 ), 0.8

ρ ε εC ε ε

dβ

ε μ ρ εε

dε d

-ì - -ïïïïïï= íï - - -ï + <ïïïïî

≥u u

u u

(8)

0.687s s

sD

s

24 (1 0.15 ), 1000

0.44, 1000

ReReC

ìïï + <ï= íïïïî ≥

Re

Re

(9)

g g g s ss

g

ρ ε dμ-

=Reu u

(10)

where ds is the particle diameter. The particle normal stress sτ attributing to the dense

collision of particles is evaluated with the correlation proposed by Harris and Crighton (1994) formulated as

s ss

cs s smax[( ), (1 )]

γP ετε ε α ε

=- -

(11)

where Ps, εcs, and γ are the model parameters. After obtaining the PDF fs, the solid concentration and

the momentum exchanging item can be evaluated as

ss s s s s

s= d d dmε f m ρ

ρòòò u

(12)

ggs s s s g p s s s

s( ) d d d

pf V ρ β V ρ

ρé ùê ú= - - -ê úë û

òòòF u u u (13)

3 Numerical settings

3.1 Computational setup

The simulated bubbling fluidized bed (BFB) of current work is an experimental apparatus reported by Hernández- Jiménez et al. (2011). The geometrical illustration of the bed geometry can be found in Fig. 1. The bed has a width, depth, and height of 0.5 m, 0.005 m, and 2 m along x, y, and z directions, respectively. The spherical particles involved have a density and diameter of 2500 kg/m3 and 0.7 mm, respectively. Before the simulation, a packed bed with a height of 0.3 m and a solid volume fraction of 0.6 is formed, giving rise to a total number of 288,000 numerical particles in the entire system. The fluidizing medium is the air in this work. The minimum fluidization velocity (Umf) representing the critical point between the stationary state and the fluidized state is 0.35 m/s for the current system explored. The superficial velocity introduced from the bed inlet is 0.62 m/s in this work.

The numerical simulation is conducted with the MPPICFoam solver in the open-source package OpenFOAM,


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which solves the gas motion on the base of the finite volume method. To numerically obtain the solution, the whole BFB is uniformly divided into 40,000 hexahedron cells. The decoupling of the pressure and velocity is achieved with the PISO algorithm (Issa, 1986). To close the governing equations, the velocity inlet boundary condition is chosen for the bed inlet; the pressure outlet boundary condition is adopted for the bed outlet; the sidewalls are assigned with the no-slip boundary condition. A total physical time of 30 s is simulated with the data of previous 10 s discarded when assessing the time-averaged information of bed hydrodynamics. Related information regarding the parameters chosen can be found in Table 1.

3.2 Model validation

Figure 2(a) illustrates the profile of the pressure drop simulated with the MP-PIC method in this work. Vividly, the pressure drop of the whole bed linearly increases with

Fig. 1 Geometry of the 3D bubbling fluidized bed explored in the current study.

Table 1 Related operating parameters adopted in the current simulation

Vessel dimension (x, y, z) 0.5 m, 0.005 m, 2 m

Gravitational acceleration z = –9.81 m/s2

Solid volume fraction at closest packing 0.6 Maximum momentum redirection

from collision 40%

Total number of grids 40,000

Total number of numerical particles 288,000

Superficial velocity of air 0.62 m/s

Bed depth 0.3 m

Particle density 2500 kg/m3

Particle diameter 0.7 mm

enlarging the flow rate in the fixed bed status. The sudden variation of the pressure drop when the superficial velocity reaches a specific value can be obtained, followed by the smooth change with further increasing the flow velocity. This critical value identified as the minimum fluidization velocity is 0.32 m/s in the current simulation, which is close to the value of 0.35 m/s reported in the literature (Hernández-Jiménez et al., 2011). Figure 2(b) presents the comparison of the lateral distribution of solid vertical velocity on the right part of the bed, z = 0.25 m. Despite the experimental data (Hernández-Jiménez et al., 2011), the numerical results evaluated with the TFM approach by Liang et al. (2014) are also included. In general, the simulation results obtained with the MP-PIC approach in this work quantitatively agree well with the experimental data reported in the literature (Hernández-Jiménez et al., 2011). Moreover, the numerical results obtained with MP-PIC are better than the numerical results simulated with the TFM regarding the general distribution tendency and the fluctuation magnitude. Based on the model verification, the multi-scale flow characteristics of the BFB together with the impact the operating parameters (e.g., bed height, superficial gas velocity, and particle diameter) are discussed in the following.

0.00 0.05 0.10 0.15 0.20 0.25-0.4

-0.2

0.0

0.2

0.4(b)

Uz (

m/s

)

x (m)

Current simulation Experiment (Hernández-Jiménez et al., 2011) TFM (Liang et al., 2014)

Fig. 2 Comparison of experimental data and the numerical results obtained with the MP-PIC and TFM method: (a) pressure drop profile; (b) horizontal distribution of the solid vertical velocity on the right part of the bed (x > 0), at the height of 0.25 m.


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4 Results and discussion

4.1 General flow behavior

Specific to the bubbling regime, plenty of the unique features in this kind of apparatus are strongly related to the gas bubble motion. Bubble distribution is extremely critical for the fluidization quality and the interaction between gas and solid phase. Figure 3(a) presents the snapshot of the spatial distribution of gas voidage in the BFB at the time instant of 20 s. In general, the bubble dynamics in the bed can be vividly observed. After introducing into the bed, the void of gas phase starts to appear due to the velocity larger than the minimum fluidization velocity. The small bubbles randomly generated in the lower part of the bed, and then grow up during the rising process as the decrease of the hydrostatic pressure (Busciglio et al., 2012) and the interaction of the neighboring bubbles. Along the vertical direction, the velocity difference combined with the elongation result in the vertical coalescence of rising bubbles along this direction. Along the horizontal direction, the rising bubbles move towards the central region of the bed due to the wall restriction, and thus coalesce along this direction. The coalescence of the neighboring bubbles enlarges the bubble volume but decreases the bubble number. The bubble motion strongly affects the solid motion in the bed. Figure 3(b) presents the instantaneous

(a)

(b)

Fig. 3 (a) Snapshot of the gas voidage and (b) particle slip velocity in the bubbling fluidized bed at the time instant of 20 s.

distribution of the slip velocity of particles in the bed. During the bubble motion, a certain negative pressure is formed in the rail region of the bubble, sucking the surrounding particles to fill the area after the pass of rising bubbles. Therefore, dragged by the rising bubbles, the particle slip velocity at the tail of the bubble is generally upward. Moreover, the particles in the dome of the bubbles rise vigorously due to the drag of gas bubble, giving rise to apparently large particle velocity in this part. After reaching the bed surface, the burst of the rising bubbles results in the vigorous motion of particles in the freeboard region of the bed.

4.2 Bubble volume

To quantitatively explore the dynamical property of the bubble phase, a bubble should be first identified. Different voidage is chosen to identify the bubble boundary in the literature (Li et al., 2011). In this work, the threshold voidage equaling of 0.7 is chosen to define the bubble boundary, followed by summarizing the total volume of all the computational cells involved in a bubble. Figure 4(a) illustrates the histogram of the probability distribution of bubble volume in the bubbling fluidized bed. Vividly, small bubbles occupy the largest fraction, while the probability decreases with the increase of the bubble volume, owing to the reduced number of bubbles in the rising process.

Figure 4(b) presents the lateral distribution of the bubble volume in the bed with different initially packed height. Obviously, the bubble volume is the largest in the central of the bed and decreases towards the bed wall, which can be attributed to the fact that the wall restriction on the bubble motion results in the movement of the rising bubbles towards the central region of the bed. Regarding the effect of bed height, it can be clearly found that enlarging the initial packed bed height enlarges the bubble volume along the lateral direction due to the enlarged growth time of rising bubbles in the bed.

Figure 5 presents the effect of initial packed bed height (a), superficial velocity (b), and particle diameter (c) on the vertical distribution of bubble volume in the bed. For all the systems explored, the bubble volume continuously increases along the bed due to the bubble coalescence. As shown in Fig. 5(a), the bed height on the bubble volume can be ignored in the lower part of the bed (z < 0.27 m). When the bubbles approach the bed surface, the bubble volume rapidly increases due to the influence of the atmospheric environment. A higher initial packed bed height gives rise to a longer residence time of bubbles in the bed. Enlarging the superficial velocity exceptionally increases the bubble volume due to more gas introduced (Fig. 5(b)). The effect of particle diameter on the bubble volume is presented in Fig. 5(c). The decrease of particle


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Fig. 4 (a) Histogram of the probability distribution of the bubble volume in the bed; (b) effect of initial bed height on the lateral distribution of bubble volume.

Fig. 5 Effect of initial packed particle height (a), superficial velocity (b), and particle diameter (c) on the vertical distribution of bubble volume in the bed.

diameter obviously enlarges the bubble volume. This is mainly due to that the introduced gas goes into interstitial flow rather than bubble flow as a higher permeability of gas flow through the larger particles (Caram and Hsu, 1986). Moreover, the system with uniform distribution of particle diameter results in a slight increase of the bubble volume as compared with the bed with the coarsest particle diameter.

4.3 Bubble chord length

Figure 6 presents the histogram of the chord length distribution of rising bubbles in the bed. Nonuniform distribution of the bubble length can be observed along both directions. Large fraction of bubble chord length along both directions can be observed. The smaller the bubble volume the large fraction appears due to the bubble coalescence of the rising bubbles.

Fig. 6 Histogram of the probability distribution of the lateral (a) and vertical (b) chord length of rising bubbles.


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Figure 7 gives the effect of initial packed bed height on the lateral distribution of the chord lengths of rising bubbles. Different distribution patterns of the lateral and vertical chord length appear. The horizontal chord length (Fig. 7(a)) is the largest in the central part of the bed, and continuously decreases towards the bed wall due to the wall restriction effect. Comparatively uniform distribution of the vertical chord length of the rising bubble can be observed along the horizontal direction (Fig. 7(b)). Specifically, the wall restriction on the bubble motion gives rise to an enlarged chord length of the bubbles near the bed wall.

Figure 8 illustrates the effect of initial packed bed height, superficial velocity, and particle diameter on the vertical distribution of chord length of the rising bubbles. For all the systems explored, the vertical bubble length continuously increases due to bubble growth. The bed height gives rise to a negligible influence on the vertical bubble length in the lower part, but results in the existence of bubbles in the upper region. As expected, the enlargement of the superficial velocity gives rise to a continuous increase of the vertical bubble length along the vertical direction as more gas flow rate introduces (Fig. 8(b)). Regarding the effect of particle diameter (Fig. 8(c)), reducing the particle diameter gives rise to a larger vertical length of rising bubbles along the whole bed height.

Fig. 7 Effect of initial packed bed height on the lateral distribution of chord length of rising bubbles in the bubbling fluidized bed: (a) lateral distribution of Lx; (b) lateral distribution of Lz.

Fig. 8 Effect of the initial packed bed height (a), superficial gas velocity (b), and particle diameter (c) on the vertical distribution of chord length of rising bubbles in the bubbling fluidized bed.

4.4 Bubble aspect ratio

In general, the bubbles in the BFB are not perfectly spherical due to the interaction of the neighboring bubbles and the strongly non-linear coupling with solid phase. To quantitatively describe the shape characteristics, the bubble aspect ratio referring to the ratio of chord lengths in different directions is adopted. In this work, the aspect ratio of Ψzx evaluated as the ratio of the vertical chord length Lz to the horizontal chord length Lx of the bubble is chosen. A large Ψzx means a slender bubble, while a small Ψzx corresponds to a flat bubble. Figure 9(a) presents the histogram of the probability distribution of the bubble


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aspect ratio Ψzx. For the current system explored, the Ψzx is distributed in the range of 0.9–4. The bubbles with a small aspect ratio have a large probability, which corresponds to the small bubbles. The bubbles with an aspect ratio larger than 1 correspond to the large bubbles in the upper part of the bed, indicating the elongated property of the large bubbles. The horizontal distribution of the bubble aspect ratio in the bed with different bed height is presented in Fig. 9(b). In the central part of the bed, no significant difference in the aspect ratio of the bubbles can be observed. The bubble aspect ratio is larger than 1, demonstrating the elongation of the rising bubbles. Particular attention should be paid on the bubbles distributed near the bed wall. Obviously elongated bubbles with an aspect ratio larger than 2 appear due to the restriction of the sidewall on the horizontal chord length of bubbles in this region.

Figure 10 shows the effect of operating parameters on the vertical distribution of the aspect ratio of rising bubbles in the bed. For all the systems explored, flat bubbles with an aspect ratio of smaller than 1 appear in the lower part of the bed. The growth of the rising bubbles gives rise to the enlargement of the aspect ratio in the upper part. The

Fig. 9 (a) Histogram of the probability distribution of the aspect ratio in the bubbling fluidized bed; (b) effect of initial packed height of solid phase on the lateral distribution of aspect ratio.

Fig. 10 Effect of initial packed bed height (a), superficial gas velocity (b), and particle diameter (c) on the vertical distribution of aspect ratio of rising bubbles in the bubbling fluidized bed.

negligible impact of the bed height on the bubble aspect ratio can be observed. Enlarging the superficial velocity increases the bubble aspect ratio in the bed as more gas introduced into the bed. The aspect ratio increases with enlarging the particle diameter as more gas leaks into the interstitial flow.

4.5 Bubble velocity

Figure 11(a) gives the probability distribution of the lateral velocity of rising bubbles. The normal distribution pattern for the lateral bubble velocity can be observed. Bubbles with


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small lateral velocity occupy the largest fraction, while a small portion of them move with an obvious large one. The histogram distribution of bubble rising velocity is presented in Fig. 11(b). As can be observed, the rising speed of the bubbles is mainly concentrated in the range of 0–4 m/s. Bubbles with a rising velocity greater than 2 m/s and less than 1 m/s account for a relatively small fraction. Bubbles with a large rising velocity occupy a small fraction, attributing to the coalescence of the neighboring bubbles in the rising process.

The effect of different packed bed height, superficial velocity, and particle diameter on the vertical distribution of the rising velocity of bubbles is illustrated in Fig. 12. As the height increases, the rising speed of the bubble increases and then decreases in all the systems explored. Negligible effect of the initial packed bed height on the bubble rising velocity appears in the lower part of the bed. Increasing the superficial velocity continuously enlarges the bubble rising velocity along the whole bed as more gas stream is introduced from the bed inlet. Enlarging the particle diameter increases the bubble rising velocity in the bed, which is in accordance with the numerical reports of Rong et al. (2012) by CFD–DEM approach as the effect of particle diameter relates to the impact of minimum fluidization velocity.

Fig. 11 Histogram of the probability distribution of the lateral (a) and vertical (b) velocities of rising bubbles in the bed.

Fig. 12 Effect of the initial packed bed height (a), superficial gas velocity (b), and particle diameter (c) on the vertical distribution of the vertical velocity of rising bubbles in the bubbling fluidized bed.

4.6 Bubble pressure

Figure 13(a) illustrates the probability distribution of bubble pressure in the bubbling fluidized bed. As can be seen from the figure, the probability is the largest for the bubbles with a large pressure, and continuously decreases with decreasing the bubble pressure. Bubbles with large and small pressure correspond to those located in the lower and upper part of the bed, respectively. In the BFB, the gas momentum is transferred to the solid phase and thus the gas pressure decreases along the bed height. Thus, the


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Fig. 13 (a) Histogram of the probability distribution of the bubble pressure; (b) effect of initial packed bed height on the lateral distribution of bubble pressure in the BFB. lowest frequency of the bubble pressure corresponds to those in the upper part of the bed. The lateral distribution of the bubble pressure is presented in Fig. 13(b). For all the cases explored, relatively small pressure exists for the bubbles near the bed wall. Enlarging the bed height clearly increases the bubble pressure across the whole column of the bed.

Figure 14 shows the effect of different initial packed particle height, superficial velocity, and particle diameter on the vertical distribution of bubble pressure in the bed. Different from the other properties of the rising bubbles, the effect of the initial packed bed height on the bubble pressure is evident. The enlargement of the initial packed bed height increases the bubble pressure along the bed height. Comparatively, a limited effect of the superficial velocity on the bubble pressure can be observed along the bed. Regarding the effect of particle diameter (Fig. 14(c)), increasing the particle diameter reduces the bubble pressure along the bed height.

4.7 Particle-scale information

The Lagrangian tracking of solid phase provides the advantage of evaluating the particle-scale level information of the solid phase in the bed. Figure 15 shows the instantaneous distribution of particles colored with the Reynolds number in the bubbling fluidized bed. In general, the distribution of the particle Reynolds number is similar to that of the particle velocity. Obviously, the Reynolds number

Fig. 14 Effect of the initial packed bed height (a), superficial gas velocity (b), and particle diameter (c) on the vertical distribution of bubble pressure in the bubbling fluidized bed.

Fig. 15 Instantaneous distribution of the particles colored with the Reynolds number in a bubbling fluidized bed at time instant of 20 s.


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of particles in the dome and tail of the rising bubbles is large, while that on both sides of the bubble is small.

Figure 16 represents the influence of various operating parameters on the particle-scale information (e.g., velocity, Reynolds number, and slip velocity) of solid phase in the BFB. As expected, the enlargement of the superficial velocity obviously increases the velocity, slip velocity, and the Rep of solid as more energy transferred from the gas phase. However, increasing the bed height diminishes the particle-scale information explored as more particles are needed to be fluidized. Meanwhile, enlarging the particle diameter obviously decreases the solid velocity, slip velocity, and the Reynolds number as a large particle has a higher moment of inertia.

4.8 Solid dispersion behavior

Solid mixing in the fluidized bed is extremely critical for

Fig. 16 Effect of initial packed bed height (a), superficial gas velocity (b), and particle diameter (c) on the distribution of particle-scale information of solid phase in the bubbling fluidized bed.

the uniform heat and mass transfer, chemical reactions in the system operating under high temperature. Understanding this aspect can benefit for the design of the fuel feeding ports in the fluidizing combustor/gasifier. The rising process of the mesoscale bubble structure results in the vertical mixing of the solid phase, while the coalescence and eruption of bubbles give rise to lateral mixing. To quantitatively assess the mixing rate of the solid phase, the dispersion coefficient is used to characterize the degree of particle dispersion in this study. Tracking the solid motion in the Lagrangian framework gives rise to the possibility of evaluating the dispersion coefficient of solid phase with the micro approach, for which details can be found in previous work (Yang et al., 2018b). Figure 17 shows the distribution profile of the instantaneous lateral and vertical dispersion coefficient of particles in the BFB from 15 to 30 s. As can be seen from the figure, the dispersion coefficient of solid phase along both the horizontal and vertical directions fluctuates around a constant value due to the chaotic and random motion of the solid phase in the bed. Meanwhile, the fluctuation of the vertical dispersion coefficient of the solid phase is more intensive as compared with the horizontal one. Furthermore, the averaged dispersion coefficient of solid phase is 2.4×105 and 4.3×105 m2/s along the horizontal and vertical directions, respectively. The larger vertical dispersion coefficient of solid phase mainly attributes to the vertical introduction of the gas phase in the bed.

To quantify the effect of operating parameters on the solid dispersion, Fig. 18 shows the mean dispersion coefficient in the bubbling fluidized bed operating with different parameters. Enlarging the bed height linearly increases the particle dispersion coefficient along both the horizontal and vertical directions. As expected, enlarging the superficial velocity and particle diameter linearly increases and decreases the horizontal and vertical dispersion coefficients of solid phase due to the expanded gas volume introduced and the large momentum inertia, respectively.

Fig. 17 Instantaneous distribution of the dispersion coefficient of solid phase in the bubbling fluidized bed.


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Fig. 18 Effect of initial packed bed height (a), superficial velocity (b), and particle diameter (c) on the vertical dispersion coefficient of solid phase in the bubbling fluidized bed.

5 Conclusions

Via multiphase particle-in-cell approach, the gas–solid flow in a bubbling fluidized bed is numerically simulated to explore the multi-scale flow characteristics of gas–solid phase together with the effect of operating parameters. The numerical results have been verified by comparing with the experimental data in the bed. Based on the numerical simulation, the findings are described as follows:

(1) A log-normal distribution pattern can be observed for the probability distribution of bubble volume due to the

bubble coalescence. Along the horizontal direction, bubble volume is the largest in the central part but small near the wall. Enlarging the bed height gives rise to a large bubble volume in the bed. The increase of the superficial velocity and particle diameter increases and decreases the bubble volume in the bed.

(2) A small chord length of rising bubbles occupies a large portion. Along the horizontal direction, the lateral chord length is the largest in the central part but the smallest near the wall due to the restriction of bed wall. However, a slightly larger vertical chord length of bubbles appears near the wall. The enlargement of the superficial velocity and diameter increases and decreases the vertical chord length, respectively.

(3) The elongation of the rising bubbles in the upper part of the bed gives rise to a large aspect ratio. Furthermore, bubbles near the bed wall behave with a large aspect ratio due to the wall restriction. Enlarging and diminishing the superficial velocity and particle diameter increases and decreases the bubble aspect ratio along the bed height, respectively.

(4) Different flow velocity can be observed for the horizontal and vertical velocity of rising bubbles. Enlarging the superficial velocity obviously increases the bubble rising velocity. In the bed, bubbles with large pressure occupy the largest fraction. Slight decrease of the bubble pressure near the wall can be observed. Enlarging the bed height obviously increases the bubble pressure along the bed height.

(5) The Reynolds number of particles in the dome and rail of the rising bubbles is the largest while small on both sides of the bubble. Moreover, enlarging the superficial velocity and particle diameter obviously increases and decreases the vertical velocity, slip velocity, and Reynolds number of the solid phase. Larger fluctuation can be observed for the solid dispersion along the vertical direction.

Acknowledgements

The authors thank the financial support from the National Natural Science Foundation of China-Yunnan Joint Fund (Grant No. U1602272) and National Natural Science Foundation of China (Grant No. 51966007).

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MP-PIC investigation of the multi-scale gas–solid flow in ... · solid phase), and the strongly non-linearly coupling of these structures (e.g., the bubble–particle interaction,