Computational study of bubble coalescence/break-up behaviors and bubble size distribution in a 3-D pressurized bubbling gas-solid fluidized bed of Geldart A particles

Teng Wang,Zihong Xia,Caixia Chen

Department of Energy and Chemical Engineering,East China University of Science and Technology,Shanghai 200237,China

Keywords:Pressurized gas-solid bubbling fluidized bed Geldart A particles Bubble size distribution Coalescence Break-up Bubble tracking algorithm

ABSTRACT A computational study was carried out on bubble dynamic behaviors and bubble size distributions in a pressurized lab-scale gas-solid fluidized bed of Geldart A particles.High-resolution 3-D numerical simulations were performed using the two-fluid model based on the kinetic theory of granular flow.A finegrid,which is in the range of 3–4 particle diameters,was utilized in order to capture bubble structures explicitly without breaking down the continuum assumption for the solid phase.A novel bubble tracking scheme was developed in combination with a 3-D detection and tracking algorithm (MS3DATA) and applied to detect the bubble statistics,such as bubble size,location in each time frame and relative position between two adjacent time frames,from numerical simulations.The spatial coordinates and corresponding void fraction data were sampled at 100 Hz for data analyzing.The bubble coalescence/break-up frequencies and the daughter bubble size distribution were evaluated by using the new bubble tracking algorithm.The results showed that the bubble size distributed non-uniformly over cross-sections in the bed.The equilibrium bubble diameter due to bubble break-up and coalescence dynamics can be obtained,and the bubble rise velocity follows Davidson’s correlation closely.Good agreements were obtained between the computed results and that predicted by using the bubble break-up model proposed in our previous work.The computational bubble tracking method showed the potential of analyzing bubble motions and the coalescence and break-up characteristics based on time series data sets of void fraction maps obtained numerically and experimentally.

1.Introduction

Bubbling fluidized beds with Geldart A particles are widely used in various industrial gas-solid processes,such as fluid catalytic cracking (FCC),coal gasification/combustion,and Fischer-Tropsch(FT) synthesis,due to their excellent solid mixing,heat,and mass transfer characteristics [1].Many of these good performances of gas-fluidized beds can be attributed to the bubbling phenomenon,since the bubble behaviors affect the solids movement and gassolid contact [2].In a bubbling fluidized bed,a large amount of small gas bubbles form at the distributor plate with a superficial gas velocity exceeding the minimum fluidization velocity.The bubbles usually grow in size while moving upward accompanied by coalescence and break-up in the bed [3,4].For Geldart A particles,the gas velocity of bubble formation is larger than the minimum fluidization velocity,and quasi-stable bubble size is achievable at certain gas velocity due to a balance between the coalescence and break-up rates [5].Traditionally,empirical correlations obtained from a range of scales experiments were used to evaluate bubble sizes and bubble velocity [6–8] as reviewed critically by Karimipour and Pugsley[9].It has been recognized that the bubble rise velocity in a gas-solid fluidized bed is scale-dependent.The difference is considered due to a sequential dynamic process:the particle-wall interaction that would consume the particle energy;the subsequent particle-particle interaction that would form clusters;and finally,the gas-particle interactions that would induce bubble coalescence and break-ups.

Compared to atmospheric pressure,fewer studies have been performed on bubble properties at higher operating pressures.King and Harrison [10] found that bubble size in beds of Geldart A particles decreases with increasing pressure.Hoffman and Yates[11] reported that bubbles distributed no-uniformly over the cross-section with increasing pressure,and the bubble rise velocity increased with increasing pressure due to the non-uniform distribution of bubbles.Since the rise velocity of bubbles is proportional to their size,an increase of bubble velocity is contradictory to decreasing bubble size in bed.Caiet al.[12]proposed a bubble size correlation as a function of operating pressure.Karimipour and Pugsley [9] compared Cai’s [12] bubble size correlation with literature data at different operating pressures,a fitting error of±11.8%was found,noting that the data of Geldart A fluidized beds were mostly from relatively shallow beds that did not cover the maximum stable bubble size region.

Several experimental studies have been carried out on bubble behavior by using sophisticated techniques.High-speed cameras and digital image analysis were conducted with a thin rectangular gas fluidized bed[13–15].As for cylindrical beds,the optical accessibility is impossible[16],and bubbles are detected and characterized using X-ray [17–19],electrical capacitance tomography [18],and magnetic resonance imaging [20,21].However,the applications of these techniques to large-scale fluidized bed are still restricted by technical limitations [19,22,23].

In recent years,benefiting from the increases in the computing power,computational fluid dynamics (CFD) has made significant progress in describing detailed flow behaviors inside the fluidized beds.High-resolution CFD simulations are capable of providing insights useful for design and scale-up of industrial fluidized beds[24–26].The virtual experimental ‘‘measurement”based on numerical simulation is much convenient to be conducted than that in the physical world.Recently,Shresthaet al.[27–28]reported studies on the effects of particle shape and van der waals force on bubble dynamics using discrete element method(DEM).They found that the bubble size and velocity decrease with the increase of van der waals force,and spherical particles have a larger bubble size and higher rising velocity than ellipsoids.Compared with the DEM approach,the two-fluid model(TFM) based on the kinetic theory of granular flow (TFM-KTGF)has been widely employed in the simulations of the gas-solid hydrodynamics of fluidized beds [19,23,26,29–31].However,bubble characteristics such as bubble size,spatial location,velocity,and aspect ratio cannot be directly extracted from simulation results.Therefore,some researchers derived bubble statistics from numerical simulations.Vermaet al.[19] and Sobrinoet al.[31] employed a tomography method to obtain bubble size and velocity from TFM simulation data of 3-D fluidized bed.They extracted voidage maps at different bed heights and employed images to reconstruct the bubbles.Luet al.[32] developed a flood-fill method which checked neighbors of every cell and used a sequence of target-grid and pending-grid approach to determine bubble boundaries.Julianet al.[22] employed the Delaunay triangulation algorithm which was filtered utilizing the α-shapes geometric method to measure individual 3D bubble size and velocity from TFM simulation data.Bakshiet al.[16] proposed a 3-D bubble statistics algorithm,namely ‘‘multiphase-flow statistics using 3-D detection and tracking algorithm”(MS3DATA).Bubbles were detected by analyzing the void fraction data from 3-D simulations,and reliable results of bubble distribution and motion were obtained.The MS3DATA has been further applied in analyzing bubbles in 2-D and 3-D,small and large-scale fluidized bed simulations successfully [16,33–35].Uddinet al.[23]proposed a 3-D face-masking algorithm to detect and track bubbles from high-resolution CFD data of different scale fluidized beds.Buchheitet al.[36] developed BubbleTree toolset to analyze bubble dynamics from CFD simulation data.

Despite various methods that have been reported on the analysis of bubble characteristics from simulation data,only a few works have addressed the coalescence and break-up behaviors of bubbles [37].Since the circulating motion of solids is driven by bubbles [16,34,36],a quantitative description of bubble size distribution and the variations is of practical use but remains a methodological challenge.By taking into account the effects of bubble coalescence and break-up,a CFD-PBM computational scheme was proposed to predict bubble size distribution in a 2-D fluidized bed in our previous work [38].However,the bubble break-up frequency was obtained by fitting literature data,and the daughter bubble size distribution was taken from a single measurement data set at a very low gas velocity.The detailed mechanism of bubble break-up and the size distribution of daughter bubbles are far from fully understand.To this end,investigating the coalescence and break-up of bubbles using detailed data from high-resolution simulations may be an effective way.In this study,a novel computational method was developed to obtain the bubble coalescence and break-up frequencies from analyzing the fine-grid TFM-KTGF simulation results data.Firstly,a lab-scale 3-D gas fluidized bed of Geldart A particles was simulated with a grid size of 3–4 times particle diameter.Secondly,the bubble sizes and the locations were detected across successive time frames by using an open-source code MS3DATA[16].Thirdly,a new bubble tracking algorithm was developed to capture the trajectories and fates of bubble movement.At last,the bubble sizes and rise velocity,the bubble break-up frequencies,and the daughter bubble size distribution were evaluated,and compared with experimental correlations.

2.Methodology

2.1.Fine-grid TFM-KTGF simulation of a lab-scale 3-D gas fluidized bed

A lab-scale cylindrical gas-fluidized bed is employed in this work to investigate bubble dynamics concerning coalescence and break-up.The dimensions of the lab-scale the fluidized-bed reactor is the same as Yanet al.[39] with 1.0 m (height) × 0.0 36 m (inner diameter).Hydrogen with a temperature of 850 °C and pressure of 3 MPa was used as fluidizing gas in the experiment of Yan.The simulation parameters are shown in Table 1.In this study,high resolution 3D numerical simulations are performed using the two-fluid model based on the kinetic theory of granular flow (TFM-KTGF) of Fluent?14.0.The grid size is in the range of 3–4 particle diameters in order to capture bubble structures explicitly without breaking down the continuum assumption for the solid phase [35,40,41].The detailed constitutive equations of TFM-KTGF and some of the key parameters are chosen and listed in Table 2.

Table 1 Simulation parameters used for pressurized fluidized bed

Table 2 Parameters setups of the KTGF model

Two superficial velocities of 3.7Umfand 5.0Umfwere used in the simulation,respectively.In each case,a uniform gas velocity was specified at the inlet.A no-slip boundary was adopted for the gas phase and a partial slip boundary for the solid phase.A time step 0.001 s and 20 iterations per time step were used to ensure CFL <1,and all residual values were set to be below 1 × 10-3as the convergence criteria.The simulation lasted for 25 s in physical time and only the last 20 s were considered inorder to remove transient startup effects.First-order Euler implicit was used for transient formulation.The least squares cell based method was used for estimating of gradients.The pressure-velocity coupling was resolved using the phase coupled semi-implicit method for pressure-linked equations (Phase-Coupled SIMPLE) algorithm.The quadratic upstream interpolation for convective kinematics (QUICK) scheme was used for volume fraction and momentum equations,and the second-order upwind scheme was specified to discretize other equations.Underrelaxation factors were set to 0.6 for the pressure,0.4 for the momentum equations and 0.5 for the volume fraction,turbulent kinetic energy and turbulent dissipation rate equations,respectively.

2.2.Raw data collection and bubble detection

As the first step,raw data of gas void fraction distribution of the simulated bubbly beds are collected for bubble detection.During the simulations,the spatial coordinates and corresponding void fraction data are sampled at 100 Hz and saved,and the data sets are input to MATLAB.The data are subsequently filtered out cells associated with void fraction less than a threshold valueto decrease the computational cost for bubble detection.In this work,is set at 0.6 following Bakshi[16],which can reduce the transfer of data volume per frame significantly without affecting the accuracy of bubble statistics.The remaining data sets are then refined by interpolation to accurately resolve bubble boundaries.Notably,every cell is identified using unique cell indices replacing spatial coordinates.The refined grid A utilized for bubble detection is expressed as

The above-collected data sets are used to detect bubbles using an open-source script MS3DATA [16].A critical void fractionas the inter-phase boundary for bubble detection is set at 0.7 following Bakshi [16] and Rudisuli [43].During bubble detection,each cell is associated with at most one bubble,and bubble properties,including the center position of each bubble and bubble diameter,can be calculated by considering the cells that form each bubble.The bubble detection algorithm is outlined in Fig.1.

2.3.Trajectory and fate of bubble movement

Once the bubbles are detected by using MS3DATA,the trajectories of bubble movements are computed using the lagrangian velocimetry technique (LVT) based on a time series consecutive frames,a similar method was used by Busciglioet al.[14] to analyze the bubbling behavior of a 2D gas fluidized beds.In present work,a unique bubble index is assigned to each bubble and a bubble linking algorithm is then used to link bubbles of two consecutive frames.For a given bubbleji-1in framei-1 and another bubblejiin framei,they will be linked as bubblejif they are the same bubble:

Fig.1.The detailed algorithm for bubble detection.

Fistly,for bubbleji-1in framei-1 (i?[2,n]),extrapolation method [16] (according to locations in framei-1,i-2 and/ori-3) is used to predict the location of the bubblejipin framei.

Secondly,the value of ΔL,which is the distance between the predicted location of bubble in frameiand the actual location of bubble in frameiis calculated.ΔLmaxis the maximum permissible value of ΔLin one time step.

Finally,for bubbleji-1in framei-1 and all other bubbles(bubbleji,ki,liandmi etc.) in framei,the bubbles in frameiwith ΔL>ΔLmaxare filtered out (bubbleli,miare filtered out for example).And of the remaining bubbles in framei(bubbleji,kifor example),the bubble (ji) whose actual location is nearest to bubblejip,which means the one with the smallest ΔLvalue (ΔL1<ΔL2),is considered to be probably linked with bubbleji-1,as shown in Fig.2.

Bubbleji-1and bubblejiwill be linked using three criteria as follows:

(1)The axial displacement of the bubble centroid across consecutive frames is positive.

(2) The distance (ΔL) between bubblejipand bubblejiis less than the maximum permissible value ΔLmax,and ΔLmax=max(Δl0,Δli,max) is chosen in present work.Δl0is the initial value of the maximum permissible distance traveled by bubble in one time-step,0.01 is chosen based on Bakshi [16].Δli,maxis the maximum of the expected displacement distance of all bubbles between framei-1 and framei.The maximum expected value(Δli,max) is chosen based on bubble speed correlation by Mori and Wen [8] and Choiet al.[44] (Δli,max=max（Δt?ub）,where,

(3) In step (2),there may be more than one bubble in frameimeet the conditions of step (1)-(2),in order to determining the bubbleji,the bubble whose actual location is nearest to bubblejipis considered to be probably linked with bubbleji-1.

Fig.2.Locate bubbles to be linked in subsequent frames. ji,the actual location of bubble in frame i. jip,the prediced location of bubble in frame i.

Once bubbleji-1and bubblejiare linked with each other,the indices of bubbles will be recorded in a matrix that represents the trace of bubbles.Theith row of bubbletrace matrix records the indices of bubbles in framei,and every column is a trace of the bubble as it moves across consecutive frames.Meanwhile,another matrix is created to record the fate of bubbles.Where,situations that the change ratio of bubble diameter is more than 10%are recorded:specifically,whendi/di-1≥1.1,coalescence occurs in framei,and the coalescence event is recorded as 1 in the bubblefate matrix;whendi/di-1≤0.9,a break-up occurs in framei,and the break-up event is recorded as -1 in the bubblefate matrix;if 0.9

By counting the number of 1 or -1 in bubblefate matrix,the number of bubble coalescence/break-up events per frame can be obtained,and the total number of bubbles per frame will be recorded in bubbletrace matrix.The frequency of break-up and coalescence of each frame can be expressed as following [45]:

Fig.3.Detailed bubble tracking algorithm for inter-frame bubble property data.

where,Δtis the sampling time,ΔNb（V）/N（V）is the fraction of bubbles break-up and ΔNc（V）/N（V）is the fraction of bubbles coalescence.

In the break-up events stored in bubblefate matrix,the indices of the mother bubble and daughter bubble can be obtained in the same positions of bubbletrace matrix.According to the bubble indices,the diameter of the mother bubble and daughter bubble can be found in the bubble property data file.Therefore,the ratio of the volume of one of the daughter bubbles and the total volume of the mother bubble can be calculated by

whereViandVjdenote the volumes of the mother bubble and daughter bubble,respectively.

3.Results and Discussion

3.1.Bubble size distributions

An example of a bubble detection outcome is shown in Fig.4.Where,Fig.4(a) shows the iso-surfaces at the void fraction of 0.7 which is the threshold of bubble boundaries.Fig.4(b)and(c)show the detected bubbles and the evaluation of bubble properties using MS3DATA [16].It is worth noting that small bubbles (roughly less than 3 to 4 cell volume) (Fig.4(b)-1) and bubbles adjacent to the freeboard (Fig.4(b)-2) are discarded when calculating bubble properties.

Fig.4.An Example of bubble detection results.(a) Iso-surface at bubble void threshold,(b)detail of detected 3D bubbles in MATLAB using MS3DATA(horizontal projection plotted in MATLAB),and(c)bubble properties evaluated using MS3DATA(horizontal projection plotted in MATLAB),the small circles denote the centroid of bubbles.

The total number of bubbles detected in both cases is 38,000 for 3.7Umfand 74,000 for 5.0Umf,respectively.When the gas velocity is increased,the number of bubbles predicted in the simulation increases,as also observed in the work of Liet al.[2].To analyze the evolution of bubbles,five regions of interest (ROI) have been selected for the whole bed:a cylinder control volume with the same diameter as a bed and a height of 2Δyhave been chosen(see Fig.5).The ROI centers are located at five different elevations above the distributor,i.e.,0.5 cm,1.5 cm,2.5 cm,3.5 cm and 4.5 cm,and an increment 0.5 cm is chosen for Δy.The number and size of ROI provide information about coalescence and break-up of bubbles from almost the whole bed.Meanwhile,the results are not sensitive to the specific position of the ROI.

Bubble size evolution as a function of the elevation above the distributor detected for two different gas velocities is presented in Fig.6.The full set of bubble size data is shown in the raw cloudy form,together with the average simulation curve with standard deviation error bars.Meanwhile,the bubble sizes evaluated by using different correlations [8,12,46,47] in literature are shown in the same figure.Qualitative agreement with the analysis by Cai is achieved,moreover,the correlation of Cai is specially for pressurized fluidized bed.However,the correlations of Mori and Wen [8],Werther [46] and Darton [47] were proposed based on the experimental data of fluidized bed under atmospheric conditions,which may be one of the reasons why the predicted values of the three correlations[8,46,47]were higher than the arithmetic average value of the raw data.In addition,special care needs to be exercised while comparing the simulation results with literature.Many data obtained in experiments of early years might overestimate the average bubble size because most of the bubble diameters detected by their probes were larger than 1–1.3 cm [9,48],and bubbles smaller than 1.3 cm could not even be detected.Some researchers [15] filtered out bubbles with a diameter less than 1.5 cm in order to compare the simulation data and experimental data.However,as can be observed in Fig.7,the bubble sizes in present simulation are mostly smaller than 1.5 cm.Due to a considerable number of small bubbles are detected in the top section of the bed,the average diameter in the upper part of the bed becomes smaller,the rising trend of bubble size becomes less pronounced.And it was also found that [19] the simulations predict a larger fraction of smaller bubbles compared to experiments.Therefore,some differences exist between the simulated data and the correlations in literature.Notably,the raw data illustrated that a significant fraction of small bubbles can be observed even at the highest elevations,which confirms the existence of bubble break-up behavior.

Fig.5.Control volume to analyze bubbles at each height.

Fig.6.Simulated bubble size evolution as a function of the elevation above the distributor,comparing with different bubble sizes evaluated by using literature correlations.(a) U0=3.7Umf and (b) U0=5.0Umf.

Fig.7 shows the full set of radial bubble size distribution data(left ordinate) presented in raw cloudy form at four different heights for two different superficial gas velocities,together with the probability density function curve (right ordinate).By definition,thex-axis of the probability density function curve covers the range 0–1.To obtain the probability density function,the radial position domain was divided into 5 intervals.The probability density function of a certain radial interval is obtained by dividing the number of bubbles in each interval by the number of all bubbles in that height range,and then by the interval range|Δ(r/R)|.|Δ(r/R)|=0.2 was chosen in this work.Because of the symmetry of the system with respect to the centerline,the probability density of-1 to 0 is the mirror image of the curve from 0 to 1 relative to the centerline.In present work,all the radial bubble distributions show the similar trend with the experimental results of the work of Movahedirad[15],which is that a double peak radial bubble distribution profile can be seen near the distributor that develops to a single peak in the top part of the bed.The reason for this phenomenon is that the bubble movement toward the centerline of the bed.When the elevation is higher,resulting from the coalescence phenomenon between bubbles,the bubbles tend to move to the center part of the bed,and hence,the double peak profile changes to a single peak.

Fig.8 shows the bubble size distribution at different heights above the distributor.The results show a single peak and the mean bubble size grows with bed height,confirming the bubble size distribution of Geldart A particles is different from the B particles at similarU0/Umf[14,35].The peak for small diameter range at the bottom of the bed slightly increases with gas velocity,which means more bubbles are formed at higher gas velocity,as physically expected.Meanwhile,it is found that the change of the bubble size distribution is not obvious when the elevation is higher than 2.5 cm,the bubble diameter evolution is in a steady-state condition,which is consistent with Fig.6.

3.2.Bubble rise velocity

The computed bubble rise velocity distribution functions are reported in Fig.9.The prediction by Davidson and Harrison [49]is compared,as well as the arithmetic average of computational results.The comparisons in both cases show that the averaged bubble rise velocities agree well with Davidson’s correlations.The full set of computational results are also presented in raw cloudy form.These data show a scattered trend similar to Busciglio’s results obtained by digital image analysis [14].According to the power law of Davidson in Eq.(4),the bubble rise velocity increases with bubble size.However,the raw data shows that the bubble rise velocity of small bubbles distributed in a wide range.Similar to Busciglio’s work [30],when the small bubbles are rising in the wake region of larger bubbles,the speed of the trailing bubble will be associated with the large leading bubble.

3.3.The frequency of bubble coalescence and break-up

Fig.10 shows the snapshots of bubble coalescence and bubble break-up.The time step between consecutive frames is 0.01 s.Fig.10(a) displays the coalescence of bubbles:bubble 1 and 2 in framei-1 collide in frameiand coalesce into bubble 3.Fig.10(b)displays the break-up of a bubble:bubble 1 in framei-1 breaks up into bubble 2 and 3 in framei.It is worth noting that the bubble coalescence behavior in Fig.10(a) is due to the collision between the wake of the leading bubble and the nose of the trailing bubble.This finding is correspondent to the conclusion of our previous work [38].

In Fig.11,the coalescence and break-up of bubbles shown in Fig.10 are tracked by the computer algorithm shown in Fig.3.Bubbles that appear simultaneously in framei-1 and frameiare linked (blue arrow).Pennet al.[21] studied the bubble dynamics in a cylindrical 3D gas-solid fluidized bed using the real-time magnetic resonance imaging.It is suggested that a time interval of 0.056 s is enough to capture and observe bubble coalescence and break-up.In this study,a time interval of 0.01 s is used and the reliability of the choice of the time interval is confirmed by Fig.11.As shown in the figure,the locations of bubbles after coalescence/break-up did not change significantly within the two consecutive frames,which makes it possible to track the coalescence/breakup behavior of bubbles for a given time step 0.01 s.

Fig.7.Radial bubble size distribution data(left ordinate)and the probability density function curve(right ordinate)at different heights:(a)U0=3.7Umf and(b)U0=5.0Umf.

Fig.8.Bubble size distribution at different heights detected using MS3DATA for(a)U0=3.7Umf and (b) U0=5.0Umf.

For each simulation case,the break-up and coalescence frequencies are computed according to Eqs.(1)and(2).Fig.12 reports the time series plots of bubble break-up and coalescence frequencies of two different cases for the last 20 s of simulation,together with the average curve.The fluctuations of bubble coalescence and break-up frequency indicate the complex bubble dynamics in the bed due to the simultaneous presence of two phenomena.The bubble coalescence and break-up are in a competition mechanism of each other,causing dynamic evolution of bubble size and its size distribution.The change of bubble size will in turn affect the rate of bubble coalescence and break-up.It is worth noting that although the amplitudes of the time series values are very large,the average frequencies do not vary much over time,suggesting a statistical law behind the dynamics.Because of the competition mechanism of bubble coalescence and break-up,the bubble diameter reaches an equilibrium value when the same coalescence and breakup frequencies occur [50].And the bubble coalescence and break-up competition mechanisms lead to a stable time-averaged trend behind the fluctuations of bubble coalescence and break-up frequencies.The observations in Fig.12 could also explain the phenomenon that the overall size of bubbles shows a continuously increasing trend,but a considerable number of small bubbles can be observed in the top section of the bed.

The distributions of bubble break-up and coalescence frequencies are shown in Fig.13.For the case of 3.7Umf,as shown in Fig.13(a),the probability density distributions of break-up and coalescence frequencies are similar to each other,which means that the equilibrium bubble size will be observed quickly.For the case of 5.0Umfshown in Fig.13(b),the probability density of bubble coalescence is higher than bubble break-up in the small value range,however,the opposite is true in the high value range.This suggests that when increasing the air velocity,the number of bubbles with high break-up frequency increases compared to the bubble coalescence frequency under the same conditions.This means that the bubble breakage frequency is more influenced by the air velocity than the bubble coalescence frequency.

Fig.9.Bubble rise velocities as function of bubble equivalent diameter for:(a)U0=3.7Umf and (b) U0=5.0Umf.

In our previous work[38],a break-up frequency correlation was proposed based on fitting the experimental data in the literature[37,46,51–53],and expressed as

where,b（Vk）is the bubble break-up frequency,U0is the superficial gas velocity andUmfis the superficial minimum fluidization velocity.qbis the volumetric bubble flux,qb=U0-0.62Umf[50].

The frequency correlation proposed in our previous work is further validated by simulated results.The overall average break-up frequencies for different superficial gas velocities are computed as 5.98 s-1for 3.7Umfand 5.34 s-1for 5.0Umf,respectively.The value ofb（Vk）×/qbis calculated using the same procedure of our previous work [38].Fig.14 shows the relationship ofb（Vk）×/qbandU0/Umfcomputed by the present tracking algorithm,comparing with the fitting curve of the previous model.Good agreement is obtained,indicating the applicability of present tracking algorithm for bubble coalescence and break-up.

Fig.10.Snapshots from simulations:(a) bubble coalescence,(b) bubble break-up.

Fig.11.Results of bubble tracking algorithm using MATLAB:(a) Detail of tracking bubble coalescence,(b) detail of tracking bubble break-up.

3.4.The distribution of daughter bubbles

The volume distribution of daughter bubbles resulting from the break-up of a mother bubble is also evaluated according to Eq.(3).The probability ofri,jfor the last 20 s break-up events tracked by the present computer algorithm is considered.To investigate the variations of the probability withri,j,the domain ofri,jis divided into certain intervals(Δri,j=0?1).The probability density function is illustrated in Fig.15.The probability of a mother bubble breaking into two bubbles of similar size is significantly larger than that of an unequal pair of bubbles.The results are similar to the experimental results of Movahediradet al.[37],where a very low gas velocity was tested.

4.Conclusions

The bubble properties in a cylindrical lab-scale pressurized gassolid bubbling fluidized bed of Geldart A particles is investigated computationally.The void fraction data from 3-D high-resolution TFM-KTGF simulations are used for bubble detection.A rigorous statistical analysis of bubble motion is performed based on the large amount of bubble data detected,and the bubble coalescence and break-up are tracked by a bubble tracing algorithm proposed in the present work.The bubble properties are analyzed based on frame-by-frame changes of bubble location and sizes.The bubble size distributes non-uniformly over the cross-section of the bed.The equilibrium bubble diameter due to bubble break-up and coalescence dynamics is obtained,and the bubble rise velocity follows Davidson’s correlation closely.The time series value and average value of bubble coalescence and break-up frequency with different bubble dimensions are computed.The results of break-up frequency are in consistent with an empirical correlation proposed in our previous work.The daughter bubbles distribution of bubble break-up is in a similar shape of the experiment result.The presented computational algorithm may be also applied to analyzing experimental data and assessing bubble coalescence and break-up frequency when frame-by-frame void fraction data is available.The algorithm developed here may well be used to fully characterized the complex break-up and coalescence behavior of freely bubbling beds,highlighting its large potential in fully understand the bubbling fluidization behavior.

Fig.12.Time series plot of computed bubble break-up and coalescence frequencies for:(a) U0=3.7Umf and (b) U0=5.0Umf.

Fig.13.Probability density function curve of bubble break-up and coalescence frequencies for:(a) U0=3.7Umf and (b) U0=5.0Umf.

Fig.14.Relationship of b（Vk）×/qb and U0/Umf.

Fig.15.Probability density function of daughter bubbles size:(a)U0=3.7Umf,(b)U0=5.0Umf.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (21908062).

Nomenclature

b(V)break-up frequency,s-1

c(V)coalescence frequency,s-1

Dbbubble diameter,cm

dibubble diameter in framei,m

gacceleration of gravity,9.81 m?s-2

H0initial packing height,m

ΔLthe distance between predictive and actual location of bubblej

Δlmaxthe maximum distance traveled by bubble during one frame,m

ΔNb(V) number of bubble break-up events

ΔNc(V) number of bubble coalescence events

N(V)total number of bubbles

ntotal number of frames

qbvolumetric bubble flux,m?s-1

ri,jratio of volume of bubbleVj

Δtsampling time step,s

Umfsuperficial minimum fluidization velocity,m?s-1

U0superficial gas velocity of bed,m?s-1

ubbubble speed,m?s-1

Vi,Vjvolume of bubbleiandj,m3

Δyheight of control volume for calculation of coalescence and break-up frequency,m

α volume fraction

Subscripts

b bubble phase

g gas phase