Modeling Agglomeration Behavior in High Temperature Gas–Solid

Article pubs.acs.org/IECR

Modeling Agglomeration Behavior in High Temperature Gas−Solid Fluidized Beds via Monte Carlo Method Qiang Shi, Zhengliang Huang, Musango Lungu, Zuwei Liao, Jingdai Wang,* and Yongrong Yang State Key Laboratory of Chemical Engineering and Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China

ABSTRACT: Particles and bubbles are important discrete elements in gas−solid fluidized beds. The interactions between them affect the evolution of agglomerates. Based on the analyses of microscopic physical processes, mathematical models of particle coalescence and agglomerate breakage caused by bubble expansion were developed. Coupling with the constant-number Monte Carlo method, the evolution of particle size distribution was simulated. Modeling results matched well with cold-mode experiment results, which demonstrated that the approach is a promising way to predict agglomeration behavior in the fluidized beds. Also, the coalescence efficiency β0 and breakage efficiency βb extracted from Monte Carlo results can provide rate constants for the kernels in population balance modeling. Further investigations of the effects of process parameters show that decreasing bed temperature or increasing superficial gas velocity reduces the coalescence efficiency and increases the breakage efficiency, which are beneficial to prevent agglomeration.

1. INTRODUCTION Agglomeration is the sticking together of solid particles to form larger particles, which exists in a wide range of applications of fluidized beds, such as fluidized catalytic cracking, drying, solid fuel conversion, granulation, direct reduction of iron ore, biomass conversion, coal combustion, and gas phase polymer production.1,2 In some processes, especially fluidized bed spray granulation, agglomeration is used to enlarge particle size. However, for processes such coal combustion and gas phase olefin polymerization, agglomeration is harmful for industrial operation. In extreme cases, agglomeration can cause defluidization and an unscheduled shutdown of the reactor. Therefore, understanding, predicting, and controlling this phenomenon are important to avoid operational problems, reduced efficiencies, reactor downtime, and thereby monetary losses.3 In past decades, microscopic mechanisms of the formation, growth, and breakage of agglomerates have been extensively investigated. Researchers proposed that particles would coalesce if the adhesive force acting on them is stronger than the particle collision force and the fluid drag force, and vice versa.4−10 As for the breakup of agglomerates, Horio et al. theoretically investigated the pressure distribution around a Davidson bubble, and found that the upper area of it exerts an © 2017 American Chemical Society

expansion force on particles or agglomerates which can lead to the breakage of them.11,12 Based on the force balance analyses of bubble expansion force and van der Waals force, they estimated the agglomerate size in a bubbling fluidized bed of group C particles. Although the agglomeration behavior in the fluidized beds can be qualitatively predicted through the analyses of microscopic physical processes, it is not enough to accurately predict the size of agglomerate and dynamically monitor the evolution of agglomerate in industrial fluidized beds. Population balance modeling (PBM) is a common and effective approach to track the evolution of particle size distribution (PSD).13 In PBM, the particle fluid system is discrete, which follows a statistics rule. The occurrence of each dynamic event such as particle collision or breakup of agglomerate possesses a probability which is related to the operating conditions of the system such as temperature, pressure, superficial gas velocity, and properties of the particle such as diameter, porosity, and roughness. The probability can Received: Revised: Accepted: Published: 1112

December 18, 2016 January 6, 2017 January 11, 2017 January 11, 2017 DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

Article

Industrial & Engineering Chemistry Research

discrete microscale processes among particles, agglomerates, bubbles, and binder droplets which occur in series and parallel.21 Also, the MC method is a powerful tool to simulate complex particulate processes by taking into account many microprocesses that can possibly affect the microscopic behaviors of the system. Therefore, in the past decade, it has been applied to simulate the evolution of PSD in fluidized bed spray granulation in which particle collision, droplet evaporation, and droplet capture occur in series and in parallel.22−25 As for gas phase olefin polymerization, agglomeration is a very tough problem. Similar to fluidized bed spray granulation, agglomeration behavior in the polyolefin fluidized beds is also controlled by many microprocesses such as particle collision, breakup of agglomerate caused by bubble expansion, and particle attrition. The kernels are hard to obtain because they are significantly influenced by the operating conditions. Under these circumstance, the CNMC method is suitable to employ in the PBM. With the aid of it, the interactions between different microprocesses can be revealed. However, there is little in the literature about it. In the present study, the physical processes of particle collision and breakup of agglomerate caused by bubble expansion are first analyzed and the criterions of particle coalescence and breakup of agglomerates are defined. After that, we apply the CNMC method to simulate the evolution of PSD and investigate the influences of process variables, aiming to give insight into the agglomeration process in polyolefin fluidized beds.

be statistically described by kernels, as shown in eq 1, which includes the rate constant and the size dependent part. Solving the population balance equation (PBE) presented in eq 2, the evolution of PSD can be achieved.14 β(t , v , w) = β0(t ) β*(v , w) S(t , v) = S0(t ) S*(v) ∂n(t , v) 1 = 2 ∂t

∫0

dw − n(t , v) +

∫v

(1)

v

∫0

β(t , v − w , w) n(t , v − w , w) n(t , w) ∞

β(t , v , w) n(t , w) dw

∞

S(t , w) b(v , w) n(t , w) dw − S(t , v) n(t , v) (2)

The solution of the PBE is the key step of PBM. The present methods of solving the PBE include the moment method, partition method, and Monte Carlo (MC) method. The former two methods are based on the Euler method, which needs to discretize the particle diameter or obtain a specific initial function of PSD in advance. The accuracy of the partition method depends on the size of the discretized interval, so it has a discretization error. If the initial function of the PSD is complicated, it is difficult to apply the moment method to solve the PBE. However, this disadvantage can be overcome by the MC method where the algorithm is easy to implement. In the MC method, discretization is not required because its inherent discrete nature adapts naturally to processes involving discrete events.1 Details of agglomerate evolution can be obtained in the simulation. Owing to this advantage, the MC method is widely applied to simulate the variation of the PSD in fluidized beds. Smith et al. first adopted the constant-number MC (CNMC) method in PBM and employed it to solve the PBE with two different kernels: constant kernel and Brownian kernel.15 They found that the CNMC method can accurately describe the evolution of agglomerates. Lee et al. also applied the CNMC method to simulate particle coalescence and breakage, and then compared the modeling results under three different conditions and found that the CNMC method can precisely simulate the agglomeration process.16 Later, Lin et al. developed mass concentration and number concentration methods to establish the connection between the simulation box and the volume of the physical system in the CNMC methods.17 Results showed that the method based on mass concentration is superior. Friesen et al. incorporated a modified version of Gillespie’s fullconditioning algorithm to simulate the kinetics of aggregating and fragmenting particles.18 They stated that, through the improvement of the algorithm, the modeling efficiency is increased. Besides, the CNMC method is also applied to agglomerative crystal precipitation or coalescence process in oil slurries.19,20 Although the above-mentioned researchers achieved accurate results, they lacked the connections between physical mechanisms and simulation results. The kernels in PBM have to be obtained from experimental results and empirical correlations, which increases the difficulty of simulation. Aiming at these problems, researchers proposed that the connections between MC and the physical particulate system can be provided by the event-driven nature of the method.1 Taking the analyses of the results of discrete events as the starting point, the simulation can be realized without aggregation and breakup kernels. In addition, agglomerate formation has been recognized to be a complex network of

2. MICROSCALE PHYSICAL PROCESSES 2.1. Particle−Particle Collisions. Particle agglomeration is the outcome of direct particle−particle collisions.26 Therefore, we first developed the mathematical model of analyzing the outcome of particle−particle collisions. According to the present models proposed by researchers, binary particle collision is considered as the main mode of particle agglomeration.3−9 Thus, in this work, we only took binary particle collision into consideration, as shown in Figure 1. In

Figure 1. Schematic diagram of binary particle collision.

high temperature gas−solid fluidized beds, interparticle forces include the van der Waals force, solid-bridge force, elastic repulsive force, and electrostatic force. The solid-bridge force mainly contributes to the formation of agglomerates, while the elastic repulsive force promotes the separation of particles.2−9 During particle−particle collision, viscous flowing on the contact surface between particles forms a solid bridge, thereby the solid-bridge force.5 According to the force balance model, if the maximum repulsive force is stronger than the solid-bridge force, particle coalescence occurs and vice versa.5−10 Assume all the particle−particle collisions are elastic, where Fcoll is the maximum elastic repulsive force and Fad is the solidbridge force. According to the Hertz contact model, when particle A with radius R1 collides with particle B with radius R2, Fcoll can be calculated by eq 3: 1113

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research ⎛ 5m*V 2 ⎞3/5 rel ⎟ Fcoll = n⎜ 4 n ⎝ ⎠

where Db is the diameter of the bubble, ρa is the density of the agglomerate, da is the diameter of the agglomerate, nk is the coordination number for the agglomerate to agglomerate contacts, P̂ s is the dimensionless particle pressure around a bubble, and g is the acceleration due to gravity. In this work, the volume average of P̂ s over the region of r = Db/2 to (3/2)Db is −0.0515.11,12 The coordination number nk is calculated according to the correlation proposed by Nakagaki and Sunada.29 The diameter of the bubble is determined by the correlations reported by Mori and Wen,30 as shown in eq 9:

(3)

where n=

4 E*(R *)1/2 , 3

R* =

R1R 2 , R1 + R 2

m* =

m1m2 m1 + m2 (4)

Vrel is the relative velocity between two particles; m1 and m2 are the mass of the particles. E* is the effective elastic modulus which is defined by eq 5: E* = (1 − v 2)/E

D b = D bm − (D bm − D bo)e−0.3H /2Dt

(5)

Here

Here E is the particle elastic modulus and is assumed to be identical for all particles in the reactor. The solid-bridge force is related to the contact time and surface properties of particles, according to the correlations proposed by Rumpf:27

D bm = 0.652[A(Ug − Umf )]0.4 ⎡A ⎤0.4 D bo = 0.347⎢ (Ug − Umf )⎥ ⎣ N0 ⎦

Fad = πσneckys‐b 2 2

(ys‐b /d p*) = [(4/5)(γs/d p*) + (1/5π )Fcoll /d p* ]

d p* =

(6)

2R1R 2 R1 + R 2

where ηs and γs are the surface viscosity and surface tension force of the particles. σneck is the tensile strength of polyethylene, and ys‑b is the diameter of the solid bridge. The contact time tmax can be determined by the equation reported by Zheng et al.28

tmax = (R1 + R 2)/Vrel

(10)

where H, Dt, and A are the distance of the bubble from the distributor, the diameter, and the cross-sectional area of the fluidized bed. Ug is the superficial gas velocity, and Umf is the minimum fluidization gas velocity. N0 is the number of cavities on the distributor. Dbo and Dbm are the minimum and maximum diameters of the bubbles, respectively. As we only consider binary particle collision in this work and the newly formed agglomerate is regarded as a new “particle”, there is only one new solid bridge between particles. Correspondingly, the agglomerate would break up into two new “particles” if the bubble expansion force is stronger than the solid-bridge force. Therefore, similar to particle−particle collision, the frequency of interactions between bubbles and agglomerates must be determined in order to calculate the breakup rate of agglomerates. According to Kunii and Levenspiel,31 for a bubble with a diameter of Db, its frequency of occurrence can be calculated by the following equation:

2

(tmax /ηs)

(9)

(7)

2.2. Bubble Expansion. Bubbles are one of the most important elements in bubbling fluidized beds. In gas−solid fluidized beds, the interactions between bubbles and particles determine the formation rate of agglomerates. Horio et al. theoretically investigated the pressure distribution around a Davidson bubble.11,12 They found that particles around the upper zone of a bubble were exerted on a strong expansion force, as shown in Figure 2. Therefore, in order to give a precise description of the evolution of PSD, breakup of agglomerates caused by bubbles must be taken into consideration.

fb = 1.5(Ug − Umf )/D b

(11)

In a fluidized bed, small bubbles form at the upper area near the distributor and coalesce to grow bigger as they rise to the bed level, as shown in Figure 3. Their diameters vary with their positions in the reactor, so their frequencies are changing at different heights. Thus, we divided the heights of the bed level

Figure 2. Schematic diagram of bubble expansion.

The bubble expansion force exerted on the agglomerate is related to the diameters of bubbles and agglomerates, which can be calculated by eq 8: Fexp =

πD bρa g ( −Pŝ)da 2 2nk

(8)

Figure 3. Bubble distribution in a fluidized bed. 1114

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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average diameter of particles, ncoal is the number of successful coalescence of particle−particle collisions, and nbreak is the number of successful breakups of agglomerates. Np is the number of sample particles, which is equal to 15 000 in this simulation. The collision frequency is determined by the correlation proposed by Buffiere and Moletta,33 as shown in eq 17

into k layers, and the thickness of each layer is equal to the diameter of the bubble within it. Then the statistical bubble frequency in the reactor can be determined, as shown in eq 12: k

fD = b, i

∑ 1.5(Ug − Umf )/Db,i i=1

D b, i = D bm − (D bm − D bo)e−0.3H /2Dt i−1

H=

∑ Db,j

⎡ ⎛ ϕexp ⎞⎤⎛ ϕexp ⎞2 ⎟⎟⎥⎜⎜ ⎟⎟ Ug fcoll = 56400⎢1 − ⎜⎜ ⎢⎣ ⎝ ϕfix ⎠⎥⎦⎝ ϕfix ⎠

(12)

j=1

Here, ϕexp and ϕfix are solid volume fractions of the expanded bed and fixed bed, respectively, while ϕexp at different times is defined as

k

H0 =

(17)

∑ Db,i i=1

Np

where H0 is the height of the bed level, Db,i is the diameter of the bubble in the ith layer, and k is the number of layers. Based on the criterion of the border of the bubble, the porosity around a bubble is larger than 0.85.32 Therefore, in the affected region of the bubble, the number of particles can be determined by the following equation:

i=1

⎞ ⎛1 ⎞ 1 ⎛⎜ 1 1 π (3D b, i)3 − πD b, i 3⎟(1 − 0.85)/⎜ πd p3⎟ ⎠ ⎝6 ⎠ 2⎝6 6 3 D b, i = 0.325 3 dp (13)

Therefore, the frequency of interactions between particles and bubbles can be determined through the combination of eqs 12 and 13 k

∑ fD ng,i i=1

b, i

Pcoag = (14)

while, for the simulation system, the real frequency fsim‑rup should be defined as fsim‐rup

V = s frup Vr

Δt =

(18)

fcoll fcoll + fsim‐rup

−ln(ζ ) fcoll + fsim‐rup

(19)

(20)

where ζ ∈ (0,1) is a random number drawn from a uniform distribution in the unit interval. 3. Carry out the selected event and update the time t to t + Δt. If particle−particle collision occurs, then randomly select two particles i and j for collision and check the following: (a) If Fcoll > Fad, particles are taken to separate after collision. Then carry out the next event. (b) If Fcoll < Fad, particles are taken to coalescence. Then randomly generate another particle from the same particle size distribution and add it to the simulation box. The formed agglomerate is regarded as a new “particle”, and its size is computed by eq 21.

(15)

where Vs and Vr are the volumes of the simulation box and the real system.

3. SIMULATION ALGORITHM The CNMC method is applied in this simulation. The concrete steps are stated as follows. 1. Initialize the simulation parameters and operating conditions, including the particle temperature, superficial gas velocity, initial particle size distribution, number of sample particles, prefactor for collision frequency, and so on. The volume of the simulation box is proportionally reduced so as to maintain the same number density of particles (nf). Because the number of individual particles is changing within the agglomeration process, the volume of the simulation box (Vs) is varying, which can be described by eq 16:

dagg

⎡ ∑n d 3 ⎤1/3 k = 1 p, k ⎥ =⎢ ⎢⎣ 1 − εa ⎥⎦

(21)

where dp,k is the diameter of an individual particle in the agglomerate and εa is the porosity of the agglomerate. The diameter of the newly formed agglomerate, the formation time tcoag,i, and the sizes of corresponding solid bridges are recorded in matrices. If interaction between the “particle” and the bubble takes place, then randomly select a “particle” around a bubble and check the following:

⎛1 ⎞ n f = ρf /⎜ πdavg 3ρp ε⎟ ⎝6 ⎠ Vs = (Np + ncoal − nbreak )/n f

1 πd p, i 3(1 − εp)/Vs 6

where εp is the porosity of polyethylene particles. Carlos et al. reported that the distribution of the relative velocity between particles obeys the Maxwell distribution.34 For simplification, the distribution of particle collision velocity is assumed to be normal distribution. 2. Select an event, particle−particle collision or interaction between particle and bubble, to take place according to their probabilities and compute the time step as described by Gillespie, shown in eq 20.35 In the process of selecting an event, we first drew a random number r from a uniform distribution. If r > Pcoag, we will select particle−particle collision; otherwise we select the interaction between particle and bubble.

ng, i =

frup =

∑

ϕexp =

(16)

where ρf is the fluidization density of the reactor, ρp is the density of polyethylene particles, ε is the bed voidage, davg is the 1115

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research

temperature of the inlet gas was controlled by the heat exchanger. In order to measure the axial temperature distribution of the bed, six temperature probes were mounted with equal intervals along the bed. Linear low density polyethylene (LLDPE) with diameters of 0.5−1.0 mm was used as the bed material. Figure 6 presents the initial particle size distribution of the bed material.

(a) If an individual particle is selected, then carry out another event. (b) If an agglomerate is selected, then compare the solidbridge force Fad′ with the bubble expansion force Fexp. As the solid bridge between particles continued to grow after the formation of agglomerate, the solid-bridge force became stronger. At time t, the solid-bridge force Fad′ should be calculated by eq 22: Fad′ = Fadt /tcoag, i

(22)

Thus, if Fexp > Fad, the agglomerate will be broken into two parts. At the same time, a random individual particle is eliminated from the simulation box. If Fexp < Fad, the next step will be carried out. The simulation algorithm is shown in Figure 4.

Figure 6. Initial particle size distribution of bed material (LLDPE).

In each experiment, the fluidized bed was preheated to the required temperature under the set superficial gas velocity, and then 460 g of LLDPE was fed into the bed immediately. When the bed temperature reached the set temperature and remained stable, the sample probe placed at a height of 11 cm from the distributor was used to remove bed material for further analyses. Eight samples with an average mass of 7−9 g were taken out for each experiment. Image analysis methodology was used to measure the particle size distributions of the samples.36 Then agglomerates within each sample were sieved and weighed. Tables 1 and 2 show the modeling and experimental parameters used in this work.

Figure 4. Simplified diagram of the simulation algorithm.

4. EXPERIMENTAL SECTION The experimental results were used for the validation of the simulation. Figure 5 shows the cold-mode fluidized bed which is made of glass, with a height of 68 cm and an inner diameter of 9.8 cm. Atmospheric air was used as fluidization gas. The

Table 1. Parameters for Simulation and Experiments variable

Ug

Tb (°C)

label

Ug

2.5Umf 2.5Umf 2.5Umf 2.0Umf 3.0Umf

90 93 96 93 93

01 02 03 04 05

Tb

5. RESULTS AND DISCUSSION 5.1. Microscale Interaction Forces. The particle collision force, solid-bridge force, and bubble expansion force exerted on particles or agglomerates are the most important microscale interaction forces in gas−solid fluidized beds. They determine the evolution of agglomerates in the reactor. Concretely, the relative values of the particle collision force and the solid-bridge force affect the generation rate of agglomerates, while the relative value of the bubble expansion force and solid-bridge force influences the breakup rate of agglomerates. Figure 7 presents the probability distributions of the particle collision force and the solid-bridge force in 20 000 times of particle− particle collision events under the operating condition label 02. The probability distributions of Fcoll and Fad present normal

Figure 5. Schematic diagram of experiment apparatus. 1116

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research Table 2. Parameters in the Simulation variable ρp (kg/m3) εa (dimensionless) v (dimensionless) E (MPa) σneck (MPa) H0 (m) ε (dimensionless)

variable γs (N/m) A (m2) N0 (dimensionless) Umf (m/s) Dt (m) ηs (1 × 1011 Pa·s) ρf (kg/m3)

920 0.302b 0.42 28.6,c 26.98,c 25.92c 30a 0.33 0.30b

0.0275a 7.36 × 10−3 428 0.215 0.0968 1.13,d 1.09,d 1.06d 595.05

a

From ref 4. bPorosities of particle and agglomerate measured by a mercury porosimeter (AutoPore IV 9510, USA). cYoung’s moduli of PE particle under temperatures of 90, 93, and 96 °C which were measured by a universal material testing machine (Zwick/Roell Z020, Germany). dSurface viscosities of PE particle under temperatures of 90, 93, and 96 °C.37

expansion force exerted on particles.11,12 Apparently, larger size bubbles are with a stronger bubble expansion force and the agglomerate is more inclined to break. Through the computation of these microscale interaction forces, agglomeration tendencies in the fluidized beds under specific conditions are obtained, which also paves the way for the quantitative characteristics of the formation of agglomerates. 5.2. Evolution of Agglomerates. 5.2.1. Model Validation. To validate the model, a “base case” was selected from the experiments which is corresponding to the operating condition label 03. Then all parameters were set to the corresponding values of label 03 and the simulation time was unified as 270 s. As shown in Figure 9, selecting the mass fraction of

Figure 7. Probability distributions of interparticle forces (label 02).

distributions, and the overlapping parts of the two curves demonstrate that, in a random particle collision event, Fad is stronger than Fcoll and particles tend to coalesce after collision. Therefore, from the probability distributions of Fcoll and Fad, we can qualitatively predict the agglomeration behavior in the fluidized beds operated under certain conditions. As stated above, the bubble expansion force is the determinative force of agglomerate breakage. Figure 8 shows the bubble size varying with its distance from the distributor under the same operating condition. It can be concluded that bubble size increases with the increase of the distance of the bubble from the distributor. According to the correlations proposed by Horio et al., we also compute the bubble

Figure 9. Variation of mass fraction of agglomerates with time (label 03).

agglomerates as the standard of assessment, it is found that the mass fraction of agglomerates increased with time and agreed well with the cold-mode experimental results. This demonstrated that this approach is qualified to predict the evolution of agglomerates in fluidized beds. It should be mentioned that sampling induced errors to the measurement of the mass fraction of agglomerates. This is because the total loading weight of LLDPE in the bed was limited. With the increase of the times of sampling, the amount of bed material decreased after agglomerates formed were taken out, which prevented them from growing. Therefore, there were no large agglomerates formed during the experiments. During 250−270 s, the mass fraction of agglomerates almost remained the same, which can be explained as above. For further characterization of agglomerate evolution, image analysis methodology was applied to measure the particle size distribution of the material sampling at 90 s, as shown in Figure

Figure 8. Variations of bubble size with increase of its distance from distributor and corresponding expansion force (label 02). 1117

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research 10. Compared with the initial particle size distribution of bed material, the distribution was wider due to the generation of

Figure 12. Variation of breakage efficiency with time.

0−10 s, the coalescence efficiency sharply increased to 0.0057. This is because, at the early stage of simulation, the number of collision events was not large enough. Every successful coalescence would result in a remarkable increase in β0(t). During t = 10−25 s, the coalescence efficiency fluctuated around 0.00245, and slowly increased to 0.0025 and remained stable after 25 s. Wang et al. reported that small particles are easy to coalesce in gas−solid fluidized beds.36 At the initial stage of agglomeration, coalescence between small−small particles or small−large particles mainly contributed to the formation of agglomerates. Therefore, at t = 0−25 s, as the mass fraction of small particles was high, successful coalescence of particles occurred frequently and the coalescence efficiency was higher. With the consumption of small particles and the formation of agglomerates, the average diameter of particles increased, and subsequently the coalescence efficiency decreased. However, owing to the breakup of agglomerates caused by bubble expansion, small particles separated from agglomerates turned into individual particles. Also, as described in section 3, once particles coalesced, another new particle generated from the initial particle distribution would be added to the system. Therefore, in the late stage of agglomeration, the coalescence efficiency stably fluctuated. Figure 12 gives the variation of breakage efficiency with time. We should note that computation of breakage efficiency was based on the formation of agglomerates. During t = 0−25 s, the breakage efficiency dropped from 0.145 to 0.014 and then slowly decreased. The solid bridge in the newly formed agglomerates was too weak to sustain the expansion of bubbles, so at the beginning agglomerates tended to break. However, solid bridges in the surviving agglomerates continued to grow, which enhanced the solid-bridge forces within them. Therefore, as time went on, agglomerates were difficult to break and the breakage efficiency decreased. This reminds us that if no effective action is carried out to prevent agglomeration formation, irreversible deterioration of fluidization quality will cause bed defluidization. 5.3. Effects of Operating Conditions. 5.3.1. Bed Temperature. In order to study the effect of bed temperature, the superficial gas velocity was kept constant (2.5Umf) in each experiment. The bed temperature varied in the range 90−96 °C, and the corresponding labels are label 01, label 02, and label 03. As presented in Figure 13, under higher bed temperature, the mass fraction of agglomerates was larger and increased more quickly with time. With increasing bed temperature, the surface viscosity (ηs) and Young’s modulus (E) of polyethylene

Figure 10. Comparison of particle size distributions at t = 90 s for simulation and experiments.

agglomerates. Owing to the errors induced by sampling, the particle size distribution of experimental materials was narrower than that in simulation. However, they both reflected the evolution of agglomerates. 5.2.2. Coalescence Efficiency and Breakage Efficiency. The acquisition of kernels is the difficulty of PBM; thus the present kernels are mostly empirical correlation or extracting constants from experiments. However, if the operating condition changes, the former kernels will not be able to be used. Therefore, in this simulation, we computed the coalescence efficiencies and breakage efficiencies under different operating conditions, which can not only provide rate constants and increase the efficiency of PBM, but also help us with the dynamic monitoring of the agglomeration behavior in fluidized beds. At given time t, if a particle−particle collision event occurs nc times and particles successfully coalesce na times, then the coalescence efficiency can be defined as β0(t) = na/nc. Correspondingly, for the breakup of agglomerates, if the total number of interactions between bubbles and particles is nb and the number of successful breakups of agglomerates is nbb, then the breakage efficiency is defined as βb(t) = nbb/nb. Figures 11 and 12 present the changes of coalescence efficiency and breakage efficiency with time. As shown in Figure 11, during t =

Figure 11. Variation of coalescence efficiency with time. 1118

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research

Figure 13. Variations of mass fractions of agglomerates under different bed temperatures.

Figure 15. Variations of breakage efficiencies under different bed temperatures.

decreased.37 It can be derived from eq 6 that the solid bridge on the contact surface between particles during collision grew more quickly, which resulted in a larger solid bridge and stronger solid-bridge force. Also, as the Young’s modulus of particles decreased with increasing temperature, the elastic repulsive force decreased, which can be derived from eqs 3−5. Therefore, particles easily coalesced after collision, which promoted the formation of agglomerates and led to the increase of coalescence efficiency. Similar to the results shown in Figure 11, Figure 14 presents the coalescence efficiencies

Figure 16. Variations of mass fraction of agglomerates under different superficial gas velocities.

Figure 14. Variations of coalescence efficiencies under different bed temperatures.

under different bed temperatures. It is seen that the stable values of β0(t) increased with increasing bed temperature. Besides, with a stronger solid-bridge force, it is hard for bubbles to break the solid bridge within agglomerates. Thus, from the modeling results shown in Figure 15, we can conclude that the variations of breakage efficiency present an opposite trend. 5.3.2. Superficial Gas Velocity. In the investigation of the effect of superficial velocity on the evolution of agglomerates, the bed temperature was set to 93 °C, while the superficial gas velocity changed from 2Umf to 3Umf. The corresponding labels are label 04, label 02, and label 05, respectively. The results are presented in Figures 16−18. Increasing superficial gas velocity led to a decrease of coalescence efficiency and an increase of breakage efficiency and consequently fewer agglomerates in the reactor. On one hand, according to the kinetic theory of

Figure 17. Variations of coalescence efficiency under different superficial gas velocities.

granular flow, particle fluctuation velocity increased with superficial gas velocity.38 As a result, the relative velocity between particles was larger. From eq 3, it is apparent that the maximum repulsive force increased with increase of particle relative velocity Vrel. Particles tended to separate after collision deducing from the above particle coalescence model, which resulted in less formation of agglomerates and lower coalescence efficiency. On the other hand, the diameter of 1119

DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

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Industrial & Engineering Chemistry Research ORCID

Jingdai Wang: 0000-0001-8594-4286 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support was provided by the Project of National Natural Science Foundation of China (91434205), the National Natural Science Foundation of China (21406194), the National Science Fund for Distinguished Young Scholars (21525627), the Natural Science Foundation of Zhejiang Province (Grant LR14B060001), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130101110063).

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Figure 18. Variations of breakage efficiency under different superficial gas velocities.

the bubble increased with increasing of superficial gas velocity (Ug) as indicated in eqs 9 and 10. The bubble expansion force exerted on particles became stronger. Solid bridges within the agglomerate were easier to break. Therefore, the breakage efficiency is higher.

6. CONCLUSIONS Based on the analyses of the interactions between particles, the force balance model was proposed to judge the outcome of particle−particle collision where the solid-bridge force and the repulsive force caused by particle collisions were taken into consideration. Considering the bubble expansion force as the determinative force of agglomerate breakup, the diameters and distribution of bubbles in the bed were first calculated. Then mathematical models of agglomerate breakage and frequency of the interactions between particles and bubbles were developed. In the last step, we applied the CNMC method to simulate the evolution of agglomerates in high temperature gas−solid fluidized beds and validated the modeling results with experiments. Results showed that the mass fraction of agglomerates increased linearly with time and agreed well with the experiments. The CNMC method coupled with microscale physical models can successfully predict the agglomeration behavior in fluidized beds. Also, variations of coalescence efficiency and breakage efficiency with time can reflect the details of agglomerate evolution and provide rate constants for PBM. Further investigations of process parameters reminded us that decreasing bed temperature or increasing superficial gas velocity can prevent agglomerates from further evolution. In summary, as the increase of the minimum fluidization velocity with increasing particle diameter was not accounted for in the simulation together with the errors induced by the sampling, deviations exist between experiments and modeling results. However, the CNMC method coupled with physical models can develop the connections between a real particulate system and simulation. It is useful to identify the sensitive variables or microprocesses that dominate the evolution of agglomerates, thus providing guidance for industrial operation.

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AUTHOR INFORMATION

Corresponding Author

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NOTATIONS Fcoll = repulsive force (N) Fad = solid-bridge force (N) Fad′ = solid-bridge force at time t (N) Vrel = relative collision velocity between particles (m/s) Vs = volume of simulation box (m3) Vr = fluidized volume of particles in reactor (m3) R = radius of particle (m) dp = diameter of particle (m) da = diameter of agglomerate (m) davg = average diameter of particles in simulation box (m) m = particle mass (kg) E = modulus of particle (Pa) E* = efficient modulus of particle (Pa) v = Poisson’s ratio of particles σneck = tensile strength of particle (Pa) ys‑b = radius of solid bridge between particles (m) γs = surface tension force of particles (N/m) ηs = surface viscosity of particles (Pa·s) tmax = contact time during particle collision (s) ρa = density of agglomerate (kg/m3) ρp = density of particles (kg/m3) ρf = fluidized density (kg/m3) g = gravity acceleration (m/s2) P̂ s = dimensionless pressure around a bubble Pcoag = probability of particle collision ε = bed voidage εa = porosity of agglomerate εp = porosity of particle Db = diameter of bubble (m) Dbm = maximum diameter of bubble in fluidized bed (m) Dbo = minimum diameter of bubble in fluidized bed (m) Dt = diameter of fluidized bed reactor (m) Db,i = diameter of bubble in ith layer (m) A = sectional area of fluidized bed reactor (m2) Ug = superficial gas velocity (m/s) Umf = minimum fluidization gas velocity (m/s) H0 = bed level (m) k = number of divided layers f b = bubble frequency f Db,i = frequency of bubble with diameter of Db,i ng,i = number of particles around bubble with diameter of Db,i f rup = frequency of interactions between bubble and particles in fluidized bed reactor fsim‑rup = frequency of interactions between bubble and particles in simulation box fcoll = collision frequency of particles DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121

Article

Industrial & Engineering Chemistry Research

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nf = number density of particles in reactor ncoal = successful coalescence of particles nbreak = successful breakup of agglomerate nk = coordination number for agglomerate to agglomerate contacts Np = number of particles in simulation box N0 = number of cavities on distributor ϕexp = solid volume fraction of expanded bed ϕfix = solid volume fractions of fixed bed r = random number from uniform distribution ζ = random number from uniform distribution tcoag = time of particle coalescence Δt = time step of simulation (s) β0(t) = coalescence efficiency βb(t) = breakage efficiency

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DOI: 10.1021/acs.iecr.6b04893 Ind. Eng. Chem. Res. 2017, 56, 1112−1121