Prediction of solids residence time distribution in cross-flow bubbling fluidized bed

Powder Technology 320 (2017) 555–564

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Prediction of solids residence time distribution in cross-flow bubbling fluidized bed Shujun Geng a,b, Yanan Qian a,b, Jinhui Zhan a, Hongling Zhang c, Guangwen Xu a,⁎, Xiaoxing Liu a,b,⁎⁎ a b c

State Key Laboratory of Multi-phase Complex System, Institute of Process Engineering, Chinese Academy of Science, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China School of Mathematics, Southwest Jiaotong University, Sichuan 610031, China

a r t i c l e

i n f o

Article history: Received 14 March 2017 Received in revised form 15 July 2017 Accepted 27 July 2017 Available online 2 August 2017 Keywords: Solids residence time distribution (RTD) Bubbling fluidized bed Computational fluidized dynamics (CFD) Semi-empirical approach Dual fluidized bed

a b s t r a c t Cross-flow bubbling fluidized beds (BFBs) have been widely used in dual fluidized bed systems such as chemical and heat looping. Understanding the residence characteristics of solids in such system is important for better design and optimization of reactors. Previously we experimentally measured the residence time distributions (RTDs) of sands in a cross-flow rectangular BFB by using coal particles as tracer. A computational investigation of solids RTD using multi-fluid Eulerian method combined with the species transport equation showed that the RTD of sands could be correctly represented by that of coal particles, and the simulation results well agreed with experimental data. Parametric studies demonstrated that, under the considered operation conditions, the influence of tracer injection time period on the predicted solid residence times was nearly ignorable. Simulation results revealed that in the investigated cross-flow BFB the solids RTD is closely related to solids inventory and solids flux. Through proper data processing, it was found that the descending part of solids RTD profile can be uniquely fitted by an empirical exponential function. A semi-empirical approach was thus developed and further validated, for the first time in the literature, to predict the entire profile of solids RTD, in which the ascending part of solids RTD profile is obtained through CFD simulation whereas the descending part is given by the fitted empirical exponential function. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Gas–solid fluidized bed has been widely adopted to handle various chemical processes due to its excellent mass and heat transfer characteristics, high gas and solids throughputs, and continuous powder handling capability. The conversation and selectivity of chemical reactions occurring in gas–solid fluidized beds are highly dependent on the solids mixing behavior in the reactors [1]. Solids residence time distribution (RTD), reflecting solids mixing behavior and flow hydrodynamics, is one of the key parameters for evaluating the reaction performance in fluidized beds. Therefore, for proper design, optimization, and scale-up of fluidized bed reactors, it is crucial to gain understanding and further accurate prediction of solids RTD in a gas–solid fluidized bed system. Experimentally, stimulus impulse technique has been widely applied to investigate the solids RTD in fluidized beds. In this technique, a small amount of tracer particles, whose properties should be as close as those of bulk bed material, are injected into the system within a short time period after the system achieves a steady state. The ⁎ Corresponding author. ⁎⁎ Correspondence to: X. Liu, University of Chinese Academy of Sciences, Beijing 100049, China. E-mail addresses: [email protected] (G. Xu), [email protected] (X. Liu).

http://dx.doi.org/10.1016/j.powtec.2017.07.085 0032-5910/© 2017 Elsevier B.V. All rights reserved.

concentration of tracer particles is then measured at some downstream position to obtain a RTD curve. Limited by detection or separation techniques, the demanded tracer properties and short tracer injection time are generally difficult to be guaranteed in experiments [2,3]. In the literature, various kinds of tracer particles have been chosen by different researchers, such as radioactive particles [2], colored particles [4], magnetic particles [5] and particles with different sizes [6]. In the past decades, many researchers have concentrated on how to improve the accuracy of experimental measurement on the choice of tracer particles and detection or separation methods of tracer particles [7,8]. For the solids, RTD in fluidized beds itself, researchers usually focused only on the influences of operation conditions such as superficial gas velocity or solids flux [8–10]. Harris et al. [8] used a fast tracer response technique to investigate the influences of superficial gas velocity and solids flux on solids RTD in a circulating fluidized bed (CFB) combustor. Chan et al. [10] investigated the solids RTD in dilute and core-annular fluidization regimes and proposed empirical correlations for them to predict the variance of RTD spread with superficial gas velocity. Van de Velden et al. [1] analyzed solids RTD and back-mixing behavior within extensive ranges of superficial gas velocity and solids flux, and examined the transition line between mixed and plug flow in a CFB riser. With fast development of high-performance computer technology, Computational Fluid Dynamics (CFD) has been increasingly employed

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as an efficient and powerful method to investigate solids RTD in fluidized beds. Compared with physical experiments, CFD simulations have the advantages that the injection conditions and properties of tracer particles can be precisely controlled. Wu et al. [11] simulated the solids RTD in fluid catalytic cracking (FCC) processes accommodated in a riser or downer reactor using the Eulerian–Lagrangian method. Their simulation results showed that the formation of particle cluster had notable influence on the spatial distribution of particle residence time. Shi et al. [12] investigated solids RTD in the riser of a CFB using a Computational Particle Fluid Dynamics (CPFD) approach and found that solids back-mixing mainly took place in the lower part of the riser. Based on the Eulerian– Eulerian models including with the species transport equation, Hua et al. [13] discussed the influences of inter-phase drag model, solids diffusion coefficient and tracer injection time on the predicted solids RTD. Liu et al. [14] investigated the influences of internals on the predicted solids RTD in FCC strippers. Li et al. [15] discussed the influences of jetting conditions on back-mixing behavior of solids in jet circulating fluidized beds. Although in the literature there are many reports about experimental and numerical studies on solids RTD in fluidized beds, most of the previous work focused on CFB riser systems. Dual fluidized bed (DFB) technology has been widely applied for efficient thermochemical conversation of fuels and chemical looping processes, which usually consists of a cross-flow bubbling fluidized bed (BFB) and a riser [16,17]. Different from conventional BFB, in the BFB of a DFB system the main flow direction of solids is perpendicular rather than parallel to that of gas. The gas enters the BFB from the bottom and flows out from the top of the bed, while raw fuel (and also the circulated heat carrier) is continuously fed into the BFB on one side and discharged from the opposite side directing to the riser. The residence time distribution of fuel particles determines the reaction efficiency in the BFB, then the stability and performance of a DFB system. To gain a basic understanding of solids RTD in cross-flow BFB systems, we previously conducted experiments to investigate solids RTD in a few of cross-flow BFBs under different operation conditions [18,19]. In our experiments, coal particles, whose density was different from that of bulk bed materials (sands), were chosen as tracer particles. Through CFD simulation, the basic objective of this research is to identify if the residence time of sands in the investigated system can be correctly represented by the measured residence time of coal particles. The residence time of solids in the investigated cross-flow BFB is rather long, making the CFD simulation extremely time-consuming. The second objective of this work is thus to develop an empirical method to correctly predict the long tail of solids RTD in the investigated

cross-flow BFBs. The latter represents the first quantification attempt in the literature, whereas this prediction is highly needed for actual reactor design and optimization of heat and chemical looping processes. This paper is organized as follows. The simulated system and adopted analysis method are described in Section 2. This is followed in Section 3 by a brief introduction of the simulation details. Experimental validation and parametric investigation are presented in Section 4. Based on the simulation results, in Section 5 a semi-empirical approach is developed to predict solids RTDs. A brief summary of conclusions is given in Section 6. 2. Simulation system The simulation system was experimentally investigated by Gao et al. [18,19]. In the experiments, solids RTDs were studied in a rectangular Plexiglas bubbling fluidized bed under ambient temperature and atmospheric pressure. Fig. 1 gives a schematic diagram of the experimental apparatus. The bed was 450 mm in width, 200 mm in height and 40 mm in thickness. The bulk bed materials were silica sand with a mean diameter of 0.2 mm and density of 2600 kg/m3. Coal particles with a mean diameter of 0.9 mm and density of 1405 kg/m3 were chosen as tracer particles. In the experiments, silica sands were continuously fed into the bed through a charging bucket over the solids inlet and flowed out of the bed from the solids outlet at the opposite side. In front of the solids outlet a vertical plat baffle with distances to both bottom and right walls of 23 mm was installed to increase solids residence time and to avoid the possible short-cutting flow of solids. The solids inventory in the bed was controlled by adjusting the position of solids outlet. The bed material was fluidized by air introduced from the bottom of the fluidized bed. After stable fluidization was achieved, tracer particles with an amount of 2% of bulk bed material weight were introduced into the system in a short time period. At the same time, the discharged bed material at the outlet was continuously collected with a time interval of 30 s to separate tracer particles from the collected samples via mtr ðtÞ sieving. The mass fraction of tracer was CðtÞ ¼ mtr ðtÞþm , and the sand ðtÞ

RTD of the tracer was then determined by the time profile of its mass fraction, as. Eðt Þ ¼

Gs C ðt Þ; M tr

ð1Þ

where Gs is the mass flux of sand particles (kg/s), Mtr is the total mass of tracer particles (kg). The unit of E(t) is then s−1. Based on E(t), the mean

Fig. 1. Schematic diagram of simulation system.

S. Geng et al. / Powder Technology 320 (2017) 555–564

residence time tm and standard variance σt were respectively calculated by, Pn t i Eðt i Þ ; t m ¼ Pi¼1 n i¼1 E ðt i Þ σ 2t

ð2Þ

Pn 2 t i Eðt i Þ −t 2m : ¼ Pi¼1 n i¼1 E ðt i Þ

ð3Þ

The coefficient of variance (dimensionless form of variance) σθ was defined as the ratio of standard variance to mean residence time: σt σθ ¼ tm

ð4Þ

Buffham and Mason [20] suggested that σθ is an appropriate parameter to measure the spreading of RTD. When σθ approaches 1, the flow state tends to mix perfectly, and when σθ approaches 0, the flow state can be considered as plug flow. The median residence time, t0.5, is defined as the time to recover 50% of the introduced tracer particles and can be calculated as follows, Z 0

t 0:5

Eðt Þdt ¼

1 : 2

ð5Þ

The detailed information of the simulated system is summarized in Table 1. 3. CFD model The multi-fluid Eulerian model in the commercial CFD software package Fluent 6.3.26 was used to model the investigated cross-flow bubbling fluidized bed system. In the Eulerian approach, both gas and solids are treated as fully interpenetrating continua. The mass and momentum conservation equations are then solved for each phase with appropriate closure relations. The kinetic theory of granular flow, which assumes the random motion of particles is analogous to the motion of molecules in gas, was used to close the governing equation for solid phase [21,22]. The inter-phase momentum exchange between gas and solid phases was estimated using the Gidaspow gas–solid drag model [23]. Our previous study showed that using this drag model the mixing behavior of coal-sand mixture in the currently investigated system could be successfully predicted [24]. To study the residence time of tracer particles in the bed, additional species transport equation was introduced, as shown in Eq. (T13). Species transport model has been widely used to simulate solids RTD [13,25]. Table 2 summarizes the conservation equations and constitutive relations used in this study. More details could be found in the Fluent 6.3.26 manual. In all the simulations, the set of governing equations were discretized by QUICK scheme and second-order upwind scheme so as to better resolve

Table 1 Summary of simulation conditions. Parameter

Value

Bed width W (m) Bed thickness T (m) Bed height H (m) ds (mm) dtr (mm) ρs (kg/m3) ρtr (kg/m3) umf (m/s) u/umf Gs (kg/h) tinj (s)

0.45 0.04 0.07, 0.10, 0.13 0.20 0.20 2600 1405, 2600 0.0368 4, 6, 8 18.75, 37.50, 55.75 1, 2, 40

557

Table 2 Governing equations and constitutive laws. Gas and solid mass continuity equation (i denotes gas g and solid p), ∂ ! ðεi ρi Þ þ ∇∙ðεi ρi u i Þ ¼ 0 ∂t Momentum conservation equation, ∂ ! ! ! ! ! ðεg ρg u g Þ þ ∇∙ðεg ρg u g u g Þ ¼ −εg ∇p þ ∇∙ τ̿ g þ εg ρg g þ βð u p − u g Þ ∂t ∂ ! !! ! ! ðεs ρs u s Þ þ ∇∙ðεs ρs u s u s Þ ¼ −εs ∇p−∇ps þ ∇∙ τ̿ s þ εs ρs g þ β i ð u g − u s Þ ∂t

T1

T2 T3

where,

h i ! ! T 2 ! ̿ τ̿ g ¼ εg μ g ∇ ug þ ð∇ ug Þ − ð∇∙ ug Þ I 3 h i ! ̿ ! !T 2 ! ̿ τ̿ s ¼ ð−ps þ εs λs ∇∙us Þ I þ εs μ s ∇us þ ð∇us Þ − ð∇∙us Þ I 3 Solid pressure, ps= εsρsΘs + 2ρs(1 + e)ε2s g0Θs

T4 T5

T6

Solid phase bulk viscosity, rffiffiffiffiffiffi 4 Θs λs ¼ εs ρs ds g0 ð1 þ eÞ 3 π

T7

Solid phase shear viscosity, rffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi i2 p sinФ 4 Θs 10ρs ds Θs π h 4 þ 1 þ εs g 0 ð1 þ eÞ þ spffiffiffiffiffiffiffi μ s ¼ εs ρs ds g 0 ð1 þ eÞ 5 5 π 96ε s g 0 ð1 þ eÞ 2 I2D

T8

Solid frictional viscosity p sinФ μ s; f ¼ s pffiffiffiffiffiffiffi 2 I2D

T9

Solid frictional pressure ðεs −εs;c Þn , where F = 0.05, n = 2, q = 5. ps; f ¼ F ðεs; max −εs;c Þq

T10

Radial distribution function, h  ε 1=3 i−1 s g 0 ¼ 1− εs; max

T11

Granular temperature equation,   ̿ ! 0= −ps I þ τ̿ s : ∇ u s −γΘs

T12

Specie transport equation, ∂ ! ðεs ρs Y tracer Þ þ ∇∙ðε s ρs u s Y tracer Þ ¼ ∇∙ðε s ρs Ds ∇Y tracer Þ ∂t

T13

Inter-phase drag coefficient, ! ! (3 ε g ρg ε s j u g − u s j εg −2:65 εg N 0:8 4 CD ds , βi ¼ ! ! εs 2 μ g ρg ε s j u g − u s j 150 þ 1:75 ε g ≤ 0:8 2 ds ε g ds  ! ! 0:678 24 Þ Res ≤ 1000 , Re ¼ εg ρg j u g − u s jds C D ¼ Res ð1 þ 0:15Res s μg 0:44 Res N1000

T14

the transient flow behavior [26]. The time step Δt was set as 0.0001 s, and the number of iterations per time step was set as 20. The detailed simulation conditions are presented in Table 3. Similar to the experiments, in all simulations sand particles were continuously fed into the bed through the solids inlet located at the left-up side of the bed and flowed out of the system through the solids outlet at the right side of the bed. At the solids inlet, constant inflow conditions were imposed according to the target solids mass flux Gs (the solids volume fraction εs was set to be 0.5). The solids outlet was set to be atmospheric pressure. A uniform velocity profile for gas phase was specified at the bottom gas inlet. No-slip wall boundary conditions were used in this study. We checked that solid-wall boundary conditions had negligible influence on the predicted solids RTD in the investigated system. Initially, certain bed material (silica sands) was packed in the bed with εs = 0.6. Gas and sand flows were continuously introduced into the system through the inlets as to establish the main flow.

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S. Geng et al. / Powder Technology 320 (2017) 555–564

4. Simulation results

Table 3 Parameters for numerical simulations. Parameter

Value

Specularity coefficient Particle-particle restitution coefficient Angle of internal fraction Initial bed voidage Initial bed height (m) Packing limit Friction packing limit Time step, Δt Discretization scheme

1 0.9 30 0.4 0.90 0.63 0.61 1e−4 s Quick

To evaluate if the steady fluidization state had been attained, for each simulation case the solids volume fraction at a vertical position of 50 mm and the solids inventory were monitored. Fig. 2 shows that the solids volume fraction achieved a steady value shortly after fluidization, while it took about 80 s before the solids feeding and discharging rates achieved a dynamic equilibrium. Therefore, in all the simulations only when the solids inventory achieved a nearly constant value were the tracer particles then injected into the bed. Fig. 3 gives the contours of tracer mass fraction in the cross-flow BFB at different moments. At the very beginning, the tracer particles were mainly concentrated in the region close the tracer injection inlet (see Fig. 3a and b). As time went on, tracer particles gradually transported to right side and flowed out of the system (Fig. 3c–f). Fig. 3c–f demonstrate that the vertical distribution of tracer mass fraction was nearly uniform, suggesting that tracer and sand particles mixed well in the investigated system. It can also be seen that the residence time of tracers was rather long. At the moment of 580 s (Fig. 3f), there were still considerable amount of tracer particles inside the BFB. The rather long solids residence time was due to the fact that the main flow direction of solids was perpendicular to that of gas. This is totally different from the case of traditional BFB systems and also CFB in which the main flow directions of solids and gas are parallel, leading to rather short solids residence time (generally about several seconds) [12–14]. The mass fraction of tracer particles at the outlet was monitored at a frequency of 50 Hz once they were injected into the system. Due to the small amount of tracer particles and also the violent movement of solid particles in this cross-flow BFB, the time-series variation of tracer mass fraction at the solids outlet presented severe fluctuations. Similar to our strategy used in experiments, the tracer mass fraction with temporal averaging time of 30 s was used to calculate E(t), i.e., C(t) in Eq. (1) was a smoothing result of 1500 monitored data points of the tracer mass fraction at the outlet.

4.1. Experimental validation Before comparing the simulation results with the experimental data, a grid-independent study was firstly performed. Both 2D and 3D simulations with uniformed grids sizes of 4 mm and 8 mm have been conducted (Case 1–Case 4). Fig. 4a gives the predicted solids RTD profiles. The corresponding quantitative results (tm, t0.5 and σθ) are presented in Table 4. It can be seen from Fig. 4a that in all simulations E(t) quickly reached a peak value and then monotonously decreased. The time when E(t) achieved its peak value was around 80 s, which was b6% of the time needed for all the tracer particles to leave the system. The long tail characteristic of solids RTD profile clearly indicates that the solids back-mixing in the investigated cross-flow bubbling fluidized bed is significant. From Table 4 one can find that the grid independence has been achieved when the grid size was 8 mm. For 2D simulations, the differences between tm, t0.5 and σθ predicted using fine and coarse grids were 0.6%, 1.4% and 1.2%, respectively. In addition, those for 3D simulations were 2%, 5.1% and 12%. The residence times predicted by 2D simulations were slightly smaller than those predicted by 3D simulations, but the differences were in an acceptable range. The value difference between tm (t0.5) predicted by 2D and 3D simulations with a grid size of 8 mm was 5.2% (12.2%), and that between σθ was 3.5%. The experimental data for H = 100 mm, Gs = 18.75 kg/h (Exp. 1) and H = 130 mm, Gs = 18.75 kg/h (Exp. 2) are also presented in Fig. 4 and Table. 4. It can be found that the simulation results are in acceptable agreement with the experimental results. For H = 100 mm, the quantitative differences for RTD characteristic parameters (tm, t0.5, σθ) between the experimental measurement (Exp. 1) and simulation results (Case 2) were 2.3%, 1% and 9.7%, and those for H = 130 mm (Exp. 2 vs Case 5) were 4.6%, 17.9% and 8.3%, respectively. The results suggest that the experimentally measured solids RTD characteristics were reasonably captured in our simulations. Note that, the simulation cases reported in this work were very time-consuming due to the long-tail nature of the solids RTD curves. For example, as to Case 2 it took N1200 s before the tracer particles completely left the bed. To realize the modeling of 1200 s fluidization in the laboratory computational platform with 24 Xeon nuclei running at 2.3 GHz, the 3D simulation with a grid size of 8 mm (totally 7410 computational cells) would take N50 days, and the corresponding 3D simulation with a grid size of 4 mm would take N4 months. Based on the results presented in Fig. 4 and Table 4, herein except for the tracer injection conditions study (2D, Δ = 4 mm), all the simulation results were obtained based on 3D simulation with a grid size of 8 mm. For quantitative comparison of RTD characteristics, all simulations were terminated when at least 90% of the injected tracer particles had left the fluidized bed. In addition, in the calculation of mean residence time tm (Eq. (2)) the time t is cut off to satisfy the condition of ∫t0E(t)dt = 0.9 for quantitative comparison. 4.2. Effect of tracer injection conditions

Fig. 2. Time variation of solid volume fraction at z = 50 mm and solids inventory (Case 1).

Traditional impulse stimulus response technique requires that the injection time of tracer particles should be far shorter than average residence time. In our experiments, tracer particles with the an amount of 2% solids inventory were introduced into the bubbling fluidized bed by opening the ball valve blow the tracer container. Depending on the bed size (Exp. 1–Exp. 6), the injection time of tracers varied from 1 s to around 20 s. How sensitive are the obtained solids RTDs to the tracer injection time? Three simulation cases (Case 3 and Case 6–7) were performed with the tracer injection time varying from 1 s to 40 s. The test injection period safely cover the possible injection time taken in experiments. Note that, in our simulations the adopted maximal tracer injection time (40 s) was still significantly smaller than the average solids residence time (359 s). The predicted RTD curves are presented

S. Geng et al. / Powder Technology 320 (2017) 555–564

559

Fig. 3. Contours of tracer mass fraction at different moments (Case 1).

in Fig. 5a. It can be seen that increasing tracer injection time led to a slight delay of E(t) to achieve its peak value. The quantitatively influences of tinj on the predicted values of tm, t0.5 are demonstrated in Table 4. With the increase of tinj from 1 s to 40 s, the predicted value of tm increased monotonously from 321 s to 360 s, and that of t0.5 from 282 s to 321 s. Though the influence of tracer injection time on solids residence times was detectable, for these cases the differences between the predicted tm and also between the predicted t0.5 were both smaller than 13%, in the general error range of experimental data. Thus, we can reasonably assumed that for the investigated cross-flow bubbling fluidized bed, the very long solid residence time makes the influence of tracer injection time period (b20 s) on the measured solids RTDs negligible. As discussed before, for solids RTD experiments the physical properties of tracer should be as close as possible to those of bulk bed materials

so that the hydrodynamic behavior of bulk bed materials could be safely represented by that of tracer. In our experiments, coal particles (ρtr = 1405 kg/m3) acted as tracer whereas sands (ρs = 2600 kg/m3) acted as the bulk bed material. A simulation case (Case 8) was thus performed with a tracer density of 2600 kg/m3 to compare with Case 3 with the same simulation conditions except for the tracer density. Fig. 5b shows that the solids RTD curves given by Case 3 and Case 8 almost completely overlapped. The quantitative differences between the predicted values of tm, t0.5, σθ using coal particles and sand particles as tracers were all smaller than 2%. It can then be safely concluded that for the operating conditions considered here the influence of density difference between tracer particles and sand particles on solids RTD was ignorable. The experimental data [18,19] and also our previous simulation results [24] have shown that coal tracer particles were well mixed with sand bed material under the investigated conditions, further justifying the negligible influence of tracer density on the obtained RTDs. 4.3. Effect of operation conditions

Fig. 4. Experiment validation and the comparison of predicted solids RTD for different grid sizes and dimensions.

In the literature there are limited researches on cross-flow BFB systems, and this section is devoted to investigating the influences of bed height, solids flux and superficial gas velocity on solids RTD in the investigated bubbling fluidized bed. Three cases with particle bed heights of 70, 100 and 130 mm (Case 9, Case 2 and Case 5) were performed. Here the bed height was controlled by the solids outlet position. Fig. 6a presents the predicted RTD curves. The peak value of E(t) decreased with the increase of bed height, which meant the larger bed height resulted in the longer residence time. Increasing bed height from 70 mm to 130 mm monotonously increased tm from 277 s to 382 s, t0.5 from 246 s to 402 s and σθ from 0.684 to 0.764. The results are consistent with the experimental findings of Babu and Setty [27] who investigated solids RTDs in columnar gas–solid bubbling fluidized beds. For the investigated particle bed heights, the solids volume fraction was all around 0.42. This meant that increasing bed height was approximately equivalent to the increase in solids inventory. Experimentally we also measured the solids residence times for bed height of 130 mm (Table 4, Exp. 2), and good agreement between simulation and experiment was again obtained. Fig. 6b shows the influence of solids flux on solids RTD profiles (Case 2, Case 10 and Case 11). The peak value of E(t) sharply increased with the increase of solids flux, leading to shorter tail of E(t) curve. Table 4 shows that by varying Gs from 18.75 kg/h to 56.25 kg/h, tm decreased from 345 s to 109 s, t0.5 from 328 s to 102 s, and σθ from 0.741 to 0.607. These variation trends were similar to the reports of Rhodes et al. [28] and Andreux et al. [29] investigating the influence of solids flux on solids residence time in CFB risers. According to Bi [30], the

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S. Geng et al. / Powder Technology 320 (2017) 555–564 Table 4 Summary of predicted solids RTD characteristics for different conditions.

Cases

Dim

Δx (mm)

tinj (s)

ρtr

hb

Gs

(kg/m3)

(mm)

(kg/h)

ug/umf

tm (s)

t0.5 (s)

σθ

Exp.1

–

–

–

1405

100

18.75

8

35 3

325

0.675

Exp.2

–

–

–

1405

130

18.75

8

365

341

0.705

Exp.3

–

–

–

1405

100

15.60

8

362

339

0.700

Exp.4

–

–

–

1405

130

15.60

8

402

396

0.636

Exp.5

–

–

–

1405

200

150.0

8

361

346

0.555

Exp.6

–

–

–

1405

400

1200

8

396

400

0.443

1

3D

4

2

1405

100

18.75

8

352

312

0.661

2

3D

8

2

1405

100

18.75

8

345

328

0.741

3

2D

4

2

1405

100

18.75

8

330

284

0.776

4

2D

8

2

1405

100

18.75

8

328

288

0.767

5

3D

8

2

1405

130

18.75

8

382

402

0.764

6

2D

4

1

1405

100

18.75

8

321

282

0.752

7

2D

4

40

1405

100

18.75

8

360

321

0.716

8

2D

4

2

2600

100

18.75

8

332

286

0.764

9

3D

8

2

1405

70

18.75

8

277

246

0.684

10

3D

8

2

1405

100

37.50

8

164

152

0.686

11

3D

8

2

1405

100

56.25

8

109

102

0.607

12

3D

8

2

1405

100

18.75

4

415

397

0.718

13

3D

8

2

1405

100

18.75

6

381

358

0.727

14

3D

8

2

1405

100

15.60

8

349

326

0.679

15

3D

8

2

1405

130

15.60

8

456

477

0.801

79

75

0.579††

16

3D

8

2

1405

100

75.00

10 79

76

0.578

101

96

0.610††

17

3D

8

2

1405

160

75.00

10 102

97

0.611

18

3D

8

2

1405

200

150.0

8

416

391

0.561

19

3D

8

2

1405

400

1200

8

426

419

0.482

††

Predicted by pure CFD simulation in Case 16 and Case 17. ⁎Blue italic types are results predicted by semi-empirical prediction approach.

decrease of solids residence time with increasing solids flux could be attributed to that the transport velocity of solid particles along the radial direction increased and back-mixing degree decreased with larger external solids flux. The effect of superficial gas velocity on solids RTD is demonstrated in Fig. 6c (Case 2, Case 12 and Case 13). Quantitatively, increasing ug/umf from 4 to 8 led to the decrease of tm from 415 s to 345 s, t0.5 from 397 s to 328 s, and σθ from 0.718 to 0.741. Although these variation trends were similar to those observed in jetting [31] and circulating [8] fluidized beds, the underlying mechanisms were different. In those fluidized bed systems, the average flow directions of solids and gas were the same. The decrease of solids residence time was mainly due to the increase in solids axial velocity which is proportional to superficial gas velocity. As to the investigated cross-flow BFB, we carefully checked that the dependence of net solids velocity in x-direction on superficial gas velocity was nearly ignorable, as a result of the different average flow directions for gas and solids. Actually, the average solids volume fraction decreased from 0.50 for ug/umf = 4 to 0.42 for ug/umf = 8. It

appeared more reasonable to attribute the decrease of solids residence time to the decrease of solid inventory. Based on the results in Fig. 6 and Table 4, one can reasonably infer that in the investigated cross-flow bubbling fluidized bed the solids residence time is closely related to solids inventory and solids flux. In fluidized beds with continuous solids feeding and discharge, the ideal plug flow time tplug has been frequently used to roughly estimate solids average residence time [8,32], t plug ¼

M ; Gs

ð6Þ

where M and Gs are solids inventory and flux, respectively. Defining the dimensionless time θ = t/tplug, the following dimensionless solids RTD distribution E(θ) could be derived from Eq. (1), EðθÞ ¼ Eðt Þ t plug :

ð7Þ

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561

Fig. 5. Comparison of predicted solids RTD for (a) different tracer injection times (b) different tracer densities.

Fig. 7 gives the profiles of E(θ). For briefness, only the profiles for different solids fluxes were shown. Though these three E(θ) curves have obvious differences, the differences showing in E(θ) are significantly smaller than those of E(t) (see Fig. 6b). The shape of descending stages for different solids fluxes was quite similar. As stated before, the CFD modeling of solids residence behavior in the investigated system is extremely time-consuming due to the long-tail nature of solids RTD. If there exists an unique function that can empirically fit the shape-similar descending part of E(θ), the CFD simulation would be significantly shortened since only the ascending stage of RTD needs to be modeled. This strategy would be developed in next section. 5. Semi-empirical approach for RTD prediction

Fig. 6. Comparison of predicted solids RTDs for different (a) bed heights, (b) solids fluxes, and (c) superficial gas velocities.

In order to derive an empirical function for the descending part of solids RTD, we defined tp as the time when E(t) achieved its peak value, td = t − tp, and ωp as the mass proportion of tracer particles flowing out of the systems in the ascending stage. The amount of remaining tracer particles for descending stage was Mtr(1− ωp). Similar to the method of obtaining the whole solids curve, the descending part of the solids RTD, E(td),can then be formulated as,

where θd = θ − θp, θp = tp / tplug and θ = t / tplug. Considering the fact that E(θp) changed with the variation of operation conditions (see Fig. 7), we further normalized E(θd) with E(θp)/(1− ωp). Based on the above data processing, in Fig. 8 we re-plot all the descending stage data presented in Fig. 6. All the data collected from different operation conditions collapsed onto a master curve and can be well fitted as,

Eðt d Þ ¼

G  s  C ðt Þ: Mtr 1−ωp

ð8Þ

EðθÞ ; 1−ωp

  E θp  expðBθd Þ; with B ¼ −1:047: 1−ωp

ð10Þ

Based on Eqs. (8)–(10), the descending part of solids RTD function E(t) can be formulated as,

The corresponding dimensionless function is, Eðθd Þ ¼ Eðt d Þ t plug ¼

Eðθd Þ ¼ 

ð9Þ

  Eðt Þ ¼ E t p expðBθd Þ; B ¼ −1:047:

ð11Þ

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Fig. 7. Comparison of predicted dimensionless solids RTDs for different solids fluxes.

Fig. 9. Detailed diagram of semi-empirical approach to predict solids RTD.

Eq. (11) suggests that, at least for the operation conditions considered here, the descending part of E(t) can be uniquely established once tp, tplug, and E(tp) were known. For the investigated cross-flow BFB system tp is generally more than eight times smaller than the emptying time of tracer particles from the same system. A semi-empirical approach was then proposed to shorten the simulation time, as schematically shown in Fig. 9. For given operation conditions, the ascending part of a solids RTD and its tp, E(tp) and tplug can be obtained through CFD simulation. As to the system investigated here (B = − 1.047), the descending part of its solids RTD curve is predicted using Eq. (11). The value of B is related to the structure of the cross-flow BFB, such as the integral shape and outlet settings (size, position, et al.). Nonetheless, the value of the empirical parameter B is independent of operation conditions, as shown in Fig. 8. We can speculate that for a cross-flow BFB system with the given structure the value of B may be unique. In turn, only one detailed simulation case or experiment is needed in order to obtain the value of B. This will be tested this in our future work. The rationality of the above approach is evaluated in Fig. 10. In Fig. 10a we compared the experimental RTD data and predicted results through the semi-empirical approach. The corresponding solids residence times are listed in Table 4. The quantitative differences of tm, t0.5 between experiment (Exp. 3) and the semi-empirical approach prediction (Case 14) were 3.6%, 3.8%, and those between Exp. 4 and Case 15 were 13.4%, 17.7%. Although there existed slight differences, the agreements were quite acceptable. For further evaluation, Fig. 10b

Fig. 8. Fitting function for descending stage acquired from simulation data.

compares the results obtained by the semi-empirical approach and pure CFD simulation results for wider range of operation conditions (Case 16 and Case 17). We can see that the descending parts of solids RTDs given by CFD simulations were successfully represented by the semi-empirical approaches. The quantitative differences between solids residence times (tm, t0.5) were smaller than 2%. For this agreement, neither experimental nor CFD simulation data presented in Fig. 10 were used to fit the empirical parameter B.

Fig. 10. Comparison of solids RTD profiles predicted by the semi-empirical approach with (a) experimental data and (b) simulation results in the investigated system.

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6. Conclusions In this study, solids RTDs in cross-flow rectangular bubbling fluidized beds were investigated via numerical simulations, with an intention of developing a simple approach to predict the entire RTD curve. Based on our previous experiments using coal as tracer, the CFD simulation showed that the density difference between tracer particles and bed material (sands) little affected the solids RTDs. Within the investigated conditions, tracer injection time had minor influence on the measured solids residence times (tm, t0.5) because the solids residence time (N300 s usually) in the investigated cross-flow rectangular BFBs was much longer than the tracer injection time period (b20 s). Simulation further verified that the descending part of solids RTD profile in cross-flow BFB can be uniquely fitted by an empirical exponential function. Only one empirical parameter appears in this exponential function and it is independent of operation conditions. On this basis, we proposed a semi-empirical approach by which the ascending part of a solids RTD profile is obtained through CFD simulation and its descending part is given by the fitted empirical exponential function. Using such an approach significantly shortened the computation time for obtaining a solids RTD profile. The approach was validated by comparing the predicted solids RTDs with those from experiments and CFD simulations in the investigated BFB system. The approach also accurately estimated the experimental solids RTDs in two large cross-flow rectangular BFBs. Thus, the work for the first time established a practical approach for the prediction of solids RTDs in cross-flow BFBs. Notation

Fig. 11. Comparison of solids RTD profiles predicted by the semi-empirical approach with experimental data obtained in two larger-size bubbling fluidized beds.

Using the above semi-empirical approach, we further investigated solids residence characteristics in two larger cross-flow rectangular BFBs. The size ratios (W:T:H) of two larger BFBs were the same as that of the above investigated BFB, whereas their sizes (W, T, H) were scaled up by two and four times, respectively. The main geometrical and operational information of these two systems are listed in Fig. 11 and Table 4 (Exp. 5 vs Case 18, Exp. 6 vs Case 19). Considering the heavy computational loads, the CFD simulations grid sizes of 16 mm and 32 mm were adopted for the middle (Case 18) and large (Case 19) beds, respectively. Correspondingly, we used the modified Gidaspow gas–solid drag model for coarse-grid simulation [33]. Fig. 11 compares the solids RTDs obtained from experiments and the semi-empirical approach. The quantitative values of solids residence times (tm, t0.5) are listed in Table 4. Again, the results of semi-empirical approach were in reasonable agreements with the experimental data. It should be stressed that, in our computational platform it would take N120 days to realize the modeling of 1100 s fluidization for Case 19 even if the coarse-grid strategy was adopted. Using the above proposed semi-empirical approach, only 25 days were needed to obtain the entire curve of solids residence characteristics. Once again, we want to address that the proposed approach is semiempirical. The value of the empirical parameter B appeared in Eq. (10) must be fitted through experimental or numerical data. Our supplementary simulations show that B is sensitive to the structure of cross-flow BFB, especially the location and the size of outlet. Nevertheless, for cross-flow BFB with given structure, our simulation results (Fig. 8) indicate that B is approximately independent of operation conditions. This means that for given cross-flow BFB system, one set of experimental or numerical solid RTD data is enough to obtain the value of B.

Symbols Gs M C t E(t) σθ u Ds p d epp g g0 Res CD I2D I̿

external flux of solid particle, kg/s solid inventory in the system, kg mass concentration of tracer particles time, s solid residence time distribution, s−1 the coefficient of variation of RTD velocity, m/s solid particles diffusion coefficient, m2 s−1 gas pressure, Pa particle diameter, m particle–particle restitution coefficient gravitational constant, m s−2 radial distribution function particle Reynolds number drag coefficient the second invariant of the deviatoric stress tensor unit tensor

Greek symbols mean residence time of each CSTR τm ρ density, kg m−3 Θ granular temperature, m2 s−2 βi gas–solid drag coefficient ζik solid–solid drag coefficient λ particle bulk viscosity, Pa s μ solid shear viscosity, Pa s ε volume fraction γΘpi collision energy dissipation

Subscripts g gas phase p solid phase tr tracer m mean

564

inj mf d

S. Geng et al. / Powder Technology 320 (2017) 555–564

injection minimum fluidization descending stage

Acknowledgments This work is financially supported by the National Natural Science Foundation of China, grant No. 21576265. Xiaoxing Liu acknowledges the financial support from the “Hundred Talents Program” of Chinese Academy of Sciences. References [1] M. Van de Velden, J. Baeyens, K. Smolders, Solids mixing in the riser of a circulating fluidized bed, Chem. Eng. Sci. 62 (2007) 2139–2153. [2] G.S. Patience, J. Chaouki, G. Kennedy, Solids residence time distribution in CFB reactors, in: P. Basu, M. Horio, M. Hasatani (Eds.), Circulating Fluidized Beds Technology, Vol. III, Pergamon Press, Nagoya, Japan 1995, pp. 599–604. [3] K. Smolders, J. Baeyens, Overall solids movement and solids residence time distribution in a CFB-riser, Chem. Eng. Sci. 55 (2000) 4101–4116. [4] J. Bi, G. Yang, T. Kojima, Lateral mixing of coarse particles in fluidized beds of fine particles, Chem. Eng. Res. Des. 73 (1995) 162–167. [5] A. Avidan, J. Yerushalmi, Solids mixing in an expand top fluid bed, AIChE J. 31 (1985) 835–841. [6] Q. Wang, J. Zhou, J. Tu, Z. Luo, X. Li, M. Fang, L. Cheng, M. Ni, K. Cen, Residence time in circulating fluidised bed, in: M. Kwauk, L. Jinghai (Eds.),Circulating Fluidised Bed Technology V 1996, pp. 128–133. [7] A.T. Harris, J.F. Davidson, R.B. Thorpe, A novel method for measuring the residence time distribution in short time scale particulate systems, Chem. Eng. J. 89 (2002) 127–142. [8] A.T. Harris, J.F. Davidson, R.B. Thorpe, Particle residence time distributions in circulating fluidised beds, Chem. Eng. Sci. 58 (2003) 2181–2202. [9] M. Stein, Y.L. Ding, J.P.K. Seville, D.J. Parker, Solids motion in bubbling gas fluidised beds, Chem. Eng. Sci. 55 (2000) 5291–5300. [10] C.W. Chan, J.P.K. Seville, D.J. Parker, J. Baeyens, Particle velocities and their residence time distribution in the riser of a CFB, Powder Technol. 203 (2010) 187–197. [11] C. Wu, Y. Cheng, Y. Ding, Y. Jin, CFD–DEM simulation of gas–solid reacting flows in fluid catalytic cracking (FCC) process, Chem. Eng. Sci. 65 (2010) 542–549. [12] X. Shi, R. Sun, X. Lan, F. Liu, Y. Zhang, J. Gao, CPFD simulation of solids residence time and back-mixing in CFB risers, Powder Technol. 271 (2015) 16–25. [13] L. Hua, J. Wang, J. Li, CFD simulation of solids residence time distribution in a CFB riser, Chem. Eng. Sci. 117 (2014) 264–282. [14] Y.J. Liu, X.Y. Lan, C.M. Xu, G. Wang, J.S. Gao, CFD simulation of gas and solids mixing in FCC strippers, AIChE J. 58 (2012) 1119–1132.

[15] T. Li, C. Guenther, A CFD study of gas–solid jet in a CFB riser flow, AIChE J. 58 (2012) 756–769. [16] J. Zhang, R. Wu, G. Zhang, C. Yao, Y. Zhang, Y. Wang, G. Xu, Recent studies on chemical engineering fundamentals for fuel pyrolysis and gasification in dual fluidized bed, Ind. Eng. Chem. Res. 52 (2013) 6283–6302. [17] J. Zhang, Y. Wang, L. Dong, S. Gao, G. Xu, Decoupling gasification: approach principle and technology justification, Energy Fuel 24 (2010) 6223–6232. [18] W. Gao, Particle Residence Time Distribution and Similar Amplification in a Bubbling Fluidized bed, China University of Ming & Technology, Beijing, Unpublished Master Dissertation, 2012. [19] J. Zhang, W. Gao, Z. Zhao, Z. Liu, M. Narukawa, T. Suda, G. Xu, Adaptability verification of scaling law to solid mixing and segregation behavior in bubbling fluidized bed, Powder Technol. 228 (2012) 206–209. [20] B.A. Buffham, G. Mason, Holdup and dispersion: tracer residence times, moments and inventory measurements, Chem. Eng. Sci. 48 (1993) 3879–3887. [21] S.B. Savage, S. McKeown, Shear stresses developed during rapid shear of concentrated suspensions of large spherical particles between concentric cylinders, J. Fluid Mech. 127 (1983) 453–472. [22] C.K.K. Lun, S.B. Savage, D.J. Jeffrey, N. Chepurniy, Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield, J. Fluid Mech. 140 (1984) 223–256. [23] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, New York, 1994 1197–1198. [24] S. Geng, Z. Jia, J. Zhan, X. Liu, G. Xu, CFD modeling the hydrodynamics of binary particle mixture in pseudo-2D bubbling fluidized bed: effect of model parameters, Powder Technol. 302 (2016) 384–395. [25] P. Khongprom, A. Aimdilokwong, S. Limtrakul, T. Vatanatham, P.A. Ramachandran, Axial gas and solids mixing in a down flow circulating fluidized bed reactor based on CFD simulation, Chem. Eng. Sci. 73 (2012) 8–19. [26] I. Hulme, E. Clavelle, L.v.d. Lee, A. Kantzas, CFD modeling and validation of bubble properties for a bubbling fluidized bed, Ind. Eng. Chem. Res. 44 (2005) 4254–4266. [27] M.P. Babu, Y.P. Setty, Residence time distribution of solids in a fluidized bed, Can. J. Chem. Eng. 81 (2010) 118–123. [28] M.J. Rhodes, S. Zhou, T. Hirama, H. Cheng, Effects of operating conditions on longitudinal solids mixing in a circulating fluid, AIChE J. 37 (1991) 1450–1458. [29] R. Andreux, G. Petit, M. Hemati, O. Simonin, Hydrodynamic and solid residence time distribution in a circulating fluidized bed: experimental and 3D computational study, Chem. Eng. Process. Process Intensif. 47 (2008) 463–473. [30] X. Bi, Gas and solid mixing in high-density CFB risers, Int. J. Chem. React. Eng. 2 (2004) 1–12. [31] W. Yang, D.L. Keairns, A study of fine particles residence time in a jetting fluidized bed, Powder Technol. 53 (1987) 169–178. [32] D.C. Guío-Pérez, T. Pröll, H. Hofbauer, Solids residence time distribution in the secondary reactor of a dual circulating fluidized bed system, Chem. Eng. Sci. 104 (2013) 269–284. [33] J. Wang, M.A. van der Hoef, J.A.M. Kuipers, Coarse grid simulation of bed expansion characteristics of industrial-scale gas–solid bubbling fluidized beds, Chem. Eng. Sci. 65 (2010) 2125–2131.