Computational Fluid Dynamics Simulation and Experimental

Article pubs.acs.org/IECR

Computational Fluid Dynamics Simulation and Experimental Measurement of Gas and Solid Holdup Distributions in a Gas− Liquid−Solid Stirred Reactor Shifang Yang,†,‡ Xiangyang Li,*,† Chao Yang,*,†,‡ Bin Ma,†,§ and Zai-Sha Mao† †

Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China ‡ University of Chinese Academy of Sciences, Beijing 100049, China § China University of Mining & Technology, Beijing 100083, China S Supporting Information *

ABSTRACT: The lack of convincing computational fluid dynamics models and basic experimental data makes it difficult to design and scale up a gas−liquid−solid three-phase stirred reactor such as in environmentally benign bioleaching processes. In this work, a steady state gas−liquid−solid Eulerian−Eulerian multifluid model was developed to predict flow field and local phase holdup distribution. The measurements of gas and solid holdup distributions in a gas−liquid−solid stirred reactor were simultaneously obtained for the first time by a new improved sample withdrawal technique. The influence of the presence of a second dispersed phase on interphase drag force was taken into account, and good agreement for the local phase holdup was obtained between the predictions and the experimental data. The solid concentration increased, and the gas holdup decreased along the radial direction from the reactor axis to the wall. A simplified bubble-size model considering the effect of turbulence dissipation rate was incorporated in the model instead of a fixed size. It was found that smaller bubbles concentrated in the impeller region, and the bubble size increased near the free surface. The mixing efficiency could be enhanced obviously when double impellers were employed.

1. INTRODUCTION Sustainable chemical, petrochemical, and metallurgical industrial processes are crucial to national sustainable and ecological development. In the recent decades, biotechnologies have received considerable attention in comprehensive and no-waste utilization of natural resources. For example, bioleaching and biooxidation have been successfully used in extraction of metals from some low grade and refractory ores. These industrial practices have proven the application of biotechnology being promising and prospective in the sustainable development of mineral and metallurgy industries. Many kinds of reactors have been tested for these processes and the stirred reactors are the most common for suspending solid particles and ensuring gas dispersion. Because the complexity in multiphase stirred reactors, the method of scientific design and scale-up of an industrial multiphase stirred reactor is far from being well developed. It is estimated that a billion dollars per annum are lost in USA alone due to shortcoming in the design of mixing processes.1 To reflect the mixing efficiency and mass transfer in a bioreactor, the local phase holdup distribution is a reliable parameter. A large number of methods have been widely used to measure the local phase holdup in two-phase reactors such as sample withdrawal,2 optical fiber,3 conductivity probe,4 particle image velocimetry,5 ultrasonic methods6 and so on. Nevertheless, due to the complex circulating and turbulent flow, it is difficult to employ these methods in a gas−liquid−solid stirred reactor. Although the sample withdrawal method has shortcomings, it has been commonly used to measure the local phase © XXXX American Chemical Society

holdup in solid−liquid, liquid−liquid, and liquid−liquid−solid mechanically stirred vessels because of its simplicity and versatility.7 However, little attention has been devoted to the application of this method in a multiphase reactor containing a gas phase up to now. It might be because bubbles moving upward could not be collected when the traditional tangential sample withdrawal method was used, and the same problem happened to the larger bubbles when the diameter of the sample tube was small. To solve these problems, a sampling tube with a bell mouth was proposed in our group to measure the local phase holdup in a gas−liquid−solid stirred reactor, and it has been verified to be reliable.8 Except for experimental measurement, the computational fluid dynamic (CFD) method has been promoted as a useful tool for it offers more extensive information on the velocity field and turbulent properties throughout the reactor and is timesaving. Murthy et al.9 and Panneerselvam et al.10,11 carried out CFD simulations to investigate gas−liquid−solid stirred reactors. In these researches, the simulations were focused only on the bulk flow properties such as the critical impeller speed for solid suspension and the effect of relevant factors. To simplify the overall model, the drag model employed in their Special Issue: Sustainable Manufacturing Received: August 27, 2015 Revised: December 17, 2015 Accepted: December 20, 2015

A

DOI: 10.1021/acs.iecr.5b03163 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research simulations was the same as that in two-phase systems and a fixed bubble size was used, which would bring significant deviations on the prediction of phase holdup distribution. Mitra-Majumdar et al.12 investigated three-phase flow through a vertical column and found that in a three-phase reactor, the presence of a second dispersed phase would influence the interphase drag force because of the large difference in density. In addition to this, the change of slurry density and viscosity should also be considered. Hooshyar et al.13 studied the influence of particles on the average rise velocity of a single gas bubble and found that the small particles (78 and 587 μm) did not directly collide with the bubble and stuck virtually on the streamlines of the liquid flowing around a bubble. In this case, the gas−liquid−solid system could be simplified to a gas−slurry system, and the influence of solid particles on the bubble coalescence and breakage was due to the change in slurry density and viscosity. These impacts should be coupled in the simulation model for a gas−liquid−solid stirred reactor to obtain an accurate flow field and local phase holdups. In addition to the drag force, the bubble size distribution is another important factor to affect the hydrodynamic characteristics in a gas−liquid−solid stirred reactor. From CFD simulations carried out on bubble behavior in gas−liquid stirred reactors, it is found that the fixed bubble diameter assumption violated the real picture and could bring significant deviations.14 Although the population balance model15,16 and the bubble number density equation17,18 have been employed in gas−liquid reactors successfully, the lack of experimental data of local bubble sizes in a three-phase reactor makes it difficult to estimate the parameters in these models. Besides, due to the complexity in a three-phase reactor, the application of multifluid models based on population balance increased the computational demand by folds. Hinze et al.19 proposed a simplified model to determine the maximum drop size in emulsion through analyzing the splitting of globules in dispersion processes. This model was successfully employed by Zhang et al.20 to predict the gas holdup in a gas−liquid stirred reactor and good agreement was obtained with experimental data. To the best of the authors’ knowledge, numerical simulation of the flow characteristics in a gas−liquid−solid stirred reactor considering the effect of the presence of a second dispersed phase on the interphase drag force and the experimental measurement of local phase holdup have not yet appeared in the literature. This work makes an attempt to fill this gap for the purpose of reliable design, scale-up, and optimization of such stirred reactors so as to make the industrial processes more sustainable and profitable. An improved sample withdrawal method was used for the experimental measurements and a steady-state gas−liquid−solid Eulerian−Eulerian multifluid model coupled with modified drag force and bubble size models was developed to predict the flow field and spatial distribution of local phase holdup and bubble size. The predictions were compared with experimental data to validate the computational model.

Figure 1. Experimental setup: 1, stirred reactor; 2, stirring shaft; 3, impeller; 4, gas sparger; 5, motor; 6, air compressor;, 7, flow meter; 8, sample tubes.

turbine downflow (PBTD) impeller of diameter D = T/3 is employed in the experiment with the impeller off bottom clearance C = T/3. A ring gas sparger (Ds = 0.06 m) with 20 holes (Dh = 0.002 m) oriented downward is used to feed gas into the reactor. Air and tap water are used respectively as gas and liquid phases. Quartz sand with sizes ranging from 75 to 150 μm and density of 2650 kg/m3 is used as the solid phase. The critical impeller speed for solid suspension was determined by the visual criterion that the solids remained motionless at the tank bottom for not more than 2 s.21 2.2. Improved Sample Withdrawal Method. The local phase holdup was measured by the sample withdrawal technique. A sample tube with a bell mouth was used. The inside diameter of the sample tube is 6 mm and the diameter of the bell mouth is 18 mm. Slurry was withdrawn from the reactor at different radial and axial positions. The gas holdup was directly read from the graduation on the sample cylinder and the solid mass fraction was determined using the pycnometric technique from eq. 122 1 αs = ⎡ 1 − ⎛ ρl ⎞ ⎤ ρ ⎝ρ ⎠ 1 + ⎢ ⎛ m ⎞ s⎛ ρ ⎞ − 1⎥ ρl ⎢ sl − l ⎥ s ⎣ ⎝ ρsVsl ⎠ ⎝ ρs ⎠ ⎦ (1) ⎜

⎜

⎟

⎟

⎜

⎟

()

where αs is the solid phase volume fraction, ρl and ρs are the liquid and solid densities, and msl and Vsl are the mass and volume of the liquid−solid slurry. To reduce the error brought by time and space related ones, each measurement was repeated six times at a prescheduled point to get the average value. More details on the sample withdrawal techniques were given by Cao et al.8

3. CFD MODELING 3.1. Model Equations. A 3D CFD simulation was performed using a newly developed Eulerian−Eulerian multiphase model in a gas−liquid−solid stirred reactor. The relative motion between the impeller and the tank region was modeled by the multiple reference frame (MRF). The phases were all treated as continua and interpenetrating with each other. The pressure field was assumed to be shared by the three phases. The Reynolds averaged mass and momentum balance equations were solved for each of the phases, and the governing equations are given below: Continuity equation (k = g, l, s):

2. EXPERIMENTAL 2.1. Experimental Setup. A stirred reactor as shown in Figure 1 with internal diameter (T) of 0.380 m is used in this work. The water height (H) is 0.675 m. To improve the suspension of solid particles, an elliptical bottom is chosen and the height (Hb) is 0.16 m. A six-bladed 45° pitched blade B


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Industrial & Engineering Chemistry Research ∇·(αkρk uk) = 0

where db is the bubble size, and CDb is the drag coefficient of a single bubble in a stagnant liquid calculated by

(2)

The momentum balance equation for each phase:

⎛ 24 8 Eo ⎞ C Db = max⎜ (1 + 0.15Reb0.687), ⎟ 3 Eo + 4 ⎠ ⎝ Reb

∇·(ρg αg u gu g) = −αg∇P + ∇·(αgμeff,g (∇u g + (∇u g)T )) + ρg αg g − FD,lg

(3)

where Eo and Reb are the Eotvos number and the bubble Reynolds number, respectively:

T

∇·(ρl αl ulul) = −αl∇P + ∇·(αlμeff,l (∇ul + (∇ul) )) + ρl αl g + FD,lg + FD,ls

(13)

Eo =

(4)

g (ρl − ρg )db2 σlg

(14)

T

∇·(ρα u u ) = −αs∇P + ∇·(αsμeff,s (∇us + (∇us) )) s s s s + ρα g − FD,ls s s

Reb =

(5)

where P is the pressure shared by all three phases, μeff is the effective viscosity, and FD represents the interphase drag force. The volume fractions satisfy the following normalization condition: αg + αl + αs = 1

(6)

C D0

⎛ d p ⎞3 = 8.67 × 10−4⎜ ⎟ ⎝λ⎠

24 (1 + 0.15Rep0.687), Rep

C D0 = 0.44,

(8)

Rep < 1000

Rep ≥ 1000

(9)

where Rep is the particle Reynolds number given by Rep =

d p|us − ul| νl

(10)

The drag force between the liquid and gas phases was calculated by eq 11, and the modified drag model28 was employed to calculate the drag coefficient CD,lg by eq. 12: 3 αgαl FD,lg = C D,lg ρl | u g − u l | (u g − u l ) 4 db (11) C D,lg − C Db C Db

⎛ d ⎞3 = 6.5 × 10−6⎜ b ⎟ ⎝λ⎠

(15)

3 αgαl FD,lg = kgC D,lg ρl | u g − u l | (u g − u l ) 4 db

(16)

kg = 1/(1 − αs0.2)

(17)

The drag coefficient CD,lg was calculated using eq 12. kg is the correction factor to account for the inhibition of the motion of gas bubbles due to the presence of solid particles and the value is greater than 1. The drag force acting on solid particles was modified to account for the lifting effect of buoyant bubbles on the movement of the particles. Different from this work, the modified factor proposed by Mitra−Majumdar et al.12 was less than 1. That is because in their work, the gas phase would carry solid particles with it, and this effect was described as a reduction of drag force on solid particles. In stirred reactors, wetted solid particles tend to follow the liquid phase. The presence of gas will promote the movement of solid and liquid phases at the same time. However, the role of lift for the liquid phase is more obvious because the lower liquid density, increasing the slip velocity between solid and liquid phases and thus the drag force, which means the modified factor in liquid− solid drag force should be greater than 1 in this work. To account for this effect, the drag force acting on the particles is calculated by

where dp is the particle size, λ is the Kolmogorov length scale, and CD0 is the drag coefficient of particles in a stagnant liquid: C D0 =

νl

Different from that in two-phase systems, the motion of gas would be affected by solid particles in a gas−liquid−solid stirred reactor. The slip velocity between gas and solid particles was large, and bubbles moved at much larger velocities due to the buoyancy effect.12 In that case, solid particles would hinder the motion of gas bubbles, which has not been considered in a three-phase reactor. To include this effect in the model, a correction factor is proposed to modify the drag acting on the gas phase.

3.2. Interphase Momentum Transfer. The forces between the phases include drag force, lift force, and virtual (or added) mass force, etc., but the main force is the drag force induced by the interphase slip velocity. Lane et al.23 compared the magnitude of these forces and found that the added mass and lift forces were much smaller than the drag force and thus could be ignored. Similar conclusions were also reported by other researchers.24−26 Therefore, only the drag force was considered in the present model. The drag force between the liquid and solid phases was calculated by eq 7, and the drag model eq 8 proposed by Brucato et al.27 was employed to calculate the drag coefficient CD,ls in turbulent flow: 3 αα FD,ls = C D,ls ρl s l |us − ul|(us − ul) 4 dp (7) C D,ls − C D0

db|ul − u g|

3 αα FD,ls = ksC D,ls ρl s l |us − ul|(us − ul) 4 dp

(18)

ks = 1 + αg0.04

(19)

As small particles did not directly collide with bubbles, another influence of the presence of solid particles was the change of viscosity and density of the mixture.13 A commonly used model for the slurry viscosity, μslu, of a suspension of particles describing the increase of μeff with increasing solid concentration, αs, is

(12) C


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Industrial & Engineering Chemistry Research Table 1. Geometrical Details of Gas−Liquid and Liquid−Solid Systems reference Micheletti et al.4 Deen et al.30 Barigou and Greaves31

μslu μ0

reactor geometry T = H = 0.29 m, D/T = 1/3, C/T = 1/3 T = H = 0.2222 m, D/T = 1/3, C/T = 1/2, CG/T = 1/16 T = H = 1.0 m, D/T = 1/3, C/T = 1/4

= (1 − αs/αm)−n

impeller

solids details

6-Rushton 6-Rushton

ρs = 2470 kg/m3 dp= 600−700 μm

6-Rushton

4. SOLUTION METHOD Steady state simulations were carried out using the CFD software FLUENT. The multiple reference frame method was used to model the motion of the impeller relative to the tank. The boundary of the rotating domain was positioned at r = 0.09 m and 0.092 m ≤ z ≤ 0.162 m (the origin of coordinates located on the center of the reactor bottom). Tetrahedral elements were used for meshing the tank and the meshing of good quality was ensured throughout the computational domain using the GAMBIT mesh generation tool. The meshes around the impeller blades and the impeller discharge region have been refined. To check the sensitivity of the simulation results on the grid size, the grid spacing was reduced by a factor of 2 until the change of the simulation was unobvious because of the reduction of the grid size. Therefore, grid elements in the range of 800 000 to 900 000 had been used for modeling of a three-phase stirred reactor. Regarding the boundary conditions, the tank wall, impeller surface, baffle surface, and gas sparger surface were all treated as the nonslip boundary condition, and the standard wall function was employed. At the liquid surface, only gas was allowed to escape and the normal liquid and solid velocities were set to zero. It was assumed that the solid particles suspended uniformly initially. The SIMPLE algorithm was used for the pressure−velocity coupling, and the convergence criterion was set at 10−4 for all the equations. The drag models and the change of bubble size were coupled using the user-defined function.

(20)

where μ0 is the viscosity of solid-free liquid. In this work, the Einstein equation for the viscosity of a dilute suspension of solid in liquid29 is used to calculate the viscosity of slurry as shown below:

μslu /μ0 = 1 + 2.5αs

(21)

The density of the slurry is the volume weighted average of solid and liquid phases: ρslu = αlρl + αsρs (22) When the density and viscosity are modified, the interphase drag force between gas and liquid phases is calculated by the following equation: 3 αgαl FD,lg = kgC D,lg ρslu | u g − u l | (u g − u l ) 4 db (23) 3.3. Turbulence Closure. The k−ε mixture turbulence model was used, and only one k and ε equations for the mixture were solved: ⎛ μt,m ⎞ (ρm u mk) = ∇·⎜ ∇k ⎟ + Gk ,m − ρm ε ⎝ σk ⎠

(24)

⎛ μt,m ⎞ ε ∇·(ρm u mε) = ∇·⎜ ∇ε⎟ + (Cε lGk ,m − Cε2ρm ε) ⎝ σε ⎠ k

(25)

where Cε1 = 1.44, Cε2 = 1.92, σk = 1.0, and σε = 1.0. The mixture density, velocity, and turbulent viscosity were calculated as follows:

5. RESULTS AND DISCUSSION 5.1. Two-Phase Flow. 5.1.1. Solid−Liquid Stirred Reactor. It is necessary to confirm the validity of the model for twophase systems. The geometrical details are shown in Table 1. For a solid−liquid stirred reactor, the drag model (eq 7) proposed by Brucato et al.27 was employed, and the simulation results were validated by the experimental data of Micheletti et al.4 The comparison of the axial solid distribution is shown in Figure 2 in the case of a radial-type impeller. The experimental data measured at a radial position r/T = 0.35 for an overall solid holdup of 9.2 vol. % at N = 700 rpm were used to compare with the simulation, and the disparity is quite small. 5.1.2. Gas−Liquid Stirred Reactor. For a gas−liquid stirred reactor, CFD simulations of velocity components of liquid and gas phases at various axial locations were validated with the experimental data of Deen et al.30 Further, the experimental data of Barigou and Greaves31 were used for comparison of radial bubble size profiles in the impeller stream and the bulk region. Instead of a fixed bubble size, eq 29 was used to describe the change of the bubble sizes in the reactor. A good agreement between the predictions and the experimental results could be found in Figures 3 and 4. 5.2. Three-Phase Flow. The hydrodynamics in a gas− liquid−solid stirred reactor was modeled and the predicted results of phase holdup distribution were compared with the

3

ρm =

∑ αiρi i=1

(26)

3

um =

∑ αiρi ui/ρm i=1

(27)

k (28) ε 19 3.4. Bubble Size Model. Hinze et al. proposed a simplified model to determine the maximum drop size considering the influence of the turbulence dissipation rate in an emulsion, and Zhang et al.20 modified this model to calculate the bubble size in a gas−liquid stirred reactor: μt,m = ρm Cμ

db = 0.68dmax dmax

⎛ σlg ⎞0.6 = 0.725⎜⎜ ⎟⎟ × εl‐0.4 ⎝ ρl ⎠

operating variables αs = 9.2 vol.%, N = 700 rpm QG = 7.2 × 10−5 m3/s, Utip = 1.4 m/s QG = 0.00164 m3/s, N = 180 rpm

(29)

(30)

This means that the bubble size varies from point to point in the reactor. D


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Industrial & Engineering Chemistry Research

Figure 2. Comparison between the predicted and experimental data of axial solid concentration profiles for 600−710 μm glass particles at N = 700 rpm (experimental data from Micheletti et al.,4 T = H = 0.29 m, Rushton, D = C = T/3)

present experimental results measured by an improved sample withdrawal technique. Besides, the bulk flow property such as the critical impeller speed for solid suspension was measured based on the criterion proposed by Zwietering.21 The method based on the value of standard deviation was employed in the model to predict the critical impeller speed.32 5.2.1. Flow Pattern. The impeller rotates clockwise and the impeller speed is 450 rpm. The initial solid loading is 4 vol. % and the superficial gas velocity is Vg = 2.21 × 10−3 m/s. Equations 18 and 23 were employed to calculate the drag force. The solid and gas flow patterns in a gas−liquid−solid stirred reactor and the phase concentration distributions are shown in Figures 5 and 6. From Figure 5 it can be found that there is a big circulation loop in the reactor and the flow pattern for solid phase. Solid particles moved upward along the wall toward the liquid surface and turned down to the bottom. Gas was fed into the reactor through the gas sparger and left from the liquid surface. For the case modeled in this work, the flow in the upper part of the reactor was too weakly stirred by a single PBTD impeller, because the ratio of the liquid height to the reactor diameter is greater than 1. The solid concentration distribution is shown in Figure 6a. It can be found that the solid particles are completely distributed at the simulation impeller speed. The solid concentration increased toward the reactor wall. As shown in Figure 6b, the air sparged into the reactor was transported downward by the impeller discharge stream and moved upward after reaching the wall. The gas concentration decreased in the radial direction, which is consistent with the experiment. There is a low gas concentration zone in the bottom of the reactor under the impeller, where the effect of the impeller is weak, and the bubbles rose under the buoyancy force, which is similar to that in a gas−liquid stirred reactor. In the impeller discharge plane as shown in Figure 6c, the gas accumulated in the low pressure region behind the impeller blades where the gas cavities were formed. During the experiment at N = 450 rpm, the free surface fluctuated wildly at high impeller speeds. A vortex was formed around the shaft where the solid and liquid phases did not exist, and air was entrained into the reactor. However, from the simulation result as shown in Figure 6a, it could be found that the solid concentration around the shaft was higher. It might be because in the steady simulation model, the free surface was set

Figure 3. Comparison between the simulated and experimental profiles of radial and axial velocities for gas−liquid phase flow at r/R = 0.37 (experimental data from Deen et al.,30 T = H = 0.2222 m, Rushton, D= T/3, C = T/2): (a) radial component of liquid velocity; (b) axial component of liquid velocity; (c) radial component of gas velocity.

to be flat. The gas left the reactor from the surface and no air was allowed to be entrained into the reactor. Thus, the vortex around the shaft was different from the actual situation. 5.2.2. Solid Suspension. One of the most important parameters for assessment of solid suspension is the minimum impeller speed required for the complete off-bottom suspension E


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Figure 4. Comparison of bubble diameters between our simulation and experimental results (experimental data from Barigou & Greaves,31 T = H = 1.0 m, Rushton, D= T/3, C = T/4): (a) impeller stream; (b) bulk region.

in the reactor. The visual observation criterion was proposed by Zwietering21 to determine Njs as the speed at which no particle can be visually observed to remain at rest on the tank bottom for more than 1 or 2 s. In the present work, the above method was employed to determine Njs. The method introduced by Bohnet et al.32 based on the standard deviation of local solid holdup was used in this work for prediction of the critical impeller speed. This method has been used successfully in liquid−solid and gas−liquid−solid stirred reactors.9−11,33 The standard deviation is defined as σ=

1 n

⎛ α ⎞2 s − 1⎟⎟ ∑ ⎜⎜ α ⎠ 1 ⎝ avg n

(31)

where n is the number of sampling locations used for measuring the solid volume fraction. The increase in the degree of homogenization was manifested as the reduction of the standard deviation value. On the basis of the quality of suspension, the range of the standard deviation was broadly divided into three ranges. For the just suspension condition, the value of the standard deviation was between 0.2 and 0.8 (0.2 < σ < 0.8). During the simulation, the impeller speeds were set equivalent to the value of the critical impeller speeds measured by our experiment. The veracity of the proposed model was confirmed through calculating the corresponding value of σ. The results are shown in Table 2. It could be noted that, at the

Figure 5. Predicted flow patterns in a gas−liquid−solid stirred reactor (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol.%): (a) solid flow pattern; (b) gas flow pattern.

experimentally measured critical impeller speeds, the corresponding predicted values of σ were 0.56, 0.54, 0.57, and 0.58, which were all in the aforementioned range. 5.2.3. Comparison of Drag Models. The comparison between the simulation results of local phase holdup and the experimental data by the sample withdrawal technique is F


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Industrial & Engineering Chemistry Research Table 2. Prediction of the Critical Impeller Speed superficial gas flow rate (m/s)

Njsg (rpm) (experiment)

standard deviation (σ) (predicted)

0 2.21 × 10−3 2.94 × 10−3 3.68 × 10−3

190 218 223 232

0.56 0.54 0.57 0.58

Figure 7. Comparison of experimental and predicted radial gas concentration profiles at z = 0.3 m (N = 450 rpm, Vg = 2.21 × 10−3 m/ s, αs = 4 vol %).

Figure 8. Comparison of experimental and predicted radial solid concentration profiles at z = 0.3 m (N = 450 rpm, Vg = 2.21 × 10−3 m/ s, αs = 4 vol %).

Table 3. Summary of Drag Models and Bubble Sizes case 1 solid−liquid drag model gas−liquid drag model bubble size

case 2

case 3

eq 7

eq 7

eq 18

eq 11 fixed (4 mm)

eq 11 variable (eq 29)

eq 23 variable (eq 29)

Figure 7 shows the comparison between the predictions and experimental radial gas concentrations. It could be found that for cases 1 and 2 the gas volume fractions were generally underpredicted and the predictions for case 3 were much closer to the experimental data. It might be because the influence of solid particles on the gas−liquid drag force was not considered in cases 1 and 2, since larger slurry density would hinder the movement of gas because of the large apparent viscosity. This effect was taken into account in case 3, which reduced the bubble velocity and hence increased the bubble residence time, leading to the increase in the gas holdup. Besides, the simulation results are a little lower than the experimental

Figure 6. Predicted phase holdup distributions (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol.%): (a) solid concentrations in the midplane; (b) gas concentrations in the midplane; (c) gas concentrations in the impeller plane.

plotted in Figures 7 and 8. The simulations were referred to cases 1, 2, and 3, as summarized in Table 3. The influences of different drag models and the models of bubble sizes were investigated. G


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Industrial & Engineering Chemistry Research results especially in the region near the shaft. It might be caused by the measuring error. The comparison between the experimental and predicted radial solid concentration profiles is shown in Figure 8. The solid concentration increased toward the tank wall because of the centrifugal force. The predicted solid holdup distribution in case 3 shows better agreement with the experimental data than other two cases. The moving bubbles with higher velocity would increase the drag force acting on solid particles as discussed before, but the discrepancy still existed between the predictions and the experimental data. It might be caused by the limitations of the sample withdrawal method. It could be concluded that it is necessary to take into account the influence of a second dispersed phase on the interphase drag force in a three-phase stirred reactor, especially when the phase holdups are high. For subsequent simulations, eqs 18 and 23 were used to calculate the interphase drag force. 5.2.4. Bubble Size Distribution. The bubble size is another important parameter for the reliable design of a gas−liquid− solid stirred reactor as it directly influences mixing and mass transfer. The predicted distributions of bubble sizes in the vertical midplane and in the horizontal impeller plane are shown in Figure 9. Smaller bubbles were generated in the

impeller discharge due to bubble breakup in the high intensity turbulence, while the bubbles became larger because of coalescence away from the impeller region. The gas accumulated around the shaft, but the bubble size near the free surface was smaller than that in other parts because there was a small circulation loop of solid particles as shown in Figure 5. The solid particles present at the gas−liquid interface could collide and interact with the gas−liquid interface, causing coalescence inhibition and a decrease in the bubble size.34 Except the shaft region, the bubble size increased in the upper part of the reactor due to the weak turbulence and the bubbles tended to coalesce into larger ones instead of breakup, which could be improved by using multiple impellers to enhance the flow turbulence in the upper region closer to the liquid surface. Besides, the static pressure decreased in the axial direction, favoring larger bubbles. To obtain the bubble size distribution quantitatively, the reactor was evenly divided into two regions (the upper and lower regions) and the volume average bubble size in the reactor was calculated. The mean bubble size in the lower region is 0.0063 m, which is close to that used in the literature for simulation of a gas−liquid−solid stirred reactor (0.003−0.005 m).9−11 However, for the upper region where the turbulence is weak, the bubble size is much larger (0.0153 m) and the mean bubble size in the whole reactor is 0.0105 m. In this case, the bubble size model should be incorporated in the simulation. 5.3. Three-Phase Flow with Double Impellers. To verify the influence of multiple impellers, double PBTD impellers were used to improve the mixing characteristics in the present gas−liquid−solid stirred reactor with a high liquid level, and the distance between the two impellers is equal to 2D. The impeller speed, initial solid loading, and superficial gas velocity are the same as those in the reactor with a single impeller. The predicted flow pattern is shown in Figure 10. The flow pattern was characterized by two independent circulation loops at the upper and lower impeller planes and there was a small circulation loop below the upper impeller. The intensity of the upper flow field was enhanced. The phase concentration distributions in the vertical section were shown in Figure 11. Similarly, the gas tended to accumulate behind the impeller blades to form gas cavities and a low gas concentration zone existed under the lower impeller. But the solid particles and gas bubbles were distributed more uniformly than that stirred by a single impeller. The bubble size distribution is shown in Figure 12, and the mean bubble size in the reactor was calculated. The mean bubble size in the upper reactor was 0.0077 m, which was much smaller than that in the reactor with a single impeller because the turbulence intensity was enhanced by the upper impeller. Bigger bubbles could be found near the wall in Figure 12 because the velocity was weak and the bubbles tended to coalesce, but they were still smaller than that predicted in the single impeller reactor. The mean bubble size in the whole reactor became much smaller (0.0058 m), indicating that using multiple impellers was advantageous to improving the mixing characteristics and mass transfer in a high liquid level reactor.

6. CONCLUSIONS In this work, a steady state gas−liquid−solid Eulerian−Eulerian multifluid model coupled with a modified interphase drag force correlation to account for the influence of a second dispersed phase was developed. A bubble size distribution model correlating the effect of local turbulent kinetic energy

Figure 9. Distribution of bubble sizes in the midplane and in the impeller discharge plane (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol %): (a) bubble sizes in the midplane; (b) bubble sizes in the impeller plane H


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Figure 11. Predicted phase concentration in a gas−liquid−solid stirred reactor with double impellers (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol %): (a) solid concentration; (b) gas concentration.

It was found that the solid concentration increased along the radial direction from the center axis to the wall and decreased in the axial direction from the reactor bottom to the free surface. The gas bubbles tended to accumulate around the shaft after being sparged into the reactor and there was a low gas concentration zone between the bottom of the reactor and the impeller. (2) The bubble size distribution in the stirred reactor was predicted using the bubble size estimation model. Smaller bubbles were generated in the impeller discharge and larger bubbles were found in the upper reactor. (3) Better agreement was obtained between the predicted and experimental data for local phase holdup, especially the gas holdup distribution, after employing the modified drag force correlation in the model coupled with the bubble size model. (4) Three-phase mixing was enhanced when double impellers were employed. It could be found that the

Figure 10. Predicted flow pattern in a gas−liquid−solid stirred reactor with double impellers (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol %): (a) solid flow pattern; (b) gas flow pattern.

dissipation on the bubble size was first used to predict the bubble size distribution. Besides, a new improved sample withdrawal technique, which was proven to be reliable for a gas−liquid−solid stirred reactor in our previous work, was employed to measure the local bubble and solid concentration data. The following conclusions were obtained. (1) The solid and gas distributions in a gas−liquid−solid stirred reactor were simultaneously obtained for the first time. I



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Figure 12. Predicted bubble size distribution in a gas−liquid−solid stirred reactor with double impellers (N = 450 rpm, Vg = 2.21 × 10−3 m/s, αs = 4 vol %).

distributions of gas and solid phases were more uniform and the bubble sizes were much smaller. Multi-impeller configurations were necessary to provide a better mixing when the ratio of liquid height to reactor diameter was greater than 1. Overall, the local phase holdup in a three-phase stirred reactor can be successfully measured by the improved sample withdrawal method and predicted by the steady state multifluid model coupled with a modified drag force. The model is promising in application to scientific design, scale up, and optimization of multiphase reactors and make the industrial processes more sustainable and profitable.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03163. Table S1, mean bubble size in the reactor with a single impeller; Table S2, mean bubble size in the reactor with double impellers; Figure S1, simulation results with different grid elements; Figure S2, computational grid of a stirred reactor PDF)

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AUTHOR INFORMATION

Corresponding Authors

*Tel.: +86-10-62554558.; Fax: +86-10-82544928. E-mail: [email protected]. *E-mail: [email protected].

Article

NOMENCLATURE C = impeller off bottom clearance, m CD,ls = drag coefficient between liquid and solid CD,lg = drag coefficient between liquid and gas CDb = drag coefficient of a single bubble in a stagnant liquid CD0 = drag coefficient of a particle in a stagnant liquid CG = distance between gas sparge and tank bottom, m Cε1, Cε2, Cμ = coefficients in turbulent equations db = bubble diameter, m dmax = maximum bubble diameter, m dp = particle mean diameter, m D = diameter of impeller, m Dh = diameter of holes in the sparger, m Ds = diameter of gas sparger, m Eo = Eotvos number FD,lg = interphase drag force between liquid and gas, N FD,ls = interphase drag force between liquid and solid, N g = acceleration due to gravity, m/s2 Gk,m = the turbulence production H = water height, m Hb = height of elliptical bottom, m k = turbulence kinetic energy, m2/s2 kg = correction factor for interphase drag between liquid and gas km = mixture turbulence kinetic energy, m2/s2 ks = correction factor for interphase drag between liquid and solid msl = mass of liquid−solid slurry, kg n = number of sampling locations N = impeller speed, rpm Njs = critical impeller speed for just suspended, rpm Njsg = critical impeller speed for just suspended in the presence of gas, rpm QG = gas flow rate, m3/s R = radius of the reactor, m Ri = radius of the impeller, m Reb = bubble Reynolds number Rep = particle Reynolds number P = pressure, Pa T = internal diameter of the reactor, m U = mean velocity, m/s Utip = velocity of impeller tip, m/s ug = local gas phase velocity, m/s ul = local liquid phase velocity, m/s um = local mixture velocity, m/s us = local solid phase velocity, m/s vg = superficial gas velocity, m/s Vsl = volume of liquid−solid slurry, L W = impeller blade width, m z = axial coordinate, m

Greek Letters

αavg = average volume fraction of solid phase αl = liquid phase volume fraction αm = the random close packing concentration of solid phase αs = solid phase volume fraction αg = gas phase volume fraction ε = turbulence eddy dissipation, m2/s3 εl = liquid phase turbulence eddy dissipation, m2/s3 λ = Kolmogorov length scale, m μ0 = viscosity of solid-free liquid, kg/m·s2 μeff,g = gas phase effective viscosity, kg/m·s2 μeff,l = liquid phase effective viscosity, kg/m·s2

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial supports from the 973 Program (2012CB224806), the National Natural Science Foundation of China (21276004, 91434126), the Major National Scientific Instrument Development Project (21427814), and the 863 Project (2011AA060704) are gratefully acknowledged. J


Article

Industrial & Engineering Chemistry Research μeff,s = solid phase effective viscosity, kg/m·s2 μslu = viscosity of slurry in gas−liquid−solid, kg/m·s2 μt,m = mixture turbulent viscosity, kg/m·s2 νl = kinematic viscosity of water, m2/s ρg = gas density, kg/m3 ρl = liquid density, kg/m3 ρm = mixture density, kg/m3 ρs = solid density, kg/m3 ρslu = slurry density, kg/m3 σ = standard deviation value for solid suspension σk, σε = coefficients in turbulent equations σlg = surface tension of gas and liquid

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Subscripts and Superscripts

eff = effective g = gas phase l = liquid phase mix = mixture r = radial s = solid phase z = axial

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