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Brazilian Journal of Chemical Engineering

Print version ISSN 0104-6632On-line version ISSN 1678-4383

Braz. J. Chem. Eng. vol.36 no.4 São Paulo Oct./Dec. 2019  Epub Jan 13, 2020

https://doi.org/10.1590/0104-6632.20190364s20180450 

Fluid Dynamics, Heat and Mass Transfer

SIMULATION AND EXPERIMENTAL INVESTIGATION OF POWER CONSUMPTION, GAS DISPERSION AND MASS TRANSFER COEFFICIENT IN A MULTI-PHASE STIRRED BIOREACTOR

1 Fujian Provincial Academy of Environmental Science, Fujian, China. E-mail: fjepi@163.com


Abstract

Bio-oxidation leaching of refractory gold concentrate was carried out in a typical gas-liquid-solid three-phase stirred bioreactor. The bacteria, which adhered to solid particle surfaces and spread over the liquid phase, require sufficient oxygen and nutrient of carbon and nitrogen, and then the mass transfer rate should be considered. However, the mass transfer coefficient, which could be used to evaluate the mass transfer rate, was determined by gas holdup, bubble diameter and power consumption. Therefore, a three-phase Eulerian equation with k-ε turbulence model and additional models of two bubble diameter, a torque and a mass transfer coefficient were applied in the simulation. The simulation results were validated with experimental data and used to analyze the spatial distribution of the mass transfer coefficient. The results demonstrated that the simulation and experimental data of Sauter mean bubble diameter and power number were in good agreement and the simulation results of the mass transfer coefficient were agree with the values calculated by an empirical correlation. The Davoody model can be strongly recommended for further simulation of the bubble diameter, and the gas holdup and mass transfer coefficient were very unevenly distributed in the lower zone. Hence, the lower impeller clearance should be considered.

Keywords: Gas holdup; Bubble diameter; Mass transfer coefficient; Multi-phase flow; Refractory gold concentrate

INTRODUCTION

Bioleaching of refractory gold concentrate was carried out in a stirred bioreactor. The stirred bioreactor is recommended for gas-liquid-solid three phase systems and it is widespread in the hydrometallurgy, fermentation, biological pharmacy, sewage treatment and allied industries (Chisti and Moo-Young 1988). Oxygen, carbon and nitrogen, which are necessary in bioleaching and aerobic process, were always provided by air. The bacteria adhered to solid particle surfaces and spread over the liquid phase, and their growth was restrained by oxygen concentration and nutrient concentration of carbon and nitrogen (Chisti and Jauregui-Haza 2002; Scargiali, et al. 2012; Karimi, et al. 2013; Scargiali, et al. 2014). To provide sufficient oxygen, carbon and nitrogen, mass transfer should be maintained for better performance (Herrera, et al. 1989; Shrihari, et al. 1991; Boon and Heijnen 1998; Nemati, et al. 1998; Hansford and Vargas 2001).

Mass transfer between phases was a limiting step of the overall process rate in most cases and sometimes positive proportional to the mass transfer coefficient (k l a), which could also be used to evaluate the mass transfer capacity and carry out the design and scale up of bioreactors (Karimi, et al. 2013; Buffo, et al. 2016). Previous documents pointed out that the mass transfer coefficient is determined by specific surface area and liquid mass transfer coefficient (Higbie 1935; Nedeltchev 2003) . As we known, specific surface area can be calculated by a = 6ϕg/db and the liquid mass transfer coefficient has a positive relation with energy dissipation () (Dhanasekharan, et al. 2005; Kerdouss, et al. 2008). Energy dissipation was proportional to power consumption. Obviously, the mass transfer coefficient was affected by the gas holdup, bubble diameter and power consumption.

In the gas-liquid-solid three phase stirred bioreactor, experiments focused on how solid suspension influences the mass transfer coefficient in the early years (Joosten, et al. 1977), and now it focuses on how to enhance mass transfer (Sedahmed, et al. 2014; Atef, et al. 2015). The simulations of the mass transfer coefficient mainly focused on the gas-liquid two phase stirred reactor (Ljungqvist and Rasmuson 2001; Kerdouss, et al. 2008), and the simulations of the gas-liquid-solid three phase stirred reactor mainly focused on the solid suspension (Murthy, et al. 2007; Panneerselvam, et al. 2008; Xu, et al. 2012). A handful of documents extended the application of the multi-phase simulation to the bubble diameter (Yang, et al. 2016; Zheng, et al. 2018; Zheng, et al. 2019). In this manuscript, the mass transfer coefficient and its influence on the power consumption, gas holdup and bubble diameter were studied based on experiments and simulations. Then, how the aeration rate and solid concentration influenced these factors in a state of just suspension were discussed based on the multi-Euler equation and standard k-ε model.

EXPERIMENTAL DETAILS

Experimental setup

Experiments were carried out in a stirred bioreactor, which was a cylindrical flat bottom vessel. The reactor was configured with 4 baffles and made of Plexiglas. Its diameter (T) was 226 mm and height (H) was 500 mm. A ring sparger, of diameter T/2, was mounted with a clearance of 40mm. A radial impeller (HEDT) of diameter 100 mm was installed with a clearance of T/3 and driven by a variable speed AC motor. The agitation speeds were noted using a tachometer. The bubble parameters were measured with a calibrated conductivity probe (BVW-2 four-channel), which was provided by the Institute of Process Engineering, Beijing, China. The torques were obtained from a torque meter (LONGLV-WTQ1050D-5Nm), which was installed vertically for connecting the agitator shaft and AC motor, and made by Shanghai Longlv Journey Electronic Technology Co. LTD. Water (ρl=1000 kg/m3) as the liquid phase, refractory gold concentrate (ρs=3548.2 kg/m3, average diameter is 45.42 μm) as the solid phase and air as the gas phase were used in the multiphase experiments. A schematic diagram for experiments is pictured in Fig.1.

Figure 1 Experimental setup (1: Reactor; 2: Baffle; 3: Impeller; 4: Ring sparger; 5: AC motor; 6: Torque sensor; 7: Gas valve; 8: Rota meters; 9: Gas buffer tank; 10: Gas bottle; 11: Conductivity probe; 12: Data collector; 13: Cameras; 14: Computer; 15: Frequency converter). 

Experimental procedure

Water was first added to the stirred bioreactor, and then 4 weights of solid particles were prepared for different concentrations of 5, 10, 15 and 20wt%. Air was admitted to the reactor via a ring sparger when the experiments were carried out as follows:

(1) First, the slurry of 5 wt% was prepared. The air from the gas bottle 10 was admitted to the reactor 1 via the ring sparger 4, and then adjusted via the gas valve 7 to control the aeration rate at 3 levels of 4, 8 and 12 L/min. The just-suspension speed was obtained by observing visually via the cameras 13, which was focused on the reactor bottom and provided that the solid particles remained at the reactor bottom for no more than 2 s. The torque was measured by the torque meter.

(2) Secondly, adjusted the probe depth, voltage signals were recorded with a BVW-2-4-CH equipment, and then the voltage signals changed to the local gas holdup and bubble diameter data with the manufacturer-supplied software.

(3) Thirdly, the slurry concentration was gradually increased to 10, 15 and 20 wt%, and the steps (1)-(2) repeated to obtain the data of just suspension, power consumption, gas holdup and bubble diameter were obtained.

Data processing

In the mass transfer process, specific surface area, which directly determines the mass transfer coefficient, is inversely proportional to the bubble diameter. Obviously, in the multiphase stirred bioreactor, the bubble diameter is one of the key factors. Calderbank (1958) proposed that the Sauter mean bubble diameter db,32could be calculated by,

db,32=db,i3db,i2 (1)

where, db is the bubble diameter.

Power consumption is another one of the key factors in a stirred bioreactor, because more power input via the impeller agitator could directly reduce the bubble diameter and improve the mass transfer coefficient. Power consumption can be obtained from torque Ta and impeller speed N,

P=2πNTa (2)

Under just suspension conditions in the gas-liquid-solid three phase stirred bioreactor, the impeller speeds were not constant. Power number (NP) is a suitable indicator to evaluate power consumption. The power number NP could be calculated from the power consumption P,

NP=PρmN3D5 (3)

where N is impeller speed, D is impeller diameter, ρm = ρ1(1 - ϕ1) + ρsϕs is slurry density.

Chandrasekharan and Calderbank (1981) suggested that the mass transfer coefficient in the three phase stirred bioreactor could be estimated as follows:

kla¯=0.0248T4PV0.551Qm0.551/Tm (4)

where T is impeller diameter, P is power consumption, V is liquid volume, Qm is aeration rate. In this equation, SI units are required.

MODEL DESCRIPTION

There is a lot of literature on simulating the gas-liquid two phase stirred bioreactor (Khopkar, et al. 2003; Gentric, et al. 2005; Ding, et al. 2010). However, CFD applications on the gas-liquid-solid three phase stirred bioreactor are still rare (Murthy, et al. 2007; Panneerselvam, et al. 2008; Xu, et al. 2012; Yang, et al. 2016). The Multi-Euler equation was confirmed as the proper way to describe flow behavior of each phase, in which gas, liquid and solid were all treated as continuous phases, penetrating and interacting with each other everywhere in the computational domain (Murthy, et al. 2007; Panneerselvam, et al. 2008; Inc 2013). Turbulence in the dispersed phases of gas and solid could be ignored in the three phase stirred bioreactor, and the standard k-ε model was proper for simulating turbulence of the liquid phase (Elghobashi and Abou-Arab 1983). The models were founded as follows:

(1) Continuity equation

tφiρi+·φiρiu¯i=0 (5)

(2) Momentum transfer equation

tφiρiu¯i+·φiρiu¯iu¯i=-φiP+·τi+φiρig+Fi (6)

(3) k equation

tφ1ρ1k1+·φ1ρ1u¯1k1-μv1+μt1δk1k1=φ1P1+ρ1ε1 (7)

(4) ε equation

tφ1ρ1ε1+·φ1ρ1u¯1ε1-μv1+μt1δε1ε1=φ1ε1k1Cε1P1+Cε2ρ1ε1 (8)

where ∑iφi = 1, τi = φiμeff,i(?u̅i + (?u̅i)T), i =g , l, s, Fg = -FD,lg, Fl =FT,l + FD,lg + FD,ls, Fs = -FD,ls - ?Ps, Cε1 = 1.44, Cε2 = 1.92, δkl = 1.0, δεl = 1.3.

The turbulence production of viscous and buoyancy forces,

Pl=μtlu:u+uT23u3μtlu+ρlkl (9)

The additional solids pressure caused by solids collision (Bouillard, et al. 1989),

Ps=G0expcφsφsmφs (10)

where, G0 =1.0Pa, c =6 00, φsm = 0.376.

The turbulence dispersion force (Barrue, et al. 2001),

FTl=CTlρlklφl (11)

The interchange forces of drag and lift (Montante, et al. 2001; Khopkar, et al. 2005),

FD,li=34CD,lidjρ1φ1u¯i-u¯1u¯i-u¯1 (12)

where, j = b,p while i = g,s respectively.

The drag coefficient could be calculated by (Bakker and Vandenakker 1994; Brucato, et al. 1998),

CD,li=CD01+Kidjλ3 (13)

where Ks = 8.67 × 10-4, Kg = 6.5 × 10-6.

Interaction coefficient between gas and liquid,

CD0=max24Re1+0.15Reb0.687, 2.667EoEo+4.0 (14)

Interaction coefficient between liquid and solid (Khopkar and Ranade 2006),

CD0=24Rep1+0.125Rep0.72 (15)

where, Eotvos number Eo = g(ρl - ρg) db 2)/σ, σ is the surface tension of the gas.

Power number and torque

Under just suspension conditions in the gas-liquid-solid three-phase stirred bioreactor, the impeller speeds were not constant. Power number (NP), which should be employed to evaluate power consumption, could be calculated by eq.(3). The torque could be calculated from each finite element of the pressure difference between the front and back of the impeller blade (∆P)i, distance from the shaft ri and the area of the finite element Ai (Zadghaffari, et al. 2009),

Ta=iΔPiAiri (16)

Bubble diameter

Although the PBM bubble diameter models are widely applied on the stirred reactor and bubble column (Wang and Wang 2007; Akbari, et al. 2015; Buffo, et al. 2017; Guo, et al. 2017), the various problems were still unresolved. Hence, in this manuscript, simple models were suggested. Hinze (1955) provided the maximum bubble diameter formula based on the hydrodynamic mechanism, as proposed by Zhang, et al. (2013),

db,max=0.725σ0.6ρ0.6ε0.4 (17)

Yang, et al. (2016) recommended that the bubble diameter could be calculated by,

db=0.68db,max (18)

where ρ is density, ε is turbulence energy dissipation rate. Zheng, et al. (2019) adopted this model to calculate the mass transfer coefficient. Zheng, et al. (2018) also suggested the model amended by Davoody, et al. (2016),

db,32=1.98φg0.5ρ0.6ε0.4σ0.6+0.0015 (19)

where φg is gas holdup, ρ = φgρg + φlρl + φsρs is slurry density, ε is energy dissipation and σ is surface tension.

Mass transfer coefficient

Higbie (1935) penetration theory showed that each point of the mass transfer coefficient could be calculated by (Nedeltchev 2003; Dhanasekharan, et al. 2005; Kerdouss, et al. 2008),

kla=4Diπερlμl1/46φgdb (20)

where Di is the gas diffusivity in water (i=O2,CO2,N2).

Variation coefficient

Bohnet and Niesmak (1979) proposed that the variation coefficient could be used to characterize the suspension quality. Barigou and Greaves (1992) extended the application of the variation coefficient to measuring uniformity of the gas dispersion. The uniformity of the mass transfer coefficient could also be measured. The equation for the variation coefficient was as follows:

σx=1ni=1nxixavg12 (21)

where i is represents each finite element, n is the total finite elements, x is the gas holdup (φg) or mass transfer coefficient (kla). The larger σx is, the more non-uniform the distribution of x.

Numerical methodology

In this manuscript, ANSYS Fluent 17.1 was applied for solving the previous CFD models. All simulation cases were carried out on the model based on the experimental setup, such as stirred bioreactor, impeller and sparger, and then validated with experimental data. The CFD control equation was derived from the three-Euler model, and solved by the finite volume method. The computer domain was divided into two regions of the impeller region (rotating reference frame) and the other region (stationary reference frame), and the MRF approaching method was applied in the computational process. The impeller region was limited in r/R ≤ 0.6 and 0.25 ≤ z/T ≤ 0.75. ICEM was employed for generating tetrahedral elements in the mesh generation process. A good quality of skewness < 0.7 was ensured and 700,000-900,000 grid elements were provided. SIMPLE was used as a pressure-velocity coupling algorithm and all equations converged with errors under 10-4. All walls, e.g., reactor wall, impeller blade surface and baffle, were treated as non-slip surfaces and the standard wall function was deployed on them. The reactor top was set as the pressure outlet boundary for the gas phase. The solid concentration was initialized as homogeneous and the region above the free surface of the liquid phase was filled with gas. For better discussion, the reactor was divided into three zones: lower zone z/T ≤ 0.20, center zone 0.20 ≤ z/T ≤ 0.45 and upper zone z/T ≥ 0.45.

RESULTS AND DISCUSSIONS

Power consumption

Power number was adapted to measuring power consumption when the impeller speeds were not constant, especially in the gas-liquid-solid three phase stirred bioreactor which was running at the just suspension status. The power numbers could be calculated from the impeller speeds and torques, which were obtained from the conditions (Qg = 4,8,12 L/min, Xs = 5,10,15,20 wt%) under the just suspension status. The torques of the experiment and simulation could be calculated by eq.(16) and eq.(3), respectively. The power numbers of the experiment and simulation were both plotted in Fig.2.

Figure 2 Comparison of simulation and experimental results of power number. 

As Fig.2(a) shows, the simulation results agreed with the experiment and, as Fig.2(b) shows, the errors between them were less than 15%. Hence, the power consumption could be well simulated in the gas-liquid-solid three-phase stirred bioreactor. The power number was not significantly changed by the aeration rate and solid concentration under the just suspension status.

The turbulent energy dissipation, which was the characteristic parameter of the local power consumption, was apparently different along with the axial direction (as shown in Fig. 3), especially in the aeration system. To validate the local power consumption of the CFD simulation, the experimental data, which were obtained from PIV technology by Geng, et al. (2012), were considered in the comparison (as shown in Fig.4).

Figure 3 The axial distribution of the dimensionless turbulent energy dissipation rate. 

Figure 4 Comparison of results for the dimensionless turblunent energy dissipation rate. 

As Fig.4 shows, the conditions, e.g. the lower, center and upper regions, and the aeration rates, were all consistent with the literature (Geng, et al. 2012). The results showed that the simulation data were in agreement with experiment. The center zone consumed a great deal of energy.

Gas holdup

Under the condition of Qg = 12 L/min, Xs = 5 wt%, the just suspension speed was 394 rpm; the experimental data of the gas holdup were measured with a conductivity probe. Then the simulation data were compared with them, as shown in Fig.5.

Figure 5 The spatial distribution of the gas holdup. 

Fig.5(a) was created for the data limits and interpolation method. Compared with the simulation result (Fig.5(b)), the cloud show the same trends. To analyze the difference between the experimental and simulation data, the data at the position of z/T=0.60,0.75 and r/R=0.56,0.73,0.82 were extracted to make a comparison, shown in Fig.6. As Fig.6 shows, the experimental and simulation local gas holdup followed the same trend.

Figure 6 Comparison of simulation and experimental results of local gas holdup. 

As Fig.5 shows, the spatial distributions of the gas holdup were very different. To investigate this difference, the reactor was divided into three zones of lower, center and upper. The gas holdup of each zone varied with the aeration rate and solid concentration under the just suspension status. The trends of these variations are shown in Fig.7.

Figure 7 Trend map of the gas holdup for different zones. 

As Fig.7 shows, the gas holdup in all zones was mainly affected by the aeration rate and little impacted by the solid concentration under the just suspension status. In each zone, the gas holdup also changed clearly. Therefore, to measure these changes, the variation coefficient of the gas holdup was taken into account in this work. The trend of the variation coefficients via the aeration rate and solid concentration are demonstrated in Fig.8. According to Fig.7 and Fig.8, the gas holdup in the lower zone was very unevenly distributed. Thus the lower impeller clearance was important.

Figure 8 Trend map of the variation of the gas holdup for different zones. 

Bubble diameter

Under the condition of Qg = 12 L/min, Xs = 5 wt%, Njs = 394 rpm, the experimental data of the bubble diameter were measured with a conductivity probe. There were two different models for calculating the bubble diameter (Yang, et al. 2016; Zheng, et al. 2018), and the simulation data were compared with them, as shown in Fig.9.

Figure 9 The spatial distribution of the bubble diameter. 

Fig.9(a) was created for the data limits and interpolation method. Compared with the simulation result (Fig.9(b)-(c)), the cloud maps of Fig.9(a) and (c) showed the same trends, and Fig.9(b) showed different trends. It appeared that the Davoody et al. (2016) model could be more applicable than the Yang et al. (2016) model. To confirm the most feasible model, the Sauter mean bubble diameters in the experimental and simulation in all conditions (Qg = 4,8,12 L/min, Xs = 5,10,15,20 wt%) were simulated by the two different models and plotted together, as shown in Fig. 10.

Figure 10 Comparison of simulation and experimental results of Sauter mean bubble diameter. 

As Fig. 10(a) shows, the simulation data were all in agreement with the trends, but the simulation data given by the Yang et al. (2016) model were significantly greater than the experimental data. According to Fig. 10(b), the Yang et al. (2016) model should be revised by the correction coefficient 0.8. As Fig. 10(b) shows, the simulation results of both the revised Yang et al. (2016) model and the Davoody et al. (2016) model agreed with experiment. In conclusion, the Davoody et al. (2016) model should be recommended in this work.

Mass transfer coefficient

There are a lot of studies of the mass transfer coefficient based on theoretical and empirical considerations (Chisti and Jauregui-Haza, 2002; Martín et al. 2008). The mass transfer coefficient could be accurately estimated. Eq.(4) suggested by Chandrasekharan and Calderbank (1981) was used to calculate the mass transfer coefficient from the aeration rate and the power comsumption. The estimation and simulation data on all conditions (Qg = 4,8,12 L/min, Xs = 5,10,15,20 wt%) were plotted in Fig.11.

Figure 11 Comparison of estimation and experimental results of the mass transfer coefficient. 

As Fig.11(a) shows, the simulation results agreed with experiment, and as Fig.11(b) shows, the errors between them are less than 30%. Hence, the simulation data could be applied for further analysis. The spatial distribution of the mass transfer coefficients, obtained under the condition of Qg = 12 L/min, Xs = 5 wt%, Njs = 394 rpm, is displayed in Fig.12.

Figure 12 The spatial distribution of the mass transfer coefficient. 

As Fig.12(a) and (c) show, the maximum values of the mass transfer coefficient were at the impeller tip and sparger, while the minimum values were under the impeller. As Fig.12(a) and (b) show, compared with the zones on and under the impeller, the mass transfer coefficients over the impeller zone were more well-distributed. To prove this point, the variation coefficient of the mass transfer coefficients via the aeration rate and solid concentration in three separated zones are plotted in Fig.13.

Figure 13 Trend map of the variation of the mass transfer coefficient for different zones. 

As Fig.13 shows, the higher aeration rate could more easily bring about an uneven distribution of the mass transfer coefficient in the lower zone (under the impeller zone). The variation coefficient of the mass transfer coefficient could significantly change in the center zone (the impeller zone), and the distribution of the mass transfer coefficient was slightly changed with the aeration rate or solid concentration in the upper zone (over the impeller zone). Well-distributed in the upper zone, a higher solid concentration could improve the uniform distribution of the mass transfer coefficient over the reactor.

CONCLUSIONS

In this work, an Eulerian multi-fluid model and the standard k-ε turbulence model were developed for simulating the gas-liquid-solid three-phase stirred bioreactor. The power consumption, bubble diameter and mass transfer coefficient models were applied for further analysis. The simulation results of the power consumption, gas holdup, bubble diameter and mass transfer coefficient were all validated by the experimental or estimation data, respectively. All cases were studied for the just suspension status. The main conclusions can be listed as follows,

(1) Good agreement was found between the simulated and measured/estimated variables, such as the power consumption, gas holdup, bubble diameter and mass transfer coefficient, in the gas-liquid-solid three-phase stirred bioreactor. The errors of the power consumption and bubble diameter were both less than 15%, and the errors of the mass transfer coefficient were less than 30%.

(2) The power number did not significantly change with the aeration rate and solid concentration under the just suspension status. The turbulent energy dissipation, which was the characteristic parameter of the local power consumption, was apparently different along the axial direction and the center zone consumes a great deal of energy.

(3) The gas holdup of each zone was mainly affected by the aeration rate and little impacted by the solid concentration under the just suspension status. The variation coefficient of the gas holdup showed that the gas holdup was very unevenly distributed in the lower zone, and the lower impeller clearance was important.

(4) Even though the Yang model was revised by the correction coefficient 0.8, the spatial trend did not agree with the experimental cloud map. The Davoody model could be more applicable and is strongly recommended for simulating the bubble diameter in further studies.

(5) The higher aeration rate could more easily bring about an uneven distribution of the mass transfer coefficient in the lower zone (under the impeller zone). The variation coefficient of the mass transfer coefficient significantly changed in the center zone (the impeller zone), and the distribution of the mass transfer coefficient was slightly changed with the aeration rate or solid concentration in the upper zone (over the impeller zone). Well-distributed at the upper zone, a higher solid concentration could improve the uniform distribution of the mass transfer coefficient over the reactor.

ACKNOWLEDGMENT

We acknowledge the financial support of the Science and Technology Department of Fujian Province, China, and the National Natural Science Foundation of China and Fujian (NO.2014R1004-3, NO.21476049, NO.21506033, NO.2015J01052, 2016H4023 and FG-2016005). We thank Fuzhou University for CFD simulation support and Xiamen Zijin Mining & Metallurgy Technology Co., Ltd. for equipment and instrument support.

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NOMENCLATURE

A

Area (m2)

C

The distance of the impeller from the reactor (tank, also called vessel) bottom, also called impeller clearance (m)

CD

Drag force constant (m)

D

Impeller diameter (m)

Di

The diffusion coefficient of i component (m2/s)

db

The gas bubble diameter (m)

dp

The solid particle size (m)

ds

The distance of two points separated by bubble, particle, eddy or bacteria (m)

F

Force per unit volume (N/m3)

g

Acceleration of gravity (m/s2)

H

Height of reactor (m)

Hl

Liquid free surface off-bottom (m)

h

The ring sparger clearance, the height from the center of the flat bottom (m)

K

An empirical constant

k

Turbulent kinetic energy (J/m3)

N

Impeller speed (rps)

NP

Power number (-)

p

Pressure of flow(Pa)

P

Power demand of the agitation (W)

Qg

Aeration rate (L/h)

Qm

Aeration rate (m3/s)

r

Radius (m)

Re

Reynolds number

T

Reactor (vessel) diameter (m)

Ta

The torque of agitation (N . m)

t

Time (s)

ui

Reynolds average velocity of the phase i (m/s)

uds

The velocity fluctuations at two points separated by ds (m/s)

udb

Fluctuation velocity of turbulent flow at bubble scale (m/s)

V

Entire liquid volume (m3)

Xs

Solid concentration, also called solid mass fraction or, solid loading (%wt)

Greek symbols

δε

Turbulent Prandtl number for the dissipation rate

δk

Turbulent Prandtl number for the turbulent kinetic energy

ε

Turbulence energy dissipation rate (m2/s2)

λ

Kolmogorov length scale (m)

μ

Viscosity (Pa . s)

ν

Kinematic viscosity (m2/s)

τ

Shear stress (Pa)

ρ

Density (kg/m3)

σl

Surface tension of liquid (N/m)

σs

The variation coefficient (normalized standard deviation) for solid

φi

Volume fraction of i phase

ω

Angular velocity (rad/s)

Subscripts

b

Bubble

D

Drag

avg

Average value

g

Gas or gassed conditions

js

Just suspended condition for solids

l

Liquid

lg

Action between liquid and gas

ls

Action between liquid and solid

m

Max

p

Particle

s

Solid

t

Turbulent

v

Viscosity

Abbreviations

CFD

Computational fluid dynamics

HEDT

Half elliptical blades disk turbine

PIV

Particle image velocimetry

Received: November 05, 2018; Revised: February 21, 2019; Accepted: April 05, 2019

* Corresponding author: Yuanfeng Chen - E-mail: fjepi@163.com

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