Numerical simulation of fluid dynamics in the stirred tank by the SSG ...

Front. Chem. Eng. China 2010, 4(4): 506–514 DOI 10.1007/s11705-010-0508-7

RESEARCH ARTICLE

Numerical simulation of fluid dynamics in the stirred tank by the SSG Reynolds Stress Model Nana QI1,2, Hui WANG1,3, Kai ZHANG (✉)1,4, Hu ZHANG (✉)2 1 State Key Lab of Heavy Oil Processing, China University of Petroleum, Beijing 102249, China 2 School of Chemical Engineering, University of Adelaide, Adelaide SA 5005, Australia 3 Beijing Aerospace WanYuan Coal Chemical Engineering Technology CO., Ltd, Beijing 100176, China 4 National Engineering Lab for Biomass Power Generation Equipment, North China Electric Power University, Beijing 102206, China

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2010

Abstract The Speziale, Sarkar and Gatski Reynolds Stress Model (SSG RSM) is utilized to simulate the fluid dynamics in a full baffled stirred tank with a Rushton turbine impeller. Four levels of grid resolutions are chosen to determine an optimised number of grids for further simulations. CFD model data in terms of the flow field, trailing vortex, and the power number are compared with published experimental results. The comparison shows that the global fluid dynamics throughout the stirred tank and the local characteristics of trailing vortices near the blade tips can be captured by the SSG RSM. The predicted mean velocity components in axial, radial and tangential direction are also in good agreement with experiment data. The power number predicted is quite close to the designed value, which demonstrates that this model can accurately calculate the power number in the stirred tank. Therefore, the simulation by using a combination of SSG RSM and MRF impeller rotational model can accurately model turbulent fluid flow in the stirred tank, and it offers an alternative method for design and optimisation of stirred tanks. Keywords stirred tank, fluid dynamics, numerical simulation, SSG Reynolds Stress Model, MRF

1

Introduction

Stirred tanks are widely used in the chemical & pharmaceutical processes, wastewater treatment and other industries. A large number of process applications involve mixing of a single phase in mechanically stirred Received January 16, 2010; accepted April 9, 2010 E-mail: [email protected], [email protected]

tanks/vessels. Nevertheless, the flow around a rotating impeller interacts with baffles in the tank, resulting in unsteady, three-dimensional, rotational turbulent flow, which is extremely complex and, therefore, has not been well understood [1]. Recently, Computational Fluid Dynamics (CFD) has been widely used to map the fluid dynamics within the stirred tanks [2–8], and it has become an effective tool for design and optimization of stirred tanks/vessels. However, one of the challenges for the CFD simulation of the flow in a stirred tank is to accurately capture the turbulence inside it. Two different methods have been developed for this purpose. One method is to perform a large eddy simulation (LES) or a direct numerical simulation (DNS). However, simulations of a fully turbulent flow with a very high Reynolds number (typically 50000 and higher) as encountered in practical applications require expensive computational resources [2]. The other approach is to apply the Reynolds-averaged Navier-Stokes (RANS) equations [3,4]. Several types of less-computational-demanding turbulence models have been used to close the RANS equations, from the zero equation model [5] to the Reynolds-stress transport models [6]. Among them, the standard k-ε model has been extensively investigated for simulating hydrodynamics in stirred tanks [7]. However, Han [8] challenged that this model was inappropriate to predict three-dimensional rotating flows with high turbulence because it is based on the assumption of isotropic turbulent transport. In contrast to the standard k-ε model, a Speziale, Sarkar and Gatski Reynolds Stress Model (hereafter referred to as SSG RSM), developed by Speziale et al. [9], is based on anisotropic turbulent transport. In this study, this model is applied to capture the turbulence details in a stirred tank agitated by a Rushton turbine. The details of the computational model as well as the results obtained are discussed in the following sections.

Nana QI et al. Numerical simulation of fluid dynamics in the stirred tank

2 System description and computational method 2.1

Stirred tank configuration

In order to verify the SSG RSM in this study, the stirred tank configuration for this simulation is the same as that used in the experiment work by Wu & Patterson [10]. As shown in Fig. 1, four baffles with width (w) of T/10 are equally arranged into a cylindrical tank with an inner diameter (T) of 0.27 m and a height (H) equal to the inner diameter (T). A six-blade Rushton turbine with a diameter (D) of 0.09 m is positioned at T/3 above the base of the vessel. Each blade has a width of D/4 and a height of D/5. The impeller is operated at a rotational speed, N, of 200 rpm, corresponding to a Reynolds number of 32000. 2.2

Grid generation

Grid generation method is essential for capturing the details of fluid flow in the stirred tank. In this study, a commercial mesh generator ANSYS ICEMCFD 10.0

507

(Ansys, Pennsylvania, USA) [11] is used to create a hybrid grid. Stirred tank is usually divided into two parts for computational purposes [1–8]. One is the inner rotating zone, and the other is the outer non-rotating zone as shown in Fig. 2. In the inner rotation zone, a densified tetrahedral, unstructured grid is applied because tetrahedral elements are able to deal with large aspect ratios and sharp element angles compared to traditional hexahedral elements. Densified elements are also required to capture details of trailing vortices and turbulence around the impeller tips. On the other hand, a hexahedral structured grid is more appropriate in the outer non-rotation zone because less densified elements can be used for the much bigger zone in contrast with the inner rotation zone, and the structured grid elements can also improve the convergence and accuracy of the calculation. Inner rotation and outer nonrotation zones are separated by an interface (Fig. 2), which is located in the middle between the impeller tip and the edge of the baffles in the radial direction. The Multiple Reference Frame (MRF) method is employed at the interface to interpolate the momentum and turbulence variables between two zones.

Fig. 1 Configuration of the stirred tank and its Rushton turbine impeller

Fig. 2 Mesh generation for the stirred tank (a) multiple reference frame; (b) structured meshes in the outer zone

508

2.3

Front. Chem. Eng. China 2010, 4(4): 506–514

↕ ↓

Governing equations for SSG RSM

At steady state, the turbulent flow of a viscous and incompressible fluid is governed by the continuity and Navier-Stokes equations as follows. Continuity equations
↕ ↓ r$
U ¼ 0 (1)

↕ ↓ ↕ ↓

In the differential stress model, u
u is made to satisfy a transport equation. A separate transport equation must be solved for each of the six Reynolds stress components of

Momentum equations (or Navier-Stokes equations)
↕ ↓ ↕ ↓
 ↕ ↓ r$
U
U – r$ r U ↕ ↓ ↕ ↓

↕ ↓

¼ – rp $ – r$ð
u
u Þ þ B ,

viscosity. B is the sum of body forces, and the fluctuating ↕ ↓ ↕ ↓ Reynolds stress contribution is
u
u. Unlike eddy viscosity models, the modified pressure has no turbulence contribution and is only related to the static (thermodynamic) pressure by   ↕ ↓ 2 p $ ¼ p þ r$ U – : (3) 3

↕ ↓ ↕ ↓

(2)

where p $ is a modified pressure, and μ is the fluid


u
u . To increase the robustness of RSM, usually, an isotropic formulation is used to replace the anisotropic diffusion coefficients. Then the differential equation for Reynolds stress transport can be simplified by

   2 κ2 2 ↕ ↓ ↕ ↓ r$ð
U
u
u Þ ¼ P þf þ r$  þ cs
ru
u – δ
ε: ε 3 3 ↕ ↓ ↕ ↓ ↕ ↓

↕ ↓

(4)

This equation becomes the normal RSM, which can be written in index notation as ∂ ∂ ðUk
ui uj Þ ¼ Pij þ fij þ ∂xk ∂xk ↕ ↓

where P is the production term, which is given by
 ↕ ↓ ↕ ↓ ↕ ↓ ↕ ↓ ↕ ↓ P ¼ –
u
u ðrU ÞT þ ðrU Þu
u :

   2 κ2 ∂ui uj 2 – δij
ε,  þ cs
ε ∂xk 3 3   ∂uj 1 ∂ui , þ Sij ¼ ∂xi 2 ∂xj

(6)

  1 ∂ui ∂uj : – Wij ¼ 2 ∂xj ∂xi

fij is the pressure-strain correlation; it can be expressed in the general form

where

fij ¼ fij1 þ fij2 ,

(7)

   ↕ ↓ ↕ ↓ ↕ ↓ 1↕ ↓↕ ↓ fij1 ¼ –
ε Cs1 a þCs2 a a – a $ a δ , 3

(8)

pffiffiffiffiffiffiffiffiffi ↕ ↓ ↕ ↓ ↕ ↓↕ ↓ ↕ ↓ fij2 ¼ – Cr1 P a þCr2
κ S – Cr3
κ S a $ a  T  ↕ ↓ ↕ ↓ T 2↕ ↓↕ ↓ ↕ ↓ ↕ ↓ þ Cr4
κ a S þ S a – a $ S δ 3   ↕ ↓↕ ↓ T ↕ ↓↕ ↓T þ Cr5
κ a W þ W a ,

ui uj 2 – δij , k 3

(11)

(12)

As the turbulence dissipation appears in the individual stress equations, an equation for ε is still required. It can be expressed as ∂
↕↓  ε
U k ε ¼ ðcε1 P – cε2
εÞ κ ∂xk    ∂ t ∂ε , (13) þ þ εRS ∂xk ∂xk where  ¼
CμRS

(9)

and aij ¼

(5)

(10)

κ2

ε

,

(14)

and the turbulent kinetic energy comes directly from 1 ↕ ↓↕ ↓ κ ¼ ui ui . 2 In this simulation, the SSG RSM, rather than the normal RSM, is employed. In this model, anisotropic diffusion coefficients are implemented into six Reynolds Stress components. In this case, the equation for Reynolds Stress transport becomes

Nana QI et al. Numerical simulation of fluid dynamics in the stirred tank

509

  ∂  ∂
κ ↕ ↓↕ ↓ ∂ui uj 2 – δij
ε: U
u u ¼ Pij þ fij þ δkl þ Cs
u k u l ε ∂xl ∂xk k i j ∂xk 3 Pij and fij are the same as the normal Reynolds Stress Model, and they can be obtained from Eqs. (6) – (12). The equation for ε is ∂
↕↓  ε
U k ε ¼ ðcε1 P – cε2
εÞ κ ∂xk   ∂
κ ↕ ↓↕ ↓ ∂ε þ δkl þ Cε
u k u l : (16) ε ∂xl ∂xk

present to the momentum equation as an add-in term. The relationship of the absolute velocity and the relative velocity is below ↕ ↓ ↕ ↓ ↕ ↓ ↕ ↓ vr ¼ v – Ω  r , ↕ ↓

CFD methodology

The numerical solution of the above governing equations in section 2.3 is obtained by ANSYS CFX 10.0 (Ansys, Pennsylvania, USA) [11], a commercial CFD software packages. In this software package, mass conservation discretization is applied on a non-staggered grid with pressure-velocity coupling based on the work of Rhie and Chow [12]. The second order upwind discretization method is used for the advection term in the momentum equation, and the solution is considered converged when the total residuals for the continuity equation dropped to below 1  10–4. In the stirred tank, the fluid hydrodynamics is quite complex due to the relative motion between the rotating impeller blades and the stationary baffles causes a cyclic variation numerical computation of the Navier-Stokes equation in the stirred tank. Several approaches have developed to model the impeller rotation including imposed boundary conditions [13], momentum source/ sink method [14], snapshot method [15], Inter-Outer method [16], Sliding Mesh (SM) model [17] and Multiple Reference Frames (MRF) model [18]. Among these models, MRF has been widely used in the stirred tank due to relative simple operation, accurate prediction and less computational demands [19,20]. In this study, steadystate calculation is performed. However, a rotation reference frame is used for the inner rotation zone, while a stationary reference frame for the outer non-rotation zone. For the interface between the rotation frame and the stationary frame, the tangential velocity must keep as zero [19]. In other words, the cross section of the interface must keep as roundness, not quadrilateral or other shapes. After importing the rotating frame, the fluid acceleration is Table 1

(17)

where Ω is the angular velocity vector, namely, the angular ↕ ↓ velocity of the rotating frame. r is the location vector in the rotating frame. In the rotating frame, the momentum equations for the absolute and the relative velocities are as follows.

↕ ↓  ∂
↕  ↓ ↕ ↓↕ ↓ ↕ ↓
v þ r$
vr v þ
Ω  v ∂t

The constants for above equations are listed in Table 1. 2.4

(15)

↕ ↓ ↕ ↓

¼ – rp þ r$ðTÞ þ
g þ F ,

(18)


 ∂
↕ ↓ ↕ ↓↕ ↓
vr þ r$
vr vr ∂t ↕ ↓
↕↓ ↕ ↓ ↕ ↓ ↕  ∂Ω ↕ ↓ ↕ ↓ ↓ þ
2 Ω vr þ Ω  Ω  r þ
r ∂t ↕ ↓ ↕ ↓

¼ – rp þ r$ðTÞ þ
g þ F ,

(19) ↕ ↓

where p is the static pressure, T is the stress tensor,
g and ↕ ↓

F are gravitation and external force, respectively. A symmetry boundary condition is defined at the surface of the liquid. The blades, disc and baffles were defined as thin surfaces, and grids are refined in the impeller region. Water with a density of 997 kg/m3 and a viscosity of 889  10–6 kg/m$s is used in this simulation. Initially, the velocities of the u, v, w are set as zero; The turbulence kinetic energy and the turbulence eddy dissipation are calculated as 3 k ¼ ½Idet maxðUs ,jUIG j,Uω Þ2 , 2

(20)

where Idef is the default turbulent intensity of 5%. Us is a minimum clipping velocity of 0.01 m/s to avoid a result of zero for the turbulent kinetic energy when an initial velocity value of zero exists. UIG is the velocity initial guess. Uω is the product of the simulation average length scale and the rotation rate. This term is designed to produce a suitable velocity scale for rotating domains.

Constants in the SSG Reynolds Stress model

CμRS

σεRS

cs

cε

cε1

cε2

Cs1

Cs2

Cr1

Cr2

Cr3

Cr4

Cr5

0.1

1.36

0.22

1.08

1.44

1.83

1.7

– 1.05

0.9

0.8

0.65

0.625

0.2

510

Front. Chem. Eng. China 2010, 4(4): 506–514

3

Results and discussion

3.1

Grid determination

The resolution of the computational grid determines the accuracy of computational results. However, fine resolution imposes a huge computational burden. A compromise should be made between “reasonable accuracy” and “reasonable computational expense” [21]. Grid elements reported in the various CFD simulations of stirred tanks range from an order of 104 to 106 in a similar geometrical dimension. To determine the grid resolution required successively refined grids, it is normally used to perform simulations until no notable differences in the predicted values for the important same variables are observed. Four different levels of grid resolution are used to assess the sensitivity of the predicted flow fields to the grid. The details of grid numbers for four levels of grid resolution are summarized in Table 2. In this simulation, the flow flied in the stirred tank is chosen to assess its sensitivity to four levels of grid resolution. The computed results for the flow field, present as the mean velocity vector is shown in Fig. 3. It can be seen that the grid has a significant influence on the calculated flow field and the structure of trailing vortices. When a coarse grid (Grid 1 and Grid 2) is chosen for the inner zone, no vortices around the tip of impellers can be seen. However, when a relatively densified grid (Grid 3 and Grid 4) in the inner zone is adopted, a couple of Table 2

vortices can be found near the impeller tip, which has been demonstrated by experimental work of Escudié et al. [22] or computational work of Deglon & Meyer [21]. The shape and structure of the vortices in Fig. 3(c) and (d) are quite similar, although Fig. 3(d) reveals more detailed features of the vortices. However, grid elements in Grid 4 significantly increase CPU time for this simulation. Three different levels of grid resolution have been used for the outer zone, from the coarsest one (Grids 2 & 3) to the finest one (Grid 4). The grid density at the outer zone is found that it does not have significant contribution to the prediction of the flow field outside the impeller zone (figures are not shown). Around 300000 elements are enough to satisfy to capture the basic shape and structure of trailing vortices at the tip of impeller. In the context, Grid 3 is employed for the following simulations. 3.2

Prediction of flow patterns

Figure 4 presents flow patterns simulated with Grid 3 under steady state by using the SSG RSM. Figure 4(a) represents the cross section of horizontal mid-plane of the impeller, while Fig. 4(a) represents the cross section of vertical mid-plane between two baffles. It is clear that the liquid is discharged from the impeller, which results in two large circulations in Fig. 4(a). One of the circulations is below the impeller, with downward flow near the tank wall and upward flow in the centre of the tank; whilst the other circulation is above the impeller with upward flow near the

Four different grid resolutions inner zone (unstructured)

outer zone (structured)

global number

Grid 1

85085

391484

476569

Grid 2

131831

299431

431262

Grid 3

244913

299431

544344

Grid 4

305774

425748

731522

Fig. 3 Mean velocity vectors near the impeller tip for different grid resolutions (a) Grid 1; (b) Grid 2; (c) Grid 3; (d) Grid 4

Nana QI et al. Numerical simulation of fluid dynamics in the stirred tank

511

Fig. 4 Velocity vector profiles in the tank (a) Vertical section in the middle of the two baffles; (b) cross section of the impeller

tank wall and downward flow in the centre of the tank. The similar simulation results are also reported by Zhu et al. [23] using the standard k-ε model for a stirred tank with geometrically similar configuration. Although both the standard k-ε model and the SSG RSM can product similar macro flow fields in the stirred tank, the standard k-ε model can not capture detailed anisotropic trailing vortices at the tip of the impeller, which are described in more details in the section 3.4. 3.3

Mean velocity prediction

To validate the predicted velocity field quantitatively, the mean velocity value in radial, tangential and axial directions is compared with experimental data obtained by Wu & Patterson [10]. The impeller and tank geometrical configuration, operated parameters and working fluid for the simulation are the same as experimental work. The velocity components in radial, tangential and axial directions are presented at the vertical line r/R = 0.5. Comparing the calculated and measured velocity profiles, it can be seen the predicted velocity component are in good agreement, quantitatively with the experimental data (Fig. 5). 3.4

Prediction of trailing vortices near the impeller

In order to display the development of trailing vortices behind the moving blade, the velocity field is drawn in Fig. 6(b) for the vertical plane at different tangential positions between two successive blades, as shown in Fig. 6(a). In Fig. 6(b) at 15° behind the leading blade, two large vortices can be seen, which have been captured by k-ε model and

other models for the stirred tank. Furthermore, two small vortices are also developed just at the tip of impeller blade, which part has been enlarged in Fig. 7. 3.5

Power number prediction

An accurate CFD model is able to predict important global parameters. Among them, the impeller power number Np has been commonly used to further validate consumption in a stirred tank [7,24,25]. Power number is defined by NP ¼

P ,
N 3 d 5

(21)

where P is the power required by the impeller, ρ is the fluid density, N is the rotation speed of the impeller, and d is the diameter of the impeller. The power is usually calculated from the total torque and rotation speed by the following equation P ¼ 2πN $Tq ,

(22)

and the total torque, Tq, required by the impeller can be obtained from X Tq ¼ Fi $Ri , (23) i

where Fi is the pressure difference between the front surface element i, and Ri is the radial distance from the axis of the shaft on which the impeller is mounted. Fi at Ri can be obtained directly form CFD simulation. Therefore, the power number from the above equations can be calculated to be 4.97, which is very close to the designed value of 5.00 [24].

512

Front. Chem. Eng. China 2010, 4(4): 506–514

Fig. 5 Velocity profiles near the impeller (a) radical velocity; (b) tangential velocity; (c) axial velocity

Fig. 6 Velocity vector distribute in different positions between two blades

4

Conclusions

The SSG RSM has successfully captured global flow patterns throughout the stirred tank and local characteristics of trailing vortices near the blade tips. The global flow field can be predicted using relatively coarse grids but predicting more subtle phenomena in the flow field, such

as the formation of trailing vortices, requires finer grids in the impeller region. The power number predicted by SSG RSM is 4.97, close to the designed value of 5.00, which show that this model can predict the power number for engineering application. Moreover, good agreement in mean velocity components is found between the experimental results and the numerical predictions by SSG RSM.

Nana QI et al. Numerical simulation of fluid dynamics in the stirred tank

513

Subscripts i, j, k

Cartesian coordinate direction vector

References

Fig. 7

Enlarged view of two trailing vortices near blade tip at 15°

Acknowledgements Financial support from the Major State Basic Research Development Program of China (973 Program, Grant No. 2005CB221205) was gratefully acknowledged. Ms. Qi would like to acknowledge financial support from the China Scholarship Council (CSC) during her stay at the University of Adelaide, Australia.

Nomenclature B

sum of body force, N/m3

cµRs，σεRS

constants in the SSG RSM

cs

constant in the SSG RSM

cε

content in the SSG RSM

cε1，cε2

constants in the SSG RSM

Cs1，Cs2

constants in the SSG RSM

Cr1~Cr5

constants in the SSG RSM

d

diameter of the impeller, m

D

diameter of Rushton turbine, m

F

external force, N/m3

H

height of the tank, m

Idef

default turbulent intensity of 5%, m2/ s2

N

rotation speed, 1/s

Np

power number

p

pressure, Pa

R

diameter of the tank, m

T

inner diameter of the tank, m

Tq

total torque, N$m

u

turbulent velocity, m/s

U

average velocity, m/s

w

baffle width, m

Greek letters δ

Kronecker delta, when i = j, δ = 1; when i≠j, δ = 0

ε

turbulence dissipation rate, m2/s3

κ

turbulence kinetic energy per unit mass, m2/s2

µ

viscosity, Pa $s

ξ

bulk viscosity, Pa $s

ρ

density, kg/m3

fij

pressure-strain correlation in the SSG model, kg $ m /s3

1. Buwa V, Dewan A, Nassar A F, Durst F. Fluid dynamics and mixing of single-phase flow in a stirred vessel with a grid disc impeller: experimental and numerical investigations. Chemical Engineering Science, 2006, 61(9): 2815–2822 2. Montante G, Lee K C, Brucato A, Yianneskis M. Numerical simulations of the dependency of flow pattern on impeller clearance in stirred vessels. Chemical Engineering Science, 2001, 56(12): 3751–3770 3. Khopkar A R, Mavros P, Ranade V V, Bertrand J. Simulation of flow generated by an axial-flow impeller: batch and continuous operation. Chemical Engineering Research & Design, 2004, 82(6 A6): 737–751 4. Li M, White G, Wilkinson D, Roberts K J. LDA measurements and CFD modeling of a stirred vessel with a retreat curve impeller. Industrial & Engineering Chemistry Research, 2004, 43(20): 6534– 6547 5. Panneerselvam R, Savithri S, Surender G D. CFD modeling of gasliquid-solid mechanically agitated contactor. Chemical Engineering Research & Design, 2008, 86(12): 1331–1344 6. Murthy B N, Joshi J B. Assessment of standard k-epsilon, RSM and LES turbulence models in a baffled stirred vessel agitated by various impeller designs. Chemical Engineering Science, 2008, 63(22): 5468–5495 7. Deglon D A, Meyer C J. CFD modeling of stirred tanks: numerical considerations. Minerals Engineering, 2006, 19(10): 1059–1068 8. Han L C. Numerical simulation of fluid flow in stirred tank reactors using CFD method. Dissertation for the Master Degree. Xiangtan: Xiangtan University, 2005 (in Chinese) 9. Speziale C G, Sarkar S, Gatski T. Modeling the pressure strain of turbulence: an invariant dynamical systems approach. Journal of Fluid Mechanics, 1991, 227(–1): 245–272 10. Wu H, Patterson G K. Laser-Doppler measurements of turbulent flow parameters in a stirred mixer. Chemical Engineering Science, 1989, 44(10): 2207–2221 11. ANSYS Incorporated. ANSYS CFX-Solver Release 10.0: Modeling. Canada: Ansys Canada Ltd, 2005 12. Rhie C M, Chow W L. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, 1983, 21(11): 1525–1532 13. Ranade V V, Joshi J B. Flow generated by a disc turbine. Part II: mathematical modelling and comparison with experimental data. Chemical Engineering Research & Design, 1990, 68A: 34–50 14. Zhang H, Lamping S R, Ayazi S P. Numerical simulation of mixing in a micro-well scale bioreactor by computational fluid dynamics. Chemical Research in Chinese Universities, 2002, 18(2): 113– 116 15. Ranade V V, Dommeti S M S. Computational snapshot of flow fenerated by axial impellers in baffled stirred vessels. Chemical Engineering Research & Design, 1996, 74: 476–484

514

Front. Chem. Eng. China 2010, 4(4): 506–514

16. Brucato A, Ciofalo M, Grisafi F, Micale G. Numerical prediction of flow fields in baffled stirred vessels: a comparison of alternative modeling approaches. Chemical Engineering Science, 1998, 53(21): 3653–3684 17. Luo J Y, Gosman A D, Issa R I, Middleton J C, Fitzgerald M K. Full flow field computation of mixing in baffled stirred vessel. Chemical Engineering Research & Design, 1993, 71: 342–344 18. Dong L, Johansen S T, Engh T A. Flow induced by an impeller in an unbaffled tank-II numerical modeling. Chemical Engineering Science, 1994, 49(20): 3511–3518 19. Koh P T L, Schwarz M P. CFD Modelling of bubble-particle attachments in a flotation cell. In: Proc. Centenary of Flotation Symposium, Brisbane, 2005, 201–207 20. Koh P T L, Schwarz M P, Zhu Y, Bourke P, Peaker R, Franzidis J P. Development of CFD models of mineral flotation cells. In: Proc.

21. 22.

23.

24. 25.

Third International Conference on CFD in the Minerals and Process Industries. Melbourne, 2003, 171–175 Deglon D A, Meyer C J. CFD modelling of stirred tanks: numerical considerations. Minerals Engineering, 2006, 19(10): 1059–1068 Escudié R, Bouyer D, Liné A. Characterization of trailing vortices generated by a Rushton turbine. AIChE Journal. American Institute of Chemical Engineers, 2004, 50(1): 75–86 Zhu X Z, Miao Y, Xie Y J. Three-dimensional flow numerical simulation of double turbine impellers. Petro-Chem Equip, 2005, 34 (4): 26–29 (in Chinese) Chen Y C. Design of agitating equipment. Shanghai: Shanghai scientific & Technical Publishers, 1990 (in Chinese) Kshatriya S S, Patwardhan A W, Eaglesham A. Experimental and CFD characterization of gas dispersing asymmetric parabolic blade impellers. Int J Chem Reactor Eng, 2007, 5(1): A1