- Email: [email protected]

ARTICLE IN PRESS Particuology xxx (2012) xxx–xxx

Contents lists available at SciVerse ScienceDirect

Particuology journal homepage: www.elsevier.com/locate/partic

CFD simulation of particle suspension in a stirred tank Nana Qi a , Hu Zhang b,1 , Kai Zhang a,∗ , Gang Xu a , Yongping Yang a a b

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China School of Chemical Engineering, The University of Adelaide, Adelaide, SA 5005, Australia

a r t i c l e

i n f o

Article history: Received 8 March 2011 Accepted 19 January 2012 Keywords: Stirred tank Smith turbine Particle suspension CFD simulation

a b s t r a c t Particle suspension characteristics are predicted computationally in a stirred tank driven by a Smith turbine. In order to verify the hydrodynamic model and numerical method, the predicted power number and ﬂow pattern are compared with designed values and simulated results from the literature, respectively. The effects of particle density, particle diameter, liquid viscosity and initial solid loading on particle suspension behavior are investigated by using the Eulerian–Eulerian two-ﬂuid model and the standard k–ε turbulence model. The results indicate that solid concentration distribution depends on the ﬂow ﬁeld in the stirred tank. Higher particle density or larger particle size results in less homogenous distribution of solid particles in the tank. Increasing initial solid loading has an adverse impact on the homogeneous suspension of solid particles in a low-viscosity liquid, whilst more uniform particle distribution is found in a high-viscosity liquid. © 2012 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

1. Introduction Mechanically stirred tanks have been widely used for dispersing solid particles into liquid, such as dissolution, crystallization, chemical synthesis with suspended catalysts and/or suspension polymerization. It is beneﬁcial for these operations to homogeneously suspend all particles in the tank. Nevertheless, such homogeneity cannot be achieved easily because of complicated ﬂuid dynamics and high-energy consumption (Yamazaki, Tojo, & Miyanami, 1986). Therefore, understanding the hydrodynamics in stirred tanks will provide considerable insight for designing the reactors and impellers as well as for determining the operational conditions of solid–liquid systems. The hydrodynamics in stirred tanks can be characterized by either experimental measurement or numerical simulation. At present, however, it is difﬁcult to measure the proﬁle of solid concentration in the whole tank, especially in the high-energy dissipation regions. Computational ﬂuid dynamics (CFD), on the other hand, has opened a window to visualize the single- and multi-phase ﬂow in stirred tanks without conducting real-time experiments, thus providing in-depth details about the ﬂuid ﬂow (Hosseini, Patel, Ein-Mozaffari, & Mehrvar, 2010; Zhang, Wang, & Fan, 2005;

∗ Corresponding author. Tel.: +86 10 61772413. E-mail addresses: [email protected] (H. Zhang), [email protected] (K. Zhang). 1 Tel.: +61 8 83033810.

Zhang, Zhang, & Fan, 2009). In the last decade, reasonable success of predicting the suspension quality has been achieved in several studies (such as Khopkar, Kasat, Pandit, & Ranade, 2006; Montante & Magelli, 2005). However, in most simulations a low-viscosity liquid and a classic impeller (usually Rushton turbine or pitched-blade turbine) were chosen. For better distributing solid particles, the Smith turbine (ST) impeller appears more efﬁcient than the Rushton turbine impeller in terms of suspension uniformity (Lup˘as¸teanu, Galaction, & Cas¸caval, 2008), impeller power number (Mishra & Joshi, 1993; Sivashanmugam & Prabhakaran, 2008) and shear rate (Folescu, Galaction, & Cas¸caval, 2007). The mixing behavior in a single-phase stirred tank with a ST impeller was characterized in our previous paper (Qi, Wu, Wang, Zhang, & Zhang, 2010). In this work, a twoﬂuid model along with a standard k–ε turbulence model is applied for elucidating the effect of particle properties (density and diameter) and effective slurry viscosity on particle suspension behavior in a solid–liquid stirred tank driven by an ST impeller. 2. Hydrodynamic model 2.1. CFD model The three-dimensional transient CFD model is employed to compute the local hydrodynamics of solid–liquid two-phase ﬂow in the stirred tank. An Eulerian–Eulerian approach is adopted to describe the ﬂow behavior of each phase. In this model, both liquid and solid are treated as continua, interpenetrating and

1674-2001/$ – see front matter © 2012 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.partic.2012.03.003

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

2

Continuity equation: Nomenclature C C0 CD CTD Cε1 Cε2 C D ds FC Fi F D

solid concentration, vol.% initial solid concentration, vol.% drag coefﬁcient momentum transfer coefﬁcient for interface drag force k–ε turbulence model constant k–ε turbulence model constant k–ε turbulence model constant impeller diameter, m particle diameter, m centrifugal and Coriolis forces, N/m3 total interfacial force, N/m3 interfacial force due to drag, N/m3

FiT g H k L N NP p P r R t T Tq u W z

interfacial force due to turbulent dispersion, N/m3 acceleration due to gravity, m/s2 liquid height, m turbulence kinetic energy per unit mass, m2 /s2 bottom clearance, m impeller rational speed, rpm power number pressure, Pa power, W radial position, m tank radius, m time, s tank diameter, m torque, N m velocity, m/s blade width, m axial position, m

i

Greek symbols ˛ volume fraction ε turbulence dissipation rate, m2 /s3 tangential position, m viscosity, Pa s liquid phase kinematic eddy viscosity, m2 /s tl tl,p particle-induced component of turbulent viscosity, kg/m/s conventional shear-induced turbulent viscosity, tl,s kg/m/s solid phase kinematic eddy viscosity, m2 /s ts v kinematic viscosity, m2 /s density, kg/m3 turbulent Prandtl number t tl liquid turbulent Schmidt number k turbulence model constant for the k equation ε k–ε turbulence model constant Subscripts liquid phase l m mixture s solid phase turbulence t tip impeller tip

∂(˛i i ) i ) = 0, + ∇ · (˛i i u ∂t

(1)

stand for volume fraction, density and velocity where ˛, and u vector, respectively. The subscript i represents liquid or solid phase. Momentum equation: i) ∂(˛i i u iu i ) = −˛i ∇ pi + ∇ · (˛i i (∇ u i + (∇ u i )T )) + + ∇ · (˛i i u ∂t FC + Fi + ˛i i g ,

(2)

where p, , and g are pressure, viscosity and gravity acceleration, respectively, FC represents the centrifugal and Coriolis forces (Ochieng & Onyango, 2008), and Fi is the interphase force acting on phase i due to the presence of the other phase, j. For closure of the Reynolds averaged momentum equation, an interface model and a turblence model are required, which are given below. 2.1.1. Interface model The interphase momentum exchange term mainly consists of drag force, virtual mass force, Basset force, lift force and turbulent dispersion force. The interfacial forces of signiﬁcance are considered to be drag force and interphase turbulent dispersion force in this study while the inﬂuence of lift force and virtual mass force on the simulated solid holdup proﬁles is relatively minor (Ljungqvist & Rasmuson, 2001) and Basset force in most cases is much smaller than the drag force (Khopkar et al., 2006). Accordingly, the interphase force can be expressed as: Fi = FiD + FiT ,

(3)

where FiD and FiT are drag force and interfacial force due to turbulent dispersion, respectively. The drag component of the solid–liquid interfacial force term is given as: 3 CD,ls s − u l |(u s − u l ). ˛s |u FlsD = 4 ds l

(4)

The drag coefﬁcient exerted by the solid phase on the liquid phase, CD,ls , is obtained by the classical Schiller Naumann drag model: CD,ls = max Re =

24 Re

(1 + 0.15Re0.687 ), 0.44 ,

(5)

s − u l| l ds |u . l

(6)

The turbulent dispersion force, FiT , is calculated by the model of Lopez de Bertodano (1991): FlsT = CTD CD tl tl

∇˛

s

˛s

−

∇ ˛l ˛l

,

(7)

where CTD stands for the momentum transfer coefﬁcient for the interphase drag force, tl for turbulent viscosity, and tl for liquid turbulent Schmidt number. ˛s and ˛l are the solid and liquid phase volume fractions, respectively. 2.1.2. Turbulence model The standard k–ε model for the liquid phase is described as:

interacting with each other in the computational domain. Based on the conservation principles of mass and momentum, the continuity and momentum equations in the stirred tank are given below:

∂ l kl ) = ∇ · ˛l l + tl (˛l l kl ) + ∇ · (˛l l u k ∂t ˛l Pl − ˛l l εl ,

∇ kl + (8)

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model

ARTICLE IN PRESS

PARTIC-405; No. of Pages 10

N. Qi et al. / Particuology xxx (2012) xxx–xxx

∂ l εl ) = ∇ · ˛l l + tl (l ˛l εl ) + ∇ · (˛l l u ε ∂t ˛l

∇ εl +

εl (Cε1 pl − Cε2 l εl ), kl

(9)

where Cε1 , Cε2 , k , and ε are parameters in the standard k–ε model and the following values are selected: Cε1 = 1.45, Cε2 = 1.9, k = 1.0, and ε = 1.3. The liquid phase turbulent viscosity is calculated on the basis of the Sato enhanced turbulence model (Sato, Sadatomi, & Sekoguchi, 1981): tl = tl,s + tl,p ,

(10)

where tl,s is the conventional shear-induced turbulent viscosity, obtained by the standard k–ε model as: tl,s = C l

k2 , ε

(11)

where C is 0.09, tl,p is a particle-induced component of turbulent viscosity given by: s − u l |. tl,p = C,p l ˛s ds |u

(12)

The solid phase turbulence is modeled using a zero equation model, in which solid turbulent viscosity is proportional to that of the liquid phase (Bartrand, Farouk, & Haas, 2009): ts =

s tl , l t

(13)

where t is a turbulent Prandtl number relating the solid phase kinematic eddy viscosity ts to the liquid phase kinematic eddy viscosity tl . 2.2. Stirred tank conﬁguration and grid topology Fig. 1 shows the stirred tank with a ST impeller. A ﬂat bottom cylindrical tank with a diameter T = 0.476 m, equal to the liquid height H, and four bafﬂes at a width of 0.1T equally spaced along the tank wall are used in this simulation. The impeller is a sixbladed ST impeller with a diameter D = 0.4T, located at T/3 above the base of the vessel. A relatively large impeller (D = 0.4T) is chosen in this study in comparison to the standard conﬁguration with D = T/3, since a large impeller can provide better suspension and dispersion of particles in the liquid (Wang, Mao, Wang, & Yang, 2006). Each blade has a width of 0.06 m and a height of 0.038 m, and other parameters of the impeller are listed in Table 1. The physical properties of the liquid and solid particles (spherical glass beads) are provided in Table 2. The impeller speed (N) is selected as 400 rpm in this paper. Table 1 Main parameters of ST impeller. Main parameters

Value (m)

D

Ddisc

Dhub

Wimp

Himp

tdisc

0.19

0.13

0.035

0.060

0.038

0.004

Table 2 Physical properties of the solid–liquid system. Liquid

Glass beads

Density (kg/m3 )

1000

Viscosity (Pa s) Particle diameter (m)

1 × 10−3 2 × 10−2 –

2500 4000 –

Volume fraction (vol.%)

–

Fig. 1. Schematic diagram of the ST impeller-driven stirred tank.

1 × 10−4 3 × 10−4 2.5 11.0

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

3

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

4

Fig. 2. Grid system used in stirred tank.

For computation, the stirred tank is usually divided into two parts (Khopkar et al., 2006; Qi, Wang, Zhang, & Zhang, 2010). As shown in Fig. 2, one is the inner rotating zone and the other is the outer non-rotating zone. An unstructured tetrahedral grid is densiﬁed to deal with large aspect ratios and sharp element angles in the inner rotation zone, while a structured hexahedral grid is employed to improve the convergence and accuracy of the calculation in the outer non-rotation zone. An interface is introduced for exchanging all parameters in the governing equations of liquid and solid between the inner and outer zones. In order to solve the relative rotation between the impeller and bafﬂes, the multiple reference frame method is employed as it provides accurate prediction and demands less computational resource (Koh & Schwarz, 2005; Qi, Wu, et al., 2010; Qi, Wang, et al., 2010).

impeller and bafﬂe, solid and liquid are treated as free-slip and no-slip boundary conditions, respectively. Initially, the desired volume fraction of solid is settled uniformly at the bottom of the tank, whilst the liquid is kept stationary in the remaining space of the tank. For the discretization scheme, high resolution is applied for the momentum equation and the convergence criterion, and time step is set as 0.01 s. By gradually increasing the number of discretized elements, 126,928 meshes can meet with the requirement of grid-independency for all simulations. 3. Veriﬁcation of modeling In order to verify the reliability of the CFD model and accuracy of the numerical method, qualitative ﬂow pattern and quantitative power number are compared with the data from the literature.

2.3. Simulation procedure 3.1. Flow pattern Boundary condition, initial condition and discretization scheme are set for the numerical simulations in the ANSYS CFX 10.0 software package (ANSYS Inc., 2005). On the walls of the tank,

For better observing particle suspension process in the tank, three-dimensional transient images of solid volume fraction are

Fig. 3. The isovolume of solid volume fraction above 1.55C0 (dark region) at different time intervals (˛s = 11.0 vol.%, = 1 × 10−3 Pa s, ds = 100 m).

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

5

solid–liquid system, the same two-loop ﬂow pattern is obtained numerically for either liquid or solid phase (see Fig. 5), although the shape of liquid ﬂow pattern in the two-phase ﬂow is more or less different with that in the single-phase ﬂow. Such result stems from the relatively small particles and the high volume of ﬂuid pumped by each concaved ST blade. Furthermore, a region with very low velocities of liquid or solid at the bottom of the tank is also observed, which may lead to higher retention time than that for other regions of the tank. The presence of such region lowers the mixing overall efﬁciency, and may produce stationary solid particles deposited on the bottom of the tank. 3.2. Power number

Fig. 4. Liquid velocity vector proﬁle in the single phase stirred tank ( = 1 × 10−3 Pa s).

presented in Fig. 3. Initially, 11.0 vol.% of solid particles are settled uniformly in the lower part of the tank with a liquid viscosity of 1 × 10−3 Pa s. The impeller starts to stir at a rotational speed of 400 rpm when time is zero. Solid particles are driven by the ﬂuid ﬂow induced by the ST impeller to the other region, which leads to a shrunk region of higher solid volume fraction (shown in dark region color) at the tank bottom. Finally, the accumulated solid particles are dispersed from the tank bottom when the time exceeds 15 s, implying that all particles are suspended in the whole tank. Solid concentration distribution depends heavily on the ﬂow ﬁeld in the stirred tank. As shown in Fig. 4, a typical twoloop ﬂow pattern is obtained at a vertical mid-bafﬂe plane in single-phase stirred tank. Such a similar ﬂow pattern with radial discharge stream, upper and lower circulation zones is also found by using another type of radial impeller, i.e. Rushton turbine impeller, by Kasat, Khopkar, Ranadeb, and Pandit (2008). For the

For overall power consumption, the impeller power number NP has been commonly used to verify quantitatively the CFD simulations of ﬂow in stirred tanks (Deglon & Meyer, 2006). As reported in our former paper (Qi, Wu, et al., 2010), the predicted power number for the single liquid phase is 3.08, which is close to the design value of 3.20 as well as the predicted value of 3.10 by Li (2007), and 3.20 by Sivashanmugam and Prabhakaran (2008). Up to now, there has been no well-established technique for determining the power number in solid suspension systems (Sardeshpande, Sagi, Juvekar, & Ranade, 2009). In this work, average slurry density is introduced for calculating the power number in the slurry phase as follows: NP =

P , m N 3 D5

(14)

where P is the power draw, and m is the average slurry density. The average slurry density and power draw are given by: m = s ˛s + l (1 − ˛s ),

(15)

P = 2 NTq ,

(16)

where s and l are the particle density and liquid density, respectively; Tq is total torque of the rotating parts, which is gained from the simulation results. By combining Eqs. (14)–(16), the predicted power number NP in the solid–liquid two-phase system for different liquid viscosities and initial solid loadings are illustrated in Fig. 6, to show that NP increases with increase with liquid viscosity and initial solid loading. Moreover, at the liquid viscosity of 1 × 10−3 Pa s,

Fig. 5. Liquid and solid velocity vector proﬁles in the solid–liquid phase stirred tank (˛s = 11.0 vol.%, = 1 × 10−3 Pa s, ds = 100 m).

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

6

Fig. 6. Predicted NP with different initial solid loadings and liquid viscosities (ds = 100 m).

NP has a proportionally linear relationship with the initial solid loading, conﬁrming the same trend reported by Bubbico, Di Cave, and Mazzarotta (1997). In this case, most of the energy from the impeller is consumed to generate liquid circulation inside the tank and solid concentration has a negligible impact on the power number. At a liquid viscosity of 2 × 10−2 Pa s, however, it is observed that NP ﬁrstly increases with an increase in the initial solid loading, and then keeps constant when the initial solid loading continues to increase. This is because that a higher viscosity leads to a relatively homogenous suspension, thus reducing the interaction between particles and liquid. The above qualitative or quantitative comparison of computational predictions with the results from the literature shows that the above-mentioned CFD model and numerical method can be utilized to simulate the particle suspension characteristics in solid–liquid stirred tank driven by a ST impeller. Fig. 7. Normalized solid concentration at r/R = 0.50 under different particle densities for ˛s = 2.5 vol.% and ds = 100 m.

4. Results and discussion To understand the factors which affect the suspension of solid particles in a liquid phase, particle sedimentation is analyzed ﬁrst in a stationary liquid, in which the terminal velocity (settling velocity) can be gained if the net force acting on the particle becomes zero when the particle falling through a ﬂuid under its own weight. There are three forces acting on the particle: buoyancy force, drag force and gravity force: F G = F B + F D, FG , FB

(17) FD

where and is gravity force, buoyancy force and drag force, respectively. Eq. (17) can be expressed alternately as: 3 d2 l v2t d s g = d3 l g + CD , 6 6 4 2

(18)

where d is particle diameter, and vt is terminal setting velocity of a spherical particle:

vt =

4 gd( − ) 1/2 s l 3

CD l

,

(19)

and CD is a drag coefﬁcient, which is a function of the terminal Reynolds number: Ret =

dvt l .

(20)

It should be noted that is the effective slurry viscosity in this solid–liquid two-phase system. For spherical particles, CD has different expressions according to Ret (Kunii & Levenspiel, 1969) as below: CD = CD =

24 , Ret 18.5 Re0.6 t

CD = 0.44,

Ret < 0.4; ,

Ret = 2–500;

(21)

Ret = 500–200, 000.

Eqs. (19) and (20) show that the terminal velocity depends on particle diameter, particle density and effective slurry viscosity when particles are assumed as spherical and liquid density is not changeable, implying that all these parameters further inﬂuence the suspension of solid particles in the solid–liquid stirred tank. 4.1. Effect of particle properties Figs. 7 and 8 exhibit the effect of particle density and diameter on the distribution of solid concentration. Simulated results conﬁrm that the uniformity of solid particles increases with a decrease in either the density difference between particle and liquid or particle diameter while the other operating conditions are kept the same. A

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

7

is achieved although few solid particles are still accumulated at the bottom of the tank. Such a phenomenon can be explained from the ﬂuid ﬂow pattern imposed by radial ﬂow impellers as shown in Fig. 5. Similar results were reported by Kasat et al. (2008) using a standard Rushton turbine with a solid volume fraction of 10 vol.% at a rotational speed of 1200 rpm, and by Fradette et al. (2007) using a down-pumping marine propeller with a solid volume fraction of 11.9 vol.% at a rotational speed of 350 rpm. For liquid viscosity of 2 × 10−2 Pa s (Fig. 9b), however, the suspension of solid particles is by far more uniformly distributed throughout the tank even after 12 s. Based on the spatio-temporal distribution of solid concentration in Fig. 9, the current reactor may be re-designed to replace the ﬂat bottom with a contoured bottom toward the impeller just below the impeller as shown in Fig. 10 to reduce the particle accumulation at the bottom. Furthermore, the quantitative solid concentration at r/R = 0.5 shows that a quite uniform distribution is found for a more viscous liquid for two different initial solid loadings, as shown in Fig. 11a and b. According to Eq. (17) or (18), the higher viscous drag and the higher ﬂow resistance in a more viscous liquid lead to smaller settling velocity of a single particle, thus preventing a particle from returning to the bottom after suspension and, also, less energy is required to maintain it in suspension. The more uniform particle distribution in a more viscous liquid can be explained in terms of the particle response time (tp ), the time that a particle takes to respond to change in the carrier ﬂow velocity (Hosseini et al., 2010): tp =

Fig. 8. Normalized solid concentration at r/R = 0.50 under different particle diameters for ˛s = 2.5 vol.% and s = 2500 kg/m3 .

higher particle density or a larger particle size results in faster settling velocity and less homogenous distribution of the particles. On the contrary, the adverse effect of higher density and larger diameter on the homogeneity of particles suspension is weakened when particles are suspended in a more viscous liquid (Figs. 7b and 8b), showing that higher liquid viscosity can be used to uniformly suspend heavier and larger particles, as was suggested by Murthy, Ghadge, and Joshi (2007). 4.2. Effect of effective slurry viscosity The effective slurry viscosity of a solid–liquid system depends on liquid viscosity and initial solid loading (Thomas, 1965). In many chemical reaction processes, such as liquid fuel synthesis (Zhang & Zhao, 2006; Qi et al., 2009), the liquid viscosity is different at different positions and times. In order to explore their independent contribution, the effect of liquid viscosity and initial solid loading are explored numerically in the following sections. 4.2.1. Liquid viscosity Fig. 9 illustrates the computed snapshots of solid concentrations at the vertical cross-section in the middle of two bafﬂes of the stirred tank. For liquid viscosity of 1 × 10−3 Pa s (Fig. 9a), the liquid pumped by the impeller gradually lifts the particles from the bottom of the tank. After 15 s, relatively homogeneous suspension

p dp2 18

.

(22)

The above equation shows the particle response time is inversely proportional to the ﬂuid viscosity. In a more viscous ﬂuid, particles have less time to respond to the varying ﬂuid velocity and the particle velocity shows less change throughout the whole tank. Such fact indicates that an increase in liquid viscosity leads to a reduction of the interactions between particles and liquid. All particles accumulated at the bottom have been lifted and the bottom effect completely vanished, as was also found by Kasat et al. (2008). 4.2.2. Initial solid loading For the same liquid viscosity, Fig. 11a and b shows relatively more uniform distribution of solid particles for lower initial solid loading of 2.5 vol.%. This result can also be conﬁrmed by the standard deviation of the normalized solid concentration, which is 1.1% at an initial solid loading of 2.5 vol.%, and 2.3% at an initial solid loading of 11.0 vol.%. While for the normalized solid concentration at a higher liquid viscosity, the standard deviation of 0.2% for a lower initial solid loading and 0.3% for a higher solid loading is obtained, indicating that a quite similar and more homogeneous solid particle distribution is obtained in a more viscous liquid. Fig. 12 compares axial liquid velocities under different initial solid loadings and radial positions in the stirred tank, showing a decreased axial liquid velocity with increasing the initial solid loading from 2.5 to 11.0 vol.% except for the impeller tip region at about z/H = 0.3—this provides a possible reason why inhomogeneous particle dispersion becomes signiﬁcant for increased initial solid loading. For the initial solid loading of 2.5 vol.%, a characteristic peak of the axial liquid velocity is found at the impeller tip, at which the maximum velocity decreases from 0.9utip at r/R = 0.42–0.7utip at r/R = 0.50 and further to 0.5utip at r/R = 0.58. For the initial solid loading of 11.0 vol.%, however, a characteristic peak of the axial liquid velocity is found at r/R = 0.42 and then becoming weaker with increasing r/R., as

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

8

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

Fig. 9. Snapshots of normalized solid concentration proﬁles (˛s = 11.0 vol.%, ds = 100 m).

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

9

Fig. 10. Schematic diagram of the new designed stirred tank with dished (or contoured) bottom.

Fig. 12. Axial liquid velocity distribution under different initial solid loadings ( = 1 × 10−3 Pa s, ds = 100 m).

Fig. 11. Normalized solid concentration at r/R = 0.50 under different liquid viscosities (s = 2500 kg/m3 ; ds = 100 m).

was shown experimentally by Wu and Patterson (1989) using a Ruston turbine with similar tank geometry. In addition, the computational axial liquid velocity proﬁles of both initial solid loadings are almost the same in a more viscous liquid, which coincides with the distributions of normalized solid concentration in Fig. 11.

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003

G Model PARTIC-405; No. of Pages 10

10

ARTICLE IN PRESS N. Qi et al. / Particuology xxx (2012) xxx–xxx

5. Conclusions Flow pattern, power number, and particle suspension characteristics in a stirred tank driven by a Smith turbine are simulated by an Eulerian–Eulerian model along with a standard k–ε turbulence model, leading to the following results. (1) Power number in solid–liquid stirred tank is predicted by introducing an average slurry density. Simulated data indicate that power number increases with increasing initial solid loading, and becomes nearly constant for initial solid loading varying from 2.5 to 11.0 vol.% for a liquid viscosity of 2 × 10−2 Pa s. (2) Typical two-loop ﬂow pattern of both liquid and solid particles is observed in the solid–liquid stirred tank. A small region with low liquid or solid velocity appears near the vertical axis of the tank below the impeller, attributable to the accumulation of solid particles at the bottom of the tank. (3) Higher particle density or larger particle size results in higher settling velocity and less homogenous distribution of the particles. However, the negative effect of higher density or larger diameter on the homogeneity of particles suspension is minimized when particles are suspended in a more viscous liquid. (4) Higher initial solid loading contributes to a non-uniform solid distribution in a less viscous liquid, while more uniform solid distribution can be achieved in a more viscous liquid at both low and high initial solid loadings. (5) A new reactor with a contoured bottom toward the impeller is designed according to the solid concentration proﬁles, which can be applied to reduce the particle accumulation at the bottom, especially just below the impeller center. Acknowledgements Financial support from National Natural Science Foundation of China (20976191 and 51025624), Program for New Century Excellent Talents in University (NCET-09-0342) and 111 Project (B12034) is gratefully acknowledged. References ANSYS Incorporated. (2005). ANSYS CFX-solver release 10.0: Theory. Pennsylvania: ANSYS Ltd. Bartrand, T. A., Farouk, B., & Haas, C. N. (2009). Countercurrent gas/liquid ﬂow and mixing: Implications for water disinfection. International Journal of Multiphase Flow, 35(2), 171–184. Bubbico, R., Di Cave, S., & Mazzarotta, B. (1997). Inﬂuence of solid concentration and type of impeller on the agitation of large PVC particles in water. Recent Progres en Genie des Procedes, 11, 81–88. Deglon, D. A., & Meyer, C. J. (2006). CFD modeling of stirred tanks: Numerical considerations. Minerals Engineering, 19(10), 1059–1068. Folescu, E., Galaction, A.-I., & Cas¸caval, D. (2007). Optimization of mixing in stirred bioreactors 3. Comparative analysis of shear promoted by the radial impellers for anaerobic simulated broths. Romania Biotechnology Letters, 12(4), 3339–3350. Fradette, L., Tanguy, P. A., Bertrand, F., Thibault, F., Ritz, J. B., & Giraud, E. (2007). CFD phenomenological model of solid–liquid mixing in stirred vessels. Computers and Chemical Engineering, 31(4), 334–345. Hosseini, S., Patel, D., Ein-Mozaffari, F., & Mehrvar, M. (2010). Study of solid–liquid mixing in agitated tanks through computational ﬂuid dynamics modeling. Industrial & Engineering Chemistry Research, 49(9), 4426–4435.

Kasat, G. R., Khopkar, A. R., Ranade, V. V., & Pandit, A. B. (2008). CFD simulation of liquid-phase mixing in solid–liquid stirred reactor. Chemical Engineering Science, 63(15), 3877–3885. Khopkar, A. R., Kasat, G. R., Pandit, A. B., & Ranade, V. V. (2006). Computational ﬂuid dynamics simulation of the solid suspension in a stirred slurry reactor. Industrial & Engineering Chemistry Research, 45(12), 4416–4428. Koh, P. T. L., & Schwarz, M. P. (2005). CFD modelling of bubble-particle attachments in a ﬂotation cell. In Proceedings of Centenary of Flotation Symposium Brisbane, Australia, (pp. 201–207). Kunii, D., & Levenspiel, O. (1969). Fluidization engineering. New York: John Wiley & Sons., pp. 64–73, 76–79. Li, Z.P. (2007). Experimental investigation and numerical simulation of ﬂow characteristics in vessels stirred by disc turbines (Unpublished doctoral dissertation, Beijing University of Chemical Technology, China) (in Chinese). Ljungqvist, M., & Rasmuson, A. (2001). Numerical simulation of the two-phase ﬂow in an axially stirred reactor. Chemical Engineering Research and Design, 79, 533–546. Lopez de Bertodano, M. (1991). Turbulent bubbly ﬂow in a triangular duct (Unpublished doctoral dissertation). Rensselaer Polytechnic Institute, Troy New York. Lup˘as¸teanu, A.-M., Galaction, A.-I., & Cas¸caval, D. (2008). Comparative evaluation of radial impellers efﬁciency for bioreactors with stirred bed of immobilized cells. 1. Disperser sawtooth and smith turbine. Roumanian Society of Biological Sciences, 13(4), 3821–3836. Mishra, V. P., & Joshi, J. B. (1993). Flow generated by a disc turbine: Part III: Effect of impeller diameter, impeller location and comparison with other radial ﬂow turbines. Chemical Engineering Research and Design, 71, 563–573. Montante, G., & Magelli, F. (2005). Modelling of solids distribution in stirred tanks: Analysis of simulation strategies and comparison with experimental data. International Journal of Computational Fluid Dynamics, 19(3), 253–262. Murthy, B. N., Ghadge, R. S., & Joshi, J. B. (2007). CFD simulations of gas–liquid–solid stirred reactor: Prediction of critical impeller speed for solid suspension. Chemical Engineering Science, 62(24), 7184–7195. Ochieng, A., & Onyango, M. S. (2008). Drag models, solids concentration and velocity distribution in a stirred tank. Powder Technology, 181(1), 1–8. Qi, N. N., Zhang, K., Song, Q. T., Gao, X. G., Zhang, H., & Lu, J. J. (2009). 2D and 3D simulations of ﬂuid dynamics in the bubble reactor for liquid fuel synthesis: Comparisons against experiments. Energy Source Part A, 31(18), 1598–1611. Qi, N. N., Wang, H., Zhang, K., & Zhang, H. (2010). Numerical simulation of ﬂuid dynamics in the stirred tank by the SSG Reynolds stress model. Frontiers of Chemical Engineering in China, 4(4), 506–514. Qi, N. N., Wu, G. Y., Wang, H., Zhang, K., & Zhang, H. (2010). CFD simulation of mixing characteristics in stirred tank by Smith turbine. Journal of Chemical Industry and Engineering (China), 61(9), 2305–2313 (in Chinese). Sardeshpande, M. V., Sagi, A. R., Juvekar, V. A., & Ranade, V. V. (2009). Solid suspension and liquid phase mixing in solid–liquid stirred tanks. Industrial & Engineering Chemistry Research, 48(21), 9713–9722. Sato, Y., Sadatomi, M., & Sekoguchi, K. (1981). Momentum and heat transfer in two-phase bubble ﬂow-I. International Journal of Multiple Flow, 7(2), 167–177. Sivashanmugam, P., & Prabhakaran, S. (2008). Simulation of an effect of a bafﬂe length on the power consumption in an agitated vessel. International Journal of Food Engineering, 4(2), 1–16. Thomas, D. G. (1965). Transport characteristics of suspension. VIII: A note on the viscosity of Newtonian suspensions of uniform spherical particles. Journal of Colloid and Interface Science, 20(3), 267–277. Wang, F., Mao, Z. S., Wang, Y. F., & Yang, C. (2006). Measurement of phase holdups in liquid–liquid–solid three-phase stirred tanks and CFD simulation. Chemical Engineering Science, 61(22), 7535–7550. Wu, H., & Patterson, G. K. (1989). Laser-Doppler measurements of turbulent ﬂow parameters in a stirred mixer. Chemical Engineering Science, 44(10), 2207–2221. Yamazaki, H., Tojo, K., & Miyanami, K. (1986). Concentration proﬁles of solids suspended in a stirred tank. Powder Technology, 48(3), 205–216. Zhang, H., Wang, W., & Fan, S. D. (2005). Power consumption and mixing in a miniaturised bioreactor. Progress in Nature Science, 15(Suppl. 1), 117–123. Zhang, H., Zhang, K., & Fan, S. D. (2009). CFD simulation coupled with population balance equations for aerated stirred bioreactors. Engineering in Life Sciences, 9(6), 421–430. Zhang, K., & Zhao, Y. L. (2006). A scale-up strategy for low-temperature methanol synthesis in a circulating slurry bubble reactor. Chemical Engineering Science, 61(5), 1459–1469.

Please cite this article in press as: Qi, N., et al. CFD simulation of particle suspension in a stirred tank. Particuology (2012), doi:10.1016/j.partic.2012.03.003