Study of SolidâLiquid Mixing in Agitated Tanks through Computational

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Ind. Eng. Chem. Res. 2010, 49, 4426–4435

Study of Solid-Liquid Mixing in Agitated Tanks through Computational Fluid Dynamics Modeling Seyed Hosseini, Dineshkumar Patel, Farhad Ein-Mozaffari,* and Mehrab Mehrvar Department of Chemical Engineering, Ryerson UniVersity 350 Victoria Street, Toronto, Ontario, M5B 2K3, Canada

Solid-liquid mixing is one of the most important mixing operations due to its vast applications in many unit operations such as crystallization, adsorption, solid-catalyzed reaction, suspension polymerization, and activated sludge processes. In this study, a computational fluid dynamics (CFD) model was developed for solid-liquid mixing in a cylindrical tank equipped with a top-entering impeller to investigate the effect of impeller type (Lightnin A100, A200, and A310), impeller off-bottom clearance (T/6-T/2, where T is tank diameter), impeller speed (150-800 rpm), particle size (100-900 µm), and particle specific gravity (1.4-6) on the mixing quality. An Eulerian-Eulerian (EE) approach, standard k-ε model, and multiple reference frames (MRF) techniques were employed to simulate the two-phase flow, turbulent flow, and impeller rotation, respectively. The impeller torque, cloud height, and just suspended impeller speed (Njs) computed by the CFD model agreed well with the experimental data. The validated CFD model was then employed to calculate the solid concentration profiles by which the degree of homogeneity was quantified as a function of operating conditions and design parameters. Introduction Mixing is one of the most widely used unit operations in polymer processing, fine chemicals, petrochemicals, biotechnology, agrichemicals, pharmaceuticals, paints and automotive finishes, cosmetics and consumer products, food, drinking water and wastewater treatment, pulp and paper, and mineral processing. It is carried out in mechanically agitated vessels for a variety of objectives, including for homogenizing multiple phases in terms of concentration gradient.1 Among various mixing processes (e.g., viscous liquid, liquid-liquid, gas-liquid, solid-liquid), solid-liquid mixing is one of the most important mixing operations because it plays a crucial role in many unit operations such as suspension polymerization, solid-catalyzed reaction, dispersion of solids, dissolution and leaching, crystallization and precipitation, adsorption, desorption, and ion exchange.2,3 The primary objectives of solid-liquid mixing are to avoid solid accumulation in the agitated vessel, to maximize the contacting area between the solids and liquid, and to ensure the system is homogeneous or solids particles are uniformly distributed throughout the vessel.4 In most of the solid-liquid processes, the solid particles are completely suspended. However, solid particles might not be distributed uniformly throughout the vessel. The performances of some processes such as crystallizers and heterogeneous photocatalytic reactors are affected by lack of the uniform distribution of solid particles. Therefore, a good understanding of the distribution of solid particles throughout the vessel is essential for design, process development, and scale up of the solid-liquid mixing systems. Some researchers have adopted a general practice for evaluation of solid-liquid mixing in agitated vessels through experimental investigation;5-12 however, it is difficult to obtain solid distribution in the whole tank through experimental measurements. Such an approach is usually time-consuming, costly, and sometimes it is impractical. Computational fluid dynamics (CFD) is a useful tool to analyze a system involving fluid flow through mathematical modeling and simulation by means of a computer based * To whom correspondence should be addressed. E-mail: fmozaffa@ ryerson.ca. Tel.: (416) 979-5000 ext 4251. Fax: (416) 979-5083.

program.13 CFD is emerging as a design tool for the development of new processes at a fraction of the cost and time of the traditional experimental and pilot-plant approaches. CFD has opened the gate to visualize the two-phase flow in solid-liquid mixing without conducting real-time experiments and provides in-depth details about the fluid flow which may not be available from practical approaches. CFD has enabled us to investigate the phase distribution in multiphase flow processes. Two main approaches are used in CFD modeling to solve the multiphase flow, namely, Eulerian-Lagrangian (EL) and Eulerian-Eulerian (EE) approaches.14 The EL approach resolves the continuous fluid phase in the Eulerian reference frame (i.e., fixed control volume for fluid) and the dispersed phase as individual particles, which move with the fluid. This approach requires a significant amount of computational time and huge memory space. This approach can provide good prediction only for low solid volume fraction (e5%).15 Some researchers have adopted the EL approach to study the solid-liquid mixing.16-20 While the EE approach considers the dispersed (particle) phase as a continuous (fluid) phase, interpenetrating and interacting with the fluid phase.21 This approach is less costly in terms of CPU time and also applicable to higher concentrations compared to the EL approach. Several models have been used to include the turbulence effect into Navier-Stokes equations such as the standard k-ε, renormalization group model (RNG k-ε), realizable k-ε, Reynolds stress model (RSM), and large eddy simulation (LES).2 Such models are generally validated by generating the velocity or concentration profile and comparing them with the experimental data.22-29 Among these, the standard k-ε model30 is most widely used model because it is robust, economical, and rapid. Besides, it gives stable calculations and reasonable results for many flow domains.23,25-27,31-38 Since mixing consumes a tremendous amount of processing time and energy,1 it is necessary to know about the various factors (e.g., impeller type, solid concentration, particle size, impeller speed, specific gravity of liquid and solid phases, and system geometry) affecting the suspension of solid particles in solid-liquid mixing processes. Some researchers have employed

10.1021/ie901130z  2010 American Chemical Society Published on Web 03/30/2010

Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010

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25,28,37

CFD to explore the effect of impeller type, particle size,27,31,32 and solid concentration25-27,31 on cloud height and just suspended impeller speed. Tamburini et al.39 employed CFD modeling to investigate the dynamic behavior of the mixing of the silica suspension from its initial stationary condition to the steady state condition for the fixed values of the impeller clearance, particle size, and solid specific gravity in a fully baffled mixing tank equipped with a radial flow impeller (Rushton turbine). However, a thorough literature review suggests that little attention has been devoted to computing the degree of homogeneity for the solid-liquid mixing processes through CFD. Thus, the objectives of this paper are to use CFD technique to explore the effect of impeller type, particle size, impeller speed, specific gravity of solid particles, impeller offbottom clearance, and power on the degree of homogeneity for the solid-liquid mixing, and to compare the CFD results with the experimental data. Experimental Setup The schematic diagram of the experimental setup used in this study is shown in Figure 1a. The mixing vessel was a transparent flat-bottomed cylindrical tank of 40 cm inner diameter (T) and 60 cm height and was filled with solid particles and water to a height (H) equivalent of the tank diameter (40 cm) occupying a total volume of about 0.050 m3. To avert vortex formation around the agitator, the tank was fitted with four equally spaced baffles having width of 3.4 cm (T/12). To prevent the accumulation of the solid particles behind the baffles, a 0.8 cm (T/50) gap between each baffle and the tank wall was considered. The tank was equipped with a top-entering impeller assembly and impeller speed was set to the desired revolutions per minute using a variable frequency drive (VFD). Three axial flow impellers, each with a 17.8 cm diameter (D), were used: Lightnin A100, A200, and A310 impellers (see Figure 1b). The pumping direction was downward for all impellers tested in this study. These axial-flow impellers produce high flow and are more costeffective in flow controlled operation like solid suspension.40 Cooke and Heggs41 reported that the hollow blade turbine is an efficient impeller for the solid-liquid mixing operations under gassed conditions. The impeller off-bottom clearance (C) was selected upon the test conditions between T/6 and T/2. A rotary-torque transducer (Staiger Mohilo, Germany) was employed to measure impeller torque and speed. In this study, glass beads and water were used as the solid phase and liquid phase, respectively. Electrical resistance tomography (Industrial Tomography Systems, Manchester, UK) was used to measure the concentration profiles within the mixing vessel as described by Hosseini et al.42 Eight tomographic sensor planes were positioned around the circumference of the mixing tank with the lowest one, numbered plane one, at a height of 4.25 cm from base, and the same interval was maintained vertically for subsequent planes, numbered accordingly in the upward direction. Each plane had 16 stainless steel sensors which were situated at equal intervals on the tank periphery. The height, width, and the thickness of the electrodes were 20, 30, and 1 mm, respectively. The reference for study was taken based on the single ground electrode located between plane four and five. More details regarding the experimental setup and procedure have been reported by Hosseini et al.42

Figure 1. Experimental setup: (a) mixing tank and (b) impellers used in this study.

conservation laws of mass and momentum for two-phase flow, the resulting conservation equations for phase k can be written as follows: Continuity equation43 ∂(akFk) f + ∇ · (akFkuk) ) 0 ∂t

(1)

Mathematical Model In the present study, the mathematical model is formulated based on the Eulerian-Eulerian multifluid model. Applying the

where F is fluid density, b u is the velocity vector, R is the volume fraction, and subscript k symbolizes phase k.

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Momentum equation43 ∂ f f f (a F u ) + ∇ · (akFkuk uk) ) -ak∇p + ∇ · (τck) + akFkb g + ∂t k k k f+F f+F f+F f) (2) (F D

Bk

Lk

νmk

where b g, p, b FD, b FBk, b FLk, and b Fvmk are gravitational acceleration, pressure, drag force, Buoyancy force, lift force, and virtual mass force, respectively. cτk is the kth phase stress-strain tensor: 2 f fT cτk ) Rk(µk + µkt)(∇µk + ∇µk ) - RkFkkkcI 3

(3)

Where µk, µkt, and kk are the shear viscosity, turbulent viscosity, and turbulent kinetic energy, respectively. Drag force is expressed by b FD )

∑ K (u lk

- uk)

f l

f

(4)

where Klk is the exchange coefficient between liquid and the kth phase. The solid-liquid exchange coefficient (Kpl) is calculated using the following equation: Kpl )

3RpRlFl 4utp2dp

( )

CD

Rer f f |u - ul | utp p

(5)

where utp, Rer, CD, and dp are terminal velocity of solid particle, relative Reynolds number (Rer ) Fldp|u bp - b ul|/µ), drag coefficient, and particle diameter, respectively. The subscripts l and p stand for the liquid and solid particles. The following empirical equations were used to calculate CD:44 CD )

24 [1 + 0.15(RlRer)0.687] RlRer

ReSR )

|ω|dp2 υ

where υ is the kinematic viscosity of the medium. It must be mentioned that b FBk, b FLk, and b Fvmk are considerably small compared to the dominant drag force and interaction forces between two phases, if the particles density ratio to liquid is more than 2.1 Using the k-ε model,46 two additional transport equations for the turbulence kinetic energy (k) and the turbulent dissipation rate (ε) are solved to compute the turbulent viscosity (µt):

((

) )

((

) )

µt ∂k ∂ ∂ ∂ (Fk) + (Fku bi) ) µ+ + ∂t ∂xi ∂xj σk ∂xj Gk + Gb - Fε - YM(13) µt ∂ε ∂ ∂ ∂ (Fε) + (Fεu bi) ) µ+ + ∂t ∂xi ∂xj σε ∂xj ε ε2 C1ε (Gk + C3εGb) - C2εFε - YM (14) k k where Gk, Gb, and YM are the generation of turbulence kinetic energy due to the mean velocity gradients, buoyancy, and the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. C1ε, C2ε, and C3ε are constants (see Table 1).13 In this table, V is the component of the flow velocity parallel to the gravitational vector and u is the component of the flow velocity perpendicular to the gravitational vector. The terms σk and σε indicate the k and ε turbulent Prandtl numbers. The turbulent (or eddy) viscosity (µt) is defined as43

if Rer e 1000 (6) µt ) FCµ

CD ) 0.44 if Rer > 1000

(7)

(12)

k2 ε

(15)

where Cµ is constant. Gk is defined as The combination of gravity force and buoyancy force is given by f+F f ) π d 3(F - F )g F B l b g 6 p p

(8)

where dp, Fp, and Fl are particle diameter, particle density, and liquid density, respectively. Virtual mass force can be derived from the following:19 f ) - 1 πd 3F d (uf - uf ) F vm l 12 p l dt p

(9)

where b up and b ul are the particle and liquid velocities, respectively. The lift force or Saffman force, or lift due to shear and rotation of the fluid, is given by18,45 f ) π d 3 Fl C ((uf - uf ) × ω) F l p 4 p 2 s l

(10)

where ω is the angular velocity and Cs ) 0.1524 for Rep g 40, Cs can be found as follows if Rep < 40:18,45 Cs )

[(

( ))

ReSR 4.1126 1.0 - 0.234 0.5 Rep ReSR

0.5

( )]

e-0.1Rep + 0.234

ReSR Rep

0.5

(11)

where Rep is the particle Reynolds number (Rep ) (Flutpdp)/µ) and ReSR is the rotational Reynolds number which can be calculated as follows:

Gk ) -Fui′uj′

∂uj ∂xi

(16)

where ui′ and uj′ are the fluctuating velocity component due to turbulence in the directions i and j, respectively, and ui′uj′ are the Reynolds stresses. CFD Simulation. In this study, a commercial CFD software package (Fluent V6.3) was used to simulate the mixing of solid suspension in an agitated vessel by solving the conservation of mass and momentum equations. The first step is grid generation, which divides the calculation domain into the discrete control volumes. Gambit 2.4 (Fluent Inc.) was used to discretize the flow domain with tetrahedral cells (unstructured grid). The advantage of using an unstructured grid is that a complex geometry (e.g., impeller) can be meshed easily. Mesh refinement near the impeller was accomplished using the mesh growth factor function. This factor controls mesh density by allowing the mesh elements to grow slowly as a function of the distance from the impeller blade to the vessel walls. The grid generated had skewness smaller than 0.6, indicating a very good mesh formation. The finer the mesh, the better the result will be achieved to capture the flow detail, but it should not be so fine, because the computational time increases with reductions in the mesh size.13 The optimum grid size was obtained by decreasing the size to a final value below which the changes in the velocity and kinetic energy profiles were less than 3%. The number of cells used for A100, A200, and A310 impellers were 311 876, 286 870, and 345 046, respectively. The standard wall functions


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13

Table 1. Standard k-ε Constants C1ε

C2ε

C3ε

Cµ

σk

σε

1.44

1.92

tanh|V/u|

0.09

1.0

1.3

Table 2. Operating Conditions descriptions

range

impeller type impeller speed (N) impeller clearance (C) particle size (dp) specific gravity (SG)

A100, A200, and A310 150-800 rpm T/6-T/2 100-900 µm 1.4-6.0

in Fluent were used for the modeling of the near wall regions. In this technique, the logarithmic law for mean velocity is known to be valid for 30 < y* < 300. The y* values calculated in this study were within these limits. The multiple reference frame (MRF) technique was employed to model the rotation of the impeller in the mixing vessel.47 A rotating frame was used for the region containing the impeller while a stationary frame was used for regions that are stationary containing the tank walls and baffles. The moving reference frame was a cylinder of 24 cm diameter and 9 cm height, and it was centered on the impeller. This frame did not include the shaft. The governing equations of the flow domain inside the rotating frame were solved in the frame of the enclosed impeller while those outside the rotating frame were solved in the stationary frame. A steady transfer of information was made at the MRF interface as the solution progressed. This method facilitates incorporation of the impeller motion even with a complex geometry. No-slip and no-penetration conditions were imposed on the transport equations on the tank walls. Symmetric boundary conditions, hence zero normal velocity and zero normal gradients for all variables, were assumed on the liquid surface. Since the shear stress is zero at a symmetry boundary, it is also called the slip wall boundary condition. A no-slip boundary condition was applied to the shaft. The local solid concentrations were monitored during the simulations. The steady state was achieved when the fluctuations of the solid concentrations in monitoring points were not noticeable. The simulation was considered converged when the scaled residuals for all transport equations were below 10-3. A single simulation run required 6000-8000 iterations to converge. The simulations were performed on a 3.0 GHz, 2 GB RAM, Pentium IV which took about 4-5 days for convergence. The simulations were carried out at the operating conditions summarized in Table 2. Results and Discussion The effects of different parameters (Table 2) on the quality of the solid-liquid mixing were investigated in this work. To validate the model, CFD results for impeller torque, cloud height, and just suspended impeller speed (Njs) were compared to experimental data. Figure 2 illustrates the impeller torque as a function of the impeller speed for the A310 impeller. These results show very good agreement between the impeller torque calculated by the CFD model and the experimentally determined value. The torque did not change significantly as the solids were suspended within the mixing tank for the solid concentration employed in this study. The power numbers computed using the CFD model for A100, A200, and A310 impellers were 0.39, 1.48, and 0.29, respectively. These results are in good agreement with those reported in the literature. Some researchers have derived models to predict the homogeneity as a solid cloud height. There is a distinct level (clear interface) to which most of the solid particles are lifted within

Figure 2. Impeller torque as a function of impeller speed (A310 impeller, C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

Figure 3. Cloud height (A310 impeller, C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5): (a) experimental result at N ) 320 rpm, (b) CFD result at N ) 320 rpm, (c) experimental result at N ) 600 rpm, and (d) CFD result at N ) 600 rpm.

the fluid at a given impeller speed. The height of this interface from the bottom of the vessel is called cloud height and above this interface there is only an occasional visit by a few solid particles.48 The CFD model developed in this study was used to estimate the cloud height. Solid concentration contours computed on a vertical plane was employed to estimate the cloud height. Figure 3 depicts the cloud height visualized using the CFD model and the digital photography at N ) 320 and 600 rpm. Both CFD and digital photography techniques were used to obtain the normalized cloud height as a function of the impeller speed for A310 impeller (Figure 4). It can be seen that the cloud height was reasonably predicted by the CFD simulation. The fluctuation of the cloud height was not significant at the steady-state. Figure 4 shows that the deviation of the CFD result from the experimental value is more pronounced at 200 rpm. The turbulent and fluid kinetic energy at the lower impeller speed lifted a small percentage of the solid particles from the bottom of the tank. However, the amount of energy imposed by the impeller was not sufficient to maintain the suspension.

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Figure 4. Normalized cloud height versus impeller speed (A310 impeller, C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

Figure 5. Using normalized solid concentration versus impeller speed to calculate the just suspended impeller speed for A310 impeller at C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5.

It must be mentioned that a ramped sequence of the impeller speed was not used for the experimental measurements of the cloud heights. In fact, we stop agitating the tank after each test and the impeller speed was increased with the same rate for all experiments. A suspension is considered as complete (off-bottom) if no solid particle remains stationary on the bottom of the tank for more than 1 or 2 s and the impeller speed at this condition is known as the just suspended impeller speed:49 Njs ) S

( ) g∆F FL

0.45

(

X0.13dp0.2υ0.1 D0.85

)

(17)

where S is constant for a given system geometry, ∆F ) Fs - Fl and g, Fs, Fl, υ, dp, X, and D are gravity, particle density, liquid density, kinematic viscosity, particle diameter, solid weight fraction, and impeller diameter, respectively. In order to calculate Njs using the CFD model developed in this study, the average solid concentration for a horizontal plane located 1 mm above the bottom of the tank was measured as a function of the impeller speed (Figure 5). The tangents to the curve were drawn at the points having maximum and minimum slopes. Njs was the impeller speed corresponding to the point at which the two tangents crossed each other. The similar method was employed by Mak4 to predict Njs through experimental data. Njs estimated by this technique was 342 rpm which is in good agreement with that calculated using eq 17 (333 rpm).

The validated CFD model can be used to monitor the distribution of solid phase inside the mixing vessel. For instance, Figure 6a illustrates the contours of the solid concentrations computed using the CFD model. This figure shows the formation of a pile of solid particles just below the impeller at N ) 250 rpm which is less than the just suspended impeller speed (Njs). These results are in good agreement with the experimental data depicted in Figure 6b.42 This figure shows the 3D image of solid concentration generated from the 2D tomography images using Slicer-Dicer (PIXOTEC, USA). Fradette et al.50 also reported the same phenomenon when they studied the suspension of glass beads in a mixing vessel equipped with a marine propeller. Kresta and Wood51 investigated the effect of the impeller clearance on the flow pattern generated by an axial-flow impeller in a mixing tank and concluded that there was a change in angle of flow discharge from the axial direction toward the radial direction as the impeller clearance increased. Once this clearance reached to C ) T/2, the counter-rotating secondary circulation loop occurred near the bottom of the tank. This secondary circulation loop (see Figure 6c) pushed the solid particles toward the center of the tank resulting in the pile-up of solid particles in the middle of the tank. However, this secondary loop did not exist for the lower impeller clearance (e.g., C ) T/6). Figure 6c also shows that the momentum directly under the impeller was not sufficient to penetrate to the bottom of the tank. Thus, a pile of solid particles was formed just below the impeller. In order to measure the degree of homogeneity, the distributions of solid concentrations for eight horizontal planes (with the lowest one numbered P1) were calculated through the validated CFD model (Figure 7). To generate the axial concentration profile, the averaged solid concentration was computed for each plane and the results were then normalized to the overall average concentration of solid particles within the tank. Figure 8 illustrates the axial solid concentration profiles as a function of impeller speed for A310 impeller. The data shown in this figure was used to calculate the degree of homogeneity within the mixing vessel:42

homogeneity ) 1 -

n

∑ (X

V

j V)2 -X

1

n

(18)

j V are the number of planes, solid volume where n, XV and X concentration, and the average solid volume concentration within the vessel, respectively. Figure 9 shows the degree of homogeneity as a function of impeller power for A310 impeller. The CFD results are in good agreement with those obtained from the tomography measurements. As expected, the homogeneity of the system increased with an increase in impeller power/ speed. Once the homogeneity reached the maximum, any further increase in impeller power/speed was not beneficial but detrimental. Other researchers also reported the similar phenomenon in solid-liquid mixing.4,6,52 In order to elucidate the relationship between the homogeneity and the impeller speed, the CFD model was used to generate the solid concentration contours as a function of impeller speed (Figure 10). It can be seen that the homogeneity within the tank improved with an increase in impeller speed and the maximum homogeneity was achieved at N ) 500 rpm. However, due to the centrifugal force inside the circulation loops at higher impeller speed, the formation of the regions with low solid concentrations was observed which eventually decreased the homogeneity of the system.52 Therefore, the optimal impeller speed has a significant effect on the degree of homogeneity and should always be between two


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Figure 7. Solid concentration contours calculated using CFD for eight horizontal planes (A310 impeller, N ) 300 rpm, C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

Figure 8. Axial solid concentration profiles computed using CFD for the A310 impeller (C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

Figure 9. Degree of homogeneity as a function of impeller power for the A310 impeller at X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

Figure 6. (a) Solid concentration contours computed using CFD. (b) 3D image of solid concentration generated from the 2D tomography images. (c) Velocity vectors (A200 impeller, N ) 250 rpm, C ) T/2, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5).

crucial impeller speeds, Njs and the impeller speed for the maximum homogeneity, preferably closer to the latter. The type of impeller has a significant effect on the extent of homogeneity. Axial impellers are preferred for solid-liquid mixing because they produce high flow and are more costeffective in flow controlled operations like solid suspension.40 In this study, the performances of three axial-flow impellers

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Figure 12. Effect of the impeller clearance on the degree of homogeneity for the A310 impeller at N ) 400 rpm, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5.

Figure 10. Solid concentration contours generated using CFD at different impeller speeds for A310 impeller (C ) T/3, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5): N ) (a) 150, (b) 200, (c) 250, (d) 280, (e) 300, (f) 320, (g) 350, (h) 400, (i) 500, (j) 600, (k) 700, and (l) 800 rpm.

Impeller off-bottom clearance also influences the degree of homogeneity achieved in solid-liquid mixing processes. The CFD model was employed to compute the extent of homogeneity at N ) 400 rpm for A310 impeller mounted at different positions (C ) T/6-T/2) from the bottom of the tank. Figure 12 illustrate the degree of homogeneity as a function of the impeller clearance. It can be seen that the highest extent of homogeneity was achieved at C ) T/3 and the CFD results are in good agreement with those measured using electrical resistance tomography.42 Hicks et al.8 studied the solid suspension at different off-bottom clearances (C ) T/2.5, T/4, and T/10), and they reported that the optimum impeller clearance was C ) T/2.5 for solid-liquid mixing operations. The particle size is one of the important physical properties of solid particles, which affects the solid suspension in mixing operations. In this study, the particle size was varied from 100 to 900 µm. The degree of homogeneity was calculated at N ) 400 rpm for A310 impeller versus the particle size (Figure 13a). As expected, the extent of homogeneity decreased with an increase in the particle size. This could be explained as per the following equation which estimates the free settling velocity (Vt) for spherical particle:53 Vt )

Figure 11. Effect of impeller type on the distribution of solid phase at P ) 15.5 W, X ) 10 wt %, dp ) 210 µm, and SG ) 2.5: (a) A100, (b) A310, and (c) A200 impellers.

(Lightnin A100, A200, and A310) on the mixing quality were explored. Figure 11 shows the contours of solid concentration computed using the CFD model for these three impellers at constant impeller power P ) 15.5 W. It can be seen that the A100 impeller was more efficient while the A200 was the least effective at obtaining a higher degree of homogeneity. The same results reported by Hosseini et al.42 who employed the electrical resistance tomography to investigate the performance of the different types of axial flow impellers in solid-liquid mixing systems.

(

4gcdp(Fs - Fl) 3CDFl

)

1/2

(19)

dp is the particle diameter, Fs is the particle density, Fl is the liquid density, and CD is the drag coefficient. The free settling velocity in solid suspension increases with an increase in the particle size.54 Thus, the larger particles settle faster than the smaller ones.2 Thus, the suspension of the large particles would be more difficult. Peker and Helvaci14 also reported that the degree of homogeneity decreases when the terminal velocity of the solid particles increases. The similar trend was also observed by Godfrey and Zhu55 using particle size ranges 212-250, 355-425, and 600-710 µm with the terminal velocities of 1.07, 2.3, and 4.35 mm/s, respectively. Similar to the particle size, the specific gravity (SG) of the solid particles plays a crucial role in the solid-liquid mixing processes. On the basis of Zwietering’s correlation (eq 17), the specific gravity of the particles affects the just suspended impeller speed (Njs). Particles with higher density display more


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extent of homogeneity achieved in solid-liquid mixing operations. Solids suspension systems with SG values of 1.4 and 2.5 are more homogeneous compared to those with higher SG values of 4.0 and 6.0 at a fixed impeller speed. The particle response time (tp) is the time that a particle takes to respond to a change in carrier flow velocity. The parameters affecting the particle response time are the particle diameter, particle density, and fluid viscosity:14 tp )

Fpdp2 18µ

(20)

Figure 14 illustrates the particle response time as a function of the particle size and solid specific gravity for the solid-liquid mixing system investigated in this study. Conclusions

Figure 13. (a) Effect of particle size on the degree of homogeneity for A310 impeller at N ) 400 rpm, X ) 10 wt %, C ) T/3, and SG ) 2.5. (b) Effect of the solid specific gravity on the degree of homogeneity for A310 impeller at N ) 400 rpm, X ) 10 wt %, C ) T/3, and dp ) 210 µm.

The computational fluid dynamics (CFD) modeling was successfully employed to explore the effect of operating conditions and design parameters on the mixing quality for the solid-liquid mixing operations. This technique provided indepth details about the distribution of the solid phase within the mixing vessel. The impeller torque and cloud height calculated by the CFD model were in good agreement with the experimentally determined values. The average solid concentration at the bottom of the tank as a function of the impeller speed was determined through CFD to estimate the just suspended impeller speed (Njs), which was close to that predicted by the Zwietering’s correlation. The validated CFD model was then utilized to obtain the axial solid concentration profiles by which the degree of homogeneity was quantified as a function of the impeller power/speed, impeller type, impeller off-bottom clearance, particle size, and specific gravity of solid particles. As expected, the homogeneity of the system increased with an increase in impeller power/speed. Once the homogeneity reached the maximum, any further increase in impeller power/speed was not beneficial but detrimental due to the formation of the regions with low solid concentrations inside the circulation loops at higher impeller speed. The CFD results for the degree of homogeneity were in good agreement with those measured by the electrical resistance tomography. It was found that the A100 impeller was more efficient in terms of homogeneity than the A310 and A200 impellers and that the optimum impeller clearance was T/3 for the solid suspension systems. The CFD results also showed that the physical properties of the solid particles such as the particle size and the specific gravity significantly affect the degree of homogeneity in solid-liquid mixing operations. Acknowledgment The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Ryerson University is gratefully acknowledged. Nomenclature

Figure 14. Particle response time as a function of the particle size (at SG ) 2.5) and the solid specific gravity (at dp ) 210 µm).

resistance to flow resulting in lower degree of homogeneity. The validated CFD model was used to compute the degree of homogeneity for four different specific gravity values (SG ) 1.4, 2.5, 4.0, and 6.0) at a fixed impeller speed (N ) 400 rpm) for the A310 impeller (Figure 13b). These results show that the specific gravity of the solid phase considerably affects the

C ) impeller off-bottom clearance (m) CD ) drag coefficient C1ε, C2ε, C3ε ) constants Cs ) Saffman force constant Cµ ) turbulent (or eddy) viscosity constant D ) impeller diameter (m) dp ) particle diameter (m) b FB ) buoyancy force (N)

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b FD ) drag force (N) b Fg ) gravity force (N) b Fl ) lift force (N) b Fvm ) virtual mass force (N) g ) gravitational acceleration (m/s2) Gk ) generation of turbulence kinetic energy due to the mean velocity gradients (kg/m · s3) Gb ) generation of turbulence kinetic energy due to buoyancy (kg/ m · s 3) H ) height of solid suspension in the tank (m) k ) turbulent kinetic energy (m2/s2) Klk ) exchange coefficient between liquid and the phase kth Kpl ) solid-liquid exchange coefficient N ) impeller speed (1/s) n ) number of planes Njs ) just suspended impeller speed (1/s) P ) impeller power (W) p ) pressure (Pa) Rep ) particle Reynolds number Rer ) relative Reynolds number S ) Zwietering constant SG ) specific gravity of the solid particles T ) tank diameter (m) tp ) particle response time (s) u ) component of the flow velocity perpendicular to the gravitational vector (m/s) b uk ) phase k velocity (m/s) b ul ) liquid velocity (m/s) b up ) particle velocity (m/s) utp ) terminal velocity of solid particle (m/s) ui′, uj′ ) fluctuating velocity component due to turbulence in the direction i and j, respectively (m/s) V ) component of the flow velocity parallel to the gravitational vector (m/s) Vt ) particle-free settling velocity (m/s) XV ) volume fraction of solids in suspension j V ) average volume fraction of solids in suspension X xi ) particle coordinates (m) X ) weight percent of the solid particles YM ) fluctuating dilatation in compressible turbulence to the overall dissipation rate (kg/m · s3) Greek Symbols Rk ) phase k volume fraction RL ) liquid phase volume fraction Rp ) particle volume fraction ε ) turbulent dissipation rate (m2/s3) µ ) viscosity (Pa · s) µk ) bulk viscosity (Pa · s) µt ) turbulent (or eddy) viscosity (Pa · s) ν ) kinematic viscosity of the liquid (m2/s) F ) density (kg/m3) Fk ) phase k density (kg/m3) Fl ) liquid density (kg/m3) Fp ) particle density (kg/m3) σk, σε ) k and ε turbulent Prandtl numbers ω ) angular velocity (rad/s)

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ReceiVed for reView July 13, 2009 ReVised manuscript receiVed February 27, 2010 Accepted March 17, 2010 IE901130Z

Study of SolidâLiquid Mixing in Agitated Tanks through Computational

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