Relation between Flow Pattern and Blending in Stirred Tanks

Ind. Eng. Chem. Res. 1999, 38, 3131-3143

3131

Relation between Flow Pattern and Blending in Stirred Tanks Ashwin W. Patwardhan and Jyeshtharaj B. Joshi* Department of Chemical Technology, University of Mumbai, Matunga, Mumbai 400 019, India

In the present work, the relationship between the flow pattern and blending has been investigated. The flow patterns generated by around 40 axial flow impellers have been examined. The impellers differed in blade angle, blade twist, blade width, impeller diameter, impeller location, and pumping direction. The mean-flow and turbulence characteristics generated by all of the impellers have been measured using laser doppler velocimetry (LDV). On the basis of available LDV data, the flow pattern throughout the vessel was established by employing computational fluid dynamics (CFD) and subsequently used for the simulation of the blending process. The predicted mixing times were found to be in excellent agreement with the experimental measurements. It has been shown that the dimensionless mixing time (θ h ) varies inversely with the secondary flow number of the impeller. Comparison of the impellers on the basis of equal power consumption per unit mass has shown that θmix ∝ NP1/3T2/3/NQS. The present CFD model has shown the possibility of reducing the eddy diffusivity to about 20% of the actual value and still achieving the same mixing time. This reduction in eddy viscosity represents substantial savings in operating costs. 1. Introduction Blending or homogenization of two or more miscible fluids is very widely encountered in a variety of physical and chemical processes. The blending process may be carried out in a batch or in a continuous mode of operation. In chemical process industries, stirred vessels are frequently employed to accomplish homogenization (mixing). The important process parameter for the blending process is the blending time, which is frequently termed as the mixing time. Experimental determination of the blending time (mixing time) involves introduction of a tracer at some location in the vessel (usually a pulse input) and measurement of the tracer concentration as a function of time with the help of a sensor. Mixing time is typically considered as the time required for the tracer concentration to reach within 95% of the completely mixed value. The blending process occurs because of the transport at three levels: molecular, eddy, and bulk (convection). Usually, the bulk motion (or bulk diffusion) is superimposed on either molecular or eddy diffusion or both. In industrial practice, many of the blending operations are carried out under turbulent conditions. In that case, the molecular diffusion can be neglected in comparison to the bulk diffusion and eddy diffusion. Over the years, several approaches have been proposed to model the blending process. These approaches are based on the above three mechanisms, and as a result all of the approaches can be classified into four categories, namely, circulation models, eddy diffusion models, network of zones models, and CFD models. 1.1. Circulation Models. These models inherently assume that the blending process is controlled by the mean flow, i.e., the bulk motion.1-5 The mean flow generated by the impeller produces circulation loops in the stirred vessel. McManamey2 proposed that the time required for the fluid to circulate once through the flow * To whom correspondence should be addressed. E-mail: [email protected]. Tel: 91-22-414 5616. Fax: 91-22-414 5614.

path should be equal to the maximum length of the circulation path divided by the average liquid velocity in the circulation path. The average liquid circulation velocity for a disk turbine was estimated from the measurements of van der Molen and van Maanen.6 The circulation time (inverse of the mixing rate constant as defined by McManamey2) was calculate as

tc )

1 2T ) K 0.85πND(D/T)7/6

(1)

The mixing time can then be taken as some multiple of (typically 5 times) the circulation time. This model was also extended for axial flow impellers. Joshi et al.3 and Rewatkar and Joshi7 have employed this model to estimate the mixing time. 1.2. Eddy Diffusion Models. These models assume that the entire blending process is controlled by eddy diffusion.8-11 Voncken et al.8 divided the vessel into two regions with different degrees of mixing. One region is close to the impeller, and the other region (circulation loop) considers the convective flow to be important. In the circulation loop, mixing is not intense and can be characterized by a dispersion coefficient DS and an average circulation velocity vj . The circulation loop was considered as a tube of length L (equal to the maximum circulation path), with the liquid flowing through the tube at an average circulation velocity vj , giving an overall circulation time of tc ) L/v. If τ ) t/tc is the dimensionless time, then the concentration at any location j ) x/L in the tube is given as

c c∞

)

x

Bo

∞

[

Bo

]

exp - (j - τ)2 ∑ 4πτj)1 4τ

(2)

If the average circulation velocity, the dispersion coefficient, and the maximum circulation path length are known, then the mixing time can be calculated. Hiraoka and Ito10 have assumed that the eddy diffusivity is proportional to the apparent friction velocity

10.1021/ie980772s CCC: $18.00 © 1999 American Chemical Society Published on Web 07/13/1999

3132 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

at the impeller tip and the size of the largest eddy. The proportionality factor was related to the impeller and tank diameter. The friction factor was correlated with the impeller power number and the vessel geometry. The values of the friction factor and impeller flow numbers were correlated empirically to the mixing time. Jahoda et al.11 have investigated multiple impellers in a tall stirred vessel. They have modeled the blending process by using a one-dimensional, unsteady-state equation for the conservation of an inert tracer. The convective terms were neglected, and the turbulent dispersion coefficient was assumed to be constant throughout the vessel. Khang and Levenspiel12 represented the blending process by a series of equal-sized well-mixed tanks with a characteristic residence time. The amplitude of the concentrationstime curve was correlated with the mean residence time and its variance. The basic drawback of both of the above classes of models is that only an approximate picture of the flow generated by the impellers is considered. The flow pattern generated by the impellers is very nonuniform. The mean velocities and turbulence structure change significantly with the location in the stirred tank. The assumption of an average circulation velocity or the assumption of constancy of eddy diffusivity leads to a very simplified picture of the blending process. 1.3. Network of Zones Models. Mann and coworkers13,14 have presented a network of zones model. Mann and Mavros13 have divided the entire vessel into a large number of interconnected zones, each of which is considered to be well mixed. Depending on the flow pattern generated by the impeller and the location of the zone, the flow into and out of each zone was assumed to be either axial or radial. The tracer was transported into or out of the zone depending on the magnitudes of the local flows. The values of the local flow rates were determined experimentally with the help of a “flow follower”. This model thus requires the experimentally determined local flow field as an input. This basic model was progressively modified to take into account the turbulent dispersion between zones (Voit and Mersmann15) and the three-dimensional network of zones (Mann and Ying14). The network of zones model is far more realistic as compared to the circulation and dispersion models, mainly because it attempts to take into account the local flow field. It forms the basis and a starting point for the CFD models. 1.4. CFD Models. CFD models enable computation of the flow field and homogenization of the tracer by a direct solution of the governing equations. The CFD modeling of flow in stirred tanks has been carried out for the last 20 years or so.16-33 These papers discuss the prediction of velocity field and turbulence characteristics. Ranade et al.34 have presented a CFD model for the prediction of velocity and turbulence characteristics in a stirred reactor. After computation of the velocity and the eddy viscosity profile, the blending process was modeled by solving the conservation equation for an inert tracer given as

∂c 1 ∂ 1 ∂ ∂ 1 ∂ ∂c + (rv c) + (v c) + (vzc) ) rΓ + ∂t r ∂r r r ∂θ θ ∂z r ∂r ∂r 1 ∂ Γ ∂c ∂ ∂c + Γ (3) r ∂θ r ∂θ ∂z ∂z

(

( )

)

( )

Here, the convective transport of tracer by mean velocities is accounted for by the three terms on the left-hand

side. The dispersive transport of the tracer due to turbulent motion in a stirred vessel is accounted for by the turbulent diffusivity, denoted by Γ. Thus, the CFD model takes into account the contribution of both bulk and eddy diffusion on a local level. This is the basic advantage of a CFD model. Patterson and Zipp35 have also presented a CFD model to describe the blending of the tracer, the residence time distribution, and the micromixing aspects. Hoffman36 and Ranade37 have applied a CFD model to estimate the jet mixing in a large (920 m3) storage tank. Togatorop et al.38 have used a CFD model for the inert and reactive mixing of tracers in a stirred vessel. The model parameters, grid sizes, and experimental boundary conditions were similar to those of Ranade et al.34 The predicted mixing time was found to be comparable with the experimental mixing time. 2. Present Work Despite many experimental and modeling efforts on homogenization/blending under turbulent conditions, there is no clear understanding in the literature on the relative contribution of bulk diffusion and eddy diffusion to the overall process of blending. In stirred vessels, the impeller is responsible for the generation of flow field (mean flow as well as turbulence; the impeller design thus plays an important role in the mixing process. To design stirred vessels based on rational principles, it would be highly desirable to understand the relationship between the flow field produced by the impeller and the mixing time. This is the basic aim of the present work. To measure the flow field in the vessel, a LDV was employed. However, measuring the flow field throughout the vessel is extremely time-consuming. Therefore, a combination of LDV measurements and CFD simulations was employed. Measurements of vr, vz, vθ, and turbulent kinetic energy (k) were made at four axial locations (about 15 points in the radial direction at each of the axial locations), and these data were used as boundary conditions for the CFD simulations. It may be emphasized that, with the present status of knowledge, CFD alone is not able to predict the flow pattern (by either sliding grid,24,32 inner-outer,27,28 or black box approach18,19,22,26). The differences are substantial in the near impeller region when a sliding mesh approach is used. In contrast, the other methods result in poor predictions in regions away from the impeller. In this paper, an attempt has been made to bring out the relation between flow pattern and the process of blending. 2.1. Experimental Section. The LDV measurements were carried out in a 0.5 m diameter transparent acrylic vessel equipped with a single impeller. The vessel was fitted with four baffles having width 1/10th that of the vessel diameter (fully baffled condition). The experimental setup had the facility of varying the impeller speed, impeller clearance, type of impeller, and the liquid level. For all of the measurements, the liquid level was kept equal to the tank diameter. The various impeller designs that were investigated are listed in Table 1. The rationale for the impeller nomenclature is as follows: PBTD stands for pitched blade turbine downflow, whereas PBTU stands for pitched blade turbine upflow. The numbers immediately following this stand for the blade angle at the hub. The letters STD indicate standard configuration implying D/T ) C/T ) 0.33 and W/D ) 0.3. When the blade width, impeller

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3133 Table 1. Geometry of the Impellers Investigated geometry impeller

D/T

C/T

W/D

PBTD30STD PBTD40STD PBTD45STD PBTD50STD PBTD60STD PBTD3020 PBTD4535 PBTD6050 PBTD3010 PBTD4525 PBTD6040 HF1 HF2 HF3 HF4 PBTD45C3H5 PBTD45CH2 PBTD45CH4 PBTD45CH6 PBTD45D02T PBTD45D05T PBTD45W20 PBTD45W40 PBTD30W3010 PBTD45W3010 PBTD60W3010 PBTD45W3050 PBTD30D05T PBTD60D05T PBTD45D06T PBTD45D07T PBTD4525D05T PBTU45CH6 PBTU45STD PBTU45CH2

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.20 0.50 0.33 0.33 0.33 0.33 0.33 0.33 0.5 0.5 0.6 0.6 0.5 0.33 0.33 0.33

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.60 0.50 0.25 0.17 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.167 0.33 0.50

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.17-0.11a 0.4a 0.52a 0.17a 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.4 reducing 0.3 to 0.1 reducing 0.3 to 0.1 reducing 0.3 to 0.1 increasing 0.3 to 0.5 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

a

measurement location with respect to the impeller center plane (mm) twistb

10 10 10 20 20 20

20

A

B

C

D

+30 +30 +30 +30 +30 +30 +30 +30 +30 +30 +30 +45 +35 +30 +30 +30 +30 +30 +30 +30 +40 +30 +40 +30 +30 +30 +40 +30 +40 +30 +45 +30 +30 +30 +30

+130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +125 +130 +130 +130 +130 +130 +90 +70 +130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +130 +70 +130 +130

-30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -45 -35 -30 -30 +230 -30 -30 -30 -30 -40 -30 -40 -30 -30 -30 -40 -40 -40 -45 -45 -30 -30 -30 -30

-140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -135 -140 -140 -140 -30 -70 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -140 -70

Projected width in the horizontal plane starting from hub to tip. b Blade twist is in degrees.

clearance, or impeller diameter is varied, it is indicated by letters W, C, and D, respectively. The numbers or letters following this indicate the variation in W, C, or D as the case may be. The nomenclature is selfexplanatory, as can be seen in Table 1. Hydrofoil impellers are increasingly becoming popular. It is claimed that these impellers have superior characteristics as compared to conventional impellers. Four hydrofoil impellers were first fabricated. These are labeled as HF1, HF2, HF3, and HF4. The HF1 impeller has three blades with blade angle of around 20°. The width of the impellers blades was gradually decreased from hub to tip. The HF2 impeller also has three blades. The blades were convex in shape, and the curvature was widthwise. The radius of curvature was approximately 100 mm. This impeller has a high solidity ratio. The HF3 impeller was similar in shape to the HF2 impeller, but it has four blades. Its radius of curvature was approximately 110 mm. The hydrofoil HF4 has three blades with blade angle of around 35°. The impeller blades have a twist in the tip region. These hydrofoil impellers are shown in Figure 1A. The characteristic feature of the various designs of hydrofoil impellers is that they employ a change in blade width, curvature, angle, or any combination of these three. With this in mind, several more impellers were fabricated with a view to systematically investigate the variation in blade angle (PBTD30STD, PBTD45STD, PBTD60STD), blade twist (PBTD3020, PBTD3010, PBTD4535, etc.), blade width (PBTD45W20, PBTD45W40, PBTD45W3010, etc.),

impeller diameter (PBTD45D02T and PBTD45D05T), etc. In addition, the effect of impeller location (PBTD45CH6,PBTD45CH4, PBTD45CH2 and PBTD45C3H5) was also investigated. All of the impellers were six bladed except the hydrofoils. The effect of impeller diameter to tank diameter (D/T) ratio was varied between 0.2 and 0.7. The impeller designs covered in the present work probably span the entire spectrum of pitched blade turbine impellers used in industrial practice. All of the impellers were operated under completely turbulent conditions (Re ∼ 105). For all of the impellers the mean velocities in the three directions (radial, axial, and tangential) and the turbulent kinetic energy were measured at four axial locations in the vessel with the help of a LDV. At each axial location the measurements were carried out from the center to the wall of the vessel at 20 mm intervals. These locations are denoted by the lines A, B, C, and D in Figure 1B. It can be seen that the lines B and D represent locations far below and above the impeller, respectively. The position of the four axial locations with respect to the impeller center plane is also given in Table 1. The positive sign indicates below the impeller, and the negative sign indicates above the impeller. The LDV system consists of a laser beam generator, beam splitter, and focusing arrangement. The laser beam can be focused at any point in the vessel with the help of the adjustment of the lenses and the traversing system. The LDV system enabled acquisition of the two velocity components simultaneously. The Doppler shift produced was measured by a photomultiplier tube in

3134 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

A

B Figure 1. (A) Hydrofoil impellers used in the present work. (B) Details of the experimental setup used in the present work for LDV measurements of flow field produced by different impellers.

the forward scatter mode. All of the data were acquired with the help of a Pentium PII personal computer. All of the LDV measurements were made midway between the two baffles. 2.2. CFD Simulations. In the present work, the k- model for turbulence was used for closure of the timeaveraged Reynolds transport equations. The equations were solved using control volume formulation on a staggered grid arrangement39 in a manner similar to that used earlier.40,41 The generalized transport equa-

tion for any variable φ is given as

1 ∂ ∂ 1 ∂ ∂φ 1 ∂ rΓ + (rv φ) + (v φ) + (vzφ) ) r ∂r r r ∂θ θ ∂z r ∂r ∂r 1 ∂ Γ ∂φ ∂ ∂φ + Γ + Sφ (4) r ∂θ r ∂θ ∂z ∂z

(

( )

) ( )

The source terms for different flow variables (Sφ) were given earlier.41 In the above equation, the variables were made dimensionless with the impeller tip velocity (Utip) and the radius of the vessel (R). Thus, the nondimen-

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3135 Table 2. Comparison of the Predicted Mixing Time with the Experimental Measurements7,42 and with the Measurements of Rewatkar and Joshi (1991) A. Experimental Measurements7,42 impeller design

tank diameter T(m)

impeller diameter D (m)

clearance

blade width

Nθmix measured

Nθmix predicted

ref

PBTD45 PBTD45 PBTD45 PBTD45 PBTD45 PBTD45 PBTD30 PBTD60 PBTU45 PBTU45 propeller

0.57 0.57 1.5 1.5 1.5 1.5 0.57 0.57 1.5 1.5 0.32

T/3 T/3 T/3 T/3 T/5 T/2 T/3 T/3 T/3 T/3 T/4

T/3 T/2 T/3 T/2 T/3 T/3 T/3 T/3 T/3 T/2 3T/8

0.3D 0.3D 0.3D 0.3D 0.3D 0.3D 0.3D 0.3D 0.3D 0.3D

42.8 35.9 40.1 35.9 146.0 17.3 61.4 35.1 41.0 44.8 90.0

42.80 36.30 42.80 36.30 139.17 14.89 55.32 38.59 30.04 56.89 77.94

7 7 7 7 7 7 7 7 7 7 42

B. Measurements of Rewatkar and Joshi (1991) (PBTD 45°, W/D ) 0.3, D/T ) 1/3, C ) H/3) θmix (s) with T ) 0.57 m

θmix (s) with T ) 1.0 m

θmix(s) with T ) 1.5 m

P/M (W/kg)

measured

predicted

measured

predicted

measured

predicted

0.1 0.2 0.3 0.4 0.7 1.0

12.8 10.0 8.9 8.1 6.8 6.3

13.70 10.88 9.50 8.64 7.16 6.36

20.0 15.0 13.1 11.8 10.0 9.4

19.94 15.83 13.83 12.56 10.43 9.26

25.2 20.0 17.5 15.9 14.1

26.13 20.74 18.12 16.46 13.66

sionalizing parameter for mean velocities was Utip, for the coordinate system, it was R, for the turbulent kinetic energy, it was Utip2, for turbulent energy dissipation rate, it was Utip3/R, and for eddy viscosity, it was RUtip. Because of symmetry, only one-quarter of the vessel was considered as the computation domain. Because of the nondimensionalization, the radial coordinate varied from 0 (at the center) to 1 (at the wall), the axial coordinate varied from 0 (at the vessel bottom) to 2 (at the liquid surface), and the tangential coordinate varied from 0 (at one baffle) to 1.57 (π/2 at the other baffle). The computation domain was divided into 30 × 68 × 12 grid points in r, z, and θ directions, respectively. The grid sizes in r, z, and θ directions were therefore 0.033, 0.029, and 0.1308, respectively. Using the predicted flow field, the values of the primary flow number and secondary flow number were calculated using the following equations:

∫0R 2πrvz dr I

NQP )

ND3

∫0R 2πrvz dr R

NQS )

ND3

(5) 3. Results and Discussion

Both the above integrals are evaluated closest to the impeller in the pumping direction of the impeller. Here, RI represents the radius of the impeller and RR represents the point of reversal of the flow. The impeller power number was calculated using the turbulent energy dissipation rates, as follows:

∫0R∫0H ∫02πFr dr dz dθ L

NP )

FN3D5

computation domain were identified. The time required for the minimum and maximum tracer concentrations to reach within 95% of the completely mixed tracer concentration was calculated. The mixing time was defined as the larger of the two times. It is well-known that CFD simulation using only one (location A) or two (locations A and C) boundary conditions gives poor predictions away from the impeller.18,19,42 To overcome this problem, the CFD model was solved with all of the available experimental data (locations A-D). This was done in order to “bind” the CFD solution to experimental values even very far away from the impeller. In other words, the CFD model was used only as a tool for interpolating between the measurement locations given in Figure 1B. This has been done mainly because LDV measurements throughout the stirred vessel are extremely time-consuming. Such a combination of the LDV data and the CFD simulations enables use of “experimentally measured flow field” for simulating the blending process.

(6)

For modeling of the blending process, eq 3 was used. In this equation, time was made dimensionless using R/Utip. The equation was integrated by an implicit method using the control volume formulation in a manner similar to that of the velocity field. The addition of the tracer pulse was simulated by specifying a very high value of concentration for one cell within the computation domain. To calculate the mixing time, the points of minimum and maximum concentration in the

3.1. Model Validation. It is important to establish the validity of the mathematical model by comparing the model predictions with experimental observations. Rewatkar and Joshi7 have reported experimental measurements of mixing time over a wide variety of impeller designs, vessel sizes, impeller diameter, impeller clearance, and blade angle. In this work, the CFD model has been validated by comparison with the experimental data of Rewatkar and Joshi7 and Kramers.43 Table 2A compares the values of Nθmix, for downflow and upflow impellers of different diameters at different impeller clearances from the vessel bottom. For PBTD 45 impeller in 0.57 and 1.5 m tanks and at different values of clearance (T/3 and T/2), the predicted mixing time is in excellent agreement with the experimental measurements (rows 1-4). In regards to the different D/T ratios of PBTD 45 impeller, the predicted mixing time is within 5% for lower D/T and within 15% for higher D/T ratios (rows 5 and 6). For PBTD 30 and PBTD 60 impellers, the predicted values differ by only

3136 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

peak is 7 times c∞ and it appears at dimensionless time of approximately 7.5. At location 3, which is very close to the top liquid surface, the response is very sluggish. It is considered that the time gap between successive peaks represents the circulation time within the vessel and the decay of amplitude of the peaks is indicative of the dispersion in the vessel.8 It is clear from Figure 2A that both transport processes occur simultaneously in the vessel. From the concentration profiles shown in Figure 2A, the circulation time was estimated. The time gaps between the successive maxima and the successive minima were computed, and an average was taken as the circulation time. The value of the circulation time was found to be 26.4 (dimensionless). It is well-known that the extent of mixing approaches uniformity in an exponential manner.9 The degree of mixing can be represented as follows:

M′ ) 1 - exp(-kt)

Figure 2. (A) Typical concentration profile at five locations in the vessel. (B) Corresponding locations in the stirred vessel.

10%. The upflow PBTU 45 impeller at clearances T/3 and T/2 show larger deviations from the experimentally observed values. The predicted mixing times are lower by 27% and higher by 25%, respectively. Even for the marine propeller whose geometry is drastically different from the conventional pitched blade impellers, the comparison is found to be within 15%. Table 2B compares the mixing times for PBTD 45 impeller in three different tank diameters (at a constant D/T ratio) and at various values of power consumption between 0.1 and 1.0 W/kg. The agreement is within 10% in all of the cases. It is well-known that, regardless of the measuring technique, the experimentally measured mixing times have an inherent standard deviation of about 5-10%. Keeping this in mind, it can be considered that the present model works well for the estimation of mixing time. This validated model can now be extended to other impellers with a fair degree of confidence. 3.2. Comparison with Circulation Models. The CFD model also gives the concentration profile as a function of time at any location in the vessel. The evolution of the concentration field for PBTD45STD impeller is shown at five locations in the vessel (Figure 2A). These locations along with the tracer input location are shown in Figure 2B. Figure 2A shows several interesting features. The concentration profiles at these locations show significant oscillations. Initially, the concentration rises very fast, and thereafter it decays, shows some oscillatory behavior, and after a long time attains a steady-state value denoted as c∞. The peak value is substantially higher than the final steady-state concentration. The location of the magnitude of the first peak and its value depends on the position in the vessel. For example, at location 1, the magnitude of the first peak is about 6.5 times c∞ and it appears at dimensionless time of approximately 4.5, whereas, at location 5, the magnitude of the first

(7)

where k represents the mixing rate. For 95% homogeneity (M′ ) 0.95), the value of kt turns out to be 3. The dimensionless mixing time for PBTD45STD impeller is 80.08. Because the circulation time is 26.4, the mixing time is about 3.03 times the circulation time. This means that 95% homogeneity is achieved in 3.03 circulation times, which is very close to the value of kt above. Thus, the mixing process can then be represented as

M′ ) 1 - exp(-t/tc)

(8)

The mixing rate k turns out to be almost equal to the reciprocal of the circulation time, that is, k ) 1/tc. This agrees very well with that of McManamey,2 who proposed the mixing rate constant to be the inverse of the circulation time. This indicates that the mixing process for this impeller may be controlled by bulk diffusion (circulation) in the vessel. McManamey2 and Joshi et al.3 have proposed that the circulation time equals the maximum length of the circulation path divided by the average circulation velocity. For a pitched blade downflow turbine, the maximum path length is typically taken as 3T.2,3 The dimensionless path length becomes 3T/R, which is equal to 6. If the average circulation velocity is based on the primary pumping capacity of the impeller, then the dimensionless average circulation velocity turns out to be vj ) 4NQP/π2. For PBTD45STD impeller, the value of the primary flow number is equal to 0.98 (Table 3). The dimensionless circulation velocity becomes 0.397. The dimensionless circulation time then becomes 6/0.397 ) 15.1. This value is much smaller than that obtained from CFD simulations, namely, 26.4. If the average circulation velocity is based on the secondary flow number of the impeller, then the dimensionless average circulation velocity turns out to be vj ) 8NQSD2/π2T2. For PBTD45STD impeller, the value of the secondary flow number is equal to 1.78 (Table 3). The dimensionless average circulation velocity becomes 0.16. Thus, the dimensionless circulation time becomes 6/0.16 ) 37.5. This is much larger than that obtained from CFD simulations, namely, 26.4. From this discussion, it is clear that, the circulation time calculated from concentration profiles is drastically different from that obtained from simple considerations of circulation path length and average circulation veloc-

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3137 Table 3. Predicted Mixing Time for Different Impellers

PBTD30STD PBTD40STD PBTD45STD PBTD50STD PBTD60STD PBTD3020 PBTD4535 PBTD6050 PBTD3010 PBTD4525 PBTD6040 HF1 HF2 HF3 HF4 PBTD45C3H5 PBTD45CH2 PBTD45CH4 PBTD45CH6 PBTD45D02T PBTD45D05T PBTD45W20 PBTD45W40 PBTD30W3010 PBTD45W3010 PBTD60W3010 PBTD45W3050 PBTD30D05T PBTD60D05T PBTD45D06T PBTD45D07T PBTD4525D05T PBTU45CH6 PBTU45STD PBTU45CH2

NP

NQP

NQS

θ h

θmix (s) at 1 W/kg, in a 0.5 m diameter vessel

1.02 1.70 1.93 2.41 3.96 0.67 2.05 3.67 0.47 1.32 2.91 0.34 2.24 3.24 0.33 2.29 2.06 2.77 3.45 2.05 2.67 1.58 2.50 0.57 1.30 1.80 2.68 0.88 5.98 2.52 2.83 0.9 1.74 2.15 1.81

0.77 0.87 0.98 1.11 1.27 0.68 0.84 1.19 0.64 0.84 1.07 0.49 0.98 1.03 0.55 1.03 0.91 0.95 0.90 0.56 0.50 0.85 1.07 0.69 0.75 0.89 1.09 0.73 0.22 0.17 0.02 0.90 0.91 1.05 0.98

1.55 1.76 1.78 2.16 2.11 1.49 1.83 2.33 1.25 1.77 2.2 0.99 2.03 2.15 1.0 1.65 1.59 1.90 1.59 2.53 0.88 1.73 2.05 1.43 1.51 1.44 2.03 0.79 0.57 0.29 0.22 1.10 1.49 2.03 1.52

109.34 88.66 80.08 66.97 69.23 115.89 74.03 62.42 124.47 80.08 64.69 168.86 67.97 75.79 158.26 65.20 70.50 73.27 76.80 162.55 42.50 84.87 60.91 126.74 94.21 104.04 66.97 60.40 33.42 43.25 35.43 71.25 69.23 59.14 106.56

5.75 5.53 5.21 4.69 5.72 5.29 4.91 5.03 5.05 4.59 4.82 6.16 4.65 5.86 5.71 4.49 4.69 5.37 6.06 7.67 4.04 5.16 4.32 5.49 5.37 6.61 4.86 3.96 4.15 4.54 4.29 4.86 4.34 3.98 6.78

Figure 3. Comparison between concentration profiles obtained from CFD and the dispersion model of Voncken et al. (1964): (A) to match the first peak; (B) to match the first valley; (C) to match the second peak. (- - -) CFD predictions. (s) Model of Voncken et al. (1964).

ity. Also, different methods of estimation of the average circulation velocity give different values of circulation time. 3.3 Comparison with Eddy Diffusion Models. The comparison of the present LDV-CFD model was carried out with the model proposed by Voncken et al.8 (eq 2). To compare the CFD model predictions with the above model, a certain value of Bo was assumed. This allowed calculation of c/c∞ as a function of θ. This profile was compared with the CFD predictions. To get a good fit, the parameter Bo was varied. A typical comparison at location 5 is shown in Figure 3A-C. From the figures it can be seen that a unique value of Bo does not characterize the entire concentration profile. If the predictions from eq 2 are made to match the first peak at location 5, then the value of Bo should be 625 (Figure 3A). If the first valley is to be matched, then Bo should be 45 (Figure 3B), whereas the value of Bo should be 25 in order to match the second peak (Figure 3C). Similar results were observed at other locations also. This indicates that there is no characteristic value of Bo which can characterize the dispersion taking place in the stirred vessel. 3.4. Effect of Impeller Design on Mixing Time. The validated model was used to predict the mixing time for all of the impellers listed in Table 1. Because of nondimensionalization, the dimensionless mixing time is defined in the following manner:

θ h ) θmixUtip/R

(Nθmix), which can be readily calculated from rearrangeh R)/(πD). To compare the ment of eq 9: Nθmix ) (θ impellers on a common basis (equal power consumption on a given scale of operation), the values of θmix corresponding to a power consumption of 1 W/kg in a 0.5 m diameter vessel are also given in Table 3. In the stirred vessel, the power consumption per unit mass is given as

P/M )

NPFN3D5 π 2 T HLF 4

(10)

Rearranging the above equation in terms of N, we get

[

]

π 2 T HL(P/M) 4 N) NPD5

1/3

(11)

Substituting the value of N from the above equation, substituting for Utip in eq 9, and rearranging gives

θmix )

θ h R[NPD5]1/3 1/3 π πD T2HL(P/M) 4

[

]

(12)

For all of the impellers investigated in this work, HL ) T. Most of the impellers were operated at D/T ) 1/3; in that case, the above equation simplifies to

(9)

It is a common practice to represent the mixing time as the product of the impeller speed and mixing time

θmix )

θ h NP1/3T2/3 12.06(P/M)1/3

(13)

3138 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

The above two relations have been used for converting the values of θ h into θmix, which are reported in Table 3. It can be seen that when compared on this basis, different impellers show dramatically different mixing times. For example, PBTD45 impeller with D/T ) 0.5 (PBTD45D05T) shows a mixing time of 4.04 s, whereas PBTD45 impeller with D/T ) 0.2 (PBTD45D02T) shows a mixing time of 7.67 s. 3.4.1. Effect of Blade Angle. To investigate the effect of blade angle, five downflow impellers, PBTD30STD, PBTD40STD, PBTD45STD, PBTD50STD, and PBTD60STD (Table 3), were simulated. From the table it can be seen that, as the blade angle increases, the primary flow number (NQP), secondary flow number (NQS), and power number (NP) increase. The NQS/NQP ratio is close to 2.0 for these impellers. However, the NQS/NP ratio decreases from 1.5 for a 30° impeller to 0.54 for a 60° impeller. Table 3 shows that the dimensionless mixing time (θ h ) decreases as the blade angle increases. However, when compared on the basis of equal power consumption, the 50° impeller shows the lowest value of mixing time θmix. 3.4.2. Effect of Blade Width. The effect of blade width was investigated in two ways. In the first set, the blade width was progressively reduced from W/D ) 0.3 at the hub to W/D ) 0.1 at the impeller tip (PBTD30W3010, PBTD45W3010, and PBTD60W3010). It can be seen that there is a drastic reduction in the power number because of the decreasing blade width. However, the reduction in NQP and NQS is smaller. The dimensionless mixing time θ h increases because of the decrease in the blade width from hub to tip. However, the effect on θmix is not straightforward. For a 30° impeller, θmix reduces, for a 45° impeller, θmix increases marginally, and for a 60° impeller, the value of θmix increases significantly. In the second set, there was no variation of the blade width from hub to tip. In addition to W/D ) 0.3, two more ratios of 0.2 and 0.4 were employed (PBTD45W20 and PBTD45W40). It is observed that the value of θ h is practically the same for W/D ) 0.2 and 0.3 impellers, but for W/D ) 0.4 the value of θ h is lower by about 25%. Correspondingly, the value of θmix is practically the same for W/D ) 0.2 and 0.3 impellers but is lower by about 20% for W/D ) 0.4 impeller. 3.4.3. Effect of Blade Twist. The effect of blade twist was investigated for 30, 45, and 60° impellers by giving a 10 and 20° twist toward the horizontal plane throughout the blade length (PBTD3020, PBTD3010, PBTD4535, etc.). For a particular blade angle, an increase in the twist decreases the values of NP and NQP. However, the values of NQS are affected to a much smaller extent. As a result of these, the mixing time, θmix, decreases with an increase in the blade twist. 3.4.4. Effect of Impeller Diameter. The effect of impeller diameter was investigated by taking five different impellers with D/T ) 0.2, 0.33, 0.5, 0.6, and 0.7 (PBTD45D02T, PBTD45STD, PBTD45D05T, PBTD45D06T, and PBTD45D07T). As the impeller diameter increases from D/T ) 0.2 to 0.5, the mixing time θmix decreases, but a further increase in the impeller diameter up to D/T ) 0.7 increases the mixing time. Thus, out of these five impellers D/T ) 0.5 is more energy efficient. A similar observation is also made for PBTD30 and PBTD60 impellers. An increase in the diameter from D/T ) 0.33 to 0.5 for PBTD30 impeller reduces the mixing time from 5.74 to 3.96. For PBTD60

impeller the mixing time reduces from 5.72 to 4.15 when the D/T ratio is increased from 0.2 to 0.5. 3.4.5. Effect of Impeller Clearance. The effect of impeller clearance was studied for a 45° impeller by changing its location in the vessel. The following values of clearance were taken C/T ) 0.6, 0.5, 0.33, 0.25, and 0.167 (PBTD45C3H5, PBTD45CH2, PBTD45CH4, etc.). Table 3 shows that, as the impeller clearance decreases from 0.6 to 0.33, the NP decreases; a further decrease in the impeller clearance increases the power number because of the proximity of the vessel bottom. The NQP and NQS values do not show any trend with respect to impeller clearance. The θmix values increase continuously as the impeller clearance decreases. 3.4.6. Effect of Pumping Direction. The effect of changing the pumping direction was studied for a 45° impeller at various values of impeller clearance. A change in the impeller pumping direction results in an upflow impeller (PBTU45STD, PBTU45CH6, and PBTU45CH2). For all of the impeller clearances, the power number decreases by making the impeller flow up. For a downflow impeller, the jet leaving the impeller interacts with the vessel bottom, causing higher levels of energy dissipation in the region below the impeller. For an upflow impeller, the jet interacts with the top liquid surface. Because this interface is free, the dissipation rates are smaller above the impeller. Therefore, the power numbers for upflow impellers are lower than downflow impellers. For example, at C ) T/6 the power number for the upflow impeller was found to be 1.74 as compared to 3.45 for the downflow impeller. At C ) T/2 the power numbers were 1.81 and 2.06 for upflow and downflow, respectively. By changing the impeller direction, the primary flow number was practically unaffected. The variation in NQS values was also not significant. When compared on the basis of equal power consumption, it can be seen that the mixing time θmix decreases by making the impeller flow up for C/T ) 0.33 and 0.167; the upflow impeller at C/T ) 0.5 shows an increased mixing time as compared to the downflow impeller at the same location. 3.4.7. Effect of Blade Shape. The hydrofoil impellers show a wide variation in blade shape. Comparison of these hydrofoil impellers is helpful in elucidating the effect of blade shape. The hydrofoil impellers show considerable differences in terms of NP, NQP, and NQS. The maximum and minimum values of NP are 3.24 for HF3 and 0.33 for HF4, respectively. In terms of NQP, these are 1.03 for HF3 and 0.55 for HF4, and in terms of NQS, they are 2.15 for HF3 and 0.99 for HF1. Out of these hydrofoils, HF2 gives the smallest value of the mixing time when compared on the basis of equal power consumption. 3.5. Relation between Flow Pattern and Blending. 3.5.1. Relation between Mixing Time and NP, NQP, and NQS. To establish the relationship between the flow pattern and mixing time, first, it was attempted to correlate the values of θmix with either NP, NQP, or NQS. It was observed that the values of θmix do not correlate well with either NP, NQP, NQS, or some simple combination of these variables. Therefore, more attention was focused on the values of θ h . It was found that the dimensionless mixing time θ h was inversely proportional to NQS. The correlation for all D/T ) 0.33 impellers is shown in Figure 4. From the figure, it was

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3139

Figure 4. Relation between the dimensionless mixing time θ h and 1/NQS.

observed that

θ h ) 152.64/NQS

(14)

Considering the wide variation in the impeller designs in terms of blade angles, blade widths, blade twists, and blade shapes, the correlation is very remarkable. The differences in the impeller designs can be characterized in terms of their hydraulic efficiency, that is, the NQP/ NP ratio. Among all of the impellers with D/T ) 1/3, the hydraulic efficiency is highest, 1.67, for the HF4 impeller and is lowest, 0.32, for PBTD60STD and the HF3 impeller. Substituting the value of θ h from the above equation into eq 13 allows a comparison of the different impellers on the basis of equal power consumption per unit mass, that is

θmix )

12.66NP1/3T2/3 NQS(P/M)1/3

(15)

The equation shows that when the impellers are being compared on the basis of equal energy input, both NP and NQS are important in determining its performance for blending. This equation forms a basis for the selection of optimum impeller design for blending. It can be seen from Table 3 that, for all of the downflow impellers having D/T ) 0.33 and located at C/T ) 0.33, PBTD4525 impeller shows the lowest mixing time when compared on the basis of equal energy input. Recently, Nienow44 has proposed the following correlation for mixing time:

NP1/3Nθmix(D/T)1/3 ) constant

(16)

Substituting for N from eq 11 into eq 16 gives (keeping HL ) T and D/T ) 1/3)

θmix(P/M)1/3/T2/3 ) C1 ) constant

(17)

The dependence of P/M and T is identical with that observed in this work (eq 15). It is clear from eq 15 that the value of the constant in eq 17 is proportional to NP1/3/ NQS. When different impeller diameters were investigated (D/T ) 0.2, 0.33, 0.5, 0.6, and 0.7), the constant in eq 14, or alternatively eq 17, shows a different value. The values of the ratio NP1/3/NQS for D/T ) 0.2, 0.5, 0.6, and 0.7 are 0.5, 1.6, 4.7, and 6.4, respectively. It is observed that the value of the constant C1 in eq 17 can be correlated with the D/T ratio using the following

Figure 5. (A) Contour plot of the radial velocity for PBTD45D05T impeller. (B) Contour plot of the axial velocity for PBTD45D05T impeller. (C) Overall flow pattern generated by PBTD45D05T impeller. (D) Overall flow pattern generated by PBTD45STD impeller.

equation:

C1 ) 0.14 exp(5.47D/T)

(18)

This observation can be explained on the basis of flow patterns generated by D/T ) 0.5, 0.6, and 0.7 impellers. The contour plots of radial velocity and axial velocity for PBTD45D05T are shown in Figure 5A,B. It is observed that the radial velocity at location A (just below the impeller) is quite large (Figure 5A). This indicates that the flow generated by this impeller does not go axially down from the impeller but at a certain angle to the vertical plane. This is confirmed by the fact that, at location B, the radial velocity is observed to be inward (Figure 5A) and the axial velocity is found to be upward (Figure 5B). This is quite different from other axial flow impellers with D/T ) 1/3. The resulting flow pattern arising out of this is shown in Figure 5C. This is responsible for the increase in the value of the

3140 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 4. Effect of Flow Field Generated by the Impeller for PBTD45STD

NP

NQP

NQS

θ h

θmix at P/M ) 1 W/kg, in a 0.5 m diameter tank

2k no change (base case) 0.5k 0.25k 0.0625k 0.01k

3.14 1.58

0.98 0.98

1.25 1.20

144.39 109.08

11.04 6.63

0.88 0.51 0.27 0.27

0.98 0.98 0.98 0.98

1.22 1.25 1.25 1.24

103.53 118.92 119.17 120.44

5.18 4.96 4.02 4.07

2vz no change (base case) 0.5vz

4.18 1.58

1.96 0.98

2.46 1.20

56.12 109.08

4.72 6.63

0.86

0.49

0.61

253.60

12.59

change from the base case

constant, C1, in eq 17. It can be seen that there is a secondary circulation loop below the impeller, which results in mixing characteristics different from the D/T ) 0.33 impeller. The overall flow pattern based on such contour plots for the D/T ) 0.33 impeller is shown in Figure 5D. The overall flow pattern for PBTD45D06T and PBTD45D07T impellers was similar to that of Figure 5C. It should be noted that the predicted mixing time for the PBTD45D05T impeller is in good comparison with the experimental measurements, as indicated by Table 2A. Thus, as the D/T ratio increases, the radial flow generated by the impeller starts becoming significant and, as a result, the overall flow pattern changes. This is responsible for the dependence of C1 in eq 17 on the D/T ratio. 3.5.2. Effect of Flow Field Generated by Impeller. Though a large number of impellers have been fabricated and investigated in this work, in terms of the hydraulic efficiency (NQP/NP ratio) the variation between the impellers is by a factor of 5, that is, from 0.32 to 1.67. To cover a much wider variation in terms of the hydraulic efficiency, the measured values of k and vz were multiplied (or divided) by a known factor at all four locations, and these modified values were supplied as boundary conditions. This allows us to investigate the role of k and vz independently of each other and other velocity components on the mixing time. The values of k are indicative of the turbulence generated by the impeller, whereas vz indicates the axial component of the mean velocity generated by the impeller. Because all of the impellers are axial flow, the vz component is the strongest of the three mean velocity components. Varying k and vz one by one allows systematic variation of the turbulence and axial flow generated by the impeller. The boundary conditions at these four locations were provided only up to r/R ) 0.33 (impeller radius). This enabled changing the hydraulic efficiency from 0.31 for the 2k case to 3.61 for the 0.01k case. The basic impeller configuration considered was PBTD45STD (base case). Any change in the impeller design from this basic configuration in terms of blade shape, angle, width, and twist was considered to modify the values of k and vz at the four boundary conditions. The multiplication factor of k was varied from 2 to 0.01, and that for vz was varied from 2 to 0.5. The variation of k and vz was carried out independently of each other. The results of these simulations are shown in Table 4. Even for all of these combinations it can be seen that the value of the constant in eq 14, that is, the product θ h NQS, turns out to be in the range 120-150.

Figure 6. Contribution of eddy diffusivity to the mixing process for (- -) PBTD30STD, (s) PBTD45STD, and (- - -) PBTD45D05T impellers.

From the table it can also be seen that reducing the value of k decreases the power number. Because the value of vz was not changed, the NQP values remain unaffected, and the NQS values vary only slightly. Thus, changing the value of k allowed variation of NP at approximately the same value of NQS. The results indicate that the mixing time θmix decreases with k, until the k level is about 6% (0.0625k) of the base case. A further reduction in k causes an increase in the mixing time. On the other hand, when the value of vz is changed, NP, NQP, and NQS all get affected. A reduction in vz from 2 to 0.5 causes a decrease in NP, NQP, and NQS to different extents. That is, a reduction in vz by a factor of 4 reduces NP by a factor of about 4.8; the reduction in NQP and NQS is about 4 times; however, the mixing time increases only by a factor of about 2.7. These simulations indicate that, for a given amount of mean flow generated by the impeller, it is desirable to reduce the turbulence generated by the impeller in order to improve the energy efficiency of the impeller. It has to be emphasized that further research work needs to be carried out which will establish a linkage between impeller shape and the k and vz levels generated. This relationship can then be used to derive an impeller shape, which will allow k and vz profiles generated by the impeller to be varied independently and kept at any desired level. 3.5.3. Contribution of Bulk and Eddy Diffusion to the Overall Mixing Process. To quantify the contribution of circulation and eddy diffusion in the overall mixing process, the values of mean velocity or eddy diffusivity in the entire vessel were multiplied by a known factor before integrating the conservation equation of the tracer (eq 3). This enabled the proportion of eddy diffusivity and the mean velocity to be changed at will. The contribution of the eddy diffusivity in the overall mixing process was investigated for PBTD30STD, PBTD45STD, and PBTD45D05T impellers. These impellers were chosen because between them they cover a wide range of NP, NQP, NQS, and as a result θmix. To compare the different impellers, the mixing times (θ h) were normalized, with the dimensionless mixing time corresponding to the multiplier of eddy diffusivity ) 1. The results are shown in Figure 6. It can be seen from Figure 6 that, for PBTD30STD and PBTD45STD impellers, the dimensionless mixing time increases nominally even if the eddy diffusivity is reduced by a factor of 5. However, the value of θ h increases sharply when the multiplier of eddy diffusivity is below 0.2 (20% of the

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3141

Figure 7. Contribution of vr and vθ to the mixing process for PBTD30STD and PBTD45D05T impellers: (s) vr for PBTD30STD; (‚‚‚) vθ for PBTD30STD; (thick line) vr for PBTD45D05T; (- - -) vθ for PBTD45D05T.

eddy diffusivity). However, for PBTD45D05T impeller, the mixing time increases rapidly only when the eddy diffusivity was reduced below 5% (multiplier ) 0.05) of its actual value. This shows that the contribution of the eddy diffusivity is smaller for PBTD45D05T impeller as compared to PBTD30STD and PBTD45STD impellers. This insensitivity of eddy diffusivity would also mean that a small error in its prediction (10-20%) would cause only a slight change in the mixing time. Therefore, even if the CFD predictions in a certain region show deviation from experimental measurements by 10-20%, the predictions of mixing time are not affected significantly. This can explain the good match between the predicted mixing time and the experimentally observed mixing time, shown in Table 2A,B. This indicates that the bulk circulation plays a dominant role in the blending process. In contrast, when the multiplier of mean velocity (all three components of velocity, vr, vz, and vθ, were multiplied) was decreased, the mixing time increases in an inverse proportion. In other words, bulk diffusion contributes to a large extent in the overall mixing process. The contribution of the three components of mean flow (vr, vz, and vθ) was also investigated. It has already been reported that the dimensionless mixing time is inversely proportional to NQS. Hence, it was decided to investigate in detail the contribution of vr and vθ to the

overall mixing process for PBTD30STD and PBTD45D05T impellers. The results of this exercise for these two impellers are depicted in Figure 7. From the figure it can be seen that the contribution of vr and vθ is larger for PBTD45D05T impeller as compared to PBTD30STD impeller. The relative contribution of vr and vθ is roughly the same for PBTD30STD impeller. On the other hand, the contribution of vr is larger than vθ for PBTD45D05T impeller. Comparison of Figures 6 and 7 shows that the contribution of vr and vθ is much more as compared to the eddy diffusivity for these impellers. 3.5.4. Identifications of Slow Mixing Zones. In the above exercise, the values of eddy viscosity and the components of mean velocity were multiplied throughout the vessel. However, it is likely that this contribution of eddy diffusivity and mean velocity is different in different regions. To get maximum energy efficiency, the optimization of flow (vr, vz, and vθ) and turbulence (νT) may be necessary in the entire flow domain. As a first step toward this, it is necessary to identify the regions in which the mixing process is slow as compared to other regions. This was achieved by simulating the mixing process in the different regions of the vessel. More specifically, this was done by eliminating the different regions of the vessel one by one. The identification of slow mixing regions was carried out for PBTD30STD and PBTD45D05T impellers. The results are tabulated in Table 5. For PBTD30STD impeller at 100% νT, there are no slower mixing regions. Even when a substantial portion of the vessel is eliminated, the mixing time reduces from 109.34 to 94.45. When νT is 10%, the mixing time reduces by about 35% (from 291.69 to 191.81), when 8 grids from the center are eliminated. However, a further increase to 10 grids from the center reduces the mixing time only by another 13% to 168.10. This indicates that there is a relatively slow moving region near the center of the vessel, but eliminating this region shows that the mixing throughout the vessel has become slower. In other words, for this impeller, except for the region near the center, the mixing process becomes slow at 10% νT. The PBTD45D05T impeller with 100% νT shows no significant slow mixing zones. The mixing time reduces by only about 13% when a large portion of the vessel

Table 5. Identification of Slow Mixing Regions dimensionless mixing time, θ h

region eliminated PBTD30STD, νT ) 100% none 2 grids from wall, 4 grids from top, 2 grids from both baffles 6 grids from wall, 4 grids from top, 2 grids from both baffles 6 grids from wall, 8 grids from top, 2 grids from both baffles

109.34 101.01 95.97 94.45

PBTD30STD, νT ) 10% none 4 grids from center, 2 grids from both baffles 8 grids from wall, 2 grids from both baffles 10 grids from wall, 2 grids from both baffles

295.73 291.69 191.81 177.94

PBTD45D05T, νT ) 100% none 2 grids from wall, center, top, bottom, and both baffles 6 grids from wall and center, 2 grids from top, bottom, and both baffles 8 grids from wall and center, 2 grids from top, bottom, and both baffles

42.50 40.73 38.46 36.95

PBTD45D05T, νT ) 1% none 2 grids from both the baffles 4 grids from center, 2 grids from bottom and both baffles 6 grids from center, 2 grids from bottom and both baffles

244.53 244.52 65.20 52.84

3142 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

from the center and the wall are eliminated. For the same impeller with 1% νT, a very slow mixing region near the center of the vessel is observed. When this region is removed, the mixing time decreases by severalfold from 244.53 to 65.20. This region near the center of the vessel corresponds to the region of secondary circulation observed in Figure 5C. The results of the above two sections indicate that there is enormous scope to optimize the local values of mean flow to achieve substantially higher energy efficiency in the mixing process. This can be brought about by changing impeller geometry, use of various internals, modification of baffles, change of vessel shape, and so forth. This aspect can be considered for future work in this area. 4. Conclusions 1. In the present work a combination of LDV and CFD measurements has been used to predict the mixing time for several axial flow impellers. It is observed that the mixing time predicted from the CFD model agrees quite well with the experimentally measured mixing time for a wide variety of impellers varying in blade angle, blade width, and impeller diameter and at several levels of power consumption. 2. It has been shown that neither of the two models explains the mixing phenomena completely. 3. From the present study, the following conclusions can be made regarding the effects of different parameters: (i) Of the different blade angles investigated, a 50° impeller shows the lowest value of θmix. (ii) With an increase in the impeller blade width, the value of θmix decreases. (iii) For PBTD45 impeller, at C ) T/3, an increase in the impeller diameter (up to D/T ) 0.5) causes a decrease in θmix; a further increase in the D/T ratio increases the value of θmix. (iv) θmix increases as the impeller clearance from the bottom decreases. (v) Under comparable conditions, the θmix value for upflow impellers is lower than that for downflow impellers. 4. It was observed that the dimensionless mixing time (θ h ) varies inversely with the secondary flow number of the impeller. For all D/T ) 1/3 impellers, it is observed that θmix ∝ NP1/3T2/3/NQS. The value of the constant of proportionality increases with an increase in the D/T ratio. This relationship was found to be valid even when the k and vz values generated by the impeller PBTD45STD were varied independently of each other and independently of other parameters. 5. In the turbulent regime, convection motion controls the mixing process to a large extent. For PBTD45D05T impeller at 1% νT, a slow mixing region is observed in the center of the vessel, and eliminating this reduces the mixing time by severalfold. 6. One of the ways to reduce the eddy viscosity levels in the vessel is to reduce the turbulence generated by the impeller. For PBTD45STD impeller it has been shown that the k levels generated by the impeller can be reduced by as much as a factor of 20. Correspondingly, the power number would reduce by a factor of around 12. This represents substantial savings in terms of operating cost. 5. Suggestions for Future Work The above results indicate that there is enormous scope to modify the local values of mean flow and eddy viscosity to achieve substantially higher levels of energy efficiency in the mixing process. To implement both the

above conclusions, further research and developmental work will have to be carried out, which will establish a linkage between impeller shape and the k and vz levels generated. This relationship can then be used to derive an impeller shape, which will allow k and vz profiles generated by the impeller to be kept at any desired level. Acknowledgment The research work is based on the project sponsored by the All India Council for Technical Education (F8017/ RD II/BOR/95/RCHE073/Rec 88). Nomenclature Bo ) Bodenstein number, vL/DS C1) constant in eq 17 CFD ) computational fluid dynamics c ) tracer concentration at any time and location, kmol/ m3 c∞ ) tracer concentration after infinite time, i.e., final steady-state value, kmol/m3 D ) Impeller diameter, m DS ) dispersion coefficient, m2/s HL) liquid height, m K ) mixing rate constant, 1/s k ) turbulent kinetic energy, m2/s2 L ) length of the circulation loop, m LDV ) laser Doppler velocimetry M ) mass of the liquid, kg M′ ) extent of homogeneity N ) impeller speed, rev/s NP ) power number of the impeller NQP ) primary flow number of the impeller NQS ) secondary flow number of the impeller P ) power consumption, W PBTD ) pitched blade downflow turbine PBTU ) pitched blade upflow turbine R ) vessel radius, m r ) radial coordinate, m Re ) Reynolds number, ND2F/µ Sφ ) source term for a generalized flow variable φ T ) tank diameter, m t ) time, s tc ) circulation time, s Utip ) impeller tip velocity, m/s VC ) average circulation velocity, m/s VL ) total liquid volume in the vessel, m3 vr ) mean velocity in the radial direction, m/s vz ) mean velocity in the axial direction, m/s vθ ) mean velocity in the tangential direction, m/s vrw ) average radial liquid velocity near the wall region, m/s vj ) dimensionless average liquid circulation velocity, v/Utip W ) impeller blade width, m z ) axial coordinate, m Greek Letters Γ ) effective diffusivity consisting of molecular plus turbulent diffusivity, m2/s  ) energy dissipated per unit mass, W/kg θ ) tangential coordinate θmix ) mixing time, s θ h ) dimensionless mixing time defined as θmixUtip/R µ ) viscosity of the liquid, Pa‚s F ) liquid density, kg/m3 τ ) dimensionless time, t/tC

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 3143 φ ) generalized flow variable

Literature Cited (1) Norwood, K. W.; Metzner, A. B. Flow Patterns and Mixing Rates in Agitated Vessels. AIChEJ. 1960, 6, 432-437. (2) McManamey, W. J. A. Circulation Model for Batch Mixing in Agitated, Baffled Vessels. Chem. Eng. Res. Des. 1980, 58, 271275. (3) Joshi, J. B.; Pandit, A. B.; Sharma, M. M.; Mechanically Agitated Gas-Liquid Reactors. Chem. Eng. Sci. 1982, 37, 813844. (4) Raghav Rao, K. S. M. S.; Joshi, J. B. Liquid Phase Mixing in Mechanically Agitated Vessels. Chem. Eng. Commun. 1988, 74, 1-26. (5) Rzyski, E. Liquid Homogenisation in Agitated Tanks. Chem. Eng. Technol. 1993, 16, 229. (6) Van der Molen, K.; van Mannen, H. R. E. Laser Doppler Measurements of the Turbulent Flow in Stirred Vessels to Establish Scaling Rules. Chem. Eng. Sci. 1978, 33, 1161-1168. (7) Rewatkar, V. B.; Joshi, J. B. Effect of Impeller Design on Liquid Phase Mixing in Mechanically Agitated Reactors. Chem. Eng. Commun. 1991, 91, 322-353. (8) Voncken, R. M.; Holmes, D. B.; Den Hartog, H. W. Fluid Flow in Turbine Stirred, Baffled TankssII. Dispersion During Circulation. Chem. Eng. Sci. 1964, 19, 209-213. (9) Brennan, D. J.; Lehrer, I. H. Impeller Mixing in Vessels Experimental Studies on the Influence of Some Parameters and Formulation of A General Mixing Time Equation. Chem. Eng. Res. Des. 1976, 54, 139-152. (10) Hiraoka, S.; Ito, R. Simple Relationship between Power Input and Mixing Time in Turbulent agitated Vessel. J. Chem. Eng. Jpn. 1977, 10, 75-77. (11) Jahoda, M.; Machon, V.; Pinelli, D.; Nocentini, M.; Fjner, D.; Magelli, F.; Homogenisation of Liquids and Fluid Dynamic Behaviour of Vessels Stirred With Multiple Axial Impellers. Proceedings of the 8th European Conference On Mixing, University of Cambridge, Cambridge, U.K., Sept 21-23, 1994; BHRA: Cranfield, U.K.; pp 113-120. (12) Khang, S. J.; Levenspiel, O. New Scale-up and Design Method for Stirrer Agitated Batch Mixing Vessels. Chem. Eng. Sci. 1976, 31, 569-577. (13) Mann, R.; Mavros, P. Analysis of Unsteady Tracer Dispersion and Mixing in a Stirred Vessel Using Interconnected Network of Ideal Flow Zones. Proceedings of the Fourth European Conference On Mixing, Leeuwenhorst, The Netherlands, April 27-29, 1982; Nordwijkerhout: The Netherlands; Vol. B3, pp 35-48. (14) Mann, R.; Ying, P. Application of 3D Network of Zones Mixing Model To a Stirred Vessel. Proceedings of the 8th European Conference On Mixing, University of Cambridge, Cambridge, U.K., Sept 21-23, 1994; BHRA: Cranfield, U.K.; pp 317-324. (15) Voit, H.; Mersmann, A. B. Calculation of Mixing Times from the field of mean and fluctuating velocity profiles. Proceedings of the 6th European Conference On Mixing, Pavia, Italy, May 2426, 1988; BHRA: Cranfield, U.K. (16) Harvey, P. S.; Greaves, M. G. Turbulent flow in an agitated vessel partsI: a predictive model. Chem Eng. Res. Des. 1982, 60, 195-200. (17) Harvey, P. S.; Greaves, M. G. Turbulent flow in an agitated vessel partsII: Numerical solution and model predictions. Chem Eng. Res. Des. 1982, 60, 201-210. (18) Ranade, V. V.; Joshi, J. B.; Marathe, A. G. Flow generated by pitched blade turbines II: Simulation using k- model. Chem. Eng. Commun. 1989, 81, 225-248. (19) Kresta, S. M.; Wood, P. E. Prediction of the threedimensional turbulent flow in stirred tanks. AIChE J. 1991, 3, 448-460. (20) Gosman, A. D.; Lekakou, C.; Politis, S.; Issa, R. I.; Looney, M. K. Multidimensional Modelling of Turbulent Two Phase Flows in Stirred Vessels. AIChEJ. 1992, 38, 1946-1956. (21) Luo, J. Y.; Gosman, A. D.; Issa, R. I.; Middleton, J. C.; Fitzgerald, M. K. Full flow field computation of mixing in baffled stirred vessels. Chem. Eng. Res. Des. 1993, 71, 342-344. (22) Armenante, P. M.; Chou, C.-c. Velocity profiles in a baffled vessel with single or double pitched blade turbines. AIChEJ. 1996, 42, 42-54.

(23) Armenante, P. M.; Luo, C.; Chou, C.; Fort, I.; Medek, J. Velocity Profiles in A Closed, Unbaffled Vessel: Comparison Between Experimental LDV Data and Numerical CFD Predictions. Chem. Eng. Sci. 1997, 52, 3483-3492. (24) Bakker, A.; Myers, K. J.; Ward, R. W.; Lee, C. K. The Laminar and Turbulent Flow Pattern of A Pitched Blade Turbine. Chem. Eng. Res. Des. 1996, 74, 485-491. (25) Ciofalo, M.; Brucato, A.; Grisafi, F.; Torraca, N. Turbulent Flows in Closed and Free Surface Unbaffled Tanks Stirred by Radial Impellers. Chem. Eng. Sci. 1996, 51, 3557-3573. (26) Harris, C. K.; Roekaerts, D.; Rosendal, F. J. J. Computational Fluid Dynamics for Chemical Reactor Engineering. Chem. Eng. Sci. 1996, 51, 1569-1594. (27) Harvey, A. D., III; Rogers, E. S. Steady and unsteady computation of impeller-stirred reactors. AIChEJ. 1996, 42, 27012712. (28) Harvey, A. D., III; Wood, S. P.; Leng, D. E. Experimental and Computational Study of Multiple Impeller Flows. Chem. Eng. Sci. 1997, 52, 1479-1491. (29) Morud, K. E.; Hjertager, B. H. LDA Measurements and CFD Modelling of Gas -Liquid Flow in A Stirred Vessel. Chem. Eng. Sci. 1996, 51, 233-249. (30) Ranade, V. V.; Dommeti, S. M. S. Computational Snapshot of Flow Generated by Axial Impellers in Baffled Stirred Vessels. Chem Eng. Res. Des. 1996, 74, 476-484. (31) Ranade, V. V. An Efficient Computational Model for Simulating Flow in Stirred Vessels: A Case of Rushton Turbine. Chem. Eng. Sci. 1997, 52, 4473-4484. (32) Bakker, A.; Laroche, R. D.; Wang, M. H.; Calabris, R. V. Sliding Mesh Simulation of Laminar Flow in Stirred Reactors. Chem Eng. Res. Des. 1997, 75, 42-44. (33) Weetman R. J. Automated Sliding Mesh CFD Computations for Fluidfoil Impellers. Recents Prog. Genie Procedes 1997, 11, 195-202. (34) Ranade, V. V.; Bourne, J. R.; Joshi, J. B. Fluid Mechanics and Blending in Agitated Tanks. Chem. Eng. Sci. 1991, 46, 18831893. (35) Patterson, G. K.; Zipp, R. P. Turbulence and Mixing: Modelling Effects on Chemical Reactions. In Mixing, Coagulation, and Flocculation; Amirthatajah, A., Clark, M. M., Trussel, R. R., Eds.; American Water Works Association: Denver, CO, 1991; pp 80-126. (36) Hoffman, P. D. Mixing in a Large Storage Tank. AIChE Symp. Ser. No. 286 1992, 88, 77-82. (37) Ranade, V. V. Towards Better Mixing Protocols By Designing Spatially Periodic Flows: The Case of a Jet Mixer. Chem. Eng. Sci. 1996, 51, 2637-2642. (38) Togatorop, A.; Mann, R.; Schofield, D. F. In an Application of CFD to Inert and Reactive Tracer Mixing in Batch Stirred Vessel. AIChE Symp. Ser. No. 299 1994, 90, 19-32. (39) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere: New York, 1980. (40) Sahu, A. K.; Joshi, J. B. Simulation of flow in stirred vessels with axial flow impellers: Effects of various numerical schemes and turbulence model parameters. Ind. Eng. Chem. Res. 1995, 34, 626-639. (41) Sahu, A. K.; Kumar, P.; Joshi, J. B. Simulation of flow in stirred vessels with axial flow impeller: Zonal Modelling and Optimisation of parameters. Ind. Eng. Chem. Res. 1998, 37, 21162130. (42) Fokema, M. D.; Kresta, S. M.; Wood, P. E. Importance of using the correct impeller boundary conditions for CFD simulations of stirred tanks. Can. J. Chem. Eng. 1994, 72, 177-183. (43) Kramers, H.; Baars, G. M.; Knoll, W. H. A comparative study on the rate of mixing in Stirred Tanks. Chem. Eng. Sci. 1953, 2, 35-42. (44) Nienow, A. W. On the Impeller Circulation and Mixing Effectiveness in Turbulent Regime. Chem. Eng. Sci. 1997, 52, 2557-2565.

Received for review December 7, 1998 Revised manuscript received April 19, 1999 Accepted April 20, 1999 IE980772S