Liquid-Phase Mixing in Stirred Vessels: Turbulent Flow Regime

Ind. Eng. Chem. Res. 2003, 42, 2661-2698

2661

Liquid-Phase Mixing in Stirred Vessels: Turbulent Flow Regime Nandkishor K. Nere, Ashwin W. Patwardhan, and Jyeshtharaj B. Joshi* Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 400 019, India

The published literature on the liquid-phase mixing in a turbulent flow regime has been critically reviewed and analyzed. Experimental techniques for mixing time have been described together with their relative merits. The effects of the impeller design (blade number, blade angle, blade and disk dimensions, and blade shape), the location of the impeller (off-bottom clearance, distance from the vessel center, i.e., eccentricity), and the vessel size on the liquid-phase mixing have been critically analyzed. The mixing performance dependency on the internals such as baffles (number, dimension, and position) and the draft tube has been presented in detail. Further, an extensive review on the mathematical models proposed for the liquid-phase mixing has been presented, and the utility of the computational fluid dynamics modeling for the mixing optimization has been illustrated. Finally, suggestions have been made for the selection of an energy-efficient impeller-vessel configuration, and directions have been given for future studies. 1. Introduction Mixing is one of the most important unit operations in chemical process and allied industries. The overall energy requirement of these processes forms a significant part of the total energy and contributes toward major expenses. Fluid mechanics prevailing in the mixers is complex, and hence the design procedures have been empirical. Empirical correlations normally lead to significant overdesign and result in inflated fixed and operating costs as well as in extra start-up times. Further, the empiricism does not give rational answers to the debottlenecking problems. Therefore, reliable procedures are needed for the design of mixing equipment. In view of this, several attempts have been made in the past, particularly during the last 35 years, to understand the mixing phenomena both experimentally and theoretically. The studies pertaining to mixing have been published in the form of research papers/monographs and books. However, clear guidelines are not available in the published literature for the selection of appropriate experimental techniques for the measurement of mixing time, the selection of mixing criteria, the effect of various internals on the flow produced by a variety of impeller designs, and the comparison between the various impeller-vessel configurations. In particular, a coherent treatise is needed to assess the quantitative contribution of all of the above parameters. Further, some of the studies show contradictory results in regards to the effect of the impeller design and the draft tube on the flow pattern and the resulting mixing time. It is important that such discrepancies are resolved. In view of these, it was first thought desirable to analyze the published literature critically. In this paper, the experimental techniques adopted by various researchers to measure the mixing time have been described with their limitations. Further, the effect of the measuring technique on the estimation of the mixing time has been discussed. A rotating impeller, which requires direct energy input, is the most important part of the typical mixing * To whom correspondence should be addressed. Tel.: 0091-22-4145616.Fax: 00-91-22-4145614.E-mail: [email protected].

equipment. The efficiency of the mixing process depends on the design of the impeller (blade number, shape, and size). Also, the location of the impeller (off-bottom clearance, distance from the vessel center, i.e., eccentricity) and its size relative to the vessel have a profound impact on the flow pattern and the mixing efficiency thereof. The effects of all of these parameters as observed by various researchers have been critically analyzed. It is well-known that any minor changes in the vessel geometry such as wall baffles have substantial effects on the flow and hence on the mixing characteristics. Hence, the mixing performance dependency on the internals such as baffles (number and position), draft tube, etc., as reported in the published literature has been discussed. Further, an extensive review on the theoretical investigations on mixing with a specific emphasis on the mathematical modeling of mixing has been presented. Various mathematical models proposed for the mixing have been described with their applicability and limitations. The results of various mixing time simulations due to the computational fluid dynamics (CFD) modeling of the mixing for various configurations have been presented, and its applicability to screen various configurations has been illustrated. Recommendations for the selection of an impeller-vessel configuration are made, and finally, suggestions have been given for future work. 2. Phenomena of Mixing The process of mixing occurs as a result of the motion at three levels: molecular, eddy, and bulk motion. The molecular motion of individual species reduces the concentration differences, and the process is known as molecular diffusion. Mixing on this level is called micromixing. Improper micromixing leads to segregation (Levenspiel1). If the stirred reactor is operated under turbulent conditions, then there is a motion of a chunk of molecules or eddies. The eddy motion also gives rise to material transport and is called eddy diffusion or eddy dispersion. The bulk motion or the convective motion also has a property of providing spread of materials needed for mixing. Usually, the bulk motion is superimposed on either molecular or eddy diffusion

10.1021/ie0206397 CCC: $25.00 © 2003 American Chemical Society Published on Web 04/22/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003

Figure 1. Circulation paths in the stirred vessel: (A) axial flow impeller; (B) radial flow impeller.

or both. Mixing on this scale is called as macromixing. In the present review, we will deal mainly with the liquid-phase turbulent mixing on a large scale, that is, macromixing. McManamey2 has reviewed the literature in regards to the circulation models and has shown the comparison of the predicted mixing time with the mixing time data of various investigators. He has proposed the circulation models for both the axial and radial flow impellers based on the flow generated by the impeller and the reactor geometry. The flow patterns generated by axial and radial flow impellers are shown schematically in parts A and B of Figure 1. The maximum length of the circulation path for axial and radial impellers can be seen to be 3T and 2T, respectively. If the average circulation velocity is VC, the circulation times are 3T/ VC and 2T/VC, respectively. His model gives reasonable estimates of the mixing times if the average circulation

velocity and the length of the longest circulation loop are taken into consideration for the calculation of the mixing time. The mixing time due to bulk motion is usually considered as 5 times the circulation time, θB ) 5tC. This follows from the observation of Holmes et al.3 that a maximum of three to five peaks appear in the concentration-time plot of a typical mixing experiment. This indicates that the corresponding number of circulations is needed to achieve adequate mixing. Hence, one can take the mixing time as approximately equal to 5 times that of the circulation time. The fluid in a turbulent flow can be considered as consisting of lumps or eddies. The largest eddies have sizes of the same order of magnitude as the largest length scale of the process equipment (vessel or impeller size, L), whereas the smallest eddies are such that viscous dissipation takes over and these smallest eddies are dissipated into heat. This scale is called the Kolmogorov length scale and is characterized by Reκ ) lκuκ/ν ∼ 1. The size of the smallest eddy is denoted by lκ. The value of lκ depends on the turbulent energy dissipation rate per unit mass and the kinematic viscosity of the fluid and is given by lκ ) (ν3/)1/4. There exists a whole range of eddy sizes between L and lκ, and this is usually represented by an energy spectrum. Tennekes and Lumley4 have given an excellent review of the structure of turbulence, the eddy sizes, their interrelations, the energy spectra, etc. Brodkey5 has reviewed various aspects of turbulent motion and its influence on mixing. Molecular diffusion takes place at the Kolmogorov scale and causes homogeneity at the scale of the smallest eddy, while eddy diffusion is responsible for the transport of material at all of the scales. The different sizes of eddies have different velocity scales and, hence, have different energies. Apart from the velocity and size, eddies also have different lifetimes. Depending on the velocity, size, and lifetime, different eddies cause eddy dispersion to different extents. All of these factors have to be taken into account while quantifying eddy diffusion, and such a detailed analysis of the eddy dispersion process is extremely complicated. To overcome these difficulties, eddy diffusion is usually characterized in terms of eddy diffusivity. Eddy diffusion or molecular diffusion, which is described in terms of diffusivity, is characterized by the corresponding diffusion time. For molecular diffusion, the characteristic time is given as tM ) lκ2/DM. Similarly, the characteristic time for eddy diffusion is tT ) lT2/DT. If the diffusivities are known, then the mixing time θM or θT can be estimated. Let us estimate the relative importance of DM, DT, and convective motion to the mixing process. Here the contribution of both the diffusivities toward the mixing time has been exemplified. 2.1. Illustrative Example. Consider a stirred reactor having a 5 m diameter, with the liquid level equal to the diameter. Assume that the reactor is equipped with a 45° downflow pitched-blade turbine (PBTD) having standard geometry as follows: D ) 1.67 m, W ) 0.5 m, six blades, C ) D. In this configuration, the values of NP, NQP, and NQS can be taken as 2.1, 1.0, and 1.8, respectively.6 Let us take the case in which the impeller is rotated at 1 rps. If we assume that the liquid properties are water-like, then based on the above data, the following estimates can be made:

Re ) ND2F/µ ) 2.8 × 106

Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2663

P ) NPFN3D5 ) 27.3 kW V ) (π/4)T2HL ) 98.2 m3 ) P/FV ) 0.28 W/kg Based on the primary flow number, the average circulation velocity can also be calculated as

VC )

NQPND3 (π/4)D2

) 2.13 m/s

The mixing time based on the liquid circulation can now be estimated as follows:

θmix ) 5tC ) 5(3T/VC) ) 35.2 s Using the values of power dissipation per unit mass, the Kolmogorov length scale and the micromixing time (θM) can be estimated as (assuming molecular diffusivity to be 1 × 10-9 m2/s):

lκ ) (ν3/)1/4 ) 4.35 × 10-5 m θM ) lκ2/DM ) 1.9 s It is well-known that the characteristic size of the largest eddy in a stirred tank is about 8% of the impeller diameter. The intensity of the turbulence is found to be around 30% of the average velocity, which is proportional to the product of the impeller speed and diameter.7 Accordingly, for the case under consideration, the velocity and the length scale can be calculated to be 0.35 m/s and 0.134 m, respectively. Hence, the eddy diffusivity, which approximately equals the product of velocity and length scale, comes out to be 0.046 m2/s. Then, the mixing time at an eddy level can be estimated as

θT ) lT2/DT ) 0.4 s Thus, the estimated values of mixing times at both the molecular and eddy levels show that the mixing time due to circulation is much larger as compared to the mixing time at the molecular or eddy level, which is to be expected at such a high Reynolds number. This suggests that the convective process may be slower than any other transport processes, which indicates that the process may be controlled by convective motion. 3. Experimental Techniques To Characterize the Mixing Process The mixing process is usually characterized in terms of the mixing time. Experimental determination of the mixing time involves giving a tracer input (usually a pulse input) at some location in the reactor and measuring the tracer concentration as a function of time by means of a suitable sensor. The tracer pulse then distributes throughout the vessel and mixes with the contents of the reactor, and finally a uniform concentration is achieved. The mixing time is defined as the time required to achieve a certain degree of uniformity. The tracer used can be a chemical species (inert or reacting), an electrolyte, or a thermal species. Correspondingly, the measurement technique varies. Over the past several years, different measurement techniques have

beendeveloped,suchas(i)visual,8-14 (ii)conductivity,3,15-19 (iii) thermal,20,21 (iv) electrical impedance/resistance tomography,22-25 (v) laser-induced fluorescence,26 (vi) liquid-crystal thermography,27 and (vii) computer tomography with coherent light.28 Of the above techniques, the simplest is the visual technique involving addition of a dye as a tracer and visual observation of the mixing of the dye throughout the reactor. It can also be assessed via the discoloration of the uniformly colored solution. For example, if the reactor contains a NaOH solution to which phenolphthalein indicator is added, it would appear uniformly pink. To start with, under these conditions, if a pulse of acid is added (slightly more than the equivalent quantity of alkalinity, approximately 10%), then as the acid blends with the contents, it neutralizes the NaOH solution, causing discoloration of the contents in the reactor. However, this technique can seldom be applied on an industrial scale because the vessels are nontransparent and a sight glass cannot cover a wide enough region of the reactor. Further, the mixing time based on the visual observation remains subjective. The conductivity technique employs a probe, which measures the conductance of the solution as a function of time that can be converted to a concentration versus time scale using proper calibration. In this case, the tracer has to be an electrolytic solution, which when mixed would cause a substantial change in the conductivity of the reactor contents. One of the characteristics of this technique is that the probe only measures the local conductivity, so it may result in an inaccurate mixing time if the mixing time indicated by the probe is a function of the probe position. This is especially true if the reactor contains multiple impellers or if dead zones exist. This drawback could be eliminated by making measurements of the mixing time at multiple locations. The effects of a number of factors, namely, the location of probe, size of the probe, and number of probes, need to be taken into consideration in order to have a proper estimation of the mixing time. Further, this technique is not applicable at higher temperatures and in the industrial reactors that process organic materials. The laser-induced fluorescence (LIF) technique (also called the light intersection imaging process) used by Hackl and Wurian27 consists of a laser sheet generator, which is directed toward the reactor. If a tracer consisting of a fluorescent indicator is added, it glows only in the plane of the laser sheet depending on its local concentration. This process can be captured on a camera to assess the mixing characteristics. The mixing time is calculated as the time required for attaining a picture with uniform color throughout. This technique offers the same advantages as those given by the visual techniques. In addition, the mixing process throughout the tank can be monitored clearly as a function of time. However, this technique requires a transparent reactor, and therefore it cannot normally be used on an industrial scale. The electrical impedance tomography technique creates a two-dimensional (2D) map of the resistivity (reciprocal of the conductivity) distribution in the measurement plane. The experimental setup consists of a series of electrodes placed along the periphery in the reactor at a particular plane. An electric current is passed (typically 5 mA) between two adjacent electrodes, and the voltage between the rest of the adjacent

2664


electrodes is measured. The voltage difference measured represents integral information over the conductivity distribution inside a tank. The local distribution of the electrical conductivity from these integral data is then reconstructed topographically with a linearized backprojection process. Thus, the typical behavior of two liquids with different electrical conductivities can be investigated. This procedure is then repeated by passing current between different sets of adjacent electrodes. The values of resistivity can be used to construct a map of resistivity in the measurement plane. The disadvantage of this technique is the high cost of instrumentation. Because the electrodes are arranged along the reactor periphery, the sensitivity is high near the reactor wall and poor at the center. To assess the mixing in the entire reactor, the whole procedure has to be repeated at different levels and requires a significantly large amount of time. Holden et al.25 have extended the 2D electrical resistivity technique and carried out electrical resistance tomography (ERT) that measures the three-dimensional (3D) electrical resistance distribution for the measurement of the dynamic mixing behavior in large-sized vessels. The spatial resolution was on the order of 5%. They have suggested that the spatial resolution can be improved by increasing the number of electrodes. They have also demonstrated the ability of the measurement technique to distinguish between the different flow patterns generated by two types of impellers. For the accurate estimation of the mixing time, the correct reconstruction algorithm should be used. They have illustrated the capability of 3D ERT to monitor plant-scale miscible liquid mixing processes experimentally with good resolution in both space and time. From the ERT data, pointwise mixing curves can be readily constructed which can give a quantitative description of the local mixing rates. Williams et al.23 have used the ERT technique for the mixing measurements in a vessel of volume 30 m3 with a disk turbine (DT) and four equispaced baffles. In this technique, 104 measurements of the voltage were taken in a short time period (∼79 ms). Li and Wie26 have analyzed the 3D ERT for the determination of the centroid of concentration distribution, its dispersion, stretching, and rotation. Further, ERT can fulfill the requirement of finer and faster measurements to discover the detailed work of a stirrer to stretch and rotate the fluid. They have proposed that the ratio of standard deviation over the mean is a more demanding measure of the uniformity than the entropy. Further, the information given by 3D ERT can be analyzed by wavelet transforms to give multiscale analysis of the structure of uniformity, which can in turn be used to determine the state of the mixture, to determine the properties of the mixer used, and ultimately to diagnose the problems encountered in mixers. The three dimensionless arrays can also be analyzed for the estimation of the average concentration, radial gradient between the center and the wall, angular gradient in a counterclockwise direction, and axial gradient. The use of this technique is again limited for lower temperatures. The scaling of the electrical resistance and the calibration for mixing is a function of temperature, and hence its feasibility should be checked at higher temperatures. For the radioactive liquid tracer technique,30 a pulse of a radioactive liquid tracer (e.g., technetium-99m, sodium-24, etc.) is injected and the profile of tracer concentration is monitored. This technique offers unique

advantages. It does not disturb the flow inside the reactor and can be employed for monitoring the mixing in nontransparent reactors. The liquid tracer used is sensitive and does not disintegrate under severe conditions. Measurement of the tracer concentration is possible at higher temperatures on the order of 300 °C, where most of the other techniques fail. These liquid tracers can be detected even at the parts per million level. Further, a very small amount of tracer is required to generate a basic dynamic response in an industrial system. The time of injection is very small (due to a very small quantity), and the addition can be made as a momentary ideal pulse without disturbing the flow at the injection point and without affecting the physicochemical properties of the fluid prevailing in the reactor. Its use is limited by its unavailability, transportation difficulty, and health hazards. The liquid-crystal thermography (LCT) technique is based on the principle that thermochromic liquid crystals (LCs) show a different color when subjected to different temperatures. The thermochromic LCs (typically, 20-µm-sized spheres) are suspended in the liquid. A thermal pulse is given, and the mixing of this thermal pulse imparts different colors to the LCs in different parts of the vessel. This can then be analyzed either visually or by a camera. LCT alleviates the disadvantages of the transducer methods (requirement of a large number of simultaneous measurements, time-consuming, and intrusive). Further, it bears the advantages offered by visual (observation) methods (mixing state at the same time and the quick identification of the dead or stagnation zones). However, it also requires transparent vessels and hence is not useful on an industrial scale. Further, accuracy of this technique is critically dependent on the calibration of the crystals, which in turn strongly depends on the lighting conditions. Interpretation of the LC images is complicated. Complexity is added because of the imperfect color response of the crystal. A careful calibration is needed for the extraction of the appropriate information of temperature from the LC color displays. The calibration needs to be repeated again because of the possibility of contamination and deterioration of the crystals. Further, values of the mixing time are frequently low as compared to the ones reported in the literature as a result of other measurement techniques.28 Computer tomography with a coherent light29 requires radiation from four lines of sight to be able to calculate the concentration field in a tank. An object marked by a dye (e.g., a liquid jet from a nozzle) is irradiated with parallel monochromatic light. A 2D image is obtained. With irradiation in different directions, the object can be reconstructed using a tomographic process in horizontal planes using mathematical techniques. The negatives of the black and white film are scanned and processed by image analysis. These images are then used for the tomographic reconstruction. The results are concentration profiles of mixing streams, from which the mixing time is calculated. Again, because transparent vessel walls are inherently required, it is not possible to measure the mixing time in the case of industrial reactors. It can be seen that many techniques have been used and reported in the literature for the assessment of mixing. Whereas with physical methods the course of the intimate mixing can generally only be monitored at a few points in the tank, with chemical methods the


happenings in the tank are completely recorded. The advantage of chemical methods is that they accurately give the end point of intimate mixing, whereas the accuracy of physical methods does not enable this. This disadvantage can, however, be circumvented in the case of physical methods by not measuring to the point of time of complete homogenization. Further, many of the measuring techniques require transparent walls, which is not possible on an industrial scale. It would be highly desirable to make measurements at different locations in the reactor to identify dead zones. Also, while selecting the conductivity probe size, one must take into consideration the sensitivity of the probe toward the smallest possible scale at which mixing by molecular diffusion occurs. In view of the best mixing criterion to be used as a better indicator of mixing extent, the measurement technique based on the 3D measurement of electrical resistance/impedance seems to be promising. Mixing measurements by such techniques are possible at industrial-scale vessels. This method has been well standardized and applied successfully by many researchers and really promises a good future. Its use is limited by higher cost and the requirement of a certain kind of expertise. Hence, because of unique advantages offered by the radioactive liquid tracer technique, such as feasibility at higher temperatures and applicability in the case of nontransparent vessels, it offers the best alternative measurement technique. However, more research is needed in the standardization of this technique, safer handling, and transport practices. To summarize, a good measurement technique should not offer obstruction/disturbance to flow; it should have good reproducibility, it should have stability in severe operating conditions, it should be applicable in the case of nontransparent vessels, and finally it should be cheaper in cost. 4. Critical Analysis of the Effects of Various Parameters on Mixing Various studies from the published literature pertaining to the liquid phase mixing have been critically analyzed. The effects of various parameters, namely, the impeller design, impeller diameter, tank diameter, impeller off-bottom clearance, impeller eccentricity, baffles, and presence of a draft tube, on the mixing time as observed by various researchers are discussed in this section. Table 1 lists the mixing studies along with the details regarding the impeller designs, vessel configuration, measurement technique, and mixing criteria used. 4.1. Impeller Design. Kramers et al.15 have studied the mixing due to upflow and downflow propellers. They have observed that the mixing time is practically independent of the mode of operation (that is, upflow or downflow). They have also compared the mixing time for a propeller and DT. They have compared the impeller designs on the basis of the number of impeller revolutions to achieve a desired degree of uniformity. It was observed that under otherwise similar conditions (impeller speed, diameter, location, etc.), the value of Nθmix is lower for a DT by up to 20% as compared to that for a propeller. However, if the power numbers are considered, then it can be seen that the power number of a DT is almost 10 times that of a propeller. In other words, when compared on the basis of equal power consumption, the speed of rotation for a propeller would 3 have to be 2.15 (x10) times that of the DT. So, at equal

power consumption, the mixing time for a DT would be 1.72 times that of a propeller. Thus, it can be seen that a propeller is more energy efficient when compared to the DT. Prochazka and Landau31 have presented the diffusion model for the course of homogenization and investigated the effect of the impeller design on the mixing time. They have measured the mixing times for three impellers, namely, a three-bladed square-pitch marine propeller, a four-bladed PBTD45, and a six-bladed DT. They have shown that, for given values of the Reynolds number and D/T ratios, the value of Nθmix is the smallest for a DT. In these cases, the diameters of the different impellers have not been kept constant and the power numbers have not been reported. One can conclude from the correlation given for dimensionless mixing time that for the same D/T ()0.33) the DT has the shortest dimensionless mixing time, which is half that of the propeller and 66% that of the PBTD. Their correlation considered the dependence of the dimensionless mixing time on the D/T ratio and the degree of homogeneity only, and its validity can be expected for similar impeller geometries or designs only. In the present work, an attempt has been made to plot the mixing time versus P/V (Figure 2), assuming appropriate values of the power numbers.32 Results of this plot are listed in Table 2 along with the results of plots obtained for other studies. It can be seen that the dimensionless mixing time varies inversely with the cube root of the power consumed per unit volume of the liquid. The constant of the proportionality was found to be the highest (7.9) in the case of the DT and the lowest (4.8) for the propeller, which again indicates that the propeller is relatively more energy-efficient when compared to the DT. The conclusion is consistent with the observation by Kramers et al.15 Biggs16 has investigated the effect of the impeller design on the mixing time. He has shown that, at a given impeller speed, the mixing times for a standard DT, straight-blade turbine, and a PBT are practically the same, but the mixing time for a standard squarepitch propeller is almost twice that of the mixing time obtained for other impellers. However, the data were plotted as mixing time versus P/V, where for equal power consumption the mixing time for the PBT is the lowest and that for the DT is the highest, as could be seen (Table 2) from the values of the proportionality constants (∼12.4 for PBTD and 15.8 for DT). The DT and the flat-blade turbine (FBT) show comparable mixing times. Thus, DT could be seen as the least efficient of all of the designs of the impellers considered. Moo-Young et al.11 have investigated the whole range of impeller Reynolds number (5-105) and the viscosity range of 1-70000 cP (laminar, transition, as well as turbulent). The efficiencies of the impellers, namely, DT, helical ribbon, and a variety of novel tubular agitators, were characterized in terms of the energy required (Pθmix) to achieve a certain degree of homogeneity. They have reported that there is no significant difference between the efficiencies of all of the impellers in turbulent conditions. They have not carried out the mixing time studies for commonly used impellers, namely, a propeller and pitched-blade type of impellers, and hence their conclusions have a limited scope. Brennan and Lehrer12 have made the observation that an increase in the blade width to impeller diameter ratio from 0.125 to 0.2 leads to a decrease in θmix by

Fox and Gex8

DT

Nishikawa et al.13

Brennan and Lehrer12

I II III

FBT

H)T

DT

C (m)

0.127, 0.244 0.114, 0.254 0.366, 0.488 0.114, 0.254

T/3 T/2 T/2 0.47T 0.5T 0.1

0.18, 0.24, 0.30 0.24, 0.3, 0.44

0.07 0.055 0.33T (0.0625-0.1)

0.05, 0.1, and 0.15 0.11

0.025-0.55

T/4

D (m)

4 4 4 4

0, 4

0, 4

4

4 4 4

4

4

0, 4 0, 4 4

0, 4

nb

0.16 0.12 0.15

0.2D

0.25D

0.2D

6

6

3 3 6 6 3 6 3

0.087D 3

6 0.75T 0.25D

6

4 3 6 6 6 3

6

6 6 3

3

LDFT np (m) W (m)

0, 2, 4 4 4 6

0.15, 0.18, 0.24, 0.24T and 0.24 < C/T < 0.73 flat bottom 0.15T, 0.1T, 0.076T, 4, 0 0.0635T, 0.102T 0.35 < C/T < 0.65 4, 0 dished bottom

p/D ) 1.5

P ) 1.25D P ) 1.01D 0.33T

D

0.5T

T/2 T/2 D

T/2

0.354T

0.36T

0.5T, 0.25T,

H ) 1.11T flat and dished bottom 0.4, 0.3 H)T 0.3 0.5T 0.4 0.5T 0.4 0.5T

0.42

FBT

1.2190

0.5590

0.24 1.8 HL ) T T T 0.3

0.22 0.6, 1.0

0.33 0.21 0.245

0.55

DT

4PBTD 3BMP DT SBT PBT propeller

0.15-0.4

FBT

Hoogendoorn and 6DT den Hartog21 3-PBT45 1,3-PBT45 3BMP 3BMP Moo-Young FBT 11 et al. tubular Khang and DT Levenspiel33 propeller DT propeller

Holmes et al.3

Biggs16

Landau31

Norwood and Metzner10 Prochazka and

0.64

DT 30° ADT propeller

T (m)

0.15-4.2 H ) 0.2T-25T

0.32

3BMP

impeller

Kramers et al.15

authors

BW (m)

0.1 0.1 0.1

0.125, 0.2

20:5:4

0.2D

20:5:4

0.125

0.2 0.2165 square pitch

0.1T

3.9/42

3.9/42

0.51T 0.51T 0.1T

0.1T

0.025 (0.0125 away from the vessel bottom)

0.08T

0.1T

clearance of 0.1T

square pitch 0.1T with wall

W/D

Table 1. Mixing Studies: Impeller-Vessel Geometry Investigated and the Measurement Techniques Used

000

conductivity technique volume of trace volume ) 25 cm3 for large tank and 5 cm3 for a smaller one; injection time ) 0.5 and 0.25 s for smaller and bigger tanks, respectively visual; decoloration technique; volume of tracer added ) 25 mL; unbaffled vessel probe diameter ) 25 mm, 75 mm away from the wall; time of tracer addition ) 0.5 s; methyl red (75 mL), red to yellow 104-3 × 105

conductivity >104

homogenization number, 1-100 cm3 95%, conductivity; proposed that the continuous tank data can be assumed equivalent to the batch mixing time data because the residence time is much much more than the θmix (terminal mixing time); correction was applied to take into consideration the time required for the tracer to flow into the tank micromolen instrument was used for the velocity measurements: 15-mm-diameter miniaturized propeller conductivity + thermal + color addition + decoloration; 90% mixing criterion

10-80 cm3 with c ) 100 mg/cm3 KCl visual, phenolphthalein, NaOH; disappearance of red color, last whisp of color was isolated and located at different positions for different momentum fluxes acid-base neutralization technique conductivity, 2-25% NaCl, 95%,

KCl, 99.9%, injection liquid

measurement technique

5-1 × 105

>170

>2 × 104

104-90

(1.6-1.8) × 105 (1.92-13.2) × 104

3.33-37 rps

45 000200 000

Re

e/D ) 0, 0.33, 1.33-5.33 rps acid-alkali decoloration 0.16, 0.5 with phenolphthalein as an indicator, HCL:NaOH ) 1.2:1

--

DDFT (m)

2666 Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003

PBU PBT I PBT II PBT III

Sano and PBT Usui18 Raghav Rao and DT Joshi19 PBT

Sano and Usui17

PBT DT

FBT 0.40 CBT hemispherical CBT bottom VEGYTERV PBT PBT-DFT PBT PBT-DFT 6DT 0.29

Shiue and Wong22

Rielly and Britter51

PBT DT

Tatterson49

0.325, 0.5 0.500 0.500 0.500 0.500 0.500 0.500 0.500 C/T ) 0.33

0.33T

0.5T

0.33T

C (m)

0.57, 1.0 Row 70 H ) T 0.167T

(3.33, 2.5, 2, 1.66, 0.3T 1.45, 1.4)D 0.33T, 0.25T,

H)T 0.5T (3.33, 2.5, 2, 1.66, 1.45, 1.4)D

0.90

0.3 H)T

FBT

Sasakura et al.34

0.312 H)T

T (m)

6DT 6FBT 6PBT45

impeller

Ogawa et al.35

authors

Table 1. (Continued)

4

4 4 4 4 4 4 4 4 6

4 4

8

4

nb

2, 4, 6, 8

6 6 4 3 4 4 2 2 4

2, 4, 8

6 6 6

np

Row 70 0.19 0.19 0.19 0.19

0.2 and 2, 4, 2-6 0.4 6, 8, 12 0.19 4 6 0.1425, 0.19, 0.25, 0.33

0.2 and 0.4

0.13-0.18 0.13 0.13 0.13 0.13 0.13 0.13 0.13 D/T ) 0.33

0.304 0.304

0.08, 0.15, 0.2, 0.24

0.33T

D (m)

LDFT (m)

0.200 0.269 0.154 0.231 0.231 0.231 0.231 0.231 0.25D 0.2D

0.1D, 0.2D, 066D, 0.36D, 0.15D, 0.125D

0.25 0.2 0.2

W/D

0.19 0.19 0.19

0.25D

0.2D 0.03, 0.04,0.053, 0.07 0.04 Row 70 0.075 0.075 0.075

0.1, 0.15, 0.05, 0.1, 0.2, 0.3, 0.15, 0.2, 0.4 0.3

W (m)

0.025T, 0.05T, 0.1T 0.1T

0.1T

0.04

0.091

0.1T

BW (m)

0.61, 0.457 Hd ) 0.457 m

DDFT (m)

conductivity, KCl; tracer injection in the region near the impeller type the vessel was divided into 20 zones; have made use of entropy-based concept for the determination of the degree of mixing electric conductivity, 8 locations, NaCl, 99%


2-3 rps (7000048000)

>100

>5 × 105

104-6.3 × 104

10% of the final concentration difference; tracer was injected at the surface; they have used two sizes of release devices (20 and 45 mm i.d.); tracer volume was varied from 5 to 120 cm3 conductivity measurement with two electrodes placed at opposite corners upstream of baffle; degree of relative deviation ) 1% for the estimation of θmix (15% reproducibility of θmix same as that of Sano and Usui17 conductivity; mixing time was defined as the time after which values of conductivity are nearly constant; no effect of the volume of the tracer pulse for VT ) 1-2% of the tank volume; no effect of the electrode location

visual; radioactive pill technique for circulation time and bromothymol blue, acid, and base neutralization technique for θmix cone bottom: 0.33 m, at bottom, and 0.165 m height 4 200 < Re < 10 thermal method, 4 Re > 10 two thermocouples; homogenization number, 95% of the total temperature difference

2.5×1041.5 × 105

turbulent regine; 1.67, 3.33, 5 and 6.67 rps

Re


PBTD PBTU DT DT PBT45 MP 30° PBT 45° PBT 60° PBT 45° PBT 45° PBT 45° PBT DT propeller A310 4-PBT45 4-PBT30 6-DT RTF/RTX-4 MIL-6 4-WMNAE

impeller

Distelhoff et al.41

DT CBT 45PBTD 60PBTD hyperboloid

Rutherford et al.40 DT

Shaw39

Ruszkowski37

Rzyski36

Rewatkar and Joshi20

authors

Table 1. (Continued) C (m)

D (m)

nb

0.57, 1.0, 1.5 T/8 to 3T/4 0.167T to 0.5T 4 in 1.5 m tank 4 4 0.6 0.33T 0.19-0.6 4, 0 H)T 4, 0 P ) D, 2D 4, 0 0.609 T/3 0.201 4 0.203 4 0.198 4 0.300 4 0.303 4 0.303 4 0.310 4 0.201 4 0.48 0.8D 0.15 H ) 0.8T 0.15 0.15 0.15 0.15 0.15 0.15, 0.175, 0.2, 0.225 0.1, 0.294 0.33T 0.3T, 0.4T 4 H)T 0.147, T/3 T/3 4 H)T

T (m)

6 6

6

6

6 6 6 6 4 3 4 4 4 4 6 4 6 3

np

0.082 0.058 0.047 0.058 0.058 0.086 0.041

LDFT (m) W (m)

thickness ) 1.5 mm

BW (m)

0.1T, baffle thickness ) 0.01T

0.1T

T/12 at 10 mm from wall

0.1T at 0.05T from wall

0.25D to 0.4D in T 0.57 m tank

W/D DDFT (m)


12 000-48 000

40000

102-103

LIF technique; fluorescein dye was used; 99% approach to complete mixing

conductivity, 95%

decoloration, alkaline solution of phenolphthalein in ethanol

visual, decoloration same as that of Rzyski (1985) conductivity, variance technique; 95% criterion

N ) 0.4-9.0 rps conductivity, 95% criterion

Re

2668 Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003


Figure 2. Plot of mixing time versus power consumption per unit volume of fluid for a propeller as observed by Prochazka and Landau.31

25% at the cost of an increase in power consumption by 60%. At a higher blade width, the discharge flow rate was found to increase. Narrower blades have a distinct advantage (as the blade width increases, the power increases, but the corresponding increase in the mixing time was marginal). They have observed that the same reduction in θmix could be achieved by increasing D/T from 0.24 to 0.28 with an increase in the power requirement by 100%. On the basis of the power requirement, it is useful to use a wider blade than a larger impeller diameter. These investigations were limited to a single impeller design. Similar studies for different impeller designs need to be carried out. Khang and Levenspiel33 have carried out mixing experiments using a DT and a propeller in a fully baffled tank. They have tried to verify the theory that the mixing rate of a tracer into a batch mixer can be characterized by a single quantity called a decay rate constant. This decay rate constant was then related to the mixing rate number. They have used this relation to derive a correlation to relate the mixing rate number with the Reynolds’ number and power number. They have developed correlations to estimate the time needed and the energy required for achieving a given degree of homogeneity. They have given mixing time correlations for the DT (for Re > 2000) and the propeller (for Re > 104) in terms of the mixing rate number. When compared on the basis of mixing time versus P/V (Table 2), the proportionality constants were found to be approximately the same, indicating the same mixing time for the same power consumption. Sasakura et al.34 have extensively carried out experimentation with flat-blade impellers of a variety of impeller diameters, different numbers of blades, and blade widths. They have observed that as the number of blades is increased the power number of the impeller increases and the mixing time decreases. Similarly, the mixing time was found to decrease as a result of an increase in the blade width. The reduction in the mixing time can be attributed to the increased power number and the corresponding higher power consumption. The data of the mixing time versus P/V were plotted using the given correlation, and the result is indicated in Table 2. It can be seen (Table 2) that the mixing time varies as -0.33 power to P/V. Ogawa et al.35 have compared the performance of a DT, a FBT, and a PBT. They have shown that an exponential relationship exists between the mixing time and the degree of homogeneity, irrespective of the type of impeller. The impellers differed only in the constants

involved in the exponential term. For a given degree of homogeneity, a FBT and a DT show similar mixing times, whereas a PBT shows a larger mixing time under otherwise similar operating conditions. When compared on an equal power consumption basis, the PBT was found to be the most energy efficient. For equal power consumption, the DT and FBT showed approximately 27% and 13% greater mixing time, respectively, as compared to the mixing time obtained for a PBTD (Table 2). Shiue and Wong22 have compared the performance of various designs of impellers in a fully baffled, dished bottom tank in terms of the parameter Pθmix2/µT3, which essentially is the energy required to achieve a certain degree of mixing.21 They found that the power and homogenization numbers are independent of the Reynolds number. They have reported that four PBTs fitted in a draft tube are the most efficient. The order of efficiency of the various impeller designs was found to be as follows: four PBTs in a draft tube > two PBTs in a draft tube > four PBTs > two PBTs > propeller > curved-blade turbine > FBT. They found that the six FBTs and six curved-blade turbine mixers are the least energy efficient, which could be seen as being in good agreement with the results of Prochazka and Landau.31 When compared on an equal power consumption basis (Table 2), it can be easily concluded that axial flow impellers show higher energy efficiency as compared to the radial flow impellers. Further, it can also be concluded that an increase in the number of blades of the impeller results in an increase in the mixing efficiency. Sano and Usui17,18 have shown that the nondimensional mixing time, the power number, and the discharge flow number in the turbulent regime are independent of the Reynolds number. They have observed that Nθmix is proportional to np-0.47 (np ) number of blades) for both the DTs and paddles. They have presented an equation for the mixing time for the case of a flat-blade impeller. They have recommended an impeller with a larger diameter and blade widths and a greater number of blades operated at low rotational speed for energy-efficient operation. This means that mixing by an impeller with a large discharge number at a given power input is preferable. This is true in view of the fact that a proper blade profile or design can avoid the excessive wastage of energy due to the shear action between the impeller and fluid. They have shown that the paddle impeller is slightly superior to the DT. Sano and Usui17 have proposed a relation between the number of circulations of the discharged flow to reach the mixing time and the dimensions of the impeller and vessel. The mixing time was found to be directly proportional to the circulation time. Their correlation for mixing time includes the impeller and vessel diameters and the number of blades of the impeller. When compared on the basis of equal power consumption, it can be seen (Table 2) that the mixing time taken by FBT is less by approximately 34% as compared to that required for DT. The power number was found to bear a proportionality relationship of the kind NP ∝ NQ1.34. Finally, they have suggested that an impeller with large diameter, blade height, and higher number of blades is good for the purpose of the mixing within the experimental range investigated [0.3 < D/T < 0.7, 0.05 < W/D < 0.3, and number of paddles (2-8) under the baffled conditions]. It can be seen from Table 2 that at equal

2670


Table 2. Listing of Constants and the Exponents in the Relation for Mixing Time of the Form θMix ) Constant × (P/ V)exponent for the Mixing Studies Reported authors

impeller

Prochazka and Landau31 Hoogendoorn and Den Hartog21 Moo-Young et al.11 Biggs14 Khang and Levenspiel33 Hiraoka and Ito44 Sasakura et al.34 Ogawa et al.35 Shiue and Wong22

Sano and Usui17 Sano and Usui18 Raghav Rao45

C/T ) 0.33 C/T ) 0.25 C/T ) 0.167


T ) 0.5 m T ) 1.0 m T ) 1.5 m

Rewatkar and Joshi20 (overall correlation) Rzyski36 Nienow42

power consumption per unit volume of liquid, the mixing time in the case of the FBT reduces by 28% and 41% if the number of blades is increased from 2 to 4 and from 2 to 6, respectively. Thus, these studies reveal the strong dependence of the impeller design (number of blades and blade dimensions) on the mixing time. Raghav Rao and Joshi19 have compared the performance of various impellers, namely, DT, downflow and upflow PBTs, and modified PBTs. They have observed that the mixing time for a given power consumption per unit volume increases gradually with a decrease in the blade width for a pitched-blade downflow impeller. Further results of the plot of the mixing time versus the power consumption per unit volume show that the mixing times for an upflow PBT are the lowest among all of the impellers investigated (Table 2). Rewatkar and Joshi20 have studied three different types of impellers, namely, DT, PBTD, and upflow PBT (PBTU). They have shown that when compared on an equal power consumption per unit mass basis, the PBTD shows the smallest mixing time. A PBTU impeller shows a marginally higher (5-10%) mixing time; a

constant

exponent

DT FBT propeller DT DT DT FBT PBTD45 DT propeller CBT propeller DT DT FBT PBTD45 DT-1 DT-2 DT-3 DT-4 CBT 3VEG•PROP 4-PBTDFT 2-PBTD 2-PBTDFT FBT DT FBT2P4B FBT4P4B FBT6P2B DT PBTU45 PBTD45 DT PBTU45 PBTD45 DT PBTU45 PBTD45 PBTD45 PBTD45 PBTD45 PBTD45

7.9 6.9 4.8 18.1 26.7 15.8 15.3 12.4 0.1 0.1 84.5 330.8 2.3 10.4 9.2 8.2 111.6 96.0 46.2 81.0 62.8 39.1 50.4 25.1 45.9 43.5 58.7 64.8 46.8 38.6 86.1 74.2 87.0 134.7 117.8 144.2 91.3 86.2 83.3 52.3 88.0 100.1 46.3

-0.333 -0.333 -0.333 -0.54 -0.33 -0.28 -0.28 -0.28 -0.33 -0.33 -0.33 -0.167 -0.33 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.333 -0.33 -0.33 -0.33 -0.33 -0.33 -0.31 -0.30 -0.43 -0.44 -0.45 -0.53 -0.39 -0.40 -0.40 -0.31 -0.33 -0.30 -0.33

PBTD45 Propeller DT all types of impellers

103.0 91.0 175.0 26.5

-0.33 -0.33 -0.33 -0.33

DT shows considerably larger mixing time (about 50%) as compared to that shown by PBTD. They have proposed a correlation for the mixing time for the case of PBTD based on the data on the mixing time reported to date as a function of the various variables. They have observed that the mixing time strongly depends on the flow pattern generated by the impeller. The authors have investigated the effects of blade angle, blade width, blade number, D/T ratio, etc., at the same power consumption. The mixing time was found to decrease with an increase in the blade angle. The PBTD45 impeller was found to be the most energy-efficient. In regards to blade width, the mixing time was found to decrease with an increase in the W/D ratio up to 0.35. However, a further increase in the W/D ratio was found to result in increased mixing time. This study clearly brings out the importance of impeller design features (diameter, blade width and angle, etc.) on the energy efficiency of mixing. Rzyski36 has presented a very simple model of liquid homogenization in a stirred tank based on the impeller pumping capacity and the degree of homogeneity. The


model was proposed for turbulent mixing conditions and was extended to the transition regime (Re ) 100-1000). He has performed the experimental studies pertaining to the effect of various parameters such as the D/T ratio, displacement of the impeller from the shaft (eccentricity), etc., for three types of impellers, namely, a sixbladed DT, propeller (with two different pitches), and six-bladed fan-type turbine, in both the baffled and unbaffled vessels in the transition regime. Power numbers were not estimated. He found that at equal power consumption the propeller shows minimum mixing time in the turbulent regime. It can be seen from the results of mixing time versus power consumption per unit volume (Table 2) that axial flow impellers show similar mixing performances while the mixing time with a DT is as much as twice that with a propeller. Ruszkowski37 has investigated the effect of impeller design on the mixing time. Impellers investigated were PBTs with varying numbers of blades and blade angles in addition to the standard six-bladed DT and a propeller. All of the impellers were found to yield equal mixing times irrespective of their designs at equal power consumption. Hence, he concluded that the mixing time is largely determined by the power input into the tank, irrespective of the impeller type. For a given impeller type, larger-diameter impellers were found to be more energy-efficient. Thus, he suggested large-diameter and low-power-number impellers for the most energy-efficient and cost-effective turbulent mixing operations. The data of the dimensionless mixing time were plotted versus the power consumption per unit volume, and the best exponential fit was imposed. It was found that the best fit had a lower regression coefficient (0.85). Thus, the conclusion that the mixing time is a function of only the power consumption seems to be inadequate. His conclusion that larger-diameter impellers are more energy-efficient indicates that an increase in the discharge flow due to an increase in the impeller diameter may be responsible for enhanced convective flow and lower mixing time. Thus, the extent of an increase in discharge at a given power consumption and subsequent lowering of the mixing time may depend on the impeller design. This is true in view of the significant effect of the impeller design on the flow number. Novak and Rieger38 have investigated mixing phenomena in unbaffled vessels. They have made use of three impellers, namely, three- and six-bladed PBTs and a six-bladed DT. They have compared the configurations on the basis of the total energy needed for homogenization. It was observed that the PBTs are more advantageous in unbaffled vessels. In addition, in unbaffled vessels, impellers with a greater number of blades are more efficient. Shaw39 has measured mixing times for eight markedly different impeller designs, namely, six-bladed DT, four-bladed PBT45 and PBT30, hydrofoil (A310), RTF-4 (four tapered cambered twisted-blade impeller), RTX-4 (four tapered low-pitched-blade impeller), MIL-6 (sixpointed deep cambered bladed impeller) and 4-WMNAF (Warman-type axial foil: three tapered cambered and twisted-blade impeller). The impellers were assessed at various power consumptions. He observed that the mixing time was the same for all of the designs investigated when compared on an equal power consumption basis. He has concluded that at low power consumptions (2-8 W/m3), the impeller type has no effect on the mixing time and the flow number does not

determine the mixing time. However, the flow was found to vary as the impeller design was changed to have the same mixing time. Hence, he speculated that the flow numbers might be different for different impeller designs to have the same mixing time. A careful analysis of the mixing time data shows that the variation in the mixing time was found to vary from 1.61% to 12.37% from the base case of the DT. Further, the largest deviation from the mean was observed to be 7%. Further, the error in the mixing time measurement is inherent in view of the discoloration technique used. Thus, the conclusions have inherent limitations and are essentially applicable with restrictions for low power consumption levels. Rutherford et al.40 have investigated the effect of the blade and disk thicknesses (tk) on the flow and power numbers. They found that the flow number, power number, fluctuating velocity, and average velocity were higher for thinner impellers. Hence, they observed lower mixing times for thinner impellers. They have observed a decrease of 16% in the mixing time when the thickness ratio (tk/D) was reduced from 0.0337 to 0.0082. This was attributed to the corresponding increase in the flow number (by almost 15%) and the consequent increase in the power consumption (∼33%). Thus, a tradeoff has been pointed out between the power reduction (which was related to lowering of the flow number) and the resulting increase in the mixing time. With comparison on an equal power consumption basis, the mixing time was found to lower by almost 13% for the thinner impeller as compared to the thicker one. Their study was limited to a single type of impeller (i.e., radial flow impeller). Such studies should be carried out for different types of impeller designs. These would prove to be useful if done altogether. Distelhoff et al.41 have extensively investigated the mixing time characteristics along with the flow and power characteristics for a variety of impellers, namely, PBTD, DT, curved-blade DT, and a hyperboloid agitator. In addition, the effect of the blade angle in the case of PBTs was also investigated. The results are tabulated in Table 3. It can be seen that the mixing time for the PBTD is less than that for the radial flow impeller. This was attributed to the higher circumferential bulk flow velocity that results in reducing the uneven concentration distribution in the tangential direction. Further, it can be seen that, at equal power consumption, the mixing times for both PBT45 and PBT60 are similar. Similar observations in regards to the mixing time were made for DT and curved-blade impellers. However, the mixing times for axial flow impellers were found to be shorter by about 1-2 s (17-34%) as compared to the radial flow agitators at equal power consumption. This study indicates the significant effect of the impeller design on the mixing time. It can further be noticed that, for the same flow numbers, the mixing times are different under otherwise similar operating conditions. Nienow42 has comprehensively discussed both the bulk flow and the turbulence-based mixing models. He has presented a mixing time correlation based on the analysis of previous studies for a variety of impeller types and sizes (over a wide range of dissipation rates) in terms of the energy dissipation rate and the relative impeller size. He concluded that the efficiencies of all of the impellers of relatively equal size are the same and depend only on the power input per unit volume of the fluid. He proposed that, if the mixing process is

2672


Table 3. Effect of the Impeller Design and Impeller and Tank Diameters on the Mixing Time author al.41

Distelhoff et (Re ) 24 000 at N ) 10 s-1) Raghav Rao45


impeller design

NP

D (m)

T (m)

D/T

P/M (W/kg)

θmix (s)

PBTD45 PBTD60 DT CBT hyperboloid PBTD

1.8 2.3 4.5 3.2 0.25 1.74 1.29 1.52 1.90

0.049 0.049 0.049 0.049 0.049 0.14 0.19 0.25 0.33 0.19 0.19 0.19 0.19 0.19 0.19 0.33 0.33 0.33 0.33 0.33 0.33 0.50 0.50 0.50 0.50 0.50 0.50

0.147 0.147 0.147 0.147 0.147 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.50 1.50 1.50 1.50

0.33 0.33 0.33 0.33 0.33 0.25 0.33 0.44 0.58 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 0.26 0.51 0.36 0.03 0.31 0.97 4.50 22.49 0.10 0.20 0.30 0.40 0.70 1.00 0.10 0.20 0.30 0.40 0.70 1.00 0.10 0.20 0.30 0.40 0.70 1.00

5.2 4.9 5.7 6.2 17.0 20.0 8.9 5.9 4.3 12.8 10.0 8.9 8.1 6.8 6.3 20.0 15.0 13.1 11.8 10.0 9.4 25.2 20.0 17.5 15.9 14.1

PBTD

controlled by convection, then the mixing time can be estimated from the liquid volume in the reactor and the volumetric flow rate generated by the impeller. On the basis of this assumption, he derived the relationship for dimensionless mixing time as

Nθmix ) A1NQP-1(T/D)3

(1A)

He assumed that the impeller pumping capacity is independent of the D/T ratio and concluded that Nmix would be proportional to (T/D)3. On the other hand, for the mixing process controlled by the rate of energy dissipation in the regions of slowest mixing, he arrived at the expression

Nθmix ) A2NP-1/3(T/D)2

(1B)

which can be alternatively expressed in terms of power per unit mass of liquid (P/M) as

θmix ) A3(P/M)-1/3(T/D)-1/3

(1C)

Relation (1B) shows that Nθmix would be proportional to (T/D)2 if the mixing process were controlled by energy dissipation in the regions of slowest mixing. On the basis of the reported dependence of the mixing time on the D/T ratio [i.e., Nθmix(D/T)1.8 to 2.6], he concluded that the mixing process can be well expressed by relationship (1B) and hence in turn is controlled by the energy dissipation rate in the regions of poor mixing and not by the convective flow generated by an impeller because it gave altogether a different dependence on T/D (3 instead of 2). However, this conclusion is based on a crucial assumption that the impeller pumping capacity is independent of the D/T ratio. Measurements of the primary pumping capacity of impeller, with the help of laser Doppler velocimetry, at different D/T ratios for a standard PBTD impeller carried out by Patwardhan and Joshi6 show that the values of the primary flow number (NQP) vary with the (T/D)0.6 ratio. If this dependence is incorporated in eq 1C, then the exponent on the term

T/D becomes closer to 2.4. Further, a relationship between the secondary flow number of the impeller, as observed by Patwardhan and Joshi,6 indicates that the secondary flow number (NQS) decreases with a reduction in the T/D ratio. The values of NQS were found to vary as (T/D)0.66. If this relation is incorporated, relation (1A) would reduce the dependence on the T/D ratio further to give the dependence of about (T/D)2.34. Thus, the exponent of the T/D term in relation (1A) appears to be fairly close to the value seen in relation (1B). Thus, both the model concepts based on the circulation and turbulence finally give similar kinds of mixing time dependences on T/D. Hence, a clear conclusion cannot be drawn based on such a dependence in regards to the controlling step for the mixing process. Further, the derivation of an equation based on the turbulence theory [relation (1B)] involved the assumption of singular dependence of the mixing time on the turbulence length scale. This turbulent length scale was assumed to be proportional to the integral length scale in the bulk of a tank on the basis that the mixing is governed by the flow in the bulk where the mixing is slow. This integral length scale was proposed to scale in accordance with the tank diameter and not the impeller diameter. Further, it assumes the independence of the energy dissipation rate distribution in the stirred vessel irrespective of the impeller design, which does not follow the experimental results available in the published literature. This assumption needs careful analysis because the turbulence pattern changes significantly as the impeller design is changed. Consequently, the assumption of constancy of the ratio of the minimum to average energy dissipation rate irrespective of the impeller design in a stirred tank stands incorrect. He has also plotted data on a mixing time versus P/M for various impellers and has shown that they fall on a line that has a slope of -1/3 [according to relation (1C)]. However, there appears to be a considerable scatter of experimental points, in the range of 30-40%. Such a wide difference needs attention and reexamination. Fentiman et al.43 have developed a new profiled hydrofoil impeller. They have studied the power and


Figure 3. Work sheet for the selection of an impeller (Zlokarnik29).

mixing time characteristics of this impeller along with the FBT and the curved-blade impeller. They have shown on the basis of the pumping/circulation efficiency that the new impeller promises to be more energyefficient when compared with those currently used in the industry. They have carried out complete characterization of the profiled blade impeller using extensive LDA measurements and power and mixing time measurements. The primary flow number was found to be 0.52, while the secondary flow number was estimated to be 1.0. It was observed that the dimensionless mixing time (Nθmix ) 48) was comparable to previously reported values for designs such as propellers, curved-blade turbines, etc. The mixing time was also correlated well by the correlations of Prochazka and Landau31 and Shiue and Wong.22 The power number for this impeller was observed to be 0.22, which is substantially smaller than that for propellers (0.4-0.6), PBTs (1.6-2.1), and Rushton turbines (5-5.5). Hence, the mixing efficiency of this novel impeller was expected to be higher. Thus, their study indicates that the mixing time can be lowered by a change in the impeller design so as to have higher discharge flow at lower power consumption. This study indicates the dependence of the mixing time and energy efficiency on the impeller design. This also indicates that improvement is possible as a result of a change in the impeller design. Zlokarnik29 has extensively discussed the homogenization in stirred tanks and suggested criteria for industrial scale-up. He stated that the scale-up criterion based on the equality of the dimensionless mixing time (Nθmix) is doubtful and proposed the use of two dimensionless numbers for the selection of the impeller for mixing. Furthermore, he has prepared a work sheet that indicates the relationship between the two dimensionless numbers [π1 ) PTF3/µ3 and π2 ) θmixµ/D2F(T/D)2] as a function of the Reynolds number for a variety of impeller-vessel configurations (Figure 3). He has suggested the use of a propeller in a baffled vessel in the turbulent regime. The impeller diameter should be kept equal to 0.33T, with the optimum impeller off-bottom clearance being 0.5T. For π2 . 0.1, π3 [)(P/T3)(θ2/µ)] is no longer constant for the stirrer types but decreases rapidly before increasing again from π2 ) 100. This means that P/V is not suitable as a scale-up criterion for homogenization operation in the turbulent regime. From the value of π2, for a given tank and impeller diameter, required mixing time, and given material properties, the corresponding value of π1 is obtained

from the plot of π1 versus π2 (Figure 3). Accordingly, the stirrer power is calculated and the Reynolds number is estimated, which gives the value of the corresponding impeller speed. Mixing optimization is based on the minimization of the work required, which is assumed to be a function of power consumption, tank diameter, molecular viscosity, and mass density. He has observed that the power characteristics for the side-entering propellers were about 33% lower than those installed vertically. Even though the studies presented in his book are extensive, they lack the following limitations: No clear-cut guidelines are given for the selection of an energy-efficient impeller in turbulent liquid-phase mixing. The work sheet presented does not include some commonly used impellers (PBTDs in particular). The limitations of measuring techniques and the definition of the mixing time are not pointed out clearly. In summary, various studies have been carried out to quantify the efficiency of a variety of impeller designs. The major emphasis has been on the most common type of impellers, namely, PBTs, FBTs, and marine-blade propellers. One can easily conclude that a unique relationship exists between the mixing time and power consumption per unit volume and is given by the following equation: θmix ) a(P/V)b. The exponent on the P/V term and the values of the proportionality constants for a variety of impellers have been summarized in Table 2. The value of b can be seen to be -1/3 in practically all of the cases provided that the condition of the turbulent regime is satisfied. The same relation appears to be valid for larger vessels as well as a large number of impeller designs. The only difference is in the values of constant a and needs to be evaluated for different impeller-vessel combinations. The dependence of the mixing time for a given power consumption on the impeller and vessel diameters can be seen as altogether different for different types and designs of the impeller. For instance, for the FBTs, an increase in the number of blades results in an increase in the mixing energy efficiency. The blade angle also plays an important role in the flow pattern and mixing. As the blade angle increases, the mean velocity components in the tangential and radial directions increase, leading to vigorous circulation and faster mixing at higher power consumption. Impellers, which are capable of producing top to bottom circulation at relatively low power, are the most efficient impellers. Hence, in general axial flow impellers are the most energyefficient when compared to radial flow impellers. In particular, the propeller has been found to be relatively more energy-efficient, while the DTs are the least energy-efficient as far as turbulent liquid-phase mixing is concerned. There are different designs of hydrofoil impellers reported in the literature. They have been characterized for their pumping efficiency. The mixing characteristics of such impellers have hardly been reported in published literature. Hence, extensive studies should be carried out in the case of hydrofoil impellers. Minor dimensions such as the blade and disk thicknesses have been found to have an influence on the flow, power, and mixing characteristics, and in some cases modification of these dimensions may prove relatively more beneficial as compared to major dimensions such as the blade length and blade width. Studies in regards to this are limited for a single kind of impeller and need to be extended to a wide variety of impeller types with a specific emphasis on the hydrofoil impel-

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lers. While drawing these conclusions, one must take into consideration the range of values of all of the parameters investigated because those could not hold true for other geometries. Various authors have reported the dependence of the mixing time on P/V for the same type of impeller. The differences can also be attributed to the differences in the geometries investigated, the measurement technique employed, and the criterion adopted to identify the degree of mixing. 4.2. Impeller Diameter (D). Fox and Gex8 have investigated the effect of the propeller diameter on the mixing time over a diameter range of 0.025-0.55 m covering a T/D ratio of 0.0714-0.33. They have shown that the mixing time varies as θmix ∝ D-1.67. They could also correlate the mixing time in terms of the momentum of the liquid jet generated by the impeller, which was expressed as θmix ∝ Mo-5/12. This implies that any combination of the impeller diameter and speed, which produces the same flux of momentum, i.e., the product (ND2), should lead to the same mixing time. However, the power requirement of such combinations will be different because the power consumed varies as N3D5. According to the above correlation, for a given mixing time under otherwise similar conditions, the larger is the impeller diameter, the smaller will be the impeller power required. Though this was the first study, it was a systematic one, and it directs us toward a rational justification for the use of larger-diameter impellers for efficient mixing operation. Norwood and Metzner10 have carried out investigations for a variety of impeller diameters (0.05-0.15 m) in stirred vessels (T ) 0.15-0.4 m) with T/D in the range of 1.33-5.66. They found that the mixing time is inversely proportional to the square of the impeller diameter for a flat-blade impeller. Biggs16 has also investigated the effect of the impeller diameter (over a range of 0.075-0.1 m) in a tank (T ) 0.245 m) and found that the mixing time varies with D-1.9(0.19, with the exponent being the same for DT, FBT, and PBT. Holmes et al.3 have measured the circulation time in stirred vessels (T ) 0.22, 0.6, and 1.0 m) agitated with a DT (D ) 0.18-0.44 m). They have reported that the circulation time and hence the mixing time is proportional to D-2, which is consistent with the previous investigations. Brennan and Lehrer12 have carried out mixing investigations over an impeller diameter range of 0.0630.102 m in a baffled vessel (T ) 0.42 m) and found that the dimensionless mixing time varies according to the relation θmix ∝ D-2. Khang and Levenspiel33 have developed a recycle mixing time model based on the mixing rate number N/K, where K ) decay rate constant, which is a function of N, F, µ, D, and T. The mixing rate number represents the rate at which the mixture approaches uniformity. They have found different dependencies of mixing times on the impeller to tank diameter ratio. The constant and the exponent involved in the correlation were found to be different for both types of agitators (DT and propeller) considered. The dependency of the mixing time on the D/T ratio in the case of DT was strong as compared to that of the propeller, which could be seen to be similar to the one observed by Prochazka and Landau.31 Hiraoka and Ito44 have analyzed the previous data on mixing times for vessels stirred by curved-blade turbines and found that the mixing time was propor-

tional to the impeller diameter (D/T ) 0.3-0.8) raised to the power -1.55 for baffled vessels. In the case of unbaffled vessels, the dependency was found to be weaker as θmix ∝ D-0.65. This is the first published unique study to investigate the effect of the impeller diameter in an unbaffled vessel. McManamey2 has derived a circulation model which gives θmix ∝ D-2.27 for radial flow impellers and θmix ∝ D-2 for axial flow impellers, which further supports the conclusions/observations of Prochazka and Landau31 and others. Sasakura et al.34 measured mixing times for impeller diameters over a range D ) 0.08-0.24 m in a vessel diameter of 0.3 m and arrived at the relation of the form θmix ∝ D-3 for flat, eight-bladed turbine agitators. This was the first study to indicate the inverse cubic dependence of θmix on the impeller diameter. This relationship can be seen to be closer to the one given by the circulation-based mixing model proposed by Nienow42 with an assumption that the flow number is independent of D/T. Shiue and Wong22 have carried out mixing time measurements in a stirred tank (T ) 0.4 m) and have also found the same dimensionless mixing time dependence on the impeller diameter over the range 0.13-0.18 m for a FBT. Sano and Usui17 have investigated the effect of the impeller diameter of 0.2-0.4 m in a fully baffled tank covering D/T ) 0.3-0.7 for a paddle-type impeller and 0.4-0.7 for the DT. They have shown that the dimensionless mixing time (Nmix) varies as D-1.8 for a DT and as D-1.67 for paddles. They also found that the mixing energy required varies with the impeller diameter as D-0.56. Sano and Usui18 confirmed the relationship for flat-blade impellers with different diameters (0.3-0.75 m) and numbers of blades (2-6). This study found similar diameter dependences for both the radial and axial flow impellers. Raghav Rao45 has investigated the effect of the impeller diameter on the mixing time and power consumption for a range of impeller types. The results are listed in Table 3. Raghav Rao and Joshi19 have reported the exponents on the impeller diameter of -1.83 for PBTD and -2.16 for DT. Rewatkar and Joshi20 have investigated the effect of the impeller diameter in the range D/T ) 0.267-0.5 on the mixing time in a vessel (T ) 1.5 m) agitated by PBTs. They observed that for a given power consumption an impeller with a D ) T/3 shows the shortest mixing time and is the most energy-efficient. Before commenting that the impeller vessel configuration with D/T ) 0.33 is the most efficient, further studies for the cases of D/T ) 0.4 and 0.45 should be carried out and the final conclusion made because there may exist a possibility of optima lying in this range. They have proposed a general correlation for mixing time by analyzing the extensive literature data. The index of the impeller diameter was found to be -1.54. Ruszkowski37 has used published mixing time data and obtained a correlation in terms of the impeller diameter. Only one vessel size (T ) 0.609 m) was studied. The mixing time was found to be inversely proportional to the square of the impeller diameter. Shaw39 has extensively investigated the effect of the impeller diameter for a variety of impellers (the range of diameters investigated for different impellers was different and is reported in Table 1) and observed that the mixing time varies in proportion to the impeller


diameter raised to the power -1.5. For these investigations, the tank diameter was kept constant. Further, he has suggested the use of a low-power-number impeller having larger diameter for the energy-efficient mixing. Thus, it can be seen that, for a given vessel, the mixing time is a function of the power drawn and the impeller diameter while at given power consumption the mixing time varies with the 3/2 power of T/D. Rutherford et al.40 have observed that the mixing time for the D ) 0.40T impeller was 64% of the corresponding value recorded for the D ) 0.33T impeller with identical blade and disk thicknesses while the power required was more by 44%. This study is useful because it directly depicts the use of large-diameter impellers for efficient mixing. Nienow42 has derived the dependency of the exponent -2, predicted by a mixing model based on the turbulence, while -3 as a result of the circulation-based model. While deriving the relationship for a circulationcontrolled model, he assumed that the flow number is independent of the impeller to tank diameter ratio. If this dependency is considered, one can end up getting the factor near to 2, as discussed earlier. Thus, the proportionality relationship, which can be obtained by considering both mechanisms, leads to the conclusion that the mixing time varies inversely with the square of the impeller diameter. Thus, the impeller diameter can be treated as the second foremost important parameter next to the impeller design, which decides mixing efficiency. It can be seen that most of the studies indicate that the mixing time is inversely proportional to the square of the impeller diameter. However, among the different types of impellers, the index of the impeller diameter is greater than -2 for axial flow impellers while the index is smaller than -2 for radial flow impellers, indicating a relatively stronger effect of the diameter in the case of the radial flow impeller. It can be seen that a wide difference exists between the reported dependence of the mixing time on the impeller diameter. The overall flow pattern is affected by the impeller diameter. It affects the primary flow produced, and also the secondary flow is affected as a result of the different extents of impeller-baffle interaction. Quantification of the change in primary and secondary flows in addition to the power numbers needs to be carried out as a result of the change in the impeller diameter. Further, these characteristics should, in turn, be related to the design objective of mixing. 4.3. Vessel Diameter (T) and Geometry. Kramers et al.15 have shown that for unbaffled vessels equipped with a propeller the mixing time increases as the tank diameter increases (0.32 and 0.64 m). Fox and Gex8 too observed the same type of relationship for vessels of diameters ranging from 0.15 to 4.2 m. Holmes et al.3 have reported that the circulation time (which is proportional to the mixing time) is proportional to T 2 for a DT. This study resulted in a relatively stronger mixing time dependence on T. Mersmann et al.46 and Brennan and Lehrer12 have found that, under similar conditions, the value of Nθmix is smaller for a flat-bottom vessel as compared to a dished-bottom tank. McManamey2 has proposed the circulation models for both the axial and radial flow impellers and shown that the mixing time is proportional to the square of the tank diameter for an axial flow impeller and a tank diameter to the power 2.3 for the radial flow impeller, indicating

relatively stronger vessel diameter dependence of the radial flow impellers. Sano and Usui17 have investigated the effect of the vessel diameter in a fully baffled tank. They have shown that the dimensionless mixing time (Nθmix) varies as T 2.31 for DT and T 2.41 for paddles versus the value of 2 reported by Mersmann et al.46 Further, the mixing energy was found to vary in proportion to T 1.11. Raghav Rao and Joshi19 have reported that the mixing time varies linearly with T for DT and T 2.2 for PBTD. Rewatkar and Joshi20 have extensively and exclusively investigated the effect of the tank diameter (0.57-1.5 m) on the mixing time over a wide range of power consumption. The results are depicted in Table 3. They have shown that the mixing time increases with the tank diameter, obeying the proportionality as Nθmix ∝ T 1.687. Further, they have observed the relationship of Nθmix ∝ T 2 for the case when the tank diameter was varied and the impeller diameter remained the same. Nienow42 has analyzed the data of Hass and Nienow47 and has reported that the mixing time is directly proportional to the square of the tank diameter. In regards to the exclusive effect of the vessel diameter, a very few good studies have been reported. Most of the studies report different dependences of the tank size on the mixing time. From most of the studies reported so far, it can be seen that the mixing time, in general, increases as the vessel diameter is increased. It can be stated that at constant D/T the mixing time varies as θmix ∝ T 2, while for a constant impeller diameter, it follows a relationship of the kind θmix ∝ T 1.67. More studies are required to investigate the effect of the tank size over a wide range for a variety of impeller designs. This can be very useful for the scaleup of the mixing process on the large scale. Further, it can be concluded that the vessel with the flat bottom is relatively more beneficial as compared to the vessels with the dished bottom. However, this needs further verification for different types of impellers because the flow produced as a result of different impellers may interact in a different way with the vessel bottom and may, in turn, have a different effect with respect to the flow pattern generated and power consumption. 4.4. Impeller Clearance (C). Kramers et al.15 have observed that for a propeller, in an unbaffled vessel, the mixing time decreases as the impeller clearance increases (from T/4 to 3T/8), while it was found to increase again with C/T up to T/2. This dependence was found to be a function of the tank diameter, as shown in Table 4. Further, the effect of impeller clearance on the mixing time was also found to be a function of the impeller eccentricity. Brennan and Lehrer12 have shown that Nθmix is the smallest in the case of a vessel agitated by a turbine impeller when the C/T ratio is close to 0.5. If the C/T ratio is either decreased or increased from this value, the mixing time was found to increase. The optimum value of the C/T ratio was found to be independent of the vessel geometry, i.e., whether the vessel has a dished or flat bottom. Shiue and Wong22 investigated the effect of C/T (0.5 and 0.325) for six FBTs and found that the homogenization number gets reduced to 34 (for C/T ) 0.325, NP ) 4.3) from 76 (for C/T ) 0.5, NP ) 4.8). It can be stated that, as the clearance of the DT decreases, the power number increases. From the data reported in their paper, it can easily be seen that in the case of the DT

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Table 4. Effect of the Impeller Off-Bottom Clearance on the Mixing Time (T ) 0.57 m, Conductivity Measurement Technique) DT

PBTU

PBTD

author

C (m)

Raghav Rao45

T/3

5.18 5.18 5.18

0.133 0.370 1.170

18.5 13.9 9.4

1.29 1.29 1.29

0.099 0.224 0.380

18.4 15.0 12.2

T/4

4.70 4.70 4.70

0.125 0.340 1.100

15.6 10.8 6.0

1.24 1.24 1.24

0.091 0.200 0.325

15.7 11.0 8.9

T/6

4.40 4.40 4.40

0.117 0.320 1.030

13.8 10.8 6.0

1.81 1.81 1.81

0.092 0.222 0.380

14.6 9.8 8.4

NP

P/M (W/kg)

θmix (s)

the mixing time can be reduced by almost 55% at the cost of only 11% more power using lower off-bottom clearance. Raghav Rao45 has carried out extensive investigations in regards to the effect of the clearance for different impeller designs at a wide range of power consumption. He has also carried out the power number measurements in each case. The results are depicted in Table 4. It can be easily pointed out that the effect of the clearance depends on the impeller design, and it is diameter-dependent in the case of PBTs. He has found that the PBTD (C ) T/3) is the most energy-efficient. To make a comparison on an equal power consumption basis, an attempt has been made to plot the mixing time as a function of the power consumption. It can be seen that a similar type of relationship is followed for the change in impeller off-bottom clearance and the corresponding power consumption and mixing time (Table 2). Raghav Rao and Joshi19 have investigated the effect of impeller clearance for PBTD and DT. They have reported that the mixing time decreases as the impeller clearance decreases in the range C ) T/3 to T/6 for DT. This result can be seen as in line with the results of Shiue and Wong.22 For PBTD, the value of the mixing time was found to increase with a decrease in the impeller clearance. For PBTU, the effect of the impeller clearance was found to be exactly opposite to that of PBTD. Rewatkar and Joshi20 have extensively investigated the effect of the impeller clearance for the tanks (T ) 0.57-1.5 m) agitated with PBTs and FBTs. They have varied the values of C/T for different impeller designs, namely, for PBTD with different blade angles and FBTs. They have observed different effects for different types of impellers. The trend of change in the mixing time was attributed to the overall flow pattern generated and the consequent change in power consumption. It can be seen that standard PBTD45 with C/T ) 0.33 performs best. This value was found to change with the impeller design, namely, the blade angle. In the case of FBT, the mixing time was the lowest at C/T ) 0.5. To summarize, the impeller off-bottom clearance plays an important role in the overall flow pattern generated by an impeller and also affects the power consumption. It has different dependences with respect to the flow pattern generated and the power dissipated for different types of impellers. Further, it can be concluded that the mixing time decreases as C/T increases up to 0.5 for an axial flow impeller. The dependence of the mixing time on C/T can be justified in view of the uniformity of the

NP

P/M (W/kg)

θmix (s)

NP

P/M (W/kg)

θmix (s)

1.29 1.29 1.29 1.29 1.35 1.35 1.35 1.35 1.61 1.61 1.61 1.61

0.098 0.315 0.535 1.030 0.010 0.316 0.495 1.034 0.118 0.378 0.520 1.160

13.0 6.6 5.4 4.4 12.0 6.4 6.1 4.5 12.6 7.9 7.0 5.0

Table 5. Effect of Baffling on the Mixing Time (T ) 0.64 m, D ) T/4) As Observed by Kramers et al.15 impeller type 3BMP

baffle condition

C/T

baffle at 0.1T from the wall

0.375 0.500 0.250 0.375 0.500 0.250 0.500 0.375 0.500 0.250 0.375 0.500 0.375 0.500

DT


30° ADT


3BMP

standard baffles

DT

standard baffles

3BMP

no baffles

NP 0.26

4.70 0.40 2.70 0.26

Nθmix 119 103 75 80 86 63 65 100 92 113 118 140 177 176

flow pattern, presence of dead zones, effective turbulent energy dissipation rate distribution, and values of minimum and maximum energy dissipation rates as a function of C/T and the impeller design. From the past investigations, it can be recommended that the C/T should be kept equal to 0.33 standard PBTD45 and DT, while it should be kept at 0.5 in the case of FBT. Further, it is suggested that power characteristics along with the mixing performance as a function of impeller off-bottom clearance should be studied for a variety of impeller types with a specific emphasis on high-efficiency axial flow impellers (hydrofoils) in order to enable the comparison of mixing performance on the basis of equal power consumption. 4.5. Baffle Geometry. Kramers et al.15 have investigated the effect of the presence and absence of baffles with and without wall clearance for a three-bladed marine propeller (3BMP). Table 5 lists the effects of baffling on the dimensionless mixing time. It can be observed that the mixing time is the smallest when the baffles are vertical and flush against the vessel walls as compared to baffles that are at some distance from the wall or baffles on the vessel bottom. An increase of 77% in the dimensionless mixing time can be seen in the case of the propeller-agitated, unbaffled vessel as compared to the baffled vessel. Further, it can be observed that the dimensionless mixing time is higher by 19% in the case of the vessel with the baffles placed away from the impeller. Further, they have observed that the presence of baffles on the vessel bottom does not affect the mixing time, unless the propeller is placed directly above the baffles. Moo-Young et al.11 found that better (more energyefficient) mixing can be achieved in baffled vessels as


compared to the mixing in unbaffled vessels in the case of FBT and tubular agitators. Brennan and Lehrer12 have observed the effect of the presence of baffles and the effect of the impeller blade width in the case of FBTs. They have found that the mixing time is independent of the impeller blade width in the unbaffled vessel while dependency was found to be present in the case of the baffled vessel. This is an interesting observation because it indicates the effect of interaction between the baffle and the impeller and hence a change in the flow pattern, which ultimately is reflected in the mixing time. Hiraoka and Ito44 have presented the correlations for the mixing time on the basis of the circulation model for both the baffled and unbaffled vessels for a vessel stirred by curved-blade turbines. These correlations were based on the data reported in the literature. They have found different dependences of the mixing time for both the baffled and unbaffled tanks. In the case of the unbaffled vessels, the mixing time was found to be independent of the parameter based on the friction factor, while it shows a dependence on the friction factor and hence strongly on the power in the case of the baffled vessels. This study leads to the conclusion that there exists a strong interaction between the baffles and the impeller and the flow produced by the impeller is significantly affected because of the presence of baffles. Nishikawa et al.13 have carried out extensive investigations of the effect of the number of baffles (0, 2, and 6) on the mixing time. They have shown that the mixing time or the mixing energy (Pθmix) is almost the same when the number of baffles (>2) for any impeller eccentricity. The mixing times for the case of two baffles and the fully baffled vessel were found to be the same. Hence, they have suggested the use of only two baffles. Further, it was suggested that the use of more baffles is not necessary because it may require more power. For a centrally mounted impeller, the mixing energy required to achieve a certain degree of mixing decreases as the number of baffles is increased. In the case of the three baffles, the mixing energy was almost equal to its value corresponding to the fully baffled condition. Sano and Usui18 have extensively studied the effect of the baffle width (BW/T ) 0.025-0.1) and the baffle number (0-12) in terms of a parameter RB ) nbBW/T in a vessel stirred by FBTs. They found that the fully baffled condition was satisfied when the value of RB is greater than 0.4. At this value, NP and NQ reach maximum values and θmix has the lowest value. The value of Nθmix was found to decrease as R increases up to 0.2. Beyond this value of the parameter (RB), there was no further effect on the mixing time. They have also shown that the number of discharged liquid circulations to reach the mixing time is independent of the number and width of the baffles. Further, they have observed that an increase in the number of baffles from 2 to 4 in the case of a vessel stirred by six FBTs results in a lowering of the mixing time by approximately 17% at equal power consumption (Table 2). Further, Novak and Rieger38 have found that agitators in an unbaffled vessel need less energy than the corresponding agitators in vessels equipped with baffles. A fairly good amount of research has been carried out in regards to the effect of the baffles on the mixing time encompassing various parameters. However, not a single study considers all of the possible baffle configuration effects for a variety of impeller-vessel designs.

Table 6. Effect of Eccentricity for 3BMP (D ) T/4) without Baffles (Kramers et al.15) Nθmix C/T

e

T/4 3T/8 T/2 T/4 3T/8 T/2 T/4 3T/8 T/2 T/4 3T/8 T/2

0.0 T/8 T/8 5T/16

T ) 0.32 m 133 121 177 182 121 117 157 135 92 112 92 90

T ) 0.64 m 177 176 (D ) 0.26 m) 218 161 142 (D ) 0.34 m) 145 140 110 (D ) 0.36 m)

This study will be useful for the sound comparison between the various vessel-baffle configurations covering a variety of impeller designs. It can be seen that the baffles affect significantly both the flow produced by the impeller and the power consumption and effectively the mixing efficiency. In view of the observations by the researchers in the past, it can be concluded that the sufficient number of baffles for the alleviation of the vortex formation and the maximum benefits of the baffling can be effectively calculated by using the factor R, as suggested by Sano and Usui18 in the case of FBTs. It is speculated that the effect of baffles would be altogether different for different types of impellers because the interaction with the baffles depends on the nature of the flow pattern generated by an impeller. The minimum baffle area required should be characterized for the optimum mixing performance. The impact of other parameters (such as the impeller diameter, impeller eccentricity, etc.) on the mixing performance has been found to depend on the extent of baffling, and hence the conclusion on the basis of singular baffle effects is difficult. Moreover, an extensive work needs to be carried out to study the effect of parameters such as the wall clearance. The effect of the baffle configuration on the flow numbers needs to be quantified in addition to the power numbers. Also the power lost due to excessive baffling should be characterized. 4.6. Impeller Eccentricity (e). Kramers et al.15 have extensively investigated the effect of impeller eccentricity on the dimensionless mixing time in the case of a 3BMP for a variety of impeller off-bottom clearance. Their results are tabulated in Table 6. It can be seen that locating the propeller more off-center in an unbaffled vessel significantly reduces the mixing time especially in the case with a high impeller clearance (C/T ) 0.5). Further, they have made an observation that the small eccentricity increases the tendency of vortex formation. They have also investigated the effect of inclination of shaft. From the results presented in their study, it can be observed that, compared to the standard vertical shaft, the configuration with the shaft at an angle of 30° is more effective. The reduction in the mixing time by about 33% was achieved in the case of the inclined shaft when compared to the vessel agitated by an impeller mounted on a centrally located vertical shaft. However, these results need to be analyzed further on the basis of an equal power consumption in order to select the proper impeller eccentricity and shaft angle. Nishikawa et al.13 have investigated the effects of impeller off-centering by carrying out the measurements of the agitation power and the mixing time in the baffles as well as unbaffled tanks for FBTs.

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They have concluded that in the case of an impeller which is set off-center in an unbaffled vessel the agitation power for e/D ) 0.5 was almost the same as that for the fully baffled condition. Despite this, the mixing time was at least twice the mixing time for the fully baffled case. Hence, they have suggested not to use an off-centering greater than 0.5 in unbaffled vessels. With two baffles, minimum mixing energy corresponds to an off-centering ratio such that the distance between the baffle and the tip of the blade on one side is equal to the distance between the tip of the blade and the wall of the tank on the other side. They have shown that the mixing time decreases with an increase in the impeller eccentricity when the number of baffles is zero or 1, in the range 0 < e/D < 3/4. If the number of baffles is more than 2, then the mixing time is practically independent of the impeller eccentricity. It was observed that for an unbaffled system the mixing energy is independent of the impeller eccentricity, which can be seen to be contradictory to the observation by Kramers15 for a propeller. Placement of an impeller away from the vessel axis leads to a significant change in the overall flow pattern and hence has a significant effect on the mixing performance. It has been observed that the impeller offcentering leads to the transition of the double loop to a single flow loop in the case of PBTs.48 Off-centering of the impeller in unbaffled vessels reduces the mixing time considerably, and its extent depends on an impeller off-bottom clearance. Further, its effect also depends on the extent of baffling. It has been observed that the mixing time is independent of the impeller eccentricity for the tanks with a number of baffles greater than 2 in the case of a marine propeller. Further, the mixing energy is independent of the impeller eccentricity for unbaffled vessels stirred by marine types of propellers. Minimum and maximum eccentricities should be avoided because they may lead to either vortex formation or the creation of dead zones. The change in the mixing time as a function of the impeller eccentricity should be quantified, and further studies should be carried out because very meager studies have been reported. From the studies reported, the optimum and most energyefficient configuration for mixing with the eccentric impeller requires one to place an impeller such that the distance between the baffle and impeller tip on one side should be equal to the distance between the wall and impeller tip on the other side (Nishikawa et al.12). Further, the effect of eccentricity should be studied for a variety of impeller designs because its effect (on both the overall flow and power consumption) inherently depends on the characteristics of the flow produced by the impeller. 4.7. Draft Tube. Hoogendoorn and Den Hartog21 have shown that the presence of a draft tube in a vessel agitated by 3BMPs in baffled and unbaffled vessels reduces the value of dimensionless mixing time (Nθmix) considerably. The extent of reduction in the mixing time was found to be by a factor of 2 or so. The power measurements have not been carried out, and hence the comparison on the equal power consumption basis is difficult. Tatterson49 has shown that the presence of a draft tube around the impeller (PBT) increases the circulation time by approximately 20%. The standard deviation in the measured circulation time was reduced, indicating that the presence of a draft tube promotes more uni-

formity in the flow. The circulation time was measured as the time between the two consecutive passages of the flow follower through the impeller while the mixing time was measured as the time required for the complete decolorization after the addition of the acid to the solution with the bromothymol blue pH indicator. It was also observed that a reduction in the size of the draft tube reduces the circulation time only marginally. The position of the draft tube with respect to the impeller also showed no effect on the circulation time. Further, the circulation time and the variance of the same were found to decrease because of the use of the conical bottom, which removes the dead zones below the impeller. He also observed that the small-diameter draft tube is slightly better than the larger-diameter draft tube in a baffled vessel. The mixing time for a standard vessel is lower than the mixing time in an unbaffled vessel. Variance of the mixing time was quite different than that of the circulation time. He has suggested that variance in the circulation time cannot be treated as the measure of mixing performance. Variance in the mixing time was found to be substantially lower than that in the circulation time. He has found that the variance in the mixing time for a standard configuration with a small draft tube in a conical bottom was lower than the one for standard configuration. Thus, such kinds of configurations can be investigated for further optimization. This was the first detailed study pertaining to the mixing phenomena in vessels fitted with draft tubes. Shiue and Wong22 have found that the presence of a draft tube around the impeller enhances the mixing efficiency of two- and four-bladed PBTs. It can be concluded from the difference between the constants (∼18.7 versus ∼50.4; Table 2) that at the same power consumption an approximately 3-fold decrease in the mixing time can be achieved by means of a draft tube. This can be seen as a remarkable result in regards to the utility of a draft tube for the enhancement of the efficacy of the mixing process. However, these results should be verified in view of the possible errors in the measurement of the mixing time. In the regions close to the draft tube, one can end up getting relatively slower mixing and hence higher mixing times if the mixing time is based on the concentration measurements in these regions. From the studies reported in the literature, it is very difficult to quantify with confidence the effect of the draft tube on the flow pattern and consequently the design objective, namely, the liquid-phase mixing. However, on the basis of the published studies, the draft tube has shown promise in the enhancement of streamlining of liquid circulation. Further, the draft tube avoids the short-circuiting of the flow, i.e., the entrainment of the flow in the impeller discharge leading to secondary loop formation. The observations reported in the literature differ in regards to the effect on the mixing time, and hence more investigations should be made in order to study the effect of the position of a draft tube relative to an impeller, its diameter, and length for different axial flow impellers. Further studies need to be carried out to quantity the effectiveness of the draft tube in the absence and presence of baffles on both the draft tube and vessel wall. Also the effect of baffles on the draft tube needs attention for its impact on the overall flow pattern and, in turn, the mixing characteristics. Further detailed characterization of such configurations, in


regards to the quantification of the flow pattern in terms of the flow number and the power consumption, should be carried out. 4.8. Effect of the Measuring Technique. Kramers et al.15 have observed that the measured value of dimensionless mixing time (Nθmix) varies linearly with the logarithm of the total mass of the tracer, i.e., Nθmix ∝ log(CiVi). They have observed that the mixing time obtained for a standard injection using 99.9% criterion (time required for the concentration to reach a value equal to 99.9% of the final averaged concentration) was the same as that obtained for the injection of twice the amount of salt with a 99.8% criterion. The studies were carried out over a limited range of the pulse volume and concentration. These effects need to be investigated for the larger volumes of the pulses over a wide range of concentrations in order to have concrete suggestions in regards to the use of the tracer volume and concentration. Biggs16 has investigated the effect of the probe location on the mixing time and observed that although the initial dispersion of the pulse injection near the impeller is faster the final (terminal) mixing time is similar to that when the tracer pulse is added near the top liquid surface. Brennan and Lehrer12 have listed the implications of different decolorizing indicators: methyl orange, methyl red, phenol red, phenolphthalein, bromothymol blue, mixed bromocresol green, and methyl red. They recommended the use of indicators of lower pH (

Liquid-Phase Mixing in Stirred Vessels: Turbulent Flow Regime

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