Screening Tool to Evaluate the Levels of Local Anisotropy of

Jun 4, 2005 - A new simplified method to evaluate levels of local anisotropy of turbulence in mixing vessels has been proposed. The method has been te...
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Screening Tool to Evaluate the Levels of Local Anisotropy of Turbulence in Stirred Vessels Elisabetta Brunazzi,† Chiara Galletti,† Alessandro Paglianti,*,‡ and Sandro Pintus† Laboratory of Process Equipment, Department of Chemical Engineering, Industrial Chemistry and Materials Science, University of Pisa, Via Diotisalvi 2, Pisa I-56126, Italy, and Department of Chemical, Mining and Environmental Engineering, University of Bologna, Viale Risorgimento 2, Bologna I-40136, Italy

A new simplified method to evaluate levels of local anisotropy of turbulence in mixing vessels has been proposed. The method has been tested both in a baffled and an unbaffled tank stirred with a Rushton turbine, by means of a detailed comparison with levels of local anisotropy of turbulence determined with the analysis of the invariants of the anisotropy stress tensor. In both configurations three-dimensional laser Doppler anemometry measurements have been performed throughout the whole vessel. The results indicated that turbulence deviates significantly from the isotropic state not only in the impeller region but also near the bottom of the vessel, for both baffled and unbaffled configurations. Larger regions with high local anisotropy of turbulence are present in the unbaffled vessel with respect to the baffled vessel. Finally, the simplified method was also tested and applied to a magnetically driven impeller that has been poorly investigated in the published literature but largely employed for industrial mixing processes. The flow field generated by this impeller is also shown. Advantages, drawbacks, and applicability of the simplified method are discussed. Introduction Mixing vessels are used in a large variety of industrial processes for different unit operations such as blending, chemical reaction, dispersion of gases or immiscible liquids into a liquid phase, solid suspension, fermentation, etc. Depending on the specific application, it comes out the issue of the best choice of the impeller geometry and vessel configuration, as well as of the operating conditions. In the 1990s it was estimated that losses of the order of $10 billion/year were due to a poor understanding of the fluid dynamics of stirred vessels and could be greatly reduced by using more accurate and detailed information for their design and development instead of “rules of thumb” and global parameters.2 Nowadays both experimental techniques and computational fluid dynamic (CFD) are available as useful tools for gaining a thorough understanding of stirred vessels. In the published literature, a large number of experimental data obtained with various measuring techniques (LDA, PIV, etc.) as well as CFD simulations of stirred vessels can be found. However most of those works refer to “standard” impellers such as the Rushton turbine3-9 or the pitched blade turbine.10-12 Therefore, as either a novel type of impeller has to be used or a standard impeller has to be fitted into a nonstandard vessel, detailed information on the fluid dynamic of the vessel is needed, and that may be gathered with either experiments or CFD simulations. Experiments are often time-consuming and usually require expensive instrumentation. Consequently, their role is foreseen to decrease over the years, whereas the role of CFD is growing enormously together with the increasing power of computers. Nowadays, CFD appears a promising tool that could well meet industrial needs. * To whom correspondence should be addressed. E-mail: [email protected]. † University of Pisa. ‡ University of Bologna.

In the industrial practice, CFD simulations are commonly based on the solution of the Reynolds-averaged Navier-Stokes equations (RANS), which allow obtaining information on mean values of the physical quantities of interest, such as velocity, pressure field, etc. Other approaches based on the direct numerical simulation (DNS) are unfeasible for the high Reynolds numbers encountered in real industrial processes. The large eddy simulation (LES), which is based on a sub-grid modeling, is also extremely time-consuming when applied to stirred vessels; therefore, nowadays this approach is unable of giving the fast responses required by the industrial world. However, only LES approaches have been proven to be capable13,14 of correctly predicting some peculiar phenomena observed recently in stirred vessels such as macroinstabilities,15-17 and they seem very promising for the future. CFD simulations based on RANS solution require a turbulence model in order to estimate the Reynolds stresses and overcome the so-called “closure problem”. The most employed turbulence models are the eddyviscosity models, which rely on the Boussinesq’s hypothesis according to which Reynolds stresses are proportional to the mean velocity gradient by means of an eddy viscosity (µt):

(

Rij ) µt

)

dUi dUj + dxj dxi

(1)

Among the eddy-viscosity turbulence models, the k- model calculates the eddy viscosity as a function of the turbulent kinetic energy and its dissipation rate (µt ) CµFk2/; Prandtl-Kolmogorov relationship), and this model has become the standard model in most CFD applications because of its simplicity and mathematical robustness. However, the model is well-known to fail for both flows of high local anisotropy of turbulence and strong streamline curvature. Those effects are directly

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related to the above formulations, which implicate local isotropic turbulence and do not take into account any body solid rotation. Such limitation of the k- model has been pointed out also for stirred vessels. For instance, some deviations between experimental and CFD data predicted with the k- model were found in the vicinity of the impeller blades of a Rushton turbine in baffled vessels18 and in that region turbulence was found to be mostly anisotropic.6-7,19 Another interesting example is offered by unbaffled vessels for which the k- model is known to completely fail the prediction even with its modified versions (i.e., RNG k-).20-22 Therefore since the k- model fails in some cases, it is important to assess the reliability of CFD predictions, especially as a nonstandard impeller/vessel geometry is used and its behavior is not known a priori. A full experimentation would be desirable; however, it is timedemanding, and there are conditions difficult to replicate in experimental tests, as with hazardous fluids. The present work is aimed at suggesting a simple method to assess whether the use of the k- model is adequate or may lead to poor predictive performances of the CFD simulation. The basic idea is to carry out just a few measurements in order to approximately indicate how far turbulence deviates from the local isotropy condition and subsequently the expected predictive performance of the k- model. The degree of local anisotropy of turbulence can be correctly estimated through the analysis of the anisotropy tensor as suggested by Lumley1 and briefly described also later in this paper. The method requires the knowledge of all Reynolds stresses, whose measurements are time-demanding and fairly difficult as they must be necessarily 2-D or 3-D measurements. Commonly other “indicators” are used to asses whether the local isotropy assumption can be reasonably adopted. Kresta and Brodkey23 suggested the following three indicators: the high local Reynolds number, the -5/3 slope in the frequency spectrum of the velocity signal, and the equality of the three rms components of velocity. It is noteworthy that it is easier to measure the rms velocities rather than, for instance, to estimate the local Reynolds number. This explains why the equality of the three rms components of the velocity has been used by many authors as a single indicator. However, such indicator leads to three different values, each derived from the comparison of one possible pair of rms velocities. In present work, it has been introduced as a method for indicating qualitatively levels of local anisotropy of turbulence, which can be seen as a restatement of the third indicator suggested by Kresta and Brodkey,23 but allows estimating the equality of the three rms velocities by means of just one parameter. Since this parameter is calculated only from the rms velocities, a 1-D measuring system would be sufficient for its determination. Such a parameter has to be considered for preliminary studies and qualitative investigations and not as a proper alternative to Lumley’s method because it does not ensure the accurate and rigorous description of local anisotropy of turbulence provided by this latter method. The method proposed has been tested for a vessel agitated with a Rushton turbine in both baffled and unbaffled configurations. Levels of local anisotropy of turbulence calculated with the novel method across the entire vessel for the two configurations have been compared to those accurately determined with the

method proposed by Lumley.1 Advantages and drawbacks of the proposed method as well as indications of its applicability are discussed. Finally the proposed method has been also applied to a magnetically driven impeller, which has been poorly investigated in the literature. The magnetic drive eliminates all problems associated with rotating seals such as leakage, contamination, and constant maintenance. Therefore, such impeller type is befitted for stirred vessels where mixing operation should be without contamination and leakage, as in pharmaceutical and food aseptic processes, and in general where toxic, hazardous, and/or sterile fluids are to be mixed. The flow field of the stirred vessel is also shown. Experimental Apparatus and Method Mixing Vessel. The mixing system comprised a flat bottom cylindrical vessel made of acrylic plastic (Perspex) of inner diameter T ) 0.290 m; it can work unbaffled or with four baffles of width B ) T/10 equally spaced around the vessel periphery. The tank was filled with distilled water up to a level H ) T and placed inside a trough also made of Perspex and filled with the same fluid in order to minimize refraction of the laser beams at the curved tank surface. The vessel was placed on a traversing system that allowed movements along the three orthogonal spatial directions with an accuracy of 0.1 mm. For some experiments the agitation was provided by a Rushton turbine of diameter D ) 0.32T and placed at tank mid height, C ) T/2. The impeller was driven by a 0.3 kW power motor, and the agitation rotational speed could be varied by means of a speed controller. Measurements were taken at an impeller rotational speed of N ) 200 rpm, which corresponded to an impeller Reynolds number Re ) 28 000 and an impeller blade tip velocity Vtip ) 1.01 m/s. Some experiments were also carried out using a sixblade impeller of diameter D ) 0.41T, placed at the tank bottom and magnetically coupled with the external motor. The impeller blades were inclined 10°, and the impeller rotational direction was such that the turbine was up-pumping, i.e., clockwise. A schematic illustration of the impeller is shown in Figure 1: the turbine (Figure 1a) is mounted on a male bearing, which is fixed and sealed to the tank and contains an inner magnet (Figure 1b). The turbine is driven by an external motor, resulting in the assembly illustrated in Figure 1c. Then the assembly is placed at the tank bottom as shown in Figure 1d. Measurements were taken in the baffled configuration with an impeller rotational speed of N ) 200 rpm (Re ) 48 000). Table 1 summarizes the geometrical details of both the Rushton and the magnetically driven turbines. Laser Doppler Anemometer. The measuring system consisted of a three-component laser Doppler anemometer. The beam from a 3 W argon-ion Spectra Physics laser was separated into green, blue, and purple (λ ) 514.5, 488 and 476.5 nm, respectively) lines. Bragg cells provided frequency shifting on all lines. The LDA system was setup in orthogonal side-scatter mode with two probes placed horizontally, therefore entering the vessel from the side walls. Two pairs of beams (green and blue) emerged from one probe, and their signals were collected from the other probe, whereas the purple pair of beams emerging from the other probe was collected from the first one. This

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Figure 1. Geometry of the magnetically driven impeller and mixing system: (a) turbine; (b) male bearing; (c) turbine and male bearing assembly; (d) assembly of magnetically driven turbine, tank, and trough. Table 1. Geometrical Details of the Two Impellers Investigated Rushton turbine D (mm) D/T (-) Nb Rb (-) Wb/D (-) Hb/D (-) tb/D (-)

magnetically driven turbine

92 0.32 6 0° 0.25 0.20 0.0108

120 0.41 6 10° 0.21 0.21 0.038

Table 2. Optical Details of the LDA System property

1

2

3

velocity component wavelength (nm) transmitting probe receiving probe focal length (mm) beam diameter (mm) beam spacing (mm) number of fringes probe volume (mm3) beam half-angle (°) LDA volume dimension - ∆x (mm) LDA volume dimension - ∆y (mm) LDA volume dimension - ∆z (mm)

axial 514.5 1 2 500 2.2 38 21 0.3646 2.176 0.149 0.149 3.921

tangential 488 1 2 500 2.2 38 21 0.3097 2.176 0.141 0.141 3.719

radial 476.5 2 1 500 1.35 60 56 0.7956 3.434 0.225 0.225 3.752

arrangement minimized the dimensions of the volume observed by the photodetectors. The green and blue pairs of beams, whose bisector coincided with the tank diameter located midway between two neighboring baffles, allowed measuring both axial and tangential velocities. The purple beam allowed measuring the radial velocity. The scattered light was transmitted by means of optical fibers to the photomultipliers and then processed by a Dantec burst spectrum analyzer. The principal optical characteristics are given in Table 2, and more details of the experimental setup can be found in Galletti et al.19 The flow was seeded with silvercoated hollow glass spheres with a mean diameter of 10 µm and a density of 1400 kg/m3.

The system could operate either in coincidence or in non-coincidence mode. When the system operated in coincidence mode, only signals scattered from particles arriving at the same time in the measuring volume and with bursts that partially overlapped were validated. Operating parameters, such as the voltage of each of the photomultipliers, were set appropriately in order to have approximately the same signal-to-noise ratio for all three component measurements. When the system operated in non-coincidence mode, all the bursts scattered from particles were validated by a single detector independently from the behavior of the other detectors. A computer program, which takes into account refraction when beams cross different media, was used to calculate the exact location of the measurement volume inside the tank. Proper alignment is mandatory and requires considerable effort. A procedure for the alignment was adopted that is described in detail in Galletti et al.19 In view of the strict coincidence criterion employed, the data rate was low (about 10 Hz). This value is be partly associated with the small control volume because, the concentration of the tracers being fixed, the number of particles that is likely to be seen is proportional to the measuring volume observed. On the contrary, higher data rates of about 50-200 Hz were observed in the non-coincidence mode because of the larger control volume. At least 1000 samples were acquired for each measurement. The repeatability of the data was checked by repeating each measurement a few times. Instantaneous velocities were acquired at different axial planes and different radial positions, and for each location the mean velocities were derived from the instantaneous velocity data. Local Anisotropy of Turbulence. (a) Analysis of Invariants of Anisotropy Stress Tensor. Lumley1 suggested a method for the accurate evaluation of the characteristics of local anisotropy/isotropy of turbulence. The method is based on the analysis of the normalized Reynolds stress anisotropy tensor:

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bij )

〈uiuj〉 1 - δij 2k 3

(2)

This tensor has real eigenvalues (λ1, λ2, λ3) and can be characterized by only the second (II) and the third (III) invariants:

II ) λ1λ2 + λ2λ3 + λ1λ3

(3)

III ) λ1λ2λ3

(4)

Lumley1 proposed representing each point and time in a turbulent flow in the (III, -II) plane; each realisable state in a turbulent flow corresponds to a point that it is located inside a “triangle” in such a plane. Figure 2 shows the Lumley’s triangle. The three bounding curves of the triangle correspond to special states. The origin O represents local isotropic turbulence. The two curves leaving from the origin (curves OP and OQ) correspond to axisymmetric turbulence, for which two components have the same energy content; the upper curve PQ represents a state in which the energy content of a component is zero. Point P represents 2-D turbulence, for which the whole energy content is shared between only two components; whereas point Q represents 1-D turbulence, with the whole energy contained in one component. To characterize local anisotropy of turbulence by means of a single parameter, Derksen et al.7 suggested using the distance L of the point from the origin mark O (isotropic turbulence) in the Lumley’s triangle:

L ) xIII2 + (-II)2

(5)

The maximum distance from O is exhibited by the point representing 1-D turbulence, and this distance is Lmax ) 0.34. (b) Comments on the Use of Lumley’s Method. The method proposed by Lumley1 requires the knowledge of all Reynolds stresses. This information necessitates the simultaneous measurement of all three components of the instantaneous velocity or at least of two of them; in the latter case measurements have to be repeated for all possible combinations (i.e., axialradial, axial-tangential, and tangential-radial) to obtain all shear stresses. The instrumentation needed to perform this kind of measurements is rather as sophisticated and expensive as a 3-D LDA system. In addition there are further issues to be considered: First, a considerable effort is needed for the system in order to be properly aligned because of the small dimensions of the measurement volume. For 3-D measurements, such a volume is determined from the intersection of three cigar-shaped volumes, one for each pair of beams (see Table 2 for dimensions ∆x, ∆y, ∆z, and volumes of each beam pair intersection). In the orthogonal side-scatter configuration, the intersection volume was calculated to be approximately 0.014 mm3, which, for example, is approximately 57 times smaller than the intersection volume of the purple beams. The system is difficult to align because three pairs of beams have to be focused into this volume as well as the receiving optics have to look at this volume. This is made worse by the complexity of the mixing tank geometry, where beams are refracted at the interface between different media.7

Figure 2. Lumley’s triangle.

Second, as mentioned previously, experiments are time-consuming because the sampling rate is low. This is partly related to the small dimensions of the measurement volume. Moreover, geometrical characteristics of the mixing system may hinder the access of the three pairs of beam in some regions and thus impede 3-D measurements. (c) Parameter A. In conclusion, the application of the rigorous method proposed by Lumley1 to stirred tanks appears challenging and requires sophisticated instrumentation. In addition, experiments are difficult to perform and require considerable time. Therefore, it would be desirable to have a preliminary screening tool capable of describing properly local anisotropy of turbulence, albeit with reduced accuracy, based on simpler instrumentation and on much more straightforward and rapid measurements. The results of such a preliminary tool could also be useful to highlight the zones of the vessel needing a more accurate and rigorous investigation. In the present work, an attempt was made in order to define a simple parameter capable of indicating levels of local anisotropy of turbulence, as an alternative to Lumley’s method. This parameter, which will be denoted in the following text as parameter A is calculated only from the rms velocities (i.e., ui′ ) x〈uiui〉); hence, non-simultaneous measurements with a 1-D measuring system would be sufficient for its determination. If a 3-D measuring system is available, measurements can be performed in a non-coincidence mode, resulting in higher sampling rates and considerable time savings, albeit with lower spatial resolution. Parameter A was defined as 3

A)

∑ i)1

|ui′ - u j ′| u j′

(6)

where u j ′ is the mean of the rms components:

u j′ )

1

3

∑ u i′

3i)1

(7)

Local isotropic turbulence implies equality of the three rms components;23 consequently, A becomes zero. Parameter A is expected to increase when the difference between the three rms components is also increased. In the case of energy distributed only along one spatial

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Figure 3. Parity plot of A/Amax vs L/Lmax for Rushton turbine in (a) baffled vessel and (b) unbaffled vessel.

direction coinciding with one of the three directions of measurement, parameter A becomes 4. This value can be assumed to be its maximum value, Amax. Parameter A is not based on invariants, and the implications of such limitation will be discussed in the next section, where the performance of A in describing levels of local anisotropy of turbulence will be compared with the analysis proposed by Lumley.1 Results and Discussion To investigate the capability of the simplified method based on A (see eq 6) of truly determining levels of local anisotropy of turbulence, A was calculated for each location of the vessel where the Lumley’s method was also applied and L was determined by performing 3-D LDA measurements. This was done in the mixing vessel stirred by the Rushton turbine for both baffled and unbaffled configurations. The results obtained with the two methods have been compared in the parity plots a and b of Figure 3 for the baffled and unbaffled vessel, respectively. Since magnitudes of L and A are different, both L and A have been normalized by their maximum achievable values (i.e., Lmax and Amax, which are of 0.34 and 4, respectively). The plot shows that all data are located around the diagram diagonal; roughly, higher values of parameter A correspond to higher values of L, which truly quantifies the degree of local anisotropy of turbulence. The correlation coefficients were 0.84 and 0.71 for the baffled and unbaffled tanks, respectively. However, there is a considerable dispersion of the data, and some are located quite far from the diagonal. To understand the reason for such a spreading in the data, it is worthwhile noting the characteristics of parameter A introduced in the present work. This parameter is calculated only from the rms velocities taken on three orthogonal directions of measurement, which in the case of stirred vessels have been chosen to be the vertical, the radial, and the tangential directions, according to the polar symmetry of the tank. Such a definition of parameter A makes it strongly dependent on the choice of the three directions of measurement. Conversely, Lumley’s method analyzes the all Reynolds stresses, including shear stresses. Such information enables the transformation of the Reynolds tensor in the coordinate system constituted by its principal axes. As a result of such a transformation, the choice of the

directions along which the velocity components are taken (i.e., the original coordinate system) is no more important and thus does not affect the estimation of the degree of local anisotropy of turbulence. The above discussion indicates the intrinsic limitation of parameter A and explains why parameter A and L, even if they agree in most regions of the stirred vessel, do not agree well in some locations. Consequently, data of Figure 3 are scattered. For instance, there are a few points in Figure 3a for the baffled vessel that are located quite far from the diagonal and on the right respect to it; for example, the point of L/Lmax ) 0.30 and A/Amax ) 0.11. In this point parameter A cannot indicate the high degree of anisotropy of turbulence that is present. This point is located in the impeller stream (r/T ) 0.173 and z/T ) 0.51), where trailing vortices24 occur and probably, because of them, the principal directions along which turbulence develops do not coincide with the measurement directions. Besides, this is a region of rod-like turbulence (see Figure 9 of Galletti et al.19), which explains why parameter A indicates a lower degree of anisotropy: since the directions of measurements do not coincide with the principal axes, parameter A is based on lower differences among the components than those that would be calculated if the measurements were taken exactly along the principal axes of the flow. Similarly, Figure 3a shows also some points located far from the diagonal and on the left side of the parity plot; therefore, parameter A indicates higher values of local anisotropy of turbulence than those quantified by means of the Lumley’s method. Some of these points refer to the vessel bottom near the vessel axis that is likely to be a region where the directions of measurement differ from the principal axes of the flow. Moreover in this region turbulence is disk-like.19 The contour plots of Figure 4 compare the descriptions of local anisotropy of turbulence throughout the whole tank estimated with L and A. The discussion of Figure 4a is reported in Galletti et al.19 It can be observed that the contour plot of A/Amax (Figure 4b) resembles that of L/Lmax (Figure 4a), which describes true levels of local anisotropy of turbulence. Also absolute values are similar. Both plots indicate high levels of local anisotropy of turbulence in the vicinity of the impeller, the vessel base, and the centers of the circulation loops; however, some differences can be observed as A seems to underestimate the extent of the region of local high anisotropy of turbulence near the impeller and overestimate that near the bottom of the vessel. Moreover, as far as the vessel bottom is concerned (z/T ) 0.05), L indicates that the highest levels of local anisotropy are located at some distance from the vessel axis (r/T ) 0.15), whereas parameter A shows that the highest levels of anisotropy are located near the vessel axis. Despite these differences, patterns given by A and L seem to be fairly in agreement. However, to understand better the reason for the differences highlighted above, it is instructive to observe Figure 5, which shows radial profiles of the six Reynolds stresses at z/T ) 0.1. The three shear stresses (see the solid symbols) are similar and close to zero in the vicinity of the vessel axis (i.e., at r/T ) 0). Moving further away from the vessel axis, the radial-tangential shear stress becomes dominant, being approximately four times higher than the other two shear stresses. As far as normal stresses are concerned (see open symbols), the radial and tangential stresses exhibit similar pro-

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Figure 4. Contour plots of (a) L/Lmax and (b) A/Amax. Rushton turbine, baffled vessel.

Figure 5. Radial profiles of Reynolds stresses at z/T ) 0.1.

files and are higher than the axial normal stress in the vicinity of the vessel axis up to r/T ) 0.3, whereas the axial stress becomes dominant near the vessel walls. It is noteworthy that near the vessel axis (i.e., at r/T ) 0), the three normal stresses are considerably different, so that one may conclude following the indicator based on the equality of the rms velocities23 for the assessment of local isotropy of turbulence, that turbulence is anisotropic. Similarly parameter A which, as mentioned earlier, may be considered as a restatement of such indicator predicts high levels of local anisotropy of turbulence (see Figure 4b). In such location the Lumley’s method (see Figure 4a) indicates low levels of anisotropy, and this may be explained by considering that this method is based on all stresses including the three shear stresses, which are all close to zero in such location. The contour plots of Figure 6 show L and A normalized over their maximum values throughout the tank for the unbaffled configuration. The contour plots have similar shapes, albeit A seems to estimate slightly higher levels of local anisotropy of turbulence than L,

as also indicated by the parity plot of Figure 3b. Both contour plots of Figure 6 indicate high levels of anisotropy of turbulence in the vicinity of the impeller, near the vessel base, and at about r/T ) 0.18 below and above the impeller. It is worthwhile remembering that the flow field in the unbaffled vessel is characterized by a strong circumferential loop, whereas both axial and radial velocity are quite low. Therefore, it is difficult to explain this latter region of high anisotropy of turbulence in terms of the circulation loops as could be done for the baffled vessel. It can be observed that turbulence seems to deviate from local isotropy throughout a larger region of the tank in the unbaffled vessel (Figure 6a) than that in the baffled vessel (Figure 4a) with the extent of the green region being larger. The baffled vessel shows highest maximum values of local anisotropy of turbulence; however, those values are confined to small regions near the impeller and vessel base, whereas turbulence can be assumed to be locally isotropic in the remainder of the tank (blue region). This may give an explanation of the poor predictive performances of CFD simulations with local isotropic turbulence models when applied to unbaffled vessels. Conversely, they provide the successful predictions for baffled configuration with the exception for the impeller region. Once the simplified method has been tested for the Rushton turbine, it has been applied to study the local levels of anisotropy of turbulence in the vessel stirred by the magnetically driven impeller. The mean flow field generated by this impeller is shown in Figure 7. The vector plot in the plane shows the resultant of the axial and radial mean velocities, whereas the tangential mean velocity is illustrated by means of the superimposed contour values. The tangential velocity is positive in the direction of the impeller rotation. The single loop pattern typical of a inclined blade type impeller set at very low bottom clearances can be observed. In addition, the tangential velocities indicate the presence of a

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Figure 6. Contour plots of (a) L/Lmax and (b) A/Amax. Rushton turbine, unbaffled vessel.

Figure 8. Parity plot of A/Amax vs L/Lmax for the vessel equipped with the magnetically driven impeller.

Figure 7. Flow field of the vessel equipped with the magnetically driven impeller: axial and radial velocity as vector plot; tangential velocity as contour plot.

double circumferential loop, as there are two regions of high Ut/Vtip: one near the walls, which extends from the vessel base up to tank mid-height, and one that seems to extend about the total vessel height at a radial distance of r/T ) 0.2. The first circumferential loop is

likely to originate from the lower six blades of turbine, whereas the second circumferential loop is maybe due to the higher part of the impeller. Local anisotropy of turbulence was determined in several points with both the Lumley’s and the proposed methods. The results are compared in the parity plot of Figure 8. The plot shows that also with this type of impeller and flow configuration most of the data are located around the diagram diagonal, thereby indicating that parameter A is capable of quantifying correctly the degree of local anisotropy. The correlation coefficient was 0.88. The proposed methodology was used to investigate the entire vessel, and the results are shown as contour plot of parameter A in Figure 9. Results indicate that high levels of local anisotropy of turbulence occur in the

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Figure 9. Contour plot of A/Amax for the vessel equipped with the magnetically driven impeller.

vicinity of the impeller and near the walls; however, there are also large regions with local isotropic turbulence. Despite the differences pointed out above between L and A, parameter A seems to show quite well the characteristics of local anisotropy/isotropy of turbulence inside the stirred tank for both baffled and unbaffled configurations. This is due to the specific configuration examined in the present work, because inside stirred vessels the polar symmetry of the flow is such that the axial, radial, and tangential directions, where LDA measurements have been taken, are likely to coincide with the principal axis of the motion. Similarly, good performance of A should be expected for all flow configurations, which show high symmetries; velocity measurements can be taken in directions that reflect these symmetries, for instance, flows in pipes, annulus, cyclones, ... For all these configurations the use of parameter A instead of the rigorous Lumley’s method would allow a rapid and easy estimation of the degree of local anisotropy of turbulence. Conclusions In the present work an attempt was made to define a simple method to estimate levels of the local anisotropy of turbulence in mixing vessels. The proposed method is based on a parameter (A) that is calculated from the rms velocities and, therefore, requires just a 1-D measuring system for its determination. The simplified method, that derives from one of the tests suggested by Kresta and Brodkey,23 was validated in a vessel equipped with a Rushton turbine in both baffled and unbaffled configurations by means of a detailed comparison between levels of local anisotropy of turbulence determined through it and those calculated through the

analysis of the invariants of the anisotropy stress tensor (Lumley1). This latter tensor was evaluated throughout the whole tank by performing 3-D LDA measurements, which present some difficulties in the system alignment and are time-consuming experiments. A was found to indicate qualitatively regions of high levels of local anisotropy of turbulence as well as those of local isotropic turbulence. However, the results showed considerable discrepancies in some locations; therefore, advantages and drawbacks of the proposed method as well as indications of its applicability have been discussed. The proposed method is aimed at providing a easy tool for determination of local anisotropy of turbulence for preliminary studies. A few and easy 1-D measurements, which have to be carried out along the directions that reflect the geometrical symmetries of the flows, could be used to determine levels of local anisotropy of turbulence with a good accuracy. The method may help indicating the regions of the vessel worthy of further investigation, and other methods as that suggested by Lumley1 could be applied only to those regions with great savings of experimental time. Results on local anisotropy of turbulence may help choosing the appropriate turbulence model for CFD simulations, which are largely employed in the industrial practice for the design and development of mixing systems. In the baffled vessel, high levels of local anisotropy of turbulence were found near the impeller, the vessel base, and the centers of the circulation loops (see also Galletti et al.19), whereas the turbulence is fairly isotropic in the remainder of the vessel. In the unbaffled vessel, the region of high local anisotropy of turbulence seems to be larger than that of the baffled vessel. This may explain the poor performances of CFD in predicting the flow field of unbaffled vessels when using local isotropic turbulence models.21 Once the method proposed was validated for a vessel agitated with a Rushton turbine in both baffled and unbaffled configurations. It was applied to a magnetically driven impeller, which has been poorly investigated in the literature but is largely used in industrial processes such as those of the pharmaceutical and food industries. The flow field of the stirred vessel was characterized by a double circumferential loop. Finally, some experimental data on the six Reynolds stresses, measured for the baffled configuration working with a Rushton turbine, have been shown. They could be used to asses turbulence models, which utilize the anisotropy information (e.g., Jovanovic´ et al.25). Acknowledgment This work was financially supported by the “Ministero dell’Istruzione, dell’Universita` e della Ricerca Scientifica”. The authors are grateful to Ing. G. Mariotti from Mariotti & Pecini, Via Sandro Pertini 41, 50019 Sesto Fiorentino (Firenze), Italia, and Mr. Cesare Merello for the technical support. Nomenclature A ) parameter for the estimation of the degree of anisotropy of turbulence, defined in eq 6 B ) baffle width, m bij ) elements of the normalized anisotropy stress tensor C ) impeller off-bottom clearance, m Cµ ) empirical coefficient usually assumed equal to 0.09

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D ) impeller diameter, m ∆x, ∆y, ∆z ) dimensions of the measurement volume, m H ) liquid height, m Hb ) impeller blade height, m II ) second invariant of the normalized anisotropy stress tensor III ) third invariant of the normalized anisotropy stress tensor k ) kinetic turbulent energy, m2‚s-2 L ) parameter for estimation of the degree of anisotropy of turbulence, defined in eq 5 N ) impeller rotational speed, s-1 Nb ) number of blades O ) origin of the Lumley’s triangle P,Q ) points in the Lumley’s triangle r ) radial coordinate measured from the vessel axis, m Re ) impeller Reynolds number, Re ) FND2/µ Rij ) elements of the Reynolds stress tensor, m2‚s-2 T ) tank inner diameter, m tb ) disk and blade thickness, m u ) instantaneous velocity, m‚s-1 u′ ) root mean square (rms) velocity, m‚s-1 u j ′ ) mean of the three rms velocities, defined in eq 7 U ) mean velocity, m‚s-1 Vtip ) impeller blade tip velocity, Vtip ) πND, m‚s-1 Wb ) impeller blade width, m x ) coordinate, m z ) axial coordinate measured from the vessel bottom, m ) temporal mean Greek Letters Rb ) angle of the blades measured from the vertical, deg δij ) Kronecker delta  ) viscous dissipation rate of turbulent kinetic energy, m2‚s-3 λ ) laser beam wavelength, m λ1, λ2, λ3 ) eigenvalues of the normalized anisotropy stress tensor µ ) dynamic viscosity, Pa‚s µt ) eddy viscosity, Pa‚s F ) density, kg‚m-3 Subscripts a ) axial r ) radial t ) tangential i, j ) indices indicating coordinate direction max ) maximum 1, 2, 3 ) indices indicating the three orthogonal coordinate directions

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Received for review November 12, 2004 Revised manuscript received April 26, 2005 Accepted May 3, 2005 IE0489103