Decomposition of Flow Structures in Stirred Reactors and Implications

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Decomposition of Flow Structures in Stirred Reactors and Implications for Mixing Enhancement Andrea Ducci,* Zacharias Doulgerakis, and Michael Yianneskis Experimental and Computational Laboratory for the Analysis of Turbulence (ECLAT), DiVision of Engineering, King’s College London, Strand WC2R 2LS, U.K.

Earlier studies have shown that macro-instability (MI) vortices in vessels stirred by radial flow impellers can be employed to improve mixing performance through guided feed insertion at selected radial positions along the path of a MI vortex, with an associated mixing time reduction of 20-30%. The present investigation provides for the first time an understanding of the physical mechanisms underlying the MI vortex phenomena, employing a proper orthogonal decomposition analysis to identify and characterize the different flow structures. It is shown that the MI structure is affected by an off-centering perturbation of the core and a stretching mechanism that interact in different ways for low, transitional, and high Reynolds numbers (Re). The different MI vortex frequencies phenomenologically identified in previous works and especially the two simultaneous frequencies encountered for transitional Re are shown to result from a competition between the two mechanisms. The implications of the findings for mixing enhancement at different Re ranges are discussed. 1. Introduction The production of a considerable proportion of chemicals is performed in stirred vessels. In 1999, the global sales value turnover amounted to around U.S. $1400 billion, with mixingrelated problems estimated to cost 0.5-3% of turnover, or around U.S. $2-11 billion per annum.1 Consequently, there is a profound need to identify ways to improve mixing performance at the initial design stage and much effort has been invested to achieve this. The objective of the present work is to identify means to aid mixing enhancement by assessing the utilization of flow structures in stirred vessels and their exploitation for improved mixing efficiency. Vortical structures in fluid flows are highly dissipative and can offer an attractive means to enhance the mixing of fluids in stirred vessels, where a number of different vortices are present, such as the trailing vortices originating at the impeller blades2 and those stemming from flow instabilities. Nikiforaki et al.3 have reviewed the different types of instabilities encountered in such reactors and identified three different types of instability. These were related to changes in the impeller offbottom clearance, changes of impeller speed, N, and/or Reynolds number (Re), and the so-called macro-instabilities (MIs). Nikiforaki et al.3 revealed the precessional-vortex-core character of the MI in vessels stirred by a radial-flow (Rushton) impeller. Roussinova et al. 4 studied MIs produced with axial-flow impellers and attributed their propagation, under very specific conditions, to a resonant frequency f ) 0.186N. MIs associated to Rushton turbines are tornado-like types of vortices that are present above and below the impeller and that precess around the impeller axis. A sketch of MI vortices is provided by Ducci and Yianneskis,5 who documented a phase shift of 180° between the upper and lower vortices. MI vortices comprise, in essence, perturbations of the instantaneous flow field that are present for specific ranges of conditions and/or locations in a stirred vessel; they may be considered as a source of energy that, unless utilized, is wasted and may not contribute * To whom correspondence should be addressed. E-mail: andrea. [email protected]. Tel.: +44 (0)20 7848 2522. Fax: +44 (0)20 7848 2932.

significantly to the mixing process. It has also been reported that in solid suspension systems the effect of precessional MIs could be important and should be investigated, as there is some evidence that such structures may be responsible for the larger dispersion observed above the impeller.6 Consequently, flow structures in stirred reactors and MIs in particular have received considerable attention in recent research. Extensive reviews of earlier research have been provided in the works cited earlier. Notwithstanding the extensive literature on the subject of MIs, it was only recently that the potential of utilizing MIs to minimize mixing time in stirred processes by 20-30% was identified by Ducci and Yianneskis,5 who provided the first detailed description of the two different precessional MI patterns at low and high Re that are characterized by f/N ratios of approximately 0.1 and 0.02 respectively. Their work indicated that accurate knowledge of the MI vortex location, extent, and track around the vessel can indeed be harnessed to improve mixing performance and provided motivation for the present work, which aims to acquire improved understanding of the structure of MI vortices and help identify how they may be further exploited in industrial practice for mixing process optimization. The experiments and analysis reported in this paper are based on a decomposition of the MI flow structures with proper orthogonal decomposition (POD) methodology. The results extend and complement the data on MI vortices provided in earlier works and offer hitherto unavailable insights on the actual flow structures present. Importantly, the MI structures present in transitional Resintermediate to the low and high Re ranges studied by Ducci and Yianneskis5sare also considered in this work, for two reasons. First, because most regions of the flow in a stirred reactor are, more often than not, neither fully laminar nor turbulent and transitional flows may be expected in most practical situations. Second, Galletti et al.7 have reported a more complex MI structure for intermediate Re that may have important practical implications but has not been satisfactorily understood to date. 2. Flow Configuration and Experimental Apparatus The experiments were carried out in a standard vessel of diameter T ) 300 mm, equipped with four equispaced baffles

10.1021/ie070905m CCC: $40.75 © 2008 American Chemical Society Published on Web 12/15/2007

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Figure 2. Percentage of energy associated with each mode (Re ) 27 200).

Figure 1. Sketch of the vessel also showing the camera location and the horizontal plane where the PIV measurements were carried out.

and stirred by a Rushton turbine. The impeller clearance and diameter were C ) T/2 and D ) T/3, respectively. More information about the mixer dimensions is provided in Ducci and Yianneskis.5,8,9 A 13 kHz time-resolved 2D Dantec particle image velocimetry (PIV) system was used to measure the velocity flow field at a horizontal plane positioned at z/T ) 0.25 (see Figure 1). Similarly to the apparatus configuration employed in Ducci and Yianneskis,10 the camera was positioned underneath the vessel and was focused through a transparent window present in the base of the vessel on a horizontal region centered around the vessel axis. The area investigated was approximately a square of side 0.8 D, but in the present experiments, a trade-off between spatial resolution and maximum number of recorded consecutive frames had to be made, as the maximum amount of memory of the PIV camera is limited, and high spatial resolution sets necessitate shorter data records and vice versa. The objective of the present study was to investigate the MI vortex with a phase-resolved POD analysis, which requires low data rate and long data records to store as many MI cycles as possible. An optimum configuration was obtained by reducing the spatial resolution to 0.025 D (grid of 38 × 31 points), approximately half of that employed in Ducci and Yianneskis,10 and doubling the number of consecutive frames to 2042. The lower spatial resolution did not affect the ability to resolve the MI scales and together with the selection of a low frame rate (1 Hz) enabled us to store in each data record a suitable number of MI precessions for an optimal phase-resolved POD analysis. The working fluid was distilled water seeded with 10 µm silver-coated hollow particles to scatter the light emitted by the double-pulsed Nd:Yag laser. The thickness of the laser sheet

Figure 3. (a) Autocorrelation coefficient of the temporal eigenfunctions associated to modes 1-4; (b) cross-correlation coefficients of a1(t) - a2(t) and a3(t) - a4(t) (Re ) 27 200).

in the investigation area was approximately 1 mm. Five experimental data sets of increasing impeller speed N were obtained in the MI transitional regime at Re ) ND2/ν ) 4400, 6000, 7000, and 8000. An additional reference data set was obtained at Re ) 27 200 in the high Re range associated to MI vortices of characteristic frequency fMI ) 0.02N. Unless otherwise indicated, in the remainder of this paper a cylindrical coordinate system (r, θ, and z) with the origin in the center of the base of the vessel will be used (see Figure 1). Details of the processing techniques employed to analyze the experimental data are provided in Section 3.

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3. Results and Discussion 3.1. Proper Orthogonal Decomposition and Phase-Resolved Analysis. In the present section, a brief description of the proper orthogonal decomposition (POD) method of snapshot, is provided; for a thorough explanation of the methodology, the reader is referred to Sirovich11 and Berkooz et al.12 POD is a linear procedure which decomposes a set of signals into a modal base. The different modes are ordered in terms of kinetic energy content, with the first mode being the most energetic and the last the least energetic. As a consequence, the lower modes are associated to large-scale structures while the higher modes are associated to smaller scale coherent vortices and random turbulence. In eq 1, the POD analysis and its energy efficient decomposition is applied to the fluctuating part of the measured velocity field, b u′:

b u(x b, t) ) U B (x b) + b u′(x b,t) )U B (x b) +

N

∑an(t)ΦB n(xb) n)1

(1)

where b u and U B are the total and mean velocity flow fields and Φ B n and an are the spatial eigenfunctions and the temporal eigenfunctions (often referred to as temporal coefficients) associated to the nth mode, respectively. It should be noted that when the POD analysis is applied to fluid mechanics, each spatial eigenfunction Φ B n corresponds to a specific velocity flow field associated to the nth mode. According to eq 1, the spatial eigenfunctions Φ B n are normalized and time independent, and their intensity and the way they contribute in time to the total flow is controlled by the associated an(t) coeffcient and the way it varies with time. The spatial, Φ B n, and temporal, an, eigenfunctions satisfy the conditions reported in eqs 2 and 3, respectively:

Bj ) Φ Bi ‚ Φ aiaj)

{

{

1 for i ) j 0 for j * i

λi for i ) j 0 for j * i

(2) (3)

The conditions of eq 2 imply that the eigenfunctions Φ B n are spatially orthogonal to each other while the conditions of eq 3 show that the temporal coefficients ai are uncorrelated in time. The spatial eigenfunctions, Φ B n, are obtained from the eigen modes of the two-point correlation tensor, C, by solving eq 4,

CΦ B n ) λnΦ Bn

(4)

In particular, Chiekh et al.,13 Oudheusden et al.,14 and Perrin et al.15 have shown that when the POD technique is applied to flows dominated by large-scale convective structures such as those present in the wake of circular and squared cylinders, the POD modes are found to occur in pairs that represent the orthogonal components of the harmonics of the periodic process investigated. In such flows, a LOM can be defined as follows:

b uLOM(x b, φ) ) U B (x b) + a1(φ)Φ B 1(x b) + a2(φ)Φ B 2(x b)

where the correlation tensor is defined in eq 5

Ckm ) b u(x bk,t) ‚ b u(x bm,t)

Figure 4. (a) Variation with MI phase angle, φMI, of the phase-resolved temporal coefficients a1, a2, a3, and a4 normalized with Vtip ; (b) loci of the points (a/1, a/2) and (a/3, a/4) for increasing phase angle φMI.

(5)

where the indices k and m refer to different points in the grid of measurement (i.e., 1 < k,m < 1178). The eingenvalue λn of eq 4 represents the energy content associated to the nth mode and its contribution to the total fluctuating energy. In the literature, POD has been used as a powerful postprocessing tool to extract dominant coherent flow patterns and to construct low-order models (LOM) that capture the largest amount of energy with the least number of modes. With this approach, it is possible to isolate the highly energetic coherent structures of interest by investigating only some specific modes and neglecting all higher modes that contain the higher-order harmonics of the coherent motion as well as the random turbulent motion.

(6)

where the periodic nature of the coherent structures associated to the first two modes is reflected in the sinusoidal variation of the coefficients a1(φ) and a2(φ) as shown in eq 7

a1(φ) ) x2λ1sin(φ), a2(φ) ) x2λ2cos(φ)

(7)

It should be noted that in eqs 6 and 7 the phase angle φ is related to the time according to φ ) 2πft, where f is the characteristic frequency of the structure of interest. In a method similar to that of the wake studies carried out by Chiekh et al.,13 Oudheusden et al.,14 and Perrin et al.,15 this work aims at identifying the modes that are associated to the MI vortex and at defining a LOM that best describes the MI precession around the vessel axis. In the first part, the phaseresolved POD analysis has been carried out on a high Re


Figure 5. Plots of the velocity field and contours of the dimensionless vorticity associated to: (a) mode 1; (b) mode 2; (c) mode 3; and (d) mode 4 (Re ) 27 200).

flow that is associated with a single precessional frequency (fMI ) 0.02N) and which is used here as reference. In the second part, the Re is gradually decreased in the transitional regime to assess what determines the occurrence of the second precessional frequency (0.1N) and to understand how the flow structures associated to the two frequencies interact with each other. 3.2. Phase-Resolved POD Applied to the High Re Case. In this section, the results for Re ) 27 200 are presented. The percentage variation of the energy associated with the first 10 modes is shown in Figure 2. In agreement with eq 1, the results shown in Figure 2 take into account only the kinetic energy of the fluctuating motions, b u′, as the mean motion has been removed before applying the POD. As expected, the energy content is decreasing as smaller flow structures (i.e., higher number modes) are considered. It is also evident that the first two modes account for 33% of the total fluctuating energy while the first four modes comprise almost 50% of the total fluctuating motion energy. Considering that the ai coefficients contain the temporal information of the flow (eq 1), an evaluation of the frequencies associated to the first four modes can be obtained by carrying out an autocorrelation analysis of each ai coefficient.10 The cross correlation coefficient defined in eq 8 becomes an autocorrelation when i ) j.

Raiaj(∆t) )

ai(t0)aj(t0 + ∆t) a′i(t0)a′j(t0 + ∆t)

(8)

The variation with ∆t/Timp of the autocorrelation coefficients associated with the first four modes is shown in Figure 3a, where Timp is the period of an impeller revolution. From this figure, it

is evident that the temporal coefficients corresponding to modes 1 and 2 fluctuate with a period of 50Timp (i.e., f ) 0.02N), while the temporal coefficients corresponding to modes 3 and 4 oscillate with a period that is half that of the previous one (25Timp w 0.04N). These considerations show that the coherent structures associated to the first two modes of the decomposition are highly interconnected to the MI structures, while the second two modes are harmonics of the first pair. The absolute values of the maxima and minima of the autocorrelation Ra1a1 and Ra2a2 are higher than those shown by Ra3a3 and Ra4a4. This difference implies that the periodicity of the coherent structures (i.e., modes) associated to a frequency of 0.02N is better defined than those exhibited by flow structures associated to a frequency of 0.04N. An estimation of the phase difference between a1 and a2 can be obtained by cross-correlating the two coefficients, where the cross-correlation is defined by eq 8. The variation of Ra1a2 with ∆t/Timp is shown in Figure 3b. The minimum value of Ra1a2 is reached for ∆T/Timp ) 12.5 which is exactly a fourth of the period of the oscillation of a1 and a2. This proves that the sinusoidal oscillations of a1 and a2 are out of phase by 90°, as was shown by eq 6, and they vary as sine and cosine waves, respectively (eq 7). The cross-correlation coefficient Ra3a4, also shown in Figure 3, indicates that the phase difference between the coefficients a3 and a4 is 20∆t/Timp, which corresponds to a 288° (i.e., -72°) phase difference between the two sinusoidal waves. The phase-resolved averages of the coefficients a1, a2, a3, and a4 are shown in Figure 4a. All the coefficients have been normalized with Vtip. The higher variations of a1 and a2 with respect to those exhibited by a3 and a4 confirm that the energy content of the first two modes is higher than that of the second

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Figure 6. Plots of the LOM velocity field and contours of the associated dimensionless vorticity for four increasing values of φMI position along the MI precession: (a) φMI ) 0°; (b) φMI ) 90°; (c) φMI ) 180°; and (d) φMI ) 270° (Re ) 27 200).

two (∆a1 > 6∆a3). Moreover, this graph also confirms the 90° and 72° phase difference between the pairs of coefficients a1 and a2 and a3 and a4, respectively. The orthogonality between two generic temporal eigenfunctions can also be assessed according to the methodology reported by Oudheusden et al.,14 who showed that a pair of eigenfunctions is orthogonal, and therefore, both parts of eq 7 are satisfied, if the points (ai(φ),aj(φ)) lie on an ellipse. In fact, eq 7 can be rearranged as follows:

a12 a22 + )1 2λ1 2λ2

(9)

which is the characteristic equation of an ellipse. From Figure 4, it is evident that the plot of the nondimensional coefficient a/1 against a/2 describes a circular shape, where a/i ) ai/(2λi)1/2. It should be noted that in Figure 4 an increase of phase angle, φMI, corresponds to circles of hues varying from blue to red. Despite the phase difference between the temporal coefficients a3 and a4 not being exactly 90°, but slightly smaller (72°), the points (a/3, a/4) are also scattered around the reference circle. As expected in this case, a cycle is completed in half of the time taken by the coefficients a/1, a/2 to close the circle, as is shown by the middle-scale green hue of the last point (triangle) shown in Figure 4. The velocity fields and vorticity contours of the spatial eigenfunctions Φ B corresponding to modes 1-4 are shown in parts a-d of Figure 5, respectively. It should be noted that the contours of ωzΦn (∂Φny/∂x - ∂Fnx/∂y) are provided to have a qualitative idea of the size and shape of the flow structures associated to the mode considered, but their absolute values are meaningless as the intensity of each mode is controlled by the

variation in time of the corresponding temporal eigenfunction an(t). From parts a and b of Figure 5, it is evident that modes 1 and 2 are related to each other; both show two distinct regions of negative and positive vorticity and they are offset by 90° in the tangential direction θ. This is in agreement with the findings of Oudheusden et al.14 in a cylinder wake flow who reported that the flow patterns associated to the first two modes are shifted in the direction of vortex convection. In the present study, the flow is rotational and therefore the convective direction is θ. Accordingly, the flow patterns associated to modes 3 and 4 (see parts c and d of Figure 5) are also similar, with a hyperbolic kind of flow pattern and two opposite regions of positive vorticity and two opposite regions of negative vorticity. In this case, the two modes are also shifted along the tangential direction (by 45°). The flow patterns of the LOM obtained by superimposing the first four modes are shown in parts a-d of Figure 6 for four increasing values of φMI. The flow field in this case is clearly dominated by the first two modes with a clockwise precession of the positive and negative vorticity regions. The impact of the fluctuating structures included in the LOM on the mean motion is shown in parts a-d of Figure 7 for four increasing values of φMI. It should be noted that the mean flow field for this Re number starts showing the rise of a solid body rotation region centered on the vessel axis (see Figure 8) whose intensity in terms of nondimensional vorticity is approximately 4 times lower than that reported by Ducci and Yianneskis5 for Re ) 3200. This is in agreement with the findings of Ducci and Yianneskis5 who reported a gradual change of mean flow pattern, from radial to tangential, as Re decreased from 32 000 to 3200. From Figure 7, it is evident that the MI structures associated to the LOM significantly deform the mean flow field


Figure 7. Plots of the superimposed LOM and mean velocity fields and contours of the total dimensionless vorticity for four increasing values of φMI position along the MI precession: (a) φMI ) 0°; (b) φMI ) 90°; (c) φMI ) 180°; and (d) φMI ) 270° (Re ) 27 200).

Figure 8. Vector plot of the mean flow pattern and contour plot of the vorticity associated to mean motion for Re ) 27 200.

as the negative (black) vorticity region of the LOM tends to accelerate the rotational mean flow on the one side while on the opposite side the positive vorticity region (light brown) attenuates the solid body rotation. The outcome of these two combined effects is the precessional vortex that has been documented in all previously cited works on MI. From the above considerations, it is possible to conclude that for this range of Re the MI is a radial off-center perturbation of the mean flow. 3.3. Phase-Resolved POD Applied to the Transitional Re Case. Galletti et al.7 were the first to show that fast Fourier transforms (FFTs) of velocity data records are characterized by MIs of different characteristic frequency depending on the Re range investigated. In particular, they identified two characteristic frequencies of approximately 0.02N and 0.1N when the

Figure 9. FFTs of the velocity record in the point (r/D ) 0.25, θ ) 180°) for four different Re.

Re number is varied between 6000 and 13 000. The FFTs of the PIV velocity records in the point (r/D ) 0.25, θ ) 180°) are shown in Figure 9 for four different Re. The FFTs clearly exhibit the frequencies reported by Galletti et al. 7 with two distinct peaks for Re ) 6000 and 7000. It should be noted that the peak energy content for Re ) 27 200 was 40 times higher than those obtained for the other three Re and to show all the peaks in Figure 9, a second ordinate scale on the right side of the plot was necessary to show the total variation of E27200. A comparison between the autocorrelations of the temporal coefficients a1, a2, a3, and a4 for four different Re (4400, 6000,

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Figure 10. Autocorrelation of the temporal coefficients a1, a2, a3, and a4 for four different Re: (a) Ra1,a1; (b) Ra2,a2; (c) Ra3,a3; and (d) Ra4,a4.

7000, and 8000) is shown in parts a-d of Figure 10. As a reference, the first four modes of the data sets corresponding to Re ) 8000 are shown in parts a-d of Figure 11. It should be noted, however, that in Figure 10 the same mode number does not necessarily correspond to the same characteristic flow structure, as for some Re numbers some specific types of modes were inverted or displayed as fifth-sixth mode. As expected, both frequencies 0.02N (i.e., ∆t/Timp ) 50) and 0.1N (i.e., ∆t/ Timp ) 10) are alternatively present in Figure 10 with the first frequency being dominant only in the first two for the upper range of the Re investigated. This behavior is expected as the 0.02N frequency is associated to the highly energetic modes only for high Re, but as Re is decreased, its periodicity attenuates, as confirmed by the gradual decrease of the local maximum of the autocorrelation (for ∆t/Timp ) 50) with decreasing Re. The extent to which the local maximum of the autocorellation decreases is more evident when comparing the autocorrelation coefficients Ra1a1 and Ra2a2 of Figure 3a for Re ) 27 200 with those shown in parts a and b of Figure 10 for 6000 eRe e 8000. It is evident that the local peak of the autocorrelation decreases consistently from values of 0.55 to 0.2. On the contrary, the four plots shown in Figure 10 bring into evidence how the 0.1N frequency is spreading from modes associated to lower energy content to those associated to higher ones as the Re is decreased. To obtain a better understanding of the interaction between the high- and low-frequency instabilities in the transitional regime, the remaining part of the discussion focuses on the results corresponding to Re ) 8000, as this data set shows the coexistence of the two MI instabilities, with a well-defined distinction between the two upper modes associated to 0.02N instabilities and the lower two associated to 0.1N

instabilities. Depending on which of the two instabilities is considered, references to a low-frequency phase angle, φLF MI, and/or a high-frequency phase angle φHF are employed in the MI remainder of the paper. The modes shown in parts a and b of Figure 11 for Re ) 8000 correspond to the off-centering instability already described in Section 3.2 for Re ) 27 200. Although its dynamics do not change, two main differences can be identified between the offcentering instabilities associated to Re ) 27 200 (see Figures 5 and 7) and Re ) 8000. The first difference is related to the distance between the positive and negative peaks of the vorticity in modes 1 and 2. The peaks’ distance is approximately 0.2D for Re ) 8000 (see parts a and b of Figure 11) and 0.375 D for Re ) 27 200 (see parts a and b of Figure 5). The shorter peak distance results in a final lower radius of precession of the vortex associated to the mean motion for Re ) 8000. The second difference is related to the mean flow fields. A comparison between the vector plots of the mean motion for Re ) 8000 (see Figure 12) and Re ) 27 200 (see Figure 8) shows that the solid body rotation is more intense for Re ) 8000 as the minimum value of the mean nondimensional vorticity is double that exhibited for Re ) 27 200. As a consequence, it is possible to conclude that the mean motion is more resilient to the external perturbations caused by the offcentering instability for Re ) 8000. The combination of these two effects is reflected in Figure 13 where superimposition of the LOMLF calculated from the first two modes and the mean motion is shown for eight positions of increasing φLF MI. It is evident that the precessional radius is smaller than the one shown in Figure 7 for Re ) 27 200. This is in agreement with the findings of Ducci and


Figure 11. Plots of the velocity field and contours of the dimensionless vorticity associated to: (a) mode 1; (b) mode 2; (c) mode 3; and (d) mode 4 (Re ) 8000).

Figure 12. Vector plot of the mean flow pattern and contour plot of the vorticity associated to mean motion for Re ) 8000.

Yianneskis,5 who were able to calculate the mean precessional radius of the MI by tracking the MI vortex along its path. Considering the high frequency instability, the modes shown in parts c and d of Figure 11 have hyperbolic patterns similar to those discussed in Section 3.2 for Re ) 27 200 (see parts c and d of Figure 5) with two opposite regions of positive vorticity and two opposite regions of negative vorticity orthogonal to the previous two. Moreover, as already pointed out for Re ) 27 200, the flow pattern exhibited by modes 3 and 4 of parts c and d of Figure 11 is offset by 45° in the tangential direction θ. However, for Re ) 8000, the periodicity associated to modes

3 and 4 is comparable to those exhibited by modes 1 and 2, with peak values of the correlation coefficients Raiai shown in Figure 10 parts a and b similar to those of parts (c) and (d). As a consequence, their impact on the final flow is more significant than that in the higher Re case, where their effect was almost negligible. Considering the flow pattern associated to modes 3 and 4 (see parts c and d of Figure 11), the central zone with almost zero vorticity level implies that the center of the solid body rotation region associated to the mean flow is only marginally affected by the pertubations induced by these two modes in this area. On the contrary, the two opposite regions of negative (positive) vorticity are locally enhancing (reducing) the corresponding outer parts of the solid body rotation region, stretching (squeezing) the mean vortex core. The flow patterns and vorticity contours of the LOMHF calculated from the combinations of modes 3 and 4 for eight values of increasing φHF MI are shown in Figure 14. As expected from the considerations made in Section 3.1 on POD decomposition, the 45° tangential offset between the flow patterns associated to modes 3 and 4 together with the phase shift between the sine and cosine waves of the temporal coefficients a3 and a4 result in a precession around the vessel axis of the two pairs of positive and negative vorticity with a frequency of 0.1N. A more effective visualization of the precessional nature of the LOMHF can be obtained by considering the clockwise rotation of a line joining the vorticity peaks of the two opposite regions of negative vorticity for the different phase angles φHF MI shown in Figure 14.

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Figure 13. Vector plots of the superimposed LOMLF and mean velocity fields and contours of the total dimensionless vorticity for eight increasing values LF LF LF LF LF LF LF LF of φLF MI along the MI precession: (a) φMI ) 0°; (b) φMI ) 45°; (c) φMI) 90°; (d) φMI ) 135°; (e) φMI ) 180°; (f) φMI ) 225°; (g) φMI ) 270°; (h)φMI ) 315° (Re ) 8000).

From the previous considerations, it is evident that the line joining the peak values of negative vorticity provides the direction of stretching of the mean vortex core for different phase angles. Similarly, the angular positions assumed by the line joining the two peaks of positive vorticity with different phase angles φHF MI correspond to the direction of squeezing of the mean vortex core. It can be concluded that the orthogonal directions of stretching and squeezing of the main vortex core are also expected to rotate with a frequency of 0.1N. This behavior is well reflected in Figure 15 where the vorticity contours and vector plots of the total flow obtained by superimposing the LOFHF and the mean flow for eight values of increasing φHF MI are shown. As expected, the mean vortex

core is stretched along directions whose orientation is gradually changing with different phase angles φHF MI . In conclusion, the transitional regime is characterized by two simultaneous instabilities, associated to two different types of perturbation of the main mean flow: an off-centering instability that results in a precession of the vortex core center with a frequency of 0.02N and a stretching instability that induces an elongation of the vortex core along a direction that is rotating with a frequency of 0.1N around the vessel axis. 3.4. Instability Interaction and Mixing Enhancement. The preceding analysis based on the POD results of the MI structures present in the stirred vessel has provided new information as well as further explained earlier, phenomenological, findings


Figure 14. Vector plots of the LOMHF and contours of the corresponding dimensionless vorticity for eight increasing values of φHF MI along the MI precession: HF HF HF HF HF HF HF (a) φHF MI ) 0°; (b) φMI ) 45°; (c) φMI ) 90°; (d) φMI ) 135°; (e) φMI ) 180°; (f) φMI ) 225°; (g) φMI ) 270°; (h) φMI ) 315° (Re ) 8000).

on the composition of such instabilities. First, the manner in which such structures are convected during an MI cycle and their relative strength has been quantified. Second, the high Re MI was shown to stem from an axial off-center perturbation of the mean flow, while the transitional Re mean vortex motion was shown to be far more resilient to external perturbations and also affected by a stretching/squeezing deformation. Third, the previously reported smaller radius of the low Re MI was confirmed. Fourth, the competition between the high and low Re MIs was shown for transitional Re values. It is important to subsequently address two questions that have considerable practical implications for mixing process optimization. First, how the trajectory of a transitional Re range MI vortex could be described on the basis of the aforementioned

considerations and second, whether and how improvements in mixing may be achieved with the new information reported here. The first question can be addressed by considering Figure 16, where a sketch showing the interaction between the highand low-frequency instabilities is provided. As pointed out in the previous section, the low-frequency modes determine an off-centering perturbation of the mean vortex core and as a consequence the offset center precesses around the vessel axis with a period of TLF MI (i.e., fMI ) 0.02N). At the same time the high frequency instability does not affect the vortex center position but determines an elongation of the vortex core along LF a direction which is rotating with a period THF MI ) TMI/5 (i.e., fMI

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Figure 15. Vector plots of the superimposed LOMHF and mean velocity fields and contours of the total dimensionless vorticity for eight increasing values HF HF HF HF HF HF HF HF of φHF MI along the MI precession: (a) φMI ) 0°; (b) φMI ) 45°; (c) φMI ) 90°; (d) φMI ) 135°; (e) φMI ) 180°; (f) φMI ) 225°; (g) φMI ) 270°; (h) φMI ) 315° (Re ) 8000).

) 0.1N) around the vortex axis. The combination of these two motions is represented in Figure 16 with a satellite type of rotation of the vortex center around the vessel axis and a simultaneous revolution of the oblong vortex core around its axis. The five sectors delimited by the five dashed blue lines of Figure 16 divide the vortex precessional path into five sections, each corresponding to a complete revolution of the vortex core around its axis. Evidence that for low Re the vortex is rotating with an approximate frequency of 0.1N around its axis can also be found in Ducci and Yianneskis5 who reported that in a low Re situation the solid body rotation region is characterized by a similar nondimensional angular speed ω/2πN (same as that for f/N). It should be noted that the sketch shown

in Figure 16 has been made to best visualize the combinations of the two instabilities and does not precisely scale the vortex diameter and the precessional radius. In particular, from the description provided in the final part of Section 3.3, it is evident that for a transitional Re that the precessional radius is smaller than the vortex diameter. In a similar way, the interaction between the perturbations of the mean vortex core associated to the 0.02N off-centering structures and the 0.04N stretching/squeezing instability is sketched in Figure 17 for a high Re case. For this range of Re. the vortex revolves around its axis with a frequency of 0.04N and therefore completes two revolutions around its axis during a total precession around the vessel axis. It should be noted


at an appropriately selected radial location could show reductions in mixing time comparable to, if not greater than, the 20-30% reduction reported for high Re MI structures by Ducci and Yianneskis.5 Such likely improvements in mixing must be the subject of future research involving mixing time measurements over a range of Re. Such experiments are considered essential as the flows in stirred vessels are seldom fully turbulent over the entire vessel volume,16 and in most mixing processes the flow over either the entire volume or at least a substantial part of it will be transitional. The MIs comprise flow mechanisms that offer additional energy for the dispersion of reactants and their exploitation, purely in terms of selection of feed location and without any further change in operating parameters, ans can involve a mixing improvement, not only at no extra cost but with considerable potential for reduction of mixing time and thus energy consumption. 4. Conclusions

Figure 16. Sketch showing the interaction of the high- and low-frequency instabilities for a transitional Re.

Figure 17. Sketch showing the interaction between the 0.02N and 0.04N instabilities for a high Re flow.

that from the considerations made in Section 3.2, for this range of Re the stretching/squeezing instability is less significant than that of the off-centering one. To address the second question, it must be first recalled that at high Re the vortex core associated to the mean motion is more susceptible to external perturbations than for low Re. This consideration together with the more complex variation expected for a transitional Re when both the off-centering and stretching instabilities have similar magnitude indicates that mixing at the highest possible Re for which both MI frequencies are present could help utilize the MI motions more effectively for mixing enhancement. The combined effect of the two types of instability described together with the more easily perturbed high Re MI being subject to an increased likelihood of instability effects could provide two flow mechanisms that may augment mixing and reactant dispersal more effectively than operation at either high or low Re alone. Therefore, insertion through feed pipes

The methodology employed for the flow decomposition and analysis presented in this paper has shown that POD is a most powerful technique that can offer detailed information on vortical structures. Such structures are highly dissipative and therefore very beneficial for mixing purposes, and the information obtained may be employed to optimize mixing processes well beyond what has been hereto possible with mean velocity and turbulence energy considerations alone. The results presented in this paper have provided a detailed decomposition of the competing MI flow structures for low, high, and transitional Re. The behavior of the MI vortex path for transitional Re values was indicated, and flow characteristics that may be conducive to mixing enhancement in practice were highlighted. Transitional Re flows are commonly encountered in stirred vessels, and consequently, the maximum possible utilization of related flow instabilities should provide significant mixing time reduction and energy savings. The application of the findings of the present work to mixing practice could be readily achieved by employing a pressure transducer in the manner of Paglianti et al.17 for the rapid and inexpensive detection of the MI vortex passage and synchronization of mixing events (e.g., tracer insertion) with it, without the need for complex PIV apparatus. The determination of the mixing enhancement that may be achievable through MI vortex utilization necessitates an extensive set of mixing time measurements that future research will address. Future work should also address the potential of better utilization of other vortical structures, such as the trailing vortices emanating from the impeller blades, to enhance mixing through the carefully selected insertion of reactants: The present work has clearly indicated that there is a plethora of information that can be extracted from mixing vessel flowfield measurements that has not been fully exploited to date, despite its potential and implications for industrial mixing practice. It would be interesting in future work to examine MI structures generated by different impeller designs and at different vessel scales so as to facilitate scale-up and scale-down criteria for mixing process design. Nomenclature AbbreViations FFT ) fast Fourier transform HF ) high frequency LF ) low frequency LOM ) low-order models

3676


MI ) macroinstability POD ) proper orthogonal decomposition PIV ) particle image velocimetry Greek Symbols ∆t ) time difference, s θ ) tangential coordinate, ° λn ) energy of the nth mode, m2 s-1 ν ) kinematic viscosity, m2 s-1 Φn ) nth POD spatial eigenfunction φMI ) phase angle, ° Roman Symbols an ) nth POD temporal eigenfunction, m s-1 C ) impeller clearance, m Cij ) correlation tensor, m2 s-2 D ) impeller diameter, m E ) energy content, m2 s f ) precessional frequency of the MI vortex, s-1 N ) number of POD modes N ) impeller rotational speed, s-1 r ) radial coordinate, m Raiai ) autocorrelation coefficient of the ith temporal eigenfunctions Re ) Reynolds number Raiaj ) cross-correlation coefficient between the ith and jth temporal eigenfunctions T ) vessel diameter, m Timp ) period of impeller, s b u′ ) fluctuating velocity field, m s-1 U B ) mean velocity field, m s-1 b u ) total velocity field, m s-1 Vtip ) velocity of the tip of the blade, m s-1 z ) axial coordinate, m Acknowledgment Financial support for the work reported here was provided by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K., Grant No. EP/D032539. Literature Cited (1) Butcher, M.; Eagles, W. Fluid mixing re-engineered. Chem. Eng. 2002, 733, 28-29. (2) Yianneskis, M.; Popiolek, Z.; Whitelaw, J. H. An experimental study of the steady and unsteady flow characteristics of stirred reactors. J. Fluid Mech. 1987, 175, 537-555.

(3) Nikiforaki, L.; Montante, G.; Lee, K. C.; Yianneskis, M. On the origin, frequency and magnitude of macro-instabilities of the flows in stirred vessels Chem. Eng. Sci. 2003, 58, 2937-2949. (4) Roussinova, V.; Kresta, S.; Weetman, R. Resonant geometries for circulation pattern macroinstabilities in a stirred tank AIChE J. 2004, 50, 2986-3005. (5) Ducci, A.; Yianneskis, M. Vortex tracking and mixing enhancement in stirred processes. AIChE J. 2007, 53, 305-315. (6) Micheletti, M.; Nikiforaki, L.; Lee, K.; Yianneskis, M. Particle concentration and mixing characteristics of moderate to dense solid-liquid suspensions. Ind. Eng. Chem. Res. 2003, 42, 6236-6249. (7) Galletti, C.; Paglianti, A.; Lee, K. C.; Yianneskis, M. Reynolds number and impeller diameter effects on instabilities in stirred vessels. AIChE J. 2004, 50, 2050-2063. (8) Ducci, A.; Yianneskis, M. Direct determination of energy dissipation in stirred vessels with two-point LDA. AIChE J. 2005, 51, 2133-2148. (9) Ducci, A.; Yianneskis, M. Turbulence kinetic energy transport processes in the impeller stream of stirred vessels. Chem. Eng. Sci. 2006, 61, 2835-2842. (10) Ducci, A.; Yianneskis, M. Vortex identification methodology for feed insertion guidance in fluid mixing processes. Chem. Eng. Res. Des. 2007, 85, 543-550. (11) Sirovich, L. Turbulence and the dynamics of coherent sturctures. Q. Appl. Math. 1987, 45, 561-590. (12) Berkooz, G.; Holmes, P.; Lumley, J. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. ReV. Fluid Mech. 1993, 25, 539-575. (13) Chiekh, B.; Michard, M.; Grosjean, M.; Bera, J. C. Reconstruction temporelle d’un champe ae´rodynamique instationnaire a` partir de mesures PIV non re´solues dans le temps. Proc. of 9e` me Congre` s Francophone de Ve´ locime´ trie Laser, Brussels, Belgium, September, paper D, 2004, 42, 93109. (14) van Oudheusden, B.; Scarano, F.; van Hinsberg, N.; Watt, D. Phaseresolved characterization of vortex shedding in the near wake of a squaresection cylinder at incidence. Exp. Fluids 2005, 39, 86-98. (15) Perrin, R.; Cid, E.; Cazin, S.; Sevrain, A.; Braza, M.; Moradei, F.; Harran, G. Phase-averaged measurements of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV and 3C-PIV. Exp. Fluids 2007, 42, 93-109. (16) Bittorf, K.; Kresta, S. Active volume of mean circulation for stirred tanks agitated with axial impellers. Chem. Eng. Sci. 2000, 55, 13251335. (17) Paglianti, A.; Montante, G.; Magelli, F. Novel experiments and a mechanics model for macroinstabilities in stirred tanks. AIChE J. 2006, 52, 426-437.

ReceiVed for reView July 2, 2007 ReVised manuscript receiVed September 17, 2007 Accepted September 18, 2007

IE070905M

Decomposition of Flow Structures in Stirred Reactors and Implications

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