Segregated regions in continuous laminar stirred tank reactors

Segregated regions in continuous laminar stirred tank reactors

Chemical Engineering Science 59 (2004) 1481 – 1490 www.elsevier.com/locate/ces Segregated regions in continuous laminar stirred tank reactors P.E. A...

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Chemical Engineering Science 59 (2004) 1481 – 1490

www.elsevier.com/locate/ces

Segregated regions in continuous laminar stirred tank reactors P.E. Arratia, J.P. Lacombe, T. Shinbrot, F.J. Muzzio∗ Department of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, NJ 08855-0909, USA Received 4 March 2002; received in revised form 30 May 2003; accepted 10 June 2003

Abstract Using visualization techniques, including acid/base reactions and UV 5uorescence, we provide experimental evidence of segregated regions (islands) during mixing of viscous Newtonian 5uids under laminar 5ow conditions in continuous stirred tank reactors (CSTRs). The e8ect of inlet/outlet stream position and Reynolds number on the dynamics of the mixing processes is examined. Numerical experiments in 3-D map were able to capture the main features of the CSTR 5ow by perturbing a Batch system using an imposed axial 5ow. Asymmetric 5ow patterns produced by o8-center positioning of inlet and outlet pipes cause a reduction in size of the segregated region, enlarging the chaotic region and leading to more e
1. Introduction An extremely pervasive misconception, perpetuated by most undergraduate textbooks, is the assumption of complete uniformity of concentration and temperature in a continuous stirred tank reactor (CSTR). Under this assumption, reactions take place at a unique concentration (and temperature), which is also the concentration of the e?uent. Since the vessel is assumed to be completely mixed, the continuity equations for the components and the energy equation can be trivially integrated over the complete volume, simplifying much of the design calculations. In addition, since temperature uniformity is assumed, under steady state, the CSTR operates isothermally, uncoupling the mass and energy balances. However, to achieve this ideal mixing state, it is a necessary condition that the feed be intimately mixed with the contents of the reactor in a time interval that is very small compared to the mean residence time of the 5uid through the vessel. As shown in this paper, under laminar 5ow conditions, the above uniformity assumption in continuous stirred tank reactors is far from warranted. ∗ Corresponding author. Tel.: +1-732-445-3357; fax: +1-732-445-6758. E-mail address: [email protected] (F.J. Muzzio).

0009-2509/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/j.ces.2003.06.003

Mixing in stirred tanks has been the subject of extensive academic and industrial research. The main goal is to understand the underlying physics of energy, mass, and momentum transport and their interactions in such reactors. Fundamental studies have used experimental, analytical, and numerical approaches, but a complete understanding of laminar mixing in stirred tanks is still lacking. Early work focused on power consumption and torque requirements as a function of impeller geometry (O’Connell and Mack, 1950). Biggs (1963) determined mixing times by measuring the conductivity of an ionic tracer fed to a mixing tank. Norwood and Metzner (1960) obtained values for mixing indexes by monitoring the extent of completion of an acid/base reaction. Most other studies measured concentrations using intrusive probes. As a direct consequence, they disturbed the 5ow pattern they intended to characterize. Since the introduction of ‘age-distribution’ function and the concept of distribution of residence time by Danckwerts (1953), continuous stirred tank reactors (usually operated in the turbulent regime) have been most often analyzed using residence time distribution (RTD) theory. In this context, the age of a 5uid element is deJned as the time elapsed since it enters the reactor until it leaves. In most cases, there is an exponential distribution of 5uid particle

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residence times in continuous reactors. The measurement of RTD can be accomplished by the injection of a tracer in the system and subsequent determination of the tracer concentration in the 5uid leaving the system. The above method is widely used in characterizing the behavior of reaction systems in turbulent regime in continuous systems (Naumann, 1969; Fernandez-Sempere et al., 1995). In most cases, the studies were concerned with turbulent systems using fast nonlinear reactions, where characterization of the 5ow Jeld is a rather di
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Fig. 1. (a) Segregated zones (toroidal regions) illuminated using UV Fluorescence in a Newtonian batch case using a Rushton impeller, (b) Segregated zone and compartmentalization in a rounded-bottom multi-impeller system using axial intermig impellers.

may be slowed and even halted before reaching completion, undesired reactions are enhanced, and product selectivity is decreased. However, little attention has been devoted to realistic 3-D laminar mixing of 5uids in stirred tanks. Experimental work such as those by Lamberto et al. (1996), Lamberto (1997), Alvarez et al. (2002), and computational studies by Zalc et al. (1999), Lamberto et al. (1999), Ranade (1997), and Harvey and Rogers (1996) have focused on batch stirred tanks. Observations and results from the above studies demonstrate the presence of highly stable segregated non-chaotic zones in a sea of chaotic 5ow. A common approach to overcome this problem has been the use of complex impeller geometries for low Re mixing. Recent studies, however, have proposed that islands of non-chaotic motion within a chaotic 5ow can often be eliminated by imposing a dynamic perturbation (Franjione et al., 1989; Liu et al., 1994a,b). One can achieve chaotic conditions in stirred vessels by using variable-speed stirring protocols (Lamberto et al., 1996), and/or by introducing geometric asymmetries (Fountain et al., 1998; Alvarez, 2000). In the case of continuous stirred tank reactors, one may use the inlet and outlet 5ow as a way to e
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was used to minimize optical distortion due to the curvature of the tank. 2.2. Experimental plan

Fig. 2. Experimental setup of a continuous ‘re5ux’ system. We investigated three inlet stream positions (A, C, P) with respect to the tank center line and one Jxed outlet stream position, which is at the bottom of the tank directly below the (P) position (7:5 cm from the tank center line).

impeller-driven 5ow is perturbed by the cross 5ow, giving rise to an asymmetric 5ow pattern that causes a reduction in size of the segregated regions, enlarging the chaotic region and leading to more e
Our goal is to examine the in5uence of the inlet stream position on the dynamics of a continuous stirred tank reactor at di8erent Reynolds numbers (Re). A constant re-circulating 5ow rate was maintained at 4:75 l=min (thus corresponding to a residence time of 3:05 min). The 5uid was re-circulated in order to maintain a constant volume system (constant tank Jlling) since the structure of the 5ow (and thus the dynamics of the mixing process) presumably changes with volume Jll. Three inlet positions were examined: Parallel (P), Center (C), and Across (A) with respect to the outlet stream. The above notation relates to the side of the inlet stream with respect to the tank centerline as seen by an observer directly in front of the tank. In separate experiments, we investigated all three inlet positions and two agitation speeds: 100 and 200 rpm, corresponding to Reynolds number values of 30 and 60, respectively, calculated as customary, i.e. ND2 ; (1) Re =  where N is the impeller agitation frequency, D is the impeller diameter,  is the 5uid viscosity and  is the 5uid density. 2.3. Flow visualization Flow visualization experiments are a useful tool to study mixing. Carefully conducted dye advection experiments reveal 5ow patterns and topological structures that serve as the starting point to analyze 5uid mixing processes from a dynamical systems viewpoint. Dye advection experiments are based on the assumption that the tracer will move with the mean velocity of the 5ow. A description of the mixing process is assessed by the location of dye as a function of space and time. Two well-known 5ow visualization techniques are employed in this study: acid/base visualization reaction and UV 5uorescence. The main advantage of these visualization techniques is that they are non-intrusive; the 5ow Jeld remains unperturbed during visualization experiments and a true assessment of the 5ow behavior can be achieved. For both techniques, 5ow patterns were recorded using a Canon EOS Elan II 35 mm with Kodak’s Elite chrome 100 Jlm and a Kodak digital camera. Acid–base reactions are very useful in unveiling the nature of the 3-D 5ow patterns (Lamberto et al., 1996). Since the reaction is intrinsically very fast, the reaction is at local equilibrium at all times. Alkaline and acidic injections dispersed in glycerin (to minimize tracer di8usion e8ects) are used to drive the pH from one side to the other of the pH transition range of an indicator. Experiments begin with an alkaline (blue) tank (see Fig. 3). Then, an acidic injection is made at the inlet at T = 0. The change in pH, and consequently color, identify well-mixed (chaotic) regions

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(a)

Fig. 3. Initial states of the acid/base reaction visualization technique. The pH indicator turns blue (a) when the solution is alkaline and yellow when the solution is acidic (b). Since the reaction is very fast, we are able to distinguish segregated zones from the main chaotic 5ow.

and segregated regions. Segregated regions (containing unreacted base) remain blue as time progresses, while chaotic regions become yellow. Eventually, acid penetrates the segregated regions due to di8usive mechanism, neutralizing the base and yielding a chemically homogeneous system. Approximately 150 ml of pH indicator solution (0.04% Bromothymol Blue) is added to the tank to visually di8erentiate between acidic (yellow) and alkaline (green-blue) conditions. The resulting solution is made alkaline by the addition of a well-mixed solution consisting of 3 ml of 1 N NaOH and 50 ml of glycerin. The system is constantly agitated and the pump is turned on until a uniform blue color is achieved. Next, a well-mixed solution consisting of 6 ml of 1 N HCl in 100 ml of glycerin solution is injected to restore the acidic conditions in the tank. As a result, the blue color disappears as acid reacts with the medium. In chaotic regions, the blue color disappears quickly. However, since the acid is injected only in the chaotic sub-domain, the acid penetrates segregated regions only by di8usion (a very slow process). As a consequence, segregated regions remain basic, and are revealed as blue-green zones in the tank. We also employ an ultraviolet 5uorescence visualization technique (Alvarez, 2000). UV 5uorescence is one of the simplest 5ow visualization techniques for 5uid mixing, requiring only a suitable light source and a 5uorescent dye (Fluorescein, Fisher ScientiJc). By using non-di8using passive tracer, which moves with the mean mass velocity of the 5uid, we are able to track material volumes and consequently illustrate the tank 3-D mechanism. Ultraviolet light can be considered to be divided into three bands: long wave UV that extends from 320 to 400 nm, medium wave UV that extends from 280 to 320 nm, and short wave UV that extends from 200 to 280 nm. In this work we utilized long wave re5ected UV light at 365 nm; long wave UV radiation can pass freely through most modern camera lenses, in contrast to the short bands that are absorbed by normal glass lenses. Two long wave UV lamps (radiation sources) are placed directly on top of the tank. A Jlter is placed on the camera lens, which in turn allows ultraviolet light to pass and

(b)

(c)

Fig. 4. Flow pattern produced by ‘base-5ow’ visualized using acid/base reaction (a,b) and UV Fluorescence (c). Note the clear formation of an asymmetric re-circulating zone at the top of the tank due to 5ow expansion.

absorbs all visible light. Injections were made of Sodium– Fluorescein in glycerin at a concentration not higher than 0:2 mg=ml. The lamps are turned on and an injection is made next to the impeller. The evolution and dynamics of the segregated areas are recorded using a camera with a UV Jlter. The evolution of the mixing processes is recorded from its initial condition to 35 min. 3. Results We begin our investigation by inspecting the underlying 5ow generated by the 5uid being circulated through the reactor without an impeller or shaft. The inlet is placed at the center of the tank and the outlet was maintained at 4:5 cm from the outer wall (Fig. 2). A distinct re-circulating zone in the axial direction at the inlet position (Fig. 4) is observed. This means that 5uid (or reactant) entering the reactor is immediately segregated from the upper region resulting in an ine
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Fig. 5. Center case: Jgure (a,b,c) represents experiments at Re = 30 while (d,e,f) displays experiments at Re = 60. Note the top re-circulating zones in both cases, which disappear shortly as the experiments continue. The lower segregated zone is quickly destroyed at Re = 30 but not at Re = 60, where the zones are more robust.

a Newtonian batch-case, a sharp separatrix between the upper and lower halves of the tank is observed along the impeller mid-plane. A slow mixing region is also observed at the very top of the tank, where the pumping action of the impeller is not strong enough to induce top–bottom 5uid circulation. Furthermore, a distinct re-circulation zone is apparent at the inlet. This is expected due to the entrance 5ow. Nevertheless, this re-circulation zone at the top of the tank indicates that the inlet produces a region of slow mixing from the start. In other words, a new segregated region is created by the inlet 5ow. As mentioned above, the inlet and outlet 5ows introduce an asymmetry in the 5ow Jeld, a8ecting the shape and dynamics of the segregated regions. For example, the lower torus is strongly perturbed by the outlet 5ow, disappearing after 10 min of operation. The upper torus, which is perturbed by the inlet 5ow, proves to be more robust as it conserves its ‘integrity’ for at least 45 min. On the other hand, for the case at Re = 60, the convective 5ow produced by the inlet/outlet stream is not strong enough to perturb the toroidal regions caused by the impeller axial 5ow, as shown in Fig. 5. The toroidal regions are no longer ‘tilted’; they are increasingly stable and robust. Next, we examine the parallel case (P) (Fig. 6). The asymmetry of the left inlet stream with respect to the tank wall has a substantial e8ect on the dynamics and shape of the upper segregated region. The inlet stream destroys the upper segregated region in just a few minutes. A fragment of the upper doughnut is still visible, but its shape is very distorted and unstable. The lower segregated region is ‘tilted’ about 7◦ (the left end of the toroidal region is pointing upwards) with respect to the impeller horizontal mid-line but it remains stable during the early part of the experiment. With time, the lower doughnut experiences a convective 5ow that erodes its outer layers, decreasing di8usive length scales, and the segregated region slowly disappears. As ob-

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Fig. 6. The ‘P’ case. The cross 5ow perturbs the system at Re = 30 (a,b) and Re=60 (d). The inlet/outlet 5ow is stopped, returning to batch system conditions (c). The distorted segregated zone is now unperturbed, and returns to its ‘batch shape’ (d). If the 5ow is re-started, the segregated zone goes back to the distorted state. A close-up of the toroidal regions in batch mode (e).

Fig. 7. Across case at Re = 30 (a,b,c) and Re = 60 (d,e,f). The pictures show that 5ow perturbation is more pronounced and beneJcial at lower speeds (c,d). At higher speeds, the segregated zones are more robust, and consequently more di
served in the center case (C), increasing agitation speed to 200 rpm (Re = 60) does not perturb the segregated regions. In this case, the top re-circulating region is still present, and is completely distinguishable from the upper segregated region. The upper and lower toroidal regions are no longer ‘tilted’ and remain stable throughout the entire experiment. Finally, we examine the across (A) conJguration (Fig. 7). At a Re = 30, the imposed 5ow does not have as dramatic an e8ect as in the parallel (P) case. Unlike the parallel case, the upper toroidal zone is present and distinguishable from the top re-circulating region. Also, the toroidal regions persist for a longer period of time. Nevertheless, we observe that once again the upper and lower segregated regions are ‘tilted’ about 7◦ (right end of

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(a)

(b)

(c)

Fractional Area Coverage, %

100 80 Re Re Re Re Re Re Re

60 Efficiency

40

30 30 30 60 60 60 30

(C) (A) (P) (C) (A) (P) Periodic (A)

20 0

(d)

0

1000

2000

3000

4000

5000

6000

Mixing Time, Revolutions

Fig. 8. Shape of the toroidal region with pump on (a) and pump o8 (b). Close-up of the toroidal region (c). Evolution of the mixing process using a periodic pumping protocol.

Fig. 9. Mixing e
toroidal regions are pointing upwards), re5ecting local 5ow asymmetry. As time progresses, the lower segregated region becomes unstable, gets distorted, and is slowly washed-out. The upper toroidal region goes through the same process, but in a slower fashion. Increasing agitation speed (Re =60) does not aid in destroying the segregated regions. As in the previous cases, the top re-circulating region and both toroidal structures remain present and became more stable. They are not tilted, nor destabilized as time progresses. An additional feature is shown in Fig. 8 for CSTRs. We observe that, similar to batch systems, the toroidal regions have complex inner structures (see Fig. 1) that are revealed slowly over time, as their outer ‘shells’ are eroded by di8usion. In continuous reactors, however, the convective nature of the inlet and outlet 5ow accelerates the process of toroidal erosion. However, as mentioned before, this erosion is only possible provided that the 5ow Jeld around the segregated zones is weaker than the perturbation 5ow Jeld. The impact of the 5ow perturbation on torus topology is examined by turning the pump on (Fig. 8(a)) and o8 (Fig. 8(b)). It is apparent that when the pump is turned o8, the torus returns to its ‘original’ (batch) position and shape. The inner core of the torus is exposed in this case revealing its essential features (Fig. 8(c)). This reversibility to the original state may prove beneJcial for practical methods of destroying segregated regions. Since the torus contracts with axial 5ow perturbation and relaxes without it, one can apply a periodic or aperiodic forcing to destroy segregated regions with inertia, since part of the segregated region will necessarily have to follow a new re-arranged trajectory, most likely a chaotic one. In fact, preliminary experiments on the across (A) conJguration (Re = 30) show that periodic pumping is able to destroy completely the upper segregated region (Fig. 8(d– g)), which remained present for the steady pumping case even after 25 min. We characterize the e
image analysis, we compute the fractional area of reacted 5uid as a function of time. Although image analysis is limited to pixel resolution and types of structures produced by the mixing protocol, area coverage data provides valuable information about the state of ‘mixedness’ at a certain time interval. We can also obtain the maximum area that can be covered by the dye and the rate at which the process occurs. We emphasize that this plot is not to be taken as an equivalent of concentration proJle, since the intensity of the pixels is not correlated with reactant concentration. For the time being, we will only concern ourselves with the size of the segregated and chaotic areas, and compare the mixing performance of each case. Since we start with an alkaline (blue) system, we measure the area of the remaining blue spots by setting a pixel threshold and di8erentiating the area of the segregated structures that fall within the threshold parameter. Once the picture is processed, a scale is set, and we measure the size of the areas of interest. The overall result is a plot of the percentage of the area of reacted 5uid versus time, as in Fig. 9 for two Re (30, 60) for all three cases (Center, Parallel, and Across) and one periodic pumping case at Re = 30. In all cases, the evolution of the segregated regions or neutralized regions can be represented by generic curves of the type (Alvarez, 2000) A = Amax (1 − e−mt );

(2)

where A is the area coverage, Amax is the maximum achievable area coverage, which is speciJc to a given geometry and Re, m is the intensive rate of coverage, and t is the time (in seconds or minutes). We can rewrite Eq. (2) as A = Amax (1 − e−KN )

(3)

to obtain a dimensionless mixing rate (K) that is dependent on dimensionless time N (revolutions). Using Eqs. (2) and (3), we are now able to obtain the coverage rate constant (m) and the mixing rate constant (K) by non-linear regression analysis (see Fig. 10). The results

Area Coverage Regression Analysis, Re= 134 (P) Fractional Area Coverage, %

100 80 Regression Data

60 40 20 0 0

500

1000

1500

2000

2500

3000

% Area Size of Upper Toroidal Region

P.E. Arratia et al. / Chemical Engineering Science 59 (2004) 1481 – 1490

30 Re 30 (C) Re 30 (A)

25

Re 30 (P) Re 60 (C)

20

Re 60 (A) Re 60 (P)

15

Re 30 Periodic (A)

10 5 0 0

1000

Mixing Time, revolutions

Table 1 Values of Amax , m (Eq. (3)), K (Eq. (4)), and regression coe
s−1

Revolutions−1

Amax (%)

R2

Re Re Re Re Re Re Re

0.0053 0.0097 0.0063 0.0027 0.0057 0.0094 0.0059

0.0032 0.0028 0.0038 0.0029 0.0034 0.0028 0.0041

88.00 86.76 90.44 91.27 85.51 85.97 92.47

0.978 0.975 0.935 0.908 0.959 0.978 0.973

30 60 30 60 30 60 30

(C) (C) (P) (P) (A) (A) (A) Periodic

of the regression analysis are presented in Table 1. We present area coverage rate constant in reciprocal seconds (s−1 ) and mixing rates in reciprocal revolutions (rev−1 ). It is interesting to notice that by using time units, the most efJcient mixing rate constant is the case of Re = 60 (Center). However, if we use dimensionless time, the most e
2000 3000 4000 Time, revolutions

5000

6000

Fig. 11. Evolution of the upper segregated area as a function of time measured by image analysis.

% Area Size of Lower Toroidal Region

Fig. 10. Regression analysis of fractional area coverage data obtained from visualization experiments. Regression is performed to obtain mixing constant and maximum area coverage for the eccentric left case (-), Re = 60.

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20 Re Re Re Re Re Re Re

16 12

30 30 30 60 60 60 30

(C) (A) (P) (C) (A) (P) Periodic (A)

8 4 0 0

1000

2000

3000 4000 Revolutions

5000

6000

Fig. 12. Evolution of the lower segregated area as a function of time measured by image analysis.

a comparison between the steady pumping case and the periodic case shows the beneJcial e8ect of the later on destroying the upper toroidal region. Similarly, increasing agitation speed reduces the impact of the perturbation on the lower segregated region (Fig. 12). However, at longer times, the dynamics of the lower segregated region is quite di8erent. While at Re = 60, the lower segregated region reaches an asymptotic value leading to a slow decay mainly driven by di8usion, at Re = 30 the behavior of these toroidal regions tend to a rapid decay, after reaching an asymptotic value, leading to their complete destruction. This may imply that the stability of such toroidal region increases considerably at higher Re. In the case of lower segregated regions, periodic pumping leads to the lowest asymptotic value and a complete destruction of the toroidal region as time evolves. As in batch stirred tank reactors, the location and size of the tori is a function of rpm (or, equivalently, Reynolds number). The initial size of the segregated regions at 100 rpm is larger than at 200 rpm. Comparing on a vertical plane, the inner edge of a tori is much closer to the impeller at 100 rpm than at 200 rpm. In addition, the position of the centers of

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the tori changes with rpm. The top torus moves downward and out, while the lower torus moves upwards and out as we increase Re (Fig. 5). In addition to the above features (present also in batch reactors), we observe that in this continuous system, the robustness of toroidal regions increases with rpm, minimizing the e8ect of the 5ow perturbation imposed by the inlet and outlet 5ow. So far, we have observed that the CSTR, under laminar conditions, behaves as a batch system perturbed by an imposed axial 5ow. In order to better understand the role of imposed 5ow perturbations on the mixing mechanisms and 5ow structures, we use a 3-D map that mimics both the unperturbed case (batch) and a CSTR. This mathematical model of the dynamics of the stirred tank is a step toward the development of an analytical method for understanding the continuous stirred tank reactor and other variants of the stirred tank in general. The most important dynamics in the stirred tank are those associated with the torus: unmixed regions unable to translate, stretch and contract that represent the primary obstacle to e
Fig. 13. Investigating toroidal dynamics in a 3-D map. The batch (unperturbed) system: (a) experiment and (b) map; asymmetric 5ow in CSTR: (c) experiment, (d) map, and (e) Poincare section (map).

(6)

When combined with a volume preserving integration system, these equations can be converted from a continuous Eulerian perspective to a discrete Lagrangian one. The dynamics of the system alone are similar to the batch tank stirred with a concentric disk. Particles placed in the 5ow create a virtually inJnite set of nested 2-tori which are by deJnition 2-D torus-shaped surfaces embedded in the 3-D space. Fig. 13 (a and b) shows the segregated regions in the batch-case in a UV visualization experiment and in a numerical simulation, respectively. One can then look upon the CSTR as a perturbation of the dynamics that take place within the simple batch stirred tank. From the perspective of the nested tori, the perturbation 5ows are both axial and translational, meaning a perturbation of the velocity in the z- and x-directions. In the mathematical model, the simplest possible forms for these perturbations are all that is necessary to visualize the types of topological bifurcation experienced by these coherent non-mixing structures. In the interest of keeping this model simple, constant velocities were chosen for the perturbation of the model system. They were imposed onto the Lagrangian behavior of the system directly via superposition. The CSTR itself has illustrated some very interesting behaviors depending on the amount of in5ow and out5ow as well as their positions. The model shows striking similarities

to the CSTR (Fig. 13(c and d)). To place these observations within the framework of the CSTR, it is helpful to imagine an impeller (a disk) rotating between the two toroidal regions (along the plane z = 0). It is noticeable that the tori are not parallel to each other or to the z-axis. Breaking the perturbation down reveals that the x-directional perturbation alone is responsible for this warping. The tori also are not simply chirally reversed images. Each has a separate and distinct structure. Beyond that simple characterization, there are di8erences in the stability of the regions. The upper region seems to be more prone to chaotic behavior than the lower region. This is evident from the Jgure in that the amount of volume enclosed by the largest stable 2-torus is greater for the lower region than the upper region. The outer layers of the lower region leave the system as if they are 5owing out of the CSTR. The particles in the out layers of the upper region evolve chaotic trajectories and eventually penetrate the lower region. The central region of the lower torus remains stable in the face of all these changes (Fig. 13(e)). 4. Summary and conclusions We presented experimental and numerical evidence that, similar to batch reactors, the mixing process in continuous

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stirred tank reactors is governed by a template of chaotic and segregated regions that coexists in the same system. The results presented here reveal the complex relationship between mixing performance and impeller pumping capacity, inlet/outlet (pump) 5ow velocity, and inlet/outlet location in a continuous stirred tank reactor operating in a laminar regime. The axial 5ow produced by the inlet/outlet streams revealed re-circulating zones at the top of the tank, driven by the 5ow from the inlet pipe into the cylindrical vessel. This observation means that the inlet 5ow is segregating reactive 5uid material right at the beginning of the operation. A slow azimuthal 5ow is produced in the ‘base-5ow’ due to the asymmetric location of the inlet and outlet streams. In this case, the recirculating zones form a well-deJned and stable structure with a series of manifolds around an elliptic ring. The presence of inlet and outlet 5ow creates an asymmetry on the mixing 5ow mechanism leading to a more efJcient mixing process than with a batch reactor. The inlet location has a major in5uence on the upper segregated region. At low speeds (Re = 30), if placed in an asymmetric position with respect to the center of the vessel, it weakens, and even destroys, the segregated region. However, in a symmetric position, it distorts the region’s shape, but it does not destroy it. The disappearance of the lower segregating zones, on the other hand, is mainly caused by the outlet 5ow, which slowly perturbs and ‘washes-out’ the outer layers of the doughnut lowering di8usional length scales. The outlet location with respect to the center of the tank creates an additional source of asymmetry that seems to perturb the lower segregated zone. Moreover, it was observed that once the 5ow is stopped, both distorted toroidal regions return to their unperturbed state. This observation led us to implement a periodic pumping protocol that was able to destroy completely the toroidal regions in a shorter amount of time than in the steady pumping case. Perhaps one of the most interesting results is that at a given and constant inlet/outlet 5ow rate, the mixing efJciency decreases at a higher agitation speed (Re). This counter-intuitive result becomes evident once one inspects the experimental pictures and the mixing curves. The parallel case at Re = 30, proved to be the most e
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5ow structure once the agitation speed is increased from 100 to 200 rpm. Furthermore, a periodic pumping protocol was implemented in order to de-stabilize the toroidal regions and consequently achieve higher mixing performance. This time-dependent forcing proved to be an e8ective manner to destroy segregated regions. Even in its preliminary form, the torus model shows promise in improving the understanding of the dynamics of CSTRs. It displays results that are geometrically and topologically similar to those seen in a real system. In the future, these types of simple dynamic models might replace or rewrite heuristic understanding of mixing systems or eliminate possibilities in factorial experimentation in real systems. Beyond mixing, there are many applications for this torus pair model that justify its further exploration. References Ali, F.S., Strizhak, P., Menzinger, M., 1999. Stirring e8ects on bistability in a CSTR. 1. experiments and simulations in the AsO3/IO3 reaction. Journal of Physical Chemistry 103 (50), 10,859–10,865. Alvarez, M.M., 2000. Using spatiotemporal asymmetry to enhance mixing in chaotic 5ows: from maps to stirred tanks. Ph.D. Thesis, Chemical & Biochemical Engineering. Rutgers University, New Brunswick, NJ. Alvarez, M.M., Zalc, J., Shinbrot, T, Arratia, P.E., Muzzio, F.J., 2002. The mechanism of mixing and creation of structure in laminar stirred tanks. A.I.Ch.E. Journal 48 (10), 2135–2148. Biggs, R.D., 1963. Mixing rates in stirred tanks. A.I.Ch.E Journal 9, 636–647. Danckwerts, P.V., 1953. Continuous 5ow systems: Distribution of residence times. Chemical Engineering Science 2, 3857–3866. Dong, L.J., Johansen, S.T., Engh, T.A., 1994. Flow induced by an impeller in an unba?ed tank - I. Experimental. Chemical Engineering Science 49 (4), 549–560. Fernandez-Sempere, J., Font-Montesinos, R., Espejo-Alcaraz, O., 1995. Residence time distribution for unsteady-state systems. Chemical Engineering Science 50 (2), 223–230. Fountain, G.O., Khakhar, D.V, Ottino, J.M., 1998. Visualition of three-dimensional chaos. Science 281 (5377), 683–686. Fox, R., Villermaux, J., 1990. Micromixing e8ects in the ClO2 minus plus I minus reaction. Perturbation analysis and numerical simulation of the unsteady-state IEM model. Chemical Engineering Science 45 (9), 2857–2876. Franjione, J.G., Leong, C.W., Ottino, J.M., 1989. Symmetries within chaos: a route to e8ective mixing. Physics of Fluids A1, 1772–1783. Harvey, A.D., Rogers, S.E., 1996. Steady and unsteady computation of impeller stirred tank reactors. A.I.Ch.E. Journal 42 (10), 2701–2712. Houcine, I., Plasari, E., David, R., Villermaux, J., 1999. Feedstream jet intermittency phenomenon in a continuous stirred tank reactor. Chemical Engineering Journal 72 (1), 19–29. Lamberto, D.J., 1997. Enhancing laminar mixing in stirred tank reactors using dynamical 5ow perturbations Ph.D. Thesis, Chemical & Biochemical Engineering. Rutgers University, New Brunswick, NJ. Lamberto, D.J., Muzzio, F.J., Swanson, P.D., 1996. Using time-dependent RPM to enhance mixing in stirred vessels. Chemical Engineering Science 51 (5), 733–741. Lamberto, D.J., Alvarez, M.M., Muzzio, F.J., 1999. Experimental and computational investigation of the laminar 5ow structure in a stirred tank. Chemical Engineering Science 54 (7), 919–942. Liu, M., Muzzio, F.J., Peskin, R.I., 1994a. E8ects of manifolds and corner singularities on stretching in chaotic cavity 5ows. Chaos Solitons & Fractals 4 (12), 2145–2167.

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