Analysis on Laminar Chaotic Mixing Based on Configuration of Streak

Apr 23, 2012 - Streak Lobes in an Impeller-Agitated Vessel. Shunsuke Hashimoto,* Ryota Osaka, and Makiko Kawamata. Division of Chemical Engineering, ...
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Analysis on Laminar Chaotic Mixing Based on Configuration of Streak Lobes in an Impeller-Agitated Vessel Shunsuke Hashimoto,* Ryota Osaka, and Makiko Kawamata Division of Chemical Engineering, Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan ABSTRACT: In the case of laminar chaotic mixing in a stirred tank, our previous study theoretically revealed that the streakline originating from the tip of the impeller provides the formation of a mixing pattern and its outline comes close to the configuration of the streakline with respect to time. In the present study, the laminar chaotic mixing in a 3-D stirred tank was investigated experimentally by use of a new index, “streak lobes”, which were equivalent to the region surrounded by the streakline. The streak lobes were created by the minute perturbation wave (traveling wave) that is generated by the rotation of the impeller blade in a stirred tank. The amplitude and frequency of their traveling waves were dominant for the size (width and area) and/or number of streak lobes. The experimental results demonstrated that the large area of streak lobes in the 3-D velocity field resulted in good chaotic mixing, even if the number of streak lobes is small. In addition, the fluid exchange between top and bottom regions in the stirred tank occurred rapidly in the case where the streak lobes could come and go between two regions. It was suggested that the streak lobes can play an important role as an index whether or not good chaotic mixing occurs in the stirred tank. The present study proposed that the control of the size and/or number of streak lobes adjusting the amplitude and frequency of the traveling wave caused by impeller rotation resulted in a shortcut to enhance laminar chaotic mixing.

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are called the “stable manifold” and “unstable manifold”, played as a template for formation of the mixing pattern (Figure 1b), and consequently chaotic mixing was produced. In this case, the unstable manifold is equivalent to the streakline that is generated from the stagnation point of the interface between two cells. The fluid particles on a turn-style lobe always exist on that as long as the system is observed periodically. Hence, chaotic mixing proceeds gradually along with the configuration of turn-style lobes. In this 2-D flow system, additionally from numerical research, Inoue and Hirata6 revealed that the profile of the mixing pattern over time infinitely came close to the configuration of the unstable manifold (streakline) (Figure 1c) regardless of initial mixing patterns, which demonstrates the role of streakline as a template for formation of the mixing pattern. In a three-dimensional (3-D) agitated vessel, it is reasonable to guess that there are time-invariant structures as in the case of the 2-D system and they cause fluid mixing in a regular manner. It is essential for the analysis of the mixing state and mechanism to visualize the flow field. In general, the timeaveraged velocity distribution (streamline patterns) and/or the instant velocity vector distribution, and the moving trajectory of one or a few test particles in the flow field (pathline) have been used to analyze mixing phenomena in an impeller-agitated vessel, where the flow field periodically changes with respect to time. However, fluid mixing is the phenomenon that is based on the long-time variation of mutual positions of many fluid particles. Hence, the analysis from a Lagrangian viewpoint such

INTRODUCTION Background. Mixing in stirred tank reactors in a wide variety of tank sizes and impeller shapes has been often utilized to homogenize different substances and to conduct chemical reactions in industrial chemical processes. Recently, in various industrial processes, a wide range of operations for stirred tanks has been required depending on purposes and conditions. Particularly in the mixing of high viscosity materials such as polymerization reactions, mixing processes are conducted under low mixing Reynolds number (Re) conditions, which causes a problem of inefficient global mixing. To enhance mixing efficiency in impeller-agitated vessels under low Re conditions, various methods are performed as follows: unsteady agitation so that the rotation direction of the impeller is periodically inverted;1 the multistage impeller that includes a few agitating impellers installed in a vessel.2 These techniques are efficient and available, but they require large-scale equipment and large agitation power. Hence, it is favorable to enhance mixing efficiency by means of simpler tools and/or methods. This means, first of all, that we must analyze the mixing mechanism in stirred tanks under laminar and/or transition flow conditions by means of not only empirical evaluation but also a phenomenological approach. Analysis of Chaotic Mixing in a Three-Dimensional Stirred Tank. To figure out mixing phenomena accurately, it is essential to analyze them based on not time-variable information such as a flow structure but on a time-invariant structure. Figure 1a shows the convective flow field with twodimensional (2-D) cellular flow, where the interface between two cells (vortex) vibrates right and left. In this 2-D flow system that changes periodically, Camassa et al.3 and Wiggins4,5 theoretically and numerically revealed that the turn-style lobe formed by the tangle of two kinds of invariant manifolds, which © 2012 American Chemical Society

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December 25, 2011 April 4, 2012 April 22, 2012 April 23, 2012 dx.doi.org/10.1021/ie203036n | Ind. Eng. Chem. Res. 2012, 51, 6939−6947

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mixing regions, and they seem to be created by folds of streaklines. That is, as in the case with the streakline in the 2-D system, the 3-D streak sheet from the tip of the impeller plays an important role as a “mixing template” that determines the mixing pattern in the stirred vessel. In addition, the experimentally visualized streakline as the vertical cross section of the 3-D streak sheet can become one of the forceful tools for the analysis of mixing in an agitated vessel. Formation Mechanism of Streak Lobes. Minute perturbations generated by the rotation of the impeller blade in an agitated vessel are key factors for fluid mixing, which creates the traveling wave spreading on the stream near the vessel wall.9 The schematic illustration of the traveling wave caused by minute perturbations of the impeller blade is shown in Figure 2a. The phenomena of chaotic mixing have been investigated systematically in previous studies from Ottino et al.10−13 Although the novel aspects of chaotic mixing by means of streamline jumping were recently reported by Christov et al.,14,15 in general, laminar chaotic mixing proceeds along the unstable manifold which is generated from the stagnation point between two cavities of the stream line. Recently, we have proposed the simplified model of the 3-D velocity field in an agitated vessel.16 In this model, as shown in Figure 2a, the stagnation point between two cavities of the stream line quivers vertically in every passage of the rotational impeller blade in the vertical cross section of the vessel. Then, the streakline as an unstable manifold is generated from the tip of the impeller blade. Figure 2b shows the time variation of the streakline in the vertical section numerically calculated from the above model. The minute perturbations of the impeller blade are so imperceptible that they are generally hard to visualize. As shown in Figure 2b, however, these perturbations induce minute folds on the streakline as the unstable manifold in the agitated vessel. These streak folds are expanded near the vessel wall, and then the streak circulates in the vessel and returns to the central part of the vessel (near the impeller). Then, the folds and expansions of the streak are repeated and consequently the streak covers the space densely in the agitated vessel. In our research, a pair of prominence-like (in other words, fold-like) structures in an upward or downward direction (colored parts in Figure 2c) is defined as a new mixing index, “streak lobes”, which are formed in the region surrounded by streaklines. As in the case with the turn-style lobe of the 2-D chaotic system, it is reasonable to guess that fluid particles move from a lobe to the external lobe one by one every rotational period of the impeller. Hence, the configuration of the space distribution of the streak lobe in a group behaves like a time-invariant template that conducts convective mixing phenomena in the agitated vessel. For example, if the steak lobe in the bottom (or top) edges into the top (or bottom) (Figure 2c), the chaotic mixing proceeds between the top and bottom regions of the vessel. The number, size, and dynamic behavior of the streak lobe, which depend on the amplitude and frequency of the traveling wave caused by the rotational impeller blade, are important factors for efficient mixing.17 Objectives of the Present Study. The present study focused attention on streak lobes as an index for the analysis of chaotic mixing in an agitated vessel. First of all, the mixing patterns were visualized by means of a decolorizing method and the mixing time was determined. Second, the vertical cross section of the 3-D streak sheet was visualized by injecting a tracer fluid from one side of impeller tip and the mixing

Figure 1. Conceptual images of laminar 2-D chaotic mixing: (a) schematic illustration of 2-D cellular cavity flow; (b) two timeinvariant manifolds and turn-style lobes; (c) the similarity between the structure of unstable manifold (streakline) and the outline of mixing patterns. They can be obtained from numerical simulation by the use of 2-D cellular flow.

as time integration of the velocity vector of a fluid-particle cluster (that is, streak) is available. In a 3-D stirred tank, of course, it is impossible to visualize the turn-style lobe directly. In addition, unlike the 2-D system, it is difficult to predict the streak structure and visualize the significant streak for mixing analysis in a 3-D stirred tank because the configuration of the streak is significantly different from those of the streamline and the pathline,7 and the trajectories of fluid particles drastically metamorphose depending on their initial positions. Therefore, it is important for the visualization of the streak for mixing analysis to select the origination of the streak appropriately. Based on our recent studies,8,9 the 3-D streak sheet that is visualized by injecting a tracer fluid (for example, methylene blue) from the tip of the impeller can be adopted as a form of unstable manifold in the 2-D system, which is static as long as it is observed at every rotational period of the impeller. As mentioned before, fluid particles near the unstable manifold always exist on the same one. In the mixing pattern, there are many streaky contrasting patterns around a pair of isolated 6940

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Figure 2. Schematic illustration of (a) traveling wave caused by minute perturbations of impeller-blade rotation, (b) time variation of streakline in vertical section, and (c) streak lobes. They were calculated from the simplified model of a 3-D velocity field in an agitated vessel.16

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EXPERIMENTAL SECTION Materials. In the present study, we used several chemical materials as follows: research grade starch (hydrosoluble), sodium thiosulfate (Na2S2O3, anhydrous, mole fraction purity 0.950), potassium iodide (KI, mole fraction purity 0.995), iodine (I2, mole fraction purity 0.998), sodium fluorescein (uranine), methylene blue (C16H18N3SCl, weight purity >70%), glycerol (glycerin, mole fraction purity 0.990), sodium hydroxide solution (NaOH, 0.001 mol/m3), and hydrochloric acid (HCl, 0.001 mol/m3). All were obtained from Wako Pure Chemical Industries, Ltd., and were used without further purifications. In addition, deionized water was produced using water-manufacturing equipment made by Nihon Millipore K. K. Apparatus. The experimental apparatus in the present study consisted of the following parts: acrylic cylindrical vessel (handcrafted), two-paddled agitating impellers (handcrafted), agitating motors (HEIDON, BL-600), torque meter (type YT, Shintou Kagaku, Co., Ltd.), syringe pump (HARVARD, PHD2000), Ar ion laser sheet control unit (Spectra Physics, 385F), and digital video camera (NAC, MLX-ST-706). The diameter of the vessel (D) was 120 mm. To reduce photographic distortion, the vessel was immersed into a square vessel of

mechanism under laminar and transition conditions (Re < 100) was analyzed based on streak lobes. Two types of impeller blade, ordinary paddle (OP) and alternative pitched paddle (AP), were adopted in the present study. Kato et al.18 have preliminarily revealed that the alternative pitched-paddle impeller exhibits higher mixing performance than the OP one, particularly under high Re (turbulent flow) conditions, while the mechanism of mixing enhancement with the AP is still unclear. Based on the above-mentiond theory, it is reasonable to guess that the AP impeller can create one traveling wave with a pair of blades, while the OP impeller generates one wave with every one of the blades. Then, it is reasonable to guess that the amplitude of the traveling wave for the AP impeller is larger than that of the OP one, while the frequency of the wave for the AP impeller is smaller than that of the OP one (for details, see Figure 3). One of the objectives in the present study is to investigate which wave is better for laminar chaotic mixing. In the present study, the difference in chaotic mixing performance between the OP and AP impellers was discussed based on the geometric structure of the streak lobe. 6941

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study a mixing state and pattern. First of all, water solution of glycerin was prepared at the concentration of 80 mass % as a working fluid. The viscosity and density of 80 mass % glycerin solution were 81.6 mPa·s and 1198 kg/m3 at 293.15 K, respectively. The high viscosity of the agitating fluid enabled us to not consider the effect of diffusive mixing. Then, appropriate quantities of I2 were dissolved in glycerin solution with the help of KI, and this solution was black in color by the dissolution of starch (iodo−starch reaction). The necessary quantity of this solution was placed into the vessel. After that, an agitating motor was operated at the desired rotating speed (rpm) that was suitable for the desired Re. After the fluid field reached a cyclostationary state in the agitated vessel, the Na2S2O3− glycerin solution prepared at the desired concentration was injected to the vessel from the tip of the impeller blade. The convective-mixing region was decolorized by the redox reaction, while the immiscible region stood out in bold relief as poor mixing regions. Consequently, mixing patterns were obtained in the agitated vessel. The experimental images were recorded by the use of a digital video camera. In addition, the time needed for complete decolorization (complete mixing) was determined as the dimensionless mixing time, nT = tm/T, where tm and T stand for the required mixing time and the rotational period of the impeller, respectively. That is, nT was equivalent to the rotational number of the impeller until the completion of decolorization. In the present study, nT was determined once the appearance of the mixing pattern (color pattern) did not change with respect to time (for details, see the section Mixing Patterns). Visualization of 3-D Streak and Its Cross Sections. In the preliminary experiment, the 3-D streak sheet was visualized by injecting a tracer fluid (methylene blue) from the tip of the impeller for both impellers (Re = 40). As mentioned below, the 3-D streak sheets have the structure of a helical envelope plane, which looks like axially symmetric in the unbaffled vessel. Hence, in the present study, the vertical cross section of the streak sheet was visualized and then the 2-D information (streakline) picked out from the 3-D information (streak sheet) was analyzed in both the OP and AP impellers. As aforementioned, water solution of glycerin that was prepared at the concentration of 80 mass % was poured to a height (H) of 120 mm in the vessel as a working fluid. The basic working fluid (80 mass % glycerin−uranine aqueous solution) was initially prepared by adding a small amount of basic solution consisting of NaOH and glycerin solution, which was used as a fluorescent tracer fluid. After the impeller reached a certain rotational speed and then sufficient time passed, the tracer fluid was continuously injected into the vessel by use of a syringe pump from the tip of the impeller blade. The injection rate of the tracer fluid was 15 mL/min, which did not affect the flow field in the agitated vessel. The streaklines in a vertical direction were induced to fluoresce using a plane sheet of Ar ion laser light, which was well-known as “laser-induced florescence” (LIF). The sequential digital images of streaklines were taken with a digital video camera. Measurements of Required Power for Mixing. In a fluid mixing processes, it is favorable to establish a homogeneous mixing state with as little power as possible.19 In the present study, the torque needed for agitating was measured by the use of a torque meter that included a mechanical spring system. The required power for mixing, P, and power number, Np, were obtained from the following equations:

Figure 3. Schematic illustration for three types of impeller and their flow patterns: (a) OP impeller; (b) AP impeller; (c) GP impeller. Each panel contains an expanded view of the impeller-blade tip.

acrylic resin filled with water. The experimental apparatus in the present study was similar to that of the previous one.17 In the present study, two types of two-bladed paddle impellers were adopted, which was the simplest among various impellers. The schematic illustration of the ordinary paddle (hereafter, OP) (panel a) and the alternative pitchedpaddle (hereafter, AP) (panel b) impeller blades is shown in Figure 3. In the case of the OP impeller, the diameter (d), length (l), and thickness of the impeller were 60 (d/D = 0.5), 12 (l/D = 0.1), and 3 mm, respectively. On the other hand, the AP impeller had the same projected area as the OP one in a vertical cross section. The pitched angle of the AP impeller blade was 45°. Just for reference, a general pitched-paddle (45°) (GP) impeller blade is shown in Figure 3c, which also had the same projected area as the other blades. Unlike the AP impeller that had one each of up-pumping and down-pumping blades, the GP one had both down-pumping blades. Hence, the GP impeller blade is familiar as a special impeller for generating strong axial flows, which is not investigated in detail here. As shown in Figure 3, one of the impeller blades was fabricated so that the tracer fluid could be injected into the vessel from the tip of the impeller blade. The impeller was installed on the centerline at a distance (h) of 60 mm from the bottom of the vessel, which was half the diameter of the vessel (h/D = 0.5). The working fluid was poured to the height of 120 mm in advance, which was equivalent to the diameter of the vessel (H/D = 1.0). The viscosity (μ) and density (ρ) of the fluid used in the present study were measured by a cone−plate type viscometer (Tokyo Keiki Co., Ltd., VISCONIC ELD) and a pycnometer, respectively. Based on these data, in the present study, the mixing Reynolds number defined as Re = ρnrd2/μ was controlled under laminar and transition conditions of Re < 100, where nr was the rotational speed (rpm) of an impeller. The experimental temperature was measured by a Pt resistance thermometer (Thermoprobe Inc., TL-1A). The uncertainties of the temperature and viscosity were 0.06 K and 1%, respectively. Procedures. Visualization of Mixing Patterns. In this method, a redox reaction of I2 with Na2S2O3 was utilized to 6942

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P = 2πnrTr

(1)

Np = P /ρnr 3d5

(2)

injected from the following two positions: the whole area (method I) and the upper-half area (method II) of the tip of the impeller blade. These methods enabled us to evaluate the mixing performance based on all three standards. In particular, in the latter method, the efficiency of fluid exchange between the top and bottom parts in the vessel can be visualized accurately. Figures 5 and 6 show typical images of time

where Tr was the rotational torque of an impeller. In addition, the obtained values of Np were correlated with the Kamei− Hiraoka equation,20 which is well-known as a correlation equation for Np in an agitating vessel without baffles.

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RESULTS AND DISCUSSION Flow Fields. Figure 4 demonstrates the patterns of the flowvelocity vector for the OP, AP (45°), and GP (45°) impeller

Figure 5. Time variation of mixing patterns in the agitated vessels obtained by the decolorizing experiment (method I, Re = 40): (a) OP impeller; (b) AP impeller; (c) GP impeller. The impeller rotates in a clockwise direction and then down-pumping flow is dominant initially.

Figure 4. Instantaneous (perfectly developed) flow-velocity vector in stirred tank (Re = 40): (a) OP impeller; (b) AP impeller; (c) GP impeller. All panels contain the vertical cross section where the impeller blades and the shaft exist. Large black arrows stand for the overall ones that summarize the small gray arrows.

blades, which were obtained from numerical simulation by use of the commercial CFD software, R-Flow (RFLOW, Co., Ltd.). In calculations, the same scale of experimental agitated vessel and flow properties were adopted (Re = 40). Unlike the case of the GP impeller blade that has uniform flow from the bottom to top regions in the stirred tank (Figure 4c), a pair of vortex flows is generated above and below the impeller blade in both the OP (Figure 4a) and AP impeller (Figure 4b) blades (shown by the use of two cyclic arrows in Figure 4a,b). The flow profiles of both the OP and AP impellers are comparable with each other except for the vertically asymmetric nature in the case of the AP impeller. Mixing Patterns. In the mixing process, there are generally three standards for analyzing mixing performance as follows: (1) the mixing time (required time for homogeneous mixing), (2) the fluid exchange degree between the top and bottom regions in the stirred tank, and (3) the existence of an isolated mixing region. In the present study, the decolorizing fluid was

Figure 6. Time variation of mixing patterns in the agitated vessels obtained by the decolorizing experiment (method II, Re = 40): (a) OP impeller; (b) AP impeller.

variation on mixing patterns in the agitated vessels at Re = 40 using OP (panels a) and AP (panels b) impellers. Figures 5 and 6 correspond to the results obtained from methods I and II, respectively. Generally, the flow patterns in an impeller-agitated vessel are dominated by the primary swirl flow and secondary discharge flow. As can be seen in Figure 5, mixing proceeds from near the vessel wall (t = 20T). Long after that (t = 30T), decolorizing proceeds gradually and expands near the shaft. Then, decolorizing proceeds in the region except for the center 6943

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of the vortices of the secondary discharged flow (above and below an impeller), which is called the “isolated mixing region” (IMR).21−27 Fluid does not circulate only near the impeller, but circulates in the whole vessel along the vessel wall. Under laminar flow conditions, although the mixing patterns of both impellers resemble each other, the mixing in the case of the AP impeller proceeds faster than that of the OP one. This is obvious from the images of t = 30T and 50T in Figure 5. Since t = 70T, the size of the IMR does not greatly change with respect to time and convective mixing may reach a steady state. In addition, the size of the IMR in the case of the AP impeller becomes much smaller than that of the OP one. Generally, under low Re conditions, it is well-known that fluid does not exchange efficiently between the top and bottom parts in the vessel. As shown in Figure 6, mixing between the top and bottom parts in the vessel is not efficient in the case of the OP impeller. On the other hand, in the case of the AP impeller, mixing between the top and bottom parts in the vessel proceeds rapidly. So far, the AP impeller exhibits a higher mixing performance than the OP one at Re = 40 based on not only the homogeneous mixing but also the exchange of fluid between the top and bottom parts in the vessel. Figure 7 shows the

Figure 8. Dependence of dimensionless volume of IMR, V*, on Re under Re < 100.

that of the OP one. The AP impeller is more efficient for global mixing in the vessel than the OP one. Incidentally, the required power number of the AP impeller averages 1.1 times as much as the OP one under Re = 10−100 (Figure 9). The required

Figure 9. Dependence of power consumption by impeller agitation, Np, on Re under Re < 100.

Figure 7. Dependence of dimensionless mixing time, nT, on Re under Re < 100.

dependence of the dimensionless mixing time, nT, on Re under laminar and transition conditions (Re < 100). Above the nT, the IMR is out of consideration because IMRs cannot be eliminated completely by convective flow and consequently remain for a long time (that is, nT means the mixing time for the decolorization of the whole region except for the IMR). As shown in Figure 7, the nT of the AP impeller is slightly smaller than that of the OP one in the region of Re < 100. In particular, the difference between them becomes large at Re < 40. Hence, in the AP impeller, the circulation of fluid into the chaotic mixing region except for the IMR is relatively fast under such low Re conditions. It is well known that the IMR core has a 3-D toroidal structure.21−27 Hence, it is possible to estimate the volume of the IMR as a volume of donut-ring structure. The size of the upper IMR was adopted in the OP impeller because of the symmetry of the IMR structure, while the size of the larger IMR was measured in the AP impeller. Figure 8 shows the dependence of the dimensionless volume of the IMR, V*, on Re. V* is obtained by dividing the IMR volume, VIMR, into the volume of the vessel, VTANK. Figure 8 suggests that the size of the IMR in the case of the AP impeller is much smaller than

power for agitation in both impellers is almost comparable with each other under the present experimental conditions, although there are a few differences between the two results. Judging from mixing patterns, despite the concordance of the intrinsic structure of the 3-D streak sheet, there is a remarkable difference in the degree of mixing between the two types of impellers. In the next section, what causes the difference in mixing efficiency between the two types of impellers is investigated. Streak Sheet and Cross Sections. As shown in Figure 10, the 3-D streak sheet, which consists of a helical rounded surface spreading to 3-D space, has a time-invariant structure every rotational period of the impeller. As mentioned previously, it contains 3-D information of the streakline that moves and deforms in vertical and horizontal directions.9,17 The 3-D streak sheet covers the whole region of the vessel except for the poor mixing region with repeating the folding and stretching. These intrinsic structures of the streak sheet in OP and AP impellers are similar to each other. However, there is only one remarkable difference of streak sheets between the two impellers. Unlike the case of the OP impeller (Figure 10a), 6944

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most of the vessel inside densely. There is the universality of shape on the streakline every rotational period of the impeller. A pair of island regions (aforementioned IMRs), which are not suffused by streaklines, are observed above and below an impeller. The streaklines are folded and stretched lots of times and wind around these isolated regions, while never breaking into these isolated regions. On the other hand, there are some crucial discrepancies between the results of OP and AP impellers. First of all, the experimental images suggest that unlike the case of the OP impeller, the streak sheet in the case of the AP impeller seems to stretch faster in the downward direction than in the upward direction. This may be due to the difference in the symmetric natures of the velocity fields between the two impellers (for details, see Figure 4). In the AP impeller, which rotates in a clockwise direction, the streak is first affected by a strong downward perturbation compared with in the OP one, and then the streak of the AP impeller is easy to stretch in the downward direction, which results in the discrepancy of the IMR size (the size of the lower IMR is larger than that of the upper one). In fact, we have already confirmed that the streak of the AP impeller is easy to stretch in the upward direction when the impeller is rotated in the reverse direction. Incidentally, only the right half of the streak cross section is discussed in the present study. The OP impeller creates a symmetric flow on the top and bottom of the impeller blade, and consequently the streak cross sections are regarded as linesymmetric structures (in fact, 3-D spirals) with respect to the agitation shaft. On the other hand, the AP impeller periodically generates an asymmetric flow on the top and bottom of the impeller blade, and then it is reasonable to guess that the left half of the streak cross section is a vertically inverted structure of its right half in the case of the AP impeller. In the case of the OP impeller, the renewed streakline is added to the section every passage of one of the impeller blades. That is, the number of streaklines equals half the passage number of the impeller blades. In the AP impeller, however, the number of streak lobes is smaller than that in the OP impeller. Instead, compared with the case of the OP impeller, the streak lobe in the case of the AP impeller is very large in width and has a larger area. We cannot identify the amplitude, frequency, and number of traveling waves directly, but they can be predicted by use of the size and number of streak lobes, as mentioned in the Introduction. Quantitatively, the area and length of the streak lobe in the AP impeller are about 2.0 times and 1.2 times as large as those of the OP one, respectively, while the number of streak lobes in the AP impeller is half that in the OP one. This may be due to the difference of amplitude and frequency in traveling waves. That is, the amplitude of traveling waves in the AP impeller is larger than that in the OP one, while the frequency in the AP one is half that of the OP one. Figure 12 shows the dependence of nT and/or V* on the area of the streak lobes (which are formed during one rotation of the impeller) at the vertical cross section normalized by the vertical cross-sectional area of the stirred tank, S*. As can be seen, both nT and V* depend on S* remarkably. In theory, although it is impossible to enhance all of the number and width and area of streak lobes simultaneously, they may be adjustable by the parameters of the number, width, and angle of the impeller blade. As aforementioned, the fluid particles on a streak sheet always exist on that as long as the system is observed periodically. In other words, fluid particles move from a lobe to the external lobe one

Figure 10. Time variation of 3-D streak sheet consisting of helical rounded surface spreading to 3-D space (Re = 40): (a) OP impeller; (b) AP impeller. The impeller rotates in a clockwise direction.

the streak sheet in the case of the AP impeller (Figure 10b) seems to stretch faster in the downward direction than in the upward direction, which is because the exit of tracer fluid exists in the side of the down-pumping blade. Under the condition of Re = 40, the vertical cross sections of the 3-D streak sheet were visualized, where the tracer fluid was injected from the whole area of the impeller-blade tip (which was equivalent to method I in the section Mixing Patterns). The experimental results at Re = 40 for the OP and AP impellers are shown in parts a and b, respectively, of Figure 11.

Figure 11. Time variations of the vertical cross sections of 3-D streak sheet (Re = 40): (a) OP impeller; (b) AP impeller; (c) GP impeller. The impeller rotates in a clockwise direction, and then down-pumping flow is dominant initially.

In the cases of both impellers, the streaklines have similar natures as the “mixing template” that determines the mixing pattern in the vessel. A streakline, which is depicted as a smooth curve, shifts to the outside after one rotational period. These streaklines discharged from an impeller blade are stretched near the vessel wall, and they circulate in the vessel and then return to near the impeller. After that, they are folded and discharged again by the impeller. Consequently, the streaklines blanket 6945

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Figure 12. Dependence of nT and V* on the normalized vertical crosssectional area of streak lobe, S*. Open symbols, OP impeller; filled symbols, AP impeller.

Figure 13. Schematic illustration of perturbation wave and vertical cross sections of streak sheets (experimental images (Re = 40, t = 20T) and schematic illustrations): (a) OP impeller; (b) AP impeller.

by one every rotational period of the impeller. Thus, the mixing in the same streak lobe occurs rapidly, while the mixing between one lobe and another lobe proceeds slowly according to the streak sheet. Furthermore, the mixing degree becomes better as Re increases because the circulation of a streak lobe becomes fast. In fact, nT becomes short as Re increases (see Figure 7). From the above, the streak lobe of the AP impeller is essentially better than that of the OP one because mixing inside the large streak lobe proceeds rapidly and the large streak lobe covers the whole region of the vessel homogeneously. In addition, as shown in the images at t = 10T, the streaklines for the OP impeller are just half-segmentalized into the top and bottom sides of the vessel and they link only near the vessel wall. On the other hand, the streak lobes for the AP impeller move down through the vessel across the top and bottom sides (as shown in the area in the white circle of Figure 11b), which results in the efficient exchange of fluid across the top and bottom sides of the vessel. This indicates that the amount of fluid exchange between the top and bottom parts in the vessel for the AP impeller is much larger than that of the OP impeller. In fact, mixing between the top and bottom parts in the vessel proceeds rapidly in the case of the AP impeller (see Figure 6b). In the case of the GP blade (Figure 11c), the streak lobe has a vertically asymmetric feature and is hard to stretch in the vertical direction, which is due to the vertical asymmetry of the flow vector (Figure 4c). Then, the mixing in the case of the general pitched-paddle blade may be somewhat bad compared with those of the OP and AP impellers under the present experimental conditions. Finally, the qualitative concept of the present study is shown in Figure 13 and summarized in this paragraph. As shown in Figure 13, it is suggested that the frequency of the perturbation wave in the AP impeller is smaller than that of the OP one. Consequently, compared with the case of the OP impeller, the number and size (containing both width and area) of streak lobes in the case of the AP impeller are small and large, respectively. In fact, in the case of the OP impeller, the streak lobe is very small in width and its number is the same as that of the impeller passage (Figure 13a). On the other hand, the streak lobe in the case of the AP impeller is very large in width and its number is half that of the impeller passage (Figure 13b). A larger area of streak lobe for the AP impeller can cover the whole region of the vessel rapidly, and may accelerate the

homogeneous mixing in the vessel. Consequently, the size of the IMR becomes small in the case of the AP impeller. In addition, the perturbation wave of the AP impeller has a vertically asymmetric structure unlike the symmetric nature of the OP one. This is why the configuration of the streak lobe in the AP impeller becomes vertically asymmetric. Actually, as shown in Figure 13b, the streak lobes for the AP impeller move down through the vessel across the top and bottom sides soon after they are folded with the impeller. However, in the case of the OP impeller, the streak lobes stretch across the top and bottom sides only near the vessel wall (Figure 13a). This implies that the amplitude of the perturbation wave derived from the rotational impeller blade in the case of the AP impeller is larger than that of the OP one, which causes the efficient exchange of fluid across the top and bottom sides of the vessel. The present study proposes that the streak lobes can become a powerful index for chaotic mixing performance. Then, the number, size (width and area), and dynamic behavior of streak lobes, which depend on the amplitude and frequency of traveling waves caused by rotational impeller blades, are important factors to develop efficient impeller blades.

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CONCLUSION In the present study, laminar chaotic mixing was investigated based on streak lobes, which play an important role as a mixing index. In both ordinary paddle (OP) and alternative-pitched paddle (AP) impellers, mixing performances were evaluated by use of the decolorization method under laminar and transition flow conditions of Re < 100. Then, the difference in mixing performances between both impellers and its mechanism were discussed based on the configuration of streak lobes. The important findings are summarized as follows: 1. In a mixing pattern, under Re < 100, mixing in the case of the AP impeller was completed more rapidly than that of the OP paddle one. In addition, the size of the IMR in the AP impeller was much smaller than that of the OP one. 2. In the case of the AP impeller, the number of streak lobes was small, while the size of streak lobes was large compared with the case of the OP impeller. In addition, the streak lobe in the AP impeller was longer than that of 6946

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Industrial & Engineering Chemistry Research

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Article

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the OP one. These support that the above difference in mixing performances between the AP and OP impellers is due to the difference of the configuration of streak lobes between them. 3. The streak lobe can become a powerful index for chaotic mixing performance. Then, the number, size (width and area), and dynamic behavior of the streak lobe, which depend on the amplitude and frequency of the traveling wave caused by the rotational impeller blade, are important factors for efficient mixing.

AUTHOR INFORMATION

Corresponding Author

*Tel./fax: +81-6-6850-6294. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by a Grant-in-Aid for Scientific Research, A (20246115), and Challenging Exploratory Research (22656176). S.H. expresses his special thanks to Dr. H. Oka (RFLOW, Co., Ltd.) for the numerical simulation by use of commercial CFD software, Rflow.

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NOMENCLATURE d = diameter of impeller blade, m D = diameter of vessel, m h = installed height of impeller, m H = height of mother liquid, m l = diameter of impeller, m nr = rotational speed of impeller, rpm nT = dimensionless mixing time Np = power number P = required power for mixing, W R = radius of vessel, m Re = Reynolds number (=ρnrd2/μ) S* = normalized area of streak lobe t = time, s T = period, s Tr = torque, N·m VIMR = volume of an IMR VTANK = volume of agitating fluid V* = dimensionless volume of IMR μ = viscosity, Pa·s ρ = density, kg/m3

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REFERENCES

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dx.doi.org/10.1021/ie203036n | Ind. Eng. Chem. Res. 2012, 51, 6939−6947