Planar Laser-Induced Fluorescence Method for Analysis of Mixing in

Ind. Eng. Chem. Res. 2004, 43, 6557-6568

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Planar Laser-Induced Fluorescence Method for Analysis of Mixing in Laminar Flows Paulo E. Arratia and Fernando J. Muzzio* Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08855-0909

We report a laser-induced fluorescence (LIF) experimental method that is able to quantify dye concentration in a laminar three-dimensional flow. The geometry of interest is a stirred tank reactor agitated by three impellers. The method is employed to reveal the spatial structure of the evolving mixing patterns and its statistical properties. For chaotic flow conditions, the statistics of the passive scalar reveals a non-Gaussian concentration probability density function with an exponential tail. We find that the concentration patterns, once established, do not change with time; i.e., they are self-similar. This LIF method also provides detailed flow structures that are of importance to the study of mixing. Flow structure varies considerably as a function of Reynolds number (Re). However, we find that the mixing patterns are structurally invariant and they slowly decay as a result of a nonequilibrium state between stretching and diffusion. 1. Introduction Fluid mixing is important to many industrial operations spanning paper and pulp, pharmaceutical, polymer, chemical, and biochemical industries, among others. Often the quality of finished products is a function of the effectiveness of mixing processes along the manufacturing sequence. When performing a mixing operation, the ultimate objective is to achieve a target level of homogeneity within the mixture and to do it in the fastest, cheapest, and (if at all possible) most elegant way. The task left to the experimentalist is to quantify or characterize this homogenization process. Carefully conducted low-concentration dye advection experiments can unveil flow patterns and structures that serve as the starting point to analyze fluid mixing in stirred tanks. One such experimental method is planar laserinduced fluorescence (LIF), where the monochromatic character of laser light allows the user to excite specific states with high precision to obtain quantitative information of a scalar, such as concentration.1-4 The vast majority of the studies involving dye concentration measurements in stirred tanks using LIF have been restricted to turbulent flows.5,6 While turbulence is often desired in mixing operations (due to its unsteady velocity fields and nonlinear inertial forces), it is not always feasible to operate in such a regime, particularly in the case of high viscosity fluids (e.g., polymers), shear-sensitive material (e.g., mammalian cells), and at small length scales (e.g., microfluidics). Mixing in such situations is inherently laminar. It has been well-established that in order to achieve efficient laminar mixing, chaos is necessary.7,8 Chaotic advection causes an initially segregated scalar to acquire complex spatial structure as fluid elements are stretched, cut, and folded by the flow. The effects of this stretching and folding process on homogenization has been explored both by theory9-11 and by experiments12-14 in idealized two-dimensional (2D) flows. One of the main observations is the presence of persistent spatial pat* To whom correspondence should be addressed. Tel.: (732) 445-3357. E-mail: [email protected].

terns whose contrast intensity decays slowly in time without change of spatial structure. These largest gradients of such persistent patterns are shown to align across regions of large stretching, generated by the globally unstable manifold of the flow. Other investigations have also explored multifractal15 and self-similar15 mixing structures in 2D flows. A recent study by Muzzio et al.16 shows that the overall evolution of the mixing process is controlled by a local orientational property of the flow, named asymptotic directionality (AD), such that once a fold reaches a certain region, it acquires a fixed orientation corresponding to such a region, maintaining a time invariant global structure. However, with few exceptions,17 the vast majority of the investigations concerning the homogenization of a passive dye and its properties in chaotic laminar flows have been restricted to 2D flows. In this paper, we describe a low-dye-concentration LIF method useful to quantify dye concentration. We use this method to investigate the statistical properties of the passive scalar of laminar flows in a common 3D flow: a stirred tank agitated by three equidistant Rushton impellers. This method provides us with a direct measure of concentration fields and detailed mixing structures, which enables us to examine the behavior of a passive scalar and length scale distribution in mixing flows. We further investigate mixing by analyzing the statistical properties and the length scale distribution of the measured concentration fields. 2. Experimental Setup 2.1. Equipment. The experiments are conducted in a custom-made Plexiglas vessel equipped with an automated driving system and an illumination-image acquisition system (Figure 1a). A 30 mJ pulsed YAG laser (NewWave Research) operated at 532 nm is used as the illumination source. Images are captured using a 1076 × 1024 pixel, 8-bit CCD camera (Dantec Dynamics, Mahwah, NJ) located at a right angle to the incident laser light. The FLOWMAP (Dantec Dynamics) software package is used for data acquisition and laser/camera synchronization. We examine the flow driven by a three

10.1021/ie049838b CCC: $27.50 © 2004 American Chemical Society Published on Web 08/27/2004

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Figure 1. (a) Laser-induced fluorescence (LIF) experimental setup and (b) tank dimensions.

equidistant Rushton impeller configuration. The working fluid is a 90:10 glycerin/water mixture (by weight), and the passive tracer is a Rhodamine B (544 MW, Fisher Scientific) glycerin/water mixture. The Rushton impellers are 7.5 cm in diameter, 9 cm apart, and placed concentrically in the tank. The distance between the lower impeller and the tank bottom is 9 cm. The distance between the highest impeller and the tank free surface is also 9 cm (Figure 1b). The vessel geometry consists of a flat-bottom cylinder, surrounded by a square Plexiglas outer box needed to minimize optical distortion. The tank height is 36 cm, and its diameter is 24 cm. The agitator is driven by a dc motor. The water/glycerin mixture is carefully prepared to avoid the presence of bubbles or foam. Water is slowly added to glycerin and the mixture is slowly mixed for 50-60 min. The tank is filled with the glycerin/water fluid once the mixture is homogeneous and bubble-free. The outer box is then filled with glycerin to eliminate lens effects from curved vessel walls to reduce light refraction on the interfaces. For all experiments, the tank is allowed to sit overnight to eliminate air bubbles that might have been entrained during pumping or filling. Tracer is prepared such that it is neutrally buoyant and has the same viscosity as the bulk fluid. The main dimensionless parameter used in this paper is the Reynolds number (Re), defined as Re ) FGD2/η. The Reynolds number is interpreted as the ratio of inertial to viscous forces and it is based on the impeller diameter D ) 7.5 cm, impeller speed G (in Hz), fluid density F (1.1 g/cm3), and fluid viscosity η (6.80 g/cm‚s). 2.2. Dye Considerations. The fluorescent dye used in this experimental procedure is prepared from solid Rhodamine B (an organic dye). This compound is selected because of its strong fluorescence centered on the red-orange part of the visible spectrum. Its absorption maximum occurs in the vicinity of 540 nm and its fluorescence maximum occurs around 590 nm. Fluorescence of this dye is induced by the green band of the laser (λ ) 532 nm), and it is differentiated from this wavelength using an optical filter.

An essential aspect of the technique is to ensure that the dye is neutrally buoyant. Rhodamine B pellets are ground to fine particles and poured into a mixture of 90% glycerin and 10% water by weight. The resulting powder is dissolved first in water. Glycerin is then added to the mixture once a complete solution of water and rhodamine is achieved. The solution is allowed to slowly mix for 50 min. Dye concentration needs to be carefully selected. Concentrated solutions are needed in order to obtain enough spatial resolution from the optical apparatus. However, highly concentrated solutions have drawbacks such as shadowing effects and the presence of particles in the solution, which affects the re-emitted light and might scatter laser photons. A much-diluted solution, on the other hand, would not be able to fluoresce to an acceptable value for experiments with long and thin striations and would be buried in noise. Thus, a tradeoff between resolution and accuracy is inevitable. Concentrations are selected in order to maximize resolution and minimize dye dissolution problems and noise. 2.3. Methods. In this section we describe the mathematical treatment used to obtain concentration measurements from fluorescent experiments. It is desired to establish a firm relationship between dye concentration and gray values or pixel values. In the case of a slowly diverging laser sheet, we can assume that the decrease of photon flux density of the beam passing through the tracer solution is due to absorption and diffusion only. We seek an equation that relates pixel value and concentration of the form

concentration ) ξΦpix

(1)

where Φpix is the pixel value and ξ is a experimental proportionality constant. The starting point of this analysis is the general relationship between intensity of excitation I and dye concentration

dIe(z) ) -1C(z) Ie(z) dz

(2)

where Ie is the intensity of the fluorescence emission at a point z along the beam path, C(z) is the dye concentration at z, and 1 is the extinction coefficient of the dye that can be experimentally calculated. At low excitation intensity, the fluorescence emission is proportional to the excitation intensity. At higher excitation intensity, on the other hand, saturation and photobleaching effects may occur, and fluorescent intensity becomes no longer linearly proportional to excitation intensity. In this paper, we examine the effects of parameters such as dye concentration and attenuation such that fluorescence and excitation intensities are proportional. Assuming that the concentration field is uniform, the resulting fluorescence emission F can be written as18

F(z,e) ) AVcI0φ1Ce-C(1b+2e)

(3)

where A is an optical calibration constant, Vc is the collection volume, I0 is the incident intensity of the laser beam, φ is the quantum efficiency, 2 is the molar extinction coefficient of the fluorescent signal, which is much lower than the coefficient 1, due to the spectral shift between the laser radiation and the fluorescent emission. The above equation takes into account the attenuation of the incident laser beam, caused by

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Figure 2. (a) Evolution of fluorescence signal for different beam paths and (b) effect of the laser beam for different dye concentration.

absorption, when the laser passes through the absorbing medium (b). In the same way, fluorescence emission is attenuated while passing along the distance (e) to the detector in the absorbing medium. If the term C(1b + 2e) is weak enough, the measured fluorescent signal turns out to be directly proportional to the concentration:

F(z,e) ) AVcI0φ1C

(4)

The measure volume per pixel (Vc) depends on the camera magnification and the laser beam width. Images are captured using a CCD camera with a 1079 × 1028 (1 109 212) pixels. Since the largest cross sectional cut had an area of 120 cm2 and the beam width is approximately 1.0 mm, the measuring area per pixel is approximately 1.08 × 10-2 mm2, and the measuring volume per pixel is approximately 1.08 × 10-2 mm3. The next step is to fit eq 3 to the calibration data in order to obtain values for 1 and 2 and ensure that the term C(1b + 2e) is small enough so that the signal is directly proportional to concentration and linearly dependent on light intensity. 4. Results 4.1. Fluorescence Parameters. The first experimental phase is to perform “static” experiments in a motionless liquid to characterize the influence of parameters such as concentration, noise, attenuation of the laser beam, and the fluorescence emission. To define accurately the range where the fluorescence signal is proportional to the concentration, one must take into account the attenuation of the laser beam as it passes through the absorbing medium. Figure 2a demonstrates the linear dependence of the fluorescent signal on concentration for different optical beam paths. Figure 2b also shows the effect of the attenuation of the incident laser beam on fluorescence. For a concentration of 100 µg/L, we are able to observe an exponential decay, as predicted by the Beer-Lambert law.

The static experiments reveal that the determination of the linear zone requires taking into account the dye concentration level and the laser beam path attenuation. Both effects are included in the exponential part of the fluorescent signal equation. From the evolution of the fluorescent signal as a function of the length of the optical beam path (Figure 2a), the molar extinction coefficient is measured by fitting eq 3 to the data. The coefficient 1 is measured to be 4.78 × 10-6 mol-1 L m-1. The molar extinction coefficient (2) for the fluorescent emission is measured to be 1.15 × 10-6 mol-1 L m-1. These values are low enough that the exponential term in the equation can be disregarded. 4.2. Calibration Procedure. The calibration procedure is perhaps the most important step in the experimental method. To obtain a linear response of pixel intensity to concentration, it is necessary to determine the linear response range for our specific laser, camera, and tank setup. A calibration curve is constructed by fixing the camera angle and distance from the tank and the laser sheet and by also fixing the laser beam intensity, angle of incident, and dye concentration. This means that all the experimental parameters are known and fixed. Dye concentration is gradually increased to a maximum or saturation value of the given camera experimental setup. The camera is operated at a gain ) 4 with a 2.8 exposure number, and the image field is 1076 × 1028 pixels. The very first step, once all the parameters are set, is to take an “empty” picture for noise reduction. The tank is filled with a rhodamine-free glycerin/water solution and the tank is allowed to operate under the experimental conditions. The image serves as background noise baseline for all experimental pictures (Figure 3a). This noise baseline is subtracted for all images during the calibration procedure and concentration image processing. To illustrate the noise-reduction procedure, we show in Figure 3b an image of the tank filled with a 50 µg/L concentration of rhodamine in glycerin/water before noise is subtracted. Figure 3c shows the image presented in Figure 3b after noise is subtracted. Once images are processed, the next step is to construct a calibration curve relating pixel value to rhodamine concentration. As shown in Figure 3d, the pixel profile throughout the noise picture is flat and nearly constant, indicating that the fluorescence response along the beam path is linear without baseline shifting. Rhodamine concentration is increased in very small steps until the pixel value versus dye concentration curve becomes nonlinear. The tank is allowed to mix for a long time (1 h) before a new step increase in concentration is added to the system. The concentration step increase is 5 µg/L from 0 to 10 µg/L and from 10 up to 200 µg/L. The curve is linear up to 95 µg of rhodamine per liter of solution (Figure 4). The calibration curve saturates at values higher than 95 µg of rhodamine/liter of solution. The smallest concentration detected by the CCD camera is 0.5 µg/L. Below this minimum concentration, noise and rhodamine pixel values are undistinguishable. Pixel values used for calibration purposes are obtained after noise is subtracted from the picture. Pixel values in the y axis are actually mean pixel values from the area of interest. 4.3. Concentration Fields. We present examples of concentration fields obtained by postprocessing intensity images. We concentrate our analysis on the right side

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Figure 3. Subtracting noise from images. (a) “Empty” images tank filled with glycerol/water mixture. (b) Image of tank with a 50 µg/L rhodamine concentration in glycerol/water. (c) Processed image resulting from subtracting image a from image b. (d) Pixel values along a horizontal line of panel c.

Figure 4. Calibration curve for pixel value as a function of dye concentration.

of the tank (relative to the laser position) between the lower and middle impeller (Figure 5a). It was convenient to select the right side due to the shadows that the impellers and the shaft would produce on the left side of the tank once the laser beam passes through the medium. Owing to flow symmetry, the left side of the tank is the mirror image of the right side. We take advantage of this flow symmetry to decrease the field of view and consequently gain spatial resolution. Images are taken and saved in intensity scale format, which stores light intensity values for each pixel on a scale from 0 (dark) to 255 (brightest light). Images are then postprocessed using in-house scripts. We assign a RGB value to each pixel according to its corresponding concentration value. The lowest concentration is shown in blue, the middle concentration in green, and highest concentration in red, as shown by the colorbars in Figure 6. In these concentration fields, and fields therein, blue regions represent a nearly 0% dye con-

centration solution while red regions represent a 100% dye concentration solution and saturation (colorbar unit is in µg/L). Intensity images of two representative cases at Re ) 40 and 60 are presented in parts b and c of Figure 5, respectively. For both cases, dye is injected at approximately 4 cm to the side and 1 cm below the central impeller midplane, as shown in Figure 5a. The location of the injection was chosen such that both the chaotic and segregated regions of the flow would receive dye, revealing the overall mixing structure. In other words, the blob of dye injected into the tank is placed partly in a region known for good convective mixing and partly in a region known for poor mixing. For the Re ) 40 case, a 4 mL injection containing 180 µg of rhodamine is made to obtain an overall 45 µg/L dye concentration in the vessel. In the second case (Re ) 60), a 4 mL injection containing 120 µg of rhodamine is made to obtain an overall 30 µg/L dye concentration in the vessel. Figures 6 and 7 show the corresponding concentration fields and their evolution with time for Re ) 40 and 60, respectively. Concentration fields, obtained by postprocessing the acquired intensity images (Figure 5b,c), show unmixed regions (characterized by intense red and blue colors). Note that the flow structure formed in the tank is well-resolved by this method. An evolution of the mixing structure is observed with time. For both cases, the overall pictures also start to approach a wellmixed state. As time evolves, the red and blue regions slowly disappear, owing to stretching and diffusion. The concentration field starts to approach the respective target concentration level of 45 and 30 µg/L for Re ) 40 and 60 cases, respectively. It is interesting to note that, although the mixing structure has changed shape, common features of laminar flows in stirred tanks still remain, such as the presence of segregated regions and compartmentalization of the flow. Note how dye is confined to the lower half of the tank, never wandering to the upper half except faintly at the tank walls (Figure 6d). The mechanism responsible for mixing in stirred tanks can be investigated by examining in detail the structures formed by the homogenization process. An example of such structure is presented in Figure 8a for the Re ) 80 case. In this case, we inject dye next to the impeller such that the dye would only invade the chaotic portion of the flow. The image was taken from the upper half side of the vessel (Figure 8b) after the system was allowed to evolve for 10 min. The mechanism of mixing in stirred tanks has been discussed in detail elsewhere.19 The increased resolution of the mixing structures accomplished with this method reveals more detailed information. For example, we observe how trains of folds are generated, ejected, and reoriented with the passage of an impeller blade. Stretching occurs as the fluid approaches parabolic and hyperbolic points (impeller or tank wall). A blowup of the picture of the region next to the impeller blades (Figure 8c) reveals this mechanism in detail. Folds (or lobes) are formed and traveling toward the tank wall while they are stretched. As the impeller blades perturb the system in a time-periodic fashion, the stable and unstable manifolds split apart. The invariant stable and unstable manifolds intersect ad infinitum to form lobe structures.20 This lobe structure is stretched, folded, and reoriented, progressively intruding the flow domain. The repetition of this stretching and folding cycle increases the intermaterial


Figure 5. (a) Region of interest and dye injection (red blob) in the stirred tank. Intensity images (b) at Re ) 40 and (c) at Re ) 60.

area exponentially, generating lamellar structures with a wide distribution of length scales (Figure 8d). However, the lobe structure fails to cover the cross section of the tank entirely, leaving unreached, segregated regions. 4.4. Statistical Analysis. Next, we illustrate the use of the technique for characterizing the statistical behavior of the passive scalar in stirred tanks operated in the laminar regime. In particular, concentration probability density function (PDF) is a very useful tool for characterizing mixing, since it provides detailed information about the tracer variability. Also, there has been considerable interest in the mechanism leading to non-Gaussian scalar PDFs. In this paper, we show how pLIF can be used to obtain relevant measurements of scalar PDFs. As an example, we center our analysis on a stirred tank operated at Re ) 40, as shown in Figure

9. Intensity images are recorded at each flow period (N) and transformed into concentration fields. The flow period, N, is defined every time we detect a recurrent pattern by cross-correlating sequential images. In other words, if the flow has a given structure at a starting point, then one period has passed when we next detect the same structure. The flow period for Re ) 40 is about 73 s. The fine structure of the concentration patterns are investigated by studying the PDF of the magnitude of the concentration gradient. An example is shown in Figure 10. The distribution is normalized by the standard deviation of the intensity field to compensate for the gradual loss of contrast. The PDF is unimodal, indicating efficient mixing between high and low concentrations present in the initial field. We note that the distribution of the gradients is exponential. Such exponential tails for distribution of concentration gradients

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Figure 6. Concentration fields of a stirred tank at Re ) 40 at t ) (a) 30 s, (b) 60 s, (c) 90 s, and (d) 120 s. Concentration bar in µg/L.

have been noted in 2D chaotic flows.13 In particular, the appearance of exponential rather than Gaussian tails implies a relatively high probability of extreme events, such as the existence of a portion of the flow field that mixes very slowly with its surrounding environment (segregated regions). Authors have shown that nonGaussian tails arise from correlations between the concentration field and its gradient and that the advection-diffusion mechanism can generate such correlation on its own. Exponential decay also indicates that the concentration amplitude decays, but the pattern is in some statistical sense invariant. It is striking to see that the PDF reaches an invariant form that is essentially identical for N ) 7 and 9 (Figure 10). If the distribution of concentration values possesses a self-similar shape, this suggests that, from a statistical point of view, the process remains “the same” as time evolves. This time invariance can be assessed by using the collapse of the distribution onto an invariant curve. As shown for N g7, concentration distributions seem to possess a selfsimilar shape as well as time-invariant properties. This concentration distribution time invariance has an important meaning: regions containing low dye concentration will remain relatively low for many periods, despite the action of convective mixing. Also, it suggests that the highly heterogeneous structures generated almost

instantly by convective mixing process tend to conserve its geometrical features.21 It has been conjectured that self-similar stretching processes characteristic of chaotic flows should give rise to self-similar scalar fields. Our results confirm this conjecture. 4.5. Structural Analysis. An interesting feature of the flows investigated in this paper is that the evolution of the mixing structure is governed by intrinsic selfsimilarity. Images presented so far (Figures 6 and 7) show that although the mixing structure changes with Re, it is invariant with time; as time progresses, the concentration field becomes more uniform, but the flow structure remains essentially the same. This has been shown before in simple 2D map-based flows.22,16 In such flows, the evolution of the structures is controlled by a multiplicative iterative operator that generates structures that are self-similar with respect to time. As time increases, the chaotic flow produces a partially mixed structure that, when recorded at periodic intervals, is essentially identical to the structure recorded at earlier periods, except that a larger number of thinner striations are found in each region (cf. Figures 6 and 9). For the 2D map-based flows, the overall evolution of the mixing process seems to be controlled by a local property of the flow, such that once a fold reaches a certain region, it acquires the orientation corresponding to such


Figure 7. Concentration fields of a stirred tank at Re ) 60 at t ) (a) 30 s, (b) 60 s, (c) 90 s, and (d) 120 s.

a region, maintaining a time-invariant manifold structure. This local property, which is named asymptotic directionality (AD),16 determines the characteristic material orientation at a given instantaneous location in the flow. Experimentally, this property can be inspected by plotting the cross-correlation coefficient between two snapshots at periodic intervals (N). We examine the evolution of the mixing structure at different flow periods (N) for the Re ) 40 case as shown in Figure 9. Cross-correlation versus flow period data (Figure 11a) shows that the coefficient rapidly approaches an asymptotic value of 91%, indicating that after a relatively short period of time, the structures generated by the flow are essentially identical to the structures recorded at earlier periods. This is consistent with previously described properties of time-periodic flows.23,24 Although the mixing structure approaches an asymptotic form, mixing continues, owing to stretching and folding of fluid elements and to diffusion. This evolution can be characterized by monitoring the concentration PDF and the decay of its normalized variance, which is

a measure of unmixedness or contrast. To emphasize our argument, we concentrate our analysis on flow periods where the structures have nearly reached an asymptotic form. The variance of the PDF (Figure 11b) shows an approximately exponential decay for longer periods, indicating that mixing is still occurring even after the mixing structure has reached an asymptotic form. This exponential decay also suggests that, following a transient adjustment state, the concentration pattern settles into an invariant form. An advantage of the low-concentration dye technique reported here is the ability to reveal, with a good degree of resolution, fine mixing structures that would be otherwise blurred by saturated optics. Hence, a quantitative assessment of the length scale of the flow structures seems possible at first. Such assessment is of great importance, since the length scale distribution of the flow structures control, for example, the rates of diffusive mixing and the rates of reactions taking place at small scales of flow. The need to determine the striation thickness distribution (STD) was made clear over a decade ago.25

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Figure 8. Stirred tank operated at Re ) 80. (a) Complex mixing structure generated by the passage of the blades. (b) Region of interest. (c) Generation of folds and the presence of a separatrix. (d) Lamellar structures.

However, the limited resolution of image analysis techniques and the erosive effect of diffusion made it practically impossible to quantify STDs using photographic techniques. We briefly re-examine this limitation in this paper. We concentrate our analysis on a particular flow structure formed and captured between the central and upper impeller at Re ) 40. This flow structure is shown at different periods in Figure 12. We record the structure at each flow period (N). At each period, a horizontal line segment is passed through the image as shown in the image corresponding to N ) 1. Intensity values along the line segment are plotted as a function of the path for different flow periods as shown in Figure 13. At a flow period N ) 1, the structure is still forming and large striations are carrying most of the dye injected, giving rise to thick, saturated striations

that are represented by the three peaks in the profile plot (Figure 13a). The thickness of the saturated striation is easily calculated by the average width of the peaks. At this early stage of the mixing process, we do not expect diffusion to play a critical role in the dynamics of the striations (profile baseline is flat). At N ) 2, a more complete and complex structure emerges with more flow filaments and consequently a wider striation distribution. Small peaks are born, indicating the presence of smaller striations that are created as large striations are being stretched into small ones (Figure 13b). This striation behavior is well predicted by purely convective models. However, at N ) 3, we observe that in addition to the added feature and filaments to the structures, the image is starting to lose detail. Small striations are


Figure 9. Evolution of the mixing structure in a stirred tank operated at Re ) 40. N corresponds to flow periods.

diffusing and collapsing into each other forming large, more uniform striations. The profile plot no longer

exhibits a flat baseline along the structure (Figure 13c). At this stage, the assessment of the striation thickness becomes highly subjective and inaccurate, since many of the peaks may overlap, masking finer striations (Figure 13d). The convection and diffusion mechanisms continues to act upon the structure (N ) 4). Eventually, due to diffusion and convection, the profile line will approach a flat line, corresponding to the asymptotic mixing state. Evidently, direct measurement of striation thickness becomes impractical as diffusion starts “smearing” the structures, even at relatively short mixing times. Nevertheless, statistical characterization of spatial structures of concentration fields can be achieved by means of the power spectrum E(k). The power spectrum, which is one of the most familiar characterizations of the spatial structure of a field, provides a useful diagnostic for the development of fine concentration structures.26 An example is shown in Figure 14 for a stirred tank at Re ) 60 at different flow periods. For 1 < N < 3, the high frequency tail of E(k) rises, as stretching and folding produce fine structures, in agreement with results using concentration profiles (see Figure 9 for N ) 1 and 3). Later, the entire spectrum declines at all frequencies with little further change in shape. The decline at high k is due to the diffusion mechanism, whereas the decline at low k is due to the continuing transport of concentration variance to high k by stretching (convection). Furthermore, the spectra collapse onto an invariant curve for N > 3, indicating structural self-similar behavior in time, providing further support for the notion that an invariant structure emerges from the mixing process due to the balance of both stretching and diffusion processes. Hence, the recurrent mixing pattern in this 3D flow seems to be a nonequilibrium state, as previously reported for an experimental 2D time-periodic flow,13 rather than a static one, as reported by map-based 2D flows.16,22

Figure 10. Probability density function (PDF) of the concentration (normalized by the standard deviation) of the images presented in Figure 9.

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Figure 11. (a) Cross-correlation coefficient and (b) variance of the concentration PDF of a stirred tank at Re ) 40.

5. Conclusions A low-dye-concentration laser-induced fluorescence technique was implemented to quantify concentration fields in a laminar 3D flowsa stirred tank agitated by three Rushton impellers. The system requires only that the mixer be transparent from two directions at right angles between the camera and the laser beam and that the working fluid is similarly transparent. The system limitations are imposed by the CCD camera, laser attenuation, and physical properties of the dye. Concentration fields were obtained by postprocessing of the LIF images using a calibration procedure and mathematical treatment. In addition, this LIF method reveals

flow structures with the necessary resolution to allow fast identification of segregated and slow mixing regions and assists in the study of the mixing mechanism. Statistical analysis of concentration fields revealed a non-Gaussian distribution along with an exponential decay of tracer variability. We find that the concentration pattern evolved in a self-similar manner, although the concentration amplitude decays as a function of flow periods (N). This suggests that the highly complex structures generated initially by the mixing process tend to conserve its features in this 3D flow. Unfortunately, we were not successful in measuring striation thickness distribution (STD). However, we


Figure 14. Power spectrum of the images presented in Figure 9.

in a self-similar fashion. We note that at the beginning of the mixing process there is a rise in high-frequency tail as stretching and diffusion produces fine structures. The spatial power spectrum evolves into an invariant after just few periods, where all the wavenumbers decline at the same rate. The decline of high wavenumbers is due to diffusion (collapses fine striations into larger ones), while the decline of low wavenumbers is due to convective mixing (stretching of larger structures). Hence, the whole decline of the spectral function and its invariance in this 3D flow is a result of a nonequilibrium state between stretching and diffusion processes. Figure 12. Evolution of a mixing structure in a stirred tank at Re ) 40.

Nomenclature Re ) Reynolds number F ) fluid density µ ) fluid viscosity G ) impeller speed D ) impeller diameter N ) flow period λ ) laser light wavelength Φpix ) pixel value ξ ) experimental proportionality constant Ie ) intensity of the fluorescent emission z ) beam path C(z) ) dye concentration along beam path z 1 ) dye extinction coefficient 2 ) molar extinction coefficient A ) optical calibration constant Vc ) collection volume I0 ) incident intensity of the laser beam F ) fluorescent emission φ ) quantum efficiency E(k) ) power spectrum k ) wavenumber

Figure 13. Direct measurement of striation thickness of the structures presented in Figure 12.

provide an alternative method to analyze length scale distribution using a power spectrum. Spatial scale analysis using a power spectrum at different flow periods (N) shows that mixing structures indeed evolved

Literature Cited (1) Walker, D. A. A fluorescence technique for measurement of concentration of mixing liquids. J. Phys. E: Sci. Instrum. 1987, 20, 217-223. (2) Bennani, A.; Lievre, J.; Gence, J. N. On the comparison of measurements methods for concentration fluctuations based on conductimetry and laser-induced fluorescence in turbulent water flows. C. R. Acad Sci. Paris 1990, Ser. II, 453-457.

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Received for review February 27, 2004 Revised manuscript received July 6, 2004 Accepted July 23, 2004 IE049838B

Planar Laser-Induced Fluorescence Method for Analysis of Mixing in

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