Characterization of mixing in a stirred tank by planar laser induced fluorescence (PLIF)

he mixing of two water feedstreams is investigated in a 0.02m 3 continuous stirred tank, from concentration measurements. These measurements are carried out with a laser sheet in a very thin vertical plane through the vessel tangent to the impeller. The proportionality between the grey level obtained by planar laser induced è uorescence and the local concentration has been previously determined. The é elds of instantaneous concentration, temporal variance and mixing index are processed from the grey level images.


INTRODUCTION
Solid particles obtained in industry by precipitation (reaction crystallization) are generally produced in mechanically stirred tanks, or in jet mixers. In these processes, two liquid streams come into contact, but the process is more or less controlled.
Precipitation supersaturations originate from a chemical reaction and thus are very high 1 . Furthermore, the effects of contacting and mixing conditions of reactants play an essential role on the nucleation and crystal size distribution 2 . Indeed, if the mixing of both reactants is imperfect in the stirred tank, the concentration pattern will not be homogeneous and supersaturation heterogeneities will be created. Consequently, nucleation and crystal growth rates will vary in the mixer from one point to another. As a mixing model, the èuorescence properties of a solution mixed with pure water in a transparent tank are used, which is crossed by a planar laser sheet. During the érst moments of mixing, the mixing of the two tracers is similar to the mixing of two reagents in ionic precipitation. Indeed, there is no signiécant consumption of the reactants during the érst seconds of mixing, owing to the small size of nuclei produced and to the time scales of the crystal growth, which are much larger than those of nucleation 3 . As a result, mixing in ionic precipitation can be characterized by studying the mixing of two inert tracers, or the mixing of a single tracer in solution with pure solvent, each èuid being injected by two separate feed tubes. The laser sheet visualization method has been used by Mahouast 4 , and developed in a 0.02 m 3 mechanically stirred tank by Houcine et al. 5 and adapted at the pilot scale by Marcant et al. 6 . The laser induced èuorescence technique (L.I.F.), which is non-intrusive like tomography 7 , can help to optimize the geometry of the mixers, the feed rates or the positions and shape of the injectors. It is based on the stimulation of èuorescence by laser and the measurement of the emitted light. For low tracer concentrations the èuorescent intensity is proportional to the local concentration and to the power of the incident laser light. Previous workers have used a CCD camera 4-6,8 , a photomultiplier 9 , and a line scan camera 10 . They measured the èuorescent light intensity, respectively in a plane, at a point, and along a line. The planar laser induced èuorescence technique (P.L.I.F.) allows characterizing mixing in any plane crossing the èow deéned by the laser sheet. This technique is based on the èuorescence of organic substances induced by a laser sheet and coupled with image analysis. The originality of this study, compared to previous works 4-6 , lies (a) in the fact that the laser sheet is very thin (down to 100 mm); (b) the camera is much more sensitive and énally; (c) more powerful tools are used for image processing.
The aim of this paper is to describe and validate the PLIF set up designed for fast and precise characterization of contacting and mixing of one inert tracer and pure water.

Principle
Several organic molecules in solution, like èuoresceine or rhodamine B exhibit a èuorescence phenomenon when they are stimulated by laser light. For low tracer concentrations the intensity of èuorescence is proportional to the local concentration of tracer and to the power of the incident laser light. The èuorescence is emitted with a wavelength (about 590 nm) higher than that of the laser (520 nm). Its intensity is éltered and monitored by a high sensitive camera.

Procedure
The érst set measurements are performed to establish the grey level images in the following cases: the tank is élled with water. It provides for the grey level éeld of the background: NG ob (x, y, t); the stirred tank is élled with a èuorescent tracer solution. Its concentration is the mixing concentration during the experiment in the completely homogeneous state C mp . It yields the grey level éeld corresponding to the perfect mixing images; énally, two water feedstreams (one containing the èuorescent tracer) supply the stirred tank, yielding the grey level éeld: NG(x, y, t).
For a diverging laser sheet Houcine et al. 5  Where NG ob x; y and NG C mp x; y are the average grey levels of the background and the perfect mixing, respectively, e i and e e are the extinction coefécients of incident laser light and emitted èuorescence light, L is the depth of penetration of the laser beam into the tracer solution, L is the optic path of the èuorescent light in the tracer solution, K is a constant grouping physical and geometrical parameters Equation (1) takes into account the attenuation of the intensity along the optical path, approximately identical for NG C mp x; y; t and NG(x, y, t). Equation (2) is intended to determine the operating conditions of tracer concentration and power favourable to obtaining a proportionality and a great sensitivity between the grey level and èuorescent dye concentration. Thus, it was possible to determine the intensity of èuorescence emission as a function of: (a) the tracer concentration: the tank is élled with different solution of homogeneous concentration, varying between 0 and 40 10 6 kg m 3 ; (b) the laser intensity: the tank is irradiated at P o increasing, varying between 0 and 2 W.

Experimental
The scheme of the apparatus used in this work is shown in Figure 1. The experiments are carried out in a standard type stirred tank of 0.02 m 3 . The cylindrical vessel of internal diameter T 0.29 m is placed inside a square vessel élled with water in order to reduce problems associated with refraction by curved surfaces. Both vessels are made of a transparent material of optical quality Altuglass (PMMA). Four bafèes of width T=10 and thickness T=100 are provided at intervals of 90 against the wall of the inner tank. The six-bladed Rushton turbine in Altuglass is coupled to a stainless steel shaft which has an external diameter of 10 mm. The impeller diameter and bottom clearance are equal to H=3, where H represents the height of the liquid in the tank. The other dimensions of the stirred tank are represented in Figure 2. The stirring speed is éxed at N 110 rpm. The optical device includes a laser diode, with an emission wavelength of 520 nm and with a maximum power P o 2 W. When crossing the diaphragm, the laser beam is focused by a mirror and is spread out into a very thin sheet (down to 100 mm) using a POWELL lens. A CCD camera, of total exposure time 35 ms, étted with a sharp cut-off high pass élter takes 9 images per second. It gives high resolutions down to ten micrometers, since it provides for images of 1280 1024 pixels encoded on 4096 grey levels. The camera and the laser are jointly assembled on a vertical and horizontal displacement system. This device is controlled by a computer with a total displacement of 0.54 m and a repeatability of 0.1 mm. Two reservoirs continuously supply the tank with equal èow rates (Q A Q B 1.45 10 3 m 3 min 1 ) by two incoming feed tubes. The tubes have an internal diameter of 6 mm and their axis are separated by a distance of 19.2 mm. The feed point locations are at a height of 2H=3 and are situated on the tangential vertical plane to the impeller. In this plane, there is no signiécant orthogonal average velocity 11 . The tracer concentration (40 10 6 kg m 3 ) is selected in the range of linearity between concentration and grey level (see Figure 3). The èuorescence images are recorded with FlowManager 1 -Dantec software and then processed in Aphelion 1 -Adcis images analysis software with speciéc developed macros.

RESULTS
Before starting the experiments in the continuous stirred tank, the range of the linear response between the grey level and the tracer concentration were determined. This was done at four points in Figure 2. It can be seen in Figure 3 that the linearity is observed up to 40 10 6 kg m 3 (8.35 10 5 mol m 3 ) with linear regression coefécient R 2 > 0.99 (least square method). For larger concentrations the response of the grey level becomes non-linear because of the exponential term of the equation (2). This result is in agreement with those obtained by Houcine et al. 5 , who used a laser power of 1.4 W. The linear response between the laser and èuorescence intensity is given on the Figure 4 (linear regression coefécient R 2 > 0.99). In the range of  An example of the instantaneous dye concentration éeld is shown in Figure 5 for the case of a continuously fed tank reactor: èuorescent tracer injected at 40 10 6 kg m 3 appears white, whereas pure water appears black. The mean concentration éeld C C x; y is obtained by averaging a number n 100 of successive instantaneous concentration éelds C(x; y; t). The reduced concentration C C x; y = C C mp x; y , Figure 6, represents the relative deviation to perfect mixing. Along a line AB (see Figure 6), the ratio varies between 0 (black) and 2 (white): Figure 7, corresponding to the feed stream of water and èuorescent tracer, respectively. Perfect mixing is reached for the value 1.
The temporal variance is expressed by the following equation: This equation characterizes the mixing dynamics. Figure 8 shows the éeld of the temporal variance corresponding to the mixing layers of both jets. A white area means high variance, while black means low variance. The persistence of two zones of intensive mixing can be observed around the injection cones at the exit of the feed point locations.
The rest of the tank is well homogenized. There is almost no deviation of concentration from the mean. All the previously shown éelds characterize different aspects of the mixing process from the tracer concentration éeld. It is also interesting to have comparison tools to study various mixing situations. Thus, a mixing index is deéned. In the case of two species A and B, Ablitzer 12 proposed: In the case of a tracer water solution (A) and pure water (B), the equation (4) becomes (see appendix), with the tracer concentration C A (x, y, t): Figure 4. Grey level variation with laser intensity.  5 where: f f Q B = Q A Q B Examples of the instantaneous and average mixing index éelds are shown in Figures 9 and 10, respectively. This index varies from 0 (black), in the feed streams jets corresponding to no contact up to a value close to 1 (white) in the rest of the tank. Figure 11 illustrates the evolution of this contacting parameter along the straight line AB (shown in Figure 10).

CONCLUSION
This work describes a powerful non-intrusive technique based on the èuorescence spectroscopy using a very thin static laser plane (down to 100 mm). The technique is validated for the case of a standard stirred tank reactor of 0.02 m 3 volume, continuously fed with a tracer solution and pure water. Images of instantaneous and average concentration, éelds of temporal variance, and mixing index are very precise and illustrate the investigative power and resolution of the PLIF technique. Thus, this device can be used to study contacting and mixing èuid patterns. This will allow, the study of inèuence of parameters, like feed rate 13 , feed position in the tank, and also allow more sophisticated mixing devices to be characterized.

APPENDIX
The aim of this appendix is to derive the mixing index a (given by equation (5)) in this case, from the one deéned by Ablitzer 12 .
Consider the mixing of two inert tracers, without consumption, shown in Figure A1. If f represents the instantaneous fraction of molecule coming from the inlet B, at point M of coordinates (x, y), the instantaneous concentrations of species A and B are respectively: Figure 7. Reduced concentration variation along a line AB (in Figure 6).    . Variation of mean mixing index along a line AB (in Figure 10). and: Thus, at the outlet: In the case of stoichiometry: When equation (A6) is divided by Q A Q B ; it becomes:

A7
Addition of equations (A1) and (A2) leads to: averaging gives: For two species A and B, Ablitzer 12 proposed the following mixing index: Taking into account the equation (A8), gives: When C B0 is substituted from equation (A7), equation (A11) leads to: