A Liquid Crystal Thermographic Technique for the Measurement of Mixing Characteristics in Stirred Vessels

0263±8762/97/$10.00+0.00 q Institution of Chemical Engineers

A LIQUID CRYSTAL THERMOGRAPHIC TECHNIQUE FOR THE MEASUREMENT OF MIXING CHARACTERISTICS IN STIRRED VESSELS K. C. LEE and M. YIANNESKIS Centre for Heat Transfer and Fluid Flow Measurement, Mechanical Engineering Department, King’s College London, UK

A

liquid crystal thermographic technique has been developed and employed to determine the transient mixing characteristics in a stirred vessel of 100 mm diameter. The technique makes use of temperature as a passive scalar and shows promise for the rapid and accurate determination of mixing time and temperature distributions in stirred vessels. Characteristic results obtained in the 100 mm vessel stirred by one and two Rushton impellers are presented and compared with previously reported measurements obtained with other techniques. Keywords: stirred vessel; mixing time measurement; liquid crystal thermography; Rushton impeller

INTRODUCTION

mixing state in the whole vessel can be determined at the same time and that in some situations `dead’ or stagnation zones may be identi® ed quickly. The most commonly used transducer methods make use of conductivity or thermocouple probes to detect the variation of the conductivity or temperature of the ¯ uid in the vessel, respectively. This variation is produced by introducing a small amount of a passive scalar of different conductivity or temperature. However, both techniques can only measure the local concentration or temperature. In order to resolve the complex concentration or temperature distributions encountered in the three-dimensional ¯ ows in stirred vessels, it is often necessary to make measurements in a large number of locations. Such measurements are nonsimultaneous, time-consuming and the probes used, however small, will interfere with the ¯ ows. One technique that shows promise for temperature/ mixing time measurements in stirred vessels is liquid crystal thermography. This utilizes the change in colour of thermochromic liquid crystals when they are subjected to different temperatures; liquid crystals exhibit a rapid and reversible response to dynamic temperature changes over a wide range of temperatures. A number of applications of liquid crystal thermography techniques for the measurement of wall temperature distributions in forced convection ¯ ows have been reported, for example by Cooper et al.6 , Goldstein and Timmers7 , Rojas et al.8 , Yianneskis9 and Lee and Yianneskis10 . Other applications have included the measurement of transient temperature ¯ uctuations in ¯ uids, for example Kuriyama et al.11 and Rhee et al.12 . The ability to suspend liquid crystal tracers in a stirred vessel can allow temperature measurements to be obtained simultaneously across the whole vessel. This renders liquid crystal thermography a very convenient and versatile technique for the measurement of mixing time in stirred

The ¯ ow® elds in stirred vessels are complex, with strongly three-dimensional characteristics, transient vortical structures which are not always well-de® ned and high turbulence levels in the vicinity of the impellers. Knowledge of the ¯ ow and mixing characteristics in a stirred vessel is therefore essential not only for the optimization of the particular process concerned, but also for the formulation of appropriate design methodologies. Single-phase non-reacting ¯ ow in stirred vessels can be adequately characterized through the measurement of the power consumption, velocity characteristics and mixing time. Power consumption is normally determined with strain gauge torque measurement techniques which are well established (see, for example, Kuboi and Nienow1 , Rutherford et al.2 ), while non-obtrusive velocity measurement techniques, such as laser-Doppler anemometry (LDA) have been extensively used to determine accurately the velocity characteristics of stirred vessels (for example, Yianneskis et al.3 , Stoots and Calabrese4 ). However, in contrast to power consumption and velocity measurement techniques, there is not a universally accepted technique suitable for the accurate determination of mixing time. Scho® eld5 reviewed the techniques employed for the measurement of mixing time in stirred tanks. For liquidliquid mixing, these can be generally classi® ed into two groupings: observation and transducer methods. In observation methods, the change in the colour of the liquid contained in the tank caused by the insertion of either a dye or a reacting liquid is observed as the mixing process proceeds. In the latter case, the addition of the liquid initiates a chemical reaction which results in a colour change. Large inaccuracies may be encountered with this technique which may lead to underestimation of the mixing time, but two advantages of observation methods are that the 746

LCT TECHNIQUE FOR MEASUREMENT OF MIXING CHARACTERISTICS tanks, using temperature as a passive scalar. Such a technique would not only present none of the aforementioned problems with probes, but could also offer all the advantages of observation methods together with considerably increased accuracy. In the present investigation, a liquid crystal thermographic technique was developed and employed for the measurement of mixing time in a vessel stirred by both one and two Rushton impellers. The principles and methodology of liquid crystal thermography, the experimental procedure and the results obtained are described and discussed below. LIQUID CRYSTAL THERMOGRAPHY (LCT) Liquid Crystals Liquid crystals are organic materials which exhibit a phase between the solid phase and the conventional isotropic liquid phase. This is termed a mesomorphic phase or mesophase. Liquid crystal thermography utilizes the temperature dependency of thermochromic liquid crystals; these have rod-like molecules, which re¯ ect light at a speci® c wavelength for a given temperature. By evaluating the colour display, accurate temperature measurements can be obtained. There are three main types of thermochromic liquid crystals: smectic, nematic and chiral nematic. Mixtures of chiral nematic crystals are used in liquid crystal thermography. In the chiral nematic phase, the molecules are arranged in thin layers. Within each plane layer the molecules are aligned with their long axes parallel like in the nematic phase, and have an average direction de® ned by a direction vector. Each layer is slightly twisted with respect to the next. The effect is cumulative and an overall helical structure is formed. The average molecular direction of each layer traces out a helix in space. The pitch length, de® ned as the longitudinal distance in which the direction vector undergoes a complete 3608 revolution, is used to quantify the degree of twist. This spiral arrangement of molecules in this mesomorphic phase is responsible for its unique optical properties. Changes in temperature affect the rotation angle between adjacent layers and hence the pitch of the helix. Light is re¯ ected from the crystals due to Bragg diffraction from molecular layers whose axes are aligned and the colour of the re¯ ected light changes with molecular orientation. The pitch decreases slightly with temperature. Chiral nematic liquid crystals usually turn from colourless to red (long wavelength) at low temperatures, passing through the colours of the visible spectrum to blue/violet (short wavelength) and ® nally to colourless again at higher temperatures. The characteristic wavelength at which a particular chiral nematic crystal scatters light is equal to the product of the mean refractive index and the pitch of the helix, and generally depends on the liquid crystal composition, the imposed electric and magnetic ® elds, pressure, shear stress, vapour present in the ¯ uid and the angles of incidence and re¯ ection of the light, as well as temperature (Chandrasekhar13 ). Moreover, as liquid crystals do not change colour abruptly at a single precise temperature, but sweep through the spectral range from red to blue over a range of temperatures, the interpretation of liquid crystal images is complicated and may be made more complex by the often imperfect colour response of the crystals. Trans IChemE, Vol 75, Part A, November 1997

747

In order to extract the temperature information from the liquid crystal colour displays, it is necessary to calibrate the liquid crystals at the outset of an experiment so as to ascribe particular colours to temperatures over the response range of the liquid crystals. Furthermore, since liquid crystals are subject to contamination and deterioration, calibration may also be necessary in regular intervals to ensure that no change of the response characteristics has occurred. Colour Measurement Monochromatic radiations with wavelengths in the range 380±780 nm are in general visible to the eye. Though wavelength is a physical quantity, which can be objectively measured with great accuracy, colours perceived by individuals depend on all kinds of properties of the human visual system. The determination of colours by humans is therefore very complex as it involves a subjective physiological process. In order to evaluate colours quantitatively, various schemes of specifying colour have been proposed and adopted. The internationally adopted system of colour speci® cation drawn up by the Commission de l’ Eclairage (CIE) is a method of trichromic decomposition of colour, which is used in television and video. By measuring the relative amount of each of the three primaries (RGB) contributing to the observed colour, its CIE chromaticity co-ordinates (u9 , v9 ) can be determined, and the hue angle, h, and the saturation, s, of the colour are given by: h

= arctan((v9 - v9 )/ (v9 - u9 )) n

n

(1)

and s = 13((v9

- v9n ) + (u9 - u9n ) ) 2

2 1/ 2

(2)

where u9n and v9n are the chromaticity co-ordinates of equitristimuli (i.e. equal amounts of R, G and B). Figure 1 illustrates the geometrical de® nition of h and s for a colour on a CIE chromaticity diagram. In the diagram, C and N are the points representing the colour considered and the reference white respectively. h is the angle between the line NC and the horizontal line drawn from N to the right, measured anti-clockwise from the horizontal line; and s is

Figure 1. v9 - u9 diagram: Graphical representation of the hue and saturation of a colour.

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LEE and YIANNESKIS

13 times the distance between N and C. The chromaticity coordinates that can be reproduced by an RGB video system are contained within the area bounded by the curve and straight line in Figure 1. The Digital Image Processing System The digital image processing system used in this work to record and interpret the video images of the liquid crystal colour displays is shown schematically in Figure 2. It comprised a CCD colour video camera with PAL colour standard composite video signal output, a S-VHS PAL video cassette recorder with still frame facility, a multisystem/RGB monitor and a computer with a video frame grabber/digitizer card and associated software. Liquid crystal colour displays are captured by the camera and the composite video signal output of the camera is input to the video cassette recorder. The images received by the video recorder are observed on the monitor to assess their quality. When satisfactory conditions are attained, the events are recorded onto video tapes at a rate of 25 frames per second. The frame rate is determined by the PAL colour standard used by the video system. The recorded events are then played back in a `frame-byframe’ mode, so that the timing of the frame to be analysed relative to a reference frame/start of the event investigated can be ascertained. The composite video signal of a frame is input to the computer via the frame grabber/digitizer card. The frame grabber decodes the composite video signal and grabs the frame as a RGB true colour image. The grabbed image is digitized into a 24-bit digital image composed of three planes of pixels where each pixel has a red, green and blue intensity, each coded on a 8-bit format (i.e. 256 levels). The digitized image is then stored on disk for off-line colour measurement to obtain the hue, saturation and lightness of each pixel.

FLOW CONFIGURATION AND MEASUREMENT TECHNIQUE The geometry of the mixing vessels and Rushton impellers used in this investigation is shown in Figure 3.

Figure 3. Cross-section of the 100 mm stirred vessel with (a) one and (b) two Rushton impellers.

It comprised a fully-baf¯ ed cylindrical vessel of internal diameter T = 0.1 m and height 0.25 m. A small vessel was used in this ® rst application of the technique as it is dif® cult to produce a suf® ciently uniform as well as wide (for use in larger vessels) sheet of white light for LCT purposes. Four equi-spaced baf¯ es of width B = 0.1T were ® tted along the internal surface of the vessel. The vessel was installed in a

Figure 2. Schematic diagram of the video image processing system.

Trans IChemE, Vol 75, Part A, November 1997

LCT TECHNIQUE FOR MEASUREMENT OF MIXING CHARACTERISTICS

trough of square cross-section. The gap between the vessel and the trough was ® lled with distilled water fed from a constant-temperature water bath to dissipate the heat generated by the light source and to control the temperature of the water inside the vessel, as well as to minimize refraction effects at the cylindrical surface of the vessel. The impellers used were six-bladed Rushton turbines of diameter D = T / 3. Both the blade thickness, tb , and the disk thickness, td , were 1 mm. A clearance C = T / 3 was used between the bottom of the mixing vessel and the impeller disk central plane for the single-impeller measurements. Clearances C1 = T / 3, between the bottom of the mixing vessel and central plane of the lower impeller disk and C2 = T / 3, between the central planes of the lower and upper impeller disks were used for the dual-impeller measurements. The working ¯ uid inside the vessel was distilled water and a liquid column height H = T was used. A transparent lid was located above the liquid surface at a height H = T so that no air bubbles were entrained into the liquid from the free surface. The effect of a lid on the ¯ ow in a stirred vessel has been previously investigated by Nouri and Whitelaw14 , who concluded that the use of a lid only affects the ¯ ow in the immediate vicinity of the lid/free surface and that the velocities elsewhere in the vessel were almost identical to those in the absence of a lid. Experiments were conducted at four impeller rotational speeds for both the single- and double-impeller con® gurations: 540, 1083, 1624, and 2165 rpm, corresponding to Reynolds numbers (Re) of 10,000, 20,000, 30,000 and 40,000, respectively. The temperature of the ¯ uid inside the vessel and of the tracer used in the experiments was 25.28 C and 278 C, respectively. Thermochromic liquid crystals encapsulated as gelatine-shell micro-spheres of a mean diameter of 20 l m were suspended in the ¯ uid in the vessel. A 5 ml tracer of liquid crystals suspended in distilled water, having the same liquid crystal concentration as the ¯ uid inside the vessel was contained in a hypodermic needle-syringe assembly, and was inserted into the vessel at r/ T = 0.25 in the h = 08 plane (located half way between two baf¯ es) through a hole on the lid surface, i.e. at z/ T = 1.00, by means of applying a dead weight to the syringe. The tracer insertion time was kept constant throughout the experiments at 0.4 s. The transient process of the mixing of the tracer in the vessel was recorded onto video tape at a rate of 25 frames per second. The tracer insertion was performed after the video recording had commenced so that the time of the start of the insertion could be accurately ascertained. The video recording of each of the processes was subsequently subjected to a frame-by-frame analysis to determine the time between the frame at which the insertion of the tracer commenced (ni ) and the frame at which the process was fully-mixed (nm ). The mixing time of the process (h m ) was then obtained from: n n h m= m- i (3) frame rate At the outset of the experiments the liquid crystal response to temperature was calibrated. The colours displayed by the liquid crystals depend on the lighting condition as well as on the image collection angle. Therefore these conditions were optimized so that during the calibration experiments the Trans IChemE, Vol 75, Part A, November 1997

749

values of hue of all pixels were within 6 5 of the mean value of hue at any particular temperature. To calibrate the variation of hue with temperature, the h = 08 plane was illuminated. The constant temperature water bath which fed water into the trough of the test section was set at the desired temperature. When steady state conditions were reached, the temperature inside the vessel was measured by a ® ne-wire thermocouple, and the ¯ ow® eld was recorded. This procedure was repeated to cover the response range of the liquid crystals. The images obtained were analysed using the digital image processing system described above to obtain the calibration curves such as that shown in Figure 4. It can be observed from Figure 4 that the value of hue varies with temperature approximately linearly between 258 C and 27.58 C. Therefore, experiments were conducted within this temperature range only, and mixing time was determined by means of analysing the hue distribution of the colours displayed by the liquid crystals. A process was considered to be fully-mixed when 95% of the pixels in each image had the same value of hue. Moreover, though the impellers and the shaft were painted matt black in order to minimize re¯ ections, a strong scatter of light was observed in the vicinity of the impellers and shaft as well as near the vessel wall, lid and base. Therefore, analysis of the hue of the images obtained was not carried out in these regions. RESULTS Single Rushton Impeller Con® guration Characteristic hue contour maps of the recorded images obtained with the single Rushton impeller at a rotational speed of 540 rpm are shown in Figures 5±8. All hue values have units of degrees of angle, but units are not shown in the ® gures in order to avoid their misinterpretation as degrees of temperature. The hue contour map of the image obtained 200 ms after the start of the insertion of the tracer is shown in Figure 5. The jet of the tracer, which was at a higher temperature than the ¯ uid in the vessel, is indicated by the region of hue values above 120. Figure 6 shows the hue contour map of the image obtained 400 ms after the start of the insertion of the tracer.

Figure 4. Typical temperature-hue calibration curve.

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LEE and YIANNESKIS

Figure 5. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by a Rushton impeller at 540 rpm, 200 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

Figure 7. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by a Rushton impeller at 540 rpm, 600 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

The region of hue values higher than 120 again indicate the jet of the tracer, which has moved further towards the vessel wall under the in¯ uence of the impeller stream and has reached the bottom of the vessel. As in Figure 5, locally high hue values (above 135) can be observed in the core of the tracer jet. Figure 7 shows the hue map of the image obtained 600 ms after the start of the insertion of the tracer. Regions of high hue values can be observed only below the impeller. At 1.76 s after the start of the insertion of the tracer (Figure 8), the hue values in most regions are between 105 and 120, indicating that the ¯ uid inside the vessel is nearly fully-mixed. Images obtained from the recordings obtained with

higher impeller speeds showed similar mixing behaviour to that described above. Hue analysis was performed on the images obtained with all impeller speeds to determine the mixing times. The mixing times were found to be 1.92, 1.04, 0.68 and 0.60 s for impeller speeds of 540, 1083, 1624 and 2165 rpm respectively.

Figure 6. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by a Rushton impeller at 540 rpm, 400 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

Figure 8. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by a Rushton impeller at 540 rpm, 1.76 s after the start of the insertion of the tracer. (See also page 2 of colour insert.)

Double Rushton Impeller Con® guration Characteristic hue contours of the images obtained with the double Rushton con® guration with a rotational speed of 540 rpm are shown in Figures 9±12. Figure 9 shows the hue contours 200 ms after the start of the insertion of the tracer. The tracer, which has a higher

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751

Figure 9. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by two Rushton impellers at 540 rpm, 200 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

Figure 11. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by two Rushton impellers at 540 rpm, 600 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

temperature than the ¯ uid inside the vessel, is again indicated by the high hue value (135±150) region in the ® gure. Figure 10 shows the hue contours of the image obtained 400 ms after the start of the insertion of the tracer. It can be observed that the tracer has now spread along the bottom of the vessel. 600 ms after the start of the insertion (Figure 11), most of the tracer is located below the lower impeller. Figure 12 shows the hue contours of the image obtained 1.12 s after the start of the insertion of the tracer. Regions with hue values below 135 can be observed over the entire ¯ ow ® eld in this ® gure, indicating that the ¯ uid inside the vessel is nearly fully-mixed. Images obtained for impeller speeds of 1083, 1624 and 2165 rpm revealed mixing patterns similar to those observed

for 540 rpm. It should be noted, however, that as the impeller speed is increased, more rigorous mixing takes place in the vessel, mixing times are shorter, and consequently, the tracer jet is dispersed progressively more in the upper part of the vessel. Although this was also observed in the single impeller experiments, it is more pronounced in this case due to the more rigorous mixing achieved by the two impellers and the time taken to reach a fully-mixed state is shorter than with the single impeller con® guration. Hue analysis of the images obtained for all the impeller speeds investigated were performed using the calibration data, to determine the mixing times. The average mixing times measured were 1.68, 0.88, 0.56 and 0.48 s, for

Figure 10. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by two Rushton impellers at 540 rpm, 400 ms after the start of the insertion of the tracer. (See also page 2 of colour insert.)

Figure 12. Hue contours of one-half of the ¯ ow® eld in the vessel stirred by two Rushton impellers at 540 rpm, 1.12 s after the start of the insertion of the tracer. (See also page 2 of colour insert.)

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impeller speeds of 540, 1083, 1624 and 2165 rpm respectively. DISCUSSION Since the number of revolutions required to achieve full mixing in an agitated tank is essentially a constant for a particular geometry (Tatterson15 ), N and h m can be related by: Nh

m

=K

(4)

where N is the impeller speed in rev/s, h m is the mixing time in seconds, and K is a constant. The h m and N h m values obtained for the single and double impeller arrangements are listed in Table 1 below. The average values of h m were obtained from 4 experiments, with the exception of the N = 2165 rpm case with the double impeller con® guration, which was obtained from 3 experiments. For each experimental condition, the h m values were repeatable to within 6 80 ms. It can be seen from Table 1 that the values of N h m for both the single- and double-impeller con® gurations are approximately constant up to a Reynolds number of 30,000. At the highest Re (40,000), the values of N h m for both con® gurations are higher. These higher values can be partly attributed to the comparatively long tracer insertion time, which was 0.40 s in all cases, corresponding to nearly 67% and 83% of the mixing times at 2165 rpm for the single- and doubleimpeller con® gurations, respectively. Clearly, shorter insertion times cannot be achieved in practice and most studies to date (for example, Armstrong and Ruszkowski16 ) have employed pulse injections of tracer lasting around 0.5±1.0 s. This represents a limitation of mixing times studies carried out in small vessels and with high impeller speeds to achieve dynamic similarity and, although the average N h m values may be overestimated, they are not strongly affected by the speed. For example, the average values of N h m determined from the experiments at all four Reynolds numbers are 18.99 and 15.84 for the single- and doubleimpeller con® gurations, respectively. The corresponding average values for the three lower Reynolds numbers are 17.97 and 15.36, respectively, i.e. they are smaller by only 5.4% and 3.1%, respectively. In contrast, it should be noted that as the velocity of the tracer insertion was relatively high, the mixing time measurements could be affected, especially during the early stages of tracer injection, when the ¯ ow in the vessel could be modi® ed by the injection process as indicated, for example, by the tracer dispersion patterns in Figures 5 and 6, resulting in an underestimation of the mixing time in the vessel. Clearly, future extension of the technique to measurements in larger vessels would Table 1. h

m

and N h

m

alleviate the need for relatively high insertion velocities and long insertion times. Finally, it should be noted that the similarity of the N h m values measured indicates that the ¯ ows in all Res studied in this work are fully-turbulent, as N h m values in the transition regime are different (Norwood and Metzner17 ). The average N h m values obtained were compared with those reported in the literature for single Rushton turbines. Prochazka and Landau18 developed a mixing time correlation for a Rushton impeller in a baf¯ ed tank in the standard con® guration for Re > 10, 000: Nh

m

= 0.905

2. 57

T D

log

9 18 27 36

Re 10000 20000 30000 40000

(5)

where X0 was an initial value of the degree of inhomogeneity which ¯ uctuated between 1.0 and 3.0, and the other symbols have their usual meanings. A value of 2.0 was recommended for X0 and the quantity Xc was the integral mean value of the local degree of inhomogeneity, de® ned as: Xc

=

C(t) - Cx Cx - Ci

(6)

where C(t) was the instantaneous concentration, Ci and Cx were the initial and ® nal concentrations respectively, and Xc was considered to be 0.05 for most con® gurations. Using the recommended values for X0 and Xc , and substituting the values of T and D used in the present work into equation (5), N h m was found to be 24.4, which is 22% higher than the value obtained for the single-impeller con® guration in the present work. Moo-Young et al.19 correlated their mixing time results using: (7) = KRe where K = 36 and a = 0 for turbines in baf¯ ed tanks for Nh

a

m

1000 < Re < 100, 000. Substituting these values into equation (7), the resulting N h m value is 36. A similar value of N h m (34) was reported by Shiue and Wong20 for a Rushton impeller located at a clearance of C = 0.325 H. Both these values are higher than those obtained in the present study for the single-impeller con® guration. Sano and Usui21 provided an expression for mixing times of tracer injection for turbines: Nh

m

= 3.8

D T

- 1. 80

W T

- 0. 51

n-p 0.47

(8)

where np is the number of blades and the other symbols have their usual meanings. Using equation (8) for the singleimpeller con® guration of the present work, N h m is 38.34.

with single- and double-impeller con® gurations. Single-impeller con® guration

N (rev/s)

X0 Xc

h

m

(s)

1.92 1.04 0.68 0.60

Double-impeller con® guration Nh

m

17.28 18.72 18.36 21.60

h

m

(s)

1.68 0.88 0.56 0.48

Nh

m

15.12 15.84 15.12 17.28

Trans IChemE, Vol 75, Part A, November 1997

LCT TECHNIQUE FOR MEASUREMENT OF MIXING CHARACTERISTICS For the double-impeller con® guration, considering the addition of an impeller as increasing the number of blades from 6 to 12, the value of N h m obtained using equation (8) was 27.68. Again, both values are higher than those obtained in the present study. For continuous ¯ ow systems, values of N h m up to 50 have been reported (Biggs22 ). There is a number of likely reasons for the differences between the N h m values determined in this work and those reported in the literature. First, the wide variation in N h m values obtained with the different published correlations (24.4±38.34) may be due to differences in experimental procedure and/or the de® nition of when the `fully-mixed’ state is achieved (e.g. 90% or 95% of the ® nal concentration of tracer). Second, tracer insertion times have not been reported in some of the aforementioned studies, but they may affect the measured N h m values. Evidence for this is provided in the work of Mahmoudi23 who reported an average mixing time of 12.8 s with a single Rushton impeller and an insertion time of 6.5 s. The corresponding N h m values are around 45±50 compared with the values of up to 38 reported by other investigators, indicating, as might be expected, that longer insertion times will result in larger N h m values. This is in agreement with the ® ndings at Re = 40000 compared with those at lower Res in the present work. A third reason might be the vessel size. Kramers et al.24 reported that mixing times are affected by scale: a larger tank required a larger number of revolutions (i.e. a higher value of N h m ) to achieve the same degree of mixing. In addition, impeller blade and disk thickness could also affect N h m . It has been established (Rutherford et al.2 ) that smaller blade thicknesses result in higher ¯ ow numbers and this might be expected to affect mixing times as well. There is a substantial effect of blade width on mixing times (Tatterson15 ), but as relevant values are not quoted by many investigators, it is dif® cult to ascertain how large this effect is. The results presented above show clearly that rapid and accurate measurements of the transient mixing characteristics and mixing time in stirred vessels are possible with liquid crystal thermography. The diversity of previously reported N h m values indicates that better de® ned procedures and more accurate whole-vessel techniques for the measurement of mixing time, such as LCT, are desirable. Two possible improvements of the present LCT technique have been identi® ed. First, the imperfect uniformity of the light sheet used for illumination necessitates at present extensive local calibration of the images. This could be ameliorated by a more uniform light source. Second, the tracer injection velocity is unavoidably high in order to achieve a relatively short insertion time; however, this could be alleviated in future applications of the technique to larger vessels. CONCLU DING REMARKS The measurements presented above have shown that the liquid crystal thermographic technique developed can be employed to provide a detailed characterization of the transient mixing processes in a stirred vessel of 100 mm diameter. Mixing times were determined at four impeller rotational speeds for both single and double Rushton impeller con® gurations. The N h m values obtained in this work were compared Trans IChemE, Vol 75, Part A, November 1997

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with those of previous studies. The comparisons indicate that vessel size and tracer injection velocity might affect the measurement of mixing time. In addition, standardization of the procedures employed is necessary if meaningful comparisons are to be made and the effect of impeller geometry, such as blade and disk thickness, must be accounted for in future work. The promise shown by the present application of the LCT technique indicates that measurements in larger vessels are required to develop further the technique and extend its application for the measurement of mixing times in stirred vessels. The technique could also be used to measure mixing times with ¯ uids other than water, provided that such ¯ uids are of similar density to the liquid crystal microcapsules so that the crystals are neutrally buoyant and that the ¯ uids do not attack chemically the microcapsules: tests are required to establish the suitability of such ¯ uids, but organic ¯ uids are unlikely to be suitable for LCT applications. NOMENCLA TURE B C C(t) Ci Cx C1 C2 C3 D h H K ni nm np N r Re s tb td T u9 , v9 u9n , v9n W X0 Xc

baf¯ e width, m clearance distance between vessel base and centre of the impeller disk, m instantaneous concentration, mol l- 1 initial concentration, mol l- 1 ® nal concentration, mol l- 1 clearance distance between vessel base and centre of the lower impeller disk, m separation distance between the centre of the upper and lower impeller disks, m submergence distance of the centre of the upper impeller disk, m impeller diameter, m hue liquid column height in vessel, m constant in equations (4) and (7) the number of the video frame at the start of the insertion of the tracer the number of video the frame when the ¯ uid in the vessel is fully mixed number of blades impeller rotational speed, rev s- 1 radial distance from the axis of the vessel Reynolds number, Re = ND2 / v saturation impeller blade thickness, m impeller disk thickness, m vessel diameter, m chromaticity coordinates chromaticity coordinates of the reference white colour impeller blade width, m initial integral mean value of the local degree of inhomogeneity in equation (5) ® nal integral mean value of the local degree of inhomogeneity in equation (5)

Greek symbols a constant in equation (7) h tangential coordinate, degrees hm mixing time, s m ¯ uid kinematic viscosity, m2 s- 1

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ACKNOWLEDGEMENT The authors acknowledge ® nancial support from the European Commission under JOULE Programme grant JOU2-CT92-0127 for the work presented in this paper.

ADDRESS Correspondence concerning this paper should be addressed to Professor M. Yianneskis, Centre for Heat Transfer and Fluid Flow Measurement, Mechanical Engineering Department, King’ s College London, Strand, London WC2R 2LS, UK. The manuscript was received 16 October 1996 and accepted for publication after revision 26 February 1997.

Trans IChemE, Vol 75, Part A, November 1997