Effect of shaft eccentricity on the laminar mixing performance of a radial impeller.
Fluid mixing in mechanically agitated tanks is a common unit operation in the chemical process industries. According to the fluid viscosity and process constraints, generally two types of impellers are encountered in mechanically agitated vessels: close clearance impellers and open impellers. Close clearance impellers are used in the laminar regime because they generate sufficient bulk movement to homogenize the fluid at an acceptable level of mixing. They are, however, disadvantaged by a high-energy consumption. On the other hand, open impellers draw less power, but they are inefficient for mixing in the laminar regime. Indeed, open impellers generate segregated regions in the laminar regime irrespective of the discharge flow type, whether radial or axial (see Figure 1). Recent studies on open impellers (Alvarez, 2000; Ascanio et al., 2002a,b; Alvarez et al., 2002; Hall et al., 2005x; Sanchez Cervantes et al., 2006) conclude that, in the laminar regime, the dynamic perturbations generated by using an eccentric shaft could help to break up the segregated regions, and increase the axial circulation in a tank even with radial agitator. This change in flow patterns prevents the formation of flow compartmentalization and helps to improve the mixing efficiency of the mixing device.
The idea of using off-centred impellers in the laminar regime is fairly recent but already pretty well established in the turbulent regime. Indeed, many authors have shown that, in turbulent flow, shaft eccentricity is equivalent to baffling (Joosten et al., 1977; Nishikawa et al., 1979; Novak et al., 1982; King and Muskett, 1985; Hall et al., 2004, 2005b; Karcz and Szoplik, 2004; Karcz et al., 2005; Montante et al., 2006). Nishikawa et al. (1979), Karcz and Szoplik (2004) and Karcz et al. (2005) have shown that for axial and radial type impellers the mixing time decreases with the increase of the shaft eccentricity. Many studies also report increased power with the shaft eccentricity with both axial and radial type impellers (Nishikawa et al., 1979; Novak et al., 1982; Karcz et al., 2005).
As only limited data is available to describe the effects of shaft eccentricity on mixing time and power consumption in the laminar regime, the objective of this article is to examine and quantify these effects when a Rushton turbine (RT) is used. Let us note that the results of this study might also be extended to axial impellers as they generate radial flow characteristics in laminar regime (Fangary et al., 2000). The purpose of this study is to determine the optimum shaft eccentricity in laminar flow.
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MATERIALS AND METHODS
The experiments were conducted using a non-baffled cylindrical polycarbonate tank with an open-top and a flat bottom, and having an internal diameter T = 0.215 m (see Figure 2). A liquid column height H = T was used yielding a liquid volume of 7.8 L. The open impeller examined in this study was a radial flow six-blade RT with a diameter of D = 6.53 x [10.sup.-2] m. The following notations will be used in the forthcoming: R denotes the radius of the tank; e, the distance from the middle of the shaft to the centreline of the tank (see Figure 2), and the eccentricity, E, is defined as E = e/R. In the different experiments E was varied between 0 and 56.2% (56.2% being the highest possible value of E for the RT without touching the tank wall), and the bottom clearance of the impeller was kept constant at C = T/3. The mixing shaft was driven by an AC motor (0.69 kW). To modify the value of the shaft eccentricity, the vessel was moved horizontally on the table, perpendicularly with the camera focal axis (see Figure 3). A torquemeter (Vibrac[TM], Model TQ-512) coupled to the agitation shaft was used for determining the power consumption in a range between 0 and 3.61 N m.
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[FIGURE 3 OMITTED]
In the experimental setup presented in Figure 3, the cylindrical mixing tank is installed in a second rectangular water-filled chamber in order to minimize optical distortions due to the curvature of the tank. To obtain homogeneous illumination, a white sheet of paper was used as light diffuser on the rectangular vessel. The mixing process was filmed with a digital mono CCD camera (Digital Handycam DCR-PC 101, Sony[TM]) linked to a computer via a 1394 IEEE (FireWire) connector.
Aqueous solutions of corn syrup (Glucose Enzose 62DE, Univar[TM]) were used as the viscous Newtonian fluids. In all the experiments the glucose solutions were allowed to settle for 24 h before starting the experiments in order to eliminate air bubbles. Knowing that the viscosity, [micro], of the viscous corn syrup solutions varies significantly with temperature, preheating of the solutions was achieved by leaving the impeller in rotation (using viscous dissipation as the source of heat). A constant temperature of 27.2[degrees]C was then reached and the temperature was monitored during the experiments to ensure it remained at this constant value. Newtonian viscosities of the solutions were determined with a viscometer (Visco 88, Bohlin[TM] Instruments) at 27.2[degrees]C with the Couette configuration. During the experiments, the viscosity of the solutions was found in the range 6.45-6.85 Pas and the liquid density, [rho], was 1360 kg/[m.sup.3].
In order to get a reliable colour change, a solution of bromocresol purple (0.08 % w/w in water) was used as the indicator. This indicator is yellow when pH < 5.2 (acid colour) and purple when pH > 6.8 (alkaline colour). Approximately 45 ml, of the solution of bromocresol purple was added in the 7.8 L of corn syrup solution. In all the experiments, we followed the colour evolution from purple (alkaline colour) towards yellow (acid colour), because it is far easier and more reliable to detect purple unmixed zones in a yellow liquid than the opposite.
At the beginning of a given experiment the glucose solution was set to purple, as close as reasonable to the colour change limit by using aqueous NaOH at 10 mol [L.sup.-1] . Then, 200 ml, of solution was sampled from the tank and mixed with 2 ml, of HCl at 10 mol [L.sup.-1]. For reproducibility purposes, the acidic solution was always rapidly and gently injected at the same distance from the shaft (5 cm) on the free surface using a syringe having a large orifice. The tracer insertion time was kept constant throughout the experiment at 1.0 s, this time being very small compared to the duration of the experiments.
Image analysis was used according to the method proposed by Cabaret et al. (2007). Each video captured by the digital camera during the acid-base colour change was sampled and the resulting images were then analyzed individually. The area of interest selected from the video frames is shown in Figure 4. For each pixel, the evolution of the level of brightness of the green colour (RGB colour model) is followed over time. By defining an individual threshold for each pixel, a pixel can be considered either mixed or unmixed by comparing its green level with the threshold. As recommended by Cabaret et al. (2007), in our experiments we used a pixel separation value (X) of 50 % between the first unmixed picture and the last fully mixed picture to define the threshold. For each sampled image, we then count the number of mixed pixels [N.sub.MixedPixels], and plot the ratio (M, %) ([N.sub.MixedPixe1s]/[N.sub.TOta1Pixe1s]) over time, to obtain the mixing curve. Obviously, M is equal to 0 at t = 0 and M goes to 100% when complete mixing of the liquid is reached. The resulting curve quantifies the colour change from the observer's point of view. The technique has been shown to be highly reproducible and yields not only an eventual mixing time value but also a complete evolution of the colour change.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Figure 5 presents mixing curves as the percentage of mixed pixels, M (%), against time, which can be seen from the observer's point of view. It is important to note that when M stabilizes at a plateau without reaching 100%, there remains at least one unmixed zone in the area of interest. The difference between 100% and the plateau represents the 2D projection surface of the unmixed zone in the area of interest and it corresponds to what an observer can see in front of the vessel without providing quantitative information on the unmixed volume. This limitation must be kept in mind in the analysis of any mixing results with an unmixed region. However the final or constant value of M can be used to compare the sizes of different segregated region(s). Moreover, let us note that the slope of the rising part in Figure 5 represents the pumping capacity of the impeller.
As the laminar region for a RT exists below a Reynolds number value of 10 (Tatterson,1991a), we performed the experiments with seven different Reynolds numbers (Re): 1.7, 2.5, 3.5, 5.1, 7.5, 10, and 13. For the shaft eccentricity, we studied seven different values of E: 0%, 14.3 %,25.7%, 34.3 %,40.9%, 50.5%, and 56.2 (56.2% being the highest possible value of E for the RT without touching the tank wall). The whole set of values of Re and E tested corresponded to the performance of 49 experiments.
RESULTS AND DISCUSSION
Effect of Reynolds Number on Mixing
The standard mixing curves obtained give the percentage of mixed pixels over time (M vs. time). However, to compare the different results, it is far more interesting to plot the percentage of mixed pixels against the dimensionless time represented in this study by the number of revolutions of the impeller ([N.sub.R]). The plots of M versus [N.sub.R] are called dimensionless mixing curves.
Figure 6 presents the effects of the Reynolds number on the mixing curves at constant eccentricity (E = 34.3%). The Reynolds number influences the mixing curves in two ways: the level of M achieved at a given NR in the rising part and the maximum value of M reached in the stable part. Indeed, as we can see with the arrows in Figure 6, both the rising and the stable part of the dimensionless mixing curves are improved when Re increases. This result is in contradiction with the theory claiming that there exists a unique dimensionless mixing curve in the laminar regime. Indeed, the dimensionless mixing time is presented as constant in this regime (Nagata, 1975; Tatterson, 1991b; Edwards et al., 1992). A thorough investigation revealed that no thermal gradient yielding a higher viscosity near the free surface (injection zone) is at the source of the varying mixing kinetics. Thus, our results corroborate those of Rice et al. (2006), showing that the net pumping capacity of the RT in the laminar regime decreases when the Reynolds number decreases. A drop in the impeller pumping capacity appears at low Re (below 10) where there are small values for the inertial forces (creeping flow). In this case, the liquid is not radially ejected due to a balance between the pressure and viscous forces.
[FIGURE 6 OMITTED]
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[FIGURE 8 OMITTED]
In order to validate that the non-constant mixing curves are independent of the eccentricity value E, we have plotted in Figure 7 the variation of M versus Re at a given number of revolutions of the impeller; in this case [N.sub.R] = 2000. The curves are presented for the seven shaft eccentricity values considered in the experiments. We see again that, for a given dimensionless time, the number of mixed pixels increases with the Reynolds number. Moreover, let us note that Re has a strong influence on M for Re < 4, which means that the pumping action of the impeller is very low for Re < 4.
Effect of Shaft Eccentricity on Mixing
The dimensionless mixing curves presented in Figure 8 show the effect of the shaft eccentricity at a given Reynolds number (Re = 7.5). We can see that the shaft eccentricity E has an evident and important impact on the maximum value of M reached. However, no effect is seen on the slope of the rising part of the curve. This means that eccentricity does not change the impeller pumping capacity but changes the mixing volume considered.
Let us note in Figure 8 that a small shaft eccentricity (E =14.3 %) has an important effect on the maximum value of M reached (compared to the centred case). A small value of E is indeed enough to significantly decrease the size of the segregated regions and increase the volume of liquid mixed. This behaviour is illustrated in Figure 9 where images are shown at different shaft eccentricities for a dimensionless time of [N.sub.R] = 1500 at Re = 7.5 (similar to the case presented in Figure 8). Figure 9 illustrates the evolution of the segregated regions shape when the shaft eccentricity increases. This Figure shows two toroidal shapes, one above and one below the impeller. The upper torus is larger than the lower one when the shaft is centred (Figure 9a). With the increasing eccentricity, the upper torus shrinks faster than the bottom one, up to the point where the lower torus still exists, while the upper one has disappeared (Figure 9d). The shaft presence in the upper part of the tank is probably at the source of a useful perturbation equivalent to baffling. Increasing further the eccentricity eventually leads to the disappearance of the lower torus (Figure 9f).
In order to evaluate the shaft eccentricity effect on the mixing efficiency for different Reynolds numbers, we have plotted the variation of M versus E for different Re at [N.sub.R] = 2000 (see Figure 10). At first glance, we observe that the mixing efficiency systematically increases with the shaft eccentricity. However, the behaviour of the increase of M with E is a function of Re. At low Re (Re = 1.7 and 2.5) we observe that a small shaft eccentricity has a low impact on M and, with the increase in Re, the shaft eccentricity has a larger impact on M. Moreover, we observe that for low Re (Re = 1.7 and 2.5) there is a critical value of E (around E = 34.3 %) after which the mixing efficiency is largely improved. This indicates that the tank wall play an important role in the laminar mixing as mentioned by Alvarez et al. (2002).
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
To summarize the results obtained on the Reynolds number and shaft eccentricity effects, we have plotted a 3D surface representing the variation of M versus Re and E at [N.sub.R] = 2000 (see Figure 11). This graph shows that a complete mixing (M = 100%) can be obtained with a RT in laminar flow only for E > 34.3 % and Re > 7.5. This zone is seen on the upper right part of the surface curve.
[FIGURE 11 OMITTED]
As the shaft eccentricity enhances the mixing efficiency, we have evaluated whether eccentricity had any detrimental consequences on the power consumption. Figure 12 shows the RT power curve for three different shaft eccentricities (E = 0%, 40.9%, and 56.2%). The impeller power consumption (P) was in the range [0.98 [+ or -] 0.07 W; 105.92 [+ or -] 0.91 W] giving a power number ([N.sub.P]) in the range [6.93 [+ or -] 0.005; 33.43 [+ or -] 0.03]. The experimental results indicate that the shaft eccentricity has no effects on the RT power consumption in the laminar regime. This result can be understood with the experiments from Rice et al. (2006), which show that just the liquid surrounding the impeller moves in laminar flow. Thus, there is no impeller power interaction with the wall compared to the turbulent flow case. Moreover, the impeller power constant ([K.sub.p]) was found to be 70.6, which is exactly the same value that we obtained for the centred configuration with four baffles having a width of w = T/10.
[FIGURE 12 OMITTED]
In this study, it has been shown experimentally that for a radial discharge open impeller operating in laminar regime, the dimensionless mixing time is not a constant, contrary to a well-accepted idea. This result is also in agreement with the latest findings by Rice et al. (2006).
It has also been shown that the Reynolds number and the shaft eccentricity both have remarkable effects on the destruction of the toroidal segregated regions surrounding the RT in the laminar regime. It was observed that the Reynolds number plays a role on the pumping action of the impeller and on the mixing volume considered, whereas the eccentricity plays a role only on the mixing volume. Moreover, results confirm that the tank wall plays an important role in laminar mixing as it is mentioned in the literature by Alvarez et al. (2002) and Doucet et al., (2005). Finally, it has been shown that the shaft eccentricity has no impact on the impeller power consumption hence the advantage of using off-centred impellers in the laminar regime.
It was thus concluded that a complete mixing can be obtained in laminar flow with an open impeller only when E > 34.3 % and Re > 7.5.
NOMENCLATURE C bottom clearance (m) D impeller diameter (m) e distance from the middle of the shaft to the centreline of the tank (m) E eccentricity (%) H liquid height (m) [K.sub.p] impeller power constant (Kp = [N.sub.p] x Re) M percentage of mixed pixel from the observer's point of view (%) N impeller rotational speed ([s.sup.-1]) [N.sub.p] power number ([N.sub.P] = P/[rho][N.sup.3][D.sup.5]) [N.sub.R] number of revolutions of the impeller P impeller power consumption (W) R radius of the tank (m) [R.sub.e] Reynolds number (Re = [rho]N[D.sup.2]/[micro]) T tank diameter (m) t time (s) V tank volume ([m.sup.3]) w baffle width (m) X pixel separation value (%) Greek Symbols [micro] Viscosity (Pa s) [rho] liquid density (kg/[m.sup.3])
The support of NSERC and of the members of the Consortium "Innovative Non-Newtonian Mixing Technologies" is gratefully acknowledged. Moreover, we wish to thank Said Bikhir for his help in the experimental work.
Manuscript received June 24, 2007; revised manuscript received September 17, 2007; accepted for publication September 17, 2007.
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F. Cabaret, L. Fradette and P.A. Tanguy * URPEI, Department of Chemical Engineering, Ecole Polytechnique, P.O. Box 6079 Station Centre-Ville, Montreal, QC Canada H3C 3A7
* Author to whom correspondence may be addressed. E-mail address: email@example.com
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|Author:||Cabaret, F.; Fradette L.; Tanguy, P.A.|
|Publication:||Canadian Journal of Chemical Engineering|
|Date:||Dec 1, 2008|
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