POWER CONSUMPTION OF POLYMER SOLUTIONS IN A STIRRED VESSEL POWERED BY AN HYPERBOLOID IMPELLER
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1 SYMPOSIUM ON "RHEOLOGY AND FLUID MECHANICS OF NONLINEAR MATERIALS" ASME International Mechanical Engineering Congress & Exposition w York, w York, November 11-16, 2001 POWER CONSUMPTION OF POLYMER SOLUTIONS IN A STIRRED VESSEL POWERED BY AN HYPERBOLOID IMPELLER Adélio S. Cavadas Fernando T. Pinho 1 Centro de Estudos de Fenómenos de Transporte Centro de Estudos de Fenómenos de Transporte Faculdade de Engenharia, Universidade do Porto Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, Porto, Portugal Rua Dr. Roberto Frias, Porto, Portugal fpinho@fe.up.pt ABSTRACT mixing in the whole vessel this agitator needs to be located close Measurements of power consumption in stirred vessel flows to the bottom. powered by a Rushton and an hyperboloid impeller were carried out. The investigations carried out so far with this agitator were of The fluids were aqueous solutions of tylose, CMC and xanthan gum two types: field tests aimed at optimization and control of field at weight concentrations ranging from 0.1% to 0.6% and also operations, where strict control of fluid properties was not included wtonian fluids. necessary (Höfken et al, 1991, 1994), and laboratory tests aimed at For the Rushton turbine flows the addition of polymer increased an in-depth knowledge of the flow hydrodynamics, which was the wton number by about 13-20% at Reynolds numbers in the carried out with wtonian fluids by Nouri and Whitelaw (1994), range 1,000-3,000, whereas with the hyperboloid impeller the Ismailov et al (1997) and Pinho et al (1997, 2000). wton number decreased about 13%. This decrease was especially Sludges in waste water treatment are multiphase and nonnoticeable for the CMC solutions and was absent from the 0.2% wtonian so the adequate use of this impeller in such systems tylose solution flow. requires an extension of research into these two fields. Preliminary Concentrated aqueous solutions of CMC (5.2%) and XG (3.6%) measurements with particle suspensions have been undertaken by were also produced to determine the characteristic impeller Pinho and Cavadas (2001) and, here, we report on recent parameter k for the hyperboloid, following the procedure of measurements of the power consumption and on the determination Metzner and Otto (1957) which was found to be 48 ±16. of the hyperboloid characteristic parameter k. The former was obtained with aqueous solutions of xanthan gum, carboxymethyl KEYWORDS: hyperboloid impeller, polymer solutions, mixing, cellulose (CMC) and methyl hydroxylcellulose (tylose), at weight wton number. concentrations ranging from 0.1% to 0.6%. For comparison purposes, the solutions were also measured in the same vessels powered by a standard Rushton impeller. 1. INTRODUCTION The determination of the characteristic parameter k of the hyperboloid impeller required the preparation of concentrated solutions of 3.6% and 5.2% of xanthan gum and CMC, respectively. Stirred vessels are extremely common in the processing industry where they fulfil a large variety of tasks from the mixing of liquids and suspensions, with and witout aeration, to heat and mass transfer and blending. This entails many different shapes and types of agitator from the bulky low rotational speed impellers required for very viscous fluids to small high speed rotating devices for low viscosity fluids. In this last category, the use of hyperboloid impellers is advantageous in waste water treatment plants, because this agitator is able to promote an overall gentle flow that avoids destruction of useful microorganisms (Höfken et al, 1991, 1994). An added advantage of this impeller is its low power consumption, although the efficiency, in terms of the ratio of circulating flow to energy expenditure, is not the best (Pinho et al, 1997). Note that to ensure The remaining of the paper is organised as follows: in the next section we describe the rig and the instrumentation. In Section 3 the fluid characteristics are presented, and this is followed by the investigation of the flow hydrodynamics. For comparative purposes we also present results with a classical Rushton impeller in the standard configuration. 1 corresponding author
2 2. EXPERIMENTAL SET-UP AND INSTRUMENTATION output, as well as that from the tachogenerator, was fed to a computer via an A/D converter. Purpose-built software gave all the Stirred vessel results, namely, the rotational speed, the torque and the power. The experimental rig is schematically shown in Fig. 1-a). It The measured torque included the torque transmitted to the fluid consisted of a 292 mm diameter T stirred vessel in acrylic, which and the torque loss absorved in the bearings. The torque loss was was mounted on a support standing directly on a 3-D milling table. subtracted from the total torque to yield the net torque, after The vessel allowed a maximum height H of liquid of 600 mm, but measurements were carried out with the water level just above the the present measurements refer to H T = 1. The vessel was mounted bottom bearing, but without touching the impeller. The uncertainty inside a square trough filled with water, that was part of a heating of the torque measurements is not constant and typically varied and cooling circuit, to help maintain a constant temperature in the from about 7.5% to 0.3% when the impeller rotated from 100 rpm bath. Within the tank, four 25 mm wide and 4 mm thick baffles to 550 rpm, which corresponded to the minimum and maximum were mounted at 90 intervals to avoid solid-body rotation of the rotational speeds used for each fluid. fluid. The baffles were attached to small triangular connectors The uncertainties of the various measurement techniques were which separated them by 6 mm from the vessel wall, to eliminate combined to produce global relative uncertainties of the Reynolds the dead zones normally found behind the baffles. The bottom of and wton numbers of ±10% and ±5%, respectively. the tank was flat and had a bearing embedded in it to support the drive shaft, thus minimizing shaft wobbling. Two different impellers of 100 mm diameter D were investigated: the hyperboloid, that constitutes the main focus of interest in this paper, and a standard Rushton impeller that was used for some comparison purposes. Note, however, that some of the non-wtonian results obtained with the Rushton impeller are also new. The geometric details of the hyperboloid agitator are presented in Fig. 1-b) and Table I, and the coordinate system used is defined in the figure. The transport ribs on the upper surface are rectangles of 7 x 3.6 mm 2 and the shear ribs at the bottom are 5 by 3.6 mm 2. More details can be found in Höfken and Bischof (1993). The hyperboloid was mounted on a 12 mm shaft, which had a small 8 mm diameter recess where the hyperboloid was fixed. The impeller was positioned with an off-bottom clearance to vessel diameter ratio of 1.3/30. The six-bladed Rushton impeller is drawn in Fig. 1-c) and it was mounted at the standard configuration of 1/3 off-bottom clearance (C H ). Table 1- Coordinates of the impeller surface of the 100 mm diameter hyperboloid impeller. x [mm] y [mm] x [mm] y [mm] Measuring equipment On top of the structure standed a 600 W DC servomotor controlled by a variable power supply unit, and the velocity could be monitored on a proper display. A tachogenerator gave an electrical impulse proportional to the speed and controlled it together with an amplifier. The analog output, from 0 to 10 V, corresponded to a speed in the range 0 to 3,000 rpm. The speed could be kept constant with an uncertainty of around ±1 rpm and it never exceeded 600 rpm, except for very short periods of time, to avoid damage to the baffles. The torquemeter, model T34FN/1 from HBM, had a full range torque of 1Nm and was free from friction losses because it consisted of two distinct components: a rotor (T34r40/1), where the strain gauge bridge was attached, was fixed to the shaft, and a stator (T34ST) which comunicated in frequency with the rotor. The output from the torquemeter fed an MGC amplifier from HBM, and its 3. FLUID PREPARATION AND RHEOLOGY Fluid Preparation Nine dilute aqueous polymer solutions, based on three different polymers, were investigated in this work. In weight concentrations, the fluids were: - 0.2%, 0.4% and 0.6% solutions of the low molecular weight (6,000 g/mole) methyl hydroxyl cellulose, brand name tylose, grade MH10000K, from Hoechst. Tylose is a small molecule with a glucose based backbone, and more details can be found in Pereira and Pinho (1994); - 0.2%, 0.3% and 0.4% solutions of moderate molecular weight (300,000 g/mole) carboxymethyl cellulose sodium salt, brand name CMC, grade 7H4C, from Hercules. CMC is a branched semi-rigid molecule, but is longer than the molecule of tylose. More details can be found in Escudier et al (2001) and Tam and Tiu (1989); - 0.1%, 0.2% and 0.25% of the high molecular weight (2x10 6 g/mole) xanthan gum, brand name Keltrol, grade TF from Kelco. Xanthan gum is a polysaccharide produced by the action of a bacteria and is also a semi-rigid, but long molecule. More details on this polymer can be found in Pereira and Pinho (2000) and in Lapasin and Pricl (1995). For the determination of the impeller parameter k two more solutions were required in order to attain the laminar regime, as will be explained latter. These were concentrated solutions of xanthan gum and CMC at concentrations of 3.6% and 5.2%, respectively. To prevent bacteriological degradation 0.02% by weight of the biocide Kathon LXE from Rohm and Haas was added to all solutions. All solutions were produced with Porto tap water following the same preparation procedure. The additives were added slowly to the water while being stirred, after which the mixture was agitated for a further 90 minutes. Then, the solutions rested for 24 hours to ensure complete hidration of the molecules, and prior to any rheological or hydrodynamic measurements the solutions were agitated again for 30 minutes to fully homogenise them.
3 a) baffle b) R6 b) R6 C H 6 6 z ribs at ribs 45 at in 45 the in horizontal the horizontal plane plane r y y D/2 T/2 25 c) R50 x R50 φ70 φ26 x φ8 24 Figure 1. Geometric representation of the stirred vessel and agitators: a) the stirred vessel; b) the hyperboloid impeller (see Table 1 for coordinates); c) the Rushton impeller. φ98 Rheology Figs. 2 to 4 show plots of the viscometric viscosity of the dilute solutions of tylose, CMC and xanthan gum, respectively. In every case we see that shear-thinning increases with polymer concentration and the least shear-thinning fluids are those made from tylose, as expected due to its low molecular weight. The xanthan gum solutions exhibit very strong shear-thinning (note the different scaling of Fig. 4) with the first wtonian plateau not yet observed at the lowest measured shear rates in contrast to what is seen for all CMC and tylose solutions. At high shear rates, these same xanthan gum solutions become the less viscous, or as viscous as the thinner 0.2% tylose solutions. Viscosity models were fitted to the measured data by a leastsquares method. For the tylose and CMC solutions the simplified Carreau model (Eq. 1) was the adopted viscosity law since the data only shows the low shear rate plateau followed by the power law region. Since the low shear rate plateau is absent from the xanthan gum solutions and here one sees a tendency for the high shear rate data to level off, the Sisko model of Eq. (2) was preferred for fitting these latter solutions. ( ) 2 η = µ 0 1+ λ c γ (1) [ ( ) 2 ] nc 1 ( λ s γ ) n s 1 + µ (2) η = µ r The parameters of the adjusted viscosity models are tabulated in Tables I and II, respectively. In order to determine the impeller constant two very viscous solutions were prepared, one based on CMC and the other based on xanthan gum at weight concentrations of 5.2% and 3.6%, respectively. The corresponding rheogram is plotted in Figure 5. Shear-thinning is now even stronger and even for the CMC solution the low shear rate plateau was not observed, within the measured range of shear rates. The procedure to determine the impeller constant of Metzner and Otto (1957), to be explained in the next Section, is facilitated
4 by fitting the viscosity data of these viscous fluids by a power law. Therefore, although in the log-log Fig. 5 we can see that the viscosity of both solutions does not follow strictly a straight line, power laws with consistency index K and power index n were fitted to the data and the parameters are those listed in Table III. both more viscous and more elastic, with G' Pa and G" 0.03 Pa at 0.1 Hz, and G' 0.8 Pa and G" 2 Pa at 10 Hz, respectively. µ [Pas] 0.2% tyl 0.4% tyl 0.6% tyl µ [Pas] 0.2% CMC 0.3% CMC 0.4% CMC γ [s -1 ] γ [s -1 ] 10 4 Figure 3- Viscometric viscosity of the CMC solutions at 25 C Figure 2- Viscometric viscosity of the tylose solutions at 25 C. Table I- Parameters of the adjusted simplified Carreau model. Fluid µ 0 [Pas] λ c [s] n c γ [s -1 ] 0.2% CMC % CMC % CMC % tyl % tyl % tyl Table II- Parameters of the adjusted Sisko model. Fluid µ r [Pas] λ s [s] µ [Pas] n s γ [s -1 ] 0.1% XG % XG % XG µ [Pas] % XG 0.2% XG 0.25% XG Regarding fluid elasticity, dynamic measurements in shear were carried out with the more concentrated tylose and CMC solutions and are reported by Coelho and Pinho (1998). Within the uncertainty of the rheometer the tylose solutions were seen to be basically viscous with G" G' ~ 3 to 5 and a value of G' of 0.12 Pa at 10 Hz. The poor accuracy of this rheometer limited the measurements to the range of 5 to 20 Hz. Escudier et al (2001), benefitting from more recent instruments, confirmed and extended the results obtained by Coelho and Pinho (1998) for 0.4% CMC. These measurements, covering the range 0.01 to 50 Hz, showed G" G' ~ 2 down to 0.7 on increasing the frequency. The fluid was γ. [s -1 ] 10 4 Figure 4- Viscometric viscosity of the XG solutions at 25 C. The larger and more branched xanthan gum molecules naturally yielded more elasticity and viscosity at low frequencies: at 0.01 HzG' 0.03 Pa and G" 0.04 Pa, but at a frequency of 10 Hz the fluid behaved almost as the 0.4% CMC with G' 1.5 Pa and G" 1 Pa.
5 µ [Pas] % CMC 3.6% XG 10 2 Eq. (7a) from Pinho et al (1997) Hyperboloid-present data Equation (4) Rushton- Hockey (1990) Rushton- Pinho et al (1997) Rushton-present data γ [s -1 ] 10 4 Figure 5- Viscometric viscosity of the viscous CMC and XG solutions at 25 C. Table III- Parameters of the power law model adjusted to the viscosity of the concentrated XG and CMC solutions Fluid K [Pas n ] n 3.6% XG % CMC RESULTS AND DISCUSSION wtonian flow The power consumption in a stirred vessel depends on fluid properties, geometrical parameters and flow quantities and the functional relationship amongst these quantities can be normalised for single-phase wtonian fluids as, P ρn 3 D 5 = f ρnd 2 η, N 2 D g, D T, C T, H T, impeller (3) where is the wton number (often called also the Power number) and the first and second numbers on the right-hand-side are the Reynolds number and the square of the Froude number, respectively. The Froude number appears because of free-surface effects which are here eliminated by the presence of the baffles. These, and the low flow velocities on the upper part of the vessel, keep the free-surface flat and without vortices. = As a check on the power measuring system we have carried out measurements of the wton number as a function of the Reynolds number for wtonian fluids using both the Rushton and the hyperboloid impellers, and the results are presented in Fig. 6. Figure 6- wton number as a function of Reynolds number for Rushton and hyperboloid impellers with H T = 1 and D T = 1 3. For the Rushton impeller the measured values are in agreement with data in the literature and in particular with those reported by Hockey (1990) as well as with those obtained by Pinho et al (1997) in the same rig. However, note that the present measurements were obtained with a more accurate torquemeter than that used earlier by Pinho et al (1997): whereas here the full scale of the torquemeter is 1 Nm, the instrument used by Pinho et al (1997) could measure up to 50 Nm with an accuracy of 10-2 Nm. The accuracy in Pinho et al's (1997) measurements was of the order of the present system full scale reading and the consequence is that the present data is far more accurate and smoother than the previous data. The hyperboloid is a low power agitator, with a consumption more than five times lower than that of the Rushton at high Reynolds numbers, and this has been confirmed again. In Fig. 6 we compare the present data with Eq. (7a) of Pinho et al (1997) which was fitted to the extensive set of data for the hyperboloid. The present data is 10% below the correlation of Pinho et al (1997) tending to = 0.8, whereas the previous data asymptoted to 0.88 at high Reynolds number. The higher accuracy of the present system and the lower uncertainties associated with the corrections to the net torque lead us to conclude that the present values are to be preferred to those of Pinho et al (1997) and, as a consequence, we fitted to the new data an expression of the same type Re Re 2 (4) which is also represented in Fig. 6, as a full line.
6 a) b) wtonian 0.2% tylose 0.6% tylose c) wtonian 0.1% XG 0.25% XG Figure 7- Variation of the wton number with the Reynolds number in stirred vessels agitated by a Rushton impeller in the standard configuration ( H T = 1, D T = 1 3 ). a) Tylose; b) CMC; c) xanthan gum. Non-wtonian flow The viscosity of non-wtonian fluids is usually variable, therefore the issue of a characteristic viscosity for the calculation of the Reynolds number in Eq. (3) arises. For stirred vessel flows wtonian 0.2% CMC 0.4% CMC this issue has been addressed long time ago by Metzner and Otto (1957) and Calderbank and Moo-Young (1959) who devised a strategy aimed at comparing - Re data obtained under different geometrical conditions and with different impellers. The adopted strategy was inspired by that used to define a characteristic viscosity in laminar pipe flow. The Reynolds number is defined Re = ρnd2 (5) η c where N is the rotational speed usually in [rps] and the remaining quantities use SI units, with η c representing the characteristic viscosity. According to the above authors, the characteristic viscosity η c, which they called apparent viscosity, is defined in such way that the -Re relationship for non-wtonian fluids coincides with the -Re relationship for wtonian fluids in the laminar flow regime, at identical rotational speeds and other flow conditions being equal. Then, Metzner and Otto (1957) proceeded to relate the apparent viscosity with other variables and assumed that the flow in the impeller region is characterised by an average shear rate which is linearly related to the rotational speed, i.e. γ = kn (6) This is arguably correct but it is a convenient simple assumption for engineering purposes and it is such objective that we pursue here by adopting it. Parameter k is strongly dependent on the geometry of the vessel and agitator, and is still unknown for the hyperboloid impeller. The determination of an indicative value for this parameter is one of the objectives of the present work. xt, we present the results measured with the Rushton impeller, where the Reynolds number is calculated using Eqs. (5) and (6) with the appropriate value of k for Rushton turbines, and this is followed by the determination of the characteristic impeller parameter for the hyperboloid after which we present results with the dilute polymer solutions.
7 Rushton impeller flow For the Rushton impeller, parameter k was determined long time ago by Metzner and Otto (1957), Calderbank and Moo-Young (1959) and Godleski et al (1962), and in 1961 Metzner et al concluded that the choice of a value between 11.5 and 13 was not critical since a 30% variation in k resulted in a smaller variation in the viscosity of shear-thinning fluids (of 12% for n= 0.5). Here, we will use a value of k= 12 and Fig. 7 compares the variation of the -Re relation for the polymer solutions and wtonian fluids. Inspection of Figure 7 shows that for the solutions of the three polymers there is an increase in wton number relative to the wtonian flow case and polymer concentration also raises the power consumption. Xanthan gum solutions exhibit the largest differences relative to the wtonian case. At low Reynolds numbers the data for CMC and tylose crosses over that of wtonian fluids. This intermediate Reynolds number range corresponds to transition from the laminar flow to the high Reynolds number turbulent flow where Hockey (1990) also found, for a 0.4% CMC solution of the same brand, a complex transitional behaviour with his data approaching the wtonian data at various places but in different ways. Our measurements here are more restricted but agree with such complex behaviour. Characteristic hyperboloid parameter The determination of the impeller parameter requires measurements of the wton number under conditions of laminar flow. To attain such conditions we progressively increased the concentration of CMC and xanthan gum, but only the results obtained with the final more concentrated solutions are reported here. For each fluid the wton number was measured and compared with similar results for wtonian fluids in a versus rotational speed plot as shown in Fig. 8. Note that for the wtonian fluid, pure glicerin, the data in Fig. 8 corresponds to values of Reynolds number below 100. For identical wton number and rotational speed, i.e. at the cross-over of curves, a condition of equality of Reynolds number is imposed. Together with Eqs. (5) and (6), for selecting the characteristic shear rate at which the viscosity is calculated, this process results in the determination of values of the impeller parameter. The intersection of the wtonian and XG curve occurs at N= 4.7 rps and = 2.3 whereas for CMC the intersection is at N= 9.95 rps and = 1.5. These data and the viscosity curve resulted in the following values of k: 31 for the 3.6% XG solution and 64 for the 5.2% CMC solution. The average is 48 and the scatter is ±16. The difference between the two values is rather large, but is not unknown in the context of the determination of parameter k. In fact, the past work with other agitators mentioned so far also show variations of the order of 30% to 40% as seen here with the hyperboloid. This variation is the result of the flow complexity and the underlying simplifications and assumptions of the method which are many and not totally justified. vertheless, as mentioned, from a practical point of view aimed at engineering practice, this definition of the Reynolds number is adequate and extensively used. An alternative would have been the use of a Reynolds number independent of the impeller and flow, such as the generalised Reynolds number of Eq. (7) which depends on the power law parameters K and n, but this would raise the question of whether data from different impellers could be directly compared. Re gen ρn 2 n D 2 (7) K In order to refine and improve the accuracy of the above value of k, this process must be repeated with more viscous solutions of these and of other polymers wtonian 3.6% XG 5.2% CMC N [rps] Figure 8- wton number versus N [rps] in the laminar flow régime for wtonian and two viscous polymer solutions. Hyperboloid impeller flow For stirred vessels powered by the hyperboloid impeller, Figures 9-a), b) and c) plots the -Re results for the tylose, CMC and xanthan gum solutions, respectively. Each figure includes the wtonian data as given by the fitted Eq. (4). A general trend in Figure 9 is a decrease in wton number for the polymer solutions relative to the wtonian fluids. The exception is the 0.2% tylose which is the less concentrated and less viscous of the non-wtonian solutions. This fluid is also the one whose rheology is the closest to the wtonian, both in terms of viscosity and elasticity. At Reynolds numbers in the range 10 3 to 10 4 the power number has the lowest values, of the order of 0.7 and less, whereas for wtonian fluids it is higher than the asymptotic high Reynolds number value of For the tylose solutions (Fig. 9-a) the wton number increases with Reynolds number at the upper limit of this range and seems to approach the wtonian curve. For the CMC, however, although there is a slight tendency for to increase with Re, this variation does not seem sufficient to
8 a) 0.2% tylose 0.4% tylose 0.6% tylose wtonian Eq. (4) b) 0.2% CMC 0.3% CMC 0.4% CMC wtonian Eq. (4) c) 0.1% XG 0.2% XG 0.25% XG wtonian Eq. (4) would result in values of wton number identical to those of water, but such high Reynolds numbers are difficult to attain as they require very high rotational speeds, unlikely to occur in practice. The hyperboloid impeller draws its power from frictional drag on its surface as well as from form drag due to flow separation in both the shear and transport ribs. At low Reynolds numbers the friction drag is more important and the drag reducing capability of these polymer solutions (Pereira and Pinho, 1999) will result in the observed 13% decrease in power consumption. Note, from the wtonian investigations in hyperboloids with and without ribs (see Pinho et al, 2000) that the ribs account for an increase in wton number in excess of 80%, from about 0.5 to As the Reynolds number increases the relative importance of friction drag drops, so drag reduction effects are reduced, with the wton number curve approaching the value for wtonian fluids. Figure 9- Variation of the wton number with the Reynolds number in stirred vessels agitated by an hyperboloid impeller impeller in the standard configuration ( H T = 1, D T = 1 3 ). a) Tylose; b) CMC; c) xanthan gum. attain the wtonian curve. We have here a decrease in the wton number in excess of 13% relative to that of wtonian fluids. The intense shear-thinning of the xanthan gum solutions leads to the highest Reynolds numbers: for the 0.1% solution a maximum value of 27,600 is reached and the corresponding power number is just under Probably, a higher Reynolds number 5. CONCLUSIONS The turbulent flow of non-wtonian fluids in stirred vessels was investigated in terms of bulk hydrodynamic characteristics. The fluids were aqueous solutions of carboxymethil cellulose (CMC), hydroxymethil cellulose (tylose) and xanthan gum (XG) at weight concentrations ranging from 0.1% to 0.6%. Two different geometries were studied: first, the agitation was provided by a standard Rushton impeller, and then a low-power consumption hyperboloid impeller was used. In both cases wtonian fluids were used for comparative purposes. One viscous solution of 5.2% CMC and one viscous solution of 3.6% xanthan gum were also used to determine the characteristic hyperboloid parameter k following the procedure outlined by Metzner and Otto in The following are the main conclusions of this work: - The Reynolds numbers of the polymer solution flows is rather low, between 1,000 and 10,000, and the flows are most likely transitional;
9 - With the Rushton turbine there was an increase in power consumption especially for the more elastic fluids based on CMC and XG, whereas for the tylose solutions the wton number was barely unchanged from that of wtonian fluids. For Reynolds numbers of about 1,000 to 4,000 the increase in wton number varied between 13 and 20% ; - With the hyperboloid impeller the opposite effect was observed. The power consumption decreased in comparison with the wtonian behaviour, at identical Reynolds numbers, and this effect was more pronounced with CMC, less so with xanthan gum and even less with tylose. Given the shape of the hyperboloid impeller and the known wtonian flo features, we speculate that this reduction in power consumption is rooted on a reduction in friction drag and has similarities to the drag reduction phenomena known to occur with these same fluids in turbulent pipe flow. At higher Reynolds numbers form drag takes over and drag reduction effects tend to disappear; - Viscous polymer solutions allowed the determination of a characteristic impeller parameter k= 48±16 following the procedure of Metzner and Otto (1957). It is necessary to extend this work to a wider range of non-wtonian fluids in order to reduce the uncertainty in the determination of this parameter. ACKNOWLEDGEMENTS The authors would like to thank financial support of JNICT- Junta Nacional de Investigação Científica- through project PEAM/C/TAI/265/93 and of the scientific committee of the Mechanical Engineering Master Course of Faculdade de Engenharia da Universidade do Porto. A. S. Cavadas has also beneffited from a research scholarship from JNICT. REFERENCES PH Calderbank & MB Moo-Young The prediction of power consumption in the agitation of non-wtonian fluids. Trans. Inst. Chem. Eng., 37, 26. PM Coelho & FT Pinho Rheological behaviour of some dilute aqueous polymer solutions (in portuguese), Mecânica Experimental, 3, MP Escudier, I Gouldson, AS Pereira, FT Pinho & RJ Poole 2001.On the reproducibility of the rheology of shear-thinning liquids. J. Non-wt. Fluid Mechanics, 97, 99. ES Godleski and JC Smith Power requiremens and blend times in the agitation of pseudoplastic fluids. A. I. Ch. E. J., 8 M Höfken, F Bischof & F Durst Novel hyperboloid stirring and aeration system for biological and chemical reactors. ASME -FED- Industrial Applications of Fluid Mechanics, 132, 47. M Höfken & F Bischof Hyperboloid stirring and aeration system: Operating principles, application, technical description. Invent GmbH report, version 1.1, Erlangen, Germany. M Höfken, K Zähringer & F Bischof Stirring and aeration system for the upgrading of small waste water treatment plants. Water Science and Technology, 29, 149. M Ismailov, M Schäfer, F Durst & M Kuroda Turbulent flow pattern of hyperboloid stirring reactors. Journal Chem. Eng. Japan, 30, R Lapasin & S Pricl Rheology of Industrial Polysaccharides: Theory and Applications. Blackie Academic and Professional, London. AB Metzner & RE Otto Agitation of non-wtonian fluids. A. I. Ch. E. J., 3 (1), 3. AB Metzner, RH Feehs, HL Ramos, RE Otto & JD Tuthill Agitation of viscous wtonian and non-wtonian fluids. A. I. Ch. E. J., 7 (1), 3. JM Nouri & JH Whitelaw Flow characteristics of hyperboloid stirrers. Can. J. Chem. Eng., 72, 782. AS Pereira & FT Pinho Turbulent pipe flow characteristics of low molecular weight polymer solutions. J. Non-wt. Fluid Mech., 55, AS Pereira & FT Pinho Bulk characteristics of some variable viscosity polymer solutions in turbulent pipe flow, COBEM 99-15th Brasilian Mechanical Engineering Congress, paper AAAAH (in CD-ROM) Águas de Lindóia SP, Brasil, November. AS Pereira & FT Pinho Turbulent characteristics of shearthinning fluids in recirculating fluids. Exp. in Fluids, 28 (3), 266. FT Pinho & AS Cavadas Power consumption and suspension criteria for two-phase flow in a stirred vessel powered by an hyperboloid impeller. COBEM Proceedings of the XVIth Brasilian Mechanical Eng. Congress, Uberlândia MG, Brasil, November. FT Pinho, FM Piqueiro, MF Proença & AM Santos Power and mean flow characteristics in mixing vessels agitated by hyperboloid stirrers. Can. J. Chem. Eng., 75, 832. FT Pinho, FM Piqueiro, MF Proença & AM Santos Turbulent flow in stirred vessels agitated by a single, low-clearance hyperboloid impeller, Chem. Eng. Sci. 55 (16), KC Tam & C Tiu Steady and dynamic shear properties of RM Hockey Turbulent wtonian and non-wtonian flows aqueous polymer solutions. J. Rheology 33 (2), in a stirred reactor. PhD thesis, University of London, UK.
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