IMECE2001/FED Proceedings of 2001 ASME International Mechanical Engineering Congress and Exposition November 11-16, 2001, New York, NY

Proceedings of 2001 ASME International Mechanical Engineering Congress and Exposition November 11-16, 2001, New York, NY IMECE2001/FED-24905 POWER CONSUMPTION OF POLYMER SOLUTIONS IN A STIRRED VESSEL POWERED BY AN HYPERBOLOID IMPELLER Ad61io S. Cavadas Centro de Estudos de Fen6menos de Transporte Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Fernando T. Pinho 1 Centro de Estudos de Fen6menos de Transporte Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal fpinho@fe.up.pt ABSTRACT Measurements of power consumption in stirred vessel flows powered by a Rushton and an hyperboloid impeller were carried out. The fluids were aqueous solutions of tylose, CMC and xanthan gum at weight concentrations ranging from 0.1% to 0.6% and also included Newtonian fluids. For the Rushton turbine flows the addition of polymer increased the Newton number by about 13-20% at Reynolds numbers in the range 1,000-3,000, whereas with the hyperboloid impeller the Newton number decreased about 13%. This decrease was especially noticeable for the CMC solutions and was absent from the 0.2% tylose solution flow. Concentrated aqueous solutions of CMC (5.2%) and XG (3.6%) were also produced to determine the characteristic impeller parameter k for the hyperboloid, following the procedure of Metzner and Otto (1957) which was found to be 48 +16. KEYWORDS: hyperboloid impeller, polymer solutions, mixing, Newton number. 1. INTRODUCTION 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 (H6fken 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 mixing in the whole vessel this agitator needs to be located close to the bottom. The investigations carried out so far with this agitator were of two types: field tests aimed at optimization and control of field operations, where strict control of fluid properties was not necessary (H6fken et al, 1991, 1994), and laboratory tests aimed at an in-depth knowledge of the flow hydrodynamics, which was carried out with Newtonian fluids by Nouri and Whitelaw (1994), Ismailov et al (1997) and Pinho et al (1997, 2000). Sludges in waste water treatment are multiphase and non- Newtonian so the adequate use of this impeller in such systems requires an extension of research into these two fields. Preliminary measurements with particle suspensions have been undertaken by Pinho and Cavadas (2001) and, here, we report on recent measurements of the power consumption and on the determination of the hyperboloid characteristic parameter k. The former was obtained with aqueous solutions of xanthan gum, carboxymethyl cellulose (CMC) and methyl hydroxylcellulose (tylose), at weight 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. 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. 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 Copyright 2001 by ASME

2. EXPERIMENTAL SET-UP AND INSTRUMENTATION Stirred vessel The experimental rig is schematically shown in Fig. l-a). It consisted of a 292 mm diameter T stirred vessel in acrylic, which was mounted on a support standing directly on a 3-D milling table. The vessel allowed a maximum height H of liquid of 600 mm, but the present measurements refer to H]T = 1. The vessel was mounted inside a square trough filled with water, that was part of a heating and cooling circuit, to help maintain a constant temperature in the bath. Within the tank, four 25 mm wide and 4 mm thick baffles were mounted at 90 intervals to avoid solid-body rotation of the fluid. The baffles were attached to small triangular connectors which separated them by 6 mm from the vessel wall, to eliminate the dead zones normally found behind the baffles. The bottom of 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-newtonian results obtained with the Rushton impeller are also new. The geometric details of the hyperboloid agitator are presented in Fig. l-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 H6fken 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. l-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 h ~erboloid impeller. x[mm] 6 6 7 "'8 10 i2 14 16 18 y[mm] 38.4 32.4 27 24.2 21 18.5 16.5 14.5 12.7 x[mm] 20 22 26 30 34 38 42 46 50 y[mm] 11.1 9.6 7.2 5.3 3.6 2.2 1.4 1.0 0.6 Measurinfl eouiament 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 l0 V, corresponded to a speed in the range 0 to 3,000 rpm. The speed could be kept constant with an uncertainty of around +l rpm and it never exceeded 600 rpm, except for very short periods of time, to avoid damage to the baffles. The torquemeter, model T34FN/l 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/l), 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 output, as well as that from the tachogenerator, was fed to a computer via an A/D converter. Purpose-built software gave all the results, namely, the rotational speed, the torque and the power. The measured torque included the torque transmitted to the fluid and the torque loss absorved in the bearings. The torque loss was subtracted from the total torque to yield the net torque, after measurements were carried out with the water level just above the bottom bearing, but without touching the impeller. The uncertainty of the torque measurements is not constant and typically varied from about 7.5% to 0.3% when the impeller rotated from 100 rpm to 550 rpm, which corresponded to the minimum and maximum rotational speeds used for each fluid. The uncertainties of the various measurement techniques were combined to produce global relative uncertainties of the Reynolds and Newton numbers of ~:10% and +5%, respectively. 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 MHI0000K, 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 (2x106 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. 2 Copyright 2001 by ASME

a) baffle b) R6 't r I X.., T/2. c) I- R50,70 k,._l v,v I O L I-" Vl ~98 L,8 v I Figure I. 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. Rheoloev 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 Newtonian 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. t'~ \It s 01 r 1 = ldrl~st) +la= (2) 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 3 Copyright 2001 by ASME

....................... 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 Ill. both more viscous and more elastic, with G'- 0.002 Pa and G"= 0.03 Pa at 0.1 Hz, and G'= 0.8 Pa and G"-- 2 Pa at 10 Hz, respectively. 10 i i I llill I i i i llill I i i i iiii11 i i i i iii lo I I I tlll I I I I IIIII I I I II1 + 0.2% tyl & 0.4% tyl X 0.6% tyl,o-, =L i + 0.2% CMC A 0.3% CMC i X 0.4% CMC X x xxx... 4.... _ 10 -I 10-2,, XXx~ X... '~... "...!--~--zx--;~... + + ++H-~++ +++++,o olo... '... ;02... 111111104 101 3 ~ [s- Figure 2- Viscometric viscosity of the tylose solutions at 25 C. Table l- Parameters of the adiusted simplified Carreau model. Fluid PO [Pas] /~ [s] n c ~' [S "l] "0.2% CMC 0.3% CMC 0.4% CMC 0.2% tyl 0.4% tyl 0.6% t~,l 0.02652 0.08533 0.14097 0.00463 0.01865 0.06268 0.03053 0.23517 0.11376 0.00377 0.00713 0.01468 0.697 0.717 0.618 0.900 0.783 0.697 7-4031 3-403l 100-4031 40-4031 10-4031 8-4031 Table II- Parameters of the adjusted Sisko model. Fluid /.t r [Pas] A s [s] /.t~ [Pas] n s )' [s-1] 0.1% XG 1.86 1970 0.001 0.543 3-2700 0.2% XG 31.47 2909 0.001 0.417 0.5-4031 0.25% XG 98.54 1700 0.001 0.326 0.1-4031 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 l0 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 i "~,A A XX x ~ i +~+~,,.i 'ha A X>~ x 10-2... ~..."..._+. ++#...~ZX..x... + A ++ F,o; IoO... 101 '... 102 '... 103 ' f... [S-1] 104 Figure 3- Viscometric viscosity of the CMC solutions at 25 C. 101[ ' ''"'"l ' ''"'"1 ' ' '"'"l ' ' '""l ' ''"' ~f I- + 0.1% XG [ A 0.2% XG 0.25% XG 10 o '...... i... 10-1...} 7 - ~ 7 ~... i...! ++ i ax~ ~ 1 2 i i+++ J irr3lv10... '... 1... '... [...,l 10 101 102 103~/Is-l] 104 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 I-IzG' = 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. / 4 Copyright 2001 by ASME

+A i i : : i + 5.20/0 =- ~%, i I A 3.6% XG 1 Lx~T% ~... t. -A...?... i,0,... i... i... +;;;i... i... 10 0! i a~ ++T i... i... i!aa++ 10-1... t_x-)~ - 103k b... I... I... I... I... I Ne ~... Eq. (7a) from Pinho et al (1997) ::1 & Hyperboloid-present data i ] ---- Equation (4) [ ~, j] X Rushton- Hockey (1990) ~. i [ + Rushton- Pinho et al (1997) [ 10: -~... i' l Rushton-present data [- +", i i i i i +",i i i i i 10 c... : 10-2 ~ ~md ~ ~md t ~m[ t ~m~l ~ ~md ~ ~L~H 10-2 10 10 2 ~ [s "l] 104 Figure 5- Viscometric viscosity of the viscous CMC and XG solutions at 25 C. Table IIl- 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 34.0 0.25 5.2% CMC 71.3 0.30 4. RESULTS AND DISCUSSION Newtonian 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 Newtonian fluids as, Ne- P f~pn~ D2 N2D D C H ) pn3d ---~ = ' g ' T' T',impeller (3) where Ne is the Newton 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 Newton number as a function of the Reynolds number for Newtonian fluids using both the Rushton and the hyperboloid impellers, and the results are presented in Fig. 6. 1001... I... [... I... I 0 101 102 103 104 Re 105 Figure 6- Newton number as a function of Reynolds number for Rushton and hyperboloid impellers with H]T= 1 and D]T = 113. 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 ars (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 Ne = 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 86.6 87.9 Ne=0.81+ Re +--Re 2 (4) which is also represented in Fig. 6, as a full line. 5 Copyright 2001 by ASME

101! Ne a) 10 l Ne b) I X %,~+++-H- 10 l Ne 0 A Newtonian + 0.2% tylose X 0.6% tylose... '1o... '1;4 ' 'R:';o, X A AAY X +++ i A Newtonian + 0.1% XG X 0.25% XG 1,,o, I I IIIll I I I I IIIII I I I I I II 2 10 3 10 4 Re 10 5 Figure 7- Variation of the Newton number with the Reynolds number in stirred vessels agitated by a Rushton impeller in the standard configuration (HIT = 1, D]T = 1/3). a) Tylose; b) CMC; c) xanthan gum. Non-Newtonian flow The viscosity of non-newtonian 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 A Newtonian + 0.2% CMC X 0.4% CMC O0 I I I 111111 I I I I IIII I I I I I 1 2 103 10 4 Re 105 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 Ne-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 = pnd2 (5) qc where N is the rotational speed usually in [rps] and the remaining quantities use SI units, with qc representing the characteristic viscosity. According to the above authors, the characteristic viscosity r/c, which they called apparent viscosity, is defined in such way that the Ne-Re relationship for non-newtonian fluids coincides with the Ne-Re relationship for Newtonian 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. = 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. Next, 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. 6 Copyright 2001 by ASME

Rnshton 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 Ne-Re relation for the polymer solutions and Newtonian fluids. Inspection of Figure 7 shows that for the solutions of the three polymers there is an increase in Newton number relative to the Newtonian flow case and polymer concentration also raises the power consumption. Xanthan gum solutions exhibit the largest differences relative to the Newtonian case. At low Reynolds numbers the data for CMC and tylose crosses over that of Newtonian 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 Newtonian 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 Newton 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 Newton number was measured and compared with similar results for Newtonian fluids in a Ne versus rotational speed plot as shown in Fig. 8. Note that for the Newtonian fluid, pure glicerin, the data in Fig. 8 corresponds to values of Reynolds number below 100. For identical Newton 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 Newtonian and XG curve occurs at N = 4.7 rps and Ne = 2.3 whereas for CMC the intersection is at N = 9.95 rps and Ne = 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. Nevertheless, 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. pn2-n D 2 Regen =- K (7) 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. 10 2 Ne 101 i AI A A i i i a! i i - i I Newtoni A 3.6% XG-- --a~x~ 5.2% CMC A i ~- 10 0 I I I l lllll i i i l l,tll 10-1 10 N [rps] 10 i Figure 8- Newton number versus N [rps] in the laminar flow rrgime for Newtonian and two viscous polymer solutions. Hyperboloid impeller flow For stirred vessels powered by the hyperboloid impeller, Figures 9-a), b) and c) plots the Ne-Re results for the tylose, CMC and xanthan gum solutions, respectively. Each figure includes the Newtonian data as given by the fitted Eq. (4). A general trend in Figure 9 is a decrease in Newton number for the polymer solutions relative to the Newtonian fluids. The exception is the 0.2% tylose which is the less concentrated and less viscous of the non-newtonian solutions. This fluid is also the one whose rheology is the closest to the Newtonian, both in terms of viscosity and elasticity. At Reynolds numbers in the range 103 to 104 the power number has the lowest values, of the order of 0.7 and less, whereas for Newtonian fluids it is higher than the asymptotic high Reynolds number value of 0.81. For the tylose solutions (Fig. 9-a) the Newton number increases with Reynolds number at the upper limit of this range and seems to approach the Newtonian curve. For the CMC, however, although there is a slight tendency for Ne to increase with Re, this variation does not seem sufficient to x 7 Copyright 2001 by ASME

................................................... 101 Ne a),, i '''"l ' ' ' ''"' + 0.2% tylose A 0.4% tylose x 0.6% tylose - - Newtonian Eq. (4) 101,, Ne b) ' ' '''"l ' ' ' '' ' + 0.2% CMC A 0.3% CMC x 0.4% CMC - - Newtonian Eq. (4) I0 ( i i X )(~ ~ ' ~ r ~. ~ ] ~... I... X ~ I I I I[lll 1 0 0 t ~ ~... 0" I I Illl I I I I I II I I I I I II 102 103 104 Re 105 / 10 2 103 10 4 Re 105 10 l Ne c) ) I I I I I ) II I I I I I I I) + 0.1% XG A 0.2% XG x 0.25% XG - - Newtonian Eq. (4) would result in values of Newton 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 Newtonian investigations in hyperboloids with and without ribs (see Pinho et al, 2000) that the ribs account for an increase in Newton number in excess of 80%, from about 0.5 to 0.8-0.9. As the Reynolds number increases the relative importance of friction drag drops, so drag reduction effects are reduced, with the Newton number curve approaching the value for Newtonian fluids. 10-12 l0, I I I I llll ) I I............. 10 3 10 4 Re I0 5 Figure 9= Variation of the Newton number with the Reynolds number in stirred vessels agitated by an hyperboloid impeller impeller in the standard configuration (H/T= I, D/T= I/3). a) Tylose; b) CMC; c) xanthan gum. attain the Newtonian curve. We have here a decrease in the Newton number in excess of 13% relative to that of Newtonian 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 0.78. Probably, a higher Reynolds number 5. CONCLUSIONS The turbulent flow of non-newtonian 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 Newtonian 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 1957. 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; 8 Copyright 2001 by ASME

- 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 Newton number was barely unchanged from that of Newtonian fluids. For Reynolds numbers of about 1,000 to 4,000 the increase in Newton number varied between 13 and 20% ; - With the hyperboloid impeller the opposite effect was observed. The power consumption decreased in comparison with the Newtonian 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 Newtonian 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-newtonian fluids in order to reduce the uncertainty in the determination of this parameter. A C K N O W L E D G E M E N T S The authors would like to thank financial support of JNICT- Junta Nacional de Investigagfio Cientifica- 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. R E F E R E N C E S PH Calderbank & MB Moo-Young 1959. The prediction of power consumption in the agitation of non-newtonian fluids. Trans. Inst. Chem. Eng., 37, 26. PM Coelho & FT Pinho 1998. Rheological behaviour of some dilute aqueous polymer solutions (in portuguese), Mecdnica Experimental, 3, 51-60. MP Escudier, I Gouldson, AS Pereira, FT Pinho & RJ Poole 2001.On the reproducibility of the rheology of shear-thinning liquids. J. Non-Newt. Fluid Mechanics, 97. 99. ES Godleski and JC Smith 1962. Power requiremens and blend times in the agitation of pseudoplastic fluids. A. L Ch.E. J., 8 617. RM Hockey 1990. Turbulent Newtonian and non-newtonian flows in a stirred reactor. PhD thesis, University of London, UK. M H6fken, F Bischof & F Durst 1991. Novel hyperboloid stirring and aeration system for biological and chemical reactors. ASME -FED- Industrial Applications of Fluid Mechanics, 132, 47. M Hrfken & F Bischof 1993. Hyperboloid stirring and aeration system: Operating principles, application, technical description. Invent GmbH report, version 1.1, Erlangen, Germany. M Hrfken, K Zfihringer & F Bischof 1994. Stirring and aeration system for the upgrading of small waste water treatment plants. Water Science and Technology, 29, 149. M Ismailov, M Sch~ifer, F Durst & M Kuroda 1997. Turbulent flow pattern of hyperboloid stirring reactors. Journal Chem. Eng. Japan, 30, 1090. R Lapasin & S Pricl 1995. Rheology of Industrial Polysaccharides: Theory and Applications. Blackie Academic and Professional, London. AB Metzner & RE Otto 1957. Agitation of non-newtonian fluids. A. 1. Ch.E.J.,3(I),3. AB Metzner, RH Feehs, HL Ramos, RE Otto & JD Tuthill 1961. Agitation of viscous Newtonian and non-newtonian fluids. A. /. Ch.E.J., 7_(I),3. JM Nouri & JH Whitelaw 1994. Flow characteristics of hyperboloid stirrers. Can. J. Chem. Eng., 72, 782. AS Pereira & FT Pinho 1994. Turbulent pipe flow characteristics of low molecular weight polymer solutions. J. Non-Newt. Fluid Mech., 55, 321-344. AS Pereira & FT Pinho 1999. Bulk characteristics of some variable viscosity polymer solutions in turbulent pipe flow, COBEM 99-15th Brasilian Mechanical Engineering Congress, paper AAAAH (in CD-ROM),/~guas de Lind6ia SP, Brasil, November. AS Pereira & FT Pinho 2000. Turbulent characteristics of shearthinning fluids in recirculating fluids. Exp. in Fluids, 28 (3), 266. FT Pinho & AS Cavadas 2001. Power consumption and suspension criteria for two-phase flow in a stirred vessel powered by an hyperboloid impeller. COBEM 2001- Proceedings of the XVlth Brasilian Mechanical Eng. Congress, Uberl~ndia MG, Brasil, 26-30 November. FT Pinho, FM Piqueiro, MF Proenqa & AM Santos 1997. Power and mean flow characteristics in mixing vessels agitated by hyperboloid stirrers. Can. J. Chem. Eng., 75, 832. FT Pinho, FM Piqueiro, MF Proenga & AM Santos 2000. Turbulent flow in stirred vessels agitated by a single, low-clearance hyperboloid impeller, Chem. Eng. Sci. 55 (16), 3287. KC Tam & C Tiu 1989. Steady and dynamic shear properties of aqueous polymer solutions. J. Rheology 33 (2), 257-280. 9 Copyright 2001 by ASME