Chemical Engineering and Processing 43 2004) 1211 1222 Mixing of complex fluids with flat-bladed impellers: effect of impeller geometry and highly shear-thinning behavior Rajeev K. Thakur, Ch. Vial 1, G. Djelveh, M. Labbafi Laboratoire de Génie Chimique et Biochimique, Université Blaise Pascal, 24 Avenue des Landais, BP 206, F-63174 Aubiere Cedex, France Received 25 June 2003; received in revised form 14 November 2003; accepted 14 November 2003 Available online 13 January 2004 Abstract Mixing of rheological complex fluids was investigated using flat-bladed impellers as close-clearance agitators in the laminar regime. Two Newtonian and six highly shear-thinning fluids were used. The non-newtonian fluids were adequately described by a power-law model with a flow index n between 0.1 and 0.4. Power draw analysis was used to explore the combined influence of pseudoplasticity and impeller geometry. Geometry was studied first by varying the column-to-impeller diameter ratio, and then by combining several similar mixing elements on the same shaft. For pseudoplastic fluid, the Rieger Novak and the power curve methods as well as an original Couette analogy were used for estimating the effective shear rate and the proportionality constant K S. A good agreement was obtained between these three methods. K S was shown to be nearly independent of n: the Metzner Otto assumption was shown to be valid for all the geometries studied. A generalized dimensionless power draw curve which took pseudoplasticity into account was obtained by shifting the non-newtonian results to the Newtonian curve. The effectiveness of flat-bladed impellers for dispersive mixing in complex fluids proved in previous works was explained by the fact that the effective shear rate remained high even when power consumption dramatically decreased with n. 2003 Elsevier B.V. All rights reserved. Keywords: Non-Newtonian mixing; Effective shear rate; Flat-bladed impeller; Power consumption; Shear-thinning fluids 1. Introduction Mechanically stirred vessels are commonly used in the process industries to fulfill a large variety of tasks, including the classical mixing of miscible fluids in single-phase flows distributive mixing), but also powder dispersion or solid blending, dispersion of an immiscible liquid phase for mass transfer or emulsification, gas dispersion into a continuous liquid phase for mass transfer or sometimes in order to form foams in food industries. Mixing operations involving several phases are generally referred to as dispersive mixing. Chemical engineers and food processors often deal with complex fluids in the laminar regime which are usually highly viscous and shear-thinning, sometimes exhibiting a highly viscoelastic behavior. It is clear that both vessel shape and impeller design have to be adapted to take these Corresponding author. Tel.: +33-473-405-055; fax: +33-473-407-829. E-mail addresses: christophe.vial@univ-bpclermont.fr Ch. Vial), djelveh@gecbio.univ-bpclermont.fr G. Djelveh). 1 Tel.: +33-473-405-266; fax: +33-473-407-829. properties into account, for distributive as well as for dispersive mixing. Generally, close-clearance impellers such as anchors, helical screws or helical ribbons, are recommended for mixing highly viscous non-newtonian fluids as the most effective mixers. However, such impellers, for example helical ribbons, are sensitive to highly shear-thinning behavior which reduces their mixing effectiveness [1]. Similarly, for gas dispersion in such fluids, their effectiveness remains unfortunately low. As an illustration, for foaming process in food industries, an ideal mixer should promote gas dispersion and at the same time mix the fluid efficiently to favor a rapid mixing and diffusion of proteins and surfactants to the gas liquid interfaces. Up to now, there is no definitive answer to the problem of gas dispersion in highly viscous fluids. Although combinations of conventional geometries have been suggested a priori as potential solutions, e.g. by combining a helical ribbon and a Rushton turbine [2], gas dispersion in highly viscous fluids remains an open question. This is the reason why, in the process industries, gas dispersion in complex fluids is generally carried out in mixing units that not necessarily the most adequate, but are generally more effective than conventional mixing tanks equipped 0255-2701/$ see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2003.11.005
1212 R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 with ribbons or anchors. These units can be divided into two groups. The first group consists of rotor stator units using a large solid rotor generally equipped with pins, as well as a narrow gap between rotor and stator, the size of which is around a few millimeters or may be lower. These rotors are driven at high rotational speed, generally between 1000 and 10,000 rpm [3]. The second group is characterized by scraping agitators and corresponds to scraped surface heat exchangers. Such mixing units can be operated using either large solid or small open i.e. small volume) rotors, both equipped with scraping blades [4,5], but rotational speed is generally lower than 1000 rpm for practical applications. Even though both groups of mixing units are widely used in the industry, they present a high power consumption level and were not originally designed for gas dispersion. Recently, Thakur et al. [6] proved that open non-scraping impellers favor gas dispersion even at low rotational speed, requiring a fairly low level of energy consumption to make food foams, as compared to solid rotors. The characteristics of impellers that have to be preferred for gas dispersion non-newtonian viscous fluids are therefore: a small gap between blades and wall close-clearance); a small volume open) rotor that favors backmixing, increasing bubble residence time; blades without holes that enforce the fluid to move through the gap. Their results proved that large rotors and high-speed rotating impellers are not necessary to form small bubbles, while scraping devices are needed only when heat transfer at the wall is the limiting step, for example, for freezing in ice cream manufacturing. In other cases, the authors recommended flat-bladed impellers that fulfill all the conditions mentioned above. Flat-bladed impellers have been already used for a wide range of industrial processes, for example, for mixing of powders [7], but generally not as close-clearance agitators. They constitute one of the simplest types of mixing elements. However, even for such simple systems, the literature contains only sketchy information at best, both in single-phase and multiphase flows. The goal of this paper is to analyze the mixing processes in terms of power consumption in laminar mixing region. This approach affords an estimation of shear rate as a function of geometry of impeller, rotation speed and fluid properties that can be used for scale-up purpose. Geometry is investigated using both the combination of several similar mixing elements and the column-to-impeller diameter ratio on the basis of power consumption measurements. Experiments are also used to gain insight into the role played by pseudoplasticity and viscoelasticity, especially to estimate the deviations from the Newtonian power input when highly shear-thinning fluids are involved. The influence of pseudoplasticity is analyzed using the classical approaches found in the literature, such as Metzner Otto concept [8], Rieger Novak method [9] and a Couette analogy developed in this work. n e =2 a) n e =3 L e b) D Fig. 1. Description of mixing unit a) and impeller geometry b). 2. Materials and methods 2.1. Experimental set-up The mechanically stirred column used in this work is similar to that in [6] for their experimental study. The inner column diameter is 35 mm for a total column length of 440 mm. Rotational speed N of impeller can be varied from 100 to 1500 rpm using a speed-controlled engine IK LaborTechnik RE-16 from Ikavisk, Germany). A stress gauge torquemeter IK MR-D1 from Ikavisk) enables the measurement of the torque T) on the shaft of the impeller up to 1 N m with an accuracy of 0.5% full scale. Torque measurements have been validated using concentric cylinders of 25, 30 and 33 mm diameter respectively. Column temperature can be controlled to maintain constant rheological properties using a water circulation in the jacket around the column. All experiments have been conducted at 25 C. Mixing has been carried out using flat-bladed elements formed by four right-angle paddles, the diameters D) were 22, 28 and 33 mm, which corresponds column-to-impeller distance of 6.5, 3.5 and 1.0 mm respectively. Element length L e )is always 65 mm. The agitators used in this work were formed either with n e = 2 or 3 consecutive identical elements, as shown in Fig. 1. This figure summarizes the geometrical characteristics of an element and shows how they are located in the column. The detailed specifications of the impellers used in this work are reported in Table 1. As preliminary Table 1 Description of the impellers used in this work Abbreviation Geometrical characteristics of the impellers D m) D/ L e m) n e FB-33-3 0.033 0.94 0.065 3 FB-28-3 0.028 0.80 0.065 3 FB-28-2 0.028 0.80 0.065 2 FB-22-3 0.022 0.63 0.065 3 FB-22-2 0.022 0.63 0.065 2
R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1213 experiments had proved that there was no difference between torque measurements in the batch and the continuous mode, measurements were conducted only in the batch mode. This was probably due to the low flow rate values generally used in such units, with a mean residence time usually between 5 and 10 min. 2.2. Fluids and rheology Two Newtonian fluids and six highly shear-thinning fluids have been prepared for this work. Steady shear viscosity as well as primary normal stress difference were measured using a stress-controlled rheometer SR-5 from Rheometric Scientific TM, USA) equipped with a Peltier circulator for temperature control. Newtonian fluids include pure and aqueous solutions of glycerol denoted GLY Acros Organics, USA), as well as a solution of polyalkylene glycol polymers Emkarox HV45 from Uniquema, The Netherlands) denoted HV45. For non-newtonian fluids, the study focuses on highly shear-thinning fluids that can be adequately fitted using a power-law model with a flow index n between 0.10 and 0.40. In this case, shear stress τ can be related to shear rate γ using the following equation where k is the consistency factor: τ = k γ n 1) Emulsions denoted E1, E2 and E3 are three model food emulsions. They have similar components as the model emulsion used in a previous work [6], but the amount of fats and starch differs between E1, E2 and E3 to modify their rheological properties. The three other shear-thinning fluids are a 2% xanthan aqueous solution XAN), a 2% guar aqueous solution GUA) and a polyacrylamide aqueous solution PAA). Fluid properties have been measured at the temperature used for mixing experiments, i.e. 25 C. At this temperature, all fluids can be adequately described by a two parameter power-law model in the range of shear rate studied in this work. However, non-newtonian fluids can be divided into two classes depending on either they exhibit a measurable primary normal stress difference N 1 or only a shear-dependent viscosity. In this study, only PAA presents non-negligible viscoelastic properties when the shear rate is higher than 100 s 1. In this case, the primary normal stress difference has been fitted using a power-law model too: N 1 = p γ q 2) Finally, the resulting values obtained by linear regression of power-law parameters for shear viscosity, primary normal difference as well as Newtonian viscosity are reported in Table 2. 3. Results and discussion 3.1. Newtonian power input Power consumption for Newtonian fluids is usually expressed by the dimensionless power number N P as a function of the rotational Reynolds number Re when no vortex is present. This procedure provides a single master curve that depends only on impeller geometry and can be used to predict power requirements for any given fluid properties density ρ and viscosity µ), impeller dimensions and rotational speed. In the laminar regime, the quantity K P is defined as follows: N P Re = K P 3) K P should be only a function of impeller geometry for any Newtonian fluid. A significant deviation from Eq. 3) may be used to detect the end of the laminar mixing region. In this work, torque data obtained with GLY and HV45 as Newtonian fluids have been transformed into power input using the classical dimensionless form of N P for all the impellers see e.g. [10,11]). The results have been expressed as a function of Re in order to obtain the master curve N P versus Re. For each impeller, K P has been estimated by performing a regression analysis on the power curve data. Experimental results confirm that the laminar regime prevails inside the column up to about Re 30 according to Eq. 3). As expected, power consumption increases both with impeller Table 2 Description of the fluids used in this work Abbreviation Rheological properties of the fluids k Pa s n ) n Shear rate s 1 ) p Pa s q ) q Shear rate s 1 ) GLY 0.78 1 0.1 1000 HV45 3.78 1 0.1 1000 PAA 52.7 0.4 10 1000 94.8 0.62 100 1000 E1 110 0.26 0.1 1000 E2 130 0.27 0.1 1000 E3 152 0.31 0.1 1000 XAN 11.2 0.18 0.1 1000 GUA 124 0.12 1 500
1214 R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 Power number NP 10000 1000 100 10 HV45 GLY Eq. 4) 1 0.1 1 10 100 Re Fig. 2. Example of power data for Newtonian fluids FB-28-3). diameter D and element number n e. K P is shown to be nearly proportional to the number of elements, which proves that there is no significant interaction between them. For all the impeller geometries used in this work, a regression analysis on K P data provides the following correlation in the laminar regime: ) ) ne L e DC D 0.42 K P = 164.8 4) D D Fig. 2 illustrates the typical Newtonian power consumption behavior depicted by Eq. 3) for impeller FB-28-3 and also includes the calculated N P versus Re curve obtained using the estimation of K P from Eq. 4). As can be seen, there is a good agreement between predictions and experimental data. As indicated above for scale-up purpose it is usual to represent the mixing efficiency of any impeller by postulating a relationship between an effective shear rate γ a around the impeller and rotational speed [1]: γ a = K S N 5) K S is the proportionality constant and is a priori a function of fluid properties. For example, it may depend on the flow index and be expressed as K S n). But ideally, K S is a constant that should be insensitive to the rheological properties of the fluids in the laminar regime for any given impeller geometry. This assumption is the basis of the Metzner Otto concept [8] which is still extensively used for scale-up purpose in the process industry when engineers have to deal with non-newtonian media [12]. As an illustration, most conventional viscosimetry techniques used for chemical, biological or food media result directly from the application of this concept [13,14]. The conventional methods for K S determination that have been developed in the literature are summarized in [14], but are also described in Section 3.2. With these methods, K S cannot be found using only Newtonian fluids, but its estimation generally requires duplicate power consumption measurements using both Newtonian and non-newtonian fluids in order to modify n. K S values obtained for non-newtonian fluids may depend on n and must be noted K S n), while the Newtonian fluids are generally used as a reference. Conversely, we propose here to develop a specific method in order to estimate K S using only Newtonian fluids. Such values will be noted K S n = 1). As this method is original, one of the objectives of this paper will also be to check whether there is a compatibility between K S n = 1) and K S n) values obtained by conventional methods. The classical Couette analogy consists in determining the equivalent diameter D of a cylinder that exhibits a power consumption equal to that of the impeller as a function of rotational speed [15]. However, D is a function of fluid properties: for power-law fluids, it is a function of flow index D n). Carreau and coworkers [15] showed that D varies only slightly with n for helical ribbon impellers, which should correspond to a K S value nearly independent of n [1]. However, this method still requires torque measurements using both Newtonian and non-newtonian fluids. In the modified method that is suggested here, one looks for a two-parameter model using only the Newtonian power input data: the first one is D = D n = 1), while the second is a radial coordinate r o in the Couette geometry for which the shear rate depends slightly on n. For any power-law fluid, the shear rate in a Couette geometry of internal diameter D and external diameter can be estimated as a function of n, N and the radial coordinate r using: γr,n) = 4πN n D ) [ 2/n D ) ] 2/n 1 1 6) 2r A relation between D and can be deduced by evaluating shear rate from Eq. 6) when r = /2, injecting Eq. 1) and using the torque balance in a Couette geometry. Details of the calculations are reported in Appendix A. The result corresponds to Eq. 7): ) D = 1 + 4πN πkne L e DC 2 1/n n 2T n/2 As a first approximation, D can be considered to be independent of n and can be estimated using power input data obtained with the Newtonian fluids only. Using the Newtonian and non-newtonian test fluids as well as Eq. 7), Table 3 confirms that D is experimentally a weak function of n for flat-bladed impellers. Experimental results also show that D is always close to D, asd /D remains between 0.94 and 0.98 in this work. Furthermore, D appears to tend to D when the ratio D/ decreases. Using again Eq. 6), itis 7)
R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1215 Table 3 Estimation of the equivalent diameter D of a virtual Couette geometry from experimental data Eq. 7)) for impeller FB-33-3 as a function of fluid properties γ r,n 1600 Fluid D m) GLY 0.031 HV45 0.031 PAA 0.030 E1 0.031 E2 0.032 E3 0.032 XAN 0.031 GUA 0.031 1200 800 n = 0.1 n = 0.2 n = 0.4 n = 0.6 n = 0.8 n = 1.0 possible to show that there is a radial coordinate for which γ/n is nearly independent of n: N, γr o,n) N γr o) N = γr o, 1) N The existence of r o needs that the following condition is fulfilled: D 2r o 9) Note that a roughly similar method was used recently by Chavez-Montes et al. [16] for other impeller geometries in order to measure the apparent viscosity. However, their analysis needs to collect torque data on the external cylinder of their tank, which limits the applicability of their results to batch units based on the principle of a strain-controlled rheometer. By contrast, the model developed here correspond to the most common situation when torque data are collected on the shaft of the impeller, which enables their application to any impeller-driven system equipped with a torquemeter, regardless of its dimension and location, either in the batch or the continuous mode. The parameter r o of Eq. 8) can be found by minimizing the function fr) defined by Eq. 10) using a least-square Levenberg Marquardt algorithm for n between 0.1 and 1, which corresponds to the flow index of the fluids used in this work: fr) = 1 [ γr,i) γr,1)] 2 10) N 0.1 i 1 The optimization procedure shows that r o always exists for any flat-bladed impeller used here. As an illustration, calculated γr, n) values are presented in Fig. 3 at a given rotational speed as a function of r and n for impeller FB-33-3. This figure confirms graphically the existence of a unique radial coordinate r o between D /2 and /2 around 0.0162 m) for which γ is nearly independent of n for this impeller. Results from the optimization procedure show that r o increases when D increases. Similarly, the ratio 2r o /D tends to 1 when D increases. The difference in r o observed respectively between FB-28-2 and FB-28-3, as well as between FB-22-2 and FB-22-3 are less than 1%, which proves that r o is clearly insensitive to n e. The relative deviation between 8) 400 0 0.031 0.032 0.033 0.034 0.035 2 r m) Fig. 3. Radial evolution of shear rate in the Couette geometry as a function of fluid properties for impeller FB-33-3 between D = 0.031 m and = 0.035 m. γr o,n)and γr o, 1) is less than ±2% for FB-33-3 and less than ±4% for FB-28-2 and FB-28-3, but it is a around ±7% for FB-22-2 and FB-22-3: as expected, shear rate deviates from the Couette model when clearance increases. The good agreement obtained with FB-33-3 is illustrated by Fig. 4 γ r 0, n 600 500 400 300 200 100 +3% -3% n=0.8 n=0.6 n=0.4 n=0.2 n=0.1 0 0 100 200 300 400 500 600 γ r o,1 Fig. 4. Comparison of γr o,n) and γr o, 1) values for FB-33-3 at the same rotational speed.
1216 R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 Table 4 Comparison of Couette analogy, slope and power curve method for K S estimation K S values FB-33-3 FB-28-3 FB-28-2 FB-22-3 FB-22-2 Couette Eq. 11)) 51.9 27.3 27.2 16.0 16.0 Slope method 51.6 30.4 29.7 19.0 18.3 PAA 71.0 28.5 E1 49.2 30.3 29.0 18.5 E2 43.5 27.8 18.9 E3 52.3 29.3 17.8 17.4 XAN 59.5 30.0 GUA 51.6 42.8 28.2 *) From the power curve method Eq. 13)); ) not in the experimental design; ) accurate estimation of K S not possible. which plots γr o,n) as a function of γr o, 1) at the same rotational speed. When r o has been found, a linear relation between γr o ) and N can be obtained. Using Eq. 6), this relation is valid for any n value between 0.1 and 1: D ) [ 2 D ) ] 2 1 γr o ) = 4π 1 N = K S N 11) 2r 0 A comparison with Eq. 5) shows that if the Metzner Otto concept is valid for the impeller used here, the K S values obtained from Eq. 11) should be at least proportional to that of Metzner Otto. In term of compatibility between both methods, the ideal situation should be that both would be equal. K S values obtained using the Couette analogy have been reported in Table 4. The data show that the number of elements is of little influence on K S, while K S increases drastically as clearance decreases. These results were expected: with flat-bladed impellers, shear rate is mainly controlled by the gap size as far as the ratio L e / is high enough to overshadow the end effects, which is the case here. As a result, the only key parameter is the ratio D/. The steep increase in K S with D/ could not be fitted using a simple power-law relation, but the following expression has been found: K S n = 1) = exp 3.25 D ) + 0.69 12) Note that Eq. 12) has been obtained using only torque data of Newtonian fluids. As a conclusion, flat-bladed impellers behave nearly as ideal impellers with Newtonian fluids: power consumption is proportional to impeller length when n e L e D and K S depends only on clearance. 3.2. Influence of shear-thinning behavior The influence of pseudoplasticty on power consumption can be analyzed using the Metzner Otto concept. As already mentioned, this approach is based on Eq. 5) as an empirical assumption. Although Metzner Otto assumed that K S was a proportionality constant independent of the rheological properties of the fluid, many authors have doubted as to whether this is true for highly shear-thinning fluids. Despite a recent contribution based on CFD that attended to give a theoretical support and to highlight the limitations of Metzner Otto concept for anchor impellers [17], its applicability still remains questionable for many mixing devices, as K S often appears to be a function of flow index for power-law fluids. As an illustrations, several authors [1,18] reported a strong dependence of K S on n for highly shear-thinning fluids using helical ribbons. This trend was confirmed for other types of impellers: for example, a strong evolution of K S with n at very low rotational speed <10 rpm) was observed with flag impellers [19]. Nevertheless, empirical K S values are extensively tabulated for various mixers in the literature and Metzner Otto method is still recommended as a standard procedure for mixing analysis with close-clearance impellers in the laminar regime [12]. In this work, we will consider that the proportionality constant can be written K S n) and depends a priori on n. The procedures used in the literature for K S n) estimation, known as viscosity matching techniques, have been summarized in [14], such as the power curve and the torque curve method. The power curve method corresponds to the most classical technique used in the literature. For any non-newtonian fluid, the average shear rate is calculated from torque data using the following expression: γ a = [ 1 k 2πT NK P D 3 )] 1/n 1 = K S n)n 13) This expression can be deduced from Eq. 3) using the definitions of N P, Re and the fact that power on the shaft is equal to the product 2πNT. The slope of the average shear rate values versus N obtained by regression analysis gives access to K S n). An alternative procedure, denoted sometimes as the slope method, was also suggested by Rieger and Novak [9]. This consists in plotting power number N P as a function of a modified Reynolds number Re m defined as follows: Re m = ρn2 n D 2 14) k Power consumption data can be used to estimate K P n) values obtained from Eq. 14): N P Re m = 2πT kn n D 3 = K Pn) 15) Considering γ a = k/µ) 1/n 1, K S is deduced by combining Eqs. 3), 5) and 15): [ ] KP n) 1/n 1 K S n) = 16) K P where K P is defined by Eq. 3) for Newtonian fluids. If K S does not depend on n, the proportionality constant can be obtained directly from the slope of the straight line resulting from the plot of logk P ) versus 1 n). This justifies the
R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1217. γ a s 1 ) 500 400 300 200 100 0 PAA XAN Couette 0 5 10 15 20 N s -1 ) Fig. 5. Typical K S determination from the power curve method and comparison with the predicted curve using the K S value from the Couette analogy impeller FB-28-3). name of slope method. Note that this method, unlike the power curve method, assumes that the Metzner Otto concept is valid. To circumvent this limitation, it is possible to apply directly Eq. 16) for estimating K S n) when the applicability of Eq. 5) is doubtful. This method will be referred to as the direct calculation of K S n). K S was first determined for each non-newtonian fluid using the power curve method Eq. 13)). A typical plot for K S determination on impeller FB-28-3 is reported in Fig. 5 for the 2% xanthan solution XAN and the polyacrylamide solution PAA. This figure illustrates the results obtained for all impellers: γ a is generally proportional to N for a certain range of N values. In Fig. 5, the plot is nearly linear for PAAuptoN equal to 20 s 1, but it deviates for XAN when N is higher than 8 s 1, which corresponds to the end of the laminar region. This figure also shows the good agreement between the values obtained from the power curve method and the modified Couette analogy respectively for impeller FB-28-3. K S n) values for each impeller have been determined in this way for all the shear-thinning fluids. The results are reported in Table 4 and compared to those of the Couette analogy. In this table, the symbol ) represents the experiments for which an accurate estimation of K S n) could not be obtained because the linear region was too short. This may correspond either to the end of the laminar region or to the limits of the power-law region, especially for PAA that presents a Newtonian region when shear rate is lower than 1s 1 followed by a transition region between 1 and 10 s 1. The results in Table 4 also show that K S n) depends only slightly on the element number n e and on the flow index n, even for highly shear-thinning fluids. By contrast, the dependence on D/ appears to be strong for flat-bladed impellers, as expected. These results tend to prove that the Metzner Otto concept may be valid and that the slope method could be applied. This also differs from recent studies cited previously on helical ribbon impellers that report a strong dependence on n for highly shear-thinning fluids. There are however some punctual deviations. The largest discrepancies are observed for PAA with impeller FB-33-3 and for GUA with impeller FB-28-3. These will be discussed later when both the slope method and the direct calculation of K S n) will be compared. In Table 4, one should also note that K S n) values from the power curve method are very close to that of the Couette analogy. These results are in agreement with those presented in Fig. 5. However, it appears that the power curve method generally provides K S n) values a bit higher, especially when clearance increases. The agreement between both methods can however be considered as satisfactory if we consider that the Couette analogy is based on power input data using only Newtonian fluids, while the power curve method requires additional experiments with non-newtonian test fluids. From experimental data, power number N P was plotted versus Re m for all impellers. Results are reported in Fig. 6 for impeller FB-28-3. K P n) values can be easily deduced from these plots Eq. 15)). As expected, the results show that K P n), and consequently power consumption, dramatically decrease when n decreases at a given Re m value. A typical application of the slope method is reported in Fig. 7 for impeller FB-28-3: K P n) has been plotted as a function of n 1 using the data from both the Newtonian and the non-newtonian fluids. This figure proves that the slope method can be applied for flat-bladed impellers and that K S n) is only slightly affected by flow index, even for highly N P 10000 1000 100 GUA PAA E1 E2 GLY & HV45 10 0.01 0.1 1 10 Re m Fig. 6. Power consumption curves: comparison of shear-thinning and Newtonian fluids for impeller FB-28-3.
1218 R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 K P n) 10000 1000 100 10 1 ln K P n) = 2127 exp[3.42n-1)] R 2 = 0.999-1 -0.8-0.6-0.4-0.2 0 n-1 Fig. 7. Typical application of the slope method for impeller FB-33-3. shear-thinning fluids. This result is valid for the five geometrical configurations used in this work. K S n) estimations using the direct calculation Eq. 16)) as well as the slope method have been reported in Table 5,in which they are compared to the results of the Couette analogy. Using Eq. 15), the largest deviation is reported for PAA and impeller FB-33-3. One should note that K S n) values for GUA are in agreement with other fluids, contrary to that had been observed previously for the power curve method, which tends to prove that either Eq. 16) or slope method give more accurate results than the power curve method. It is also true that the highest deviations from the straight line of the slope method or from the direct estimation Table 5) appear for the FB-33-3 agitator. However, this may be explained because measurements are particularly difficult with this impeller, as the gap is only 1 mm: vibrations at high rotational speed can drastically affect power consumption measurements, while they are of little influence when the gap is larger for 3.5 and 6.5 mm). The results remain how- Table 5 Comparison of Couette analogy, slope and direct calculation methods for K S estimation K S values FB-33-3 FB-28-3 FB-28-2 FB-22-3 FB-22-2 Couette Eq. 11)) 51.9 27.3 27.2 16.0 16.0 Slope method 51.6 30.4 29.7 19.0 18.3 PAA 67.4 30.8 E1 52.6 32.0 32.1 20.5 E2 47.6 30.5 19.0 E3 52.9 30.3 18.5 18.0 XAN 63.7 30.9 GUA 52.9 29.6 28.5 *) From the direct calculation Eq. 16)); ) not in the experimental design; ) accurate estimation of K S not possible. ever quite satisfactory, even for this impeller. The high K S n) value observed with PAA and impeller FB-33-3 cannot be attributed to viscoelastic forces because these are known to decrease K S n) [12,15]. Forflat-bladed impellers driven under high-speed conditions, it seems that the effect of viscoelasticity is overshadowed by shear-thinning, which is in agreement with the conclusions of Doraiswamy et al. [12]. This result is particularly important for gas dispersion, as bubbles introduced in non-newtonian media tend to increase their viscoelastic character [6]. The influence of impeller geometry on K P n) and K S n) can be analyzed as a function of element number and diameter. Even for non-newtonian fluids, K P n) seems to be nearly proportional to n e, which is in agreement with the results obtained for K P with Newtonian fluids Eq. 4)). As shown in Fig. 6, K P n) values exhibit a strong dependence on n and decrease significantly with n. By contrast, the influence of the gap size on K P n) decreases as shear-thinning behavior increases. As already seen for Newtonian fluids, K S n) depends only slightly on n e. Power curve method, slope method as well as the direct method are in good agreement and show that K S n) is nearly independent of n between 0.1 and 1. As a conclusion, the Metzner Otto concept seems to be valid for flat-bladed impellers. K S n) depends therefore only on the gap size. Using the direct calculation, K S n) can be fitted using the following expression: K S n) = K S = exp 3.41 D ) + 0.86 17) Note that this relation only slightly differs from Eq. 12) which had been obtained using only Newtonian fluids. Eq. 17), as well as K S n) values reported in Tables 4 and 5, show that the slope method and the direct approach generally lead to slightly higher values than the Couette method. The results are however quite close if we consider that they stem from different fluids and experiments. As a conclusion, it appears that K S values from the modified Couette analogy and other techniques can be considered as equal for flat-bladed impellers and that there is a good compatibility between the Couette analogy developed in this work and more conventional methods. Finally, K P n) can therefore be predicted by combining Eqs. 4), 16) and 17): ) ) ne L e DC D 0.42 K P n) = 164.8 D D [ exp 3.41 D n 1 + 0.86)] 18) Eq. 18) allows the calculation of a unique generalized dimensionless power curve for any flat-bladed impeller. This requires the definition of an apparent Reynolds number Re a expressed as Re a = ρn2 n D 2 kk n 1 19) S
R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1219 N P 10000 1000 100 GUA PAA E1 E2 GLY & HV45 Eq. 18) 10 0.1 1 10 100 Re a Fig. 8. Generalized dimensionless power consumption curve for impeller FB-28-3. Fig. 8 illustrates the plot of N P versus Re a for impeller FB-28-3: all power draw data are brought together into a single dimensionless curve, regardless of flow index n. This figure can also be compared to the set of original power draw curves Fig. 6). Similar plots are obtained with other impeller geometries. 3.3. Relative mixing effectiveness For homogenization or distributive mixing, the mixer effectiveness depends on two main factors: power consumption and mixing time. However, defining mixing effectiveness is a difficult task and it is generally easier to define an effectiveness parameter relative to a reference or a parameter which compares the performance of two impellers [20]. For dispersive mixing, the aim of which is to decrease the characteristic size d) of the dispersed phase, it is generally admitted that the mean bubble or droplet size formed by mechanical stirring in the laminar regime is adequately described by the dimensionless laminar Weber number We defined as follows: We = τd σ = k K SN) n d 20) σ where σ is the surface tension for bubbles and the interfacial tension for droplets. The key parameter for dispersive mixing in Eq. 20) is therefore the effective shear stress applied to the dispersed phase. As previous works have shown that flat-bladed impellers are efficient for gas dispersion in non-newtonian fluids [6], it may be interesting to compare the stress level applied by flat-bladed impellers as a function of n to that applied by other more conventional agitators. This comparison can be conducted as a function of rotational speed, but it seems preferable to consider power consumption rather than rotational speed. A good impeller for dispersive mixing should indeed provide a high K S values at the lowest power consumption. The following paragraph is devoted to the definition of an indicator able to take both kinds of parameters into account. In order to compare the mixing characteristics of flat-bladed impellers to more conventional agitators, two conventional impellers denoted as reference systems were selected. The first one is a cylinder which corresponds to a classical Couette device. The main advantage is that this particular configuration can be described analytically. The second reference impeller is an helical ribbon impeller that has been studied in many works summarized in reference [1]. The authors report both experimental data and correlations able to predict K P as a function of the geometrical ratios of these impellers. A power ratio R P n) based on power consumption can therefore be defined as the ratio of the power constant K P n) of a standard reference impeller to the power constant of a standard flat-bladed element estimated for the same fluid at the same rotational speed. The standard geometry that has been retained to compare these impellers are n e L e = D and D/ = 0.88. As a result, R P n) can be expressed as follows: R P n) = K Preference; n e L e = D; D/ = 0.88; n) K P flat-bladed; L e = D; D/ = 0.88; n) 21) R P decreases when additional power is required by the flat-bladed impeller as compared to the reference agitator. Similarly, one can define a ratio based on effective shear stress R S n) that represents the difference in effective shear stress between two impellers for the same fluid at the same rotational speed: [ ] KS flat-bladed; n e L e = D; D/ = 0.88; n) n R S n) = K S reference; L e = D; D/ = 0.88; n) 22) The power n in Eq. 22) stems from Eq. 20). Arbitrarily, one suggests to define the relative effectiveness parameter Effn) which takes both power consumption and stress level into account as follows: Effn) = R a S R1 a P 23) The exponent a 0 <a<1) used to weight each factor is chosen arbitrarily: when a is higher than 0.5, Eq. 23) tends to emphasize the influence of the effective shear rate as a key parameter, while a similar effect is observed for power requirements when a is lower than 0.5. For practical applications, Eff evaluates the expense of a difference in effective shear stress between two impellers in terms of power consumption. High values of Eff at a given n value are therefore an indicator of an impeller adequate for dispersive mixing.
1220 R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1.5 1.25 1 0.75 0.5 0.25 0 0 0.2 0.4 0.6 0.8 1 n Fig. 9. Evolution of R P, R S and Eff with concentric cylinders as a reference. For dispersive mixing, a must be chosen higher than 0.5: it is usually 1 Eff = R S ) in biological and food applications for which the only key parameter is the characteristic size of the dispersed phase. As an illustration, the results presented below use a = 0.75. Eq. 4) gives K P = 380.5 for the standard flat-bladed impeller n e L e = D; D/ = 0.88). R P and R S values can be estimated using the relations provided by the literature for helical ribbon impellers [1] and the analytical solution of the Couette flow for concentric cylinders. As a result, Eff values can be evaluated. R P, R S and Eff for a = 0.75 are reported, respectively in Fig. 9 for concentric cylinders and in Fig. 10 for the helical ribbon as a reference. Fig. 9 shows that R S decreases slightly when n increases, while R P rises slightly with n when the Couette geometry is the reference. Actually, the effective shear rate produced by flat-bladed impellers is always around 80% of that of concentric cylinders. R S tends to 1 when n tends to 0.1, which proves that 1.5 1.25 1 0.75 0.5 0.25 0 R P R S Eff 0 0.2 0.4 0.6 0.8 1 n Fig. 10. Evolution of R P, R S and Eff with helical ribbon as a reference. flat-bladed impellers tend to behave as concentric cylinders when pseudoplasticity increases. Consequently, Eff is nearly independent of n in the range 0.1 1 and remains close to 1Fig. 9). As a conclusion, the indicator Eff shows that flat-bladed impellers and Couette device are equivalent in terms of dispersive mixing effectiveness for a = 0.75. Conversely, when the helical ribbon impeller is taken as a reference Fig. 10), R P presents a steep increase when n decreases below 0.4. R P becomes even higher than 1 when n is lower than 0.18. As the effective shear rate is higher for flat-bladed impellers, apparent viscosity remains lower and power consumption dramatically decreases for flat-bladed impellers as compared to helical ribbons when n is low. R S also appears to increase when n decreases Fig. 10), but only slightly due to the exponent n that is used for the calculation of the effective shear stress Eq. 22)). R S remains however always higher than 1. Finally, Eff increases when n decreases and is far higher than 1 when n is lower than 0.3. As expected, flat-bladed impellers have to be preferred for dispersive mixing in highly shear-thinning fluids. Note that this is important for chemical and food applications involving complex media. For example, most food emulsions prepared based on milk or meat present a flow index value between 0.25 and 0.3. Finally, the results provided by the indicators R S and Eff for both reference impellers are found to be in agreement with the literature. First, helical ribbon are known to be poorly effective for gas dispersion in highly shear-thinning fluids, which is confirmed by the high Eff and R S values obtained in Fig. 10. Conversely, flat-bladed impellers have been proved to form bubbles with a characteristic size similar to that of solid rotors in food emulsions, between 40 and 50 m [6]. This is in agreement with R S and Eff value around 1inFig. 9. 4. Summary and conclusions In this work, flat-bladed impellers have been described quantitatively in terms of power consumption and effective shear rate as a function of impeller geometry and pseudoplasticity of the fluid in the laminar mixing region. The influence of highly shear-thinning behavior has been highlighted. Power constant K P can be predicted as a function of impeller length and clearance. K P n), as well as the proportionality constant K S n) for effective shear rate estimation have been determined. K S values found using the modified Couette analogy, the power curve method, the direct calculation as well as Rieger and Novak s approach are in good agreement and support the hypothesis that K S is nearly independent on flow index n even for highly shear-thinning fluids and depends only on the gap dimension. As a result, the Metzner Otto concept is valid for all the geometries studied here, regardless of impeller length and clearance. A unique dimensionless power draw curve has been obtained for each impeller geometry. The indicators Eff and
R.K. Thakur et al. / Chemical Engineering and Processing 43 2004) 1211 1222 1221 R S defined for estimating the ability of impellers to apply high shear stress at low power requirements both confirm that flat-bladed impellers could be valuable tools for dispersive mixing in highly shear-thinning fluids, as they maintain a high shear rate level, regardless of pseudoplasticity. This agrees with the results of gas dispersion experiments using flat-bladed impellers [6]. Viscous effects seem to be predominant over elasticity, which is also favorable to gas dispersion processes, as a gas phase in form of bubbles can add a non-negligible elastic component to a non-newtonian viscous fluid. Flat-bladed impellers can also be recommended for use in mixer viscosimetry techniques, especially for rheologically evolving systems. Owing to the applicability of the modified Couette analogy, the simultaneous estimation of the effective shear rate and the apparent viscosity is now possible, even when the rheological properties of the fluid are unknown. This opens new perspectives for the analysis of batch and continuous dispersion processes. Appendix A In a Couette system, torque is known to be independent of the radial coordinate r. As a result, torque balance on a Couette system can be written as r, T = 2πr 2 n e Lτr) A.1) Injecting Eq. 1) gives T = 2πr 2 n e L e k[ γr,n)] n When r = /2, one obtains 2T = πdc 2 n el e k [ γ /2,n) ] n A.2) A.3) Combining Eq. A.3) and Eq. 6) provides 2T = πdc 2 n el e k 4πN D ) [ 2/n D ) ] 2/n 1 1 n n A.4) D diameter of impeller or cylinder m) D equivalent impeller diameter in a Couette geometry m) diameter of the column or shell m) Eff effectiveness factor Eq. 23)) fr) objective function Eq. 10)) k consistency factor Eq. 1)) Pas n ) K P power constant Eq. 3)) K P n) generalized power constant Eq. 15)) K S shear rate proportionality constant Eq. 5)) K S n) generalized shear rate proportionality constant Eq. 16)) L e length of a mixing element m) n flow index Eq. 1)) n e number of mixing elements N rotational speed s 1 ) N 1 primary normal stress difference Eq. 2)) Pa) N P power number or Newton number p viscoelasticity factor Eq. 2)) Pas q ) q viscoelasticity index Eq. 2)) r radial coordinate m) r o fitted parameter of Couette analogy m) R P n) power constant ratio Eq. 21)) R S n) effective shear rate ratio Eq. 22)) R 2 correlation coefficient Re rotational Reynolds number Re a apparent Reynolds number Eq. 19)) Re m modified Reynolds number Eq. 14)) T torque measured on the shaft of the impeller N m) We laminar Weber number Eq. 20)) Greek letters γ shear rate s 1 ) γ a effective shear rate Eq. 5)) s 1 ) µ fluid viscosity Pa s) ρ fluid density kg m 3 ) σ surface/interfacial tension N m 1 ) τ shear stress Pa) Eq. A.5) can also be expressed as DC D ) ) 2/n 1 = 4πN πd 2 1/n C n e L e k A.5) n 2T References which finally gives Eq. A.6), identical to Eq. 7). ) D = 1 + 4πN πkne L e DC 2 1/n n/2 n 2T Appendix B. Nomenclature a weighting parameter Eq. 23)) d diameter of gas bubble or droplet m) A.6) [1] E. Brito-De La Fuente, L. Choplin, P.A. Tanguy, Mixing with helical ribbon impellers: effect of highly shear thinning behaviour and impeller geometry, Trans. Inst. Chem. Eng. 75A 1997) 45 52. [2] T. Espinosa-Solares, E. Brito-De La Fuente, A. Tecante, L. Medina-Torres, P.A. Tanguy, Mixing time in rheologically evolving fluids by hybrid dual mixing systems, Trans. Inst. Chem. Eng. 80A 2002) 822. [3] W. Hanselmann, E. Windhab, Foam generation in a continuous rotor stator mixer, in: G.M. Campbell, C. Webb, S.S. Pandiella, K. Niranjan Eds.), Bubbles in Food, Eagan Press, MN, USA, 1999. [4] D.R.G. Cox, A.J. Gerrard, L. Wix, Power consumption and backmixing in horizontal scraped surface heat exchangers, Trans. Inst. Chem. Eng. 71C 1993) 187 193.
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