Viscoplastic fluid mixing in a rotating tank

Viscoplastic fluid mixing in a rotating tank

Chemical Engineering Science 62 (2007) 2290 – 2301 www.elsevier.com/locate/ces Viscoplastic fluid mixing in a rotating tank Frédéric Savreux, Pascal J...

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Chemical Engineering Science 62 (2007) 2290 – 2301 www.elsevier.com/locate/ces

Viscoplastic fluid mixing in a rotating tank Frédéric Savreux, Pascal Jay ∗ , Albert Magnin Laboratoire de Rhéologie, Université de Grenoble, (UJF, INPG) et CNRS (UMR 5520), 38041 Grenoble, Cedex 9, France Received 11 July 2006; received in revised form 27 October 2006; accepted 11 January 2007 Available online 25 January 2007

Abstract The laminar 2D mixing of viscoplastic fluids with an anchor impeller in a stirring tank is studied numerically. The flow structures, and particularly the static and moving rigid zones induced by the plasticity such as the presence of vortices, are examined in detail. Calculations are made for a wide range of yield and inertia values. It is shown that the application of a small yield stress can eliminate vortices. On the other hand, the increase in rotation velocity, i.e. the inertia, is not generally sufficient to reduce the size of the rigid zones. Power consumption is calculated and the mixing quality is determined by particle tracking. The calculations are made for different types of impeller configurations and an optimum configuration is proposed. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Mixing; Yield stress; Rigid zones; Viscoplasticity; Complex fluid; Simulation

1. Introduction The manufacture and development of materials very often involves agitation or mixing, whether at industrial level or in the laboratory. In the case of complex fluids, the design and choice of an agitation system are often empirical. This is particularly noticeable for viscoplastic fluids. This type of fluid also called “yield stress fluid” starts to flow only when the applied stress is above a threshold value called the yield stress. Many fluids, such as gels, slurries, filled polymers, emulsions, doughs, etc. (Bird et al., 1983) have a viscoplastic behavior. This typical property drastically modifies the agitation performance, the quality of mixing and the power consumed by creating rigid zones in the mixing device. In spite of the industrial interest, viscoplastic fluid mixing has not been studied to any great extent either experimentally or numerically. For this type of fluid, Solomon et al. (1981) showed experimentally that fluids mixed with a Ruhston turbine contain well-agitated caverns surrounded by completely stagnant zones because of the yield stress. Tanguy et al. (1994) and Bertrand et al. (1996) showed the influence of viscoplasticity on the flow pattern and on power consumption in the case of an anchor impeller. They compared the Newtonian ∗ Corresponding author. Tel.: +33 04 76 82 51 54; fax: +33 04 76 82 51 64.

E-mail address: [email protected] (P. Jay). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.01.020

case (Od = 0), where “Od” is the dimensionless number representing the yield effect) to a particular viscoplastic case corresponding to Od = 20. Kaminoyama et al. (1994) carried out a numerical analysis of the flow of a Bingham fluid in an anchor impeller. They calculated the flow pattern for particular cases and compared their results with experimental findings. For the same type of configuration, Marouche et al. (2002), also by numerical analysis, showed the changes in flow pattern for low inertia and high yield stress. Poullain et al. (2001) in the case of a concrete mixer using an anchor impeller, determined the stress field tensor both experimentally and numerically. For the power consumption (Np) of viscoplastic fluid mixing, Nagata et al. (1970) applied the Metzner and Otto (1957) concept to determine the power consumption of fluid needed with four different types of impeller. Numerical work by Tanguy et al. (1994) confirmed the earlier experimental results found by Nagata et al. (1970). Tanguy et al. (1996) performed a literature review on the Np calculation and studied the effect of the power law index on Np both experimentally and numerically. Bertrand et al. (1996) reviewed and clarified Nagata’s work. They pointed out the effect of Bingham number on power number. In all these papers, the effects of both plasticity and inertia on the flow structure and on power consumption are not clearly shown or in some cases only partially. Consequently, the aim of this work is to try to clarify numerically these effects in the

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case of 2D mixing with an anchor impeller in a stirring tank. As will be seen later, the plasticity creates rigid zones which are highly detrimental to mixing quality and power consumption. So, from a practical standpoint, these zones must be identified and minimized. These zones will be accurately determined as a function of the governing parameters. Moreover, their influence on mixing quality will be assessed by particle tracking. 2. Basic equations The steady-state flow of a viscoplastic fluid in a rotating tank with a stationary chamfered anchor impeller in isothermal conditions is considered. The 2D computation domain is shown in Fig. 1. The calculation is performed from the point of view of the impeller. The Bingham model is used for viscoplastic behavior:    o ij = 2 K + Dij if I I > o , (1) ˙ Dij = 0 if I I o , where Dij is the strain rate tensor defined by   juj 1 jui Dij = . + 2 jxj jxi

(2)

˙ the second invariant of the strain rate tensor:  ˙ = 2Dij Dij

(3)

and I I the second invariant of the stress tensor:  1 I I = ij ij . 2

(4)

N

2291

In regions where I I is less than the yield stress o , the fluid exhibits no deformation and a solid structure is formed. These regions are called unyielded or rigid zones. For values greater than o , the fluid is sheared and exhibits nonlinearly viscous shear-thinning behavior. The equations are made dimensionless by using for the lengths the larger side of the impeller l, for the velocities the linear velocity at the center of the impeller V1 = 2R1 N , and for the stresses and pressure K(V1 / l). After a study on the influence of the impeller diameter to tank diameter ratio (Savreux, 2004, see also Delaplace et al., 2006, for shearthinning fluids) the tank and anchor geometries chosen correspond to R/ l = 5, R1 / l = 3.5 and e/ l = 0.15 where e is the impeller thickness. The two dimensionless numbers governing the flow are then • The Reynolds number, which represents inertia effects: Re =

V1 l . K

(5)

Note that the Re number has been calculated using the length of the impeller. By using the radius R of the tank, a new Re number could be obtained multiplying the previous one by a 25 factor of 3.5 . • The Oldroyd number, which represents the yield stress effects: 0 Od = . (6) K(V1 / l) Non-slip conditions are chosen on the tank and on the impeller. Consequently, at the vessel wall, v+ = R/R1 and at the impeller wall, v+ = 0. Boundary conditions are of the Dirichlet type. The power consumption per unit length can be determined from the equation  P = tr(ij : Dij ) dv. (7) v

3. Numerical techniques R

Impeller Y X

e

O l

R1

Rotating tank

Fig. 1. Geometry of the problem.

The finite element solver “Polyflow/Fluent” is used in this study. This code is based on a pressure–velocity formulation and the resolution on an iterative Newton scheme coupled with Picard iterations (Crochet et al., 1984). Convergence is assumed when the norm of the change in solution vector between successive iterations is less than 10−5 . As can be seen from Eq. (1), the Bingham model shows a discontinuity for  = 0 . To remove this singularity, different ways to regularize these equations have been proposed. Some regularized models have been evaluated in detail by Burgos et al. (1999) and by Frigaard and Nouar (2005). The regularization of Papanastasiou (1987) introduces an exponential part in the model which becomes (in the dimensionless form):   Od ij = 2 1 + (1 − e−m˙ ) Dij . (8) ˙

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Fig. 2. The computational domain and the mesh. (a) The whole domain; (b) Zoom on the impeller.

This regularization has been used in numerous recent papers such as Alexandrou et al. (2001), Smyrnaios and Tsamopoulos (2001), Liu et al. (2002), Zisis and Mitsoulis (2002), Mitsoulis (2004). However, a very small regularization parameter (m in the previous equation) has to be used which drastically increases the CPU time. It has been shown (Savreux et al., 2005) that an accurate determination of the rigid zones needs a value of m greater than 10+4 . The value m = 10+5 was used for this study. It is also worth noting another type of regularization, suggested by Bercovier and Engelman (1980), used by Beris et al. (1985) and very recently by Liu et al. (2002, 2003). One of the alternatives to regularization is the “augmented Lagrangian method” developed by Fortin and Glowinski (1983). In a recent paper, Moyers-Gonzalez and Frigaard (2004) showed the great advantages of this method for single and also multifluid problems. Vola et al. (2004), Vinay et al. (2006, 2005) have also shown that this method can be used for very difficult engineering calculations. The regularization method assumes that there are no strictly rigid zones. It is not therefore possible to locate the rigid zones as areas where the strain rate is zero. However, as shown by Burgos et al. (1999) and also explained by Dimakopoulos and Tsamopoulos (2003), these zones can nonetheless be located using the criterion II < 0 . When the second invariant is greater than the threshold value 0 , the region is considered to be fluid. The mesh used in this study is shown in Fig. 2. It is composed of 10,599 nodes. The influence of the mesh on the results, and in particular on the hydrodynamics inside the tank and on the power, was studied systematically (Savreux, 2004). The rigid dead zones are calculated in this study. To obtain a precise determination of these zones and to make the results mesh-independent the mesh must be very accurate. Therefore, near the plate, the mesh was highly refined. On the other hand, determination of the power consumption needs far fewer nodes in order to be mesh independent. So the rest of the mesh has larger elements, thereby making it possible to maintain a reasonable CPU time. This CPU time ranges typically between 300 s (Od = 0, Re = 0.01) and 18,000 s (Od = 10, Re = 0.01).

4. Results 4.1. Stream lines The pattern of stream lines is shown in Fig. 3 for the Newtonian case (Od = 0) and for a viscoplastic fluid (Od = 1 and 10). When inertia is low, Re = 0.01, the lines close to the wall describe concentric circles. In the center of the tank, the trajectories form a closed loop, the shape of which is comparable to a figure eight, a form that is observed in the literature, especially by Tanguy et al. (1994) and Bertrand et al. (1996). When the Reynolds number increases (Re = 50) and the fluid is Newtonian, the shape of the central trajectory changes and looks more like an oval inclined in the rotation direction. When Od increases, this effect becomes increasingly less visible. Around the impeller, with low inertia, the trajectories circumvent the obstacle. When inertia is increased, a vortex is created downstream of the anchor. The presence of the wall prevents a second symmetrical vortex from forming as would be the case in an infinite medium (Dennis et al., 1993; Savreux et al., 2005). A zoom around the impeller, when Re = 50, shows the variation in stream lines when Od increases. As soon as plasticity effects are present, the vortex disappears and the stream lines tend to become symmetrical. To obtain a new vortex, the inertia will have to be considerably increased. In the mixing process, it is advantageous to know whether there is a vortex or not. The map of Re versus Od, Fig. 4, defines the zones with and without any vortex. Here again, note that when yield stress increases, more inertia is needed to obtain a vortex. In other words, for all the values of Re studied, a low yield stress value makes vortices disappear. In the same figure, the no-vortex/vortex limit has been drawn for flow around a plate in an infinite medium (Savreux et al., 2005). In the case of a mixer, this limit is pushed towards high Re values because the fluid is confined, thus delaying the appearance of the vortex.

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Fig. 3. Stream lines in a section of tank and around the impeller.

0.5

0.4

0.3 Od

No-vortex

0.2

Fig. 5. Typical rigid zones for Od = 15, Re = 0.01.

0.1

4.2. Rigid zones

Vortex

0 40

60

80

100

Re Fig. 4. Map of vortex occurrence. The dotted line represents the no-vortex/ vortex limit in the case of flow around a plate in an infinite medium.

When a viscoplastic fluid is agitated with an impeller, various unyielded zones can be observed. Three main zones can be distinguished (Fig. 5): a moving rigid core, a peripheral rigid zone and some static rigid zones located on both sides of the impeller. Mixing is very difficult in these zones: in the rigid static zones, the fluid is never mixed, and in the moving zones, the fluid moves like a solid. The location and shape of these

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Fig. 6. Influence of Re and Od values on unyielded zones.

28

24

Unyielded Surface %

rigid zones, according to the dimensionless parameters controlling the flow (Re and Od), are described in Fig. 6. When the Oldroyd number increases, the area of the unyielded zones increases drastically (Figs. 6 and 7). Two solid cores appear on either side of the tank center. They are detected even for low values of Od. These zones rotate in a block. As can be seen in Fig. 8, their rotation speed increases with Od and decreases with Re. Their size increases considerably with Od and the length separating them decreases (Fig. 6). The angle between the rotation center of these solid cores and the horizontal axis increases with Re. It can also be noted that the two parts of the peripheral rigid zone also increase drastically with Od. In some specific configurations (see Fig.13 for example) or for high Od numbers, these two zones can join together to form a continuous annulus. This configuration is certainly very bad for mixing. Qualitatively, the Reynolds number effects are less evident. The overall area of unyielded zones (Fig. 7) depends on the Od number: a decrease in area is observed with increase in Re for high values of Od, but it can also be seen that this area increases for values of Re = 10 and Od1. This can be explained physically: for high values of Od, the rigid zones are large; Inertia creates shear and thus tends to decrease their size. For very small values of Od(Od = 0.1), the rigid zones are very small. When Re increases, the wake behind the blades also increases, and thus the size of low stress

20

16

12

Od=0.1 Od=0.5 Od=1

8

Od=5 Od=10

4

10

20

30

40

50

Re Fig. 7. Influence of Re and Od on unyielded zone area.

zones where the rigid zones can develop. As the Re values continue to increase (Re = 30), a decrease in the size of the rigid zones is observed as a result of an increase in shear. For

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0.35 Ld+ Re=0.0 Ld+ Re=50 Lu+ Re=0.01 Lu+ Re=50

0.3

0.25 0.2 Ld Lu

0.2 Ω

Od=1 Od=5

0.15

Od=10 0.1

0.05

0.1 0.01

0 0.1

1 Re

0

10

Fig. 8. Influence of Re and Od on solid core rotation velocity.

2

4

6

8

10

Od Fig. 10. Dimensionless lengths of rigid static zones vs. Od number for two Reynolds number values.

4.3. Power consumption The variation in the power consumption is represented using the dimensionless number Np, called the power number. With our notations, this number is written as Np = Fig. 9. Influence of Re on rigid static zones for Od = 10.

intermediate values of Od (Od = 0.5 and 1), a decrease in these zones is observed by shear effect until dimensions become very small, followed by an increase by the wake effect, as noted for Od = 0.1. By increasing Re even more, there would certainly be a reduction in dimensions as for the previous case. For the rigid dead zones only, the observed changes are simpler: regardless of the Od number, the size of the downstream rigid dead zone increases with Re whereas the upstream zone becomes smaller as noted in Figs. 9 and 10. The increase in rotation velocity, i.e., the Reynolds number, is not then sufficient to create an significant reduction in the unyielded zones. For this, other solutions must be found. This aspect will be discussed in Section 4.4. Other minor rigid zones can be found in the tank and around the impeller. For the case Od = 0.1, Re = 50 there is a rigid zone (Fig. 6) near the center of the vortex which is an area of low stress. Near the blade, some rigid zones are detected (Od = 10; Re = 0.01) located either side of the impeller edge. These zones are similar in appearance to the “Island” observed by Zisis and Mitsoulis (2002) and by Deglo de Besses et al. (2003) in other configurations.

P , N 3 l 5

(9)

where P is the power consumption per unit length. Fig. 11 represents the variation in Np × Re versus Re. For Od = 0 (Newtonian case), three modes are identified and can be connected to kinematics. For Re < 1, there is no variation. The stream lines circumvent the obstacle without detachment. Then, up to a value of approximately Re = 10, the curve has a gentle slope, representing separation of the stream line, and leading to an increase in energy consumption. For higher Re values, a vortex appears. In this case, a steeper slope can be observed, indicative of an increase in power due to the presence of the vortex. These three modes gradually disappear as Od increases. For Od = 10, Np × Re does not seem to depend on Re any longer. This result is in agreement with the fact that the introduction of plasticity prevents the formation of vortices. The power consumption increases slightly with Re but very strongly with Od, as shown in Fig. 11. This power is proportional to the drag coefficient of the blades for a given value of the rotation velocity. In the absence of plasticity, it has been found that Np varies proportionally to Re−1 , a law which can be related to the change in a cylinder or a plate drag coefficient. Hence, the product Np × Re remains close to a constant  when Re varies (Fig. 11). The results obtained for Od = 0 by Tanguy et al. (1996) are also plotted in Fig. 11 for different Reynolds

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1.5

1.4

Od=0. Newtonian case. Np.Rep/α

Np.Re

Od=0.1 106

Od=10 Tanguy et al.(1996). Newtonian case.

1.3

Re=0.01 Re=1

1.2

Re=50

1.1

1

105 0.01

0.1

1

0.01

10

numbers. There is rough agreement with the results of this study although the work of Tanguy et al. (1996) was in three dimensions and for a slightly different impeller. When plasticity is introduced, the Reynolds number must be changed. Purely viscous forces K(V1 / l) must be changed by adding typical viscous forces and a fraction of the yield value 0 . This value of  is unknown but can easily be calculated with our values. The variation in Np × Re according to Od is close to a linear relation of the type (10)

Thus, by modifying the Reynolds number in Re . 1 +  Od

10

Fig. 12. Change in Np × Rep vs. Od.

Fig. 11. Change in Np × Re vs. Re.

Rep =

1 Od

Re

Np × Re = (1 +  Od).

0.1

(11)

Np × Rep / can be plotted as a function of Od (Fig. 12). For Re values less than or equal to 10, a good approximation to the relationship mentioned above is observed. In this range of Re,  = 1.9 × 105 and  = 1.53. For higher Reynolds numbers this relation is no longer linear because there is an interaction between the tank wall and the impeller and as a result of the creation of a vortex. In this case, the Od number must be increased to obtain a linear relation once again. 4.4. Improvement of the agitation system One of the aim of this work is to reduce the size of unyielded zones, in particular the rigid moving cores, obviously without too drastic an increase in power consumption. Section 4.2 showed that the increase in inertia was insufficient to reduce these zones. Another possibility is to change the impeller configuration. The three configurations shown in Fig. 13 were

tested: the first consisted of the previous impeller with a rotating axis, while for the second, two aligned blades were added and for the last, two perpendicular blades. The choice of distance was guided by the knowledge of the shear zone length around a blade in an infinite medium (Savreux et al., 2005). The effect on the rigid zones is presented in this same figure for Re = 0.01. The numerical values correspond to the percentage of the rigid zone area (ZR) and to the power consumption (P ). For Od=1 and the tank with four aligned blades, the cores are lengthened, less high and they become almost crescentshaped. Between the two blades, two small satellite islands are formed at the same height as the cores. It can be noticed that the existence of these small islands is not totally verified and has been predicted but also refuted by different authors (Liu et al., 2003; Zisis and Mitsoulis, 2002; Roquet and Saramito, 2003). For the impeller with four perpendicular blades, the cores are cut into two parts on either side of the blade. Around the two new blades, a rigid zone was formed. It appears that the four perpendicular blade case is the most effective configuration for reducing the rigid zones. Power consumption is obviously not so high for the four perpendicular blades as for the four aligned blades. However, the variation is not very large (approximately 8%). For Od = 10, the tendency is not the same. The four aligned blades seem more suitable for reducing the rigid zones. However, the main change is the peripheral rigid zone which goes all round the tank, without interruption, and this is not good for mixing efficiency. For the four perpendicular blades, the satellite cores and small islands become larger. The rigid cores located on either side of the blade also grew bigger. The peripheral zone is discontinuous opposite the blades. This calculation was also made for a higher Re number (Re =10). The same variations were noted. Many other configurations have been tested without any success (Savreux, 2004).

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Fig. 13. Variation in the unyielded zones for the three configurations. ZR= Rigid zone area/tank area. P = power consumed.

This reduction now has to be checked to see whether it provides satisfactory agitation. One possible checking method is particle tracking. The following part of this section describes this method. 5. Particle tracking and quality of mixing

Fig. 14. Variation in unyielded zones when the blades scrape the wall. Re = 0.01. Od = 10. ZR= Rigid zone area/tank area. P = power consumed.

For example, a configuration (Fig. 14) in which the blades scrape the tank wall in an attempt to reduce the peripheral rigid zone was calculated. But, as can be observed, this drastically increases the area of the unyielded zones as well as the power consumption. Consequently, the best configuration for reducing the rigid zones would appear to be that with two perpendicular blades. In this case, the rigid zones are much smaller and power consumption remains of the same order of magnitude as for the initial tank.

One way of determining the quality of agitation and mixing is to quantify the capacity of the flow to deform the matter and to generate interfaces. To quantify this change, infinitely small elongations of vectors can be evaluated in a large number of points dispersed throughout the field. The movement of these particles in the flow is reflected in changes in vector elongation. The elongation and rate of elongation vary point by point in the flow. This method can be used to evaluate the quality of mixing by a simple visual approach by finding, for example, an area of the fluid field depleted in particles. To obtain a more quantitative approach, Ottino (1989), Fan et al. (2000), the rate of elongation  can be calculated for each particle, or a more overall variation can be determined by taking the average in the entire tank defined by log  =

N 1  log i , N

(12)

1

where N is the total number of particles and I the rate of elongation of particle “i”. In our case, as the fluid is mixed by two

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Fig. 15. Particle tracking. Influence of the Od number.

separate blades, the estimated value of  is fairly representative of the quality of mixing. Fig. 15 shows the change in 4000 particles initially distributed homogeneously, at the end of 5, 20 and 45 revolutions for different Od numbers and for Re = 0.01. The color of the particles indicates the log  value. There is a significant difference between the Newtonian case (Od = 0, 45 revolutions) and the viscoplastic case (Od = 10, 45 revolutions). For the latter, the particles located in the rigid zones are not deformed very much and are then colored green or blue corresponding to a small log  value and it is possible to find the shape of these rigid zones again (Fig. 6). The more deformed particles are situated in the wake of the impeller. Still in the viscoplastic case, a crown with a low density of particles can be found. All these phenomena, very detrimental to mixing quality, do not exist in the Newtonian case. Mixing quality is therefore significantly modified by the plastic effect.

To compare the efficiency of the different configurations described above (Section 4.4), 2000 particles were introduced in the tank along a line (Fig. 16*). The change in these particles is shown at the end of 5, 10 and 20 revolutions for Od = 10 and Re = 0.01 and for the three previous configurations. Regardless of the studied geometry, the change in particles is very different from one agitator to another, for a fixed number of revolutions. Particle distribution is better for the case where the two additional blades are located in the rigid cores. Note also that the peripheral zone turns with the tank. The particles entering this zone are not subject to deformation. Therefore, in order to have good mixing, this peripheral zone must be “open” so that particles can leave and come into contact with the blades. In a more quantitative way, Fig. 17 represents the variation in log  according to the number of revolutions. The four aligned blades have the smallest value of log  in spite of

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Fig. 16. Particle tracking for the three configurations. Re = 0.01; Od = 10.

a smaller rigid zone area compared to the four perpendicular blade case. Consequently, the reduction in the static rigid zones is not the only criterion involved in obtaining good agitation quality. It is preferable to cut the rigid areas rather than to reduce them. This graph confirms that the case with two perpendicular blades leads to better quality agitation. The same calculations were made with Re = 10 and practically the same variations were found. 6. Conclusion The aim of this paper was to investigate numerically the detailed flow morphology and the 2D mixing of a viscoplastic fluid in a mixer consisting of a rotating tank and a static anchor impeller. The aim was also to propose and analyze different impeller configurations for increasing mixing quality without a drastic increase in power consumption. First, a complex flow structure was shown to occur in the tank with

different unyielded zones which can be static or in movement. For high Reynolds numbers, some vortices are observed which disappear as soon as a yield stress is introduced. The rigid zones are strongly dependent on the Od number and can occupy a large part of the tank for high Od numbers. To reduce these zones, it was shown that increasing the Reynolds number is not sufficient. Consequently, two other impeller configurations were proposed. For each configuration, the power consumption, the rigid zone area and the mixing quality obtained by particle tracking were calculated. The mixing quality is significantly affected by the viscoplasticity. Finally, it seems that the configuration with two perpendicular blades was the best for agitating viscoplastic fluids. Moreover, it was noted that the reduction in rigid zone size is not the only criterion involved in improving mixing quality. It is preferable to obtain discontinuous rigid zones so that the particles can migrate outside the rigid zones and mix with other zones.

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References 3.5 Od=10 3

Re=0.01



2.5

2

1.5

1

4 perpendicular blades Initial configuration

0.5

4 aligned blades

0 0

5

10 Number of revolutions

15

20

Fig. 17. Estimation of mixing quality for the three configurations. Od = 10; Re = 0.01.

Notation Dij e l Ld Lu m N Np Od P p R1 R Re Rep tr u V1 ZR

rate of strain tensor, s−1 impeller thickness, m larger side of the impeller, m downstream length, m upstream length, m Papanastasiou index, s rotation speed, tr s−1 power number Oldroyd number power per unit length, W m−1 pressure, Pa center of impeller at center of vessel, m vessel radius, m Reynolds number modified Reynolds number trace velocity vector, m s−1 velocity at the center of the impeller, m s−1 percentage of rigid zones = rigid zone area/area of tank

Greek letters ˙ K   ij II 0

shear rate, s−1 viscosity, Pa s rate of elongation density, Kg m−3 stress tensor, Pa second invariant of the stress tensor yield stress, Pa

Subscript +

dimensionless value

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