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Energy efﬁciency of different impellers in stirred tank reactors Houari Ameur* ^ma, 45000, Algeria Institute of Science and Technology, University Center Ahmed Salhi, Naa

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 February 2015 Received in revised form 19 October 2015 Accepted 21 October 2015 Available online xxx

The ﬂow energy efﬁciency of different impellers for stirring rheologically complex ﬂuids (yield stress ﬂuids) in cylindrical tanks has been investigated in this paper. Four impellers have been used: a Maxblend, an anchor, a gate and a double helical ribbon impeller. Our investigations were achieved via numerical simulations with the help of a CFD (computational ﬂuid dynamics) computer program CFX 13.0. It was found from the predicted results that the Maxblend impeller gives the best performance. The effects of some design parameters on the ﬂow energy efﬁciency and the power consumption have been also studied; it concerns the Maxblend impeller: its grid size and its paddle curvature. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Energy efﬁciency Maxblend impeller Yield stress ﬂuid CFD simulations Stirred tank

1. Introduction The efﬁcient mixing of liquids is very important in most industrial technologies. When designing a stirred system it is desirable to provide the required transport characteristics (i.e. mixing intensity, heat and mass transfer, gas hold-up, etc.) at the lowest possible energy consumption. The performance of such systems depends on many variables, such as the geometry of the equipment, physical properties of the liquid and operating conditions [1]. With the development of computer performance in the last decades, many researchers have turned to CFD (computational ﬂuid dynamics) for several applications. Yu et al. [2] described, with CFD simulations, the suspension and settling in the bio-waste particles ﬂow in a bioreactor. Baigmohammadi et al. [3] studied numerically the behavior of methane-hydrogen/air pre-mixed ﬂame in a micro reactor. Benajes et al. [4] studied the exhaust gas recirculation and miller cycle strategies for mixing-controlled low temperature combustion in a diesel engine. Hashimoto and Shirai [5] performed numerical simulations of mixed combustion of bituminous and sub-bituminous coals. Sharma et al. [6] predicted syngas composition in a continuous stirred-tank reactor by using extents of major reaction. Zhang et al. [7] studied the mixing phenomena in a fermenter of starch to bio-ethanol. Isfahani et al. [8] studied the effect of micromixing on the performance of a membrane-based

* Tel.: þ213 770343722. E-mail address: [email protected] http://dx.doi.org/10.1016/j.energy.2015.10.084 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

absorber. El-Askary et al. [9] investigated the hydrodynamics characteristics of the hydrogen-generation process through electrolysis. Jin et al. [10] explored the ﬂuid ﬂow and heat transfer characteristics in a solar air heater duct having multi V-shaped ribs on the absorber plate. Modlinski [11] performed numerical simulations of ﬂow with combustion in a coal-ﬁred grate boiler. Ma et al. [12] studied the feasibility of using longitudinal vortex generators to enhance the heat transfer in a thermoelectric power generator. Ye et al. [13] explored numerically the effects of the blade tip grooving on the efﬁciency on an axial ﬂow fan. For stirred tank reactors, Achouri et al. [14] simulated numerically the ﬂuid ﬂows in a in a vessel stirred by a PBT (Pitched Blade Turbine). For a Newtonian ﬂuid, Ammar et al. [15] achieved a numerical investigation of turbulent ﬂows generated in bafﬂed stirred vessels equipped with three different turbines in one and twostage systems. Aided by CFD simulations, Ameur and Bouzit [16] developed a new correlation for predicting the power required for stirring shear thinning ﬂuids by two-blade impellers in cylindrical tanks. For Newtonian ﬂuids, Rao and Sivashanmugam [17] presented experimental and simulation investigations on the power consumption for a new energy saving turbine agitator. These authors performed modiﬁcations by introducing single and double rectangular and V cuts in the conventional turbine agitator. Bao et al. [18] established correlations for calculate the mixing time and the power consumption for mixing CMC (Carboxy Methyl Cellulose) solutions by four coaxial mixers. They combined with either Pfaudler or CBY turbine as the inner stirrer, and helical ribbon or anchor as the outer one. Liu et al. [19] studied the power

H. Ameur / Energy 93 (2015) 1980e1988

consumption of a coaxial mixer formed by an outer wall-scraping frame and double inner turbines. Their predicted results indicated that the power consumption of the outer frame increases in counter-rotation mode and decreases in co-rotation mode. A new impeller design called Maxblend (see Fig. 2), is one of the most promising new generation wide impellers due to its good mixing performance, lower dissipation rates and simple geometry. However, only few studies have been performed and reported in the literature, especially for the agitation of shear thinning ﬂuids with yields stress. The majority of non-Newtonian ﬂuids are naturally opaque; therefore, visualizing the ﬂow ﬁelds of these ﬂuids inside a mixing vessel is a challenging task. Patel et al. [20] interested to the Rushton turbine and the Maxblend impeller and demonstrated an efﬁcient method to visualize the non ideal ﬂows such as dead zones in stirred tank reactors. Fontaine et al. [21] studied with experiments the ﬂow dynamics of Newtonian, shear thinning and viscoelastic ﬂuids in a Maxblend impeller system. They reported that elasticity in the laminar regime produces a reversal ﬂow and a solid body rotation in the bottom region of the tank. With Newtonian ﬂuids, Iranshahi et al. [22] investigated experimentally and numerically the ﬂow and mixing in a vessel equipped with a Maxblend impeller in the laminar and transition regimes for bafﬂed and unbafﬂed conﬁgurations. Ameur et al. [23] performed 3D numerical simulations of a Maxblend impeller system with viscoplastic ﬂuids possessing yield stress. They focused on the effect of ﬂuid rheology, impeller rotational speed, impeller clearance from the tank bottom and impeller blade size on the ﬂuid ﬂow and power consumption. Devals et al. [24] performed a CFD characterization of the hydrodynamics of the Maxblend impeller with viscous Newtonian and non-Newtonian inelastic ﬂuids. They found that the bottom clearance plays a signiﬁcant role on the power consumption, and that the value of the Reynolds number and the power law index strongly affect the axial pumping efﬁciency and the shear rate proﬁle. The best performance was obtained when the impeller Reynolds number is superior to 10. Physical investigations have been carried out by Fradette et al. [25] to characterize the power consumption and the mixing time in a vessel equipped with a Maxblend impeller. These authors investigated laminar, transitional and turbulent ﬂow regimes using viscous Newtonian and non-Newtonian shear-thinning ﬂuids. Based on data from the literature, a comparison with many

1981

Fig. 2. Characteristics of the stirred system.

different large impellers indicated that the Maxblend technology makes a better use of the mixing power in the upper part of the laminar regime and in the low turbulent regime. The power consumption was found almost identical as that of an anchor but far lower than that of a double helical ribbon impeller. Aided by numerical simulations, Iranshahi et al. [26] investigated and compared the mixing characteristics of the Ekato Paravisc with those of an anchor and a double helical ribbon. For Newtonian and shear thinning ﬂuids, they found that the Paravisc mixer characteristics lie between that of the other impellers at low Reynolds number. For shear thinning ﬂuids possessing yield stress, Patel et al. [27] showed by experiments that the Maxblend impeller is more efﬁcient than the axial-ﬂow impeller (Lightnin A320), radial-ﬂow impeller (Scaba 6SRGT), and close-clearance impeller (anchor) in terms of reducing the extent of dead zones inside the mixing vessel with a lower power consumption. Stobiac et al. [28] investigated the accuracy of the extrapolation method for the lattice Boltzmann simulation of Newtonian ﬂuid

Fig. 1. Geometries of all impellers studied.

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H. Ameur / Energy 93 (2015) 1980e1988

ﬂows in a Maxblend impeller system. Their results showed that the standard bottom clearance is not optimum in the transitional regime. Stobiac et al. [29] examined numerically the pumping mechanisms generated by the Maxblend impeller for Newtonian ﬂuids (Re ¼ 2 e 140) and strongly shear-thinning ﬂuids (Reg ¼ 0.1 e 50). Their results indicate a small pumping capacity in the deep laminar regime followed by its sharp increase in the transitional regime. In the case of the strongly shear thinning ﬂuids, the ﬂow ﬁelds and socalled pumping volumes revealed, similarly to the Newtonian case, a change in the structure of the axial and (secondary) radial ﬂow when the Reynolds number is increased. Guntzburger et al. [30] introduced a new experimental method to determine the global pumping capacity of four impellers, namely: a Maxblend impeller, a RT (Rushton turbine), a three-blade HP (hydrofoil propeller) and a four PBT (pitched blade turbine). Their results showed that the Maxblend impeller performs better than the other three turbines in the transitional regime, and it has a similar pumping capacity than that of the RT and the PBT in the turbulent regime. Liu et al. [31] investigated the micro mixing characteristics of a novel LDB (Large-Double-Blade) impeller based on Maxblend and FZ (Fullzone) impellers. They compared the performance of LDB impeller with that of FZ and DHR (double helicon ribbon) impellers. The LDB stirrer is found to be the most efﬁcient for the same power consumption per unit volume. It is often advantageous to increase ﬂuid ﬂow velocities at the tank walls or in the bulk ﬂow in order to improve the heat transfer, to reduce the rate at which scale grows at the tank wall surface, and to increase the metal extraction rate [32]. The best challenge is to increase ﬂuid ﬂow velocities within constraints of the available power input, as it is expensive to alter larger scale motors to obtain an additional power. Therefore, it seems important to improve the impeller ﬂow energy efﬁciency. From our search in the literature, it was found that most studies focused on the ﬂow structure, the mixing time and turbulence parameters, with less attention to the ﬂow energy efﬁciency. Therefore, the main purpose of this paper is to study the ﬂow energy efﬁciency of different impellers for stirring shear thinning ﬂuids possessing yield stress. The impellers considered are: a Maxblend, an anchor, a gate and a DHR (double helical ribbon). Some design parameters of the Maxblend impeller are also investigated. The four impellers studied belong to the same kind, is that of the close clearance impellers. These stirrers are widely used for mixing highly viscous and non-Newtonian ﬂuids. But, which is best mixer? 2. Stirred system The mixing system used in this paper is a ﬂat-bottomed cylindrical unbafﬂed vessel equipped with a rotating impeller. Four impellers are used: a Maxblend (Fig. 1a), an anchor (Fig. 1b), a gate (Fig. 1c) and a double helical ribbon impeller (Fig. 1d). We note that the following parameters are taken with all geometric conﬁgurations: the vessel diameter D ¼ 130 mm, the vessel height H/D ¼ 1, the impeller clearance from the tank bottom c/ D ¼ 0.2, the impeller blade height h/D ¼ 0.6, impeller blade diameter d/D ¼ 0.6, the shaft diameter ds/D ¼ 0.4. For the Maxblend impeller (Fig. 2), the paddle height is h1/D ¼ 0.23 and the grid height is h2/D ¼ 0.38. In the purpose to examine the effect of impeller design, four geometrical conﬁgurations were realized, it concerns the Maxblend grid size: d1/D ¼ 0.30, 0.45, and 0.60, respectively. The design of the Maxblend paddle has also been modiﬁed by realizing three shapes as will be shown later (on Fig. 2).

3. Theoretical considerations Shear thinning ﬂuids with yield stress were modeled in this work, the Xanthan gum solution was considered. Rheological properties of the materiel simulated are summarized on Table 1, which are based on measurements conducted by Galindo and Nienow [33]. It's rheology can be described by the Herchel Bulkkley model [34]: :n

t ¼ ty þ Kg

(1)

where ty is the yield stress, K is the consistency index, g_ is the shear rate and n is the ﬂow behavior index. According to the Metzner and Otto's correlation [35], the average shear rate can be related to the impeller speed by: :

gavg ¼ Ks N

(2)

The average shear rate can be used to evaluate the apparent viscosity (h) of the solution, which is a HerscheleBulkley ﬂuid.

h¼

:

t

gavg

¼

ty þ KðKs NÞn t ¼ ks N Ks N

(3)

The Reynolds number can be given as:

Rey ¼

Ks N 2 D2 r ty þ KðKs NÞn

(4)

The impeller pumping ﬂow rate could be expressed as (refer to Wu et al. [36]):

NQ

Q¼

!

1=3

P0

P 4 D r

1=3 (5)

where Q is the ﬂow rate, r the ﬂuid density and P the power consumption. The power number Np is determined according to this equation:

NP ¼

P rN 3 D5

(6)

The ﬂow number NQ is calculated as:

NQ ¼

Q ND3

(7)

The factor NQ/P1/3 0 is deﬁned as the impeller ﬂow efﬁciency index [32]: Based on Eq. (5), the averaged velocity at impeller exit can be calculated as:

V¼

! 1=3 4 NQ 2=3 P 4 D D p P 1=3 r 0

(8)

To estimate the velocities at the tank walls at a given power input, we may normalize Eq. (8), to obtain a non-dimensional efﬁciency coefﬁcient h: Table 1 Rheological properties of Xanthan gum solution. Concentration (%)

K [Pa sn]

n [e]

ty [Pa]

3.5

33.1

0.18

20.6

H. Ameur / Energy 93 (2015) 1980e1988

NQ V h ¼ 1=3 ¼ j 1=3 2 P0 P rD

(9)

where j is a non-dimensional parameter dependent on impeller and tank diameters and locations of the impeller(s). The signiﬁcance of h is that the increase of h yields higher ﬂow velocity per unit power input. Thus, the energy efﬁciency of various impeller designs for a given vessel size should be compared in terms of this coefﬁcient (h) [32].

1983

Np

Exp [27] Num [Present work]

100

10

4. Numerical simulation Simulations were performed by using the computer code (CFX 13.0) which is based on the ﬁnite volume method to solve the equations of momentum and energy. Since the stirred tank is not provided with bafﬂes, a RRF (rotating reference frame) approach was used. Here, the impeller is kept stationary and the ﬂow is steady relative to the rotating frame, while the outer wall of the vessel is given an angular velocity equal and opposite to the velocity of the rotating frame. This technique has been used by other researchers [37,38] for different mixing systems and accurate results were found. For solving the equations of momentum and to perform pressureevelocity coupling, a pressure-correction method of the type SIMPLEC (Semi Implicit Method for Pressure Linked Equations Consistent) is used. Geometries of the mixing systems were realized with the help of the preprocessor (Ansys ICEM CFD 13.0). The ﬂow domain was divided into tetrahedral meshes. When creating meshes, an increased mesh density was performed near the impeller and the tank walls to capture the ﬂow details. Mesh tests were performed by checking that the numerical results were grid independent. Mesh density was increased until the additional cells did not change the energy efﬁciency and the power consumption in regions with high gradients by more than 2.5%. Calculations were run in a platform with Core i7 CPU 2.20 GHz with 8.0 GB of RAM. Numerical results were considered converged when the residual target drop below 107. Most simulations required 2500e3000 iterations and 5e6 h for convergence. 5. Results and discussion First, we have checked the validity of our numerical results by a comparison with the experimental data given by Patel et al. [27].

Np

1 1

100

Rey

Fig. 4. Power number for the anchor impeller.

Both numerical and experimental results for the power number are depicted on Figs. 3 and 4; the comparison shows agreement. 5.1. Flow energy efﬁciency of the four impellers The ability of an impeller to convert its power input to the ﬂuid ﬂow is referred to as its hydraulic efﬁciency h. The higher this quantity, the higher the energy transferred by the impeller to the agitated material, resulting in higher intensity of the ﬂow [39]. As a comparison, the energy efﬁciency of four impellers operating in viscoplastic ﬂuids with yield stress is considered. For a Reynolds number Rey ¼ 100 and an angular position q ¼ 90 (i.e. the plane orthogonal to the impeller), simulation results of the ﬂow energy efﬁciency along the tank wall surface are depicted on Fig. 5. We note that the origin of the angular position (q ¼ 0 ) is taken at the prolongation of the blade for each impeller. We remark from this ﬁgure that the ﬂow energy efﬁciency proﬁles follow the same trend for the anchor, the gate and the Maxblend impellers. However, the maximum is reached with the Maxblend agitator. For the double helical ribbon impeller (DHR), we remark that the maximum of h is reached in the area swept by the blade arms but with small amplitude when compared with the Maxblend. On Fig. 6 we present the variations of the coefﬁcient h along the vessel radius for two locations: (a) Z* ¼ Z/D ¼ 0.075 near the vessel base, (b) Z* ¼ 0.83 near the free surface of liquid. For more visibility, we present on Figs. 7 and 8 the spatial distribution of the ﬂow energy efﬁciency on horizontal cross section planes for different vertical positions.

Exp [27] Num [Present work]

100

10

Z

*

Anchor impeller Gate impeller Helical ribbon impeller Maxblend impeller

1.0 0.8 0.6

10

0.4 0.2 0.0

1 1

10

Fig. 3. Power number for the Maxblend impeller.

100

Rey

0.00

0.01

0.02

0.03

0.04

0.05

η

Fig. 5. Flow energy efﬁciency near the vertical vessel wall, Rey ¼ 100, q ¼ 90 .

1984

H. Ameur / Energy 93 (2015) 1980e1988

η

η

Anchor impeller Gate impeller Helical ribbon impeller Maxblend impeller

0.12 0.11 0.10

Anchor impeller Gate impeller Helical ribbon impeller Maxblend impeller

0.18 0.16

0.09

0.14

0.08

0.12

0.07 0.06

0.10

0.05

0.08

0.04

0.06

0.03 0.02

0.04

0.01

0.02

0.00

0.00

-0.01 0.0

0.2

0.4

0.6

0.8

1.0

*

R

0.0

0.2

(a) Z* = 0.075

0.4

0.6

0.8

1.0

R

*

(b) Z* = 0.83

Fig. 6. Flow energy efﬁciency, Rey ¼ 100, q ¼ 90 .

Fig. 7. Contours of the ﬂow energy efﬁciency for Rey ¼ 100, at Z* ¼ 0.3.

The conclusions drawn from the analysis of these ﬁgures are as follows: In the whole vessel volume, the anchor impeller is the less efﬁcient impeller (Figs. 6e8). Adding a vertical arm to this impeller for obtaining a gate can improve mixing in the lower part of the vessel, but it seems insufﬁcient for the upper part of the vessel (Fig. 8). The anchor and gate impellers create tangential ﬂows, as a result, a vortex is formed at the free surface of liquid and the ﬂuid begins rotating with the stirrer rather than being pumped. At higher stirrer rotational speeds, the vortex increases in size and the rotational movement entrains the ﬂuid in the whole vessel volume, consequently, the pumping effect of the stirrer is reduced. The paddle design of the Maxblend impeller seems the best efﬁcient in the lower vessel volume (Fig. 8). The shape of the paddle creates large pressure variations inside the vessel and it can be considered responsible for the efﬁcient ﬂow motion, as reported by

Fradette et al. [25]. The front side of the paddle creates an overpressure near the vessel base and pushes the ﬂuid close to the wall in the upward direction. The ﬂuid at the upper part of the vessel is sucked by the low pressure created at the bottom of the tank; the ﬂuid goes downward along the impeller shaft. Mixing is achieved and enhanced by axial pumping. However, the grid mounted on the Maxblend cannot provide sufﬁcient homogenization energy near the free surface of liquid. The DHR impeller can remedy this issue as remarked on Fig. 8 and this is due to the size of the arm blade. Electric energy consumption is an important parameter for the stirred tank design. It is deﬁned as the amount of the electric energy necessary (in a period of time) to generate the ﬂuid circulation within a tank by means of mechanical agitation [40]. The costs associated with power drawn contribute signiﬁcantly to the overall operation costs of industrial plants. Therefore it is desirable to

Fig. 8. Spatial distribution of the ﬂow energy efﬁciency for Rey ¼ 100.

H. Ameur / Energy 93 (2015) 1980e1988

Np

η

Maxblend impeller Anchor impeller Gate impeller Double Helical ribbon impeller

14 13 12 11 10 9 8 7 6 5 4 3 2 1

1985

d /D = 0.30 d /D = 0.45 d /D = 0.60

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

10

20

30

40

50

60

Rey

-0.05 0.0

Fig. 9. Power number for vessel size ¼ 0.6.

0.2

0.4

0.6

0.8

1.0

1.2

*

R

1.4

Fig. 11. Flow energy efﬁciency at Z* ¼ 0.75, Rey ¼ 100, q ¼ 90 .

Table 2 Values of Np in the deep laminar regime.

Np

Re

Maxblend

Anchor

Gate

DHR

26

0.1 1 10

706.90 71.69 7.300

853.75 84.88 8.59

951.72 95.95 9.734

1304.73 130.92 13.13

25 24

perform efﬁciently the mixing process with a minimum expense of energy [41]. Here, we estimated the energy consumption for the four impellers. As shown on Fig. 9, the lower the Reynolds number, the greater the energy required to achieve the agitation operation with any kind of impellers. The energy required diminishes continuously with increasing Reynolds number. This reduction is strong in the deep laminar regime (Re < 10), however a fairly reduction is remarked in the transitional regime. The trend of the power number curve obtained in this study is similar to that reported in the literature [27]. It has been shown that the Np*Re is constant in the laminar regime and the power consumption (P) changes to some extent in the transitional regime. Fig. 3 and Table 2 show that at Re < 10, the power curve with a slope of 1 ﬁts the data, which implies that Np*Re is constant and the ﬂow is in the laminar regime. In the transitional regime, the power number changes slightly with Re. Fradette et al. [25] and Iranshahi et al. [26] reported that the laminar region ends for Re in the range between 35 and 40 for the Maxblend and anchor impellers. *

Z

1.4

23 22 21 20 19 0.30

0.35

0.40

0.45

0.50

0.55

0.60

d1/D

Fig. 12. Power number for Rey ¼ 10.

As a comparison, the Maxblend impeller seems to be the best efﬁcient in term of power consumption. Following a descending order in the power consumption for the four agitators: DHR > gate > anchor > Maxblend. The DHR impeller was found to be efﬁcient to improve mixing near the free surface of liquid but with more power drawn.

d /D = 0.30 d /D = 0.45 d /D = 0.60

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

η

Fig. 10. Flow energy efﬁciency near the vertical vessel wall, Rey ¼ 100, q ¼ 90 .

Fig. 13. The different geometrical conﬁgurations realized to examine the effect of the Maxblend paddle design.

1986

H. Ameur / Energy 93 (2015) 1980e1988

Z

mixing in this region. In this work, we tried to examine the effect of the impeller grid diameter. For this purpose, three geometric conﬁgurations are realized and which are: d1/D ¼ 0.30, 0.45 and 0.60, respectively. Variations of the circulation energy efﬁciency along the tank wall surface are followed and depicted on Fig. 10. As observed, the maximum ﬂow energy efﬁciency is h ¼ 0.037 for d1/D ¼ 0.60 compared to h ¼ 0.032 for d1/D ¼ 0.45 and h ¼ 0.029 for d1/D ¼ 0.30. The circulation energy efﬁciency proﬁles follow the same trend for the three geometric conﬁgurations; however this parameter (h) spreads more in the upper part of the vessel with increasing impeller grid size. More illustration is given on Fig. 11 where h is presented along the vessel radius for a vertical position of Z* ¼ 0.75 (area swept by the grid). It can be said from these results that, the higher the impeller grid size, the higher the energy efﬁciency with additional power drawn (Fig. 12).

*

Geometry A Geometry B Geometry C

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045

η

Fig. 14. Flow energy efﬁciency near the vertical vessel wall, Rey ¼ 100, q ¼ 90 .

η

Geometry A Geometry B Geometry C

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

*

R

Fig. 15. Flow energy efﬁciency at Z* ¼ 0.22, Rey ¼ 100, q ¼ 90 .

5.2. Effect of the Maxblend grid diameter In the case of the Maxblend impeller, we remarked that the ﬂow energy efﬁciency is pretty low at the top of the vessel, which corresponds to a quasi-solid body rotation. Devals et al. [24] reported that the distance between the upper part of the impeller and the liquid surface in the vessel must be adjusted if one wants to ensure

5.3. Effect of the Maxblend paddle design When mixing highly viscous non-Newtonian shear thinning ﬂuids with yield stress, velocities in the tank tend to decrease with an increase in the viscosity and the ﬂow behavior index. It is very important to increase ﬂuid ﬂow velocities through impeller modiﬁcations at a constant power input, where it is often not feasible to increase the motor power for vessels with large geometrical dimensions [42]. In this part of the paper, another modiﬁcation to the Maxblend impeller design is done. It concerns the impeller bottom shape. Three geometries are realized in the purpose of maximizing the ﬂow energy efﬁciency at the lower part of the vessel (see Fig. 13). Predicted results of the ﬂow energy efﬁciency are presented along the tank wall (Fig. 14) and along the vessel radius at a vertical position Z* ¼ 0.22 (near the vessel base) (Fig. 15). To give a different perspective on the type of deformations induced by the impeller, we show in Fig. 16 the spatial distribution of the ﬂow energy efﬁciency h at an horizontal cross section corresponding to Z* ¼ 0.18 of level. As clearly illustrated, the geometry denoted C seems to be the best design in term of maximizing the ﬂow energy efﬁciency near the vessel walls. However, an additional power is required to achieve the mixing operation in the laminar regime (Fig. 17). 6. Conclusion This paper provided a 3D numerical simulation of yield stress ﬂuid ﬂows generated by different impellers within cylindrical vessels. The ﬂow energy efﬁciency of four impellers is investigated,

Fig. 16. Spatial distribution of the circulation energy efﬁciency at Z* ¼ 0.18, Rey ¼ 100.

H. Ameur / Energy 93 (2015) 1980e1988

Np

Vz Vq Vr R R* Z Z*

27 26 25 24

22 A

B

C

Geometry

Fig. 17. Power number for Rey ¼ 10.

namely: a Maxblend, an anchor, a gate and a double helical ribbon impeller. Following a descending order in the power consumption for the four agitators: DHR > gate > anchor > Maxblend. The DHR impeller was found to be efﬁcient to improve mixing near the free surface of liquid but with more power drawn. As a comparison in term of circulation energy efﬁciency, it was found that the particular shape of the Maxblend impeller (paddle at the bottom surmounted by a grid) proved to be very efﬁcient for generating efﬁcient mixing at a low power consumption. The modiﬁcations performed to the Maxblend impeller design allowed to choose the geometry denoted C as the best design in the purpose of maximizing the ﬂow energy efﬁciency. To ensure great ﬂow energy efﬁciency, the design of the lower part of the impeller must follow the shape of the vessel base. For a low impeller rotational speed, a low clearance from the tank bottom is also needed. A better mixing quality and more energy savings can be achieved by considering these concluding remarks in the design of continuous-ﬂow mixing of shear thinning ﬂuids with yield stress. More studies on the subject may be very interesting, especially for viscoelastic ﬂuid ﬂows with the heat transfer phenomenon. Nomenclature

h

Rey V

axial velocity (m/s) tangential velocity (m/s) radial velocity (m/s) radial coordinate (m) dimensionless radial coordinate, R* ¼ 2R/D vertical coordinate (m) dimensionless vertical coordinate, Z* ¼ Z/D

Greek letters : average shear rate (1/s) gavg r ﬂuid density (kg/m3) t shear stress (Pa) ty suspension yield stress (Pa)

23

c d d1 d2 h1 h2 h1 n D ds H K Ks N P Np Q NQ

1987

impeller off-bottomed clearance (m) impeller diameter (m) Maxblend paddle diameter (m) Maxblend grid diameter (m) paddle height (m) grid height (m) impeller height (m) ﬂow behavior index (dimensionless) tank diameter (m) shaft diameter (m) vessel height (m) consistency index (Pa sn) MetznereOtto's constant (dimensionless) impeller rotational speed (1/s) power (W) power number (dimensionless) ﬂow rate (m3/s) ﬂow number (dimensionless) ﬂow energy efﬁciency (dimensionless) yield stress Reynolds number (dimensionless) velocity (m/s)

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