CFD simulation of

Editorial Manager(tm) for Central European Journal of Engineering Manuscript Draft Manuscript Number: Title: CFD simulation of the laminar flow in stirred tanks generated by double helical ribbons Short Title: and double helical ribbons screw impellers Article Type: Topical Issue: CDF Benchmark Section/Category: Mechanical Engineering Keywords: CFD, finite volume, modelling, laminar flow, Double helical ribbons, screw, stirred tank. Corresponding Author: Zied Driss, Ph.D. Corresponding Author's Institution: National Engineering School of Sfax First Author: Zied Driss, Ph.D. Order of Authors: Zied Driss, Ph.D.;Sarhan Karray, M.D.;Hedi Kchaou, Ph.D.;Mohamed Salah Abid, Ph.D. Abstract: In this paper, the mixing performance of double helical ribbons and double helical ribbons screw impellers mounted on stirred tanks is numerical investigated. The computer simulations are conducted within a specific computational fluid dynamic (CFD) code, based on resolution of the Naviers-Stokes equations in the laminar flow with a finite volume discretization. The field velocity and the viscous dissipation rate are presented in different vessel planes. The global characteristics and the power consumption of these impellers are also studied. The numerical results showed that the velocity field is more active with the double helical screw ribbons impeller. In this case, the effectiveness of the viscous dissipation and the pumping flow has been obviously noted. Also, the pumping and the energy efficiency reach the highest values at the same Reynolds number. The good agreement between the numerical results and the experimental data quietly confirmed the analysis method. Suggested Reviewers: Mohamed Haddar Ph.D. National School of Engineers of Sfax [email protected] Mohamed kharrat Ph.D IPEIS [email protected] Mohamed Sadok Guellouz ENIM [email protected] Opposed Reviewers: Mounir Baccar Ph.D National Engineering School of Sfax [email protected] Prof. Mounir Baccar belongs to our research group at National Engineering School of Sfax

*Manuscript Click here to download Manuscript: Manuscript.doc

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CFD simulation of the laminar flow in stirred tanks generated by double helical ribbons and double helical ribbons screw impellers

Zied Driss *, Sarhan Karray, Hedi Kchaou, Mohamed Salah Abid Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), B.P. 1173, km 3.5 Soukra, 3038 Sfax, TUNISIA

* Corresponding author. Tel.: + 216 74 274 409; Fax: + 216 74 275 595. E-mail address: [email protected], [email protected] (Z. Driss).

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Abstract In this paper, the mixing performance of double helical ribbons and double helical ribbons screw impellers mounted on stirred tanks is numerical investigated. The computer simulations are conducted within a specific computational fluid dynamic (CFD) code, based on resolution of the Naviers-Stokes equations in the laminar flow with a finite volume discretization. The field velocity and the viscous dissipation rate are presented in different vessel planes. The global characteristics and the power consumption of these impellers are also studied. The numerical results showed that the velocity field is more active with the double helical screw ribbons impeller. In this case, the effectiveness of the viscous dissipation and the pumping flow has been obviously noted. Also, the pumping and the energy efficiency reach the highest values at the same Reynolds number. The good agreement between the numerical results and the experimental data quietly confirmed the analysis method.

Key Words: CFD, finite volume, modelling, laminar flow, Double helical ribbons, screw, stirred tank.

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1 Introduction Viscous dispersive mixing is an important unit operation in polymerization, food and other industrial processes. To respond to the needs of industrial processes, various impellers have been developed and a number of studied of the various impeller characteristics have been reported; useful reviews have been given in Nagata [1]. Among the different impellers available, the helical ribbon and the screw impellers are considered to be more efficient for the agitation of highly viscous liquids. In the literature, many works have interested to these impellers. For example, EspinosaSolares et al. [2] have carried an ungassed power measurements in a dual coaxial mixer composed of an helical ribbon and a Rushton turbine in laminar mixing conditions for Newtonian and non Newtonian shear thinning fluids. For the Newtonian case, the power draw constant for the hybrid geometry was not the sum of the individual. This was explained by considering the radial discharge flow in the turbine region as well as the top-to-bottom circulation pattern of the helical ribbon impeller. For the non-Newtonian fluids, the results showed that, at a given Reynolds number, power consumption decreases as the shear thinning behaviour increases. Tanguy et al. [3] investigated the mixing performance of a new dual impeller mixer composed of a disc turbine and an helical ribbon impeller mounted on the same axis but rotating at different speeds. The methodology is based on a blend of experimental measurements and 3D numerical simulations in the case of Newtonian and non-Newtonian shear-thinning fluids. It is shown that the dual impeller mixer outperforms the standard helical ribbon in terms of top-to-bottom pumping when the fluid rheology evolves during the process. The power consumption of this mixer is also studied which allows to derive a generalized power curve. Bertrand et al. [4] elucidated the role of elasticity on power draw, though studies

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with helical ribbon impellers indicate that elasticity increases torque, in the case of viscoelastic fluids. The objective is to show that in the case of second-order fluids, the use of a simple constitutive equation derived from a second-order retarded-motion expansion succeeds in predicting a rise in power draw owing to elasticity. The equations of change governing fluid flow are solved using a finite element method combined with an augmented Lagrangian method for the treatment of the non-linear constitutive equation. They presented how the underlying non-linear tensor equations can be solved directly using a spectral decomposition of the related matrix operator. Kaneko et al. [5] analysed the three-dimensional motion of particles in a single helical ribbon agitator by the Discrete Element Method (DEM). To validate the computed results experiments were carried out with a cold scale model of 0.3 m inside diameter. Circulation time of particles in the agitator and the horizontal particle velocity distribution in the core region predicted by the simulation agreed well with those obtained by experiments. Based on DEM simulation, the particle circulation and mixing characteristics in the agitator vessel were investigated. Vertical mixing of particles was found rather poor during upward and downward flows through the blade and core regions, respectively. Yao et al. [6] analyzed the local and total dispersive mixing performance of large type impellers, a standard type of MAXBLEND and double helical ribbons impellers using two indices, local dispersive mixing efficiency and NPD function. The results indicated that a standard type of MAXBLEND has a satisfactory local dispersive mixing performance, especially in the grid region where the local dispersive mixing efficiency is high to near 1. However, when the Reynolds number Re is low, the total dispersive mixing performance is not as satisfactory as that operated under a moderate Reynolds number. The double helical ribbon impeller can not provide a promising local mixing

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performance. Although, it can induce a good total circulation throughout the stirring tank. Wang et al. [7] proposed a simple correlation to predict the Metzner-Otto constant. Through a comparison of different methods for predicting of Metzner-Otto constant from the viewpoint of numerical analysis, they introduced a new algorithm for estimating this constant. Niedzielska and Kuncewicz [8] examined the effect of impeller diameter modification and the pitch ratio on power consumption. In the course of experiments of power consumption, rotary frequence of impeller was changed and for this frequence value of moment of rotary was read. A change of liquid viscous was gained in result of water addition. Experiments were continued in range, in which liquid movement was laminar. In the present study, an attempt was made to develop a three-dimensional computational fluid dynamics (CFD) code to compare the laminar flow results of the double helical ribbons with the double helical ribbons screw in agitator vessel. In fact, a general trend of using CFD codes has evolved in these years [9-10]. To evaluate the accuracy of the numerical method, the calculated results were compared with those obtained by the experimental results of Nagata [1] in the same geometry and size.

2 Geometric Arrangement Figure 1 show the detailed of the stirred tanks equipped by double helical ribbons and double helical ribbons screw. The first impeller (figure 1.a) has the same characteristics defined by the Nagata application [1]. The second is an association of a double helical ribbons and a screw impeller (figure 1.b). These impellers dimensions are defined by b=0.1 d and h=d. The tip to tip impeller diameter ratio d/D is a 0.95. The shaft is placed concentrically with a diameter ratio s/D of 0.04. In this arrangement, D is the diameter

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of the cylindrical stirred tank having a plated bottom with a height-to-diameter ratio H/D of 1.

3 Model formulation

3.1 Mathematical modeling The simulation of the laminar flow field of the double helical ribbons and the double helical ribbons screw impellers in a stirred tank is governed by continuity and NavierStokes equations [11]. The continuity equation for incompressible fluids is given in the following form:  div V  0

(1)

Navier-Stokes equations are written in a rotating frame reference. Therefore, the centrifugal and the Coriolis accelerations terms are added. These equations, written in cylindrical coordinates (r,θ,z), are expressed in the general conservation form which can be written as follows:   2 U 2  d  1   +divV U   ηgradU t π D Re  

      2η  U + 1 V + 1  r ηU  2   2 2   p 2  d  1  r θ  r r  r   r =  +    r π D Re     V    W ηr    + η  +  r θ  r  r   z  r  

   2 + V + r+2V   r   

(2)

  1 u  V 1   U     η       r2 θ r  r  r r η θ V  2 2      V 2 d 1 p 2 d  1       UV2U +divV V   ηgradV         t π D Re rθ πD Re     V 2U    W  r        η    η  r θ  r θ r  z  r θ        (3)

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 1    U    V   r η  + η   2 2       z   1    r  r  z r  θ     W 2 d 1 p 2 d 1      + +div VW   ηgradW = +     Fr         t π D Re  z π D Re    W      η +      z   z  

(4)

The equations system is solved using the three-dimensional CFD code developed in our Laboratory [11-14]. The dimensional analysis enables us to characterize power consumption in a stirred tank through the power number Np [15-16] defined as follows: Np 

P ρ N3 d5

(5)

In case of the laminar flow, the total power consumption was calculated from the general relationship: P     v dv P = Vc  v dv

(6)

The viscous dissipation function v can be expressed in cylindrical coordinates in the following form:  U2  V U2 W2   V V U  2  W V 2  U W 2  v  2                         r  rθ r   z    r r rθ  rθ z   z r  

(7)

The pumping flow number was calculated using the following equation: NQp 

Qp N d3

(8)

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The numerical data of mean axial velocity below the impeller, in the radial direction were used for the calculation of pumping flow Qp: zB h

Qp 



d/2

π d  U rd / 2 dz 

zB



2 π r  W zz dr B

(9)

0

The pumping efficiency [17] can be defined by dividing the pumping flow number by the power number: Ep 

NQp Np

(10)

However, the energy efficiency can be obtained as follows: Ee 

NQp3 Np

(11)

3.2 Numerical method Our computational fluid dynamics (CFD) code is based on solving the continuity and the Navier-Stokes equations using a finite volume method. The transport equations are integrated over its own control volume using the hybrid scheme discretization method. The discretized equations were solved iteratively using the SIMPLE algorithm for pressure-velocity coupling [18]. The algebraic equation solutions are obtained in reference to the fundamental paper published by Douglas and Gunn [19]. The discretization method and numerical solution procedure used have been described in detail elsewhere [11-14]. To simplify calculation and to avoid re-meshing, a steady flow field on the rotating frame fixed on the impellers is adopted. In these conditions, a no-slip condition on the non-moving impeller and a rotational speed on the tank walls are considered. In order to

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take into account the presence of the impeller, all radial as well as tangential and axial velocity mesh nodes, which intersect with impeller, were taken equal to zero. However, at the internal wall tank we have to set the angular velocity component equal to the rotating speed because of the rotating frame. In the part of the boundary where the fluid leaves the computation domain, zero velocity gradients are assumed. For the meshing, we used the design software Solid-Works to construct the impeller shape. Then, we defined a list of nodes belonging the interfacing separating the solid domain of the flow domain. Using this list, the meshes in the flow domain are automatically generated for the three-dimensional simulations. Therefore, the region to be modeled is subdivided into a number of control volumes defined on a cylindrical coordinates system (r,θ,z). A staggered mesh is used in such a way that four different control volumes are defined for a given node point, one for each of the three vector components and one for the pressure. The flow field was computed using a grid size of NR=30, Nθ=60 and NZ=60. The solution was obtained when the total residuals for the equations dropped to below 10-6.

4 Numerical results Our computer simulations results, such as the flow patterns, the viscous dissipation rate and the evolution of the pumping flow number NQp, the pumping efficiency number Ep and the energy efficiency Ee the power number Np, offer local and global information about the laminar mixing within double helical ribbons and double helical ribbons screw impellers mounted on stirred tanks. They give a more precise understanding of the hydrodynamic mechanism than those obtained by experimental studies. The flow conditions are represented by the Reynolds number equal to Re=4 and the Froude

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number equal to Fr=0.19. These results obtained by our code are compared with the ones found by Nagata application [1].

4.1 Flow patterns 4.1.1 Flow patterns in r-θ plane Figure 2 shows a velocity vector plot of the primary flow (U,V) presented in r-θ plane defined by the axial coordinate equal to z=1. It appeared that the flow was strongly dominated by the tangential component. Far from the region swept by the double helical ribbons, it has been noted a progressive slowing of the flow. Also, it has been noted that the velocity vector was directly affected by the proximity impellers type. Indeed, within the double helical screw ribbons it has been observed that the velocity field is very active in the two opposite sides that are confounded with the impeller tip.

4.1.2 Flow patterns in r-z planes Figures 3, 4 and 5 show a velocity vector plot of the secondary flow (U,W) presented in r-z planes. These presentation planes have been defined respectively by the angular coordinate equal to θ=122°, θ=136° and θ=316°. These positions have been chosen in order to show the velocity field evolution. With a double helical ribbons, it’s noted that the flow have a centrifugal radial movement in the middle of the tank in the first plane (Figure 3.a). This movement decrease in the second plane (Figure 4.a). However, it becomes a centripetal type in the third plane (Figure 5.a). In the swept domain localised in the top of the tank, the flow have a descending axial movement. However in the bottom, this movement reverses and becomes an ascending and oblique character. While approaching to the lateral surface of the tank, the flow becomes weak. Within a

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double helical screw ribbons, the same observations are noted. But, it’s clear that the velocity field is more active than the double helical ribbons.

4.1.3 Radial profiles of the dimensionless velocity components Figure 6, 7 and 8 illustrate the predicted radial profiles of the dimensionless radial U(r), tangential V(r) and axial W(r) velocity components of the double helical ribbons and the double helical screw ribbons. These profiles are presented in two different r-θ planes defined by the dimensionless axial coordinates equal to z=0.45 and z=0.8. These figures adequately portray the swirling radial, tangential and axial jets character. In these figures, the numerical results of the two impellers were superposed to compare the local characteristics. Figure 6.a show that the radial velocity component U(r) reaches its maximal value equal to U=0.036 in the horizontal plane situated in the tank bottom. The radial position corresponding to this maximal value is defined by r=0.33. This result is also observed for tangential V(r) and axial W(r) velocity components. In fact, the tangential velocity component V(r) reaches its maximal value V=0.3 in the radial position equal to r=0.5 (Figure 7.a). Whereas, the axial velocity component W(r) reaches its maximal value W=0.03 in the radial position equal to r=0.58 (Figure 8.a). The minimal value W=-0.012 is reached in the radial position equal to r=0.16 (Figure 8.b). The negative value of the axial component corresponds to a downward movement toward the tank bottom. In the tank top, it’s noted a progressive reduction of the flow movements. In fact, the radial movement decreases and the radial velocity component U(r) reaches very weak values. The tangential velocity component V(r) shows a parabolic pace. It reaches very low values in the neighborhood of the axis and the tank walls. For the axial velocity component W(r), it’s noted a more intense downward

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movement. Globally, it’s clear that the Double helical ribbons screw present a more active velocity field than the Double helical ribbons.

4.2 Viscous dissipation rate Figures 9, 10 and 11 show respectively the viscous dissipation rate in different r-θ planes defined respectively by the axial positions equal to z=0.1, z=0.95 and z=1.15. Figure 12 shows the viscous dissipation rate in r-z plane defined by the angular position equal to θ=108°. These presentations planes have been chosen in order to show the viscous dissipation rate evolution in the stirred tank. Seen the symmetry of the problem, two symmetry wake shape are observed in these r-θ planes. Globally, the maximal value of the viscous dissipation rate is reached in the meeting of the double helical ribbons with these presentation planes. Out of this domain, the viscous dissipation rate becomes rapidly very weak. All of these observations are available for the two impellers. But, it’s noted that the viscous dissipation rate for the double helical screw ribbons is more important than the double helical ribbons.

4.3 Global characteristics To compare the global characteristics of the double helical ribbons in closed stirred tanks to the double helical ribbons screw, the pumping flow number NQp, the pumping efficiency number Ep and the energy efficiency Ee are calculated from the CFD code. The dependence of these characteristics on Reynolds number Re was presented respectively in figures 13, 14 and 15. Globally, these parameters were proportional to the Reynolds number Re in the laminar flow regime. At the same Reynolds number, it’s

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clear that the double helical ribbons screw characteristics values are superior to the double helical ribbons.

5 Comparison with experimental results Figure 16 show the power number Np variation on Reynolds number Re in the laminar flow range of the double helical ribbons and the double helical ribbons screw impellers. In these conditions, the power number was inversely proportional to the Reynolds number and the value of the product Kp= Np Re remained constant. According to these results, it’s noted that these curves present the same linear variation. Moreover, it’s noted that the power number of the double helical ribbons screw is superior to the double helical ribbons at the same Reynolds. To verify our computer results, the power number calculated from the CFD code were compared with the experimental results found in the literature in the case of the double helical ribbons. In these conditions, the numerical value of the power number Np is slightly inferior to the experimental value when compared at the same Reynolds number Re. These experimental results founded by Nagata [1] were superposed over an average error of 10%. The good agreement between the experimental results and the numerical results quietly confirmed the analysis method.

6 Conclusion Using our specific Computational Fluid Dynamics (CFD) code, three-dimensional simulation of the laminar flow generated by the double helical ribbons and the double helical ribbons screw impellers was investigated. Numerical results concerning velocity fields, viscous dissipation rate and global characteristics are presented in this paper.

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These results showed that the velocity field is more active with the double helical screw ribbons impeller. In this case, the effectiveness of the viscous dissipation and the pumping flow has been obviously noted. The pumping and the energy efficiency have reached the highest values at the same Reynolds number. The comparison of the power number has been presented to be compared with ones found by other researchers. The good agreement between the numerical results and the experimental data validate the numerical method. In the future, we intend characterising the hydrodynamic structure of this impeller by the particle image velocimetry (PIV) technique.

Nomenclature d

impeller diameter, m

D

internal diameter of the vessel tank, m

Ee

energy efficiency, dimensionless

Ee

energy efficiency, dimensionless

Ep

pumping efficiency number, dimensionless Fr 

Fr

Froude number, dimensionless, Fr

g

gravity acceleration, m2 .s-1

h

impeller height, m

H

vessel tank height, m

N

velocity of the impeller, rad.s-1

Np

power number, dimensionless

 2π N 2 d g

NQp pumping flow number, dimensionless P

power, W

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p

pressure, dimensionless

Re

Re  Reynolds number, dimensionless, Re

r

radial coordinate, dimensionless

s

shaft diameter, m

t

time, s

U

radial velocity components, dimensionless

V

angular velocity components, dimensionless

W

axial velocity components, dimensionless

z

axial coordinate, dimensionless

 N d2 

Greek symbols 

angular coordinate, rad



fluid viscosity, Pa.s



fluid viscosity, dimensionless



fluid density, kg.m-3

v

viscous dissipation rate, dimensionless

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References : [1] S. Nagata, Mixing: principles and applications, John Wiley & Sons: Halstead press, Japan, 1975 [2] Espinosa-Solares T., Britto-De La Fuente E., Tecante A., Tanguy, P.A., Power consumption of a dual turbine-helical ribbon impeller mixer in ungassed conditions, Chemical Engineering Journal, 1997, 67, 215-219 [3] Tanguy P.A., Thibault F., La Fuente E.B., Espinosa-Solares T., Tecante A., Mixing performance induced by coaxial flat blade-helical ribbon impellers rotating at different speeds, Chemical Engineering Science, 1997, Vol. 52, N. 11, 1733-1741 [4] Bertrand F., Tanguy P.A., Britto-De La Fuente E., Carreau P., Numerical modeling of the mixing flow of second-order fluids with helical ribbon impellers, Comput. Methods Appl. Mech. Engrg., 1999, 180, 267-280 [5] Kaneko Y., Shiojima T., Horio M., Numerical analysis of particle mixing characteristics in a single helical ribbon agitator using DEM simulation, Powder Technology, 2000, 108, 55-64 [6] Yao W., Mishima M., Takahashi K., Numerical investigation on dispersive mixing characteristics of MAXBLEND and double helical ribbons, Chemical Engineering Journal, 2001, 84, 565-571 [7] Wang J.J., Feng L.F., Gu X.P., Wang K., Hu C.H., Power consumption of innerouter helical ribbon impellers in viscous Newtonian and non-Newtonian fluids, Chemical Engineering Science, 2000, 55, 2339-2342

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[8] Niedzielska A., Kuncewicz Cz., Effect of impeller geometry on the power consumption for helical ribbon impellers, Tatranské Matliare, Slovak Republic, 2003, 138 [9] Imai Y., Aoki T., A higher-order implicit IDO scheme and its CFD application to local mesh refinement method, Computational Mechanics, 2006, Volume 38, Number 3, 211-221 [10] Xu C., Amano R.S., Computational analysis of pitch-width effects on the secondary flows of turbine blades, 2004, Volume 34, Number 2, 111-120 [11] Driss Z., Contribution in studies of the turbines in an agitated vessel, PhD thesis, National School of Engineers of Sfax, University of Sfax, Tunisia, 2008 [12] Driss Z., Kchaou H., Baccar M., Abid M.S., Numerical investigation of internal laminar flow generated by a retreated-blade paddle and a flat-blade paddle in a vessel tank, International Journal of Engineering Simulation, 2005, Vol. 6, Number 3, 10-16 [13] Driss Z., Karray S., Kchaou H., Abid M.S., Computer simulations of fluid-structure interaction generated by a flat-blade paddle in a vessel tank, International Review of Mechanical Engineering 6 (2007) 608-617. [14] Driss Z., Bouzgarrou G., Chtourou, W., Kchaou H., Abid M.S., Computational studies of the pitched blade turbines design effect on the stirred tank flow characteristics. European Journal of Mechanics B/Fluids, 2010, 29, pp. 236-245. [15] Abid M.S., Xuereb C., Bertrand J., Modeling of the 3D Hydrodynamics of 2-blade impellers in stirred tanks filled with a highly viscous fluid, Can. J. Chem. Eng., 1994, Vol. 72, 184-193 [16] Xuereb C., Bertrand J., 3-D Hydrodynamics in a tank stirred by a double-propeller system and filled with a liquid having evolving rheological properties, Chemical Engineering Science, 1996, 51, 1725-1734

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[17] Bouzgarrou G., Driss Z., Abid M.S., CFD simulation of mechanically agitated vessel generated by modified pitched blade turbines, International Journal of Engineering Simulation, 2009, Volume 10, Number 2, 11-18 [18] Patankar S.V., Numerical heat transfer and fluid flow, Series in Computational Methods in Mechanics and Thermal Sciences, Mc Graw Hill, New York, 1980 [19] Douglas J., Gunn J.E., A general formulation of alternating direction implicit methods, Num. Math., 1964, Vol. 6, 428-453.

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Figure

(a) Double helical ribbons

(b) Double helical ribbons screw

Figure 1: Stirred tank equipped by proximity impellers

1



Figure 2: Flows patterns induced in r-θ plane defined by z=1

2



Figure 3: Flows patterns induced in r-z plane defined by θ=122°

3




4




5

U

a .z=0,45

0,04

Composante radiale de la vitesse U(r)

Hélicoïdehelical ribbons Double Combinaison Double helical ribbons screw

0,02

r

0 0

0,5

1

Composante radiale r Radial position

-0,02

r (a) Axial position of the presentation plane equal to z=0.45 U

b. z=0,8

Composante radiale de la vitesse U(r)

0,04

Double helical ribbons Hélicoïde Combinaison Double helical ribbons screw

0,02

r

0 0

0,5

1

-0,02

-0,04

Composante radiale r Radial position

r (b) Axial position of the presentation plane equal to z=0.8 Figure 6: Radial profiles of the radial velocity U(r)

6

V

a. z=0,45

0,4

Composante tangentiele de la vitesse V(r)

Double helical ribbons Hélicoïde Combinaison Double helical ribbons screw

0,2

r

0 0

0,5

1

Composante radiale rr Radial position -0,2

(a) Axial position of the presentation plane equal to z=0.45 V

b. z=0,8

0,4

Hélicoïde Double helical ribbons

Composante tangentiele de la vitesse V(r)

Combinaison Double helical ribbons screw

0,2

r

0 0

-0,2

0,5

1

Composante radiale rr Radial position

(b) Axial position of the presentation plane equal to z=0.8 Figure 7: Radial profiles of the tangential velocity V(r)

7

W

a . z=0,45

0,04

Hélicoïdehelical ribbons Double

Composante axiale de la vitesse W(r)


0,02

r

0 0

0,5

1

Composante radiale r r Radial position -0,02

(a) Axial position of the presentation plane equal to z=0.45 W

b. z=0,8

0,04

Hélicoïde Double helical ribbons

Composante axiale de la vitesse W(r)


0,02

r

0 0

-0,02

0,5

1

Composante radiale rr Radial position

(b) Axial position of the presentation plane equal to z=0.8 Figure 8: Radial profiles of the axial velocity W(r)

8



Figure 9: Dissipation rate induced in r-θ plane defined by z=0.1

9




10




11



Figure 12: Dissipation rate induced in r-z plane defined by =108°

12

NQp 1

Nombre de pompage NQp

Double Hélicoïdehelical ribbons Combinaison Double helical ribbons screw

0,1

0,01 1

10

100

Nombre de Reynolds Re

Reynolds number Re Figure 13: Pumping flow number variation NQp=f(Re)

13

Ep 0,08


0,07

Efficacité de pompage Ep

0,06

0,05

0,04

0,03

0,02

0,01

0 0

10

20

30

40

50

60

70

Re


Reynolds number Re Figure 14: Pumping efficiency number variation Ep=f(Re)

14

Ee 1


0,1

Efficacité energétique Ee

0,01

0,001

0,0001

0,00001

0,000001

0,0000001

0,00000001

0,000000001 1

10

100

Re

Nombre de Reynolds Re Re Reynolds number

Figure 15: Energy efficiency variation Ee= f(Re)

15

Np 100

Nombre de Puissance Np

Hélicoïdehelical ribbons Double Hélicoïdehelical Nagataribbons [1] Double Combinaison Double helical ribbons screw

10

1 1

10

100

Re


Reynolds number Re Figure 16: Power number variation Np=f(Re)

16