2006 chem eng sci

Upload: vekbalam

Post on 07-Apr-2018

229 views

Category:

Documents


0 download

TRANSCRIPT

  • 8/4/2019 2006 Chem Eng Sci

    1/13

    Chemical Engineering Science 61 (2006) 2895 2907www.elsevier.com/locate/ces

    Mixing analysis in a coaxial mixer

    Christian Riveraa, Stephane Foucault a, Mourad Henichea,Teodoro Espinosa-Solaresb, Philippe A. Tanguya,

    aURPEI, Department of Chemical Engineering, cole Polytechnique, P.O. Box 6079, Station Centre-Ville, Montral, Que., Canada H3C 3A7b Departamento de Ingeniera Agroindustrial, Universidad Autnoma Chapingo, P.O. Box 161, Chapingo 56230, Edo. de Mxico, Mxico

    Available online 19 January 2006

    Abstract

    The performance of a coaxial mixer in the laminar-transitional flow regime was numerically investigated with Newtonian and non-Newtonianfluids. These mixers comprised two shafts: a central fast speed shaft mounted with an open turbine, and a slow speed shaft fitted with a

    wall scraping anchor arm. To model the complex hydrodynamics inside the vessel, the virtual finite element method (POLY3D TM software)

    coupled with a Lagrange multiplier approach to cope with the non-linearity coming from the rheological model was employed. Co-rotation

    and counter-rotation mode were compared, based on several numerical criteria, namely, mixing time, power consumption and pumping rate. It

    was found that co-rotating mode is more efficient than counter-rotating mode in terms of energy, pumping rate and homogenization time.

    2005 Elsevier Ltd. All rights reserved.

    Keywords: Mixing; Hydrodynamics; Simulation; Fluid mechanics; Coaxial mixer; Virtual finite element method

    1. Introduction

    The characteristics of a mixer are particularly critical for the

    process economics and the quality of the end product. Usually,

    the design is based on process objectives taking into account

    many variables. For example, the high viscosity of phases usu-

    ally restricts the mixing to the laminar-transitional regime due

    the inefficient task to generate turbulent instabilities in such

    conditions.

    Nowadays, the industry needs impellers that can work in

    laminar, transitional or turbulent regimes with minimum mod-

    ifications. Standard agitators like close clearance and open im-

    pellers exhibit some limitations with this aspect. On one hand,

    close clearance impellers such as helical ribbons have a goodmixing performance in laminar regime (Yap et al., 1979; De

    la Villeon et al., 1998). However, this situation is completely

    reversed when the condition changes from laminar to transi-

    tional or turbulent (Hoogendoorn and Den-Hartog, 1967). On

    the other hand, open impellers like the Rushton turbine are

    known to be very efficient at high Reynolds number but in lam-

    inar regime, segregated zones are produced (Salomon et al.,

    Corresponding author. Tel.: +1 514340 4017; fax: +1 514340 4105.

    E-mail address: [email protected] (P.A. Tanguy).

    0009-2509/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

    doi:10.1016/j.ces.2005.11.045

    1981). The situation becomes critical if along the process timethe phases to be mixed develop non-Newtonian rheological

    properties such as shear-thinning or thixotropy.

    Recently, several innovative strategies have been proposed

    to tackle this problem, based for instance on coaxial mix-

    ers (Tanguy et al., 1997; Espinosa-Solares et al., 1997, 2001;

    Thibault and Tanguy, 2002; Foucault et al., 2004, 2005), plan-

    etary mixers (Tanguy et al., 1999) or conical mixers (Dubois

    et al., 1996). The main idea is simple, association of differ-

    ent agitators rotating at different speeds. In this way it is pos-

    sible to create a mixer that achieves the process objectives,

    blending the capabilities of several agitators. At the end, a

    dynamic mixing unit that adapts with the process necessities is

    obtained.Coaxial mixers have been shown as a good alternative to gen-

    erate particles suspension and Newtonian and non-Newtonian

    mixing (Foucault et al., 2004, 2005). A standard configuration

    consists of the combination of a dispersive turbine and a wall

    scrapping anchor. The anchor blades rotate at low speed in or-

    der to scrap the vessel wall and bring back into the bulk the ma-

    terial to be mixed. The dispersing turbine rotates at high speed

    to produce a distributive and dispersive effect throughout the

    bulk. The superposition of both effects is known to produce a

    very efficient mixing.

    http://www.elsevier.com/locate/cesmailto:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-mailto:[email protected]://-/?-http://www.elsevier.com/locate/ces
  • 8/4/2019 2006 Chem Eng Sci

    2/13

    2896 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    From this perspective, the association of a close clear-

    ance impeller with an open agitator seems reasonable but the

    question of optimal operation conditions arises. In general,

    dimensionless analysis on pilot rigs has been utilized to design

    mixing systems (Tatterson, 1992). However, for this kind of

    mixers, the presence of different impellers rotating at differ-

    ent speeds makes ambiguous the choice of the characteristiclength and velocity. Consequently, the definition of dimen-

    sionless parameters like the Reynolds number becomes not

    obvious. In addition, there are several factors to take into ac-

    count, namely; number and type of impellers, speed ratio, fluid

    rheology, rotation mode and geometric position. This makes

    the design of this equipment a challenging task. Ideally, a

    compromise must be found among all these variables so that

    the mechanical energy is enough to ensure a good distributive

    and dispersive action yet not too large to avoid large power

    consumption. Foucault et al. (2005) in their work clearly show

    using dimensionless analysis that a slight modification in the

    Reynolds number definition can be meaningful to correlate

    experimental data as mixing time and power consumption

    with operating conditions in a generalized fashion. In this

    way, these authors were able to build power consumption

    and homogeneity time master curves valid for both Newto-

    nian and non-Newtonian fluids. Co-rotating mode was found

    to be better than counter-rotating mode from a mixing view-

    point. However, the hydrodynamic causes were not completely

    elucidated.

    Torquemeter

    Motor 2.23 kW

    (without gearbox)

    Tank

    Torquemeter

    Gearbox (15 : 1)

    Motor 373 W

    High speed shaft

    = 0.0254 m

    Dc = 0.38 m

    Da = 0.36 m

    Wa = 0.0318m

    Cw = 0.0095m

    Low speed shaft

    = 0.0381 m

    Temperature

    sensors

    Computer

    Wa

    Cw

    Da

    Dc0.5

    8m

    0.6

    6m

    Fig. 1. Schematic coaxial mixer configuration.

    Both numerical and experimental studies have been re-

    ported on the hydrodynamics of complex mixing systems

    involving multiple independent impellers (Tanguy et al., 1997;

    Espinosa-Solares et al., 1997, 2001). Due to the complex ge-

    ometry of these systems, it is usually very difficult to analyze

    in depth their hydrodynamics from an experimental stand-

    point. The objective of the present investigation is to carefullyassess the hydrodynamic conditions of coaxial mixing in co-

    rotating or counter-rotating operation when Newtonian and

    non-Newtonian fluids are employed in the laminar-transitional

    regime. The methodology is based on a numerical approach

    experimentally validated following the philosophy of our re-

    search group.

    2. Mixing system and numerical methodology

    The mixing system consisted of a centered Rushton turbine

    with an anchor rotating at a speed ratio equal to 10. Fig. 1

    illustrates the mixing system and presents tank and impellers

    dimensions. In this work, we consider a Newtonian fluid with

    a viscosity of 10 Pa s and a non-Newtonian fluid described by

    the power law model with flow index value (n) of 0.5 and a

    consistency index (m) of 8.3 P a sn. The fluid density for the

    Newtonian fluid was 1350 kg/m3 and for the non-Newtonian

    liquid 1010 kg/m3.

    Co-rotating and counter-rotating modes were investi-

    gated for two speed couples, 20020 and 10010 RPM, with

    http://-/?-
  • 8/4/2019 2006 Chem Eng Sci

    3/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2897

    Fig. 2. Computational domain: (a) three-dimensional unstructured mesh for the vessel, (b) virtual impellers and (c) three-dimensional mesh with the immersed

    virtual impellers.

    Newtonian and non-Newtonian fluids. These operating condi-

    tions avoid the formation of a vortex at the free surface. Further-more, for comparison purposes, a configuration that consists

    of a rotating Rushton turbine with a static anchor was studied.

    The experimental power consumption and mixing time were

    obtained following the work of Foucault et al. (2004, 2005).

    To predict numerically the three-dimensional flow field in a

    stirred tank, the momentum and mass conservation equations

    were solved with help of standard 3D Galerkin finite element

    method.

    j

    jt+

    + p + = f, (1)

    = 0. (2)

    The boundary conditions are as follows:

    No normal velocity at the fluid surface (z = 0).

    No slip condition at the vessel wall ( = 0).

    Constant angular velocity at the impeller surface (Ni= constant).

    To deal with the non-linear rheological model an augmented

    Lagrangian approach was utilized as proposed by Tanguy et al.

    (1984). With this method, the rheological non-linearity is re-

    moved from the momentum equation and handled separately

    in an auxiliary equation with help of a Lagrange multiplier.

    The Lagrange multiplier can be viewed as a kernel that con-

    nects both equations. Finally, the variational formulation ofthe momentum equation along with the incompressibility con-

    straint and Lagrange multipliers are solved in a set of Uzawa

    iterations (Bertrand and Tanguy, 2002) with the help of Bi-

    CGSTAB method as a linear solver (Van der vorst, 1992). This

    loop is immersed into a Newton scheme to tackle the convective

    term.

    Since both impellers rotate at different speeds, the use of a

    Lagrangian frame of reference does not help the simplification

    of the problem. Then, in this work, a Eulerian frame of reference

    was used resorting to the fictitious domain method (Glowinski

    et al., 1994) to reproduce the unsteady rotation of the impellers.

    This method has been extended to mixing problems by Bertrandet al. (1997) and Tanguy et al. (1997). Briefly, this approach is

    based on the imposition of the impeller kinematics by means

    of a set of control points distributed along the surface of the

    impeller (this is done using Lagrange multiplier and penalty

    techniques). At each time step, the velocity and position of the

    control points is updated and a new problem is solved. One of

    the main advantages of this method is that only a single mesh

    is needed avoiding the necessity of re-meshing at each time

    step. To take into account the unsteady nature of the flow, a

    Gear scheme was employed (Fortin et al., 1986). A total of 120

    time steps per revolution for the anchor and 12 for the Rushton

    turbine were used. Computations were carried out until periodic

    solutions were obtained.

    http://-/?-http://-/?-http://-/?-
  • 8/4/2019 2006 Chem Eng Sci

    4/13

    2898 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    In VFEM, the meshing is an important issue as shown by

    Rivera et al. (2004). In this work an intelligent mesh was built

    composed of different partitions near the anchor and turbine

    blade to allow a good definition of the virtual objects. Fig. 2

    illustrates the constructed mesh with the virtual impeller. Due

    to the intrinsic complexity of the geometry, tetrahedral 8 nodes

    elements P+

    1.

    P0 that approximate the velocity with a super-linear polynomial and consider constant the pressure inside

    each element was employed. This type of elements ensures

    a rigorous stability and convergence for complex fluid flow

    problems (Bertrand et al., 1992). The final computational mesh

    required approximately 90,000 elements and 200,000 nodes

    producing a system of approximately 1,250,000 equations.

    All the described numerical features are available in the

    commercial 3D finite element software POLY3DTM (Rheosoft,Inc.). The total memory requirements for the Newtonian andnon-Newtonian solutions were 1.3 and 1.6 Gbyte, respectively.

    All simulations were run on an IBM computer cluster. The

    intelligent meshing was generated on I-DEAS (EDS) soft-

    ware and the visual post-processing was carried out with En-

    sight (CEI).

    3. Results and discussion

    3.1. Power consumption

    Following the work of Foucault et al. (2005) the Reynolds

    number of a coaxial mixing system can be defined by Eq. (3).

    It is worth noting here that along all this work, N is defined

    employing the same definition as Foucault et al. (2004, 2005);

    co-rotating N = Nt Na , counter-rotating N = Nt + Na andsingle Rushton turbine N = Nt.

    Recounter-rotation =(Nt + Na)D

    2t

    ;

    Reco-rotation =(Nt Na)D

    2t

    . (3)

    For the case of non-Newtonian fluids, the generalized Reynolds

    number is defined as

    Recounter-rotation =(Nt + Na)

    2nD2t

    k;

    Reco-rotation =(Nt Na)2nD2t

    k. (4)

    Table 1

    Power consumption for the investigated scenarios in the coaxial mixer

    Operating conditions ReMod NpExp NpNum PowerExp (W) PowerNum (W) P / V (W/m3)

    Co-rotating Newtonian fluid 16.20 7.00 6.94 81.64 81.00 2041.0

    Rushton impeller only Newtonian fluid 18.00 5.50 5.25 88.00 84.00 2200.0

    Counter-rotating Newtonian fluid 19.80 5.30 5.07 112.86 108.00 2821.5

    Co-rotating non-Newtonian fluid 8.94 3.25 3.43 3.54 3.75 88.5

    Rushton impeller only non-Newtonian fluid 10.47 3.25 3.02 4.86 4.52 121.5

    Counter-rotating non-Newtonian fluid 12.08 3.25 3.06 6.47 6.10 161.8

    Furthermore, for a given velocity field, the power consumption

    can be computed by Eq. (5) (Tanguy et al., 1997).

    P =

    : d. (5)

    For analysis purposes, the obtained power consumption can be

    translated into the power number Np (Nagata, 1957) which canbe determined using Eq. (6).

    Np =P

    N3D5t. (6)

    In the same manner as the Reynolds number, the power number

    can be redefined as Eq. (4) (Foucault et al., 2004, 2005).

    Npcounter-rotation =P

    (Nt + Na)3D5t

    ;

    Npco-rotation =P

    (Nt Na )3D5t

    . (7)

    Table 1 summarizes the power number with their respectiveReynolds number obtained by experimental and numerical ap-

    proaches. It can be readily seen that a quite good agreement is

    obtained between the data. Furthermore, the observed higher

    power consumption and specific energy for the counter-rotation

    mode is in accordance with the work of Foucault et al. (2004,

    2005).

    3.2. Flow patterns

    To ease the analysis of the three-dimensional hydrodynam-

    ics, the velocity was projected onto two-dimensional planes.

    Fig. 3 shows the flow patterns for the Newtonian case. Werestrict ourselves to show only the Newtonian patterns since

    the non-Newtonian ones exhibit the same particularities. In the

    plane XZ, the flow is mainly dominated by the well-known toristructures typical of the Rushton turbine. However, when the

    operating mode changes from co-rotating to counter-rotating

    there is a shift in the position of the center of such structures.

    In counter-rotating mode, the center is located closer to the im-

    peller. In the XY-plane, a dominant angular motion is readily

    observed. A secondary circulation region can be noted around

    the anchor when the impellers are in counter-rotating mode.

    This irregularity is reduced by the single turbine and completely

    removed in co-rotating mode..The global effect is a contraction in the size of the well

    mixed region. This can be better addressed with the velocity

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/4/2019 2006 Chem Eng Sci

    5/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2899

    Fig. 3. Flow patterns (velocity, m/s) for planes XZ and XY (at Z = 0.2 m): (a) co-rotating mode, (b) counter-rotating mode and (c) single Rushton turbine.

    iso-contour presented in Fig. 4 For the case of co-rotating

    mode the size of the well mixed region is larger. With the non-

    Newtonian fluid, the contraction in the cavern for the counter-

    rotating case is more evident. This is produced by the fact that

    in counter-rotating mode the shear is higher than for the other

    two operation modes. This generates a low viscosity zone close

    to the central turbine that diminishes the pumping in the axial

    direction as we will show.

    To quantitatively address the impact of the operation mode

    over the hydrodynamics, we present in Fig. 5 the velocity pro-

    files along the tank height for the six cases considered. These

    plots clearly illustrate the fact that co-rotating mode generates

    an overall increase in axial and tangential velocities close to

    the impeller for both Newtonian and non-Newtonian cases. In

    Fig. 5a, it is interesting to note a considerable axial velocity re-

    duction in the upper part of the tank (z/H= 0.7.1) for the case

    of non-Newtonian turbine and counter rotating scenarios. Fur-

    thermore, in Fig. 5b, we can observe that the counter-rotating

    mode generates a tangential flow in both positive and negative

    directions. As a consequence, the flow segregation is greater in

    counter-rotating mode. Finally, we plot the radial velocity pro-

    file in Fig. 5c. As must be expected, the co-rotating mode in

    both Newtonian and non-Newtonian cases generates a higher

    radial velocity close to the impeller. However, at the upper and

    lower parts of the vessel (0.4 >z/H > 0.7), the situation is re-

    versed due to the modification in the tori structures as was al-ready pointed out.

    To better understand the flow inside the coaxial mixer, the

    shear rate norm was plotted in the radial discharge zone of

    the Rushton turbine in Fig. 6a for the Newtonian fluid and

    Fig. 6b for non-Newtonian fluid. The most interesting feature

    is the peak at r/R = 0.75 that can be found for counter-

    rotating mode and single Rushton turbine. This fact illustrates

    the high shear region generated between the anchor and the

    turbine when they operate in counter-rotating mode. However,

    for single turbine the shear rate decreases near the wall. We

    can also observe that for the case of the non-Newtonian fluid

    the peak becomes magnified. This has a strong repercussion

    over the viscosity, since it is shear rate dependent. We plot-

    ted the viscosity profile for the non-Newtonian scenario in

  • 8/4/2019 2006 Chem Eng Sci

    6/13

    2900 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    Fig. 4. Velocity iso-contours at 0.3 m/s for the Newtonian case and at 0.15 m/s for the non-Newtonian case: (a) co-rotating, (b) counter-rotating and (c) single

    Rushton turbine.

    Fig. 7. The viscosity differences generated by the Rushton

    turbine case can be readily observed, where a small cavern

    of low viscosity is surrounded by a high viscosity zone. In

    the counter-rotating mode this situation is less prominent

    but a region of high viscosity still persists at the top of the

    vessel.

    3.3. Pumping rate

    The computation of the total horizontal contribution in the

    radial (r) and tangential () pumping rate was done solvingnumerically the stream function (Rivera et al., 2004; Heniche

    et al., 2005). The pumping rate can be computed by

    Qr (z) =1

    H

    H0

    (rmax rmin) dz,

    Q(z) =1

    H

    H0

    (max min) dz. (8)

    In addition, the axial pumping rate was computed by means

    of

    Q+

    z (z) =

    1

    HH

    0

    A V z+

    dA =

    1

    HH

    0

    A V z

    dA, (9)

    where the + stands for the positive axial velocities. Since the

    obtained velocity field ensures mass conservation, the corre-

    sponding values ofQ+z is of the same magnitude as Qz (where

    the stands for the negative axial velocities).

    For analysis purposes, the dimensionless flow number is

    helpful which was determined with Eq. (10).

    Nqi =Qi

    N D3, i = r,, z. (10)

    Table 2 illustrates the comparison of axial volumetric flow

    rate in the radial (r), tangential () and axial (z) components

    for the different investigated scenarios. It is noted that axial and

    tangential pumping throughout the vessel is improved in the

    case of co-rotating mode. In the case of the Newtonian fluid,

    they are 71% and 26% higher than the counter-rotating mode,

    respectively. For the non-Newtonian scenario, differences of

    41% and 29% were obtained. In agreement with the velocity

    profiles, the radial pumping is higher for the counter-rotating

    mode (100% Newtonian and 22% non-Newtonian).

    3.4. Pressure patterns

    Even though in the mixing analysis of mechanically agi-

    tated systems the pressure parameter is usually neglected, we

    http://-/?-
  • 8/4/2019 2006 Chem Eng Sci

    7/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2901

    Fig. 5. Velocity profiles along a line parallel to the Z-axis at x = 0.1m and

    y = 0 m: (a) axial velocity, (b) tangential velocity and (c) radial velocity.

    Fig. 6. Shear rate norm in the radial discharge zone: (a) Newtonian fluid and

    (b) non-Newtonian fluid.

    cannot forget that an impeller is a kind of pump. Fluid mo-

    tion is basically powered by pressure forces. Then, based on

    this analogy we can better explain why the co-rotating mode

    performs better than the counter-rotating mode. In Fig. 8, the

    pressure patterns are presented. It is noticed in the XZ-plane

    that for co-rotating mode, a larger low-pressure zone forms

    around the centered turbine. In the XY-plane a low-pressure

    zone forms behind the blades of the turbine and the anchor. If

    the impellers are in co-rotating mode, the pressure gradient is

    in the forward direction; thus in this case, the pressure forces

  • 8/4/2019 2006 Chem Eng Sci

    8/13

    2902 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    Fig. 7. Non-Newtonian viscosity profile (Pa s): (a) co-rotating mode, (b)

    counter-rotating mode and (c) single Rushton turbine.

    are additive. The turbine drags the anchor in the flow direc-

    tion and as a consequence a lower power is obtained. On the

    other hand, in counter-rotating mode the pressure gradient of

    Table 2

    Flow number for the investigated scenarios in the coaxial mixer

    Operating conditions Nqr Nq Nqz

    Co-rotating Newtonian fluid 0.119 0.901 0.125

    Rushton impeller only Newtonian fluid 0.136 0.635 0.111

    Counter-rotating Newtonian fluid 0.236 0.715 0.073

    Co-rotating non-Newtonian fluid 0.163 0.739 0.093Rushton impeller only non-Newtonian fluid 0.073 0.441 0.068

    Counter-rotating non-Newtonian fluid 0.201 0.570 0.065

    both impellers are in opposite direction, then, the flow com-

    ing from the turbine blade suddenly faces a higher-pressure

    zone generated by the anchor. This produces a repulsive ef-

    fect that increases the total power consumption. These find-

    ings are in agreement with the experimental work reported by

    Foucault (2005). Furthermore, the flow is segregated in two re-

    gions: an enclosed volume close to the turbine and a recircula-

    tion zone at the wall clearance in agreement with the previousdiscussion.

    To quantify the above observations, we compute the differ-

    ence of pressure in front of and behind the blades of the im-

    pellers (anchor and turbine). Then, we were able to determine

    a representative total driven force in the vessel by

    pTotal = pTurbine + pAnchor . (11)

    Table 3 presents the computed pressure drops. It appears

    that the co-rotating mode offers the higher values which are in

    agreement with the latter discussion. Based on all the facts, we

    finally can represent a coaxial mixer as two centrifugal pumps

    working in series (Fig. 9). In co-rotating mode the discharge ofthe first pump is connected to the input of the second one, gen-

    erating a balanced flow. For counter-rotating mode, the output

    of the first pump is connected with the output of the second.

    3.5. Distributive and dispersive mixing

    The fact that the counter-rotating mode offers more shear-

    ing is commonly related with a better mixing performance.

    However, we must take into account that mixing is a compro-

    mise between distributive and dispersive actions. The former

    is directly related to the pumping, while the latter is related to

    the shearing effects. Thus, a parameter that quantifies the bal-

    ance between the dispersive and distributive mixing would be

    helpful.

    In mixing analysis, a dimensionless number called head num-

    ber can be related to the shearing action. This is defined by

    Nh =2gH

    2N2D2, (12)

    where H is a theoretical height created by the impeller that

    comes from the analogy between an agitator and a centrifugal

    pump

    H=

    P

    Qg=

    Np

    Nq=

    ND2

    g2 . (13)

  • 8/4/2019 2006 Chem Eng Sci

    9/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2903

    Fig. 8. XZ and XY (at Z =0.2 m) planes for the pressure (Pa) patterns in the Newtonian case: (a) co-rotating, (b) counter-rotating and (c) single Rushton turbine.

    Table 3

    Pressure drop for the investigated scenarios in the coaxial mixer

    Operating conditions PAnchor (Pa) PTurbine (Pa) PTotal (Pa)

    Co-rotating Newtonian fluid 350.96 2049.568 2400.52

    Counter-rotating Newtonian fluid 1293.16 1645.82 352.66

    Rushton impeller only Newtonian fluid 580.63 2064.51 1483.88

    Co-rotating non-Newtonian fluid 65.20 260.12 325.32

    Counter-rotating non-Newtonian fluid 230.14 213.67 16.47

    Rushton impeller only non-Newtonian fluid 67.64 238.86 171.22

  • 8/4/2019 2006 Chem Eng Sci

    10/13

    2904 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    Rushton turbine

    P0- P0-

    P0+ P0+P1+

    P1+P1-

    P1-

    Counter-rotating mode

    Anchor

    Rushton turbine

    Anchor

    Co-rotating mode

    Fig. 9. Pumps analogy for the coaxial mixer.

    Table 4

    Head number and flow number for the investigated scenarios in the coaxial

    mixer

    Operating conditions Nq Nh Nh/Nq

    Co-rotating Newtonian fluid 0.917 1.546 1.685

    Rushton impeller only Newtonian fluid 0.658 1.474 2.477

    Counter-rotating Newtonian fluid 0.756 1.631 1.948

    Co-rotating non-Newtonian fluid 0.761 0.865 1.136

    Rushton impeller only non-Newtonian fluid 0.452 1.082 3.222

    Counter-rotating non-Newtonian fluid 0.608 1.457 1.778

    Furthermore, the total pumping rate can be computed as

    Q =

    Q2r + Q

    2

    + Q2z . (14)

    A parameter to quantify the relationship between shearing

    and pumping actions can be obtained with the ratio betweenthe head number and the flow number. A large value indicates

    a good dispersive action, while a low one corresponds to a

    Fig. 10. Tracer dispersion for the Newtonian fluid: (a) co-rotating mode at 15 s, (b) co-rotating mode at 150 s, (c) counter-rotating mode at 15 s and (d)

    counter-rotating mode at 150s.

    distributive dominant flow. Table 4 presents the results for the

    different studied scenarios. As it can be seen, the flow produced

    by the co-rotating mode is more balanced since we obtain a

    distributive action with not excessive shearing. On the other

    hand, as could be expected from the last discussions on shearand pumping rate, counter-rotating mode offers good dispersive

    mixing properties but poor distributive flow.

    3.6. Mixing time

    To determine the mixing time, the most important task was

    the computation of tracers trajectories. This has been usually

    based on the velocity integration over time with high order

    schemes (Ottino, 1989; Souvaliotis et al., 1995). The challenge

    is to find an optimal time step to predict accurately the trajec-

    tories. In this work, to overcome this difficulty, an element by

    element tracking of massless particles as proposed by Henicheand Tanguy (2005) was employed. In this work, it was ob-

    served that the obtained velocity field is periodic over time.

    http://-/?-http://-/?-http://-/?-
  • 8/4/2019 2006 Chem Eng Sci

    11/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2905

    Fig. 11. Intensity segregation evolution along the time: (a) Newtonian fluid and (b) non-Newtonian fluid.

    Table 5

    Mixing time and dimensionless mixing energy for the studied scenarios

    Operating conditions TmExp (s) TmNum (s) N TmExp NT mNum Np (NT m) (Dimensionless mixing energy)

    Co-rotating Newtonian fluid 125 105 375 319 2631

    Rushton impeller only Newtonian fluid 281 225 936 777 5152

    Counter-rotating Newtonian fluid 1589 5828 30,888

    Co-rotating non-Newtonian fluid 341 210 512 389 1664

    Rushton impeller only non-Newtonian fluid

    Counter-rotating non-Newtonian fluid

    For a better numerical efficiency, the time periodic velocity field

    was approximated by a Fast Fourier Transform (FFT) with 20

    harmonics.

    The intensity of segregation was selected as the homogeneity

    criterion, which is defined as

    Iseg =1

    C(1 C)

    1

    VTotal

    Mi=1

    (Ci C)2Vi . (15)

    The final computation was based on the trajectory of 6000 trac-

    ers injected from the top. Fig. 10 illustrates the mixing action

    of both rotation modes. After 15 s an unmixed region appears

    with counter-rotating mode. At the same time, the co-rotating

    mode exhibits a larger mixing zone. In Fig. 11 the evolution

    of intensity of segregation is illustrated. Table 5 summarizes

    the numerical and experimental mixing times. It is readily seen

    that the co-rotating mode gives the shorter mixing times, the

    single Rushton turbine stays as an intermediate case, and the

    counter-rotating mode exhibits the longer ones. The numerical

    results follow the same trend as the experimental ones. Finally,

    we conclude by computing the dimensionless mixing energy

    as follows:

    Emix = Np N Tm. (16)

    From all the above discussion, it is clear that co-rotating

    mode offers low energy consumption, making the counter-

    rotating mode the most inefficient in terms of energy consump-

    tion.

    4. Conclusions

    The objective of this work was to analyze by means of CFD

    the hydrodynamics of a coaxial mixer in the case of Newtonian

    and non-Newtonian shear thinning fluids. It was shown that co-rotating mode is more efficient than counter-rotating mode in

    terms of energy, pumping rate and homogenization time. The

    fact that the anchor and the turbine rotate in the same direction

    allows a collaborative action that have a strong influence on

    the mixing performance. The co-rotating mode yields a flow

    where both distributive and dispersive mixing capabilities are

    balanced. Reversely, the counter-rotating mode produces a less-

    balanced system where high shearing and unmixed regions co-

    exist. The situation becomes worse for non-Newtonian fluids

    where the viscosity is shear rate dependent, resulting in smaller

    well-mixed zones. Finally, we expect that the generated infor-

    mation would be helpful for further investigations of this inter-

    esting and versatile device for difficult mixing applications.

  • 8/4/2019 2006 Chem Eng Sci

    12/13

    2906 C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907

    Notation

    C concentration of tracers

    C average concentration of tracers

    D diameter, m

    Emix mixing energy, dimensionless

    f body force, N/m3

    g gravity, m/s2

    H tank height, m

    Iseg intensity of segregation, dimensionless

    k consistency index, Pa sn

    M number of finite elements

    N angular speed, rev/s

    N shear thinning index, dimensionless

    Nh head number, dimensionless

    Np power number, dimensionless

    Nq pumping number, dimensionless

    p pressure, PaP power, W

    Q pumping rate, m3/s

    Re Reynolds number, dimensionless

    Tm mixing time, s

    velocity, m/s

    VTotal total volume

    Greeks letters

    deformation rate, s1

    Newtonian viscosity, Pa s

    density, kg/m3

    viscous stress, Pa

    stream function

    Mathematical symbols

    j partial derivative

    difference

    gradient

    divergence

    Subindices

    a anchor

    r radial direction

    t turbine Rushton

    z axial direction

    tangential direction

    Acknowledgements

    The financial assistance of NSERC is gratefully acknowl-

    edged.

    References

    Bertrand, F., Tanguy, P.A., 2002. Krylov-based Uzawa algorithms for the

    solution of the Stokes equation using discontinuous-pressure tetrahedral

    finite elements. Journal of Computational Physics 181, 617638.

    Bertrand, F., Gadbois, M.R., Tanguy, P.A., 1992. Tetrahedral elements for

    fluid flow. International Journal for Numerical Methods in Engineering

    33, 12511257.Bertrand, F., Tanguy, P.A., Thibault, F., 1997. A three-dimensional fictitious

    domain method for incompressible fluid flow problems. International

    Journal for Numerical Methods in Fluids 25, 719736.

    De la Villeon, J., Bertrand, F., Tanguy, P.A., Labrie, R., Bousquet, J.,

    Lebouvier, D., 1998. Numerical investigation of mixing efficiency of helical

    ribbons. A.I.Ch.E. Journal 44, 972977.

    Dubois, C., Thibault, F., Tanguy, P.A., Ait-Kadi, A., 1996. Characterization of

    viscous mixing in a twin intermeshing conical helical mixers. Institution

    of Chemical Engineers Symposium Series 140, 249258.

    Espinosa-Solares, T., Brito-De La Fuente, E., Tecante, A., Tanguy, P.A., 1997.

    Power consumption of a dual turbine-helical ribbon impeller mixer in

    ungassed conditions. Chemical Engineering Journal 67, 215219.

    Espinosa-Solares, T., Brito-De La Fuente, E., Tecante, A., Tanguy, P.A., 2001.

    Flow patterns in rheologically evolving model fluids produced by hybrid

    mixing systems. Chemical Engineering Technology 24, 913918.

    Fortin, A., Fortin, M., Tanguy, P., 1986. Numerical simulation of viscous

    flows in hydraulic turbomachinery by the finite element method. Computer

    Methods in Applied Mechanics and Engineering 58, 337358.

    Foucault, S., Ascanio, G., Tanguy, P.A., 2004. Coaxial mixer hydrodynamics

    with Newtonian and non-Newtonian fluids. Chemical Engineering

    Technology 27 (3), 324329.

    Foucault, S., Ascanio, G., Tanguy, P.A., 2005. Power characteristics in coaxial

    mixing Newtonian and non-Newtonian fluids. Industrial & Engineering

    Chemistry Research 44, 50365043.

    Glowinski, R., Pan, T.W., Periaux, J., 1994. A fictious domain method

    for Dirichlet problem and applications. Computer Methods in Applied

    Mechanics and Engineering 111, 283.

    Heniche, M., Tanguy, P.A., 2005. A predictorcorrector shooting scheme for

    tracer trajectory calculations. In: Proceedings of the Fourth InternationalConference on Computational Fluid Dynamics in the Oil and Gas,

    Metallurgical Process Industries, Trondheim, Norway, 68 June.

    Heniche, M., Reeder, M.F., Tanguy, P.A., Fassano, J., 2005. Numerical

    investigation of blade shape in static mixing. A.I.Ch.E. Journal 51, 4458.

    Hoogendoorn, C., Den-Hartog, A., 1967. Model studies in the viscous flow

    region. Chemical Engineering Science 22, 16891699.

    Nagata, S., 1957. Mixing Principles and Applications. Wiley, USA.

    Ottino, J.M., 1989. The Kinematics of Mixing: Stretching, Chaos and

    Transport. Cambridge University Press, UK.

    Rivera, C., Heniche, M., Ascanio, G., Tanguy, P.A., 2004. A virtual finite

    element model for centered and eccentric mixer configurations. Computers

    and Chemical Engineering 28, 24592468.

    Salomon, J., Elson, T.P., Nienow, A.W., Pace, G.W., 1981. Cavern sizes in

    agitated fluids with yield stress. Chemical Engineering Communications11, 143164.

    Souvaliotis, A., Jana, S.C., Ottino, J.M., 1995. Potentialities and limitations

    of mixing simulations. A.I.Ch.E. Journal 41, 16051621.

    Tanguy, P.A., Fortn, M., Choplin, L., 1984. Finite element solution of dip

    coating, II: Non-Newtonian fluids. International Journal for Numerical

    Methods in Fluids 4, 459475.

    Tanguy, P.A., Thibault, F., Brito-De La Fuente, E., Espinosa-Solares, T.,

    Tecante, A., 1997. Mixing performance induced by coaxial flat blade-

    helical ribbon impellers rotating at different speeds. Chemical Engineering

    Science 52, 17331741.

    Tanguy, P.A., Bertrand, F., Dubois, C., Ait-Kadi, A., 1999. Mixing

    hydrodynamics in a planetary mixers. Transactions of the Institution of

    Chemical Engineers 77, 318324.

    Tatterson, G., 1992. Scale Up and Design of Industrial Mixing Processes.

    McGraw-Hill, New York, USA.

  • 8/4/2019 2006 Chem Eng Sci

    13/13

    C. Rivera et al. / Chemical Engineering Science 61 (2006) 28952907 2907

    Thibault, F., Tanguy, P.A., 2002. Power draw analysis of coaxial mixer with

    Newtonian and non-Newtonian fluids in the laminar regime. Chemical

    Engineering Science 57, 38613872.

    Van der Vorst, H.A., 1992. BI-CGSTAB: a fast and smoothly converging

    variant of BI-CG for the solution of nonsymmetric linear systems. SIAM

    Journal on Scientific and Statistical Computing 13, 631644.

    Yap, C.Y., Patterson, W.I., Carreau, P.J., 1979. Mixing with helical ribbon

    agitators, Part III: Non-Newtonian fluids. A.I.Ch.E. Journal 25, 516521.