Engineering Science and Technology, an International Journal 19 (2016) 189–196
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Engineering Science and Technology,
an International Journal
j o u r n a l h o m e p a g e : h t t p : / / w w w. e l s e v i e r. c o m / l o c a t e / j e s t c h
Press: Karabuk University, Press Unit
ISSN (Printed) : 1302-0056
ISSN (Online) : 2215-0986
ISSN (E-Mail) : 1308-2043
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Available online at www.sciencedirect.com
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Full Length Article
Agitation of yield stress fluids in different vessel shapes
Houari Ameur *
Institut des Sciences et Technologies, Centre Universitaire Salhi Ahmed CUN-SA, BP 66, Naama 45000, Algeria
A R T I C L E
I N F O
Article history:
Received 18 February 2015
Received in revised form
20 June 2015
Accepted 20 June 2015
Available online 20 August 2015
Keywords:
Mechanical agitation
CFD
Yield stress fluid
Flow pattern
Power consumption
A B S T R A C T
The Agitation of yields stress fluids with a six-curved-blade impeller (Scaba 6SRGT) is numerically investigated in this paper. The xanthan gum solution in water which is used as a working fluid is modeled
by the Herschel–Bulkley model. The main purpose of this paper is to investigate the effect of vessel design
on the flow patterns, cavern size and energy consumption. Three different vessel shapes have been performed: a flat bottomed cylindrical vessel, a dished bottomed cylindrical vessel and a closed spherical
vessel. The comparison between the results obtained for the three vessel configurations has shown that
the spherical shapes provide uniform flows in the whole vessel volume and require less energy consumption. Effects of the agitation rate and the impeller clearance from the tank bottom for the spherical
vessel are also investigated. Some predicted results are compared with other literature data and a satisfactory agreement is found.
© 2015, Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under
the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Agitation is of a great importance and it is used in many industrial processes. In the various applications, stirred tanks are required
to fulfill several needs like suspension of solid particles, dispersion of gases into liquids, heat and mass transfer, etc. Although many
experimental as well as numerical studies on liquid flows in cylindrical vessels have been published, very little attention has been
paid to the study of flow fields and energy consumption in fully
closed vessels.
Nagata [1] proposes the use of hemispherical bottomed cylindrical reactors, envisaging probably the improvement of the reactor
efficiency, due to the cancellation of the profile discontinuity at the
wall-bottom junction. Other researchers [2–4] studied the power
consumption in closed vessels for Newtonian fluids; they noticed
a discrepancy between their results (closed vessels) and the same
type of results for open vessels.
For a Newtonian fluid, Medek and Fort [5] studied experimentally the distribution of pressure along the lid of fully filled closed
cylindrical vessels at mixing by high-speed impellers (a Rushton
turbine or a pitched bladed turbine). They found that the power input
is not greater than that of an open cylindrical reactor, and the
pumping number of the same impeller is increased by approximately 10% probably due to the uniformization of the flow pattern
at the top end of the reactor.
* Tel.: +213770343722, fax: +21349797640.
E-mail address: houari_ameur@yahoo.fr
Peer review under responsibility of Karabuk University.
Armenante et al. [6] determined, by experiments and CFD simulations, the velocity profiles and the turbulent kinetic energy
distribution for the flow generated by a pitched-blade turbine in
an unbaffled, flat-bottom, cylindrical tank provided with a lid, and
completely filled with water.
By numerical simulations, Ciofalo et al. [7] studied the Newtonian turbulent flow in closed and free surface unbaffled tanks stirred
by flat-bladed impellers. For a low viscous Newtonian fluid and turbulent flow regime, Taca and Paunescu [8] studied experimentally
the power input in a spherical closed vessel stirred by a Rushton
turbine or six pitched blade impeller. Taca and Paunescu [9] reported that the optimum shape a vessel used for the suspension
of solid particles should have is the spherical one. These authors
reported also that some anomalies have also been noticed for the
fully filled lidded cylindrical reactors, as compared to the open reactors: the decrease of the power number for turbulent regime
(Re > 75,000), with the increase of Reynolds number and the impeller diameter.
By CFD simulations, Yapici and Basturk [10] studied the conjugate heat transfer and homogeneously mixing two immiscible
different fluids in a stirred and heated hemispherical vessel. For a
Newtonian fluid and turbulent flow regime, Ammar et al. [11] studied
by numerical simulations the effect of vessel design on the flow
pattern generated with a pitched blade turbine.
Agitation of yield stress fluids results in the formation of a cavern
(well mixed region) around the impeller [12,13] and isolated regions
far away. Some works have been published using curved blade impellers to evaluate the cavern size as a function of the power drawn
by yield stress fluids including those by Galindo and Nienow [14,15]
for Lightnin A315 and Scaba 6SRGT impellers; Amanullah et al. [16]
http://dx.doi.org/10.1016/j.jestch.2015.06.007
2215-0986/© 2015, Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://
creativecommons.org/licenses/by-nc-nd/4.0/).
190
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
Fig. 1. Agitation system.
for axial flow SCABA 3SHPI impeller; Serrano-Carreon and Galindo
[17] for four different impellers (Rushton turbine, Chemineer He3, CD-6 and Scaba 6SRGT) in individual and dual arrangements;
Pakzad et al. [13,18,19] and Ameur et al. [20] for SCABA 6SRGT impellers. Pakzad et al. [21] were interested with the agitation of
viscoplastic fluids by Scaba-anchor coaxial mixers. For stirring shear
thinning fluids, Ameur and Bouzit [22] studied the effect of curvature blade on the power consumption. They found that the curve
bladed impeller requires less power consumption compared with
the flat bladed impeller.
Our search in the literature shows that a little space has been
reserved to the agitation of viscoplastic fluids by curved-bladed impellers within closed vessels. Therefore, the main purpose of this
paper is to investigate the flow fields and the energy required for
the agitation of viscoplastic fluids by a Scaba 6SRGT impeller. We
focus on the effects of vessel design on the flow fields, cavern size
and energy consumption. Three geometric configurations are realized to perform the test: a flat bottomed cylindrical vessel, a dished
bottomed cylindrical vessel and a spherical vessel. Effects of the agitation rate and the impeller clearance within a closed vessel are
also investigated.
2. Mixing system
Effects of the vessel design are investigated in this paper by realizing three types of vessels: a flat bottomed cylindrical vessel, a
dished bottomed cylindrical vessel and a spherical vessel (Fig. 1).
Each vessel is equipped by a Scaba 6SRGT impeller (Fig. 2) which
consists of six curved blades fixed on a disc with 8 mm of thickness.
The disc is attached on a cylindrical central shaft of diameter ds/
D = 0.05. The vessel height (H) is equal to its diameter (D),
D = 400 mm. All other parameters are listed in Table 1.
Effects of the impeller clearance from the tank bottom are also
studied. Three different configurations are realized for this purpose
and which are: c/D = 0.2, 0.35 and 0.5, respectively.
3. Mathematical background
The xanthan gum solution in water used in this study has a yield
stress behavior modeled by the Herschel–Bulkley model [18].
Thus, its apparent viscosity (η) is given by:
η=
τy
+ K γ n−1
γ
(1)
where τy is the yield stress, γ is the shear rate, and K and n are the
consistency index and the flow behavior index, respectively.
According to the measurements conducted by Galindo and
Nienow [15], the rheological properties of the xanthan gum solution used were summarized in Table 2.
The Herschel–Bulkley model used causes a numerical problem
during the CFD simulations because the non-Newtonian viscosity
becomes unbounded at small shear rate. This behavior causes instability during computation [23]. Thus, the modified Herschel–
Bulkley model was employed to avoid the numerical instability. It
was assumed that the xanthan gum solution acts as a very viscous
fluid with viscosity μo at τ ≤ τy and the fluid behavior is described
by a power law model at τ > τy [23]:
η = μ0 at τ ≤ τ y
n
n
⎡
⎛τ ⎞ ⎤
τ y + K ⎢γ n − ⎜ y ⎟ ⎥
⎝ μ0 ⎠ ⎦
⎣
η=
at τ > τ y
γ
(2)
Table 1
Vessel parameters.
[mm]
D
H
d
ds
dt
bt
400
400
60
20
8
6
Table 2
Rheological properties of the xanthan gum solution used in this work.
Fig. 2. Geometrical parameters of the impeller.
Xanthan gum concentration %
(in mass content)
K [Pa sn]
n [−]
τy [Pa]
3.5
33.1
0.18
20.6
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
The Metzner–Otto correlation [24] was employed to calculate
the modified Reynolds number for the Herschel–Bulkley fluids:
Re y =
ρNd
ρNd
ρNd
ρN d ks
ρN d ks
=
=
=
=
n
η
τ γ avg τ ks N
τ
τ y + K (k s N )
2
2
2
2
2
2
2
(3)
∫
ηQdv
(4)
vessel volume
The power number is calculated according to this equation:
NP =
P
ρN 3d 5
V*z
Num [present work]
Exp [13]
0.08
0.06
where γ avg is the average shear rate and ks is a Metzner–Otto
constant.
The power consumption is a macroscopic result obtained by integration on the impeller surface of the local power transmitted by
the impeller to the fluid:
P=
191
(5)
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
0.0
0.2
0.4
0.6
0.8
1.0
4. Numerical method
In this section, only a brief description of the computational techniques used in this work is presented. Further details can be found
in a previous paper [25].
Once the discretization of the flow domain (tetrahedral mesh)
has been performed using ICEM-CFD, the CFD solver CFX 13.0, Ansys
Inc., is used to obtain values for the velocity components and pressure at each of the node points. The solver uses a full finite volume
formulation to solve the fully-coupled mass and momentum conservation equations. The flow fields are solved in a rotating frame
of reference in which the flow is time-independent.
In this study, the Reynolds number is varying from 1 to 3000.
Even if the flow is not fully into the turbulent regime, we should
simulate the flow as being turbulent. Fortunately, the turbulent properties vanish when in the laminar flow, so the correct way of
simulating in the transitional regime is using a turbulence model.
In our study, we have used the SST model with specified intermittency of turbulence.
In each simulation, the solution is considered converged when
the normalized residuals for the three velocity components and the
pressure all fall below 10−6. Calculation of the velocity and pressure fields takes between 90 and 300 iterations to converge and each
run takes less than five hours of CPU time. The computations were
run in Core i7 CPU 2.20 GHz with 8.0 GB of RAM.
5. Results and discussion
Fig. 3. Axial velocity for Rey = 80.9, n = 0.12.
was ejected horizontally from the impeller and it was directed in
one of 2 ways. Either the fluid flowed up the wall toward the top
surface and then back down the impeller shaft, forming a recirculation zone above the impeller, or the material was directed down
to the dish center and was pulled up through the center of the dish
back to the blade, forming a second recirculation zone below the
impeller. In a comparison between the three cases, the recirculation loop center is very close to the blade tip for the spherical vessel,
and this is due to the vessel design.
The pressure distribution inside the three vessel shapes is also
presented (Fig. 6). As remarked, the pressure is strong in the area
swept by the impeller and highest value is reached at the blade tip
for the three vessels. In a comparison between the three cases, the
spherical shape yields the stronger pressure on the blade tip.
For a flat bottomed cylindrical vessel (Fig. 7a), we remark that
the liquid tends to move mainly along circular trajectories, resulting in weak radial flows directed toward the tank walls and small
relative velocities between the impeller and the fluid. This results
in a poor axial mixing and in the formation of a vortex on the free
Np
Exp [26]
Num [Present work]
10
At the beginning, it is necessary to check the validity of the CFD
code and the numerical method performed. For this objective, we
referred to the experimental work presented by Pakzad et al. [13].
With the same geometry (i.e. a baffled cylindrical vessel with a flat
bottom), variations of the axial velocity along the vessel height are
predicted and presented on Fig. 3. Another validation is made with
the experimental data of Woziwodzki and Słowiński [26]. These
authors used other impellers but the same type of fluid. For a
Rushton turbine and a Xanthan gum solution with 1% of concentration, values of the power number versus Reynolds number are
presented on Fig. 4. As observed on these figures (Figs. 3 and 4),
the comparison between our numerical results and the other experimental data shows a satisfactory agreement.
5.1. Effect of vessel shape
Figure 5 presents the flow patterns in a vertical plane containing the blade. It can be observed that for any vessel shape, the fluid
Z*
1
10
100
Rey
Fig. 4. Power consumption for a Rushton turbine and 1% of Xanthan gum
concentration.
192
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
Fig. 5. Flow patterns for Rey = 2500, c/D = 0.5, (a) flat bottomed cylindrical vessel, (b) dished bottomed cylindrical vessel, (c) spherical vessel.
surface of the liquid, whose depth depends on the impeller rotational speed. This is due to the absence of baffles.
At the same impeller rotational speed, if we change the vessel
shape by using a dished bottom, it can reduce the size of the vortex
which is formed on the free surface of the liquid (Fig. 7c).
The agitation of viscoplastic fluids results in the formation of a
well stirred region (cavern) near the impeller and stagnant zones
elsewhere. The stagnant zones yield poor heat and mass transfer
rates, high temperature gradients, etc. [27]. Therefore, the cavern
size should be increased as well as much as possible.
Figure 8 presents the cavern size generated by the Scaba for different vessel shapes. As observed, the cavern is limited in the area
swept by the turbine. Another remark is that the difference is slight
between the three cases studied for low Reynolds number. The
Fig. 6. Pressure distribution for Rey = 2500, c/D = 0.5, (a) flat bottomed cylindrical vessel, (b) dished bottomed cylindrical vessel, (c) spherical vessel.
Fig. 7. Flow patterns for Rey = 1000, c/D = 0.2, (a) flat bottomed cylindrical vessel, (b) dished bottomed cylindrical vessel, (c) spherical vessel.
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
193
Fig. 8. Cavern size for c/D = 0.5, (a) flat bottomed cylindrical vessel, (b) dished bottomed cylindrical vessel, (c) spherical vessel.
increasing Reynolds number yields a wider cavern for cylindrical
vessels than the spherical vessel (Fig. 8). But and if the impeller is
placed very close to the vessel base (Fig. 7), toroidal vortices will
appear near the free surface of liquid within cylindrical vessels. These
vortices may cause huge vibrations and consequently a damage of
the rotating impeller. This issue can be eliminated by using a spherical closed design.
At the vessel mid-height (Z* = 0.5), the radial velocity component is presented along the vessel radius for the three types of vessel
(Fig. 9a). We remark that the flat bottomed cylindrical vessel generates a powerful radial jet of flow when compared with the two
other cases.
The axial velocity component (V*z) is also presented for more description of the hydrodynamics induced (Fig. 9b). The V*z profiles
are plotted along the vessel height (Z*) for a radial position near
the blade tip (R* = 0.6). The minus sign of velocity indicates the existence of a recirculation zone. At the lower part of the vessel, the
axial velocity magnitude is intense for the case 1. However, at the
upper part of the vessel, this velocity component becomes more
intense within the spherical vessel.
The uniformization of the flow pattern at the vessel’s upper part
(for a closed spherical vessel) results in the recovery of centrifugal
forces and the increase of impeller flow rate.
As reported by Taca and Paunescu [8], the closed vessels are
highly useful in several mixing processes, especially if steep pressures are needed and when the operation of agitation without the
presence of dispersed gases (air) are needed. A closed vessel is also
favorable as the whole vessel can be used since it may be filled all
the way to the top. Moreover, the vessel bases are not required for
preventing the vortex formation.
Vr*
Z*
Flat bottomed cylindrical vessel
Dished bottomed cylindrical vessel
Spherical vessel
0.30
5.2. Effect of impeller clearance
The impeller location has an important effect on the performance of a mixing system. In this section, we investigate the effect
of impeller clearance from the tank bottom. Three geometric
Flat bottomed cylindrical vessel
Dished bottomed cylindrical vessel
Spherical vessel
1.2
1.0
0.25
0.8
0.20
0.6
0.15
0.4
0.10
0.2
0.05
0.0
0.00
0.2
0.4
0.6
(a)
0.8
1.0
R*
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
(b)
Fig. 9. Velocity profiles for Rey = 1000, θ = 0°, c/D = 0.2, (a) Radial component at Z* = 0.5 (b) Axial component at R* = 0.6.
0.03
0.04
Vz*
194
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
Fig. 10. Flow patterns for Rey = 70.
configurations have been tested for a spherical vessel and which
are: c/D = 0.5, 0.35 and 0.2. As observed on Fig. 10a, the fluid flow
is blocked near the vessel base when the impeller clearance is very
small, resulting in higher power consumption (Fig. 11). On the other
hand, the rest of the vessel is poorly stirred (Fig. 10b).
Knowledge of the power consumption is very important for the
choice of the system installed. The power consumption depends on
all parameters characterizing the impeller geometry, the vessel geometry and the impeller rotational speed.
Figure 11 presents the numerical values of the power consumption required for different vessel shapes and different clearances from
the vessel base. The numerical investigations showed the superior
performances of spherical reactors, as compared to the classical flat
bottomed and dished bottomed cylindrical reactor. The improved
Flat bottomed cylindrical vessel
Dished bottomed cylindrical vessel
Spherical vessel
Np
4.8
4.7
4.6
efficiency of the spherical reactor is due to its shape, having non
contour discontinuities at the wall-bottom and wall-lid junctions,
as would be the case for a cylindrical vessel. That demonstrates the
existence of the so-called “sphere effect” [9].
5.3. Effect of Reynolds number
The flow patterns generated by the Scaba for different agitation rates are presented on Fig. 12. With a low Reynolds number
(Case 1: Rey = 50), we remark that the radial flow impinging from
the impeller blade is weak, and centers of the recirculation loops
which are formed above and below the impeller are located in the
area swept by the blade. With increasing Reynolds number (Case
2: Rey = 200), these recirculation loops are displaced horizontally
toward the tank sidewalls. The center of each loop is located at the
blade tip in this case. With a greater Reynolds number (Case 3:
Rey = 3000), these loops are detached from the blade tip. This is due
to the higher radial jet which can increase the size of these structures.
For more details on this phenomenon, we presented on Fig. 13
the variation of axial velocity along the vessel radius for two locations: Z* = 0.26 (below the impeller) and Z* = 0.44 (above the
impeller). The minus sign of the velocity indicates the existence of
a recirculation loop. As observed, the increase of Reynolds number
yields a larger recirculation zone and a wider cavern (Fig. 13b).
6. Conclusion
4.5
4.4
4.3
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 11. Power consumption for Rey = 4.
0.50
c/D
The flow fields and the energy required for the agitation of
viscoplastic fluids by a Scaba 6SRGT impeller have been numerically investigated. Three different vessels were used to study the
design effect on the mixing system characteristics, namely: a flat
bottomed cylindrical vessel, a dished bottomed cylindrical vessel
and a spherical vessel.
For any vessel shape, the flow impinging from the Scaba is radial,
resulting in the formation of two recirculation loops. The comparison between the results obtained for the three vessel configurations
has shown that the spherical shape provides uniform flows in the
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
195
Fig. 12. Flow patterns for a spherical tank, c/D = 0.35.
whole vessel volume and it requires less energy consumption, recommending this type of vessel for a more frequent usage in the field
of the processing industries equipment.
Effects of the impeller clearance from the vessel bottom are also
studied for the case of a spherical vessel. It was found that the middle
of the vessel is the most appropriate position.
The agitation rate has also an important effect. With a curvedbladed impeller and low Reynolds number, the cavern size is
very limited. However, a sufficient impeller rotational can promote
the axial circulation of fluid, enlarge the cavern size and enhance
mixing.
Vz
*
Nomenclature
bt
c
d
ds
dt
h
n
D
blade thickness, m
impeller off-bottomed clearance, m
blade diameter, m
shaft diameter, m
disc thickness, m
blade height, m
flow behavior index, dimensionless
tank diameter, m
Vz*
Rey = 40
Rey = 600
Rey = 1200
Rey = 2500
0.05
0.04
0.03
Rey = 40
Rey = 600
Rey = 1200
Rey = 2500
0.02
0.01
0.02
0.00
0.01
-0.01
0.00
-0.02
-0.01
-0.02
-0.03
-0.03
-0.04
-0.04
0.0
0.2
0.4
(a)
0.6
0.8
*
1.0 R
0.0
0.2
0.4
0.6
0.8
1.0
(b)
Fig. 13. Radial profiles of the axial velocity component for a spherical vessel, c/D = 0.35, (a) below the impeller (Z* = 0.26), (b) above the impeller (Z* = 0.44).
R*
196
H
K
Ks
N
P
Np
Qv
R, Z
R*
Z*
Rey
Vz, Vθ, Vr
V*
H. Ameur / Engineering Science and Technology, an International Journal 19 (2016) 189–196
liquid level, m
consistency index, Pa sn
Metzner–Otto’s constant, dimensionless
impeller rotational speed, 1/s
power, W
power number, dimensionless
viscous dissipation function, 1/s2
radial and axial coordinates, respectively, m
dimensionless radial coordinate, R* = 2R/D
dimensionless axial coordinate, Z* = Z/D
Reynolds number for a yield stress fluid, dimensionless
axial, tangential and radial velocities, respectively, m/s
dimensionless velocity, V* = V/πND
Greek letters
γ avg
average shear rate, 1/s
τ
τy
ρ
η
η0
θ
ω
shear stress, Pa
suspension yield stress, Pa
fluid density, kg/m3
apparent viscosity, Pa s
yielding viscosity, Pa s
angular coordinate, degree
angular velocity, rad/s
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