Ability of URANS approach in prediction of unsteady turbulent flows in an unbaffled stirred tank

Accepted Manuscript

Ability of URANS Approach in Prediction of Unsteady Turbulent Flows in an Unbaffled Stirred Tank Ali Zamiri, Jin Taek Chung PII: DOI: Reference:

S0020-7403(16)30987-0 10.1016/j.ijmecsci.2017.08.008 MS 3831

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

4 December 2016 8 June 2017 1 August 2017

Please cite this article as: Ali Zamiri, Jin Taek Chung, Ability of URANS Approach in Prediction of Unsteady Turbulent Flows in an Unbaffled Stirred Tank, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.08.008

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Highlights • URANS approach for prediction of impeller interaction with fluid inside

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the vessel is proposed.

• Numerical simulation is validated by the experimental data in terms of velocity and turbulent kinetic energy profiles.

• Pressure waves with different convective velocities and periodic unsteady separation, are captured in the time/space domain along the circumferen-

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tial direction close to impeller.

• It is expected that the URANS can be used as a prediction tool not only

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for mixing performance, but also for transient characteristics.

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Ability of URANS Approach in Prediction of Unsteady Turbulent Flows in an Unbaffled Stirred Tank Ali Zamiri, Jin Taek Chung∗

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School of Mechanical Engineering, Korea University, Seoul, Rep. of Korea

Abstract

Three-dimensional, unsteady Navier-Stokes equations are numerically solved to

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investigate the turbulent flows in a stirred vessel. The computational domain

consists of an unbaffled, cylindrical vessel with a pitched-blade turbine impeller. An Eulerian-Eulerian multiphase flow model is applied to determine the shape of the free-surface vortex core. This numerical method is validated by comparing its results with laser Doppler velocimetry measurements in terms of velocity

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distribution and turbulence kinetic energy profiles at different positions. In the present study, URANS approach with a hybrid zonal turbulence model, k − ω

SST and SST-SAS, is used to predict the unsteady pressure and velocity fluc-

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tuations within the vessel. Pressure and inward-outward radial velocity waves are generated by the impeller rotation and are captured in the time/space do-

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main close to the impeller trailing edge. The pressure and velocity spectra are computed to characterize the blade passing frequency as the main source of unsteadiness in the turbulent flow within the vessel. The results indicate that the

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current URANS approach with a proper turbulence model and well-resolved grids can be used as a predictive tool for the flow field and large turbulence scales in the stirred tanks.

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Keywords: CFD, URANS, Pitched blade impeller, Stirred vessel, Blade passing frequency

∗ Corresponding

author Email address: [email protected] (Jin Taek Chung)

Preprint submitted to Elsevier

August 9, 2017

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1. Introduction Stirred tanks play an important role in a wide range of industrial applications such as chemical, biochemical and mineral industries. Mixing, crystallization,

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gas dispersion and heat transfer are significant examples of industrial operations performed by mechanically agitated vessels. One of the most frequently

applications of agitators is homogenization of miscible liquids in mixing pro-

cesses which has an important effect on both the quality of the product and the production costs. Due to three-dimensional, turbulent and complex flow structures generated by the impeller in stirred tanks, a detailed understanding of the fluid flow within the agitated vessel is needed to be able to improve the

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mixing performance as well as the quality of the mixture.

Extensive experimental studies have been carried out to better understand the flow structures and the fundamental properties of liquids in stirred vessels [1-4]. Reliable information can be obtained by taking measurements via hot-wire anemometry, laser Doppler anemometry (LDA) [5] and particle image velocime-

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try (PIV) [6]. Fort et al. [7] investigated the pumping capacity of pitched blade impellers in a cylindrical pilot plant vessel with four standard radial baffles at

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the wall under a turbulent flow regime. Ranade et al. [8] and Roy et al. [9] investigated trailing edge vortices by considering the turbulent kinetic energy 20

distribution around the pitched blade impellers. Joshi et al. [10] carried out

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detailed laser Doppler velocimetry (LDV) on down-pumping pitched blade turbines to predict the influence that the impeller geometry and size had on the

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flow patterns in the vessel. Mavros et al. [11] used LDV to compare the flow fields of radial and axial flow impellers in both Newtonian and non-Newtonian fluid.

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Although advanced experimental methods have resulted in significant contri-

butions to the evaluation of complex turbulent flows in stirred tanks, but these methods still present some limitations. For example, PIV and LDA techniques cannot be applied with opaque fluids, in nontransparent vessels, or when the

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system is sensitive to laser radiation. Meanwhile, the development of high per-

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formance computers and mathematical models has resulted in computational fluid dynamics (CFD) becoming a powerful tool to determine comprehensive fundamental conceptions on turbulent mixing processes, circulation patterns,

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vortical structures and Reynolds stresses in agitated tanks which would be expensive to measure or impossible to obtain experimentally [12, 13].

Many investigations have used CFD techniques to present detailed descrip-

tions of the flow field and velocity distribution in a mixing vessel [14-17]. In CFD simulations of a stirred tank, various parameters should be taken into ac-

count, including the appropriate grid resolution, discretization scheme, impeller rotation model and turbulence closure. Many researchers have presented as-

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sessments of different turbulent models in stirred vessels [10, 18-20] and one of the option is to conduct a direct numerical simulation (DNS) that resolves all turbulent time and length scales. However, a DNS is not a feasible method for stirred tank flow in industrial application. The most common turbulent models are the Reynolds-averaged Navier-Stokes (RANS) approach and the large eddy simulation (LES) method.

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Hartmann et al. [18] used the standard k −  turbulent model, Reynolds

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stress model (RSM) and LES model to study the turbulence characteristics

in a baffled stirred tank driven by a Rushton impeller, and they compared 50

their results with experimental measurements. They found that the transient

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k −  RANS simulation was able to provide an accurate representation of the

flow field, but it failed to predict turbulent kinetic energy and discharge flow

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in the impeller region where most of the mixing takes place. Alcamo et al. [21] conducted a transient simulation in combination with the LES turbulence

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model for a single phase liquid in an unbaffled stirred tank driven by a Rushton

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turbine impeller and they found a good agreement between numerical results and experimental measurements. However, the LES approach is out of reach in the most practical settings because it requires expensive computational resources. Wu [19] implemented six different turbulence RANS models and LES with three

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sub-grid scale models in an agitated vessel with an axial impeller, and found that the global flow patterns for all RANS simulations were similar to those obtained 5

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with LES. Further studies have investigated the transient simulations of stirred tanks, but to the best of the authors knowledge, no publication has investigated the circumferential pressure fluctuations, pressure and velocity fluctuations close to impeller tip, and unsteady impeller surface pressure wave propagation.

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In the present work, a prediction method based on URANS (unsteady Reynoldsaveraged Navier-Stokes) approach is utilized to predict the flow field and tur-

bulent characteristics of an unbaffled stirred vessel with a pitched blade turbine impeller consisting of six inclined blades. The three-dimensional, incompress70

ible, multiphase unsteady simulations were carried out using the ANSYS CFX

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V16.2 software. Numerical simulations were then conducted using different tur-

bulence models and different grid resolutions to validate the velocity profiles that were predicted by comparing the results to experimental data [22].The current paper is aimed in investigation of the three-dimensional flow field to identify 75

physical unsteady flow structures caused by the impeller rotation. Therefore, further details of the pressure and velocity fluctuations within the stirred ves-

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sel were captured using transient simulations. Spectral analysis, Fast Fourier Transform (FFT) of the pressure fluctuations, showed that the BPF (blade

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passing frequency) is the dominant source of fluctuations within the vessel. The numerical results obtained in this study show that the current URANS approach with a proper turbulence model and the well-resolved grids provides promising

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results that are comparable to those obtained with experiments and requires

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fewer computational resources than DNS or LES. 2. Numerical Method 2.1. Geometry and computational model

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Figure 1 depicts the schematics of the experimental apparatus proposed by

Armenante et al. [22], which was simulated in this study. The stirred vessel consists of a standard configuration, unbaffled, cylindrical, flat-bottomed vessel with a diameter of 0.293 m (T), filled with water at room temperature (density

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= 997 kg/m3 , dynamic viscosity = 0.001 P a.s) to a height equal to the tank 6

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Figure 1: Left: schematic of the vessel and impeller configuration (positions of the monitoring lines are indicated by the dashed lines); right: geometry of the pitched-blade turbine impeller.

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diameter (H=T). In the experimental model, a lid is positioned at a height of 0.293 m to prevent the formation of a central free vortex that is typically observed in unbaffled stirred vessels. However, the vessel height in the numerical

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simulations was extended to 0.4 m to be able to capture the free surface vortex core at the top of the vessel.

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The agitation system consists of a downward-pumping, six blade, 45-degree

pitched blade turbine with a diameter of 0.098 m (D=T/3), as shown in Fig.1.

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The impeller was attached to a centrally-located shaft and was mounted 0.073 T [m]

H/T

T/D

h [m]

hb [m]

lb [m]

t [m]

0.293

1

3

0.073

0.0138

0.0196

0.002

Table 1: Geometric parameters and dimensions of the vessel and the pitched-blade impeller

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Figure 2: Total view of computational domain consisted of stationary and rotating domains.

m off the tank bottom, as measured from the bottom of the impeller. The basic 100

information regarding the dimensions of the vessel and impeller is summarized

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in Table 1. The impeller rotational speed was set to 450 rpm, corresponding to the impeller Reynolds number of 7.1 × 104 (Re = (ρN D2 )/µ, where ρ is the

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fluid density, µ is the fluid dynamic viscosity and N is the impeller rotational speed in rps), with a tip speed of 2.3 m/s.

The three-dimensional computational fluid domain for the numerical simu-

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lation is shown in Fig.2, in which the domain is extended in the axial direction more than in the real one to capture the free surface vortex. The computational

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model consists of: 1) a stationary vessel domain and 2) a rotating domain included of the PBT impeller.

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The total grid used for the computational domain consists of 5,100,000 and

3,200,000 grid points for the rotating and stationary zones, respectively. To ensure a good quality for the mesh, very fine meshes are generated near the wall surfaces to preserve the boundary layer effect. Although, our calculation is 8

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intended for incompressible flow, the fully turbulent and separated flows close 115

to the impeller, result in a value of y + , average dimensionless wall distance, for the first grid point above the wall of the impeller that is less than 3 (y + < 3).

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Figure 3 illustrates the y + contour on the impeller walls which confirms that y + is less than 3 everywhere even for the sharp edges of the impeller blades. Moreover, a finer mesh was used at the water-air interface to accurately track 120

the free surface vortex formation. 2.2. Governing Equations

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The computation was conducted by solving the three-dimensional, incom-

pressible, two-phase URANS equations, Eqs. (1) and (2), using the commercial finite volume ANSYS CFX V16.2 solver,

(1)

∂ (ρU ) + O.(ρU ⊗ U − µ(OU + (OU )T )) = SM − Op ∂t

(2)

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∂ (rα ρα ) + O.(rα ρα Uα ) = 0 ∂t

where α is denoted for each phase, U is the averaged velocity and r is the volume

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fraction. The density ρ and viscosity µ for each phase are calculated by Eqs.

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(3) and (4), respectively.

Figure 3: y + contour at the surfaces of the impeller blades.

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ρ=

NP X

rα ρα

(3)

NP X

rα µα

(4)

µ=

α=1

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α=1

An Eulerian-Eulerian multiphase model was used to calculate the flow field [23] and the MRF (multi reference frames) approach was used to simulate the 130

rotation of the impeller inside the vessel [23, 24]. The non-slip boundary condition was applied for all solid walls, and the hydrostatic pressure distribution

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inside of the vessel was specified with a mathematical equation to define the initial height of the water (0.293 m). At the top of the computational domain

in the air phase region, the opening boundary condition was chosen to allow the 135

vortices to pass the boundary in and out at the same time.

For the turbulence closure model, the k − ω shear stress transport model

(SST) was used which is known as the most accurate and efficient turbulence

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model for separated flows [25-27]. In addition, CEL functions (CFX expression language) and user-defined routines were specified to active the SST-SAS (scale adaptive simulation) turbulence model for the rotating domain. Scale adaptive

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simulation model is an improved URANS approach with ability to adopt the length scale to resolved turbulent structures. This model is based on the von

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Karman length scale and resolves turbulence structures in a URANS simulation, which results in a LES-like behavior in unsteady flow regions, while provides 145

standard RANS capabilities in stable regions. Therefore, the SAS model cannot

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be forced to go unsteady by grid refinement same as DES or LES. Moreover, the surface tension and free surface model were applied to resolve the water-air

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interface and to capture the free surface vortex core. A high-resolution scheme [28] consisting of a hybrid method between the first order upwind and a second

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order upwind schemes was used to discretize the convection terms in order to reduce the numerical diffusion errors. For steady simulations, a frozen rotor model was applied at the interface of the impeller and vessel, in which the frame of reference is changed, but 10

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the relative orientation of the components across the interface is fixed. In the 155

case of transient simulations, the transient rotor-stator interface was used, and the second order backward Euler scheme was applied for time discretization.

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Considering the impeller grid rotation corresponding to the rotational speed of the impeller, a time step with 2-degree increments in the angle was selected. The transient simulation was thus carried out until the fluctuations of the flow 160

field become time periodic. 2.3. Parametric Sensitivity Studies

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The influence of grid resolution on the accuracy of the computational results is studied by conducting a numerical test using different grid resolutions and different turbulence models and comparing the results with those obtained 165

through an experiment. In the case of the mesh independence test, three different grid resolutions, including fine (12,100,000 cells), moderate (8,300,000 cells) and coarse (3,500,000 cells), were computed with the k − ω SST turbulence

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model. Table 2 shows a comparison of the torque values captured on the impeller blades for different mesh resolutions. The torque values of the moderate and fine meshes are almost the same as each other, and there is no significant

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change in the results by increasing the number of grid cells. The radial velocity profiles along the axial direction close to the trailing edge of the impeller are

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also compared for different grid resolutions, as shown in Fig. 4. The figure shows that there is a good agreement between the results obtained for the fine 175

and moderate meshes. However, in the case of the coarse mesh, the velocity

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profile is different, especially in the range of 0.1 < Y < 0.25. Therefore, the

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moderate mesh was selected for all simulations of the present study to reduce

Number of grid points Torque [N.m]

3.5 million

8.3 million

12.1 million

(Coarse)

(Moderate)

(Fine)

0.0514

0.0560

0.0572

Table 2: Mesh independence test

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Figure 4: Radial velocity profiles along the axial direction close to the impeller trailing edge for different grid resolutions.

the computational time.

Furthermore, three different turbulence models, (k − , RNG k −  and k − ω

SST), were evaluated for the moderate mesh to show the influence of the tur-

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bulence closure on the accuracy of the numerical solution. Figure 5 compares

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the numerical results to the measurements obtained for axial and radial velocity profiles normalized by the impeller tip speed in the plane under the impeller region (corresponding to Z=0.071 m)[22]. In the experiments, the tridimensional average and fluctuating velocities close to the impeller region were deter-

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mined by LDV at 20 different radial distances between the impeller shaft and

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the vessel wall [22]. Therefore, to obtain a reasonable comparison and achieve pseudo-stationary results, it is necessary to compute the time-averaged and spatial-averaged velocities for the numerical results. The time-averaged results were calculated for more than 10 impeller revolutions, and spatial-averaged re-

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sults were obtained for more than 51 different positions in the radial and axial directions. As depicted in Fig. 5, the results of the simulation based on the k − ω SST model were a better match for those obtained through experiments than the results of other turbulence models. Thus, the moderate mesh and k −ω

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Figure 5: Comparison of the normalized velocity profile at the bottom plane of the volume

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swept by the impeller (Z=0.071 m) for the numerical simulation and experiment; top: radial velocity; bottom:axial velocity [positive values indicate upward velocity in the axial direction, or outward velocity in the radial direction].

SST turbulence model were applied for all simulations in this study.

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3. Results and Discussions

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3.1. Time-averaged data Figures 6 and 7 compare the time-averaged computed axial and radial veloc-

ities with the results obtained through experiments four different axial levels (Z

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equal to 0.053, 0.071, 0.088 and 0.160 m, see Fig.1), respectively. As observed in Fig. 6, the axial velocities right below and above the impeller are all negative due to the effect of the downward pumping impeller. Moreover, the magnitude of the axial velocities above the impeller are lower than those below the impeller, 13

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Figure 6: Comparison of the normalized axial velocity between the experimental measurements via LDV [22] and computational results at four different horizontal locations (Z equal to 0.053,

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0.071, 0.088 and 0.160 m).

which indicates a strong downward pumping flow close to the impeller tip. A 205

large radial velocity was observed in the plane below the impeller close to the

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impeller tip, and this was caused by the wake region of the impeller tip jet flow (Fig. 7). Also the influence of the impeller jet flow on the radial velocity was

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observed at the bottom of the vessel in the range of 2r/T > 0.4, for which the magnitude of the radial velocity increased toward the vessel wall. Figures 6 and

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7 show a very good agreement in the velocity fields obtained from the numerical and experimental results. The comparison of the computed turbulence kinetic energy with experimental data of Armenante et al. [22] at different axial levels corresponding to Figs. 14

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Figure 7: Comparison of normalized radial velocity between experimental measurements via LDV [22] and computational results at four different horizontal locations (Z equal to 0.053, 0.071, 0.088 and 0.160 m).

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6 and 7 is presented in Fig.8. A good agreement is obtained for the turbulence kinetic energy at the upper parts of the impeller (Z values equal to 0.088 and

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0.160 m). Although, the computed turbulence kinetic energy is underestimated

compared to that of the experiment at down parts of the impeller (Z values

equal to 0.053 and 0.071 m), the trend of turbulence kinetic energy profile of

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numerical simulation is quantitatively well matched with the experimental data.

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The power number can be calculated by two different methods in the numer-

ical simulations. One is obtained using the torque applied on the impeller and the other is calculated by integrating the turbulence energy dissipation rate  over the vessel volume. Table 3 summarizes the prediction of the power number

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Figure 8: Comparison of dimensionless turbulence kinetic energy between experimental measurements via LDV [22] and computational results at four different horizontal locations (Z equal to 0.053, 0.071, 0.088 and 0.160 m).

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calculated by these two different methods. It is observed that the power number determined from the torque on the impeller over-predicted by 4.62% compared

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to experimental data. However, the predicted power number by integrating the turbulence energy dissipation rate is close to the measured power number of

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0.67 (1.79% error).

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Figure 9 shows the time-averaged pressure, velocity magnitude and vorticity

contours for the mixture of water and air on a mid-plane of the tank. Figure 9(a) confirms that our mathematical model works well to predict the hydrostatic pressure of the water column inside the vessel. Moreover, the influence of the impeller rotation on the static pressure is clearly observed in the pressure

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Experiment

CFD based on torque

CFD based on 

0.670

0.701

0.658

Power Number

Table 3: Comparison of calculated power number using different methods with experimental

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data

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Figure 9: Time-averaged plots at the cutting mid-plane of the vessel: (a) pressure, (b) velocity magnitude in stationary frame, and (c) vorticity contour (red and blue colors show

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counterclockwise and clockwise rotation in the vorticity contour, respectively).

distribution contour levels (curved contour levels in the axial direction). The red circles in Fig. 9(b) show that higher velocity regions generated by the jet

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flows are observed close to the impeller tips. The higher velocity at the top of the vessel close to the rotating shaft is generated by the rotation of air at the

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top of the vessel due to the free-surface vortex formation. Figure 9(c) presents two contra-rotating vortices generated by the impeller jet flow, and these are

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convected to the bottom of the vessel by impeller rotation. In addition, there

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are vortical structures at the top of the vessel caused by rotational air flow. Figure 10 shows the instantaneous velocity vectors of the gas and liquid

phases in the mid-plane. Both the gas and liquid phases have recirculating flows inside the vessels. The impeller jets are found to be strongly directed

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outward in the radial direction, making two main contra-rotating vortices, one

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Figure 10: Time-averaged of tangential velocity vectors at the cutting mid-plane of the vessel (two contra-rotating vortices depicted by red rectangle).

above and the other below the impeller. The discharge flow from the impeller impinges on the bottom of the vessel at 45 degrees, and it moves to the side walls and extends up to 2/3 of the vessel height, as observed by Bittorf and Kresta

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[29]. Then it returns to the impeller stream through the core of the tank. This process comprises the primary circulation loop. Moreover, a weak reverse

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flow below the hub of the impeller generates a relatively strong reverse flow (radially inward) at the bottom of the vessel. This counter-rotating secondary loop leads to deflection of the discharge angle of the primary circulation loop, as

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was reported by Kresta and Wood [1] and Bruha et al. [30]. Therefore, the flow is redirected in the radial direction and the axial velocity component decreases. In the region above the impeller (core of the tank), the flow stream is mainly

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directed in the axial direction toward the impeller. The contour of the volume fraction on the mid-plane of the vessel is shown in

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Fig. 11(a), in which the air-water interface profile can be clearly identified. The

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iso-surface of the water volume fraction is shown in Fig. 11(b), which confirms our free surface modeling works well in capturing the central vortex core. In addition, to better understand and visualize the flow structure within the

stirred tank, the iso-surface of the second invariant of the velocity gradient Qcr (Q criterion) colored with static pressure was used to show the instantaneous 18

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Figure 11: (a) Contour of the volume fraction on the mid-plane of the vessel (red and blue colors indicate water and air, respectively); (b) iso-surface of water volume fraction (value is 0.95).

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flow structure (Fig. 12). Qcr is used to determine the vortical structures, and it indicates that the vorticity magnitude prevails over the strain rate magnitude

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in a vortical structure. There are two vortex rings close to the bottom of the vessel due to two contra-rotating vortices generated by the impeller jet flows.

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Moreover, the vortex filaments that are captured clearly show the vortical flows

Figure 12: Instantaneous iso-surface of Qcr inside the vessel superimposed by pressure contour.

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Figure 13: Time/space contour map of unsteady pressure at the impeller mid-plane circumferential direction close to the impeller trailing edge (separation waves are indicated by rectangle).

within the vessel, and the fluid elements are stretched and moved along these filaments. 3.2. Time-series data

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Figure 13 depicts the static pressure contour on the mid-span circumferential direction close to the impeller trailing edge with respect to the fraction of the 275

revolution (one period of impeller rotation). Pressure waves are clearly observed

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around the circumferential direction, and the slope of these waves represents the convective velocity. The main flow structures are convected toward the vessel

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wall at this speed. As shown in the figure, six clear blade pressure waves are generated by the interaction of the impeller trailing edges with the fluid inside

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the vessel. Furthermore, there are other pressure waves between the blade

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pressure waves with lower magnitude and shorter wave lengths caused by the separation and wake flows of the impeller trailing edges. These waves are also convected to the vessel wall with the same convective velocity.

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The instantaneous radial velocity vectors of the mixture along the monitoring

line are shown in Fig. 14(a) and the inward and outward radial velocity are

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Figure 14: (a) Instantaneous radial velocity vectors on the axial monitoring line close to impeller trailing edge (inward and upward radial velocities are indicated by red circles); (b) Time/space contour map of radial velocity of the mixture along the monitoring line.

indicated using red circles. Figure 14(b) presents the radial velocity contour of

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the mixture on the monitoring line corresponding to Fig. 14(a) with respect to the time of one impeller revolution. Six clear outward radial velocity waves generated by the impeller rotation can be seen just below the impeller, which is in good agreement with the velocity profiles shown in Fig. 14(a). Moreover,

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six inward radial velocity waves are observed in the contour plot with a lower magnitude close to the top of the impeller blades. Therefore, the fluid is sheared

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and dispersed by these inward and outward radial velocities generated by the impeller inclined blades. In the range of 0.12 < Y < 0.3, the radial velocity is almost constant which is due to axial flow direction at the top of the impeller

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(see Fig.7). The radial velocity increases in the region of Y > 0.3 which shows the existence of rotational air flow at the top of the vessel due to the vortex

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core.

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The flow field behavior in the stirred tanks is fully transient due to the three-

dimensional turbulent flow and impeller rotation. Figure 15 presents the time evolution of the pressure and velocity fluctuation versus the normalized time at two different monitoring points, one close to the impeller trailing edge (A) and

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the other close to the vessel wall (B) (the monitoring points are indicated at the top of Fig. 15). The recorded pressure and velocity signals show the turbulent 305

fluctuations including the periodic components and the random turbulence. The

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fluctuation with a large magnitude are related to the periodic component caused by the impeller blade passage (six peaks for one period T due to six blades of the impeller), and the fluctuations with small magnitude superimposed with pe-

riodic components are generated by the random turbulence. Moreover, the time 310

periodic signals indicate the convergence of the unsteady solution at different locations within the vessel. The pressure signals at two different locations are

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almost similar to each other. However, the velocity signal at monitoring point B shows that the velocity magnitude is reduced close to the wall compared to that near the trailing edge due to the diffusion effect.

Turbulent flow is consisted of a range of time and spatial scales. Therefore, it

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is necessary to compute the energy spectrum to describe how the energy of one of the components is distributed over different frequencies. The one-dimensional

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power spectral density (PSD) can be estimated by taking FFT on the velocity and pressure signals. The power density spectra corresponding to the pressure and velocity signals at monitoring point A (close to the trailing edge of the

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impeller) are shown in Fig. 16. The expected fundamental frequencies, sharp peaks in the spectra corresponded to the BPF and its harmonics (the first BPF

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is 45 Hz), can be clearly identified in Fig. 16, which are in good agreement with the theoretical BPF. The power density magnitude of BPF peaks is about one or two orders larger than the power density of the low frequencies which

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indicates that a large amount of kinetic energy due to coherent velocity and

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pressure fluctuations is caused by the blade passages. The straight lines with a slope of −5/3 in the spectra represent the energy

distribution predicted using the Kolmogorov hypothesis from a dimensional sim-

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ilarity analysis for the turbulence inertial subrange [32]. The agreement between the baseline of the energy spectra calculated by the current URANS approach and the −5/3 slope confirms the reasonably good prediction of the turbulent ki-

netic energy of this study. This agreement indicates the existence of the inertial 22

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Figure 15: Pressure and velocity fluctuations at various monitoring points versus normalized time, T is a period for one revolution of the impeller.

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subrange and some isotropic properties in the stirred tank.

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In addition, the red circles in Fig. 16 show peaks around 15 Hz in the

velocity and pressure spectra that are related to the large scale structures of the flow [31]. This low frequency fluctuation of the signals is commonly referred as the macro-instability fluctuations which was also observed by Bruha et al. [30]

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Figure 16: Power spectral density spectrum of; left: pressure fluctuations, right: velocity fluctuations (red line reports the -5/3 law of Kolmogorov).

and Kresta [31]. In the present study, this frequency can be associated with the 340

oscillation of the primary circulation loop that rotates slower compared to the

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flow region close to the impeller.

Figure 17: Wavelet analysis for monitoring point A near the impeller trailing edge.

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Time-frequency analysis, wavelet analysis, is most commonly performed by segmenting a signal into short periods and estimating the spectrum. Wavelet analysis is able to determine the dominant modes of variability and to show how they vary in time by decomposing the time into time-frequency space.

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345

Therefore, the wavelet analysis has been used for detecting and identifying the

coherent structures in turbulent flows [9, 33]. Figure 17 shows the contour

plot of the wavelet spectrum obtained by the wavelet analysis performed on

the pressure and velocity signals, which captured near the trailing edge of the 350

impeller blade (monitoring point A). The vertical and horizontal axes show the

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frequency and time, respectively. The wavelet analysis identifies frequencies

with high amplitude fluctuation over the entire sampling time which are related to the BPF (45 Hz) and its harmonics which are in good agreement with those of identified in Fig. 16. Moreover, as indicated by the red rectangles in Fig. 355

17, wavelet analysis clearly detects the peak with lower frequency f=15 Hz and

4. Conclusions

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higher amplitude which is belonged to the macro-instability oscillations.

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This paper provides a detailed description of the principal flow pattern found in the stirred tank with a pitched-blade turbine impeller. A three-dimensional 360

numerical study based on the URANS approach was carried out to simulate the

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multiphase fluid flow and to analyze the flow structure within the stirred tank. The computational results were in good agreement with the experimental data

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in terms of the velocity distribution and turbulence kinetic energy profiles at different locations inside the vessel. Transient computations were performed to

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show the unsteadiness of the flow field inside the vessel generated by the impeller

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rotation. The time history of the velocity and pressure distributions confirmed the flow unsteadiness that is periodically caused by the impeller interaction with the working fluid inside the vessel.

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The time/space contour of the pressure and radial velocity close to the im-

peller trailing edge showed that high unsteady pressure and velocity waves pri-

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marily occur close to impeller trailing edge. These waves are convected to the vessel walls with the convective velocity as shown by the slope of the pressure waves. In the time/space contour of the radial velocity, a shear flow was clearly

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observed close to the impeller and was generated by the inward and outward radial velocities.

Regarding the unsteadiness inside the stirred tank, it is important to realize

that turbulence fluctuations are greatly dominated by the impeller rotation. Therefore, velocity and pressure fluctuation signals were captured close to the

impeller trailing edge for use in the spectral analysis, in which the BPF and its

harmonics were well predicted. In addition, a comparison of the Kolmogorov

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380

−5/3 law with the power spectrum results indicates that the current URANS

approach can resolve the large turbulence scales in the stirred tanks slightly

into the inertial subrange. These results indicate that the URANS approach with a proper turbulence model and well-resolved grids for the full model of 385

the stirred tank, can provide quite promising results, not only in predicting the

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blade passing frequency but also the baseline of the spectrum, requiring fewer

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computational resources than LES or DNS simulations. Acknowledgment

This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant

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funded by the Korea government Ministry of Trade, Industry & Energy (No.

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20144010200770), and by the Korea University Grant.

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