An experimental and computational investigation of vortex formation in an unbaffled stirred tank

Chemical Engineering Science xxx (2017) xxx–xxx

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An experimental and computational investigation of vortex formation in an unbaffled stirred tank S.S. Deshpande ⇑, K.K. Kar, J. Walker, J. Pressler, W. Su The Dow Chemical Company, Midland, MI 48674, USA

h i g h l i g h t s
Vortex depth depends upon Froude, Reynolds numbers, and impeller submergence.
New correlation proposed for vortex depth, significantly increasing Reynolds range.
New correlation has correct limiting behavior, and seems to be scale insensitive.

a r t i c l e

i n f o

Article history: Received 25 January 2017 Received in revised form 30 March 2017 Accepted 2 April 2017 Available online xxxx Keywords: Vortex depth Stirred tanks Experiments Computational fluid dynamics Reynolds number Froude number

a b s t r a c t The present article focuses on the quantification of the depth of a depression (i.e. a vortex) formed when stirring liquids in an unbaffled stirred vessel. Based on several experiments, we show that while the vortex depth may be described very well by Nagata’s (1975) inviscid model for large Reynolds number (ReD ¼ ND2 m J 104 ) this model does not apply for smaller Reynolds numbers. A number of researchers (e.g. Zlokarnik (1971), Rieger et al. (1979)) have addressed this difficulty by including Reynolds number in their correlations. However, those correlations produce unphysical estimates for ReD K 103 – a situation still of considerable industrial relevance. To this end, we have developed a new correlation based on 100 experiments where viscosity, impeller size, agitation speed, and impeller submergence were independently varied. The new correlation significantly extends the range of Reynolds numbers over which it can be applied (ReD 2 ð102 ; 105 Þ), and has the appropriate inviscid behavior, unlike the prior models. Using validated computer simulations in OpenFOAM, and additional experiments the correlation has been validated at various scales (0.046 m to 0.92 m tank diameters). The computational results also provide insights into the effect of fluid viscosity on the overall flow structure within the tank. In particular, the flow velocity magnitude rapidly decays away from the impeller for ReD K 103 , and the flow is no longer dominated by tangential motion of the fluid. Consequently, under viscous conditions, the surface motion and vortexing diminish rapidly as impeller submergence is increased. Ó 2017 Published by Elsevier Ltd.

1. Introduction Blending fluids is a key process in the chemical process industry, with applications such as fermentation (Silva et al., 1996; Atkinson and Mavituna, 1991), liquid-solid blending (Schober and Fitzpatrick, 2005), crystallization (Torbacke and Rasmuson, 2001), waste water treatment (Mali and Patwardhan, 2009), liquid-liquid, and gas-liquid mixing (Paul et al., 2004). A convenient method to bring about homogenization in large vessels is by mechanically agitating its contents using a rotating stirrer. The resulting flow is characterized by strong swirling in the mixing tank, particularly when baffles are absent, often leading to a ⇑ Corresponding author. E-mail address: [email protected] (S.S. Deshpande).

depression in the liquid surface. This phenomenon is referred to as ‘vortex formation’ (Paul et al., 2004; Nagata, 1975). Fig. 1 shows the formation of a vortex when water is stirred in a cylindrical vessel with a standard Rushton impeller (Rushton et al., 1950). For the setup shown in this figure, the lowermost portion of the depression is above the impeller location (Fig. 1a–c) until a speed of 150 rpm. When the agitation speed is further increased (Fig. 1(d–f)), the depression becomes deep enough to intercept the impeller. At this point, some portion of the impeller is in contact with the liquid phase while the remaining portion is in contact with the gas. This has several undesirable effects including loss of mixing power, mechanical stressing of the shaft, noise and vibration during mixing, and formation of large amplitude circumferential waves in the tank. As is clear from the images, vortex touching the impeller also leads to ingestion of gases from the headspace

http://dx.doi.org/10.1016/j.ces.2017.04.002 0009-2509/Ó 2017 Published by Elsevier Ltd.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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Nomenclature c C C D Fr D Fr T g GaD I N n NP P p r RC ReD T T t

off-bottom clearance [m] dimensionless bottom clearance c=D [–] fit coefficient [–] impeller diameter [m] impeller Froude number N2 D=g [–] tank Froude number, N 2 T=g [–] acceleration due to gravity [m/s2] Galileo number Re2D =FrD [–] identity matrix [–] agitation speed [Hz] normal vector [–] power number [–] power [W] pressure [Pa] radial vector [m] radius of rotating zone in Nagata’s model [m] impeller Reynolds number ND2 =m [–] tank diameter [m] dimensionless tank diameter [–] time variable [s]

u velocity field [m/s] ur ; uh ; uy radial, tangential and axial velocity components [m/s] x space variable [m] a fit coefficient [–] b fit coefficient [–] D overall vortex depth [m] d difference between initial liquid level and lowermost point on vortex [m] Dx computational cell size [m] C liquid phase volume fraction [–] c fit coefficient [–] j interface curvature [m1] l dynamic viscosity [Pa s] m kinematic viscosity [m2/s] x angular velocity [rad/s] r coefficient of surface tension [N/m] f dimensionless impeller submergence Z S =D [–]

Fig. 1. Stirring of the fluid in an unbaffled tank leads to the formation of a vortex, whose depth increased with agitation speed. At high enough speeds, vortex touches impeller, drawing gas bubbles into the liquid.

into the liquid, which may be undesirable in applications such as aerobic fermentation (Atkinson and Mavituna, 1991), mixing of pigment and binder in paints. On the other hand, gas draw from vortexing can be conveniently utilized to bring about gas-liquid

contacting when no external means of gas sparging are employed. Some examples where vortexing is necessary include solids drawdown, waste water treatment, and ethoxylation (Mali and Patwardhan, 2009). While the desirability of gas draw through vor-

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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texing depends upon the process needs, its quantification is often an important step in designing an agitation system for parameters such as impeller type and size, impeller submergence (or alternatively number of impeller stages), and agitation speed. Estimating, a priori, the vortex depth has been a topic of numerous prior investigations, starting with Nagata’s (Nagata, 1975) inviscid model, which has subsequently been revised (Smit and Düring, 1991; Busciglio et al., 2013; Wong and Hayduk, 1987; Rao et al., 2009) in order to improve prediction of interface profile. All these analyses lead to a clear identification of a Froude number based on impeller diameter as the key dimensionless parameter for describing the vortex depth. In Section 3, we demonstrate that this model does indeed very closely represent the phenomenon when viscous effects are negligible. However, under viscous conditions, Nagata’s analysis fails to produce the experimentally observed vortex depths. This is an important point, particulary when scaling up a chemical blending application from lab-scale apparatus ( 0:15 m beaker or smaller) to a production scale vessel ( 1:5 m diameter or larger). It is a common experience that while there is no vortex formation in the lab-tests, a large vortex accompanied by aeration results at the production scale (at the same Froude number). It is for this reason that a reasonably accurate description of viscous effects on vortex formation is essential for process design. We note that the effect of Reynolds number has been considered by a number of researchers including Zlokarnik (1971), and Rieger et al. (1979), and reviewed by Markopoulos and Kontogeorgaki (1995). While the correlations presented by these authors differ in functional form, they all seek to capture the decrease in vortex depth with decrease in Reynolds number and increase in impeller submergence. However, as will be discussed in Section 3, these correlations did not describe our experimental data which covers a much larger Reynolds number range than the previous studies. Upon closer examination, we found that the correlations of Zlokarnik and Reiger et al. can indeed produce unphysical predictions for a large range of industrially relevant flow conditions. Additionally these correlations are shown to not converge to Nagata’s theoretical model in the inviscid limit. In the present article, our attempt is to develop a correlation to describe the vortex depth such that it applies over a wider Reynolds number range, covering nominally laminar (ReD  102 ) and turbulent (ReD J 104 ) regimes. The study consists of experiments as well as computer simulations based on a volume of fluid method described in Section 2. For quantifying the effect on vortex formation, we covered industrially relevant Reynolds and Froude number ranges by varying fluid viscosity, impeller diameter, agitation speed, and impeller submergence. Our correlation, which is designed to have an appropriate behavior at the limits of these ranges, is then presented in Section 3.1. Its applicability in scaleup or scale-down calculations is then tested by performing additional experiments as well as computer simulations in Section 3.1. Conclusions from this study are then presented in Section 4.

2. Methods 2.1. Experiments Fig. 2 shows the experimental setup and physical parameters of interest in the present work. Experiments were performed in a T ¼ 0:45 m clear acrylic vessel, supported on a leveled base. Agitation was accomplished using a vertically mounted 1/2 hp motor. The motor is equipped with a variable frequency drive allowing the agitation speed, N, to be varied conveniently. As show in Fig. 2, a laser pointer mounted on a vertical traverse with graduations is used to probe the vortex depth, D. In the image shown, the

pointer is at the topmost location on the vortex. The tank is placed in a clear rectangular enclosure and the space between the two is filled with water in order to reduce the measurement error due to refraction of the laser beam at the tank wall. For ensuring that error due to possible tilting of the laser beam in the vertical plane does not influence D measurements, we verified using a level gauge that laser pointer was horizontally oriented for each experiment. Based on these control measures, and several repeat experiments, it was clear that the approach taken for measuring the vortex depth was sufficiently accurate and highly repeatable. An experiment begins with filling up of the cylindrical vessel with fluid of desired viscosity l. Liquid level was kept constant at 0.33 m from lower tangent line across all experiments. Impeller submergence, Z S (see Fig. 2b) was varied by simply lowering or raising the platform on which the tank is mounted. While this does alter the off-bottom clearance, c, as well, based on our experiments as well as (Markopoulos and Kontogeorgaki, 1995) the clearance does not have any measurable influence on vortex depth. For each fluid, several experiments were conducted by varying the agitation speed N, impeller submergence Z S and impeller size D. After starting agitation, it takes between 5 and 30 s (at the scale of our interest) for the vortex to develop to its largest, steady state depth. All measurements were performed after the vortex had stabilized. 2.1.1. Scaling and design of experiments Based on the schematic shown in Fig. 2, we identify the following dimensionless groups to characterize the flow in an unbaffled stirred tank,

ReD ¼ qND2 =l; Fr D ¼ N2 D=g; f ¼ Z S =D; T ¼ T=D; WeD ¼ qN2 D3 =r; C ¼ c=D:

ð1Þ

Consistent with the Reynolds and Froude numbers, and typical dimensionless submergence values typically encountered industrially, our experiments spanned the parameter space,

FrD 2 ð0:01; 0:4Þ;

ReD 2 ð101 ; 105 Þ;

f 2 ½0:7; 1:2:

ð2Þ

Note, Fr D and ReD have been varied independently by employing fluids with viscosity l 2 ð1; 2600Þ cP. For this, corn-syrup and water mixtures of various concentration were employed. In addition to viscosity, we changed also the impeller diameter in steps D ¼ 0:15; 0:18 and 0.23 m, impeller submergence Z S = 0.7D, 0.95D, and 1.2D as well as agitation speed, 0:83 < N < 5 rps, thereby also changing the other dimensionless groups of Eq. (1). In what follows, we have ignored WeD because the flow is inertia dominated and surface tension effects are negligible. The dimensionless bottom clearance, C, is neglected because in our experiments this parameter did not contribute measurably to D. Furthermore, as long as T J 2, one can show (Busciglio et al., 2013) that this parameter does not influence the vortex depth significantly. This is also supported by our experiments. As a result, it suffices to characterize vortex depth as

D ¼ FðFr D ; ReD ; fÞ; D

ð3Þ

where Fð. . .Þ represents the functional form of a correlation. Procedurally, we load the tank with a corn syrup-water mixture of the desired viscosity. Changing the viscosity is a time consuming process, involving blending of the necessary amount of corn-syrup with water. Therefore, it is preferable to complete a set of experiments at certain viscosity level before changing to the next (lower) viscosity value. The next level of ‘hardness’ consists in varying the impellers. For a given viscosity, and an impeller diameter, the submergence and agitation speed can be varied relatively easily. For the levels of variation considered for all the parameters, a full set of experiments consists of 225 runs, each producing one value

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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Fig. 2. Schematic diagram showing (a) the experimental setup and (b) definition of various quantities.

for D. Instead of running this full set, a design of experiments (DOE) was developed to reduce the number of experiments to 100 and still span the ðReD ; Fr D ; fÞ space adequately. The DOE consisted of a split-split-plot design to account for the restrictions on randomization of two variables l and D. The choice of specific 100 experiments was based on the D-Optimality criterion, which seeks to maximize the determinant of the information matrix (see chapter 5.5.2.1 in Natrella (2010)) for a detailed discussion). The JMP software (Institute, 1990) was used to determine the final design consisting of 100 experiments (marked by red crosses in Fig. 3). For reference, the full set of experiments (225, blue circles) is also plotted in this figure. 2.2. Computer simulations In addition to the experimental results to be presented here, we have also performed detailed two-phase flow simulations of vortex formation in an unbaffled stirred vessel. The purpose of these simulations is twofold: (i) to gain qualitative insights into how viscous effects and submergence influence the vortex depth, and (ii) to evaluate the applicability of our correlation (Section 3.1) at different scales, where experiments cannot be performed due to practical reasons (see Table 2). All the simulations presented here are performed using the solver interDyMFoam from the open-source computational fluid dynamics (CFD) toolkit, OpenFOAM (Open, 2011) (version 2.3.1). The solver employs an algebraic volume of fluid (VOF) method

Fig. 3. The  design of experiments captures very well the designed parameter space of Re 2 10; 2  105 ; Fr 2 ð0:01; 0:45Þ; f 2 ½0:7; 1:2.

(Deshpande et al., 2012; Rusche, 2003) to simulate flow of two incompressible immiscible fluids. The VOF method (Hirt and Nichols, 1981; Tryggvason et al., 2011) employs a liquid fraction field, Cðx; tÞ, defined over a computational cell of volume jVj as

Cðx; tÞ ¼

1 jVj

Z

Iðx; tÞdV;

ð4Þ

V

where I ¼ 1 in liquid phase and zero elsewhere. Naturally, C varies between 1 in the liquid phase to 0 in the gas phase. The interface between the two phases is not explicitly known, but implicitly captured over a region 2Dx to 3Dx as C transitions from 1 to 0. The solution procedure starts with mesh motion, which is necessary in order to capture the rotation of the impeller. This is readily accomplished in OpenFOAM by defining a cylindrical zone around the impeller (see Fig. 4) and imposing a solid-body rotation with a prescribed angular velocity x on all the cells and boundaries within this zone. The rotating and stationary cell zones are connected through an arbitrary mesh interface (AMI), where the cell faces on the two sides of this interface need not be conformal (Farrell and Maddison, 2011). The mesh class dynamicFvMesh handles the necessary mesh motion and correspondingly updates the boundary conditions. Of particular interest is the fluid velocity at the impeller surface, which in the present work is described using the boundary condition movingWallVelocity (Open, 2011). A no slip boundary condition is imposed on the tank wall. Next, the volume fraction field is evolved in time by solving Eq. (5) using explicit Euler time stepping,

Fig. 4. Computational setup.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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@C þ u  rC ¼ 0: @t

ð5Þ

In order to keep the interface region relatively sharp while keeping the computational cost relatively low, OpenFOAM employs a high-order face interpolation of the C field (Deshpande et al., 2012) using the flux-corrected transport (FCT) methodology (Boris and Book, 1973) embodied within the MULES solver (multi-dimensional limiter for explicit solution). Once the liquid fraction field at the updated time level is available, fluid properties are calculated using

qðx; tnþ1 Þ ¼ ql Cðx; tnþ1 Þ þ qg ð1  Cðx; tnþ1 ÞÞ lðx; tnþ1 Þ ¼ ll Cðx; tnþ1 Þ þ lg ð1  Cðx; tnþ1 ÞÞ;

ð6Þ

where ql and qg are gas and liquid densities while ll and lg are the corresponding dynamic viscosities. The momentum equation for a two phase flow (Tryggvason et al., 2011) (Eq. (7)) is next solved.



    @u q þ u  ru ¼ rp þ r  l ru þ ruT þ FS ; @t

ð7Þ

where FS is the surface tension force which is modeled using the continuum surface force model of Brackbill et al. (1992) as

FS ¼ rjrC;

ð8Þ

and j ¼ r  ðrC=jrCjÞ is the local interface curvature. Eq. (7) is solved by first constructing a predicted velocity field, and then correcting it using the pressure implicit splitting of operators (PISO) correction procedure (Issa, 1986). This yields the time advanced solutions for pressure and a solenoidal velocity field. 2.2.1. Validation-1: Vortex formed in liquid undergoing solid-bodyrotation As a first validation case, we compare our computational results pertaining to depth of the depression formed when a liquid stored in a cylindrical container of radius R undergoes a solid-body rotation at angular velocity of x ¼ xey , as shown in Fig. 5. This situation is directly relevant to the subject of this article i.e. formation of a depression (or a vortex) in liquids undergoing tangential motion in cylindrical tanks. In order to computationally simulate the situation, we initialize the cylindrical tank with a liquid level H ¼ 4R, by prescribing the volume fraction field as

a¼

1 y 6 4R 0 y > 4R:

5

ð9Þ

In order to speed up convergence to a steady state (in the unsteady simulation), the initial velocity field is defined as

uðx; t ¼ 0Þ ¼ ðux ; uy ; uz Þ ¼ xðz; 0; xÞ:

ð10Þ

The flow evolves over time to reach a quasi-steady state, at which point we measure the vortex depth, D. Note, consistent with our definition of D in Fig. 2(b), the vortex depth here is the distance between the topmost and the bottommost point of the liquid surface, and not the difference between initial liquid surface and bottom of the vortex. Fig. 5 shows the interface profiles for three different rotation speeds corresponding to tank-based Froude numbers of Fr T ¼ ðRxÞ2 =ð2gRÞ ¼ 0:2; 0:46 and 0.81. Clearly, the interface takes a parabolic shape as is expected from Eq. (11) (see e.g. Panton, 2006)

YðrÞ  Yð0Þ ¼

R2 x2  r 2 ; 2g R

ð11Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ z2 is the radial distance. The YðrÞ profiles were extracted from the three simulations, and are plotted in Fig. 6a. Also plotted on top is the prediction from theory (Eq. (11)), showing that the computational vortex shapes are in excellent agreement with the theory. Finally, the vortex depth (D ¼ YðRÞ  Yð0Þ) is obtained from the computational data and plotted in Fig. 6b against the Froude number defined as FrT ¼ R2 x2 =2Rg. The expected linear relation between the two is recovered very well in our simulations. This is a very important observation, and is the foundation of Nagata’s inviscid model (Nagata, 1975), as will be discussed later. 2.3. Validation-2: Power number of a Rushton impeller The second validation test we present pertains to quantifying the power draw by a Rushton impeller. The drawn power depends directly upon agitator geometry, rotation speed, and fluid viscosity, and is typically described by

  P ¼ N P  qN 3 D5 ;

ð12Þ

where NP , the power number, is a characteristic of an impeller and depends upon impeller geometry as well as ReD . For standard impellers such as the Ruston impeller (which is used in the present study) these power numbers are tabulated (Oldshue, 1983; Paul et al.,

Fig. 5. Vortex formation in solid body rotation of liquids in a cylindrical vessel.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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Fig. 6. Normalized free-surface profiles and vortex depths for the validation exercise of Fig. 5.

2004). In this test, we compare the power number obtained computationally with these tabulated values. Accurate estimation of power draw requires that the solver satisfactorily captures the resulting flow. Here we consider the computationally challenging case of turbulent flow (ReD ¼ 87  105 ), where it is of utmost importance to capture such turbulent shedding of vortices in the wake of impeller blades (Cutter, 1966; Lee and Yianneskis, 1998) as it directly influences the torque experienced by the impeller (and also the shaft power). We have computationally treated turbulence by using the implicit large eddy simulation (implicit LES) approach (Grinstein et al., 2007). To this end, we have used a monotonicity preserving (vanLeer Van Leer, 1979) scheme, already implemented in OpenFOAM, for the convective term u  ru. Such an approach has produced quantitatively meaningful results when simulating turbulent two phase flows (Deshpande et al., 2012; Deshpande and Trujillo, 2013; Deshpande et al., 2015; Fedina et al., 2011), as well as flows with moving boundaries (Bensow and Liefvendahl, 2008) in OpenFOAM. In order to identify a minimal spatial resolution needed to adequately capture the turbulent motions, we have also performed a grid refinement study with 46, 76 and 114 cells across the tank diameter. Computationally, the agitator power is computed as

P¼xs¼x

Z Si

  r  pI þ l ru þ ruT  ndS;

ð13Þ

where pðx; tÞ and uðx; tÞ fields are obtained as a part of the numerical solution, x ¼ 2pNey is the angular velocity in rad/s and integration is carried out over the surface of the impeller. Power number is then calculated using Eq. (12). Fig. 7 shows the evolution of NP over time for unbaffled and baffled tanks. In the case of baffled vessels, the impeller power number reaches a nominally stationary value of NP ¼ 5 within 10 impeller rotations. This is in a very good agreement with experimental data (Paul et al., 2004). In the case of an unbaffled vessel operating at a high Reynolds number, we obtain N P  0:55, which is again in a good agreement with the value of 0.64 obtained from Scargiali et al. (2014)

NP  19:5  Re0:3 D

ð14Þ

It should be noted that in the unbaffled case, a fair scatter in power numbers has been reported (0.6 Rushton et al., 1950, 0.65 Scargiali et al., 2013; Alcamo et al., 2005, 0.7 Musik and Talaga, 2016, 0.75 Steiros et al., 2017). This is likely because, in the unbaffled case, the power number is fairly sensitive to the geometric details of the experimental setup (Paul et al., 2004), as well as to

Fig. 7. Power number, N P , for turbulent flow (ReD ¼ 87  103 ) due to a Rushton impeller in baffled and unbaffled vessels.

whether or not a gas-liquid interface is present (i.e. effect of Froude number) (Scargiali et al., 2014; Laity and Treybal, 1957). In any case, the approach taken here appears to adequately represent the physics of fluid flow generated in a stirred tank by a Rushton impeller. We note, based on Fig. 7, that a computational resolution of T=Dx J 46 is sufficient for the purposes of describing the flow. In all the simulations presented hereafter, we have used T=Dx ¼ 50. The LES simulations shown here took  Oð102 Þ processor-hours for one second of simulated time. Depending upon the particular flow situation, these simulations ran for between 60 and 180 h on 20–40 processors. 3. Results As discussed earlier, description of vortex depth has been a topic of several investigations (Nagata, 1975; Busciglio et al., 2013; Zlokarnik, 1971; Markopoulos and Kontogeorgaki, 1995; Rao et al., 2009; Rieger et al., 1979). Nagata (1975) introduced the first mechanistic model for describing the vortex depth when viscous effects are unimportant. The model is based on an assumed velocity profile in the stirred tank, given by

(

u ¼ ður ; uy ; uh Þ ¼

ð0; 0; rxÞ ð0; 0; R2C

for r < RC

x=rÞ for RC < r < T=2:

ð15Þ

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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Here RC < D=2 is an undetermined parameter, obtained by fitting velocity profiles using Eq. (15). The key point to note here is that the flow within the tank is assumed to be purely tangential, with negligible axial and radial velocity components. This situation is similar to free-surface evolution in a liquid undergoing solid body rotation (see Section 2.2.1), where the vortex depth was shown to scale linearly with Froude number. In the case of stirred tanks as well, a similar analysis results in the following relation for vortex depth (Ciofalo et al., 1996) (see Appendix A for the derivation)

( 2  2  2 !) D p2 2RC 2RC D : 2 ¼  Fr D  T D 2 D D

ð16Þ

In order to verify the applicability of this model, we simulated the flow resulting from rotation of a 0.23 m Rushton impeller in an T=D ¼ 2 tank filled with water (ReD ¼ 87  104 ; Fr D ¼ 0:06). Fig. 8a shows an instantaneous view of the vortex (at a nominally steady state) and the resulting flow profiles. Notably, the velocity magnitude increases radially from the center of the vessel, until some radius RC < D=2. Beyond this radius, the velocity magnitude drops, reaching a zero value at the tank surface, r ¼ T=2. Streamlines of the instantaneous velocity also point to a dominant tangential flow, as one would expect from Nagata’s model. The time averaged velocity magnitudes are plotted in Fig. 8b to make this point clearer. First, the flow is dominated by a swirling motion, and axial and radial components of velocity are negligible everywhere within the tank. The tangential velocity does not vary appreciably with axial distance and is demonstrably independent of the angular coordinate. Furthermore, the radial variation of tangential velocity is indeed well described by Eq. (15). Based on the velocity profile of Fig. 8b, we obtained RC  0:7D=2. This, along with the condition D=T 6 0:5 (as in all the present experiments and simulations), implies that terms in curly braces of Eq. (15) evaluate to  1, allowing the vortex depth relation to be simplified to

D p2  Fr D : D 2

ð17Þ

In Fig. 9, we plot Eq. (17) along with all the experimental data. The excellent agreement between the simple Froude number scaling and dimensionless vortex depth seen in this figure further confirms the applicability of Nagata’s model, in the cases where viscous effects are unimportant. It is also clear that vortex size progressively decreases as viscous effects become increasingly important.

7

Fig. 10a shows the flow field generated when the conditions of Fig. 8a are reproduced with a 1000 cP fluid. The resulting ReD ¼ 87, and the viscous effects are expected to be significant. As is evident from the instantaneous streamlines, the flow is characterized by a dominant axial motion with a tangential flow superimposed on it. Time averaged velocity profiles at different axial locations are plotted in Fig. 10b. Firstly the tangential velocity magnitudes in the case of ReD ¼ 87 are a much smaller fraction of the tip speed compared to the ReD ¼ 87  103 case. The impeller is seen to generate significant axial suction, with axial velocities uy ðr; yÞ comparable to (or even larger than) the tangential velocity. All velocity components decay away from the impeller and therefore, the farther the impeller is from the free surface, the weaker the motion of the surface is expected to be. With respect to the present work, the observation that while D is practically independent of f for ReD J 104 (Fig. 8), but is strongly affected by f at lower Reynolds numbers, is crucial to the choice of a functional form of the new correlation, as will be discussed in Section 3.1. Functionally, the vanishing of flow farther from the impeller can have negative impact to such operations as solids draw-down (Paul et al., 2004; Khazam and Kresta, 2008), where a vortex is desirable. Given the complexity of the flow, a theoretical model for vortex depth has not yet been formulated. Nevertheless, several attempts have been to correlate the vortex depth with Reynolds and Froude numbers, and submergence. Here we will discuss in brief the correlation proposed by Zlokarnik (1971) to motivate our need for a more appropriate correlation (Section 3.1). In that work, experiments were performed with fluids of viscosity up to 530 cP, in tanks of diameters between 0.3 m and 0.6 m. Impeller diameter to tank diameter ratio, T=D ¼ 3 and impeller clearance c=D ¼ 1 were kept constant. Experiments covered the operating range GaD ¼ Re2D =Fr D 2 ð2:7  106 ; 1010 Þ, where Ga is the Galileo number. A fitting function of the form given in Eq. (18) was then used to describe the distance d between the initial liquid level and the final vortex depth,

 B1 n o d ZS  C 1  GaD1 : ¼ A1 FrD D D

ð18Þ

The coefficients A1 ¼ 62; B1 ¼ 0:16; C 1 ¼ 0:1; D1 ¼ 0:18 were found to describe Zlokarnik’s data satisfactorily. In Fig. 11, we compare our experimental data with prediction from Eq. (18). Experiments falling within the Galileo number range prescribed by Zlokarnik are reasonably well described by the correlation, with

Fig. 8. The flow at large Reynolds number is largely tangential, and Nagata’s model (Nagata, 1975) applies.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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Fig. 9. Experimental data on vortex depths, D, for all 100 experiments.

a consistent overprediction of 20% which we believe might be due to minor geometric differences between our apparatus and

Fig. 11. Evaluating Zlokarnik’s model (Eq. (18) for the d values obtained in the present experiments. Filled symbols: experiments satisfying Ga 2 ð2:7  106 ; 1010 Þ. Hollow symbols: experiments outside the range.

that used in Zlokarnik (1971). For GaD < 3:5  105 , the correlation produces not only inaccurate, but unphysical (negative) d values. This is undesirable because the corresponding impeller Reynolds

More importantly we note that Eq. (18) does not converge to the theoretical estimate of Nagata (1975) in the limit of vanishing viscosity. Instead, Zlokarnik’s correlation approaches

numbers ReD  Oð102  103 Þ are still very relevant in viscous industrial applications as well as lab-scale operations.

ð19Þ

 B1 d ZS ¼ ðA1 C 1 ÞFr D : D D

Fig. 10. The flow at small Reynolds number is not dominated by pure tangential motion. The impeller drives significant axial and radial flow as well.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

S.S. Deshpande et al. / Chemical Engineering Science xxx (2017) xxx–xxx

As a consequence, the predicted d values increase as the impeller submergence is decreased at a fixed Froude number. For this to be realistic, the tangential velocity must be a function of axial distance even at large ReD ; a situation not supported by our simulations (Fig. 8b), experiments (data for water in Fig. 9) or Nagata’s model (Nagata, 1975). Another experimental fit was developed by Rieger et al. (1979) (Eq. (20)), where experimental range was extended to include lower Reynolds numbers than those considered by Zlokarnik.

d D ðGa ÞE2 ðT=DÞG2 ¼ A2 ðGaD ÞB2 ðTÞC2 FrD2 D ; D

ð20Þ

where the coefficients A2 ; B2 ; C 2 ; D2 ; E2 depend upon the Galileo number range. Regardless of the range, the proposed value for GaD exponent is B2 > 0, which implies d ! 1 under inviscid conditions. It is clear that a simple polynomial correlation may not adequately capture the viscous vortex depth. 3.1. Development of correlation In order for the new correlation to reasonably capture D over relevant Reynolds number spectrum (10 K ReD K 105 ), the chosen functional form must satisfy at least the following conditions: 1. The relation must simplify to the well validated inviscid model by Nagata (1975) in the limit of vanishing viscosity. This means

D ¼ FðFr D Þ ¼ aFr D ; when ReD ! 1: D

ð21Þ

2. The correlation must be applicable regardless of physical dimensions of the apparatus. This condition does not determine the choice of a functional form, but is used to verify developed correlation over a range of lengthscales. To this end, we will consider a functional form given by Eq. (22) to describe the vortex depth. The fit parameters are determined using non-linear analysis in JMP software. The correlation is then tested at different scales (0.046, 0.15, 0.46 and 0.92 m) using addi-

Table 1 Fitting parameters obtained using Analytic Gauss-Newton algorithm in JMP software. Parameter

Value

Approx. std. error

a

12.9 0.11 0.17 4.27

0.55 0.02 0.03 0.52

b

c C

9

tional computational and experimental means to evaluate its performance in extrapolation.

( ! ) D 1 ¼ Fr D  a 1  b fc þ C D ReD

ð22Þ

Here a; b; c; C are undetermined coefficients, to be obtained from data fitting. As stated previously, the ReD and f parameters are considered in a multiplicative combination following the observation that f does not influence D unless ReD is small enough (see Figs. 8a and 10a). The Nonlinear platform in JMPÒ (Bruin, 2011) statistical software was used to fit the nonlinear model. The analytic GaussNewton algorithm was selected to estimate the parameters. Since the smallest vortex depth that can be reliably measured with our apparatus is 0.01 m, all D readings smaller than this value are neglected while developing the correlation. The coefficient values thus obtained are listed in Table 1. The resulting fit is reproduced in Fig. 12 alongside all the experimental data. Here the experimental data points are colored by Reynolds number and their size is proportional to D. Small vortex depths (identified by small markers) constitute most of the scatter, since (i) the measurement uncertainties most significantly affect small vortex measurements, and (ii) division of D=D by Fr D amplifies these errors. Nevertheless, the fit captures very well the overall behavior of D=D  Fr 1 D . This correlation covers a much wider Reynolds number range (ReD 2 ð10; 105 Þ) than any of the previous correlations known to the authors. From Fig. 12, first we point out that neither the experimental data nor the fit show have truly yielded the scaling D=D  Fr D even at ReD  Oð105 Þ. This can be explained from the fact that b is small, but not zero. It is only in the limit ReD ! 1 that we expect to recover a truly linear scaling with Fr D alone. Evaluating Eq. (22) at ReD ! 1, the asymptotic vortex depth is

  D ¼ ða þ CÞFr D  8:57Fr D : D ReD !1

ð23Þ

This is different from the prediction based on inviscid analysis (Eq. (17)), where the constant of proportionality is evaluated to be p2 =2. The difference might be attributed to the fact that derivation of Eq. (17) requires specification of an arbitrary constant, RC , which is based on experiments where ReD – 1. The reason why our experiments with water, shown in Fig. 9, bear a nearly perfect agreement with Eq. (17) is that we recover

" (

a 1

fc RebD

)

#

þC 

p2 2

ð24Þ

Fig. 12. Demonstration of fit (Eq. (22)) produced from statistical analysis. The parameters a; b; c; C correspond to Table 1. Marker size is proportional to D.

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

10

S.S. Deshpande et al. / Chemical Engineering Science xxx (2017) xxx–xxx

of operating conditions (ReD 2 ð10; 105 Þ; Fr D 2 ð0:01; 0:4Þ; Z S =D 2 ð0:7; 1:2Þ, and D=T 2 ð0:3; 0:6Þ). In order to achieve the parameter space, we varied N; D; l and Z S where the exact details of experiments were arrived at through a statistical design of experiments. The experimental results show that vortex depth sharply decreased at small Reynolds numbers, scaling very poorly with an impeller based Froude number alone. At the other extreme (i.e. ReD ! 1, the vortex depth asymptotically approaches a linear scaling with Fr D , in agreement with the inviscid theory of Nagata (1975). We described this behavior by correlating our experiments using the following functional form given by Eq. (22), designed specifically to satisfy these expected limiting behaviors.

( ! ) D 1 c ¼ FrD  a 1  b f þ C ; D ReD

ða ¼ 12:9; b ¼ 0:11; c ¼ 0:17;

C ¼ 4:27; Þ:

Fig. 13. Validating the correlation (Eq. (22) and Table 1) over several tank diameters.

over the Reynolds number range 104 K ReD K 105 corresponding to those experiments. For the sake of visualization, we plot the line 2 Fr1 D D=D ¼ p =2 in Fig. 12. This line passes through the data from water experiments. Another important point regarding the exponent of Reynolds number is that since b < 1, we recover a vanishing vortex depth as N vanishes (i.e. DðN!0Þ ! 0). Furthermore, the fit produces c > 0, which is necessary to capture reduction in vortex depth as submergence is increased. Unfortunately, even our correlation produces negative D values, similar to Zlokarnik’s correlation. Nevertheless, this problem occurs only for ReD K 10 – a range unlikely to be experienced in an industrial application of a stirred tank, and certainly small enough for any noticeable vortex to develop. Next we evaluate the utility of Eq. (22) by comparing the predictions with measurements for various tank diameters. For this, several multiphase flow simulations as well as experiments in a 0.15 m diameter beaker were performed. A summary of these tests at different scales is provided in Appendix B (Table 2). These additional results are plotted along with data from our 0.46 m diameter tank and the correlation in Fig. 13. A reasonable agreement between the prediction and measurements over an impeller Reynolds number range covering laminar and turbulent regimes, a number of different tank diameters, and typical Froude numbers suggests that Eq. (22) can be used to determine the vortex depth over a wide range of operating conditions. In summary, the scaling is linear with the conventionally

defined Froude number (Fr D ¼ N 2 D=g) only in the limit of ReD ! 1. Deviations from this condition result in deviations from linear scaling with Fr D . Nevertheless, for simplicity of interpretation of these results, we suggest that a linear scaling between D=D and a Froude number still holds using a modified Froude number defined as

n   o c Fr ¼ Fr D  a 1  Reb þC : D f

ð25Þ

4. Conclusion In the present work, we investigated the effect of fluid viscosity on development of a ‘vortex’ (or a depression in the liquid level) in an unbaffled stirred tank. Several experiments were performed under laboratory condition (T ¼ 0:46 m), spanning a wide range

While this functional form itself does not explain the physics of a viscous vortex, it does seem to describe its behavior very well, over the wide range of parameters studied, as well as under scale-up and scale-down. Using validated computer simulations, we demonstrated that the flow under inviscid and viscous situations is qualitatively different (see Figs. 8a and 10a). When viscous effects are negligible (ReD J 104 ), the flow is dominated by a tangential fluid motion, and axial and radial velocity components are unimportant. Furthermore, the velocity field itself is largely insensitive to the axial coordinate. It is under these conditions that Nagata’s analysis, which leads to a simple linear Froude number scaling, is applicable. Once viscous effects are no longer negligible, a qualitative change happens in the flow, where a strong axial pumping towards the impeller dominates the tangential motion. As a result, all velocity components rapidly decay away from the impeller. The effect of impeller submergence, naturally, becomes noticeable under these conditions. This observation has direct implication on solids draw down applications (Schober and Fitzpatrick, 2005; Khazam and Kresta, 2008), limiting the maximum submergence one can have in such applications. Finally, based on the functional form of our validated correlation (Eq. (22)), we suggest that the vortex depth still scales linearly with a modified Froude number defined as Fr ¼ Fr D   c a 1  Reb þC . D f Appendix A. Derivation of total vortex depth under inviscid conditions As discussed in Section 3, Nagata’s (Nagata, 1975) model of an inviscid vortex consists of two regions – a forced vortex (r < RC ) and a free vortex (RC < r < T=2). The velocity profiles in the two regions, given by Eq. (15), are reproduced here.

(

u ¼ ður ; uy ; uh Þ ¼

ð0; 0; rxÞ ð0; 0; R2C

for r < RC

x=rÞ for RC < r < T=2:

Starting with this velocity profile, Eq. (16) is obtained by solving the momentum equation in cylindrical coordinates (Panton, 2006). The radial component of the momentum balance can be simplified to

@p qu2h ¼ ; @r r

ð26Þ

where u ¼ ður ¼ 0; uz ¼ 0; uh ðrÞÞ has been assumed. The axial momentum equation, then, simplifies to

@p ¼ qg: @y

ð27Þ

Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002

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S.S. Deshpande et al. / Chemical Engineering Science xxx (2017) xxx–xxx

Within the forced vortex region (i.e. r < RC ), the velocity is given by uh ðrÞ ¼ r x. Eqs. (26) and (27) solved with this uh yield

DY forced ¼ Yðr ¼ RC Þ  Yðr ¼ 0Þ ¼

R2C x2 2g

ð28Þ

as the vortex depth within the region r < RC . This situation is identical to that in the solid body vortex of Fig. 5. Similar treatment in the free vortex region (i.e. uh ðr > RC Þ ¼ R2C x=r) gives

DY free

R2 x2 ¼ Yðr ¼ T=2Þ  Yðr ¼ RC Þ ¼ C 2g

(

 2 ) 2RC 1 : T

ð29Þ

The total vortex depth, D, is then simply given by

D ¼ DY forced þ DY free

R2 x2 ¼ C 2g

(

 2 ) 2RC ; 2 T

ð30Þ

or in the dimensionless form, by Eq. (15), reproduced below

D p2 ¼ Fr D  D 2

( 2  2  2 !) 2RC 2RC D 2 : T D D

Appendix B. Additional data for validating Eq. (22) at different scales Table 2.

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Table 2 Additional experiments and simulations to test performance of Eq. (22) at different scales. T (in)

D (in)

Z S (in)

l (cP)

6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

3.0 3.0 3.0 3.0 1.5 1.5 1.5 4.5 4.5 4.5 3.0 3.0 3.0 4.5 4.5 4.5

520 520 520 520 520 520 520 520 520 520 125 125 125 125 125 125

18 18 36

9 9 18

9 6.3 18

36 1.8 1.8

18 0.9 0.9

18 0.63 0.63

N s1

ReD (–)

Experiments in a 0.15 m beaker 7.7 94 10.2 125 9.1 111 6.6 81 4.6 57 6.3 77 3.4 41 6.9 85 10.7 131 13.4 165 5.2 267 7.0 355 4.3 217 7.2 367 9.9 506 4.3 217

FrD (–)

D=D (Measurement)

D=D (Eq. (22))

0.5 0.8 0.6 0.3 0.2 0.3 0.1 0.4 0.9 1.4 0.2 0.4 0.1 0.4 0.8 0.1

0.3 0.8 0.6 0.2 0.2 0.5 0.2 0.0 0.4 1.0 0.3 0.7 0.2 0.6 1.2 0.1

0.4 0.8 0.6 0.2 0.2 0.5 0.1 0.1 0.5 1.0 0.3 0.7 0.2 0.6 1.3 0.1

0.35 0.16 0.68

0.32 0.10 0.71

0.62 0.3 0.12

0.68 0.19 0.08

Multiphase flow simulations in T = 0.46, 0.92, and 0.046 m vessels 1 1.667 87,000 0.065 275 1.667 400 0.065 1 1.667 0.13 35  105 2500 2.916 290 0.40 1 5.27 2750 0.065 10 5.27 275 0.065

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Please cite this article in press as: Deshpande, S.S., et al. An experimental and computational investigation of vortex formation in an unbaffled stirred tank. Chem. Eng. Sci. (2017), http://dx.doi.org/10.1016/j.ces.2017.04.002