Free-surface shape in unbaffled stirred vessels: Experimental study via digital image analysis

Chemical Engineering Science 104 (2013) 868–880

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Free-surface shape in unbaffled stirred vessels: Experimental study via digital image analysis A. Busciglio a,n, G. Caputo b, F. Scargiali a a b

Dipartimento di Ingegneria Chimica, Gestionale, Informatica, Meccanica, Universitá degli Studi di Palermo Viale delle Scienze, Ed. 6, 90128 Palermo, Italy Dipartimento di Ingegneria Industriale, Universitá degli Studi di Salerno, Via Ponte Don Melillo, 1-84084 Fisciano (SA), Italy

H I G H L I G H T S








Free surface measurement in uncovered unbaffled stirred vessel via image analysis. Development of an original two parameters model for free-surface description. Robust fitting of model equation to free surface profile by means of a self-developed algorithm. Free-surface data relevant to several vessel geometries are presented and discussed. Data presented provide useful design information and viable benchmark data for CFD validation.

art ic l e i nf o

a b s t r a c t

Article history: Received 15 April 2013 Received in revised form 7 September 2013 Accepted 15 October 2013 Available online 22 October 2013

There is a growing interest in using unbaffled stirred tanks for addressing a number of processing needs such as low shear damage (sensitive biocultures), low attrition (solid–liquid applications), deep-cleaning/ sterilization (pharmaceutical applications). The main feature of uncovered, unbaffled stirred tanks is highly swirling motion of the fluid that results in a deformation of the free liquid surface. At sufficiently high agitation speeds the resulting whirlpool reaches the impeller and gives rise to a gas–liquid dispersion, so leading to the formation of a dispersion without the use of gas-sparger; the so-called selfinducing operation of the vessel. In this work, digital image analysis coupled with a suitable shadowgraphy-based technique is used to investigate the shape of the free-surface that forms in uncovered unbaffled stirred tanks, when different stirrer geometries are considered. The technique is based on back-lighting the vessel and suitably averaging over time the recorded free surface shape. For each investigated geometry, the deformed free-surface was analyzed at different impeller speeds. Different geometries of the vessel were analyzed, by varying impeller distance from vessel bottom as well as agitator type (Rushton turbine, Lightnin A310, Pitched Blade Turbine). It is shown that impeller design strongly affects the free surface profile, and in turn the impeller speed at which the free surface reaches the impeller. A model was developed to fully describe free-surface profile at all agitation speeds and for all investigated geometries, suitable for being adapted to experiments by means of physically consistent parameters adjustment. & 2013 Elsevier Ltd. All rights reserved.

Keyword: Mixing Unbaffled vessels Image analysis Free-surface shape

1. Introduction The main feature of unbaffled stirred tanks is the strongly swirling liquid motion, that due to centrifugal effects leads to the formation of a vortex on the liquid free-surface. Also, impeller– liquid relative velocities are rather small, so resulting in a comparatively small pumped flow rate and in turn in poorer axial

n

Corresponding author. Tel.: þ 39 091 238 63 779. E-mail addresses: [email protected] (A. Busciglio), [email protected] (G. Caputo), [email protected] (F. Scargiali). 0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.10.019

mixing with respect to baffled tanks. The adoption of baffles effectively destroys the circular liquid patterns, hence inhibits main vortex formation and strongly increases the pumped flow rate, leading to improved mixing rates. For these reasons baffles are almost invariantly adopted in stirred tanks. There are cases, however, in which the use of unbaffled tanks may be desirable. Baffles are usually omitted in the case of very viscous fluids (Re o 20) as they can give rise to dead zones that are bound to worsen the mixing performance (Nagata, 1975). Unbaffled tanks may be advisable in crystallizers, where baffles may promote particle attrition (Mazzarotta, 1993), and give rise to higher fluid–particle mass transfer rates for a given power consumption (Grisafi et al., 1998;

A. Busciglio et al. / Chemical Engineering Science 104 (2013) 868–880

Brucato et al., 2010), even if baffles promote solids suspension and circulation. Vortex existence may be useful when floating particles are to be drawn down (Freudig et al., 1999) or when gas bubbles have to be rapidly removed from the liquid (Smit and During, 1991). As concerns bioreactor applications, for aerobic fermentations and plant or animal cell cultivations liquid agitation is required in order to ensure oxygen and nutrient transfer and to maintain cells in suspension. When shear sensitive cells are involved, both mechanical agitation and sparging aeration can cause cell death, which makes it advisable the adoption of unbaffled, unsparged vessels (Scargiali et al., 2012b). In addition, unbaffled stirred vessels can be operated as selfingesting reactors, showing interesting characteristics (Scargiali et al., 2012b), even when compared with both sparged (Scargiali et al., 2010) and self-ingesting (Conway et al., 2002; Scargiali et al., 2012a) baffled stirred vessels. Clearly, in all these cases the ability of correctly predicting free surface shape plays an important role in the vessel design. The whirlpool in unbaffled mixing tanks is formed in the central part of the vessel as a result of centrifugal forces acting on the rotating liquid, hence its shape mainly depends on the flow field. Some of the works dealing with unbaffled tank modelling (Nagata, 1975; Smit and During, 1991) pointed out that liquid tangential velocity mainly depends on the distance from the shaft. Axial variation is small, with the exception of the zone near vessel bottom, where the tangential velocities are decreased by bottom wall friction. On this basis, a simplified potential flow model can be adopted for vortex geometry description (Nagata, 1975; Rieger et al., 1979; Smit and During, 1991; Ciofalo et al., 1996). In fact, by assuming that velocity in the vessel is purely tangential (ur ¼ uy ¼ 0) and only depends on the radial coordinate (uθ ¼ uθ ðrÞ), Navier–Stokes equations in cylindrical coordinates (neglecting viscous forces) reduce to: 8 u2 > > > ∂p ¼ ρ θ > < ∂r r ð1Þ ∂p > ¼  ρg > > > ∂z : At the liquid surface h pressure is constant (pðr; zÞ ¼ p0 ), hence its derivative must be identically nil: dp ¼

∂p ∂p dr þ dz ¼ 0 ∂r ∂z

By substituting Eq. (1), the following differential equation is obtained describing vortex profile as a function of the radial coordinate: u2 dh ¼ θ dr rg

ð2Þ

Hence, the free-surface profile h(r) can be derived once uθ ðrÞ is known. Eq. (2) may be suitably made dimensionless by introducing the following dimensionless quantities: uθ uθ uθ ¼ ¼ U tip ωD=2 π DN r ξ¼ D=2 h ψ¼ D N2 D Fr ¼ g

θ¼

To determine free-surface profile, Nagata (1975) suggested that the whole flow field can be subdivided into an inner region ξ r ξc (forced vortex region) exhibiting a rigid body motion with angular velocity ω and an outer region ξ 4 ξc (free vortex region) where the angular momentum uθ r is constant. This leads to the following tangential velocity profile: 8 if r r r c < ωr uθ ¼ ωr 2c ð4Þ : if r 4 r c r In a dimensionless form: 8 if ξ r ξc > <ξ θ ¼ ξ2c > if ξ 4 ξc :

ð5Þ

ξ

By substituting Eq. (5) into Eq. (3), integrating and imposing the continuity of free-surface profile, the following equation can be obtained as a function of the liquid level at the vessel wall ψ w : ! 8 > π 2 ξ2c ξ2c ξ2 > > ψw  2   if ξ r ξc > > 2 < ξ2w ξ2c ! ð6Þ ψ¼ > π 2 ξ2c ξ2c ξ2c > > > ψw   ξ 4 ξ if c > : 2 ξ2 ξ2 w

where ξw ¼ T=D and ξc ¼ 2r c =D. By imposing total liquid mass conservation, liquid levels at the vessel wall and at the vessel axis ψ b can be written as a function of the liquid level at rest ψ 0 : ! π 2 ξ2c ξ2c 1 ξ2c ln ψw ¼ ψ0 þ ð7Þ 2 ξ2 2 ξ2 w

w

π 2 ξ2c ξ2c 3 ξ2 ξ2  ln 2c 2 2c ψb ¼ ψ0 þ 2 2 2 ξ ξw ξw w

2

ð3Þ

! ð8Þ

By simple manipulation the following expression for overall vortex depth is obtained: ! 2 2

Δψ v ¼ ψ w  ψ b ¼

π 2 ξc 2

2þ

ξc Fr ξ2w

ð9Þ

In this model, the only free parameter is the dimensionless radius ξc that was found from vortex depth measurements to be in the range 0.55–0.65 for systems agitated by Rushton turbines with D=T ¼ 0:3  0:7 and Re ¼ ρND2 =μ above 4  104 (Nagata, 1975). Smit and During (1991) investigated vortex shape in an unbaffled tank equipped with 4 pitched-blade turbine having D=T ¼ 0:8, finding that for Fr o 0:4, Δψ v ¼ 3:33Fr. On the basis of LDA measured tangential velocities these authors suggested that the following velocity profile has to be adopted for free surface profile calculation: 8 if r r r c < 0:825ωr ωr 0:6 uθ ¼ ð10Þ c : 0:825r c if r 4 r c r and in dimensionless form: 8 0:825ξ if ξ r ξc > <  0:6 θ¼ ξc > if ξ r ξc : 0:825ξc ξ

the following general dimensionless equation for the description of free-surface profile is obtained: dψ θ ¼ π 2 Fr dξ ξ

869

ð11Þ

where ξc ¼0.675. This is significantly different from Nagata's model, as for the forced-vortex-region rigid body motion is still assumed but with a revolution frequency smaller than N, while in the free vortex region a velocity profile not obeying to angular momentum conservation is adopted, i.e. the product vθ r 0:6 is kept constant instead of vθ r. Notably, these velocity profiles are relevant to a

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completely different vessel (4 pithed blade impellers, high D/T ratio) with respect to that adopted by Nagata, therefore making direct parameter comparison almost meaningless. In addition, the assumption of non-constant angular momentum in the free vortex region was not theoretically justified by the authors, but used to fit the experimental data on velocity profiles in their apparatus. Nevertheless, the Authors have shown the basic assumptions of the Nagata model (i.e. inviscid fluid, purely tangential motion and tangential velocity component mainly dependent on radial coordinate alone) are substantially correct, at least for the purpose of free-surface characterization in unbaffled vessels. Rieger et al. (1979) adopted a different form of dimensionless momentum equations to get the linear dependence of vortex depth on the Fr number and the Ga ¼ Re2 =Fr number. The linear relation between the Fr number and the vortex depth can be also theoretically justified assuming that the difference between liquid levels at the vessel walls and at the vortex bottom (resulting in a difference in potential energy) is maintained by the continuous conversion of kinetic energy into potential energy (Tsao, 1968; Rao et al., 2009).

In addition, several works can be found aimed at experimentally characterizing free surface shape and/or reviewing scale-up criteria (Zlokarnik, 1971; Rieger et al., 1979; Markopulos and Kontogeorgaki, 1995; Rao et al., 2009). In all these works, liquid levels at both the vessel axis and the vessel wall were found to linearly depend on the Fr number, in agreement with the aforementioned theoretical analyses. In the past, several works employed Computational Fluid Dynamics to model flow field and surface deformation (amongst other fluid-dynamic features) in free-surface unbaffled tanks (Ciofalo et al., 1996; Haque et al., 2006; Cartland Glover and Fitzpatrick, 2007; Torré et al., 2007; Lamarque et al., 2010), which seems to indicate a growing interest in both research and application of these last. Clearly availability of accurate data on free surface shapes is important in the realm of CFD model validation. The present work is therefore aimed at providing accurate measurements of free-surface shape in unbaffled vessels characterized by different geometries (impeller type and position, as well as liquid height).

Fig. 1. Snapshots of the vessel agitated by Rushton turbine, C ¼ T=3, H 0 ¼ T at different impeller speeds: (a) 200 rpm, (b) 500 rpm, (c) 800 rpm, and (d) 1000 rpm.

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On the basis of the new data obtained, a novel (simple yet accurate) predictive model for free-surface shape description is finally proposed.

2. Experimental set-up The experimental system here investigated was an unbaffled cylindrical vessel (diameter T ¼0.19 m) stirred by a Lightnin TS2010 drive with velocity control, equipped with several impeller types (i.e. Rushton 6 blades turbine, Pitched blade turbine, Lighting A310 propeller) all having diameter D¼ T/3. System geometry was also varied by changing the initial liquid height H0 (from 0.75 to 1.25 times the vessel diameter) and the impeller clearance from vessel bottom C (from T/6 to T/2). For each geometry, several impeller velocities in the range 100–1000 rpm were investigated, finally resulting in 173 experiments, covering the Re number range from 2:1  105 to 1:05  106 . The liquid phase was always deionized water. The vessel was back-illuminated by two lamps shielded by a 5 mm Nylon sheet as a light diffuser. For each experiment, 100 images were collected by means of a MVblueFOX C2514-M CCD camera operated at 5 frames per second. In Fig. 1 some typical snapshots are reported for one of the investigated systems at different agitation speeds. As it can be

observed, the back-lighting technique employed clearly results in a neat observation of free-surface shape. Notably, the liquid surface is far from being smooth, since some rippling is present, due to the small instabilities of liquid surface and eruption of small bubbles ingested by the vortex itself at the highest impeller speeds. At the highest velocities shown (800 and 1000 rpm) vortex bottom has already reached the impeller leading to bubble injection in the liquid phase (clearly visible in the images) so resulting into the formation of a gas–liquid dispersion. This fluid-dynamic regime will be hereafter referred to as super-critical regime, while the impeller speed at which vortex bottom reaches the impeller plane will be referred to as critical impeller speed Ncr. Notably, bubbles ingested by the liquid phase are radially entrained by the impeller stream towards vessel wall. Then, while moving upwards under the effect of gravity, they undergo a centripetal acceleration towards the central vortex, due to their smaller density with respect to the liquid phase. Instantaneous free-surface shape measurement from the images in Fig. 1 is not straightforward, due to (i) surface rippling, (ii) the circumstance that surface shape is subject to more or less pronounced oscillations, and (iii) the presence of bubbles in supercritical conditions. Therefore, in order to get meaningful results, images have to be suitably time-averaged. To this end, all 100 images obtained for each data set were pixelwise averaged, so resulting in the time-averaged images reported in Fig. 2.

0.25

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Fig. 2. Averaged images of the vessel agitated by Rushton turbine, C ¼ T=3, H0 ¼ T at different impeller speeds: (a) 500 rpm and (b) 1000 rpm.

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It is worth noting that practically no-noise is present in the sub-critical conditions image (2.a), while some noise is visible in the case of super-critical conditions (2.b) due to bubbles presence. This is bound to adversely affect the following free-surface recognition algorithms based on pixel luminance value. In order to overcome this problem, further image manipulation was found to be advisable. At first, the collected images were manipulated in order to obtain a modified image in which pixels occupied by the liquid surface were assigned larger values than all others. To this end, the squared deviation of each raw image from the average was computed and pixelwise averaged, finally resulting in a deviation image in which moving objects were highlighted. Typical deviation images so obtained are reported in Fig. 3. As it can be seen, the rippling surface gives rise to a very bright region in the deviation images. The deviation images are well suited for directly assessing freesurface shape. To this end a self-developed Direct Fitting on Image (DFI) algorithm was used. This is based on the possibility of fitting a (general) model equation to an object within an image. The algorithm is based on the following steps:


chose a model equation and relevant parameters describing vortex shape;


compute the vortex profile (i.e. in metrical units);


identify image pixels exactly lying on the computed vortex


profile: clearly when these coincide with the actual vortex edge, relatively high luminance values are found. sum the luminance values of the pixel lying along the curve describing the vortex profile.

It is clear that for each parameter set, a luminance sum value is found. The larger this value, the closer the fit between model equation and actual free-surface profile, and therefore the best fitting parameters can be found by a simple optimum search.

3. Vortex modelling Let us consider the hypotheses already discussed in the Introduction section as valid (purely tangential flow field, inviscid fluid, tangential velocity only dependent on the distance from the vessel axis). Hence, Eq. (3) linking the tangential velocity profile to the free surface profile is valid. In this section, a modified flow field model is presented. It involves only one additional parameter α (for forced vortex velocity correction) with respect to Nagata's model (in which α

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z [m]

0.25

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0.1

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0.05

0

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0

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Fig. 3. Deviation images of the vessel agitated by Rushton turbine, C ¼ T=3, H 0 ¼ T at different impeller speeds: (a) 500 rpm and (b) 1000 rpm.

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3

ξ

ξ

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2 ψ

3

ψ

3

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0 0

1

1

2 ξ

3

0 0

1 ξ

Fig. 4. Averaged deviation images of the system agitated by Rushton turbine, C ¼ T=3, H 0 ¼ T at different impeller speeds, and relevant analytical profiles obtained by Nagata's model (dashed blue line, only in sub-critical cases) and 2ZM (solid red line): (a) 200 rpm, (b) 400 rpm, (c) 600 rpm and (d) 800 rpm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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is in practice set to unity): 8 if r rr c < αωr uθ ¼ αωr 2c : if r 4r c r

θ¼

8 αξ > <

ξ2c > :α ξ

ð12Þ

if ξ r ξc ð13Þ

if ξ r ξc

all velocities vanish. This may be expected to result in a rather poor description of free-surface near vessel bottom. However, given the small liquid volume involved, this only marginally affects liquid surface profile predictions in all other regions. By means of the already discussed DFI algorithm, the best fit values of both α and ξc can be directly found. Free-surface profiles obtained with this model will be hereafter referred to as TwoZones Model (2ZM) profiles.

4. Results and discussion The model proposed assumes conservation of angular momentum in the free vortex zone as in the Nagata (1975) model, but introduces a simple correction factor for the velocity in the forcedvortex region as done by Smit and During (1991). By substituting the dimensionless form of Eq. (13) in Eq. (3) and subsequent integration, the following equations are obtained, for all cases in which the deformed surface does not reach the impeller: ! 8 > ξ2 ξ2 > > ψw β 2 c  Fr if ξ r ξc > > < ξ2T ξ2c ! ð14Þ ψ jψ Z ψ c ¼ > ξ2c ξ2c > > > ψ  β  ξ 4 ξ Fr if c > : w ξ2 ξ2 T

where ðαπξc Þ2 2

ð15Þ

By imposing mass conservation (i.e. liquid volume is the same at rest and during agitation) and taking advantage from system symmetry, one can analytically derive formulas for the liquid level at vessel wall and at vortex bottom: ! ξ2c 1 ξ2c  ln 2 Fr ψw ¼ ψ0 þβ 2 ð16Þ ξ 2 ξ w

w

!

ξ2 3 ξ2 ξ2  ln 2c  2 w2 Fr ψ b ¼ ψ 0 þ β 2c ξw 2 ξw ξc

ð17Þ

The critical impeller speed Ncr and relevant critical Froude number (Frcr) can also be analytically obtained by resolving Eq. (17) for the impeller speed at which vortex bottom height equals impeller plane height (ψ c ), finally resulting in: Fr cr ¼

2 1 ξw ψ0 ψc β ξ2c 2ξ2w  3  lnξ2w

ξ2c

ð18Þ

ξ2c

2

4

When vortex bottom falls below the impeller plane it was found that the free-surface portion placed above the impeller plane can still be described by Eq. (14). The free-surface portion placed below the impeller plane was found to be conveniently described on the basis of the vortex bottom height, resulting in the following expression (derived from Eq. (3) integration):

ψ jψ o ψ c ¼ ψ b þ βFr

ξ2 ξ2c

ð19Þ

Again, vortex bottom height can be computed by using Eq. (17), as long as the computed ψ b is positive, while a nil value is assumed otherwise. In this last case a free-surface profile discontinuity arises over the impeller plane, in agreement with experimental observations. It is worth noting that a basic assumption of the model (tangential velocity is independent of z) should not be extended down to vessel bottom, where in the real system

3.5 3 2.5 ψ

β¼

In Fig. 4a–d, free surface profiles observed for the Rushton turbine (C ¼ T=3, H 0 ¼ T) at various agitation speeds are reported. As expected, vortex depth increases while increasing impeller speed. The critical agitation speed falls somewhere between 500 and 600 rpm and in the two super-critical cases (Figs. 4c and d) the free-surface extends below the impeller plane finally reaching vessel bottom. In the same figure, predictions obtained by means of the Nagata (1975) model are reported as dashed blue lines. Notably, Nagata's predictions are reported only in the case of subcritical impeller speeds, this model being derived for such conditions only. As it can be seen, Nagata's model is found to be in reasonable agreement with the present experimental data provided that an optimized ξc value equal to 0.68 is adopted, i.e. a value slightly higher than those proposed by Nagata (in the range 0.55–0.65, see Nagata, 1975). As regards 2ZM, the above-described DFI algorithm was run for all 19 investigated impeller speeds, and the overall best fit values of α ¼ 0:898 and ξc ¼ 0:780 were obtained (not far away from those pertaining Nagata's model, α ¼ 1:00 and ξc ¼ 0:68). The relevant 2ZM vortex shape predictions are reported in Fig. 4 as solid red lines and, as it can be seen, they are in good agreement with experiment not only at sub-critical, but also at super-critical agitation speeds, to which Nagata's model is not meant to be applied. The ability of the proposed model to correctly assess freesurface profiles in unbaffled vessels can also be appreciated by looking at Fig. 5, where the experimental values of liquid height at vessel wall ψ w and at vortex bottom ψ b are reported versus Fr, together with relevant predictions obtained with 2ZM. As a first comment, it is possible to observe that both liquid levels follow a linear trend with Fr number, as it could have been expected on the basis of Eqs. (9), (16) and (17). A change in the slopes is found when free-surface reaches the vessel bottom (Fr  0:75), because

ψw ψb

2

2ZM predictions

1.5 1 0.5 0

0

0.5

1 Fr

1.5

2

Fig. 5. Comparison between experimental and predicted liquid levels as a function of the impeller Froude number (RT, C ¼ T=3, H 0 ¼ T).

A. Busciglio et al. / Chemical Engineering Science 104 (2013) 868–880

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Fig. 6. Averaged deviation images of the system agitated by A310 impeller, C ¼ T=3, H0 ¼ T at different impeller speeds, and relevant analytical profiles obtained by Nagata's model (dashed blue line) and 2ZM (solid red line): (a) 200 rpm, (b) 400 rpm, (c) 600 rpm and (d) 800 rpm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

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of the resulting constrained liquid surface deformation. As it is possible to observe, the 2ZM is able to correctly follow quite well both levels over the entire Fr range. As a consequence, the 2-zones model here proposed appears to be a promising tool for freesurface profile description. Clearly, one of the most important features of a wellformulated model is its general applicability. Under this respect, one may wonder how the model under analysis behaves in the case of different system geometries. To start answering this question, experimental data obtained with a completely different stirrer, namely the well-known Lightnin A310 impeller (axial flow, aerofoil shaped, C ¼ T=3, H 0 ¼ T D ¼ T=3) are now discussed. In Fig. 6 the deviation images obtained in this case, at the same impeller speeds as in Fig. 4, are reported. By comparing Fig. 6 with 4 one may note that quite strong differences exist between the free-surface profiles obtained with the two different impellers. As a matter of fact, not only the A310 gives rise to quite higher Ncr values than the RT, but also the resulting whirlpool shapes are different, with the former being quite narrower and shallower than the latter at all impeller speeds. A preliminary analysis on this system, not reported here for the sake of brevity, showed that there is no way to accommodate the observed differences by acting only on the sole parameter of the original Nagata model (ξc ), since it was found impossible to simultaneously adjust whirlpool shape and depth. In order to show this feature, Nagata's model predictions were optimized to fit vortex depths (ξc ¼ 0:35). The resulting profiles, reported in Fig. 6 as blue solid lines, clearly completely miss vortex shape. Conversely a good agreement is achieved at all agitation speeds if the 2-Zones Model is adopted (see the solid redlines in Fig. 6). In the case of the A310 impeller, the optimized 2ZM parameters are α ¼ 0:540 and ξc ¼ 0:654. These values are significantly different from that obtained for the Rushton turbine. Therefore, parameters of the 2ZM are geometry dependent. Again, the ability of the proposed model in catching the main features of free-surface profile at all investigated agitation speeds can also be appreciated by looking at Fig. 7, where the experimental values of liquid height at vessel wall ψ w and at vortex bottom ψ b are reported versus Fr, together with relevant predictions obtained with 2ZM. Notably, the optimal ξc value (ξc ¼ 0:654) is not too far from that obtained in the case of RT (ξc ¼ 0:780), while a larger difference is found between the relevant α values, 0.540 and 0.898. These differences are likely due to the different flow fields,

4

radially directed in the case of the RT and axially directed in the case of the A310 propeller. In order to further explore the 2ZM adaptability to impeller geometry, free-surface shape data were also collected with a downward pumping pitched blade turbine (PBT), a well known mixed flow impeller. Modelling results reported in Figs. 8 and 9, despite being slightly poorer than in the previous cases (especially in the near axis zone), are still adequate for all practical purposes. In any case they are better than that obtained with Nagata's model. The DFI optimized parameter values α ¼ 0:710 and ξc ¼ 0:812 are again different from those obtained for the other impellers, confirming their dependence on system geometry. Overall, one may observe that the main difference between the impellers under analysis lies in the α value (RT, α ¼ 0:898; PBT, α ¼ 0:743; A310, α ¼ 0:540), while somewhat smaller differences are observed for the critical radius (RT, ξc ¼ 0:780; PBT, ξc ¼ 0:739; A310, ξc ¼ 0:654). Model adaptability to different vessel configurations was finally explored by repeating the above procedures (for all impellers) for two more stirrer positions and two more initial liquid fillings, in addition to the base case of H 0 ¼ T and C ¼ T=3. Notably, for all investigated geometrical configurations, it was always possible to find constant (independent of impeller speed) values for the two 2ZM parameters that gave rise to good agreement between model and experiment. For the sake of shortness all figures supporting the above statement were omitted and in the following two sections only the optimal 2ZM parameter values are discussed. 4.1. Influence of initial liquid filling In Fig. 10, the best fit α and ξc values obtained for various initial liquid fillings (H 0 =T ¼ 3=4; 1; 5=4) are reported as a function of suitable vessel shape factor ψ 0  ψ c (corresponding to the dimensionless distance between the impeller plane and the liquid surface at rest, as suggested by Markopulos and Kontogeorgaki (1995)). In all cases impeller clearance was maintained at a fixed C ¼ T=3 value. As it is possible to observe in Fig. 10a, the ξc values are well correlated by the adopted shape factor: for each impeller, a slightly decreasing trend is found. The impeller design also plays a role in determining the critical radius value. In particular, if one sorts the impellers according to the axiality of their discharge flow (A310 4 PBT 4 RT) a consistent decrease of ξc values is obtained. A similar (but more marked) dependence on the impeller design can be observed in Fig. 10b, in which the α values obtained are reported. In this case a slight increasing trend may be guessed as regards the dependence on shape factor.

3.5 4.2. Influence of impeller clearance

3

ψ

2.5 2 1.5 ψw

1

ψb

0.5

2ZM predictions 0

0

0.5

1 Fr

1.5

2

Fig. 7. Comparison between experimental and predicted liquid levels as a function of the impeller Froude number (A310, C ¼ T=3, H0 ¼ T).

In Fig. 11, the best fit α and ξc values are reported as a function of the same vessel shape factor used for Fig. 10, but in this case the shape factor variation is obtained by changing impeller clearance from vessel bottom (C ¼ T=6 and C ¼ T=2) while liquid filling was maintained at a fixed H 0 ¼ T height. In this case, the critical radius is found to be practically independent of the shape factor while as regards the dependence on the impeller design the considerations previously made obviously apply once again. The α values, reported in Fig. 11b, show a slightly increasing trend when increasing shape factor, but again the main role is played by impeller shape. It is also worth noting that for each impeller, both the ξc and α values are quite close to the relevant values reported in Fig. 10, making Figs. 10a and 11a, as well as Figs. 10b and 11b, quite similar to each other.

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0

1

ξ

2

3

2

3

ξ

3

2

2 ψ

ψ

3

1

1

0

0

1

2 ξ

3

0

0

1 ξ

Fig. 8. Averaged deviation images of the system agitated by PBT, C ¼ T=3, H0 ¼ T at different impeller speeds, and relevant analytical profiles obtained by Nagata's model (dashed blue line, only in sub-critical cases) and 2ZM (solid red line): (a) 200 rpm, (b) 400 rpm, (c) 600 rpm and (d) 800 rpm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

878

A. Busciglio et al. / Chemical Engineering Science 104 (2013) 868–880

It is worth observing how the model parametrization adopted is able to successfully catch the effects that the change in the flow pattern (from radial to axial impellers) have on liquid surface

profile. This is accomplished without changing model hypotheses, but only adjusting model parameters in a physically consistent way. In Table 1, the fitted parameters are reported for the sake of readability.

4 3.5 5. Concluding remarks

3

ψ

2.5 2 1.5 ψw

1

ψb

0.5 0

2ZM predictions 0

0.5

1 Fr

1.5

2

Fig. 9. Comparison between experimental and predicted liquid levels as a function of the impeller Froude number (PBT, C ¼ T=3, H0 ¼ T).

In this paper, extensive data on liquid surface shape in uncovered unbaffled stirred vessels are reported and analyzed by means of an original two parameters model for free-surface profile description. The proposed model was found to be able to quite satisfactorily catch free-surface shape in all investigated cases. It was also found that the model parameters only depend on system geometry, and not on impeller speed, even if regime transition occurs from sub-critical to super-critical one, therefore leading to quite easy use of the model itself. The proposed 2ZM was therefore found to be a simple yet effective tool for free-surface shape description, prone to being adapted to experiments by means of physically consistent parameters adjustment. As a final remark, it can be stated that 2ZM predictions accuracy is surely adequate for all CFD code validation purposes.

1

0.8

ξc

0.6

0.4

RT

0.2

PBT A310 0

1

1.5

2 ψ0−ψc

2.5

3

1.5

2 ψ0−ψc

2.5

3

1

0.8

α

0.6

0.4

RT

0.2

PBT A310 0

1

Fig. 10. Influence of initial liquid filling on 2ZM parameters: best fit values as a function of vessel shape factor and impeller type: (a) ξc and (b) α.

A. Busciglio et al. / Chemical Engineering Science 104 (2013) 868–880

879

1

0.8

ξc

0.6

0.4

RT

0.2

PBT A310 0

1

1.5

2 ψ0−ψc

2.5

3

1.5

2 ψ0−ψc

2.5

3

1

0.8

α

0.6

0.4

RT

0.2

PBT A310 0

1

Fig. 11. Influence of impeller clearance on 2ZM parameters: Best fit values as a function of vessel shape factor and impeller type: (a) ξc and (b) α.

Nomenclature

Table 1 Fitted parameters. H 0 =T

α

ξc

ξc;Nag

Rushton turbine 2.00 1/3 1.25 1/3 2.75 1/3 1.50 1/2 2.50 1/6

1 3/4 5/4 1 1

0.898 0.870 0.969 0.870 0.945

0.780 0.790 0.738 0.780 0.788

0.680 0.651 0.687 0.660 0.705

A310 impeller 2.00 1/3 1.25 1/3 2.75 1/3 1.50 1/2 2.50 1/6

1 3/4 5/4 1 1

0.540 0.531 0.541 0.520 0.550

0.654 0.700 0.607 0.650 0.650

0.350 0.350 0.350 0.350 0.350

Pitched blade turbine 2.00 1/3 1.25 1/3 1.50 1/2 2.50 1/6

1 3/4 1 1

0.743 0.723 0.714 0.755

0.739 0.754 0.735 0.717

0.530 0.532 0.517 0.527

SF

C/T

C D Fr

impeller clearance (m) impeller diameter (m)

Ga

Galileo number Re2 Fr  1 (–) acceleration due to gravity (m s  2) liquid level at rest (m) vortex profile distance above the vessel bottom (m) impeller rotational speed (s  1) pressure (Pa) pressure acting on the liquid free surface (Pa)

g H0 h N p p0 Re r T ur uθ z

α

Froude number N2 Dg  1 (–)

Reynolds number ρND2 μ  1 (–) radial coordinate (m) vessel diameter (m) radial velocity (m s  1) tangential velocity (m s  1) axial coordinate (m) critical region velocity corrector (–)

880

Δψ v μ ξ ξc ξw ρ ψ ψb ψc ψw ψw ω 2ZM RT A310 PBT

A. Busciglio et al. / Chemical Engineering Science 104 (2013) 868–880

dimensionless vortex depth hw =D  hb =D (–) fluid viscosity (–) dimensionless radial coordinate r=ðD=2Þ (–) dimensionless critical radius r c =ðD=2Þ (–) dimensionless vessel radius ðT=2Þ=ðD=2Þ (–) fluid density (kg m  3) dimensionless axial coordinate (–) dimensionless height of vortex bottom hb =D (–) dimensionless impeller clearance C/D (–) dimensionless liquid level at the vessel wall hw =D (–) dimensionless liquid level at rest h0 =D (–) angular velocity (rad s  1) 2 zones model Rushton turbine Lightnin A310 impeller Pitched blade turbine

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