14th European Conference on Mixing Warszawa, 10-13 September 2012

ON VORTEX SHAPE IN UNBAFFLED STIRRED VESSELS AS MEASURED BY DIGITAL IMAGE ANALYSIS

Antonio Busciglioa, Franco Grisafia, Francesca Scargialia, Alberto Brucatoa

a University of Palermo, Dipartimento di Ingegneria Chimica, Gestionale,

Informatica e Meccanica (DICGIM), Viale delle Scienze, Ed. 6, 90128 Palermo (Italy)

antonio.busciglio@unipa.it Abstract In this work, digital image analysis coupled with a suitable shadowgraphy-based technique is employed to investigate the shape of the free-surface vortex that forms in an uncovered unbaffled tanks stirred by either a D=T/3 Lightnin A310 or a D=T/3 Rushton turbine. The technique is based on back-lighting the vessel and suitably averaging vortex images over time. Data obtained show that the two different impellers give rise to quite different vortex shapes. A novel 2-parameter model is proposed that successfully describes vortex shapes obtained with both impellers. Keywords: Mixing, unbaffled stirred tanks, vortex shape, modelling, shadowgraphy.

1. INTRODUCTION Unbaffled stirred tanks are seldom employed in the process industry as they are considered poorer mixers than baffled tanks. However, they may provide significant advantages in a number of applications, including some biochemical, food and pharmaceutical processes, where the presence of baffles is undesirable. The main feature of unbaffled stirred tanks is the highly swirling liquid motion, which leads to the formation of a central vortex on the liquid free surface, when the vessel is operated without top-cover (Unbaffled Uncovered Stirred Tank, UUST). In these last systems, vortex shape determined by the fluid flow field, and as such it is a simple convenient information in the realm of CFD model validations [1]. Apart from that, vortex shape aknowledge is important for design purposes, as the liquid side wall rise under stirring clearly depends on vortex shape. Also quantification of its surface area may be important for gas liquid mass transfer considerations. For these reasons vortex shape in unbaffled tanks has received some attention in the past [2, 3, 4]. In this work, digital image analysis applied to shadowgraps is employed to investigate the shape of free-surface vortexes formed in UUST stirred by either a D=T/3 Lightnin A310 or a D=T/3 Rushton turbine. A quantitative description of vortex shapes obtained with the two impellers is given by means of a novel two-parameter flow field model.

2. EXPERIMENTAL SET UP AND METHODS The experimental system here investigated was an unbaffled cylindrical vessel (T =

0.19m) stirred either by a standard Rushton turbine (D = T/3) or a D=T/3 Lightnin A310, both placed at C=T/3 from vessel bottom. The liquid phase was deionized water at 25 °C. The cylindrical vessel was immersed in a water filled rectangular tank in order to minimize optical distortions due to vessel cylindrical geometry. The gas-liquid dispersion 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.

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In Fig.1 a typical snapshots sequence is reported for one of the systems investigated. The back-lighting technique clearly allows best observation of vortex shape. Notably, the vortex 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.

Figure 1. Raw images recorded with Rushton turbine at 800rpm (Rushton turbine, C = T/3, H0 = T, N = 800rpm).

In order to assess vortex shape, an original direct-fitting-on-image algorithm was developed. Images were suitably manipulated in order to obtain a modified image in which pixels occupied by the vortex were assigned larger values than all others. To this end, all 100 images obtained for each data set were suitably pixelwise averaged, so resulting in a time-averaged image. The deviation of each raw image from the average was computed, and the final deviation image computed.

Figure 2. Rushton turbine vortex as shown in deviation images at 300, 500 and 800 rpm.

It is possible to observe that at the lowest agitation speed the vortex is highly defined, while it becomes more blurred while increasing impeller speed. At the highest velocity it is also possible to observe some noise into the liquid, due to the presence of bubbles generated by the interaction between the vortex and the impeller.

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3. VORTEX MODELLING The vortex in unbaffled mixing tanks is formed in the central part of the vessel as a result

of centrifugal forces acting on the rotating liquid. Some of the works dealing with unbaffled tank modelling [2, 3] pointed out that liquid tangential velocity mainly depends on the distance from the shaft with little axial variations, with the exception of the zone near the vessel bottom, where the tangential velocities are decreased by bottom wall attrition. On this basis, a simplified potential flow model can be adopted for vortex geometry description [1, 2, 3, 4]. In fact, by assuming that velocity in the vessel is purely tangential (ur = uy = 0) and only depends on the radial coordinate (uθ = uθ(r)), Navier-Stokes equations in cylindrical coordinates neglecting viscous forces reduce to:

rp

ru

∂∂

=ρ θ2

(1)

zpg∂∂

=ρ− (2)

By imposing p(r,z) = p0 at the vortex surface h, i.e. dp(r,z) = 0, one has:

rgu

drdh 2

θ= (3)

Hence, the free surface profile h(r) can be derived once uθ(r) is known. By introducing the following dimensionless quantities

DNu

Du

Uu

tip π=

ω==θ θθθ

2/

2/Dr

=ξ Dh

=ψ g

DNFr2

= (4)

the following general dimensionless equation for the description of vortex profile is obtained:

ξψ

=ξθ

πddFr

22 (7)

To determine vortex shape, Nagata (1975) suggested that the whole flow field can be subdivided into an inner region ξ≤ ξc (forced vortex region) exhibiting a rigid body motion with the impeller (angular velocity = ω) and an outer region ξ > ξc (free vortex region) where the angular momentum uθr is constant: this leads to the following tangential velocity profile, solely based on the rc parameter :

⎪⎩

⎪⎨

⎧

>→ω

≤→ω=θ

cc

c

rrrr

rrru 2 (8)

In the paper by Smit and During (1991), a different velocity model was proposed where two additional corrective factors (the exponent 0.6 and the factor 0.825) were added:

⎪⎪⎩

⎪⎪⎨

⎧

>→⎟⎟⎠

⎞⎜⎜⎝

⎛ω

≤→ω

=θc

cc

c

rrrrr

rrr

u 6.0

825.0

825.0

(9)

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In the present work a somewhat different model is proposed, that involves only one additional parameter α with respect to Nagata’s model, that corrects tangential velocity in the forced vortex region only:

⎪⎩

⎪⎨

⎧

>→αω

≤→αω=θ

cc

c

rrrr

rrru 2

⎪⎩

⎪⎨

⎧

ξ>ξ→ξ

αξ

ξ≤ξ→αξ

=θc

c

c2 (10)

This is similar to Smit and During model, except that conservation of angular momentum is here assumed in the free vortex zone, as in Nagata’s model. By substituting Eqn.10 in Eqn.7 and subsequent integration, the following equations describing vortex shape are derived, provided that vortex bottom does not reach the impeller:

⎪⎪

⎩

⎪⎪

⎨

⎧

ξ<ξ→⎟⎟⎠

⎞⎜⎜⎝

⎛

ξξ

−ξξ

−β−ψ

ξ≥ξ→⎟⎟⎠

⎞⎜⎜⎝

⎛

ξξ

−ξξ

β−ψ

=ψψ≥ψ

ccT

cw

cT

ccw

c

2

2

2

2

2

2

2

2

2

where: ( )Frc

2

2απξ=β (11)

On this basis, given the mass conservation (i.e. liquid volume is the same at rest and when the vortex is formed) and given the symmetry of the investigates system, one can analytically derive the liquid height at vessel wall:

⎟⎟⎠

⎞⎜⎜⎝

⎛

ξξ

−ξξ

β+ψ=ψ 2

2

2

2

0 ln21

w

c

w

cw (12)

The impeller speed at which vortex bottom reaches the impeller will be hereafter referred to as the critical impeller speed (Ncr). The relevant critical Froude number (Frcr) can also be analytically obtained by substituting Eqn.12 in Eqn.11 and resolving for impeller speed at which vortex bottom height equals impeller plane height (ψc), finally resulting in:

( )

2

2

2

20

2

2

2ln

232

2

c

w

c

w

c

c

w

ccrFr

ξξ

−−ξξ

ψ−ψξξ

απξ= (13)

The case of super-critical systems, i.e. those where N≥Ncr or Fr≥Frcr,(e.g Fig.1 or Fig.2 right) will not be treated here. Hence in the followings only sub-critical systems will be analyzed.

4. RESULTS Typical variance images obtained are reported in Fig.3 for the two investigated impellers (Rushton turbine Fig.3 a and b, Lightnin A310 Figs.3 c and d) at two different agitation speeds (300 and 500 rpm respectively). As it can be seen, the two impellers at the same agitation speed (hence same Re and Fr) give rise to very different vortex shapes, clearly showing that vortex shape is strongly impeller dependent. In the same Figure, vortex shape predicted by the Nagata model (equations 8, or present model with α = 1, ξc = 0.57) is shown as a solid cyan line. As it can be seen, not only

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Nagata’s model completely fails in the case of the A310 impeller, an obvious result in the light of the above observation on the dependence of vortex shape on impeller type, but also in the case of the Rushton turbine Nagata’s model is found to substantially miss the experimental datum. By fitting for each case parameters α and ξc of the model here proposed (equations 10) to vortex shape, an almost perfect agreement was obtained at all agitation speeds, with both the Rushton and A310 impellers (dashed red lines in Fig.3), though clearly with different values of α and ξc.

Figure 3. Comparison between vortex images and model prediction in the case of Rushton turbine at 300 and 500 rpm; A310 impeller at 300 and 500rpm respectively. Cyan solid line, prediction of the Nagata model (alpha = 1, ξc = 0.57); red line: present model with fitted ξc and α; blue solid line: present model with averaged α and ξc values. In particular the fitting was performed with an original Direct Fitting On Image Algorithm (not detailed here for the sake of brevity) able to automatically find the model parameters that give rise to the best fit with vortex image . The resulting α and ξc values are reported in Fig.4 a and b for the Rushton and A310 turbines respectively.

Figure 4. critical radius and α measurements for the case of Rushton turbine (left) and A310 impeller (right) as a function of Froude Number in sub-critical regime. As it can be seen, for each impeller the two best fit parameter values are almost constant at the various agitation speeds. The model here proposed can therefore be simplified by assuming constant values of

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the two parameters for each geometrical configuration, namely α=0.89 , ξc =0.75 and α=0.55 , ξc =0.68 for the Rushton and A310 impellers respectively. The results are shown in Fig.3 as solid blue lines, and as it can be seen they practically coincide with the red dashed line of the best fit. The adequacy of the present model (with constant α and ξc ) for vortex shape description can be appreciated by observing its ability to predict the two main vortex geometry features: vortex height (liquid height at vessel wall) sand vortex bottom (liquid height at vessel axis). The relevant dimensionless experimental values (ψw = hlw/D , ψb = hlb/D) are compared in Fig.5 with present model predictions. The excellent agreement between model predictions and experiment there observed at all agitation speeds, may be regarded as a good validation of the simplified model here proposed.

Figure 5. Measured liquid levels at vessel wall and at vortex bottom, compared with predicted values for the case of Rushton turbine (left) and A310 impeller (right) as a function of Froude Number in subcritical regime.

5. CONCLUSIONS Vortex shape in unbaffled vessels was found to strongly depend on impeller type. A novel simple model involving two parameters dependent on geometry only, was proposed. Model predictions were compared with experiment and an excellent agreement was found. The model proposed can therefore be regarded as being validated, at least for the cases here investigated.

6. REFERENCES [1] Ciofalo M., Brucato A., Grisafi F., Torraca N., 1996. “Turbulent flow in closed and free-

surface unbaffled tanks stirred by radial impellers”, Chem. Eng. Sci., 51(14), 3557-3573. [2] Nagata S., 1975. Mixing: principles and applications, Wiley, New York. [3] Smit L., During J., 1991. Vortex geometry in stirred vessel, in: Proceedings of the 7th

European Congress of Mixing, 2, Bruges, Belgium, pp. 633-639. [4] Rieger F., Ditl P., Novak V., 1979. “Vortex depth in mixed unbaffled vessels”, Chem.

Eng. Sci., 34, 397-403.

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