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Oscillation dynamics of free vortex surface in uncovered unbaffled stirred vessels
Chemical Engineering Journal 285 (2016) 477–486
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Oscillation dynamics of free vortex surface in uncovered unbaffled stirred vessels A. Busciglio a,⇑, F. Scargiali b, F. Grisafi b, A. Brucato b a b
Dipartimento di Chimica Industriale ‘‘Toso Montanari”, Alma Mater Studiorum – Universitá di Bologna, via Terracini 34, 40131 Bologna, Italy Dipartimento di Ingegneria Chimica, Gestionale, Informatica e Meccanica, Universitá degli Studi di Palermo, Viale delle Scienze, Ed. 6, 90128 Palermo, Italy
h i g h l i g h t s Periodic oscillations of the free surface in unbaffled stirred vessels were measured. Different vessel sizes, liquid fillings, impeller types and speeds were adopted. Oscillation frequencies mainly depend on impeller speed, and vessel scale. Resonant sloshing leading to high amplitude oscillations arise in selected conditions.
a r t i c l e
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Article history: Received 7 July 2015 Received in revised form 2 October 2015 Accepted 5 October 2015 Available online 10 October 2015 Keywords: Mixing Unbaffled stirred tanks Macro-instabilities Surface-instabilities Sloshing
a b s t r a c t 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 (Uncovered Unbaffled Stirred Tanks, UUST). One of the main drawbacks of such vessels, that limits their industrial applicability, is the possible onset of low-frequency sloshing of the free surface. In this work, original data on oscillation dynamics in UUST are presented. In particular, data focus on the oscillation amplitude as well as on their frequency. Data were obtained by means of a novel experimental technique based on digital image analysis. The effect of impeller geometry (Rushton turbine, pitched blade turbine, Lightnin A310 propeller) and that of scale-up were finally investigated. It was found that the natural oscillation frequencies only depend on vessel scale and geometry, while impeller type plays a role in the way the free surface oscillation frequencies change with impeller speed. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction 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 topcover (Unbaffled Uncovered Stirred Tank, UUST). In the latter systems, vortex shape is determined by the fluid flow field, and as such it is a conveniently simple information that may be exploited for CFD models validations [1]. Apart from that, vortex shape knowledge is important for design purposes, as the side wall liquid rise under stirring conditions clearly depends on vortex shape [2]. Unbaffled stirred tanks are seldom employed in the process industry due to their poorer mixing performance with respect to baffled tanks. Nonetheless, they may bring about significant advan⇑ Corresponding author. E-mail addresses: [email protected] (A. Busciglio), francesca.scargiali@ unipa.it (F. Scargiali), [email protected] (F. Grisafi), [email protected] (A. Brucato). http://dx.doi.org/10.1016/j.cej.2015.10.015 1385-8947/Ó 2015 Elsevier B.V. All rights reserved.
tages in a number of applications, including a number of biochemical, food and pharmaceutical processes, where the presence of baffles may be undesirable for several reasons [3]. For instance UUSTs might be conveniently employed as bioreactors for growing shear sensitive biomasses. As a matter of fact, by operating the agitator at speeds sufficiently small for the free vortex bottom to remain above impeller blades, no bubbles are injected in the liquid phase, while a limited oxygen transfer towards the culture medium occurs through the only surface available for oxygen transfer, i.e. the free-surface itself [4]. This certainly limits the maximum viable biomass concentration in the bioreactor, but on the other hand the very high stresses related to bursting bubbles, and relevant cell damage, are avoided; also, the liquid velocity gradients are significantly smaller than in baffled vessels operated at the same agitation speed, so resulting in an overall gentler stirring. As a result UUST may well be particularly suited for growing shear sensitive cells, such as mammalian cells, as suggested by Scargiali et al. [5].
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When dealing with robust cells and non-foaming systems, gas– liquid mass transfer can be further improved by promoting gas ingestion from the liquid surface or by using a self-ingesting device [6–8]. There are other cases in which the use of unbaffled vessels may be desirable. Baffles are usually omitted in the case of very viscous fluids (Re < 20) where they can give rise to dead zones, that badly affect mixing performance [9]. Unbaffled tanks may also be advisable in crystallization operations, where the presence of baffles may promote particle attrition due to the larger impeller– fluid relative velocities [10], or for drawing down floating particles [11]. In bioslurry reactors for soil remediation processes, the use of a sparger might create practical problems, as the solid phase may cause sparger holes blockage [12]. Finally, a strongly vortexing flow field may be employed for quickly removing gas bubbles from a liquid phase [13]. On the other hand there are limitations that make baffle removal unadvisable in many instances. For instance, the longer mixing times exhibited may adversely affect operations requiring fast mixing rates, though it has recently been found that, by suitably selecting the agitation speed, mixing efficiency may be made comparable with that of baffled tanks [14]. Little information is still available on heat transfer through the sidewalls, though smaller heat-transfer coefficients than in baffled vessels may be anticipated, a feature that might make it difficult to extract/provide the needed amount of heat for heat-intensive operations. Amongst the drawbacks that significantly limit UUST adoption in industrial applications there certainly is a free surface instability phenomenon, that under certain conditions makes the free surface shape undergo periodic shape changes, that may have severe adverse mechanical effects on the agitation train and/or vessel supports. It is a phenomenon germane to that called sloshing in transportation engineering, where it has been the subject of a number of studies due to the catastrophic consequences it may have over ships, trucks and even rockets or space vehicles, carrying relatively large quantities of liquids (e.g. [15,16]). As a difference from the characterization of the stationary vortex shape, which has received some attention in the past (see [9,17], among others), free surface instability in unbaffled stirred vessels has not received significant attention so far. Sloshing phenomena in cylindrical vessels received some attention in the past. In the paper by Bauer [18], the oscillation frequencies arising in a cylindrical container are studied for translational and pitch excitation. Rotary sloshing induced by bottom blown liquid jets in cylindrical vessels containing one [19] or two liquids [20] was also studied. Some other references to rotary sloshing can be found in the review by Ibrahim et al. [16], but in that case the rotary sloshing is induced by lateral harmonic excitation. Hence, in the authors’ knowledge, no paper dealing with free surface oscillations in uncovered stirred vessels can be actually found in the literature. From what precedes, it can be concluded that this is an aspect of UUST fluid dynamics that needs to be addressed in order to gain confidence in assessing their suitability for given industrial applications. The present work is actually aimed at gathering initial information on periodic instabilities of vortexing free-surface in unbaffled, uncovered stirred vessels.
2. Experimental set-up and methods The experimental systems investigated here were two unbaffled cylindrical vessels of different sizes (T ¼ 0:19 m;T ¼ 0:48 m). The smaller vessel was agitated by impellers of three different types (six bladed Rushton turbine, Pitched Blade turbine, Lightnin A310 propeller) all of D ¼ T=3 size. As it regards the larger vessel, only the six bladed Rushton turbine (D ¼ T=3), placed at a clearance C ¼ T=3, was explored, in order to start assessing scale-up
effects. In the T ¼ 0:19 m vessel, each impeller type was placed at three different off-bottom clearance values, namely C ¼ T=6; C ¼ T=3 and C ¼ T=2. The minimum impeller speed investigated for all impellers was equal to 100 rpm (Re ¼ 7500 for the smaller vessel, Re ¼ 47; 900 for the larger vessel), while the maximum impeller speed investigated was varied for each impeller depending on the relevant critical impeller speed. The critical impeller speeds were computed by means of suitable correlations [2,3] and reported in Table 1. The liquid phase was deionized water at 25 C. The liquid level within the vessel is equal to the vessel diameter in all cases investigated (H ¼ T). Images of the vessel were collected by means of a MVblueFOX C2514-M CCD camera. In Fig. 1 close-ups of the liquid surface for two different cases (N ¼ 150 rpm and N ¼ 200 rpm) are reported. As it can be seen, the vortex top surface is not flat in any of the two systems, though this effect is much more pronounced at the higher agitation speed. Several surface shape departures from perfect axial symmetry were observed, with different angular periodicities, which also showed a sort of precession motion about the vessel axis, so resulting into more or less complex level oscillations over any vertical line on the vessel wall. In order to assess the characteristic frequencies of such oscillations, image acquisition rates of at least 50 Hz were found to be needed. With the available apparatus this implied the need to reduce the size (in pixels) of the acquired images. The reduced image was focused on a small region near the liquid level at the vessel wall. Each experiment lasted at least 60 s, in order to have sufficient information for assessing low frequency oscillations of the order of 1 Hz. A strong front-illumination device (two 500 W halogen lamps placed at 0:5 m from the vessel wall) was employed to compensate for the small exposure time of the camera. For these experiments, 500 ppm of milk were also added to make the liquid phase opaque and therefore allowing sufficient contrast between the liquid phase and air, without otherwise affecting liquid physical properties. In Fig. 2(a), an example of an instantaneous image as taken by the set-up previously described is reported. As it can be seen, the contrast between the liquid phase (white region) and air (dark region) is neat. In Fig. 2(b) the average value of pixel luminance for each y-location is reported. Notably its derivative (reported in Fig. 2(c)) was found to be very effective for automatic recognition of the instantaneous liquid level. This was obtained in practice via least square fitting of a Gauss function (red line in Fig. 2(c)), whose peak location provided a robust assessment of the instantaneous liquid level. All quantities directly measured in pixel units were subsequently translated into metric length units.
Table 1 Summary of critical condition for the systems investigated. Fr cr
N cr ½s1
Recr
N max ½s1
Rushton turbine 0.19 1/6
0.534
9.12
4:1 104
9.17
0.19
1/3
0.482
8.66
3:9 104
8.33
0.19
1/2
0.385
7.74
3:5 104
7.50
0.48
1/3
0.565
5.89
2:1 105
7.48
A310 impeller 0.19 1/6
2.222
18.60
8:4 104
15.83
0.19
1/3
1.823
16.85
7:6 104
15.00
0.19
1/2
1.491
15.24
6:9 104
12.50
Pitched blade turbine 0.19 1/6
0.988
12.41
5:6 104
10.83
0.19
1/3
0.774
10.98
4:9 104
10.00
0.19
1/2
0.635
9.95
4:5 104
8.33
T
C=T
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Fig. 1. Snapshots of oscillating liquid surface for a vessel equipped with Rushton turbine.
230
220
y (mm)
210
200
190
180
0
0.5 L
1 −0.2
0 dL/dy
0.2
Fig. 2. Example of instantaneous liquid level measurements: (a) raw image (vertical scale is referred to the vessel bottom); (b) average luminance along y direction; (c) derivative of the average luminance and fitting with a Gaussian function, whose maximum position coincides with liquid level height. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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3. Results and discussion As reported in Section 2, by means of the previously described automated procedure it is possible to accurately measure the time evolution of sidewall liquid level at a given angular position. The particular angular position chosen clearly has no role in this measurement, due to system axial symmetry. In the system under consideration, liquid level can only oscillate in time about some average value, with more or less complex oscillation frequencies and amplitudes. To perform a frequency analysis of the liquid level dynamics, the relevant deviation quantity y is first computed, as:
y ðtÞ ¼ yðtÞ
1 T
Z ð1Þ
yðtÞdt T
A typical oscillating liquid level data series, obtained in the small vessel equipped with Rushton turbine agitated at N ¼ 200 rpm , is reported in Fig. 3. Only ten seconds are reported for the sake of clarity. In this case, a well defined oscillating behavior is clearly visible. On average, the liquid level shows oscillations with an amplitude DH 8:5 mm, i.e. about 0:13D. The oscillating liquid level function can therefore undergo standard FFT transform, so obtaining the relevant power spectrum (in the frequency domain) reported in the lower part of Fig. 3. The analysis of Fig. 3 clearly shows the presence of different peaks: a major peak at 3:43 Hz, and other peaks at 2.47, 6.90, 10.33, 13.77 Hz respectively. The major peak has a frequency not far away from, yet not coinciding with, that of the forcing frequency acting on the liquid because of impeller rotational frequency (3:33 Hz). The frequencies of all other peaks do not show simple relationships with stimuli frequencies. However, some of the measured peak frequencies appear to be in simple relation with each other, specifically, the peaks at f i ¼ 6:90; 10:33 and 13:77 Hz are approximately equal to 2f max ; 3f max ; 4f max , where f max ¼ 3:43 Hz is the maximum energy peak frequency. If the same analysis is repeated for the larger vessel equipped with the Rushton turbine at the same agitation speed (i.e. at the
same stimulus frequency), the results reported in Fig. 4 are obtained. Notably, the liquid level time evolution shows again a clearly oscillating behavior, but in this case much larger values of oscillation amplitudes DH ¼ 0:32D are found. Analysis of the relevant power spectrum, reported in the lower part of Fig. 4 shows again the presence of different peaks (a major peak is at 2:43 Hz, while other peaks are at 1.67, 4.83, 7.27, 9.67 Hz respectively). Once again, none of the measured peaks frequencies can be simply related to impeller rotation or blade passage frequency. This fact is a clue that free-surface resonance has to be ascribed to non-trivial interactions between forcing frequencies and system natural frequencies. However, in the case of the large vessel, a pattern similar to that obtained in the smaller vessel is obtained, though peaks location is clearly different, being those pertaining to the larger vessel compressed in a smaller frequency range. As observed for the peak data in the small scale vessel, some of the observed peaks appear to be in simple relation with the major peak: in this case, f i ¼ 4:83; 7:27; 9:67 Hz are approximately equal to 2f max ; 3f max ; 4f max , where f max ¼ 2:43 Hz. These facts point out that the observed frequency components of the power spectrum should be ascribed to the liquid oscillation natural frequencies (main frequencies and relevant harmonics). These may be expected to depend (at least) on liquid inertia and therefore on liquid mass, as well as on the fluid rotational motion. 3.1. Oscillation amplitude In order to assess the operating conditions at which free-surface resonance occurs, in Fig. 5 the average amplitudes measured in all experiments are reported as a function of impeller speed. The average oscillation amplitude was computed by calculating the upper (yu ðtÞ) and lower (yl ðtÞ) envelope of liquid level vs. time curve, and computing the relevant (time averaged) difference. This allows to better show the maximum deviation from average condition.
DH 1 ¼ D T
Z yu ðtÞ yl ðtÞ dt D D T
ð2Þ
5 40
y* [mm]
20 y* [mm]
0
0 −20
−5 0
2
4
time [s]
6
8
10
−40 0
2
4
time [s]
6
8
10
−3
4
x 10
0.025 0.02 |Y(f)|
|Y(f)|
3 2 1 0
0.015 0.01 0.005
2
4
6
8 10 Frequency (Hz)
12
14
Fig. 3. Liquid level oscillations measured at 200 rpm (small vessel equipped with Rushton turbine) and relevant FFT power spectrum in frequency domain.
0
2
4
6
8 10 Frequency (Hz)
12
14
Fig. 4. Liquid level oscillations measured at 200 rpm (large vessel equipped with Rushton turbine) and relevant FFT spectrum in frequency domain.
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Δ H/D [−]
0.2 Rushton, T = 0.19m
0.15 0.1
C = T/6 C = T/3 C = T/2
0.05 0 0
2
4
6
8
10
12
14
16
Δ H/D [−]
0.4
0.2
C = T/6 C = T/3 C = T/2
0.1 0 0
Δ H/D [−]
PBT, T = 0.19m
0.3
0.1
2
4
6
C = T/6 C = T/3 C = T/2
8
10
12
14
16
14
16
A310, T = 0.19m
(i.e. with N 2 ) until in-phase regime is considered, while in this case, the onset of resonance phenomena indicates that direct data comparison is not possible. It is also worth noting that in several cases, an increase of oscillation amplitude is observed at larger impeller velocities than those previously listed. In all cases, these occur when the impeller speed approaches the critical conditions [2], i.e. when the freesurface vortex bottom approaches the impeller plane. Oscillation amplitude enhancement occurs at N ¼ 7—9 rps for the RT, at N ¼ 10—11 rps for the PBT and at N ¼ 14—16 rps for the A310. This phenomenon is clearly shown in Fig. 6 by simply reporting DH=D data relevant to small scale vessel as a function of the normalized impeller velocity N=N cr . In the large vessel equipped with a Rushton turbine, amplitude enhancement generally occurs in a wider impeller speed range, i.e. at N between 1.7 and 5.8rps. Maximum amplitude values of DH=D ¼ 0:47D were measured, hence of the order of four times those observed in the smaller scale geometrically similar vessel, as it is clearly shown in Fig. 7.
0.05 0 0
2
4
6
8
10
12
0.2 Δ H/D [−]
Fig. 5. Mean oscillation amplitudes.
In all cases, a negligible influence on impeller clearance was found in terms of rotational speeds at which resonance occurs, while impeller clearance seems to have an effect on the oscillation amplitude. Axial impellers in fact show considerably smaller oscillation amplitudes when the lowest clearance values (C ¼ T=6) are adopted. A possible reason could be a change in the general flow pattern: these phenomena are well documented in baffled vessel with radial turbines, where the small off-bottom clearance [21], or the proximity of the liquid surface with the impeller [22], radically changes the general flow within the vessel from the wellknown double-loop to the single loop typical of axial impellers. The clarification of which flow modification actually takes place in unbaffled vessels could be the basis for future investigations in the field. It is also possible to resemble some similarities between the oscillating behavior observed in this work with that typical of shaken bioreactors [23–25]. However, in shaken vessels a monotonic increase of the value of DH=D is observed with the Fr number
C = T/6 C = T/3 C = T/2
Rushton, T = 0.19m
0.05 0.2
0.4
0.6
0.8
1
Δ H/D [−]
0.4 0.3
1.2
C = T/6 C = T/3 C = T/2
PBT, T = 0.19m
0.2 0.1 0 0
Δ H/D [−]
In the small scale vessel equipped with the Rushton turbine, amplitude enhancement occurs between 2.5 and 5 rps, with maximum amplitude values of about DH=D ¼ 0:13D. In the same vessel equipped with a pitched blade turbine, amplitude enhancement occurs between 3.3 and 7 rps, with maximum amplitude values of about DH=D ¼ 0:26D. in the same scale vessel equipped with an A310 propeller, amplitude enhancement generally occurs between 5 and 7.5 rps, with maximum amplitude values of about DH=D ¼ 0:08D.
0.1
0 0
0.1
0.2
0.4
0.6
0.8
1
1.2
A310, T = 0.19m C = T/6 C = T/3 C = T/2
0.05 0 0
0.2
0.4
0.6 N/Ncr [−]
0.8
1
1.2
Fig. 6. Mean oscillation amplitudes in the small scale vessel as a function of the N=N cr ratio.
0.6
Run 1 Run 2
0.5 Rushton, T = 0.48m Δ H/D [−]
As it is possible to observe, in all investigated cases a range of impeller speeds exists in which relatively large oscillations occur, as it may be expected when resonance phenomena take place. It is worth noting that in all cases the average fluctuation height is smaller that 2% of impeller diameter, while it reaches much larger values when a certain range of impeller speeds is reached. Notably this range shifts towards higher frequencies the stronger the axial impeller action while system size (see Fig. 7) seems to mainly affect oscillations amplitude. In particular:
0.15
0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 N/Ncr [−]
0.8
1
1.2
Fig. 7. Mean oscillation amplitudes in the large scale vessel as a function of the N=N cr ratio.
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The resonant impeller speeds appear to depend on the type of impeller action (impellers with more intense axial action appear to have larger resonant velocities), while the oscillation amplitude seems to strongly increase with vessel scale. This fact highlights how a clear comprehension of the phenomena leading to surface instability is of crucial importance to predict and avoid catastrophic failures during operation of industrial equipments.
fi Rushton (T = 0.19m) [Hz]
15
3.2. Oscillation frequencies A further attempt to gain insights on the oscillation phenomena under examination can be made by analyzing the power spectra obtained at various agitation speeds. As an example, in Fig. 8 the spectral analysis of liquid level oscillations observed at various agitation speeds is reported. As it can be seen, different peak families can be recognized, i.e. peaks appear in different spectra in related positions. The resonant behavior in this case appears at an impeller speed of N ¼ 3:33 rps, as it was shown in Fig. 5. At this impeller speed, a very large peak near f ¼ 3:46 s1 appears, together with several other peaks at smaller and larger frequencies, as already discussed. The analysis of other spectra measured at larger impeller speeds shows that the same peaks are also found. Notably, slightly different peaks locations and relative heights are found. Both these aspects are worth further discussion. In fact, an interesting feature of characteristic peak frequencies can be observed by reporting the measured peak locations vs. impeller speed, as done in Fig. 9 for the smaller vessel stirred by a Rushton turbine, placed at various clearances from tank bottom. As it can be seen there: Peak locations are practically independent of impeller clearance (data obtained with differently placed impellers are almost perfectly overlapped). Peak families are quite well defined and easy to distinguish; in particular eighth peak families were easily identified. An ninth family, at about 18 Hz could also be plotted but was not reported for the sake of clarity. For each peak family, a neat linear dependence on impeller speed is found. Each peak family may be expected to be related to some natural frequency of the system under investigation. As peak frequencies linearly increase with agitation speed, one may expect that their
10
5
0
0
1
2
3
4
5
6
2
|Y(f)|
1.5
1
0.5
3
5
7
9
11
9
10
intercept at zero impeller speed should result into a frequency list simply related to fluid mass and shape, possibly to some base natural oscillation frequency and its harmonics. Notably, the evidence that a change in impeller vertical location has no effect on the measured peak frequencies encourages such exercise. Also peak relative height changes with impeller speeds, as it is possible to observe in Fig. 8: if the impeller speed is below N 5 rpm, the peak belonging to the second family has the largest height (and give rise to the large oscillations measured at N ¼ 3:33 rps), while at larger N values, the peak belonging to the first family is higher (and give rise to the large oscillations measured at N ¼ 8:33 rps). One may now think that when a periodic stimulus with frequency close enough to one of the natural frequencies of the system is applied, a resonance phenomena should take place, as it happens in mechanics. The resulting resonance would give rise to large fluctuation amplitudes. Considering that peak frequencies change with impeller speed, i.e. with average fluid velocity within the system, one should expect that strong resonance phenomena appear every time the impeller frequency (reported as a dashed line in Fig. 9) equals one of the harmonics frequencies, i.e. passes by a symbol.
−3
1
8
Fig. 9. Measured peak frequencies as a function of impeller speed in the case of small vessel equipped with Rushton turbine, for different impeller clearances (C ¼ D=6 D=2), together with relevant linear best fits. Data relevant to peaks of the same family obtained from systems with different impeller clearance were plotted with the same symbols.
x 10
0
7
N [1/s]
13
15
17
19
8.33 rps 7.50 rps 6.67 rps 5.83 rps 5.00 rps 4.17 rps 3.33 rps 2.50 rps 1.67 rps
Frequency (Hz) Fig. 8. Power spectra recorded at different impeller speeds (small vessel equipped with Rushton turbine, C ¼ T=3).
A. Busciglio et al. / Chemical Engineering Journal 285 (2016) 477–486
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Unfortunately, things are not so plain and while in certain cases this is actually observed (see for instance the second peak at N ¼ 3:33 rps) in other cases it is not (see for instance the fifth peak at 5:8 rps.) Clearly these observations entail a behavior complexity that reinforces doubts on the possibility of finding simple models to explain all observed effects.
3.3. Influence of impeller type Data obtained with the smaller vessel stirred by different impeller types, namely with a Pitched blade turbine or with an A310 propeller, are reported in Fig. 10 as measured peak locations vs. impeller speed. Data show quite similar trends in terms of both peak location (as those reported in Fig. 9) and relative height (data similar to those reported in Fig. 8 were obtained but not reported here for the sake of brevity), so that quite similar conclusion can be drawn. The influence of impeller design on peak frequencies can be better described analyzing data relevant to the undisturbed system frequencies f 0;i , i.e. the extrapolated system frequencies at N ¼ 0 rps (reported in Fig. 11(a) for each peak family) and the slopes describing the oscillation frequency variation with rota-
fi PBT (T = 0.19m) [Hz]
15
10
5
0
0
2
4
6
8
10
12
N [1/s]
15
fi A310 (T = 0.19m) [Hz]
Fig. 11. Measured natural system frequencies and relevant slopes in the case of small vessel equipped with different impellers. 10
5
0
0
2
4
6
8
10
12
14
16
18
20
N [1/s]
Fig. 10. Measured peak frequencies as a function of impeller speed in the case of small vessel at different impeller clearances (C ¼ D=6 D=2), together with relevant linear best fits. Data relevant to peaks of the same family obtained from systems with different impeller clearance were plotted with the same symbols.
tional speed (reported in Fig. 11(b)) as a function of relevant undisturbed system frequencies. The analysis of Fig. 11(a) shows that undisturbed system frequencies do not practically depend on impeller type (as well as on impeller clearance, as previously pointed out), as it should have been expected. Conversely, the impeller type (but not the impeller clearance) plays an important role in the observed oscillation frequency vs. impeller speed slopes. Notably, the measured slope generally increases with the harmonic considered (only 4th and 5th harmonics behave differently in all cases). Moreover, it must be noted that the slope increases with increasing the radial action of the impeller. This is likely due to the different tangential flow field generated by different impellers [2]. Summarizing, the comparison of data obtained with different impellers shows that:
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In all investigated cases, once the impeller geometry is chosen, impeller clearance from vessel bottom plays no role in peak frequency determination. For each impeller type, the same peak families are found. In all cases, the peak family frequencies linearly depend on impeller speed. If the straight lines that best fit all peaks in each peak family are extrapolated to zero impeller speed, a group of f 0;i frequencies is found (referred to as undisturbed or natural system frequencies) independent of impeller clearance and impeller design. The slope si ¼ dfi =dN of the fitting line depend on the relevant f 0i (peak families at lower frequencies show generally smaller slopes), as well as on impeller type, namely increasing with the impeller radial action (in fact, for each peak family, si;Rush > si;PBT > si;A310 ). Increasing the impeller radial action, resonance occurs at progressively smaller impeller rotational speeds. Resonance may occur approximately at N ¼ 0:3 0:6N cr (slightly different values are found for different impellers) by excitation of the second harmonic and at N ¼ 0:8 1:0N cr by excitation of the first harmonic.
impeller), the only difference being the vessel scale, in order to isolate its effect on free surface instabilities. As it can be seen, the plot obtained is similar to that obtained in the small vessel. However, the system natural frequencies (f 0;i ) appear to involve a quite narrower range of values, as it can be appreciated in Fig. 13(a), where the average natural frequencies in the small vessel case are compared with those measured in the large vessel. The slopes describing the oscillation frequency variation with rotational speed (reported in Fig. 13(b) as a function of relevant natural system frequencies) show a similar trend when changing the vessel diameter, therefore confirming that such slopes mainly depend on the flow field generated by the impeller. The f 0;i data trend is exactly similar when changing the vessel scale, but smaller frequencies are found in the larger vessel, as expected. The number of natural frequencies ready to be excited
The substantial lack of dependence of oscillation frequencies at zero impeller speed f 0i on impeller geometry and position shows that these quantities only depend on liquid mass in the vessel and its aspect ratio, i.e. may be regarded as the natural sloshing frequencies of the rotating liquid. Conversely, the flow field has to play some role in shifting the natural frequencies. Clearly, a suitable mathematical modeling of the system aimed at providing full physical interpretation of the above findings is needed if results extrapolation to other fluids or other system sizes is desired. This is planned to be the aim of future works. For the time being, on the basis the above results, one can only guess that it is not going to be a simple and straightforward task.
3.4. Influence of vessel scale In Fig. 12 the measured peak locations as a function of impeller speed in case of large vessel equipped with Rushton turbine are reported, together with relevant linear fitting line. The vessels under investigation have the same geometry (same D=T ¼ 1=3; H=T ¼ 1 and C=D ¼ 1 ratios, geometrically similar
12
fi Rushton (T = 0.48m) [Hz]
10
8
6
4
2
0 0
1
2
3
4
5 N [1/s]
6
7
8
9
10
Fig. 12. Measured Peaks frequencies as a function of impeller speed in the case of large vessel equipped with Rushton turbine (C ¼ T=3), together with relevant linear best fits.
Fig. 13. Measured natural system frequencies and relevant slopes in the case of differently sized vessels equipped with six-blade Rushton turbines (C = T/3, D = T/3).
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in the range 1:4—3:7 Hz is likely to be responsible for the highly oscillating behavior of large scale vessel. On average, natural frequencies f 0;i in large scale vessel are 0:67 times those measured in small scale vessel. This scale-up value can be predicted if the Miles’ solution for the sloshing analysis in cylindrical vessel is considered [26]. According to this model, the dominant mode of oscillation of a liquid in a vertical cylindrical tank can be computed as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2g 2H tanh 1:84 f 0 ¼ 0:216 T T
ð3Þ
When applied to the vessel under investigation, a value of f 0 ¼ 2:19 Hz is found for the small vessel (equal to f 0;1 ¼ 2:18) and f 0 ¼ 1:38 Hz for the large vessel (quite similar to the measured value, f 0;1 ¼ 1:43): the first peak family is therefore related to the dominant sloshing frequency of the liquid in the vessel. Notably, when passing from small to large scale vessel, in Eq. (3) all factors remains constant with the exception of the T value (the H/T value remains constant), so resulting in the following scaling law:
f 0;i T 0:5 ¼ const:
ð4Þ
The validity of Eq. (4) for all the measured harmonics is apparent in Fig. 14. Last point to discuss, it was shown that some of the peaks appeared to be in simple relation with the second peak, and that during resonance phenomena either second (at smaller N values) of first peak (near to critical conditions) became the largest peaks. These two peak families are playing a major role in determining the resonant behavior. Hence, it was decided to report undisturbed system frequencies in dimensionless form, as f 0;i =f 0;1 or f 0;i =f 0;2 , to highlight any simple relation between natural frequencies. As it is possible to observe in Fig. 15, the 4th, 7th and 8th natural frequencies are multiples of the 1st natural frequency, while the 6th, 7th are multiples of the 2nd natural frequency. While 1st natural frequency (and its multiples) has a clear origin in the dominant oscillation mode discussed by [26], the origin of the second natural frequency (and its multiples) is not yet clear, as well as that of other natural frequencies. Only hypotheses can be made, such as the relation with macro-instabilities similar to those observed in baffled vessels [27,28], but a thorough explanation of the above observations was not achieved at this stage. 5 4.5 4
Fig. 15. Dimensionless natural system frequencies.
Clearly it will involve a complex in depth analysis of the involved phenomena, for which probably further data, as well as different experimental techniques, will be needed.
f0,i T0.5 [Hz]
3.5 3 2.5
4. Conclusions
2 1.5 1 0.5 0
1st
2nd
3rd 4th 5th 6th Natural harmonic
7th
Fig. 14. Scaled frequencies for systems at different scales.
8th
In this work, oscillation amplitudes and frequencies of free surface in unbaffled stirred vessels were measured by means of a novel technique based on image analysis. The influence of impeller design, impeller clearance and vessel scale was assessed. It was found that the natural oscillation frequencies only depend on vessel scale and geometry, while impeller type plays a role in the way the free-surface oscillation frequencies change with impeller speed. The resonant impeller speeds appear to depend on the type of impeller action (impellers with more intense axial action appear
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to have larger resonant velocities), while the oscillation amplitude seems to strongly increase with vessel scale. This fact highlights how a clear comprehension of the phenomena leading to surface instability is of crucial importance to predict and avoid catastrophic failures during operation of industrial equipments. The relation between natural system frequencies, stimuli frequencies and oscillation amplitudes was not clarified here, due to the significant complexities involved. The present work should be regarded as a first paper addressing the free surface oscillation phenomenon and reporting on quantitative experimental observations that may guide future modeling efforts as well as provide a data base for model validations. Acknowledgements This work was financially supported through the Project BIO4BIO – Biomolecular and Energy valorization of residual biomass from Agroindustry and Fishing Industry led by the Cluster Sicily Agrobio and Fishing Industry and funded by the Italian Research Fund (PON R&C 20072013, DD 713/Ric. – PON02 00451 3362376. References [1] M. Ciofalo, A. Brucato, F. Grisafi, N. Torraca, Turbulent flow in closed and freesurface unbaffled tanks stirred by radial impellers, Chem. Eng. Sci. 51 (14) (1996) 3557–3573. [2] A. Busciglio, G. Caputo, F. Scargiali, Free-surface shape in unbaffled stirred vessels: experimental study via digital image analysis, Chem. Eng. Sci. 104 (2013) 868–880. [3] F. Scargiali, A. Busciglio, F. Grisafi, A. Tamburini, G. Micale, A. Brucato, Power consumption in uncovered-unbaffled stirred tanks: influence of viscosity and flow regime, Ind. Eng. Chem. Res. 52 (42) (2013) 14998–15005. [4] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Mass transfer and hydrodynamic characteristics of unbaffled stirred bio-reactors: influence of impeller design, Biochem. Eng. J. 82 (2014) 41–47. [5] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Free surface oxygen transfer in large aspect ratio unbaffled bio-reactors, with or without draft tube, Biochem. Eng. J. 100 (2015) 16–22. [6] K. Conway, A. Kyle, C. Rielly, Gasliquidsolid operation of a vortex-ingesting stirred tank reactor, Chem. Eng. Res. Des. 80 (Part A) (2002) 839–845. [7] F. Scargiali, R. Russo, F. Grisafi, A. Brucato, Mass transfer and hydrodinamic characteristics of a high aspect ratio self-ingesting reactor for gas–liquid operations, Chem. Eng. Sci. 62 (5) (2007) 1376–1387. [8] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Gas–liquid–solid operation of a high aspect ratio self-ingesting reactor, Int. J. Chem. Reactor Eng. 10 (1) (2012) A27, http://dx.doi.org/10.1515/1542-6580.3011.
[9] S. Nagata, Mixing: Principles and Applications, Wiley, New York, 1975. [10] B. Mazzarotta, Comminution phenomena in stirred sugar suspension, AICHE Symp. Ser. 89 (293) (1993) 12–117. [11] B. Freudig, S. Hogekamp, H. Schubert, Dispersion of powders in liquids in a stirred vessel, Chem. Eng. Process. 38 (1999) 525. [12] A. Tamburini, A. Brucato, A. Busciglio, A. Cipollina, F. Grisafi, G. Micale, F. Scargiali, G. Vella, Solid–liquid suspensions in top-covered unbaffled vessels: influence of particle size, liquid viscosity, impeller size, and clearance, Ind. Eng. Chem. Res. 53 (2014) 9587–9599. [13] L. Smit, J. During, Vortex geometry in stirred vessel, in: 7th European Congress of Mixing, vol. 2, Bruges, Belgium,1991, pp. 633–639. [14] A. Busciglio, F. Grisafi, F. Scargiali, A. Brucato, Mixing dynamics in uncovered unbaffled stirred tanks, Chem. Eng. J. 254 (2014) 210–219. [15] H. Abramson, The dynamic behavior of liquids in moving containers (with application to space vehicle technology), National Aeronautics and Space Administration, Washington DC, nASA SP-106, N67–15884, prepared under contract no. NASr-94(07), 1966. [16] R. Ibrahim, V. Pilipchuk, T. Ikeda, Recent advances in liquid sloshing dynamics, Appl. Mech. Rev. 54 (2) (2001) 133–199. [17] J. Markopulos, E. Kontogeorgaki, Vortex depth in unbaffled single and multiple impeller agitated vessels, Chem. Eng. Technol. 18 (1995) 68–74. [18] H. Bauer, Natural frequencies and response of spinning liquid column with apparently sliding contact line, Acta Mech. 97 (1993) 115–122. [19] Y. Ueda, T. Ishii, M. Iguchi, Numerical simulation on a horizontal flow pattern of jet induced rotary sloshing in a cylindrical container, ISIJ Int. 47 (2007) 940– 944. [20] M. Iguchi, T. Ishii, Y. Ueda, Prediction of the occurrence of swirl motion at interface between stratified two liquids, Mat. Trans. 46 (3) (2005) 651–657. [21] G. Montante, K. Lee, A. Brucato, M. Yianneskis, Numerical simulations of the dependency of flow pattern on impeller clearance in stirred vessels, Chem. Eng. Sci. 56 (2001) 3751–3770. [22] S. Motamedvaziri, P. Armenante, Flow regimes and surface air entrainment in partially filled stirred vessel for different fill ratios, Chem. Eng. Sci. 81 (2012) 231–250. [23] H. Weheliye, M. Yianneskis, A. Ducci, On the fluid dynamics of shaken bioreactors – flow characterization and transition, AIChE J. 59 (1) (2013) 334– 344. [24] G. Rodriguez, W. Weheliye, T. Anderlei, M. Micheletti, M. Yianneskis, A. Ducci, Mixing time and kinetic energy measurements in a shaken cylindrical bioreactor, Chem. Eng. Res. Des 91 (2013) 2084–2097. [25] A. Ducci, H. Weheliye, Orbitally shaken bioreactors – viscosity effects on flow characteristics, AIChE J. 60 (11) (2014) 3951–3968. [26] J. Miles, On the sloshing of liquid in a cylindrical tank, Tech. Rep. Report No. AM 6-5, AD 605958, The Ramo-Wooldridge Corporation, 1956. [27] V. Roussinova, S. Kresta, R. Weetman, Low frequency macroinstabilities in a stirred tank: scale-up and prediction based on large eddy simulations, Chem. Eng. Sci. 58 (2003) 2297–2311. [28] L. Nikiforaki, G. Montante, K. Lee, M. Yianneskis, On the origin, frequency and magnitude of macro-instabilities of the flows in stirred vessels, Chem. Eng. Sci. 58 (2003) 2937–2949.
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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej
Oscillation dynamics of free vortex surface in uncovered unbaffled stirred vessels A. Busciglio a,⇑, F. Scargiali b, F. Grisafi b, A. Brucato b a b
Dipartimento di Chimica Industriale ‘‘Toso Montanari”, Alma Mater Studiorum – Universitá di Bologna, via Terracini 34, 40131 Bologna, Italy Dipartimento di Ingegneria Chimica, Gestionale, Informatica e Meccanica, Universitá degli Studi di Palermo, Viale delle Scienze, Ed. 6, 90128 Palermo, Italy
h i g h l i g h t s Periodic oscillations of the free surface in unbaffled stirred vessels were measured. Different vessel sizes, liquid fillings, impeller types and speeds were adopted. Oscillation frequencies mainly depend on impeller speed, and vessel scale. Resonant sloshing leading to high amplitude oscillations arise in selected conditions.
a r t i c l e
i n f o
Article history: Received 7 July 2015 Received in revised form 2 October 2015 Accepted 5 October 2015 Available online 10 October 2015 Keywords: Mixing Unbaffled stirred tanks Macro-instabilities Surface-instabilities Sloshing
a b s t r a c t 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 (Uncovered Unbaffled Stirred Tanks, UUST). One of the main drawbacks of such vessels, that limits their industrial applicability, is the possible onset of low-frequency sloshing of the free surface. In this work, original data on oscillation dynamics in UUST are presented. In particular, data focus on the oscillation amplitude as well as on their frequency. Data were obtained by means of a novel experimental technique based on digital image analysis. The effect of impeller geometry (Rushton turbine, pitched blade turbine, Lightnin A310 propeller) and that of scale-up were finally investigated. It was found that the natural oscillation frequencies only depend on vessel scale and geometry, while impeller type plays a role in the way the free surface oscillation frequencies change with impeller speed. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction 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 topcover (Unbaffled Uncovered Stirred Tank, UUST). In the latter systems, vortex shape is determined by the fluid flow field, and as such it is a conveniently simple information that may be exploited for CFD models validations [1]. Apart from that, vortex shape knowledge is important for design purposes, as the side wall liquid rise under stirring conditions clearly depends on vortex shape [2]. Unbaffled stirred tanks are seldom employed in the process industry due to their poorer mixing performance with respect to baffled tanks. Nonetheless, they may bring about significant advan⇑ Corresponding author. E-mail addresses: [email protected] (A. Busciglio), francesca.scargiali@ unipa.it (F. Scargiali), [email protected] (F. Grisafi), [email protected] (A. Brucato). http://dx.doi.org/10.1016/j.cej.2015.10.015 1385-8947/Ó 2015 Elsevier B.V. All rights reserved.
tages in a number of applications, including a number of biochemical, food and pharmaceutical processes, where the presence of baffles may be undesirable for several reasons [3]. For instance UUSTs might be conveniently employed as bioreactors for growing shear sensitive biomasses. As a matter of fact, by operating the agitator at speeds sufficiently small for the free vortex bottom to remain above impeller blades, no bubbles are injected in the liquid phase, while a limited oxygen transfer towards the culture medium occurs through the only surface available for oxygen transfer, i.e. the free-surface itself [4]. This certainly limits the maximum viable biomass concentration in the bioreactor, but on the other hand the very high stresses related to bursting bubbles, and relevant cell damage, are avoided; also, the liquid velocity gradients are significantly smaller than in baffled vessels operated at the same agitation speed, so resulting in an overall gentler stirring. As a result UUST may well be particularly suited for growing shear sensitive cells, such as mammalian cells, as suggested by Scargiali et al. [5].
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When dealing with robust cells and non-foaming systems, gas– liquid mass transfer can be further improved by promoting gas ingestion from the liquid surface or by using a self-ingesting device [6–8]. There are other cases in which the use of unbaffled vessels may be desirable. Baffles are usually omitted in the case of very viscous fluids (Re < 20) where they can give rise to dead zones, that badly affect mixing performance [9]. Unbaffled tanks may also be advisable in crystallization operations, where the presence of baffles may promote particle attrition due to the larger impeller– fluid relative velocities [10], or for drawing down floating particles [11]. In bioslurry reactors for soil remediation processes, the use of a sparger might create practical problems, as the solid phase may cause sparger holes blockage [12]. Finally, a strongly vortexing flow field may be employed for quickly removing gas bubbles from a liquid phase [13]. On the other hand there are limitations that make baffle removal unadvisable in many instances. For instance, the longer mixing times exhibited may adversely affect operations requiring fast mixing rates, though it has recently been found that, by suitably selecting the agitation speed, mixing efficiency may be made comparable with that of baffled tanks [14]. Little information is still available on heat transfer through the sidewalls, though smaller heat-transfer coefficients than in baffled vessels may be anticipated, a feature that might make it difficult to extract/provide the needed amount of heat for heat-intensive operations. Amongst the drawbacks that significantly limit UUST adoption in industrial applications there certainly is a free surface instability phenomenon, that under certain conditions makes the free surface shape undergo periodic shape changes, that may have severe adverse mechanical effects on the agitation train and/or vessel supports. It is a phenomenon germane to that called sloshing in transportation engineering, where it has been the subject of a number of studies due to the catastrophic consequences it may have over ships, trucks and even rockets or space vehicles, carrying relatively large quantities of liquids (e.g. [15,16]). As a difference from the characterization of the stationary vortex shape, which has received some attention in the past (see [9,17], among others), free surface instability in unbaffled stirred vessels has not received significant attention so far. Sloshing phenomena in cylindrical vessels received some attention in the past. In the paper by Bauer [18], the oscillation frequencies arising in a cylindrical container are studied for translational and pitch excitation. Rotary sloshing induced by bottom blown liquid jets in cylindrical vessels containing one [19] or two liquids [20] was also studied. Some other references to rotary sloshing can be found in the review by Ibrahim et al. [16], but in that case the rotary sloshing is induced by lateral harmonic excitation. Hence, in the authors’ knowledge, no paper dealing with free surface oscillations in uncovered stirred vessels can be actually found in the literature. From what precedes, it can be concluded that this is an aspect of UUST fluid dynamics that needs to be addressed in order to gain confidence in assessing their suitability for given industrial applications. The present work is actually aimed at gathering initial information on periodic instabilities of vortexing free-surface in unbaffled, uncovered stirred vessels.
2. Experimental set-up and methods The experimental systems investigated here were two unbaffled cylindrical vessels of different sizes (T ¼ 0:19 m;T ¼ 0:48 m). The smaller vessel was agitated by impellers of three different types (six bladed Rushton turbine, Pitched Blade turbine, Lightnin A310 propeller) all of D ¼ T=3 size. As it regards the larger vessel, only the six bladed Rushton turbine (D ¼ T=3), placed at a clearance C ¼ T=3, was explored, in order to start assessing scale-up
effects. In the T ¼ 0:19 m vessel, each impeller type was placed at three different off-bottom clearance values, namely C ¼ T=6; C ¼ T=3 and C ¼ T=2. The minimum impeller speed investigated for all impellers was equal to 100 rpm (Re ¼ 7500 for the smaller vessel, Re ¼ 47; 900 for the larger vessel), while the maximum impeller speed investigated was varied for each impeller depending on the relevant critical impeller speed. The critical impeller speeds were computed by means of suitable correlations [2,3] and reported in Table 1. The liquid phase was deionized water at 25 C. The liquid level within the vessel is equal to the vessel diameter in all cases investigated (H ¼ T). Images of the vessel were collected by means of a MVblueFOX C2514-M CCD camera. In Fig. 1 close-ups of the liquid surface for two different cases (N ¼ 150 rpm and N ¼ 200 rpm) are reported. As it can be seen, the vortex top surface is not flat in any of the two systems, though this effect is much more pronounced at the higher agitation speed. Several surface shape departures from perfect axial symmetry were observed, with different angular periodicities, which also showed a sort of precession motion about the vessel axis, so resulting into more or less complex level oscillations over any vertical line on the vessel wall. In order to assess the characteristic frequencies of such oscillations, image acquisition rates of at least 50 Hz were found to be needed. With the available apparatus this implied the need to reduce the size (in pixels) of the acquired images. The reduced image was focused on a small region near the liquid level at the vessel wall. Each experiment lasted at least 60 s, in order to have sufficient information for assessing low frequency oscillations of the order of 1 Hz. A strong front-illumination device (two 500 W halogen lamps placed at 0:5 m from the vessel wall) was employed to compensate for the small exposure time of the camera. For these experiments, 500 ppm of milk were also added to make the liquid phase opaque and therefore allowing sufficient contrast between the liquid phase and air, without otherwise affecting liquid physical properties. In Fig. 2(a), an example of an instantaneous image as taken by the set-up previously described is reported. As it can be seen, the contrast between the liquid phase (white region) and air (dark region) is neat. In Fig. 2(b) the average value of pixel luminance for each y-location is reported. Notably its derivative (reported in Fig. 2(c)) was found to be very effective for automatic recognition of the instantaneous liquid level. This was obtained in practice via least square fitting of a Gauss function (red line in Fig. 2(c)), whose peak location provided a robust assessment of the instantaneous liquid level. All quantities directly measured in pixel units were subsequently translated into metric length units.
Table 1 Summary of critical condition for the systems investigated. Fr cr
N cr ½s1
Recr
N max ½s1
Rushton turbine 0.19 1/6
0.534
9.12
4:1 104
9.17
0.19
1/3
0.482
8.66
3:9 104
8.33
0.19
1/2
0.385
7.74
3:5 104
7.50
0.48
1/3
0.565
5.89
2:1 105
7.48
A310 impeller 0.19 1/6
2.222
18.60
8:4 104
15.83
0.19
1/3
1.823
16.85
7:6 104
15.00
0.19
1/2
1.491
15.24
6:9 104
12.50
Pitched blade turbine 0.19 1/6
0.988
12.41
5:6 104
10.83
0.19
1/3
0.774
10.98
4:9 104
10.00
0.19
1/2
0.635
9.95
4:5 104
8.33
T
C=T
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Fig. 1. Snapshots of oscillating liquid surface for a vessel equipped with Rushton turbine.
230
220
y (mm)
210
200
190
180
0
0.5 L
1 −0.2
0 dL/dy
0.2
Fig. 2. Example of instantaneous liquid level measurements: (a) raw image (vertical scale is referred to the vessel bottom); (b) average luminance along y direction; (c) derivative of the average luminance and fitting with a Gaussian function, whose maximum position coincides with liquid level height. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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3. Results and discussion As reported in Section 2, by means of the previously described automated procedure it is possible to accurately measure the time evolution of sidewall liquid level at a given angular position. The particular angular position chosen clearly has no role in this measurement, due to system axial symmetry. In the system under consideration, liquid level can only oscillate in time about some average value, with more or less complex oscillation frequencies and amplitudes. To perform a frequency analysis of the liquid level dynamics, the relevant deviation quantity y is first computed, as:
y ðtÞ ¼ yðtÞ
1 T
Z ð1Þ
yðtÞdt T
A typical oscillating liquid level data series, obtained in the small vessel equipped with Rushton turbine agitated at N ¼ 200 rpm , is reported in Fig. 3. Only ten seconds are reported for the sake of clarity. In this case, a well defined oscillating behavior is clearly visible. On average, the liquid level shows oscillations with an amplitude DH 8:5 mm, i.e. about 0:13D. The oscillating liquid level function can therefore undergo standard FFT transform, so obtaining the relevant power spectrum (in the frequency domain) reported in the lower part of Fig. 3. The analysis of Fig. 3 clearly shows the presence of different peaks: a major peak at 3:43 Hz, and other peaks at 2.47, 6.90, 10.33, 13.77 Hz respectively. The major peak has a frequency not far away from, yet not coinciding with, that of the forcing frequency acting on the liquid because of impeller rotational frequency (3:33 Hz). The frequencies of all other peaks do not show simple relationships with stimuli frequencies. However, some of the measured peak frequencies appear to be in simple relation with each other, specifically, the peaks at f i ¼ 6:90; 10:33 and 13:77 Hz are approximately equal to 2f max ; 3f max ; 4f max , where f max ¼ 3:43 Hz is the maximum energy peak frequency. If the same analysis is repeated for the larger vessel equipped with the Rushton turbine at the same agitation speed (i.e. at the
same stimulus frequency), the results reported in Fig. 4 are obtained. Notably, the liquid level time evolution shows again a clearly oscillating behavior, but in this case much larger values of oscillation amplitudes DH ¼ 0:32D are found. Analysis of the relevant power spectrum, reported in the lower part of Fig. 4 shows again the presence of different peaks (a major peak is at 2:43 Hz, while other peaks are at 1.67, 4.83, 7.27, 9.67 Hz respectively). Once again, none of the measured peaks frequencies can be simply related to impeller rotation or blade passage frequency. This fact is a clue that free-surface resonance has to be ascribed to non-trivial interactions between forcing frequencies and system natural frequencies. However, in the case of the large vessel, a pattern similar to that obtained in the smaller vessel is obtained, though peaks location is clearly different, being those pertaining to the larger vessel compressed in a smaller frequency range. As observed for the peak data in the small scale vessel, some of the observed peaks appear to be in simple relation with the major peak: in this case, f i ¼ 4:83; 7:27; 9:67 Hz are approximately equal to 2f max ; 3f max ; 4f max , where f max ¼ 2:43 Hz. These facts point out that the observed frequency components of the power spectrum should be ascribed to the liquid oscillation natural frequencies (main frequencies and relevant harmonics). These may be expected to depend (at least) on liquid inertia and therefore on liquid mass, as well as on the fluid rotational motion. 3.1. Oscillation amplitude In order to assess the operating conditions at which free-surface resonance occurs, in Fig. 5 the average amplitudes measured in all experiments are reported as a function of impeller speed. The average oscillation amplitude was computed by calculating the upper (yu ðtÞ) and lower (yl ðtÞ) envelope of liquid level vs. time curve, and computing the relevant (time averaged) difference. This allows to better show the maximum deviation from average condition.
DH 1 ¼ D T
Z yu ðtÞ yl ðtÞ dt D D T
ð2Þ
5 40
y* [mm]
20 y* [mm]
0
0 −20
−5 0
2
4
time [s]
6
8
10
−40 0
2
4
time [s]
6
8
10
−3
4
x 10
0.025 0.02 |Y(f)|
|Y(f)|
3 2 1 0
0.015 0.01 0.005
2
4
6
8 10 Frequency (Hz)
12
14
Fig. 3. Liquid level oscillations measured at 200 rpm (small vessel equipped with Rushton turbine) and relevant FFT power spectrum in frequency domain.
0
2
4
6
8 10 Frequency (Hz)
12
14
Fig. 4. Liquid level oscillations measured at 200 rpm (large vessel equipped with Rushton turbine) and relevant FFT spectrum in frequency domain.
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Δ H/D [−]
0.2 Rushton, T = 0.19m
0.15 0.1
C = T/6 C = T/3 C = T/2
0.05 0 0
2
4
6
8
10
12
14
16
Δ H/D [−]
0.4
0.2
C = T/6 C = T/3 C = T/2
0.1 0 0
Δ H/D [−]
PBT, T = 0.19m
0.3
0.1
2
4
6
C = T/6 C = T/3 C = T/2
8
10
12
14
16
14
16
A310, T = 0.19m
(i.e. with N 2 ) until in-phase regime is considered, while in this case, the onset of resonance phenomena indicates that direct data comparison is not possible. It is also worth noting that in several cases, an increase of oscillation amplitude is observed at larger impeller velocities than those previously listed. In all cases, these occur when the impeller speed approaches the critical conditions [2], i.e. when the freesurface vortex bottom approaches the impeller plane. Oscillation amplitude enhancement occurs at N ¼ 7—9 rps for the RT, at N ¼ 10—11 rps for the PBT and at N ¼ 14—16 rps for the A310. This phenomenon is clearly shown in Fig. 6 by simply reporting DH=D data relevant to small scale vessel as a function of the normalized impeller velocity N=N cr . In the large vessel equipped with a Rushton turbine, amplitude enhancement generally occurs in a wider impeller speed range, i.e. at N between 1.7 and 5.8rps. Maximum amplitude values of DH=D ¼ 0:47D were measured, hence of the order of four times those observed in the smaller scale geometrically similar vessel, as it is clearly shown in Fig. 7.
0.05 0 0
2
4
6
8
10
12
0.2 Δ H/D [−]
Fig. 5. Mean oscillation amplitudes.
In all cases, a negligible influence on impeller clearance was found in terms of rotational speeds at which resonance occurs, while impeller clearance seems to have an effect on the oscillation amplitude. Axial impellers in fact show considerably smaller oscillation amplitudes when the lowest clearance values (C ¼ T=6) are adopted. A possible reason could be a change in the general flow pattern: these phenomena are well documented in baffled vessel with radial turbines, where the small off-bottom clearance [21], or the proximity of the liquid surface with the impeller [22], radically changes the general flow within the vessel from the wellknown double-loop to the single loop typical of axial impellers. The clarification of which flow modification actually takes place in unbaffled vessels could be the basis for future investigations in the field. It is also possible to resemble some similarities between the oscillating behavior observed in this work with that typical of shaken bioreactors [23–25]. However, in shaken vessels a monotonic increase of the value of DH=D is observed with the Fr number
C = T/6 C = T/3 C = T/2
Rushton, T = 0.19m
0.05 0.2
0.4
0.6
0.8
1
Δ H/D [−]
0.4 0.3
1.2
C = T/6 C = T/3 C = T/2
PBT, T = 0.19m
0.2 0.1 0 0
Δ H/D [−]
In the small scale vessel equipped with the Rushton turbine, amplitude enhancement occurs between 2.5 and 5 rps, with maximum amplitude values of about DH=D ¼ 0:13D. In the same vessel equipped with a pitched blade turbine, amplitude enhancement occurs between 3.3 and 7 rps, with maximum amplitude values of about DH=D ¼ 0:26D. in the same scale vessel equipped with an A310 propeller, amplitude enhancement generally occurs between 5 and 7.5 rps, with maximum amplitude values of about DH=D ¼ 0:08D.
0.1
0 0
0.1
0.2
0.4
0.6
0.8
1
1.2
A310, T = 0.19m C = T/6 C = T/3 C = T/2
0.05 0 0
0.2
0.4
0.6 N/Ncr [−]
0.8
1
1.2
Fig. 6. Mean oscillation amplitudes in the small scale vessel as a function of the N=N cr ratio.
0.6
Run 1 Run 2
0.5 Rushton, T = 0.48m Δ H/D [−]
As it is possible to observe, in all investigated cases a range of impeller speeds exists in which relatively large oscillations occur, as it may be expected when resonance phenomena take place. It is worth noting that in all cases the average fluctuation height is smaller that 2% of impeller diameter, while it reaches much larger values when a certain range of impeller speeds is reached. Notably this range shifts towards higher frequencies the stronger the axial impeller action while system size (see Fig. 7) seems to mainly affect oscillations amplitude. In particular:
0.15
0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 N/Ncr [−]
0.8
1
1.2
Fig. 7. Mean oscillation amplitudes in the large scale vessel as a function of the N=N cr ratio.
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The resonant impeller speeds appear to depend on the type of impeller action (impellers with more intense axial action appear to have larger resonant velocities), while the oscillation amplitude seems to strongly increase with vessel scale. This fact highlights how a clear comprehension of the phenomena leading to surface instability is of crucial importance to predict and avoid catastrophic failures during operation of industrial equipments.
fi Rushton (T = 0.19m) [Hz]
15
3.2. Oscillation frequencies A further attempt to gain insights on the oscillation phenomena under examination can be made by analyzing the power spectra obtained at various agitation speeds. As an example, in Fig. 8 the spectral analysis of liquid level oscillations observed at various agitation speeds is reported. As it can be seen, different peak families can be recognized, i.e. peaks appear in different spectra in related positions. The resonant behavior in this case appears at an impeller speed of N ¼ 3:33 rps, as it was shown in Fig. 5. At this impeller speed, a very large peak near f ¼ 3:46 s1 appears, together with several other peaks at smaller and larger frequencies, as already discussed. The analysis of other spectra measured at larger impeller speeds shows that the same peaks are also found. Notably, slightly different peaks locations and relative heights are found. Both these aspects are worth further discussion. In fact, an interesting feature of characteristic peak frequencies can be observed by reporting the measured peak locations vs. impeller speed, as done in Fig. 9 for the smaller vessel stirred by a Rushton turbine, placed at various clearances from tank bottom. As it can be seen there: Peak locations are practically independent of impeller clearance (data obtained with differently placed impellers are almost perfectly overlapped). Peak families are quite well defined and easy to distinguish; in particular eighth peak families were easily identified. An ninth family, at about 18 Hz could also be plotted but was not reported for the sake of clarity. For each peak family, a neat linear dependence on impeller speed is found. Each peak family may be expected to be related to some natural frequency of the system under investigation. As peak frequencies linearly increase with agitation speed, one may expect that their
10
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4
5
6
2
|Y(f)|
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10
intercept at zero impeller speed should result into a frequency list simply related to fluid mass and shape, possibly to some base natural oscillation frequency and its harmonics. Notably, the evidence that a change in impeller vertical location has no effect on the measured peak frequencies encourages such exercise. Also peak relative height changes with impeller speeds, as it is possible to observe in Fig. 8: if the impeller speed is below N 5 rpm, the peak belonging to the second family has the largest height (and give rise to the large oscillations measured at N ¼ 3:33 rps), while at larger N values, the peak belonging to the first family is higher (and give rise to the large oscillations measured at N ¼ 8:33 rps). One may now think that when a periodic stimulus with frequency close enough to one of the natural frequencies of the system is applied, a resonance phenomena should take place, as it happens in mechanics. The resulting resonance would give rise to large fluctuation amplitudes. Considering that peak frequencies change with impeller speed, i.e. with average fluid velocity within the system, one should expect that strong resonance phenomena appear every time the impeller frequency (reported as a dashed line in Fig. 9) equals one of the harmonics frequencies, i.e. passes by a symbol.
−3
1
8
Fig. 9. Measured peak frequencies as a function of impeller speed in the case of small vessel equipped with Rushton turbine, for different impeller clearances (C ¼ D=6 D=2), together with relevant linear best fits. Data relevant to peaks of the same family obtained from systems with different impeller clearance were plotted with the same symbols.
x 10
0
7
N [1/s]
13
15
17
19
8.33 rps 7.50 rps 6.67 rps 5.83 rps 5.00 rps 4.17 rps 3.33 rps 2.50 rps 1.67 rps
Frequency (Hz) Fig. 8. Power spectra recorded at different impeller speeds (small vessel equipped with Rushton turbine, C ¼ T=3).
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Unfortunately, things are not so plain and while in certain cases this is actually observed (see for instance the second peak at N ¼ 3:33 rps) in other cases it is not (see for instance the fifth peak at 5:8 rps.) Clearly these observations entail a behavior complexity that reinforces doubts on the possibility of finding simple models to explain all observed effects.
3.3. Influence of impeller type Data obtained with the smaller vessel stirred by different impeller types, namely with a Pitched blade turbine or with an A310 propeller, are reported in Fig. 10 as measured peak locations vs. impeller speed. Data show quite similar trends in terms of both peak location (as those reported in Fig. 9) and relative height (data similar to those reported in Fig. 8 were obtained but not reported here for the sake of brevity), so that quite similar conclusion can be drawn. The influence of impeller design on peak frequencies can be better described analyzing data relevant to the undisturbed system frequencies f 0;i , i.e. the extrapolated system frequencies at N ¼ 0 rps (reported in Fig. 11(a) for each peak family) and the slopes describing the oscillation frequency variation with rota-
fi PBT (T = 0.19m) [Hz]
15
10
5
0
0
2
4
6
8
10
12
N [1/s]
15
fi A310 (T = 0.19m) [Hz]
Fig. 11. Measured natural system frequencies and relevant slopes in the case of small vessel equipped with different impellers. 10
5
0
0
2
4
6
8
10
12
14
16
18
20
N [1/s]
Fig. 10. Measured peak frequencies as a function of impeller speed in the case of small vessel at different impeller clearances (C ¼ D=6 D=2), together with relevant linear best fits. Data relevant to peaks of the same family obtained from systems with different impeller clearance were plotted with the same symbols.
tional speed (reported in Fig. 11(b)) as a function of relevant undisturbed system frequencies. The analysis of Fig. 11(a) shows that undisturbed system frequencies do not practically depend on impeller type (as well as on impeller clearance, as previously pointed out), as it should have been expected. Conversely, the impeller type (but not the impeller clearance) plays an important role in the observed oscillation frequency vs. impeller speed slopes. Notably, the measured slope generally increases with the harmonic considered (only 4th and 5th harmonics behave differently in all cases). Moreover, it must be noted that the slope increases with increasing the radial action of the impeller. This is likely due to the different tangential flow field generated by different impellers [2]. Summarizing, the comparison of data obtained with different impellers shows that:
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In all investigated cases, once the impeller geometry is chosen, impeller clearance from vessel bottom plays no role in peak frequency determination. For each impeller type, the same peak families are found. In all cases, the peak family frequencies linearly depend on impeller speed. If the straight lines that best fit all peaks in each peak family are extrapolated to zero impeller speed, a group of f 0;i frequencies is found (referred to as undisturbed or natural system frequencies) independent of impeller clearance and impeller design. The slope si ¼ dfi =dN of the fitting line depend on the relevant f 0i (peak families at lower frequencies show generally smaller slopes), as well as on impeller type, namely increasing with the impeller radial action (in fact, for each peak family, si;Rush > si;PBT > si;A310 ). Increasing the impeller radial action, resonance occurs at progressively smaller impeller rotational speeds. Resonance may occur approximately at N ¼ 0:3 0:6N cr (slightly different values are found for different impellers) by excitation of the second harmonic and at N ¼ 0:8 1:0N cr by excitation of the first harmonic.
impeller), the only difference being the vessel scale, in order to isolate its effect on free surface instabilities. As it can be seen, the plot obtained is similar to that obtained in the small vessel. However, the system natural frequencies (f 0;i ) appear to involve a quite narrower range of values, as it can be appreciated in Fig. 13(a), where the average natural frequencies in the small vessel case are compared with those measured in the large vessel. The slopes describing the oscillation frequency variation with rotational speed (reported in Fig. 13(b) as a function of relevant natural system frequencies) show a similar trend when changing the vessel diameter, therefore confirming that such slopes mainly depend on the flow field generated by the impeller. The f 0;i data trend is exactly similar when changing the vessel scale, but smaller frequencies are found in the larger vessel, as expected. The number of natural frequencies ready to be excited
The substantial lack of dependence of oscillation frequencies at zero impeller speed f 0i on impeller geometry and position shows that these quantities only depend on liquid mass in the vessel and its aspect ratio, i.e. may be regarded as the natural sloshing frequencies of the rotating liquid. Conversely, the flow field has to play some role in shifting the natural frequencies. Clearly, a suitable mathematical modeling of the system aimed at providing full physical interpretation of the above findings is needed if results extrapolation to other fluids or other system sizes is desired. This is planned to be the aim of future works. For the time being, on the basis the above results, one can only guess that it is not going to be a simple and straightforward task.
3.4. Influence of vessel scale In Fig. 12 the measured peak locations as a function of impeller speed in case of large vessel equipped with Rushton turbine are reported, together with relevant linear fitting line. The vessels under investigation have the same geometry (same D=T ¼ 1=3; H=T ¼ 1 and C=D ¼ 1 ratios, geometrically similar
12
fi Rushton (T = 0.48m) [Hz]
10
8
6
4
2
0 0
1
2
3
4
5 N [1/s]
6
7
8
9
10
Fig. 12. Measured Peaks frequencies as a function of impeller speed in the case of large vessel equipped with Rushton turbine (C ¼ T=3), together with relevant linear best fits.
Fig. 13. Measured natural system frequencies and relevant slopes in the case of differently sized vessels equipped with six-blade Rushton turbines (C = T/3, D = T/3).
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in the range 1:4—3:7 Hz is likely to be responsible for the highly oscillating behavior of large scale vessel. On average, natural frequencies f 0;i in large scale vessel are 0:67 times those measured in small scale vessel. This scale-up value can be predicted if the Miles’ solution for the sloshing analysis in cylindrical vessel is considered [26]. According to this model, the dominant mode of oscillation of a liquid in a vertical cylindrical tank can be computed as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2g 2H tanh 1:84 f 0 ¼ 0:216 T T
ð3Þ
When applied to the vessel under investigation, a value of f 0 ¼ 2:19 Hz is found for the small vessel (equal to f 0;1 ¼ 2:18) and f 0 ¼ 1:38 Hz for the large vessel (quite similar to the measured value, f 0;1 ¼ 1:43): the first peak family is therefore related to the dominant sloshing frequency of the liquid in the vessel. Notably, when passing from small to large scale vessel, in Eq. (3) all factors remains constant with the exception of the T value (the H/T value remains constant), so resulting in the following scaling law:
f 0;i T 0:5 ¼ const:
ð4Þ
The validity of Eq. (4) for all the measured harmonics is apparent in Fig. 14. Last point to discuss, it was shown that some of the peaks appeared to be in simple relation with the second peak, and that during resonance phenomena either second (at smaller N values) of first peak (near to critical conditions) became the largest peaks. These two peak families are playing a major role in determining the resonant behavior. Hence, it was decided to report undisturbed system frequencies in dimensionless form, as f 0;i =f 0;1 or f 0;i =f 0;2 , to highlight any simple relation between natural frequencies. As it is possible to observe in Fig. 15, the 4th, 7th and 8th natural frequencies are multiples of the 1st natural frequency, while the 6th, 7th are multiples of the 2nd natural frequency. While 1st natural frequency (and its multiples) has a clear origin in the dominant oscillation mode discussed by [26], the origin of the second natural frequency (and its multiples) is not yet clear, as well as that of other natural frequencies. Only hypotheses can be made, such as the relation with macro-instabilities similar to those observed in baffled vessels [27,28], but a thorough explanation of the above observations was not achieved at this stage. 5 4.5 4
Fig. 15. Dimensionless natural system frequencies.
Clearly it will involve a complex in depth analysis of the involved phenomena, for which probably further data, as well as different experimental techniques, will be needed.
f0,i T0.5 [Hz]
3.5 3 2.5
4. Conclusions
2 1.5 1 0.5 0
1st
2nd
3rd 4th 5th 6th Natural harmonic
7th
Fig. 14. Scaled frequencies for systems at different scales.
8th
In this work, oscillation amplitudes and frequencies of free surface in unbaffled stirred vessels were measured by means of a novel technique based on image analysis. The influence of impeller design, impeller clearance and vessel scale was assessed. It was found that the natural oscillation frequencies only depend on vessel scale and geometry, while impeller type plays a role in the way the free-surface oscillation frequencies change with impeller speed. The resonant impeller speeds appear to depend on the type of impeller action (impellers with more intense axial action appear
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to have larger resonant velocities), while the oscillation amplitude seems to strongly increase with vessel scale. This fact highlights how a clear comprehension of the phenomena leading to surface instability is of crucial importance to predict and avoid catastrophic failures during operation of industrial equipments. The relation between natural system frequencies, stimuli frequencies and oscillation amplitudes was not clarified here, due to the significant complexities involved. The present work should be regarded as a first paper addressing the free surface oscillation phenomenon and reporting on quantitative experimental observations that may guide future modeling efforts as well as provide a data base for model validations. Acknowledgements This work was financially supported through the Project BIO4BIO – Biomolecular and Energy valorization of residual biomass from Agroindustry and Fishing Industry led by the Cluster Sicily Agrobio and Fishing Industry and funded by the Italian Research Fund (PON R&C 20072013, DD 713/Ric. – PON02 00451 3362376. References [1] M. Ciofalo, A. Brucato, F. Grisafi, N. Torraca, Turbulent flow in closed and freesurface unbaffled tanks stirred by radial impellers, Chem. Eng. Sci. 51 (14) (1996) 3557–3573. [2] A. Busciglio, G. Caputo, F. Scargiali, Free-surface shape in unbaffled stirred vessels: experimental study via digital image analysis, Chem. Eng. Sci. 104 (2013) 868–880. [3] F. Scargiali, A. Busciglio, F. Grisafi, A. Tamburini, G. Micale, A. Brucato, Power consumption in uncovered-unbaffled stirred tanks: influence of viscosity and flow regime, Ind. Eng. Chem. Res. 52 (42) (2013) 14998–15005. [4] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Mass transfer and hydrodynamic characteristics of unbaffled stirred bio-reactors: influence of impeller design, Biochem. Eng. J. 82 (2014) 41–47. [5] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Free surface oxygen transfer in large aspect ratio unbaffled bio-reactors, with or without draft tube, Biochem. Eng. J. 100 (2015) 16–22. [6] K. Conway, A. Kyle, C. Rielly, Gasliquidsolid operation of a vortex-ingesting stirred tank reactor, Chem. Eng. Res. Des. 80 (Part A) (2002) 839–845. [7] F. Scargiali, R. Russo, F. Grisafi, A. Brucato, Mass transfer and hydrodinamic characteristics of a high aspect ratio self-ingesting reactor for gas–liquid operations, Chem. Eng. Sci. 62 (5) (2007) 1376–1387. [8] F. Scargiali, A. Busciglio, F. Grisafi, A. Brucato, Gas–liquid–solid operation of a high aspect ratio self-ingesting reactor, Int. J. Chem. Reactor Eng. 10 (1) (2012) A27, http://dx.doi.org/10.1515/1542-6580.3011.
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