**4.1 Change in circulation pattern**

232 Hydrodynamics – Advanced Topics

samples) as the number of samples increases. The empty triangles indicate the presence of further low frequency peaks. The main fact to be aware of is that low time intervals conceal the

Both FFT and LSP inform how much of each frequency component exists in the signal, but they do not tell us when in time these frequencies occur in the signal. For transient flows it may be of interest the time localisation of the spectral component. The Wavelet Transform (WT) is capable of providing the time and frequency information simultaneously, hence it gives a time-frequency representation of the signal (Daubechies, 1990, Torrence and Compo, 1998). The WT breaks the signal into its "Wavelets", that are functions obtained from the scaling and the shifting of the "mother Wavelet" ψ. The WT has been proposed for the investigation of stirred vessels by Galletti et al. (2003) and subsequently applied by Roy et

POD is a linear procedure, based on temporal and spatial correlation analysis, which allows to decompose a set of signals into a modal base, with the first mode being the most energetic (related to large-scale structures thus trailing vortices and flow instabilities) and the last being the least energetic (smaller scales of turbulence). It was first applied for MI characterisation by Hasal et al. (2004) and latterly by Ducci & Yianneskis (2007). An in-depth

A possible classification of flow instabilities in stirred vessels is reported in Fig. 3. The graph is not aimed at imposing a classification of flow instabilities, however it suggests a way of interpretation which may be regarded as a first effort to comprehend all possible

flow instabilities by covering only a portion of the fluctuations.

explanation of the methodology is given in Berkooz et al. (1993).

Fig. 3. Possible classification of flow instabilities in stirred vessels.

**3.2 Time-frequency analysis** 

**3.3 Proper orthogonal decomposition** 

**4. Classification of flow instabilities** 

al. (2010).

instabilities.

A first kind of flow instability (see left-hand side of the diagram of Fig. 3) manifests as a real change in the circulation pattern inside the tank. Two main sources of such a change have been identified: a variation of the Reynolds number (Re) or a variation of the impeller/vessel geometrical configuration.

In relation to the former source, Nouri & Whitelaw (1990) reported a transition due to Re variations in the flow pattern induced by a 60° PBT with D = T/3 set at C = T/3 in a vessel of T = 0.144 m. For non-Newtonian fluids a flow pattern transition from a radial to an axial flow was observed as the Re was increased up to Re = 4,800. For Newtonian fluids the authors observed that the flow pattern transition occurred at about Re = 650. This value was also confirmed by the power number measurements through strain-gauges carried out by Distelhoff et al. (1995). Similar investigations on such transition may be found in the works of Hockey (1990) and Hockey & Nouri (1996).

Schäfer et al. (1998) observed by means of flow visualisation the flow discharged by a 45° PBT to be directed axially at higher Re and radially at lower Re. The flow stream direction was unstable, varying from radial to axial, for Re = 490-510. A similar flow transition was also indicated by Bakker et al. (1997) who predicted with CFD techniques the flow pattern generated from a 4-bladed 45° PBT of diameter D = T/3 and set at C = T/3 inside a tank of T = 0.3 m. The regime was laminar, the Reynolds number being varied between 40 and 1,200. The impeller discharge stream was directed radially for low Re numbers, however for Re larger than 400 the flow became more axial, impinging on the vessel base rather than on the walls.

A second source of instabilities, manifesting as a flow pattern change, is associated with variations of the impeller/vessel geometrical configuration, which means either variations of the distance of the impeller from the vessel bottom (C/T) or variation of the impeller diameter (D/T) or a combination of both variations.

This kind of instabilities were firstly noticed by Nienow (1968) who observed a dependency on the clearance of the impeller rotational speed required to suspend the particles (Njs) in a solid-liquid vessel equipped with a D = 0.35T RT. He observed that for C < T/6 the pattern was different (the discharge stream was directed downwards towards the vessel corners) from the typical radial flow pattern, providing low Njs values. Baldi et al. (1978) also observed a decrease of the Njs with the impeller off-bottom clearance for a 8-bladed turbine. Conti et al. (1981) found a sudden decrease of the power consumption associated with the change in the circulation pattern when lowering the impeller clearance of a 8-bladed turbine. The aforementioned authors concluded that the equation given by Zwietering (1958) for the calculation of the Njs should be corrected in order to take into account the dependency on C/T.

The dependency of the power number on the impeller off-bottom clearance was also observed by Tiljander & Theliander (1993), who measured the power consumption of two PBT of different sizes, i.e. D = T/3 and D = T/2, and a high flow impeller of D = T/2. The visual observation of the flow pattern revealed that at the transition point between the axial and the radial flow patterns, the circulation inside the vessel appears chaotic.

Ibrahim & Nienow (1996) investigated the efficiency of different impellers, i.e. a RT, a PBT pumping either upwards or downwards, a Chemineer HE3 and a Lightnin A310 hydrofoil pumping downwards and a Ekato Intermig agitator, for solids suspension. For the RT, the aforementioned authors observed a sudden decrease of both the impeller speed and the mean dissipation rate required to just suspend the particles as the clearance was decreased

Flow Instabilities in Mechanically Agitated Stirred Vessels 235

circulation from the vessel axis towards the walls. For the higher clearance the impeller stream turned upwards before reaching the base of the vessel, generating also a reverse flow

Kresta & Wood (1993) measured the mean flow field of a vessel stirred with a 4-bladed 45° PBT for two impeller sizes, i.e. D = T/3 and D = T/2, and varying the impeller clearance systematically from T/20 up to T/2. They observed that the circulation pattern underwent a transition at C/D = 0.6, and for the larger impeller (D = T/2) such a transition was accompanied by a deflection of the inclination of the discharge stream toward the

Ibrahim & Nienow (1995) measured the power number of different impellers for a wide range of Reynolds number, i.e. 40 < Re < 50,000, in Newtonian fluids. For a D = T/3 RT they observed that the power numbers with clearances of C = T/3 and C = T/4 was the same for all Re; however for C = T/6 the discharge flow was axial rather than radial and the associated power number was considerably lower (by about 25%) for all the range of Re investigated. For a D = T/2 RT a radial discharge flow was still observed at C = T/6 for all

Rutherford et al. (1996a) investigated the flow pattern generated by a dual Rushton impeller and observed different circulation patterns depending on the impeller clearance of the lower impeller and the separations between the two impellers, observing three stable flow

Mao et al. (1997) measured with LDA the flow pattern generated from various PBT of different sizes in the range of 0.32 < D/T < 0.6 and number of blades varying from 2 to 6 in a stirred vessel in turbulent regime (Re > 20,000). They used two impeller off-bottom clearances, C = T/3 and C = T/2, observing a secondary circulation loop with the higher

Montante et al. (1999) provided a detailed investigation of the flow field generated by D = T/3 RT placed at different off-bottom clearances varying from C = 0.12T to C = 0.33T. They found that the conventional radial flow pattern (termed "double-loop" pattern) occurred for C = 0.20T, but it was replaced by an axial flow pattern (termed "single loop" pattern) as the clearance was decreased to C = 0.15T. A reduction of the power number from 4.80-4.85 for C/T = 0.25-0.33 down to 3.80 as the clearance was decreased to C/T = 0.12-0.15 was reported, so that the power consumption was reduced by about 30% as the flow underwent

Galletti et al. (2003, 2005a, 2005b) studied the flow pattern transition for a D = T/3 RT and identified a kind of flow instabilities, which will be denoted as CIs (clearance instability). The authors found that the flow pattern transition (single- to double-loop pattern) occurred for C/T = 0.17-0.2, thus within an interval of clearances of about 0.03T. Such C interval was dependent on the fluid properties, lower clearances being observed for more viscous fluids. At clearances of flow pattern transition the velocity time series indicated flow pattern instabilities as periods of double-loop regime, single-loop regime and "transitional" state that followed each other. When the flow underwent a change from one type of circulation to another, the transitional state was always present and separated in time the single- from the double-loop flow regime. Nevertheless, a flow pattern could change firstly into the transitional state and afterwards revert to the original flow regime, without changing the type of circulation. The occurrence of the three flow regimes was shown to be random, and

directed radially from the walls towards the vessel axis at the base of the vessel.

Re except for those with the highest viscosity (1 Pa·s).

a transition from the double- to the single-loop pattern.

**4.1.1 Clearance instabilities (CIs)** 

patterns: "parallel flow", "merging flow" and "diverging flow" patterns.

horizontal.

clearance.

from C = T/3 down to C = T/6 for the impeller having D = T/3; such a clearance corresponded to the transition from the radial flow pattern to the axial.

Subsequently, a strong influence of the clearance on the suspension of particles was confirmed also by Myers et al. (1996) for three axial impellers. If the clearance was sufficiently high the discharge flow impinged on the vessel wall rather then the base, leading to a secondary circulation loop which was directed radially inward at the vessel base and returned upwards to the impeller at the centre of the vessel. Such a reversed flow occurred for C > 0.45T for a PBT of diameter D = 0.41T and for C > 0.25T for a straight-blade turbine of the same diameter, whereas only for very high clearances (C > 0.95T) for a high efficiency Chemineer impeller having the same diameter.

Bakker et al. (1998) reported that the flow pattern generated by either a PBT or a three-blade high efficiency impeller depended on C/T and D/T, influencing the suspension of the particles.

Armenante & Nagamine (1998) determined the Njs and the power consumption of four impellers set at low off-bottom clearances, typically C < T/4. For radial impellers, i.e. a RT and a flat blade turbine, they observed that the clearance at which the change in the flow pattern from a radial to an axial type occurred was a function of both impeller type and size, i.e. D/T. In particular the flow pattern changed at lower C/T for larger impellers. This was in contrast with previous works (see for example Conti et al., 1981) which reported a clearance of transition independent on D/T. For instance Armenante & Nagamine (1998) found the flow pattern transition to occur at 0.16 < C/T < 0.19 for a Rushton turbine with a diameter D = 0.217T and at 0.13 < C/T < 0.16 for a D = 0.348T RT. For the flat blade turbine the clearances at which the transition took places were higher, being of 0.22-0.24 and 0.19- 0.21 for the two impeller sizes D = 0.217T and D = 0.348T, respectively.

Sharma & Shaikh (2003) provided measurements of both Njs and power consumption of solids suspension in stirred tanks equipped with 45° PBT with 4 and 6 blades. They plotted the critical speed of suspension Njs as a function of C/T distinguishing three regions, according to the manner the critical suspension speed varied with the distance of the impeller from the vessel base. As the impellers were operating very close to the vessel base, the Njs was observed to be constant with C/T (first region); then for higher clearances Njs increased with C/T because the energy available for suspension decreased when increasing the distance of the impeller from the vessel base (second region), and finally (third region) for high clearances the Njs increased with C/T with a slope higher than that of the second region. The onset of third region corresponded to the clearance at which the flow pattern changed from the axial to the radial flow type. In addition the aforementioned authors observed that as the flow pattern changed the particles were alternatively collected at the tank base in broad streaks and then suddenly dispersed with a certain periodicity. They concluded that a kind of instabilities occurred and speculated that maybe the PBT behaved successively as a radial and axial flow impellers.

The influence of C on the flow pattern has been intensively studied also for single-phase flow in stirred tanks. Yianneskis et al. (1987) showed that the impeller off-bottom clearance affects the inclination of the impeller stream of a Rushton turbine of diameter D = T/3. In particular the discharge angle varied from 7.5° with respect to the horizontal plane for C = T/4 down to 2.5° for C = T/2.

Jaworski et al. (1991) measured with LDA the flow patterns of a 6-bladed 45° PBT having a diameter D = T/3 for two impeller clearances, C = T/4 and C = T/2. For the lower impeller clearance, the impeller stream impinged on the vessel base and generated an intensive radial

from C = T/3 down to C = T/6 for the impeller having D = T/3; such a clearance

Subsequently, a strong influence of the clearance on the suspension of particles was confirmed also by Myers et al. (1996) for three axial impellers. If the clearance was sufficiently high the discharge flow impinged on the vessel wall rather then the base, leading to a secondary circulation loop which was directed radially inward at the vessel base and returned upwards to the impeller at the centre of the vessel. Such a reversed flow occurred for C > 0.45T for a PBT of diameter D = 0.41T and for C > 0.25T for a straight-blade turbine of the same diameter, whereas only for very high clearances (C > 0.95T) for a high

Bakker et al. (1998) reported that the flow pattern generated by either a PBT or a three-blade high efficiency impeller depended on C/T and D/T, influencing the suspension of the

Armenante & Nagamine (1998) determined the Njs and the power consumption of four impellers set at low off-bottom clearances, typically C < T/4. For radial impellers, i.e. a RT and a flat blade turbine, they observed that the clearance at which the change in the flow pattern from a radial to an axial type occurred was a function of both impeller type and size, i.e. D/T. In particular the flow pattern changed at lower C/T for larger impellers. This was in contrast with previous works (see for example Conti et al., 1981) which reported a clearance of transition independent on D/T. For instance Armenante & Nagamine (1998) found the flow pattern transition to occur at 0.16 < C/T < 0.19 for a Rushton turbine with a diameter D = 0.217T and at 0.13 < C/T < 0.16 for a D = 0.348T RT. For the flat blade turbine the clearances at which the transition took places were higher, being of 0.22-0.24 and 0.19-

Sharma & Shaikh (2003) provided measurements of both Njs and power consumption of solids suspension in stirred tanks equipped with 45° PBT with 4 and 6 blades. They plotted the critical speed of suspension Njs as a function of C/T distinguishing three regions, according to the manner the critical suspension speed varied with the distance of the impeller from the vessel base. As the impellers were operating very close to the vessel base, the Njs was observed to be constant with C/T (first region); then for higher clearances Njs increased with C/T because the energy available for suspension decreased when increasing the distance of the impeller from the vessel base (second region), and finally (third region) for high clearances the Njs increased with C/T with a slope higher than that of the second region. The onset of third region corresponded to the clearance at which the flow pattern changed from the axial to the radial flow type. In addition the aforementioned authors observed that as the flow pattern changed the particles were alternatively collected at the tank base in broad streaks and then suddenly dispersed with a certain periodicity. They concluded that a kind of instabilities occurred and speculated that maybe the PBT behaved

The influence of C on the flow pattern has been intensively studied also for single-phase flow in stirred tanks. Yianneskis et al. (1987) showed that the impeller off-bottom clearance affects the inclination of the impeller stream of a Rushton turbine of diameter D = T/3. In particular the discharge angle varied from 7.5° with respect to the horizontal plane for C =

Jaworski et al. (1991) measured with LDA the flow patterns of a 6-bladed 45° PBT having a diameter D = T/3 for two impeller clearances, C = T/4 and C = T/2. For the lower impeller clearance, the impeller stream impinged on the vessel base and generated an intensive radial

corresponded to the transition from the radial flow pattern to the axial.

0.21 for the two impeller sizes D = 0.217T and D = 0.348T, respectively.

successively as a radial and axial flow impellers.

T/4 down to 2.5° for C = T/2.

efficiency Chemineer impeller having the same diameter.

particles.

circulation from the vessel axis towards the walls. For the higher clearance the impeller stream turned upwards before reaching the base of the vessel, generating also a reverse flow directed radially from the walls towards the vessel axis at the base of the vessel.

Kresta & Wood (1993) measured the mean flow field of a vessel stirred with a 4-bladed 45° PBT for two impeller sizes, i.e. D = T/3 and D = T/2, and varying the impeller clearance systematically from T/20 up to T/2. They observed that the circulation pattern underwent a transition at C/D = 0.6, and for the larger impeller (D = T/2) such a transition was accompanied by a deflection of the inclination of the discharge stream toward the horizontal.

Ibrahim & Nienow (1995) measured the power number of different impellers for a wide range of Reynolds number, i.e. 40 < Re < 50,000, in Newtonian fluids. For a D = T/3 RT they observed that the power numbers with clearances of C = T/3 and C = T/4 was the same for all Re; however for C = T/6 the discharge flow was axial rather than radial and the associated power number was considerably lower (by about 25%) for all the range of Re investigated. For a D = T/2 RT a radial discharge flow was still observed at C = T/6 for all Re except for those with the highest viscosity (1 Pa·s).

Rutherford et al. (1996a) investigated the flow pattern generated by a dual Rushton impeller and observed different circulation patterns depending on the impeller clearance of the lower impeller and the separations between the two impellers, observing three stable flow patterns: "parallel flow", "merging flow" and "diverging flow" patterns.

Mao et al. (1997) measured with LDA the flow pattern generated from various PBT of different sizes in the range of 0.32 < D/T < 0.6 and number of blades varying from 2 to 6 in a stirred vessel in turbulent regime (Re > 20,000). They used two impeller off-bottom clearances, C = T/3 and C = T/2, observing a secondary circulation loop with the higher clearance.

Montante et al. (1999) provided a detailed investigation of the flow field generated by D = T/3 RT placed at different off-bottom clearances varying from C = 0.12T to C = 0.33T. They found that the conventional radial flow pattern (termed "double-loop" pattern) occurred for C = 0.20T, but it was replaced by an axial flow pattern (termed "single loop" pattern) as the clearance was decreased to C = 0.15T. A reduction of the power number from 4.80-4.85 for C/T = 0.25-0.33 down to 3.80 as the clearance was decreased to C/T = 0.12-0.15 was reported, so that the power consumption was reduced by about 30% as the flow underwent a transition from the double- to the single-loop pattern.
