The Tangential Force Affecting the Radial Baffles in a Stirred Vessel : Analysis of the Macro-instability Related Component

Experimental data obtained by measuring the tangential component of the force affecting radial baffles in a flat-bottomed cylindrical mixing vessel stirred with a Rushton turbine impeller is analysed. Spectral analysis of the experimental data demonstrated the presence of its macro-instability (MI) related low-frequency component embedded in the total force. Two distinct dimensionless frequencies (both directly proportional to the impeller speed of rotation N) of the occurence of the MI component were detected: a lower frequency of approximately 0.025N and a higher frequency of about 0.085N. The relative magnitude Q MI of the MI-related component of the total tangential force was evaluated by a combination of proper orthogonal decomposition (POD) and spectral analysis. The values of magnitude Q MI varied in the interval [rom approximately 0.05 to 0.30. The magnitude Q MI takes maximum values at low Reynolds number values (in laminar and transitional regions). In the turbulent region (Re M >20000) the Q MI value is low and practically constant. The dependence oj the Q MI values on vertical position in the vessel is only marginal. The results suggest that the magnitude of the MI component of the force is significantly influenced by the liquid viscosity and density.


Introduction
The liquid flow in mechanically stirred vessels has been studied intensively in recent decades.Numerous theoretical and experimental studies have been performed concerning various aspects of liquid flow in stirred vessels, e.g., the mean flow velocity, the intensity of turbulence, the energy dissipation rate, spatial and temporal scales of the turbulent velocity field, etc.The liquid flow in a stirred vessel operated under steady operational conditions may be considered as a pseudo--stationary high-dimensional dynamical system constituted by hierarchically ordered unsteady flows (vortices and eddies).The temporal and spatial scales of these flows span several decimal orders of magnitude.Recently, a pseudo-periodic large-scale flow has been identified in stirred vessels occurring with very low frequency and manifesting itself on spatial Scales cornpaiable to the size of the mixing vessel [-8].This flow was named the marm-irutnbihry MI) of tht llow fuftern, and its existence has been confirmed by.a special.me-chanical measuring device [6], by laser-Doppler velocimetry (LDV) t?l, and by im4ge analysis methods [8].The presence of the MI in the flow pattern is typically displayed by a distinct peak in the low-frequency part ofthe power spectrum ofthe local liquid velocity or other liquid,flow-related experimental data.Comprehensive reviews of a broad spectrum of the experimentally observed phenomena related to the macro- -instability were given in our previous PaPers [6, 7, 9-13].'l he macro-instability of the flow pattern yields strong impacts on mixing processes closely linked to fluid motion, e.g., on local mass-and heat-transport rates, local gas hold-up, the homogenisation rate, etc. [3, 5].Bittorf and Kresta [14] iden- tified the macro-instability of the flow pattern as a mechanism responsible for liquid mixing outside the active volume of the primary liquid circulation loop.Although the existence of the macro-instability of the flow pattern in stirred vessels has been proved many times in experiments and CFD simula- tions its origin and development is not completely clear yet.Three possible reasons of the existence of the MI in stirred vessels with a Rushton turbine impeller have been hypothesised in the literature [15-17]: the first kind of insta- bility is associated with variations in the impeller ofl-bottom clearance.Montante et al [6] reported a variation in the flow at different clearances from the vessel bottom' The radial flow pattern was observed at high off-bottom clear- ances (C > 0.2 7) whilst the axial pattern was observed at C < 0.15 T Intermittent switching between axial and radial flow was detected at 0.15 T < C < 0.2 T. The changes in the flow pattern in the unstable cases occurred over time scales of the order of l0 min.The second kind of flow pattern instabil- iry is related to variations in the impeller rotational speed, when the flow pattern switches from mainly radial to mainly axial at a certain (critical) impeller speed (or Re") value.The third kind of macro-instabiliry arises from a "whirlpool" type of vortex (or coupled vortices) moving around the mixing vessel axis with the frequency of around 2 7o of the impeller fiequency N (see Nikiforaki et al [17])' The origin and the development of the MI of the flow pat- tern in the mixing vessels with the PBI was described by Brriha et al [6] and by Montes et al [7], who visually detected an axial flow originating in a pseudo-periodic manner at the vessel bottom and ascending along the baffle towards the liquid surface.
The fiequency of the occurrence of the macro-instabiliry in all experiments reported in the literature [e.g., 6, 7, 9-l l, l7l was directly proportional to the fiequency of the impeller rotation.Howeve6 considerable scatter of the Proportionality constant values (the dimensionless MI fiequency) is apparent from the published data.Nikiforaki et al [17] have explained a part of this scatter by inadequacies in the experimental data treatment and interpretation methods adopted by different authors.Howeveq it is evident from the published data that at least two different macro-instability frequencies, correspond- ing to different mechanisms of MI origin, exist in the mixing vessels stinedwith a Rushton turbine' Nikiforaki et al [17] ob- served that higher MI frequency (about l0 7o of ltl) occurs at low impeller Reynolds number values, and that the lower 50 frequency.(about 2 Vo of N) prevails at'high Rer.Both fre- quencies were simultaneously present in the power spectra of the data measured at medium Reu values.The lower frbquency was assigned to intermittent switching between the single-and the double-loop flow pattern, while the higher frequency was assigned to precessional vortex motions.
The macro-instability of the flow-pattern also exerts strong forces affecting the solid surfaces immersed in the stirred liquid, for example on the baflles, the draft tubes, the cooling and heating coils, etc. [2].These forces may signifi- cantly affect the performance of the mixing vessel and, in certain cases, can even cause serious failures of the mixing equipment.
An axially located impeller in a standard cylindrical mixing vessel equipped with radial baffles exhibits, in general, three force effects: axial, radial and tangential.The axial and the radial forces only slightly affect the radial baffles.Con- versely, it is the tangential force that exhibits the most of the dynamic pressune affecting them.The vertical distribution (along the ba{fle) of the tangential force in a standard mixing vessel was measured by Kratdna et al [12, 13] over a wide interval of the impeller Reynolds number value and in dependence on the impeller off-bottom clearance.Tivo kinds of the impeller were used: the pitched blade impeller with four or six blades, and a standard Rushton turbine' The pitched blade impellers elicited maximum force at the vessel bottom, and the magnitude of the force gradually decayed in the direction towards the liquid level.The Rushton turbine impellers yielded the maximum force at the impeller level, and the force rapidly vanished in both the below-impeller and the above-impeller region.
Few attempts have been reported in the literature [18, l9] to separate the deterministic (low frequency and Ml-related) and the stochastic (turbulent) components of mixing phe- nomena in stirred tanks.Letellier et al [18] adopted a Hilbert transform-based procedure for separating the low-dimen- sional deterministic part of the experimental time series.Kovacs et al [19] used the Fourier transform for the same Pur- pose.Recentlywe have established a new technique (based on a combination of spectral analysis and proper orthogonal decomposition) for detecting the macro'instability of the flow pattern from the local velocity data, evaluating its relative magnitude and reconstructing its temporal evolution [9-l 1].
In this paper we apply this technique to the experimental data obtained by Kratdna et al [12, 13] by measuring the tangential comPonent of the force exerted on the radial baffles in the mixing vessel with a standard Rushton turbine impeller in order to identify and quantify its macro-instability related component.We also analyse the vertical distribution of the MI component in the vessel and the effects of the frequency of the impeller revolution and of the liquid viscosity on its magnitude.The frequency of occurrence of the macro- -instability related force component is also analysed.

Experimental
A cylindrical flat-bottomed vessel with four radial baffles was used in all experiments rePorted in this paper.The di- mensions of the vessel and of the Rushton turbine impeller are marked in  solutions (viscosity 3 and 6 mPas, respectively) were used as working liquids.The fiequency of impeller rotation N was varied from 3.17 s-t to 6.0 s-r.The corresponding interval of Re" values (approximately from 6000 to 60000) covers the laminaq transition and turbulent regions.The tangential force affecting the baffles was measured by means of a trailing target (target height ht= l0 mm, target width B = 28 mm) located in a slit made in the baffle and enabled to rotate around an axis parallel to the vessel axis.The target was balanced by a couple of (calibrated) springs, and its angular displacement was scanned via a photo-electronic sensor (see Kratdna et al [2, 13] for details of the measuring equipment).The sampling period was Ts= 20 ms.
The signal from the sensor was recorded on a PC (after Ay'D conversion) and subsequently used for evaluation ofthe tan- gential force affecting the target.The duration of a single experiment was 20 minutes and N5 -60000 samPles were typically stored.The local force measured at the target posi- tion was converted to a dimensionless force according to the relation

F+F (l) ON'D'
The time series of the dimensionless force at distinct locations HIH of the target along the baflle (see Thble l) were used for detection and analysis of the macro-instability related component of the total force.At each target position and relative impeller off-bottom clearance ClH, the value of Ren, was varied over the entire attainable inten'al.The total number of processed data sets was 3 10.   4 Nurnerical analysis of experimental data The numerical procedures used for experirrental dara analysis are here described oniy in basic features, as details can be found either in our previous reports [9-l l] or in the original papers [20, 2l].
The power spectra.of the time records of the measured tangential force were used to detect the presence of its low- -frequency component(s) generated by the macro-instabiiity of the flow pattern.The power spectral densities were evalu- ated by means of an algorithm based on the fast Fourier transform.
A procedure based on an application of the Proper or- thogonal decomposition (POD) technique was used for extracting the Ml-related low-frequency component fiom the experimental data.Iirst, the raw data is centred by sub- tracting the mean force value.Then, by application of the so-called N-window [21] to the data and after some algebraic manipulations the so-called trajectory matrix W is obtained.Then, the auto-covariance matrix R of the trajectory matrix W is then evaluated by matrix multiplication R:WWT (2) and its (non-negative) eigenvalues clp and eigenvectors Vp (k=1,..., Af are determined.The eigenvalues, sorted in decreasing order, form the spectrum ofPOD eigenvalues and each ak expresses the magnitude of the contribution of the A-th eigenmode to the total force.'fhe eigenvectors V; are used for evaluating the eigenmodes a1(l) of the measured data [0].Then, the power spectra of the eigenmodes a1(l) are evaluated and the only eigenmodes with significant peaks in their power spectra, located exactly at the macro-instability frequency (or frequencies), are used for evaluating the rela- tive magnitude Qy1 of the macro'instability related compo- e.ur =--6-, t zro j j=t where Ky1 is the set of indices of the eigenmodes contributing to the macro-instability.When two MI fiequencies (cf.Fig. 2) were present in the analysed data all POD eigenmodes con- tributing to both of them were included in the Kp11 set in Eq. (3), i.e., the total macro-instability related component of the force a{fecting the baffle was evaluated.The POD eigenmodes very fiequently captured simultaneously both MI frequencies, and therefore it was impossible to discriminate between their contributions to either of the individual MI fiequencies.Only a rough estimate of the relative ProPortion of the lower fiequency and the higher frequency sub-compo- nents of the total MI force component was possible in a few cases (see section Results and Discussion). o.
The value of the relative magnitude Q"' defined by Eq. ( 3) is expressed by means of the eigenvalues of the POD eigen- modes.Q", is therefore equal to the value of the ratio of the variance of the Ml-related force component to the total force variance, as shown by Lumley [20] or Aubry et al [21] o .25 o Fig. 2: The normalised power spectral densities of the measured tangential force affecting the baffle.a) CIH = 0.35, HrlH = 0.50' N = 3.333 r'', R"" = go05; b)clH = 0.35,HrlH = 0.50,N = 6.0 s-', Re" = 15889' ,9 errlr=}4!= o "MI, riot rial +si-l,,tt 5 Results and drscusslon There are nvo primary factors characterising the macro- -instability related component of the tangential force aflecting the ba{Iles: the frequency (or frequencies) of its occurrence/", and its relative magnitude Q"' (see Eq. 3).
An analysis of the experimental data obtained in the mixing vessel with the pitched blade impeller has shown [l l] that the frequency of MI occurrence /", is directly proPor- tional to the impeller frequency N, i.e., the dimensionless fiequency/"/N attains a constant value.The ProPortionality constant (or dimensionless frequency) value was 0.074.The dimensionless frequency in the case of the PBI was independb)  Qualitatively distinct behaviour of the macro-instability was observed in the mixing vessel stirred with a Rushton turbine impeller.'Iivo frequencies of a MI occurrence were typically detected in the laminar and transitional regions of liquid flow (see Frg. 2a) and a single MI frequency was typi- cally detected in the turbulent flow region (Fig. 2b).This observation agrees well with the experimental results of Nikiforaki et al [l7], and we can conclude that two mechanisms of MI formation were present in the mixing vessel studied in this paper.The lower dimensionless frequency of MI occurrence /r, varies about 0.02, and the upper dimensionless frequency value is close to 0.085.Both frequencies, however; are inde- pendent of the Re" value and of the position of the measuring targetHrlH, as documented in Figs. 3 and 4. It can be de- duced from ligs.2-4 that the MI occurrence frequency is slightly lower at higher impeller off-bottom clearance, i.e., at CIH =0.5.The mean values of the dimensionless fiequencies at different vessel configurations are summarised in Thble 2'   In general, there is considerably higher scatter of the MI frequency data at the lower impeller position (ClH=0.35).
The reason for this obseryation consists probably in the in- stability of the flow pattern arising due to the asymmetry of Tab.2: Dimensionless frequencies of the macro-instability generated by a Rushton turbine impeller the below-impeller and the above-impeller liquid circulation loops and due to their irregular (erratic) interactions.
The vertical distributions of the relative magnitude of the Ml-related component of the force affecting the baffles Qnn, (evaluated according Eq. 3) are shown in Fig. 5. Generally, higher values of Qvr are observed at low ReM values and a quite steep decrease is observed at ReM values just be- low approx.20000.The Qn,,, values are almost constant in the highly turbulent region of liquid flou', despite the impeller off-bottom clearance and vertical position in the vessel.By comparing Figs.5a,c and 5b,d it is easily recognisable that the dependence of the Q", value on Re' qualitatively diflers in the below-impeller and the above-impeller regions: the decrease of the Qnn, value with increasing Renn is more pro- nounced in the above-impeller region.Generally, the magnitude of the Ml-related component of the tangential force gains slightly higher values in the above-impeller region at both impeller off-bottom clearances.The range of observed Qn,, values is quite broad, approximately from 0.05 to 0.30.The low-frequency component of the force affecting the ra- dial baflles thus represents a significant part ofthe total force, especially at low Re\r values (Renn<20000).Vertical distributions of the Q", value in the vessel are depicted in Ftg.6, where averaged Qn', values are plotted against HrlH (fot a fixed vertical position and a given working liquid all available Qn,, valuesdiffering in corresponding Re*, -were averaged).

Impeller off-bottom clearance Liquid
Dimensionless MI frequency /y/,4  a) CIH =OS:b-rhe above impeller region;-b; CIH =0.35 -rhi below impeller region; c)-ClH =o.50 -the above impeller region; d) C/H= 0.50 -the below impeller region.Working liquids:l ... cold glycerol solution; O .'. hot glycerol solution; O ... water.baflle are averaged) are plotted in trig. 7against the Reynolds number values.The Ml-related component of the force fol- lows the general trend observed in lig.5: the maximum (Qv)   value is observed in the laminar region' then (with the in- creasing Re") the (Qr,') value monotonously falls until it reaches a practically constant value in the region of fully turbulent flow This trend repeats in all plots in lig.7, except for trig.7b, where (due to the enhanced data scatter an almost linear spline approximation is used).No effects of liquid vis- cosity or density on the (Q",) values are apparent from Flg ' 7'   As we have noted in the section on Numerical analysis of experimental data it is, in general, impossible to distin- guishbetween the contributions of the lower-frequency and The vertical (axial) profiles of the averaged Q", values are quite flat and the (Q"r) values are not apparently influenced by the impeller off-bottom clearance. (Qur) takes its highest values when cold glycerol is used as the working liquid and .decreaseswith decreasing liquid viscosity.Howeve4 this trend reflects the fact that the impeller Reynolds number value increases with decreasing viscosity, so the differences in the (Q",) values observed in Fig. 6 are primarily due to the in- creasing Re" value (cf.Iig. 5)' To elucidate the influence of the impeller off-bottom cleamnce on the (Q",) values and to emphasise the difference between the above-and below-impeller regions, sPatially av- eraged (Q",) values (at a fixed Re" value all Q",'s along the Ret the higher-frequency sub-components to the total Ml-related force.Such a dissection was, howeve4 possible for several data sets obtained at the lowest Re" values, i.e., with the cold glycerol solution as a working liquid.The ratio of the lower  f) a amount of data available for the analysis, howeveq did not allow of any plausible conclusions to be deducted on the effects of the operational parameters of the vessel and the impeller on the ratio value.

Conclusions
The analysis of the macro-instability related component ofthe tangential force affecting the radial baflles in a stirred vessel with a Rushton turbine impeller performed in this paper enables us to determine the frequency of its occurrence u.ra to evaluate its relative magnitude.The frequencies are linearly dependent on the ftequency of revolution of the impeller.'TivoMI fiequencies were observed at ReM< 20000' ihe Ml-related component of the force constitutes a highly significant contribution to the total force until the mixing is laini.ta.or weakly turbulent.The results of our study agree well with the findings of other authors obtained, howeve6 by analysing experimental data of a qualitatively distinct nature (e.g., Iiquid velocity components).
Fig. I and their values are listed in Thble l.
Fig. l: The mixing vessel and the impeller geometry.The label TGT indicates the moving target positioned in the ba{Tle slit.
Fig. b: Relative magnitude of the Ml-related component of the tangential force affecting the baffles.

Fig. 7 :
Fig.7:The averaged magnitude of the Ml-related component of the tangential force (Qur) as a function of the impeller Reynolds number.a) CIH = 0.35, the above impeller region; b) CIH = 0.35, the below impeller region; c) CIH =0.50, the above impeller region; '