Dynamics of the Flow Pattern in a Baffled Mixing Vessel with an Axial Impeller

A mechanically agitated system under a turbulent regime of flow with or without internals (radial baffles, draft tubes, coils etc.) consists of a broad spectrum of eddies from the size of the main (primary) circulation loop (PCL) of the agitated batch down to the dissipative vortices corresponding to micro-scale eddies. This study deals with an experimental and theoretical analysis of the behaviour of the flow pattern, and mainly the PCL in an agitated liquid in a system with an axial flow impeller and radial baffles. Some characteristics of the investigated behaviour are considered to be in correlation with the occurrence of flow macroinstabilities (FMIs), i.e. the flow macro-formation (vortex) appearing periodically in various parts of a stirred liquid. The flow macroinstabilities in a mechanically agitated system are large-scale variations of the mean flow that may affect the structural integrity of the vessel internals and can strongly affect both the mixing process and the measurement of turbulence in a stirred vessel. Their space and especially time scales considerably exceed those of the turbulent eddies that are a well known feature of mixing systems. The FMIs occur in a range from several up to tens of seconds in dependence on the scale of the agitated system. This low-frequency phenomenon is therefore quite different from the main frequency of an incompressible agitated liquid corresponding to the frequency of revolution of the impeller. Generally, experimental detection of FMIs is based on frequency analysis of the oscillating signal (velocity, pressure, force) in long time series and frequency spectra, or more sophisticated procedures (proper orthogonal decomposition of the oscillating signal, the Lomb period gram or the velocity decomposition technique) are used to determine the FMI frequencies. A theoretical method for finding FMIs could contribute significantly to a deeper understanding of fluid flow behaviour in stirred vessels, e.g. a description of the circulation patterns of a agitated liquid, application of the theory of deterministic chaos, knowledge of turbulent coherent structures, etc. [1–10]. This study investigates oscillations of the primary circulation loop (the source of FMIs) in a cylindrical system with an axial flow impeller and radial baffles, aiming at a theoretical description of the hydrodynamical stability of the loop. However, no integrated theoretical study of the MI phenomenon has been presented up to now.


Introduction
A mechanically agitated system under a turbulent regime of flow with or without internals (radial baffles, draft tubes, coils etc.) consists of a broad spectrum of eddies from the size of the main (primary) circulation loop (PCL) of the agitated batch down to the dissipative vortices corresponding to micro-scale eddies.
This study deals with an experimental and theoretical analysis of the behaviour of the flow pattern, and mainly the PCL in an agitated liquid in a system with an axial flow impeller and radial baffles.Some characteristics of the investigated behaviour are considered to be in correlation with the occurrence of flow macroinstabilities (FMIs), i.e. the flow macro-formation (vortex) appearing periodically in various parts of a stirred liquid.The flow macroinstabilities in a mechanically agitated system are large-scale variations of the mean flow that may affect the structural integrity of the vessel internals and can strongly affect both the mixing process and the measurement of turbulence in a stirred vessel.Their space and especially time scales considerably exceed those of the turbulent eddies that are a well known feature of mixing systems.The FMIs occur in a range from several up to tens of seconds in dependence on the scale of the agitated system.This low-frequency phenomenon is therefore quite different from the main frequency of an incompressible agitated liquid corresponding to the frequency of revolution of the impeller.
Generally, experimental detection of FMIs is based on frequency analysis of the oscillating signal (velocity, pressure, force) in long time series and frequency spectra, or more sophisticated procedures (proper orthogonal decomposition of the oscillating signal, the Lomb period gram or the velocity decomposition technique) are used to determine the FMI frequencies.A theoretical method for finding FMIs could contribute significantly to a deeper understanding of fluid flow behaviour in stirred vessels, e.g. a description of the circulation patterns of a agitated liquid, application of the theory of deterministic chaos, knowledge of turbulent coherent structures, etc. [1][2][3][4][5][6][7][8][9][10].
This study investigates oscillations of the primary circulation loop (the source of FMIs) in a cylindrical system with an axial flow impeller and radial baffles, aiming at a theoretical description of the hydrodynamical stability of the loop.However, no integrated theoretical study of the MI phenomenon has been presented up to now.

Experimental
The experiments were performed in a flat-bottomed cylindrical stirred tank of inner diameter T = 0.29 m filled with water at room temperature with the tank diameter height H = T.The vessel was equipped with four radial baffles (width of baffles b = 0.1 T) and stirred with a six pitched blade impeller (pitch angle 45°, diameter D = T/3, width of blade W = D/5), pumping downwards.The impeller speed was adjusted n = 400 rpm = 6.67 s -1 and its off-bottom clearance C was T/3 (see Fig. 1).

Dynamics of the Flow Pattern in a Baffled Mixing Vessel with an Axial Impeller
O. Brůha, T. Brůha, I. Fořt, M. Jahoda This paper deals with the primary circulation of an agitated liquid in a flat-bottomed cylindrical stirred tank.The study is based on experiments, and the results of the experiments are followed by a theoretical evaluation.The vessel was equipped with four radial baffles and was stirred with a six pitched blade impeller pumping downwards.The experiments were concentrated on the lower part of the vessel, where the space pulsations of the primary loop, originated due to the pumping action of the impeller.This area is considered to be the birthplace of the flow macroinstabilities in the system -a phenomenon which has been studied and described by several authors.The flow was observed in a vertical plane passing through the axis of the vessel.The flow patterns of the agitated liquid were visualized by means of Al micro particles illuminated by a vertical light knife and scanned by a digital camera.The experimental conditions corresponded to the turbulent regime of agitated liquid flow.It was found that the primary circulation loop is elliptical in shape.The main diameter of the primary loop is not constant.It increases in time and after reaching a certain value the loop disintegrates and collapses.This process is characterized by a certain periodicity and its period proved to be correlated to the occurrence of flow macroinstability.The instability of the loop can be explained by a dissipated energy balance.When the primary loop reaches the level of disintegration, the whole impeller power output is dissipated and under this condition any flow alteration requiring additional energy, even a very small vortex separation, causes the loop to collapse.The visualized flow was scanned by a digital camera; series of shots were generated with a time step of 0.16 s and were analyzed by appropriate graphic software.The total length of the analyzed record was 30.24 s.The characteristics observed and analyzed were: a) the shape and sizes of the PCL, b) the size of the PCL core, c) the positions of the top of the flow macro formation and the corresponding functions were obtained: a) the height of the PCL h ci = h ci (t) (see Fig. 3), b) the width of the PCL s ci = s ci (t) (see Fig. 3), c) the equivalent mean diameter of the PCL core r c,av i = r c,av i (t) (see Fig. 3), d) the height of the top of the flow macro formation h FMI,i = h FMI,i (t) (see Fig. 4).

Results of experiments
The analysis of the experiments provided the following findings: The PCL can be described as a closed stream tube with a vertical section elliptical in shape with a core.The flow in the elliptical annular area is intensive and streamlined, while the core is chaotic and has no apparent streamline characteristics.The flow in the remaining upper part is markedly steadier.This is in agreement with the earlier observations of some authors [6,7].
The flow process is characterized by three stages.In the first stage, the PCL grows to a certain size (its shape can be approximated as elliptical).Then a quasi-steady stage follows, when the PCL remains at a constant size for a short time.In the next stage the PCL collapses into very small flow formations (vortices) or disintegrates into chaotic flow, see Fig. 5a-c The process (oscillation of the PCL) is apparently characterized by a certain periodicity.The function of the incidence of the PCL in time is illustrated in Fig. 6, which shows that the PCL incidence ratio approaches 80 % of the considered time.
The graph in Fig. 7 (vertical dimension of the PCL as a function of time) illustrates this process for the whole analysed time of 30.24 s.The calculated average time of one cycle (the time between the origin of the PCL and its collapse) is t c,av = 1.59 s.This means that the frequency of the PCL oscillations is f osc = 1/t c,av = 0.63 Hz and dimensionless frequency F osc = f osc /n = 0.094.This dimensionless frequency is markedly higher than the value detected earlier in the interval 0.02-0.06[8][9][10].This corresponds to our finding that only some of the flow formations generated by one PCL cycle result in macro--flow formation causing a surface level eruption, detected as "macroinstability".
The function h c = h c (t) for one cycle is shown in Fig. 8.This function was obtained by regression of the experimental data h ci (t), where the individual cycle intervals were recalculated for an average time cycle t c,av = 1.59 s.Fig. 8 shows that the mean maximum height of the PCL hc,max reaches a value of 0.181 m, which is 62 % of surface level height H.This agrees with earlier findings [11] that h c,max »2/3 H.The function h c = h c (t) was used for calculating the PCL rising velocity, see Fig. 9.
Finally, the average rising time of flow macro formation generated by PCL was calculated from the time function h FMI = h FMI (t) (time between generating and disintegration), and the value obtained was t av, FMI = 1.0 s.
The regression curve h FMI,i = h FMI,i (t) corresponding to the experimental data for one cycle (development of one macro-formation, the time recalculated for average rising time t av,FMI ) is illustrated in Fig. 10.
To specify the PCL characteristics more deeply, the volume of the PCL (active volume of the primary circulation) was calculated.This was assumed as a toroidal volume around its elliptical projection in the r-z plane of the mixing vessel.The function of the ratio of the PCL volume to vessel volume

V t T h t s t T r t
is calculated from the corresponding regression curves h c = h c (t), s c = s c (t) and r c,av (t).Fig. 11 shows that the mean maximum ratio of the PCL volume to the vessel volume V c,max /V reaches a value of 0.326.

Theoretical calculation of energy balance
An energy balance for the PCL was carried out with a view to explaining the reasons for the flow field behaviour (predominantly PCL oscillations).As mentioned above, there are three different stages in the flow process.The energy balance was carried out for the quasi-steady stage under conditions when the PCL reaches its top position, i.e. h c = h c, max , see Fig. 12.
This stage directly follows (precedes) the collapse and is assumed to have a critical influence on the flow disintegration.
The impeller power input N is dissipated in the PCL by the following components: I) Mechanical energy losses: 1) N turn -lower turn of the primary circulation loop about 180°from downwards to upwards.2) N wall -friction of the primary circulation loop along the vessel wall.
II) Turbulence energy dissipation: 1) N disch -dissipation in the impeller discharge stream just below the impeller.2) N up -dissipation in the whole volume of the agitated liquid above the impeller rotor region, i.e. in the space occupied by both the primary and secondary flow.
It is expected that under quasi-steady state conditions (just before the PCL collapses) these components are in balance with the impeller power output.This means that no spare power is available, and that no alternative steady state formation is possible.The individual power components and quantities can be calculated when the validity of the following simplifying assumption for the flow in the PCL is considered: 1) The system is axially symmetrical around the axis of symmetry of the vessel and impeller.
2) The primary circulation loop can be considered as a closed circuit.3) The cross section of the primary circulation loop (stream tube) is constant.4) The conditions in the primary circulation loop are isobaric.5) The liquid flow regime in the whole agitated system is fully turbulent.6) The character of the turbulence in the space above the impeller rotor region is homogeneous and isotropic.
The impeller power output can be calculated for known impeller hydraulic efficiency hh (hh=0.48 for a four 45°p itched blade impeller [14]).
Impeller pumping capacity Q p can be calculated from the known flow rate number N Qp (N Qp = 0.94 for Re M >10 4 [13]).
The mass flow rate is m Q p p = = r 566 .kg s -1 (r = 1 kg l -1 ). ( The axial impeller discharge velocity is equal to the average circulation velocity of the PCL (assuming a constant average circulation velocity in the loop): where pD 2 /4 is the cross sectional area of the impeller rotor region, i.e. the cylinder circumscribed by the rotary mixer.
Then the primary circulation loop is a closed stream tube consisting of the set of stream lines passing through the impeller rotor region.

Calculation of turbulent dissipation below the impeller rotor region, N disch
The energy dissipation rate per unit mass is [14] where and the integral length of turbulence [14] is The kinetic energy of turbulence per unit of mass is According to [14] the average value of q in the impeller discharge stream is q 0 = 0.226 m 2 s -2 for the conditions described in [14]: a four pitched blade impeller with a pitch angle of 45°, D 0 = 0.12 m, T 0 = 0.24 m, n 0 = 6.67 s -1 , Po 0 = 1.4,Re M,0 = 44×10 4 , P 0 = 10.3W, L 0 = 0.012 m.We have from Eq. ( 7) e 0 = 6.029 m 2 s -3 .
The energy dissipation per unit mass e related to the experimental system used in this study is and after substitution V = 0.0192 m 3 , P 0 = 10.3W, V 0 = 0.0109 m 3 we obtain e = 1.384 m 2 s -3 . Using where the dissipation volume below the impeller rotor region V disch = 0.000330 m 3 (see Fig. 13).We obtain N disch = 0.46 W.

Calculation of dissipation in the whole volume above the impeller region, N up
According to [15] N P where the portion of the impeller power input dissipated above the impeller rotor region h up = 0.14 for a six 45°pitched blade impeller (D/T = C/T = 1/3, H = T) and Re M >10 4 .
When calculating the quantity N up , the validity of introduced assumption No. 6 is considered.Moreover, because of the momentum transfer between the primary flow and the secondary flow out of the PCL the rate of energy dissipation above the impeller rotor region is related to the whole volume, consisting of both the primary and secondary flows.

Calculation of the dissipation in the lower turns of the PCL, N turn
According to [16] N w m Substituting the sum of loss coefficients z j j å = 0.50 for bend of turn of the PCL b = 180°as well as the values of the average circulation velocity of the PCL w c,av , and the mass flow rate m p , we obtain N turn = 0.84 W.

Calculation of the dissipation by the wall friction along the PCL, N wall
Using the relation for mechanical energy loss due to the friction along the wall, from [17] where equivalent diameter is defined as and the friction factor for a smooth wall  16)-(18) to Eq. ( 15), we get N wall = 0.14 W.

Energy balance in the quasi-steady stage of the PCL
The sum of all dissipated power components considered here is The value in Eq. (19a) is in good agreement with the impeller power output from Eq. ( 3), N = 2.04 W, though the data used for the calculations come from six independent literature sources.This result means that in the quasi-steady phase of the PCL oscillation cycle, all the impeller power output is consumed by dissipation.No power is available, either for increasing the PCL kinetic energy or for any changes in flow formation resulting in a higher energy level.However, it is well known that vortex disintegration in smaller formations or even a very small vortex separation from a primary flow is a process that consumes energy (according to the law of angular momentum conservation).The experiments proved that vortex separations (small and large) occur in all the stages, thus also in the quasi-steady stage.This seems to provide an explanation for the PCL collapse: in the quasi-steady stage, the energy necessary for vortex separation or for other changes in flow formation cannot be supplied by the impeller, but energy is exhausted from the ambient flow field.And then, even a very small energy deficit can result in a qualitative flow field change appearing as the PCL collapse.It should be noted that the PCL collapse can be followed by a noticeable rise in the surface level, classified as a macroinstability.However, not every collapse reaches the level and is observed as a macroinstability.This corresponds to this disagreement between the PCL oscillation frequency determined experimentally in this study and the macroinstability frequency presented.
The phenomenon observed here affects the processes taking place in an agitated charge , especially on the macro level, i.e., when miscible liquids blend and when solids suspend in liquids.Oscillations of the macroflow contribute to attain a so called "macroequilibrium" in an agitated batch, i.e., better homogeneity both in a pure liquid (blending) and in a solid--liquid suspension (distribution of solid particles in a liquid).These processes can be observed predominantly in the subregion of low liquid velocity above the impeller rotor region, corresponding to approx.One third of the volume of the agitated charge.Oscillations of the primary circulation of an agitated liquid can contribute to the forces affecting the body of the impeller as well as the mixing vessel and its internals.This low frequency phenomenon probably need not have fatal consequences, because the standard design of industrial mixing equipment should have sufficient margins for unexpected events during processes running in an industrial unit.

Conclusions
a) The average circulation velocity in the primary circulation loop is more than one order of magnitude higher than the rising velocity of the loop.
b) The frequency of the primary loop oscillations is about one order of magnitude lower than the revolution frequence of the impeller.c) The top of the disintegration of the primary circulation loop is a birthplace of macroinstabilities in the region of secondary flow.d) The primary circulation loop collapses owing to disequilibrium between the impeller power output and the rate of dissipation of the mechanical energy in the loop.A small change (e.g. a small turbulent vortex or a small increase in the primary circulation loop) can have a great effect.

Fig. 7 :Fig. 8 :
Fig. 7: Dependence of the vertical dimension of PCL h ci on time for the whole analyzed time Fig. 10: Dependence of the height of the top of the flow macro formation h FMI, i on the time for one cycle

Fig. 12 :
Fig. 12: Schematic view of the PCL under its top position (h c = h c, max )

Fig. 13 :
Fig. 13: Dissipation volume below the impeller rotor region V disc before the first turn of the loop (D = T/3 = 0.0967 m, z = 0.045 m)

3 Vm 3 W 1 ¢ 1 e
c,max maximum value of calculated volume of the PCL, m 3 V disch volume of dissipation below the impeller rotor region, width of blade, m w c,av axial impeller discharge velocity, m s w z ¢ w r , ¢ w q turbulent velocity fluctuations in individual axis directions, m s -1 z height of cylindrical volume of dissipation below the impeller rotor region, m b bend of turn of the PCL, energy dissipation rate per unit mass,