Local velocity scaling in T400 vessel agitated by Rushton turbine in a fully turbulent region

The hydrodynamics and flow field were measured in an agitated vessel using 2-D Time Resolved Particle Image Velocimetry (2-D TR PIV). The experiments were carried out in a fully baffled cylindrical flat bottom vessel 400 mm in inner diameter agitated by a Rushton turbine 133 mm in diameter. The velocity fields were measured in the zone in upward flow to the impeller for impeller rotation speeds from 300 rpm to 850 rpm and three liquids of different viscosities (i.e. (i) distilled water, ii) a 28% vol. aqueous solution of glycol, and iii) a 43% vol. aqueous solution of glycol), corresponding to the impeller Reynolds number in the range 50 000 < Re < 189 000. This Re range secures the fully-developed turbulent flow of agitated liquid. In accordance with the theory of mixing, the dimensionless mean and fluctuation velocities in the measured directions were found to be constant and independent of the impeller Reynolds number. On the basis of the test results the spatial distributions of dimensionless velocities were calculated. The axial turbulence intensity was found to be in the majority in the range from 0.388 to 0.540, which corresponds to the high level of turbulence intensity.


Introduction
It is important to know the flow and the flow pattern in an agitated vessel in order to determine many impeller and turbulence characteristics, e.g. impeller pumping capacity, intensity of turbulence, turbulent kinetic energy, convective velocity, and the turbulent energy dissipation rate. The information and data that are obtained can also be used for CFD verification.
In PIV studies, the spatial distributions of various properties are often presented. However, the inspection analysis shows that the validity of the spatial distribution of any dimensionless property for arbitrary process conditions in geometrically similar agitated vessels requires independence of a given dimensionless property from the impeller Reynolds number. The spatial distribution that is obtained is in general valid only when this independence is both theoretically predicted and experimentally verified. Unfortunately, in many studies the results are presented only for one impeller speed and, in addition, when a Rushton turbine is used as an impeller, the experiments are often carried out in the transient flow regime.
Secondly, the flow discharging from an impeller has been investigated mainly, while the regions outside the impeller region have not been treated with the same level of interest ( [1]). In our previous work [2] the local velocity profiles in the zone in upward flow to the impeller for three impeller rotation speeds in a vessel 300 mm in the inner diameter filled by a water and agitated by a Rushton turbine were presented.
The aim of this work is to study scaling of the velocity field outside the impeller region in a vessel 400 mm in inner diameter mechanically agitated by a Rushton turbine in a fully turbulent region for three liquids of different viscosities. The independency of dimensionless spatial velocity distribution on the impeller Reynolds number was tested statistically. The hydrodynamics and the flow field were measured in an agitated vessel using Time Resolved Particle Image Velocimetry (TR PIV).

Inspection analysis of flow in an agitated vessel
The flow of a Newtonian fluid in an agitated vessel has been described by the Navier -Stokes equation: where the dimensionless properties are defined as follows: x dimensionless instantaneous velocity: x dimensionless instantaneous pressure: x dimensionless space gradient: x impeller Reynolds number x impeller Froude number: g and where n & is a unity vector. Similarly, the equation of continuity for stationary flow of a non-compressible fluid is given and one can be rewritten into the dimensionless form as follows: The following relations can be obtained for the dimensionless velocity components and the dimensionless pressure, respectively, by inspection analysis of Eqs. (2) and (5), as follows: ) Re, , , ( where * x & is a dimensionless location vector. For stationary flow with a periodic character the velocity time dependence can be eliminated by substituting the velocity and pressure by time-averaged properties. For highly turbulent flow in a baffled vessel, the viscous and gravitational forces can be neglected and finally the time-averaged dimensionless velocity components and the pressure are independent of the impeller Reynolds and the Froude numbers, and depend on location only: Reynolds decomposition of instantaneous velocity components has been applied for the velocity profiles studied in this work.

Mean and fluctuation velocity
Using PIV, the instantaneous velocity data set U i (t j ) in the i th direction for j = 1, 2, … N R at observation times t j with equidistant time step 't S (i.e. 't S = t j+1 -t j ) was obtained in a given location. Assuming the ergodic hypothesis the time-averaged mean velocity i U was determined as the average value of velocity data set U i (t j ): where i U is mean velocity in the i th direction, U i (t j ) is instantaneous velocity in the i th direction at the observation time t j , and N R is the number of data items in the velocity data set. Consequently, the fluctuation velocity in the i th direction u i (t j ) at observation time t j is obtained by decomposition of the instantaneous velocity: where u i (t j ) is the fluctuation velocity in the i th direction at observation time t i , i U is mean velocity in the i th direction, U i (t i ) is instantaneous velocity in the i th direction at the observation time t i . The root mean squared fluctuation velocity is determined as follows: where i u is the root mean squared fluctuation velocity, and u (t i ) is the fluctuation velocity at observation time t i .

Experimental
The hydrodynamics and the flow field were investigated in an agitated vessel using Time Resolved Particle Image Velocimetry (TR PIV). The experiments were carried out in a fully baffled cylindrical flat bottom vessel 400 mm in the inner diameter. The tank was agitated by a Rushton turbine 133 mm in diameter, i.e. the dimensionless impeller diameter D/T was 1/3. The dimensionless impeller clearance C/D taken from the lower impeller edge was 0.75. The tank was filled with degassed liquid, and the liquid height was 400 mm, i.e. the dimensionless liquid height H/T was 1. The dimensionless baffle width B/T was 1/10. To prevent air suction the vessel was covered by a lid. Three liquids of different viscosities were used as the agitated liquid: i) distilled water (Q = 9.35x10 -7 m 2 /s), ii) a 28% vol. aqueous solution of glycol (Q = 2x10 -6 m 2 /s), and iii) a 43% vol. aqueous solution of glycol (Q = 3x10 -6 m 2 /s). The velocity fields were measured at an impeller rotation speed in the range from 300 rpm to 850 rpm, which covers the impeller Reynolds number range from 50 000 to 189 000. This Re range secures the fullydeveloped turbulent flow. The operational conditions are summarized in Table 1. The position of middle point P of the investigated area corresponds to dimensionless radius 2r/T = 0.224 and dimensionless height z/T = 0.138. The scheme of the experimental apparatus and the investigated area is depicted in Fig. 1. The camera was positioned orthogonally to the laser sheet. The investigated area corresponds to region F according to the classification given by FoĜt et al. [4]. The method of ensemble averaged velocity field can be used due to the axisymmetric character of the flow in this region.

Experimental data evaluation
According to the inspection analysis, the dimensionless velocities normalized by the product of impeller speed N and impeller diameter D should be independent of the impeller Reynolds number.
The effect of impeller Reynolds number on dimensionless velocities was tested by hypothesis testing ( [5]). The statistical method of hypothesis testing can estimate whether the differences between the predicted parameter values (e.g. according to some proposed theory) and then the parameter values evaluated from the measured data are negligible. In this case, we assumed dependence of the tested parameter on the impeller Reynolds number, described by the simple power law correlation parameter = B.Re E , and the difference between predicted exponent E pred and evaluated exponent E calc was tested. The hypothesis test characteristics are given as t = (E calc -E pred )/ s E where s E is the standard error of parameter E calc .  If the calculated _t_ value is less than the critical value of the t-distribution for (m-2) degrees of freedom and significance level D, the difference between E calc and E pred is statistically negligible (statisticians state: "the hypothesis cannot be rejected"). In our case, the independence of dimensionless velocities of the impeller Reynolds number was tested as the hypothesis, i.e.  Table 2 by the percentage of points in which the above-formulated hypothesis parameter = const. is satisfied and by the percentage of points in which the hypothesis parameter = const. can not be accepted. For illustration, the average values of calculated _t_ value are presented here also.
The dimensionless radial mean velocities were found to be close to zero, in the range from 0.008 to 0.074. These findings correspond to characteristics of the given zone, according to FoĜt et al. [4]. This region contains predominantly ascending flow along the vessel axis towards the impeller. The dimensionless axial mean velocity was found to be in the range from 0.317 to 0.511. These values are higher than the dimensionless values for radial mean velocity as expected for this region. The tested hypothesis can be accepted in the majority of profile points of the investigated area.
The dimensionless radial r.m.s. fluctuation velocities were found to be in the range from 0.189 to 0.323. The tested hypothesis can be accepted in the majority of profile points, as is signalized by the very low calculated _t_ values. The dimensionless axial r.m.s. fluctuation velocities were found to be in the range from 0.153 to 0.295. The tested hypothesis can be accepted in the majority of profile points as is again signalized by the low calculated _t_ values. On the basis of the results of this hypothesis test, we assume that all dimensionless velocities can be statistically taken as constant and independent of the impeller Reynolds number. The spatial distributions of dimensionless velocities for seven operational conditions were averaged and are presented in

Intensity of turbulence
The axial turbulence intensity was calculated for each point in the investigated area as follows: where ax u is axial r.m.s. fluctuation velocity, ax U is axial mean velocity. For dimensionless velocities independent of the impeller Reynolds number, the turbulence intensity should also be independent of this quantity. The independence of dimensionless velocities from the impeller Reynolds number was again tested by hypothesis testing. The t-distribution coefficient t (m-2), D for seven impeller Reynolds numbers and significance level D = 0.05 is 2.5706. Hypothesis testing was done for each point in the investigated area. The hypothesis test results are presented in Table 2, as well. The tested hypothesis can be accepted in the majority of profile points On the basis of these hypothesis test results, we assume that the axial turbulence intensity can be statistically taken as constant and independent of the impeller Reynolds number, as expected. As shown, the calculated values are in the majority of points in the range from 0.388 to 0.540. These values correspond to high turbulence intensity. As expected, the highest turbulence intensity was found close to the impeller axis in the ascending flow core. The radial profiles of axial turbulence intensity are presented in Figure 10 for selected horizontal positions z* = 0.120, 0.135 and 0.150. The spatial distribution of quantity T.I. ax is shown in Figure 11.

Experimental data evaluation
The following results have been obtained in this study: The hydrodynamics and flow field were measured in a vessel 400 mm in the inner diameter agitated by a Rushton turbine using the 2-D Time Resolved Particle Image Velocimetry (2-D TR PIV). The velocity fields were measured in the zone in upward flow to the impeller for impeller rotation speeds from 300 rpm to 850 rpm and three liquids of different viscosities, corresponding to the impeller Reynolds number in the range 50 000 Re 189 000.
The dimensionless radial mean velocities were found to be close to zero. These findings correspond to the characteristics of the given zone according to FoĜt et al. [4]. This region contains predominantly ascending flow along the vessel axis towards the impeller.
In accordance with the theory of mixing, the dimensionless mean and fluctuation velocities in measured directions were found to be constant and independent of the impeller Reynolds number. On the basis of the test results the spatial distributions of dimensionless velocities were calculated.
The axial turbulence intensity was calculated and was found to be in the majority of points in the range from 0.388 to 0.540, which corresponds to high level of turbulence intensity. As expected, the highest turbulence intensity was found close to the impeller axis in the ascending flow core. It was found that the axial turbulence intensity can be statistically taken as constant and independent of the impeller Reynolds number.