Acta Polytechnica doi:10.14311/AP.2014.54.0430 Acta Polytechnica 54(6):430–438, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap LOCAL VELOCITY PROFILES MEASURED BY PIV IN A VESSEL AGITATED BY A RUSHTON TURBINE Radek Šulc∗, Vít Pešava, Pavel Ditl Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Process Engineering, Technická 4, 166 07 Prague, Czech Republic ∗ corresponding author: Radek.Sulc@fs.cvut.cz Abstract. 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 300 mm in inner diameter. The tank was agitated by a Rushton turbine 100 mm in diameter. The velocity fields were measured for three impeller rotation speeds: 300 rpm, 450 rpm and 600 rpm, and the corresponding Reynolds numbers in the range 50000 < Re < 100000, which means that fully-developed turbulent flow was reached. 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 rotational speed. The velocity profiles were averaged, and were expressed by Chebyshev polynomials of the 1st order. Although the experimentally investigated area was relatively far from the impeller, and it was located in upward flow to the impeller, no state of local isotropy was found. The ratio of the axial rms fluctuation velocity to the radial component was found to be in the range from 0.523 to 0.768. The axial turbulence intensity was found to be in the range from 0.293 to 0.667, which corresponds to a high turbulence intensity. Keywords: mixing, Rushton turbine, Particle Image Velocimetry, flow, velocity profile, mean velocity, fluctuation velocity. 1. 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. Drbohlav et al. [1] made an experimental investiga- tion of the velocity field in the stream discharging from a Rushton turbine at Reynolds numbers Re = 146000 and Re = 166000. They described the axial profiles of the mean velocity components in this region using a phenomenological three-parameter model based on a tangential cylindrical jet, proposed by [2]. Obeid et al. [3] used the proposed model to describe the velocity field in the discharged flow produced by various types of turbine impellers. The flow in a mechanically agitated vessel should be divided into several regions where the flow behaviour is quite different. Fořt et al. [4] divided the flow in a vessel agitated by a Rushton turbine into the following regions: (1.) region O, in which the stream discharges from the impeller; (2.) region A, close to the wall, in which the flow direction changes from radial into axial; (3.) region B, which contains the predominantly as- cending and descending sections of flow along the vessel wall; (4.) region C, in which the flow direction changes from axial to radial at the vessel bottom or at the liquid surface; (5.) region D, which contains the prevailing radial flow at the vessel bottom or at the liquid surface; (6.) region E, in which the flow at the bottom or at the surface turns into the direction of the vessel axis; (7.) region F, which contains the predominantly as- cending or descending flow along the vessel axis towards the impeller; (8.) region G, in which the streamline pattern is stag- nant and unstable, where neither of the mean ve- locity components is significant. The flow pattern characterized by the mean velocity components is described by the proposed theoretical model based on the Stokes stream function. Although the flow discharging from an impeller has been investigated by many authors, the regions outside the impeller region have not been treated with the same level of interest ([5]). The aim of this work is to study scaling of the velocity field outside the impeller region in a vessel mechanically agitated by a Rushton turbine in a fully turbulent region at a high Reynolds number in the range of 50000 < Re < 100000. The hydrodynamics and the flow field were measured in an agitated vessel using Time Resolved Particle Image Velocimetry (TR PIV). The radial profiles of the mean and fluctuation velocities are expressed in terms of Chebyshev polynomials. 430 http://dx.doi.org/10.14311/AP.2014.54.0430 http://ojs.cvut.cz/ojs/index.php/ap vol. 54 no. 6/2014 Local Velocity Profiles 2. Theoretical background 2.1. 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: % (∂~U ∂t + ~U ·∇~U ) = −∇p + µ∇2 ~U + %~g. (1) This equation can be rewritten into dimensionless form, as follows (e.g. [6]): ∂~U∗ ∂t∗ + ~U∗ ·∇∗~U∗ = −∇∗p∗+ 1 Re ∇∗2 ~U∗+ 1 Fr ~n∗, (2) where the dimensionless properties are defined as fol- lows: • dimensionless instantaneous velocity ~U∗ = ~U ND ; • dimensionless instantaneous pressure p∗ = p %N2D2 ; • dimensionless space gradient ∇∗ = ∇ D ; • Reynolds number Re = ND 2 ν ; • Froude number Fr = N 2D g ; and where ~n is a unity vector. Similarly, the equation of continuity for stationary flow of a non-compressible fluid given as ∇· ~U = 0 (3) can be rewritten into the dimensionless form: ∇∗ · ~U∗ = 0. (4) The following relations can be obtained for dimen- sionless velocity components and dimensionless pres- sure, respectively, by inspection analysis of Eqs. (2) and (4), as follows: ~U∗ = f1(~x∗, t∗, Re, Fr ), (5) p∗ = f2(~x∗, t∗, Re, Fr ), (6) where ~x∗ is a dimensionless location vector. For stationary flow with a periodic character, the velocity time dependence can be eliminated by sub- stituting the velocity and pressure by time-averaged properties. For highly turbulent flow in a baffled vessel, the viscous and gravitational forces can be ne- glected and finally the time-averaged dimensionless velocity components and the pressure are independent of the Reynolds number and the Froude number, and depend on location only: U∗i = f1(~x ∗), p∗ = f2(~x∗), (7) Reynolds decomposition of the instantaneous velocity components has been applied for the velocity profiles studied in this work. 2.2. Mean and fluctuation velocity Using PIV, the instantaneous velocity data set Ui(tj) in the ith direction for j = 1, 2, . . . ,NR at observation times tj with an equidistant time step ∆tS (i.e., ∆tS = tj+1 −tj) was obtained in a given location. Assuming the so-called ergodic hypothesis, the time-averaged mean velocity Ui was determined as the average value of velocity data set Ui(tj): Ui = 1 NR NR∑ j=1 U(tj), (8) where Ui is mean velocity in the ith direction, Ui(tj) is instantaneous velocity in the ith direction at obser- vation time tj, and NR is the number of data items in the velocity data set. Consequently, the fluctuation velocity in the ith direction ui(tj) at observation time tj is obtained by decomposition of the instantaneous velocity: ui(tj) = Ui(tj) −Ui for j = 1, 2, · · ·NR, (9) where ui(tj) is the fluctuation velocity in the ith di- rection at observation time ti, Ui is mean velocity in the ith direction, Ui(tj) is instantaneous velocity in the ith direction at observation time tj. The root mean squared fluctuation velocity is de- termined as follows: ui = ( 1 NR NR∑ j=1 ui(tj)2 )1/2 , (10) where ui is the root mean squared fluctuation velocity, and u(tj) is the fluctuation velocity at observation time tj. 3. Experimental The hydrodynamics and the flow field were measured 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 300 mm in inner diameter [7]. The tank was agitated by a Rushton turbine 100 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 distilled water, and the liquid height was 300 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. Velocity fields were measured for three impeller rotation speeds: 300, 450 and 600 rpm, at which fully developed turbulent flow was reached. Distilled water at a temperature of 23 °C (density % = 997.4 kg m−3, dynamic viscosity µ = 0.9321 mPa s) was used as the agitated liquid. The time resolved LITRON LDY 304 2D-PIV sys- tem (Dantec Dynamics (Denmark)) consists of a 431 R. Šulc, V. Pešava, P. Ditl Acta Polytechnica Neodyme-YLF laser (light wave length 532 nm, im- pulse energy 2×30 MJ), a SpeedSence 611 high speed PIV-regime camera (resolution 1280 × 1024 pixels) with a Sigma MacroDg objective equipped with an optical filter with wave length 570 nm. Rhodamine B fluorescent particles of mean diameter 11.95±0.25 µm were used as seeding particles. The fluorescent parti- cles lit by 532 nm light emit 570 nm light. In this way, non-seeding particles such as impurities and bubbles are separated and are not recorded. The operating frame rate was 1 kHz (1000 vector fields per second), i.e., the sampling time ∆ts was 1 ms. The measured vertical plane was located in the center of the vessel and in the middle of the baffles. The plane was illuminated by a laser sheet 0.7 mm in thickness. The investigated area was 43×27 mm. The position of the right top apex E [rE; zE] was [10; 55], i.e., r∗ = 2r/T = 20/300 and z∗ = z/T = 55/300, i.e., the right top edge was located 20 mm below the impeller paddle edge and 10 mm from the impeller axis. The scheme of the experimental apparatus and the investigated area are depicted in Fig. 1. The cam- era was positioned orthogonally to the laser sheet. The experiments were conducted in the framework of cooperation with Dr M. Kotek (Technical University Liberec) and Dr B. Kysela (Institute of Hydrodynam- ics, Czech Academy of Sciences). The velocity profiles in four horizontal planes located in the investigated area in the positions z∗ = 0.1114, 0.1294, 0.1474 and 0.1654 are presented in this paper in the dimensionless radius range r∗ = 〈0.0702÷0.3509〉. The investigated area corresponds to region F according to the clas- sification given by Fořt et al. (1982). The ensemble averaged velocity field method can be used due to the axisymmetric character of the flow in this region. For all experiments, 5000 images were taken at a sampling interval of 0.001 s, i.e., the total record time length was 5 s. Unfortunately, a recording time of only 3.865 s was available for the 300 rpm measurement, due to damage to the storage disk. 4. 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 Reynolds number. For a single impeller size and a single liquid, this dimensionless velocity should be independent of the impeller rotational speed. The effect of impeller rotational speed on dimen- sionless velocities was tested by hypothesis testing [8]. The statistical method for hypothesis testing can esti- mate whether the differences between the predicted parameter values (e.g., predicted by some proposed theory) and the parameter values evaluated from the measured data are negligible. In this case, we assumed dependence of the tested parameter on the impeller rotational speed, described by the simple power law parameter = BNβ, and the difference between pre- dicted exponent βpred and evaluated exponent βcalc Figure 1. Scheme of the experimental apparatus and the investigated area. was tested. The hypothesis test characteristics are given as t = (βcalc−βpred)/sβ where sβ is the standard error of parameter βcalc. If the calculated |t| value is less than the critical value of the t-distribution for m − 2 degrees of freedom and significance level α, the difference between βcalc and βpred is statistically negligible (statisticians state: “the hypothesis cannot be rejected”). In our case, the independence of dimen- sionless velocities from the impeller speed was tested as the hypothesis, i.e., parameter = BN0 = const., i.e., βpred = 0. The t-distribution coefficient tm−2,α for three impeller rotational speeds and significance level α = 0.05 is 12.706. Hypothesis testing was per- formed for each point in the investigated profiles. The hypothesis test results are presented for each profile in Table 1 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. cannot be accepted. For illustration, the average calculated |t| values are also presented here. 432 vol. 54 no. 6/2014 Local Velocity Profiles Hypothesis: parameter = BN0 Percentage; t-characteristics |t| Parameter Ur/(ND) ur/(ND) Uax/(ND) uax/(ND) Profile z∗ = 0.1114 acceptable 92.4 %; 4.1 100 %; 0.7 100 %; 3.4 100 %; 0.9 not acceptable 7.6 %; 46.7 0 % 0 % 0 % Profile z∗ = 0.1294 acceptable 98.7 %; 0.76 100 %; 0.6 95 %; 2.9 96.2 %; 1.2 not acceptable 1.3 %; 185 0 % 5 %; 57 3.8 %; 47.7 Profile z∗ = 0.1474 acceptable 100 %; 0.9 100 %; 0.6 92.4 %; 4.1 93.7 %; 1.1 not acceptable 0 % 0 % 7.6 %; 21.5 6.3 %; 22.3 Profile z∗ = 0.1654 acceptable 95 %; 2.4 100 %; 0.73 86.1 %; 2.81 97.5 %; 1.44 not acceptable 5 %; 28.1 0 % 13.9 %; 51.3 2.5 %; 18.1 Table 1. Dimensionless velocities – effect of impeller speed. Figure 2. Dimensionless radial mean velocity profile – effect of impeller speed; z∗ = 0.1474. Figure 3. Dimensionless radial fluctuation velocity profile – effect of impeller speed; z∗ = 0.1474. 433 R. Šulc, V. Pešava, P. Ditl Acta Polytechnica Figure 4. Dimensionless axial mean velocity profile – effect of impeller speed; z∗ = 0.1474. Figure 5. Dimensionless axial fluctuation velocity profile – effect of impeller speed; z∗ = 0.1474. For illustration, the profiles of the dimensionless velocities for three impeller speeds and their average are presented in Figures 2–5 for the line z∗ = 0.1474. The dimensionless radial mean velocities were found to be close to zero. As shown in Fig. 2, some ve- locity values are positive, while other velocity val- ues are negative. These findings correspond to char- acteristics of the given zone, according to Fořt et al. (1982). This region contains predominantly ascend- ing flow along the vessel axis towards the impeller. The dimensionless axial mean velocity was found to be in the range from 0.323 to 0.488. These values are higher than the dimensionless values for the ra- dial mean velocity, as expected for this region. The tested hypothesis can be accepted in almost all profile points. The dimensionless radial rms fluctuation velocities were found to be in the range from 0.194 to 0.326. The tested hypothesis can be accepted in all profile points, as is signalized by the very low calculated |t| values. The dimensionless axial rms fluctuation velocities were found to be in the range from 0.139 to 0.204. The tested hypothesis can be accepted in the majority of profile points as is again signalized by the low calculated |t| values. Because the selected position is relatively far from the impeller and outside the impeller discharge flow, we expected the local isotropy state defined on the length-scale level corresponding to integral length scale to be an equality of the fluctuation velocity components. However, this expectation was not con- firmed. The ratio of the axial rms fluctuation velocity to the radial component was found to be in the range from 0.523 to 0.768. 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 rotational speed. The velocity profiles were averaged, and were expressed by the polynomial approximation written in Chebyshev form. 434 vol. 54 no. 6/2014 Local Velocity Profiles Profile a0 × 10+2 a1 × 10+2 a2 × 10+2 a3 × 10+3 a4 × 10+3 Iyx† δr ave/δr max‡ Radial mean velocity Ur/(ND) z∗ = 0.1114 −0.8872 8.407 −2.716 −2.484 4.457 0.9988 30.7/1459 z∗ = 0.1294 −2.893 6.289 −2.276 −2.649 4.652 0.9987 30.8/1249 z∗ = 0.1474 −4.727 5.097 −1.772 −3.677 3.809 0.998 6.5/22 z∗ = 0.1654 −6.457 3.907 −1.455 −4.148 3.322 0.9957 3.8/10.5 Axial mean velocity Uax/(ND) z∗ = 0.1114 36.37 7.362 0.7714 3.053 4.342 0.9967 0.82/2.2 z∗ = 0.1294 39.74 7.974 −0.6153 8.866 0.1507 0.9975 0.7/1.7 z∗ = 0.1474 41.36 8.17 −1.387 7.645 2.016 0.9953 0.9/4.6 z∗ = 0.1654 42.31 7.801 −2.142 8.041 7.179 0.9975 0.6/3.2 Radial rms fluctuation velocity ur/(ND) z∗ = 0.1114 27.84 −8.426. 1.025 8.843 3.06 0.9983 1/1.8 z∗ = 0.1294 27.08 −7.825 0.1065 8.521 5.46 0.9987 0.73/2.6 z∗ = 0.1474 26.44 −7.295 −0.5566 8.669 3.414 0.9986 0.8/3.8 z∗ = 0.1654 25.5 −6.929 −0.3675 7.561 6.272 0.9969 1.1/2.8 Axial rms fluctuation velocity uax/(ND) z∗ = 0.1114 14.65 −1.313 1.212 0.2547 1.802 0.977 1.14/4.7 z∗ = 0.1294 15.29 −2.063 1.151 1.144 1.028 0.9759 1.8/4.1 z∗ = 0.1474 16.07 −2.347 1.317 0.2177 2.836 0.9828 1.6/4.6 z∗ = 0.1654 16.78 −2.84 1.746 −3.173 1.911 0.994 0.91/6.5 † Correlation index. ‡ Relative error of velocity: average/maximum absolute value. Table 2. Profiles of dimensionless velocity components – coefficients of polynomial approximation in Chebyshev form. 4.1. The profile as a function of the dimensionless radius The velocity profiles were described as a function of di- mensionless radius r∗ in the range r∗ ∈ 〈r∗lower; r ∗ upper〉 using Chebyshev polynomials of the 1st order (see [9], based on the original work of Chebyshev [10]), as follows: parameter = 4∑ k=0 akTk(xCH), (11) where ak are coefficients of polynomial approximation in Chebyshev form, consisting of four terms, xCH is the Chebyshev polynomial variable, and Tk(xCH) are Chebyshev polynomials, defined as follows: T0(xCH) = 1, (12) T1(xCH) = xCH, (13) T2(xCH) = 2x2CH − 1, (14) T3(xCH) = 4x3CH − 3xCH, (15) T4(xCH) = 8x4CH − 8x 2 CH + 1. (16) The four terms of polynomial approximation were found to be sufficient for a quality description of the velocity profiles in this region. The Chebyshev poly- nomial variable xCH was calculated as follows: xCH = 2r∗ − (r∗lower + r ∗ upper) r∗upper −r∗lower , (17) where r∗ is dimensionless radius in a given point de- fined as the ratio of radius r in a given point and tank radius T/2, r∗lower and r ∗ upper are the lower and upper limit values of the dimensionless radius. The evalu- ated parameters of the Chebyshev polynomials are presented in Table 2. A comparison of the averaged velocity profiles and the regression polynomials is pre- sented in Figures 6–9 for all four horizontal positions. Extremely high relative error values obtained for the dimensionless radial mean velocity were observed for values close to zero. For faster calculation of the velocity in a given point, Eq. (11) can be rewritten into the following form: parameter = b0 + b1xCH + b2x2CH + b3x 3 CH + b4x 4 CH, (18) where b0 = a0 −a2 + a4, (19) b1 = a1 − 3a3, (20) b2 = 2a2 − 8a4, (21) b3 = 4a3, (22) b4 = 8a4. (23) 4.2. Intensity of turbulence The axial turbulence intensity was calculated as fol- lows: TIax = Uax/uax, (24) 435 R. Šulc, V. Pešava, P. Ditl Acta Polytechnica Figure 6. Dimensionless radial mean velocity profile Ur/(N D) = f (r∗). Figure 7. Dimensionless radial fluctuation velocity profile Uax/(N D) = f (r∗). Figure 8. Dimensionless axial mean velocity profile ur/(N D) = f (r∗). Figure 9. Dimensionless axial fluctuation velocity profile uax/(N D) = f (r∗). 436 vol. 54 no. 6/2014 Local Velocity Profiles Hypothesis: parameter = BN0 Percentage; t-characteristics |t| Parameter Profile z∗ = 0.1114 z∗ = 0.1294 z∗ = 0.1474 z∗ = 0.1654 acceptable 96.2 %; 1.91 100 %; 1.2 92.4 %; 1.64 93.7 %; 1.6 not acceptable 3.8 %; 207 0.00 % 7.6 %; 183 6.3 %; 28.1 Table 3. Axial turbulence intensity – effect of impeller speed. Profile a0 × 10+2 a1 × 10+2 a2 × 10+2 a3 × 10+3 a4 × 10+3 Iyx† δr ave/δr max‡ TIax z∗ = 0.1114 41.64 −12.34 3.817 −4.749 1.637 0.9928 1.8/5.3 z∗ = 0.1294 40.06 −13.94 5.261 −12.62 6.243 0.9933 2.2/4.4 z∗ = 0.1474 40.49 −14.58 6.237 −14.92 7.515 0.9968 1.7/5.4 z∗ = 0.1654 41.37 −15.59 7.88 −24.23 1.472 0.9981 1.2/10.1 Area 40.89 −14.11 5.799 −14.13 4.217 0.9979 1.3/4.6 † Correlation index. ‡ Relative error of velocity: average/maximum absolute value. Table 4. Profiles of axial turbulence intensity – coefficients of polynomial approximation in Chebyshev form. Figure 10. Axial turbulence intensity profile TIax = f (r∗). where uax is axial rms fluctuation velocity, and Uax is axial mean velocity. For dimensionless velocities independent of the impeller rotational speed, the tur- bulence intensity should also be independent from the impeller rotational speed. The independence of dimensionless velocities from impeller speed was again tested by hypothesis testing. The t-distribution coeffi- cient tm−2,α for three impeller rotational speeds and significance level α = 0.05 is 12.706. Hypothesis test- ing was performed for each point in the investigated profiles. The hypothesis test results for each profile are presented in Table 3. The tested hypothesis can be accepted in almost all profile points, as is again signalized by the low calculated |t| values. On the basis of these hypothesis test results, we assume that the axial turbulence intensity can be sta- tistically taken as constant and independent of the impeller rotational speed, as expected. The radial pro- files of the axial turbulence intensity were averaged, and were expressed by the polynomial approximation in Chebyshev form. The evaluated coefficients are presented in Table 4. The radial profiles are presented in Fig. 10 for given horizontal positions. As is shown, the calculated values are in the range from 0.293 to 0.667. These values correspond to high turbulence in- tensity. As expected, the highest turbulence intensity was found close to the impeller axis in the ascending flow core. The calculated axial turbulence intensity values were found to be approximately the same in each horizontal profile, see Fig. 10. The effect of the dimensionless profile height on the turbulence inten- sity was therefore tested by hypothesis testing for each 437 R. Šulc, V. Pešava, P. Ditl Acta Polytechnica point in the investigated profiles. The t-distribution coefficient tm−2,α for three impeller rotational speeds and significance level α = 0.05 is 4.3027. The percent- age of points in which the hypothesis TIax = const is satisfied was found to be 73.4 %, and the average cal- culated |t| value was 1.6. The percentage of points in which the hypothesis TIax = const cannot be accepted was found to be 26.6 %, and the average calculated |t| value was 8.41. On the basis of this hypothesis test result, we as- sume that the axial turbulence intensity can be sta- tistically taken as constant and independent from the dimensionless profile height in the investigated area. The radial profiles in four horizontal planes were aver- aged and were expressed by the polynomial approxi- mation in Chebyshev form. The evaluated coefficients denoted as “area” are presented in Table 4. 5. Conclusions The following results have been obtained: (1.) The hydrodynamics and the flow field were mea- sured in a vessel 300 mm in inner diameter agitated by a Rushton turbine using 2-D Time Resolved Par- ticle Image Velocimetry (2-D TR PIV). The velocity fields were measured in the zone in upward flow to the impeller for three impeller rotation speeds: 300, 450 and 600 rpm, corresponding to a Reynolds num- ber in the range 50000 < Re < 100000. (2.) The dimensionless radial mean velocities were found to be close to zero. These findings correspond to the characteristics of the given zone according to Fořt et al. (1982). This region contains predomi- nantly ascending flow along the vessel axis towards the impeller. (3.) 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 rotational speed. Consequently, the velocity profiles were averaged and were expressed by Chebyshev polynomials of the 1st order. (4.) Because the investigated area is relatively far from the impeller and outside the impeller discharge flow, we expected a local state of isotropy defined on the length-scale level corresponding to the integral length scale by equality of the fluctuation velocity components. This expectation was not confirmed. The ratio of the axial rms fluctuation velocity to the radial component was found to be in the range from 0.523 to 0.768. This state will affect the deter- mination of the turbulent energy dissipation rate in a given region. (5.) The axial turbulence intensity was calculated and was found to be in the range from 0.293 to 0.667, which corresponds to high 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 rotational speed. The radial profiles of the axial turbulence intensity were averaged and were expressed by Chebyshev polynomials of the 1st order. The calculated values for axial turbulence inten- sity were found to be approximately the same in each horizontal profile. The effect of dimensionless profile height on turbulence intensity was therefore tested by hypothesis testing for each point in the investigated profiles. It was found that the axial turbulence intensity can be characterized using a single formula in the investigated area. Acknowledgements This research has been supported by Grant Agency of the Czech Republic project No. 101/12/2274 “Local rate of turbulent energy dissipation in agitated reac- tors & bioreactors” and by CTU in Prague project No. SGS14/061/OHK2/1T/12. References [1] Drbohlav, J., Fořt, I., Máca, K., Ptáček, J.: Turbulent characteristics of discharge flow from the turbine impeller. Coll. Czechoslov. Chem. Comm., 1978, Vol. 43, pp. 3148-3162 [2] Drbohlav, J., Fořt, I., Krátký, J.: Turbine impeller as a tangential cylindrical jet. Coll. Czechoslov. Chem. Comm., 1978, Vol. 43, pp. 696-712 [3] Obeid, A., Fořt, I., Bertrand, J.: Hydrodynamic characteristics of flow in systems with turbine impellers. Coll. Czechoslov. Chem. Comm., 1983, Vol. 48, pp. 568-577 [4] Fořt, I., Obeid, A., Březina, V.: Flow of liquid in a cylindrical vessel with a turbine impeller and radial baffles. Coll. Czechoslov. Chem. Comm., 1982, Vol. 47, pp. 226-239 [5] Kysela, B., Konfršt, J., Fořt, I., Kotek, M., Chára, Z.: Study of the turbulent flow structure around a standard Rushton impeller. Chem. and Process Eng., 2014, Vol. 35, No. 1, pp. 137-147. DOI: 10.2478/cpe-2014-0010. [6] Novák, V., Rieger, F., Vavro, K.: Hydraulické pochody v chemickém a potravinářském průmyslu, SNTL, Praha, 1989 [7] Kotek, M., Pešava, V., Kopecký, V., Jašíková, D., Kysela, B.: PIV measurement in a vessel of D = 0.3 m agitated by Rushton turbine. Research report for project No. 101/12/2274. Liberec, 2012 [8] Bowerman, B.L., O’Connell, R.T.: Applied statistics: improving business processes. Richard D. Irwin, USA, 1997, ISBN 0-256-19386-X [9] Rivlin, T.J.: The Chebyshev polynomials. Pure and Applied Mathematics, Willey, New York, 1974 [10] Chebyshev, P.L.: Théorie des mécanismes connus sous le nom parallélogrammes. Mémoires des savants étrangers présentés a l´Académie de Saint-Pétersbourg, 1854, Vol. 7, pp. 539-586 438 Acta Polytechnica 54(6):430–438, 2014 1 Introduction 2 Theoretical background 2.1 Inspection analysis of flow in an agitated vessel 2.2 Mean and fluctuation velocity 3 Experimental 4 Experimental data evaluation 4.1 The profile as a function of the dimensionless radius 4.2 Intensity of turbulence 5 Conclusions Acknowledgements References