Radek Šulc

Estimating the local turbulent energy dissipation rate using 2-D PIV measurements and a 1-D energy spectrum function

A simple method for estimating the local turbulent energy dissipation rate for a non-isotropic state is proposed. The proposed method is based on an estimation of the isotropic dissipation rate under the assumption of local isotropy, which is subsequently corrected for the actual conditions. The level of anisotropy is characterized by the fluctuation velocity components. The isotropic dissipation rates are obtained by energy spectrum function fitting. A data block averaging technique is used to smooth the spectrum. The effect of the data number within block on the calculated turbulent energy dissipation rate is taken into account. The proposed method has been tested on data obtained by a 2-D time-resolved PIV method. The effect of the spatial resolution of PIV on the estimation of the dissipation rate is also taken into account, using the correction proposed by Delafosse et al. (2011). The estimate of the local turbulent energy dissipation was found to be the same irrespective of the fluctuation velocity component that was taken for calculating the energy spectrum. This accords with the scalar character of the dissipation rate. The method also enables an estimate to be made of the integral length scale components.

Local turbulent energy dissipation rate in an agitated vessel: New approach to dimensionless definition

Using theory of turbulence, particularly using turbulence spectrum analysis, the relations ε* = ε/(u4/ν) = const., vK/u = const. and Λ/ηK = const. were derived. Assuming that u ∝ (Nd) from this it follows that the widely used dimensionless local turbulent energy dissipation rate defined as ε/((N3d2) is directly proportional to impeller Reynolds number, i.e. ε/((N3d2) ∝ Re, and length scale ratio Λ/d is indirectly proportional to impeller Reynolds number, i.e. Λ/d ∝ Re–1, in an agitated vessel at high Reynolds number. The relations obtained by turbulence spectrum analysis were used for estimation of local turbulent energy dissipation rates experimentally measured by Ståhl Wernersson and Trägårdh (1998, 1999) covering the range of Re = 87600–910200 and own experimental data covering the range of Re = 50000–189000. The experiments have been performed in tanks of 300 mm and 400 mm in the inner diameter for three different viscosities and for various impeller rotational speeds.

LOCAL VELOCITY PROFILES MEASURED BY PIV IN AN VESSEL AGITATED BY RUSHTON TURBINE

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 50 000 < Re < 100 000, which means that the 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 1 st 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.

The effect of process conditions on the flocculation process occurring in an agitated vessel

The effect of flocculation time, flocculant dosage, pollutant concentration, mixing intensity and tank size on flocculation performance was experimentally studied in a mechanically agitated vessel. The macroscopic approach was applied for fl occulation modelling. The simple semi-empirical generalized correlation for flocculation kinetics proposed by Šulc (2003a) and the simple semi-empirical generalized correlation quantifying the effect of flocculation time and flocculant dosage on fl occulation proposed by Šulc, Ditl (2007) were used for data treatment.

Local Turbulent Energy Dissipation Rate in an Agitated Vessel: Experimental and Turbulence Scaling

The hydrodynamics and the flow field in an agitated vessel were measured using 2-D time resolved particle image velocimetry (2-D TR PIV). The experiments were carried out in fully baffled cylindrical flat bottom vessels 300 and 400 mm in inner diameter. The 300 mm inner diameter tank was agitated by a Rushton turbine 100 mm in diameter, and the 400 mm inner diameter tank was agitated by a Rushton turbine 133 mm in diameter. Three liquids of different viscosities were used as the agitated liquid: (i) distilled water (ν = 9.35 × 10–7 m2/s), (ii) a 28 vol % aqueous solution of glycol (ν = 2 × 10–6 m2/s), and (iii) a 43 vol % aqueous solution of glycol (ν = 3 × 10–6 m2/s). The velocity fields were measured at an impeller rotation speed in the range from 300 to 850 rpm, which covers the Reynolds number range from 50000 to 189000. This means that fullydeveloped turbulent flow was reached. The experiments were performed to investigate the applicability of the following relations: ε* = ε/(u4/ν) = const, vK/u = const, Λ/ηK = const, τΛ/τK = const, ε* = ε/((Nd)4/ν) = const, Λ/d ∝ Re–1, ηK/d ∝ Re–1, vK/(Nd) = const, NτΛ ∝ R–1, NτK ∝ Re–1, and ε/(Nq) ∝ Re. These formulas were theoretically derived in our previous work, using turbulence theory, in particular, using turbulence spectrum analysis. The correctness of the proposed relations is investigated by statistical hypothesis testing.

Local velocity scaling in an impeller discharge flow in T400 vessel agitated by tooth impeller in a fully turbulent region

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 tooth impeller 133 mm in diameter. The velocity fields were measured in the impeller discharge flow for impeller rotation speeds from 300 rpm to 700 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 68 000 < Re < 221 000. 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 radial turbulence intensity was found to be in the majority in the range from 0.3 to 0.9, which corresponds to the high level of this quantity.

Time evolution of the drop size distribution for liquid–liquid dispersion in an agitated tank

Agitating two immiscible liquids or a solid–liquid suspension is an operation frequently performed in the chemical and metallurgical industries, for example, in suspension/emulsion polymerization, heterogeneous/phase-transfer catalytic chemical reactions, and hydrometallurgical solvent extraction. For emulsification, suspension polymerization, solid particle dispersion, and crystallization, it is essential to be able to predict the mean drop/particle size and the drop/particle size distribution. A simple model was proposed for predicting the time evolution of drop size distribution during drop breaking, and was successfully tested on data published by Ruiz and Padilla (Hydrometallurgy 72:245–258, 2004) and by Sathyagal et al. (Chem Eng Sci 51: 1377–1391, 1996) and on our own data. The time evolution of DSD was investigated in a baffled tank agitated by a Rushton turbine for a liquid–liquid dispersion. The tests were carried out on a silicone oil–water dispersion (oil in water) with a dispersed-phase fraction of 0.00047. The drop sizes were determined by image analysis.