THE DETAILS OF THE TURBULENT FLOW FIELD IN THE VICINITY OF A RUSHTON TURBINE

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1 14 th European Conference on Mixing Warszawa, September 2012 THE DETAILS OF THE TURBULENT FLOW FIELD IN THE VICINITY OF A RUSHTON TURBINE Harry E.A. Van den Akker Kramers Laboratorium, Dept. of Multi-Scale Physics, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, Netherlands h.e.a.vandenakker@tudelft.nl Abstract. Discrepancies between DNS results and experimental data on k and ϵ values in the near vicinity of a Rushton turbine have been investigated. To this end, new data have been extracted from the existing DNS database. Differences in Reynolds number, in associated differences in position of trailing vortices and turbulence levels, and in limitations in spatial resolution may explain the discrepancies under investigation. Keywords: DNS, turbulent kinetic energy, energy dissipation rate, Rushton, LDA, PIV 1. INTRODUCTION Whenever physical operations such as dispersing gases and liquids as well as chemical reactions are carried out in stirred vessels under turbulent-flow conditions, power draw via the revolving impeller is a very important and often only available operating variable. Several typical quantities such as average bubble or drop size, or selectivity in the case of multiple reactions, are dependent on power draw P. Upscaling or downscaling is often done on the basis of keeping the specific power draw ϵ (in W/kg) constant, where ϵ = P/ρV, with ρ denoting liquid density, and V the volume of the stirred liquid. Under steady-state conditions, turbulence theory teaches that the power drawn at the impeller (blade) scale is continuously dissipated into heat at the smallest turbulent time and length scales, i.e. in the so-called Kolmogorov eddies. Specific power draw therefore also denotes the rate at which the turbulent kinetic energy k (in m 2 /s 2, of J/kg) is dissipated. The Kolmogorov length scale η depends on the liquid s kinematic viscosity ν and the local ϵ, according to η = (ν 3 /ϵ) ¼. The larger the Reynolds (Re) number, the smaller the Kolmogorov eddies, the wider the spectrum of turbulent eddies. In stirred vessels, turbulence is most intense in the near vicinity of the impeller where the turbulence is created. Many of the physical operations or chemical processes in stirred vessels are sensitive to the spatial distributions of turbulence levels. Usually, this fact of life is expressed in terms of spatial fields of k and ϵ with maximum values in the near vicinity of the impeller blades. In the case of a Rushton turbine, an important role is played by the trailing vortices shed away from the corners of the blades, with opposite direction of rotation. The centrifugal forces imposed by the blades induce a radially outward flow which advects these vortices and the accompanying turbulence away from the impeller swept domain. As a result, there is vivid interest in the field of mixing and stirring as to k and ϵ. Computational Fluid Dynamics (CFD) might be instrumental in reproducing their spatial distributions. 485

2 The parameters k and ϵ figure predominantly in the so-called k-ϵ turbulence model which is widely used in CFD software for simulating turbulent flow fields. The k-ϵ model makes part of the family of Reynolds Averaged Navier-Stokes (RANS) based CFD simulations. Starting point of the RANS concept is a clear separation of scales: the time scales of the turbulent fluctuations, or eddies, are very different from those of the average velocity field and the periodic blade passages. The idea behind RANS based simulations is to reproduce the average flow field owing to a proper model for the dispersing action of the turbulent eddies. When and where the turbulent flow is isotropic, the effective transport of momentum in any direction due to the action of the turbulent eddies is modelled by means of a single eddy viscosity. In any k-ϵ model, this eddy viscosity is calculated from local values of k and ϵ which are calculated by means of additional transport equations. In RANS based CFD simulations, turbulence levels are usually underpredicted to a substantial degree. This is not very helpful when dealing with operations or processes which are highly dependent on turbulence levels. It is therefore tempting to take refuge to more sophisticated CFD techniques such as Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS). Both in LES and DNS, k and ϵ and their spatial distributions, however, are neither needed nor well defined. In fact, k and ϵ can be estimated or calculated a posteriori from turbulent flow fields produced by LES and DNS, in a similar way as done from experimental data obtained by Laser Doppler Anemometry (LDA) or Particle Image Velocimetry (PIV). In the pertinent data processing, the resolved and/or (partly) unresolved velocity fields have to be translated such as to satisfy the separation of time scales underlying the concepts of k and ϵ. In addition, both LDA and PIV suffer from (severe) limitations in spatial resolution and in the number of measurement positions. This paper addresses the reliability of a DNS carried out for a stirred vessel, provided with a Rushton turbine, at an impeller Reynolds number of Details about this DNS can be found elsewhere [1]. The focus of this paper is on local k and ϵ values as determined from this DNS flow field and on an assessment of these data with the help of k and ϵ values derived from the best experimental velocity data available [2-6]. 2. SPATIAL RESOLUTION OF OUR DNS A big advantage of DNS over RANS and LES is that no models are invoked in solving the flow equations, i.e. the turbulent flow field is resolved completely. The major restriction on the validity of the outcome of a DNS is related to the ratio of η to Δ, where Δ denotes the size of the computational grid cells. Several authors (e.g. [7], [8]) found that η/δ>1 is not strictly required to accomplish complete resolution of a turbulent flow field. In the current DNS, the most critical location is at the boundary of the impeller swept domain. In a plane 15 behind an impeller blade (ϕ=π/4, see Figure 2) η/δ values as low as 0.3 are found which could be indicative of an insufficient spatial resolution. Figure 3, however, shows the average radial velocity component and the pertinent rms value of the radial velocity as a function of angular position ϕ slightly outside the impeller domain (x = 0, r/t = 0.20 see Figure 1). It is obvious that the profiles are very smooth; in addition, the rms is maximum where the slope of the average radial velocity is positive. These profiles therefore are rather reassuring as to the spatial resolution of the simulation. The profiles of Figure 3 compare favourably with the typical velocity profiles (averaged and rms, measurement position unspecified) obtained by LDA [9]: see Figure 4. The LDA profiles exhibit a non-smooth appearance, probably due to a (too) low sampling rate. Another aspect to be taken into account is that the DNS of interest was carried out by means of a Lattice Boltzmann technique which requires an orthogonal grid. While the cylindrical vessel is positioned within this grid, the usual bounce back rule is applied to 486

3 satisfy the no-slip condition at the stationary and revolving curved vessel and impeller walls, while the usual forcing technique is applied to impose the revolving motion of the immersed blades. Locally, this could give rise to minor irregularities or inaccuracies. For the rest, the DNS of Gillissen and Van den Akker [1] should be reliable anyhow. Figure 1. Impeller geometry and position of planes and measuring points [1] Figure 2. Ratio η/δ in a plane 15 behind an impeller blade (ϕ=π/4, see Figure 1) Figure 3. Non-dimensional radial velocity profiles vs angular position ϕ at x = 0, r/t = 0.20; left: average velocity; right: rms. - extracted from our DNS data Figure 4. Typical radial velocity profiles vs angular position ϕ (measurement position not specified) obtained by LDA; left: average velocity; right: rms. - reproduced from [9] 487

4 3. ASSESSMENT OF k DATA FROM DNS Gillissen & Van den Akker [1] compared local k values derived from their DNS with local k values derived from experimental velocity data due to both Schaefer [2] and Lee & Yianneskis [3]. This comparative study is extended here. First of all, it is noted that both our DNS and Schaefer s LDA data relate to Re = At this Re, however, the flow is not yet fully turbulent. The LDA data of [3] relate to fully turbulent flow conditions (Re = 40,000). Figure 5 (left) from [1] presents k-values derived from our DNS in excess of k-values based on Schaefer s LDA data. Such a comparison at a fixed position, however, starts from the assumption that the overall, or averaged, flow field is the same in the DNS and the experiments. Schaefer s data - see Figure 8 in [1] - show the velocity field in the ϕ=π/4 plane being rather asymmetric with respect to the plane of the impeller disk, while the cores of the two trailing vortices in that same plane are not on a vertical line. This is different from the symmetric velocity field found via our DNS. The reason for the lack of symmetry in Schaefer s data is unclear. As a result, it may not be very surprising that also k in Figure 5 (left) exhibits rather distinct values. Figure 5. Phase resolved profiles of non-dimensionalised k; dots denote k-values from LDA data due to Schaefer [2]; solid lines: from DNS [1]; dashed line: from LES [1]; left: as a function of x/t, at the position specified; right: as a function of ϕ, at the position specified. Figure 6. Phase averaged profiles of k vs vertical position with respect to impeller disk level (x=0), for the radial positions indicated; dots denote k-values from LDA data due to Schaefer [2]; solid lines: from DNS [1]; dashed line: from LES [1] 488

5 Figure 5 (right) presents phase resolved k-values derived from both DNS and LDA data at the position indicated. At this position, the k-levels are quite comparable. This may be indicative of the potential of the DNS technique for correctly predicting turbulence levels. A similar conclusion may be drawn from the phase averaged k-profiles for two different radial positions presented in Figure 6. Similar data for r/t=0.175 have been presented in [1]. The discrepancies between k-values derived from DNS and LDA data decrease with increasing r. While the k-values based on Schaefer s LDA data exhibit asymmetrical positions of the trailing vortices, the DNS based data are indicative of an almost perfect symmetry. The phase averaged k-values at r/t=0.175 due to Lee & Yianneskis [3] - see Figure 9(a) in [1] - are lower than Schaefer s at the same radial position. This again may be due to a different averaged flow field at Re = 40,000. With increasing Re, the trailing vortices appear to lie radially further outside [3], around r/t=0.2, resulting in lower k-values at r/t= ASSESSMENT OF ϵ DATA FROM DNS Gillissen & Van den Akker [1] also assessed their DNS with the help of ϵ-estimates based on PIV measurements [4, 5]. They found non-dimensionalised phase-averaged maximum ϵ- values of 0.32 at r/t=0.225 on the basis of their DNS (for Re = 7300); their values were lower - by a factor of some 2 to 3 - than ϵ-values derived from LDA data for Re >20,000. Their analysis is extended here as well. Figure 7 (left) presents ϵ data which again show trailing vortices starting well within the impeller swept domain at Re = 7300, while they may be much more at the rim of the impeller swept domain for higher Re [3]. This may explain why our Re = 7300 data result in lower ϵ- values at r/t=0.225 than the PIV based data at much higher Re. The same remark may apply to the higher scaled ϵ-values [6]. The DNS based ϵ-values at r/t=0.17 presented in Figure 7 (right) are higher than those at r/t=0.225; this suggests that values lower than expected might not be caused by the low η/δ ratio at this position (cf. Figure 2). Figure 8 (top) presents one of the ϵ-components vs the radial coordinate, as calculated from our DNS data. The smoothness of this profile once more proves the spatial resolution to be sufficient. It compares favourably with the spread in Figure 8 (bottom) in the same ϵ- component derived from both two-point LDA data [6] and PIV data [4] in spite of all extreme care taken in these studies. Note the data in Figure 8 (bottom) were nondimensionalised differently (resulting in values differing almost exactly a factor 10) and were obtained at higher Re numbers. Figure 7. Non-dimensionalised ϵ data derived from our DNS data; left: phase resolved, in the ϕ=π/4 plane [1]; right: phase averaged at r/t=

6 5. CONCLUSIONS On the basis of a more detailed comparison for more data extracted from our DNS, the conclusion is that the discrepancies between the DNS based k and ϵ values and experimental data from the literature are not caused by a locally insufficient spatial resolution, but may be related to differences in averaged flow fields, mainly due to differences in Re numbers. The smooth curves obtained in and derived from the DNS compare favourably with the noisy curves based on experiments which seem to suffer from inevitable experimental inaccuracies, data processing issues, and modelling uncertainties. ACKNOWLEDGEMENTS The author wishes to thank J.J.J. Gillissen, M.Yianneskis and M.J. Tummers for their help in extracting the data from the DNS databank and for fruitful discussions. 6. REFERENCES i Figure 8. Profiles of one of the ϵ- components vs r/t, made nondimensional (differently); top: from our DNS data; bottom: from experimental data, reproduced from [6] [1] Gillissen J.J.J., Van den Akker H.E.A., DNS of the Turbulent Flow in a Baffled Tank Driven by a Rushton Turbine, AIChE J., published online, DOI: /aic [2] Schaefer M., Hofken M., Durst F., Detailed LDV measurements for visualization of the flow field within a stirred-tank reactor equipped with a Rushton turbine, Chem.Eng.Res.Des.,75, [3] Lee K.C., Yianneskis M., Turbulence Properties of the Impeller Stream of a Rushton Turbine, AIChE J., 44, [4] Baldi S., Yianneskis M., On the quantification of energy dissipation in the impeller stream of a stirred vessel from fluctuating velocity gradient measurements, Chem.Eng.Sci., 59, [5] Escudié R., Liné A., Experimental Analysis of Hydrodynamics in a Radially Agitated Tank, AIChE J., 49, [6] Ducci A., Yianneskis M., Direct Determination of Energy Dissipation in Stirred Vessels with Two- Point LDA, AIChE J., 51, [7] Moin P., Mahesh K., Direct Numerical Simulation: A Tool in Turbulence Research, Annu. Rev. Fluid Mech., 30, [8] Donzis D.A., Yeung P.K., Pekurovsky D., Turbulence Simulations on O(10 4 ) Processors, Proc. TeraGrid 2008 Conference, Las Vegas, NV. [9] Schaefer M., Charakterisierung, Auslegung, und Verbesserung des Makro- und Mikromischens in gerührten Behältern, PhD Thesis (in German), Erlangen, Germany. 490

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