Detailed Simulations of Liquid and Solid-Liquid Mixing

Detailed Simulations of Liquid and Solid-Liquid Mixing Turbulent agitated flow and mass transfer

DETAILED SIMULATIONS OF LIQUID AND SOLID-LIQUID MIXING Turbulent agitated flow and mass transfer

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr ir J. T. Fokkema, voorzitter van het College voor Promoties, in het openbaar te verdedigen op vrijdag 18 november 2005 om 10.30 uur

door

Hugo HARTMANN

natuurkundig ingenieur geboren te Zuidelijke IJsselmeerpolders

Dit proefschrift is goedgekeurd door de promotor: Prof. dr ir H. E. A. van den Akker Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof. dr ir H. E. A. van den Akker, Prof. dr M. Yianneskis, Prof. dr C. D. Rielly, Prof. dr ir P. Wesseling, Prof. dr D. J. E. M. Roekaerts, Dr ir J. J. Derksen, Prof. dr ir M. M. C. G. Warmoeskerken,

Technische Universiteit Delft, promotor King’s College London, Verenigd Koninkrijk Loughborough University, Verenigd Koninkrijk Technische Universiteit Delft Technische Universiteit Delft Technische Universiteit Delft Universiteit Twente

The project ’OPTIMUM’ was financially supported by the Commission of the European Union under the program ’Promoting Competitive and Sustainable Growth’, Contract G1RD-CT-2000-00263. Printed by: Ponsen & Looijen B.V. Nudepark 142 6702 DX Wageningen The Netherlands http://www.p-l.nl ISBN 90-6464-234-6 Keywords: stirred tank, simulations, turbulence, scalar mixing, solids suspension, mass transfer.

c

2005 by Hugo Hartmann. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system without written permission from the publisher.

Zoek uw kracht in de Heer, in de kracht van zijn macht. Efeziërs 6:10 (De Nieuwe Bijbelvertaling)

Aan Mirjam en Léon

Contents

Summary

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Samenvatting

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1 Introduction 1.1 Mixing in the process industries . . . . . . . 1.2 Two mixing applications; OPTIMUM project 1.2.1 Motivation, aim and project partners . 1.2.2 Suspension polymerization . . . . . . 1.2.3 Particle coating . . . . . . . . . . . . 1.3 Mixing and LBM at the Kramers Laboratory . 1.4 Motivation and aim . . . . . . . . . . . . . . 1.5 Stirred tank geometry . . . . . . . . . . . . . 1.6 Outline of this thesis . . . . . . . . . . . . .

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2 Assessment of large eddy simulations on the flow in a stirred tank 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stirred vessel configuration . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Experimental method . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Computational method . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 RANS simulation . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . 2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Behavior of the subgrid scale models . . . . . . . . . . . . . . 2.5.2 Phase-averaged flow field and turbulence levels . . . . . . . . . 2.5.3 Phase-resolved flow field and turbulence levels . . . . . . . . . 2.5.4 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Turbulence anisotropy . . . . . . . . . . . . . . . . . . . . . .

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2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Macro-instability uncovered in a Rushton turbine stirred tank 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Flow phenomena in stirred tanks . . . . . . . . . . . . . . . . 3.1.2 MI significance to multi-phase chemical processes . . . . . . 3.1.3 RANS simulation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . 3.2 Flow system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . 3.3.2 Simulation aspects . . . . . . . . . . . . . . . . . . . . . . . 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Single flow field realizations . . . . . . . . . . . . . . . . . . 3.4.2 Time-series . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Effect of Reynolds number and impeller off-bottom clearance 3.4.5 Pseudo turbulence determination procedure . . . . . . . . . . 3.4.6 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.7 Grid size effects . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Development and validation of a scalar mixing solver based on finite volume discretization 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Finite volume discretization of the convection diffusion equation . . . . . 4.3 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The treatment of convection . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Monotone schemes . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 TVD discretization scheme . . . . . . . . . . . . . . . . . . . . . 4.4.3 TVD multidimensional application . . . . . . . . . . . . . . . . 4.5 The fully discretized convection diffusion equation . . . . . . . . . . . . 4.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Neumann boundary condition implementation in 1-D . . . . . . . 4.6.2 Extension to multiple dimensions . . . . . . . . . . . . . . . . .

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41 44 47 47 47 48 49 50 50 51 51 52 54 54 54 58 60 61 65 67 69 72

75 75 76 77 78 78 80 82 83 85 86 87

4.7

Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Laminar cavity flow in a square box . . . . . . . . . . . . . 4.7.2 Laminar cavity flow in a box with inclined side walls . . . . 4.7.3 Mixing behavior in a cylindrical tank with a side-entry mixer 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 A parameter study of the mixing time in a turbulent stirred tank means of LES 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Experimental research on turbulent scalar mixing . . . . . . . . 5.1.2 Potential of CFD on scalar mixing . . . . . . . . . . . . . . . . 5.2 Flow system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Large eddy simulation . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Scalar transport . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Immersed boundary technique . . . . . . . . . . . . . . . . . . 5.3.4 Simulation aspects . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Snapshots of the scalar concentration field . . . . . . . . . . . . 5.4.2 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Mixing time and coefficient of mixing . . . . . . . . . . . . . . 5.4.4 Mixing time correlations . . . . . . . . . . . . . . . . . . . . . 5.4.5 The Ruszkowski (1994) mixing time correlation . . . . . . . . 5.4.6 Performance immersed boundary technique, mass conservation 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

by 105 . 105 . 105 . 107 . 109 . 109 . 109 . 110 . 112 . 113 . 115 . 115 . 116 . 121 . 125 . 127 . 131 . 134 . 136

6 Numerical simulation of a dissolution process in a stirred tank reactor139 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Flow system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Flow solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Particle transport solver . . . . . . . . . . . . . . . . . . . . . . 144 6.3.3 Mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.3.4 Scalar mixing solver . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3.5 Simulation aspects . . . . . . . . . . . . . . . . . . . . . . . . . 147

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Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Snapshots of the particle distributions and concentration fields 6.4.2 Stages in the dissolution process . . . . . . . . . . . . . . . . 6.4.3 Evolution of particle size distribution in time . . . . . . . . . 6.4.4 Dissolution time . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Time series of concentration . . . . . . . . . . . . . . . . . . 6.4.6 Mass conservation finite volume scheme . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Conclusions and perspectives 7.1 General discussion . . . . . . . . . . . . . 7.2 Turbulent flow phenomena . . . . . . . . . 7.3 Scalar mixing . . . . . . . . . . . . . . . . 7.4 Solid-liquid mixing including mass transfer 7.5 Perspectives and recommendations . . . . .

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Bibliography

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Dankwoord

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List of publications

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About the author

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Summary Mixing operations involving turbulent flow are widely applied in the process industries for bringing two or three phases into intimate contact to achieve inter-phase heat and mass transfer, chemical reactions, etc. Although mixing is only part of the whole process, it often has a significant impact on the product quality, process reliability and economy. An inadequate understanding of mixing costs a few billion dollars per year in the USA alone. Furthermore, the lack of insight in the hydrodynamic phenomena involved and their coupling with chemistry, and heat and mass transfer must be overcome to achieve competitiveness improvement. In the last decades, experiments have contributed significantly to the understanding of the flow phenomena and processes, and to better design and operation methods. However, parameters dominating the smallest scales are still hardly accessible for experimental techniques. A promising alternative gaining more attention in the recent years, is to resolve the relevant turbulence quantities by simulation. This work was part of the European project ’OPTIMUM’ in which experimental and numerical tools have been developed and employed to increase the understanding of a suspension polymerization and a particle coating process. Through addressing this topic at a European level, know-how on the specific chemical reactions and products of the particular processes, fundamental information on hydrodynamic phenomena, and expertise on experimental and numerical techniques were combined. The main aim of the present work arose from the ’OPTIMUM’ project and was formulated as a contribution to reliable and accurate numerical predictions of complex, multi-phase processes. For this purpose, three issues were treated successively: (i) turbulent flow phenomena, (ii) scalar mixing, and (iii) solid-liquid mixing including mass transfer. The flow geometry used throughout this work is a baffled tank equipped with a Rushton turbine operated under turbulent conditions. A detailed and accurate representation of the flow field is the first step toward a successful simulation of a complex, multi-phase process. For this purpose, a large eddy simulation (LES) was used. The flow solver based on the lattice-Boltzmann discretization scheme was critically assessed in order to get confidence in the method. Two subgrid-scale mod-

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els were compared. The focus was on the accuracy of the LES flow field predictions (in terms of velocity, turbulent kinetic energy, energy dissipation, and turbulence anisotropy), and the recovery of coherent velocity fluctuations induced by large-scale precessing vortices (i.e. a so-called macro-instability). The important result from this assessment was that LES predicted the turbulence levels to a high level of accuracy, which contrasts with the underprediction of 50% due to a simulation based on the RANS (Reynolds-Averaged Navier-Stokes) approach. This result is of high importance as e.g. the mixing patterns, the motion of particles, and inter-phase mass transfer are highly affected by the velocity fluctuations. In order to describe the mixing performance of the turbulent flow generated by the impeller or inter-phase mass transfer between the continuous and disperse phases, information of a scalar concentration in the continuous phase is required. Therefore, a solver for the description of scalar transport was developed. An Eulerian approach based on the finite volume method has been followed. Subsequently, attention was paid to the implementation of complexly shaped boundaries which are in general off-grid. The zero-gradient constraint at these walls was imposed by a newly developed immersed boundary technique. The scalar mixing solver was assessed in various laminar and turbulent flow cases. Subsequently, LES including scalar transport have been carried out on a blending process. The focus was on the mixing time as a function of impeller size and injection position of the passive tracer. An important observation was that in the literature there is no standardization of mixing time experiments. In literature, measured or simulated mixing times are often not carefully compared with data and correlations. Our simulations, however, make it possible to relate all data and correlations properly. The predicted mixing times agreed within 30% with experimental results and values obtained from two correlations presented in the literature. According to the simulations, the mixing time scales with the tank over impeller diameter ratio to the power 2.5. The mixing time was not significantly influenced by the position of the feed point. Finally, a particle transport solver was coupled to the flow and scalar mixing solvers to carry out a simulation of a solid-liquid suspension including mass transfer. The particle transport solver is based on a Lagrangian approach in which spherical, solid particles are tracked in the Eulerian flow field through solving the dynamic equations of linear and rotational motion of the particles. Particle-particle and particle-wall collisions were included. The force exerted by the fluid on the particle was fed back to the fluid (two-way coupling). A solubility process was chosen to serve as the benchmark case. The simulation was restricted to a lab-scale vessel with a volume equal to 10 liter. A set of 7 million spherical particles was released in the top part of the vessel. At the moment of release, the local volume fraction amounted to 10%, which corresponds to a vessel averaged volume fraction of

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1%. The particle properties were such that they resemble those of calcium-chloride beads. The focus was on the dispersion of the particles throughout the vessel, the solubility rate in terms of solids and scalar concentration distributions, particle size distributions, and the solubility time. For the particular process considered, various stages were identified, and the solubility time was found to be at most one order of magnitude higher than the time needed to disperse the solids throughout the vessel. Initially, the particles organized in streaky patterns. Due to decreasing inertia of the particles, they were behaving more like fluid tracers. Till the point the particles were dispersed throughout the vessel, the spatial solids and scalar concentration distributions were very inhomogeneous. Subsequently, the distributions were becoming more homogeneous.

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Samenvatting In de procesindustrie wordt uitgebreid gebruik gemaakt van menging om intiem contact tussen twee of drie fasen te bewerkstelligen ten behoeve van stofoverdracht, chemische reacties en warmtetransport. Hoewel menging maar een onderdeel is van het gehele proces, heeft menging een belangrijke invloed op de kwaliteit van het produkt, de betrouwbaarheid van het proces en de winstgevendheid. Onvoldoende inzicht in menging kost alleen al in de Verenigde Staten een aantal miljard dollar per jaar. Daarnaast kan vergroting van het inzicht in de interactie tussen hydrodynamica en chemie, massa- en warmtetransport bijdragen aan de verbetering van de concurrentiepositie. In de laatste eeuw hebben experimenten significant bijgedragen aan het begrip van de hydrodynamica en van de processen, en aan betere ontwerp- en handelingsmethoden. De belangrijke parameters op de kleine schalen zijn echter nog steeds moeilijk te verkrijgen met behulp van experimentele technieken. Een veelbelovend alternatief dat steeds meer aandacht krijgt in de laatste jaren is het verkrijgen van de relevante turbulente grootheden door middel van simulaties. Dit werk maakte deel uit van een Europees project getiteld ’OPTIMUM’ waarin experimentele en numerieke technieken ontwikkeld en aangewend werden om de kennis te verbeteren van een suspensiepolymerisatieproces en van een deeltjescoatingproces. Door dit project uit te voeren op Europees niveau konden knowhow van de specifieke chemische reacties en produkten, fundamentele informatie over de hydrodynamica, en expertise betreffende experimentele en numerieke methoden gecombineerd worden. De belangrijkste doelstelling van dit werk is voortgekomen uit het ’OPTIMUM’ project en was geformuleerd als een bijdrage tot betrouwbare en nauwkeurige numerieke voorspellingen van complexe, meerfaseprocessen. Daartoe zijn drie kwesties achtereenvolgens behandeld: (i) turbulente stromingsfenomenen, (ii) scalaire menging, en (iii) het suspenderen van vaste deeltjes in een vloeistof inclusief stofoverdracht. In dit werk is gebruik gemaakt van een tank met keerschotten voorzien van een Rushton turbine welke opereert onder turbulente condities. Een gedetailleerde en nauwkeurige representatie van het stromingsveld is de eerste stap naar een succesvolle simulatie van complexe, meerfase processen. Hiervoor werd een

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large-eddy simulatie (LES) gebruikt. De stromingscode, gebaseerd op een rooster Boltzmann discretizatieschema, was kritisch beoordeeld om vertrouwen te winnen in de gebruikte methode. Twee subgrid-schaal modellen zijn vergeleken. De focus lag op de nauwkeurigheid van de LES voorspellingen van het stromingsveld (in termen van de snelheid, de turbulente kinetische energie, de energie dissipatie en de turbulente anisotropie), en het oplossen van coherente snelheidsfluctuaties veroorzaakt door grote precesserende vortexen (een zogenaamde macro-instabiliteit). Het belangrijke resultaat van deze studie is dat een LES de turbulente kinetische energie nauwkeurig voorspelt, in tegenstelling tot de 50% onderschatting in een simulatie gebaseerd op de RANS (Reynolds-Averaged Navier-Stokes) benadering. Dit resultaat is van groot belang, omdat bijvoorbeeld de mengpatronen, de beweging van deeltjes, en de stofoverdracht in grote mate beïnvloed worden door de snelheidsfluctuaties. Om bijvoorbeeld het menggedrag van de turbulente stroming gegenereerd door een roerder, of de stofoverdracht tussen de continue en disperse fases te beschrijven, is informatie nodig over de scalaire concentratie in de continue fase. Hiervoor is een code ontwikkeld die scalair transport beschrijft. Een Euleriaanse benadering is gebruikt die gebaseerd is op de eindige volume methode. Vervolgens is er aandacht besteed aan de implementatie van complexe wanden die in het algemeen niet gelijk lopen aan het grid. De nul-gradiënt restrictie op deze wanden is opgelegd door middel van een nieuw ontwikkelde techniek. De code voor scalair transport is getest voor drie eenvoudige gevallen van laminaire en turbulente stromingen. Daarna zijn pas de large-eddy simulaties met scalair transport uitgevoerd voor een mengproces. De focus lag daarbij op de mengtijd als functie van roerdergrootte en injectiepositie van de passieve tracer. Een belangrijke observatie daarbij was dat er geen overeenkomst bestaat in de literatuur over standaard mengtijdexperimenten. Gemeten of gesimuleerde mengtijden worden vaak onzorgvuldig vergeleken met data en correlaties in de literatuur. Onze simulaties zijn geschikt om data en correlaties zorgvuldig te vergelijken. De voorspelde mengtijden kwamen binnen 30% overeen met de experimentele resultaten and waarden verkregen uit twee correlaties in the literatuur. Volgens de simulaties schaalt de mengtijd met de ratio van de tank- en roerderdiameter tot de macht 2.5. De mengtijd hangt niet significant af van de positie van het injectiepunt. Tenslotte is een deeltjestransportcode gekoppeld aan de stromings- en scalair transportcodes om een simulatie uit te voeren van een geroerde vast-vloeibare suspensie met stofoverdracht. De deeltjestransportcode is gebaseerd op een Lagrangiaanse benadering waarin ronde vaste deeltjes gevolgd worden in een Euleriaanse benadering van het stromingsveld door de dynamische vergelijkingen van beweging en rotatie van de deeltjes op te lossen. Deeltje-deeltje en deeltje-wand botsingen zijn opgenomen in de code. De kracht uitgeoefend door de vloeistof op de deeltjes is weer teruggekoppeld naar de vloeistof (two-way

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koppeling). Als toepassing is gekozen voor een oplosproces in een vat op laboratorium schaal met een volume van 10 liter. Een verzameling van 7 miljoen deeltjes is losgelaten in het bovenste gedeelte van het vat. Op het moment van loslaten was de lokale volumefractie gelijk aan 10%, wat overeenkomt met een vat-gemiddelde volumefractie van 1%. De eigenschappen van de deeltjes komen overeen met die van calciumchloride. De simulatie betreft de dispersie van de deeltjes in het vat, de oplossnelheid in termen van tijdsafhankelijke distributies van de deeltjes en de scalaire concentratie, de deeltjesgroottedistributie in de tijd en de oplostijd. Voor het proces dat bestudeerd is, zijn verschillende regimes geïdentificeerd, en de oplostijd was hoogstens een orde van grootte hoger dan de tijd benodigd voor het volledig dispergeren van de deeltjes door het hele vat. Aanvankelijk groeperen de deeltjes zich in streepvormige patronen. Door het afnemen van de deeltjesenertie gingen de deeltjes zich meer gedragen als vloeistoftracers. Tot aan het tijdstip waarop de deeltjes volledig gedispergeerd waren in het vat waren de ruimtelijke deeltjes en scalaire concentratiedistributies erg inhomogeen. Vervolgens werden de distributies meer homogeen.

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1 Introduction

1.1

Mixing in the process industries

The concept of mixing is of high importance in many (chemical) applications. For fluid systems, mixing operations are necessary to mix either miscible fluids (e.g. blending of petroleum products) without chemical reaction or mass transfer, or immiscible fluids (e.g. emulsification) with inter-phase mass transfer. Mixing operations are also widely applied in multiphase contacting processes where they are necessary for bringing two (gas and liquid; liquid and solid; gas and solid) or three phases (gas, liquid and solid) into intimate contact to achieve inter-phase heat and mass transfer, chemical reactions, etc. Mixing tasks are mainly carried out in stirred tanks or static mixers, where the mixing is brought about by revolving impellers (of various shapes and sizes) or internals, respectively. Also bubble columns are frequently applied in view of gas/liquid mixing. Gas-solid mixing is often achieved in a fluidized bed, where a gas passes through a bed of solid particles at a high speed causing a certain degree of fluidization of the bed. As a result, inter-phase mass transfer or segregation of light and heavy particles is achieved. Mixing is a central and essential feature of many processes in the food, pharmaceutical, paper, plastics, ceramics and rubber industries. Industrial applications of mixing are ranging from very simple cases to very complex processes such as polymerization, crystallization, and precipitation. Although in such processes mixing is just part of the whole process, it often has an important impact on product quality, process reliability, and process economy. As a result, the financial investment in both the capital and running costs of mixing processes, when viewed on international scale, is considerable. It has been estimated that the cost to process industries due to an inadequate understanding of mixing is a few billion dollars per year in the USA alone (Tatterson, 1994). Also, rules of thumb for scaling up contacting processes cannot be applied with confidence. This requires expensive pilot tests and/or results in either process failures or too conservative process designs. The major challenges for process industry to improve their competitiveness are short-

2

Chapter 1. Introduction

ening of process development time, enhancement of product quality, waste reduction and usage of existing plants up to full capacity. These challenges are triggered by the desire of industrial companies to make products at the lowest possible cost, and by environmental constraints imposed by national and European laws. For these processes the lack of insight into the hydrodynamic phenomena involved and their coupling with chemistry and heat and mass transfer must be overcome to achieve competitiveness improvement. All this has resulted in the need for more detailed insights in the turbulent flow phenomena and processes and for better design and operation methods. The role of experiments in order to meet this need has shifted over the years. The characterization of the flow in a stirred tank by means of global parameters, such as the power drawn by the impeller (Rushton et al., 1950), the pumping capacity of the impeller, and circulation times within the tank (Holmes et al., 1964), made way for local, mostly optical, flow measurements. Detailed measurement techniques, such as laser Doppler anemometry (LDA; e.g. Schäfer et al., 1997), particle image velocimetry (PIV; e.g. Sharp & Adrian, 2001) and laser induced fluorescence (LIF; e.g. Distelhoff et al., 1997) have great significance in resolving the large-scale as well as small-scale flow structures. Furthermore, parameters that dominate the smallest scales (e.g. energy dissipation rates (Micheletti et al., 2004), spectral information at the micro-scale and shear rates) are becoming more accessible for experimental techniques. A promising route to assess the small scales, which is becoming feasible with the availability of large computational resources, is to resolve the relevant turbulence quantities by simulation. Computational fluid dynamics (CFD) has proven to be a versatile tool for studying a wide range of hydrodynamic phenomena. CFD codes based on finite volume discretization have been introduced in the eighties (FLUENT in 1983 and CFX in 1987). Next to commercial packages, numerous in-house codes exist that are based on existing discretization schemes or novel schemes. One scheme that is gaining more attention in the academic as well as in the industrial world is based on the Lattice-Boltzmann Method (LBM), which is a solver for the Navier-Stokes equation. There is a commercial code (Power FLOW, EXA Corporation) available, that can be used for flow problems related to e.g. the automotive and aerospace industries. The Navier-Stokes solver used in this work is an LBM-based academic research tool. Today, CFD is incapable of solving all flow scales in industrial applications, within a reasonable amount of time and memory space. A direct numerical simulation (DNS) of the turbulent flow at industrially relevant Reynolds numbers (Re≥ 104 ) would require an enormous amount of grid cells and time steps to capture all relevant time and length scales in the flow. In a large eddy simulation (LES), the smallest scales are modeled with a subgrid-scale model, thereby reducing the amount of computational resources. While a

1.2. Two mixing applications; OPTIMUM project

3

LES is still computationally expensive, attention to this method is increasing in the last years. Nowadays, most CFD codes solve the Reynolds-Averaged Navier-Stokes (RANS) equations in combination with a closure model for the Reynolds stresses, which compared to a LES or DNS is computationally less expensive. However, turbulence models in a RANS approach rely on empirical parameters obtained in standard flow problems (e.g. turbulent channel flow), and therefore do not perform that well in more complicated flow problems. Furthermore, there is no clear distinction between the part of fluctuations that is explicitly resolved and the part that is taken into account by a turbulence model. The available experimental and numerical techniques of today are of great help to broaden our understanding of the flow phenomena, in order to improve process scale-up, process design, product quality, etc. The drive toward larger degrees of sustainability in the process industries has urged for lower amounts of solvents and for higher yields and higher selectivities in chemical reactors. These goals are often achieved via cooperation of industrial companies and universities in European projects. The next section focuses on the EU project ’OPTIMUM’ in which experimental and numerical tools have been employed to increase the understanding of a suspension polymerization process and a particle coating process.

1.2

Two mixing applications; OPTIMUM project

This research was part of a European project called OPTIMUM (OPTimization of Industrial MUlti-phase Mixing), funded by the European Community under the ’Competitive and Sustainable Growth’ Program. The project start date was December 1, 2000, and the closure date was November 30, 2003. The total budget amounted to approximately 4 million euros, of which 10.5% was on the account of TU Delft. In the subsections below, motivation and aim of the project are described, together with the two chemical processes under consideration. This description is somewhat concise, because of confidentiality imposed by the industrial partners. A significant research effort has been made in this project, and my relevant contributions will be briefly addressed in the next subsections.

1.2.1 Motivation, aim and project partners Chemical manufacturing currently constitutes one of the major European industries on which many communities depend economically. A trend to increase chemical production capacities in non-European countries with lower costs (less strict protection of environ-

4


ment, labor costs) is witnessed. Therefore, sustained innovation of products and processes is absolutely necessary to keep Europe a chemical manufacturing region of significance. This is the main reason why the OPTIMUM project was initiated. Recent progress in CFD research and development brought considerable progress in numerical fluid dynamics and reaction engineering, to the effect that complex multi-phase mixing and reaction processes come within reach of predictive computations. Therefore, major parts of conventional process design methods can be replaced by predictive computations, which in turn result in optimized processes and in decreasing development time and costs. The reliability of the computations need to be verified and validated against detailed experimental data. The purpose of the project is the development of computer tools especially for multi-phase mixing. This ambitious goal requires know-how on the specific chemical reactions and products of the particular processes under investigation, fundamental information on hydrodynamic phenomena (e.g. turbulence modeling, micro-mixing, break-up and coalescence models) and expertise on experimental measuring techniques, and a software company that integrates the information into a CFD code. Therefore, this project was addressed at European level. Two chemical processes, a polymerization process forming a solid granular product (suspension polymerization), and a process for making coated particles of various functional properties (particle coating), have been selected for the OPTIMUM project for a number of reasons that are summarized below. • Suitability: Both processes are highly complex, multi-phase and mixing sensitive. • Generalization: Each process constitutes a separate class of problems accessible to CFD for multi-phase flow, giving the project the broadness needed for application to other processes. • Socioeconomics: The processes are mature and already employed in large scale European production facilities. Scaling-up difficulties and cost effectiveness hamper new investment in large European production facilities. Consequently, secure and growing employment and waste product reduction are important issues in the project. • Potential: New products developed, particularly nano-materials, can be produced by particle coating. The computer tools developed within the project will greatly facilitate research and contribute to the European technical progress. The seven partners with complementary expertise from three EU member states form the consortium:


5

• Merck KgaA, Germany; Particle coating process. • Dow Benelux NV, The Netherlands; Suspension polymerization process. • INVENT Umwelt- und Verfahrenstechnik GmbH & Co KG, Germany; Mixing technology. • AEA Technology, United Kingdom (Today merged with Ansys Inc., USA); CFX code supplier. • TU Delft, The Netherlands; Modeling expertise, experience large eddy simulations. • LSTM Erlangen, Germany; Experimental and numerical competencies on stirred tank flows. • King’s College London, United Kingdom; Expertise in design and execution of experiments in a large range of complex engineering processes. A detailed study on turbulence models employed in the CFD codes is an important issue, since the simulation accuracy of multi-phase flows will heavily rely on correct turbulence parameter prediction. The flow agitated in a stirred tank is inherently timedependent, due to the motion of the impeller. The existing engineering turbulence models implicitly assume steady-state behavior, and are notorious for their underprediction of the turbulence kinetic energy. Therefore, these RANS models (e.g. k − ) need to be evaluated through experimental data and with respect to numerical results based on large eddy simulations. The large eddy simulations performed within this context are described in chapters 2 and 3. In the next two subsections, a brief description is given of the two processes considered in the OPTIMUM project.

1.2.2 Suspension polymerization The Dow company is one of the major engineering plastics manufacturers. The suspension polymerization process of styrene is employed in the production of one of the plastics products (e.g. insulation foam, consumer electronics, rigid packaging, food service disposables). Today, the major market opportunity growth is in Asia, and consequently the competition of this market segment from Asian manufacturers is increasing rapidly. In order to withstand this competition, one needs to utilize existing facilities in Europe better to increase the profitability and capacity of existing installations. As a result, the OPTIMUM

6


project aims at a more competitive process, thereby enabling opportunities to expand production capacity in Europe. Suspension polymerization of styrene is a batch process in which monomer(s), relatively insoluble in water, is (are) dispersed as liquid droplets with steric stabilizer or the so-called Pickering stabilizer and under vigorous stirring (which is maintained during polymerization) to produce polymer particles as a dispersed solid phase. The suspension of monomer in water has volume fractions up to about 50%. The turbulence generated by the revolving impeller induces break-up and coalescence of droplets, and consequently the formation of a droplet size distribution. The rate-limiting break-up and coalescence processes take place at the small scales (comparable to the droplet size), and are controlled by the rate at which turbulent kinetic energy is dissipated. Suspension stabilizers are used in order to prevent uncontrolled coalescence in the final stage of the process and to control the particle size distribution. The process takes place in a jacketed stirred vessel. At the start of the process an initiator is added. The initiator molecules decompose into free radicals, which start the polymerization process. During polymerization the monomer is slowly converted into polymer. As a result, the viscosity in the disperse phase increases. The process can globally be divided into three stages. The first stage is called the low-viscosity stage with relatively small monomer/polymer droplets. With increasing conversion in the dispersed phase, the viscosity increases sharply (caused by the so-called Trommsdorff- or gel-effect), the properties of the inter-facial layer and the density change slightly, causing a change in equilibrium of breakage and coalescence toward coalescence. The particle size distribution shifts to a broader distribution with larger particle size. This stage is called the sticky phase. If the suspension remains stable (due to suspension stabilizers), the particle size distribution reaches its final state, which is called the identity point. A schematic picture of the vessel geometry, accompanied by an averaged velocity field in the baffle plane (obtained by an LES computation), and a schematic course of the Sauter mean diameter in time (including the three stages mentioned) is shown in Figure 1.1. The average velocity field shows that, because of the low mounting position of the impeller, one large and one small recirculation zone is created above and below the impeller, respectively. The impeller pumps in a radial direction. The particle size distribution in the suspension was modeled based on a population balance approach (according to Ramkrishna (1985)). Initially, the evolution (in space and time) of the particle size distribution was based on two parameters, the degree of polymerization and the particle size. The characteristic response times for heat transfer phenomena in the tank have been analyzed and presented by Hartmann (2001) in a confidential report. Main conclusion from that report was that the process time scales are an order of


7

Sauter diameter

(b)

‘low viscosity stage’

identity point ‘sticky’ stage

final stage

time (a)

(c)

Figure 1.1: (a) Schematic representation of the vessel geometry with styrene droplets suspended in water with stabilizer. The vessel has a heated jacket to regulate the temperature of the contents. (b) Averaged flow field (LES) in the vertical baffle plane, the well-mixed region shown is depicted in (a) with a dashed box. (c) Schematic time trace of the Sauter mean diameter, including the three stages.

magnitude larger than the particle response times. Consequently, effects of non-uniform temperatures due to heating up of the process could be ignored. Since the degree of polymerization (i.e. conversion) is directly related to the temperature, it could be assumed uniform throughout the reactor vessel. As a result, the population balance equations describe the evolution (in space and time) of a single parameter being the particle diameter. The population balances are conservation equations which account for transport of

8


droplets into and out of the reactor volume, and generation and disappearance of droplets through break-up and coalescence processes. The population balance model is in the CFD code CFX often referred to as the MUSIG (MUltiple SIze Group) model. Coalescence and break-up correlations are required to provide the rates at which the droplets change size during the course of the simulation. The popular break-up model of Luo & Svendsen (1996) and coalescence model of Prince & Blanch (1990) are valid for gas-liquid systems (bubbly flows). Therefore, they are not not applicable for liquid-liquid dispersions such as the suspension polymerization process. The break-up and coalescence correlations highly depend on the rheology of the dispersed phase. Two models have been investigated within this project (Hartmann, 2002b). The first model is introduced by Vivaldo-Lima et al. (1998), and assumes the rheological properties of the dispersed phase (styrene/polystyrene mixture) to be similar to a powerlaw fluid. The other model has been introduced by Alvarez et al. (1994) and Maggioris et al. (2000), and assumes the fluid of the dispersed phase to behave as a Maxwell fluid with visco-elastic properties. From a confidential report of Hartmann (2002b), the following observations based on these two models are summarized. • The power-law model is poorly documented, and Vivaldo-Lima et al. (1998) use a power-law index equal to one. As a result, they model the disperse phase as a Newtonian fluid. • The power-law model has 10 adjustable constants. • Visco-elastic effects are discarded in the power-law model. • The Maxwell model is poorly documented by Alvarez et al. (1994) and Maggioris et al. (2000), the papers containing numerous typographical errors in the equations. Most of these errors and other inconsistencies were solved in the report (Hartmann, 2002b), partly through a personal communication with Kotoulas, a PhD successor of Maggioris. • The Maxwell model needs 5 adjustable constants. • The Maxwell model takes into account visco-elastic effects in the dispersed phase. Main conclusion from the report is that the Maxwell model was found to be most appropriate for the suspension polymerization process. This model has been implemented in the CFX code, and has been selected for the ultimate particle size simulations with the MUSIG model.


9

NaOH ‘sprinkling’

TiOCl 2 feed

mica solution (a)

(b)

Figure 1.2: (a). Schematic representation of the vessel geometry in the vertical baffle plane. (b) Averaged flow field (LES) in a vertical plane which is perpendicular to the baffle plane. The well-mixed region shown is depicted in (a) with a dashed box.

1.2.3 Particle coating The particle coating process is executed by Merck, which is the only European company in this field. Similar to the Dow company, Merck is challenged by Asian competitors. In order to participate in the market growth, Merck is forced to meet ever more stringent quality requirements from its main customers from the plastics, paint and cosmetics industries. In the particle coating process, natural mineral mica particles are covered with a thin layer of metal oxides (e.g. titanium dioxide). Through an interplay of transparency,

10


(a)

(b)

Figure 1.3: (a) Illustration of the mica particles. (b) Illustration of a coated mica particle. refractive index, coating and multiple reflections, a variety of color effects is obtained. The overall coating process can be described as a titration method. An aqueous, strong acid T iOCl2 solution is continuously dosed to a mica suspension. The mass fraction of the mica particles is around 6%. The mica partices have the shape of plates (i.e. particles with a high width/height over thickness aspect ratio. The precipitation of the titanium oxidehydroxide occurs at a certain pH value. The pH value is kept constant by continuous addition of N aOH according to the following reaction scheme: T iOCl2 + 2N aOH + mica → T iO(OH)2,mica + 2N aCl

(1.1)

A schematic representation of the mixing vessel geometry accompanied by an averaged flow field obtained by a LES computation is shown in Figure 1.2. The impeller pumps axially, thereby creating a large and a small recirculation zone. The species react most likely in a homogeneous reaction, and form nano particles. Subsequently, these nano particles agglomerate on the mica particles to form the coating. This process, and consequently the product quality, is strongly dependent on parameters such as the location of the feed point, and the pH. Also heterogeneous reactions (i.e. reactions at the mica surface) take place. The situation of heterogeneous reactions is akin to growth in crystallization, which is mainly determined by the rate at which supersaturation can be transported to the crystals. The turbulence generated by the revolving impeller plays an important role in the formation of the crystals. Similar to break-up and coalescence phenomena mentioned with a view to the suspension polymerization process, the transportation rate of supersaturation to the crystals is rate-limiting and controlled by the energy dissipation rate. Figure 1.3 illustrates the mica particles at the start of the process and the coated layer on a mica particle at the end of the process. In the final computations of the particle coating process, the particles are tracked in the

1.3. Mixing and LBM at the Kramers Laboratory

11

turbulent flow field. Since the mica particles have a high width/height of thickness aspect ratio, standard correlations for the drag force are not valid. A theoretical investigation has been carried out by Hartmann (2002a) to provide an estimation of the drag force experienced by the mica particles. The main conclusions will be briefly described. The non-spherical mica particles immersed in the fluid experience a drag force that acts against their direction of motion. The orientation of the particles is highly dependent on the particle Reynolds number and shape, and so is the drag force. Based on the report of Hartmann (2002a), the mica particles do not have preferred orientations throughout the tank (i.e. in both the turbulent and quiescent regions). Therefore, a drag force correlation has been proposed, based on a mean resistance of translation. The drag coefficient proposed depends on the aspect ratio being the thickness to height/width (the shape) and is inversely proportional to the particle Reynolds number.

1.3

Mixing and LBM at the Kramers Laboratory

Subsequent to the processes described within the OPTIMUM project, this section deals with earlier successful research in the Kramers Laboratory on (multi-phase) complex process applications with the help of the previously mentioned lattice-Boltzmann NavierStokes solver (see section 1.1). Since the introduction in 1996 of the lattice-Boltzmann method (LBM) in our Laboratory, a significant research effort has been undertaken to study the flow phenomena in a large range of real-life applications. One of the lattice-Boltzmann discretization schemes used in the Kramers Laboratory (in particular in the work presented in this thesis) originates from the scheme presented by Somers (1993). The first LES study in a stirred tank geometry with LBM was reported by Derksen & Van den Akker (1999). The results revealed a range of flow field characteristics (e.g. the so-called trailing vortex structure, kinetic energy, dissipation data). A newly developed immersed boundary technique, that imposes the no-slip velocity constraint at the off-grid walls, is successfully applied. Various subgrid-scale models have been explored. Derksen (2001) assessed the performance of the structure function model (Metais & Lesieur, 1992) and the widely used Smagorinsky (1963) model in the prediction of a turbulent flow agitated by a pitched blade impeller. Furthermore, the Voke (1996) model was assessed by Hartmann et al. (2004b), which is a subject of chapter 2. Turbulently agitated solid-liquid suspensions are encountered in various industrial processes (e.g. catalytic slurry reactors, crystallization). Derksen (2003) reports on solidliquid simulations of the turbulent flow in a stirred tank geometry according to an Eu-

12


lerian/Lagrangian approach. Particle-particle collisions proved to play a critical role in order to have the particles in suspension in accordance with the just-suspended criterion (Zwietering, 1958). Another industrial application is a crystallization process. Agglomeration is most often an unwanted phenomenon, as it tends to broaden the crystal size distribution. Hollander et al. (2001) have investigated the agglomeration behavior in a stirred tank and observed a huge discrepancy between the agglomeration kernel obtained by Mumtaz et al. (1997) in simple shear flow and stirred tank flow. This discrepancy was attributed to hydrodynamic effects. A similar industrial application was studied by Ten Cate et al. (2001). They studied crystal-crystal collisions in a draft tube baffled industrial crystallizer by means of LES. The information on the local turbulence in the crystallizer was then used to determine local collision frequencies and intensities through a detailed DNS of an isotropic turbulent field in a box with thousands of particles. Gas-solid cyclone separators are simple devices (no moving parts) to separate solid particles from gas streams. These devices are used in various applications (e.g. in the oil industry). Derksen & Van den Akker (2000) studied the turbulent flow in a reverseflow cyclone by means of LES. They recovered the average flow field with a high level of accuracy. Vortex core precession was observed in the simulations, its frequency being close to the experimentally found value. The lattice-Boltzmann discretization scheme, known as the LBGK (Lattice BhatnagarGross-Krook) method (Qian et al., 1992), is definitely the most well-known and widely used approach for LBM. Kandhai et al. (2003) used this scheme for investigating the average drag that acts on particles in a fluidized bed reactor under steady-state conditions. The applicability of existing drag models has been checked by simulations through varying Reynolds number, number of spherical particles and particle volume fraction. Van Wageningen et al. (2004) used LBGK to investigate the flow in a Kenics static mixer for a range of Reynolds numbers. They assessed their results with results based on simulations with FLUENT. An important observation was that, for Re> 300, the LBM simulations were faster and computationally less demanding than the FLUENT simulation.

1.4

Motivation and aim

When we look back to the processes mentioned in the previous sections, it is clear that a detailed knowledge of the flow phenomena involved is only a first step in gaining more

1.4. Motivation and aim

13

understanding of complex multi-phase processes. The previous section illustrated the amount of research already undertaken in our laboratory to improve the knowledge of (multi-phase) process applications. In this section we again consider particle coating (see section 1.2.3), in order to make clear the issues this thesis aims to focus on. The ultimate goal is to contribute to reliable and accurate predictions of processes such as particle coating. Inter-phase mass transfer from the continuous phase to the mica particles (solid phase) plays an important role in the heterogeneous reactions. If this process is to be solved via simulation, a scalar concentration distribution in the continuous phase is required for determining the mass transfer rate. The transport of a scalar is governed by the convectiondiffusion equation, and solution of this equation can either be obtained in an Eulerian (e.g. finite volume, finite difference methods) or a Lagrangian framework (e.g. Monte Carlo modeling). The work of Van Vliet (2003) has shown that the Lagrangian approach is feasible but expensive. Therefore, an Eulerian approach based on the finite volume formulation is attempted in this work for scalar mixing calculations. Furthermore, in the absence of second-order (or higher-order) chemical reactions in the liquid phase we do not need to take concentration fluctuations into account. A simple blending process is used as a reference case in chapter 5. Scalar mixing calculations, eventually in combination with particle transport, are not possible without a detailed knowledge of the flow phenomena occurring in the continuous phase. For instance, particles will not feel the average flow but the turbulent eddies with size comparable to the particle size. Furthermore, mass transport is not likely to be controlled by the large-scale flow. As a result, the growth rate of the layer thickness on the mica particles is expected to be in-homogeneously distributed throughout the tank, which induces the formation of a particle (mica) size distribution in the tank. These arguments motivate a detailed resolution of the flow field (i.e. the first step mentioned in the first paragraph). In this thesis, this is accomplished by performing large eddy simulations. Large eddy simulations need to be critically assessed against experimental and numerical data based on the RANS approach. This aspect is the subject of chapter 2. Furthermore, in a LES fluctuations are resolved down to the size of the numerical grid. One may wonder if all these fluctuations are of a purely turbulent nature. Experimental data have confirmed that part of the fluctuations are induced by so-called macro-instabilities that severely affect the turbulence levels (Nikiforaki et al., 2002). Large eddy simulations have been executed to determine whether it is possible to determine these instabilities in a numerical simulation. Chapter 3 deals with the recovery of the macro-instability by means of large eddy simulations. The implementation of complex boundaries (e.g. cylindrical tanks, curved impeller

14


blades) is a difficult task. Nowadays, commercial codes make use of unstructured meshes and body fitted grids. However, in structured codes (such as LBM or CFX-4) the complex boundaries are in general off-grid. Within LBM on a Cartesian mesh, Derksen & Van den Akker (1999) have successfully developed an immersed boundary technique to impose the no-slip velocity constraint at the walls that are off-grid. With a view to the finite volume scalar mixing solver developed in chapter 4 and used in chapters 5 and 6, a novel immersed boundary technique needs to be developed. The stair-case approximation is often applied with a fine grid resolution, that is an inaccurate representation of the wall and which is computationally expensive. Several other techniques have been developed and tested in the literature for a wide range of applications. In this thesis, a novel technique is developed (see chapter 4) in order to describe a zerogradient boundary condition for the scalar concentration at off-grid boundaries. In summary, the focus in this thesis is on • Assessment of the lattice-Boltzmann Navier-Stokes solver (LES) • Development of a scalar mixing solver in the finite volume framework in combination with an immersed boundary technique • Assessment of coupled flow (LES) and scalar mixing solvers through simulation of a blending process • Study on mass transfer and solids suspension through simulation of a dissolution process with coupled flow (LES), scalar mixing and particle transport solvers.

1.5

Stirred tank geometry

Fluid dynamics research on stirred tanks has adopted a few de facto standard impeller and tank geometries, for which quite a few experimental as well as numerical data are available. The tank geometry is mostly a flat-bottomed cylindrical tank, with a diameter equal to the liquid height. Four baffles positioned along the perimeter of the tank prevent a solid body rotation of the fluid, and consequently enhance the mixing performance of the tank. In Figure 1.4, two tanks equipped with a standard impeller are shown. The first impeller consists of four, 45o -pitched blades mounted on a shaft (Figure 1.4a). The pitched blade impeller pumps axially downward or upward, depending on the direction of rotation. The second impeller consists of six flat, vertically oriented blades mounted on a disk (the so-called Rushton turbine, see Figure 1.4b). This impeller pumps the fluid in the

1.6. Outline of this thesis

(a)

15

(b)

Figure 1.4: Standard stirred tank configurations. (a) Cylindrical tank equipped with four baffles and a pitched blade turbine. The four blades are under a 45o angle. (b) Cylindrical tank equipped with four baffles and a disk (Rushton) turbine with six, flat blades. radial direction. The Rushton turbine has been used for all the simulations presented in this thesis.

1.6

Outline of this thesis

In view of the motivation of this work described in section 1.4, the thesis is structured in the following order. Turbulent flow phenomena The first two chapters of this thesis deal with single-phase flow simulations in a Rushton turbine stirred tank geometry. These large eddy simulations are performed with the lattice-Boltzmann Navier-Stokes solver based on the scheme published by Somers (1993). Both chapters have been published in scientific journals.

16


In chapter 2, a large eddy simulation is assessed against a transient RANS simulation based on the commercial software package CFX-5, and LDA experiments performed by Schäfer et al. (1997). As the Reynolds number was relatively low (Re=7300), the flow is fully resolved in some parts of the tank. Therefore, next to the standard Smagorinsky model, an alternative subgrid-scale model introduced by Voke (1996) has been tested. This latter model blends smoothly between a DNS in the bulk flow and a LES in the impeller region and discharge flow. Chapter 3 deals with the so-called macro-instability that is present in stirred tank flows. A large-scale vortical structure precessing about the tank centerline has been identified experimentally in the literature. The frequency of this precession lies approximately at 0.02 times the impeller speed. Large eddy simulations have been performed with a focus on the flow structures, and the spectral characteristics of the velocity components. The flow has been simulated at various Reynolds numbers and grid resolutions. Scalar mixing Chapter 4 treats the development of a finite volume code in order to solve the mixing of a passive scalar in a (turbulent) flow field. The code is based on an explicit formulation of the convection-diffusion equation. Numerical diffusion is minimized through a second-order TVD scheme for the discretization of the convection term. A first test is performed on laminar flow inside a cavity. In order to model complex (off-grid) boundaries, an immersed boundary technique is developed. This technique is tested in an inclined cavity geometry (laminar flow) and a cylindrical tank equipped with a side-entry mixer (turbulent flow). Subsequently, detailed scalar mixing simulations are performed in a Rushton turbine stirred tank geometry. Chapter 5 describes the simulation procedure and the results in terms of snapshots of the concentration field, simulated time series of the scalar concentration at various monitoring points, and the mixing time. The developed boundary technique is critically assessed (mass conservation). Solid-liquid mixing including mass transport A particle transport solver has been developed by Derksen (2003). The code solves the transport equation for the particle motion, and takes into account particle-wall and particleparticle collisions. Simulations in a Rushton turbine stirred tank geometry have shown that the exclusion effect (i.e. the particles are not allowed to physically overlap) brought about by the particle-particle collisions plays a crucial role to keep the particles suspended. A combination of the lattice-Boltzmann Navier-Stokes solver, the scalar mixing solver, and

1.6. Outline of this thesis

17

the particle transport solver allows for detailed simulations of solid-liquid mixing with mass exchange between the liquid and solid phases (as occurring in e.g. crystallization or dissolution processes). Chapter 6 describes a simulation of a dissolution process. Suspended particles in an unsaturated liquid dissolve under the action of a concentration gradient. Results are presented through snapshots of the particle distribution, scalar concentration field, and the dissolution time (i.e. the time needed for all particles to fully dissolve in the continuous liquid). Conclusions and outlook Finally, chapter 7 reflects on the work presented in the previous chapters. Main conclusions are summarized, and an outlook for future work is presented. Note from the author Chapters 2 and 3 have been published in scientific journals, and chapters 5 and 6 have been submitted as journal papers. Therefore, some parts of the text in this thesis may appear more than once.

2 Assessment of large eddy simulations on the flow in a stirred tank Large eddy simulations and RANS calculations were performed on the flow in a baffled stirred tank, driven by a Rushton turbine at Re = 7300. The LES methodology provides detailed flow information as velocity fluctuations are resolved down to the scale of the numerical grid. The Smagorinsky and Voke subgrid-scale models used in the LES were embedded in a numerical latticeBoltzmann scheme for discretizing the Navier-Stokes equations, and an adaptive force-field technique was used for modeling the geometry. The uniform, cubic computational grid had a size of 2403 grid nodes. The RANS calculations were performed using the computational fluid dynamics code CFX 5.5.1. A transient sliding mesh procedure was applied in combination with the Shear-Stress-Transport (SST) turbulence closure model. The mesh used for the RANS calculation consisted of 241464 nodes and 228096 elements (hexahedrons). Phase-averaged and phase-resolved flow field data, as well as turbulence characteristics, based on the LES and RANS results, are compared both mutually and with a single set of experimental data. Key words: stirred tank, mixing, turbulence, fluid dynamics, simulations, LDA This chapter has been published in Chem. Eng. Sci. 59(12), 2419-2432 (2004).

2.1

Introduction

The flow structures in a turbulently stirred tank are highly three-dimensional and complex with vortical structures and high turbulence levels in the vicinity of the impeller. Under the

20 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank action of a revolving impeller, the fluid is circulated through the tank. Mixing is enhanced by baffles along the tank wall preventing a solid-body rotation of the fluid. As stirred tanks have always played an important role in industry, the understanding of the flow phenomena encountered in such tanks have been subject of research studies. These often document power number - Reynolds number relationships, and circulation times, characterizing the global flow field. Also integral investigations for selected stirrer/vessel combinations were used to derive scaling rules. In modern chemical engineering, however, there is an increasing demand for local flow information. For instance, yield prediction of chemically reacting fluids for a specific reactor is difficult, since most industrial flows are highly turbulent, inhomogeneous, time dependent, and cover a wide range of spatial and temporal scales (Van Vliet et al., 2001). In other applications, mixing at the small scales is important since rate-limiting phenomena take place at these scales. For example, droplet breakup is controlled by the dissipation rate of turbulent kinetic energy at the scale of the drop size, and the sizes of the turbulent eddies at this scale (Tsouris & Tavlarides, 1994), (Luo & Svendsen, 1996). Many investigations to date have been concerned with the flow in tanks stirred by a Rushton impeller. This type of impeller is much-studied, and a significant amount of experimental and numerical (RANS) data are available in the literature that can be applied for verification. The Rushton impeller induces a strong radial discharge stream. In the wakes behind the impeller blades, three dimensional vortices are formed. The variation of the radial velocity in the impeller discharge flow due to the passage of the impeller blades was found to cause an increase of the turbulence levels (Van ’t Riet & Smith, 1975). This non-random part of the turbulent fluctuations is sometimes presented as ’pseudoturbulence’. Detailed flow measurement techniques, such as Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV) (Sharp & Adrian, 2001) are suitable for the investigation of highly vortical flows, and are able to resolve large-scale as well as small-scale flow structures. LDA investigations have identified the formation of the trailing vortex pair behind each impeller blade and provided information about the flow periodicity in the impeller vicinity and its associated increase of the turbulence levels (e.g. Yianneskis et al., 1987; Schäfer et al., 1998). Laser Doppler Anemometry has proven to be more accurate in the measurement of flow fields in stirred tank reactors than other techniques (e.g. Pitot-tubes, hot-wire anemometers), since it provides flow information even in unsteady and highly turbulent flow regions as well as in the return flow areas of the tank and it operates without fluid contact. Furthermore, parameters that dominate the smallest scales (e.g. energy dissipation rates (Micheletti et al., 2004), spectral information at the micro-scale and shear rates) are becoming more accessible for experimental techniques.

2.1. Introduction

21

Since computational resources increase year-by-year, detailed computational information about the flows at the scales relevant to mixing, chemical reactions and physical processes (e.g. bubble break-up, particle collisions) can be obtained at a fraction of the cost of the corresponding experiment. Consequently, computational modeling of the flow has become an alternative route of describing flows in stirred-tanks. One of the options is a Direct Numerical Simulation (DNS), which resolves all turbulent length and time scales by directly integrating the Navier-Stokes equations using a very fine grid. A direct simulation of stirred tank flow at industrially relevant Reynolds numbers (Re≥ 104 ) is not feasible, as the resolution of all length and time scales in the flow would require enormous amounts of grid cells and time steps. Even at the lower limit of the turbulent regime (Re= 7300) considered here, a true DNS would require very fine grids in the critical flow regions (such as the impeller swept volume) in order to fully resolve the flow. Bartels et al. (2001) have attempted a DNS for the same model system operating at Re=7300 using a clicking/sliding mesh technique for the coupling of the rotating and fixed frame of reference. They assumed two-fold periodicity of the solution domain, to keep the computational cost at an acceptable level, and the finite thickness of the blades and disk was not taken into account. We have used the technique of Large Eddy Simulation (LES). In LES, the NavierStokes equations are low-pass filtered. This way, the range of the resolved scales is reduced by the numerical grid, and the effect of the unresolved subgrid scales on the resolved large scales is taken into account with a subgrid-scale model. It has been proven to be a powerful tool to study stirred-tank flows (Eggels, 1996), as it accounts for its unsteady and periodic behavior and it can effectively be employed to explicitly resolve the phenomena directly related to the unsteady boundaries (the impeller blades). The LES methodology embedded in the lattice-Boltzmann solver has been successfully applied to single phase flow phenomena (macro-instabilities in a stirred tank by Hartmann et al. (2004a)) as well as multi phase systems (solids suspension in a stirred tank by Derksen (2003)). With LES, the computational effort is high but smaller than that of a DNS. Most of the Computational Fluid Dynamics (CFD) simulations solve the ReynoldsAveraged Navier-Stokes (RANS) equations in combination with a closure model for the Reynolds stresses. Flow periodicity and unsteadiness are taken into account with the sliding mesh procedure, in which the geometry of the individual blades is modeled using a grid rotating with the impeller, whereas the bulk flow is calculated with a stationary frame. In contrast to LES, in the RANS approach it is difficult to distinguish the part of fluctuations, that is explicitly resolved and the part represented by the Reynolds stresses. In contrast to the DNS work of Bartels et al. (2001), the full solution domain has been simulated in the present LES and RANS simulations, and the blades and disk were of finite thickness. Nowadays, the RANS approach is commonly applied for the calculation of the com-

22 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank plex flows in large-scale industrial reactors. As computational resources are expected to increase further in future, detailed local flow information of industrial, chemical reactors becomes within reach. As a result, the LES methodology is believed to become an engineering tool. Therefore, extensive comparisons with experimental data and RANS predictions are necessary, which was the purpose of the work presented in this paper. The flow geometry considered in this research was a Rushton turbine stirred vessel. In the LES, the standard Smagorinsky subgrid-scale model and the Voke subgrid-scale model were adopted for turbulence modeling. For the RANS calculation, the Shear-Stress-Transport (SST) model was used. First, the behavior of the LES subgrid scale models is checked. Subsequently, comparisons of the velocity and turbulent kinetic energy fields are made; phase-averaged and phase-resolved (near impeller). Attention is also paid to the computed energy dissipation field and the (an)isotropy of the flow field. These latter predictions are hardly accessible experimentally, but they are of interest for many mixing applications.

2.2

Stirred vessel configuration

The stirred vessel reactor used in this research was a standard configuration cylindrical vessel of diameter T = 150 mm, with four equi-spaced baffles of width 0.1T with a small clearance of 0.017T and a liquid column height of H = T , as shown in Figure 2.1. A lid was positioned at height H, to prevent entrainment of air bubbles. The working fluid was silicon oil of density 1039 kg/m3 and dynamic viscosity 15.9 mPas. The impeller rotational speed was 2672 rpm, resulting in a flow Reynolds number (Re=ρN D 2 /µ, ρ is the fluid density, µ is the fluid dynamic viscosity and N is the impeller rotational speed in rev/s) of 7300 and a tip speed of 7 m/s.

2.3

Experimental method

As a standard LDA system only supplies flow information at a single point, data acquisition was automated to enable a rapid determination of the entire flow field. A multifunctional stirrer test rig has been developed at the Institute of Fluid Mechanics (LSTM), and the entire measuring circuit was refractive index matched. The rotational speed of the impeller was measured by means of an optical shaft encoder, which was coupled to the impeller shaft. The detailed experimental results reported in this paper are taken from (Schäfer, 2001). More detailed description of the experimental setup can be found in the paper of Schäfer et al. (1997), and the experimental procedure is described by Schäfer et al. (1998).

2.4. Computational method

23

0.017 T

T/10

D = T/3 0.75D 0.04D

0.2D

H=T

0.04D 0.16D T/3

0.32D 0.25D T

Figure 2.1: Cross-section of the tank (left). Plan view and cross-section of the impeller (right). At the top level there is a lid. The impeller is a Rushton turbine. The vertical measurement plane was located mid-way between two baffles. The complete phase-averaged flow field was measured with a high resolution of 4 mm in axial and radial directions. Phase-resolved measurements in a range of 0o − 60o behind the blade were made in the vicinity of the impeller stream with a resolution in axial and radial directions of 1 mm and 2 mm inside and outside the impeller swept volume, respectively. The measurements were performed for the axial, radial and tangential mean velocity components and their corresponding RMS levels.

2.4

Computational method

2.4.1 RANS simulation The three dimensional simulations were performed using the computational fluid dynamics code CFX version 5.5.1, a finite-volume based computational fluid dynamics analysis program which solves the Navier-Stokes equations. Turbulence was modeled using the SST (Shear-Stress-Transport) model (Menter, 1994), which is a combination of the k-ω model near the wall and the k-ε model away from the wall. In this way, both models are used in areas where they perform best. The reason for making use of this model was that the mesh was well resolved, resulting in y + values that tended to be lower than 11. By using the SST model, CFX-5 uses an automatic wall function, which blends smoothly between a low Reynolds formulation and a standard log wall function. This gives a more accurate representation of the friction at the wall in the areas where y + is smaller than 11. For the mixing vessel simulations performed, the computational mesh is made up of

24 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank

(a)

(b)

Figure 2.2: Vertical (a) and horizontal (b) view of the mesh in the RANS calculations. The interface between the rotating and sationary meshes was located at r/T = 0.25, and it extents from the bottom of the tank to the free surface. two parts: an inner rotating cylindrical volume enclosing the impeller, and an outer, stationary, annular volume containing the rest of the vessel. The sliding mesh procedure was applied, which is in fact a transient method. At each time step the inner grid is rotated with a small incremental angle, and the flow field is recalculated taking into account the additional velocity due to the motion of the grid. The location of the interface between the two volumes was such that the region of flow periodicity was contained within the sliding mesh (i.e. at r/T = 0.25 and extending all the way from the bottom of the tank to the free surface). The mesh used for the RANS simulation consists of 241464 nodes and 228096 elements (hexahedrons). An illustration of the computational mesh is given in Figure 2.2. In the sliding mesh procedure, the transient rotor-stator frame change model was used, which predicts the true transient interaction of the flow between a stator and rotor passage. This model forms part of the General Grid Interfaces (GGI), which refers to a class of grid connection technology (CFX-5, 2002). As a consequence of the sliding mesh technique, the computer resources are relatively large, in terms of simulation time, disk space and quantitative post processing of the data. Upon simulation, a blended advection scheme for the momentum and continuity equations was used to minimize effects of false diffusion. This scheme is based on upwinding along a streamline with the addition of Numerical Advection Correction (NAC). With a


25

n e,V/ n (-)

10

1 b=0 b = 2/9 0.1 0.1

1 n e / n (-)

10

Figure 2.3: The Voke subgrid scale model, Eq.(2.4). While at large νe /ν numbers the Voke eddy viscosity (solid line) approaches the Smagorinsky eddy viscosity (dashed line), at small numbers it is significantly reduced. blending factor of 1, the scheme is fully second-order but not bounded. Therefore, the high resolution scheme has been used for the turbulence model equations. This scheme is also a blend between a first-order and second-order scheme, but here the blending factor is calculated based on the solution. The high resolution scheme is always bounded, and therefore does not cause overshoots or undershoots in the solution. Hence, this scheme is suitable for variables that are always positive (like k, ε and ω) or bounded (e.g. volume fractions). First a run was made with a relatively large time step of 1.25 ms (corresponding to 20 degrees of impeller rotation), in order to allow the solution to reach a quasi steady state. The run has then been carried on with a smaller time step of 250 µs (i.e. 4 degrees of impeller rotation). Per time step 10 iterations were performed. Typically within every time step, the RMS residuals at the end of the time step are three orders of magnitude smaller than at the beginning of the time step, which indicates a reasonable level of convergence. The simulation was run on Xeon dual processors (Pentium IV) machines (Dell) with 2 Gb of memory, 2 GHz clock frequency and a Linux operating system. The final run was done on 6 processors in parallel.

2.4.2 Large eddy simulation In a Large Eddy Simulation (LES), the small scales in the flow are assumed to be universal and isotropic (i.e. independent of the specific flow geometry). The effect the small scales

26 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank have on the larger scales is modeled with a subgrid-scale model. In this research, the well-known Smagorinsky model (Smagorinsky, 1963) was used. This is an eddy-viscosity model with a subgrid-scale eddy viscosity, νe , which is related to the local, resolved deformation rate: √ (2.1) νe = λ2mix S 2 where λmix is the mixing length of the subgrid-scale motion and S 2 is the resolved deformation rate, defined as: 2 1 ∂ui ∂uj 2 S = + (2.2) 2 ∂xj ∂xi with ui the resolved ith velocity component. Note the summation convention over the repeated indices i, j. In the standard Smagorinsky model, the ratio between the mixing length and the lattice spacing, ∆, is constant, called the Smagorinsky constant (c s ). However, near the walls, the subgrid-scale stresses should vanish, which is not automatically guaranteed in the standard Smagorinsky model. In general, this is accomplished by a reduction of the length scale λmix toward the wall. In this research, the Van Driest’s wall damping function (Van Driest, 1956) is used at the tank walls, which determines the mixing length as: y+

λmix = cs ∆ 1 − e− A+

(2.3)

where y + denotes the distance from the wall in viscous wall units (y + = yu∗ /ν, uτ is the so-called friction velocity) and A+ is a constant taken equal to 26. A value of 0.10 was adopted for cs , which is a typical value used in LES computations of shear-driven turbulent flows (Piomelli et al., 1988). At relatively low Reynolds numbers (Re = 2000 − 10000) and a fine grid, as is the case in this research, the flow in quiescent regions in the tank is completely resolved. Still, the Smagorinsky model predicts a nonzero eddy viscosity in these regions (Eq. (2.1)). According to Voke (1996), the Smagorinsky model is valid for high νe /ν numbers, that is for high Reynolds numbers. Based on a form of the dissipation spectrum proposed by Pao (1965), a modified Smagorinsky model was introduced by Voke (1996), which is also applicable for relatively low νe /ν numbers: νe νe,V = νe − βν 1 − e− βν (2.4) with β = 2/9, νe is the Smagorinsky eddy viscosity and νe,V is the Voke eddy viscosity. Figure 2.3 illustrates the deviation of the Voke eddy viscosity with respect to the


27

Smagorinsky eddy viscosity. For large νe /ν numbers the Voke eddy viscosity approaches the Smagorinsky eddy viscosity and for small νe /ν numbers (i.e. more or less resolved flow) the Voke eddy viscosity is greatly reduced. E.g. for νe /ν = 10 and νe /ν = 1 the eddy viscosity is reduced to 2.2% and 22%, respectively. With a view to the flow system studied at a Reynolds number at the lower limit of the turbulent regime, we compared the performance of the Voke subgrid scale model (because of its adequacy at ’low’ Reynolds numbers) with that of the commonly used Smagorinsky model. Hence, more advanced models like the mixed-scale models and dynamic models (see e.g. Lesieur & Metais, 1996) have not been attempted in this work. Derksen (2001) has assessed the structure function model with the same numerical scheme as has been used here, but this model did not lead to significant changes in the results. For the large eddy simulations, a uniform, cubic computational grid of 2403 lattice cells was defined. The diameter of the tank corresponds to 240 lattice spacings, resulting in a spatial resolution of T /240 = 0.625 mm. As a result, the mean Kolmogorov length scale (based on the specific power input, and a power number equal to 5) equals 90 µm or 0.145∆. A Lattice-Boltzmann numerical solver has been used for the finite difference solution of the filtered momentum equations (Derksen & Van den Akker, 1999). This scheme is based on a microscopic model of many particles that can shuffle and collide on the numerical grid according to completely local collision rules. It can be shown that in the incompressible limit the macroscopic behavior of the particles obeys the NavierStokes equations (Chen & Doolen, 1998). In the current work, N = 1/2900, that is, the impeller makes a full revolution in 2900 time steps (one time step equals (2900N ) −1 = 7.74 µs). The diameter of the impeller equals 80 lattice spacings. As a result, the tip speed, vtip , of the impeller was 0.09 (in lattice units, i.e., lattice spacings per time step), which is sufficiently low for meeting the incompressibility limit in the Lattice-Boltzmann discretization scheme. At the bottom of the tank a no-slip boundary condition was imposed, whereas at the top a free-slip boundary condition was set to mimic the free surface. Inside the computational domain, the cylindrical tank wall, the baffles, the impeller, and the impeller shaft were defined by sets of points. A forcing algorithm (Derksen & Van den Akker, 1999) takes care of the no-slip boundary conditions at these points. The algorithm calculates forces acting on the flow in such a way that the flow field has prescribed velocities at points within the domain. As a geometry point does not need to coincide with the lattice nodes, second-order interpolation of the flow velocities at the surrounding lattice sites is used to obtain the flow velocity at that point. The spacing between the geometry points has to be set such (i.e. smaller than a lattice spacing), that the fluid cannot penetrate at the surface of e.g. an impeller blade. This technique is very flexible, as there is no need for building a


(a) LES: Smagorinsky model

(b) LES: Voke model

Figure 2.4: Phase-averaged plot of the eddy viscosity. new computational mesh when adapting the geometry. On every grid node 21 (18 directions for the LB particles and 3 force components) single-precision, real values need to be stored. The memory requirements of the simulation are proportional to the grid size, resulting in an executable that occupies 240 3 × 21 × 4 ≈ 1.16 GByte of memory. The simulations were performed on an in house PC cluster with Intel 500 MHz processors using a MPI message passing tool for communication. The simulations were run on 4 or 6 processors in parallel. Since the entire tank was simulated, statistical information of the flow was stored in four vertical planes mid-way between two baffles. Hence, four statistically independent realizations of the flow contributed to the averaging procedure. The results were averaged over 15 impeller revolutions.

2.5

Results

2.5.1 Behavior of the subgrid scale models Because the Reynolds number investigated in the present work is at the lower limit of the turbulent regime (Re=7300), the performance of the two subgrid scale models applied in

2.5. Results

29

the large eddy simulations should be checked. As a fine grid resolution was used, the results of the large eddy simulations performed might nearly resemble the results of a Direct Numerical Simulation (DNS). Figure 2.4 shows phase-averaged plots of the eddy viscosity distribution in a vertical plane mid-way between two baffles for both subgrid scale models applied. The results of the eddy viscosity distribution for both subgrid scale models clearly show that the eddy viscosity in the bulk flow is less than 10% of the kinematic viscosity. From that it can be concluded that the Kolmogorov scale in the bulk flow becomes of the order of the grid spacing, and as a consequence the results would resemble those of a direct numerical simulation. However, in the impeller region and the discharge flow the eddy viscosity becomes of the order of the kinematic viscosity. Moreover, instantaneous realizations showed eddy viscosities of 2-4 times the kinematic viscosity. As a result, the present grid resolution would have been too coarse to capture all the turbulent scales. A direct numerical simulation with the present grid resolution may therefore not accurately capture the flow phenomena occurring in the impeller region and the discharge flow where most of the mixing takes place. The authors note that for further evaluations of the subgrid scale models, the present results should also be compared to those of a direct numerical simulation with the same grid resolution. Further evaluations of the Smagorinsky model at a higher Reynolds number (i.e. Re=29000) with the same numerical scheme as is used in the present work are presented in the paper by Derksen & Van den Akker (1999). A direct comparison between Figure 2.4a and Figure 2.4b shows a reduction of the Voke eddy viscosity with respect to the Smagorinsky eddy viscosity. The relative reduction of the eddy viscosity in the bulk flow is more pronounced than in the impeller outflow region, as expected (see Figure 2.3).

2.5.2 Phase-averaged flow field and turbulence levels In comparing the LES results with experimental and RANS based data, the focus is first on the global, phase-averaged flow field. Figure 2.5 shows the experimental, RANS and LES based results of the global, phase-averaged flow field and its associated turbulent kinetic energy levels, mid-way between two baffles. The dominant flow feature are the large two circulation loops; one below the impeller with downward flow near the tank wall and upward flow near the axis and one above the impeller with upward flow near the tank wall and downward flow near the axis. The upper circulation loop does not extend to the top of the tank, and consequently the fluid volume in the upper part of the tank (about 14% − 18% of the total tank volume) is badly mixed. The LES for both subgrid-scale models (see Figures 2.5c,d), as well as the RANS


(a) Experiment, phase-averaged turbulent kinetic energy was only measured in the impeller region

(b) RANS, not all vectors are plotted for clarity

(c) Smagorinsky model, not all vectors are plotted for clarity

(d) Voke model, not all vectors are plotted for clarity

Figure 2.5: Phase-averaged plot of turbulent kinetic energy (based on random velocity fluctuations) and velocity vector field in a mid-way baffle plane. The spacing between the vectors in all subfigures is approximately the same.

2.5. Results

31

(a) Experiment

(b) RANS

(c) LES: Smagorinsky model

(d) LES: Voke model

Figure 2.6: Phase-averaged plot of turbulent kinetic energy (based on random velocity fluctuations in the vicinity of the impeller.

simulation show good correspondence with respect to the form and center of the lower circulation loop. With respect to the upper circulation loop, a deviation of the separation point is observed between the LES and experimental results. The experimental result shows an average upward flow near the tank wall till z/T = 0.8, whereas the LES results with the Smagorinsky and Voke model show an upward flow till z/T = 0.73 and z/T = 0.7, respectively. The separation location of the upper circulation loop (z/T = 0.8) has been well captured by the RANS simulation, although the latter shows some spurious flow at the top of the tank. It is believed that this spurious flow is associated with the fact that the run has not fully reached its pseudo-stationary state, since it is more pronounced in the early stages of the simulation (not shown). The circulation in the lower circulation loop is stronger compared to the experimental result. We further note that the k − ε and DNS


0.4 0.38

r/T=0.183

r/T=0.25

LDA RANS LES(S) LES(V)


r/T=0.317

z/T (-)

0.36 0.34 0.32 0.3 0.28 0.26

0

0.25

0.5

0.75

0

0.25

vr /v tip (-)

0.5

0.75


0

0.25

vr /v tip (-)

0.5

0.75

vr /v tip (-)

(a) Radial velocity component at radial locations r/T = 0.183 (left), r/T = 0.25 (middle) and r/T = 0.317 (right).

0.4

r/T=0.183

0.38

r/T=0.25

r/T=0.317



z/T (-)

0.36 0.34 0.32 0.3


0.28 0.26

0

0.25

0.5

0.75

0

vt /v tip (-)

0.25

0.5

0.75

0

vt /v tip (-)

0.25

0.5

0.75

vt /v tip (-)

(b) Tangential velocity component at radial locations r/T = 0.183 (left), r/T = 0.25 (middle) and r/T = 0.317 (right).

0.4

r/T=0.183

0.38

r/T=0.25

r/T=0.317



z/T (-)

0.36 0.34 0.32 0.3


0.28 0.26

0

0.025

0.05

k/v 2tip (-)

0.075

0

0.025

0.05

k/v 2tip (-)

0.075

0

0.025

0.05

k/v 2tip (-)

0.075

(c) Kinetic energy of the random velocity fluctuations at radial locations r/T = 0.183 (left),r/T = 0.25 (middle) and r/T = 0.317 (right).

Figure 2.7: Phase-averaged axial profiles of the radial (a) and tangential (b) velocity components and the kinetic energy of the random velocity fluctuations (c) at three radial locations.

2.5. Results

33

results of Bartels et al. (2001) also predict the wall separation point at a too low position (z/T = 0.7). Velocity fluctuations in a turbulently stirred tank are partly periodic (directly related to the blade passage frequency) and partly random (turbulence). As a result, the kinetic energy can be divided in a random part and a coherent part. The total kinetic energy, k tot , in the velocity fluctuations is: ktot = kcoh + kran =

1 2 ui − u i 2 2

(2.5)

where kcoh and kran are the coherent and random contributions to the total turbulent kinetic energy, respectively. Note the summation convection over the repeated index i. The (time) averages are over all velocity samples, irrespective of the angular position of the impeller. The random part of the kinetic energy can be determined if phase-resolved average data are available: 1 kran = (2.6) < uiθ2 > − < uiθ 2 > 2

with uiθ the ith resolved velocity component sample at the angular position θ. Consequently, the time averages are over the velocity samples at the angular position θ. The notation < > denotes averaging over all angular positions. The kinetic energy in a large eddy simulation is partly resolved and partly unresolved. The kinetic energy residing at the subgrid levels can be estimated based on isotropic, localequilibrium mixing-length reasoning (Mason & Callen, 1986). A comparison between the grid scale and subgrid scale kinetic energy (instantaneous realization) in a mid-way baffle plane showed that about 0.1% of the kinetic energy is at the subgrid scale level near the impeller and less in the rest of the tank (not shown), and the subgrid scale contribution is therefore negligible. In the following results, the resolved kinetic energy is presented. The turbulent kinetic energy distributions plotted in Figures 2.5 and 2.6 are based on the random velocity fluctuations. In case of the measurements and LES (Figures 2.5a,c,d and 2.6a,c,d), the random part of the turbulent kinetic energy was obtained by averaging the phase-resolved experimental/numerical results, Eq (2.6). In case of the transient RANS simulation (Figures 2.5b and 2.6b), the coherent fluctuations are resolved in the transient calculations, but they are not included in the turbulent kinetic energy solved for by the RANS model. As a result, the RANS turbulent kinetic energy is regarded here to be based on random velocity fluctuations. The results show low turbulence levels in the bulk flow. The result of the Voke subgrid scale model shows a slight decrease in turbulent kinetic energy in the bulk flow, with re-

34 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank spect to the result based on the Smagorinsky model (compare Figure 2.5c and Figure 2.5d). This observation contrasts to what is expected; a reduction of the eddy viscosity in the bulk would lead to increased levels of turbulent kinetic energy. High turbulent kinetic energy levels are encountered in the impeller outflow region, see Figure 2.6. The structure of the kinetic energy distribution of the LES calculations agrees better with the experimental results than the agreement between the RANS and experimental results. Figure 2.7 shows axial profiles of the radial and tangential velocity components and the kinetic energy based on random velocity fluctuations at three radial locations. The agreement of the radial velocity component between simulations and experiments is good, although the upward directed radial impeller outflow is more pronounced in the LES with the Voke subgrid scale model (see Figure 2.7a). All simulations overestimate the tangential velocity component at the impeller tip (i.e. r/T = 0.183; z/T = 0.32 − 0.35). While the agreement of the tangential velocity component for the LES is good at r/T = 0.25 and r/T = 0.317, the RANS predictions differ from the experimental results. At r/T = 0.183 the turbulent kinetic energy is underpredicted by all simulations, but the agreement of the LES predictions with experimental data is better than that for RANS. Also at r/T = 0.25 and r/T = 0.317 the RANS predictions significantly underpredict the turbulent kinetic energy, whereas LES shows an overestimation between z/T = 0.32 and z/T = 0.37. However, the spreading rate (i.e. the width of the profiles) of turbulent kinetic energy predicted by LES is in accordance with the experimental data. Except for the radial velocity component, the differences between the two subgrid scale models applied in the LES are small. Mesh refinement and/or time-step refinement are not expected to improve substantially the predictions of the turbulent kinetic energy in the transient RANS simulation (Ng et al., 1998). It is believed that the discrepancies stem from the turbulence model employed. The SST eddy viscosity model used here assumes locally isotropic turbulence. However, if the local flow field is anisotropic (i.e. u0 6= v 0 6= w0 ) the application of such a turbulence model may lead to an inaccurate prediction of the local turbulent kinetic energy.

2.5.3 Phase-resolved flow field and turbulence levels In a phase-resolved experiment or simulation the position of the impeller blade, with respect to the plane where experimental and numerical results were extracted, is recorded. This allows for a reconstruction of the mean flow field and its fluctuations as function of the impeller angle. Phase-resolved, mean flow fields in the vicinity of the impeller are shown in Figures 2.8

2.5. Results

35

(a) Experiment

(b) RANS

(c) LES: Smagorinsky model, not all vectors are plotted for clarity

(d) LES: Voke model, not all vectors are plotted for clarity

Figure 2.8: Phase-resolved plot of turbulent kinetic energy (based on random velocity fluctuations) and velocity vector field at 15o behind the impeller blade.

and 2.10, at angles 15o and 45o , respectively. An important flow phenomenon is the trailing vortex system, that develops in the wake of the turbine blade (Figure 2.8a), and is then advected by the impeller stream into the bulk of the tank. The trailing vortices are shown in more detail in Figure 2.9. They die out, and the radial impeller outflow is directed slightly upward (Figure 2.10a). The LES prediction with the Smagorinsky subgrid scale model (Figures 2.8c and 2.10c) and the RANS prediction (Figures 2.8b and 2.10b) show a good representation of the trailing vortex system, although the upward directed radial outflow was not predicted by RANS. The Voke subgrid scale model (Figures 2.8d and 2.10d) predicts the lower trailing vortex as being much stronger than the upper vortex at 15 0 , and a remnant of that lower vortex at 450 , which does not correspond to the experimental


(a)

(b)

(c)

(d)

Figure 2.9: Phase-resolved flow field at 15o behind the impeller blade in the domain 0.07 ≤ r/T ≤ 0.2; 0.29 ≤ z/T ≤ 0.38. The experimentally measured flow field is depicted in (a), and the flow field simulated by RANS, LES (Smagorinsky) and LES (Voke) in (b-d), respectively.

findings. It can be seen that the vortex structure is accompanied with high turbulent kinetic energy levels. At 15o behind the impeller blade at r/T values in between about 0.24 and 0.28 a region is observed with high turbulent kinetic energy levels. This region of high turbulent kinetic energy is caused by the vortex system related to the previous blade passage. With the LES model, this region is slightly overpredicted. The reduction of the Smagorinsky eddy viscosity in that region by the Voke subgrid scale model is relatively small (see Figure 2.4), and consequently differences of the kinetic energy between both subgrid scale models are marginal. At 45o , the structure and levels of turbulent kinetic energy are well resolved. For the RANS calculation, the turbulent kinetic energy is mostly underestimated in areas where the trailing vortices are observed (i.e. near the tip of the blade). This underestimation may again be associated to the turbulence model used here. Figure 2.11 shows profiles of the turbulent kinetic energy at r/T = 0.183, r/T = 0.25 (15 degrees behind impeller blade) and r/T = 0.217 (45 degrees behind the impeller blade), which cover high regions of turbulent kinetic energy caused by the trailing vortices. In general, this figure shows a better agreement between LES and experimental results, compared to that between RANS and experiments. The RANS results significantly and systematically underpredict the turbulent kinetic energy. The agreement between the LES and experimental results of the width of the profiles is good, whereas Figure 2.11a shows an underestimation of the turbulence levels between z/T = 0.33 and z/T = 0.36 and Figure 2.11c shows an overestimation between z/T = 0.32 and z/T = 0.36.

2.5. Results

37

(a) Experiment

(b) RANS

(c) LES: Smagorinsky model, not all vectors are plotted for clarity

(d) LES: Voke model, not all vectors are plotted for clarity

Figure 2.10: Phase-resolved plot of turbulent kinetic energy (based on random velocity fluctuations) and velocity vector field at 45o behind the impeller blade.

2.5.4 Energy dissipation The dissipation rate of turbulent kinetic energy (ε) is important in many mixing applications, since it controls the flow at the micro-scale and acts as a controlling parameter in, e.g, break-up and coalescence processes. Unfortunately, it is hardly possible to directly measure the dissipation rate. Indirect ways of measuring ε via turbulence intensities and length scales require assumptions about the nature of turbulence (e.g. isotropy or equilibrium) which are not appropriate in stirred-tank flow. Since the predictions of the turbulent kinetic energy correspond fairly well to the experimental results, it seems fair to assume that the energy dissipation rate distribution is predicted with acceptable accuracy by the simulations. By assuming local equilibrium between production and dissipation at and

38 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank 0.4 0.38

15°; r/T=0.183

15°; r/T=0.25

45°; r/T=0.317

z/T (-)

0.36 0.34 0.32 0.3 0.28 0.26


0 0.025 0.05 0.075 0.1 0.125 k/v 2tip(-)


0 0.025 0.05 0.075 0.1 0.125 k/v 2tip(-)


0 0.025 0.05 0.075 0.1 0.125 k/v 2tip (-)

Figure 2.11: Phase-resolved profiles of turbulent kinetic energy (based on random velocity fluctuations) at 15o degrees behind the impeller blade, r/T = 0.183 (left) and r/T = 0.25 (middle), and at 45o degrees behind the impeller blade at r/T = 0.217 (right).

below subgrid-scale level, the energy dissipation rate can be coupled to the deformation rate: ε = (ν + νe )S 2 (2.7) Figure 2.12 shows that the dissipation rate distribution (i.e. the sum of the resolved scale and subgrid scale contributions) in the tank is very inhomogeneous, stressed by the logscale used for the gray-scale coding. The energy dissipation rate in the impeller region is several orders of magnitude higher than in the bulk region. The transient RANS result of the energy dissipation rate reveals higher energy dissipation rates compared to the LES results. Furthermore, the RANS prediction of the spreading rate of energy dissipation in the discharge flow is smaller than that predicted by LES. Beyond that, the structure of the distributions are similar. A region at the top of the tank and the two lobes in the impeller outflow region show increased energy dissipation rates, which are thought to be due to the fact that the run had not yet fully reached its pseudo-stationary state. In comparing the results of the Voke model to those of the Smagorinsky model (see Figures 2.12b,c), the energy dissipation rate is reduced in the bulk flow, whereas the energy dissipation rate in the impeller region is more or less unaffected. This effect is expected, since the eddy viscosity in the Voke model is more suppressed in the bulk region (more or less resolved flow) of the tank, leading to a lower energy dissipation rate in that region, see Eq. 2.7.

2.5. Results

39

(a) RANS

(b) Smagorinsky model

(c) Voke model

Figure 2.12: Phase-averaged plot of dissipation rate of turbulent kinetic energy in a midway baffle plane.


(a) Smagorinsky model

(b) Voke model

Figure 2.13: Phase-averaged plot of | A | in a mid-way baffle plane.

2.5.5 Turbulence anisotropy The k-ε model is still widely used for simulations in stirred tank configurations, also in the present work. This model is an eddy-viscosity model and it locally assumes isotropic turbulent transport. In rotating and/or highly three-dimensional flows, the k-ε model is known to be inappropriate (Wilcox, 1993). LES at least predicts the resolved part of the Reynolds stresses and thus may provide an assessment of the isotropy assumptions in the kε model. A treatment in terms of (an)isotropy is beneficial for a meaningful interpretation of the Reynolds stresses (Derksen et al., 1999). The Reynolds stress data will be presented in terms of the anisotropy tensor aij and its invariants. The anisotropy tensor, defined as: aij =

ui uj 2 − δij k 3

(2.8)

has a first invariant equal to zero by definition. The second and third invariant respectively are A2 = aij aji and A3 = aij ajk aki . The range of physically allowed values of A2 and A3 is bounded in the (A3 , A2 ) plane by the so-called Lumley triangle (Lumley, 1978). In order to characterize anisotropy with a single parameter, the distance from the isotropic

2.6. Conclusions

41

Figure 2.14: The locations within the Lumley triangle of the phase-averaged points in a plane mid-way between two baffles. Not all points are plotted for clarity. The borders represent different types of turbulent flows: 3-D isotropic turbulence, 2-D axisymmetric turbulence, 2-D turbulence and 1-D turbulence (Lumley, 1978). p state | A |= A22 + A23 was defined. Phase-averaged results of | A | in a plane mid-way between two baffles are presented in Figure 2.13. The strongest deviations from isotropy, i.e. the highest levels of | A |, occurred in the discharge flow, the inflow regions (i.e. below and above the impeller), the boundary layers and at the separation points. The turbulence anisotropy at the separation points is advected into the bulk flow. In the recirculation loops, turbulence is nearly isotropic. Figure 2.14 shows the distribution of the invariants in the (A3 , A2 ) plane obtained with the Smagorinsky model. The points are mostly clustered in the lower part (i.e. almost isotropic turbulence) of the so-called Lumley triangle. However, Figures 2.13 and 2.14 both show that the assumption of mean isotropic turbulence in stirred vessel configurations is questionable. This may explain the underestimation of the turbulent kinetic energy levels in the transient RANS results observed in sections 2.5.2 and 2.5.3.

2.6

Conclusions

In this paper, results of Large-Eddy Simulations (LES) and RANS simulations were assessed by means of detailed LDA experiments globally throughout the tank and locally near the impeller. Two subgrid scale models were investigated, the standard Smagorinsky model and a modified Smagorinsky model (i.e. the Voke subgrid scale model). With a view to chemical mixing processes, the LES methodology has clear, distinct advantages

42 Chapter 2. Assessment of large eddy simulations on the flow in a stirred tank over a RANS type of approach, as velocity fluctuations, and consequently the Reynolds stresses and turbulent kinetic energy, are resolved down to the scale of the numerical grid. As a result, there is a clear (spectral) distinction between the resolved and unresolved scales that is indefinite in a RANS approach. As the experiments were conducted at the lower limit of the turbulent regime, the behavior of the subgrid scale models applied was checked. In the bulk flow, all scales were more or less resolved, whereas in the impeller region and discharge flow the Smagorinsky and Voke eddy viscosity became of the order of the kinematic viscosity. As a result, a direct numerical simulation with the LES grid resolution will not accurately capture the smallest scales in the impeller (outflow) region. The simulated phase-averaged flow fields were in good agreement with the experimental result. The RANS flow field showed deviations in the tangential velocity component in the discharge flow, whereas the wall separation point of the upper circulation loop in the LES flow field differed from the experimental result for both subgrid scale models. All simulations overpredicted the tangential velocity component at the center of the impeller tip. The disagreement between the experimental and LES results of the wall separation point is thought to be due to the wall damping function used in this work. Therefore, other wall damping functions will be explored in future investigations. The development of the trailing vortex system was well represented by the RANS simulation and LES with the Smagorinsky subgrid scale model. The LES with the Voke subgrid scale model showed the lower vortex as being much stronger than the upper vortex, which did not correspond to the experimental findings. The upward directed radial impeller outflow was well represented with LES, and was not found in the RANS simulation. In general, the structure and levels of the turbulent kinetic energy in the impeller discharge flow were better represented by LES than by the RANS simulation. The levels predicted by the RANS simulation systematically and significantly underpredicted the experimentally obtained levels. The levels of turbulent kinetic energy obtained by LES were overestimated between z/T = 0.32 and z/T = 0.37, except for the averaged levels at r/T = 0.183 that showed an underestimation between z/T = 0.26 and z/T = 0.4. In order to improve the LES kinetic energy predictions in the impeller outflow region, either the Smagorinsky model should also be evaluated at higher Reynolds numbers (e.g. Derksen & Van den Akker (1999)), or more advanced subgrid scale models (e.g. mixed scale model, dynamic Smagorinsky model, see Lesieur & Metais (1996)) might be attempted. The dissipation of turbulent kinetic energy is very inhomogeneously distributed throughout the tank, with high levels in the impeller outflow region and low levels in the bulk of the tank. The spreading rate of energy dissipation predicted by the RANS simulation was

2.6. Conclusions

43

smaller than that by LES; this was also observed for the spreading rate of the turbulent kinetic energy (i.e. by the widths of the profiles). The levels obtained by the RANS simulation were higher than those obtained by LES. As velocity fluctuations are resolved by the numerical grid, LES enables an estimation of the turbulence (an)isotropy throughout the tank. Nearly isotropic turbulence was observed in the circulation loops. However, in the impeller stream, the boundary layers, and at the separation points turbulence was found more anisotropic, probably caused by the local, high shear rates. Consequently, the choice for making use of an isotropic eddy viscosity turbulence model should be treated with care. For the Reynolds number being in the lower limit of the turbulent regime, the Voke subgrid scale model was explored that blends smoothly between a DNS in the bulk flow and a LES in the impeller region and discharge flow. This has been illustrated by the Voke eddy viscosity distribution, that showed a more pronounced relative reduction of the Voke eddy viscosity in the bulk of the tank compared to that in the impeller region and discharge flow. Despite the potential of the Voke subgrid scale model for application at relatively low Reynolds number flows, this work did not show significant improvements in the flow field results. In conclusion, a transient RANS simulation is able to provide an accurate representation of the flow field, but fails in the prediction of the turbulent kinetic energy in the impeller region and discharge flow where most of the mixing takes place. In order to have a fair comparison with phase-averaged and phase-resolved experimental data, the transient RANS simulation has been executed in a full flow domain. Similar to a LES, such a simulation is time consuming as well. The large eddy simulations performed have shown that LES provides an accurate picture of the flow field, and reasonable data of the turbulent kinetic energy directly obtained from the resolved velocity fluctuations. As LES resolves the velocity fluctuations locally, information of the turbulence (an)isotropy in e.g. a stirred tank geometry is now available.


Nomenclature Roman aij A+ A2 A3 cs D H k kcoh

N r S T ui uiθ uτ y y+ vr vt vtip z

Description anisotropy tensor constant in Eq. (2.3) second invariant of anisotropy tensor third invariant of anisotropy tensor Smagorinsky constant impeller diameter height of the tank turbulent kinetic energy turbulent kinetic energy based on coherent velocity fluctuations turbulent kinetic energy based on random velocity fluctuations impeller speed radial coordinate resolved deformation rate tank diameter velocity component i velocity component i linked to an angular position friction velocity distance from wall distance from wall in viscous wall units radial velocity component tangential velocity component impeller tip speed axial coordinate

s−1 m s−1 m m.s−1 m.s−1 m.s−1 m m.s−1 m.s−1 m.s−1 m

Greek β ∆ ε θ λmix µ

Description constant in Eq. (2.4) lattice spacing energy dissipation rate angle mixing length dynamic viscosity

Unit m m2 .s−3 rad m kg.m−1 .s−1

kran

Unit m m m2 .s−2 m2 .s−2 m2 .s−2

2.6. Conclusions ν νe νe,V |A|

45 kinematic viscosity Smagorinsky eddy viscosity Voke eddy viscosity distance from isotropic state

m2 .s−1 m2 .s−1 m2 .s−1 -

3 Macro-instability uncovered in a Rushton turbine stirred tank Low-frequency mean flow variations, identified experimentally in various stirred tank geometries, are studied by means of large eddy simulations in a Rushton turbine stirred tank. The focus is on flow structures and the spectral characteristics of the velocity components. The lattice-Boltzmann Navier-Stokes solver and the Smagorinsky subgrid-scale model are adopted for solving the stirred tank flow, and boundary conditions are imposed by means of an adaptive force-field technique. Simulations performed at Reynolds numbers 20, 000 and 30, 000 on grid sizes of 1203 , 1803 and 2403 grid nodes confirm the experimentally found flow variations at various monitoring points in the bulk flow. The period of the flow variations found in the bulk of the tank corresponds to approximately 250 blade passage periods. A simulation performed at a Reynolds number of 12, 500 showed pronounced flow variations with a time scale of about 65 blade passage periods, which is consistent with experimental observations. Transient flow field results in the bulk flow of the tank uncover a whirlpool type of precessing vortex of which the impact on the kinetic energy of the velocity fluctuations is further analyzed. Key words: stirred tank, mixing, turbulence, macro-instability, LES This chapter has been published in AIChE J. 51(10), 2419-2432 (2004).

3.1

Introduction

3.1.1 Flow phenomena in stirred tanks Stirred tanks have always played an important role in industry, because of their mixing abilities brought about by the agitation of the impeller together with baffles along the tank

48

Chapter 3. Macro-instability uncovered in a Rushton turbine stirred tank

wall preventing a solid body rotation of the fluid. As a result, a large number of studies focused on the flow phenomena encountered in such tanks. Rushton et al. (1950) and Holmes et al. (1964) were the among the first to report on power consumption and circulation times for various impeller and tank configurations, which characterize the global flow field. More recently, various experimental and numerical studies reported on local flow information, with a view to optimizing mixing processes (e.g. Yianneskis et al., 1987; Schäfer et al., 1998; Derksen et al., 1999; Derksen & Van den Akker, 1999). For instance, the flow structures are identified to be highly three-dimensional and complex with trailing vortices and high turbulence levels (and dissipation rates) in the vicinity of the impeller. This knowledge of the local flow is of importance as rate-limiting phenomena such as droplet break-up (Tsouris & Tavlarides, 1994; Luo & Svendsen, 1996) or chemical reactions (Van Vliet et al., 2001) are influenced by the small-scale mixing. One of the complications in stirred tank flows is the presence of Macro-Instabilities (MI, i.e. low-frequency mean flow variations) that affect the flow patterns and consequently mixing performance. Different types of instabilities have been identified experimentally, and were reviewed in the paper of Nikiforaki et al. (2002). One of these types was identified as a whirlpool type of vortex moving around the tank centerline (Yianneskis et al., 1987). A large number of studies has been concerned with this instability and the frequency of its precessing motion at different operating conditions, tank and impeller types, and impeller clearances (e.g. Nikiforaki et al., 2002; Haam et al., 1992; Hasal et al., 2000; Myers et al., 1997). However, the frequency was found difficult to reproduce. Nikiforaki et al. (2002) have experimentally pointed out that the whirlpool type of vortex can contribute significantly to the kinetic energy contained in the velocity fluctuations. This effect has similarities to the coherent contribution of the flow periodicity near the impeller blades to the kinetic energy (see e.g. Yianneskis et al., 1987; Derksen & Van den Akker, 1999).

3.1.2 MI significance to multi-phase chemical processes In chemical processing, the velocity field is used to accomplish objectives such as heat transfer, mass transfer and chemical reaction. Various investigations provide evidence of the significance of the MI to multi-phase chemical processes. Haam et al. (1992) measured the local heat transfer flux and temperature on the tank wall. Time variations in the local heat transfer coefficient were observed, and the authors hypothesized that these time variations were due to the precession of an axial vortical structure. Feed stream intermittency was studied by Houcine et al. (1999) by means of laser induced fluorescence (LIF). The feed stream jet was injected at the baffle in the upper part of the tank. Three stages of

3.1. Introduction

49

the feed stream were observed, of which one of them was characterized as effective intermittency with a period of 1-2s. This phenomenon can be linked to the existence of an MI. Derksen (2003) provided another example of the presence of precessing vortices in a solids distribution system. The lower vortex may contribute to resuspension of the solids lying on the bottom of the tank, and consequently mass transfer is enhanced. But, the voidage in the vortex itself may be regarded as a lost volume. Finally, the upper precessing vortex may cause surface aeration which often is an unwanted process.

3.1.3 RANS simulation The presence of low-frequency coherent fluctuations associated with macro-instabilities in stirred tanks have important implications for computational flow models. Commercial software packages generally solve the Reynolds-Averaged Navier-Stokes (RANS) equations in combination with a closure model for the Reynolds stresses. The stationary and rotating components are handled by means of interface models, such as the multiple frames of reference (MFR) technique (CFX-5, 2002). Precessing vortices moving around the tank centerline contribute to the flow unsteadiness, and therefore exclude models that allow for a steady-state approximation and domain reduction through geometrical symmetries (e.g. frozen-rotor and circumferential-average models). The transient sliding mesh interface, on the contrary, is able to simulate the unsteady fluid motion as the flow variation both in time and space is taken into account. However, it is unclear what the effect is of the location of the interface between the rotating and stationary meshes on the simulated unsteadiness of the mean flow. Subsequently, the turbulence modeling is a point for discussion. The k- model is often used to account for the turbulent fluctuations. This eddy-viscosity model, which assumes isotropic turbulent transport, is known to be inappropriate in rotating and/or highly three-dimensional flows (Wilcox, 1993). Assessment of the isotropy assumptions by LDA experiments (Derksen et al., 1999) and large eddy simulations (Hartmann et al., 2004b) has indicated turbulence anisotropy in the impeller stream. Furthermore, in a transient RANS simulation it is not a priori clear which part of the fluctuations is temporally resolved, and which part is taken care of by the turbulence model. This especially applies to flows with no clear spectral separation between the low-frequency, coherent fluctuations on one side, and turbulent fluctuations on the other. Experiments in turbulently stirred tanks (e.g. Nikiforaki et al., 2002), as well as our LES show no clear separation between MI-related fluctuations, and random fluctuations.

50

Chapter 3. Macro-instability uncovered in a Rushton turbine stirred tank T/10

D = T/3 0.16D

0.75D

r H=T

0.04D

0.04D

T/2

0.2D

z T

0.25D

0.24D

Figure 3.1: Cross-section of the tank (left). Plan view and cross-section of the impeller (right). At the top level there is a lid. The impeller, with diameter D, is a Rushton turbine.

3.1.4 Large eddy simulation We studied the flow macro-instability numerically by means of Large Eddy Simulation (LES). This methodology is embedded within the lattice-Boltzmann scheme for discretizing the Navier-Stokes equations, and the use of an interface model is avoided by modeling the geometry with an adaptive force field technique (Derksen & Van den Akker, 1999). In LES a clear distinction is made between the resolved and unresolved scales, as the range of resolved scales is limited by the numerical grid. The effect of the unresolved subgrid scales on the resolved large scales is taken into account with a subgrid-scale model. The LES methodology has proven to be a powerful tool to study and visualize stirred tank flows (Eggels, 1996), because it accounts for the unsteady and periodic behavior of these flows and can effectively be employed to explicitly resolve phenomena directly related to the unsteady boundaries. In terms of velocity fields and kinetic energy contour plots in the vicinity of the impeller and in the bulk of the tank, assessment of LES proves it to be an accurate method for simulating stirred tank flow (Hartmann et al., 2004b). Roussinova et al. (2003) have shown that the circulation pattern instability (i.e. another type of macroinstability at a different time scale) can effectively be resolved by means of LES. In this work, the LES methodology will be adopted to identify the whirlpool type of precessing vortex and its spectral characteristics (e.g. characteristic frequency and fluctuation levels).

3.2

Flow system

The stirred vessel reactor used in this research was a standard configuration cylindrical vessel of diameter T , with four equi-spaced baffles of width 0.1T placed at the perimeter

3.3. Simulation procedure

51

of the tank, and with liquid height H = T . A lid was positioned at height H. A standard Rushton turbine disk impeller with six blades was mounted at mid-height of the tank. Flow visualization results of Yianneskis (2004) indicated that the macro-instabilities were more pronouned and more clearly defined with the impeller located at mid-height of the tank. A schematic representation of the geometry is given in Figure 3.1. Provided that geometric similarity is maintained, the flow system can be fully characterized by the flow Reynolds number, defined as: N D2 (3.1) ν with N the impeller speed, D the impeller diameter and ν the kinematic viscosity of the working fluid. Re =

3.3

Simulation procedure

3.3.1 Large Eddy Simulation Many industrially relevant flows are associated with high Reynolds numbers. A Direct Numerical Simulation (DNS) would require enormous amounts of grid cells and time steps, in order to capture all length and time scales present in the flow. In a Large Eddy Simulation (LES), the small scales in the flow are assumed to be universal and isotropic, and the effect the small scales have on the larger scales is modeled with a subgrid-scale model. In this research, the well-known Smagorinsky model (Smagorinsky, 1963) was used. This is an eddy-viscosity model with a subgrid-scale eddy viscosity, νe , which is related to the local, resolved deformation rate: √ νe = λ2mix S 2

(3.2)

where λmix is the mixing length of the subgrid-scale motion and S 2 is the resolved deformation rate, defined as: 1 S = 2 2

∂uj ∂ui + ∂xj ∂xi

2

(3.3)

with ui the resolved ith velocity component. Note the summation convention over the repeated indices i, j. The ratio between the mixing length and the lattice spacing, ∆, is constant, called the Smagorinsky constant (cs ). However, near the walls, the subgrid-scale stresses should vanish, which is not automatically guaranteed in Eq. (3.2). In general, this

52


Table 3.1: Numerical setup. The grid spacing ∆ and time step δt are given in the first two columns. Subsequently, the Reynolds number, the number of grid nodes and the number of subdomains are given. The number of simulated impeller revolutions times the (wall-clock) time to simulate one impeller revolution, τ , gives the duration of the simulation. Case ∆ δt Re Res. Subd. # Imp. revs. τ I T /120 (1200N )−1 20, 000 1203 4 570 0.69 h II T /180 (1600N )−1 12, 500 1803 6 570 2.03 h III T /180 (1600N )−1 20, 000 1803 6 570 2.03 h IV T /180 (1600N )−1 30, 000 1803 6 570 2.03 h V T /240 (2500N )−1 30, 000 2403 8 150 5.73 h is accomplished by a reduction of the length scale λmix toward the wall. The Van Driest wall damping function (Van Driest, 1956) is applied at the tank walls, which determines the mixing length as: y+ λmix = cs ∆ 1 − e− A+ (3.4) where y + is the distance to the wall in viscous wall units and A+ is a constant taken equal to 26. A value of 0.10 was adopted for cs , which is a typical value used in LES computations of shear-driven turbulent flows (Piomelli et al., 1988). A lattice-Boltzmann method (Chen & Doolen, 1998) was used for solving the filtered momentum equations. The specific scheme we used was developed by Somers (1993). The entire tank was simulated on a uniform, cubic computational grid. Inside the computational domain, the no-slip boundary conditions at the cylindrical tank wall, the baffles, the impeller, and the impeller shaft were imposed by an adaptive force-field technique (Derksen & Van den Akker, 1999).

3.3.2 Simulation aspects The MI was studied in detail at Re=30, 000. Additional simulations have been performed at Re=20, 000 and Re=12, 500. The size of the computational grid was varied in order to study the influence of the grid-size on the solution. The settings for the simulations are given in Table 3.1. The computer code runs on a parallel computer platform by means of domain decomposition: the computational domain was horizontally (i.e. perpendicular to the tank centerline) split in a number of equally-sized subdomains.


53

0.88 0.76 0.63

0.5

0.77

2r/T (-)

0.64 0.58 0.50 0.43 0.36

z/T (-)

0.37

0.24 0.12

Figure 3.2: Locations of the horizontal and vertical lines on which the three velocity components are stored. The vertical plane is located mid-way between two baffles.

On every grid node, 21 (18 directions for the LB-particles and 3 force components) single-precision, real values need to be stored. The memory requirements of the simulation are proportional to the grid size, resulting in an executable that occupies in case of the 180 3 grid 1803 × 21 × 4 ≈ 0.5 GByte of memory. The simulations were performed on an in house PC cluster with Athlon 1800+ MHz processors using an MPI message passing tool for communication within the parallel code. Within the lattice-Boltzmann framework the time step and lattice spacing are nondimensional numbers, which are equal to one. Hence, dimensions are expressed in lattice spacings and velocities in lattice units (i.e. lattice spacings per time step). In Case I for example, the tank diameter corresponds to 120 lattice spacings and one impeller rotation corresponds to 1200 time steps. During the simulations, the axial, radial and tangential velocity components were stored on six horizontal lines and four vertical lines. Three horizontal lines were located below the impeller and three above the impeller. The four vertical lines were located in the impeller outflow stream at different radial positions. The locations of the lines are shown in Figure 3.2. In case of the 1803 -grid, the radial resolution at the horizontal lines was 0.022T , with a total number of 23 monitoring points at each line and the first monitoring

54


point at 2r/T = 0. The axial resolution at the vertical lines is 0.006T , with a total number of 25 monitoring points at each line and the first monitoring point at z/T = 0.433. This results in a total number of 238 monitoring points. The sampling frequency, f s , equals one in the lattice-Boltzmann framework, as the data are stored at each time step. The simulations were started from either a zero-velocity field or from a flow field of a previous simulation. After roughly 20 − 30 impeller revolutions a quasi steady state was reached. This was checked by monitoring the total kinetic energy as function of time. A flow field was transferred from one grid to the other by means of linear interpolation.

3.4

Results

Most of the results presented in this section are based on case IV; Re=30, 000. This case serves as a reference case.

3.4.1 Single flow field realizations The flow field in the vicinity of the tank centerline is shown Figures 3.3 and 3.4 at axial positions z/T = 0.12 and z/T = 0.88, respectively. The contours represent the ratio of the axial vorticity component and the impeller angular speed; a positive value corresponds to fluid rotation in the (clockwise) direction of the impeller rotation. In Figure 3.3a, the main vortical structure is located in the upper-right corner of the graph. When viewing the graphs in sequence, with a time interval of 11 impeller revolutions between each graph, a precessional motion of the vortex in a clock-wise direction (i.e. in the same direction as the impeller rotates) around the tank centerline is observed. Note the long period of the precessional motion, being roughly 40 − 50 impeller revolutions. Compared to Figure 3.3, Figure 3.4a shows a weaker vortical structure in the upper-left corner of the graph. The black circle represents the impeller shaft. Again, the sequence of the graphs shows the vortex precessing around the impeller shaft. A direct comparison between Figures 3.3 and 3.4 indicates a phase difference between the upper and lower moving vortices.

3.4.2 Time-series The transient behavior of the flow strongly depends on the position in the tank. To illustrate the broad range of frequencies encountered in stirred tank flows, the simulated time series of the radial velocity component is depicted in Figure 3.5 at two typical positions, respectively. At the impeller tip, Figure 3.5a, most of the fluctuations are periodic as a

3.4. Results

55

0.2

1

0.6 0.2 0

0.2

0.5 vtip ωz /ω0 1

0.1

2y/T (-)

0. 1

2y/T (-)

0.2

0.5 vtip ωz /ω0

0.6 0.2 0 0.2 0.6

0.6 › 0.1

› 0.1

1

PSfrag replacements

1

PSfrag replacements › 0.2 › 0.2

› 0.1

0.1

0

› 0.2 › 0.2

0.2

0

0.1

0.2

2x/T (-)

(a) Snapshot at the level z/T = 0.12.

(b) As in (a) after 11 impeller revolutions.

0.2

0.2

0.5 vtip ωz /ω0 1

0.6 0.2 0 0.2

0.5 vtip ωz /ω0 1

0.1

2y/T (-)

0.1

2y/T (-)

› 0.1

2x/T (-)

0.6 0.2 0 0.2 0.6

0.6 › 0.1

1

PSfrag replacements

› 0.1

1


› 0.1

0

0.1

0.2

2x/T (-) (c) As in (a) after 22 impeller revolutions.

› 0.2 › 0.2

› 0.1

0

0.1

0.2

2x/T (-) (d) As in (a) after 33 impeller revolutions.

Figure 3.3: Impressions of the flow field (Case IV) about the tank centerline at four different points in time. The gray scales represent the axial vorticity component, ω z , made dimensionless with the impeller angular speed, ω0 = 2πN .


0.2

1

0.1

2y/T (-)

0.2

0.5 vtip ωz /ω0 0.6 0.2

0 0.2

0.5 vtip ωz /ω0 1

0.1

0.6

2y/T (-)

56

0.2 0 0.2 0.6

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(c) As in (a) after 22 impeller revolutions.

(d) As in (a) after 33 impeller revolutions.

Figure 3.4: Impressions of the flow field (Case IV) about the tank centerline at four different points in time. The gray scales represent the dimensionless axial vorticity component. The black circle represents the impeller shaft.

3.4. Results

57

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5 PDF (a.u.) (c)

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Figure 3.5: Traces of the simulated (i.e. Case IV) radial velocity component at positions z/T = 0.5; 2r/T = 0.367 (a and b) and z/T = 0.12; 2r/T = 0 (d and e). Please note the different record lengths of 500 (a and d) and 5 (b and e) impeller revolutions, respectively. The PDF’s of the time series shown in (a) and (d) are given in (c) and (f), respectively.

result of the regular blade passage. This can be clearly seen in the enlarged time series (Figure 3.5b); the tangential and axial velocity components show a corresponding behavior (not shown). No clear low-frequency oscillations are observed. In the bulk of the tank, Figure 3.5d, a clear low-frequency, coherent behavior of the radial velocity component is observed. It shows a clear, cyclic variation of the mean flow superimposed on the turbulent fluctuations. The period of the oscillations correspond with the approximated period extracted from the flow field snapshots, around 50 impeller revolutions. This periodic behavior was found less pronounced in the axial velocity component, but the periodicity in the axial velocity component has been observed at various other positions in the tank (not shown). The enlarged time series (Figure 3.5e) shows the random behavior of the fluctuations, which are superimposed on the low-frequency flow oscillation.

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Figure 3.6: Power Spectral Density functions (Case IV) of the radial velocity component in the impeller outflow region (a and b) and the bulk region (c and d). (a and c) are log-log plots. (b and d) are frequency enlarged lin-log plots.

3.4.3 Frequency analysis The determination of the characteristic macro-instability frequency is essential in order to accurately assess the time scale of the flow motion. Therefore, the power density spectra of the velocity components were analyzed at all monitoring points. The time series recorded, with a length of 570 impeller revolutions, were first detrended and multiplied with a Hanning window (Oppenheim et al., 1983). Subsequently, the power density spectra were obtained by means of the FFT technique. The frequency resolution of the power

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Figure 3.7: Histogram of maximum frequency peaks found in the power density spectra of the velocity components.

density spectra equals 570−1 N ≈ 0.00175N .

As an illustration, the power density spectra of the presented time series of the radial velocity component in Figures 3.5a and 3.5d are shown in Figure 3.6. Note that only a selection of frequencies in the power density spectra are presented, the maximum detectable frequency is 0.5fs = 0.5 (the Nyquist frequency), which equals f = 800N . The spatial resolution of the simulations, however, cannot keep up with this high frequency. A more meaningful estimate of the highest frequency that contains flow information is based on (twice) the grid-spacing and typical values of the RMS velocity. If we take 0.4v tip for the latter, we get f = 32N . In the impeller outflow region, the turbulent fluctuations are mostly periodic due to the passage of the six impeller blades (see Figure 3.5a). This is confirmed in the power density spectrum of the radial velocity component, Figure 3.6a; a clear and distinct peak is observed at f = 6N . Furthermore, the power density of this frequency peak is about three to four orders of magnitude larger than the power density of the lower frequencies. No other distinct peaks are observed in the spectra. These observations indicate that a large amount of the kinetic energy (i.e. the integral of the area under peak at f = 6N ) is due to coherent velocity fluctuations caused by the blade passages only. The time series of the radial velocity component at the monitoring point located in the bulk region (z/T = 0.12, 2r/T = 0) clearly shows a cyclic variation of the mean flow superimposed on the turbulent fluctuations (Figure 3.5d). This cyclic variation is

60


confirmed in the power density spectrum of the radial velocity component. A clear peak is observed at frequency f = 0.0228N , which corresponds to a period of about 44 impeller revolutions. The power density of this peak is more than 10 times larger than that of the other frequencies. More important, the power density of the lower frequencies, f < 0.1N is significantly higher with respect to that obtained at the monitoring point in the impeller outflow region. Referring to the discussion about the turbulence modeling within the RANS framework in the introduction section, this power density spectrum clearly illustrates the absence of a spectral gap between the MI-induced fluctuations, and the random fluctuations (consistent with Nikiforaki et al. (2002)). The power density spectra were analyzed at all monitoring points and the results of the principal frequencies (i.e. the frequency with the maximum power density) are presented in Figure 3.7a. These results indicate that around 50% of all monitoring points give the principal frequency at f = 0.0228N . Also a significant amount of monitoring points give the principal frequency at f = 6N , which is the blade passage frequency. The characteristic frequency of f = 0.0228N was found in the simulated time series of the axial component as well, albeit at less monitoring points (30%, not shown). This may indicate that the axis of the vortex is essentially oriented in the axial direction. A similar spectral analysis of the velocity components was performed on the basis of the simulation at Re = 20, 000 (Case III). The results of the spectral analysis indicate, in addition to the distinct blade passage frequency, a clear MI-frequency peak at f = 0.0255N . Nikiforaki et al. (2002) observed in a similar Rushton turbine stirred tank with clearance T /2, a characteristic frequency of f = 0.013N − 0.018N in the Reynolds number range of Re= 16, 000 − 48, 000. The frequencies we found are encouraging in the sense that LES proves to be capable of reproducing the precessing vortex phenomenon. The observed deviation from the experimentally found frequencies (which is not that significant in view of the experimental and numerical accuracy) challenges further improvement of the subgrid-scale model and/or the numerical settings.

3.4.4 Effect of Reynolds number and impeller off-bottom clearance Galetti et al. (2004) confirmed the established linear dependence of the macro-instability frequency by Nikiforaki et al. (2002) of f ∼ = 0.02N , however they found that macroinstabilities exhibited a different behavior when considering the laminar, transitional and turbulent Re regions. This lead to different values of f /N . Three regions were defined; Region 1 (400 < Re < 6, 300), with a single peak at f = 0.106N , Region 2 (6, 300
− < uiθ 2 > 2

(3.6)

with uiθ the ith resolved velocity component sample at the angular position θ. Consequently, the time averages are over the velocity samples at the angular position θ. The notation < > denotes averaging over all angular positions. Each velocity sample v j at time tj = jδt (or angle θj = jδθ = j · 2πN δt) is ascribed to an angular position with respect to an impeller blade as follows: θ = mod(2πjN,

π ) 3

(3.7)

With an angular velocity of (1600δt)−1 exactly 800 velocity samples cover the angular range (0 − π). With Equation 3.7, the angular positions within this range are reflected in the range (0 − π3 ), with an angular resolution of 23 πN δt. A velocity time series of 570 impeller revolutions results in a number of 1140 velocity samples at each angular position, which is sufficient for meaningful flow statistics. The above mentioned angle averaging practice is straightforward for the extraction of the kinetic energy content related to the blade passage frequency, as this frequency and its phase are fixed and imposed to the flow. The precessing vortex on the other hand is a flow

3.4. Results

63

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(b) PDF of the time series shown in (a).

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N t (−) (c) Time series of the radial velocity component after removal of the low-frequency content.

P DF (a.u.) (d) PDF of the time series shown in (c).

Figure 3.9: Results of the low-pass filtering procedure on the time series of the radial velocity component shown in Figure 3.5d. vrad,f is the filtered radial velocity component and vrad,−f is the time series of the radial velocity component after removal of the low-frequency content. property and although the frequency of its precessional motion is well defined, its motion is not. The vortex certainly precesses in a clock-wise direction as is indicated in Figures 3.3 and 3.4, but due to the turbulent behavior of the flow, the vortex position fluctuates around its mean motion. Therefore, the described angle averaging practice cannot be applied for determining the kinetic energy in the velocity fluctuations related to the precessing vortex

64


2 k/vtip (−)

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Figure 3.10: The development of the kinetic energy as function of time series record length (Case IV). The monitoring points at positions z/T = 0.12;2r/T = 0 and z/T = 0.5;2r/T = 0.367 are labeled ’bulk’ and ’impeller’, respectively.

phenomenon. The kinetic energy content related to the MI can be estimated by means of a filtering procedure. The difficulty is which frequency range and what amount of power contained within this range should be accounted to the MI since (as we discussed above) the MI is not spectrally separated from the rest of the fluctuations. Roussinova et al. (2000) applied a moving average filter to their measured LDA signals with different time window sizes, related to a number of blade passages. We used a low-pass filter, and (as a first approximation) attributed the power content below the cut-off frequency to the coherent velocity fluctuations related to the precessing vortex. The influence of the filter bandwidth is shown in Figure 3.8. At the position in the bulk flow, the kinetic energy content of the filtered signal, kf , sharply increases at a cut-off frequency close to fM I . This sharp increase is not found in the filtered signal obtained in the impeller outflow region. Moreover, the filtered kinetic energy content in the bulk of the tank is found to be much larger than in the impeller outflow region. Henceforth, the kinetic energy content related to the MI is attributed to the power content in the frequency range: 0 ≤ f ≤ 0.0228N . As an illustration of the filtering process, the time series of the radial velocity component shown in Figure 3.5d is filtered and shown in Figure 3.9a, with its pdf in 3.9b. Figure 3.9c gives the time series without the low-frequency content and its pdf is shown in Figure 3.9d. The pdf shows that the velocity fluctuations in the filtered time series are more or less Gaussian distributed. Therefore, it can be concluded that the coherent low-frequency content is almost completely removed from the original time series by the filtering procedure.

3.4. Results

65

k kbpf

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Figure 3.11: Axial profiles of the kinetic energy at several radial positions (Case IV). A distinction between kinetic energy and kinetic energy related to random velocity fluctuations is made.

3.4.6 Kinetic energy The kinetic energy in a large eddy simulation is partly resolved and partly unresolved. The kinetic energy residing at the subgrid scales can be estimated based on isotropic, localequilibrium mixing-length reasoning (Mason & Callen, 1986). A comparison between the grid scale and subgrid scale kinetic energy (instant realization) in a mid-way baffle plane resulted in the conclusion that the subgrid scale kinetic energy was at least one order of magnitude less than the grid scale kinetic energy (Derksen, 2003). Therefore, the subgrid scale contribution to the total kinetic energy was neglected, and in the following results

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Figure 3.12: Radial profiles of the kinetic energy at several axial positions (Case IV). A distinction between kinetic energy and MI kinetic energy is made.

the total kinetic energy is the resolved total kinetic energy. The precessing vortex motion has important implications for the determination of the kinetic energy contained in the velocity fluctuations. For an accurate prediction of the kinetic energy, a couple of precessing vortex periods needs to be captured, see Figure 3.10. The line labeled ’bulk’ is obtained from the time series presented in Figure 3.5d. More than 100 impeller revolutions (i.e. roughly two precessing vortex periods) are needed for a more or less constant value of the kinetic energy. When the influence of the precessing vortex decreases (at increasing 2r/T ) fewer impeller revolutions are needed to arrive at a

3.4. Results

67

constant value of the kinetic energy (e.g. for 2r/T = 0.88 around 100 impeller revolutions are needed). In the impeller outflow region the influence of the precessing vortex on the value of the kinetic energy is negligible, which is shown by the line labeled ’impeller’ in Figure 3.10. Here, around 10 impeller revolutions are sufficient for an accurate prediction of the kinetic energy. Four axial profiles of the total kinetic energy are shown in Figure 3.11. These results are based on record lengths of the velocity components corresponding to 570 impeller revolutions. Close to the impeller tip at 2r/T = 0.367, a large amount of the kinetic energy is found to be related to the coherent blade passage frequency. This blade passage contribution to the kinetic energy decreases rapidly as the distance from the impeller tip becomes larger; from 2r/T = 0.633 radially outward this contribution becomes negligible. The contribution of the precessing vortex to the kinetic energy was found negligible in the impeller outflow region (less than 1% of the total kinetic energy). Figure 3.12 shows four typical radial profiles of the total kinetic energy over record lengths of 570 impeller revolutions. This number of revolutions is sufficient for a constant value of the kinetic energy. Next to the radial profiles of the kinetic energy, the radial profiles of the kinetic energy of the macro-instability related velocity fluctuations are shown. The coherent blade passage contribution to the kinetic energy was found negligible (less than 0.1% of the total kinetic energy) at all monitoring points located on the horizontal lines. The radial profiles of the kinetic energy are approximately flat in the region 0.2 < 2r/T < 0.8. Increased k-levels are observed along the tank wall as a result of the impeller outflow. Also around the tank centerline, increased k levels are found. This effect is found more pronounced in the domain below the impeller. About the tank centerline there is a significant contribution to the kinetic energy due to the passage of the precessing vortex, being 44% of the total kinetic energy at z/T = 0.12, 26% at z/T = 0.36, 14% at z/T = 0.64 and 21% at z/T = 0.76. These percentages illustrate in the first place that the MI is more pronounced in absence of the impeller shaft. In the second place, the contribution of the MI to the kinetic energy is most significant in the top and bottom parts of the tank at 2r/T < 0.2. At radial positions larger than about 2r/T = 0.2, the MI effect on the turbulence levels is less than 10%. These latter observations imply that the precessing vortex moves in the area 2r/T < 0.2, which was already suggested by Figures 3.3 and 3.4.

3.4.7 Grid size effects The results presented were based on simulations performed on the 1803 -grid. It is interesting to study the effect the grid size may have on the (spatial) resolution of the precessing


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Figure 3.13: Histograms of maximum frequency peaks found in the power density spectra of the velocity components.

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vortex motion. Therefore, the stirred tank flow at Re= 20, 000 has been simulated on two grid sizes; a coarser 1203 -grid and the standard 1803 -grid. The frequencies containing the maximum power density were captured and put into a histogram, see Figure 3.13. The

3.5. Conclusions

69

histograms show that the blade passage frequency is accurately found at both grid sizes. A broad range of low frequencies (i.e. f < N ) is found at the 1203 -grid, but a clear dominant peak due to the MI phenomenon is not observed. Contrary to this, the histogram for the 1803 -grid does show a clear dominant frequency peak at f = 0.0255N . Next, a second flow simulation at Re=30, 000 has been performed on a finer grid (i.e. 2403 grid nodes, Case V). Because of the higher computational demand, 150 rather than 570 impeller revolutions were simulated. This number of revolutions is not enough to determine accurately the characteristic frequency and the consequent MI impact on the fluctuation levels. Nevertheless, the time series of the velocity components indicated the existence of the precessing vortex motion. As an illustration, the time series of the radial velocity component is shown for two typical positions; one in the bottom part of the tank and one in the top part of the tank (in Figure 3.14). They clearly show consistent behavior compared to the results presented for Case IV in terms of the principal frequency, phase difference between upper and lower vortices, and the strength of the mean flow variation. The time series confirm the precessing vortex motion with a characteristic frequency of f = 0.02N − 0.025N (based on two periods). As grid refining did not lead to a better agreement between the experimental and numerical result of the characteristic frequency, the conclusion for the time being is that improvement of the subgrid-scale model may lead to a better correspondence. In view of the simulations performed, a grid of 180 3 nodes is necessary and sufficient to resolve the precessing vortex motion at the Reynolds numbers investigated. From Figures 3.3 and 3.4 the size of the precessing vortex can be estimated. With a view to the discussion above, it appears that on the 1803 -grid about 10 − 20 grid cells are required to accurately resolve the precessing vortex. This leads to a diameter of the vortex of about T /18 − T /9.

3.5

Conclusions

In this paper, we have investigated the precessing vortex phenomenon (i.e. a macroinstability) numerically in a Rushton turbine stirred tank by means of large eddy simulations. Such an investigation is worthwhile for the determination of the integral time scale of the flow motion, and the significance of the precessing vortex with respect to the fluctuation levels and the flow patterns. The presence of precessing vortices has important implications for numerical simulations of stirred tank flows. The vortex motion is a transient phenomenon which cannot be captured in a quasi steady-state simulation, and reduction of the computational domain

70


because of symmetry properties of the geometry is not possible. We have performed large eddy simulations instead of RANS simulations, because within the LES methodology the scales of interest are directly resolved, and there is a clear distinction between the resolved and unresolved scales. Furthermore, we avoided the interface models commonly applied for the calculation of stirred tank flows in commercial codes, since it is unclear how significant the (spatial) resolution of the precessing vortex motion is affected by the location of the interface between the rotating and stationary meshes, and by the interface model itself. The sequences of snapshots of the flow field in a horizontal plane below and above the impeller showed a vortical structure moving around the tank centerline in the same direction as the impeller. The vortex below the impeller was found more pronounced in strength and size compared to the vortex above the impeller, and both vortices move with a mutual phase difference. The presented time series of the velocity components reveal, besides the random turbulent fluctuations, two types of coherent fluctuations with time scales separated by more than two orders of magnitude. The fluctuations in the impeller outflow region are dominated by the passage of the impeller blades, whereas fluctuation levels close to the tank centerline are dominated by the low frequency motion of a vortical structure. A frequency analysis provided the characteristic frequencies of f = 0.0255N and f = 0.0228N at flow Reynolds numbers of 20, 000 and 30, 000, respectively. The determined values of the characteristic MI frequency are in good agreement with reported (turbulent) frequencies found experimentally in a similar flow geometry. At a Reynolds number of 12, 500 a second frequency peak at f = 0.092N was observed, which is consistent with the experimentally observed frequency, the so-called laminar frequency, in the literature. Meaningful flow statistics can only be extracted if the flow is calculated in a time span covering several integral time scales. In the impeller outflow region with 2r/T < 0.63, the flow is dominated by the passage of the impeller blades. A time span covering several blade passage periods is sufficient here. In the bulk flow region with 2r/T < 0.2, the flow is dominated by the low-frequency precessing vortex, and at least 100 impeller revolutions need to be captured for an accurate prediction of the kinetic energy contained in the velocity fluctuations. By means of a low-pass filtering procedure, it has been observed that in the top and bottom parts of the tank at 2r/T < 0.2 a significant amount (up to 44%) of the kinetic energy is related to the precessing vortex. The stirred tank flow has been simulated using three different lattice node densities. The main conclusion is that a grid size of 1803 lattice nodes is necessary and sufficient for resolving the vortex motion at the Reynolds numbers investigated. LES has proven its significance in the prediction of the precessing vortices present in a stirred tank flow. The detailed results gave insight in the structure and characteristic

3.5. Conclusions

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frequency of the vortices, and drew attention to the precessing vortex as a point for improvement in the existing modeling techniques. Furthermore, the results form a promising perspective for the application of the LES methodology in future studies considering the significance of the precessing vortex in multi phase mixing processes.

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Nomenclature Roman A+ cs D f fco H k kbpf kf kM I kran N r S t tj T ui uiθ u0i vj vrad vrad,f vrad,−f vtip x, y y+

Description constant in the Van Driest (1956) wall damping function Smagorinsky constant impeller diameter frequency low-pass filter cut-off frequency height of the tank kinetic energy in the velocity fluctuations kinetic energy in the velocity fluctuations related to blade passage frequency kinetic energy in the velocity fluctuations after lowpass filtering kinetic energy in the velocity fluctuations related to the macro-instability kinetic energy in the velocity fluctuations related to turbulence impeller speed radial coordinate resolved deformation rate time discrete time tank diameter velocity component i velocity component i linked to an angular position fluctuating velocity component i velocity sample radial velocity component radial velocity component after low-pass filtering radial velocity component after removal of lowfrequency content impeller tip speed Cartesian coordinates distance from wall in viscous wall units

Unit m s−1 s−1 m m2 .s−2 m2 .s−2 m2 .s−2 m2 .s−2 m2 .s−2 s−1 m s−1 s s m m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 m -

3.5. Conclusions

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z

axial coordinate

m

Greek ∆ θ θj λmix ν νe τ

Description lattice spacing angle discrete angle mixing length kinematic viscosity Smagorinsky eddy viscosity wall-clock time needed for calculating a single impeller revolution axial vorticity component impeller angular speed of revolution

Unit m rad rad m m2 .s−1 m2 .s−1 s

ωz ω0

rad.s−1 rad.s−1

4 Development and validation of a scalar mixing solver based on finite volume discretization

4.1

Introduction

In the preceding chapters the focus was on the performance of a numerical tool to improve the understanding of the flow phenomena occurring in stirred tanks. The flow solver combined with the LES methodology has proven its potential in accurately recovering the global and local flow features. Furthermore, the turbulence levels were more accurately represented by a LES, compared to a RANS based solver. By means of LES it was demonstrated that the turbulent fluctuations are partly random and partly coherent. The latter fluctuations are induced by the passage of impeller blades, and slowly precessing, large-scale vortices (a so-called macro-instability) about the tank centerline. The influence of turbulence on the mixing performance and the description of twophase processes including mass transfer (e.g. the influence of turbulent velocity fluctuations on the motion of particles and on the local mass transfer rate) is extremely important. The LES flow solver has demonstrated its capability in accurately recovering the turbulent scales down to the size of the grid. However, characterizing the mixing performance of a stirred tank (e.g. in terms of a mixing time) or describing two-phase processes including inter-phase mass transfer (e.g. particle coating, dissolution processes) requires additional information on scalar transport in the continuous phase. In this chapter, the development of a scalar transport solver is described. The solver is based on the finite volume method, which is an Eulerian approach. Particular attention is paid to the finite volume schemes for discretizing the various terms in the convectiondiffusion equation. Furthermore, an immersed boundary technique is described for imposing the zero-gradient constraint at off-grid walls. The performance of the solver and the

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Chapter 4. Development and validation of a scalar mixing solver

newly developed immersed boundary technique are assessed on the basis of two laminar flow cases and one turbulent flow case.

4.2

Finite volume discretization of the convection diffusion equation

Consider the convection-diffusion equation for a general scalar φ ∂ρφ ∂ρui φ ∂ + = ∂t ∂xi ∂xi

ρΓ

∂φ ∂xi

+ Sφ

(4.1)

where Γ is the scalar diffusion coefficient and Sφ is the source term. Note that total mass conservation is part of this class as can be seen by taking φ=1, Γ=0 and Sφ =0. Finite volume integration of the above model equation over an arbitrary control volume Ω, and using the Gauss-theorem leads to Z

Ω

∂ρφ dΩ + ∂t

Z

ni ρui φdS =

S

Z

ni ρΓ S

∂φ dS + ∂xi

Z

Sφ dΩ

(4.2)

Ω

where S is the surface of the control volume Ω and ni is the ith component of the normal vector of S. From now on we suppose an uniform, cubic grid (i.e. identical latticespacing in all directions, ∆) and ignore the source term (Sφ =0). The above equation can be written as follows

∂(ρφ)i,j,k ∆V = ∂t

i+ 21 ,j,k ∂φ ρAΓ − FC φ + ∂x1 i− 12 ,j,k i,j+ 21 ,k ∂φ − FC φ + ρAΓ ∂x2 i,j− 21 ,k i,j,k+ 21 ∂φ − FC φ ρAΓ ∂x3 i,j,k− 1

(4.3)

2

i+ 1 ,j,k

where e.g. (.)|i− 21 ,j,k = (.)i+ 21 ,j,k −(.)i− 12 ,j,k , x1 , x2 , x3 are the spatial coordinates in 2 the directions i, j, k, respectively, ∆V is the volume of cell i, A is the area of the relevant cell face and FC is a convective prefactor, e.g.

4.3. Time discretization

77

FC;i+ 21 ,j,k = (ρu1 A)i+ 21 ,j,k

(4.4)

where u1 is the velocity normal to the cell face (or the x1 component of the velocity vector), obtained by linear interpolation of the surrounding nodal values. The diffusive fluxes are handled by second-order accurate central approximations, e.g. φi+1,j,k − φi,j,k ∂φ = (4.5) ∂x i+ 1 ,j,k ∆ 2

The treatment of the convective fluxes and the time discretization are the subjects of the next two sections.

4.3

Time discretization

The finite volume integration of equation 4.3 over a control volume Ω must be augmented with a further integration over a finite time step ∆t. Considering a first-order accurate Euler forward time integration, the 1-D version of equation 4.3 reads ∆V (ρφ)n+1 − (ρφ)ni = i

Z

t

t+∆t

ρAΓ

i+ 12 ∂φ dt − FC φ ∂x i− 1

(4.6)

2

where φn refers to the scalar at time t and φn+1 refers to the scalar at time t + ∆t. To evaluate the right-hand side of the above equation we need to make an assumption about the variation of φ with time. We could use the scalar at time t or at time t + ∆t to calculate the integral or, alternatively, a combination of the scalar values at time t and t + ∆t. We may generalize the approach by means of a weighting parameter θ between 0 and 1 and write the integral of scalar φ with respect to time as Z

t+∆t t

ρφdt = θ(ρφ)n+1 + (1 − θ)(ρφ)n ∆t

(4.7)

The exact form of the eventual discretized equation depends on the value of θ. When θ is zero, only scalar values at the old time level t are used at the right-hand side of equation 4.6 to evaluate φ at the new time; resulting in an explicit scheme. When 0 < θ ≤ 1 scalar values at the new time level are used on both sides of the equation; the resulting schemes are called implicit. The extreme case of θ = 1 is termed fully implicit and the case corresponding to θ = 0.5 is called the Crank-Nicolson scheme. The latter scheme is second-order accurate in time.

78


The convection-diffusion equation is solved on the LES grid, and the time step is the same as the LES time step. The LES time step was found to be sufficiently small in order to satisfy the Courant stability criterion. Therefore, an explicit scheme for the scalar calculations will be considered in the following discussions. The explicit scheme has a first-order Taylor series truncation error accuracy with respect to time. This scheme can be made second-order accurate in time by means of the Adams-Bashford time integration (i.e. linear interpolating polynomial): (ρφ)n+1 = (ρφ)ni + i

∆t (3Qni − Qin−1 ) 2

(4.8)

where Qi is Qi = ∆V

4.4

−1

i+ 12 ∂φ − FC φ ρAΓ ∂x i− 1

(4.9)

2

The treatment of convection

In this section special attention is paid to various existing convection schemes. Issues of importance are stability aspects and the way to minimize numerical diffusion.

4.4.1 Monotone schemes Consider the one-dimensional scalar convection-diffusion equation in absence of a source term ∂φ ∂ + ∂t ∂x

∂φ uφ − Γ ∂x

=0

(4.10)

Any linear general 3-point discretization can be written in the following way n+1 n = bi φn+1 ai φn+1 i+1 + ci φi−1 + φi i

(4.11)

where ai = bi +ci +1 for consistency. This scheme is called positive or monotone when its coefficients bi , ci are positive. Positivity implies that whenever the initial conditions are positive, the solution will remain positive during time stepping. It also means that no spurious wiggles will occur. A non-monotone scheme may allow wiggles to occur in regions where steeps gradients exist. When the grid density is sufficiently fine to accurately resolve these gradients, the size of these wiggles may not be very disturbing in most cases.

4.4. The treatment of convection

79

Mathematical models that solve turbulent flow, multi-phase flow or scalar transport contain, however, a number of variables that are inherently positive, e.g. the turbulent kinetic energy and dissipation, phase fractions, concentration, etc. Oscillations in the solution of these equations cannot be tolerated, for they could result in negative values of these variables, which in turn may result in divergence of the algorithm. Therefore, some form of monotone discretization is essential for the solution of these physically positive variables. Small oscillations may slow down or prohibit convergence in nonlinear systems and the usage of monotone schemes can be of help to overcome this problem. Many convection schemes have been proposed in the past. The first-order upwind scheme (Patankar, 1980) φi+ 21 =

(

φi , φi+1 ,

ui+ 12 ≥ 0 ui+ 21 < 0

(4.12)

for the evaluation of the face value has long been very popular. This scheme is unconditionally positive and the implicit discretization leads to a linear system with favorable solution properties (diagonally dominant). Furthermore, it takes into account flow direction when determining the value at a cell face, which is very important in strongly convecting flows. On the other hand, the scheme is first-order accurate only, which is insufficient for most practical applications. On top of this, it is notorious for its numerical diffusion (or false diffusion), especially when the flow is skewed with respect to the mesh. Consequently, the upwind scheme is inapplicable for convection dominated (high Schmidt number) flows. In spite of the deficiencies, it is still widely applied in present-day CFD. The central difference scheme, also widely used, is based on a linear interpolation of the surrounding grid nodes φi+ 21 =

1 (φi+1 + φi ) 2

(4.13)

Although this scheme has second-order accuracy, it fails in identifying flow direction. Furthermore, this scheme is positive under the condition that the cell Peclet number defined as Pei+ 12 =

ui+ 12 ∆ Γi+ 21

(4.14)

does not exceed the value of 2. Combinations of the above schemes have been proposed, such as the hybrid and powerlaw variants (Patankar, 1980). These schemes behave as the central difference scheme at

80


low Peclet numbers and switch to first-order upwinding for higher values. These variants are unconditionally positive, but share the property of numerical diffusion when the flow is convection dominated. As mentioned above, second-order accuracy is desirable, especially when the problem is convection dominated. Moreover, Godunov (1959) has shown that linear monotone schemes can have an order accuracy of at most 1 (order barrier). This barrier does not apply to nonlinear discretizations. Higher-order upwind schemes have been developed in the past, such as the second-order upwind scheme and the QUICK scheme (Leonard, 1979). These schemes are, however, only conditionally positive. More recently, there have been strong developments on nonlinear discretizations in the area of compressible flows. These techniques are gaining attention in sciences outside the aeronautics related fields.

4.4.2 TVD discretization scheme For the last two decades there has been a sustained effort by the CFD community to develop robust high resolution schemes for the simulation of convection dominated flows. The main ingredients common to these schemes are a higher-order profile for the reconstruction of the cell face values from cell averages, combined with a monotonicity criterion. To satisfy monotonicity, a number of concepts have been proposed over the years. In the flux corrected transport (FCT) approach (Zalesak, 1979; Boris & Book, 1973), a first-order accurate monotone scheme is converted to a high resolution scheme by adding limiting amounts of anti-diffusive flux. In the monotonous upstream-centered scheme for conservation laws (MUSCL) of Van Leer (1979), monotonicity is enforced through a limiter function applied to a piecewise polynomial flux reconstruction procedure. The concept of TVD schemes was introduced by Harten (1983). For certain types of equations, these algorithms can ensure that the total variation over the whole computational domain does not increase with time (i.e. total variation diminishing, TVD); thus no spurious numerical oscillations are generated: X X n φi+1 − φni φn+1 − φn+1 ≤ i+1

i

i

(4.15)

i

Since by numerical schemes only the value of the cell average is available, with the concept of TVD the cells are reconstructed in the way that no spurious oscillation is present near a discontinuity or a zone with steep gradients and high-order accuracy is retained simultaneously. In the following discussion, we will consider ui+ 12 > 0 without loss of generality. The usual methodology to define a second-order TVD discretization scheme is through

4.4. The treatment of convection

81

the introduction of a so-called flux-limiting function Ψ controlling the amount of antidiffusion on a first-order upwind approximation (Roe, 1981) 1 φi+ 12 = φi + Ψ ri+ 12 (φi+1 − φi ) (4.16) 2 As wiggles originate from steep changes in gradients of the scalar φ, it is natural to let the flux-limiter depend on the local ratio of gradients ( ri+ 21 ): ri+ 21 =

φi − φi−1 φi+1 − φi

(4.17)

By using a flux limiter, different first-order and second-order schemes can be written in the form of equation 4.16. For example, for Ψ equal to zero, the first-order upwind scheme is obtained. Other schemes can be similarly formulated: • First-order downwind scheme: Ψ(r) = 2 • Second-order central difference scheme: Ψ(r) = 1 • Second-order upwind scheme: Ψ(r) = r • Second-order QUICK scheme: Ψ(r) =

3+r 4

Following Sweby (1983), these schemes may be plotted along with the TVD monotonicity region on an r − Ψ diagram (see Fig. 4.1). Using this diagram, it is easy to understand the formulation of TVD schemes: any flux-limiter function, formulated to lie within the TVD monotonicity region yields a TVD scheme. Sweby (1983) has also shown that to get second-order accuracy in smooth regions, the condition Ψ(1) = 1 should be met. The Superbee flux-limiter, formulated as h i Ψ(r) = max 0, min[2r, 1], min[r, 2]

(4.18)

obeys the two previously mentioned criteria. Wang & Hutter (2001) have used the Superbee limiter in their work and reported that this limiter is most favorable in treating convectively dominated problems. Therefore, we have used this limiter in our work for simulating the scalar transport in high Schmidt number systems. Usually one restricts Ψ to be positive for r ≥ 0. In addition, it is set to zero for r < 0 (e.g. the Superbee limiter). Thus, at extrema, the discretization reduces to first-order upwinding. By fulfilling certain additional restrictions, uniformly accurate limiters can be constructed which do not degrade to first-order accuracy near extrema (Zijlema, 1996).

PSfrag replacements 82


Ψ(r)

DW

2

SOU QUICK

1

SUPERBEE

CD

0

1

r

UW

Figure 4.1: TVD monotonicity regions (gray area) on the Sweby (1983) diagram. The flux-limiters of the upwind scheme (UW), downwind scheme (DW), central difference scheme (CD), QUICK scheme, second-order upwind scheme (SOU) and TVD scheme (Superbee) are drawn in the diagram. We will not consider limiters of this kind in this thesis, as their additional effect was found to be minimal (Lathouwers, 1999). TVD schemes can also be formulated in a form slightly different from the one given in equation 4.16, as in: 1 0 (φi − φi−1 ) φi+ 21 = φi + Ψ0 ri+ 1 2 2

(4.19)

−1 0 where ri+ . The relation between the two formulations is given by 1 = r i+ 1 2

2

1 0 ri+ 1 2

0 Ψ0 ri+ = Ψ ri+ 21 1

(4.20)

2

Taking for example the second-order upwind scheme where Ψ(r) = r, the equivalent flux-limiter function becomes Ψ0 (r0 ) = r 0 r = 1. It can be shown that the Superbee fluxlimiter is the same for both formulations.

4.4.3 TVD multidimensional application Almost all theory about TVD schemes has been developed for 1-D flow. According to Goodman & LeVeque (1985), a straightforward extension of TVD definitions to multiple dimensions implies that a conservative TVD scheme is first-order accurate at most. Clearly, a less strict criterion is needed for monotonicity than TVD in multiple dimensions. On the other hand, experience is that plain extensions of the 1-D TVD concepts

4.5. The fully discretized convection diffusion equation

83

work well and lead to accurate oscillation-free solutions in multiple dimensions (Hirsch, 1990). The extension of equation 4.16 in three dimensions and considering arbitrary convection speeds reads

φi+ 21 ,j,k

 +  (φi+1,j,k − φi,j,k ) ,  φi,j,k + 21 Ψ ri+ 1 2 ,j,k =   φi+1,j,k − 1 Ψ r− 1 (φi+1,j,k − φi,j,k ) , 2 i+ ,j,k 2

ui+ 12 ,j,k ≥ 0

(4.21)

ui+ 12 ,j,k < 0

where the limiter is controlled by the upwind and local gradients of the solution: + ri+ 1 = 2

φi − φi−1 ; φi+1 − φi

− ri+ 1 = 2

φi+2 − φi+1 φi+1 − φi

(4.22)

Other face values are approximated by similar expressions, found by shifting and transposing indices.

4.5

The fully discretized convection diffusion equation

The starting point for the derivation of the fully discretized convection-diffusion equation is equation 4.3. For simplicity reasons, we only consider the 1-D version of this equation. Extensions to multiple dimensions is straightforward. The 1-D version of equation 4.3, after handling the diffusive fluxes by second-order central approximations (equation 4.5), reads

∆V

d (ρi φi ) = − (FC φ)i+ 12 + (FC φ)i− 12 dt + FD;i+ 21 (φi+1 − φi ) − FD;i− 12 (φi − φi−1 )

(4.23)

where FD is a diffusive prefactor, e.g. FD;i+ 12 =

ρAΓ ∆

(4.24) i+ 21

If we now assume first-order Euler forward time integration (equation 4.6) and consider an explicit formulation (θ = 0 in equation 4.7) equation 4.23 reads

84


∆V (ρi φi )n+1 − (ρi φi )n = − (FC φ)ni+ 1 + (FC φ)ni− 1 2 2 ∆t n n n + (FD )i+ 1 (φi+1 − φi )

(4.25)

2

n

− (FD )i− 1 (φni − φni−1 ) 2

Finally the convective fluxes are approximated by means of TVD discretization with the Superbee flux-limiter. After some rearrangements the fully discretized convection diffusion equation is given by: (bi φi )n+1 = (ai φi )n + (ai−1 φi−1 )n + (ai+1 φi+1 )n + cni

(4.26)

where the coefficients bi ,ai ,ai−1 ,ai+1 are given by:

bi ai−1 ai+1 ai

=

∆V ρi ∆t

h i = FD;i− 21 + max FC;i− 21 , 0 h i = FD;i+ 12 + max − FC;i+ 12 , 0 = bi − ai−1 − ai+1 − FC;i+ 21 − FC;i− 21

(4.27) (4.28) (4.29) (4.30)

The coefficient ci in equation 4.26 gives the amount of anti diffusion given by the TVD scheme as

ci =

h i h i 1 + − max FC;i+ 12 , 0 Ψ ri+ − max − FC;i+ 12 , 0 Ψ ri+ (φi+1 − φi )+ 1 1 2 2 2 i i h h 1 + − max − FC;i− 21 , 0 Ψ ri− − max FC;i− 21 , 0 Ψ ri− (φi − φi−1 ) 1 1 2 2 2 (4.31)

If we set Ψ(r) = 0 (upwind flux limiter), equation 4.26 reduces to a first-order accurate (spatial upwinding and temporal Euler forward) explicit finite volume formulation of the convection diffusion equation. If we consider second-order Adams-Bashford time integration, the scalar update is through equation 4.8. The fully discretized Qi reads

4.6. Boundary conditions

85

Qi = ∆V −1 (ai φi + ai−1 φi−1 + ai+1 φi+1 + ci )

(4.32)

where ai−1 and ai+1 are given by equations 4.28 and 4.29, respectively, and ai is given by ai = −ai−1 − ai+1 − FC;i+ 21 − FC;i− 12

4.6

(4.33)

Boundary conditions

The equations described in the previous sections require boundary conditions on the complete boundary of the domain. Several boundary types may be distinguished. In the first place, inflow boundaries where e.g. mass fluxes, phase fractions, and scalar concentration need to be specified. If available, these values are obtained from interpolation of experimental data. Specification of scalar variables at inflow boundaries in a numerical method is done via a Dirichlet boundary condition: φ = φI

(4.34)

where φI is the scalar value at the inflow boundary. On the other hand, at outflow boundaries usually no information about the local flow variable is available at all. In this case, zero-gradient conditions (i.e. a Neumann boundary condition) for scalars are usually adopted, assuming fully developed flow: ∂φ =0 ∂n

(4.35)

were n is the outward pointing normal. The Neumann boundary condition may be applied for scalars at solid walls as well, since there is no scalar transport through these walls. In a Cartesian grid, complex boundaries generally lie off-grid. Various approaches have been proposed in the past to model these boundaries accurately (Tucker & Pan, 1999; Calhoun & LeVeque, 2000). With the cut cell technique, the cells are cut by the boundary leaving fragments with a volume smaller than the control volume. In the explicit formulation of the convection diffusion equation, the time step scales with the volume of the cell. In order to guarantee stability of the scheme, an infinitesimal time step is required for an infinitesimal control volume. Therefore, the cut cell approach is not appropriate here for modeling (moving) boundaries that are not aligned with the mesh.

86


A simple approach for modeling off-grid boundaries is by approximating the boundary with staircases. The boundary technique based on the staircase approximation is applicable in the explicit finite volume discretization, and it is easy to implement. However, at a coarse grid resolution with respect to the dimension of the boundary, the staircase boundary is an inaccurate representation of the real boundary. Furthermore, the flow is solved in a lattice-Boltzmann framework, in which the Dirichlet boundary conditions for the velocity components at the off-grid boundaries are imposed by an immersed boundary technique (i.e. an adaptive force field technique developed by Derksen & Van den Akker (1999)). As a result, by making use of the staircase technique to impose the Neumann boundary constraint at the off-grid boundaries in the finite volume framework, mass conservation is not guaranteed. In line with the immersed boundary technique in the lattice-Boltzmann framework to impose the Dirichlet boundary conditions for the velocity components at the off-grid walls, we propose an immersed boundary technique in the finite volume framework to impose a Neumann boundary condition for the scalar. This technique makes use of ghost cells, and is consequently referred to as the ghost cell technique. In the next subsections, the ghost cell technique is explained.

4.6.1 Neumann boundary condition implementation in 1-D Consider three cells in a 1-D domain (see Figure 4.2a). The spacing between the cell nodes is ∆. The cell with index i = −1 is the ghost cell, the cells with indices i = 0, 1 are flow cells. The physical wall is in between the ghost node and the first flow node (−∆ ≤ xw < 0). We consider a zero-gradient constraint (Neumann condition) at the wall. Via second-order interpolation the scalar value at the ghost node reads φ−1

2x0 φ0 − x0w + = w x0w − 12

1 2

φ1

(4.36)

where x0w = xw /∆. In the special case where the boundary is positioned halfway between two grid nodes (xw = −0.5∆), the ghost cell value equals φ−1 = φ0

(4.37)

Here, we only need to copy the near wall scalar values to the ghost nodes (in this case so-called mirror nodes) to satisfy the zero-gradient constraint. As a result, representing a physical boundary located halfway two grid nodes by means of mirror nodes results in a

4.6. Boundary conditions

87

φ1 dφ dn

PSfrag replacements

φ−1

-1

dφ dn

=0 PSfrag replacements φ0

xw 0 ∆

1

=0

φ−1

φ0 ∆

φ1 ∆

∆ (a)

(b)

Figure 4.2: Neumann boundary condition implementation in 1-D (a). Schematic representation of the determination of the scalar ghost cell value in 2-D (b).

second-order approximation of the Neumann constraint.

4.6.2 Extension to multiple dimensions For the determination of the scalar ghost cell value (resulting from a Neumann constraint) at walls that are skewed with respect to the mesh we consider for simplicity a 2-D problem. Extension to three dimensions is straightforward. Figure 4.2b shows a curved wall immersed on a Cartesian grid. The gray cells represent the ghost cells as their cell centers lie outside the flow domain. Black cells are exterior cells as they do not occupy flow domain. We reduce the determination of the scalar ghost cell values in a multi dimensional problem to a one dimensional problem described in the previous section. For that purpose we determine the positions of the scalar values φ0 and φ1 through a fictitious line perpendicular to the wall. The spacing between φ−1 , φ0 and φ1 is chosen equal to the lattice spacing. The procedure for the determination of the scalar ghost cell value φ −1 is shown in Figure 4.2b. The scalar values φ0 and φ1 are determined by bi-linear interpolation of the surrounding cell nodes. For the determination of φ0 at least one of the surrounding nodes is the ghost node. We can either use bi-linear interpolation by taking the old values of the surrounding nodes (explicit) or solve the scalar ghost cell value implicitly. Test calculations showed no significant change by solving the scalar ghost node value implicitly, and therefore the scalar ghost cell value determination was treated explicitly.

88


For the scalar concentration, the Neumann boundary condition is imposed at the walls. In this case, equation 4.36 is used for the determination of the scalar concentration ghost cell value. This method is unconditionally stable as −1 ≤ x0w < 0.

4.7

Validations

With the help of three benchmark cases, the performance of the scalar mixing solver and the novel immersed boundary technique will be assessed. These benchmark cases are laminar cavity flow in a square box, laminar cavity flow in a box with inclined side walls, and turbulent flow in a cylindrical tank with a side-entry mixer.

4.7.1 Laminar cavity flow in a square box In this section, we study the mixing behavior of two miscible liquids having the same (Newtonian) viscosity in a cavity flow geometry. The focus is on the performance of the different discretization schemes with respect to numerical diffusion. The reference geometry studied here is based on a mixing experiment reported by Hoefsloot et al. (2000). The experiments were performed in a rectangular cavity of plexiglass. The experimental set-up with the relevant dimensions is shown in Figure 4.3a. In the experiments the two liquids have different rheological parameters. The rheology of the dye solution is similar to that of a Newtonian liquid, whereas the polyacrylamide in glycerin with 2w% water solution behaves like a non-Newtonian liquid. Based on the kinematic viscosity of the dye solution (4.78·10−4 m2 /s), the Reynolds number (Re=Ulid L/ν, where Ulid is the velocity of the band and L is the width/height of the cavity) equals 7.5. In addition to experiments, Hoefsloot et al. (2000) report results obtained by simulations with Fluent 5.1.1. These simulations are based on the finite volume formulation of the Navier-Stokes equations and a Volume Of Fluid approach (VOF) for the location of the interface. Both experiments and simulations started with a zero-velocity field. More details on the simulation procedure are reported in the paper of Hoefsloot et al. (2000). The cavity flow was simulated with our code in 2-D, with 256 lattice spacings in the xand y-directions. A no-slip boundary condition for the flow and a zero-gradient boundary condition for the scalar concentration have been imposed at the walls. Since the walls are aligned with the grid, the no-slip boundary condition for the flow has been applied with a second-order, half-way bounce-back scheme. The zero-gradient boundary condition has been applied via mirror nodes (second-order accurate, see equations 4.36 and 4.37). The velocity of the lid and the viscosity of the fluid have been set such, that the Reynolds

4.7. Validations

band

89

Ulid = 0.036 m/s 0.08 m

symmetry plane

L = 0.1 m y z x

Ulid

L

0.15 m L = 0.1 m

(a)

L

(b)

Figure 4.3: The experimental set-up, a 3-D cavity (a) and the symmetry plane (b).

number equals 10. At the start of the simulations, ’fluid 1’ (containing a zero scalar concentration) and ’fluid 2’ (containing a non-zero scalar concentration) both occupy 50% of the cavity volume. ’Fluid 1’ (white color) is located on top of ’fluid 2’ (black color), see Figure 4.3b. Contrary to the experiments and simulations reported by Hoefsloot et al. (2000), the mixing calculations were started in the fully developed flow field and both liquids have the same kinematic viscosity. In order to test the performance of the different discretization schemes, the diffusion coefficient is set to zero (i.e. the Schmidt number equals infinity; Sc= ∞). Figure 4.4 shows snapshots of the concentration in the cavity. The flow rotation is clock-wise as the band is moving from the left to the right. As a result, ’fluid 1’ is lifted upward at the left side of the cavity. Figure 4.4a shows the experimental mixing pattern after 10 seconds (Hoefsloot et al., 2000), and Figure 4.4b is their simulated result (Hoefsloot et al., 2000). The agreement between the experiment and simulation is good. Due to a bad illumination in the left top corner, the small white region observed in Figure 4.4b cannot be observed in the experimental result. Figures 4.4c, 4.4d and 4.4e show the mixing pattern after 8.5 seconds obtained with our simulations with three different discretization schemes. Although the Reynolds number is somewhat higher (10 instead of 7.5) and the simulations were started in a fully developed flow field, the results look qualitatively the same compared to Figures 4.4a and 4.4b. The white region at the top left corner is accurately resolved. The present simulations are a little ahead compared to the results of Hoefsloot et al. (2000). Furthermore, the black ’tongue’ observed at the top of the cavity is thicker.

90


(a)

(c)

(b)

(d)

(e)

Figure 4.4: The mixing pattern in the symmetry plane after 10 seconds: Hoefsloot et al. (2000) experiment (a), Hoefsloot et al. (2000) simulation. The mixing pattern in the symmetry plane after 8.5 seconds: Power-law scheme (c), QUICK scheme (d) and TVD scheme (e). The dashed arrows correspond with the profiles in Figure 4.5.

The differences mentioned may be caused by the following discrepancies between the settings in the present simulations, and the experiment/simulation of Hoefsloot et al. (2000). In the first place, the present simulations were started in a fully developed flow field, whereas the experiment/simulation of Hoefsloot et al. (2000) was started in a zerovelocity field. Secondly, the Reynolds number in the present simulation is somewhat

91

2.4

2.4

2

2

2

1.6

1.6

1.6

1.2

1.2

1.2

0.8

c/c¥ (-)

2.4

c/c¥ (-)

c/c¥ (-)

4.7. Validations

0.8

0.8

0.4

0.4

0.4

0

0

0

-0.4

-0.4 0

0.2 0.4 0.6 0.8 y/L (-)

(a)

1

-0.4

0

0.2 0.4 0.6 0.8 y/L (-)

(b)

1

0

0.2 0.4 0.6 0.8 y/L (-)

1

(c)

Figure 4.5: Profiles of the concentration (normalized with the final concentration c ∞ ) after 8.5 seconds at x/L = 0.5 (i.e dashed arrows in Figure 4.4). Power-law scheme (a), QUICK scheme (b) and TVD scheme (c).

higher compared to that in the experiment/simulation of Hoefsloot et al. (2000). Thirdly, in the present simulations the rheology of the fluids is the same, which is not the case in the experiment/simulation of Hoefsloot et al. (2000). Figure 4.4c shows the mixing pattern after 8.5 seconds obtained with the power-law discretization scheme. Although the diffusion coefficient is set to zero, the interface is smeared out. Since the streamlines are not aligned with the Cartesian mesh, the upwind values of the concentration are not accurately obtained. As a result, the interface smears out due to numerical diffusion. Figures 4.4d and 4.4e show the mixing pattern under the same conditions obtained with the QUICK and TVD schemes. These higher-order schemes significantly reduce the effect of numerical diffusion; the interface remains sharp. Figure 4.5 shows profiles of the concentration at position x/L = 0.5, which corresponds with the dashed arrows in Figures 4.4c, 4.4d and 4.4e (the direction of the arrow is from y/L = 0 to y/L = 1). The profile shown in Figure 4.5a shows a smooth transition between low and high scalar concentrations, which is attributed to the numerical diffusion associated with the power-law scheme. We have seen in Section 4.4.2 that the QUICK scheme is conditionally stable (the QUICK flux-limiter is not located solely in the TVD monotonicity region, see Figure 4.1),

92


Ulid

L 76

0

L

Figure 4.6: Cross-section of the inclined cavity at 76o with respect to the horizontal axis. whereas the TVD scheme is unconditionally stable. This is clearly demonstrated in Figures 4.5b and 4.5c. The profile obtained by the QUICK scheme shows wiggles and negative values of the concentration at the extrema, whereas in the profile obtained by the TVD scheme only sharp step changes (no wiggles) are observed. As a result, the TVD scheme is favorable compared to the QUICK and power-law schemes and it is used for the simulations in the following chapters. Since the mass change in the simulations is smaller than 0.1%, the finite volume scheme may be regarded as mass conservative.

4.7.2 Laminar cavity flow in a box with inclined side walls In this section, the performance of the developed immersed boundary technique is assessed against the performance of the staircase technique. In this case, we again study the lid driven cavity flow. The side walls of the cavity are inclined at 76o with respect to the horizontal axis, see Figure 4.6. ’Fluid 1’ (represented by the white color) contains a zero scalar concentration, and ’fluid 2’ (represented by the black color) contains a non-zero scalar concentration. Both fluids occupy 50% of the cavity volume, and ’fluid 1’ is located on top of ’fluid 2’. The cavity flow was simulated in 2-D. A no-slip boundary condition for the flow and a zero-gradient boundary condition for the scalar concentration have been imposed at the walls. The no-slip boundary constraint at the top and bottom walls are imposed by the halfway bounce-back boundary condition, and the side walls are modeled with the immersed boundary technique proposed by Derksen & Van den Akker (1999). The zero-gradient constraint for the concentration is enforced by means of the newly developed immersed boundary technique (see section 4.6).

4.7. Validations

93

1.10

1.10 L/ 1.08 L / L/ 1.06 L /

25 50 100 200

M/M0 (-)

M/M0 (-)

L/ 1.08 L / L/ 1.06 L / 1.04

1.04

1.02

1.02

1

1

0.98

25 50 100 200

0.98 0

10

20 (Ulid /L)t (-)

(a) Staircase technique.

30

40

0

10

20 (Ulid /L)t (-)

30

40

(b) Ghost cell technique.

Figure 4.7: Simulated time traces of the total mass in the system normalized with the total mass at t = 0 (M0 ). The lines represent simulations at four different grid resolutions. The velocity of the lid and the viscosity of the fluid have been set such, that the Reynolds number equals 10. The scalar mixing in the inclined cavity has been simulated at four different grid resolutions in order to check its effect on the mass change and the near-wall scalar concentration distribution. The different grid resolutions adopted, are L/25, L/50, L/100 and L/200, respectively. The mixing calculations were started in the fully developed flow field. The Schmidt number equals 1000, which is a typical value in liquid systems. Figure 4.7 shows simulated time series of the normalized total mass at four grid resolutions. Both the staircase and ghost cell techniques (Figures 4.7a and 4.7b, respectively) do not conserve mass. With increasing grid resolution (from L/25 to L/200) the total (positive) mass change decreases (from 9% to 4% at (Ulid /L)t = 40). Based on the time series of the mass change, the ghost cell technique performs similar to the staircase technique. In Figure 4.8, snapshots of the scalar concentration field are shown for the coarse (L/25) and fine (L/200) grid resolutions at (Ulid /L)t = 5. The walls represented by staircases and by the ghost cell technique are shown by means of the solid lines. With the staircase technique used at the coarse grid (Figure 4.8a), the zero-gradient constraint is not satisfied at the side walls. Application of the ghost cell technique (Figure 4.7b) results in a more accurate representation of the zero-gradient constraint at the side walls (the concentration iso-lines tend to end perpendicular to the inclined side walls). Both techniques, however, have a problem in recovering the low concentration region in the top

94


(a) Resolution L/25, staircase technique.

(b) Resolution L/25, ghost cell technique.

(c) Resolution L/200, staircase technique.

(d) Resolution L/200, ghost cell technique.

Figure 4.8: Instantaneous realizations of the scalar concentration distribution at (Ulid /L)t = 5. The walls are represented with the solid lines.

left corner of the cavity due to a lack of resolution. At a fine grid resolution (see Figures 4.8c and 4.8d), both techniques perform similar. A close inspection of the concentration iso-lines at the right wall reveals the level of accuracy of the zero-gradient boundary constraint. This constraint is more accurately recovered by means of the ghost cell technique compared to the staircase technique. In conclusion, the ghost cell technique does not guarantee mass conservation. The

4.7. Validations

95

change in mass with the ghost cell technique is comparable to that of the staircase technique for the grid resolutions tested. Grid refinement is a remedy to resolve this problem, but this implies higher computational effort. With the ghost cell method, the zero-gradient constraint is applied at a number of points on the boundary which scales with the grid resolution. A coarse grid resolution implies a lower number of points on the boundary compared to the number at a high grid resolution, and consequently a less accurate zerogradient boundary constraint at the wall. Nevertheless, in contrast to the staircase method, the ghost cell method takes into account the shape of the boundary regardless of the grid resolution. With a view to future calculations in a stirred tank geometry, it is expected that the tank’s outer wall will be sufficiently accurately represented (in terms of mass conservation and accuracy of the zero-gradient constraint) by the ghost cell method, as, with respect to the tank’s outer wall, the grid resolution is comparable to L/200. With respect to the impeller blade length, the grid resolution is comparable to L/25. As a result, the performance of the ghost cell method with respect to an impeller blade (not to speak about the blade edges) may be comparable to the performance observed in the results of the above simulation at grid resolution L/25. The change of mass should therefore be critically checked in future calculations. The performance of the ghost cell method in a cylindrical tank geometry is checked in the next section.

4.7.3 Mixing behavior in a cylindrical tank with a side-entry mixer The third validation case focuses on the turbulent flow in a cylindrical tank with a sideentry mixer. Side-entry mixers are often used to mix the contents of large cylindrical vertical storage tanks. The mixing time of these tanks is large as the diameter of the impeller is two orders of magnitude smaller than the tank diameter. This validation case is based on the geometry presented in the paper by Van Looy et al. (2004). A typical layout of the storage tank is shown in Figure 4.9. An extensive experimental program was carried out at Shell Research in 1969-1970 in order to determine the mixing time in storage tanks. Based on the experimental results, an empirical correlation was derived that can be used to calculate the mixing time for various tank sizes and density differences of the two miscible liquids. Recently, CFD has been employed to model this problem (Van Looy et al., 2004). In this section, results of the mixing performance of the turbulent flow generated by a side-entry mixer in a storage tank are presented, based on an LES including passive scalar transport. This validation case is of help to address issues such as the implementation of

96


8

D = 0.053T

o

fluid 1 D T (a) Top view.

0.0417T

D fluid 2 T (b) Cross-section.

Figure 4.9: Schematic representation of the storage tank with a side-entry mixer. T and D are the tank and impeller diameters, respectively. subgrid-scale diffusion and assessment of mass conservation in the finite volume scheme (the tank’s outer wall is modeled with the newly developed ghost cell technique). Furthermore, the performance of the scalar mixing solver is compared to a LES using Fluent. The Schmidt number in liquid systems is of the order of 1000. As a result, the Batchelor scale (i.e. the smallest length scale of the scalar field) is a factor 30 smaller than the Kolmogorov scale. As a result, the effect of the scalar subgrid-scale fluctuations on the resolved-scale scalar field may be larger than that of the subgrid-scale velocity fluctuations on the resolved-scale velocity field. From this perspective, it seems reasonable to take the turbulent Schmidt number (Sct ; relating the eddy diffusion coefficient (Γe ) to the eddy viscosity via Γe = νe /Sct ) to be smaller than unity. We have chosen a turbulent Schmidt number of 0.7. A total diffusion coefficient (Γ = Γmol + Γe , with Γmol the molecular diffusion coefficient) is calculated and applied in the diffusive prefactor (see equation 4.24). ’Fluid 1’ contains a zero scalar concentration, and ’fluid 2’ contains a non-zero scalar concentration. Both fluids occupy 50% of the tank volume, and ’fluid 1’ is located on top of ’fluid 2’ (see Figure 4.9b). As buoyancy effects are not included in the finite volume ρ N 2D scheme yet, a simulation has been carried out at infinite Froude number (Fr= ∆ρ g , with g the magnitude of the gravitational acceleration vector and ∆ρ the density difference which equals zero). The simulation starts with a zero-velocity field. For the LES, the time step needed can be estimated by realizing that the large scale structures of the impeller jet have to be resolved. The impeller stream velocity Ujet is typically 5 m/s and the impeller diameter 0.125 m. At the very least, the temporal resolution should therefore be D/Ujet = 0.025 s.

4.7. Validations

97

With an impeller speed (N ) of 50 s−1 , the dimensionless temporal resolution equals 1.25. In the LES, the dimensionless temporal resolution is set at 0.01. The Reynolds number, defined as N D 2 /ν, is equal to 8.75 · 105 . The impeller is modeled as a swirling jet in the simulation. The profiles of the tangential and axial velocity components (vtan and vax , respectively) inside a cylindrical region with diameter and length equal to D read vtan = 1.02N rj ;

0 ≤ rj ≤ D/2

vax = 0.77N D

(4.38) (4.39)

with rj the radial distance. This schematic approach for the mixer modeling is justified since the mixing time is much larger than the impeller revolution time, and there is no interest in the flow in the impeller region. Only the net effect of the impeller has been taken into consideration. The numerical grid used has 106 grid nodes, which is a factor of 20 higher than the number of nodes used by Van Looy et al. (2004). The simulation was conducted on 4 parallel CPU’s, instead of 8 CPU’s by Van Looy et al. (2004). While the number of grid nodes per CPU is much larger in the present simulation, the calculation per impeller revolution is similar. Figure 4.10 shows a qualitative picture of the scalar mixing in the storage tank in terms of instantaneous realizations of the scalar concentration field in a vertical symmetry plane. The interface between low and high scalar concentration is first sucked into the impeller. A jet issuing from the side-entry mixer in the bottom left corner appears to cross the domain and hits the opposing wall, where it is deflected upward. Subsequently, the jet hits the top wall and is directed to the left. At N t = 300 mixing is taking place in a large part (about 70%) of the domain. The snapshots reveal the mixing time in the storage tank being of the order of 1000 impeller revolutions. Following Van Looy et al. (2004), the mixing time is calculated based on the coefficient of mixing, defined as:

cmix

v ! u P c −c 2 i u ∆Vi i c t P = i ∆Vi

(4.40)

where index i stands for the location of the monitoring node with volume ∆V , and i = 1, ..., NV , where NV is the total number of control volumes in the flow domain. The volume averaged mean concentration, c, is calculated as:

98


Nt = 100

c / c¥ 2 Nt = 200

1 Nt = 300

Nt = 400

0

Nt = 500 Figure 4.10: Snapshots of the concentration field in a vertical symmetry plane.

4.7. Validations

99 1 Fluent LBFV

cmix (-)

0.8 0.6 0.4 0.2 0 0

400

800 1200 Nt (-)

1600

Figure 4.11: The coefficient of mixing as a function of dimensionless time. The symbols represent simulated data points by Fluent (not shown in the paper of Van Looy et al. (2004)). The solid line is the simulated time trace of the coefficient of mixing obtained with our lattice-Boltzmann and finite volume (i.e. LBFV) solvers.

c=

P i (ci ∆Vi ) P i ∆Vi

(4.41)

For industrial applications, a value of cmix = 0.01 is typically used as a criterion for well-mixed conditions. In the work of Van Looy et al. (2004), the mixing time is defined as the time needed for reaching the above specified value of cmix . However, for the simulation at infinite Froude number (not shown in Van Looy et al. (2004)) a value of 0.05 has been used. In Figure 4.11, a simulated time trace of the coefficient of mixing is shown (solid line). The symbols represent the simulated coefficient of mixing by Fluent. The results agree very well. Our simulation estimates a mixing time of about 1600 impeller revolutions, which is 35% higher than the Fluent prediction. As a result, the mixing performance predicted by Fluent and our code compare well. The experimental value of the mixing time at infinite Froude number corresponds to about 4500 impeller revolutions (see Figure 5 in the paper of Van Looy et al. (2004)). The predicted mixing times are significantly below this experimental value (up to 75%). However, it is unclear which criterion was used to determine the mixing time in the experiments. Furthermore, the measurement uncertainty was not documented. As a result, the predicted mixing times cannot be equally compared with the experimentally measured mixing time.

100

Chapter 4. Development and validation of a scalar mixing solver 1.03

M/M0 (-)

1.02

1.01 1

0.99 0

400

800 Nt (-)

1200

1600

Figure 4.12: The evolution of the total mass in the system in time.

Nevertheless, the underprediction of the mixing time is significant. A possible explanation on the numerical side may be the Smagorinsky (1963) subgrid-scale model being too dissipative in the first stage of the simulation. While initially the flow field is fully resolved in at least large parts of the tank (the simulation starts with a zero-velocity field), the Smagorinsky (1963) subgrid-scale model artificially generates extensive values of the eddy viscosity, resulting in more mixing than experimentally observed. In this respect, it would be interesting to study the performance of the Voke (1996) subgrid-scale model (see chapter 2). This model significantly suppresses the eddy viscosity in regions where the flow is fully resolved. The performance of the ghost cell technique, applied for modeling the tank’s outer wall, is checked by monitoring the total mass (M ) in the system as a function of time. The time trace of the total mass normalized with the total mass at t = 0 (M0 ) is shown in Figure 4.12. Till N t = 180, the total mass in the system remains nearly constant, for N t > 180 the total mass increases gradually. At N t = 1600 the total mass has increased by 2.5%, and the average mass increase per impeller revolution equals 0.0016%. The reason for the mass increase may be understood from a close inspection of the concentration snapshots. Till N t = 180 the interface between high and low scalar concentration near the walls moves slowly downward at the left side and upward at the right side of the tank (see e.g. the upper snapshot in Figure 4.10). At N t = 180 the jet of high concentration collides on the wall, and is forced upward (not shown). From this point in time, the total mass starts to increase gradually. These observations may indicate that the ghost cell technique has problems with keeping up the fast changes in scalar concentration near the wall which occur for N t > 180.

4.8. Conclusions

4.8

101

Conclusions

In this chapter, we have developed a scalar mixing solver which can be used as a tool for studying blending and two-phase mixing processes including mass transport (such as particle coating, crystallization and dissolution processes). For reasons of computational memory usage, the scalar mixing solver is based on the finite volume method. The discretization of the convection-diffusion equation is treated fully explicitly, which makes implementation relatively simple. The discretization of the temporal and spatial terms in the convection-diffusion equation is second-order accurate. Time discretization is performed with the Adams-Bashford scheme. The diffusion and convection terms are discretized with the central differencing and flux-limiting (TVD) schemes, respectively. The latter scheme significantly suppresses numerical diffusion, and is bounded (in contrast to the QUICK scheme). The effect of subgrid-scale diffusion is taken into account with an eddy diffusion coefficient, which scales with the eddy viscosity. The zero-gradient boundary constraint for a scalar at off-grid walls is imposed by means of a newly developed immersed boundary technique, which is in line with the immersed boundary technique developed by Derksen & Van den Akker (1999) for imposing the no-slip velocity constraint. In contrast to the commonly applied staircase technique in a Cartesian grid environment, the immersed boundary technique provides a more accurate representation of the off-grid walls. Simulations of the scalar mixing in a square cavity have demonstrated the potential of the flux-limited scheme (unconditionally stable, negligible numerical diffusion) in favor of the upwind and QUICK schemes. Furthermore, it is proven that the scheme itself is mass conservative. The scalar mixing in an inclined cavity has been simulated by making use of the staircase and the newly developed immersed boundary technique. Both techniques are not mass conserving. At various grid resolutions, the total change in mass is for both techniques comparable. However, the ghost cell technique provides a more accurate representation of the boundary compared to the staircase technique. An LES of the mixing performance of a turbulent flow in a cylindrical tank with a sideentry mixer has successfully reproduced the mixing time obtained by a LES with Fluent. Furthermore, the mass change resulting from the application of the immersed boundary technique was not significant because of the fine grid resolution adopted. The errors induced by the ghost cell technique may become more significant if the size of an object (e.g. an impeller blade) is not accurately covered by the grid: a coarse grid resolution results in less ghost nodes for imposing the zero-gradient constraint compared

102


to a fine grid. Therefore, the ghost cell technique should be critically assessed in future simulations.

4.8. Conclusions

103

Nomenclature Roman ai , bi ci A c cmix c¯ c∞ D Fr FC FD g L M M0 n ni N NV Pe Q rj r, r+ , r− r0 Re S Sφ Sc Sct t T u

Description coefficients in finite volume discretization scheme coefficient in finite volume discretization scheme representing the amount of anti diffusion surface area concentration mixing index spatial average of the concentration final concentration diameter propeller Froude number convective prefactor in equation 4.3 diffusive prefactor in equation 4.3 magnitude of the gravitational acceleration vector width/height of the lid driven cavity total mass total mass at t = 0 normal vector normal vector component i impeller speed number of control volumes Peclet number variable defined in equation 4.8 radial distance ratio of consecutive scalar gradients reciprocal of r Reynolds number surface of total control volume source term in equation 4.1 Schmidt number turbulent Schmidt number time diameter of the cylindrical tank velocity

Unit kg.s−1 kg.s−1 .[φ] m2 kg.m−3 − kg.m−3 kg.m−3 m − kg.s−1 kg.s−1 m.s−2 m kg kg − − s−1 − − kg.m−3 .s−1 .[φ] m − − − m2 kg.m−3 .s−1 .[φ] − − s m m.s−1

104


ui Ujet Ulid vax vtan x, y, z xi xw x0w

velocity component i impeller stream velocity velocity of the band driving the flow in the cavity axial velocity component tangential velocity component coordinate directions coordinate i wall position wall position normalized with the lattice spacing

m.s−1 m.s−1 m.s−1 m.s−1 m.s−1 − m m −

Greek Γ Γe Γmol ∆ ∆t ∆V ∆ρ θ ν νe ρ φ φI φ−1 , φ0 , φ1

Description scalar diffusion coefficient eddy diffusion coefficient molecular diffusion coefficient lattice spacing time step control volume density difference weighting parameter in equation 4.7 kinematic viscosity eddy viscosity density general scalar variable value of a general scalar variable at an inlet values of a general scalar at the ghost node, and two points in the flow domain flux-limiting function total control volume

Unit − m2 .s−1 m2 .s−1 − s m3 kg.m−3 − m2 .s−1 m2 .s−1 kg.m−3 [φ] [φ] [φ]

Ψ Ω

− m3

5 A parameter study of the mixing time in a turbulent stirred tank by means of LES The mixing performance of the turbulent flow generated by a Rushton turbine has been studied numerically for a wide range of conditions by means of large eddy simulations (LES) including passive scalar transport. The starting point is a simulation of an experimental benchmark case of a Rushton turbine stirred tank. The lattice-Boltzmann Navier-Stokes solver and the Smagorinsky subgrid-scale model have been adopted for solving the turbulent stirred tank flow. A finite volume scheme for solving the convection-diffusion equation for a passive scalar has been coupled to the LES. The simulation results in this benchmark case agree very well with the experimental data from the literature. Subsequently, a parameter study has been carried out varying impeller size and injection position of the passive scalar. All mixing times in these simulations differ from various experimental data from the literature by less than 30% only. The empirical mixing time correlation due to Ruszkowski is evaluated in greater detail. Key words: stirred tank, turbulence, mixing time, LES, scalar transport This chapter has been submitted to AIChE J.

5.1

Introduction

5.1.1 Experimental research on turbulent scalar mixing Stirred tanks play an important role in the chemical, pharmaceutical, food and water treatment industries. The quality of paints, polymers, detergents, drugs and foodstuffs depend

106

Chapter 5. A parameter study of the mixing time by means of LES

on the geometry and operating conditions of the stirred tank. As a result, the mixing performance is of high importance to achieve process optimization. The emphasis of past experimental research for the flow configurations and impellers has been directed to understanding the velocity characteristics of the turbulent flow (e.g. Yianneskis et al., 1987; Schäfer et al., 1998) together with the power required to drive the stirrer (e.g. Rushton et al., 1950; Holmes et al., 1964). The essential requirement of mixers in liquid systems is, however, to bring together two or more fluids, which are initially separate and this implies the need for information of the scalar mixing characteristics of the stirrers. One of the most crucial parameters is the mixing time, which is the time to achieve complete homogenization of inserted passive scalars. A large number of experimental studies focused on mixing performance in terms of the mixing time for different tank and impeller configurations. Kramers et al. (1953) were among the first to report on mixing time in a propeller agitated tank as a function of the baffle position and impeller rotational speed. Moo-Young et al. (1972) investigated the influence of Newtonian and nonNewtonian fluids in different flow configurations on the mixing time, and other researchers focused on the impeller configurations and/or operating conditions in transitional and turbulent flow regions (Hoogendoorn & Den Hartog, 1967; Shiue & Wong, 1984; Sano & Usui, 1985; Bouwmans et al., 1997; Distelhoff et al., 1997). Results of several studies have been combined into empirical correlations, which can be of use for industrial applications (Prochazka & Landau, 1961; Sano & Usui, 1985; Ruszkowski, 1994). The most common methods for mixing time measurements are the conductivity probes (e.g. Shiue & Wong, 1984; Sano & Usui, 1985; Ruszkowski, 1994), or decolorization techniques (Moo-Young et al., 1972). The disadvantage of the conductivity probe is that it disturbs the flow field (intrusive). The decolorization technique is based upon visual observation and relies on a subjective decision by the worker. Another technique more recently used by Distelhoff et al. (1997) is based upon laser induced fluorescence (LIF) for measuring the scalar concentration in continuously operated stirred tanks. The advantages of this technique are that the flow remains undisturbed, the measurement volume is an order of magnitude smaller compared to conductivity probes, and the ability of LIF to measure time traces of the concentration throughout the tank. In literature, there is no standardization of a mixing time experiment. In the first place, mixing time experiments are executed in tanks of different geometries and operating conditions. Secondly, the passive tracer is injected at various injection speeds, thereby changing the influence of jet mixing and stirred tank mixing effects, and positions in the tank. Thirdly, there is no agreement on the position and number of measurement points. And finally, the definition of the mixing time varies from study to study. Most of the studies

5.1. Introduction

107

report the mixing time based on a concentration % (e.g. Bouwmans et al. (1997) and Distelhoff et al. (1997)), others define mixing time by the variance of concentration fluctuations (e.g. Ruszkowski (1994)). In particular the variation of injection speed, number and position of the measurement points, and the definition of the mixing time in the experimental studies result in unequal comparisons between a measured and/or simulated mixing time, and a mixing time or a correlation reported in literature. As a result, a measured and/or simulated mixing time should be viewed with great care. This also applies to mixing time correlations in the literature.

5.1.2 Potential of CFD on scalar mixing Another way of describing and understanding the complex turbulent flow in stirred tanks is through computational modeling. Simulations based on the Reynolds-Averaged NavierStokes (RANS) equations (contained in various commercial software packages) provide a reasonable picture of the overall mixing patterns in the whole tank. The mean flow is directly resolved, whereas the Reynolds stresses (related to turbulent velocity fluctuations) are modeled with a turbulence model. The k- eddy-viscosity model is often used, which assumes isotropic turbulent transport. It is however inappropriate in rotating and/or highly three dimensional flows (Wilcox, 1993). While RANS models predict the mean flow reasonably, it invariably underpredicts the turbulent kinetic energy in the impeller discharge stream by about 50% (e.g. Hartmann et al., 2004a). It is expected that this will certainly affect the predictions of the mixing patterns and mixing time in stirred tanks. There have been many modeling studies of turbulent flow characteristics in stirred tanks, but only a few investigations of mixing patterns have appeared in the literature (e.g. Osman & Varley, 1999; Jaworski et al., 2000; Bujalski et al., 2002). Bouwmans et al. (1997) used a particle tracking technique in a RANS flow field for blending liquids with different densities. Most of these studies have used the RANS approach within a sliding mesh or multiple frames of reference (MFR) framework, which is believed to provide fully predictive simulations of the mixing time. The former (sliding mesh) approach is a fully transient approach, while the latter provides a steady-state approximation of the flow field. It is evident that the sliding mesh approach is more accurate but it it also much more time consuming than the multiple frames of reference approach. In spite of the use of the sliding mesh approach, the predicted mixing times were found, in general, 2 − 3 times higher than the measured values (e.g. Osman & Varley, 1999; Jaworski et al., 2000; Bujalski et al., 2002), which is in line with the underprediction of the turbulence levels (Ng et al., 1998; Hartmann et al., 2004a). While in case of Rushton

108


turbines the mixing time is usually overpredicted due to poor mixing across the central plane of the discharge stream, RANS simulations of pitched blade turbines sometimes yield underpredicted mixing times. This may be due to concentration fluctuations being not really included. The main difficulty with the RANS approach in combination with a closure model is that there is no clear distinction between the part of the turbulent fluctuations that is directly resolved and the part which is represented by the Reynolds stresses. While the mean flow is predicted reasonably well, the turbulence levels are underpredicted by 50%, and this has a negative effect on the mixing characteristics and time. In contrast, an approach that is capable of resolving the turbulent nature of the flow is the large eddy simulation (LES), which is getting more attention in the last years. In LES there is a clear distinction between resolved and unresolved scales, as the range of resolved scales is limited by the numerical grid. LES has shown in the past that it is a powerful tool to study stirred tank flow (Eggels, 1996; Derksen & Van den Akker, 1999; Hartmann et al., 2004a,b; Yeoh et al., 2004; Bakker & Oshinowo, 2004), because it accounts for the unsteady behavior of these flows and it can be effectively employed to explicitly resolve the phenomena directly related to the unsteady boundaries. The LES methodology has been recently applied for a mixing time simulation in the study reported by Yeoh et al. (2005). Their simulation was designed to match the experimental set-up of Lee (1995). The results focus on the mixing patterns, time traces of the concentration and comparison of the predicted mixing time with correlations presented in the literature. This work aims at a thorough analysis (by means of LES) of the mixing time in a stirred tank geometry as a function of impeller size and injection position of the passive scalar. In contrast to the work of Yeoh et al. (2005), the passive scalar is injected at zerospeed to avoid the influence of jet mixing effects on the mixing time. The starting point is a numerical reproduction of the mixing time experiment of Distelhoff et al. (1997). Subsequently, large eddy simulations have been performed with different impeller sizes and injection positions. With a view to the various definitions of the mixing time found in the literature, careful comparisons are made between the simulated mixing times in this work and those obtained by two experimentally obtained mixing time correlations reported by Sano & Usui (1985) and Ruszkowski (1994). The Navier-Stokes equations are solved with a lattice-Boltzmann discretization scheme (Somers, 1993; Derksen & Van den Akker, 1999). This scheme is easy to implement and parallelize, and one does not need to solve the Poisson equation for pressure as in finite volume solvers. Furthermore, we have coupled to the Navier-Stokes solver a finite volume discretization scheme for the scalar transport, which is less memory intensive compared

5.2. Flow system

109

T/10 D = T/3 3D/4

r

0.04D 0.04D

H=T

D/5 0.16D

T/3

z

D/4

T

Figure 5.1: Cross-section of the tank (left). Plan view and cross-section of the impeller (right). At the top level there is a lid. The impeller is a Rushton turbine mounted at height T /3 and has a diameter T /3. In the simulations presented in this work, impeller sizes of T /4 and T /2 have been used as well. to a lattice-Boltzmann discretization. In this way we make use of the advantages of both schemes.

5.2

Flow system

The stirred tank used in this work was a standard configuration cylindrical tank of diameter T = 147 mm, with four equi-spaced baffles of width 0.1T mounted along the perimeter of the tank (Distelhoff et al., 1997). The liquid height was set equal to the tank diameter, H = T . The impeller was a six-bladed Rushton turbine with standard dimensions, mounted at height T /3. A schematic representation of the flow system is shown in Figure 5.1. The flow system can be fully characterized by the flow Reynolds number (Re=N D 2 /ν) if geometric similarity is maintained. With an impeller speed (N ) of 10 s−1 , T /D = 3, and a kinematic viscosity of 1.0 · 10−6 m2 /s (tap water at room temperature) the Reynolds number yields 24, 000 (as also used in Distelhoff et al., 1997).

5.3


5.3.1 Large eddy simulation For the simulation of flow at industrially relevant Reynolds numbers (i.e. at strongly turbulent conditions), direct simulation of the flow is not feasible and turbulence modeling

110


is required. In a Large Eddy Simulation (LES), the small scales in the flow are assumed to be universal and isotropic, and the effect the small scales have on the larger scales is modeled with a subgrid-scale model. The filtering of the small-scale motion is based on the assumption that the motion of the smallest scales is isotropic in nature and that the subgrid-scale energy is dissipated via an inertial subrange that has a geometry independent character. The LES model applied in this research is a standard Smagorinsky model (Smagorinsky, 1963). For more details on the LES methodology used we refer to Hartmann et al. (2004a,b) and Derksen & Van den Akker (1999). A lattice-Boltzmann method (Chen & Doolen, 1998) was used for solving the filtered momentum equations. The specific scheme we used was introduced by Somers (1993), and is described in more detail in Derksen & Van den Akker (1999). The entire tank was simulated on a uniform, cubic computational grid. Inside the computational domain, the no-slip boundary conditions at the cylindrical tank wall, the baffles, the impeller, and the impeller shaft were imposed by an adaptive force-field technique (Derksen & Van den Akker, 1999).

5.3.2 Scalar transport In order to describe scalar transport, the convection diffusion equation needs to be solved. Eggels & Somers (1995) have performed scalar transport calculations on free convective cavity flow with the lattice-Boltzmann discretization scheme. This scheme, however, is more memory intensive than a finite volume formulation of the convection-diffusion equation. In a finite volume discretization we only need to store two or three (depending on the time integrator; Euler-forward or Adams-Bashford, respectively) double precision concentration fields, whereas in the lattice-Boltzmann discretization typically 18 single-precision variables need to be stored. Therefore, we have coupled to the lattice-Boltzmann flow solver a finite volume discretization scheme for the scalar transport. The convection-diffusion equation in compressible form reads ∂ρφ ∂uj ρφ ∂ + = ∂t ∂xj ∂xj

∂φ ρΓ ∂xj

+ Sφ

(5.1)

where ρ is the density of the continuous phase, φ is a general scalar, uj is velocity component j, Γ is the diffusion coefficient (i.e. the sum of the molecular and eddy diffusion coefficients: Γ = Γmol + Γe ) and Sφ is a source term. Since the lattice-Boltzmann scheme is a compressible scheme, we have implemented the discretized form of the compressible convection-diffusion equation. In this context it should be noted that the maximum


111

velocity (which is approximately the tip speed) is set sufficiently low for meeting the incompressibility limit in the lattice-Boltzmann scheme. For reasons of simplicity, we consider in the following discussion the 1-D version of the above equation. Extension to multiple dimensions is straightforward. Integration over a finite volume ∆V and over time ∆t, and using the Gauss theorem leads to ∆V (ρi φi )n+1 − (ρi φi )n = − (FC φ)ni+ 1 + (FC φ)ni− 1 2 2 ∆t n n n + (FD )i+ 1 (φi+1 − φi )

(5.2)

2

n

− (FD )i− 1 (φni − φni−1 ) 2

where FC and FD are convective and diffusive prefactors, e.g. FC;i+ 12 = (ρuA)i+ 1

(5.3)

(5.4)

2

and FD;i+ 12 =

AρΓ ∆

i+ 21

where u is the normal velocity through the cell face with surface A. The density, normal velocity components and the diffusion coefficient at the cell faces are approximated by linear interpolation between the surrounding cell nodes values. For reasons of simplicity, the time discretization in equation 5.2 is first-order Euler forward. We have implemented a second-order accurate Adams-Bashford time discretization scheme. The convection-diffusion equation is solved on the LES grid, and the time step is the same as the LES time step. The finite volume formulation presented is fully explicit; the update of the scalar value at cell node i is determined by the scalar values, velocity components and density at cell node i and its surrounding nodes at time instant n − 1. An explicit scheme has severe restrictions on the time step in order to guarantee stability. However, the time step used for the LES is very small, and no stability problems were encountered in the scalar transport calculations. The molecular Schmidt number (Sc=ν/Γmol ) is typically of the order of 1000, which means that the smallest scale of mass transfer (i.e the Batchelor scale) is a factor 30 smaller than the smallest scale of turbulence (i.e. the Kolmogorov scale). Hence, it is expected that the influence of the scalar subgrid-scale fluctuations on the resolved-scale scalar field is larger than the influence of the subgrid-scale velocity fluctuations on the resolved-scale

112


velocity field. From this, it is reasonable to take a turbulent Schmidt number (Sc t = νe /Γe ) to be smaller than unity (we chose 0.7). The cell face values of the general scalar φ, needed for the convection terms (the first two terms on the right hand side of equation 5.2), have been approximated with a secondorder TVD (Total Variation Diminishing) scheme, that has been introduced by Harten (1983). This scheme belongs to the family of high resolution schemes and does not suffer from numerical diffusion (in contrast to first-order upwind, power-law and hybrid schemes (Patankar, 1980)), and it is unconditionally stable (contrary to central differences, QUICK (Leonard, 1979)). Following Roe (1981), the face value φi+ 21 is written as the sum of a diffusive first-order upwind term and an anti-diffusive one. 1 φi+ 21 = φi + Ψ ri+ 12 (φi+1 − φi ) 2

(5.5)

The anti-diffusive part is multiplied by the flux limiter function Ψ(r), which is a nonlinear function of r, which is the upwind ratio of consecutive gradients of the solution. According to Wang & Hutter (2001), the so-called Superbee flux limiter is the least diffusive of all acceptable limiters. Therefore, we have used the Superbee limiter throughout our simulations. The Superbee limiter is defined as: h i Ψ(r) = max 0, min[1, 2r], min[r, 2]

(5.6)

5.3.3 Immersed boundary technique The Neumann boundary condition at the non-square or moving (off-grid) objects in the flow domain are imposed by means of a newly developed immersed boundary technique, which is applicable in the explicit formulation of the finite volume scheme. With our novel technique, the off-grid walls are more accurately represented (especially the impeller walls) compared to the well-known technique approximating the walls by staircases. The algorithm makes use of so-called ghost cells (i.e. boundary cells with their cell centers outside the flow domain) in order to impose a zero-gradient scalar constraint (i.e. Neumann boundary condition) at the walls that are off-grid. The scalar update is executed only for the cells with their centers inside the flow domain. In order to update the scalar in a cell near a wall, the scalar value in the neighboring ghost cell is needed. The procedure for the determination of the scalar ghost cell value is shown in Figure 5.2. We consider here a curved wall immersed in a Cartesian grid. Two scalar values on a line through the ghost cell center and perpendicular to the wall are estimated through bi-linear interpolation of the surrounding cell nodes. The spacing between


113

Figure 5.2: Schematic representation of the determination of the scalar ghost cell value. White cells are flow cells, gray cells are the ghost cells, black cells are exterior cells. the positions of the two estimated scalar values and the ghost cell center is taken equal to the lattice spacing. The scalar value at the ghost cell is determined via second-order extrapolation with the two scalar values and the zero-gradient constraint at the wall. The ghost cell method is unconditionally stable, which is desirable in the explicit finite volume formulation. Other techniques that make use of body-fitted cells (so-called cut cells, Tucker & Pan (1999); Calhoun & LeVeque (2000)) will certainly fail, as the time step scales with the cell volume. A drawback of the proposed technique is that it does not automatically guarantee scalar mass conservation, and therefore this needs to be checked during the simulations.

5.3.4 Simulation aspects Mixing time simulations have been performed in a Rushton turbine stirred tank. The experimental benchmark case studied, is based on the work of Distelhoff et al. (1997). They observed for Re> 104 only a slight increase of the dimensionless mixing time with increasing Reynolds number. A key parameter dominating the mixing time is the impeller diameter according to the correlation proposed by Sano & Usui (1985) and Ruszkowski (1994). We have investigated the influence of the impeller diameter on the mixing time by means of three flow simulations, each with a different impeller diameter. The tank over impeller diameter ratios equal T /D = 2, T /D = 3 and T /D = 4. The Reynolds number in each of the simulations equals 24, 000. Furthermore, we have varied the injection position of the passive scalar. Four injection positions have been chosen, and as a result, four scalar mixing calculations have been performed simultaneously per flow simulation. The different cases studied are listed in Table 5.1. Please note that θ increases in the direction

114


Table 5.1: Numerical setup. The parameters varied are the impeller diameter and the feed location. Please note that θ increases in the direction of the impeller rotation, and θ = 0o is a mid-way baffle plane. rp is the feed pipe radius. Case T /D Feed: r/T Feed: z/T Feed: θ Feed: rp /T 2A 2 0.17 1 0o 0.0238 2B 2 0.1875 0.483 0o 0.0125 2C 2 0.35 0.333 0o 0.0125 2D 2 0.475 0.8 320o 0.0125 3A 3 0.17 1 0o 0.0238 3B 3 0.125 0.467 0o 0.0125 3C 3 0.211 0.333 0o 0.0125 3D 3 0.475 0.8 320o 0.0125 4A 4 0.17 1 0o 0.0238 4B 4 0.09375 0.458 0o 0.0125 4C 4 0.225 0.333 0o 0.0125 4D 4 0.475 0.8 320o 0.0125

of the impeller rotation. Case 3A resembles the Distelhoff et al. (1997) experiment. The feed position is located at the top of the tank in a mid-way baffle plane (i.e. θ = 0o ) at 0.17T from the tank centerline. Cases 2A and 4A have the same feed location and dimension of the feed pipe, but with tank over impeller ratio’s T /D = 2 and T /D = 4, respectively. Cases 2B, 3B and 4B have the feed pipe location at 0.1T above the top of the impeller blade, and at a radial position halfway along the impeller blade. Cases 2C, 3C and 4C have the feed pipe located at disk height at 0.1T from the impeller tip. In cases 2D, 3D and 4D the passive scalar is injected in the wake of a baffle (i.e. at 5o angle with respect to the baffle) at 0.8T axial height. The tracer injection time in the mixing time experiment was chosen to be less than 1% of a typical mixing time (Distelhoff et al., 1997). As a result, the injection time was set at half an impeller revolution (i.e. N t = 0.5). This injection time was set in all simulations performed. Time traces of the scalar concentration were recorded at various monitoring points positioned in accordance with Distelhoff et al. (1997). A total of 32 monitoring points were set in the four vertical planes mid-way between two baffles. The points were located at 0.19T and 0.67T axial heights at r/T = 0.126, 0.252, 0.361, 0.469, respectively. The code runs on a parallel computer platform by means of domain decomposition: the computational domain was horizontally (i.e. perpendicular to the tank centerline) split

5.4. Results

115

in eight equally-sized subdomains. For the coupled LES/scalar mixing simulations, a cubic, Cartesian grid of 240 3 lattice cells was defined. The diameter of the tank equals 240 lattice spacings, and hence the spatial resolution equals T /240 which corresponds to 0.6125 mm in the experiment. The temporal resolution is limited by the lattice-Boltzmann method. In order to meet the incompressibility limit in the lattice-Boltzmann discretization scheme, the tip speed of the impeller should approximately be set at 0.1 lattice spacings per time step. As a result, the impeller speed is different for the three impeller types. For the T /D = 2, 3, 4 cases the impeller makes a full revolution in 3600, 2400 and 1800 time steps, respectively. The corresponding temporal resolutions in physical units are 62.6 µs, 41.7 µs and 31.3 µs (at N = 4.44 s−1 , 10 s−1 and 17.77 s−1 , respectively). For the LES, 21 (18 directions for the LB-particles and 3 force components) singleprecision, real values need to be stored. For the scalar mixing simulation twelve (four concentration fields at the time instants n + 1, n and n − 1) double-precision, real scalar values needs to be stored. The memory requirements of the simulation are proportional to the grid size, resulting in an executable of about 2 GByte. The simulations were performed on an in-house PC cluster with eight Athlon 1, 800+ MHz processors using an MPI message passing tool for communication within the parallel code. To calculate one time step takes about 58 seconds wall-clock time, hence an impeller revolution takes 1 − 2 days. The scalar was injected in a quasi steady state flow field that was obtained from a zerovelocity flow field. The quasi steady state was reached after 10 − 40 impeller revolutions (depending on the impeller size), which has been checked by monitoring the total kinetic energy of the flow as a function of time.

5.4

Results

5.4.1 Snapshots of the scalar concentration field Figure 5.3 gives an impression of the scalar mixing process in the Rushton turbine stirred tank (Case 3A; T /D = 3) during the course of the simulation. In the first stage of the mixing process, high concentration, macroscopic structures are identified that are advected by the flow from the tracer injection point (at the top of the tank) toward the impeller region. These structures are vigorously mixed by the turbulence generated by the revolving impeller. After about 30 impeller revolutions, the scalar concentration is heading toward a homogeneous distribution. The impact of the impeller diameter on the duration of the mixing process is illustrated

116


c/c¥ 5 4

Nt = 3

Nt = 9

Nt = 15 3 2 1 0

Nt = 21

Nt = 27

Nt = 33

Figure 5.3: Case 3A; T /D = 3. Six instantaneous realizations of the scalar field in a midway baffle plane. The scalar is injected at the top of the tank (black dot in the graphs). in Figure 5.4. In this case, the passive scalar is injected 0.1T above the impeller disk. Again, high concentration macroscopic structures are identified in all cases presented. The impeller diameter significantly influences the mixing process: the mixing time decreases at increasing impeller diameter. While in case 4B (T /D = 4) more than 32 revolutions are needed to reach a more or less homogeneous scalar distribution, in case 2B (T /D = 2) only 8 revolutions are necessary to reach a similar situation.

5.4.2 Time series Case 3A: comparison simulation and experiment Simulated and experimentally measured time series are shown in Figure 5.5 at four monitoring points (i.e. two radial positions and two axial heights) in a vertical mid-way baffle plane at θ = 90o . The experimentally measured time series are taken from Distelhoff et al. (1997). A qualitative comparison between the simulated and experimentally mea-

5.4. Results

117

Nt = 8

Nt = 16

Nt = 32

c/c¥ 5 4 3 2 1

Nt = 6

Nt = 12

Nt = 24 0

Nt = 4

Nt = 8

Nt = 16

Figure 5.4: Cases 4B (T /D = 4); upper graphs, 3B (T /D = 3); middle graphs and 2B (T /D = 2); lower graphs. Three instantaneous realizations of the scalar field in a mid-way baffle plane. The scalar is injected 0.1T above the impeller (black dot in the graphs).

sured time series results in the conclusion that the time series compare well in terms of the path of the curves and the mixing time scale. The time step used in the simulations is a factor of 20 smaller than the sampling time in the experiments (Distelhoff et al., 1997). As a result, the simulated time series show more concentration fluctuations compared to the experimentally measured time series. The simulated time series at z/T = 0.19 show a delayed response on the tracer injec-

118


5

5 r/T = 0.252 0.469

r/T = 0.252 0.469

4

3

c/c¥ (-)

c/c¥ (-)

4

2 1

3 2 1

0

0 0

10

20

30 40 Nt (-)

50

60

0

(a) Simulation, z/T = 0.67

20

30 40 Nt (-)

50

60

(b) Experiment, z/T = 0.67

5

5 r/T = 0.252 0.469

4

r/T = 0.252 0.469

4

3

c/c¥ (-)

c/c¥ (-)

10

2 1

3 2 1

0

0 0

10

20

30 40 Nt (-)

(c) Simulation, z/T = 0.19

50

60

0

10

20

30 40 Nt (-)

50

60

(d) Experiment, z/T = 0.19

Figure 5.5: Case 3A, tracer injection point at the top of the tank. Simulated (a,c) and experimental (b,d) time series at various monitoring points in a vertical midway baffle plane (θ = 90o ). The experimentally measured time series are taken from Distelhoff et al. (1997).

tion (0 ≤ N t ≤ 0.5) compared to the time series at z/T = 0.67, which is to be expected as the tracer is injected at the top of the tank. The experimental time series show an earlier response on the tracer injection compared to the simulated time series.

5.4. Results

119

5

5 z/T = 0.67 0.19

z/T = 0.67 0.19

4

c/c¥ (-)

c/c¥ (-)

4 3 2

3 2 1

1

0

0 0

4

8

12 Nt (-)

16

20

0

10

20

40 30 Nt (-)

50

60

(b) Case 3B, T /D = 3, r/T = 0.126.

(a) Case 2B, T /D = 2, r/T = 0.126. Data for N t < 2 is missing.

5 z/T = 0.67 0.19

c/c¥ (-)

4 3 2 1 0 0

20

40

60 Nt (-)

80

100

(c) Case 4B, T /D = 4, r/T = 0.126.

Figure 5.6: Cases 2B, 3B and 4B, tracer injection point at 0.1T above impeller blade. Simulated time series at various monitoring points in a vertical mid-way baffle plane (θ = 90o ). Cases 2B, 3B, 4B: influence impeller size In Figure 5.6 simulated time series are shown for the three different impeller dimensions at two monitoring points. In this case, the tracer is injected at 0.1T above the impeller blade. Due to a processing error, the data for N t < 2 is missing in Figure 5.6a. According

120


5

5 z/T = 0.67 0.19

z/T = 0.67 0.19

4

3

c/c¥ (-)

c/c¥ (-)

4

2 1

3 2 1

0

0 0

10

20

40 30 Nt (-)

50

0

60

(a) Case 3A, r/T = 0.126.

20

40 30 Nt (-)

50

60

(b) Case 3B, r/T = 0.126.

5

5 z/T = 0.67 0.19

4

z/T = 0.67 0.19

4

3

c/c¥ (-)

c/c¥ (-)

10

2 1

3 2 1

0

0 0

10

20

40 30 Nt (-)

(c) Case 3C, r/T = 0.126.

50

60

0

10

20

40 30 Nt (-)

50

60

(d) Case 3D, r/T = 0.126.

Figure 5.7: Cases 3A, 3B, 3C and 3D (i.e. four different feed point positions). Simulated time series at various monitoring points in a vertical mid-way baffle plane (θ = 180o ).

to Sano & Usui (1985) and Ruszkowski (1994), mixing time strongly depends on the impeller size. This is confirmed by the simulated time series. For T /D = 2 only 14 impeller revolutions are needed to reach more or less the final concentration. About 45 and 100 impeller revolutions are needed for T /D = 3 and T /D = 4, respectively, to reach a similar stage.

5.4. Results

121

Cases 3A, 3B, 3C, 3D: influence injection position Figure 5.7 shows simulated time series with the T /D = 3 impeller type. The position of the tracer injection point has a small effect on the mixing time scale. The time traces corresponding to the cases 3A and 3D show that after roughly 45 impeller revolutions the concentration approximates the final level. In cases 3B and 3C a similar situation is reached after roughly 35 impeller revolutions.

5.4.3 Mixing time and coefficient of mixing The mixing time, θm , is the time required to mix the added passive tracer with the contents of the tank until a certain degree of uniformity is achieved. The location in the tank where the passive tracer concentration is measured, has to represent the state of mixing of the entire vessel (i.e. the mixing time measured at any point in the tank should be the same). Distelhoff et al. (1997) measured mixing times where the concentration variations were smaller than 10%, 5% and 1% of the fully mixed concentration. These concentrations are called the 90%, 95% and 99% concentration. The mixing times defined by the 90% and 95% concentration varied by up to 27% and 21%, respectively, between different regions of the tank. These uncertainties can be reduced by averaging the measurements obtained simultaneously at several locations. The variation between the mixing times for the 99% concentration was found substantially smaller. Based on these observations, Distelhoff et al. (1997) based their mixing time on the 99% concentration. Another way of defining a degree of mixing is through the variance of the concentration measured at various positions in the tank. In order to have a precise mathematical formulation of mixing criteria that can be used for estimation of mixing times, the coefficient of mixing is introduced. The coefficient of mixing is defined as:

cmix

v ! u P c −c 2 i u ∆V i i Pc =t i ∆Vi

(5.7)

where index i stands for the location of the monitoring point with volume ∆V , and i = 1, ..., K, where K is the total number of monitoring points. The volume averaged mean concentration, c, is calculated as: P i (ci ∆Vi ) (5.8) c= P i ∆Vi The method provides the concentration variance of points in different regions of the

122


4000

0.25

number (-)

cmix (-)

0.2 0.15 0.1

3000 2000 1000

0.05 0 70

fit 1 fit 2

0 75

80 85 90 % concentration (a)

95

100

0.004

0.005 0.006 cmix (-)

0.007

(b)

Figure 5.8: Calibration graph (a) of coefficient of mixing vs. concentration %. The histogram of the coefficient of mixing at 99% concentration is shown in (b), together with two Gaussian fits. tank. In order to link the coefficient of mixing to a concentration % (e.g. the 99% concentration) we perform the following numerical experiment. Consider a set of K statistically independent, dimensionless concentrations ranging between 0.99 and 1.01. The dimensionless concentrations are obtained at the monitoring points. With these numbers, a coefficient of mixing can be calculated. If we repeat this experiment L times (i.e inclusion of time as a variable), we obtain a distribution function of the coefficient of mixing for the 99% concentration. The experiment is repeated for other concentration percentages. Figure 5.8a shows a calibration graph of the coefficient of mixing vs. the concentration %. The total number of monitoring points (K) equals 32 and the number of repetition (L) was set at 105 , which resembles a time interval comparable to the length of the simulated time series. The graph shows that with increasing concentration %, the average coefficient of mixing decreases, and the distribution becomes narrower as expected. Figure 5.8b shows the histogram of the coefficient of mixing for the 99% concentration. The distribution function is nearly Gaussian, as shown by the two Gaussian fits. Gaussian fit 1 is fitted in the range 0.0045 − 0.006 and fit 2 in the range 0.005 − 0.0065. The average coefficient of mixing that can be linked to the 99% concentration equals 0.00566. For the 95% and 90% concentrations, the average coefficient of mixing equals 0.0283 and 0.0566, respectively. The coefficient of mixing as a function of time calculated from the concentrations at

5.4. Results

123

101

101

10

0

10

-1

90% 95%

10-2

3A 3B 3C 3D

0

cmix (-)

cmix (-)

10

2A 2B 2C 2D

10-1 90% 95% -2

10

99%

99% 10

-3

0

5

10 15 Nt (-)

20

25

(a) Cases 2A, 2B, 2C and 2D; T /D = 2. Lines fitted in the range 6 ≤ N t ≤ 14.

10

-3

0

10 20 30 40 50 60 70 80 Nt (-)

(b) Cases 3A, 3B, 3C and 3D; T /D = 3. Lines fitted in the range 20 ≤ N t ≤ 35.

101

cmix (-)

10

4A 4B 4C 4D

0

10-1 90% 95% 10

-2

99% -3

10

0

20

40

60 80 100 120 140 Nt (-)

(c) Cases 4A, 4B, 4C and 4D; T /D = 4. Lines fitted in the range 40 ≤ N t ≤ 80.

Figure 5.9: The coefficient of mixing as a function of the dimensionless time. The horizontal lines labeled 90%, 95% and 99% represent the corresponding concentration percentage.

the 32 monitoring points is shown in Figure 5.9, where all cases are summarized in three graphs. In Figure 5.9a the data is missing for N t < 2, due to a processing error. The three graphs show similarities. In the first stage of the mixing process, the macroscopic

124


Table 5.2: The mixing times for the 90%, 95% and 99% concentrations, respectively. Case N θm,90% N θm,95% N θm,99% 2A 13.6 16.0 21.3 2B 13.0 15.1 20.2 2C 12.1 14.2 19.3 2D 16.4 18.5 23.5 3A 46.8 54.7 73.0 3B 38.1 44.2 58.5 3C 33.0 39.8 55.7 3D 44.5 51.1 66.5 4A 85.0 100.2 135.4 4B 84.5 98.7 131.9 4C 81.8 96.5 130.8 4D 81.4 96.6 131.9

high-concentration structures are broken up and mixed into smaller structures with lower concentration. The coefficient of mixing strongly fluctuates with a value higher than one, and no clear trend is observed. In the second stage, all three graphs reveal an exponential decay of the coefficient of mixing. This has also been stated by Ruszkowski (1994). He based, similarly to our work, the degree of mixedness on the variance of concentration fluctuations. In the final stage of the simulations, the coefficient of mixing stabilizes at the value of approximately 0.02. This value corresponds with the 96% − 97% concentration, which means that concentration fluctuations of 3%−4% remain. This is clearly unphysical, and these fluctuations are attributed to the numerics (e.g. the compressibility of the latticeBoltzmann and finite volume schemes used, the introduction of numerical errors at the walls by the our novel immersed boundary technique). Table 5.2 lists the value of the 90%, 95% and 99% mixing times, respectively for all cases. These mixing times are evaluated by intersection of the fitted straight lines (implying exponential decay) and the horizontal lines corresponding to the 90%, 95% and 99% concentration, respectively (see Figure 5.9). For the mixing case studied by Distelhoff et al. (1997), it took 57 impeller revolutions to reach the 99% concentration. The simulation of case 3A predicts that about 47, 55 and 73 impeller revolutions that are needed to obtain the 90%, 95% and 99% concentrations, respectively. As a result, the simulation overestimates the 99% mixing time by 28%. The spread of the mixing times with respect to the injection points for the T /D = 3 cases, however, is relatively large compared to the T /D = 2 and T /D = 4 cases. The 99% mixing time for the other

5.4. Results

125

injection positions all lie close to the 99% mixing time reported by Distelhoff et al. (1997). The mixing time in a stirred tank geometry reported by Yeoh et al. (2005) resembles the 95% mixing time, and equals roughly 33 impeller revolutions. Our results are 65%, 33%, 20% and 54% higher for the cases 3A, 3B, 3C and 3D, respectively. While case 3A is most similar to the case studied by Yeoh et al. (2005) (injection position at r/T = 0.17 instead of 0.25), the difference between the predicted mixing times is the largest of all cases (i.e. 65%). This observation can be explained as follows. In the first place, Yeoh et al. (2005) used only typically 500k cells for their LES. As a result, their flow and scalar fields are much less resolved compared to those presented in this work. Less resolution leads to a reduction of the predicted mixing time. In the second place, in the simulation of Yeoh et al. (2005) the tracer is injected at an injection speed of 15.9 m/s for about 50% of their predicted mixing time, whereas in our work the tracer is injected at zero-speed in accordance to the experiment by Distelhoff et al. (1997). A higher injection speed introduces jet mixing effects, the passive tracer reaches the impeller region at an earlier time, and the mixing time is reduced.

5.4.4 Mixing time correlations Numerical simulations, such as presented in this work, contain all information needed to have an equal comparison between the simulated mixing times and experimentally obtained mixing times and correlations from the literature. This has been illustrated in this work by repeating the mixing time experiment of Distelhoff et al. (1997), and it will be demonstrated in this section. Sano & Usui (1985) measured the mixing times under turbulent conditions in agitated tanks equipped with either a paddle or turbine impeller. The operating parameters they varied were the impeller diameter, the height of the impeller blade (w) and the number of impeller blades (nb ). The mixing time was determined by the point on a recorder chart at which the signal reaches the final concentration after the tracer liquid is poured into the tank, with a degree of relative deviation of concentration < 1% (i.e. the 99% mixing time). From the measurements the following correlation was proposed: w 0.1 T 2.5 N θm = 12.2P o−3/4 nb T D

(5.9)

In the geometry under consideration in this work, w = 0.2D and nb = 6. The power number (Po) has been measured by Distelhoff et al. (1997) for the particular case of interest and equals 4.5. As a result, the Sano & Usui (1985) correlation reduces to

126


150

99% Sano & Usui (1985)

Nqm (-)

120 90 60 30 0

0

1

3

2

4

5

T/D (-) Figure 5.10: Simulated 99% mixing times as a function of T /D. The symbols refer to the cases mentioned in table 5.1. The thick, solid line resembles the Sano & Usui (1985) mixing time correlation.

N θm = 4.02

T D

2.4

(5.10)

It is noted that the power number is not independent of T /D according to e.g. Shiue & Wong (1984). In the measured range 2.2 < T /D < 3, the power number varies between 5.5 (at T /D = 2.2) and 4.3 (at T /D = 3). With respect to the mixing time, this effect will be less significant (i.e a deviation of 10% − 20%) compared to the effect of T /D. As a result, the relationship between power number and T /D will be ignored in the following discussions. In Figure 5.10 the simulation results of the 99% mixing time and the Sano & Usui (1985) correlation (equation 5.10) are shown as a function of T /D. The mixing time calculated of all cases with T /D = 2 and the cases 3B and 3C (T /D = 3) compare very well with the correlation proposed by Sano & Usui (1985). The mixing time for case 3A is overestimated by 30%, and for the cases 3D, 4A-D by 18%. The overestimation is relatively small compared with the deviations (100% and more) reported in earlier CFD work based on the RANS approach (e.g. Osman & Varley, 1999; Jaworski et al., 2000; Bujalski et al., 2002). Furthermore, it should be noticed that √ the correlation proposed by Sano & Usui (1985) is based on measurements in the range 2 ≤ T /D ≤ 2.5, which only partly overlaps the T /D range simulated in this work.

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127

Table 5.3: The fit parameters a0 and a1 of the mixing time correlation given in equation 5.11. The fit parameters are given for the 90%, 95% and 99% mixing times, respectively. Concentration % a0 a1 90% 2.46 2.54 95% 2.80 2.56 99% 3.60 2.60

The simulation results are fitted based on a similar correlation: N θm = a 0

T D

a1

(5.11)

where a0 and a1 are the fit parameters. The values of the fit parameters are listed in Table 5.3. The value of the fit parameter a1 shows an increased dependency of T /D with increasing concentration %. The 99% mixing time obtained with our corresponding correlation lies about 9% above the experimental value measured by Distelhoff et al. (1997). Next to the Distelhoff et al. (1997) mixing time experiment, the experimentally obtained mixing times of Hoogendoorn & Den Hartog (1967) and Bouwmans et al. (1997) are compared with the mixing time obtained from our correlations. The 90% mixing time has been measured at various Reynolds numbers (up to 104 ) by Hoogendoorn & Den Hartog (1967). At high Reynolds numbers the mixing time becomes Reynolds number independent and requires about 30 impeller revolutions. The mixing time obtained with our 90% correlation equals 40 impeller revolutions, which is an overestimation of about 30%. Bouwmans et al. (1997) measured a 95% mixing time of 25 impeller revolutions in a Rushton turbine stirred tank with T /D = 2.5. The value of the 95% mixing time (i.e 29 impeller revolutions) obtained with our correlation lies 17% above the experimental value of Bouwmans et al. (1997).

5.4.5 The Ruszkowski (1994) mixing time correlation Ruszkowski (1994) measured the mixing times under turbulent conditions for a Rushton turbine and other impellers and suggested the following correlation: N θm = 5.3P o

−1/3

T D

2

= 3.21

T D

2

(5.12)

Our 90%, 95% and 99% mixing times listed in Table 5.2 have been plotted, together

128


150

90% 95% 99% fit 90% fit 95% fit 99% Ruszkowski (1994)

Nqm (-)

120 90 60 30 0

0

1

3

2

4

5

T/D (-) Figure 5.11: Simulated 90%, 95% and 99% mixing times as a function of T /D, together with the fitted lines based on equation 5.11. The symbols refer to the cases mentioned in table 5.1. The thick, solid line resembles the Ruszkowski (1994) mixing time correlation.

with the fit lines (equation 5.11), as a function of T /D in Figure 5.11. For comparison the Ruszkowski (1994) correlation (equation 5.12) is shown in the figure. The simulation results for the 90%, 95% and 99% mixing times significantly deviate from the Ruszkowski (1994) correlation. If the Ruszkowski (1994) correlation would correspond to e.g. the 90% mixing time, the power number should equal 1 at T /D = 4. This value is significantly lower than the range of experimentally measured power numbers by Shiue & Wong (1984). Based on this remark and the results shown in Figure 5.11, the Ruszkowski (1994) correlation is likely to correspond to a concentration % lower than 90%. We will now investigate this possibility in greater detail. The Ruszkowski (1994) correlation is based on a mixing index defined by

IM

v u 4n+7 u 1 X ci,rms − c∞ 2 =1−t 8 i=4n c∞

where n is a time index, c∞ is the final concentration and ci,rms is defined by

(5.13)

10

2

10

1

10

0

10

-1

10

-2

10

-3

129

3A 3B 3C 3D 1-IM (-)

1-IM (-)

5.4. Results

-4

2

10

1

10

0

10

-1

10

-2

10

-3

3A 3B 3C 3D

-4

10 -5 10 10

10

10 -5 10

-6

10

0

10

20

30 Nt (-)

40

50

(a) Time window 8 points, in accordance with equation 5.13.

60

-6

0

10

20

30 Nt (-)

40

50

60

(b) Time window 400 points, i.e. one blade passage period.

Figure 5.12: The value of 1 − IM as a function of time for the T /D = 3 cases. The dashed line represents IM = 0.95.

ci,rms

v  u M u u 1 X 2  =t c N j=1 i,j

(5.14)

where ci,j is the concentration at a monitoring point j at time t = i∆t, with ∆t the sample time in the experiment. The mixing index expresses the root mean square concentration fluctuation as a fraction of the mean concentration in the tank after addition of the tracer. IM can vary from −∞ for an ’infinitely unmixed’ system to 1.0 for a perfectly mixed system. The mixing index is calculated for one section with eight points, the section is moved four data points along the time history and recalculated, and so on. The mixing time was defined as the time for IM to reach 0.95. Figure 5.12a shows the simulated time trace of 1−IM for the cases 3A, 3B, 3C and 3D. The value of IM = 0.95 is represented by a dashed line. All four cases show that the mixing time equals approximately 29.5 impeller revolutions (i.e. average over the four cases). Increasing the width of the time window to a blade passage period (Figure 5.12b) results in less fluctuations of IM , but it has only a marginal influence on the path of the curves, and consequently the mixing time. Despite the mixing vessel geometry of Ruszkowski (1994) was a bit different compared to our system (e.g. dished bottom instead of a flat

130


150

Nqm (-)

70% fit 70% 120 Ruszkowski (1994) 90 60 30 0

0

1

3

2

4

5

T/D (-) Figure 5.13: 70% Mixing time as a function of T /D. The Ruszkowski (1994) correlation and the fitted correlation (i.e. equation 5.11) are shown for comparison. The symbols refer to the cases mentioned in table 5.1.

bottom, baffle diameter T /12 instead of T /10), the mixing time predicted by our simulations compares very well with the value of 31 impeller revolutions reported by Ruszkowski (1994). The mixing time based on IM = 0.95 for the cases 3A, 3B, 3C and 3D equal 32, 30, 24 and 32 impeller revolutions, respectively. With the help of Figure 5.9b, these mixing times correspond with a coefficient of mixing ranging between 0.12 and 0.22. Figure 5.8a shows that this range of the coefficient of mixing corresponds with a concentration % of about 70%. As a result, the definition of the mixing time by Ruszkowski (1994), based on IM = 0.95, corresponds to the time required for the concentration fluctuations to become less than 30% of the fully mixed concentration (i.e. the 70% mixing time). In order to check whether this statement also holds for the T /D = 2 and T /D = 4 cases, the 70% mixing time has been calculated based on the average coefficient of mixing of 0.17. The results are shown in Figure 5.13. The Ruszkowski (1994) correlation now fits the data for T /D = 2 and T /D = 3. For T /D = 4 the simulations overestimate the mixing time by about 15%. The data have been fitted with the correlation given in equation 5.11, resulting in the fit parameters a0 = 1.92 and a1 = 2.48.

5.4. Results

131

c/c¥ 5 4 3 2 1 0

(a) Complete concentration field.

(b) Enlarged view of the concentration field near the impeller.

(c) Enlarged view of the concentration field near the upper left baffle in (a).

Figure 5.14: Performance immersed boundary technique. Snapshot of the concentration in a horizontal plane at z/T = 0.33 (i.e. disk height) and N t = 15. The dots represent the reconstructed position of the zero-gradient boundary condition, the lines represent the walls of the geometry.

5.4.6 Performance immersed boundary technique, mass conservation In this section we check the performance of the immersed boundary technique which imposes a zero-gradient at walls that are off-grid (see Section 5.3.3). It is, however, not


1.1

1.1

1.05

1.05

M/M0 5.

M/M0 5.

132

1 0.95 0.9 0

10

20

30 40 Nt (-)

(a) Cases 3A, 3B, 3C and 3D

3A 3B 3C 3D 50

1 0.95

2B 3B 4B

0.9 60

0

20

40 60 Nt (-)

80

100

(b) Cases 2B, 3B and 4B

Figure 5.15: The dimensionless total mass as a function of time. M0.5 is the total mass after half an impeller revolution (i.e. after the addition of the full amount of tracer).

expected that a violation of the mass conservation constraint by the immersed boundary technique will have a significant impact on the mixing time. Possible mass leakage occurs only at the walls, and a local increase or decrease of mass will not affect the decay rate of the coefficient of mixing (observed in Figure 5.9), and consequently the mixing time. In order to check the performance of the immersed boundary technique, the position of the zero-gradient constraint has been reconstructed with the ghost cell values and two estimated scalar values in the concentration field (i.e. similar to the routine described in Section 5.3.3. Figure 5.14a shows a snapshot of the concentration field at z/T = 0.33 (i.e. disk height) after 15 impeller revolutions. The reconstructed zero-gradient positions are represented by (overlapped) dots. Figure 5.14b shows an enlarged view of the impeller region. The dots are clearly identified. The lines represent the walls of the impeller geometry. The dots and the lines overlap, which means that the zero-gradient boundary constraint coincides with the geometry walls, as should be expected. The same conclusion is drawn based on Figure 5.14c that shows an enlarged view near the baffle. The total mass in the system has been monitored in order to check the mass conservation constraint. The total mass, M , is calculated as follows:

5.4. Results

133

M=

Vtot X

ci ∆Vi

(5.15)

i=1

where Vtot is the amount of control volumes in the flow domain. The dimensionless total mass (with M0.5 the total mass after half an impeller revolution, i.e. after stopping the injection) as a function of the dimensionless time for the cases with the T /D = 3 impeller is shown in Figure 5.15a. The time traces of the total mass show after some time a more or less linear increase of the total mass. At the end of the simulations, the total mass increase has increased by 5% − 8%. The slope of the mass increase depends on the impeller size as can be seen in Figure 5.15b. An increase of impeller size results in a steeper slope of the mass as a function of time. This observation implies that mass conservation at the impeller blades is a problem; larger blades result in a larger mass increase. The increase of mass may be attributed to the blade thickness (i.e. three lattice spacings). The blades are thin with respect to the resolution of the grid, and consequently the blade edges are not accurately represented by the immersed boundary technique. As a result, the blade edges may be considered as the main sources of mass leakage. Furthermore, increasing the impeller size results in a larger surface of the impeller blade edges, which explains the steeper increase of the total mass observed in Figure 5.15b. Also the edges of the baffles and impeller disk may contribute to the mass increase, as their edges have the same thickness as the impeller blade edge. That the blade and baffle thickness with respect to the grid resolution is a problem in conserving the total mass is observed in the time traces shown in Figure 5.15a. The mass increase of case 3D starts in the early stage of the mixing process, because of the vicinity of a baffle. In case 3C the injection point is in the impeller stream, where the concentration is strongly advected radially outward toward the baffles. In case 3B the injection point is 0.1T above the impeller and the mass increase starts at roughly N t = 2. In case 3A, the mass injection point is at the top of the tank, and it takes some time for the concentration being advected toward the impeller region. As a result, the mass increase is expected at a later moment in time (i.e. N t = 10) compared to the other cases (see e.g. the instantaneous concentration fields in Figure 5.3). In summary, the immersed boundary technique developed positions accurately the zero-gradient constraint at the geometry walls (Figure 5.14), but the mass conservation constraint is not satisfied (Figure 5.15). The mass increase is significant (5% − 8%), and in future research the current technique needs to be improved. It is expected that an increase of the grid resolution or the usage of a cylindrical grid in the scalar mixing solver

134


will result in a better approximation of the mass conservation.

5.5

Conclusions

In this chapter, we have investigated the influence of injection position and impeller size on the mixing time in a Rushton turbine stirred tank by means of a LES including scalar mixing. In literature, there is no standardization of a mixing time experiment. Mixing times are reported for tanks of different layouts and operating conditions, tracer injection positions and injection speeds, number and positions of measurement points. Furthermore, there is no agreement on the definition of the mixing time. As a result, a measured and/or simulated mixing time should be compared with great care as is done in this work. This also applies to mixing time correlations in the literature. We have coupled the mixing time to a coefficient of mixing, that provides a concentration variance of points in different regions in the tank. Simulated time traces of the coefficient of mixing revealed after some impeller revolutions an exponential decay. At the end of the calculations, the coefficient of mixing stabilized at about 0.02. This is unphysical, and this effect is attributed to numerical errors, induced by e.g. the compressibility effect of the schemes used or the scalar concentration errors introduced at the walls by the immersed boundary technique. The exponential decay has been extrapolated through a fitting procedure. The mixing times were defined by the intersection of the fitted lines and the calibrated values of the 90%, 95% and 99% concentration criterions, respectively. The mixing time is significantly influenced by the impeller size. The simulated results of the 90%, 95% and 99% mixing times show an impeller size dependency of typically (T /D)2.5 . The simulated mixing times agree within 30% with experimentally obtained values. Furthermore, the simulated 99% mixing times compared very well with the Sano & Usui (1985) mixing time correlation, within an overestimation up to 30%. Our work has indicated that the Ruszkowski (1994) mixing time correlation corresponds to the 70% concentration criterion. The position of the tracer injection point does not significantly affect the mixing time. While a spread of the simulated mixing time was observed in the T /D = 3 cases, it was found to be clearly less significant in the T /D = 2 and T /D = 4 cases. The predicted 99% mixing time in the case similar to the experiment of Distelhoff et al. (1997) overestimates the experimentally obtained mixing time with 26%. The scalar transport has been solved with the finite volume approach. In contrast with a lattice-Boltzmann discretization of the convection diffusion equation, the former approach yields a less memory intensive scheme. False diffusion effects in the finite volume scheme

5.5. Conclusions

135

are minimized through the use of a high-resolution TVD scheme. Furthermore, an immersed boundary technique is developed that imposes a zero-gradient boundary constraint for the scalar at the walls that are off-grid. Our novel immersed boundary technique does not guarantee mass conservation. The average mass increase per impeller revolution depends on the impeller size being 0.25%, 0.125% and 0.05% for the impeller sizes T /D = 2, T /D = 3 and T /D = 4, respectively. The unphysical mass increase is attributed to the mass leakage at the edges of the impeller blades, impeller disk and baffles as they are not accurately represented by the grid resolution. It is expected that (local) grid refinement or the usage of a cylindrical grid in the scalar mixing solver will result in less leakage of mass. The coupled LES and finite volume scalar mixing simulations have provided a detailed insight in the scalar mixing characteristics of the stirred tank. The detailed simulations presented in this work enabled a careful comparison of the simulated mixing times with those experimentally and/or numerically measured, and with those obtained via correlations in the literature. Furthermore, this research opens a promising perspective for the coupling of the developed code to a particle transport code developed by Derksen (2003) in order to study either crystallization or dissolution processes.

136


Nomenclature Roman a0 ,a1 A A+ c ci,rms cmix c¯ c∞ cs D FC FD H IM K L n nb N M M0.5 Po r r rp Re S Sφ Sc Sct

Description fit parameters of the mixing time correlation in equation 5.11 surface area constant in the Van Driest (1956) wall damping function concentration the rms concentration at time i∆t coefficient of mixing spatial average of the concentration final concentration Smagorinsky constant impeller diameter convective prefactor in equation 5.2 diffusive prefactor in equation 5.2 height of the tank mixing index defined by Ruszkowski (1994) number of monitoring points value of repetition of experiment described in section 5.4.3 time index number of impeller blades impeller speed total mass injected mass after half an impeller revolution Power number radial coordinate upwind ratio of consecutive gradients radius of the feed pipe Reynolds number resolved deformation rate source term in convection-diffusion equation molecular Schmidt number turbulent Schmidt number

Unit m2 kg.m−3 kg.m−3 kg.m−3 m kg.s−1 kg.s−1 m s−1 kg kg m m s−1 kg 2 .m−6 .s−1 -

5.5. Conclusions

137

t T u ui Vtot w xi y+ z

time tank diameter velocity velocity component i amount of control volumes in flow domain height of the impeller blade coordinate i distance from wall in viscous wall units axial coordinate

s m m.s−1 m.s−1 m3 m m m

Greek Γ Γe Γmol ∆ ∆t ∆V θ θm λmix ν νe ρ φ Ψ

Description diffusion coefficient eddy diffusion coefficient molecular diffusion coefficient lattice spacing time step finite volume angle mixing time mixing length kinematic viscosity Smagorinsky eddy viscosity density general scalar variable flux limiter

Unit m2 .s−1 m2 .s−1 m2 .s−1 s m3 o

s m m2 .s−1 m2 .s−1 kg.m−3 -

6 Numerical simulation of a dissolution process in a stirred tank reactor

A dissolution process of solid particles suspended in a turbulent flow of a Rushton turbine stirred tank is studied numerically by large eddy simulations including passive scalar transport and particle tracking. The latticeBoltzmann flow solver and the Smagorinsky subgrid-scale model have been adopted for solving the stirred tank flow. To the large eddy simulation (LES), a finite volume scheme is coupled that solves the convection-diffusion equation for the solute. The solid particles are tracked in the Eulerian flow field through solving their dynamic equations of linear and rotational motion. Particleparticle and particle-wall collisions are included, and the particle transport and fluid flow are two-way coupled. The simulation has been restricted to a lab-scale tank with a volume equal to 10−2 m3 . A set of 7 · 106 spherical particles 0.3 mm in diameter is released in the top part of the tank, resulting in a local initial solids volume fraction of 10%. The particle properties are such that they resemble calcium-chloride beads. The focus is on solids and scalar concentration distributions, particle size distributions, and the dissolution time. For the particular process considered, the dissolution time is found to be at most one order of magnitude larger than the time needed to fully disperse the solids throughout the tank. Key words: stirred tank, turbulence, dissolution, suspension, simulation, particle transport Parts of this chapter have been submitted to Chem. Eng. Sci.

140

6.1

Chapter 6. Numerical simulation of a dissolution process

Introduction

Processes in which turbulently agitated solid-liquid suspensions are involved, have a large share in various industrial applications. Examples are crystallization (Hollander et al., 2001), suspension polymerization (initially a liquid-liquid dispersion that in the course of the process turns into a solid-liquid mixture, see e.g. Kiparissides (1996)), particle coating and catalytic slurries. Such processes are very complex multi-phase. In order to improve competitiveness, there is an industrial drive for research on the hydrodynamic phenomena, and their coupling with chemistry and heat and mass transfer. One of the key aspects in the (dynamic) behavior of the processes mentioned is the role of hydrodynamics. On a macroscopic scale, the hydrodynamic conditions control e.g. residence time and circulation time in the specific flow system (e.g., a stirred tank). On a microscopic scale, rate-limiting processes such as mass transfer (needed for nucleation and growth in crystallization), agglomeration and attrition (a major source for secondary nucleation), collisions, and the yield of a chemical product (in case of competitive chemical reactions) are largely determined by the smallest-scale flow phenomena. It is difficult to determine a priori the impact of the smallest-scale flow phenomena on the overall outcome of the particular process (e.g. the layer thickness in a particle coating process, the final particle size distribution in a crystallization or suspension polymerization process, dissolution time of salts in a dissolution process). Next to the role of hydrodynamics, there are many unresolved issues with respect to what is actually going on at the particle scale. One may think of heat and/or mass transfer between the particle and the continuous phase, mechanical load on the particle (which determines the occurrence of an attrition event) as a result of particle-particle or particleimpeller collisions, and the influence of the presence of particles on the local and global flow features in the flow system. Furthermore, physical mechanisms are often controlled by non-linearities. For instance, in crystallization processes the collision rate depends on the square (i.e. non-linear) of the particle number concentration (Hollander et al., 2001), and in chemical reactive flows the reaction rate constants depend in a non-linear way on the concentrations of the reactants. Next to various experimental and theoretical investigations reported in the literature, numerical modeling is an alternative route to investigate the issues mentioned. In this work, the focus is on a dissolution process under strongly turbulent conditions induced by a Rushton turbine in a stirred tank. Because of limited computational resources, the strongly turbulent flow cannot be fully captured by the computational grid. Consequently, direct solving the flow system under consideration goes beyond the present and future foreseeable computational possibilities. As a result, we need to revert to model-

6.1. Introduction

141

ing. The first step in realistic modeling of solids suspension in turbulently stirred tanks is an accurate representation of the single-phase flow field. An approach widely used today for solving the turbulent flow is based on the solution of the Reynolds-averaged NavierStokes (RANS) equations. Although various investigations (Ng et al., 1998; Hartmann et al., 2004b) have shown that a simulation based on a RANS approach is able to reasonably represent the average flow field, the turbulence levels are underpredicted by 50%. This certainly affects the mixing patterns and consequently solid-liquid mixing. As stated above, the motion of particles, mass transfer, and collisions are determined by the small-scale flow phenomena, and consequently small-scale flow information comparable to the particle size is needed. Therefore, the basis of the simulation discussed here is a representation of the continuous flow by means of large eddy simulation (LES). Compared to a RANS approach, turbulence modeling by means of LES leaves less room for speculation in modeling the turbulence and the motion of the solids immersed in the flow. Various numerical investigations have demonstrated that LES can accurately represent the single-phase flow in a stirred tank, including the turbulence levels (Eggels, 1996; Derksen & Van den Akker, 1999; Hartmann et al., 2004a,b; Yeoh et al., 2004; Bakker & Oshinowo, 2004). The confidence achieved in using the flow solver based on the LES methodology has set a solid basis for setting up a simulation on solid-liquid processes including mass transport, which is the focus of this work. In the LES flow field, spherical particles are released in the top part of the tank. The motion of the particles and collisions are handled by a solver developed by Derksen (2003). For the inter-phase mass transfer between the disperse and continuous phases, a single-particle correlation of Ranz & Marshall (1952) is applied. Information on the local scalar concentration field, needed for the determination of the mass transfer rate, is obtained by solving the convection-diffusion equation for the species involved. Exactly as in a LES, the latter equation is solved in an Eulerian framework through a finite volume discretization. The finite volume scheme used in this work has been described in Chapter 5. The aim of this work is to provide insight in the physical mechanisms occurring in a dissolution process by means of detailed LES including passive scalar transport and particle tracking. In this work, we focus on the different stages occurring in the dissolution process. Furthermore, we pay attention to the evolution of the particle size distribution in time. Through monitoring the total amount of particles in the suspension in time, a dissolution time is determined.

142


T/10 D = T/3 3D/4 0.04D 0.04D

H=T

D/5 0.16D

T/3 D/4 T

Figure 6.1: Cross-section of the tank (left). Plan view and cross-section of the impeller (right). At the top level there is a lid. The impeller is a Rushton turbine mounted at height T /3 and has a diameter T /3.

6.2

Flow system

The stirred tank used in this work was a standard configuration cylindrical tank of diameter T , with four equi-spaced baffles of width 0.1T mounted along the perimeter of the tank. The liquid height was set equal to the tank diameter, H = T . The impeller was a six-bladed Rushton turbine with standard dimensions, mounted at height T /3. A schematic representation of the flow system is shown in Figure 6.1. If geometric similarity is maintained, the single-phase flow can be fully characterized by the Reynolds number (Re=N D 2 /ν, with N the impeller speed, D the impeller diameter and ν the viscosity of the continuous phase). In this work, the Reynolds number amounts to 105 . The tank volume (V ) is set to 10−2 m3 , which implies an impeller diameter of D = 7.78 · 10−2 m. The continuous phase is water (with viscosity ν = 10−6 m2 /s and density ρl = 103 kg/m3 ). A set of Np0 = 7 · 106 mono-disperse spherical particles with diameter dp0 = 0.3 mm, and density ρp = 2150 kg/m3 is released uniformly distributed over in the upper part (0.9T − T ) of the tank. The saturation concentration (csat ) is set to 600 kg/m3 , and the molecular diffusion coefficient (Γmol ) equals 0.7 · 10−9 m2 /s, which yields a Schmidt number of about 1400. The settings of the particle diameter, the saturation concentration and the diffusion coefficient are typical for calcium-chloride beads in water. Since calcium ions are larger than chloride ions, they have a lower diffusion coefficient. As a result, the diffusion of calcium ions is rate-limiting, and therefore the diffusion coefficient in the simulation is that of calcium ions. The initial solids volume and mass fractions of the set of particles released in the top part of the tank amount to 10% and 21.5%, respectively (the tank average volume and mass fractions are 1% and 2.15%, respectively).


143

The initial macroscopic Stokes number (i.e. the ratio of the Stokesian particle relaxation 2 ρ d N

p0 time and the time of one impeller revolution; Stk= ρpl 18ν ) yields Stk=0.1774. The initial microscopic Stokes number which relates the Stokesian particle relaxation time to the time scale of the smallest turbulent fluctuations (i.e. the Kolmogorov time scale based on the tank-averaged energy dissipation rate) equals 600. As a result, the particles will initially follow the large-scale turbulent fluctuations. During the course of the simulation, the particle size decreases and the motion of the particle will be influenced by an increasing part of the turbulence spectrum (i.e. the particle will behave more and more as a tracer). The impeller speed is chosen to be above the (initial) just-suspended impeller speed (Njs ) according to the Zwietering (1958) correlation:

0.45

Njs = s

0.1 d0.2 (g∆ρ) Φ0.13 p0 ν m0 ρ0.45 D0.85 l

(6.1)

with g the magnitude of the gravitational acceleration vector, ∆ρ = ρp − ρl , Φm0 the initial solids mass fraction in %, and s is a constant that equals 8 for the particular configuration. The impeller speed is set to 16.5 rev/s, whereas the (initial) just-suspended speed is 11.4 rev/s. Since the particles dissolve during the course of the simulation, particle size and the solids mass fraction decreases in time, and consequently the just-suspended speed decreases.

6.3


6.3.1 Flow solver The Reynolds number presented in the previous section (Re=105 ) implies a strongly turbulent flow that cannot be fully resolved by the computational grid. As a result, turbulence modeling is required. We apply large eddy simulation (LES), in which the small scales in the flow are assumed to be universal and isotropic. The flow resolved by the grid can be interpreted as a low-pass filtered representation of the true flow. The effect the small scales (i.e. the part that has been filtered out) have on the larger scales is modeled with a subgridscale model. The LES-model applied here is the Smagorinsky (1963) subgrid-scale model, where the action of the subgrid-scale motion is considered to be purely diffusive through an eddy viscosity. For more details on the LES methodology used we refer to Hartmann et al. (2004a,b) and Derksen & Van den Akker (1999). A lattice-Boltzmann method (Chen & Doolen, 1998) was used for solving the filtered momentum equations. The specific scheme we used was introduced by Somers (1993),

144


and is described by Derksen & Van den Akker (1999). The entire tank was simulated on a uniform, cubic computational grid. Inside the computational domain, the no-slip boundary conditions at the cylindrical tank wall, the baffles, the impeller, and the impeller shaft were imposed by an adaptive force-field technique (Derksen & Van den Akker, 1999).

6.3.2 Particle transport solver The particle transport solver used for the simulation of the dissolution process makes use of the Eulerian-Lagrangian approach and has been developed by Derksen (2003). Next to the solution of the solid particles dynamics, particle-wall, particle-impeller and particleparticle collisions are considered in the solver. The latter mechanism proved to be crucial in order to obtain a realistic particle distribution throughout the tank (Derksen, 2003). This improvement was primarily caused by the exclusion effect brought about by the collision algorithm. We will describe here the global features of the solver, the details can be found in the paper of Derksen (2003). The motion of the particles is controlled by the drag, added mass, gravity, lift and stress gradients forces. The inclusion of the lift force requires knowledge of the vorticity of the fluid and the particle rotation. The latter is obtained from the solution of the dynamic equation for the particle angular velocity. The time step in the discrete version of the dynamic equations equals the time step in the LES. The LES provides the resolved part of the fluid motion. Solid particles released in the LES flow field, will also feel the subgrid-scale part of the velocity fluctuations. Derksen (2003) has shown that for the particular case studied here, the resolved flow dominates the particle motion. In the first place, the time step is able to keep up with the particle relaxation time for a large range of particle diameters (i.e 0.05 ≤ dp /dp0 ≤ 1). In the second place, the resolved velocity fluctuations are an order of magnitude larger than the (estimated) subgrid-scale fluctuations (Derksen, 2003). Only for the determination of the drag force, a subgrid-scale part has been included. The initial solids volume fractions are such that two-way coupling effects may be important (Elgobashi, 1994). As a result, they have been included in the solver. In the particle collision algorithm, particle-wall, particle-impeller and particle-particle collisions are considered to be fully elastic and frictionless. The efficiency of the collision algorithm has been improved by limiting the computational effort in handling particleparticle collisions. This has been achieved by grouping the particles in a so-called link-list (see Chen et al., 1998; Derksen, 2003). A proper functioning of the collision algorithm is limited by the time step, or the particle volume fraction. Derksen (2003) has indicated that


145

the number of missed collisions increases with increasing solids loading. The increase is most pronounced in highly turbulent regions (i.e. in the impeller region). As in our case the initial solids volume fraction equals 10% in the top part of the tank, the number of missed collisions may not be significant compared with the number of correctly handled collisions. In the first place, the turbulence intensity is low in the top part of the tank. Secondly, the local solids volume fraction decreases rapidly as the particles are dispersed throughout the tank.

6.3.3 Mass transfer The particles lose/gain weight due to a continuous process of inter-phase mass transfer between the particles and the liquid (i.e. the continuous phase) or vice-versa. Mass transfer is controlled by a mass transfer coefficient (k) and a driving force, which is the difference between the mass saturation concentration (csat ) and the mass concentration of the surroundings (c): m ˙ int = kA (csat − c)

(6.2)

where m ˙ int is the inter-phase mass transfer rate between the particle and liquid, and 2 A = πdp is the surface of the spherical particle. The mass transfer coefficient is defined as: k = Sh

Γmol dp

(6.3)

where Sh is the Sherwood number. The concentration of the surroundings in equation 6.2 is assumed to be the average concentration in the control volume of the scalar transport solver. The effect of concentration fluctuations at the subgrid-scale level on the mass transfer rate has been ignored, since the mass transfer rate linearly scales with the concentration (see equation 6.2), and the effect of fluctuations tends to average out. The cell average concentration is determined via the discrete solution of the convection-diffusion equation. This is achieved by coupling a scalar mixing solver (described in the next subsection) to the LES and particle transport solvers. Mass transfer between solid particles and continuous phase depends on the motion of the solid surface relative to the liquid. Apart from linear velocities, particle rotation might play a role in mass exchange. Derksen (2003) has shown the spatial distribution of linear and rotational slip velocities in terms of the respective Reynolds numbers. For

146


particles with a diameter equal to 0.3 mm, the rotational Reynolds numbers are one order of magnitude smaller than translational Reynolds numbers. As a result, it is expected that mass transfer is likely to be controlled by translation of the particles, and the contribution of rotation to the overall mass transfer rate is small and can be neglected. In this work, the Ranz & Marshall (1952) correlation is used for the calculation of the Sherwood number: 1

1

Sh = 2.0 + 0.6Rep2 Sc 3

(6.4)

where Rep is the particle Reynolds number, defined as: Rep =

vsl dp ν

(6.5)

where vsl is the slip velocity (i.e. the magnitude of the velocity of the particle relative to the fluid).

6.3.4 Scalar mixing solver In the simulation of passive scalar transport, we assume one-way interaction with the flow field, i.e. the instantaneous velocity affects the scalar evolution, while the passive scalar does not change the flow characteristics. In this way, the scalar field is mathematically decoupled from the dynamical equations that govern the flow field. Thus, the solution of the flow field is a prerequisite to the solution of the scalar field. Eggels & Somers (1995) have performed scalar transport calculations on free convection cavity flow with the lattice-Boltzmann discretization scheme. This scheme, however, is more memory intensive than a finite volume formulation of the convection-diffusion equation. Therefore, we have coupled a compressible finite volume discretization scheme for the scalar transport to the lattice-Boltzmann flow solver. The convection-diffusion equation in compressible form reads ∂ ∂ρl c ∂ui ρl c = + ∂t ∂xi ∂xi

∂c ρl Γ + Sc ∂xi

(6.6)

where ρl is the density of the continuous phase, c is the scalar concentration, ui is velocity component i, Γ is the diffusion coefficient and Sc is a source term. Note the summation over the repeated index i. The diffusion coefficient is the sum of the molecular and eddy diffusion coefficients: Γ = Γmol +Γe . The eddy diffusion coefficient is related to the eddy viscosity via a turbulent Schmidt number (Sct ) through Γe = νe /Sct . For the turbulent Schmidt number we have chosen a value of 0.7. Since the lattice-Boltzmann scheme


147

is a compressible scheme, we have implemented the discretized form of the compressible convection-diffusion equation. In this context it should be noted that the maximum velocity (which is approximately the tip speed) is set sufficiently low for meeting the incompressibility limit in the lattice-Boltzmann scheme. The finite volume scheme used in the context of this work has been described in detail in Chapters 4 and 5. The source term Sc in equation 6.6 describes the mass transfer between the particles and the continuous phase. The discretized source term Sc,i reads Np,i

Sc,i = ρl ∆V −1

X

m ˙ int,j

(6.7)

j=1

where ∆V is the volume of cell i (which is the same for all cells in the uniform, cubic grid), Np,i is the number of particles with their center of mass in cell i and m ˙ int,j is the inter-phase mass transfer rate (see equation 6.2) between particle j and the surrounding continuous phase. The diameter of particle j is adjusted each time step as follows: dn+1 p,j

=

r 3

dnp,j

3

−

6 −1 ˙ int,j ∆t ρp m π

(6.8)

where ∆t is the time step. The zero-gradient constraint for the scalar concentration at the off-grid (moving) walls has been imposed by the novel immersed boundary technique described in sections 4.6 and 5.3.2.

6.3.5 Simulation aspects The particles are released in a quasi steady-state flow field. The scalar concentration at the start of the simulation equals zero throughout the tank. The set-up of the dissolution process has been described in section 6.2. The code runs on the parallel computer platform ’Aster’ located at SARA in Amsterdam. Aster is an SGI Altrix 3700 system, consisting of 416 CPU’s (Intel Itanium 2, 1.3 GHz, 3 Mbyte cache each), 832 Gbyte of memory and 2.8 Tbyte of scratch disk space. The total peak performance is 2.2 Teraflop/sec. Every node in Aster is a CC-NUMA machine (Cache-Coherent Non Uniform Memory Access). All processors are integrated into one machine, which significantly speeds-up the communication between the processors. Within the parallel environment of ’Aster’ the code makes use of domain decomposition: the computational domain was horizontally split in 30 equally-sized subdomains (i.e. normal to the tank centerline). The presence of particles in a stirred tank significantly com-

148


plicates the parallelization of the computer code. In the first place, load-balancing needs to be reconsidered since the computational load of a sub-domain now depends on the number of particles inside the sub-domain in a quite unpredictable manner. Furthermore, there is much more communication between subdomains compared to a single-phase simulation, as particles cross sub-domain borders. Dynamic load-balancing was not implemented in the code. This initially results in a poor parallel efficiency and a low calculation speed, as all particles are located in the upper subdomains. Gradually, the calculation speed increases as the particles travel toward the impeller region and become more or less homogeneously distributed throughout the tank (bearing in mind that particles dissolve during simulation, and are consequently more easily resuspended from the bottom). The MPI message passing tool has been used for communication between the subdomains. The simulation has been executed on a cubic, Cartesian grid of 2403 lattice cells. The diameter of the tank equals 240 lattice spacings, resulting in a spatial resolution of T /240 = 0.973 mm. The temporal resolution is limited by the lattice-Boltzmann method. The tip speed of the impeller is set to 0.1 lattice spacings per time step in order to meet the incompressibility limit. This results in a temporal resolution of 1/(2400N ) = 25 µs. For the LES, 21 (18 directions and 3 force components) single-precision, real values need to be stored. For the scalar transport, 3 (concentration fields at time instants n + 1, n and n−1) double-precision, real values need to be stored. Finally for the particle properties a memory space for 16 (i.e. particle properties; particle coordinates, velocity and angular velocity components, diameter, etc) times the number of particles (7·106 ) double precision, real values is needed. In total, the memory requirements of the simulation result in an executable of about 2.5 GByte. The full simulation of 100 impeller revolutions took about 6 weeks.

6.4

Results

6.4.1 Snapshots of the particle distributions and concentration fields Figures 6.2 and 6.3 give an impression of the dissolution process during the course of the simulation. The scalar concentrations are normalized with the final concentration c ∞ defined as: c∞

Mp0 2 = = Np0 ρp V 3

dp0 T

3

(6.9)

6.4. Results

149

Nt=2

Nt=5

Nt=7

Nt=10

Nt=20

Nt=40

Figure 6.2: Snapshots of the particle distribution in a vertical plane mid-way between two baffles. In the graphs, the particles in a slice with thickness T /240 have been displayed. The diameter of the particles is 3 times enlarged for clarity. The respective concentration distributions are shown in Figure 6.3.

150


Nt=2

Nt=5

c/c¥ Nt=7

Nt=10

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

Nt=20

Nt=40

0.2

Figure 6.3: Snapshots of the concentration distribution in a vertical plane mid-way between two baffles. The respective particle distributions are shown in Figure 6.2.

6.4. Results

151

where Mp0 = Np0 ρp π6 d3p0 is the total particle mass at N t = 0. During the first five impeller revolutions, the particles are transported toward the impeller region. High particle concentrations are accompanied by high scalar concentrations, and a clear interaction between the particles and turbulence is identified. The particles are swept radially outward by the revolving impeller, and are subsequently entrained in the lower recirculation loop. In the next 5 − 15 impeller revolutions, the particles are distributed throughout the tank. During the first 10 − 20 impeller revolutions (till the point that the particles are distributed throughout the tank), clear macroscopic scalar concentration structures are identified. The scalar transport matches the particle transport. As the dissolution process proceeds, a number of features are identified from the snapshots shown in Figure 6.2. In the first place, the snapshots of the particle distributions at N t = 10 and N t = 20 show high particle concentrations at the bottom and outer walls. Secondly, the particles are organized in streaky patterns. Because of the particle inertia and the density of the particles being higher than that of the liquid, the particles are swept out of the high-vorticity regions. Thirdly, the size of the particles becomes smaller (compare the snapshots at N t = 20 and N t = 40). As the solid particle mass decreases during the process, the particle inertia decreases. As a result, the streaky patterns die out. Furthermore, smaller particles are more easily picked up from the bottom and resuspended in the flow. The high particle concentration at the bottom, observed in the snapshots at N t = 10 and N t = 20, has disappeared in the snapshot at N t = 40. A decrease of the particle mass, accompanied by a decrease of the particle inertia and resuspension of particles, results in the particle distribution to become more and more homogeneous. From the point the particles are distributed throughout the tank, the macroscopic structures in the scalar concentration have disappeared (see Figure 6.3, the snapshots at N t = 20 and N t = 40). A decrease of the particle mass observed in the snapshots in Figure 6.2 is associated with an increase of scalar concentration. At the wall separation points and at the bottom of the tank spots of high concentration are observed that correspond with local high solids concentrations. In Figure 6.4 two snapshots of the particle distribution at N t = 26.5 and N t = 60 are shown together with the overall particle size distributions. The particles are 10 times enlarged and colored by their size. The particle distribution at N t = 26.5 shows an axial gradient in the average particle size. The particles transported in the lower recirculation zones are generally smaller than the particles transported in the upper recirculation zone. The reason for this effect is that compared with the upper recirculation loop, the lower recirculation loop is stronger and this induces an increased mass transfer rate. A comparison between the upper and lower graphs clearly shows the difference in particle size. The spatial particle distribution at N t = 60 is nearly homogeneous.

152


Np /Np0(-)

0.01

dp /dp0

0.0075

0.72 0.7

0.005 0.0025

0.66

0 0 0.25 0.5 0.75 1 dp /dp0 (-)

0.62 0.58 0.54 0.5

Nt = 26.5 Np /Np0(-)

0.01

dp /dp0

0.0075 0.5 0.005 0.45 0.4

0.0025 0 0 0.25 0.5 0.75 1 dp /dp0 (-)

0.35 0.3 0.25 0.2 0.15 0.1 0.05

Nt = 60 Figure 6.4: Snapshots of the particle distribution in a vertical mid-way baffle plane accompanied by the respective overall particle size distributions. In the graphs, the particles in a slice with thickness T /240 have been displayed. The diameter of the particles is 10 times enlarged for clarity, and the colors represent the dimensionless particle diameter.

6.4. Results

153

dp /dp0 0.82 0.79 0.76 0.73 0.7 0.67

c/c¥ 0.8 0.72 0.64 0.56 0.48 0.4

Figure 6.5: Snapshot of the particle distribution at N t = 15 (upper graph) in a horizontal plane at disk height accompanied by the respective concentration distribution (lower graph). In the upper graph, the particles in a slice with thickness T /240 have been displayed. The diameter of the particles is 10 times enlarged for clarity, and the colors represent the dimensionless particle diameter.

154


The snapshot of the particle distribution at N t = 26.5 shows a region void of particles extending from the bottom to closely underneath the impeller, slightly right from the tank centerline. In this region there is a manifestation of a slowly precessing vortex that crosses the cross-section at N t = 26.5. In this region of high vorticity, the particles having a larger density than that of the liquid are swept out due to their inertia. This precessing vortex (a so-called macro-instability), here in interaction with the particles, was studied in detail in Chapter 3 (Hartmann et al., 2004a). Figure 6.5 shows instantaneous realizations of the particle distribution and the scalar concentration in a horizontal plane at disk height at N t = 15. In the horizontal crosssection of the particle distribution, large voids behind, and high solids concentrations in front of the blades are observed (in accordance with the simulation results of Derksen (2003)). The streaks of particles keep their identity of quite a long radial distance. Also high solids concentrations near the walls are observed, as a result of the centrifugal forces. Spots of high concentration are observed near the walls that are associated with locally high solids concentration. Furthermore, the scalar concentration distribution shows increased concentration levels in front of the blades as a result of the high solids concentrations.

6.4.2 Stages in the dissolution process Figure 6.6a shows time-traces of the number of particles in ten axial slices of height 0.1T , with a focus on the top and bottom slices. From the start of the process, the particle number in the upper slice decreases rapidly as the particles are transported toward the impeller region. After approximately 10 impeller revolutions, the first particles reach the bottom of the tank. Because of a density ratio of the disperse and the continuous phases being larger than unity, the largest number of particles is observed in the bottom slice for N t > 15. Based on the observations in the previous subsection and the path of the curves shown in Figure 6.6a, five stages can be identified: • Stage I (0 ≤ N t < 12): Mixing and dispersing • Stage II (12 ≤ N t < 24): Quasi steady-state • Stage III (24 ≤ N t < 42): Resuspension • Stage IV (N t ≥ 42): Dissolution • Stage V (N t ≥ 58): Homogeneous suspension

6.4. Results

155

1

I

II

III

IV

Np /Np0 (-)

0.8 0.6 0.9T - T 0.4 0 - 0.1T

V

0.2 0 0

20

40

60

80

100

Nt (-) (a)

1 a = 0.9T - T b = 0.8T - 0.9T c = 0.7T - 0.8T d = 0.6T - 0.7T e = 0.5T - 0.6T f = 0.4T - 0.5T g = 0.3T - 0.4T h = 0.2T - 0.3T i = 0.1T - 0.2T j = 0 - 0.1T

Np /Np0 (-)

0.8 a

0.6 0.4 b

0.2

c d

0 0

3

j e f g h

i

6

9

12

15

Nt (-) (b)

Figure 6.6: Evolution of particle number in ten axial slices of height 0.1T . In (a) the focus is on the top and bottom slices. Five stages in the dissolution process can be defined: Mixing and dispersing (I), Quasi-steady-state (II), Resuspension (III), and Dissolution (IV). At time instant N t = 58 a more or less homogeneous suspension is reached (Stage V). In (b) the focus is on the first stage (i.e. mixing and dispersing).

156


In stage I (Mixing and dispersing), the solids are distributed throughout the tank till a quasi steady-state situation is reached (stage II). During stage II, the dissolution process proceeds, but the number of particles in all slices remains approximately constant. The number of particles in the bottom slice is about 1.6 − 2.3 times higher than the number of particles in the other slices. At N t = 24 the number of particles in the bottom slice starts to decrease significantly (stage III). At this point in time, the mass of the particles residing at the bottom has decreased to a level where the turbulent motions are strong enough to resuspend the particles. After about 42 impeller revolutions, the first particle is fully dissolved (the onset of stage IV). Stage V is a part of stage IV and starts at the time instant N t = 58 when a fully homogeneous suspension is reached. Based on Figure 6.6a it can be concluded that (for the particular case studied here), the time to dissolve all particles (i.e. the dissolution time) is at most one order of magnitude larger than the time to distribute the particles throughout the tank. Figure 6.6b presents an enlarged view of the first stage. The number of particles normalized with the total amount of particles in slice a (i.e. the top slice) decreases rapidly from 100% to about 11%. Subsequently, the normalized number of particles in slices b to i increase successively to 8% − 11%. The normalized number of particles in the bottom slice (j) is significantly higher (i.e. 18%) representing the high solids concentration at the bottom.

6.4.3 Evolution of particle size distribution in time In Figure 6.7 the particle size distributions are shown at nine instants in time. The first six distributions correspond to the particle distributions shown in Figure 6.2. During the course of the simulation, the particle size distribution broadens and shifts toward smaller particle sizes. At N t = 60, 80, 100 a peak is observed in the first bin, representing the number of dissolved particles (i.e. 16.8%, 97.3%, 99.9% of the total amount of particles released). A peculiar feature of the particle size distributions at N t = 60, N t = 80 and N t = 100 is its steep slope at the smallest particle sizes. This can be explained by equations 6.2 and 6.3. As dp → 0 the Sherwood number approaches 2. Consequently, the mass flux scales as d−1 p , and the mass transfer rate as dp . A mass conservation equation describing the inter-phase mass transport results in the time derivative of the particle diameter to scale with d−1 p . As a result, the particle size distribution collapses faster at low particle sizes than at higher particle sizes. Figure 6.8 shows the time trace of the normalized Sauter mean diameter (d32 = P Np 3 P Np 2 dp,i )/( i=1 ( i=1 dp,i )). The Sauter mean diameter decreases as a function of time till

6.4. Results

157

0

Np /Np0 (-)

10

Nt = 2

Nt = 5

Nt = 7

Nt = 10

Nt = 20

Nt = 40

Nt = 60

Nt = 80

Nt = 100

10-2

10-4

Np /Np0 (-)

10-6 100 10-2

10-4

Np /Np0 (-)

10-6 100 10-2

10-4

10-6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

dp /dp 0 (-)

dp /dp 0 (-)

dp /dp 0 (-)

Figure 6.7: Instantaneous realizations of the particle size distribution (lin-log plots). 1

d32/dp0 (-)

0.8 0.6 0.4 0.2 0 0

20

40

60

80

100

Nt (-)

Figure 6.8: Evolution of the Sauter mean diameter in time. The crosses correspond with the particle size distributions shown in Figure 6.6.

158

Chapter 6. Numerical simulation of a dissolution process 1

Np /Np0 (-)

0.8 0.6 0.4 0.2

Nqs,99%

0 50

60

70

80 Nt (-)

90

100

Figure 6.9: Evolution of number of particles throughout the tank in time, starting at N t = 50. N t = 70. In the final stage of the dissolution process, the Sauter mean diameter increases slightly. This effect is explained by the fact that the particle size distribution collapses faster at low particle sizes than at high particle sizes.

6.4.4 Dissolution time Compared to the various definitions in the literature of the mixing time in e.g. blending processes (see Chapter 5), there is not much debate on the definition of a dissolution time. The dissolution time is the time needed to fully dissolve all the solids. In a numerical simulation including particle tracking, the total number of solid particles is monitored, and hence the dissolution time is the time when the total number of particles has reached the zero value. In Figure 6.9a the evolution of the total number of particles normalized with the number of released particles as a function of time is shown. The dissolution process starts at N t = 42.5 (not shown), and from N t = 50 a steep decrease in the number of particles is observed till N t = 70. In the final stage of the dissolution process a long tail is observed in the time series of the particle number. The simulation was stopped at N t = 100 when less than 0.1% of the total amount of released particles was left. With a view to a sensible use of computational effort, it is not believed that any valuable information apart from the dissolution time is to be found for N t > 100. The time instant where 99% of the particles

159

2

2

1.6

1.6

1.2

1.2

c/c¥ (-)

c/c¥ (-)

6.4. Results

0.8 0.4

0.8 0.4

0

0

0

20

40

60

80

100

Nt (-)

(a) z/T = 0.67; r/T = 0.25

0

20

40

60

80

100

Nt (-)

(b) z/T = 0.19; r/T = 0.25

Figure 6.10: Simulated time traces of the concentration normalized with the final concentration c∞ .

is dissolved is θs,99% = 84N −1 .

6.4.5 Time series of concentration In Figure 6.10 time series of the scalar concentration are shown, recorded at two monitoring points. One monitoring point is located in the upper recirculation loop (i.e. in a mid-way baffle plane at z/T = 0.67 and r/T = 0.25), and the other in the lower recirculation loop (i.e. in a mid-way baffle plane at z/T = 0.19 and r/T = 0.25). In the first 12 − 15 impeller revolutions strong concentration fluctuations are observed, which correspond with the macroscopic structures seen in Figure 6.3. The concentration fluctuations registered at the monitoring point at z/T = 0.67 (Figure 6.10a) are stronger compared to those at the monitoring point at z/T = 0.19 (Figure 6.10b). This is because the particles were released in the top part of the tank. The strong concentration fluctuations only occur in the mixing and dispersing stage (I). Subsequently for N t > 15, the concentration recorded at both monitoring points gradually increases to a final level of about 1.12c∞ . The reason for the significant deviation of 12% is subject of the next subsection.

160

Chapter 6. Numerical simulation of a dissolution process 1.2

M/Mp0 (-)

1 0.8 0.6 0.4 Mp 0 - Mp

0.2

Mc 0 0

20

40

80

100

Nt (-)

Figure 6.11: The dissolved particle mass (M = Mp0 − Mp ; solid line), and the tracer mass (M = Mc ; dashed line), made dimensionless with the total particle mass Mp0 as a function of time.

6.4.6 Mass conservation finite volume scheme We have used the newly developed immersed boundary technique for the scalar concentration to impose a zero-gradient constraint at the walls that are off-grid (see Sections 4.6 and 5.3.2). This technique is unconditionally stable in the explicit formulation of the finite volume scheme used. However, mass conservation is not guaranteed. The total mass in the system has been monitored in order to check the mass conservation constraint. The total mass of the solids (Mp ) is evaluated through: Mp = ρ p

Np πX 3 d 6 i=1 p,i

(6.10)

The total mass dissolved equals (Mp0 − Mp ). In theory, the total mass dissolved equals the total scalar mass (Mc ), which reads Mc =

NV X

ci ∆Vi

(6.11)

i=1

where ∆V is a control volume and NV is the number of control volumes. In Figure 6.11 the time trace of the total mass dissolved (i.e. solid line) and the total scalar mass divided by the fraction of dissolved particle mass (dashed line) normalized

6.5. Conclusions

161

with the total mass of the solids are shown. The total mass dissolved increases rapidly in the first part of the dissolution process. At the end of stage I (Mixing and dispersing; N t = 12) approximately 50% of the total solids mass is already dissolved, and at the start of stage IV (Dissolution; N t = 42) 90% of the solids mass is dissolved. At the end of the simulation, the total scalar mass equals 1.12 times the total dissolved mass. The significant deviation of 12% has already been observed in the time series of the scalar concentration in the previous subsection. The average mass increase per impeller revolution equals 0.12% which corresponds to the average mass increase observed in the simulations of the blending process (Chapter 5). In Chapter 5 the increase of mass has been attributed to the lack of resolution to accurately represent the edges of the impeller blades, impeller disk and baffles. Consequently, the mass conservation constraint is not satisfied, and in future research the current technique needs to be improved. Possible solutions for better conserving scalar mass are (local) grid refinement, and the use of a cylindrical grid by the scalar transport solver.

6.5

Conclusions

A simulation of a dissolution process (i.e. a solid-liquid suspension) according to an Eulerian-Lagrangian approach has been presented in this chapter. In this approach, 7 million particles were tracked in an unsaturated liquid flow field that was represented by a large eddy simulation (LES). The impeller speed exceeds the just-suspended speed (Zwietering, 1958) to keep the particles suspended in the liquid. Under the action of a driving force (i.e the difference between the saturation concentration and the concentration of the surroundings) and a mass transfer coefficient (depending on the particle slip velocity) the particles slowly dissolved during the course of the simulation. The effects of particle rotation and subgrid-scale concentration fluctuations on the mass transfer rate have been ignored. Typical features with respect to the solids distribution and the associated concentration distribution have been observed. In the first stage of the process (typically the first 12 impeller revolutions), the particles are distributed throughout the tank. The particles organize in streaky patterns, by leaving high-vorticity regions. High solids concentrations are found below the impeller, at the bottom of the tank, in front of the impeller blades, at the outer walls, and at the wall separation points. High solids concentrations are accompanied by high scalar concentration (i.e. macroscopic structures). Large regions void of particles are observed behind the impeller blades. The decrease of the particle mass in the course of the process has two important implications. In the first place, the high particle concentration

162


at the bottom of the tank disappears as the particles get resuspended. Secondly, the streaky patterns disappear due to decreasing inertia of the particles. Through monitoring the number of particles in axial slices of width 0.1T , various stages of the dissolution process were identified. In the first stage (0 − 12 impeller revolutions), the particles are dispersed throughout the tank. Subsequently, the system is in a quasi steady-state situation (i.e. the second stage, 12 − 24 impeller revolutions). A significant decrease of the number of particles in the bottom axial slice indicates that the particles at the bottom of the tank get resuspended (i.e. the third stage, 24 − 42 impeller revolutions). The fourth and final stage covers the dissolution of the particles. At roughly 58 impeller revolutions a homogeneous solids suspension is reached. The non-homogeneous mixing and consequently mass transfer in a stirred tank reactor is expressed in the evolution of the particle size distribution. During the course of the simulation, the particle size distribution broadens and moves to lower particle diameters. The peculiar steep slope observed at the smallest particle sizes is due to the mass transfer rate increasing at decreasing particle diameter for very small particles. This latter effect causes the slight increase of the Sauter diameter in the final stage of the process: the particle size distribution collapses at a higher rate for small particle sizes compared to larger particle sizes. For computational and physical reasons the simulation was stopped at N t = 100, covering 99.9% of the dissolution process. The 99% dissolution time has been determined by the time series of the particle number (i.e. the number of particles with dp > 0) and equals 84 impeller revolutions. Relatively more particles are fully dissolved in the impeller region and below. Time traces of the concentration show in the first 15 impeller revolutions strong concentration fluctuations associated with the macro-scopic structures observed in the concentration distribution. Subsequently, the concentration measured at the monitoring points increases gradually to 1.12c∞ . The significant 12% deviation observed, corresponds to the deviation between the total scalar mass and the total dissolved mass. The reason for this unphysical disagreement is the newly developed immersed boundary technique for imposing the zero-gradient boundary constraint at the off-grid walls. The lack of resolution in recovering the edges of the impeller blades, the impeller disk and the baffle causes an inaccurate representation inducing leakage of mass into the system. The artificially higher concentration (due to mass leakage) provides a reduced mass transfer rate, and consequently an overprediction of the dissolution time. In the present case studied, the final concentration is a factor 30 lower than the saturation concentration. As a result, the overprediction of the dissolution time is expected to be marginal (only 0.5% based on a back-of-the-envelope calculation).

6.5. Conclusions

163

There are still quite some issues open for improvement with respect to the modeling attempt here. The volume fraction occupied by the particles, which is significant in the first stages of the dispersion, has not been taken into account. For further issues related to the particle transport solver we refer to the paper of Derksen (2003). The mass leakage into the system induced by our (novel) immersed boundary technique may be repaired by (local) grid refinement or the usage of a cylindrical grid in the case of the scalar transport solver. The simulation of a dissolution process through LES including passive scalar transport and particle tracking gave a detailed insight in the complex phenomena (i.e. the intricate interplay of turbulence, mass transfer, motion of particles, collisions, etc) occurring during the process. For the process considered in this work, it has been shown that the dissolution time is at most one order of magnitude longer than the dispersion time scale. About 50% of the total particle mass dissolves in the first stage (i.e. the mixing and dispersing stage; 0 ≤ N t ≤ 12) of the process. The current successful application of the coupled LES/particle transport/scalar mixing solvers opens worthwhile future directions of numerical research like crystallization processes.

164


Nomenclature Roman A c cs csat c∞ dp dp0 d32 D g H k m ˙ int M Mc Mp Mp0 N Np Np0 NV r Re Rep s Sc Sct Sc Stk Sh t T ui

Description surface of the spherical particle concentration Smagorinsky constant saturation concentration final concentration particle diameter initial particle diameter Sauter mean diameter impeller diameter magnitude of the gravitational acceleration vector height of the tank mass transfer coefficient inter-phase mass transfer rate total mass total scalar mass total particle mass initial total particle mass impeller speed number of particles initial number of particles total amount of control volumes radial coordinate Reynolds number particle Reynolds number constant in equation 6.1 Schmidt number turbulent Schmidt number source term in equation 6.6 Stokes number Sherwood number time tank diameter velocity component i

Unit m2 kg.m−3 − kg.m−3 kg.m−3 m m m m m.s−2 m m.s−1 kg.s−1 kg kg kg kg s−1 − − − m − − − − − kg 2 .m−6 .s−1 − − s m m.s−1

6.5. Conclusions

165

vsl V xi z

slip velocity tank volume coordinate i axial coordinate

m.s−1 m3 m m

Greek Γ Γe Γmol ∆ ∆t ∆V ∆ρ

Description diffusion coefficient eddy diffusion coefficient molecular diffusion coefficient lattice spacing time step control volume difference between the densities of the particle and continuous phase dissolution time kinematic viscosity Smagorinsky eddy viscosity density of the continuous phase density of the particle phase initial mass fraction

Unit m2 .s−1 m2 .s−1 m2 .s−1 − s m3 kg.m−3

θs ν νe ρl ρp Φm0

s m2 .s−1 m2 .s−1 kg.m−3 kg.m−3 −

7 Conclusions and perspectives

7.1

General discussion

The goal of the present work was to contribute to reliable and accurate numerical predictions of complex, multi-phase processes occurring in stirred tank configurations. In order to achieve this ambitious goal, three issues were treated successively: • Turbulent flow phenomena • Scalar mixing • Solid-liquid mixing including mass transfer In order to improve our understanding of the complex, multi-phase processes, the first ingredient needed is detailed information of the turbulent flow phenomena encountered in a stirred tank. For that purpose, the large eddy simulation (LES) was chosen. A thorough assessment of the LES flow solver based on the lattice-Boltzmann discretization scheme has been presented in chapters 2 and 3. The main findings have been summarized in section 7.2. The description of inter-phase mass transfer requires information of the local species concentrations. In order to describe passive scalar transport, a scalar mixing solver (based on the finite volume approach) has been developed. The solver has been described and assessed in chapter 4. An LES including passive scalar mixing is well suited to determine the mixing performance of the turbulent flow in a stirred tank, which was the focus of chapter 5. The main conclusions regarding scalar mixing are summarized in section 7.3. The motion of suspended particles is handled with a particle transport solver, that was already available. The coupled flow, scalar mixing and particle transport solvers allow for detailed simulations of processes dealing with solid-liquid mixing including mass transfer in accordance with the ultimate goal described above. We decided to study a dissolution process, which is the subject of chapter 6. The main conclusions based on the final simulation are summarized in section 7.4.

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7.2

Chapter 7. Conclusions and perspectives

Turbulent flow phenomena

The lattice-Boltzmann Navier-Stokes solver based on the LES methodology has been critically assessed in order to gain confidence in the method. Two subgrid-scale models have been attempted. The focus is on the accuracy of the LES flow field predictions (in terms of velocity, turbulent kinetic energy, energy dissipation, and turbulence anisotropy), and the recovery of coherent fluctuations induced by large-scale precessing vortices (i.e. a so-called macro-instability). For the LES assessment, results of LDA experiments and a transient RANS simulation were used. The most important conclusions are as follows: • Both the LES and RANS models predict the average velocity field to a high level of accuracy. Furthermore, the LES and RANS results show, in the absence of experimental results, an inhomogeneous distribution of the energy dissipation rate throughout the tank. While a LES resolves the turbulence levels to a high level of accuracy, in a RANS simulation they are underestimated by some 50%. The high levels of turbulence anisotropy (e.g. in the impeller stream, and in the boundary layers along the walls) observed in a LES suggest the need of a more sophisticated closure model than a k- type of model in the RANS approach. • Next to the widely used Smagorinsky subgrid-scale model, a more sophisticated subgrid-scale model (the so-called Voke model) has been tested. The latter model is a blend between a DNS in fully resolved regions and a LES with the Smagorinsky subgrid-scale model in regions where the flow is not fully resolved. Despite the potential of the Voke model, the results did not show any significant improvements. • Large eddy simulations have successfully recovered coherent velocity fluctuations caused by vortices precessing about the tank centerline. The frequencies of these fluctuations at various values of the Reynolds number agree within 30% with frequencies found experimentally in a similar stirred tank geometry. Furthermore, the LES results have shown that the coherent fluctuations induced by the precessing vortices contribute significantly to the turbulence levels (up to 44%) in the top and bottom parts of the tank near the tank centerline, which is in accordance with experimental findings.

7.3

Scalar mixing

The scalar transport is modeled by means of the finite volume approach. Within this approach, an immersed boundary technique was developed to impose a zero-gradient con-

7.3. Scalar mixing

169

straint at off-grid boundaries. The scalar mixing solver has been assessed in two laminar and one turbulent flow case. The focus is on the performance of the immersed boundary technique, and the mixing time (in case of the blending process). The main conclusions of the assessment are summarized below. • A description of scalar transport by means of the finite volume approach is less memory intensive than a lattice-Boltzmann approach. The time step used for the LES including passive scalar transport was found to be small enough to attain a stable algorithm in the explicit formulation of the finite volume scheme. • Direct numerical simulations of the scalar mixing by a laminar flow in a square cavity have lead to the conclusion that the flux-limited TVD scheme performs best compared to the first-order upwind and second-order QUICK schemes. The TVD scheme significantly reduces numerical diffusion and is unconditionally stable. • In case of the scalar mixing in an inclined cavity, the newly developed immersed boundary technique to impose a zero-gradient constraint at off-grid walls performs similar to the existing staircase technique in terms of mass conservation. Both techniques do not conserve mass. In contrast to the staircase technique, the immersed boundary technique takes into account the shape of the boundary regardless of the grid resolution. • According to the results of the scalar mixing in an inclined cavity, the grid resolution has a significant effect on the scalar mass change. This is confirmed by the blending cases in the storage and the Rushton turbine stirred tanks. In the former case, the average scalar mass increase amounts to 0.0016% per impeller revolution. Since the cylindrical wall is accurately represented by the grid, the mass leakage is not significant. In the latter case, the average scalar mass increase per impeller revolution equals 0.05%, 0.125% and 0.25% of the amount of scalar mass injected at impeller sizes equal to one-fourth, one-third and one-half of the tank diameter, respectively. The significant scalar mass increase is ascribed to the lack of resolution to accurately recover the thickness of the impeller blades, impeller disk and baffles. • In the literature there is no standardization of mixing time experiments. Mixing times are reported for tanks of different layouts and operating conditions, tracer injection positions and injection speeds, and number and positions of the measurement points. Furthermore, there is no agreement on the definition of the mixing time. As a result, simulated and/or measured mixing times should be compared with great care. This also applies to mixing time correlations in the literature.

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• An LES of the scalar mixing in a cylindrical tank equipped with a side-entry mixer resulted in a mixing time that agrees within 35% with the mixing time predicted by a LES using Fluent. The mixing time in a Rushton turbine stirred tank, obtained by a LES including passive scalar transport, agrees within 30% with experimental results from the literature. Furthermore, the mixing time obtained at different impeller sizes and injection positions also are within 30% of values calculated from two correlations presented in the literature. The mixing time is not significantly affected by the tracer injection position. On the other hand, impeller size has a significant effect on the mixing time. According to the simulations, the mixing time scales with the tank over impeller diameter ratio to the power 2.5.

7.4

Solid-liquid mixing including mass transfer

A dissolution process has been simulated in detail by means of a LES including passive scalar transport, coupled to a Lagrangian description of spherical, solid particles immersed in the turbulent flow driven by a Rushton turbine. The focus is on the transient phenomena observed, in terms of solids and scalar concentrations, particle size distributions, and the dissolution time. The main conclusions are formulated below. • The dissolution process has been successfully simulated until the point that 99.9% of all particles have been fully dissolved. For the particular dissolution process simulated, the dissolution time was found to be at most one order of magnitude larger than the time needed to fully disperse the solids throughout the system. • With settings used in the simulation, four stages have been identified. In the first stage the particles are dispersed throughout the system. In the second stage, the distribution of solid particles throughout the tank is more or less steady. In the third stage, the high solids concentration at the bottom of the tank decreases significantly due to resuspension of particles. During the fourth stage, the particles are fully dissolved, and the solids distribution is heading toward a homogeneous distribution. • Initially, the particles organize in streaky patterns due to the solids density being higher than the density of the liquid, and the inertia of the particles. The particles are swept out the high vorticity regions, which is in particular nicely illustrated by the presence of the slowly precessing vortex below the impeller. During the course of the process, the inertia of the particles decreases in time, and consequently the particles will behave more like fluid flow tracers. As a result, the streaky patterns disappear.

7.5. Perspectives and recommendations

171

• Until the point the particles are fully distributed throughout the tank, the spatial solids and scalar distributions are very inhomogeneous. High scalar concentrations are associated with high solids concentrations. Subsequently, the spatial solids and scalar concentration distributions become more or less homogeneous. The concentration increases gradually throughout the tank. • The influence of non-homogeneous mixing effects on the mass transfer is expressed in the development of a particle size distribution. During the course of the simulation, the particle size distribution broadens and shifts toward smaller particle diameters. • The unphysical change of scalar mass due to the newly developed immersed boundary technique, is significant (i.e. an average increase per impeller revolution of 0.12% of the initial solids mass), and resembles that determined in the blending process in the Rushton turbine stirred tank. The higher levels of scalar concentration due to leakage of scalar mass into the tank result in a lower driving force of inter-phase mass transfer. This consequently leads to an overestimation of the dissolution time. However, the impact of the mass leakage on the mixing and dissolution times is expected to be marginal. In case of the dissolution process studied, a back-of-the-envelope calculation resulted in an overestimation of the dissolution time by about 0.5%.

7.5

Perspectives and recommendations

In view of the work presented in the successive chapters, the ultimate goal of contributing to reliable and accurate predictions of complex, multi-phase processes has been met. The basis of a simulation is the solution of the flow field. In order to have an accurate description of the various, complex mechanisms occurring in these processes, their interplay with the turbulent velocity fluctuations present in the flow field should be represented to a high level of detail. An attractive, and widely used approach today (in terms of run times, memory use, user-friendly interface) is CFD based on the Reynolds-averaged Navier-Stokes equations (RANS). With a view to the complexity of a multi-phase process, the high level of detail is not reached by a steady-state approximation. Although a transient RANS simulation is able to predict the flow field reasonably, we have seen that it fails in providing detailed information of turbulent length and time scales. A k − type of closure model, which is widely used today, invariably underpredicts the turbulence levels by 50%, and this will

172


certainly affect the mixing performance and the overall outcome of the particular process of interest. In the last decade, large eddy simulations (LES) have demonstrated their ability to provide accurate details of the flow field and the turbulence levels. Application of LES to complex, multi-phase processes in industrial scale reactors is not yet within reach, because of limited computational resources. However, studying such processes in lab-scale reactors by means of LES in conjunction with e.g. scalar mixing and particle transport solvers (as in this work) has become a promising possibility. The studies as reported in chapters 5 and 6 provide detailed insight in the mixing performance of the turbulent flow in an agitated tank, and the physical mechanisms occurring in a dissolution process. Because of the continuing growth of computer resources, scaling-up studies are a matter of time. Although we have been able to achieve the goal posed, there is quite some room for improvements. When reading the thesis, several weak spots in the modeling attempts can be discerned. In the first place, the applicability of the particle transport code is limited by the size of the particles and the time step in relation to the collision algorithm. Secondly, the hydrodynamic forces and the mass transfer rate are based on single-particle correlations; the hydrodynamic interactions between particles are not included. Thirdly, the immersed boundary technique introduced in the finite volume approach is not mass conservative, the mass leakage is expected to be most pronounced at the edges of the impeller blades. We propose here some suggestions for improvement. The performance of the collision algorithm may be improved by reducing the time step. The single-particle correlations might be adapted such that hydrodynamic interactions are included through the inclusion of e.g. the local solids volume fraction. In this respect, a more advanced approach based on direct numerical simulations of a particle suspension in a simple geometry may be used to derive relations for hydrodynamic interaction forces. Increasing the spatial resolution (notably near the walls by means of local grid refinement) results in a better performance of the immersed boundary technique. Other boundary techniques may be explored, such as the cut-cell technique (see e.g. Tucker & Pan (1999)) in combination with an implicit formulation of the convection-diffusion equation. In order to increase the confidence in a multi-phase simulation, detailed experiments are needed. These experiments should provide averaged as well as angle-resolved data of the particle concentration and scalar concentration fields, local flow velocities in the presence of particles, and information on collisions and mass transfer rates. The numerical results obtained in this thesis demonstrate the capability of the current solvers to simulate a complex, multi-phase process in detail. As a result, other worthwhile future directions of numerical research, like industrial crystallization processes, are opened. Various phenomena common in crystallization can already be recovered with

7.5. Perspectives and recommendations

173

the current solvers available. For instance, the mass transfer needed for crystal growth, brought about by the fluid flow, is determined by the species concentrations (e.g. supersaturation) which is provided by the scalar transport solver. Also, information on crystalcrystal collisions that are induced by local velocity gradients is directly available. The knowledge of the local supersaturation in combination with the collision rate of the crystals enable the description of crystal agglomeration. Furthermore, attrition events (a major source of secondary nucleation) may be predicted with a relative ease, since the frequency and intensity of crystal-crystal and crystal-impeller collisions are provided by the particle transport solver. We have seen that realistic modeling of solid-liquid suspensions including mass transfer is not an easy task. While it is demonstrated that an accurate representation of the flow field by means of a LES is the first step toward success, we still need to revert to modeling particle/crystal related issues. For instance, the occurrence of a nucleation event is highly determined by concentration fluctuations at the subgrid-scale level. Consequently, these fluctuations should be taken into account by means of a subgrid-scale model. Furthermore, the performance of the current collision algorithm is limited by particle size, and consequently the simulation of a crystallization process is restricted to low volume fractions. In addition, the description of chemical reactions is an open problem as well. In case of reaction schemes with order higher than one, a closure model is needed for the reaction rate term of the scalar components that arises in the filtered convection-diffusion equation. Two major approaches may be adopted in handling the closure of the chemical reactions; the mechanistic micro-mixing approach (which has been exploited in the Kramers laboratory by Bakker (1996)) or the more advanced stochastic approach in terms of probability density functions (PDF’s) through Monte Carlo simulations (exploited by Van Vliet (2003)). While, in the latter approach the causality problems encountered in the former approach are avoided, the computational demand is higher. The work presented in this thesis has set a firm basis for accurate numerical predictions of complex, multi-phase processes. Hopefully, this thesis will contribute to new developments and encourages other researchers to tackle the challenges and problems (such as mentioned above) that are encountered in the future directions of numerical research on transport phenomena and chemically reacting flows.

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Dankwoord Het werk is voor mij nu af. Een periode van vier jaar lijkt als je begint erg lang, maar niets is minder waar. De jaren zijn voorbij gevlogen en nu kan ik terugkijken op een tijd die voor mij erg leerzaam is geweest. Naast de wetenschappelijke verdieping heb ik veel geleerd van de samenwerking in het Europese project ’OPTIMUM’ en van de contacten met ’lotgenoten’ in de wetenschappelijke wereld tijdens conferenties. Het resultaat van mijn onderzoekswerk dat opgetekend staat in dit boekwerk is niet louter door mij alleen verkregen. Alhoewel je geacht wordt zelfstandig onderzoek te kunnen verrichten, is het slagen hierin te danken aan de steun van vele mensen in mijn (directe) omgeving. Ik heb nu de mogelijkheid deze mensen te bedanken en hun namen op de laatste bladzijden van mijn proefschrift vast te leggen. In de eerste plaats wil ik mijn promotor Harrie van den Akker bedanken. Ik weet nog goed dat je in de tijd van mijn afstudeerperiode vertrouwen in mij had om een promotieonderzoek succesvol af te kunnen ronden. Halverwege mijn promotietijd hebben we samen nog wel eens nagedacht over de richting waarin mijn onderzoek zou gaan. Het ’OPTIMUM’ project gaf mij niet veel vrijheid een eigen smaak te geven aan mijn onderzoek. Gelukkig heb ik dat het laatste jaar helemaal goed kunnen maken, mede dankzij jouw steun en inspiratie in een voor jou moeilijke tijd. Harrie, bedankt voor alle goede discussies en voor de leuke en leerzame momenten tijdens de conferenties in Indianapolis en Lake Placid. Mijn onderzoek was niet van de grond gekomen zonder de steun van mijn begeleider Jos Derksen. Jos, de door jou ontwikkelde LES code vormde de basis van mijn werk. Ik ben je dankbaar voor alle input die je mij geleverd hebt. We hebben vele discussies gevoerd binnen, maar zeker ook buiten het lab. Voor het ’OPTIMUM’ project zijn we samen vaak op pad geweest. Weet je nog dat we in een fantastisch hotel in Erlangen overnacht hebben en de hachelijke verkeersituaties in Engeland die we overleefd hebben? Op deze plaats wil ik jou en je gezin ook van harte bedanken voor de week dat ik bij jullie mocht logeren in Princeton. De discussies die we daar gehad hebben, resulteerden uiteindelijk in een significant gedeelte van mijn proefschrift.

186

Dankwoord

I will now switch to English for a moment to thank all partners of the ’OPTIMUM’ project. In the next sentences I will thank some of them in particular. Bernd Genenger, you were the chief in charge. I would like to thank you for the different discussions we had, in particular about the macro-instabilities. Gerrit Hommersom, thanks for the discussions on the suspension polymerization process, the Maxwell model, and the nice conversations we had during our stay in a small hotel in England. Ian Hamill, Christiane Montavon and Jane Pearson, your help was of high importance in case of the preparation of chapter 2 and the paper published in Chemical Engineering Science. Christiane, we had fruitful (e-mail) discussions considering the Maxwell model, thank you for that. Michael Yianneskis, Loukia Nikiforaki and Martina Micheletti, thank you for the discussions and the great moments we spent together during the ’OPTIMUM’ project. Your input was valuable in the realization of chapter 3 and the paper published in AIChE Journal. Mike, I appreciate your effort in reading my dissertation and for taking place in the commission for my PhD defense. Weer terug in het Nederlands ben ik aangekomen bij mijn collega-promovendi. Ondanks dat ik vaak weg was voor ’OPTIMUM’ vergaderingen en conferenties, heb ik toch het grootste gedeelte van mijn tijd doorgebracht in het lab. Een aantal mensen wil ik in het bijzonder bedanken: mijn W220 kamergenoten Andreas ten Cate, Wouter Harteveld, Martin Rohde, Ruurd Dorsman, Marco Zoeteweij en Jos van ’t Westende. Wouter en Martin, met jullie heb ik enorm veel lol gehad! Ik denk met veel plezier terug aan al de momenten (daar kan je zelfs een proefschrift van schrijven!) die ik met jullie beleefd heb. Ik kan zeker niet voorbijgaan aan de dames van het secretariaat: Karin van de Graaf en Thea Miedema. Bedankt voor alle gesprekken die we gehad hebben en de steun die ik van jullie heb mogen ervaren tijdens de ziekte van Mirjam. Ik mis jullie al vanaf de eerste dag dat mijn contract verlopen was, onvervangbaar! Ook de drie Japen, Thea, Jan, Wouter en Ab bedankt voor de leuke gesprekken aan de koffietafel! Ik kan als promovendus die volledig afhankelijk is van het rekencluster niet voorbijgaan aan de systeembeheerder (alias computermuis) Peter Bloom: bedankt voor het fungeren als vraagbaak en je bijdrage aan de significante uitbreiding van het rekencluster! Nu kom ik aan bij de mensen die mij vanuit de privesfeer enorm tot steun zijn geweest. Allereerst mijn vrienden Arjan en Mirjam Grevengoed en Han en Ella Knoeff. Jullie zijn mij heel dierbaar geworden. Meer dan de motivatie tijdens mijn promotiewerk waardeer ik jullie ondersteuning op geestelijk vlak. Jullie hebben mij elke keer weer laten zien om Wie het draait in het leven. Heel veel dank ben ik jullie verschuldigd! Pa en ma, ik wil jullie danken voor wie ik ben zoals ik nu ben. Jullie zorg en motivatie tijdens mijn school- en universiteitsperiodes hebben hiertoe bijgedragen. Ook mijn schoonouders (laat het woordje ’schoon’ maar weg!) wil ik bedanken. In de acht jaar dat

187 ik jullie dochter nu ken, zijn ook jullie voor mij ouders geworden. Bedankt dat ik mijn frustraties en onzekerheden bij jullie kwijt kon tijdens mijn promotie. Meer dank nog voor jullie zorg voor Mirjam en Léon tijdens Mirjams ziekteperiodes. Eigenlijk is dit niet in woorden uit te drukken. Tot slot mijn vrouw, Mirjam. Wat jij voor mij betekent is onbeschrijflijk. We hebben hele moeilijke periodes gekend de laatste twee jaar. Tijdens de momenten dat jij zo ziek was, dacht je nog aan mij en Léon. Jij hebt tijdens mijn afstuderen en promoveren heel erg veel geduld met mij gehad. Bedankt voor jouw gebeden en dat je mij elke keer er weer op gewezen hebt dat ik alles bij God neer mag leggen. Zonder jou had ik het niet gered!

List of Publications • H. Hartmann, J.J. Derksen and H.E.A. van den Akker (2003). An LES investigation of the flow macro-instability in a Rushton turbine stirred tank. Proceedings of the 11th European Conference on Mixing, Bamberg, Germany • H. Hartmann, J.J. Derksen, C. Montavon, J. Pearson, I.S. Hamill and H.E.A. van den Akker (2004). Assessment of large eddy and RANS stirred tank simulations by means of LDA. Chem. Eng. Sci., 59(12), 2419-2432 • H. Hartmann, J.J. Derksen and H.E.A. van den Akker (2004). Macro-instability uncovered in a Rushton turbine stirred tank. AIChE J., 50(10), 2383-2393 • H. Hartmann, J.J. Derksen and H.E.A. van den Akker (2005). LES of the mixing time in a Rushton turbine stirred tank. Submitted to AIChE J. • H. Hartmann, J.J. Derksen and H.E.A. van den Akker (2005). Numerical simulation of a solubility process in a stirred tank reactor. Submitted to Chem. Eng. Sci.

About the Author Hugo Hartmann was born on August 8th 1977 in Zuidelijke IJsselmeerpolders (Lelystad), the Netherlands. In 1995, he graduated from pre-university education (VWO) at the ‘Interconfessionele Scholengemeenschap Arcus’ in Lelystad. From 1995 to 2001, he studied Applied Physics at Delft University of Technology. During this period he was a member of a Christian students’ association ‘C.S.R.’ (Civitas Studiosorum Reformatorum). He did his graduation research at the ’Kramers Laboratorium voor Fysische Technologie’ (Delft University of Technology), and on February 15th 2001 he graduated on his thesis entitled ’Investigation of flashing induced instabilities in boiling water reactors’. On March 1 st 2001, he started working on his PhD research at the same department. Till December 2003, he took part in a European project called ’OPTIMUM’ (Optimization of Industrial Multiphase Mixing). From June 1st 2005, he is working as a separation technologist at Frames Process Systems in Zoeterwoude. The author married Mirjam Kolk on May 18 nd 2001, and on April 22nd 2003 he became father of a son called Léon.