10th European Conference on Mixing

European Conference on Mixing Proceedings of the lOth European Conference, Delft, The Netherlands, July 2-5, 2000

lOth European Conference on Mixing Proceedings of the lOth European Conference, Delft, The Netherlands, July 2-5, 2000

This Page Intentionally Left Blank

lOth European Conference on Mixing Proceedings of the 10th European Conference, Delft, The Netherlands, July 2-5, 2000

edited by

H.E.A. van den Akker and J.J. Derksen Kramers Laboratorium

v o o r Fysische T e c h n o l o g i e ,

Delft University of Technology, Delft, The Netherlands

2000

ELSEVIER A m s t e r d a m - Lausanne - N e w Y o r k - O x f o r d - S h a n n o n - S i n g a p o r e - Toky 9

E L S E V I E R S C I E N C E B.V. S a r a B u r g e r h a r t s t r a a t 25 P.O. B o x 211, 1000 A E A m s t e r d a m , T h e N e t h e r l a n d s

9 2 0 0 0 E l s e v i e r S c i e n c e B.V. A l l r i g h t s r e s e r v e d .

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CONTENTS Preface Mixing: Terms, Symbols, Units European Federation of Chemical Engineering - Working Party of Mixing 1999

XV

TURBULENCE CHARACTERISTICS IN STIRRED TANKS

Trailing vortex, mean flow and turbulence modification through impeller blade design in stirred reactors M. Yianneskis Turbulence generation by different types of impellers M. Sch&fer, J. Yu, B. Genenger and F. Durst Limits of fully turbulent flow in a stirred tank K.]. Bittoff and S.M. Kresta

17

MEASUREMENTS IN CHEMICALLY REACTING FLOWS

Spatially resolved measurements and calculations of micro-and macromixing in stirred vessels M. Buchmann, K. Kling and D. Mewes

25

Characterisation and modelling of a two impinging jet mixer for precipitation processes using laser induced fluorescence N. B~net, L. Falk, H. Muhr and E. Plasari

35

Four-dimensional laser induced fluorescence measurements of micromixing in a tubular reactor E. van Vliet, ].]. Derksen and H.E.A. van den Akker

45

MODELLING OF MICRO-MIXING

Simulation with validation of mixing effects in continuous and fed-batch reactors G.K. Patterson and J. Randick

53

A computational and experimental study of mixing and chemical reaction in a stirred tank reactor equipped with a down-pumping hydrofoil impeller using a micro-mixing-based CFD model O. Akiti and P.M. Armenante

61

vi Mixing with a Pfaudler type impeller: the effect of micromixing on reaction selectivity in the production of fine chemicals I. Verschuren, J. Wijers and J. Keurentjes

69

Comparison of different modelling approaches to turbulent precipitation D. Marchisio, A.A. Barresi, G. Baldi and R. Fox

77

Application of parallel test reactions to study micromixing in a co-rotating twin-screw extruder A .Rozen, R.A. Bakker and J. Baldyga

85

Solid liquid mixing at high concentration with SMX static mixers O. Furling, P.A. Tanguy, L. Choplin and H.Z. Li

93

EFFECTS OF VISCOSITY AND RHEOLOGY ON MIXING

Influence of viscosity on turbulent mixing and product distribution of parallel chemical reactions J. Baldyga, M. Henczka and L. Makowski

101

Mixing of two liquids with different rheological behaviour in a lid driven cavity H. Hoefsloot, S.M. Willemsen, P.J. Harnersma and P.D. ledema

109

Mobilization of cohesive sludge in storage tanks using jet mixers M.R.Poirier, H. Gladki, M.R. Powell, and Ph.O. Rodweil

117

SLURRY SYSTEMS

CFD simulation of particle distribution in a multiple-impeller high-aspect ratio stirred vessel G. Montante, G. Micale, A. Brucato and F. Magelli

125

Power consumption in slurry systems A. Barresi and G. Baldi

133

LIQUID-LIQUID DISPERSIONS .....

Drop break-up and coalescence in intermittent turbulent flow W. Podgdrska and J. Baldyga

141

Measurement and analysis of drop size in a batch rotor-stator mixer R.V. Calabrese, M.K. Francis, V.P. Mishra and S. Phongikaroon

149

vii

The impact of fine particles and their wettability on the coalescence of sunflower oil drops in water A.W. Nienow, A.W. Pacek, R. Franklin and A.J. Nixon

157

Influence of impeller type and agitation conditions on the drop size of immiscible liquid dispersions M. Musgrove and S. Ruszkowski

165

Experimental findings on the scale-up behaviour of the drop size distribution of liquid/liquid dispersions in stirred vessels G. W. Colenbrander

173

Investigations of local drop size distributions and scale-up in stirred liquid-liquid dispersions K. Schulze, J. Ritter and M. Kraurne

181

GAS-LIQUID SYSTEMS

Gas-liquid mass transfer in a vortex-ingesting, agitated draft-tube reactor C. Leguay, G. Ozcan-Taskin and C.D. Rielly

189

Modelling of the interaction between gas and liquid in stirred vessels G. Lane, M.P. Schwarz and G.M. Evans

197

Experimental investigation of local bubble size distributions in stirred vessels using Phase Doppler Anemometry M. Sch~fer, P. W~chter and F. Durst

205

Void fraction and mixing in sparged and boiling reactors Z. Gao, J.M.Smith, D. Zhao and H. MOIler-Steinhagen

213

PARTICLE COLLISIONS IN CRYSTALLISATION

A numerical investigation into the influence of mixing on orthokinetic agglomeration 221 E.D. Hollander, J.J. Derksen, O.S.L. Bruinsrna, G.M. van Rosmalen and H.E.A. van den Akker An experimental method for obtaining particle impact frequencies and velocities on impeller blades K.C. Kee and C.D. Rielly

231

ADVANCED CFD

Comparison between direct numerical simulation and ~:-~ prediction of the flow in a vessel stirred by a Rushton turbine C. Bartels, M. Breuer and F. Durst

239

viii

The use of large eddy simulation to study stirred vessel hydrodynamics A. Bakker, L.M. Oshinowo and E.M. Marshall

247

Compartmental modelling of an 1100L DTB crystallizer based on large eddy flow simulation A. ten Cate, S.K. Bermingham, J.J. Derksen and H.M.J. Kramer

255

POSTERS

Detailed CFD prediction of flow around a 45 ~ pitched blade turbine J.K. Syrj~nen and M.-F. Manninen

265

Comparison of CFD methods for modelling of stirred tanks G.L. Lane, M.P. Schwarz and G.M. Evans

273

Predicting the tangential velocity field in stirred tanks using the Multiple Reference Frames (MRF) model with validation by LDA measurements L. Oshinowo, Z. Jaworski, K.N. Dyster, E. Marshall and A.W. Nienow

281

Numerical simulation of flow of Newtonian fluids in an agitated vessel equipped with a non standard helical ribbon impeller G. Delaplace, C. Torrez, C. Andre, N. Belaubre and P. Loisel

289

A contribution to simulation of mixing in screw extruders employing commercial CFD-software M. Motzigemba, H.-C. Broecker, J. PrEnss,D. Bothe and H.-J. Wamecke

297

Experimental and CFD characterization of mixing in a novel sliding-surface mixing device J.M. Rousseaux, Ch. Vial, H. Muhr and E. Plasari

305

An investigation of the flow field of viscoelastic fluid in a stirred vessel W. Ju, X. Huang, Y. Wang, L. Shi, B. Zhang and J. yuan

313

Flow of Newtonian and non-Newtonian fluids in an agitated vessel equipped with a non-standard anchor impeller G. Delaplace, C. Torrez, M. Gradeck, J.-C. Leuliet and C. Andr~

321

Characterization of convective mixing in industrial precipitation reactors by real-time processing of trajectography data B. Barillonand P.H. J~z~quel

329

Characterization of flow and mixing in an open system by a trajectography method P. Pitiot and L. Falk

337

Characterization of the turbulence in a stirred tank using particle image velocimetry M. Perrard, N. Le Sauze, C. Xuereb and J. Bertrand

345

ix

Turbulent macroscale of the impeller stream of a Rushton turbine R. Escudi~,, A. Ling and M. Roustan

353

Analysis of macro-instabilities (MI) of the flow field in stirred tank reactor (STR) agitated with different axial impellers V. Roussinova and S.M. Kresta

361

Local dynamic effect of mechanically agitated liquid on a radial baffle J. Krat~,na, I. Fort, O. Bruha and J. Pavel

369

Interpretation of macro- and micro-mixing measured by dual-wavelength photometric tomography M. Rahirni, M. Buchmann, R. Mann and D. Mewes

377

Effect of tracer properties (volume, density and viscosity) on mixing time in mechanically agitated contactors A. Pandit, P.R. Gogate and V.Y. Dindore

385

Mixing, reaction and precipitation : an interplay in continuous crystallizers with unpremixed feeds N.5. Tavare

395

Simulation of a tubular polymerisation reactor with mixing effects E. Fournier and L. Falk

407

Mixing equipment design for particle suspension - generalized approach to designing F. Rieger and P. Ditl

415

Characterization and rotation symmetry of the impeller region in baffled agitated suspensions Z. Yu and A. Rasmuson

423

Solids suspension by the bottom shear stress approach M. Fahlgren, A. Hahn and L. Uby

431

A phenomenological model for the quantitative interpretation of partial suspension conditions in stirred vessels G. Micale, F. Grisafi, A. Brucato and L. Rizzuti

439

A self-aspirating disk impeller- an optimization attempt C. Kuncewicz and J. Stelrnach

447

A novel gas-inducing agitator system for gas-liquid reactors for improved mass transfer and mixing E.A. Brouwer and C. Buurman

455

Hold-up and gas-liquid mass transfer performance of modified Rushton turbine impellers S.C.P. Orvaiho, J.M.T. Vasconcelos and S.S. Aires

461

A simple method for detecting individual impeller flooding of duaI-Rushton impellers A. BombaY, and I. Zun

469

Numerical simulation of gas-liquid flow in a parallelepiped tank equipped with a gas rotor-distributor E. Waz, C. Xuereb, P. Le Brun, B. Laboudigue and J. Bertrand

477

Experimental and modelling study of gas dispersion in a double turbine stirred tank S. Alves, C.I. Maia, S.C.P. Orvalho, A.J. Serralheiro and J.M.T. Vasconcelos

485

Local heat transfer in liquid and gas-liquid systems agitated by concave disc turbine J. Karcz and A. Abragimowicz

493

Effect of the viscosity ratio ~ld/TIcon the droplet size distributions of emulsions generated in a colloid mill C. Dicharry, B. Mendiboure and J. Lachaise

501

Experimental measurement of droplet size distribution of a MMA suspension in a batch oscillatory baffled reactor of 0.21 m diameter G. Nelson, X. Ni and !. Mustafa

509

Power consumption in mechanically stirred crystallizers R. Bubbico, 5. Di Cave and B. Mazzarotta

517

Fluid dynamic studies of a large bioreactor with different cooling coil geometries H. Patei, C.M. Kao, W. Bujalski, P. Mohan, J. McKemmie, C.R. Thomas and A.W. Nienow

525

Author index

533

xi

Preface

In September 1974, the first of the European Conferences on Mixing was held in Churchill College in Cambridge/UK. On that first occasion, just eighteen papers were presented. Since then, eight further conferences took place in successively Mons/B (1977), York/UK (1979), Noordwijkerhout/NL (1982), WCirzburg/D (1985), Pavia/I (1988), Brugge/B (1991), Cambridge/UK (1994), and Paris/F (1997). Now, at the turn of the century, the 10~ Conference in this successful series will be held in DelftJNL on July 2 - 5, 2000. This 10th Conference marks the 50th Anniversary of the Kramers Laboratorium voor Fysische Technologie of Delft University of Technology that over the years contributed significantly to the research on mixing in stirred vessels. This may best be illustrated by the long list of Ph.D. theses prepared in the Kramers Laboratorium in the field of mixing: Van de Vusse (1953), Westerterp (1962), Voncken (1966), Reith (1968), Van Heuven (1969), Stammers (1970), Van 't Riet (t975), Warmoeskerken (1986), Frijlink (1987), Bartels (1988), A. Bakker (1992), Bouwmans (1992), R.A. Bakker (1996) and Venneker (1999). The aim of the European Conferences on Mixing is to bring together scientists and chemical engineers working in the field of fluid mixing and multiphase contacting processes carried out in stirred vessels and static mixers. The topic of solids mixing, e.g. in fluidised beds, is usually notcovered in these conferences however. In the course of time, these Conferences turned out to be excellent carriers for disseminating the advances of scientific work in the field of mixing and their applications to process industries. Attendance is high, nowadays in the order of 200 participants, with many groups contributing one or more papers to each conference. Traditionally, fluid mixing and the related multiphase contacting processes have always been regarded as an empirical technology. It was surrounded by a great deal of romanticism cautiously fostered by a few gurus who collected their wisdom during life-long experience with practical mixing problems. Their expertise comprised dimensionless numbers as well as rules of thumb for scaling up contacting processes, along with almost private insights as to the relative performances of a wide variety of impeller types. Many aspects of mixing, dispersing and contacting were related to power draw, but understanding of the phenomena was limited or qualitative at the most. This is nicely illustrated in the book "Mixing of Liquids by Mechanical Agitation", edited by J.J. Ulbrecht and G.K. Patterson, and published by Gordon and Breach Science Publishers (1985) in their Series 'Chemical Engineering: Concepts and Reviews'. In particular during the last decade, however, plant operation targets have tightened and product specifications have become stricter, as a result of increasing pressure from shareholders for higher profits. The public awareness as to safety and environmental hygiene has increased. The drive towards 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. All this has resulted in a market pull: the need for more detailed insights in flow phenomena and processes and for better verifiable design and operation methods.

xii

Fortunately, dazzling developments in miniaturisation of sensors and circuits as well as in computer technology have rendered leaps possible in computer simulation and animation and in measuring and monitoring techniques. This development could be denoted as a technology push which took place in the same decade as the above market pull due to changing company policies. This technology push really turned mixing from an empirical technology into a scientific activity, and replaced the romanticism of empiricism by the promises and challenges of modern computer aided experimentation and simulation. Studying local flow and transport phenomena, their spatial variations and their dynamics, and local transfer and chemical processes has become quite feasible. Black-box models, mean residence times, and residence time distributions have turned into concepts that now are caught up by much more sophisticated models. The latter often take the form of large sets of partial differential equations in the local variables that quite efficiently can be solved by rigorous and robust numerical tools. The advances simultaneously attained in exploiting and interpreting non-intrusive laser and radiation diagnostics, and in miniaturising sensors or probes make it possible to study and analyse the complicated pathways volume elements and/or second-phase particles follow inside the stirred vessels or static mixers under consideration. In brevity: macro-balances and macro-scale rules now clear the passage for micro-balances and distributed parameters. One cannot really pinpoint when and where this revolution in methods and techniques started. It may have started hesitatingly, distributed over some years, and initially almost unnoticed by many of the mixing experts. So large was the gap between the traditional chemical engineering approach in the mixing field and the scientific methods developed and practised in the fields of fluid mechanics, turbulence theory and multiphase flow. A search for the first papers presenting such allegedly sophisticated experimental and computational techniques may end up in the proceedings of the 7th Conference in this series (Brugge, 1991). Later on, the best papers presented in Brugge were collected in a special volume edited by Roger King and published by Kluwer Academic Publishers. This volume was ominously entitled "Fluid Mechanics of Mixing: Modelling, Operations and Experimental Techniques" and contained a number of interesting papers on Computational Fluid Dynamics (CFD), on Laser Induced Fluorescence (LIF), on Laser Doppler Anemometry (LDA), and on Phase Doppler Particle Analysis. The next Conferences in Cambridge (1994) and Paris (1997) showed a gradual but steady increase in the number of papers dealing with local phenomena inside stirred vessels. This observation pertains to both numerical simulations and experimental techniques. In view of this 10th Conference in Delft, papers were preferably solicited on improved modelling, numerical simulations and on sophisticated (non-intrusive) measuring techniques rather than on global and empirical approaches in terms of averaged quantities such as power draw. Generally, oral presentations have been reserved for original contributions reporting on real advances in experimental and computational techniques. My first impression is that at this 10th Conference transient flow phenomena are going to be discovered, due to the exploitation of Large-Eddy Simulations and of fast laser and radiation diagnostics among which tomographic techniques.

xiii

The idea of the organisers of this 10th European Conference on Mixing was to encourage a leap forward in the field of mixing by exposing the mixing community to the current, overwhelming wealth of sophisticated measuring and computational techniques. This leap may be made possible by the blessings of modern instrumentation, signal and data analysis, field reconstruction algorithms, computational modelling techniques and numerical recipes. That is why in the reviewing procedure quite some emphasis was put on trying and identifying novel and promising generic techniques rather than novel results which by nature are mostly specific to particular processes or circumstances. The reader may judge for himself whether or not the organisers have succeeded in this intention. The members of the Organising and the Advisory Committees are gratefully acknowledged for their efforts in refereeing first the almost 130 abstracts submitted and, in a later stage, the 85 papers submitted. Their highly appreciated assistance was a great help in selecting the 69 papers now contained in this Volume. Harry E.A. van den Akker, Chairman

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European Federation of Chemical Engineering

Working Party of Mixing 1999

Mixing: Terms, Symbols, Units

xvi

Mischen" Begriffe, Formelzeichen, Einheiten Mixing" Terms, Symbols, Units Melange: Termes, Symboles, Unitds Literatur References Rdfdrences

[1] Units of measurement, lSO standards handbook 2. 1979. [2] AIIgemeine Formelzeichen. Deutsche Norm DIN 1304. November 1971. [3] RL~hrerfur RShrbeh~lter. Deutsche Norm DIN 28131. Februar 1979. [4] RL~hrbehw

Deutsche Norm DIN 28136. Entwurf Juli 1979.

[5] Recommended Standard Terminology and Nomenclature for Mixing. The Institution of Chemical Engineers. Rugby, England. 1980. [6] Michaci Zarizeni. CSSR Standards ON 691000-691039, 1969. [7] AIChE Standard Testing Procedure. Mixing Equipment (Impeller Type). January 1965. [8] AIChE Equipment Testing Procedure. Paste and Dough Mixing Equipment. [9] Ullmanns Encyklop~die der technischen Chemie. 4., neubearbeitete und erweiterte Auflage. Band 2, Verfahrenstechnik I (Grundoperationen). Verlag Chemie, Weinheim/Bergstr., 1972. [10] Thermische Trennverfahren in der Verfahrenstechnik: Begriffe, Formelzeichen und Einheiten. VDI-Richtlinie 2761. VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen/European Federation of Chemical Engineering. Dezember 1975. [11] Recommended Standard Terminology and Nomenclature for Mixing. Eighth European Conference on Mixing. Institution of Chemical Engineers, 1994. [12] Fliessbilder verfahrenstechnischer Anlagen. Deutsche Norm DIN 28004. Mai 1988.

xvii

1. l n d i z e s Subscripts Indices

and superscripts

In diesem Kapitel ist der Buchstabe A als Beispiel for ein beliebiges Formelzeichen eingesetzt, um verschiedene Indizes zu zeigen. In this section the letter A is taken to represent an arbitrary symbol to indicate various subscripts and superscripts. La lettre A est utilisee dens ce chapitre comme exemple d'un quelconque symbole pour repr6senter de differents indices.

Einheit unit unit6

Banennung name nom

101

Gr6sse A quantity A quanitd A

102

Strom flow rate ddbit

auf die Zeit bezogen with respect to time par unit6 de temps

[A]/s [A]/h

Stromdichte

auf die Zeit und Querschnittsfl&che bezogen with respect to time and area of cross section par unite de temps et de section de passage

[A] / s m = [A] / hm2

auf Masse, Volumen .... bezogen with respect to mass, volume .... par unite de masse, de volume ....

[A] / kg, [A] / m s....

103

Definition, Erkliirung definition, description d6finition, description

Formelzeichen symbol symbole

Nr. no. no.

[A]

flux flux 104

bezogene GrSsse i related quantity quantite relative

1 0 5 ' molare Grosse ; molar quantity quantit~ molaire

Am

A m - .~

....

106

kritischer Wart critical value valeur critique

107

Gleichgewichtswert equilibrium value valeur a 1'6quilibre

, A*

108

gasf6rmige Phase gas phase phase gazeuse

i AG

109

fliJssige Phase liquid phase phase liquide

110

[A] / kmol [A] / tool

iI

i festa Phase solid phase phase solide

111 i kontinulerliche Phase continuous phase _ ~ phase continue 112

disperse Phase dispersed phase phase dispersee

Ao

113

1., 2...... i ...... n.Wert 1st, 2nd ..... ith ..... nth value 1re, 2e ..... iibme ..... nibmevaleur

A~, A 2.... , A i..... A.

11";'

Minimalwert

115

minimum value valeur minimum

,! . . . . . . .

Bemorkung remark remerque

Maximalwert maximum value valeur maximum

i

,, Ami.

'1

,! Amn

I !

A -. 101, n -, 209

xviii

116"

Totalwert total value valeur totale

n

&o,-Z &

A,=

Aj ~ 113, n -,, 4 0 8

i-1

117

arithmetischedzeitlicher Mittelwert arithmetic/time mean value valeur moyenne arithm6tique / par unitd de temps

118

Bezugs-/Referenzwert reference value valeur de rMdrence

Ao

119

Wert am Eingang/Anfang value at inlet/start valeur ~ I'entrde/initiale

Ao. (Ai.)

120

Wert am Ausgang/Ende value at outlet/end valeur b la sortie/finale

A,,,, (A=.,)

121

Wert an der Wand value at wall valeur ~ la paroi

Aw

122

Weft der Mischung value of mixture valeur du mdlange

123

laminar laminar laminaire

124

turbulent turbulent turbulent

125

in x-, y-, z-Richtung in x, y, z direction dans les directions x, y, z

A-.* 101, A ~ - ~ 116 n ~ 408, t -~201

I

AM

A,. Ay. A,

xix

2. Grundbegriffe, Fundamental Termes

Konstanten terms,

constants

fondamentaux,

constantes

201

Zeit time temps

202

thermodynam ische Tern perat ur thermodynamic temperature tempdrature thermodynamique

T = t + 273.15

T,B

203

Celsius-Temperatur Celsius temperature dagre Celsius

t - T - 273.15

t,O

204

i

s,h

t -~ 203 ,

Druck pressure pression

205

L~nge length iongueur

206

Fl~che area surface

210

Kraft force force

211

W~rme heat quantitd de chateur

212

Leistung power puissance

213

Erdbeschleunigung acceleration due to gravity acc61dration due a la pesanteur

214

Avogadro-Konstante, Loschmidt~Zahl Avogadro constant constante d'Avogadro

215

Universelle/allgem=ine Gaskonstante molar gas constant constante molaire des gaz

Pa bar

1 P a - 1 N/m 2 I bar - 10s Pa

m3

Volumen volume volume . . . . . 208 Masse mass masse Stoff-/Molmenge amount of substance quantite de substance

...... T --, 202

m2

207

209

~

n, (N)

kmol reel

6

,,

~Wh

' ~ kc= - '186.8 J

I kWh ,,- 860 kcal

g - 9.81 m/s =

N A- 6.023 10=* kmo1-1 - 6.023 10~ reel -1 J/kmolK J/molK

R - 8314 J/kmolK - 8.314 J/molK

XX

3. Stoffeigenschaften Properties of substance Proprietds d'une substance

M

30i ....I M'olmasse molar mass masse molaire

302 ~Schmelz-/Ersta rrungspu nkt melting/freezing point tempdrature de fusion/congelation 3o3

Taupunkt dew point temperature de rosee

305

Dampfdruck vapour pressure tension de yap.cur

306t

K, ~

T.~ e . , , t.,, 0 .

i,,

~ Siedepunkt , boiling point temperature d'dbullition

304.

i kg/kmol

Tb, e~ ~ Ob

Ta, Od, t~, Od

i !

K,~

K, ~

Pa bar

.

spezifische Schmelzenthalpie/-wiirme specific enthatpy of fusion chaleur latente specifique de fusion

&hsL"

J/kg

307

spezifische Sublimationsenthalpie/-wSrme specific enthalpy of sublimation chaleur latente specifique de sublimation

i&hr~

J/kg

308

spezifische Verdampfungsenthal pie/-w~rme specific enthalpy of eva porization chaleur latente spdcifique de vaporisation

~g'ILa

J/kg

309

spezifische Wiirme bei konstantem Druck specific heat at constant pressure chaleur specifique ;~ pression constante

310

W~rmeleiff~higkeit thermal conductivity conductivite thermique

311

Tern peraturleitf&higkeit thermal diffusivity diffusivitd tharmique

312

Dichte density masse volumique

J~gK

i W/inK

X p Cp

a, (=)

p

~, (~)

314

~,, (~,)

scheinbare Viskositlit apparent viscosity i viscosit(~ apparente v = ---~-

1 kcal/hmK 1.163 WlmK

m~

kg/m =

313 ' dynamische Viskosit~t dynamic viscosity viscositd dynamique

315 I kinematischeViskositw kinematic viscosity viscosit6 cinfmatique

1 Pa = 1 N / m z

1 bar - 105 Pa

Pas

1 Pas

i Pas

1 Pas

- 1 Ns/m 2

, m=/s

lmZ/s

=10 4St

i

= 1 Ns/m z =10P 1 mPas - 1 cP -10P 1 m Pas - I cP

xxi

316

Diffusionskoeffizient diffusion coefficient coefficient de diffusion

317

Gren'zenfl~chen'-/Oberfl~chenspannung interfacial/surface tension tension interfaciate/superficielle

D, (.~)

m3/s

N/m

O'GL, GM., Ol.S

,,

Benetzungswinkel wetting angle angle de mouillage 319

320

rad'~

....

H, H~

Henrykoeffizient Henry's law coefficient constante de Henry spezifische Mischungsenthalpie

specific enthalpy of mixing

enthaipie specifique de melange

Ah M

i 1" - ~-~ red

Pa kmol/kmo~ i Pa tool/tool

bezogene Enihalpie, die I)e'im Vermischen der reinen Komponenten bei konstanter Temperatur und konstantern Druck aufgenommen (+) oder abgegeben (-) wird related enthalpu consumed (+) or evolved (-) in mixing pure components at constant temperature and constant pressure effet thermique relatif Iors du m~lange de composants puts tempdrature et pression constantes: (+) Iorsque la chaleur est absorbee, (-) Iorsqu'elle est d6gag6e

' ' 1 mN/m"'--'l dyn/cm

J/kg

J

xxii

4. M i s c h u n g , Homogenit/it Mixture, homogeneity Mdlange, homogdnditd

401

Teilgr6sse dutch Gesamtvolumen partial quantity per unit total volume quantit6 panielle par unitd de volume total

Konzentration concentration concentration

402

Anteil, Gehalt fractional concentration fraction

Teilgr6sse dutch Gesamtgr6sse ratio of partial to total quantity quantite partielle rapport6e ~ la quantitd totale

kg/m= kmol/m= m=/m=

x,y

kg/kg kmol/kmol tool/tool maim z

X,Y

kg/kg kmol/kmot mol/moi

X,Y

kg/kg i bevorzugter Bekmol/kmol gdff in der Extrakmol/mol tion$- und Trocknungstechnik preferred term in extraction and drying technology terme pr6f6r6 an extraction et sdchage

in FI/Jssig-/1-'_xtraktphase in liquid/extract phase dane la phase liquide/l'extrait

,03

in Dampf-/Raffinatphase in vapour/raffinate phase dens. la phase gazeuse/le raffinat Mengenverh<nis quantity ratio rapport de quantite

Teilmenge durch Bezugsmenge ratio of partial to reference amount quantitd partielle rapportde a la i quantite de rdfdrence in Fl~ssig-/Extraktphase in liquid/extract phase dane Is phase liquide/l'extrait in Dampf-/Raffinatphase in vapour/raffinate phase dane la phase gazeuse/le raffinat

404

Beladung mass ratio on solute-free basis fraction rapportee au solvant

Teilmenge durch Tr~germenge ratio of partial amount to carrier amount quantite partielle rapportde ~ la quantit6 de solvant in FIQssig-/Extraktphase in liquid/extract phase dans la phase liquide/l'extrait in Dampf-/Raffinatphase in vapour/raffinate phase dens la phase gazeuse/le raffinat

405 ~Blasen-, Tropfen-, Teilchengr6sse bubble, droplet, particle size taille des bulles, gouttes, particules .... ; gemessene Gr6sse measured quantity quantitd mesurde i

407 =ii Mischzeit

'

i mixing time -

temps de mdlange -1. . . .

. . . . t Zeit, um geforderte Homogenitw zu erzielen time to achieve required homogeneity : temps requis pour atteindre I'homog6ndite ddsirde

408 ! Anzahl der Proben/Messungen i number of samples/measurements i hombre d'dchantillons/de mesures ,

..........

Einhaiten jR nach gemessener Gr6sse units according to . measured quantity I uniMs felon quantitd mesur6e

!

! tM

I I

-

!

xxiii 409

Standardabweichung standard deviation d6viation standard

'

1'I"

'

/ ~ : (xi-~) 2

,,..~/+-!__ Y

n-1

Einheiten je nach gemessener Gr6sse units according to measured quantity unitds selon quantit6 mesur6e n ~

4 0 8 , x --+ 4 0 6

410

Variationskoeffizient coefficient of variation coefficient de variation

x -~ 4 0 6 , o -~ 4 0 9

411

relative Standardabweichung relative standard deviation ddviation standard relative

(7 -,' 409

+12

Segregationsgrad degree/intensity of segregation degrO de segrdgation

S ,,.

0"2 O'o:t

--, '409

xxiv 5. Mischer Mixers Mdlangeurs

501

Anzahl der statischen Mischelemente statischer Mischer static mixer number of static mixing elements m61angeur statique nombre d'dldments m,tlangeurs statiques

502

L~nge eines statischen Mischelementes length of a static mixing element Iongueur d'un 61dment mdlangeur statique

503

L~nge des statischen Mischers length of static mixer Iongueur du mdlangeur statique

504

Rohrinnendurchmesser inside pipe diameter diam~tre interieur du tube

Innendurchmesser R0hrbeh~lter ROhrbehlilter inside diameter of stirred tank stirred tank diambtre interieur de la cuve agitee cuva agit6e j .............. . 5O6 Aussendurchmesser des ROhrers overall agitator diameter diam~tre ext6rieur de I'agitateur

505

507

I

T

m

' D (D1,

i Rfihrerabstand vom Boden agitator distance from i bottom of vessel distance entre I'agitateur et le fond de ia cuve t Breite des Strombrechers 509 baffle width largeur de la chicane ! . . . . . Abstand des Strombrechers sl0 yon der Wand clearance of baffle from wall dcartement de la chicane par rapport la paroi

512

Totale FlOssigkeitsh6he Total liquid depth Profondeur totale du liquide

J

m

m

m

i

i

m

i

m

i

i

Anstellwinkel des R0hrblattes zur Horizontalen blade pitch angle to the horizontal angle d'inclinaison de la pale i par rapport ~ !'horizonta!e

514 ' Winkeigeschwindigkeit i angular velocity vitesse angulaire i 515 ; Drehmoment torque moment d'un couple 516

h (hi, h: .... )

Anzahl der ROhrbl~tter number of blades on agitator nombre de pales de I'agitateur

513 R~hrerdrehzahi agitator speed vitesse de rotation de l'agitateur

.... )

w (w I, w=.... )

Breite der ROhrbliitter width of blades on agitator largeur des pales de I'agitateur

508

511

D2

r162

n, (N) ! 1 i ~ = 2~rn

i

rad ,"

I

i

..

!

1~ "1"~" red

S-I

rad/s !

,

,

Nm

m

' n -* 513 ,

!

Xxv

6. Str6mungstechnik Fluid dynamics Dynamique des flu/des 601

v=, vv, v,

Str6mungsgeschwindigkeit flow velocity vitesse d'dcoulement

m/s

M,V,W

602

turbulente Schwankungsgeschwindigkeit turbulent fluctuating velocity fluctuation de vitesse turbulente

V'= V - ( /

603

Turbulenzgrad turbulence intensity intensit~ de turbulence

Tu~3V,'

mittlere Strbmungsgeschwindigkeit

Volumenstrom durch Leerrohrquerschnitt volume flow rate divided by unobstructed pipe cross section ddbit volum6trique par rapport la section du tube vide

m/s

Hohlraum durch Gesamtvolumen ratio of void volume to total volume rapport entre le volume du vide et le volume total

m=/m3

604

superficial flow velocity vitesse d'ecoutement superficieile 605

relatives L~ickenvolumen void fraction (holdup) fraction de vide

606

hydrauiischer Durchmesser hydraulic diameter diametre hydraulique

607

Druckabfall pressure drop perte de charge

m/s

Tu

v--* 601

v--, 601

I

v

vierfacher freier Str~mungsquerschnitt geteitt durch benetzten Umfang fourfold of unobstructed flow cross section divided by wetted periphery quatre fois la section iibre de passage d/vise par le p6rimetre mouille statischer Mischer static mixer melangeur statique v

&p

Pe bar

i 1 Pa

- 1 N/m =

1 bar - 10sPa

i D ~ 504, v ~ 604

Ne -~ 807

2

~(~v) ~

~P" f --'g- 15"~'h"We pv, 5 608

Reibungsbeiwert friction factor coefficient/facteur de friction

609

Mischleistung mixing power puissance de mdlange/d'agitation

610

Spezifische Mischleistung specific mixing power/mixing power per unit mass/energy dissipation per unit mass puissance d'agitation specifique

f, (Co)

statischer Mischer static mixer P = ~/Z~p melangeur statique R[Jhrbeh~lter } stirred tank P = Po pn3Ds cuve agit~e R(Jhrbeh~ilter stirred tank cuve agitee ~M= -Pro

Ap .-, 607

Po --. 807, n -* 513 D -~ 506 W/kg

P "* 6 0 9

xxvi

611

Schubspannung shear stress contrainte/tension de cisaillement

If

laminar laminar laminaire 1-xz=

Pa

1 Pa- 1 N/;~"

z "* 904

dz

turbulent turbulent turbulent .._...... 1-'=~ =-

p v', v'y

1-'= =- p v'~'v, T'yz - - p VSyv', 612

Scherrate shear rate vitesse/taux de cisaillement

6i3'

mittlere Verweilzeit mean residence time temps de sdjour moyen

614 -615

616

Faxialer Dispersionskoeffizient axial dispersion coefficient coefficient de dispersion axiale , Anzahl idealer R~hrbehiilter number of ideal stirred tanks .nombre de cuves parfaitement . agitees i F-Faktor F-factor facteur F

1-

5,.-1 i

,

t~

I

s,h

V

~'D=, (.~,J

i

mZ/s

802

Bo

j , , ~ j = -2v-* 604

xxvii

7. Wiirme- und Stoffaustausch Heat and mass transfer Transfert de chaleur et de mati~re

701

W~rme0bergangskoeffizient heat transfer coefficient coefficient de transfert de chaleur

702

W~rmadurchgangskoeffizient overall heat transfer coefficient coefficient global de transfert de chaleur

703

Mittlere Temperaturdifferenz log mean temperature difference moyenne iogarithmique de la difference de tempdrature

704

Zahl der 0bertragungseinheiten number of transfer units nombre d'unit~s de transfer

~ = = A AT,,

a, (h)

W/m=K

1 kcal/hmaK = 1.163 W/m2K A -~ 206.AT m-~ 703

' Cl = k A ATm ; I

k, (U)

W/m=K

1 kcal/hm=K 1.163 W/mZK A -* 206, AT,,-* 703

r

~T~

Y~ NTUG= f

NTU

y -* 402

dy

x~

x - . 402

NTUL= f dx x= x - ~ 705

H6he/LSnge einer Ubertragungseinheit height/length of a transfer unit hauteur/Iongueur d'une unite de transfert

706 StoffObergangskoeffizient mass :ransfer coefficient coefficient de transfert de matiere

HTU HTU =

L NTU

rh = l ~ A ~ c fi = kGA (Y*- V) = kLA ( x - x * )

m/s kmol/sm 2

A--- 707 c [kg/m =) --. 401 fi --* 209 x --. 4 0 2 , y -~ 4 0 2

707

Phasengrenzfi~iche, Austauschfl&che interracial area aire interfaciale

708

Anteil der dispersen Phase (holdup) fraction of dispersed phase (holdup) fraction volumique de la phase dispersee

709

............. Blasen-, Tropfendurchmesser nach Sauter bubble, droplet diameter according to Sautar diametre des bulles, gouttes salon Sauter .....

m2

Vo r = V D+ V c ~o' =

v0

n ;Z nid P3 d3== i= 1_._._..~= 6V..~ n A Z n~de2 / i=1

~4, s

(P'er d3=

A ~ 707. n ~ 408

xxviii 8. Dimensionslose Kennzahlen Dimensionless numbers Nombres adimensionnels

801

Archimedes-Zahl Archimedes number nombre d'Archim6de

R0hrbeh~lter stirred tank cuve agit6e Ar -P

802

Bodenstein-Zahl Bodenstein number ! nombre de Bodenstein

-

Statischer Mischer static mixer mdlangeur statique

Ar

! i

D --- 5 0 6

Bo

615

v L Bo.

803

804

Euler-Zahl Euler number nombre d'Euler

'r' Froude-Zahl Froude number hombre de Froude

v --, 604

D=x

Eu - &--~P= Ne L pv= D

Eu

statischer Mischer "1 fL static mixer .~. Eu = melangeur statique R0hrbeh~lter stirred tank cuve agit6e Fr = n2--~D g

805

modifizierte/densiometdsche Froude-Zahl modified/densiometric Froude number hombre de Froude modifid/ densiom6trique

D -* 504, Ap --, 607, v ~ 604

statischer Mischer static mixer melangeur statique

n --, 513, D - - 5 0 6

Fr"

v -. 604

Ap

-~- g Dh

R0hrbehilter stirred tank cuve agitde Fr' = n=--.--~D

806

Galilei-Zahl Galilei number nombre de Galiide

R0hrbehi~iter stirred tank cuve agitde G a . g D= i. . . .

n - . 513, D -,' 5 0 6

Ga

D --, 5 0 6

xxix

807-- Newton-Zahl/Leistungszahi Newton/power number hombre de Newton/de puissance

Ne

=

~

=L

PV5

Eu =-'C"

Ne, Po

5

D -~ 504, Ap .-, 607, v - * 604

statischer Mischer static mixer mdlangeur statique fD N e = ~z Dh 2~ V

D - * 504

R/3hrbeh~lter stirred tank cuve agitee P Po ,= n---~-~D p 808

N"usselt-Zahl ' Nusselt number nombre de Nusselt

n --* 513, P -* 609 D - , 506 Nu

statischer Mischer : static mixer mdlangeur statique aD Nu,.~ X

D--* 504, = - , 701

R~hrbeh<er stirred tank cuve agitee sT Nu=--

T --, 505, a --, 701

Pe = Reo Pr - v.._.DD a

a -* 311, D -* 504, v -* 604

x

Pdclet number nombre de Peclet

!

810

Prandtl-Zahl Prandtl number nombre de Prandtl

Pr = .~-~ = ~

811

Reynolds-Zahl Reynolds number hombre de Reynolds

Leerrohr empty pipe tube vide

Pr

Re~ = v D

] a-..* 311

D --, 504, v--, 604

v

statischer Mischer static mixer melangeur statique v Dh Re = ~v

v -~ 604

RCihrbeh~lter stirred tank cure agitee Re = nD2

n --, 513, D --* 506 i

XXX

812

Schmidt-Zahl Schmidt number n o m b r e de S c h m i d t

813

Sherwood-Zahl Sherwood number n o m b r e de S h e r w o o d ............ Weber-Zahl W eber n u m b e r n o m b r e de W eber

814

D--'316

Sc=-v D

Sh - ~--~" " D ,, , statischer Mischer static mixer melangeur statique

we

(7

Sh

"'

D-318

....

We

v --* 604, ~r --- 317

ROhrbeh~lter stirred tank cuve agitde We --

p n z d~ (7 ..,

n -* 513, r --* 317

9. Verschiedenes Miscellaneous Divers

901

Differenz difference diff6rence

902

Summe sum somme

903

nat~rlicher Logarithmus natural logarithm Iogarithme natural ....... kartesische Koordinaten Cartesian coordinates coordonn6es cart6siennes

904

In ,, 2.303 Ig

X, yo Z

This Page Intentionally Left Blank

I 0th European Conference on Mixing H.E.A. van den Akker and 3'.,I. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

Trailing vortex, mean flow and turbulence modification through impeller blade design in stirred reactors M. Yianneskis Centre for Heat Transfer and Fluid Flow, Department of Mechanical Engineering, King's College London, Strand, London WC2R 2LS, U.K. The effect of the trailing vortices produced around impeller blades on the turbulence, pumping, mixing and power characteristics of stirred vessels is considered. It is suggested that the relation of impeller design and such parameters is more complex than generally thought and its assessment is complicated further by the different methodologies employed for impeller and mixing characterisation. Suggestions are made for more uniform and reliable approaches which should improve understanding of mixing processes. 1. INTRODUCTION The flowfields in stirred reactors have received considerable attention in recent years and a number of informative and interesting experimental and/or numerical investigations have been reported. However, while some researchers (for example [1,2]) have shown that different impeller designs can generate similar flow rates with lower power consumption, some have indicated that the mixing time (tM) at equal power draw, although affected by impeller size, it may not be affected by impeller type [3,4]. It was concluded in [4] that the most significant effect on mixing time is that of the power draw (P), followed by the effect of impeller diameter (D) and that the most energy efficient impellers for turbulent blending are large D, low power number (P0) impellers. An interesting assessment is given in [5]; relevant experimental and theoretical work was reviewed and it was found that an equation for mixing time and a basic turbulence model imply that all impellers of equal D/T ratio are equally energy efficient, while impeller efficiency based on the flow generated at equal measured power suggests a significant difference between impeller types. Although the studies reported to date have highlighted some interesting features of the flows produced by different impeller blade designs, a thorough understanding of the characteristics of different impeller designs is still far from complete. The aim of the present paper is to review such effects on impeller efficiency and indicate the complexity of the influence of impeller and blade design on the trailing vortices produced at the blade tips and their effect on mean flow and turbulence generation in a stirred vessel. Various commonly-employed impeller/blade shapes are considered in terms of the modification of the extent and/or intensity of the trailing vortices and the associated mean flow and turbulence levels produced and the implications of the findings for impeller, vessel and process design. Differences in the data reported and methodology employed and difficulties in drawing more general conclusions in the absence of consistent measurement approaches are outlined and remedies are suggested.

2. TIP VORTICES A N D BLADE D E S I G N

The pressure gradient in the region near the tip of a finite span wing produces a transverse flow which winds around the tip to form a tip vortex. The vortex is continuously fed by the roll-up of the vortex sheet issuing from the trailing edge of the wing. Changes in the vortex structure may be caused by modification of the wing crosssection and these can be traced to viscous effects which affect the bound circulation at mid-span, the boundary layer thickness and the tip vortex radius [6]. Tip vortices can be highly three-dimensional and unsteady and tip roughness can increase the size of the vortex core and reduce the maximum tangential velocity in the vortex [7]. Similar phenomena occur around the blades of an impeller in a stirred vessel. The vortices produced near the blade break down as they move away from the tip into a region of strong turbulence where fluid friction produces the highest energy dissipation. The forces produced in the vortex region aid the disintegration and breakage of droplets, flocs and particles, as well as homogenisation. Impellers which produce high pumping and low shear might be expected to generate weak vortices, while high-shear impellers form strong vortices and have lower pumping capacity. Blade design is critical in this respect: reducing the impeller blade angle to the horizontal in a stirred tank will result in less power absorption by the blade, in a manner analogous to propeller propulsion. Consequently, the power number increases with blade tip angle to the horizontal (a): e.g. the P0 of a 3-blade D = T/3 Maxflo T impeller increases nearly linearly with 0~ and for (x = 18~ 22 ~and 26", P0 = 0.37, 0.50 and 0.67 respectively [8]. The trailing vortices around impeller blades have been studied for a number of impeller designs such as Rushton [9-13] pitched-blade [14-17] and hydrofoil [18] turbines. In all cases, a region near the blades which is dominated by the trailing vortices has been identified, and therefore the flow field in the vessel may be considered as consisting of 2 regions. The first is a cylindrically-shaped region around the impeller, within which the flow is periodic due to the influence of the crossing of the blades and an angle- or phase-resolved treatment of the mean flow and turbulence is necessary. The second region comprises the remainder of the flow field, where turbulence levels as well as mean velocities are lower, the periodicity induced by the blade crossing has decayed substantially or entirely and 360 ~ensemble-averaged results may be used to describe the flow quantities. Inside the region dominated by the vortices, turbulence levels are an order of magnitude higher than outside. The vortices provide a major and potentially very useful source of turbulence in stirred tanks and information on the vortex location and extent could be exploited, for example to locate feed pipes, in order to reduce t M and enhance mixing and process efficiency. It has also been shown that even parameters normally assumed to have a negligible effect on the mean flow and turbulence, such as blade thickness (t), can affect power consumption as well as the mean velocity and turbulence levels near the blades [19-20] and should be taken into account, as discussed later. The number of blades (z) can also affect mixing performance in a manner which is not easily predictable due to the complex flow modifications that can occur: for example it is reported in [21] that for disk turbines with 2, 4, 6 a n d 8 blades the axes of the vortices initially extended further out into the vessel with increasing z but the vortex size did not vary in the same manner: interaction between vortices from neighbouring blades was stated as the main reason for such differences. Both the average deformation

rates and k/Vtip 2 in the vortices were highest for the 4-blade turbine, followed in decreasing order by those for the 6-, 2- and 8-blade turbines: this was observed for both constant speed and constant power input per blade conditions. Tatterson et al. [22] also observed that the trailing vortices from a six-blade PBT were weaker than those produced by a 4-blade PBT and they attributed this to the closer construction of the 6blade PBT. Clearly, trailing vortex size is only one factor to be considered, as the increase in blade surface area with z affects for example, P0 which e.g. for a D = T/3 Maxflo T impeller with (~ = 22 ~ and 3, 4 and 6 blades is 0.50, 0.56 and 0.68 respectively [8], i.e. P0 is proportional to z ~ In general, impellers that produce weak vortices might be expected to generate higher pumping, while impellers with lower P0 are likely to exhibit lower turbulence levels, both in terms of the local maxima near the impeller blades as well as in the bulk of the vessel. For example, the maximum turbulence kinetic energy (normalised by the square of the blade tip speed, k/Vtip 2) produced by different impeller designs is an important variable for mixing operations, and c a n v a r y by as much as an order of magnitude: for example, a Rushton impeller (P0= 4.9) exhibits k/Vtip 2 levels of around 0.2 near the blade and 0.02 in the bulk flow [10] a 45 ~ PBT with P0 around 1.3 shows levels of around 0.06 and 0.006 respectively [16] while a hydrofoil with P0 = 0.22 has levels of around 0.02 and 0.002 respectively: a tenfold decrease in k levels from the nearblade to the bulk region is observed for all three designs and the decrease of k with P0 is evident. It might be possible therefore to adjust the desired or allowable maximum k levels by careful blade design. However, inspection of some of the data reported in the literature reveals inconsistencies, for example comparing data for impellers of similar sizes and flow numbers, a propeller [23] and a Chemineer HE3 [2], a 30% increase in P0 results in a 300% increase in the maximum turbulence level, while by comparing a Maxflo T [2] and a propeller [24] a 77% increase in P0 does not affect the maximum turbulence level. Additional evidence on the complex manner in which flow affects k~a x and power consumption is provided by the data of [21], summarised in Figure 1. The total power input and P0 of a disk turbine increase almost linearly with z under both equal power draw and equal rotational speed conditions, while both kmax and kmax/Vtip 2 increase with an increase of z from 2 to 4 and decrease as z is increased further to 6 and 8. The maximum and average deformation rates and the average k/Vtip 2 levels exhibited similar trends as well. This was attributed to the trailing vortex development which varies in a complex manner with z [21]. The relation between the flow number, F1, and kmax is also complex: data for a Scaba 3HSP1, a PBT and a Rushton turbine [25] are summarised in Figure 2. It is observed that the local average turbulence fluctuation (LTF), average k (kay) and kav/Vtip 2 in the impeller stream increase in the order Scaba/PBT/RT as might be expected from the related increase in P0 (not shown), while F1 is highest for the PBT. It should be expected that power consumption is related to impeller design and pumping capacity. Establishing a relation between P0 and the pumping capacity of impellers - as expressed by F1 - is however difficult, primarily because of the manner in which most available data have been obtained: as ensemble-averages over 360 ~ of revolution and in different distances from the blade tip in different investigations. This

is illustrated in Figure 3, where the variation of P0 with F1 for 30 impellers from published data is shown. The data in Figure 3 exhibit considerable scatter. A more appropriate way to characterise impeller pumping capacity is the secondary or circulation flow number, F1c, which accounts for the entrainment into the impeller stream caused by blade design. F1c values are rarely reported: in Figure 3 the variation of P0 with F1c for 8 D / T = 0.33 impellers is also shown. It can be seen that there is better correlation with P0 than in the case of F1 but there is still scatter and more data are required to ascertain trends. Some of the data are at equal power draw and the increase of F1c with P0 contradicts the expectation that low P0, low shear impellers generate higher pumping. The scatter may be partly caused by the different distances from the blade where F1c has been calculated by the various investigators and mainly by the fact that more parameters such as D, kay, t M etc. must be used to correlate impeller performance. However, reported t M data vary significantly, as discussed later. It is instructive to consider therefore the r / R distance from the blade where F1c should be calculated. Figure 4 shows F1 and F1c data for a Rushton impeller [26]. F1c = 1.0 at r / R = 1.525, i.e. approximately R/2 away from the tip: this location is both sufficiently near the impeller so that its influence is felt and sufficiently far so that entrainment is well accounted for, and it is suggested that a distance of R/2 should be used in future work for F1c determination. Similar guidelines are required for axial flow impellers, but sufficiently detailed and accurate data is not available at present. 3. IMPELLER EFFICIENCY & SCALING One of the most important criteria in impeller selection is the power consumption (energy expenditure) required for a particular mixing performance. Consideration will be restricted here to blending of water-like 'thin' miscible fluids only. It is well known that in general tM varies approximately with p-0.33 [4,27]. This is well established and supported by the findings of most if not all investigators. It has also been reported that at equal D / T and power draw blend or mixing time is only a function of the power input and is independent of impeller type [4,28]. This might be considered contrary to physical insight into the mixing process as an impeller ensuring more rapid convection to all parts of the vessel should also facilitate more rapid terminal mixing by efficiently dispersing the inserted fluid throughout the vessel. Lack of uniform and clearly established procedures of measuring mixing time complicate analysis of the results obtained to date. For example, 95 or 99% mixing times are interchangeably determined (which can be considerably different), the location of the probes and probe interference with the flow may affect readings, and decolourisation methods may often yield subjective results. An non-intrusive LIF method was used recently [27] to measure 99% mixing times in a batch vessel with different impellers. The results show conclusively that the variation of mixing times with axial flow impellers (PBTs) is proportional to p-0.32 and with radial flow impellers (Rushton and 'bucket') to p-0.29. They found that indeed mixing time differences between radial impellers at equal power draw were small, as were differences between different axial flow impellers: however, mixing times with axial flow impellers were consistently shorter than those with radial flow ones by 25% to 65% at low and high power draws respectively. Other investigators have also identified

similar effects: [29] used flow/power to characterise impeller performance, [2] found that hydrofoils are superior in generating a circula~g flOW for a given power input and [1] developed a low power hydrofoil with a mixing number similar to those of impellers with substantially higher P0. Clearly, more accurate, detailed and methodical mixing time investigations, preferably utilising non-intrusive techniques, are urgently required but the above benchmark LIF data indicate that the effect of impeller design on mixing time is more complex than is often thought and that it is possible to achieve more efficient blending through appropriate impeller selection. The physical reasons for differences in impeller performance are related to the fact that with axial flow fields mixing is faster because of the better large scale fluid motion which can be achieved at constant power by varying, for example, D/T, blade size and/or blade design. As observed for the F1 definition in the previous section, standard methodology for t M measurements is urgently required. 99% mixing times should be more indicative of mixing performance but in view of measurement difficulties, it may be more instructive to employ an average t M estimated from, e.g. 95% to 99% t M measurements, in a manner similar to that employed by Mahmoudi [30], who found that calculating the arithmetic average of the mixing times for 92.5% to 98.5% of the terminal value in 0.5% intervals gave more repeatable and representative mixing time data when using a conductivity probe. Of course, LIF techniques should be preferred to conductivity probes as they can provide data without affecting the flow across many points in the vessel simultaneously. In addition, the subject of mean flow and turbulence scaling in stirred vessels has attracted considerable debate and this could stem from the difficulties in comparing data from different investigations as mentioned above. It has been recently established by many investigators [20,31-33] that the fluid mean and fluctuating velocity components in stirred vessels scale with impeller tip speed and impeller diameter and/or tank size. However this is only true if all impeller and vessel dimensions are precisely scaled, which is rarely the case in practice and may be the source of many inconsistencies in findings; for example, it has been established [19-20] that an increase of 2.5 mm in t / D results in a decrease of 33% % in P0, 15% in the mean velocities and 20% in the turbulence levels in the impeller stream. The presence of bolts etc. to attach blades to the impeller hub or disk may affect strongly the flow over the blades, trailing vortices etc. but it is often neglected and could also be a source of inconsistencies in findings. Finally, extensive and detailed mean flow and turbulence data across the vessel (as e.g. given in [17]) are urgently required which should include kmax, kmin, areax, Emin and in the absence of of such data the structure of the flows may often be over-simplified a n d / o r potentially important effects are neglected: for example, tangential flows in the opposite direction to that of impeller rotation are encountered [10,31] above or below a Rushton impeller, and lack of such knowledge may inhibit understanding of the effects of flow on mixing performance. The flows around impellers can also exhibit substantial instabilities [30] which may introduce additional complexities and must be accounted for in modelling and experimentation. Close inspection of the published literature reveals that many of the existing uncertainties stem from a lack of thorough knowledge of the flow structures and of the effects of geometrical impeller/vessel parameters on them. Similar observations to those made earlier for kmax and kay should also apply to the k dissipation rates (E) and it has been suggested that local E rates are affected by the k contained in the trailing vortices [33,34] and that, for constant mean a, turbulence is dependent on impeller geometry [33], while

to maintain constant ~-maxscale-up should be based on D 2, not D 5 or constant P/volume which is commonly used [35]. 4. C O N C L U D I N G REMARKS

The above observations indicate that the complex, three-dimensional and periodic flows in stirred vessels are strongly affected by impeller and blade design and the associated trailing vortices produced by the blades. Understanding of the flows and mixing performance is hindered by lack of universally-accepted methodology for the measurement of global parameters such as flow number and mixing time. Suggestions for their improved determination have been made and it has been shown that careful design taking into account all geometrical details of impellers and vessel internals is necessary for a reliable assessment of mixing performance. The results outlined indicate that flow effects should be taken fully into account if mixing processes are to be better understood and characterised: consequently simpler approaches to mixing analysis may only offer an approximate estimate of impeller performance. The presence of trailing vortices and associated steep velocity gradients may also be mainly responsible for discrepancies observed between measurements and CFD predictions of turbulence quantities near the impeller blades. Although in general quantitatively good predictions are obtained of the mean flow in all flow regions and the turbulence energy k in the bulk of the vessel, the consistent underprediction of k near the blades with RANS methods and turbulence models [36,37] may be partly due to the inability of such models to predict the strong gradients in and around the vortices where streamline curvature is severe. In contrast, predictions with LES methods show excellent agreement between measured and predicted k near the blades [38] and such methods are clearly better suited to resolve large-eddy structures and the trailing vortices and should be more widely utilised in future work. REFERENCES

1. N.J. Fentiman, N. St. Hill, K.C. Lee, G.R. Paul and M. Yianneskis, Trans. I. Chem. E., 76 (1998) 835. 2. Z. Jaworski, A.W. Nienow and K.N. Dyster, Can. J. Chem. Eng., 74 (1996) 3. 3. M. Cooke, J.C. Middleton and J.R. Bush, 1988,. 2nd Int. Conf. Bioreactor Fluid Dynamics, R. King (Ed.), BHR Group, Cranfield, U.K., 37. 4. S. Ruszkowski, 8th Eur. Conf. Mixing., I.Chem.E. Syrup. Series 136 (1994) 283. 5. A.W. Nienow, Chem. Eng. Sci., 52 (1997) 2557. 6. D.H. Fruman, P. Cerrutti, T. Pichon and P. Dupont, J Fluids Engng, 117 (1995), 162. 7. D.R. Stinebring, K.J. Farrell and M.L. Billet, J Fluids Engng, 113 (1991) 496. 8. K. Myers, M.F. Reeder, A. Bakker and D.S. Dickey, Recents Progres en Genie des Procedes, 11 (1997) 115. 9. K. van't Riet, W. Bruijn W. & J.M. Smith, Chem. Eng. Sci., 31 (1976) 407. 10. M. Yianneskis, Z. Popiolek & J.H. Whitelaw, J. Fluid Mech., 175 (1987) 537. 11. M. Yianneskis & J.H. Whitelaw, Trans. I.Chem.E. Part A, 71 (1993) 543. 12. K.C. Lee and M. Yianneskis, A.I.Ch.E. Journal, 44 (1998) 13. 13. J.J. Derksen, M.S. Doelman and H.E. van den Akker, 9th Int. Syrup. Appls. Laser Techniques Fluid Mechanics, Lisbon (1998) 14.5.1. 14. A.M. Ali, H.H.S. Yuan, D.S. Dickey and G.B. Tatterson, Chem. Eng. Commun., 10 (1981) 205.

i5. J.B. Fasano, A. Bakker and W.R. Penney, ChemicalEngineering, August 1994. 16. S.M. Kresta and P.E. Wood, Chem. Eng. Sci., 48 (1993) 1761. 17. M. Sch~ifer, M. Yianneskis, P. W~ichter and F. Durst, A.I.Ch.E.J., 44 (1998) 1233. 18. N.J. Fentiman, K.C. Lee, G.R. Paul and M. Yianneskis, Trans. I. Chem. E., (1999). 19. W. Bujalski, A.W. Nienow, S. Chatwin and M. Cooke, Chem. Eng. Sci., 42 (1987) 317. 20. K. Rutherford, S.M.S. Mahmoudi, K.C. Lee & M. Yianneskis, Trans. I.Chem.E., 74 (1996) 369. 21. W.-M. Lu and B.-S. Yang, Can. J. Chem. Engng. 76 (1998) 556. 22. G.B. Tatterson, H.-H.S. Yuan and R.S. Brodkey, Chem. Eng. Sci, 35 (1980) 1369. 23. P. Plion, J. Costes & J.P. Couderc, 5th Europ. Conf. Mixing, Wurzburg, (1985) 341. 24. V.V. Ranade, V.P. Mishra, V.S. Saraph, G.B. Deshpande & J.B. Joshi, Ind. Eng. Chem. Res., 31 (1992) 2370. 25. P. Tiljander, B. R6nmnmark & H. Theliander, Can. J. Chem. Engng. 75 (1997) 787. 26. M. Sch/ifer, PhD Thesis, Univ. Erlangen-Niimberg (2000). 27. M.F.W. Distelhoff, A.J. Marquis, J.M. Nouri & J.H. Whitelaw, Can. J. Chem. Engng. 75 (1997) 641. 28 J.A. Shaw, Chem. Eng. Progress, February (1994) 45. 29. R.J. Weetman and J.Y. Oldshue, 6~ Eur. Conf. Mixing, Pavia (1988) 43. 30. S.M.S. Mahmoudi, Ph.D. Thesis, London University 1994. 31. M. Petersson and A.C. Rasmuson, A I Ch E J., 44 (1998) 513. 32. C.W. Wong and C.J. Huang, 6~ Eur. Conf. Mixing, Pavia (1988) 29. 33. J.J. Ducoste, M.M. Clark and R.J. Weetman, AIChE J., 43 (1997) 328. 34. K. van den Molen & H.R.E. van Maanen, Chem.Eng. Sci., 33 (1978) 1161. 35. G. Zhou and S.M. Kresta, A.I.Ch.E.J., 42 (1996) 2476. 36. K. Wechsler, M. Breuer and F. Durst, ASME J Fluids Engng., 121 (1999) 318. 37. K. Ng, N.J. Fentiman, K.C. Lee and M. Yianneskis, Trans. I.Chem.E., 76 (1998) 737. 38. J. Derksen and H.E.A. van den Akker, A.I.Ch.E J., 45 (1999) 209. 8

0.25 9 9

,.;:,-,

o b~

9 9

9

[]

"-<>.,

9

,q, 9

,.. -0.2 %

t

6-

M

4-0.15

i

I

O

2-

o 0

151

I

I

I

2

4

6

8

P0 constant

N

...... 9 9 .......

Total P const N

.... O ....

P0

- - - - o ....

Total P const P / b l a d e

---O--.

kma x constant N

....

kma x constant P / b l a d e

c o n s t a n t P/blade

,.~

...

0.1 10

N u m b e r of blades

Figure I Effect of the number of blades of a disk turbine on P, P0 and turbulence levels in the impeller stream (data from [21].

p===~ r r t_._a

1.25

----G---- LTF/Vti p

1-

! t.....a r

r162162162~'r oN

~'~" ,,.,,~,,,,~,,~

0.75-

........O........ kav/vtip2

Zo

0.5-

om

0.25-

....

I

I

I

HFI

PBT

RT

.... 9....

kav

.... /~....

FI

! t._._a

Impeller type

Figure 2 F1, LTF, kay and kav/Vtip 2 for a Scaba 3HSP1 (HFI), a pitched-blade and a Rushton turbine. Data from [251.

2.5

m_

6 5

_

4-

ra

O

_

3-

1.5-

o

2-

1-

10 0

I

I

I

I

0.5

1

1.5

2

2.5

I

I

I

1

1.5

2

I

2.5

r/R

Fl, Flc [-] Figure 3 Po versus F1 (0) and Flc (o) for impellers.

0.5t 0.5

30

Figure 4 Variation of F1 with r / R in the i m p e l l e r s t r e a m of a Rushton turbine. Data from [26].

10th European Conference on Mixing H.E.A. van den Akker and 3,.,]. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

Turbulence generation by different types of impellers M. Sch~ifer, J. Yu, B. Genenger and F. Durst Institute of Fluid Mechanics, Friedrich-Alexander-University Erlangen-Ntirnberg, Cauerstrasse 4, D-91058 Erlangen, Germany http;//www.lstm.un.i-erlangen.de

The turbulence generation by three different types of impellers, (1) a Rushton turbine, (2) a 4/45 ~ pitched blade impeller and (3) a hydrofoil impeller, was investigated by using a refractive index matched, automated LDA-technique. The peak values of turbulence kinetic energy are associated with the presence of trailing vortices. It was found that the extent and intensity of trailing vortices is determined by the type of impeller. The intensity of the trailing vortices can be determined by the vorticity, which is very intense within and low outside the vortex regions. The trailing vortices contain the highest turbulence kinetic energy values, which are k/V,~ = 0.158 for the Rushton turbine, k/V,~ = 0.078 for the 4/45 ~ pitched blade impeller and k/V~ = 0.028 for the hydrofoil impeller. In addition it was found that the distribution of turbulence kinetic energy in the discharge flow depends significantly on the type of impeller. While high values of turbulence kinetic energy are present along the entire discharge profile of the Rushton turbine, the discharge flow of the hydrofoil impeller exhibits only two peaks at the inner and outer edges of the blade, where vortices are generated. The data obtained contributes to a better understanding of the mixing performance of different types of impellers. In addition, the validation of numerical simulations of the flows in stirred tank reactors with CFD requires more detailed information on the flow field generated by different types of impellers than is available to date so that the data gained in the present study can make a meaningful contribution to the ongoing developments in CFD. 1.

INTRODUCTION

Stirring and mixing processes depend to a large extent on the magnitude and local distribution of turbulence that is generated by the stirrer element. The trailing vortex system generated near impeller blades has, in particular, been identified as the major flow mechanism responsible for mixing and dispersion in stirred vessels, and high turbulence levels in the vortices have an important impact on such phenomena as drop break-up and cell damage in stirred (bio-) reactors. If we consider liquid blending, the high turbulence kinetic energy contained in the trailing vortex system can contribute considerably to mixing down to molecular scale, which is important when very fast and complex reactions are involved [1]. However, the generation of trailing vortices is also associated with the energy losses of an impeller and, in processes in which only macromixing is important, this can result in longer macromixing times for the same power input. To illustrate this, hydrofoil impellers, optimised in terms of energy losses, have been found to be much more efficient than Rushton turbines or

10 pitched blade impellers [2]. It is therefore important to clarify the details of trailing vortices and the way turbulence is generated and distributed within the stirred reactor by different types of impellers. In the present study the trailing vortex systems and the turbulence generated by (1) a Rushton turbine (RT), (2) a downpumping pitched blade impeller (PBT) and (3) a hydrofoil impeller (HI) developed at the Institute of Fluid Mechanics in Erlangen were investigated in detail. 2.

EXPERIMENTAL SET-UP

A fully automated test rig for detailed LDV-measurements in stirred tank reactors has been developed at the Institute of Fluid Mechanics in Erlangen within the framework of several research projects. The set-up included three main parts, (1) the measuring section, (2) the traversing equipment for automation and (3) the LDV measurement system consisting of a diode fiber laser Doppler anemometer operating in backscatter mode, a traversable probe and a frequency counter. Two different tank scales were employed for the measuring sections. The Rushton turbine and the pitched blade impeller were mounted in a cylindrical stirred vessel with a diameter of T = 152 mm. The liquid height was equal to the vessel diameter (H = T). Four equally spaced baffles of width B = T/10 and thickness of 3 mm were mounted along the inner wall of the cylinder at a distance of 2.6 mm. The vessel could be rotated about its axis which facilitated the adjustment of the vessel for measurements in different vertical planes. The hydrofoil impeller was investigated in a larger scale vessel, T = 400 mm. The baffle arrangement was geometrically matched to the T = 152 mm vessel. The only difference in the vessels was that the top of the smaller vessel was closed with a lid to avoid air entrainment into the liquid from the free surface. Nouri and Whitelaw [3] showed that the lid has only an influence on the flow field in the immediate vicinity of the lid/free surface. Figure 1 shows the geometry of the mixing vessel and the coordinate system used. The details of the impellers used are given in Figure 2. All three impellers were mounted at a clearance of h = T/3. The blade thickness was 1.75 mm for the Rushton turbine (also disk thickness), 0.9 mm for the pitched blade impeller and 1 mm for the hydrofoil impeller. As pointed out further above the flow field within the impeller region (i.e. between the blades) is an important flow region because trailing vortices are generated there. Through exact matching of the refractive index of the stirred medium to the refractive index of the measuring section, it was possible to gain optical access to the inner part of the impeller without any distortion of the laser beams. For this purpose, the vessel walls, the baffles and the impeller blades (for Rushton turbine and pitched blade impeller ) were constructed from transparent Duran glass with the same refractive index as the working fluid (n = 1.468), a mixture of silicone oils. For the hydrofoil impeller complete refractive index matching could not be achieved since the blades were not constructed from Duran glass. A high-resolution measuring grid was realized in the entire flow field by automation of the data acquisition, whereby the optical probe was mounted on a 3-D traversing unit controlled by a PC via a CNC-controller. If the LDV-measurements are processed as 360 ~ ensemble-averaged measurements, the fluctuating quantities contain both periodic and turbulence contributions, which can lead to significant overestimation of turbulence quantities in the impeller stream [4]. Such measurements are not suited for validation of numerical simulations. Therefore, angleresolved LDV measurements are required in which the flow information is assigned to the

11

Fig. 1. Vessel configuration

Fig. 2. Geometry of Impellers

corresponding angle of the impeller. For this purpose an optical shaft encoder was used providing 1,000 pulses and a marker pulse per revolution. The marker pulse corresponded to the angle ~ = 0 ~ of an impeller blade and was set at the middle of a blade at the radial tip. The results shown in the next section were conducted at a stirrer speed of N = 2,672 r.p.m. (Wti p --- nDN = 7 m/s) for the Rushton turbine and pitched blade impeller and N = 550 r.p.m Wti p -- 4.32 m/s) for the hydrofoil impeller corresponding to a Reynoldsnumber of Re = ND2/v = 7,300 and Re = 13,523, respectively. 3.

RESULTS

3.1. Impeller Flow Fields In order to capture all the details of the flow in the region of the trailing vortices, a high resolution of the measuring grid was chosen for the impeller flow measurements. The radial and axial step widths were between 1 and 2 mm in all three test cases. The azimuthal resolution was set to 1~ This resulted in altogether 173,520, 218,880 and 414,720 individual measuring points for the Rushton turbine, pitched blade impeller and hydrofoil impeller, respectively. The results will be presented in form of an animation in which planes at each degree between two blades (l~ _ 60 ~ for the Rushton turbine, 1~ - 90 ~ for the pitched blade impeller and 1~ - 180 ~ for the hydrofoil impeller) are shown step-by-step, leading to a rotating impeller movie. With the help of this animation, the generation, development and break-down of the trailing vortex system can be characterised exactly. By way of example Figures 3a - 3c show the flow fields for selected planes located at ~ = 10~ behind a blade for all test cases. The blades must be considered as moving out of the page towards the reader. The radial and axial coordinates were normalised with the tank diameter T in order to facilitate a comparison between the results gained for different vessel sizes. The mean velocity components were

12

Fig. 3a. Impeller flow field at t) = 20 ~ for RT

Fig. 3b.Impeller flow field at 0 = 30~ for PBT Fig. 3c. Impeller flow field at r = 60 ~ for HI normalised with the blade tip velocity Vttpand are denoted U/V,p, V/Vapand W/V,p for the radial, tangential and axial components, respectively. The Rushton turbine generates a pair of trailing vortices behind each blade which leave the impeller region with the radial jet and dissipate within a distance of one impeller radius from the outer tip of the blade. The diameter of each trailing vortex is approximately equal to half of the blade height. The pitched blade turbine generates one trailing vortex behind each blade. The diameter of the vortex is in maximum about the projected height of the blades. The trailing vortex leaves the impeller region in axial direction and dissipates within two blade heights from the lower tip of the blades. The flow field of the hydrofoil impeller reveals a clear tip vortex at the outer edge of the impeller blade which extents over one blade to blade distance (180~ The vortex moves slightly inwards in radial direction and dissipates within an axial distance of 0.05T from the lower tip of the blade. The maximum diameter is about 0.03T or 1.5 times the projected height of the blade. In addition, a smaller tip vortex was obtained at the inner edge of the blade.

3.2. Vorticity Although the depicted flow fields clearly show the trailing vortices and the vortex centres can be determined accurately due to the high density of the measuring grid, it is more difficult to locate exactly the vortex edges, which are needed to give a full description of the trailing vortices. As vortices are characterised the intensity of vortex motion was found to be a

13

valuable parameter for determining the extension of the trailing vortices in a more quantitative manner. For this purpose the dimensionless vorticity ( ~ in ~planes was calculated as an estimation of the vortex intensity using the following equation:

ff = O(z/T-------~-

O(r/T'------~

(1).

The vorticity is very intensive within and very low outside trailing vortices. A limiting value of vorticity can be found that indicates the edges of the trailing vortices. For the Rushton turbine a value of ~ = 13, for the pitched blade turbine ~ = 7 and for the hydrofoil impeller ~ = 10 was determined. It was then possible to visualise the trailing vortices with a contour plot in which contours lower than the limiting value of the vorticity (cut-off value) were erased. This is shown in Figures 4a - c. It must be noted that the vortex of the hydrofoil impeller could not be visualised completely since measurements within the impeller swept region were not possible. From Figure 4 the total extent of the trailing vortices generated behind each stirrer blade of each impeller type can be determined. Some details of the trailing vortices are summarised in table 1, which contains the total length, the diameter, the radial (from outer tip), axial (from lower tip) and tangential (from ~ = 0 ~ extents of the vortices. To characterise the path of the vortex the change of the radial and axial positions of the vortex centres with the blade angle is given, too.

Fig. 4b. Trailing vortex PBT

Fig. 3c. Trailing vortex HI

14 It is important to point out that Figure 4 gives not only a qualitative view of the trailing vortices. It is also possible to provide quantitative data at each point within the trailing vortices, such as mean and fluctuating velocities and turbulence kinetic energy (see next section). This data will also be valuable for a more detailed comparison with computations that are also capable to simulate the trailing vortex system, as recently shown by Wechsler et. al [5]. Table 1 Characteristics of trailing vortices

Length Diameter Radial Tang. extent Extent

Axial extent

~)(z/T) ~)q~

O(r/T)

RT

0.375T

0.039T0.059T

0.099T

100 ~

Slight upward movement

Slight upward movement

PBT

0.434T

0.052T0.066T

-

160 ~

0.125T

Const.: (~176176

const.: Lower vortex is (o.oolar] shifted slightly in ~ J radial direction compared to upper vortex 0

HI

0.552T

0.030T0.035T

Slight inward movement

180 ~

0.045T

Const.: Slight (0-0003r/ inward movement

Imp.

Remarks

r

3.3. Turbulence Distribution The turbulence produced by impellers is essential to achieve mixing down to molecular scale. However, it contributes to the energy losses of impellers and more energy has to be introduced to the reactor. Therefore, it is important to gain more knowledge on how impellers distribute turbulence within the reactor. The trailing vortex system contains the highest values of turbulence kinetic energy [6, 7] and it is interesting to compare the turbulence values generated by the different types of impellers. In Figures 5 and 6 the distribution of turbulence kinetic energy is shown for profiles which cut through the trailing vortices. The dimensionless variable x/T indicates the distance from the middle of the impeller blade. As the RT is a radial flow type impeller a vertical profile is shown in Figure 5 and x/T is the axial distance from the middle of the blade at the outer tip, which is i.e. the impeller clearance h. Since PBT and the HI have an axial outflow a horizontal profile was chosen in this case and x/T describes the radial distance from the middle of the blade length at the lower tip. The turbulence kinetic energy k was calculated from the fluctuating velocity components u ', v' and w' according to 2 and normalised with the square of the stirrer tip velocity vt~. Figure 5 refers to a position within the trailing vortex, at which the highest turbulence kinetic energy was obtained for each impeller. The discharge flow of the RT contains the highest values of turbulence kinetic energy with values up to k / V,~ =0.158. The two peaks in

15 the distribution result from the pair of trailing vortices, whereby the lower vortex has a significant lower turbulence intensity than the upper one. The distributions for the PBT and HI reveal a peak at the outer edge of the impeller where trailing vortices are generated. The maximum values obtained were k/V~r =0.078 for the PBT and k/V,~ =0.028 for the HI. For the HI a clear second peak of less intensity is evident at the inner edge of the blade. The two peaks are associated with the presence of two vortices, which are comparable to the typical tip-wing vortices of airfoils. These tip-wing vortices of the HI do not produce that much turbulence as the vortex systems generated by PBTs and RTs. In addition, it is interesting to note that the discharge flow of the RT has high values of turbulence kinetic energy over the entire discharge profile, whereas the HI has very low turbulence values between the two peaks. The turbulence distribution of the PBT shows one peak which is associated with the presence of the trailing vortex, but in contrast to the distribution of the HI the turbulence kinetic energy in the rest of the discharge flow does not fall down to values which are comparable to those outside the impeller region. In general the turbulence kinetic energy is a factor of 3 higher for the PBT compared to the HI. Figure 6 shows the same profiles as in Figure 5, but at a position which refers to one half of the total length of the trailing vortices. The profiles indicate the decay of turbulence along the path of the trailing vortices. The profile of the Rushton turbine shows clearly that the vortices are moving upwards since the peak values are shifted slightly to the right in Figure 6. It is interesting to note that the peak value of the profile for the PBT has decreased, whereas the profile between the vortex and the shaft is at the same level as at the position shown in Figure 5.

Fig. 5. Turbulence distribution I 4.

Fig. 6. Turbulence distribution II

CONCLUDING REMARKS

The flow fields, the trailing vortex systems and the turbulence characteristics of three different types of impellers were investigated in detail by means of high resolution LDAmeasurements within and in the vicinity of the impellers. Hydrofoil impellers produce a vortex system which is similar to that of tip wing vortices obtained for airfoils. These vortices

16 contain considerably less turbulence kinetic energy than those generated by Rushton turbines and pitched blade impellers. This is in good agreement with the observation that hydrofoil impellers have in general lower power-numbers than other types of impellers [2]. However, the vortices produce higher turbulence than obtained in other regions of the reactor. These high values of turbulence kinetic energy could be harnessed to aid mixing down to molecular scale, if micromixing phenomena have an impact on the process result, but the locations of the insertion points of reactants, such as feed pipes, must be selected with care to ensure utilisation of these high levels of turbulence. The detailed information on the path of the trailing vortices given in this paper is therefore valuable for locating such insertion points. The turbulence distribution has an impact on the mixing performance of impellers. In a future communication the results will be further evaluated and combined with integral measurements such as power-number and mixing time to lead to a better understanding of how the turbulence distribution around the impeller affects the mixing efficiency in stirred tanks. ACKNOWLEDGEMENTS The authors acknowledge financial support provided by the Commission of the European Union under the BRITE EURAM Programme, Contract number BRPR-CT96-0185 (Further partners in this research project are: NESTE OY, FIN, EniChem, I, PFD, IRE, BHR-Group, UK, AEA-Technology, UK and INVENT-UV, D) and the Deutsche Forschungsgesellschaft DFG (Gz: Du 101/45). REFERENCES 1. Villermaux, J., 1986, Micromixing Phenomena in Stirred Reactors, Encyclopedia of Fluid Mechanics, Ch. 27, pp. 707 - 771, Gulf Publishing Company. 2. Fentiman, N. J., Hill St. N., Lee, K. C., Paul, G.R. and Yianneskis, M., 1998, A Novel Profiled Blade Impeller for Homogenization of Miscible Liquids in Stirred Vessels, Trans IChemE, Vol. 76, Part A, pp. 835 - 842. 3. Nouri, J. M. and Whitelaw, J. H. 1990, Effect of Size and Confinement on the Flow Characteristics in Stirred Reactors, Proc. Fifth Int. Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, pp. 23.2.1 - 23.2.8. 4. Yianneskis, M. and Whitelaw, J. H. 1993, On the Structure of the Trailing Vortices around Rushton Turbine Blades, Trans. I.Chem.E., vol. 17, Part A, pp. 543 - 550. 5. Wechsler, K., Breuer, M. and Durst, F., 1999, Steady and Unsteady Computations of Turbulent Flows Induced by a 4/45 ~ Pitched-Blade Impeller, J. of Fluids Eng., Trans. ASME, Vol. 121, pp. 318-329. 6. Sch~ifer, M., H6fken, M. and Durst, F. 1997, Detailed LDV Measurements for Visualization of the Flow Field within a Stirred-Tank Reactor Equipped with a Rushton Turbine, Trans. I.Chem.E., vol. 75, Part A, pp. 729 - 736. 7. Sch~ifer, M., Yianneskis M., W~ichter, P. and Durst, F., 1998, Trailing Vortices around a 45 ~ Pitched-Blade Impeller, AIChE-Journal, Vol. 44, No. 6, pp. 1233-1246.

10th European Conference on Mixing H.E.A. van den Akker and s Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

17

Limits of Fully T u r b u l e n t Flow in a Stirred T a n k Kevin J. Bittorf and Suzarme M. Kresta Department of Chemical and Materials Engineering, University of Alberta Edmonton, Alberta Canada, T6G 2G6 In a stirred tank, the flow has been considered fully turbulent for all Reynolds numbers greater than 2• 104. In fully turbulent flow, the power number is constant, and increases in the Reynolds number do not affect the shape of the dimensionless velocity profiles. Both of these conditions are met for Rei>2• 104 close to the impeller, where the velocity profiles scale with D/2 and VTIp. This paper shows a new method for scaling velocities in a stirred tank in which the velocity profiles in the bulk of the tank are scaled with the characteristic velocity and length scale in the wall jet that is formed along the baffle of the tank. By means of this scaling argument it was determined that fully turbulent flow in the top third of the tank does not exist for ReI=2X 104. This is important for design of vessels where H>T, since the lack of fully turbulent flow means that the velocity profiles will be affected both by the characteristic velocity scale and the fluid viscosity. It also provides some insights required for the characterization of turbulence and the application of computational fluid dynamics (CFD) to conditions in the bulk of the tank. 1. INTRODUCTION Turbulence in a stirred tank was defined by Rushton and co-authors (1946 & 1950) in terms of the power consumed by the impeller. Rushton's analysis was based solely on the impeller region. When D<
Inertial Forces Viscous Forces

__ fla.U c x. A

_ -

Lc . U c v

(1)

Fully developed turbulence occurs when the inertial forces in the system are so large that the viscous forces become negligible and the accurate definition of this point is critically dependent on appropriate choice of the characteristic length (Lc) and velocity (Uc) scales.

18 The Reynolds number length and velocity scales are well defined for classical flows, like pipe flow and jet flow. For the impeller region in a stirred tank, the characteristic velocity and length scalesare the impeller diameter (D) and impeller tip speed (ND): N.D 2 Re I = ~ (2) V

The flow is considered fully turbulent in the impeller region of a stirred tank when the power number, [Po=P/(pN3DS)] becomes constant with increasing Rei (Rushton, et al. 1950). This is analogous to pipe flow, in which fully developed turbulent flow occurs when the friction factor becomes constant with increasing Re. For an impeller it is generally accepted that fully developed turbulence exists for all Rei > 2x 104. In some instances, the fully turbulent condition persists to lower Rei, (Rushton et al., 1950). The power number and friction factor are used to define the onset of fully turbulent flow for their respective systems; however, the onset of fully turbulent flow can be more accurately determined using dimensionless velocity profiles. In fully turbulent flow, the dimensionless velocity profiles will collapse to a single similarity profile if the proper characteristic velocity and length scales are used. For pipe flow, the characteristic length is the pipe radius and the characteristic velocity is the maximum velocity at the center of the pipe. In the impeller discharge stream the characteristic velocity is the impeller tip speed (riND), and the characteristic length is the impeller radius (D/2). This dimensionless velocity profile is self-similar if the flow is fully turbulent for all common impellers (Ranade and Joshi, 1989 and Nouri et al. 1987) independent of the fluid in the tank. In the transitional regime, both inertial and viscous forces influence the velocity profiles and similarity no longer holds. For the purpose of this study, the onset of transitional flow is defined as the point when the dimensionless velocity profile deviates from the turbulent profile. The Rex required to sustain fully turbulent flow in the outer regions of the tank is shown to be significantly higher than that for the impeller region. 1.1. Scaling in the Bulk of the Tank The mean flow in the bulk of the tank was examined to determine the dominant flows and from this possible characteristic velocity and length scales were extracted. Bittorf (2000) showed that scaling fluid velocity with impeller characteristics (impeller tip speed and radius) does not work in the bulk of the tank; however, scaling can be accomplished using wall jet principles because a three-dimensional wall jet is formed along each of the baffles (Bittorf and Kresta, 1999). It is the only mean flow structure along the wall of the tank and it has been shown to dominate both the active zone of mean circulation and the cloud height for solids suspension (Bittorf, 2000). Turbulent scaling for wall jets has been widely examined (Glauert 1956, Newman et al. 1972, Padmanabham & Gowda 1991), and the characteristic velocity and length scales

19 are the local maximum velocity (Urn) and the half width of the jet (bl/2). The half width of the jet (bl/2) is the point at which U/Um=0.5 on the dimensionless velocity profile (u/urn). The distance from the wall is made dimensionless with the half width (rl=y/bl/2), and the similarity profile for the wall jet is reported as U/Umverses rl. The dimensionless velocity profile retains similarity at any streamwise (z/T) position as long as flow remains fully turbulent. This scaling approach was used to examine the limits of fully developed turbulence in the bulk of the tank. 2. EXPERIMENTAL A fully baffled cylindrical tank with a diameter of T=240mm agitated by a pitched blade turbine with 4 blades (PBT) is examined in detail in this paper. The fluids examined were water, Bayol, and solutions of triethylene glycol (TEG) in water, with viscosities ranging from 1cP to 16 cP. The full range of experimental conditions are shown in Table 1 but due to the limited space in this paper only the PBT D=80mm will be presented in detail (see Bittorf 2000 for other results). Using LDV, velocity profiles were measured throughout the tank (see Bittorf and Kresta, 1999). Each traverse passes through the three-dimensional wall jet as shown in Figure 1 in both plan and profile views. Table 1" Experimental Variables Impellers PB "P Lightnin A310 Chemineer HE3 Clearances z traverses Liquids

= T/3 * and D = 120mm =T/2 D = 80mm = T/3 and D = 120mm =T/2 D = 60mm = T/4 C1= "1"/3 & C2= D/2 z - - 110, 140, 170, 200,220mm or z/T=0.46, 0.58, 0.71, 0.83, 0.92

D = 80mm

Water Bayol TEG and Water (Mix 1)

(Mix 2)

v=1.0 x l 0 "6 m2/s v=3.0 x 10"6 m2/s v=6.2 x l 0 "6 m2/s v=14.5 x l 0 "6 m2/s

Figure 1" Experimental Set-Up and Traverse Locations

20 3. RESULTS The wall jet was used to determine the location where the flow changes from fully turbulent to transitional flow for two reasons. First, the velocities in the top of the tank are much smaller than those at the impeller. In a jet, the velocities are made dimensionless with the local maximum velocity; this scales the velocity so the same scale can be used throughout the jet decay regardless of axial location. By using a local velocity scale, a more accurate comparison can be made between the velocity profiles with higher and lower Um. Second, the jet model forces the profiles to collapse at U/Um=l and at rl=l. If a profile deviates from similarity in spite of a forced fit at 11=1, it is clear that similarity is violated and the flow is no longer in the fully turbulent regime.

3.1. Criteria for Assessing Fully Developed Turbulence Figure 2 shows the dimensionless similarity profile for the three-dimensional wail jet in a stirred tank. In this figure, the x-axis is the dimensionless distance from the wall (rl=y/bl/2) and the y-axis is the axial velocity made dimensionless with the local maximum velocity in the wall jet (u/urn). The classical exponential profile (Glauert 1956) could not be used here because the velocity in this wall jet passes through zero, whereas the classical profile, developed for a stagnant surrounding fluid, asymptotically approaches zero. A cubic regression is required to account for both the inflection point and the maximum at U/Um=l. Over 200 data points from various experimental configurations were used in order to ensure an accurate similarity profile. The resulting cubic regression has an R2=0.99: u/urn=0.925 + 0.77911 - 1.778rl 2 + 0.574rl 3

(3)

Figure 2: Similarityprofile for the three dimensionalwalljet. This profile is used to test local velocityprofiles for scalability and thus for fully turbulent flow. Note that all data is force to fit at u/um=l.0 and u/u,,=0.5 This regression is valid from the maximum velocity to the point at which the dimensionless velocity passes through zero. Beyond this point, the jet is no longer examined for similarity because it is affected by the impeller intake and discharge streams. The boundary layer region from 11=0 to 11=0.28 is also neglected in this analysis. It is

21 expected to exhibit similarity but limited data was taken in this region and a regression could not be accurately completed. The regression residuals were used to determine the 99% confidence interval of the similarity profile. Any measurement with a deviation from the regression curve larger than +0.010 indicates one of two things: a) the point is an outlier or b) the velocity profile violates similarity and the flow is no longer turbulent. If multiple points deviated from the 99% confidence interval, the traverse was classified as transitional. 3.2. Limits of Fully Developed Turbulence Beginning close to the impeller at z/T=0.46, all 7 traverses in Figure 3a obey

similarity from Rei=2xl04 to 1.7x105 and up to rl=l.5. Beyond rl=l.5, the influence of the impeller becomes apparent, and the appropriate scaling changes to the impeller tip speed and radius. Continuing upwards in the tank, similarity is obeyed for all Rei up to z/T=0.58, as shown in Figure 3b. Figure 4a shows the first failure of similarity at z/T=0.71. In this case, 2 traverses must be discarded, and similarity is obeyed for Rei>3xl 04, as shown in Figure 4b. The same condition exists for z/T=0.83, as shown in Figures 5a and b. The last traverses examined are at z/T=0.92, where the limiting ReI is 6.6xl 04. Note that the first occurrence of transitional flow for an impeller Reynolds number in the fully turbulent range occurs between 0.58
Table 2: Summary of Last Known Turbulent Height (z/T) at Various to Ret for a D=T/3 PBT at Two Clearances Fluid

Water

Bayol

2.0x10 4

2.0 x l 0 4

C/D=0.5

0.58

C/D=1.0

0.58

Rel

TEG & Water

Bayol

Bayol

Water

Water

3.0 x l 0 4

6.4 x l 0 4

6.6 xl04

1.0 x l 0 s

1.7 xl0 s

0.71

0.83

0.92

0.92

0.92

0.92

0.58

0.83

0.83

0.92

0.92

0.92

22 4. CONCLUSIONS The objective of this work was to carefully examine the limits of fully developed turbulence in the bulk of a stirred tank. Similarity profiles of axial velocity were examined in the wall jet at the baffle, and itwas shown that for a D=T/3 PBT at C/D=I.0 and 0.5, the upper third of the tank drops into the transitional flow regime at Rex=2xl 04. This result agrees well with a previous study which examined the active volume of mean circulation and determined that the limit of this volume occurred close to a z/T=0.667. This result has implications for many industrial applications with surface feed, or with dip pipes in the top third of the tank. It shows that similarity rules will not necessarily be obeyed in the top of the tank, even if ReI>2Xl 04. The appearance of transitional flow should also be an important consideration for those researchers wishing to model the flow and turbulence in the bulk of the tank on a small scale. ACKNOWLEDGEMENTS Support of the Natural Sciences and Engineering Research Council of Canada and Lightnin.

Figure 3a: Fully turbulent profiles at z/T=0.46. All profiles collapse onto the jet profile

Figure 3b: Fully turbulent profiles at z/T=0.58. All profiles collapse onto the jet profile

Legend for Figures 3-6

23

Figure 4a: Velocity profiles at z/T=0.71

Figure 5a: Velocity Profiles at zff=0.83

Figure 6a: Velocity Profiles at a/T=0.92

Figure 4b: Fully turbulent velocity profiles at z/T=0.71 with profiles 6 & 7 eliminated

Figure 5b: Fully turbulent velocity profiles at z/T=0.83 with profiles 6 & 7 eliminated

Figure 6b: Fully turbulent velocity profiles at z/T=0.92 with profiles 4, 5 6 & 7 eliminated

24 NOMENCLATURE A area (m2) bl/2 distance to um/2 (m) C clearance (m) D impeller diameter (m) H liquid depth Lc characteristic length (m) rh mass flow rate (kg s"l) N impeller speed (s -~) P power (W) Po power number Po=P/N3D 59 Re1 Reynolds number Re=ND2/v T tank diameter (m)

y z

characteristic velocity (m sl) axial velocity component (m S"l) local maximum velocit7 (m s"a) impeller tip speed(m s") distance from tank wall (m) axial coordinate (m)

Greek rI v 9 x

dimensionless distance (rl=y/bl/2) kinematic viscosity (m2 s1) density (kg m "3) shear stress (Pa)

Uc u Um

VTip

REFERENCES Bittorf K.J. and S.M.Kresta, 2000, Active Volume of Mean Circulation for Stirred Tanks Agitated with Axial Impellers, Chem. Eng. Sci., 55, 1325-1335. BittorfK.J., and S.M. Kresta, 1999, Wall Jets in Stirred Tanks, accepted AIChE J. Bittorf K.J., 2000, Wall jets in stirred tank with applications to turbulent limits and solids distribution, University of Alberta, Ph.D. Thesis, Edmonton, Canada. Glauert M.B. 1956 The wall jet, J. Fluid Mech. 1,625-643. Newman B.G., R.P. Patel, S.B. Savage and H.K. Tjio, 1972, Three-dimensional wall jet originating from a circular orifice, Aero. Quarterly 23, 188-200. Nouri J.M., J.H. Whitelaw, and M. Yianneskis, 1987, The scaling of the flow field with impeller size and rotational speed in a stirred reactor. Second International Conference on Laser Anemometry - Advances and Applications, Strathclyde, UK, (Sept. 1987) Padmanabham G. and B.H. Gowda, 1991, Mean and turbulence characteristics as a class of three-dimensional wall jets - Part 1: Mean flow characteristics, J. of Fluids Engineering 113, 620-628. Per P., L. Magnus, and J. Lennart, 1996, Measurements of the velocity field downstream of an impeller, J. of Fluids Engineering 118, 602-610. Ranade V.V. and J.B. Joshi, 1989, Flow generated by pitched blade turbines I: measurements using Doppler anemometer, Chem. Eng. Comm., 81,197-224. Rushton J.H., 1946, Technology of Mixing Can. Chem. Process. lnd., 30, 55-61. Rushton J.H., E.W. Costich, and H.J. Everett, 1950, Power characteristics of mixing impellers, Chem. Eng. Progress, 46, 395-467

10t~European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

25

SPATIALLY RESOLVED MEASUREMENTS AND CALCULATIONS OF M I C R O - A N D M A C R O M I X I N G I N S T I R R E D V E S S E L S Mathias Buchmann 1, Kerstin Kling 2, Dieter Mewes 3 Institut fttr Verfahrenstechnik, Universit/at Hannover, Callinstrasse 36, 30167 Hannover, Germany, e-mail: [email protected], [email protected], [email protected] The local intensities of segregation are measured at a multitude of points inside a stirred vessel by using the tomographical dual wavelength photometry. A mixture of an inert and a reacting dye are injected into the vessel. The inert dye serves as a tracer for the macromixing, whereas the vanishing of the reacting dye shows the micromixing. The concentration fields of the dyes are measured simultaneously by transilluminating the vessel from three directions with superimposed laser beams of different wavelengths. The light absorption by the dyes is measured with CCD-cameras and these projections are used for the tomographic reconstruction of the concentration fields. Low Reynolds' number measurements with a Rushton turbine show better macro- and micromixing for a dye injection closer to the stirrer shaft compared to a position closer to the main vortex. During the theoretical examinations the flow field and the convective transport through the vessel are calculated with commercial CFD software. The CFD software does not resolve structures smaller than the grid size of the Finite Volume Grid, so that the molecular mixing can not be calculated. Therefore a new model is derived that describes the deformation of small fluid elements and the molecular exchange between these elements and the surrounding liquid. This model is implemented through two transport equations into the CFD software and yields the local mixing quality at a multitude of points through the vessel. The calculated results are in agreement with the experimental measurements. 1. INTRODUCTION The various aspects of micro- and macromixing have been addressed by a wealth of scientists. The first suggestion for a differentiation between mixing on the macro- and the microscale came from Danckwerts [4]. He introduced the scale of segregation as a measure of the size of the regions of non-homogeneity. The quality of the non-homogeneity between the regions can be described by the intensity of segregation Is(t) -

cr 2 ( t )

,

(1)

O''0

which is defined as the mean square deviation of the composition from the mean for a given time t compared to the one for t = 0.

26 The direct local determination of these two values is difficult, because this would require measurements on the molecular scale. Hence, a variety of techniques measuring characteristic features of the micromixing indirectly have been developed. K~ippel [6], Hiby [5] and Mann et al. [7] developed color change measurement techniques to determine the global intensity of segregation. Baldyga and Bourne [1] and Villermaux et al. [12] use the yield of a mixing sensitive reaction to determine the micromixing time. In the case of slow laminar mixing the reaction cloud travels through the vessel, so that the spatial resolution of this technique is rough. Ottino [8], [9] suggested a lamella model to describe the laminar mixing and developed mathematical methods to describe the stretching and folding of this lamellas by the flow field. His methodology gives insight into the fundamentals of the mixing process itself, but it can not be used yet to describe the complex process inside stirred vessels. With the tomographical dual wavelength photometry, it is possible to measure the local intensity of segregation at a multitude of points simultaneously. With this aim in view a mixture of an inert and a reacting agent is injected into the vessel. The local concentration ratios of the two dyes yield the local intensities of segregation. During the theoretical examinations the flow field and the convective transport through the vessel are calculated with commercial CFD software. The CFD software does not resolve structures smaller than the grid size of the Finite Volume Grid, so that the molecular mixing can not be calculated. Therefore a new model is derived that describes the deformation of small fluid elements and the molecular exchange between these elements and the surrounding liquid. This model is implemented through two transport equations into the CFD software and yields the local mixing quality at a multitude of points through the vessel. 2. EXPERIMENTAL TECHNIQUE 2.1 Injection of inert and reacting dye The course of the laminar mixing process is schematically depicted in Figure 1. During the first step the length scale of segregation - e.g. the thickness of a lamella - decreases due to the stretching and folding of the injected material. Also diffusion occurs but its influence is negligible because of the large length scale of segregation. That is why the intensity of segregation remains nearly unchanged, which means that hardly any micromixing occurs. Therefore a dye that discolors with the vessel contents remains mostly unaffected, because the discoloration reaction requires molecular mixing. When the length scale of segregation falls below a certain value, the stronger diffusion levels out local concentration gradients and the intensity of segregation decreases. This second step leads to a reduction of the reacting dye whereas the inert dye remains unchanged from a macroscopic viewpoint. After complete micromixing is achieved only the inert dye remains. The local ratio of the concentration of the reacting dye cj compared to the concentration of the inert dye Cp describes the intensity of Fig. 1: Course of the laminar mixing process and its segregation. K~ippel [6] defined the influence on inert and reacting dye

27 global degree of deviation A in analogy to Danckwerts' [4] intensity of segregation (equation (1)), but using the first centered moment of the concentration distribution instead of the mean square deviation. He also showed that it is equal to

A(t)

cj(t) = ~ . C j,0

(2)

That is the concentration ratio of a discoloring dye cj at a given time t compared to its initial concentration cj,0, when the dye and the discoloring agent are fed stoichiometrically. This method only allows global measurements, because according to this definition the initial concentration of an injected droplet at a given location is not known. The local concentration is namely affected by two mechanisms: the first effect is a dilution of the droplet due to the macro transport without micromixing, the second effect is the discoloration after the micromixing. To overcome this problem and to separate these two effects, a mixture of an inert dye and a reacting dye is fed into the vessel. The dilution affects both dyes whereas the micromixing only affects the concentration of the reacting dye. It is now possible to define a local degree of deviation by comparing the local concentration of the reacting dye cj with the one of the inert dye Cp:

A(Y,t) = cj (Y~,t) K with K =Cp(t~ = O)

cj(,-o)

(3)

The degree of deviation equals the conversion rate and shows the same behaviour over time as the intensity of segregation as it decreases from one (totally segregated) to zero (completely mixed). On certain assumptions it is possible to convert one measurement to the other, but to keep the measured results comprehensible only the degree of deviation is used in the figures presented here.

2.2 Experimental setup The flat-bottomed vessel has a diameter of 100 mm. It is filled to a height of 125 mm with a non-Newtonian (shear thinning) liquid. The Rushton turbine is placed in the center of the vessel. The stirrer speed of 250 min 1 leads to a Reynold's number of 20. A mixture of a red inert dye and a blue reactive dye is injected at two different positions. One position is located close to the main vortex, whereas the other position is situated closer to the stirrer shaft. Details about the composition of the liquid and the two dyes can be found in [2].

2.3 Tomographic dual wavelength photometry The tomographic dual wavelength photometry enables the simultaneous measurement of the two dye concentrations at multiple points in the vessel. The optical set-up is shown in Figure 2. Two laser beams of different wavelengths are superimposed in a beamsplitter and then divided into three superimposed beams. The three beams are expanded and are used to transilluminate the mixing vessel. The beams give projections of the dye concentrations on RGB-CCD cameras. The projections are used to determine the integrated dye concentrations along the paths of the beams. For that, a reference picture of the vessel without the dye is taken by every camera and is then compared with pictures of the dye projections. The absorption of the laser beams is described by Lambert Beer's law and depends on the

28 absorption coefficients of the dyes, the length of the path through the vesseland on their concentrations along the beams. The inverse wavelength dependence of the absorption coefficients of the two dyes enables precise concentration measurements: the red laser beam is strongly absorbed by the blue reacting dye but hardly by the red inert dye whereas the green beam is strongly absorbed by the red dye. The tomography is used to reconstruct the three dimensional concentration fields from the measured concentrations along the paths. For the reconstruction the vessel Fig. 2: Optical setup for the tomographical is divided into 30 slices each divided again dual wavelength photometry into 1220 elements yielding a total of 36600 elements in the whole vessel. The concentration in each of these 36600 elements is calculated. The reconstruction is done with the iterative algebraic reconstruction technique (ART), as shown by Ostendorf and Mewes [ 10]. Details about the experimental setup and the tomographical reconstruction can be found in [2]. Finally the concentration ratio of reacting and inert dye is calculated for every element to yield the field of the degree of deviation. 3. EXPERIMENTAL RESULTS AND DISCUSSION Figures 3 and 4 show the course of the macro- and micromixing for the two different injection positions with time. The macromixing is depicted by an isosurface surrounding all volume elements with a local concentration of the inert dye Cp higher 1% of the initial concentration Co,0. Elements outside this dye cloud are basically uncolored and therefore not macromixed with the injected droplet. The micromixing is depicted by an isosurface surrounding all volume elements with a local degree of deviation A higher 10 %. Elements inside this cloud are not sufficiently micromixed yet. Figure 3 represents the results for the injection close to the main vortex. After 1 second the droplet is only stretched one-dimensionally along a part of the main vortex. Therefore the length scale of segregation is still high. That results in a slow diffusive transport and a high degree of deviation in most of the vortex. After 3 seconds a part of the dye remains close to its old position. The other part got transported through the upper half of the stirrer. The dye cloud keeps its tubular shape, which is due to the fact that it got mostly stretched in one direction. This ineffective stretching causes regions that are still not sufficiently micromixed after 3 seconds. Figure 4 represents the results for the injection closer to the shaft. This position in the main suction zone of the stirrer yields more effective mixing. After 1 second the droplet is stretched in radial and circumferential direction, which results in a fast dilution and a fast decrease of the length scale of segregation. The steep gradients enable a strong diffusive transport and as

29

Fig. 3: Normalized Ponceau concentration and Degree of Deviation l s and 3s after the injection into the vortex

Fig. 4: Normalized Ponceau concentration and Degree of Deviation ls and 3s after the injection close to the shaft

a result the degree of deviation is smaller. After 3 seconds the dye cloud still has its toroidal shape and moves upward on the outside of the main vortex. Most of the injected fluid is micromixed by that time and only a small region has a degree of deviation higher 10 %. This injection position yields better mixing, but no dye is transported through the stirrer, so that the macromixing is limited to the upper half of the vessel. 4. MODEL FOR THE LAMINAR MIXING PROCESS

During the theoretical examinations of the laminar mixing process the flow field and the convective transport through the vessel are calculated with commercial CFD software. The vessel is divided into 36700 elements. The exchange between the grid ceils is calculated by transport equations for mass and momentum. The CFD software does not resolve structures smaller than the grid size of the Finite Volume Grid, so that the molecular mixing can not be calculated. Therefore a new model is derived that describes the deformation of small fluid elements and the molecular exchange between these elements and the surrounding liquid. This model is implemented through two transport equations into the CFD software.

30 4.1 Calculation of the deformation of injected fluid elements A droplet injected into the vessel is deformed by the flow field into a lamella. If diffusion is neglected the initial volume of the lamella V0 remains unchanged. The thickness of the lamella can then be calculated by

s = -A

(4)

with A is the cross sectional area of the lamella. The normal vector ~ is perpendicular to A with the magnitude equal to its size. Therefore, it is sufficient to define ~ as a transport variable and to calculate the local thickness of the lamella by

ih~,t~.

s(Y,t) =

(5)

The deformation of the lamella, represented by its normal vector h, in the flow field is schematically depicted in Figure 5. The deformation includes stretching or compression and rotation. It depends on the orientation of the vector and on the flow field. At each position this is described by the local velocity gradients. Therefore, for the rate of change of the normal vector fi the transport equation D~

---T

~=-L Dt

.h.

(6) ---T

can be derived as shown in [3]. In equation (6) L is the transpose of the velocity gradient m

tensor L. This equation is defined in cylindrical coordinates for each of the components n r,

Fig. 5" Lamella represented by its normal vector inside a grid element

31 n:and n, of the vector ~. It is implemented into the CFD-code and as a result the local thickness of the lanaella is calculated according to equation (5). 4.2 Calculation of the molecular mass transfer between fluid elements

In addition to the deformation of the injected fluid elements there is a molecular mass transport between these elements and the surrounding liquid. In analogy to the experiments this molecular diffusion is described by the local degree of deviation, for which a new transport equation is derived. For that, two one dimensional unsteady mass transfer equations for the components A and B, which undergo a second order chemical reaction, are solved simultaneously. The integral of the concentration of component A over the thickness of the lamella yields the number of moles per cross sectional area of the lamella N A. The dimensionless number of moles N A(Y,t) = NA (~,t) NA,O

(7)

is the number of moles at a given location and time N A(2,0 compared to the initial value

Nn,o. In [3] it is approximated by an empiric equation. In analogy to the experiments it is set to the local degree of deviation A(Y,t) defined in equation (3). The decrease of the local degree of deviation is then calculated by the transport equation

DA = 3 DA Dt so

t

Ds(t)

-s(t)

Dt s(t) 2

A

(8)

with DA is the diffusion coefficient of component A and so is the initial value of the thickness of the lamella. The derivative of the thickness of the lamella Ds(t) is deduced from Dt equation (5) and (6). The complete derivation of equation (8) can be found in [3]. Since the middle part of equation (8) is negative, the local degree of deviation decreases with time. As expected the decrease is stronger for higher diffusion coefficients DA and for lower initial values for the thickness of the lamella so. 5. CALCULATED RESULTS The following results have been calculated for a stirrer speed of 250 min l . In analogy to the experiments two different injection positions have been simulated. One position is close to the center of the main vortex and one is close to the stirrer shaft. In Figure 6 the concentration profile 0.5s after the injection is shown. The calculated results agree well with the experiments. The macromixing is more effective for the injection in the region of higher shear stress close to the stirrer shaft as shown in Figure 6 b. The concentration is smaller than for the injection in the main vortex. In this case the dye is not distributed into other regions of the vessel but remains in the main vortex. The concentration is still high as shown in Figure 6 a. The concentration profile does not give insight into the form of the injected fluid elements since the lamellas are smaller than the grid size. This means that the concentration is not

32 homogeneous inside one grid cell. Therefore the deformation of the injected fluid elements and the molecular exchange between them and the surrounding liquid have been calculated as described above. Figure 7 shows the distribution of the thickness of the lamella 0.125s after the injection. For the injection in the main vortex (Figure 7 a) a plane above Fig. 6: Distribution of the dye 0.5s after the injection in the the Rushton turbine is main vortex (a) and closer to the shaft (b) displayed. The thickness of the lamella consistently decreased to a value of 0.3mm to 0.4mm from its initial value of lmm. The reason is the homogeneous flow field inside the main vortex. The velocity component in circumferential direction is predominant so that the lamella is stretched only in this direction. This yields a bad micromixing. After 0.125s the local degree of deviation did not significantly decrease, which is in well agreement with the experimental results. For the injection close to the shaft the same plane is displayed in Figure 7 b. After 0.125s the local degree of deviation also did not significantly decrease. The concentration gradients are not steep enough yet to enable substantial molecular exchange. But because of the inhomogeneous flow field near the stirrer the lamella is stretched and folded more than in the main vortex. This yields better micromixing later on. Near the blades the thickness of the lamella is decreased most. The lamella is even cut into two pieces while passing the blades. The flow around the blades can also be seen in Figure 8, in which the distribution of the component of the normal vector in circumferential direction n~ is shown. On the suction side of the blade, situated on the left side of the figure, the fluid elements are tilted to the left side.

Fig. 7: Thickness of the lamella 0.125s after the injection in the main vortex (a) and after the injection close to the shaft

33

Fig. 8: Distribution of the circumferential component of the normal vector n~ 0.125s after the injection closer to the shaft (b) This results in negative values for n~. For the pressure side of the blade, situated on the right sight, the fluid elements are tilted to the right side which yields positive values for n~. 6. CONCLUSIONS The tomographical dual wavelength photometry gives new insight into the mixing process. With this technique it is possible to measure the local intensity of segregation at a multitude of points inside the stirred vessel. This is done by injecting a mixture of an inert and a reacting dye into the vessel and by measuring its concentration fields. The inert dye serves as a tracer for the macromixing, whereas the vanishing of the reacting dye shows the micromixing. All measurements were performed with a Rushton turbine. The injection in the main vortex of the Rushton turbine leads to slow macro- and micromixing. The injection near the shaft yields a fast decrease of the microsegregation, but the mixing is also limited to the upper half of the vessel. The flow field and the convective transport through the vessel are calculated with commercial CFD software. The CFD software does not resolve structures smaller than the grid size of the Finite Volume Grid, so that the molecular mixing can not be calculated. Therefore a new model is derived that describes the deformation of small fluid elements and the molecular exchange between these elements and the surrounding liquid. This model is implemented through two transport equations into the CFD software and yields the local mixing quality at a multitude of points through the vessel. The calculated results are in agreement with the experimental measurements. NOMENCLATURE A C

/s K

= cross sectional area of the lamella (m2) = concentration (mol/m3) = intensity of segregation, defined in (1) = concentration ratio, defined in (3)

m

L n

= velocity gradient tensor = normal vector, defined in (5)

34 s t V

= thiclmess of the lamella, defined in (4) = time (s) = volume of the lamella (m3) = position vector (m)

Greek letters A = degree of deviation = mean square deviation

Subscripts j p 0

= reacting dye = inert dye = initial state

REFERENCES [1] J.R. Bourne. and J. Baldyga, "Simplification of Micromixing Calculations. II. New Applications", The Chemical Engineering Journal, 1983, 42, pp. 93-101. [2] M. Buchmann and D. Mewes, "Measurement of the Local Intensities of Segregation with the Tomographical Dual Wavelength Photometry", Can. J. Chem. Eng., 1998, 76, pp. 626630. [3] M. Buchmann, "Simultanes Messen des laminaren Mikro- und Makromischens mit Hilfe der tomographischen Zweiwellenl/ingenphotometrie", VDI-Fortschrittbericht Nr. 611, Reihe 3, VDI-Verlag, 1999, [4] P.V. Danckwerts, "The effect of incomplete mixing on homogeneous reactions", Chem. Eng. Sci., 1958, 8, pp. 93-102. [5] J.W. Hiby, "Definition und Messung der Mischgtite in fliissigen Gemischen", Chem. Ing. Tech., 1979, 51, pp. 704-709. [6] M. K/ippel, "Entwicklung und Anwendung einer Methode zur Messung des Mischungsverlaufs bei Fltissigkeiten", PhD Thesis, University Munich, Germany, 1976. [7] R. Mann, S. K. Pillai, A. M. E1-Hamouz, P. Ying, A. Togatorop, R. B. Edwards, "Computational fluid mixing for stirred vessels: progress from seeing to believing", Chem. Eng. J., 1995, 59, pp. 39-50 [8] J.M. Ottino, "Mixing and Chemical Reaction- A Tutorial", Chem. Eng. Sci.,1994, 94, pp. 4005-4027. [9] J.M. Ottino, S. C. Jana and A. Souvaliotis, "Potentialities and Limitations of Mixing Simulations", AIChE J., 1995, 41, pp. 1605-1621. [10] W. Ostendorf and D. Mewes, "Measurement of three-dimensional unsteady temperature profiles in mixing vessels by optical tomography", Chem. Eng. Techn61Qgy, 1988, 11, pp. 148-155. [11] J. Villermaux, L. Falk and M. C. Fournier, "Scale-up on Micromixing Effects in Stirred Tank Reactors by a Parallel Competing Reaction System", in "8th European Conference on Mixing", Cambridge, UK, BHRA, 1994, pp. 251-258.

I0 th European Conference on Mixing H.E.A. van den Akker and J.,l. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

35

CHARACTERISATION AND MODELLING OF A TWO IMPINGING JET MIXER FOR PRECIPITATION PROCESSES USING LASER INDUCED FLUORESCENCE N. B6net, L. Falk, H. Muhr, E. Plasari Laboratoire des Sciences du G6nie Chimique-CNRS, Ecole Nationale Sup6rieure des Industries Chimiques-INPL, 1 rue Grandville BP 451 54001 Nancy Cedex, France A two impinging jet mixer for precipitation processes is investigated by means of the laser induced fluorescent technique. Neutralisation reactions with a fluorescent pH-indicator are used to reveal the mixing process up to the molecular level. On the basis of the experimental results, a phenomenological mixing model is derived. The mixing model involves a macrodilution process and a coalescence-redispersion process through which the fluids are contacted at the molecular level. The mixing model is applied to the precipitation of barium sulfate, and it is found that its predictions are very close to experimental results. 1. INTRODUCTION In precipitation, the nucleation step is typically induced by the mixing of reactants and is generally very fast. Besides, the number of nuclei produced (which determines the end particle size) is highly sensitive to the reactant concentration. As a consequence, the control of the nucleation step is very hard in classical single jet or double jet stirred tank precipitators, because of backmixing. Thus, a new strategy of precipitation consists in using a continuousflow mixer as an independent nucleator. The Two Impinging Jet (TIJ) mixer is developed in such a prospect. It is specially designed for precipitation as the mixing process takes place in a space free from any material surfaces. Indeed, for most materials, classical continuous-flow mixers (essentially tubular mixers) cannot be used at the industrial scale because of encrustation problems. The principle of impingling streams, was initially developed for heterogeneous phase contacting and combustion. It was first applied to precipitation by Midler et al.2.3, for pharmaceutical compounds, with evidences of an improvement of the product quality. Impinging jets are used nowadays in the pharmaceutical industry. However, no comprehensive description and modelling of the mixing process, with the view of precipitation application, has been reported up to now. The main difficulty lies in the fact that it is basically a three fluid mixing problem, the third fluid being the surrounding fluid. It should be mentioned nevertheless, the study of Becker et al.4, who measured the concentration fields in an impinging air jet mixer. Concerning the velocity field, it has been investigated by Witze and Dwyer,5 who performed a comparative study between the radial jet resulting from free impinging jets and the true radial jet (that is the jet emerging from a radial

36 nozzle). An interesting result was that the impinged radial jet spreads at a rate about three times greater than the true radial jet. The aim of this study is to build a phenomenological mixing model, which allows the simulation of a precipitation process taking place in the TIJ mixer. The parameters of the model are directly identified from laser induced fluorescence experiments and correlated to the operating parameters. This provides a complete description of the TIJ mixer. At last, the validity of the model is illustrated by the application to the precipitation of barium sulfate. 2. D E S C R I P T I O N

OF THE TIJ MIXER

The TIJ mixer consists of two equal opposed round free jets, that impinge in a boundary free space (see figure 1). It is operated in submerged mode. In precipitation applications, the TIJ mixer can be either immersed in a stirred tank reactor, where controlled growth, agglomeration or ageing take place, or in a small separated vessel. High jet velocities (5 to 20 m/s) are used, both to promote intensive mixing and to prevent disturbances due to external stirring. The jets are delivered by free pulsation gear pumps, using pipe nozzles of 1 mm internal diameter and 5 cm long. The adjustment of jet axis, a critical point, is achieved by means of a micro-screw system, watching the slant of the sheet resulting from the impingement of the jets in air. This study is restricted to an internozzle spacing of 1 cm. In a previous study6, we have pointed out two distinct cases concerning the internozzle spacing: the closely spaced jets (L/do < 5) and the widely spaced jets (L/do > 10), where do is the nozzle internal diameter and L the half internozzle spacing. This distinction corresponds to the transition, at the impingement point, from the potential core region to the fully developed turbulent round jet. In the first case, the relevant length scale is the nozzle internal diameter, whereas the internozzle spacing is relevant in the second ease. Hence, the geometrical factor L/do, is only a qualitative parameter, although the influence of the internozzle spacing has not been investigated with a great accuracy for the closely spaced jets. In this study, only the closely spaced jets are considered (L/do = 5). This is the most interesting case, as backmixing resulting from the incorporation of the surrounding fluid in the round jets is reduced.

Reactant2

Reaetantl

~ surroundingfluid Radialjet

Round

freejet ~

Streamlineofthe

~.i

"-"- Impingement_ ~ne

surrounding fluid

:

~'x

Figure 1. Flow pattem displayed by the TIJ mixer.

The TIJ mixer displays three distinct zones (see figure 1): - the two round free jets, which incorporate a part of the surrounding fluid. For the closely spaced jets, the amount of incorporated surrounding fluid is only about 20 % (for 80% of fresh fluid) at the impingement point. - the impingement zone, where the two feed streams are contacted and radially distributed. This reduces the initial scale of segregation (the jet diameter) by a factor 2 (from the volumetric flowrate conservation in the impingement zone).

37 -the out-flow, where the fluids mix and which has the feature of a turbulent radial jet. The TIJ mixer is developed in order to carry out very fast reactions (precipitation). Consequently, the surrounding fluid is in practice, comparatively to the jet feed streams, highly diluted (at least as regards the reactants). Thus, two mixing processes are basically involved: the mixing of the jet feed streams together and their dilution in the surrounding fluid. 3. LASER INDUCED FLUORESCENCE MEASUREMENTS

3.1 Description of the experimental technique and setup The laser induced fluorescence (LIF) technique is used to investigate the mixing process. The mixing zone is crossed by a laser beam and the fluorescence emitted is collected by a range of 512 photodiodes distributed on 12.8 mm long (see figure 2). This allows the measurement of a fluorescent tracer concentration along a given portion of the laser beam. Beforehand, the laser beam is focused by a 180 mm lens. Like this, the beam diameter is about 0.5 mm and almost constant on the portion seen by the range of photodiodes. Prior to collection, the light is high-pass filtered in order to remove the scattered laser light. Each measurement is followed by a blank measurement (no fluorescent tracer) and a standard measurement (homogeneous known concentration of fluorescent tracer). The experiments are carried out in a great cubic vessel (30 liters) so as to prevent wall disturbances. The resolution of the measurement system does not allow the detection of concentration fluctuations. Thus, only mean concentrations are measured, the acquisition time being 4s. Consequently, in order to capture the mixing process up to the molecular level, neutralisation reactions are carded out with a fluorescent pH-indicator (pyranin). It should be mentioned that the detection of concentration fluctuations up to the smallest scales of turbulence in liquid flows (e.g. 10 ~tm) is still beyond reach, or at least with a high power laser and a very sophisticated optical system. Although the measurement of concentration fluctuations is required for the investigation of fundamental aspects of mixing (correlation functions such as turbulent diffusivity and spectra), the use of a reactive tracer (which allows to reveal mixing up to the molecular level through the measurement of mean tracer concentrations) is sufficient for the requirements of chemical engineering science. At last, it should be remembered that, globally, mixing up to the molecular level can be limited by the maeromixing and/or the micromixing. Figure 2. Experimental setup.

Pyranin is a very good fluorescent pHindicator 7. It is almost perfectly coloured

38 above pH = 9 and colourless below pH = 5, contrary to fluorescein which remains slightly coloured up to about pH = 3. Its maximum excitation wavelength is 450 nm while its maximum emission wavelength is 512 nm. Thus, it can be excited by the 488 nm line of an argon ion laser. Just like fluorescein, it is unstable (photobleaching), but this is not a problem as long as the fluid is continuously renewed.

3.2 Operating conditions and results The LIF experiments are performed in such a way to investigate separately the two mixing process, that is, the mixing of the jet feed streams together and the dilution in the surrounding fluid. The operating conditions of the various experiments are detailed below. 3.2.1. Mixing of the jet feed streams together a. Macromixing One jet is marked by an inert tracer at a concentration Co, while the surrounding fluid is at the concentration Co/2. The inert tracer is pyranin, all the fluids being at pH = 12. The results are expressed in terms of the normalised deviance from the concentration after mixing, that is I C- Co/21 / (Co/2), where C is the mean concentration of the inert tracer. It should be noticed that the concentration deviance is inherently reduced by the dilution. In facts, to get rid of the dilution reduction, it is enough to divide the normalised deviance by the mean fraction of fresh fluid. On the other hand, a pertinent mixing criteria would be the global normalised deviance flowrate, which itself is conserved by dilution and only decreases through macromixing of the jet feed streams together. (see Figure 3.a) b. Neutralisation reaction One jet is acid and the other alkali, while the surrounding fluid is neutral (pH=4). Each fluid is marked by pyranin at the same concentration. Like this, the nominal concentration of the tracer is not affected by the dilution and the decolorization only results from the mixing of the jet feed streams together. Various excess of acid are used (E=I 0%, 20%, 50% and 200%) in order to follow accurately the whole course of the mixing process. The measurements are performed in a transitory state, for the surrounding fluid to be maintained at neutral pH. The measure is started 10 seconds after injection, for stabilisation of the flow. Due to the large volume of the vessel, the entrained surrounding fluid is supposed to remain at neutral pH during the measure (see the streamline of the entrained surrounding fluid in figure 1). This point is proved true by checking the reproductibility of the measure during about 30 seconds of a single run. The effect of the pH of the surrounding fluid (slightly acid for colourless) has also been tested: no differences were observed between pH = 5 and pH = 4, whereas pH = 3 results in a significant reduction of the plume. The normalised mean concentration of the pH-indicator is equal to the proportion of alkali aggregates (as all fluids are marked with the pH-indicator). Consequently, if we call A the fluid of the acid jet and B that of the alkali jet, then the measured concentration of the tracer provides the proportion of fluid aggregates containing B and whose relative fraction fB,A= fB/(fA+ fB) is greater than fpH--7= (I+E)/(2+E), where E is the excess of acid. (see figures 3.b and c)

39

Figure 3. Mixing of the jet feed streams together.

Figure 4. Macrodilution. (mean fraction of fresh fluid)

40

Figure 5. Photographs obtained with planar LIF. (the intemozzle spacing is 2 cm) 3.2.2. Dilution process a. Macrodilution The two jets are marked and the surrounding fluid is pure. (see figure 4) b. Neutralisation reaction The two jets are alkali and the surrounding fluid is acid. Each fluid is marked with pyranin. Two acid concentrations are used: one corresponding to a dilution factor at molecular level of 2 for complete neutralisation (that is one volume of fresh fluid is neutralised by one volume of the surrounding fluid) and the other to a dilution factor of 4 (that is one volume of fresh fluid is neutralised by three volumes of surrounding fluid). The experimental setup can be modified to produce a laser sheet. This allows the visualisation of the whole mixing pattern. Three photographs obtained with planar LIF are presented (see figure 5). 4. A PHENOMENOLOGICAL MIXING MODEL

4.1 Description of the model The aim is to model, with a phenomenological approach, the two mixing processes taking place in the radial jet from the impingement point. From the unidimensional hypothesis, the flow in the radial jet can be considered as a plugflow (if the velocity distribution is averaged). In fact, this point should be discussed. Indeed, the very large spread of the radial jet, which appears in figure 5.c, cannot result from only turbulent diffusion. It is mainly due to flow instabilities, arising from jet flowrate fluctuations and not synchronised large scale turbulence of the two round free jets. This gives rise to fluctuations of the impingement plane along the x-axis ( visually, about 0.5 mm large) and fluctuations of the flow direction at the exit of the impingement zone. On the other hand, as can be seen on figure 1 (streamline of the surrounding fluid), there is a reverse flow at the edge of the radial jet, where the surrounding fluid is sucked up. If the flow is perfectly stable, local backmixing due to the reverse flow is insignificant (as can be seen from CFD simulations). On the contrary, it is clear that if the radial jet fluctuates, a part of the fresh fluid

41 will be reincorporated by the reverse flow. Such a local backmixing can be observed visually. The instability of the flow, also reported by Becker et al.4, appears clearly in figure 5.b. This explains the difference observed by Witze and Dwyer 5, between the spread of an impinged radial jet and that of a true radial jet. Nevertheless, as backmixing has not been quantified, it is not taken into account in the model, except implicitly in the estimation of the convective time. Thus, a plug-flow is assumed and consequently, the only space variable is the radial coordinate. For convenience, the mixing model is expressed in a lagrangian frame, through the convective time. The convective time is taken as the mean residence time of the fresh fluid, from the impingement point to a given point in the radial direction. It is given by the ratio of the volume of fresh fluid between these two points, to the volumetric flowrates at the nozzles. It is calculated from the macrodilution results, deducting the volume corresponding to the two round jets alone, while the convective time in the impingement zone is extrapolated. The mixing model assumes that the three fluids are contacted at the molecular level in the radial jet, through a single mixing process. On the other hand, the amount of surrounding fluid in the radial jet is controlled by the macrodilution. The mixing model is represented schematically in figure 7. The assumption of the model is based on the observation of figure 8. Indeed, figure 8 shows that the dilution up to the molecular level (revealed by the neutralisation reaction) is late with comparison to the macrodilution. This indicates a limitation by the micromixing, although there can also be an influence of the flow instability. Such a limitation by the micromixing also seems to be true for the mixing of the jet feed streams together. Indeed, as can be seen in figure 3, macromixing of the jet feed streams is almost instantaneous, whereas neutralisation reactions reveal imperfect mixing (there can also be an influence of the flow instability resulting in an imperfect distribution of the two jet feed streams at the impingement zone exit). Thus, the contacting process of the model is referred to micromixing, which, indeed, indifferently mix the three fluids together. At last, it should be noticed that, even if the assumptions at the basis of the model are only partially exact, what is finally important is the formal truth of the model. The contacting process of the model is represented by the coalescence-redispersion (CD) model. The choice of the CD model lies in the fact that it provides very realistic distributions of concentration (see figure 9). The CD model, first introduced by Curl 8, assumes an equal probability of mixing between all the fluid aggregates. Its characteristic parameter is the mean frequency of coalescence co of the fluid aggregates. In practice, it is simulated by the Monte Carlo method.

Figure 7. Schematic representation of the mixing model.

Figure 8. Dilution profiles in the plane x=0.

42 4.2 Calibration of the mixing model.

The parameters of the model are not determined from theoretical assumptions, but directly identified from the LIF experiment results. For this, we have used the concentration profiles measured in the plane x=0 (impingement plane). The amount of surrounding fluid in the radial jet (expressed below in terms of the macrodilution factor txd) is thus directly provided by the measured mean fraction of fresh fluid in the plane x=0. Concerning the Figure9. Concentration distributions frequency co, a power law of dependence on the provided by the CD model. radial coordinate is assumed. The two parameters of (two fluids in equal proportions) the law are then identified by fitting of the neutralisation reaction experiments results. As can be seen in figures 10 and 11, given a simple law for co, the model fits very well the mixing experiment results. This proves the ability of the CD model to represent formally the whole course of the mixing process. The parameters of the model are next correlated to the operating parameters. From a dimensional analysis, it appears that the system only depends on the Reynolds number Re=uodo/v and the geometrical factor L/do (the Schmidt number is not relevant when Sc<<40009). But we have seen that the geometrical factor is only a qualitative parameter. Besides, the experimental results show that the Reynolds number has no influence (see figure 12). Differences between the three profiles can be observed around the point r - 0 (figure 12), but in fact, these differences are only due to very small deviations of the laser beam from the true impingement plane, and to the sharp transition of the pH-indieator concentration, from the alkali jet to the acid jet. The correlations are thus trivial. They are presented below:

tcU~ = do

_0.0007(~o_o )4 + 0.0056(~oo)a+ 0.33(~oo)2 -0.07

r do

ad = --0.0003(~o)' +0.01(~o)3--0.1(~oo)2+0-71 ---+1.2 r do

Figure 10. Comparison between model and experiment. (mixing of the jet feed streams)

(1) (2)

Figure 11. Comparison between model and experiment. (dilution process)

43

So 2000 Experiments 3.4 ~tm Model 4.7 ~tm Figure 12. Influence of the jet velocity. (neutralisation reaction with E=50%)

If mr < 1.5 then do r > 1.5 then If .d o

codo = 0.37 uo COdo do)l.7 ~ = 0.37(1.5 uo r

Table

4000 0.35 gm 0.54 gm

1. Precipitation results. (mean particle size)

(3) (4)

where tc is the convective time, Uo the jet velocity at the nozzle and ad the macrodilution factor. 5. APPLICATION TO THE PRECIPITATION The mixing model is applied to the precipitation of barium sulfate. The results of the simulation are compared to results from experiments carried out in a semi-batch mode6. We have used for the nucleation, the kinetic fitted by Dirksen and Ring 1~ from Nielsen results. The growth kinetic is also taken from Nielsen ll results.

J(Nb/(m 3s))

=

1036

- 2440. exP(ln 2(S))

G(m / s) = 5.8 x 10 -12 (S - 1) 2

(5) (6)

Two concentrations of reactants, corresponding to initial levels of supersaturation of 4000 and 2000 have been tested. As can be seen in table 1, the predictions of the model are very close to the experimental results. CONCLUSION It is proved that, from a simple phenomenological approach based on LIF experiments, a relatively complex mixing process can be accurately modelled. The mathematical description of the mixing process relies on the coalescence-redispersion model, which predicts very realistic distributions of concentration. The mixing model is easy to perform. Besides, it provides a global dynamical description of the process, from which the influence of the operating parameters becomes transparent. This phenomenological approach is an alternative to CFD simulation, as, due to instabilities of the flow, this latter approach can provide less

44 accurate results. The mixing model can be advantageously used to investigate the capabilities of the TIJ mixer when applied to precipitation processes, and it can serves as a basis for the design.

NOMENCLATURE C Co do E

fA fB,A

G J L r

S So tr Uo X (~d (~dn CO

mean concentration of the inert tracer concentration of the inert tracer at the nozzle nozzle internal diameter acid excess volume fraction of fluid A relative fraction of fluid B growth rate nucleation rate half internozzle spacing radial coordinate supersaturation ratio initial supersaturation ratio convective time jet velocity at the nozzle axial coordinate macrodilution factor (inverse of the mean fraction of fresh fluid) dilution factor for complete neutralisation mean frequency of coalescence of the fluid aggregates

REFERENCES 1. Tamir A., 1992, Advances in Transport Process VIII, Elsevier. 2. Midler M., Paul E.L., Whittington E.F., 1989, Engineering Foundation Conf. on Mixing, Potosi, MO. 3. Midler M., Paul E.L., Whittington E.F., Furtran M., Liu P.D., Hsu J., Pan S., 1994, US patent No. 5,314,506. 4. Becker H.A., Cho S.H., Ozum B., Tsujikawa H., 1988, Chem Eng Comm, 67, 291-313. 5. Witze P.O. and Dwyer H.A., 1976, J. Fluid Mech., 75,401-417. 6. Brnet N., Falk L., Muhr H., Plasari E., 1999, 14th Intemational Symposium on Industrial Crystallisation, Cambridge. 7. Wolfbeis O.S., Fiirlinger E., Kroneis H., Marsoner H., 1983, Fresenius Z anal Chem, 314, 119-124. 8. CurlR.L., 1963, AIChEJournal, 9, 175-181. 9. Baldyga J. and Bourne J.R., 1989, Chem. Eng Sci., 42, 83-92. 10. Dirksen J.A.and Ring T.A., 1991, Chem. Eng Sci., 46, 2389-2427. 11. Nielsen A.E. and Tort J.M., 1984, Journal of Crystal Growth, 67, 278-288.

10th European Conference on Mixing H.E.A. van den .4kker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

Four-dimensional micro

mixing

45

Laser Induced

in a tubular

Fluorescence

measurements

of

reactor

E. van Vliet, J.J. Derksen and H.E.A van den Akker Kramers Laboratorium voor Fysische Technologie, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands

Abstract A Laser Induced Fluorescence experiment is presented that measures the time evolution of a three dimensional concentration field. This way, the mixing of a passive scalar into the turbulent flow of a tubular reactor is studied at Re = 4,000. In the experiment a fluorescent dye is radially injected in the flow system. The mixing process is visualised by exciting the dye molecules with a sheet of laser light. The resulting fluorescence, being proportional to the local dye concentration, is imaged with a high speed digital camera, yielding two dimensional images of the concentration field. By sweeping the laser sheet in the depth direction, parallel to itself, a three dimensional realization of the concentration field can be measured from the successive, closely spaced two dimensional images. As such 3D concentration fields can be captured successively in time, the technique will be denoted as four dimensional laser induced fluorescence (4D-LIF).

1

Introduction

Mixing is the process of bringing initially separated fluid elements into close contact, usually with the purpose to achieve a certain mass exchange of reactants. Especially the molecular diffusion at the smallest dynamical scales of the turbulent flow is important, since at these scales the chemical reactions take place. The small scale flow locally behaves as a laminar flow, consisting of vortices that are driven by the larger scale motion. The vortices mix the fluid by rolling up, thereby creating multiple layers of fluid. Because the vortices are also stretched axially, these layers tighten. Due to this shrinking of layers, concentration gradients increase, which enhances molecular diffusion. In case of competing, diffusion limited chemical reactions, it is well known that mixing strongly influences the final yield. Various micro mixing models have been developed (e.g. Baldyga and Bourne, 1984; Bakker, 1996), which all have in common that in some way the effect of shrinking layers on the predicted yield is incorporated. Experiments are necessary to validate the assumptions made on shape and temporal evolution of these shrinking layers. In this paper we present the design and first results of an experimental set up that can critically assess modelling assumptions. Planar Laser Induced Fluorescence (PLIF) is an established technique for non-intrusive

45

Figure 1. Schematic (a) and three dimensional (b) representation of the tubular reactor at the position of the injector. The dimensions of the internal reactor diameter Dr, the external injector diameter D / a n d injector feed opening Di are 0.10m, 0.02m and 2mm, respectively. To avoid image deformations due to the cylindrically shaped tube wall, a square, transparent box filled with water has been installed around the tube.

measurement of the concentration field of a mixing scalar. A fluorescent dye, added to the turbulent flow of interest, is excited by means of a thin laser sheet. A digital CCD camera is used to capture images of mixing structures. As the images are essentially two dimensional, it is not possible to distinguish between a real shrinking of layers due to deformations and a virtual shrinking of layers due to rotation and translation of the vortex in the laser sheet. In our approach, the laser sheet is swept in the depth direction, parallel to itself. Each time that the laser sheet sweeps through the flow, multiple images are taken by means of a high speed digital camera. A reconstruction of the 3D scalar field can be made out of the closely spaced subsequent images within a single data volume. Repeating this process yields the time evolution of the 3D scalar field, hence the technique will be denoted as four-dimensional laser induced fluorescence (4D-LIF). 4D-LIF has been applied earlier by Dahm et al. (1990) and Maas et al. (1994). In both cases the flow facility concerned was a very large, square vessel containing motionless water in which turbulence was created by injecting a jet of fluorescent dye. Due to the size of the flow facility, time scales of the turbulent micro structures are relatively large and hardly advected. By using recently developed, high speed CCD cameras, we are able to apply the technique in the turbulent flow of a tubular reactor, a flow facility that is actually used in industry to mix fluids.

2

T h e flow facility

The reactor consists of a perspex tube, Dt -- 0.10 m in diameter, in which a stationary, moderately turbulent flow of water is established. The injector, having an external diameter D / = 0.02 m, was used to add the fluorescent dye to be mixed with the main flow.

47 Measurements were carried out at several distances down stream the injector. In order to avoid image deformation of the flow structures due to the cylindrically shaped tube wall, a square perspex box filled with water has been installed around the TR. The Reynolds number based on the mean bulk velocity (U), kinematic viscosity u and internal reactor diameter Dt, defined as Reb -- -UDt///, was about 4,000. The smallest dynamical length scale (the Kolmogorov length scale) is given by ?7 __ ( / / 3 / s

(1)

where e is the mean local energy dissipation. At the centreline of a pipe flow, the dimensionless energy dissipation eDt/(2u3,) (u, being the friction velocity) can be estimated to be about 2.2 (Hinze, 1959, fig. 7-67). The energy dissipation at the centre line of the TR can therefore be written as 3 it,

//3

3

c = 4.4~t = 4.4~4t4Re ,,

(2)

where the Reynolds number is now based on the friction velocity u,. The friction velocity u, and mean velocity U are related according to (u,/-U) 2 = CI (Bradshaw, 1976), where the friction coefficient C I is given by the Power-law applied on pipe flows" C I = O.079/Red/4. Combining this relates Red and Re,:

= 0.3R4/

(3)

By substituting equation (3) into (2), the energy dissipation can be estimated for Red = 4,000 to be about 3.10-6m2s -3. Substituting this value into equation (1), yields the Kolmogorov length scale to be about 0.8 mm. The same value of c can be used to to estimate the Kolmogorov time scale ~- - (u/e) 1/2 to be about 0.6 s. The Kolmogorov advection time scale "ra = rl/U, defined as the time required for a Kolmogorov eddy to pass a fixed point when advected by the mean velocity, is estimated to be about 20 ms. Another relevant length scale is the Batchelor length scale, which can be interpreted as the diffusion length of a scalar within one Kolmogorov eddy life time:

=

= V/

=

(4)

where /D is the scalar diffusivity and Sc is the Schmidt number defined as Sc =_///1D. The Schmidt number of the fluorescent dye (disodium fluorescein) is 1,930, thus the Batchelor scale can be estimated using equation (4) to be of the order of 20 #m. The radially inserted injector will complicate the flow field significantly. Large scale flow structures are formed in its wake, and mixing will be enhanced. It takes some time, however, for the mixing dye to disperse in the flow, hence mixing structures will only reach the Kolmogorov scales at some distance down stream the injector. It is therefore not expected that the injector considerably influences the size of the smallest mixing flow structures.

48

Figure 2. Overview of the key elements of the experimental setup, consisting of a laser illumination source, sheet forming optics, a high speed digital CCD camera, a PC to capture the measurement data and some electronics to slave the galvano mirror used to sweep the laser sheet in the depth direction.

3

T h e e x p e r i m e n t a l setup

A schematic representation of the optical arrangement is given in figure 2. The blue line of an argon ion laser (continuously emitting 1.4 W) is used as a light source. The laser beam is converted into a horizontal sheet by means of a negative cylindrical lens (f - -25.4 mm). A galvano mirror positioned in the focal point in front of this cylindrical lens is used to scan the laser beam in the depth direction. The galvano mirror is slaved by the CCD camera in a way that every Nz frames taken by the CCD camera, the laser sheet scans the flow ones from its uppermost position downwards. After each scan, the sheet is instantaneously stepped back to its initial position. A positive spherical lens (f = 300 mm) has been positioned at its focal distance from the galvano mirror. This way, the scanning laser sheet moves parallel to itself after being directed through the lens. Moreover, as the focal points of the cylindrical and the spherical lens coincide, the sheet's divergence in horizontal direction is compensated for. Finally, the spherical lens has been positioned at such a distance from the flow facility, that the thickness of the laser sheet reaches its minimum of about 100 #m at the measuring position in the flow. The dye in the TR that is excited by the sweeping laser sheet is imaged by means of a high speed digital CCD camera positioned above the TR with its optical axis perpendicular to the laser sheet. The CCD camera captures 1000 frames per second on a light sensitive area of 256 times 256 pixels. Each single pixel represents the fluorescence intensity in 256 levels of grey values (1 byte/pixel). The scalar concentration can be obtained from the fluorescence intensity as both are proportional in the low concentration limit. The image data flow of about 62 Mb/s was stored in RAM of an NT workstation. As one Gb of memory was available, measurement series up to 15 seconds could be performed.

49

~

Z

1

1

Ill

.

Ly

Lx (a)

(b)

Figure 3. (a) Schematic representation of a data volume consisting of Nz closely spaced data planes of Np times Np pixels each. (b) The (4D) data space consisting of (3D) data volumes captured successively in time, mapping the conserved scalar concentration field in the L~ x Ly x Lz volume in object space. The data volumes are sampled at a temporal resolution AT on a Np x Np x N~ grid of Ax x Ay x Az sample volumes.

4

Spatial and temporal resolution

The 4D-LIF experimental set up has been designed to resolve the smallest flow structures in both space and time. This means that the spatial and temporal resolution had to be small compared to the Kolmogorov length scale ~ and the Kolmogorov advection time scale Ta, which were estimated to be of the order 0.8 mm and 20 ms, respectively. Figure 3 shows a schematic of the 4D-LIF data space. The spatial resolution in the depth direction (Az) is determined by the distance between two successive data planes within one data volume, which is about 160 #m. The thickness of the laser sheet should be smaller than this distance. Therefore the average laser sheet thickness over a single data plane has been set to be about 100 #m. The number of data planes per data volume (Nz) is 31, hence the depth Lz of the data volume is Nz. Az ---- 5 mm. The pixel size of the array sensor of the CCD camera is lp x lp - 10 x 10 #m 2. The magnification m of the collection optics (defined as the ratio lp/AX) equals 0.10. Hence, each pixel corresponds to an area Ax x Ay = 100 x 100#m 2 in a data plane in the flow. As each image consists of 256 x 256 pixels, the size L~ x Ly of each data plane is 25.6 x 25.6 mm 2. A data volume of 31 data planes, combined with a camera frame rate fCCD of 1000 fps leads to a time AT between two successive data volumes of N z / f c c D = 31 ms. Altogether, expressed in term of Kolmogorov scales 7-/and ~'~i the 4D-LIF system measures the concentration field of a data volume of 32r/x 327 x 6ri, on a grid consisting of 256 x 256 x 31 data points. Each data point represents the average concentration of a sample volume of 0.137"/x 0.137"/x 0.2r/. The data volumes are captured at a temporal resolution of 1.6~-a. Therefore it can be concluded that Kolmogorov length scales can be resolved. The Batchelor scales 77b, however, estimated to be of the order of 20 #m, are still too small to be detected. The temporal resolution of the system is slightly too low

50 to fully resolve the Kolmogorov advection time scale. 5

T h e d e p t h of view

The spatial resolution of the measurement system, as estimated in the previous section, really is an upper limit. As the image plane moves with respect to the camera, care has to be taken to capture sharp images. The highest detectable frequency at the sensor array, denoted as the array limited cut off frequency, depends on the pixel to pixel distance lp according to fc,array = 1/(2/p), which corresponds to m/(2lp) in the flow field. Next to the sensor array, however, also the lenses of the collection optics impose a constrained to the maximum detectable frequency, which is known as the diffraction limited cut of frequency fc,l~ns. The cut off frequency of the entire system, f~,totat, is then determined by f-l~,tot~l= f~,,,.,.ay-1+ f-lc,le,~s.For an in focus object, fc,le~8 is much larger then f~,~ay, hence in this case the cut off frequency of the system is totally determined by the latter. Focussing errors, however, reduce the diffraction limited cut off frequency severely. For a lens with finite aperture and focussing error ~ (the absolute value of the difference between the "in-focus" object distance and the actual ditance), the diffraction limited cut off frequency can be written as (Hopkins, 1955) A,t,ns = 1.22 f~(1 + m) '

(5)

where f~ is the lens f-number defined as the ratio of the focal length and the aperture diameter of the lens, f/d. The maximum focussing error 5 is typically as large as half the the depth of the measuring volume Lz/2. Hence, increasing Lz/2 at some point starts to affect the cut off frequency of the entire system due the decreasing diffraction limited cut off frequency. The decrease of fc,le,s can be compensated by choosing a high f-number (i.e. choosing a smaller diaphragm), however this also limits the light gathered by the optical system proportional to ~ 1/f~ and hence degrades the signal to noise ratio. A good balance between spatial resolution and the signal quality occurs when the array limited and diffraction limited cut off frequency are approximately equal (Paul et al., 1990): f~,l~ns-~ f ~ , ~ y , hence

1/(21p) =

f~(1 + m) 1.22

m2Lz/2 ,

(6)

yielding an expression for the f-number in terms of magnification m, pixel to pixel distance lv and the depth of the measuring volume Lz:

m2Lz

= 0.20 (1 +

(7)

In our case (m = 0.1, lp - 10 #m and L~ - 5 mm), it implies that the f-number had to be about unity. 6

Example measurement

Figure 4 shows six consecutive time steps (AT = 31 ms) of the 3D concentration field from a time series of 255 time steps in total. Each single data volume of 25.6 • 25.6 • 5 mm 3

51

Figure 4. Typical example of six successive time steps of the mixing concentration field in a data volume Lz x Ly x Lz = 25.6 x 25.6 x 5 mms. The concentration (same legend as figure 5) at the faces of the data volume is shown. To show the interior of the data volume an insertion has been cut out.

Figure 5. Typical examples of mixing, vortical structures extracted from the measurement data (dimensions in ram). The laser sheet was scanned in the direction of the z-axis, which implies that the planes of figure (a) are oriented perpendicular to the data planes captured by the CCD camera. The positive z-axis indicates the mean flow direction.

52 consists of 31 layers of 256 • 256 pixels each. The spatial resolution in object space is 100 #m in the spanwise and 161 #m in the depth direction. This measurement was carried out 10 mm downstream of the feed pipe. The successive time steps hardly show any motion of flow structures, giving an indication that the temporal resolution, which was estimated to be ].6~-a, is adequate to resolve the flow. Two typical small scale laminar flow structures from the data sets are shown in figure 5(a) and 5(b). Similar to figure 4, the laser sheet was scanned in the direction of the positive z-axes. Figure 5(a) therefore demonstrates that the spatial resolution in the depth direction is adequate to resolve the small scale of the flow, as the planes in the graph are oriented perpendicular to the laser sheet. Figure 5(b) is an example of a cylindrical stretched vortical structure. These structures play a key role in the micromixing model developed by Bakker (1996). 7

Conclusions and o u t l o o k

In this paper, the design and some preliminary results with a 4D-LIF setup for measuring 3D concentration fields are presented. It is shown that the small scale flow structures can be well resolved in both space and time. In the near future, 4D-LIF measurements will be systematically performed at several positions with respect to the injector. Besides, image processing routines will be developed to extract quantitative flow data from the measurements. The three dimensional character of the data allows to assess orientation dependent quantities such as the layers thickness of the small scale vortical structures and the growth of interfacial area of the mixing fluids. References Bakker, R. A. (1996). Micromixing in chemical reactors: Models, experiments and simulations. PhD thesis, Delft University of Technology. Baldyga, J. and Bourne, J. R. (1984). A fluid mechanical approach to turbulent mixing and chemical reaction part II Micromixing in the light of turbulence theory. Chem. Eng. Commun., 28:243-258. Bradshaw, P., editor (1976). Turbulence, volume 12 of Topics in applied physics. SpringersVerlag. Dahm, W. J. A., Southerland, K. B., and Buch, K. A. (1990). Four-dimensional laser induced fluorescene measurements of conserved scalar mixing in turbulent flows. Proc. 5th Int. Symp. on Appl. of Laser Techniques to fluid Mechanics, Lisabon. Hinze, J. O. (1959). Turbulence. McGraw-Hill, second edition. Hopkins, H. (1955). The frequence response of a defocused optical system. Proc. R. Soc. Inst. Set. A, 231:91-106. Maas, H. G., Stefanidis, A., and Gruen, A. (1994). From pixels to voxels: tracking volume elements in sequences of 3-D digital images. ISPRS Comm. III Intercongress Symposium, Munich. Paul, P. H., Van Cruyningen, I., Hanson, R. K., and Kychahoff, G. (1990). High resolution digital flowfield imaging of jets. Exp. in Fluids, 9:241-251.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

53 84

Simulation with Validation of Mixing Effects in Continuous and Fed-Batch Reactors Gary K. Patterson and Jeffrey Randick Chemical Engineering Department, University of Missouri-Rolla, Rolla, MO 65401, USA Abstract This paper examines by numerical simulation the effects of scale-up of stirred reactors on product yield for competitive reactions with various rate constant ratios, for two impeller types, and for continuous and fed-batch modes of operation. Many of the chemical reactions that we have simulated have been studied experimentally, so comparisons under identical reaction conditions were used to validate the numerical method. The simulations were done using FLUENT as the.transport computation engine with added subroutines to simulate mixing rates and chemical reaction rates as a function of the degree of mixing. The closure used for the chemical component mass balances was the Paired-Interaction (P-I) Model. The P-I model treats component segregation (lack of mixing) as a conserved quantity requiring a balance equation for its production, dissipation and transport. The study examines typical strategies for scaling-up the mixing power for stirred chemical reactors: constant power per unit volume and constant impeller tip speed. Most of the simulations were for continuous flow reactions since they require about one-tenth of the time to compute than simulations for fed-batch reactions. Some of the cases simulated are for fed-batch operation, which generally give higher yields than continuous flow operation for the same mixing rates, concentrations, etc., but directional effects for changes in size, feed location, and impeller rotation rate are usually the same for the two modes. 1. Basic Simulation Equations The basic equations for component mass (concentration) and segregation used in the modeling of a second-order reaction are as follows: Component Mass:

DC_

0

Dt

OXj

( v, _ ~ ) _ KR(C,Ck + c~ck) O'c O X j

Segregation (si = ci2): m

Ds~ Dt

E m 0 (v__z_,Os, ) v,(OC,) ~ _ ~2si - KR(C~* c~ck +Ck OXjas~Xj +C8' C~(/et)

9s~ + S~Ck)

54 The left-hand sides of these equations are the substantial derivatives describing time dependent and convective effects. The right-hand sides contain terms for turbulent diffusion and rates of decrease due to chemical reaction. The segregation equation contains the Spalding (1971) term for rate of segregation production and the Corrsin (1964) term for rate of segregation decay or mixing without the Schmidt number term, which is usually small. The complete form m

of the Corrsin term is [ 2 s J / [ C c ( k / 6 ) + ( v / 6 ) l / ~ l n ( N s c ) ] . T h e constant Cc was found to be 2.2 in order to obtain a best fit to the Vassilatos and Toor (1965) data. Corrsin predicted a value of Cc of about 4.1, but he used the term (Ls2/e)l/3 instead of (k/8). The term (k/t) is considered to be about twice the as large as (Ls2/e)1/3 (Pope,1985), which probably leads to the smaller constant. The value of 2.2 was then used in the simulations of mixing with reaction in the stirred tank reported in this paper. A value of 4.0, instead of the 2.7 recommended by Spalding (1971) and the 2.8 recommended later by Elghobashi, Pun and Spalding (1977), was used for Cgl because the larger value gave closer results for segregation in mixing without reaction. The Paired-Interaction closure for c~ck for the reaction terms in both equations is m

c~ck = "si * s k / C i

* Ck 9

The term s i c k in the segregation equation is sick --- Z E R O . T h e result for CiCk has been presented previously (Patterson, 1975 and 1985), which is derived from a three-spike probability distribution function: pure component-i, pure component-k, and an ik-mixture with the same ,,,

ratio as the final completely mixed ratio. The assumption of si ck equal to zero is based on the idea that since si is always positive, its correlation with Ck should be much smaller in magnitude than the correlation of ci with Ck. 2. Simulation Method and Mixing Vessels Simulated

The simulations were done by linking a new subroutine containing the segregation balance equation and the P-I closure into FLUENT version 4.51, which provided the hydrodynamic simulation of flows, turbulence, and chemical component balances. The vessel simulated was a stirred tank with liquid height (H) equal to the diameter (T) and a flat bottom. The impellers (6-blade disk turbine and 6-blade 45 ~ pitched blade turbine) were located 1/3-tank diameter (T/3) from the bottom and their diameters (D) were both 1/3 of the tank diameter. The simulation grid for the finite-volume method used is shown superimposed on the tank with the disk turbine in Figure 1. The grid is cylindrical and consists of 20 angularly-spaced r-z grids with 20 nodes in the r-direction (radial) and 26 nodes in the z-direction (vertical). The r-z nodes are spaced closer near the impeller and the angular spacing is closer near baffles. Some simulations were done with halved node spacing (doubled grid) in all three directions in order to test grid independence. There was little difference in the results.

55 The vessel sizes simulated were 0.785 liter (T=I 0 cm) and 6280 liters (T=2 m), giving a scale-up ratio of 8000. In all cases presented here except one the power per unit volume was maintained constant for comparisons of small and large vessels. Rather than using a sliding mesh method, in order to obtain faster computations, impeller flows were input via the FLUENT FIX facility, whereby velocities at chosen nodes may be fixed and not recalculated. For the disk turbine, only the impeller tip velocity was fixed at all nodes tangent to the sweep of the blade tips. For the pitched blade turbine, the downward (axial) and angular velocities from measurements by Fort, Votruba and Medek(1999) were fixed at the nodes swept by the bottoms of the impeller blades. In both cases turbulence levels within a few percent of experimental values resulted and velocity patterns were very close to experimental values.

Figure 1. Schematic of Simulated Vessel Showing the r-z Grid.

Figure 2. Reactant Concentration at Selected Points within the Vessel: Points 1-8 are Upper Recirculation; Points 9-18 are Impeller Stream; Points 19-26 are Effluent Region

3. Validations of the Simulation Method

Zipp and Patterson(1998) showed that the P-I closure gave simulated segregation and conversion results close to experimental values for two single chemical reactions: acid-base neutralization with a rate constant of approximately 1012 L/mol s and oxidation-reduction with a rate constant of 23,000 L/mol s. The results for the reactant concentrations in the oxidationreduction case are shown in Figure 2 above. Some experimental points are missing in the impeller stream because there was an optical path obstacle there. Since the measurements were by remote fluorescence measurements, those points could not be measured. The experimental work was done long before they were needed for validation of this simulation.

56 During some earlier work and during the present study, some of the experimental conditions used by Paul and Treybal (1971) in their fed-batch reactor (T=29 cm) were simulated. Comparisons of the final yields (Tyrosine-I produced/Tyrosine reacted) predicted by those simulations with the experimental results are shown in Figure 3. The parallel-consecutive reactions at 25 ~ C are as follows, where the KRi-Values are the rate constants: Tyrosine + I2 ") Tyrosine-I + HI ; KR1 = 35 L/mol s Tyrosine-I + I2 --) Tyrosine-I2 + HI ; KR2 = 3.8 L/mol s The correspondence between experiment and simulation was very good. In the Paul and Treybal reactor a large difference between top feed and impeller feed resulted because the vessel height was much greater than its diameter (H = 1.71 T).

Figure 3. Yield of Tyrosine-I in the Paul and Treybal Stirred Vessel- Comparisons of Simulations with Experimental Data. Xs are from Simulations using the P-I closure.

Figure 4. Comparison of the Baldyga, Bourne, and Hearn Data for Mixed Reaction in a Static Mixer with Results of the P-I closure shown as Xs on the graph.

Finally, simulations were done using the P-I closure for the reaction carried out in a twisted-ribbon static mixer by Baldyga, Bourne and Heam (1997). The reaction scheme was as follows, where the KRi-values are the rate constants: A + B --)p-R ; KRlp= 12238 L/mol s A + B - - ) o - R ; KRlo-" 921 L/mol s p-R + B --) S ; Kvap = 22.25 L/mol s

57

o-R + B --) S ; KR2o= 1.835 L/mol s AA + B --) Q ;KR3 =

125L/mols

The identification of each of these as real chemicals may be done with the Baldyga, Bourne and Heam paper. In order to simulate the hydrodynamics and mixing of the static mixer, the initial scale of segregation was set equal to one-quarter of the inside diameter of the mixer pipe, which is approximately the hydraulic diameter of the pipe with the twisted ribbon dividing it in half. The rate of turbulence energy dissipation was determined from the pressure gradient in the mixer, as did Baldyga, Bourne and Hearn, using the expression e = (fD/2~)(4/n3)(Qf 3/rD7), where d~,the fraction open area of the pipe, is 0.9 and fD, the effective friction factor, is 1.9. The inside diameter of the pipe, D, is 0.04 m. For example, at Qf = 1.0 dm3/s, e = 11.0 m2/s3. The Corrsin mixing term in the segregation balance equation using Ls is 2si/Ce(Ls2/e)l/3, where Cc is 4.1 as discussed above. The segregation of A from B was assumed to be complete where they were fed to the static mixer so no segregation production term was included. The calculations were done assuming plug flow, so the balance equations for mass and segregation were onedimensional. Since that resulted in a set of ordinary differential equations, POLYMATH, an ODE solver, was used. The simulation results, expressed as yield of Q from B reacted, matched the experimental results closely as shown in Figure 4, except at the lowest flow rate, 0.5 dm3/s. It must be emphasized that no parameters were adjusted to make the simulation results fit the experimental results in each of the cases described above. The values of the constants in the k-e formulation for the hydrodynamics were left at their standard values, and the Corrsin and Spalding constants for segregation decay and production were kept constant except where Ls was used instead of k for the mixing time constant in the static mixer simulations. Based on these considerations, the results seem to show reasonable validation of the closure method being used and warrant its extension to a study of industrial competitive reactions where yield is an issue. 4. Results of Simulations for an Isothermal CSTR with a Disk Turbine

Even for very low impeller rotation rates, most of the chemical reaction occurs close to the impeller where the mixing is most rapid. All the simulation results given in the rest of the paper use the following chemical reactions with rate constants KRi: A + B --)' C ; KR~ A + C --) D;KR2 A large number of simulations of the reaction scheme shown above were done for continuous flow operation. Feed location in all these cases was into the center of the impeller, which was a disk turbine, with one component above the disk and the other below. The simulations were done using various reaction rate constants and residence times in order to study the effects of those variables on yield when the reactor vessel is scaled up from 0.785 L to 6280 L. Table I shows comparisons of those results:

58 Table I. Results of Continuous Flow Simulations for the Large (6280 L) and Small (0.785 L) Vessels; Power/Volume = 0.0085 kW/m 3 ; Nsma. = 180 rpm; Nlarge= 24.4 rpm Size

KR1

KR2

small small Small -large large

18,0 36.0 180 3.6 18.0

18.0 36.0 180 3.6 18.0

.

.

.

.

tresidenee Teftluent

.

Yield of C

1000 100 . . . . . 1000 1000 1000

298 298 298 298 298

0.360

1000 1000 1000 1000 1000 1000

298 298 298 298 298 298

0.840 0.778 0.739 0.652 01630 0.575

.

0.2~3 0.338 0.348 0.266

.,,

small small small large large large

18.0 1.80 36.0 3.60 180 18.0 18.0 1.80 36.6 3.60 180 ......18.0

Two main observations may be made from Table I: For KR1 = Kru, a reduction in yield of C occurs when the rate constants increase and yield decreases for a reduction in residence time. At the same rate constants and power per unit volume, yield decreases significantly when the vessel volume is increased by a factor of 8000. For KRI > KR2, yields of C are higher than for equal rate constants, but the same trend, reducing yield, is seen when both rate constants are increased the same ratio. This is seen in cases for kl = 10 k2. Reductions in yield of C are again significant when the vessel is scaled up from 0.785 L to 6280 L.

Table II. Comparisons of Yields for Feed at Impeller Center and at Tank Top for Continuous Flow; Large = 6280 L; Small = 0.785 L; KRl = 36.0 L/mol s; K ~ = 3.6 L/mol s Size

Feed Loc.

N

Power/Vol.

tresidence Teffluent Yield of C

small small large large large large

impeller top impeller top impeller top

180 180 24.4 24.4 9.0 9.0

0.0085 kW/m 3 0.0085 0.0085 0.0085 0.00043 ..0.00043

1000 1000 1000 1000 1000 1000

. _

298 298 298 298 298 298

0.778 0.780 0.594 0.474

0.600 0.486

....

59 Table II shows some comp~isons of yields for feed at the tank top and into the impeller center for the disk turbine at various impeller rotation rates for the 0.785 L tank and the 6820 L tank. The cases for 24.4 rpm in the large tank are at the same power per unit volume as the small tank at 180 rpm. At 9 rpm the large tank impeller is at the same blade tip speed as for the small tank. A significant reduction in yield occurs for feed at the top in the large tank. 5. Results of Simulation of Fed-Batch Reactors

Only a few fed-batch simulations have been done so far because they are very time and computer consuming. Time dependent solutions of the hydrodynamic, mass and segregation balance equations requires about the same time for each major time step where a full data set is stored as for one continuous flow (steady-state) solution. Because of that most of the comparative studies for effects of vessel size, reaction rate, and heat effects have been done in the CSTR. The fed-batch reactor should produce greater yield for a given reaction system than the continuous flow reactor. This is because the reactant that produces the secondary product can be fed at a low enough rate to keep its concentration at a very low value relative to the other reactant which is resident in the reactor. Therefore, much less of the secondary product can be produced. This did not seem to happen with the reaction rates simulated here, as seen in Table III. Perhaps they are too low for the concentration effect tp be evident. Table III shows comparisons of the yield of component C for the 0.785 L vessel and the 3785 L vessel, all at approximately the same power per unit volume. In contrast with the CSTR there is very little effect on yield with a scale-up from 0.785 L to 3785 L at the same power per unit volume. Also, even more surprising, there is very little difference in yield between the disk turbine and the pitched-blade turbine at the same impeller rotation rates even though the power per unit volume is different. Table III. Yield of C in a Fed-Batch Reactor, Isothermal Case; Power per Unit Volume for the Disk Turbine = 0.091 kW/m3; for the Pitched Blade Turbine = 0.025 kW/m 3 (60 rpm for the large tank; 397 rpm for the small tank for both impellers)

.

size/Impeller

KR1

Kp,2

Large/Disk Small/D[sk Large/Pitched Small/Pitched

36 36 36 36

3.6 3.6 3.6 3.6

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

A-Feed A-Feed Time, s Conc.,mol/L 50 1.0 ~0 . 1.0 50 1.0 50 1.0

B-Tank Yield of-C Conc.,mol/L .... 0.1 0.697 0.1 0,703 0.1 0.675

0.1

0.702

6. Conclusion

Simulation of stirred chemical reactors has progressed to the point that application of the method to industrial problems should yield good results. The Paired-Interaction Closure has been validated in several ways and seems to give results close to experimental in all tests so far. Incorporation of the P-I closure into the FLUENT code for computational fluid mechanics makes

60 its use convenient for almost any type of chemical reactor where rate of mixing of the reactants has an effect on the outcome. Studies of scale-up using the simulation method show how much various operating parameters of the reactor affect the yield of complex reactions. In these studies the effect of chemical kinetics, activation energy and heat of reaction, impeller type, and power per unit volume were determined by various comparisons for continuous flow operation. Fedbatch simulations show only a weak effect of scale-up on the yield. Effects of other variables in fed-batch reactors are similar in magnitude to those in continuous flow reactors. References

Baldyga, J., J. R. Bourne, and S. J. Hearn, "Interaction between Chemical Reactions and Mixing on Various Scales", Chem. Eng. Sci., 52, 457-466 (1997). Corrsin, S., "The Isotropic Turbulent Mixer: Part II. Arbitrary Schmidt Number", AIChE J., 10, 870-877 (1964). Elghobashi, S. E., W. W. Pun, and D. B. Spalding, "Concentration Fluctuations in Isothermal, Turbulent Confined Jets", Chem. Eng. Sci., 32, 161 - 166 (1977). Fort, I., Votruba, P., Medek, J., "Turbulent Flow of Liquid in Mechanically Agitated Closed Vessel", Ms. from I. Fort, 1999. Patterson, G. K.,"Simulating Turbulent Field Mixers and Reactors", a chapter in Application of Turbulence Theory to Mixing Operations, ed. By R. S. Brodkey, Academic Press, New York (1975). G. K. Patterson, "Modeling of Turbulent Reactors", in Mixing of Liquids by Mechanical Agitation, ed. By J. J. Ulbrecht and G. K. Patterson, Gordon and Breach (1985). Paul, E. L., and R. E. Treybal, "Mixing and Product Distribution for a Liquid-Phase, SecondOrder, Competitive-Consecutive Reaction", AIChE J.,17, 718-731 (1971). Pope, S. B.,"PDF Methods for Turbulent Reactive Flows", Prog. Energy Combust. Sci., 11, 119192(1985). Spalding, D. B., "Concentration Fluctuations in a Round Turbulent Free Jet", Chem. Eng. Sci., 26,95-108(1971). Vassilatos, G., and H. L. Toor, "Second-Order Chemical Reactions in a Nonhomogeneous Turbulent Field", AIChE J., 11,666-672 (1965). Zipp, R. P., and G. K. Patterson, "Experimental Measurements and Simulation of Mixing and Chemical Reaction in a Stirred Tank", Can. J. Chem. Eng., 76, 657-669(1998).

I 0th European Conference on Mixing H.E.A. van den Akker and J.d'. Derksen (editors) 2000 Elsevier Science B. E

61

A Computational and Experimental Study of Mixing and Chemical Reaction in a Stirred Tank Reactor Equipped with a Down-pumping Hydrofoil Impeller using a Micro-Mixing-Based CFD Model. Otute Aldti and Piero M. Armenante ~ Department of Chemical Engineering, Chemistry and Environmental Science New Jersey Institute of Technology, University Heights, Newark NJ 07102-1982, USA

Abstract. In this work the following fast parallel competing reactions scheme (Bourne and Yu, Ind. Eng. Chem. Res. 33, 41-55, 1994): A+B~P+R A+C~Q+S was modeled in a stirred tank reactor using CFD, and the results compared with original experimental data. The experimental system comprised a cylindrical stirred tank reactor fitted with an axial down-pumping hydrofoil impeller (Chemineer HE-3) operated in semi-batch mode, with the limiting reagent being slowly added to the contents of the reactor. The final yield, Xs, of the undesired product S was experimentally measured. The flow field in the reactor was sinmlated using the Reynolds Stress turbulence model. The full impeller geometry was incorporated in the CFD simulation using the Multiple Reference Frames (MRF) model. The reaction zone was modeled in a Lagrangian way using a multi-phase model (Volume of Fluid (VOF) model). The interaction of turbulence and reaction was accounted for by means of the engulfment model for micro-mixing (Baldyga and Bourne, Chem. Eng. J., 42, 83-92, 1989). The agreement between previously published experimental velocity distribution data (Jaworski, Nienow and Dyster, Can. J. Chem. Eng., 74, 3-15, 1996) and the results of the simulations was generally good. The micro-mixing model, in conjunction with CFD, predicted a final value of Xs in close agreement with the experimental data, demonstrating that the proposed approach can be successfully used to model turbulent reactive systems without the need for experimental inputs.

1

Introduction

In many industrial synthesis reactions, the kinetics of product formation can be quite complex and involve a number of intermediate steps as well as fast parallel reactions. Some of these reactions often result in the formation of undesired by-product(s) that consume the reactants, thus reducing the overall conversion efficiency of the process. The formation of these undesired byproducts typically also requires additional purification steps to produce a final product that meets the required specifications. When applied to such systems, traditional computational fluid dynamics (CFD) approaches based on solving a turbulence model coupled with mass balance equations for each reacting species often prove inadequate because of their inability to resolve the microscales of the concentration field. Alternative methods are needed. I Corresponding author. E-mail: [email protected].

62 In this work, a novel means of simulating mixing and chemical reaction in stirred tank reactors is presented. The method uses CFD to obtain macroscopic information about the turbulent flow field, and then links this information to a mixing sensitive reaction model to simulate mixing and chemical reaction. The model is general, easy to implement, and requires no experimental input.

2

Experimental System and Method

2.1

Reaction System

The parallel competing reaction set studied is: NaOH + HC1 (,1) (B) NaOH +CH2CICOzC2H5 (A) (C)

>NaC1 + H20 (?) (R) k2

>CH2C1CO2Na +CzHsOH (Q) (s)

The rate constant kl and k2 are, respectively, 1.3-108 and 0.0257 m3mol'I/s at 23~ (Bourne and Yu, 1991). The final product distribution is sensitive to hydrodynamic effects (Baldyga and Bourne, 1989b; Bourne and Yu, 1991; Bourne and Yu, 1994), making this reaction set suitable for studying the effect of turbulence on chemical reactions. Of interest is the yield of S from A. When mixing is perfect, no segregation exists, and the yield of S, Xs, is given by: Xs =

k2 Cc k2 Cc +kl CB

(1)

When mixing is perfect, Xs is essentially zero for the above system of reactions because kl>>k2. When the segregation is intense (i.e., the reactions take place independent of each other), the yield of S is not a function of the kinetics and is given by: Xs =

Cs = C~---------2-C e +C s C~ +C c

(2)

In such as case, if equal quantities of A, B and C are reacted (as in this work), Xs=0.5. Intermediate degrees of mixing intensity yield results between these two extremes. In such cases a micromixing model is required to describe the system.

2.2

Experimental Apparatus and Procedure

The apparatus used for the fast competitive parallel reactions is shown in Figure 1. The reactor was first charged with two of the reagents, HC1 and CHzCH3CO2C1, both with an initial concentration of 18 mol/m 3. The limiting reagent, NaOH (feed concentration=900 mol/m3; CUlrmlative feed volume=l/50 of reactor volume) was then fed to the reactor with a variable speed pump over a period of 60 minutes. The yield Xs was experimentally determined by measuring the concentration of ethanol, Cs, in the final product mixture using gas chromatography. The concentration of ethyl-chloroacetate, Cc, was also measured to verify the mass balance. Experiments were conducted using two feed points (Fs and Fi; Figure 1), at four different impeller speeds (150, 200, 300 and 400 rpm).

63

Figure 1: Experimental Apparatus. 3 N u m e r i c a l Simulation of the Flow Field Numerical simulation of the flow field was obtained using a commercially available CFD package (FLUENT v. 4.5.1). The full tank geometry (360 ~ was incorporated into the simulation. The computational domain consisted of 362,810 cells built from a 146 x 35 x 71 grid generated using the MIXSIM v. 1.5 preprocessor (which is part of the FLUENT software). The turbulence model used was the Reynolds Stress Model (Rodi, 1984; Fluent, 1994). Pressure coupling was achieved using the PISO algorithm (Fluent, 1994). The impeller geometry was incorporated into the simulation by means of the Multiple Reference Frames (MRF) model (Luo et al., 1994; Luo et al., 1993).

3.1 Numerical Simulation of Mixing and Chemical Reaction The approach taken in this work to resolve the interaction of mixing and kinetics is through the use of a micromixing mode namely, the Engulfment Model (E Model) developed by Baldyga and Bourne (Baldyga and Bourne, 1989a and 1989b). Accordingly, the general equation for species i undergoing engulfment is as follows: dt

i

where E is the engulfment parameter given by (Baldyga and Bourne, 1989a):

64

E = ln___22= 0.058

(4)

"/'v

r,, is the lifetime of a vortex. The engulfment parameter depends on the state of turbulence through the local energy dissipation rate per unit liquid mass, ~. 3.2 Interaction between Micro- and Macro-Mixing Visualization studies have shown that for a system of the type described in this work, the reaction zone moves away from the feed point while the reaction takes place as a result of bulk motion and macromixing (Bourne et al., 1995). Consequently, the moving reaction zone experiences varying levels of turbulent intensity, which, in turn, affect the reactions (Eq. 3 and 4). Here, a multiphase model was used to represent the reacting fluid element as a separate and distinct phase. The Volume of Fluid (VOF) model, a multi-phase model designed for two or more fluids (Hirt and Nichols, 1981), was chosen because of its simplicity and modest computational requirements. 3.3

Use of VOF Model and E-Model to Account for Turbulence-Chemistry Interaction

The location of the reaction zone was tracked with the VOF model. Whereas in the E-model the reaction volume grows because of fluid engulfment, in the present model it was assumed that no volume growth occurred. The VOF model was used to determine the location of the cells in which the added fluid volume dispersed, in order to capture the change in turbulence intensity with time at each cell location. At each position in time, the value of the specific rate of dissipation of turbulent kinetic energy, s, which is needed to compute the engulfment parameter, E, was obtained by calculating the volume average of z over all the cells occupied by the reacting fluid. Thus at any position in time, such a volume average value g was evaluated as follows:

I ,z dV ~- ~ Ie -

=~

c dV

(5)

For computational purposes, the feed solution was discretized into cr equal parts, with each part being fed into the system in sequence, thereby simulating slow feed addition. For each cr addition, the E-Model equations were integrated in the reaction zone (which was treated as a completely segregated zone while the surrounding fluid was completely mixed) until the limiting reagent was consumed. Then, the concentration of all the species in the entire tank was updated via a mass balance, and the process repeated until all the feed additions had been made. 4

Results

4.1 Flow Field Simulation The results of the CFD simulation of the flow field in the reactor were compared with experimental data from the literature (Jaworski et al., 1996), appropriately scaled to match the agitation speeds used in this work. Only data obtained at 200 rpm are shown here (Figure 2). The simulation and experimental data are in reasonable agreement.

Figure 2" Comparison between CFD flow field simulation (this work) and experimental velocity and turbulent kinetic energy measurements (after

t3~

66 0.35 0.30 0.25 0.20 0.15

0.10

0.05

0.00 0

i

i

i

20

40

60

. . . .

I

80

i

1O0

"i

120

40

Feed Time (minutes)

Figure 3: Variation of Xs with Feed Time (N=150 rpm; feed location near the surface, Fs) 4.2

Simulation of Parallel Competing Reaction Process

4.2.1 Influence of Feed Time on Xs The feed time used in the experimental work must exceed a critical feed time (tom) in order to ensure that the experiments are carried out in a micromixing-controlled regime (Baldyga and Bourne, 1989a and 1989b). tom should be determined for each set of experimental conditions. This is not realistic. A solution to this problem is to experimentally determine tcrit for the worst possible set of experimental conditions, i.e., lowest agitation speed, highest initial concentration for CA, and feed location near the surface (Bourne and Yu, 1991), and then use this value as t~m for all the other less demanding experimental conditions. Figure 3 shows a plot of the experimentally determined Xs as a function of feed time (150 rpm; feed location near the surface, Fs). For feed times longer than -60 minutes, Xs was found to be independent of feed time. Thus tcm was taken to be 60 minutes. This was the feed time used in all experiments.

4.2.2

Influence of Agitation Rate on Xs

Figures 4 and 5 show a comparison between the experimentally derived and the model predicted values of Xs as a function of the agitation speed, N, for two different feed locations. The agreement between experiments and predictions is satisfactory for both eases and for all N's. Better predictions were obtained when the simulations were conducted with the feed location in the impeller region (Figure 5). X~ was found to decrease with increasing N values, as also reported in previous studies (Bourne and Yu, 1994; Bakker and van den Akker, 1994; Bakker and van den Akker, 1996). When the feed was located in the highly turbulent region in the impeller suction stream (Fi) as opposed to near the liquid surface (Fs), Xs was generally lower, at constant N (Figures 4 and 5).

67

0.35

0.30

9

0.25

Experimental Predicted

0.20

0.15

0.10

0.05 .

0.00

0

,

~00

.

.

.

,

200

.

!

aoo

.

.|

400

soo

N (rpm) Figure 4" Variation of Xs with N (feed location: Fs). 5 Conclusions The results of this work indicate that a model based on CFD simulation of the macroflow, coupled with a suitable micromixing model through the use of the VOF model can successfully predict the product distribution of reactions exhibiting complex chemistry in a fed-batch system. This formulation has distinct advantages over other methods presented in the literature: * by incorporating the full geometry of the impeller into the simulation, there is no need for experimentally-derived velocity boundary conditions for the impeller region. Novel impeller designs may readily be investigated with this approach; , turbulence modeling is improved by using the MRF model, which captures the major 0.35

0.30

9 0.25

Experimental Predicted

0.20

0.15

0.10

0.05 ,.

0.00 0

100

200

300

400

N (rpm) Figure 5: Variation of Xs with N (feed location: Fi).

500

68 unsteady properties of the flow field into the simulation; by using the MRF model to simulate the flow field, it is possible to carry out simulations that involve all parts of the reactor, including the impeller region. This is not possible when impeller boundary conditions are used since the impeller region is not properly modeled; 9 because the reaction zone is modeled as a separate phase, properties such as density, viscosity, and surface tension may be readily incorporated into the model for multi-phase systems simply by defining them for the phase of interest. This work highlights the need to use different modeling tools in conjunction with CFD as a means of predicting the performance of complex turbulent reacting systems. 9

Acknowledgment. This work was partially supported by a grant from the Emission Reduction Research Center sponsored by the Bristol-Myers Squibb Pharmaceutical Research Institute (thanks to Dr. San Kiang) and Schering-Plough Corporation (thanks to Perry Lagonikos). Their contribution is gratefully acknowledged. We would like to thank Dr. Liz Marshall (Fluent, Inc.) and Chemineer, Inc. for providing the HE-3 impeller geometry definition used in the simulation.

References Bakker, R. A. and H. E. A. van den Akker. A computational study of chemical reactors on the basis of micromixing models, Trans. IChemE 72, 733-738 (1994). Bakker, R. A. and H. E. A. van den Akker. A lagrangian description of micromixing in a stirred tank reactor using l d-micromixing models in a CFD flow field. Chem. Eng. Sci. 51(11), 2643-2648 (1996). Baldyga, J. and J. R. Bourne. Simplification of micromixing calculations I: Derivation and application of new model. Chem. Eng. J. 42, 83-92 (1989a). Baldyga, J. and J. R. Bourne. Simplification of micromixing calculations II: New applications. Chem. Eng. J. 42, 93-101 (1989b) Bourne, J., R. V. Gholap, and V. B. Rewatkar. The influence of viscosity on the product distribution of fast parallel reactions, Chem. Eng. J. 58, 15-20 (1995). Bourne, J. R. and S. Yu. An experimental study of micromixing using two parallel reactions, 7th Europ. Conf. on Mixing, Volume I, Brugge, Belgium, pp. 67-75 (1991). Boume, J. R. and S. Yu. Investigation of micromixing in stirred tank reactors using parallel reactions, Ind. Eng. Chem. Res. 33(1), 41-55 (1994). Fluent Inc. Fluent v4.3 Manual, Fluent Inc, Lebanon, New Hampshire (1994). Hirt, C. W. and B. D. Nichols. Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39, 201-225 (1981). Jaworski, J., A. W. Nienow, and K. N. Dyster. An LDA study of the turbulent flow field in a baffled vessel agitated by an axial, down-pumping hydrofoil impeller. Can. J. Chem. Eng. 74, 3-15 (1996). Luo, J., A. Gosman, R. Issa, J. Middleton, and M. Fitzgerald. Full flow field computation of mixing in baffled stirred vessels, Trans. IChemE 71(Part A), 342-344 (1993). Luo, J. Y., R. I. Issa, and A. D. Gosman. Prediction of impeller induced flows in mixing vessels using multiple frames of reference, Inst. Chem. Eng. Syrup. Ser. No.136, pp. 549-556 (1994). Rodi W. Turbulent models and their application in hydraulics--A state of art review, 2 nd Edition, International Association for Hydraulic Research, Delft, The Netherlands (1984).

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

MIXING WITH A PFAUDLER TYPE IMPELLER; THE EFFECT OF MICROMIXING ON REACTION SELECTIVITY THE PRODUCTION OF FINE CHEMICALS

69

IN

Iris Verschuren, Johan Wijers and Jos Keurentjes Eindhoven University of Technology, Department of Chemical Engineering and Chemistry, Process Development Group, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ABSTRACT Stirred tank reactors with a Pfaudler type impeller are frequently used in the production of fine chemicals, but information on the mixing performance of this type of impeller is limited. This information is required to predict selectivities of mixing sensitive processes. For low feed rates, which are often used in the production of fine chemicals, the mixing process is controlled by micromixing. To investigate the micromixing in stirred tank reactors with a Pfaudler type impeller the product distributions of mixing sensitive reaction sets are determined. These experiments show that in a large part of the reactor the product distribution is not a function of feed point position. With a micromixing model (E-model) using the average energy dissipation rate, the product distribution is calculated. The calculated product distributions are in reasonable agreement with the measured product distributions for partially baffled reactors with a Pfaudler type impeller for a broad range of process conditions. 1. INTRODUCTION Before a chemical reaction can take place between two or more reactants, the reactants have to be mixed on a molecular scale. When reaction is slow compared to the mixing process, the solution will be homogeneously mixed before reaction takes place and the product distribution will only depend on the chemical kinetics. However, when the time scale for reaction is of the same order of magnitude as the time scale for mixing, the selectivity of the process for competitive reactions will depend on the mixing rate. An example of a mixing sensitive process is the addition of an acid or base to a solution of an organic substrate, which degrades in the presence of a high or low pH. Slow mixing will limit the neutralisation reaction and allow the organic substrates to react in the presence of an acid or base, thus producing unwanted by-products [1 ]. Stirred tank reactors with a Pfaudler type impeller are frequently used in the chemical industry for the production of fine chemicals or pharmaceuticals. The shape of a Pfaudler type impeller (also called retreat curve impeller) permits a glass lining to be applied. Therefore, this type of impeller is particularly useful in environments where corrosive substances are present. Although the Pfaudler type impeller is widely used in the chemical industry, information on the mixing performance of this type of impeller is limited.

70 The mixing in a stirred vessel consists of three processes: macromixing, mesomixing and micromixing. Macromixing is the convection of fluid by the average velocity. Mesomixing refers to the turbulent dispersion of a feed stream by large-scale turbulent motions. Micromixing is mixing inside small-scale turbulent motions by engulfment, deformation and diffusion. In the production of fine chemicals usually low feed rates are used to prevent a thermal runaway and to control the product distribution. For sufficiently low feed rates the product distribution is controlled by micromixing [2]. In this study micromixing in a stirred tank reactor with a Pfaudler type impeller is investigated and is described with a micromixing model. 2. MICROMIXING MODEL

Micromixing by diffusion within shrinking laminated structures formed by engulfment is described by the EDD (engulfment, deformation and diffusion) model [3]. Baldyga and Bourne have shown that for systems having a Schmidt number less than 4000, engulfment is the rate-determining step of the micromixing process and the EDD-model is simplified to the engulfment model (E-model) [4]. The growth of the micromixed volume according to the Emodel is described by: dVmi dt

=

EVmi

E -- 0.058~/-~

(1)

where E is the engulfment rate, o is the kinematic viscosity and ~ is the local energy dissipation rate. The E-model is used to calculate the selectivity of a chemical process in a semibatch reactor in which a reagent is slowly added to a reagent already present in the vessel [4]. A fluid element added to the reactor will grow according to equation 1. The mass balance for a component i in this fluid element is: dc----2-=E(< Ci > - - C i ) + R i dt

(2)

in which Ci is the concentration of i in the micro mixed volume, is the concentration of i in the surrounding fluid near the micro-mixed volume and Ri is the specific reaction rate. No macroscopic concentration gradients are included in the model. In the model the total feed volume is divided into c=l 0 equal parts. Increasing the number of feed parts will result in just a minor increase in the amount of ethanol. After the reagent present in a part of the feed has reacted, the reactor content is homogenised. 3. EXPERIMENTS

The stirred tank reactor used is shown schematically in figure 1. The internal tank diameter was 0.2 m. The impeller was a glass-lined Pfaudler type impeller. Details of the impeller and the reactor are given in table 1. The tank was partially baffled with a baffle and a feed pipe. Six different feed point locations were used. The feed point locations are given in figure 2 and table 2. The internal feed pipe diameter was 5 mm.

71

l

| l m < T!

....

I -:

. . . . . . . .

1.... r[ ...... 1

-"-> S ~

I /

H

.

II

l~

w, X

Fig. 1. Geometry of the vessel

Fig. 2. Top view of the vessel with feed point locations.

Table 1 Reactor and impeller dimensions and co-ordinates illustrated in figure 1 T=H D H w ot s 1 0.2m 0.120m 0.018 m 0.035 m 10 ~ 0.022m 0.036m Table 2 Co-ordinates feed point a b c d

of the feed R [m] 0.50 0.29 0.50 0.78

points illustrated in figure 2. 13 x/T 0.44 90 ~ 0.19 90 ~ 0.19 45 ~ 0.19 29 ~ 0.19

0.62

The third Bourne reaction [5] was used to investigate the micromixing: NaOH + HC1 ~ NaCI + H20 kl>l 0 s m 3 mol l s1 at 298 K NaOH + CH2CICOOC2Hs ~ CHzCICOONa + C/HsOH kz=0.031 m 3 mol q sq at 298 K For each experiment the vessel was filled with a solution of 0.09 M ethyl chloroacetate (ECA) and 0.09 M hydrochloric acid (HC1). The feed stream was a solution of 1.8 M sodium hydroxide (NaOH). The feed volume was 1/20 of the initial volume of 5.74.10 -3 m 3 present in the vessel. When the mixing is fast, all NaOH will be neutralised by HC1 and no ethanol will be produced. More ethanol will be produced at lower mixing rates. The product distribution was measured for various feed point locations, feed times and stirrer speeds. The average turbulent energy dissipation rate ( g ) was estimated with:

=

PoN3D s Vvessel

(3)

72 in which Vvessei is the liquid volume in the vessel, N is stirrer speed, D is impeller diameter and Po is the power number of the impeller. The power number of an impeller is defined as" P Po = ~ pN3D 5

(4)

with p is density and P is the total power input. The total power input was determined from torque measurements. 4. RESULTS AND DISCUSSION In figure 4 the measured power numbers in the partially baffled reactor and power numbers provided by the Pfaudler company for a fully baffled and unbaffled reactor are given as a function of Reynolds number. For the partially baffled reactor and Reynolds numbers between the 1.4.104 and 105 the power number is 0.64. For higher Reynolds numbers the power number decreases with increasing Reynolds number. The decrease in the power number is caused by vortex formation at high stirrer speeds as it can be observed visually [7].

10 ------data Pfaudler baffled data Pfaudler unbaffied measurements o

1

IP II o~ o

t

1,0E+01

i

1,0E+02

i

1,0E+03 Re

. . . .

t

1,0E+04

,

1,0E+05

1,0E+06

Fig. 4. Power numbers measured in a partially baffled reactor and power numbers provided by the Pfaudler company for a fully baffled and unbaffied reactor as function of Reynolds number. In figure 5 the product distribution, defined as the amount of ethanol at the end of an experiment divided by the amount of ECA present at the beginning of an experiment (XEtoH), is given as a function of stirrer speed for three different heights of the feed point above the impeller. The feed point at the highest location above the impeller yields a larger amount of ethanol compared to the lower feed points. Nevertheless, the differences between the amount of ethanol formed for the different feed point locations are relatively small.

73 0,3 0,25

9

x x/T=0,19 o x/T=0,44 9 , x/T=0,62 -. E-model

.

0,2 0,15 0,1 0,05

,,,

0

2

4

6

8

N [Hz] Fig. 5. Ethanol yield as a function of stirrer speed for feed points at different heights above the impeller. Curve is calculated ethanol yield with the E-model. In figure 6 the product distribution (XEtoH)is given as a function of stirrer speed for four different feed point locations all at a height of 0.19 times the vessel diameter above the upper edge of the impeller. From these results it follows that no significant differences between the amounts of ethanol obtained for the different feed point locations can be observed.

0,25

oa

0,2

ob

ac

xd

0,15 0,1 0,05

0

i

i

s

2

4

6

N [Hz]

8

Fig. 6. Ethanol yield as a function of stirrer speed for different feed point locations. Seen the small variation in the product distribution with feed point position in a large part of the reactor, it is plausible to assume a homogeneous distribution of the energy dissipation rate. Therefore, the average energy dissipation rate obtained from the experimentally determined power numbers is used to calculate the product distribution with the E-model described in paragraph 2. In figure 5 the calculated product distributions are plotted together with the results from the experiments as a function of stirrer speed. From figure 5 it can be concluded that a reasonable agreement is obtained between the model and the experiments. However, the model overestimates the effect of the stirrer speed on the ethanol yield.

74 To verify the E-model for a broader range of experimental conditions, measurements of micromixing in a commercial scale reactor with a Pfaudler impeller [8] are used. In this study the first Bourne reaction [9] was used. The reaction mechanism, along with the second order kinetic constants [6], is given below. A + B --->p-R kip = 12238 m 3 mol l s"l at 298 K and pH = 9.9 A + B --->o-R klo = 921 m 3 mol 1 s "1 at 298 K and pH = 9.9 p-R + B --->S k2o = 1.835 m 3 mol "1 s "1 at 298 K and pH = 9.9 o-R + B --->S k2p = 22.25 m 3 mol "1 s"1 at 298 K and pH = 9.9 where A is 1-naphtol, B is diazotized sulfanilic acid, o-R is ortho monoazo dye from 1naphtol, p-R is para monoazo dye from 1-naphtol and S is bisazo dye from 1-naphtol. For this purpose, a tank with a working volume of 0.63 m 3 was used. The product distribution was measured for a partially baffled reactor and an unbaffled reactor, respectively. In table 3 the measured product distributions in the commercial scale reactor are given. Table 3 Product distribution of the 1e bourne reaction measured in a commercial scale reactor [8] and calculated with the E-model. position above the impeller [x/T] E-model position feed pipe e [m2/s3] tank base 0.025 0.175 0.425 0.715 baffled: % mono unbaffled: % mono

0.07 0.05

65 57

97 94

97.5

96.5 93

92 91

97.2 96.7

No significant variation in the product distribution with feed point position is observed for the feed point positions just above the impeller and below a height of 0.42 times the vessel diameter above the upper edge of the impeller. To calculate the product distribution with the E-model, average energy dissipation rates are used as given in [8]. The calculated product distributions are given in table 3. The agreement between the model and the experiments is reasonable for the partially baffled reactor with feed points located between the upper edge of the impeller and 0.42 times the vessel diameter above the upper edge of the impeller. The deviation between the model and the experiments is larger for the unbaffled reactor. In this study a reasonable agreement is obtained between measured product distributions in a partially baffled reactor for a broad range of process conditions and calculated product distributions with the E-model, using an average energy dissipation rate. The scale-up criterion for constant product distribution, which follows from the E-model, is a constant energy dissipation rate in the reaction zone [10]. This indicates that a constant energy input per unit volume will be a reasonable scale-up rule for a partially baffled stirred tank reactor with a Pfaudler type impeller. 5. CONCLUSIONS In reactors stirred with a Pfaudler type impeller and a feed point between the upper edge of the impeller and 0.6 times the vessel diameter above the impeller the variation in the product distribution with feed point position is small. The E-model, in which an average energy dissipation rate is used, proved to be well suited to predict the product distribution of mixing sensitive reaction sets for a broad range of process conditions in these reactors. Therefore, to obtain a constant product quality, a constant energy input per unit volume will be a reasonable scale-up rule for a partially baffled stirred tank reactor with a Pfaudler type impeller.

75 REFERENCES

1. E.L. Paul, J. Mahadevan, J. Foster, M. Kennedy and M. Midler, Chem. Eng. Sci., 47 (1992) 2837-2840 2. J.R. Bourne and C.P. Hilber, Trans. I. Chem. E., 68 (1990) 51-56 3. J. Baldyga and J.R. Bourne, Chem. Eng. Commun., 28 (1984) 243 4. J. Baldyga and J.R. Bourne, Chem. Eng. J., 42 (1989) 83-92 5. J.R. Bourne and S. Yu, Ind. Eng. Chem. Res., 33 (1994) 41-55 6. J. Baldyga and J.R. Bourne, Turbulent mixing and chemical reactions. John Wiley, Chichester (1999) 7. J.H. Rushton, E.W. Costich and H.J. Everett, Chem. Eng. Prog., 46 (1950) 395 8. W. Angst, J.R. Bourne and F. Kozicki, Proc. third Europ. Conf. on Mixing, Paper A4, York, U.K., (April, 1979) 9. J.R. Bourne, F. Kozicki and P. Rys, Chem. Eng. Sci., 36 (1981) 1643-1648 10. J.R. Bourne and P. Dell'Ava, Chem. Eng. Res. Des., 65 (1987), 180

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I 0th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

77

Comparison of different modelling approaches to turbulent precipitation D. L. Marchisio at, A.A. Barresi", G. Baldi a, and R.O. Foxb aDip. Scienza dei Materiali e Ingegneria Chimica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy bChemical Engineering Dept., Iowa State University, 2114 Sweeney Hall, Ames, IA 50011, USA The aim of this work is to evaluate the predictive ability of three different modelling approaches for turbulent precipitation reactors. The first approach is the study of the reactor performance solving the population balance equation for a plug flow with axial dispersion. In this model the degree of segregation is controlled by the parameter Pe, which describes the mixing in the age domain. The second model is a presumed beta-PDF model applied to a simplified hydrodynamic field, while the last one is a presumed finite-mode PDF, coupled with CFD (FLUENT). The three different approaches are compared with experimental data from a barium sulphate precipitation study using a Couette reactor. 1. INTRODUCTION The role of mixing at various scales in precipitation is a well-studied problem but contradictory results were found and different interpretations have been proposed. A good model is useful for understanding the role of the different phenomena involved and necessary for the design the scale up of precipitator reactors. The models used to study precipitation are complicated due to the coupling of the micromixing term with the population balance, which is necessary in order to consider both nucleation and growth of crystals. The oldest approaches used phenomenological models (i.e., E-model, IEM-model, EDD-model). However in the recent years, research on turbulent reactive flows has focused on the probability density function approach, developed in the study of the gas phase combustion, and recently successfully applied to liquid reactive systems (Pipino and Fox, 1994, Pipino e t al., 1994). The aim of this work is to evaluate the ability of different models to predict the precipitation reactor performance, by comparison with experimental data. The experimental data used in this work are obtained from two different Couette reactors, the first one working in continuous mode, and the second one in semi-batch conditions. The Couette reactor is made of two coaxial cylinders with the inner one rotating. The fluid is contained in the annular gap between the two cylinders, and depending on the operative conditions several fluid dynamical regimes can be t Correspondingauthor: e-mail: [email protected];fax: +390115644699. The researchhas been financially supported by a National researchproject (MURST40% - Multiphase reactors: hydrodynamicanalysisand solid-liquid analysis).

78 achieved (Kataoka, 1986). For high Taylor number (turbulent vortex flow and fully turbulent regimes) the system is sufficiently mixed in the radial and azimutal direction to be treated as a plug flow with axial dispersion. Several authors have proposed relationships between the overall Peclet number (Pe) and the operating conditions (Re, Rez). In this work the correlation obtained on the same apparatus in a previous work (Marchisio et al., 1998) will be used. Further details of the experimental set-up are given in Barresi et al. (1999). 2. THE PLUG FLOW WITH AXIAL DISPERSION MODEL A first approach to the problem could be the solution of the population balance using the plug flow with axial dispersion model. For this system the population balance equation is as follows:

On ~L(Gn)=D=-~i 02n

Vx-~ +

(1)

with the following boundary conditions: On[

(2)

._ ~ --0 Ir

(3)

L

where vx is the axial velocity, L the crystal size (the length of the longest size is considered), n the numerical crystal size distribution and D~x the axial dispersion coefficient, evaluated from: Pe -~ = 0.0059Re~

RedO.49

(4)

Note that the parameter Dax imposes the degree of mixing in the age domain and implicitly imposes the degree of segregation. The correlation between the Pe number and the degree of segregation implicitly assumed for the case of premixed feed has been discussed by Vatistas (1991). The equation can be solved in term of the moments of the numerical crystal size distribution, simply multiplying by UdL, and integrating from L=0 to L=oo (Pdvera and Randpolph, 1978). The solution of the population balance in terms of the moments leads to a set of ordinary differential equations. For the simple case of homogenous flow in batch condition the equation is as follows:

dm dt where B is the nucleation rate, G is the growth rate and mj is the j-th moment of the crystal size distribution. The dimension of the ODE set is equal to the number of moments of the crystal size distribution that are computed. In order to close the mass balance and to estimate the mean crystal size, the first five moments are sufficient. The mass balance equation can be written as follows: CAo ~ CA

~m~ M

(6)

79

where cA is the reactant concentration, CAo is the inlet reactant concentration, 9 is the crystal density, kv is the crystal shape factor, and M is the crystal molecular weight. The mean crystal size can be calculated using the following expression: m4

d,3 = ~

(7)

1713

For plug flow with axial dispersion there are two boundary conditions at the limit of the domain, thus the problem can be solved using an implicit discretisation method. Let us consider the results of the model illustrated applied to the continuous Couette reactor using the precipitation of barium as a test reaction. The kinetic law of barium sulphate precipitation, for nucleation and growth, can be found in Baldyga et al. (1995), while the volume shape factor for several morphologies has been previously evaluated (Pagliolico et al., 1999). The influence of the volume shape factor is clear from Figure 1. For the more common morphologies (tabular, tabular with pyramidal growth, simple twin) the results are very close, while for the rose crystal the predicted values of mean dimension and residual concentration are quite different. As the experimental data showed that rose crystals are very rare, and that when they are present the concentration is very low, a mean value, calculated from the first three morphologies can be used (kv=0.06). In Figure 2 the mean crystal size is plotted against the Pe number, for a supersaturation ratio S = 104: the line shows the predictions of the model, and the points are the experimental data. The limit of Pe tending to infinity (plug flow limit) in the figure is also reported. The effect of increasing Pe is to decrease the mean crystal size. In fact increasing Pe the degree of segregation of the system increases, and thus nucleation is favoured. This causes the formation of more, but smaller crystals. The dispersion of the data is due to the different feeding modes that affect the results but are not taken into account by the model. ta~

pyramid kv =0,06 ~--~

./

tabular dendritric

.ff

J

8

8

~6

P~

7 PF

6

2 5

rose kv =0,35

4

L , 0

,

,

,

i

,

,_,

5

, 10

Pe

Fig. 1. Mean crystal size versus Pe number for tR=81 s and log(S)=4. Continuous Couette reactor.

0

5

10

15

Pe

Fig. 2. Comparisonbetween experimental data and model predictions for tR=81 s and log(5)=4.

3. P R E S U M E D PDF M O D E L S

For the simple case of two non-premixed feed streams, it is easy to define a conserved scalar quantity called the mixture fraction, equal to unity in one feed stream and to zero in the other: CA - C ~ +CBo =

(8)

CAo + CBo

80 where CAo and @o are the concentrations of the two reactants in their feed streams. In the case of non-reactive systems (only mixing of two solutions in a non-premixed feed system) the relation between the mixing fraction and the reagents concentration (cA~ cB~ is" o

cA = ~:

(9)

C Ao

while in the case of instantaneous reaction the two reagents cannot exist in the same region. Thus the relation between the mixing fraction and the reagents concentration (cA~ cB~~ becomes: oo

cA = 0 for ~' < ~'~t

(10)

C Ao

c2 CAo

-

~ -~

:'-for

1 -- ~s,

~: > ~st

where:

~-

% CAo+CB~

(11)

As recommended by Baldyga (1989) the PDF can be described as a beta function in terms of the mixture fraction:

(12)

B(u,w) where:

B(u,w) = }f(~')d~'

(13)

0

-1-z~ u=r

(14)

Is

w=(1 -~) 1-Is

(15)

Is and where Is is the intensity of segregation and the overbar indicates the mean value. Using a different approach the presumed PDF can be described using a finite mode model (Fox, 1998). Setting the first scalar to be the mixture fraction, the presumed PDF has the following form:

f (~ x,t) = Y~Pn (x,t

_~(n) (x,t

n

where n is the number of modes and p, is the probability of the mode n. The use of either method requires the knowledge of the flow field and its turbulent properties. Using the commercial code FLUENT, the semi-batch Couette cell was simulated with two different models, the k-e and the Reynolds stress model. The k-e is a semi-empirical model that has been proven to provide engineering accuracy in a wide variety of turbulent shear flows with planar shear layer such as a planar jet. The isotropic description of the turbulence in not well suited to

81 the prediction of highly non-isotropic turbulence such as swirling flows. For this type of flow, the RSM is more appropriated (Fluent inc., 1995). 3.1. P r e s u m e d b e t a - P D F m o d e l

The mixing process can be defined as the process of variance (or intensity of segregation) reduction. Baldyga (1989) proposed a model in which the intensity of segregation is divided into three contributions: the inertial-convective 11, the viscous-convective 12, and the viscousdiffusive 13. The equation that governs the spatial distribution of the intensity of segregation has the form: 01i O[ alil (17) + - Dt = Rp, - Rai Ot & ] ujlj Oxj ) where Rp; and Re, are the rate of production and dissipation of the intensity of segregation at the different subranges. The turbulent properties of the flow, needed to compute Eq. (17), are calculated using the RSM in FLUENT, and the equation is solved for the Couette reactor with a reduction of the dimensionality of the problem by averaging over the radial direction. To simulate the continuous reactor, the axial flow component is added to the axial velocity profile supposing that it does not interact with the flow field. Let us consider the population balance written in the moment form. The moment can be considered as a function of the crystal size distribution, which is affected by the turbulent fluctuations. Averaging using the beta PDF and assuming the solid is moving in the reactor with the axial velocity v~, considering proportionality between the crystal size distribution and the solid concentration (c~) and deriving the relation between cA and the mixing fraction with a linear interpolation between the two limit cases (instantaneous reaction and non-reacting system), the following equations are obtained (Baldyga and Orciuch, 1997):

v~ d-mj dx

=

Oj ~ + j m--j_, , ~ I G(4')c~ (~)f(4)d4 cC o

(18)

vx ardA _3 k,v,D~?n21 . . . . . IG(~X)cc(~)f(~)d~ dx M-do

(19)

In Figures 3-4 the model results are compared with the experimental data. The figures show

lO

/ . .//

lO ::k

8

8 4

-o 9 6

2 0 0

1.E+02

. . . . . . . .

'

1.E+03

. . . . . . . .

i

. . . . . . . .

1.E+04

,

1.E+05

i

,

......

i

1.E+06

i

,

......

i

20000

i

|

i

I

~

i

i

70000

i

I

i

t

120000

t

t

170000

1.E+07

s Fig. 3. Effect of the initial nominal supersaturation for Re=86.000, tR=86s, feed tube diameter 2 mm.

Re Fig. 4. Crystal mean size versus Re for tR =86 S and log(S)=4, feed tube diameter 2 mm.

82 that in spite of the more accurate micromixing model the assumption of plug flow in the moment equation affects the ability of prediction. The beta-PDF model is able to predict the increase of the mean crystal size with increasing nominal supersaturation (Fig. 3). The effect of the velocity of the inner cylinder probably caused by the interaction of micro- and macromixing is predicted for low value of Re, while for high value the maximum is not described (Fig. 4). The effect of the feed tube diameter is very low, while experimentally the role of this parameter in the mesomixing dynamics is clearly evident (Barresi et aL, 1999). 3.2. Presumed Finite-Mode PDF In this work we will take n=3 environments (Fox, 1998). The first environment corresponds to the first stream entering into the reactor (V,=I), the second environment corresponds to the second stream entering the reactor (~=0), while the third corresponds to the environment in which the two inlet streams are mixed together and react. The three environment (mode) probabilities are defined by their scalar transport equations:

Ot + -~,

-~ (F + F,)

Op2 0 (
+ Y,P3-7,P, (1 -Pl )

(20) (21)

+r,p~-r,p=(1-p=)

p2, and the transport equation for the weighted mixture fraction in

=

rt)-:-

r,p,

-p,

-

~

(22) where S(3)--~(3)P3, ~1~ (1)= 1, ~2~

F is the molecular diffusivity, Ft is the turbulent diffusivity, U, is the/-velocity, the overbar is the Reynolds average value and

7, = 0.25C~ k

(23)

2(F + F,) 0g/'/~$: (~) &i ~Xi ?',=

l_2~:b)(1

(24)

$:(3))

The last term is a correction term, deriving from the extension of the model from homogeneous to inhomogeneous flow. The role of this term is to ensure that the mixture fraction variance obeys the correct transport equation. Introducing the moments of the crystal size distribution a nine scalars model is obtained. The model can be implemented in FLUENT as user defined scalars as described by Piton et al. (2000). The chemical source term has been closed in terms of the reaction progress variable Y. In the simple case of a one-step reaction the governing equations are:

83

(25)

cAo

cA

(26)

cAo p3kvm2 G

(27)

~t CAoM The results are compared with experimental data obtained from a semi-batch Couette reactor. The reactor has been filled with a sodium sulphate solution, and barium chloride has been injected as shown in Fig. 5. The system was modelled assuming axial symmetry. For all the cells in the computation domain p2 was initially set equal to zero, and at the injection point pl was set equal to one.

Fig. 5. Experimental set-up for the semi batch Couette reactor and model initial condition,

Fig. 6. Comparisonbetween model (line) and experimental data for the semi-batchCouette reactor; log(S)=4; tR=200 s

The simulations have been carried out with three different values of the mean mixture fraction using the same injection velocity, and thus different injection time. The inlet concentration of the barium chloride has been varied in order to keep constant the ratio between the two reactants. The model seems to under-predict the mean crystal size, while a relatively good agreement for the outlet concentration has been found (Fig. 6). If we compare these results with the limit case of a perfectly mixed batch reactor, we find out that the under-prediction of the mean crystal size is due to the strong segregation of the system. The contour plots of the reactant concentrations show that the vortex structure of the flow predicted by FLUENT results in relatively slow axial mixing (i.e., every vortex behaves as a perfectly mixed reactor). Moreover, the slow axial mixing results in values of the local ratio of the reactant concentration (cA/cB) far from unity. The effect of this parameter on the kinetic rates has been evaluated by Aoun et al. (1996), and an increase of the growth rate has been found with increasing or decreasing the reactants concentration ratio. 4. CONCLUSION Three different modelling approaches have been tested to validate their predictive ability. In the first one the hydrodynamics of the system has been modelled simply considering a plug

84 flow plus dispersion model, and the Peclet number has been calculated using an experimentally based power law: satisfactory agreement with the experimental data has been obtained in spite of the limited interpretation of mixing to the macro-scale of the reactor. The second approach is based on the beta-PDF; it has been applied to the continuous system, evaluating the turbulent properties by means of a CFD code, but assuming a simplified hydrodynamics (one dimensional model). The third one is based on the finite-mode PDF, and in this case a bidimensional flow field has been considered. The results show that the use of the presumed PDFs is very promising, but the accuracy of the prediction is strongly affected by the quality of the CFD simulation. The flow field prediction of a commercial code (FLUENT) for the turbulent vortex flow is not accurate enough, and thus a deeper study of the flow field prediction, coupled with experimental validation is required. A comparison of the performances of the two presumed-PDF models and the full PDF model will be the next step of our research. REFERENCES

Baldyga J., 1989, "Turbulent mixer model with application to homogeneous, instantaneous chemical reaction", Chem. Eng. Sci., 44,1175-1182. Baldyga J., Orciuch W., 1997, "Closure problem for precipitation", Trans. 1ChemE, 75 A, 160170. Baldyga J., Podgorska W., Pohorecki R., 1995, "Mixing-precipitation model with application to double feed semibatch precipitation", Chem. Eng. Sci., 50, 1281-1300. Barresi A.A., Marchisio D., Baldi G., 1999, "On the role of mesomixing in a continuous Couette type precipitator", Chem. Eng. Sci., 54, 2339-2349. Fluent inc., 1995. "FLUENT user Manual, version 4.5", Fluent inc, Lebanon, New Hampshire, USA. Fox R.O., 1998, "On the relationship between Lagrangian micromixing models and computational fluids dynamics", Chem. Eng. Process., 37, 521-535. Kataoka K., 1986, "Taylor vortices and instability in circular Couette flows", in: Encyclopaedia of Fluid Mechanics (N.P. Cheremisinoff, Ed.), vol. 1, pp 237-273, Gulf Publishing, Houston. Marchisio D., Barresi A.A., Baldi G., 1998, "Influence of the hydrodynamics on the solids characteristics in a Couette type precipitator", Proc. 6th Int. Conf. on Multiphase Flow in Ind. Plants, Milan, Italy, 335-346. Pagliolico S., Marchisio D., Barresi A.A., 1999, "Influence of the operating conditions on BaSO4 crystal size and morphology in a continuous Couette precipitator", J. Therm. Anal. Cal., 56, 1423-1433. Pipino M. and Fox R.O., 1994, "Reactive mixing in a jet tubular reactor: A comparison of PDF simulation with experimental data", Chem. Eng. Sci., 49,5229-5241. Pipino M., Barresi A.A., Fox R.O., 1994., "A PDF approach to the description of homogeneous nucleation", Proc. 4 th Int. Conf. on Multiphase Flow in Ind. Plants, Ancona, Italy, 248-259 Piton D., Fox R.O., Marcant B., 2000, "Simulation of fine particle formation by precipitation using Computational Fluid Dynamics", Canad. J. Chem. Eng., in press. Rivera T., Randolph A.D., 1978, "A model for the precipitation of Pentaerythritol tetranitrate (PETN)", Ind. Eng. Process Des. Dev., 17, 183-188. Vatistas N., 1991, "Danckwerts' degree of segregation for the axial dispersion model", Chem. Eng. Sci., 46, 307-311.

10 th European Conference on Mixing H.E.A. van den Akker and.]..1. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

85

Application of Parallel Test Reactions to Study Micromixing in a Co-Rotating Twin-Screw Extruder. A. Rozefia, R.A. Bakkerb, J. Ba|dygaa a Department of Chemical and Process Engineering, Warsaw University of Technology, Waryfiskiego 1, 00-645 Warszawa, Poland. b DSM Research, P.O.Box 18, 6160 MD Geleen, The Netherlands. Laminar mixing of Newtonian and completely miscible liquids was studied experimentally in a co-rotating twin-screw extruder by means of parallel test reactions. The side stream of the concentrated solution of base (limiting reactant) was mixed with the main stream of the diluted pre-mixture of acid and ester. A neutral thickening agent was applied to alter the viscosity of both solutions. The selectivity of the test reactions being related to the course of micromixing was affected by operating conditions, the screw geometry and differences in viscosities of the mixed liquids. An explanation of the observed phenomena was proposed. 1. INTRODUCTION The co-rotating twin-screw extruder (CoTSE) is widely applied in the polymer industry for continuous processing of very viscous materials [ 1]. Melting, addition of side streams of catalysts or fillers, mixing, complex reactions of polymer synthesis or modification, removal of volatiles and extrusion of the final product can all be performed in a single CoTSE. Each unit operation takes place in a separate section of the extruder; the mean residence time in each section is short and the extruder has to perform these operations efficiently. Fulfilling these conditions is especially important when mixing is followed by fast or very fast chemical reactions of non-linear kinetics proceeding between the mixed reactants (mixing of a catalyst or an initiator with monomers of polymers). The flow in the extruder is laminar and the coefficients of molecular diffusion of the reactants are very low, which may result in controlling the chemical reaction by mixing on the molecular scale (micromixing) [2]. Non-ideal mixing on the molecular scale may promote slow side reactions and affect the product quality (composition, properties, purity). Such factors as: the screw speed, the screw geometry, the physical properties of the processed materials (e.g. viscosity, miscibility) and the way in which reactants are fed into the extruder can influence mixing conditions in CoTSE [3+6]. In this work the authors present experimental results obtained when two parallel reactions: N a O H + HCl --->NaCI + H 2 0 ,

(1)

(2) used as a test reaction system and a neutral agent increasing the viscosity of aqueous solutions of the reactants (polyethylenepolypropylene glycol) were applied to study the effects of various process conditions on micromixing in CoTSE. The first reaction is practically instantaneous and so completely controlled by micromixing, while the second reaction proceeds at rates comparable to the rate of micromixing in the laminar flow [7]. Hence, the final selectivity of the test reactions (1-2) is directly dependent on the course of micromixing of initially unN a O H + CH2CICOOC2H 5 --->CH2CICOONa + C2HsOH ,

86 mixed reactants [7,8]. Using this method the authors showed that the screw speed, the screw geometry, the ratio of the flow rates of the reactant solutions and differences in viscosities of the mixed solutions affect micromixing in CoTSE. The model of laminar mieromixing was used to determine the average rate of deformation in the reaction zone when the mixed liquids were equally viscous. Results obtained for mixing of liquids differing in viscosity allowed to identify several phenomena, which can affect mixing on the molecular scale in such a case. 2. EXPERIMENTAL METHOD Experiments with the parallel reactions (1+2) were conducted in the co-rotating twinscrew extruder shown in Fig. 1. The extruder screws were mounted in a transparent perspex barrel at centreline distance of 0.021 m . Each screw comprised up to 23 blocks of 0.0246 m diameter and 0.037 m length. Three screw configurations were applied. In the first set-up the

Figure 1. Diagram of a co-rotating twin-screw extruder. screws consisted of double-flighted transport elements (TE) only. In the second set-up neutral kneading discs ( N ) replaced the 6th, 7th and 8th transport blocks. In the third set-up (Fig. 1) turbine mixing elements (TME) replaced the kneading blocks. The screw blocks are presented in Fig. 2. The diluted pre-mixturc of HC1 (reactant B) and CH2CICOOC2H5 (reactant C) was continuously fed under the atmospheric pressure to the main feed port and then conveyed by the screws along the extruder. The concentrated solution of NaOH (reactant A) was continuously injected into the intermeshing region of the screws via one of small ports localised downstream. The chemically equivalent amounts of reactants were used (VA'CAo= Vnc'CBo = VBc.Cco ) . The ratio of the feeding flows, VBc/V A , was equal either to 7.33 or to 24 and appropriately the initial reactant concentrations were either set to cao/24=cso=cco=lO mole.m 3 or CAO/7.33=CSO=eco=lO.9 mole'm "a. In all the experiments the ratio of the extruder throughput, V, to the maximum drag flow, V ~ , measured for an open discharge, was maintained at 0.3. The extruder was completely filled with the mixed Figure 2. Screw blocks, solutions. The viscosity of the main stream was increased to 0.27 Pa-s. The viscosity of the side stream was varied between 0.001 and 2.3 Pa.s. The temperature of the mixed solutions was adjusted to 298 K. Samples of the post-reaction mixture were taken after the time of the process longer than two mean residence times to reach a steady-state and analysed by means of High Pressure Liquid Chromatography to determine their composition. The final selectivity of the test reactions were calculated from X = [Vsc.Cco- V.cc ]/(VA.cAo ) .

(3)

87 In the case of the perfect mixing this selectivity equals zero [7,8]. If both reactions are fully controlled by mixing the selectivity should reach its m ~ i m u m value, which depends on the relation between diffusion coefficients of the reactants (e.g. Xm,=--0.5 for DA=D B=D c ) [7+9]. 3. MIXING OF LIQUIDS OF EQUAL VISCOSITIES The flow of the processed material along each extruder is secured by the transport elements (Figs. 1 and 2). The short sections consisting of the kneading discs or the turbine mixing elements are used to improve mixing downstream a melting zone or the injection point of a side stream [3+5]. Therefore, it was important to find first a proper position for injection of NaOH solution allowing for the X 9 NKD 9 TME maximum use of homogenisation A 9 , 0.25 capabilities of few mixing blocks. Figure 3 shows how the selectivity changed when the feeding ,11 0.20 point was moved from the transport section to the mixing section II and further to the next transport 0.15 section. In the case of the neutral o o kneading discs, the lowest selecn n tivities could be observed after transport o mixing transport 0.10 i blocks blocks blocks >< 1/3 length of the mixing section till its end. In the case of turbine 1 7 113 1 215 311 37 mixing elements the selectivity injection port number decreased sharply just before the Figure 3. Effect of the position of the injection point of beginning of the mixing section base solution on the selectivity; n = 100 rpm, VBc/VA= 24. and remained very low till the 2/3 length of this section. Hence further experiments with the investigated mixing elements were conducted with the injection point localised near the centre of the region comprising these screw blocks; injection ports 17 or 19 were used to feed the base solution into the extruder. A comparison of the seleetivities, X, obtained for the different screw elements at various screw speeds, n, and for two valscrew block TE NKD TME ues of the ratio of the feeding 0.30 flows, VBc/I;'~, is presented in 9 9 symbol o 9 A A m Fig. 4. It can be seen that all these g B 9 9 0.25 O O 9 9 9 three factors affect the course of r"L ~ O 9 9 mixing in CoTSE. A comparision 0.20 of the selectivities obtained for D A 9 9 9 [] 9 the same ratios of the feeding 0.15 flows and the same (or very [] 9 A 9 similar) screw speeds indicates [] that the turbine mixing elements 0.10 [] secure the fastest micromixing. 20 100 140 180 The neutral kneading discs create n[rpml slightly worse mixing conditions Figure 4. Effect of the screw speed, the screw geometry (higher selectivities). The highest and the ratio of the feeding flows on the selectivity. selectivities indicating very slow o

i

i o | i !

|

|

10

l

88 micromixing were observed for the transport elements. These results remain in good agreement with the results of other mixing studies based on the observations of the morphology and properties of the extruded polymer blends [3+5]. For each tested screw geometry increasing the screw speed, n, usually resulted in decreasing the selectivity, X . This result proves that the average rate of deformation of liquid elements in the reaction zone is strongly influenced by the screw speed. Decreasing the ratio of the feeding flows, VBc/VA, also improves micromixing for all the investigated screw geometries when the screw speed and the amounts of the mixed substrates remain unchanged. This is so because it is easier to mix the same reactant amounts when the volume ratio of the reactant solution is closer to one [8]. An new version of the micromixing model originally presented by Batdyga et al. [7] was used to identify the average values of the rate of deformation of liquid elements in the reaction zone of the parallel reactions. In the model [9] the local material balance of ith reactant c3c, + 0t

c3u._._~c3c_..2_= D ~ " m--1 n=l

~"" o~. o{ m

~-~,2,+R'

(4)

.__,

is integrated with the weight functions "1" and "~k'~t" over the space coordinates. These integral transformations give two equations: dM~

aM,.. (ou. .M,x , + D,.M, ) + I I I

.__.._x_. = 2. at

t.a~

_.0

R''~2 d~ld~2d~3 '

(6)

where M~ and M~.,, denote zero and second order concentration moments, respectively +oo

+oo

M,,•= I II(ci-ci.x.)~:d~ld~2d~3.

M,= I II(c,-c,| -or)

(7)

-oo

According to Eq. (6) molecular diffusion and chemical reaction can be described in the local frame of reference, as occuring in a simple stagnation flow with the deformation rates, otk Uk = OU,[

.gk = r

0 ~ , I~__~

9{k ~

k = 1,2,3

(8)

,

In this flow the fluid element shrinks in at least one direction and extends in the remaining one(s) due to the continuity condition, Otl+(Z2+ot3=0; e.g. if Otl<0 and or2, or3 >0 the fluid element takes a shape of a thinning slab. If molecular diffusion proceeds at speeds lower than or comparable to the rate of viscous deformation (only then mechanical mixing is important) then molecular diffusion becomes significantly accelerated in the direction of the fastest shrinking and slowed down in other directions. As the result, a three-dimensional initially local concentration field degenerates to a one-dimensional one [7]. Hence, it can be assumed that a drop of the reactant solution eventually forms an elongated striation. If the local axis (e.g. {l) has been made perpendicular to the symmetry plane of this slab of thickness s then au~ 1 ds C~1 --'~ . . . . . . . (9) ~1

S

dt

The model equations (5+6) contain integrals with the reaction terms, R~, which depend on the local concentration profiles. To approximate these profiles functions similar to those proposed by Tryggvasson and Dahm [10] were used:

89

=

f

c,.

(Ci=-Ci=)[0.5-tl/(~l,A,,~li,F.,)]+C,:

c,|

tlS({, A, 8,.c,) = ~ +____~1(21;, - 1)(~l '

q~

'

4

"

I~,1_
-- A , ) 3 +

38~({1 - A,)

[I~,(~l _Ai) 2 q_~213/2

0.50 l 0.25 0.00 -0.25 -0.50 .0

-0.5

.,

0.0

05

Figure 5. Gradient profile function.

R, =

f

i = A(base) - klcAc 8 - k2cAc c = B(acid) - klcAc B =C(ester)

,

(10)

(11) where Ai and 5i denote the displacement of the inflexion point and the half width of the gradient profile. The reactant concentration at the slab centre is denoted by Cim, while the far field concentration by ci~ The coefficient, ei, determines the shape of the profile - Fig 5. 1.0 Application of the smooth algebraic profiles of the adjustable shape to evaluate reaction terms for the parallel reactions: (12)

- k2cAc c

allows the closure of the system of model Eqs. (5+7), which can be then solved for the half width, 5i, and the displacement, Ai, with the initial conditions: A, = s0/2, 8, = 0 , i=A,B,C. (13) The concentrations at the origin and in the environment remain constant during integration: CAI=CAo, CA| CB,=0, CB| Ccm=O, Cc==Cco. (14) until the half width of the profile, 8i, becomes equal to its displacement, Ai. Since that moment Ai is replaced with 5i in Eqs. (10+ 11) and the concentration at the origin, C~m, is parameterised. The shape coefficients, ei, are pre-set to certain values specific for each reactant, which ensure the best accuracy of the model solution [9]. In this form the model can be used to predict the selectivity of the parallel reactions occurring between the substrates of different molecular diffusivities with a high accuracy (--98+99%) [9]. According to correlations reported in [7] the molecular diffusivities of the reactants in solutions of 0.27 Pa.s viscosity were equal to DA=6.7 10"1~m2.s-1, D~r=l 1.6 10-l~ m2.s -1 and Dc=l.4 101~ m2.s -1. The average rate of deformation of liquid elements in the reaction zone, <~1>, was determined for each experiment by fitting of the selectivity predicted by the model to the measured selectivity. The initial striation thickness, so, applied in the condition (13) was constant and equal to 0.5 ram. The reason for using the constant value of so was the fact that when the ratio of the throughput to the maximum drag flow, V/Vma x , is kept constant then dimensionless

90 sb~~o~~

T,~

NKD

TM'~

flow field, v / ( n . D ) , in the extruder is unchanged and so is symbol o p t i so/D. Furthermore, it was found E q . ( 1 5 ) . . . . . . . Qt/n that for s0<0.5 mm the initial rate of molecular diffusion was higher then the rate of deformation de9 i::3 . . j . . ~ -'~" ~4....~..--I[ termined'in the fitting procedure. 0 ...-13~ ~'" 9 ..lf "~" A O "'~"~ A Z~ I "~'~ LX Only for s0>__0.5mm the molecular diffusion was significantly 9 ~..,111 ~ o ~ 0,1 accelerated by mechanical mixing and the final selectivity of the / 0 o o 9 parallel test reactions, X, was no o more dependent on the initial striation thickness, so. The values of the dimensionless rate of de0.0120 l 100 200 formation, /n, identified in nlrpml this case are the maximum ones. Figure 6. Average rate of deformation determined for As it is shown in Fig. 6, the different operating conditions and screw geometries. highest values of the rate of deformation were obtained for the turbine mixing elements. The intermediate values of the rate of deformation were determined for the neutral kneading discs, while the lowest values were found for the double-flighted transport elements. For all screw geometries the values of /n, obtained for different ratios of the feeding flows, VBC/VA, depend on the screw speed in a similar way. This indicates that the model correctly accounts for changing the ratio of the feeding flows. Correlations between the screw speed and the average rate of deformation = a . n b (15) are also plotted in Fig. 6. The values of coefficients a and b are reported in Table I. Table I. Coefficients a and b.

Screw block

TE

NKD

TME

a

8.1.10 -4

2.4.10 "3

1.0.10"2

b

1.99

2.00

1.81

4. MIXING OF LIQUIDS OF DIFFERENT VISCOSITIES The viscosity difference of the materials contacted in CoTSE is an important factor influencing the degree of deformation of the dispersed phase and the morphology of the resulting mixture. Lee and White [6] discovered that when polymer pair is blended in the extruder the smallest drops of the dispersed phase are created when the blended polymers have equal viscosities. Both increasing and decreasing of the viscosity of the dispersed polymer results in increasing the average diameter of drops of the dispersed phase [6]. Consequently the rates of generation of intermaterial surface area and molecular diffusion of the mixture components can also be affected by viscosity differences. To verify this hypothesis a few series of experiments with the parallel test reactions were carried out. In these experiments the viscosity of the main stream (equal to 0.27 Pa.s) was different than that of the side stream. The extruder

91

0.35

0.30

m

9

0.25

0.20

6

8 z~

o

A

+

+

9

i

9

9

+

+ I

9

0.15

o

l 120

II

160

n[rpm] Figure 7. Effect of the screw speed and the viscosity ratio on the selectivity; VBc/VA= 7.33.

screws consisted of the transport elements only. Figure 7 shows the results obtained for VBc/V,4 = 7.33. In this case decreasing of the viscosity of the side stream to 0.086, 0.025 and finally to 0.0009 Pa's resulted in higher selectivities of the test reactions than those obtained for mixing of equally viscous solutions. On the contrary, when the viscosity of the side stream was increased to 0.89 and 2.34 Pa.s then the selectivities decreased below the values obtained for mixing of solutions of equal viscosities. However, when the viscosity of the side stream was equal to 0.89 Pa.s the selectivities were also lower than those obtained when this viscosity was equal to 2.34 Pa.s.

A different picture was observed when VBc/VA was increased to 2 4 - see Fig. 8. Then decreasing of the viscosity of the side stream to 0.024 Pa.s resulted in lower or comparable selectivities to those determined for equally viscous solutions. When the viscosity of the side stream is higher than that of the main stream then deformation of the liquid elements of the side stream by less viscous environment becomes more difficult. The rate of generation of the contact surface area between the mixed liquids is slowed down. Additionally due to the high viscosity of the side stream the molecular diffusivity of NaOH becomes considerably decreased. As the result, molecular diffusion and chemical reactions between the contacted substrates can not proceed until the segregation scales in the system are considerably reduced, which may take a long time. In this time, however, small portions of the more viscous base solution can be eluted and entrained by the less viscous environment containing the local excess of the acid and the ester. Hence, small amounts of the base contained in these "eroded" portions will rather react with the acid than with the ester. The results shown in Fig. 7 indicate that if the viscosity difference is not too high than this 0.35 symbol last mechanism prevails and the selectivity of the test reactions decreases. De0.30 creasing of the viscosity of the side stream below that one of the main stream + + may in principle accelerate deformation 0.25 of the liquid elements of the side stream [] + + by the more viscous environment. How0.20 0 0 ever, flow destabilisation and formation of periodic segregated structures can sig0.15 nificantly retard the process of deformation in this case, as shown by Batdyga , 610 , I , I , 20 100 140 180 and Ro~efi [11]. The higher differences in n[rpml viscosity of the mixed liquids, the higher Figure 8. Effect of the screw speed and the chances for the flow instabilities origiviscosity ratio the selectivity; VBc/V,4=7.33. nating from the discontinuity of the ve-

92 locity gradient at the contact surface to occur. The flow disturbances should eventually disappear when the difference in viscosity of the mixed liquids becomes smaller due to erosion of small portions of the more viscous environment into the less viscous base solution. In this case, the test reactions proceed in the local excess of the base and the ester conversion increases. The results shown in Fig. 7 seem to confirm that these phenomena took place. On the other hand, decreasing of the viscosity of the side stream increases the molecular diffusivity of NaOH, which enlarges its local penetration range so that only the first test reaction (neutralisation) is diffusion controlled. The role of this mechanism increases with increasing differences in the volumes of reactant solutions and the initial reactant concentrations. Eventually this last mechanism should prevail over those mechanisms, which tend to increase the ester conversion, as it probably happened for VBc/VA= 7.33 - Fig. 8.

Acknowledgements The authors express their gratitude to Dr. P.H.M. Elemans for his initiative to conduct the experimental part of this study and for organizing a financial support of the DSM Research in Geleen, The Netherlands. NOTATION Concentration of reactant i Ci Screw diameter D Molecular diffusivity of reactant i Di k~ reaction rate constant Zero order concentration moment M~ Mt,kt Second order concentration moment Screw speed n Total extruder throughput V Maximum flow rate Vo.x e, Volumetric feeding rate Reaction rate Ri Striation thickness S

t ui

Time Velocity in local frame of reference Velocity in extruder channel X Selectivity ~i Rate of deformation Time averaged rate of deformation Ai Displacement of gradient profile ~i Half width of gradient profile 8i Shape coefficient of gradient profile [.l,i Dynamic viscosity of feeding stream ~i Coordinate in Lagrangian frame

REFERENCES 1. P.H.M. Elemans, H.E.H. Meijer, in: N.P. Cheremisinoff (Ed.), Encyclopedia of Fluid Mechanics, Vol. 9, Gulf Publishing, Houston (1990) 361+371. 2. K.J. Ganzeveld, L.P.B.M. Janssen, Polym. Eng. Sci., 32 (1992) 457+466. 3. M.H. Mack, T.F. Chapman, SPE ANTEC Tech. Papers, 33 (1987) 136+139. 4. S. Lim, J.L. White, Int. Polym. Process., 8 (1993) 119+ 128. 5. T.P. Vainio, A. Harlin, J.V. Sepp/il/i, Polym. Eng. Sci., 35 (1995) 225+232. 6. S.H. Lee, J.L. White, Int. Polym. Process., 12 (1997) 316+322. 7. J. Ba/dyga, A. Ro2efi, F. Mostert, Chem. Eng. J., 69 (1998) 7+20. 8. J. Batdyga, J.R.Bourne, Turbulent Mixing and Chemical Reactions, John Wiley & Sons, Chichester, New York, Weinheim, Brisbane, Singapore, Toronto (1999). 9. A. Rozefi, J. Batdyga, R.Bakker, "Application of an Integral Method to Modelling of Laminar mixing", prepared for publication in Chem. Eng. J. 10. G. Tryggvason, W.J.A. Dahm, Combustion and Flame, 83 (1991) 207+220. 11. J. Baidyga, A. Rozefi, "Investigation of Micromixing in Very Viscous Liquids", 8th European Conference on Mixing, Cambridge, England, September (1994) 267+274.

I 0th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

93

Solid liquid mixing at high concentration with SMX static mixers O. Furling a, P.A. Tanguy a, L. Choplin b and H.Z. Li b aURPEI, Dept. of Chemical Engineering, Ecole Polytechnique, P.O. Box 6079, Stn CV, Montreal, H3C 3A7 Canada bGEMICO, ENSIC, BP 451, 54001 Nancy Cedex, France

Abstract The performance of SMX static mixers was evaluated in the context of the preparation of highly concentrated slurries. The effect of the powder feeding rate and solid content on the power consumption and dispersion quality was measured and compared with the results of batch processes. The experiments were carried out with kaolin clays up to 72 wt. %. It was found that static mixers are able to disperse high solids slurries very efficiently with a significantly lower power consumption as compared with in-tank dispersers, making them a promising tool for the inline preparation of paper coating colors. 1. INTRODUCTION Pigment dispersions are used in many formulations based industries, like the paint, food and paper industries. In the paper industry, these dispersions are mixed together with binder and additives to make the coming fluids used in the surface treatment of paper and paperboard. As the coating solid content can reach 70 wt. %, strong interactions develop between the components making the theology extremely complex (shear thinning, shear thickening, viscoelasticity and time effects) (Yziquel et al., 1999). The dispersion of powder in liquids can be seen as a two-step process. During the wetting stage, the water penetrates into the interstices of agglomerates and removes air from the surface. In a second stage, the dispersion process leads to breakage or erosion of the agglomerates depending on the stress applied on the particle. The preparation of the pigment slurry in the paper coating industry is far more complex than it might appear at first glance. Indeed, the hydrophobieity of many coating pigments makes the wetting stage difficult due to the formation of lumps. Moreover, the strong increase in viscosity at the end of pigment feeding at high solids impedes efficient dispersion and the destruction of the lumps.

94 The size reduction is the objective of the dispersing step. This is usually achieved in the paper industry using high shear dispersing tools like the Cowles dissolver (saw tooth turbine) or the Kady mill (rotor-stator) in semi-batch mode. These systems are undoubtedly good dispersers, although their power consumption is very high due to their high rotational speed, even more so when the suspensions exhibit a shear-thickening behavior (Boersma et al., 1990). New impellers like the Sevin and the Deflo turbines (VMI-Rayneri) have been introduced for pigment slurrying at a lower energy cost. Their performance has been shown very promising for paper coating make-down (Tanguy et al, 1999). The development of specialty paper products and just-in-time production schedule favor naturally the use of continuous processes. In this context, in-tank based dispersion processes are not the best answer. Static mixers would be more suitable, provided they can reach the same level of dispersion efficiency. Barresi et al. (1996, 1997) have succesfuly done the size reduction of fine ceramics powders with SMX static mixers for low concentration suspension. The aim of this article is to explore the ability of the SMX static mixer to produce inline concentrated suspensions. SMX static mixer consists of an in entangled blade network as shown in Figure 1.

Figure 1" SMX static mixers Our methodology will be purely experimental. The slurry viscosity, and power consumption will be measured vs. the feeding rate, the flow rate and the pigment properties, and the results will be compared with those obtained in an on-going study on dispersing turbines. 2. EXPERIMENTAL CONDITIONS A static mixing loop has been built as shown in Figure 2. The setup is composed of a small tank of 60 liters, a progressive cavity pump connected to a 3 HP motor that can deliver up to 50 liters/rain, and a static mixing section. 6 to 24 SMX elements can be fitted in the system although only 6 elements were used in the present experiments. A differential pressure transmitter is installed to measure the pressure drop through the static mixers.

95 A volumetric feeder consisting of a conical hopper and a vibrating channel is used to deliver the powder in the vessel. The control of the pump, ~ mixer and feed system as well as data acquisition and processing are performed with Labview software (National Instalments). The typical operating mode of this setup is as follows. We start by filling the tank with water. Then the tank is fed with the powder at a predefmed rate at the surface. Small rotating rods (Duquesnoy et al., 1997) are used to partially wet and draw down the powder so that no powder accumulation takes place at the free surface. The maximal rotational speed of the rods is 60 rpm at the end of the feeding stage for the high powder feeding rates. Kaolin hopper Volumetric channel

DifferentialTransmitterPressure

Static Mixers

/ = = = ~ ] Tank

Progressive cavitypump

Figure 2: Experimental setup Three different kaolin clays were used in the experiments namely delaminated clay (Nuclay), the most used clay in North America, with a platelet like shape, a hydrous predispersed clay (HT), with good flow properties (low viscosity suspensions) and a calcined clay (Ansilex 93) with a high hydrophilic character. They were all provided by Engelhard (Iselin, NJ). Slurries with a solid content up to 72 wt. % were made with Nuclay and HT. Only 40 wt. % was achieved with purely calcined clay. No surfactant was added in the experiments. The volume of water at the beginning of the experiment was 17 liters, and it increased up to 35 liters for the more concentrated suspensions. Samples were collected regularly downstream the static mixers. The characterization of the solid content was carried out by drying samples in a kiln, following the usual practice in the paper industry. The dispersion quality was controlled with rheological measurements (flow curves) on a Bohlin stress controlled rheometer (CVO). The measurements were done 1 to 6

96 hours after the make-down. A good re-homogeneisation of the sample was performed before each rheological experiment. The surface of all samples was coated with silicon oil to prevent dewatering by evaporation during the experimentation. 3. RESULTS AND DISCUSSION 3.1. Influence of feeding rate

We show in Figures 3 and 4 a typical pressure drop-time curve obtained with the HT clay. The feeding rate is 0.0126 kg/s in Figure 3 and 0.0179 kg/s in Figure 4. The flow rate is 0,57 m3/h. The solid content is 71.9 wt.%. It can be seen that the power is increasing while the clay is fed. The maximum pressure drop is obtained near the end of incorporation stage (2600 s in Figure 3 and 3400 s in Figure 4). This increase is more pronounced at fast incorporation rate (Figure 4). The oscillations observed on the pressure drop signal correspond to the dispersion of lumps in the static mixers, which creates a high local concentration of powder in the mixer and therefore a viscosity singularity. There is a period of time after the end of the feeding stage before the pressure drop decreases to a f'mal plateau value. This lapse corresponds to the destruction of some remaining lumps floating in the tank that needed more time to be drawn down and sucked in by the pump.

Figure 3: HT incorporation curve

Figure 4: HT fast incorporation

Similar results were obtained with Nuclay, but the intensity of the peaks were higher, due to the presence of agglomerates more difficult to disperse. The pressure drop increased rapidly to 2 bars with a solid content of 70 wt. % under fast feed conditions(0.02 kg/s). With HT, the solid content reachable was 72 wt. % without much difficulties. We attribute this difference in behavior to the presence of surfactants in the HT clay. It is important to highlight the role of the feeding rate for the make-down of high solids slurries. A high feeding rate will create big lumps. These lumps ranging in size from 0.5 cm to 2 cm are hard to disperse because the external surface is partially wetted, generating strong agglomerates. The size and the amount of lumps depend on the feeding rate and the rotational

97 speed of the rods at the free surface. In our experiments, the speed was maintained sufficiently low so as to ensure the formation of lumps and therefore evalaute the ability of the static mixers to destroy these agglomerates. The comparison of the viscosity curves of the above slurries with that obtained with dispersing turbines is shown in Figure 5. It can be seen that the viscosity curves are fairly similar. The difference arises from the solid content of the two suspensions, i.e. 71.4 wt. % for the slurry made with the dispersion turbine and 71.9 wt. % with the static mixers.

Figure 5: Comparison of viscosity. From an energetic standpoint, the SMX needs only a higher pressure furnished by a pump. By comparing the incorporation with a dispersion turbine at a same feeding rate, the maximal torque reaches at the end of incorporation is 20 N.m at 700 RPM in the same tank, leading to a power consumption of 1500 W. With static mixers, the value for a pressure drop of 2 bars gives 40 W for a low rate of 0.57 ma/h. Integrating the incorporation curve with time, the total energy consumption is 195 W-h for the turbine and 17 W-h for 6 SMX elements. The energy consumption is therefore 11 times lower for the static mixer than for a dispersion turbine with the same dispersion quality. A quick calculation shows that the number of SMX elements required to perform an inline dispersion in one pass will vary from 12 to 24 elements, depending on the solid content and the kaolin type. For example with the Nuclay, the dispersion step is achieved in 300 s at 50 wt. % and 800 s at 70 wt. % using 6 SMX elements. With a volume of 35 liters and a flow rate of 0,57 m3/h. In this case, we would then need 12 SMX elements in the first case and 24 in the second case to disperse in one pass.

98 Experiments were also carried out with calcined clay. The pressure drop increased drastically when the solids reached 40%, as seen in Figure 6. 2,5

m .Q

2

v

o. 1,5

2

'u

.o 0,5 .

0

.

.

J

.

_

500

.

.

,

1000

i

_ ,,..J

1500

2000

i

2500

tI 3000

Time (s) Figure 6: Incorporation of calcined clay In fact, the problem is a local overconcentration of powder in the static mixer, together with the development of a very high shear-thickening behavior. The three little peaks at 2500s come from a small addition of 10 to 20g of powder. This small m o u n t of powder creates a shock, leading to an abrupt increase in power consumption. This phenomenon is similar as those observed during the preparation of coating colors where the development of physicochemical interactions between the components enhance the viscosity. The dispersion after 40 wt. % is similar. A significant increase in viscosity appears due to the difficulties of wetting the kaolin particles and particle-particle interactions. The same increase exists in batch process, but the peak of power consumption begins at 45 wt. %. 3.2. Influence of solid content on the dispersion quality

The interest of this section is to measure the influence the flow rate, ie the shear capacity provide by the SMX, on the dispersion quality for the Nuclay at different solid content (50, 60, 65 wt. %). Figure 7 shows the pressure drop-time relation for Nuclay (solids target = 50 wt. %) at 0.57 m3/h according to the displayed flowrate history. The end of incorporation was at 1300s. Samples were taken aider 1700 s, 2000 s and at the end of the experiment. This procedure was repeated for the preparation of 60 wt. % and 65 wt. % slurries. The rheological curves (Figure 8) show that the dispersion quality seems independent of the flow rate at 50 wt. %. However, at 60 wt. % (Figure 9), some difference appears between the dispersion at 0.57 m3/h and 0.85 m3/h although nothing can be seen at 65 wt. % (Figure 10). It seems that attrition phenomena take place, so that the dispersion is achieved for every flow

99 rate. Indeed, a flow rate of 0.85 m3/h is sufficient to obtain a good dispersion for every solid content.

Figure 7:50 wt.% Nuclay incorporation curve

Figure 9: Dispersion quality at 60 wt.%

Figure 8: Dispersion quality at 50 wt.%

Figure 10: Dispersion quality at 65 wt.%

At 50 wt. %, the rheological flow curves will not give information about the dispersion quality of the suspension. In fact, the interactions between the particle are not sufficient to increase the viscosity. For more concentrated suspensions, the volume fraction of the particle increases with respect to the suspending phase. A poor dispersion, meaning the presence of large agglomerates, will lead to a higher viscosity than a good dispersion.

4 CONCLUSION This study has shown that it is possible to prepare high solids slurries with inline static mixers with the same quality as that obtained with high shear dispersers in tanks. The power draw to achieve the dispersion is significantly lower with static mixers. The solid content plays an important role as far as the quality of the dispersion is concerned. Attrition seems to begin at 6fi wt. %, leading to a lower flow rate to achieve the dispersion.

100 SMX static mixers appear very promising for the inline preparation of coating colors.

Acknowledgements The f'mancial support ofNSERC, Papdcan and Engelhard is gratefully acknowledged.

References Barresi A.A., Pagliolico S., Pipino M., Mixing of slurries in static mixer: evaluation of a lower energy alternative ro simultaneous eomminuition and mixing for production of composite ceramic material, Proc. 5th Int. Conf. Multi Phase Flow in Ind. Plants, Amalfi, Italy, Sept. 26-27, 302-313, 1996. Barresi A.A., Pagliolico S., Pipino M., Wet mixing of fine ceramic powders in a motionless device, R6c. Prog. G~nie Proc., 11, no. 51, 291-298, 1997. Boersma W.H., Laven J., Stein H.N., Shear Thickening (Dilatancy) in Concentrated Dispersions, AIChE Journal, Vol. 36, No3., pp 321-332, 1990. Duquesnoy J.A., Tanguy P.A., Thibault F., Leuliet J.C., A New Pigment Disperser for High Solids Paper Coating Colors, Chemical Engineering Technology, Vol. 20, pp 424-428, 1997. Tanguy P.A., Furling O., Choplin L, A New Dispersing Disk for Non-Newtonian Concentrated Suspension, 3r~ International Symposium on Mixing in Industrial Process, Osaka, Japan, pp 417-424, 1999. Yziquel F., Moan M., Carreau P.J., Tanguy P.A., Nonlinear Viscoelastic Behaviour of Paper Coating Colors, Nordic Pulp and Paper Research Journal, Vol 14, no. 1, pp 37-47, 1999.

10th European Conference on Mixing H.E.A. van den Akker and,l.3". Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

101

Influence of viscosity on turbulent mixing and product distribution of parallel chemical reactions J. BaIdyga, M. Henczka, L. Makowski Department of Chemical and Process Engineering Warsaw University of Technology, ul. Waryflskiego 1, PL 00-645 Warsaw, Poland The effects of reactants viscosity on turbulent mixing and related selectivity of parallel chemical reactions are studied experimentally using an unpremixed feed tubular reactor, interpreted theoretically and simulated using a model. The model is based on the multipletime-scale turbulent mixer model and a closure procedure developed previously by the authors. The closure was validated using the method of asymptotic behaviour. Model predictions and experimental data show that increasing of reactant viscosity but keeping the flow rate constant may result in increasing as well as decreasing of the selectivity of parallel reactions. 1. INTRODUCTION An increase of viscosity of components of liquid mixture in a single phase turbulent mixer or reactor affects the flow field structure, the structure of concentration fluctuations, the rate of mixing, and influences this way the course of chemical reaction. In the ease of tubular reactor that we consider in what follows, an increase of viscosity increases frictional resistance of the flow. This increases obviously both: the rate of energy dissipation (s) and the flux of momentum from the flow to the wall. As the turbulent transport of momentum results from velocity fluctuations, one can expect significant increase of the kinetic energy of turbulence, k. One should not expect large effects of viscosity on the integral scale of turbulence, L, as in the case of developed turbulence this value depends mainly on the system geometry. One can thus expect that combined effect of increase of v, (e) and k will increase the Taylor microscale of turbulence ~,g, and decrease the Reynolds numbers Re~ and ReL (based on the Taylor microscale ~.g and on the scale of the large energy containing eddies, L, respectively). Turbulent mixing in liquids can be interpreted as dissipation of concentration fluctuations. A structure of the concentration fluctuation field can be well described by using the concentration spectrum Ec(r,). At large Reynolds numbers there is an inertial subrange of energy spectrum where molecular viscosity effects are negligible. In liquids of Sc >> 1 the effects of molecular diffusivity must be much smaller than the viscosity effects, so related E c depends only on the rate of concentration fluctuation dissipation (6c) and the rate of energy dissipation (6) ; this constitutes the inertial-convective subrange [1][2], with Ec(K)=Co(e.c)(S)-l/3K

-5/3

(1)

where Ce is the Obukhov-Corrsin constant and eq. (1) is valid for r, oc << lc << lcr, where 1ok, is the Kolmogorov wave number r / . r, oc is the characteristic wave number of large eddies (large spots of the tracer) and it is proportional to the reciprocal of the eddy size.

102 In the inertial-convective subrange mixing (equivalent to the spectral transfer) results from convection, which reduces the scales of turbulence and increases the wave numbers. At wave numbers comparable with the Kolmogorov wave number ~:r viscosity becomes important, whereas molecular diffusivity does not yet affect the concentration spectrum due to Sc >> 1. The scales of concentration fluctuations are reduced by viscous deformation resulting from the strain-rate field, forming the viscous-convective subrange that extends between the Kolmogorov wave number ~:r and the Batchelor wave number ~:8, r'B = Kx "Scl/2 (2) At wave numbers K > ~:B we observe the viscous-diffusive subrange and mixing in this subrange occurs by molecular diffusion in deformed slabs. An estimate of the concentration spectrum in the viscous-convective and the viscous-diffusive subranges can be expressed by single equation [3]:

Ec(r,)=Cs,,(v,c) -~

exp--CBa

(3)

Eqs (1), (2) and (3) show two opposite effects of the viscosity increase at constant flow rate through the system: increase of (e) decreases Ec in all subranges, whereas a direct effect of viscosity (through v and Sc) is observed in the viscous-convective and viscous-diffusive subranges (increase of Ec). Depending on the relative importance of these effects one can expect an increase as well as a decrease of the selectivity of complex reactions with increasing the viscosity. One problem, however, arises. The models of turbulent mixing that are used together with CFD codes (including the one employed in this work) are usually based on the assumption of developed turbulence, i.e. they assume that Rex and ReL take high enough values to make the model constants independent of the Reynolds number. When increasing the viscosity a question arises about the model limitations and the range of its applicability. One can expect either complete change of the model structure or, at least, dependence of the proportionality constants on the Reynolds number. A possible way for introducing such dependency is suggested by Sreenivasan [4] for the "longitudinal" concentration spectrum:

Ec(tr x) = C* (e,c)(e)-1/3 xxS/3(xxL) -r+'~

(4)

where C* =C'o.Re -38/4 ~5 is a Reynolds number dependent constant, and 3' is the intermittency exponent. In this paper we investigate effects of viscosity on selectivity of parallel chemical reactions and discuss the model predictions. 2. MODEL OF REACTIVE MIXING In what follows the effects of mixing on the course of homogeneous chemical reactions are modelled using the k-e model to calculate the flow field and the multiple-timescale turbulent mixer model [5] to calculate distributions of concentration variances of the nonreaeting tracer. The employed closure scheme is based on the beta probability distribution of concentration of the nonreacting tracer and interpolation of local instantaneous values of reactant concentrations between the ones characterising the instantaneous and infinitely slow chemical reactions. 2.1. Multiple-time-scale turbulent mixer model In the model we employ the local instantaneous values of chemically and hydrodynamically passive tracer that are expressed in the dimensionless normalised form of "mixture fraction" f

103 0

f = c~ (5) CAO The turbulent mixer model [5] enables to calculate distributions of concentration variances of 2 viscous-convective, (122 and the mixture fraction, f, characterising the inertial-convective, G,, 2 subranges of turbulent spectrum. The model interprets the process of viscous-diffusive, (13, mixing as convection, dispersion and finally dissipation of the concentration variance (is2 (1S - -

--

<:>)'>

,

---(3"I + ( Y 2

,

(6)

"4"(13

The distribution of the average value of the mixture fraction (f) is described using the gradient diffusion approximation

a[

a<s-A+(uj>o<:> : a; Oxj

Ot

+s>'

)o<:>] ~xj I

(7)

The same assumption is used for calculation of the concentration variance components

Ot

q-
/ ff~xj

-""

~j

(Om -Jr.O T )

jj

.4-Rpi - RDi

(8)

for i = 1, 2, 3. In eq. (8) Rei and Roi stand for production and dissipation terms: 9 the inertial-convective variance component G,2 is produced from gradients of (f) by velocity fluctuations

Rei =-2(u'J') 0 ( f ) = 2Dr(O(f}] 2 - -

(9)

and decays due to the inertial-convective reduction of the scales of concentration fluctuations, producing this way c 22 / \ 13 2

Ro~ = Rp2 = ~ = ( 1 ~ R

; R=2

(10)

1; S

9 the viscous-convective variance component c 2 decays due to the viscous-convective shrinkage of slabs forming the viscous-convective part of the spectrum; the variance is transformed this way into c 32

RD2 = Re3 = EG~; E = 0.058

(11)

9 the viscous-diffusive part of the spectrum is dissipated due to the mixing on the molecular scale by molecular diffusion in deforming slabs Rz~3=G6~;

(

G = E 0.303+

17050] Sc )

(12)

2.2. The probability density function (PDF) and the closure procedure PDF of the concentration of nonreacting tracer is approximated by the Beta function at any point of the system fv-l(l_ f)--I 9( f ) = , (13) ~y,,-i (1 - y)"-' dy 0

104

where v =

>[

(f (f)(1-2 ) - I G'S

J

and w =

(I-(f))

I

]

(f)(Ic 2-
Once we are able to express all the local instantaneous concentrations of reactants in terms of the mixture fraction, f, we can use the PDF from eq. (13) to estimate the local average values of the reaction kinetics terms. The mixture fraction f behaves like the inert type composition variable formed with the difference of reactant concentrations. For example in the case of single reaction A + B --+ R one gets [6] f = ca - c B + cB0

(14)

CAo + CBO

and for the parallel reactions A + B + R, A + C --+ S we have [7] f = ca - ( c s +Cc)+CB~ +Cc~

(15)

CAo + CBO+ Cco

It is easy to express the local concentrations as functions o f f in the case of instantaneous reactions when the reactants can not coexist within the same fluid elements [for example when cA > 0, cs = 0 in eq. (14)] as each concentration can be expressed then directly from eqs (14) and (15). When reaction is not infinitely fast reactants can coexist and although eqs (14) and (15) are still valid, they do not determine the reactant concentrations unequivocally. In such a case one can employ the method of linear interpolation of q ( f ) between values for instantaneous reaction c~' ( f ) and for very slow reaction c~( f )

>

(c~)-(c 7' ) [co ( f ) _ c7' (f)]

(16)

The interpolation method is described in detail elsewhere [7][8][9][10]. 2.3. Validation of closure using the method of asymptotic behaviour Consider a system of the competitive-parallel reactions with the average reaction rate terms resulting from Reynolds averaging:

A +B

k, >R; (G)= kl[(cA)(cB)+oo

(17)

C+B

k= >S;

k2[(cB>(Cc)+(c'BC'c)]; k~ -+0

(18)

(r=)=

Let the homogeneous chemical reactions (17) and (18) be carried out in the one-dimensional unpremixed feed reactor (premixture of A and C is mixed with B solution). The reactive mixing has turbulent character and turbulence is homogeneous. Following idea of Toor [11 ] but employing here different notation we express the unclosed terms in averaged kinetic equations (covariances) as follows:

(C'AC'B}

c~176

x,)] = (C~o)-~2o)

cov.ctt

x,)] =

(19)

(20)

Similarly for mixing of the passive, conserved scalar tracer we denote

d t,

(f,),-

_

<s>O -is>) -

<:>0 <:>) O'S

(21)

105 For single instantaneous reaction A + B ~ R in the unpremixed feed system with stoichiometric ratio equal to one (r/A8 = 1) and assuming equal molecular diffusion coefficients for all species and Gaussian PDF forfduring the process, Toor [12] found that -COVaB[t (or xl)]=d2[t (or xl) ] (22) The initial PDF is however not Gaussian [13][14] and takes the bimodal shape initially, that only relaxes asymptotically [t (or x l) ~ oo] to the Gaussian form. This relaxation of PDF was considered by Kos~ily [14] yielding coy As 2 . . . . (23) d ~n Batdyga and Bourne [10] have shown that eq. (23) is only valid for initially symmetrical PDF, which is equivalent to the reactant volume ratio equal to one (or the reactant volume fraction lira

t (or x,)--~oo

(f) = 0.5). However, the case with the volume ratio equal to one is hardly observed in practice; Batdyga and Bourne [10] have found that for any (f) value and for PDF that becomes asymptotically Gaussian, the asymptotic solution for r/An= 1 reads COVAB _ 1 lim d2 - -

2n(f)(1-(f))

t(orxl)--~oo~

(24)

Using the same method as employed Batdyga and Bourne [10] one can extend the solution to characterise the covariances in the case of complex reactions. In the case of the parallel chemical reactions system [eqs (17) and (18)] and for r/,~8= 1, qnc = 1, eq. (24) holds for the first reaction, whereas for the second one [eq. (18)] we get lim ''~

COV

1

~c =

d 2

lim

(25)

2(f)

COVi j --

'-

0.0

c~

t(orxl)~ d 2

-

-- 0,5 2

-"

--

COVAB

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

............

-x-

i

. . . . . . . . . . .

lO -

#

Y

" '~

:

/

...............

.............;"i

j, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II

-2.0 --

~ ~.-';"~ . . . . . .

%. % . . . . .. .. . . . . .

tt

/

', t |

I I

.

-3.0

! !

I I

!

!

!

I

I

I

tI

~

0.0

I 0.2

I 0.4

I 0.6

I 0.8

1.

Fig. 1. Volume fraction effect on the unclosed terms in average kinetic equations.

Relations (24) and (25) are plotted in Fig. 1. Both covariances, COVAB and covBc strongly depend on the volume ratio of reactant solutions, but in a different way. This explains the effect of the reactant volume ratio on the selectivity of parallel reactions [9] and shows why it is so difficult to describe the course of complex reactions using moment closures; the relation between covariances is very complex. Presented theoretical results can be used for validation of mixing models. Fig. 2

106 shows that the closure based on the Beta PDF and presented in Section 2.2 yields proper asymptotic behaviour. It is easy to show that the spiky distributions consisting of some Diracdelta functions do not pass this test. C~ d2

O m

COVBc

2

= 0.5 1 "<'ff>- ~.801

-1

d2

O = 0.5

-1--

-2-

_s___ 0.97t

-2--

-3-

4-

-3lO

-5-

0

T~T

1

2

I

3

I--'--V"---

4

5

0

6

0

1

2

3

4

5

6

0

7

Fig.2. Validation of the closure scheme. In the calculation d 2 = exp(-| | = t/Xm, "l;m being the mixing time constant. Solid lines result from the closure and dashed ones represent asymptotic values at t ~ oo. 3. EXPERIMENTAL Experimental investigation for parallel reactions in a solution with increased viscosity were carried out in a tubular reactor with an i.d. of Dt = 0.032 m equipped with a concentrically located injector tube with an i.d. of dj. = 0.00181 m, the same as employed before by Baldyga and Henczka [7]. Polyethylene-polypropylene glycol was applied to increase the viscosity of the aqueous solutions. A premixture of hydrochloric acid (A = HC1) and ethyl chloroaeetate (C = CH2C1COOC2H5) was introduced over the entire cross-sectional area of the reactor (c~0 = cco = 0.014 mol/dm 3) whereas the solution of sodium hydroxide (B = NaOH) was fed through the concentrically located injector (CBo = 0.45 mol/dm3). High values of the concentration ratio cBO/CAOand CBO/CCOresulted in spreading of the reaction zone in axial direction over a few reactor diameters. During experiments the average mixture velocity varied between 1.25 and 1.87 m/s. The mean velocity in the injector was the same as the mean velocity in the reactor. The viscosities of both mixed solutions were equal to each other in each experiment; adding of the polyglycol increased the viscosity to the values shown in Fig. 3. Experiments were carded out at T = 293 K with kl-+oo and k2 = 23 dm3/(mol's). It was checked experimentally that the rate constant was not affected by presence of the viscosity increasing agent. Similarly it was checked that in the applied range of the polyglycol concentrations its presence hardly affects the molecular diffusion coefficients, so increase of the Schmidt number is in practice proportional to the viscosity increase. The ester concentration was measured before and after the experiments chromatographically (HPLC) and the final selectivity of forming S rather than R from B was calculated from X s = Cc~ -Cc.~ (26) CAO where Cc,o,t represents the average ester concentration at the reactor outlet and cA0 and Cco are average feed concentrations.

107 4. RESULTS AND D I S C U S S I O N The experimentally determined values of selectivity are compared with model predictions in Figs 3 and 4. The 10 9 experimental u" = 1 . 2 5 0 m l s results of simulation were obtained by 9 experimen_tal ~ = 1.875 mls Xs[%] using the model presented briefly in 9 - - - p r e d i c t e d u = 1.250 m / s Sections 2.1 and 2.2 (see also [7][8] p r e d i c t e d u = 1 . 8 7 5 m l s 9 9 8[9][10]). The CFD package FIDAP 8.5 employing the k-e model of turbulence was used to compute 6the mean velocity, turbulent kinetic energy and turbulent energy 4dissipation fields together with balances for the concentration variance components and reactants. A 2grid consisting of 9200 nodes was used in computations. The results presented in Figs 3 and 4 show that 9 I " ! " I " i " there is an agreement of model 0 1 2 3 4 5 predictions with experimental data; although the scatter of experimental data is large, the main trends observed Fig. 3. Effect of viscosity on selectivity of parallel in both experiments and modelling reactions. agree. Fig. 3 shows that at high 10 velocity (u=1.875 m/s) in the 9 e x p e r i m e n t a l v = 2 . 0 4 2 " 1 0 4 m 2 IS X e x p e r i m e n t a l v = 4 . 0 1 4 " 1 0 4 m 2 Is Xs[%] X reactor, effects of the viscous9 p r e d i c t e d v : 2 " 1 0 4 m 2 Is convective and the viscous-diffusive 9 9 . . . . p r e d i c t e d v = 4 " 1 0 4 m 2 Is 8% % micromixing are negligible and the effect of increased rate of energy dissipation dominates. On the other hand at u=1.250 rn/s effect of decreasing of the mixing rate in the viscous-convective and viscousdiffusive subranges by increase v and Sc (and resulting increase of the 2"----.... concentration variance in these subranges) increase significantly the selectivity. Fig. 4. shows that for 0 . , f " ...... I " I " i " viscosities of the solution 1.2 1.4 1.6 1.8 2.0 2.2 v = 2.10 .6 m2/s and v = 4.10 -6 m2/s respectively this change of the trend Fig. 4. Effect of the velocity in the reactor on the is observed roughly at u = 1.75 m/s. selectivity of parallel reactions. The results presented in Figs 3 and 4 show that good prediction of trend was achieved without any correcting of the model parameters. Results of computations obtained with the k-e model enable estimation of the local values of Re~ o

/

\

/Re~ = 2 . 5 8 k / ( ~ ) ; in the considered ranges of the viscosity and velocity variations Re~ values are usually in the range between 40 and 120 reaching locally in some cases as small

108 value as Re~ = 20. In spite of this one does not see any need to correct the model constants although this may result in part from large scatter of experimental data. Interesting that Sreenivasan [4] has plotted the values of the Obukhov-Corrsin constant against Rez in the range of Rez between-40 and 4.104 and concluded that "the Reynolds number trend, if exist at all, is weak and difficult to discern with any certainty from these data". This may explain in part our conclusions. NOTATION

A,B, C,R,S- reagents CBa the Batchelor constant C& C'o the Obukhov-Corrsin constants molecular diffusivity turbulent diffusivity E engulfment parameter Ec(~:) concentration spectrum f dimensionless concentration of nonreacting tracer G molecular diffusion parameter k kinetic energy of turbulence kl, k2 second order rate constant L scale of large energy containing eddies R velocity-to-scalar time-scale ratio Re Reynolds number Re~ Taylor microscale Reynolds number ReL Reynolds number of large scale eddies

Dm Dr

Sc u Xs Xj ~c 11 ~c ~cB ~cK ~c0c ~,g v ~2S~ Xm xs

Schmidt number velocity product distribution Cartesian coordinates rate of energy dissipation per unit mass concentration variance dissipation rate stoichiometric ratio wave number Batchelor wave number Kolmogorov wave number characteristic wave number of large eddies for passive scalar Taylor microscale kinematic viscosity cr~dimensionless concentration variances tithe constant for mixing time constant for inertial-convective mixing dimensionless time

| REFERENCES [1] A.M. Obukhov, Izv. Akad. Nauk SSSR, Ser. Geogr. and Geophys., 13 (1949) 58. [2] S. Corrsin, J. Appl. Phys., 22 (1951) 469. [3] H. Tennekes and J.L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, MA (1972) [4] K.R. Sreenivasan, Phys. Fluids, 8 (1) (1996) 189 [5] J. Batdyga, Chem. Eng. Sci., 44 (1989) 1175 [6] K.T. Li and H.L. Toor, Ind. Eng. Chem. Fundam.,25 (1986) 719 [7] J. Baidyga and M. Henczka, Recent Progress en Genie des Procedes, 11 (1997) 341 [8] J. Batdyga, Chem. Eng. Sci., 49 (1994) 1985 [9] J. Batdyga and M. Henczka, Chem. Eng. J., 58 (1995) 161 [10] J. Batdyga and J.R. Bourne, Turbulent Mixing and Chemical Reactions, Willey, Chichester (1999) [11 ] H.L.Toor, in Turbulence in Mixing Operations, ed. R.S. Brodkey, Academic Press, New York (1975) [12] H.L. Toor, Ind. Eng. Chem. Fundam., 8 (1969) 655 [13] J. Batdyga and S. Rohani, Chem. Eng. Sci., 42 (1987) 2597 [14] G. Kosaly, AIChEJ, 33 (1987) 1998 Acknowledgement- This work was supported by the Scientific Research Committee (of Poland) under Grant No. 3 T09C 012 13

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

109

Mixing of two liquids with different rheological behaviour in a lid driven cavity. H.C.J. Hoefsloot, S.M. Willemsen, P.J. Hamersma and P.D. Iedema. Department of Chemical Engineering, Universiteit van Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands The mixing of two liquids in a lid driven cavity is studied. In the three experiments performed the volume of the liquids is equal. In the base case experiment a 100 ppm polyacrylamide in glycerol, with two weight percent water, is put at the bottom and glycerol with two weight percent water but without polymer is put at the top. In the second experiment the initial positions of the solutions with regard to the base case, is reversed. In the third experiment the 100 ppm solution of the base case is replaced with a 50 ppm solution. To the glycerol water mixture without polymer a fluorescent dye was added to monitor the experiments with a laser induced fluorescence technique. The two-phase flow is simulated with FLUENT using a volume of fluid method. It was found that the experiments and simulations are in good agreement. From both experiments and simulations it followed that mixing is enhanced if the motion of the lid is applied to the most viscous liquid. A second conclusion is that mixing is faster if the apparent viscosities of both liquids are closer. 1. Introduction and previous work The mixing of two liquids with different viscosity is often encountered in polymer production or polymer processing. Especially if the liquids are highly viscous, mixing can be a difficult task. We investigate the early stages of the mixing of two liquids, where the mixing results in the increase of the interfacial area. The lid driven cavity is taken as an example because it can be seen as a crude model for a melt extruder. An extruder is a common type of mixing device for the mixing of highly viscous liquids. To our knowledge the first papers dealing with lid driven cavity flow of two liquids is by Bigg and Middleman [1]. They study both experimentally and computationally the mixing of two fluids in a rectangular cavity with a viscosity ratio of the two fluids between 1 and 30. The cavity is filled with equal volumes of both fluids. The lid of the cavity was moved with a constant speed, creating an increase in interfacial area. The mixing behaviour was studied experimentally by taking still photographs of the fluid containing tracer particles. A marker and cell method described by Harlow and Amsden [2] was used in the computer simulations. Chakravarty and Ottino [3] studied the effect of viscosity ratios and fluid placement on the stretching and deformation of two viscous immiscible liquids, interfacial tension effects are neglected. Two situations are studied: steady lid forcing and timeperiodic forcing. In the article of Chella and Vifials [4] the mixing of two liquids was calculated, the liquids having the same viscosity but having a non zero surface tension. The above-mentioned articles only have investigated Newtonian liquids. Polymer solutions and polymer melt are very often non-Newtonian. This paper is to our knowledge the first to describe the mixing of non-Newtonian liquids in a lid driven cavity.

110 The objective of this work is to describe both experimentally and computationally the mixing behaviour of two liquids with different rheological behaviour. It will be shown that the rheology of both liquids and the spatial distribution at the start of the experiment is important in the mixing process. Results are primarily presented in the form of still video pictures and contour plots of volume fractions. The remainder of this paper is organised as follows. First we give a problem description with the definitions. Then we describe the experiments and the numerical method. In section 4 the results of both experiments and simulations are presented and we end with discussion and conclusions.

2. Experiments All the experiments where performed in a rectangular cavity of plexiglass. The dimension of the cavity are, a width of 0.15 m, a height and a depth of 0.1 m, see figure 1. The band driving the flow in the cavity has a width of 0.08 m. The band had a constant velocity of 0.036 m/s during all experiments.

Figure 1, the experimental set-up, a 3_D cavity and the symmetry plain at x=0.075. The liquids used were 50-ppm or100-ppm polyacrylamide in glycerine with 2w% water and glycerine with 2w% water containing Creyl Violet Perchlorate, which is fluorescent dye. For the preparation of the fluorescent solution, 10 mg of the dye was dissolved in 25 ml of de-mineralised water and then mixed into the glycerine. Rheological parameters for the polyacrylamide solution were determined with a Haake rheometer. A Carreau model was used to describe the viscosity [5].

u---u|

u|

+ (A)~)2]~

The parameters for the solutionswere: Zero shearviscosity Infiniteshearviscosity Power-Law index Time constant

pm poly. yl.midr01tion

3.4 mZs"~ 0.66 m2s"~ 0.8 275 s

50 ppm polyacrylamide 1.4 m2s "~ 0.66 m2s-1 0.6 200 s

111 The dye solution turned out to be Newtonian with a viscosity of 0.60 Pa s. The density of all the three liquids that were used is 1254 kg m"3. The cavity was filled halve with one of the liquids and the second liquid was put on top carefully to get a horizontal interface between the two liquids. When the cavity was filled completely the band was lowered onto the upper fluid and the motor driving the band was started. At that moment the camera was already running. With the optical set up a laser sheet is created, the laser used is a 15x10"3 Watt He-Ne laser with its maximum emission at 632 nm. The dye absorbs in the area of the laser emission wavelength and has a fluorescence wavelength at 650 nm. The laser sheet is positioned in the plane of symmetry. A CCD camera, with dismounted red filter, placed perpendicular to the laser sheet is used to record the mixing process.

Figure 2 the optical set-up The amount of dye is crucial for the recordings, if too much dye is used there are areas where the laser light can not penetrate because already all the light has been absorbed. When too little dye is used the contrast between the two liquids is bad. A frame grabber is used to record the images on a Personal Computer for further analyses. The liquid with the dye (and without the polyacrylamide) could be seen as red on the pictures. These pictures are transformed into black and white, white representing the liquid with the dye. 3. Numerical method The Navier-Stokes equations are solved with the finite volume formulation. The location of the interface between the two fluids is calculated using a Volume of Fluid (VOF) method. For the calculations Fluent 5.1.1 is used. In the two dimensional simulations the domain of interest is a square. The grid is refined towards the top of the cavity, and cells at the bottom have a height of 1.05 mm and at the top a height of 0.2 ram. The simulation were performed With a ~;rid size of 102 by 152 cells, the time step is set to 2.5 10"3s. Convergence criteria of 5.0 10-~ for continuity and 5.0 10"4 for velocities are used. Calculation times for a grid of 102 by 152 cells is about 500 time steps per hour with a 450 MHz Pentium II with 256 MB memory. Several tests are performed to check the numerical accuracy. The number of grid cells in both directions is doubled and time step as well as convergence criteria are decreased. With all these test only minor differences could be detected in the contour plots of the volume fraction.

112 The three dimensional simulation was run with a grid in x-, y- and z-direction of respectively 50, 150 and 16 cells. In the z-direction the symmetry of the problem is used. Grid refinement in y-direction is the same as in the two-dimensional case. All the other simulation parameters were identical to the two-dimensional case. The simulation for 20 seconds real time, took about one week. 4. Results

Three different experiments were performed. The base case is that the liquid on top is glycerine with 2w% water with dye solution and at the bottom a 100 ppm polyacrylamide in glycerine with 2w% water. In the figures the solution containing the dye is represented by a white colour. The experimental and numerical results are presented in the same figure, the left picture is the still video and the fight picture is the simulation result, both at the same time. There is a reflection from the top of the apparatus and in the experiment this can be seen as a line in the top of the picture of the experimental results. The agreement between experiment and simulation is good. From the experiment it can be seen that illumination on the fight top comer is a problem. This is because the illumination is from below and much of the light is already absorbed, because of the spatial distribution of the dye, before it reaches the upper fight comer.

Figure 3, the mixing pattern after 10 seconds, on the left the experiment and on the right the calculations

Figure 4, the mixing pattern after 20 seconds, on the left the experiment.

113 Qualitatively there is a good agreement between experiment and simulation, but there are differences. The simulation is ahead of the experiment. But the thickness of the striations seems also to be affected. But it is difficult to judge if this is caused by the fact that the simulation is ahead or by some other effect. Despite of the problem with the illumination in the upper right comer it can be seen in figure 5 that the simulation is ahead of the experiment.

Figure 5, the mixing pattern after 30 seconds for the base case, on the left the experiment.

Figure 6, the mixing pattern after 40 seconds for the base case, on the left the experiment. The last comparison between experiment and simulation is done after 40 seconds, figure 6. The similarity is still good, further simulations show (not presented here) that this holds for even longer times. The second case was performed to investigate the influence of the rheology. The difference with the base case is that the 100-ppm solution was replaced by a 50-ppm polyacrylamide solution. The parameter most affected is the zero shear viscosity of the polymer solution. The results can be seen in figures 7 and 8. The similarity between experiment and simulation is excellent with the same observation as in the base case, that the simulation is somewhat ahead of the experiment. Comparing the second case with the base case it can be seen that the shape of the interface is not effected very much. The simulation result of the second case after 15 en 30 seconds resemble remarkably the experimental results of the base case after 20 and 40 seconds. The

114 same holds for the simulation result after 30 seconds of the base case and the experimental result after 30 second of the second case.

Figure 7, the mixing pattern alter 15 seconds for the second case (black represent a 50 ppm polyacrylamide solution), on the left the experiment and on the fight the calculations

Figure 8, the mixing pattern after 30 seconds for the second case, on the lef~ the experiment. In the third case the effect of spatial distribution is investigated. The ordering of the layers was reversed with respect to the base case, at the start of the experiment the liquid at the top is the 100-ppm polyacrylamide solution and the liquid on the bottom is the dye solution. The agreement between experiment and simulation is not so good as in the two other cases. Not only is the simulation running ahead but also the predicted shape, especially after longer simulation times is not as good as in the other two cases. In the simulation of the third case the dye solution comes much closer to the bottom of the cavity than in the experiments. The thickness of the striations is predicted better then in the first two experiments. The thickness of the horizontal striation of the polymer solution near the top of the cavity is almost equal in the experiment and the simulation. For the first two experiments this was not the case. Some parts seem to come loose from the main part of the polymer solution; this is probably due to the fact that the second fluid is not placed carefully enough on the second one.

115

Figure 9, the mixing pattern after 10 seconds for the third ease (reversed ordering of the layers), on the left the experiment and on the fight the calculations.

Figure 10, the mixing pattern after 15 seconds for the third case, on the left the experiment. As a further test a simulation for the base case of a three dimensional domain was performed in order to see if three dimensional effect could be the cause for the simulation being faster then the experiment. In Figure 11 the mixing pattern at the plane of symmetry is presented for the three dimensional simulation.

Figure 11 the simulated mixing pattern at the plane of symmetry after 5, 10, 15 and 20 seconds using a three dimensional grid. Comparing figure II with figure 3&4 it can be seen that the results using a three dimensional grid do not differ much from the results from the two dimensional grid. In the

116 three dimensional simulation the effect of larger grid cells can be observed. Especially at the left wall the shape of the interface between the two fluids is very rugged. 5. Discussion and Conclusions

The simulations are in good agreement with the experiments although the simulations run ahead somewhat. Several effects could cause this. First at the start of the experiment, the velocity is not immediately at a constant value like in the simulations. Secondly the measured velocity of the band has an error of about 3%. And thirdly the error in the viscosity model has an influence on the simulations as can be seen from the results of the base and the second case. Both simulation and experiments show that the mixing in the base case is slowest. The 50ppm polyacrylamide solution has a lower zero shear viscosity than the 100-ppm solution and therefor in the second case the mixing is faster compared to the base case. Applying force to the most viscous liquid results in better mixing. This is the reason that the third case where the polyacrylamide solution is on top mixes faster than the base case. The volume of fluid method gives a good performance as long as there is a sufficient number of grid cells within one striation. If the number of cells in a striation is to low break up occurs in the simulation while this is not seen in the experiment. This type of research will help to gain insight if and when CFD software is useful in mixing problems. It also will help to understand the mixing process of two non-Newtonian liquids in industrial equipment like extruders. For the mixing at later stages of the mixing process sub grid models are necessary. It seems strange that in the laminar mixing regime the methodology to incorporate mixing models with reaction is not as advanced as in the turbulent regime. We foresee a development where CFD calculations are used in lamellae mixing model to predict reactor performance. Further an increase in the use of so-called mesoscopie modelling tools like latticeBoltzmann (LB) and dissipative particle dynamics (DPD) will play a roll in the future. It is already shown that for some applications these simulation tools are performing better then a finite element method [6].

Acknowledgement We thank R. E. van Vliet, J.M. Bergink, D. Vidal, M. Springer, D.C. Visser and S. de Lindt for their help. 6. References

1. Bigg, D. and Middleman, S., 1974, Laminar mixing of a pair of liquids in a rectangular cavity. Ind Engng Chem. Fundam. 13, 184-190. 2. Harlow, F.H. and Amsden, A.A., 1970, The SMAC method. Los Alamos Scientific Laboratory. 3. Chakravarthy, V.S. and Ottino, J.M., 1996, Mixing of two viscous liquids in a rectangular cavity, Chem. Eng. Science. 51, 3613-3622. 4. Chella, R. and Vifials, J., 1996, Mixing of a two-phase fluid by cavity flow. Phys. Rev. E. 53, 3832-3840. 5. Fluent Manual, July 1998. 6. D. Kandhai, D.J.-E. Vidal, A.G. Hoekstra, H.C.J. Hoefsloot, P. Iedema and P.M.A. Sloot, 1998, A comparison between Lattice Boltzmann and Finite-Element Simulations of Fluid Flow in Static Mixer Reactors, Int. J. of Modern Phys. C., 9, 1123.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

117

MOBILIZATION

OF COHESIVE SLUDGE IN STORAGE JET MIXERS Michael R. Poirier a, Hanna Gladki b, Mike R. Powell ~, Phillip O. Rodwell ~

TANKS

USING

a Westinghouse Savannah River Company, Aiken, South Carolina b ITI" Flygt Corporation, Trumbull, Connecticut c MesoSystems Technology, Inc., Richland, Washington d Bechtel SavannahRiver, Aiken, South Carolina Summary Sludge resides on the floors of many U.S. Department of Energy (DOE) radioactive waste storage tanks. Sludge is composed of fine-grained particles cohesively bonded together by both colloidal and mechanical forces. Retrieval of sludge from these tanks is difficult - access is limited and radiation exposure concerns require remote operation. This paper is a study of the relationship between sludge properties and the mixing system requirements in order to improve the design of a sludge retrieval system. Tests were performed in which simulated radioactive sludge that was suspended and mixed using submerged, horizontal fluid jets. The fluid jet was generated by a free jet flow agitator (FJFA). Tank diameters used for the tests were 0.45 m, 1.8 m and 5.7 m. Sludge shear strengths ranged from 10 Pa to 2000 Pa. The shear stress needed to mobilize most of the sludge simulants is correlated to the undisturbed sludge strength. The average wall shear stress generally required to mobilize 80% of the sludge is approximately 5.0% of the measured shear strength. Mixing system performance is predicted by estimating the fraction of the tank floor where the applied shear stress exceeds the sludge's estimated critical shear stress. Computational modeling of the floor's shear stress distribution in our 5.7 m diameter tank test yields sludge mobilization predictions reasonably consistent with our results. Introduction The Pacific Northwest National Laboratory (PNNL), Savannah River Site and ITT Flygt Corporation conducted a joint mixer testing program sponsored by the DOE to evaluate the applicability of Flygt mixers to nuclear tank waste retrieval. Testing was done in three different size tanks so that a scaling method could be evaluated, validated and the results used to make a full-scale mixer performance prediction [1]. Flygt Submersible Mixers consist of an electrically powered 3-bladed propeller with a close-fitting shroud. (See Fig. 1.) Depending on the size of the tank, different sized or propeller numbers were used. A propeller spinning rapidly will create a turbulent jet. !

EXPERIMENTAL TESTING This presentation focuses on testing the mobilization of the sludge simulant composed of fine-grained particles Fig. 1 I T T F l y g t M i x e r cohesively bonded together by both colloidal and mechanical forces. To achieve the right property of the simulant, kaolin clay was used. In some cases, bentonite clay was

118

added to increase the strength of the sludge fractionally due to cohesion rather than friction. Shear strength zs is the maximum shear stress a material can withstand without rupture. For sludge, this is most readily measured using a shear vane. The cohesive sludge suspension tests revealed the relationship between the average wall stress to [2] and the stress required to induce mobilization (i.e., the critical shear stress "co). When shear stress is applied to the sludge surface by a turbulent jet which exceeds the zc value of sludge, erosion occurs. The acceptable degree of mobilization was arbitrarily chosen at the 80% level. The fraction of mobilized sludge was computed based on a mass-balance calculation and measured slurry density. During testing, the mixer speed was gradually increased while the density was monitored by a Krohne Model 2000-300P meter. Periodically, the samples were withdrawn using a 100.0ml pycnometer. When the solids mobilization reached 80%, the thrust and average wall shear stress was determined. At the end of each test, the remaining sludge was collected in a steel drum to determine its volume and weight. The tank diameters used for the tests were 0.45m, 1.8 m and 5.7m. The tests were done in two Phases, A and B. In each phase, the number of runs varied with the sludge/simulant theological properties such as the strength and yield stress. In the entire test, the sludge shear strengths ranged approximately from 10 Pa to 2000 Pa. The small scale test, Phase A (Runs #1 to #7), was performed in a 0.45cm diameter Plexiglas tank with the water level inside the tank adjusted to 17.7cm or 25.7cm. The small scale mixer has a 3-bladed propeller with a 7.8cm diameter. A load cell was used to measure the axial thrust produced by the small-scale Flygt mixer. The mixer was positioned horizontally, close to the wall and as close as possible to the bottom, aimed towards the opposite wall but about 10~ to the left of the tank's centerline. For the purpose of data analysis, a subset of the mixer thrust vs. mixer speed data were fitted to the mixer thrust affinity law. The test Phase B (Runs #1 to #3) was performed in two different tank sizes. The smaller tank was 1.8m in diameter and larger tank was 5.7m in diameter. In the 1.8m diameter tank, one Flygt mixer (Model 4640) was used (Fig. 1). At full load, the mixer motor provides 4hp at the shait and generates 810N of thrust. The mixer propeller-blades are 37cm in diameter and have a blade angle of 11 degrees. The three blades are mounted inside a jet ring, which directs the flow of liquid through the blades. The mixer is mounted to "a floor mounted mixer stand". In the 1.8m diameter tank, the mixer configuration corresponds to the arrangement in the small scale. In the smaller 1.8m diameter tank, the measured simulant shear strength was 405+83Pa (Test 1B)and 2070~600Pa (Test 2B). Shear strength measurements were made several days after the test, but the shear strength of the kaolin or kaolin/bentonite based simulant does not change with time provided that the water content is maintained [3, 4]. The relatively large uncertainty in the shear strength measurements results primarily from the variations of measurement taken from various locations in the tank. In the 5.7m diameter tank (Test Phase 3B), three Flygt Model 4640 mixers were used. Their characteristics are as depicted above. The configuration of these three mixers corresponds to the expected deployment orientation in the actual waste tank. The mixers are positioned with their backs 2.65m from the tank center and at60 ~ 120~ and 300 ~ The 120 ~ and 300 ~ mixers are directed 30 ~ to the left from the diagonal line between them. The 60 ~ mixer is about 10~ away from the centerline. In this case, the sludge simulant strength within the tank was 363 + 50Pa.

119

SLUDGE MOBILIZATION TEST RESULTS Small Scale Test Phase A.The sludge mobilization test results are shown in Table 1. There is some uncertainty in the sludge properties for TestslA and 2A. In Tests 1A and 2A, the target shear strength and yield stress were both 50Pa. A 43.5 % concentration by weight kaolin clay slurry was prepared for these tests. Previous testing has shown that this concentration of kaolin has a shear strength and disturbed yield stress of 50Pa [3]. The sludge strength was likely decreased to some unknown extent. Judging by the thickness of the sludge layer just prior to testing (3cm), the average weight percent solids throughout the layer is estimated to have been 31.7wt%. Previous testing has shown that a kaolin slurry with this concentration has a yield stress of about 7Pa. It is likely, however, that the fraction of sludge nearest the tank floor had a concentration greater than 31.7wt% but less than or equal to 43.5wt%. Table 1 Small Scale Slud$e Mobilization Test phase A (Tank Diameter 0.45m) Xo for 80% Sludge Properties " Test no. Mobilized x_~ ~ ~._o.80%) / x_~ (Pa) (Pa) (Pa) 1A 4.8 7<Xy<50 7<xs<50 0.1 to 0.7 2A

<9.5

~ 2.4

~ 2.4

<4.0

3A

9.5 to 15

100

100

0.095 to 0.150

4A

4.7

100

100

0.047

5A

50

100

800

0.063

6A

69

150

2000

0.035

7A

<4.7

10

10

<0.47

Notes: Xo is the average wali shear stress, which equals total mixer thrust divided by wetted area. x y is the sludge dynamic yield stress and xs is the vane shear strength. Test no.2A was performed by letting the fully mixed slurry from Test 1A settle to form a layer roughly 3.8cm thick. The average kaolin concentration in this slurry layer is " ~ lOO v estimated at 26wt%. The yield stress of a ~5 26wt% kaolin slurry is approximately 2.4Pa. Again, there may have been a -8 ~ 6o significant vertical concentration gradient through the layer, so the yield stress of ~ ,o the simulant nearest the floor was $ probably higher than 2.4Pa. Tegt The data given in Table 1 are o9 o plotted in Fig.2. Fig.2 shows the effect of ~• o ) o o simulant shear strength on the average Shear Strength (Pa) wall shear stress required to reach roughly [3

0

0

Phase A T e t l

I"1

Phase B

, ,

Fig. 2 Effectof Shear Strengthon Sludge Mobilization

120

80% of the sludge mobilized. It appears that the required wall shear stress is roughly 5% of the sludge simulant shear strength. This estimate is very rough, however, because so few data points are present. By comparison, the effect of disturbed simulant yield stress on the average wall shear stress required to mobilize 80% of the sludge shows that the disturbed simulant yield stress does not provide as good a measure of sludge mobilization resistance as does the undisturbed shear strength. This is consistent with previous mobilization testing from literature [4, 5]. It can be expected that by allowing more time at given mixer speed a larger amount of sludge would be mobilized than was measured. The magnitude of this increase, however, is not known. Larger Scale Test Phase B. The sludge mobilization test results from the Phase B are shown in Table 2. Test 1B and 2B (1.$m diameter tank): In both cases, an 8cm thick layer of kaolin clay or kaolin with addition of bentonite clay was placed on the tank bottom and covered with water until the liquid level reached 70.6 + 1 cm. The 405 Pa shear strength simulant was nominally 57% kaolin clay and water. The 2070Pa shear stress simulant was composed of kaolin clay and bentonite clay. The bentonite clay was added to the mix to increase a fraction of the sludge strength due to cohesion rather than friction.

800

1.04

eO0 ~ .

g

1.03

u, c ~

g

0.8

t ~

120

+•

700

o ,,,s ~

0.6

~

O.4

u

0.2

m

1.02 - 4OO t.01

J 0

~

i

3OO

J

' i

t

~

50

100

150

200

w

w

250

300

100 350

Time (minutes)

Fig. 3 SlurryDensity & Mixer Speed (Test 1B) (Tank l.Sm, Sludge405Pa)

0.0

i

r

,

,

i

i

l

0

50

100

150

200

250

.....

O

i 300

I$0

Tirol (mtnmmm)

Fig. 4 Fraction Sludge Mobilized and Average Wall Shear Stress (Test 1B) (Tank, 1.8m,Sludge405Pa)

The mixer speed was gradually increased while the density was monitored (Krohne Model 2000-300P). The density and mixer speed versus time data are given in Figures 3 and 5. The mixer was allowed to run in each case for an extended period at some speed to ensure that the sludge mobilization had effectively stopped. In Fig.5 (for 2070 Pa sludge), a disruption in the continuity of the density data can be noticed due to excessive air entrainment in the slurry. Agreement between the pycnometer and the Krohne density meter is excellent during most of the test. Figures 4 and 6 give the fraction of sludge mobilized (as computed from the slurry density) and the average wall stress versus time. From this plot, it is estimated that a mixer speed of about 500 + 40 RPM was needed to mobilize 80% of the 405Pa shear strength sludge, and about 800RPM was needed to mobilize 80% of the 2070Pa shear strength sludge. The average wall stress at this mixer speed is shown in Table 2. These results are similar to those observed in the 0.45m small tank during Phase A.

121

Fig. 5 SlurryDensity& Mixer Speed (Test 2B) Tank 1.8m,Sludge2070Pa

Fig. 6 Fraction SludgeMobilized & Average Shear Stress(Test 2B) Tank 1.8m, Sludge2070Pa

Test 3B To evaluate the effects of scale-up on sludge mobilization using Flygt mixers, a kaolin-clay based sludge simulant was placed in the 5.7m diameter tank. The shear strength and composition of this simulant were nominally the same as the one used for Test 1B in the 1.8m diameter tank. The actual shear strength and solids concentration were slightly lower than that used in Test lB. The mobilization behavior of kaolin clay in this strength range has shown to not be a strong function of shear strength [4]. The shear strength measured for this sludge was at 363 + 50Pa and the solids concentration was 56.1% by weight. When the sludge was spread on the bottom, water was added slowly to avoid disturbing the sludge until the liquid level reached 2.0 + 0.02m.

Fig. 7 SlurryDensity & Mixer Speed (Test 3B) Tank 5.7m,Sludge363Pa

Fig. 8 SlurryFraction of Sludge Mobilized & Average Shear Stress(Test 3B) Tank 5.7m,Sludge363Pa All three mixers were operating at 430RPM. Little increase in the slurry density was observed over 30 minutes, so the mixer speed was increased to 570RPM. Again, little increase was observed, so the mixer speed was adjusted to 720RPM after an additional 25 minutes. Operation at this speed continued until 6 hours had elapsed from the start of the test. The slurry density was still

122 increasing slowly, but only about 20~ of the sludge had been mobilized. The mixer speed was then set to 860RPM. The mixer speed was maintained for 24 hours and the slurry density stabilized at about 1.009g/cm3. This slurry density corresponds to approximately 55% of mobilized sludge. The density and mixer speed data for this test versus time are plotted in Fig. 7. The fraction of mobilized sludge and the average wall stress data are provided in Fig. 8. After 24 hours of mixing at 860 RPM, the liquid level was decreased from 2.0 to 1.0m. This reduction in fluid volume increases the applied shear stress. Full speed mixer operation continued under this new condition for an aditional 15 hours. During this time, the density increased to nearly 1.022g/cm 3 and the fraction of mobilized sludge reached about 85% (based on the total amount of sludge simulant initially placed in the tank). With the mixers continuing to operate, the retrieval pump was activated and remaining slurry pumped out. The remaining sludge was collected in the steel drum to determine its volume and weight. It was estimated that 90% of the sludge was mobilized and removed from the tank. The discrepancy between the 85% value indicated by the density measurements and the 90% value is likely due to a combination of experimental uncertainty and the additional sludge mobilized as the liquid level decreased when the slurry was pumped out of the tank. The average wall shear stress required to result in 80% of the sludge mobilized is estimated to be about 53 Pa. The shear strength of the sludge simulant used in this test was measured at 363 + 50 Pa. Assuming an uncertainty in the average wall stress estimate about 10 Pa, the ratio between applied average wall stress and sludge shear strength required to achieve 80% mobilization is 14.6 + 3.4%. This ratio is somewhat higher than the ratio observed in the 1.8-m tank. All the test results from the phase B are shown in Table 2. Table 2 Large Scale Sludge Mobilization Test Phase B ii

' Test no. IB _ 2B 3B

' Tank Dia. (m) 1.8 1.8 5.7

Xofor 80~ Mobilized (Pa) .... 42 + 7 85 53 + 10

Shear Strength xs (Pa) 405 +_83 2070 + 600 363 +_50

(Xo 80%) / x s (Pa) 0.10 + 0.03 0.041 + 0.012 0.146 _+0.034

Final Remarks

The sludge mobilization process is qualitatively different than the suspension of rapidly settling solids. Therefore, a different analysis approach is used. For our purposes, the sludge is composed of very small, cohesive particles. To mobilize a given piece of sludge away from the sludge interface, the local fluid velocity (or shear stress) must exceed some minimum value for some minimum amount of time. The exact value of the required shear stress and time is a function of the sludge properties and to some extent the slurry properties. Once these criteria are met, a particle or group of particles (flake) is removed from the sludge surface thereby exposing the particles underneath to the erosive action of the flowing slurry. Then the process is repeated. The dislodged sludge pieces tend to be small, usually millimeter sized and smaller. These flakes of sludge are rapidly ground down to particles and clumps of particles with sizes on the order of tens of microns and smaller, as the flakes circulate through more strongly mixed regions of the

123

tank (i.e., through the mixer propeller). From there, the particles slowly settle and only a small amount of agitation is required to maintain them in suspension off the tank floor. Thus, once a piece of sludge is dislodged from the bulk, it effectively no longer presents a difficult challenge from a mixing perspective, if sufficient agitation exists in the tank to ensure that the dislodged pieces are broken down into very small, slow settling particles that do not flocculate. For sludge mobilization, the challenging step in the process is dislodging small pieces of the sludge from the bulk of the sludge layer. The subsequent mixing is usually comparatively easy and is ignored in the analysis. A more rigorous analysis of the Phase A and B sludge mobilization test data required that the shear stress distribution be determined for each of the tests and for the planned full-scale application. In general, such an effort was beyond the scope of the present work. However, in one case only (for Test 3B, Tank diameter 5.7m) the floor's shear stress distribution from computational modeling was done [7]. From the results on Figs. 9 and 10, it is apparent that for two different water levels (lm and 2m), sludge mobilization predictions are reasonably consistent with test results. According to these diagrams, the percentage of area covered by a shear stress of 1.5Pa and higher (for 2m and lm water depths in the tank) complies with the percentage of mobilization in a 5.7m tank (about 55% for the 2m depth and 80% for the l m depth).

Fig. 9 Bottom Shear Stress in the Tank with 3 Mixers and 2 m Liouid Level (TankDia. 5.7m)

Fig. 10 BottomShear Stress in the Tank with 3 Mixers and 1 m Li0uid Level (Tank Dia. 5.7m)

CONCLUSION Sludge mobilization occurs whenever the applied shear stress at the surface of the sludge exceeds some minimum value. This minimum shear stress for mobilization is called the critical shear stress and is related to the physical properties of the sludge. The effect of the decreasing liquid level results in an increased average wall shear stress, which will result in a greater degree of sludge mobilization. Previous work [4, 6] with sludge simulants has shown that critical shear stress can be

124 correlated with sludge shear strength. As a rough approximation, the critical shear stress is on the order of 1/200th of the shear strength. The approach used in the Phase A and B sludge mobilization analyses relates the sludge shear strength to the average wall shear stress (the average shear stress xo is equal to the total mixer thrust divided by the wetted surface area of the tank) required to achieve 80% of the sludge mobilized. The 80% level was arbitrarily chosen to represent an acceptable degree of mobilization. The same criterion was applied to all the tests for a more direct comparison. This approach does not consider the changes in shear stress distribution on the sludge's surface, or the extent of sludge mobilized with changes in mixer speed and number of mixers. Tests using Flygt mixers to mobilize and mix cohesive sludge simulants were performed in both Phase A and B tests. Phase A testing indicated that the average wall stress xo required to achieve 80% of the sludge mobilized was 5.0 + 1.5% of the sludge's shear strength when the strength was greater than 100Pa. The findings of the Phase B tests are consistent with those of Phase A (all results are in Fig. 2). The 2B test used kaolin/bentonite/water sludge simulant with a shear strength of 2070 + 600 Pa. The "co required to achieve 80% mobilization was found to be 4.1 +_ 1.2% of the shear strength. This result is consistent with the 6A test in a small tank in which kaolin/plaster sludge simulant with a shear strength estimated to be 2000Pa required a xo of 3.5% of the sludge's strength. Phase B Tests 1B and 3B used kaolin/water sludge simulant with shear strengths in the 400Pa range (Test IB was 405 _+83 Pa and Test 3B was 363 + 50 Pa). The average wall shear stress required to reach 80% mobilization in Test 1B (1.8m diameter tank) was 42 + 7 Pa and an estimate Xo value in Test 3B (5.7m diameter tank) was 53 Pa. Thus, in Test 1B, the average wall stress required to achieve 80% mobilization was 10 + 3% of the sludge's strength. In Test 3B, the required stress was estimated at 14.6 + 3.4% of the sludge's strength. A more accurate and complete understanding of the mixer sludge mobilization process can likely be attained by considering the shear stress distribution within the tanks used for testing and full-scale tanks. In the meantime, a rough estimate of the total mixer thrust required to achieve 80% sludge mobilization is given by multiplying the sludge shear strength by 0.10 _+0.05. REFERENCES

1.PoweUM.R., FarmerJ.R, GladkiH., Hatchell B.K., JohnsonM.A, Porier M.R., RodwellP.O., Evaluationof Flygt Mixers for Application in SavannahRiver Site Tank (Summaryof Test results from Phase A, B, and C Testing), Pacific NorthwestNational Laboratory, Richland, Washington,PNNL-12168, 1999. 2. Gladld H. PowerDissipation, Thrust Force and Average Shear Stress in the MixingTank with a Free Jet Agitator, Industrial MixingFundamentals, AIChE, New York, Vol. 91, p. 146, 1995. 3. PoweUM.R., Golcar G.R., Geeting J.G.H., RetrievalProcess Developmentand EnhancementsWaste Simulant Compositions and Defensibility. PNNL-11685, Pacific NorthwestNational Lab., Richland, Washington, 1997. 4. Powell M.R., Gates C.M., Hymas C.R., SprecherM.A., Morter N.J., Scale SludgeMobilizationTesting, PNNL-10582, Richland Washigton, 1995. 5. Graf W.H., Hydraulicsof:SedimentTransport, McGraw-Hill, New York, 1971. 6. NearingM.A., Parker S.C.,Bradford J.M., Elliot W.J., Tensile Strengthof Thirty-Three SaturatedRepacked Soils, Soil Sci. Am. J. 55:1546-1551, 1991. 7. Saunders S., CFD as a Mixing Application Tool, Scientific Impeller, ITT IndustryFLYGT, no.5, 1998.

10 th European Conference on Mixing H.E.A. van den Akker and d.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

125

CFD Simulation of Particle Distribution in a Multiple-impeller HighAspect-Ratio Stirred Vessel G. Montante ~b, G. Micale b, A. Brucato b, F. Magelli a a Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali Universith di Bologna - Viale Risorgimento 2- 40136 Bologna (Italy) b Dipartimento di Ingegneria Chimica dei Processi e dei Materiali Universit~ di Palermo - Viale delle Scienze - 90128 Palermo (Italy) Fully predictive simulations of solid-liquid suspensions in a high-aspect-ratio, multiple-impeller stirred tank are presented. These were performed by using the Inner-Outer impeller modelling technique coupled with the Multi Fluid Model (MFM) for the treatment of the dispersed phase. The strongly simplified Settling Velocity Model (SVM) was also tested. The effects of free-stream turbulence on the drag coefficient CD and particle settling velocity were accounted for by means of a recently proposed correlation. Comparison of simulation results with experimental particle concentration profiles shows that the MFM approach leads to fair agreement with experimental data. Results obtained by the much simpler SVM are found to be in similar agreement with experiment, but at the cost of a much smaller computational effort. It is finally shown that one of the main sources of discrepancy between experiment and simulations lies in the underestimation of the pumped flow rate, a feature usually observed also in fully predictive single-phase simulations. This implies that improvements in the reliability of fully predictive two-phase simulation can be expected from advances in the single-phase simulation capability. Apart from that, the presently available two-phase models appear to be sufficiently accurate for engineering purposes, at least for the dilute two-phase systems here considered. I. INTRODUCTION Multiphase flows are frequently encountered in many unit operations of the process industry. Despite this, little information is still available on their CFD simulations, a particularly unfortunate lack of knowledge if one considers that multiphase operations are a major source of troubles in the design of industrial units and the ability to run realistic simulations would be very beneficial. As a difference with the case of single-phase flows, for multiphase flows reliable and extensively tested models for CFD simulations are not available yet. Apart from the difficulties arising from the Reynolds decomposition of the relevant transport equations, which results into more terms that need to be modelled, the complexities of particle-fluid and particle-particle interactions are to be taken into account. As a matter of fact, in the case of

126 very dilute slurries, particles move in response of the fluid motion without practically affecting it ("one-way" coupling). For denser systems the influence of particles on the turbulence structure and/or particle-particle interactions should be taken into account ("twoway" and "four-way" coupling effects, [1 ]). Clearly modelling difficulties and computational effort increase with particle concentration and at present the computations are limited in practice to relatively low particle concentrations. For the industrially important case of solid-liquid stirred tanks, flow models and numerical methods based on either a Eulerian or Lagrangian treatment of the solid phase have been proposed. Bakker et al. [2] investigated the relationship among the liquid flow field and the solid particle distribution in a stirred tank. The single-phase (liquid only) flow field was simulated first by imposing experimentally derived velocity and turbulence quantities at the impeller periphery. The flow field information so obtained was then employed for the computation of particle spatial distribution by solving particle conservation equations. Particle velocity was simply assumed to coincide with the particle free-settling velocity in the still liquid. Good qualitative agreement with experiment was claimed for the predicted particle distribution, but no quantitative comparison was attempted. Later on Myers et al. [3] found that this simulation strategy was not quantitatively accurate, while a fully-coupled Eulerian approach gave rise to better predictions. Decker and Sommerfeld [4] simulated the particle distribution in a stirred tank by employing a Eulerian-Lagrangian approach. The fluid flow field was predicted by the Eulerian approach and the trajectories of a certain number of particles were calculated by a Lagrangian approach, while neglecting "two-way" and "four-way" couplings. The resulting fluid flow field was found to be in good agreement with the experimental data, but a similar comparison was not made for the predicted particle distribution. Brucato et al. [5, 6] presented CFD simulations obtained with both an approach termed "Settling Velocity Model" (SVM), similar to that used in [4], and a more rigorous "Multi Fluid Model". The investigated cases were a standard stirred vessel [5] and a multiple impeller stirred tank [6]. In both works the IBC method was employed for the stirrer simulation and satisfactory agreement with experimental particle concentration profiles was observed. The most recent contribution to CFD simulations of solid-liquid distribution in stirred tanks are due to Sha et al. [7] and Barrue et al. [8] who adopted similar approach as Brucato et al.

In the present work, the fully predictive "Inner-Outer" method for simulating the impeller is applied to the same high-aspect ratio multiple-impeller vessel previously investigated, in order to ascertain whether fully predictive simulations of two-phase stirred vessels are viable. Also, a new correlation for the estimation of the effects of turbulence on the particle drag factor is tested. 2. CFD SIMULATIONS In the present work, a Eulerian framework was adopted for the solid phase, a simulation strategy often referred to as "Eulerian-Eulerian", to point out the Eulerian treatment of both phases. In particular the Multi Fluid Model (MFM) [9] was employed in

127 conjunction with the "Inner-Outer" (IO) fully predictive impeller simulation strategy [10] to carry out the two-phase simulations. Details on these simulation methods can be found in the quoted references and are not reported here for the sake of brevity. The only difference with single-phase IO simulations is that in the present case the information exchanged between the Inner and Outer simulations included also the velocities of the solid phase. For comparison, the extremely simplified Settling Velocity Model [5] was tested also in conjunction with single-phase IO simulations for the simulation of particle distribution. The computational domain covered one quarter of the vessel, since a periodicity condition along the azimuthal direction can be invoked for the time averaged flow field. The computational grid adopted to run all simulations consisted of 175"20"16 cells in the inner domain and 196*30* 18 in the outer domain, along the axial, radial and tangential coordinates respectively. The momentum, continuity and turbulent transport equations were numerically solved by the "SIMPLEC" algorithm and by using the CFX 4.2 commercial CFD code. Each "Inner" or "Outer" simulation involved 1000 SIMPLEC iterations. Seven innerouter swaps were carried out for each case and required, on the SUN-Ultral workstation employed, total CPU times of about 120 hours in the case of MFM and about 60 hours in the case of SVM. 2.1. M u l t i - F l u i d M o d e l ( M F M )

In the present case of turbulent flows, the mass and momentum balance equations have to be time averaged, which gives rise to new unknown terms that involve the fluctuating velocities and volume fractions of both phases. As a difference with the single-phase case, no well-tested turbulence model is available for turbulent multiphase problems. A possible solution is to employ a two-phase extension of the standard k-e model. In this case the modelling of the k and e transport equations is more complex, as new unknown "inter-phase transfer terms" arise. These terms allow, in principle, to account for turbulence promotion or damping due to the presence of the solid phase. In the absence of reliable information on such terms, the simple 'homogeneous' approach has been employed here [9] where both phases are assumed to share the same values of k and e, and the inter-phase turbulence transfer is not considered. A turbulent Schmidt number of 0.8, which is in the range (0.5-1) of literature values [11 ] was employed for the solid phase in all MFM and SVM simulations.

Inter-phase drag terms The momentum transfer between the two phases was modelled by including the drag force contribution only. Indeed it has been shown [12, 13] that for the solid-liquid systems of interest here, the other forces (added mass, liil, Basset terms) are practically negligible. The inter-phase force per unit volume is expressed as:

,.(d) ,.(d) where "~dc = '~cd are the inter-phase drag terms. These are computed on the basis of the drag coefficient (CD) that has to be provided as an input datum. The explicit equation for these terms in the case of spherical particles is:

cd - ~ dp rd Pc

-

128 As a result, proper estimate of the particle drag coefficient in a turbulent fluid is critically important in order to correctly model the momentum transfer by means of eqn. 1. It is well known that free-stream turbulence may significantly affect the value of particle drag coefficient and the relevant particle settling velocity. In the present work, the values pertaining to turbulent conditions were computed on the basis of a recently proposed correlation [ 14]:

o o0: o0

(3)

where ~ is the Kolmogoroff length scale, and CD0is the drag coefficient for the same particle in the still liquid.

2.2. Settling Velocity Model (SVM) In this approach no momentum balance is solved for the particles which are assumed to be transported in the vessel space just like a passive scalar. The only deviation from the passive scalar condition is the assumption that, in the mass balance equations for the particle phase, a suitable "sedimentation" flow is to be added to the usual convective and turbulent diffusion flows. In particular it is assumed that the slip velocity between particles and fluid is purely axial and equals the particle settling velocity at any point inside the vessel. As a consequence of the passive scalar assumption, particles cannot affect the liquid flow field; therefore the application range is limited to the case of dilute suspensions. The fluid phase flow field in the vessel can therefore be independently simulated first by assuming that only the liquid phase is present. The passive scalar balance equation is assumed to hold for the solid particles, with no change of the turbulent diffusivity (the molecular diffusivity contribution is usually negligible in the case of turbulent fields and was assumed to be zero for the solid particles. The particles, in addition to being transported in the same way as molecular species, are assumed to "rain-in" from the cell immediately above, through the separation surface A, and to "rain-out" through the cell floor towards the cell below. As a difference with previous works [5,6] where a purposely written FORTRAN post-processing code had been employed, in the present work the CFX internal solutors were exploited. To this end the effect of sedimentation was taken into account by imposing, in each cell, a suitable value for the particle source term S: Si = Us Ai Cabove - ~ Ai Ci

(4)

where Ci is the particle concentration in the cell under consideration (cell "i"), Caboveis the particle concentration in the cell immediately above cell "i" and Ai is the area of the floor of cell "i" which, due to the "rectangular" grid employed here, coincides with that of the cell immediately above cell "i". Eqn. (4) applies to a genetic cell. In the case of bottom cells (as well as for the cells immediately above the impeller disk) the second term at the R.H.S. is not included while for

129 the top surface cells (as well as for the cells just below the impeller disk) the first term at the R.H.S. was omitted. Finally, Us is the average particle settling velocity in the turbulent field existing in the stirred vessel; it is worth noting that it does not coincide with the terminal settling velocity of the same particles. 3. EXPERIMENTAL The multiple impeller stirred tank employed was a high aspect-ratio stirred baffled tank (diameter T=23.6 cm, height H=4T), agitated by four Rushton turbines (diameter D=T/3); the turbine spacing was T and the lowest turbine was placed at T/2 from the vessel bottom. This geometrical configuration can be regarded as a particularly severe benchmark for solid particle dispersion models, as it gives rise to pronounced solid concentration profiles. Axial particle concentration profiles measurements were obtained by means of a light attenuation technique, whose details can be found elsewhere [ 15]. Three different solid phases (glass, bronze and Poly-Methyl-Metacrylate (PMMA) were employed with either water or a more viscous water solution of Poly-Vinyl-Pyrrolidone (PVP). The physical properties and the agitation speeds of the three experimental runs simulated in the present work are reported in Table 1. In all cases, particles were fully suspended in the vessel. System N (rpm)

1.water-glass

975 2. P~;P sol..-bronze 1770 3.water-PMMA 388

DI p s (kg m "3)

[.1,1

(mPa s) ( tm) 137

998 2450 998 8410 998 1150

9.31

327

0.873

327

C (kgmb

vt

Part. vol. fract.

y,(em/s)

1.67

1.25

0.0007 3.58 0.00O4

1.10 4.07 3.90 0.802 0.62

,,

1.97

0.0017

CD0

..CD 16.5 21.0 19.8 21.8 10.4 17.2 ,

Table 1" experimental runs

In Table 1, the still-fluid terminal velocity value, vt, is that observed experimentally and CD0 is the relevant drag factor value. The values in turbulent conditions (vs and CD) were estimated by means of eqns.3, with the Kolmogoroff length scale (~) computed on the basis of the average power dissipation in the tank. This last was estimated at all agitation speeds by assuming a power number of 5 for each impeller. 4. RESULTS AND DISCUSSION Particle concentration profiles predicted by MFM simulations (solid lines) are compared with the relevant experimental data (solid circles) in Figs. 1 (a-c). As can be seen, though a significant agreement is obtained, a systematic overestimation of particle concentration gradients can be observed in all three cases. A similar behaviour had not been observed in the case of single impeller tanks [16]. This is not

130 surprising, however, as in the present case particle concentration gradients are repeated four times thus giving rise to much more pronounced profiles and therefore to a much more stringent test for simulation procedures. Since the drag coefficients employed are already corrected for turbulence effects, such a lack of agreement is believed to stem from other causes. In particular it was found that the Pumping Number predicted by the present IO simulations for each stirrer is about 0.65. Though experimental anemometric data are not available for the multiple impeller tank here employed, it seems reasonable that the actual Pumping Number be close to that experimentally observed in single impeller stirred tanks (0.77). Hence, the pumped flow rate is likely to be underestimated by about 15%, a drawback commonly observed in fully predictive simulations of single-phase stirred tanks, with both the IO and SG simulation approaches. In order to estimate the influence of this underestimation on the predicted particle concentration profiles, further 1000 SIMPLEC iterations were run for the "Outer" domain, in which the radial velocities imposed as impeller boundary conditions were suitably increased by applying a multiplying factor equal to the ratio between the actual and the predicted Pumping Numbers to the values stemming from the last inner simulation. The particle concentration profiles obtained in this way are compared with the experimental data in Figs 1 (d-e), where a very good agreement between the two can be observed. This result implies that the main reason of the previously discussed lack of agreement of MFM- IO simulations is the underestimation of the discharge flow generated by the impellers. The importance of further developments simulating single-phase flows, in order to be able to set-up effective two-phase simulations, is therefore ascertained. On the other hand, the fact that once the single-phase simulations are improved, particle concentration profiles are well predicted implies that the two-phase models already available are well grounded and can be confidently employed, at least for the diluted solidliquid systems investigated here. 4.1. SVM Simulation Results

SVM simulations were also run for the case of system 1. The particle settling velocity adopted was that pertaining to the turbulent fluid. As can be observed in Fig.2a, the SVM simulation gives rise to results very similar to MFM results and the lack of agreement between simulation and experiment is essentially the same. If the radial velocities imposed as boundary condition to the SVM simulation are suitably increased, in analogy to what was done with MFM simulations, the results shown in Fig. 2b are obtained. As can be seen, like with MFM simulations, also SVM simulations are significantly improved in this way and the results are in good agreement with experimental evidence, though slightly poorer than that obtained with MFM results. It is therefore confirmed, also in this very stringent test case, that the main feature affecting solids distribution in stirred tanks is the axial slip velocity among the two phases. It is worth reminding that SVM simulations are much simpler and quicker than MFM simulations. The slightly poorer agreement observed depends on the fact that the different inertia of liquid and solid phases is neglected in SVM simulations, but it is a matter of judgement assessing whether the increase of the precision of results is worth the additional effort.

131

ACKNOWLEDGEMENT

The authors wish to thank AEA Technology for the CFX special license agreement. This work was carried out with the financial support of the Italian Ministry for University and Research (MURST), selected project on "Fluid Dynamics in Multiphase Reactors". NOTATION Cav

- average solid particle concentration [kg/m3] drag coefficient in turbulent liquid [-] CD0 - drag coefficient in still liquid [-] dp - particle diameter [lain] ~, = Kolmogoroff length scale [lam] Ps, Pl - densities of solid and liquid phases [kg/m3] l.tl - liquid viscosity [mPa s] N - agitation speed [rpm]; T - vessel diameter [cm] Vs - particle terminal settling velocity in turbulent fluid [cm/s] vt - particle terminal settling velocity in still fluid [cm/s]

CD

REFERENCES

1. J.A.M.Kuipers and W. P. M. van Swaaij, 1997, Rev. in Chem. Eng., 13 (1997) 1. 2. A. Bakker, J.B. Fasano and K.J Myers, I.Chem.E. Syrup. Ser., 136 (1994) 1. 3. K.J. Myers, A.Bakker and J.Fasano, A.I.Ch.E. Syrup. Ser., 91 (1995) 139. 4. S. Decker and M.Sommerfeld, I. Chem. E. Syrup. Ser. 140 (1996) 71. 5. A. Brucato, M.Ciofalo, J. Godfrey, F. Grisafi and G. Micale, Proc. of"The 5th Int. Conf. on Multiphase Flow in Ind. Plants"; Amalfi, 26-27 Sept. 1996, 323. 6. A. Brucato, M. Ciofalo, F. Grisafi, F. Magelli, G. Micale, AIDIC Conf. Ser., 2 (1997) 287 7. Z. Sha, S. Palosaari, P. Oinas and K. Ogawa, Proc. "Third Int. Syrup. on Mixing in Industrial Processes", Osaka, 19-22 Sept. 1999, 29 8. H. Barrue, J. Bertrand, B. Cristol and C. Xuereb, Proc. "Third Int. Syrup. on Mixing in Industrial Processes", Osaka, 19-22 Sept. 1999, 37 9. CFX Release 4.2 Solver, AEA Technology, UK, (December, 1997). 10. A. Brucato, M. Ciofalo, F. Grisafi, F. and G. Micale, Chem. Eng. Sci., 53 (1998) 3653. 11. V. Yakhot, S.A. Prsza and A. Yakhot, Int. J. of Heat & Mass Transl., 30 (1987) 15. 12. A.D. Gosman, R.I. Issa, C. Lekakou, M.K. Looney, S. Politis, A.I.Ch.E.J., 38 (1992) 1946 13. M. Ljungqvist, PhD. Thesis, Chalmers Univ. ofTech., Goteborg (Sw) (1999) 14. A. Brucato, F. Grisafi and G. Montante, Chem. Eng. Sci., 53 (1998), 3295 15. F. Magelli, D. Fajner, M. Nocentini and G. Pasquali, Chem. Eng. Sci., 45 (1990) 615. 16. G. Micale, G. Montante, F. Grisafi, A. Brucato, J. Godfrey, Chem. Eng. Res. Des, in press

132

0

2

1

3

0

C’Cav

2

1

3

CICav

0

1

2

3

C’CW

-

Fig. 1 Comparisonbetween MFM predictions (lines) and experimental data (circles). Pumping Number: (a), (b) and (c) as predicted; (d), (e) and (f) increased to 0.77.

Fig2 - System 1: comparison between SVM predictions and experimental data. Pumping Number: (a) as predicted (b) increased to 0.77.

0

1

C’CW

2

3

0

1 c/cav

3

I 0thEuropean Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

133

Power consumption in slurry systems Antonello A. Barresi* and Giancario Baldi Dip. Scienza dei M a t e r i a l i e I n g e g n e r i a Chimica, P o l i t e c n i c o di Torino

C.so Duca degli Abruzzi 24, 10129 Torino, Italy The dependence of the power consumption on particle size and concentration in the stirring of slurries has been investigated; a pitched blade and a Rushton turbine have been studied in a vessel with flat bottom. Suspensions of glass spheres of different size (volumetric fraction up to 30%) have been considered. The use of an average slurry density for the prediction of the power drawn by the impeller has been shown to be conservative in conditions of sufficient suspension except with the largest particles.

1. INTRODUCTION Mechanically agitated solid-liquid contactors are widely used in chemical processes; if only the dispersion of the two phases or mass transfer operations have to be carried out, the system is operated at the minimum stirrer velocity for complete suspension, but it has been shown that if fast complex chemical reactions occur, the selectivity can depend on the liquid hydrodynamics and on the solid distribution in the apparatus, and higher stirrer velocities can be required [1, 2]. Previous works have evidenced that the power required to stir a slurry is higher than that required for the clear liquid; the use of an average slurry density in the correlationP = N p o p , z , , ~ N 3 D 5 , where Npo is the power number measured in the liquid, has been generally proposed and considered conservative, even if it has been observed that depending on solid characteristics and apparatus geometry, in some cases the power required can be underestimated [3-10]. Herringe [ 11 ] proposed to use the slurry density in the zone of the stirrer; this approach seems more correct, but it is difficult to estimate this density. A few data are available on the dependence of the power transferred to the slurry on the solid concentration and physical properties and on the vessel and stirrer geometry [11-15], and in some cases are limited to the just-suspended condition [ 13-15]. In this paper the dependence of the power draught by the impeller in a slurry on the stirrer type and stirrer clearance will be discussed; suspensions of glass beads in water at high solid loading (up to 30% by volume) and two different stirrers (a Rushton type turbine and a four pitched blade turbine) will be considered. Some data obtained in a similar system, but in a vessel with rounded bottom have been presented in a previous paper [ 1]. * To whom correspondence should be addressed; fax 9+39-011-5644699, e-mail: [email protected]

134 2. EXPERIMENTAL SET UP The experiments were carried out in a tank reactor with fiat bottom (T = 200 mm, H = 213 mm), equipped with four baffles (w/T = 0.1) andclosed by a lid in order to prevent vortex formation and air entrainment at the high stirrer velocity required to suspend large size and/or high load solid particles. The presence of the lid has a very small influence on the flow field, especially in the lower part of the vessel, and does not modifies significantly the conditions for complete suspension [16, 17], even if it can affect the value of the power number as a consequence of the increase in area available for skin friction [ 18, 19]. The vessel is in glass, to allow the measurement of the cloud height, and is jacketed for temperature control; experiments have been carried out at 25~ Two different impellers have been investigated, with a relative stirrer to tank diameter ratio approximately equal to 1/3 and a clearance equal to the stirrer diameter: a six-blade Rushton turbine (D = 73 mm; disk diameter = 53 mm; blade: height 15 mm, width 19 mm, thickness 2 mm) and a 45 ~ four-pitched-blade turbine pumping downward (D = 70 mm; blade: height 14 mm, width 22 mm, thickness 2 mm). A beating fixed on the bottom of the vessel avoid stirrer wandering. The stirrer is rotated by a variable speed DC motor; the power transmitted to the suspension is calculated from the stirrer torque and rotation velocity, which have been measured by a torque transducer (Vibro meter TG 0.1, nominal value 1 Nm, error +5 mNm) and a tachometer dynamo, respectively. Net torque was obtained from gross torque after correction for dynamic friction, that was in any case very low, and averaging (as fluctuations in the measured torque were observed). Suspensions of glass spheres in water were used; two particle classes, obtained by sieving (small, d = 0.302 mm, p = 2500 kg/m3; large, d = 0.544 mm, p = 2450 kg/m3), and volumetric solid fractions varying in the range 5-30% were tested; the stirrer speed varied between 350 and 950 rpm, covering the range of uncompleted and complete off-bottom suspension.

3. RESULTS AND DISCUSSION The power number of the two impellers in clear liquid and in the chosen configuration has been measured experimentally, and is constant in the range investigated (Re = 4.105 - 1.1.106); as it depends on the stirrer clearance, a wider range than that actually used has been considered, in order to estimate the effect of a layer of unsuspended solid on the bottom on the power draught. In fact, the particles on rest on the bottom create a false bottom, reducing the actual clearance of the stirrer. Figure 1 shows that the power number of the pitched

Npo

9

0

i

0

m

9

I

i

0.5

m

i

1 C/D

Fig. 1 - Dependence of power number on stirrer clearance. O, Rushton turbine; I , pitched blade turbine.

135 blade turbine is weakly affected by the clearance, while the Rushton turbine is more sensitive to variation in the distance from the bottom, and differently from the other impeller, the power number increases with C/D. The power measured with the slurry at high solid loading was influenced by the procedure followed: in fact, at rest the stirrer is partially immersed in the solid with a 10% or 20% volumetric fraction (at C/D = 0.5 and C/D = 1 respectively). With higher particle loading the initial torque is very high, and a high rotation velocity is also necessary to start solid suspension, but an hysteresis phenomenon is observed, and a lower torque is measured if the velocity is first raised until the solid is completely suspended and then decreased; the profile assumed by the solid at rest on the bottom, in case of uncompleted suspension, is also affected by the procedure followed. Thus the measures reported have been done bringing the system at the highest stirrer velocity and then decreasing it. At relatively low rotation speed, a large fraction of the solid can be unsuspended, creating a false bottom; this fact can significantly affect the power, because both the stirrer clearance and the bottom shape is modified. In addition, the suspension is largely inhomogeneous, and the presence of the solid can be limited to the lower part of the vessel. The fraction of solid in suspension could not be detected in this work, but the height reached by the solid cloud in the vessel, and thus the volume fraction occupied by the slurry was measured. Some results are shown in Figures 2-3, which show the

~7i

1

h/H

0.8

i:-':

o~~ &

0.6 [] 0.4 0.2

9Cv = 5%

[]

10%

&

15%

o

20%

I

25%

0 500

3OO

700

900

N , rpm

Fig. 2 - Dependence of cloud height on solid loading; Rushton turbine, C/D = 1, coarse particles.

9

h/H

[]

[]

0.8

[]

O

0.6

A

[]

[]

0.4

A

I I Rushton C=D DRushton C=0.5 D &Pitched blade C=D /kPitched blade C=0.5 D

0.2

400

600

800

1000 N, rpm

Fig. 3 - Cloud height with different stirrer configurations; small particles, Cv = 15%.

P/P0

1.6

1.4

1.2 1 a00

9 [] ~ .4__ .

. 400

[3 ~

[] o 9 9 .

.

c 1 h/H 0.8 0.6 0.4

9 0.2 0

. 500

600

N, rpm

influence of the stirrer type and clearance Fig. 4 - Power consumption (filled symbols) and and of the solid size and concentration, cloud heigth (open symbols) at high solid loading; The cloud height increases with stirrer Rushton turbine, C = D, small particles. speed and is lower with larger particles and 9 O, Cv- 20%; II !::1, Cv = 30%. at higher solid hold up; it can be noted (see

136 Fig. 2) that with coarse particles, when the solid fraction is high, the bed does not expand up to an high rotation speed, and then a steep increase occurs. Figure 3 evidences that the behaviour of the impellers is very different: in no case (except with the smallest particles and a 5% solid loading) the particles were dispersed in the whole volume using the pitched blade turbine; the clearance is not very important with this respect, while for the Rushton turbine the most efficient configuration is that with C = D. 1.4 As a consequence of the uncompleted PfPo o o o o o o suspension and of solid distribution in the 1.2 vessel, at low rotation velocity the power draught by the impeller in the slurry (P) can o C v = 5% r-I lO% be up to 30-40% lower than that measured & 15 % 0 20% 0.8 in the clear liquid (Po), but this difference 9 25% 0 30% progressively reduces and at higher rotation 0.6 velocity P/Po is generally higher than unity 500 600 700 800 900 1000 and independent of N. In some cases, and in N , rpm particular with the highest solid hold up, P/Po is very high at low rotation speed and Fig. 5 - Rushton turbine; C = D, small particles. then shows a minimum; an example can be Dependence of power consumption on solid seen in Figure 4. loading. A comparison of the dependence of P/Po on stirrer speed at different solid loading for the two impellers is shown in Figures 5-9. 1.4 P/P0 The data have been reported plotting the 1.2 9 9 9 9 9 9 ratio P/Po, where P is the actual power measured in the slurry and P0 is the power 1 drawn in clear liquid in the same operative o C v = 5% lr-I 10% conditions, o.8 & 15 % 0 20% With both stirrers an asymptotic 9 25% condition is reached, at which the ratio P/Po 0.6 remains constant, but the stirrer velocity at 5oo 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 N , rpm which this condition is obtained depends on the stirrer type and on the geometric Fig. 6 - Rushton turbine; C = D, coarse particles. configuration. It can be evidenced that this Dependence of power consumption on solid asymptotic behaviour is not strictly related loading. with the attainment of the sufficient suspension condition: in fact, P/Po becomes independent of N when the cloud occupies only a part of the liquid volume (this happens with the Rushton turbine with the larger particles or the smaller clearance, and almost always with the pitched blade turbine). Comparing the data in Figure 5-7 relative to the Rushton turbine, it can be noted that the power consumption depends not only on the solid loading, but also on the particle size: even if coarser particle are more difficult to suspend and to disperse in the vessel, the power consumption is lower than that measured using the same volumetric fraction of smaller particles. The stirrer clearance has a very weak effect on P/Po, but this configuration is more favourable considering the energy dissipation, as the power number is lower with a smaller clearance. i

9

I

I,,

I

I

i

I

I

137 The behaviour observed with the pitched 1.4 blade turbine is significantly different, and P/Po can be explained with the differences in the 9 o 8 ~ $ ~ 8 1.2 ~ ~ ~ hydrodynamics and in solid distribution; P/Po is higher with coarser particles, and is affected by the clearance. Lower values are 9 = 5% [] 10% observed with C/D = 1, and at low solid 0.8 15 % 0 20% hold up the power draught by the pitched 25% blade turbine can be lower than in clear 0.6 liquid (see Figures 8-9). 700 800 900 1000 500 600 N , rpm Previous authors assumed that the power number measured in clear liquid can Fig. 7 - Rushton turbine; C = 0.5 D, small be used also in slurries, provided the value particles. Dependence of power consumption on of the local slurry density around the stirrer solid loading. is taken into account; but, as this one is hardly available, they used the bulk mixture 1.4 density [9, 11] or the mean suspension P/l'o density (calculated considering only the 1.2 o o o o o o o o volume occupiedbythesuspension) [12]to , ~ .~ ~ ~ ~ ~. ~. calculate an apparent power number for 1 suspension, Npo*, and plotted Npo*/Npo . suggesting this is also a measure of the 0.8 .cv= 2o/~ [] lO~ relative local slurry density. A 1~ ro o 2o~ Figure 10 shows a comparison between 0.6 .............. the power calculated considering an 500 600 7oo 800 9o0 lOOO average slurry density, P*, and that actually N, rpm i

I

!

!

I

i

I

I

measured in the slurry: the power is Fig. 8 - Pitched blade turbine; C = 0.5 D, small generally overestimated, but in some cases, particles. Dependence of power consumption on with larger particles, it can be solid loading. underestimated. The difference depends on the particle size and concentration, and on the geometry of the apparatus; the effect of 1.4 particle size and stirrer clearance is larger P / P o when the pitched blade turbine is adopted. 1.2 o o o o o o 0 o The results indicate that the effect of the o 9 suspended solid on the power consumption is not only due to the increase in the t 0.8 average density, but the dispersed phase affects the liquid hydrodynamics. 0.6

,

500

,

600

,

I

700

,

L

800

,

I

900

,

1000

N , rpm

4. CONCLUSIONS

Fig. 9 - Pitched blade turbine; C = D, small The results previously discussed indicate particles. Dependence of power consumption on clearly that the dependence of the power solid loading. Symbols as in Fig. 8.

138 dissipated on the physical characteristics of 1.2 the solid is quite complex, and cannot be I R ushto n [ accounted by the simple consideration of an t:1 ta average slurry density, even when the solid ~ .g 1.1 is completely suspended and distributed in the whole volume. 1 o c o a r s e C=D The ratio P/Po has been considered in o--o., o I this work, because by this way the effect of , , , ~ I ~ ~[ Ocoarse,, ,, , C=0, ,.5 D the suspended solid is immediately evident, 0.9 30 10 20Cv and no assumption is necessary: actually the increased average density of the suspension is partially responsible for the increase in the measured power, but modifications in 1.2 I P i t c h e d blade~i I the liquid circulation and changes in the bottom shape and impeller clearance, as a ~ 1.1 D consequence of the presence of the O [3 ra unsuspended solid, are also relevant. 0 [] . s m a , , C=O F'] 0 The stirrer clearance affects both the 1 e c o a r s e C=D 0 power consumption and the distribution of IE]srn all C=0.5 o c o a r s e C=0.5 the solid; for the Rushton turbine the 0.9 configuration with the lowest power 0 10 20 Cv 30 demand is that with C/D = 0.5, while for the axial turbine is that with C=D. By Fig. 10- Comparison of estimated and real power comparison with previous results, it can be consumption for different clearances and particle concluded that with both stirrers the power sizes (aymptotic values). consumption is lower using a fiat bottom instead of a rounded one. On the other hand, with a rounded bottom, the suspension occupies a larger volume.

9

I.,ma,,

I

Acknowledgements - This work has been financially supported by the Italian Ministry of University (MURST 40% - Fluidodmamica Multifase). The contribution of M. Botta to the experimental work is gratefully acknowledged.

139 NOTATION C

c~ D d H h N

N~o P p, Po T W

p

stirrer clearance volumetric solid fraction stirrer diameter particle average diameter liquid height cloud height stirrer rotation speed impeller power number power drawn by the impeller, in slurry power drawn by the impeller, calculated with average slurry density power drawn by the impeller, in clear liquid tank diameter baffle width density

REFERENCES

[1]

[2] [3]

[4] [5] [6] [7]

[8] [91

Barresi, A.A., Baldi, G., "Hydrodynamics of solid-liquid suspensions in stirred vessels. Influence of the solid on power number and selectivity of fast reactions". Proc. 6th International Conference on Multiphase Flow in Industrial Plants (ANIMP, AIDIC, Polytechnic of Milan), Milan, Italy, 347-358, 1998. Barresi, A.A., "Selectivity of mixing-sensitive reactions in slurry systems". Chem. Eng. Sci. 55, 1929-1933, 2000. Kohler, R.H., Estrin, J., "Power consumption in the agitation of solid-liquid suspensions".A.1.Ch.E.J. 13, 179-181, 1967. Nagata, S.,Mixing. Principles and applications. Wiley, New York, p. 63-65, 1975. Cliff, M.J., Edwards, M.F., Ohiaeri, I.N., "The suspension of settling solids in agitated vessels". 1.Chem.E. Symp. Ser. 64, M1-M11, 1981. Pasquali, G., Fajner, D., Magelli, F., "Effect of suspension viscosity on power consumption in the agitation of solid-liquid systems". Chem. Eng. Commun. 22, 371375, 1983. Bubbico, R., Mazzarotta, B., "Effetto della geometria del recipiente sulla potenza richiesta per la miscelazione liquido-solido". Atti 4 ~ Convegno lnternazionale: Fluidodinamica Multifase nell'lmpiantistica Industriale, Ancona, Italy, 401-412, 1994. Bubbico, R., Mazzarotta, B., "Effect of the solid concentration on the power dispersed in solid-liquid mixing". Abstracts 2nd ltalian Conference on Chemical and Process Engineering, 1CheaP-2 (AIDIC), Firenze, Italy, 485-489, 1995. Bubbico, R., Di Cave, S., Mazzarotta, B., "Influence of solid concentration and type of impeller on the agitation of large PVC particles in water" in: Rdcents Progr~s en Gdnie des Procddds, Vol. 11, Nr. 51 (Mixing 97), pp. 81-88. Ed. Lavoisier, Paris, 1997.

140 [101 Bujalski, W., Takenaka, K., Paolini, S., Jahoda, M., Paglianti, A., Takahashi, K., Nienow, A.W., Etchelles, A.W., "Suspension and liquid homogenization in high solid concentration stirred chemical reactors". Chem. Eng. Res. Des'., Trans. IChemE 77A, 241-247, 1999. [11] Herringe, R.A., "The behaviour of mono-size particle slurries in a fully baffled turbulent mixer". Proc. 3rd European Conference on Mixing (BHRA), Univ. York, U.K., 199216, 1979. [12] Chudacek, M.W., "Formation of unsuspended solids profile in a slurry mixing vessel" Proc. 4th Europ. Conf. Mixing (BHRA), Cranfield, U.K., 275-287, 1982. [13] Raghav Rao, KS.MS., Joshi, J.B., "Liquid-phase mixing and power consumption in mechanically agitated solid-liquid contactors". Chem. Eng. J. (Lausanne) 39, 111-124, 1988. [14] Drewer, G.R., Ahmed, N., Jameson, G.J., "Suspension of high concentration solids in mechanically stirred vessels". Proc. 8th European Conference on Mixing, Cambridge, U.K., 1.Chem.E. Symp. Ser. 136, 41-48, 1994. [151 Armenante, P.M., Uehara-Nagamine, E., "Solid suspension and power dissipation in stirred tanks agitated by one or two impellers at low off-bottom impeller clearance" in: Rdcents Progrds en Gdnie des Procddds, Vol. 11, Nr. 51 (Mixing 97), pp. 73-80. Ed. Lavoisier, Paris, 1997. [161 Nouri, J.M., Whitelaw, J.H., "Effect of size and confinement on the flow characteristics in stirred reactors". Proc. 5th International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, paper 23.2, 4 pp., 1990. [17] Ljungqvist, M., Rasmuson, A., "Hydrodynamics of open and closed stirred vessels" in: Rdcents Progrds en Gdnie des Procddds, Vol. 11, Nr. 51 (Mixing 97), pp. 97-104. Ed. Lavoisier, Paris, 1997. [18] Nienow, A.W., Miles, D., "Impeller power numbers in closed vessels". 1rid. Eng. Chem. Process Des. Develop. 10, 41-43, 1971. [19] Papastefanos, N., Stamatoudis, M., "Effect of vessel geometry on impeller power number in closed vessels for Reynolds numbers between 40 and 65000". Chem. Eng. Res. Des. 67, 169-174, 1989.

I 0th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

141

Drop break-up and coalescence in intermittent turbulent flow W. Podg6rska and J. Ba~dyga Department of Chemical and Process Engineering, Warsaw University of Technology ul. Waryfiskiego 1, PL 00-645 Warszawa, Poland Effects of turbulent agitation on the transient drop size distributions are considered. Drop dispersion and coalescence phenomena depend on the fine-scale properties of turbulence, including its intermittency. A multifractal formalism describes intermittency and is used to derive drop break-up and coalescence functions for drops, whose diameter falls within the inertial subrange of turbulence. The transient drop size distributions are predicted by solving the population balance equation for the single-circulation-loop model of agitated tank; the predictions are in good agreement with reported experimental data. 1. INTRODUCTION Liquid-liquid dispersions are of importance to industry, as they are involved in many engineering operations including chemical reaction, extraction, emulsion and suspension polymerization, emulsion preparation, etc. Drop size distributions and dynamics of their evolution are important characteristics of such dispersions, because they can affect the quality of many products. A distribution of droplet sizes and its time evolution depend on the relative break-up and coalescence rates. These rates depend obviously on the conditions of turbulent agitation (scale and geometry of the vessel and the impeller, agitation rate) as well as on the physical properties of the mixed phases. An understanding of these effects should unable us to model the process and to control the drop size distribution. A drop breaks-up if the local instantaneous stresses generated by turbulent flow exceed the stabilizing forces due to interfacial tension and the drop viscosity. Regarding coalescence, turbulence governs the drop collision frequency, force and duration time and affects this way drop interface deformation, , film drainage and rupture, and resulting confluence of drops. To interpret these phenomena we need to employ such theory of turbulence that gives detailed information about the finescale features of turbulence. In this paper the authors are interested in the turbulent dispersion and coalescence of droplets of low viscosity, whose diameter falls within the inertial subrange of turbulence, i.e. for (rl> << d << L. 2. INTERMITTENT CHARACTER OF TURBULENCE Traditionally the theories of drop break-up and coalescence employ the classical theory of turbulence that is based on Kolmogorov's dimensional analysis (Kolmogorov, 1941) and Obukhov's concept of energy cascade (Obukhov, 1941). Such approach neglects intermittent character of turbulence as it is based on local average values of the rate of energy dissipation,

142 root-mean-square velocity fluctuations, rates of strain, turbulent stress, etc. In fact the distributions of these quantities display strong fluctuations and do not scale as predicted by Kolmogorov (1941). Due to large variability (in time and space) of the local instantaneous value of the rate of energy dissipation e, obviously ( e ~ 1 6 2(e) p , which affects the form of equations describing average stresses, average rates of drop break-up and coalescence, etc. The fluctuations of e about its local mean value, (e), increase with increasing the Reynolds number (Meneveau and Sreenivasan, 1991). This phenomenon is called "internal intermittency", "local intermittency" or "fine-scale intermittency". The multifractal theory of turbulence proposed by Frisch and Parisi (1985) accounts for the fine-scale structure of turbulent flow resulting from effects of the "fine-scale intermittency". Batdyga and Bourne (1993,1995) suggested that a drift of transient drop size distributions at very long agitation times, a drift of the exponent on the Weber number from -0.6 (resulting from the Kolmogorov theory) to -0.93, and the effects of the system scale on the rate of drop break-up result from the fine-scale intermittency. Batdyga and Podg6rska (1998) applied the multifractal method to derive the breakage function and to predict the drop size distributions in the system with negligible coalescence. In this paper we extent this approach to the system with significant coalescence effects. The multifractal approach (Frisch and Parisi, 1985) is based on the scaling invariance of the Navier-Stokes equation for length scale r in the inertial subrange of turbulence. From this approach it follows that the velocity increment u r over a distance r scales as

U r =

U L

-(r/L)~ = c .r

L

(1)

where L is an integral scale, u L is the velocity increment over distance L (or the velocity fluctuation of the large energy containing eddies) and r is the scaling exponent. L and u L are related through the rate of dissipation of kinetic energy, (e) = u 3/L L . The turbulent events labelled by cx appear with probability dependent on a , r and L. For developed turbulence there is a range of possible scaling exponent cx and for each cx value there is a specific fractal dimension. The fractal set with scaling exponent between ct and cx +&x has a fractal dimension fd(O0 in a ds-dimensional physical space. For relatively large r values, but still within the inertial subrange of turbulence the probability density function for cx is given by Chhabra et al. (1989). P(cz) =

p(a~ln(L/r)(r/L) a'-f"{~)

(2)

Sensitivity of p(a) to cx is relatively small and usually neglected in applications. Distributions of fd(a) one can find in Batdyga and Bourne (1999) and Batdyga and Podg6rska (1998). Using eq. (1) one gets (Batdyga and Bourne, 1993, 1995) an equation describing the instantaneous, local normal pressure stresses acting upon objects of size r in the inertial subrange of turbulence (rl << r << L )

143 p(ct, r) = CpPc [u r (1")]2 : CpPcU ~ (r/L)~ = Cpp c ((e>" r) 2/3(r/L)} (~-')

(3)

Similarly we get characteristic frequency and rate-of-strain for eddies of scale r that are labelled by the scaling exponent or: g(a, r ) : S((~,r) = U r ((~, r)/r : (~)'/3 r-2/3 (r/L)3 -I

(4)

The multifractal theory represents an extension of the classical approach and gives us all possible local instantaneous values of the velocity increment, rate of energy dissipation, rate of strain, etc. (labelled by (x) together with probability of their appearance. Notice also that the multifractal theory introduces an influence of the scale of the system to description of the process through the integral scale L. 3. T R A N S I E N T

DROP SIZE DISTRIBUTIONS

The time evolution and space distribution of the drop size distribution can be predicted by solving the population balance equation: On(v, x, t ) c+3 [ u i (x,t) 9n(v,x, t) ] =

l h(v - v , v', x ,'t

-

- v , v', '

,tn

- v , x',

', x , t

tn

v' - g

20

(v-)(v-) ,x, tn ,x,t+

(5)

-n(v,x,tlfh(v,v,x,t) (v,v,x,t)n(v,x,t) + 0

v

after incorporating appropriate breakage and coalescence relationships. 3.1. D r o p b r e a k - u p r a t e

The relation for drop break-up rate including the impact of intermittency was derived by Batdyga and Podg6rska, 1998. We consider here only drops of low viscosity of diameter d within the inertial subrange of turbulence - other possible cases are analysed elsewhere (Batdyga and Podg6rska, 1998 ). Because the drops are unstable when the local instantaneous stresses p(d, (x) exceed the stabilising forces due to the interfacial tension, p(d, ct)> 4or/d, we can identify the eddies vigorous enough to cause break-up as the ones labelled by the values of the scaling exponent c~ from the range Ctm~n _
d2/3

dot ~mln

(6)

144 with d=(6vhr) u3 and f({x) estimated for d s = 1 by fitting to experimental data by Meneveau and Sreenivasan (1991) - see Batdyga and Podg6rska (1998) or Batdyga and Bourne (1999) for details. The number of daughter drops formed per breakage, v, is assumed to be 2 and for the probability density of forming drops of volume v from breakage of drops of volume v', fl(v, v'), we employ the U-shaped distribution proposed by Tsouris and Tavlarides (1994). 3.2. Drop coalescence rate

The local average value of the coalescence rate is expressed traditionally (see Chesters, /

\

1991) usa product of the drop collision frequency hid j, d k, x) and the coalescence efficiency 1

/

\

\

L[dj,dk,X t, but the expressions for h[dj,dk,X ) and E[dj,dk,X / are new in this paper. Following Chesters (1991) the continuous phase flow is split into the external flow that governs the collision frequency, force and duration, and the internal flow, related to the drop interface deformation and film drainage. We start from the drop collision frequency in the inertial subrange of turbulence. In the case of intermittent turbulence and assuming orthokinetic collisions, the collision frequency for an event characterised by exponent reads: tt-I

f(dj'dk'~'x)=Cin(E(x))'/3/dj+dk)7/3(dj+dk/-:-2 ' ~ 2L

nj(x). nk(X)

(7)

where Ci. = ~ n - / 3 , which gives the average collision rate

2

(, 2L

nJ(x)"n'(x)

(8)

Eq. (7) has been derived using the method of Kuboi et al. (1972) and for cx = 1 eq. (7) becomes equivalent to the equation given by these authors. It is seen from eq. (8) that intermittency lowers slightly the rate of collision comparing to the case of no intermittency. Such small effect is observed only in the case with no repulsive colloidal forces; in the case of such energy barrier effects of intermittency are expected to be more significant. The /

\

efficiency of coalescence ;L/dj,dk,X ) can be expressed in terms of the average coalescence time te and the average contact time t (Chesters, 1991)

+j,dk ,X)--exp[-tc d,,

dk,x)]

(9)

The contact time, t, depends on two phenomena: convection of drops by the continuous phase flow and bouncing of drops (Chesters, 1991). Assuming that only convection governs the collision and using eq. (1) for Ur we get

t~`t =djk/Ur

't~/3(~:)-'/3 (djk/L) '3~ --"jk

djk -(dj +dk)/2

(10)

145 The collision time resulting from droplet bouncing can be derived under assumption that the whole kinetic approach energy is transformed into excess surface energy. The relative velocity of droplets results from turbulent movements and reads Ur ~- (1~)1/3"jkCll/3(djk/L~Ct-1)/3 and the individual velocities are uj = m k 9ur/(m j + m k)and u k = mj .u r/(mj + m k), respectively. Assuming that both initially spherical drops are deformed in such a way that both of them have flattened regions of the same film radius a and the spherical regions of radius Rs and RL, respectively and employing otherwise the method proposed by Chesters (1991) one gets for the collision time

ti

Z S +Z L = 2 R3(pD/Pc +g)pc(g+l) 2 ,/2 = Ur "3-" a 0 +G3X1 +q2) ,

1

Rs

g= R---7

11,

where Rs and RL are radii of the small and large drop, respectively. However, in reality larger droplet is less rigid and thus larger drop is much more deformed than the small one. Assuming that only larger drop is deformed we get z

ti ~ V=k

[2 R](pD/p c 3,)pc] '/2 --" +3 3 13'0W~ )

(12)

The ratio of ti from eqs (11) and (12) is not larger than 21a; taking into account that the method of ti estimation is rather crude one can use any of eqs (11) or (12). In what follows ti from eq (12) is employed. A mechanism controlling the time of interaction of drops can be indicated by comparing t i and t~xt yielding t --- min(t i , t~xt). To evaluate the time of drainage, to, we consider the case of drainage between deformablepartially-mobile drop interfaces (DPMI); the DPMI model assumes that drainage is controlled by the quasi-creeping flow created in the dispersed phase. We start from the relation for thinning rate proposed by Chesters (1991) for drops of unequal size dh dt

_-

2(2xo'/R~q )3/2

h2

(13)

nl.tD Fl/2

The average coalescence time is a period of time necessary for drainage process to decrease the film thickness from the initial value, h0, to the critical rupture thickness he. The critical film thickness proposed by Chesters (1991) is given by:

hc ~ [A" Req/(8~o)~/3 ,

Req =dj

.dk/(d j +dk)

(14)

where A is the Hamaker constant. Chesters (1991) considered the case with constant value of the interaction force F exerted by one drop on another. In this paper interaction force is assumed to result from reaction of larger drop to its surface deformation, as the large drop as less rigid is more deformed during collision

146

F -_-rm 2

20

= 2X~

RE

(8"R](PD/PCoR-r-+ T)P+q3'x c(g')2/3cl2/311/2L~ l

J

"TJk

(15)

The critical film thickness is given by eq. (14), and the initial film thickness was estimated by comparing the turbulent velocity with the drainage rate. The resulting value of h o reads: ho = lx~2(e)l/4a'/41~ 3 / Sr"'[ P3/acl~("'s PD/P"jk

c +T)/(96(~5(I + g3))]'/s

(16)

The coalescence time is then equal to

t~=p.D(e.)'/6,4]/--.q6o3/--s2o3/4[pc(pD/pc+T)/(96crS(l+q3))]]/4(ho-h. c)/hohr .jk

(17)

Having expressed coalescence time, to, and contact time for collision, t, one gets coalescence efficiency defined by Eq (9). As t c and t contain some crude approximations we replace the ratio to/t in Eq (9) by C. t~/t, C being a correction factor of order unity. 3.3. Breakage and coalescence in stirred tank. The general model described above is employed in this section to interpret the transient drop size distribution in stirred tank using a simple, one-dimensional, singlecirculation-loop, plug-flow model. The model assumes that along the loop there are zones differing in properties of turbulence (rate of energy dissipation calculated from correlation by Okamoto et al. (1981) and the integral scales of turbulence as measured by Wu and Patterson, (1989). The population balance equation (5) is Reynolds averaged, which after neglecting the

turbulent diffusion term u'in' yields: 0n(v,x,t) 0n(v,x,t) +Ux = 0t 0x v

l!h(v 2

v',v',x)X(v oo

v',v',x)n(v

v',x,t)n(v',x,t)dv'

g(v,x)n(v,x,t)+

(18)

oo

-n(v,x,t)j'h(v,v , ,x)X(v, v ,' x ) n- ( v ' ,x, t)dv'+ j'13(v,v')v(v')g(v',x)n(v',x,t)dv' 0

v

The proposed model contains two constants of universal character Cx and Cg. Both constants are connected with drop break-up. The constant Cx characterises the maximum stable drop size and was determined from the results of Lagisetty et al. (1986) as C x ~ 0.23. The constant Cg characterises the breakage rate and was estimated in our previous paper (Batdyga and Podg6rska, 1998) as Cg = 0.0035. In calculations a typical value of Hamaker constant A = 10-2~ is applied; comparing the model predictions with experimental data of Konno et al. (1988) we get for the correction constant C = 0.5. Konno et al. (1988) measured drop size distributions during the process of drop coalescence after a stepwise reduction of

147 impeller speed. In experiments a mixture of o-xylene and carbon tetrachloride was dispersed in distilled water containing 0.1 mol/dm 3 NaCI. Comparison of model predictions with experimental data is shown in Fig 1; the model adequately predicts the effect of stirrer speed on transient drop size distribution.

1,0 ~

a)

/

I,,

iv

.~

/,"~

#

/

0,8

f

//~ // #, '

:,'/

,,~ l

N tt>- - v o = 2 . 3 s "]

7/

:t

/,' . . . .

o

experimental Ko.nno et al., 1988

/'

//.,

i

, I

,

,

0,0

,. I l"

0

.i, ,'

,'"1 ,t

:

I

, ;

r,.) 0,2

/,' /,,' /,' ,'

I

:

+~ 0,4

t"

'

,:/

,1

,'..1:1 ,'.7

'.'1

I

Nt_ o= 7 s

:q

, ;

0,6

,"-

Z':"

l

~

:o.os

,,,J

,

o

t./

#./

,~

,;'

,

~t.

<-/.I

'.7

100

calculated (imPeller zone) calculated (bulk zone)

~-

T = 0.186 m, T / D = 2 400

300

200

......

500

600

700

800

Drop diameter, larn 1,0

,

~

,

o,8

9

"/

0,0

e ~, ' a' .'

i

ii[ i f / f ~

I

I:

t~

I

:7 i / 1 ! /

:

I r t

r..)

' ~ #i t"

i

0,61

0,=b

,

i if/ !/I//I : l i///iii ! , ,7;7/t; ! I

': :I

I"

I

I

i

,"/ii:';i1!i

I

:

I'

I"

l

I

i/,,7/,,/

#;

i

:/I !1,21

t:

[

I ,7,r

0

100

I

#

N._o:7~,

//

-I

. . . .

........

,

.

i'

.

I

N,> -3.ss --

"i

o

1

experimental

~onno et al., 1988 ............ calculated (impeller zone)

~

calculated

(Uu~k zone)

i

#

200

.

~-0.05

,

[

o 0,4 F t

.,7,.-qg.,--,r

,i

:" Ir : <~.7 ~7.r ,';,"7 o , ,,,..7,~i,~,s ; -"1 c,g q #,it

IO

T = 0.186 m, T / D = 2

300

400

500

600

700

800

Drop diameter, lam Fig. 1. Comparison between calculated and experimental (Konno et al., 1988) transient drop size distributions (ix c = 6.9.10-4Pas, IXD = 8.9.10-4Pas, Pc = 1000 kg/m 3 , I t ) = 1 0 3 0 k g / m 3 , er = 0.034 N / m ).

148

NOMENCLATURE a = film radius, m A = Hamaker constant, J C, Cg, Cin, Cp, Cx =constants d = drop diameter, m D = impeller diameter, m f(dj, dk) = collision rate, m'3 s.l F = interaction force, N g(d), g(v) = breakage function, s"l g(d,a) = characteristic frequency of eddies of scale d, s"~ h(v,v') = drop collision function, m 3 s"i L = integral scale of turbulence, m mj, mk = masses of colliding drops, kg n(v,t) = number density of drops of volume v at time t, m "~ N = impeller rotational speed, s"~ p(d) = pressure stress, Pa P(a) = probability density of a r = size of an eddy, m R = drop radius, m RE = radius of larger drop, m Rs = radius of smaller drop, m S(a,r) = rate of strain, s"~ t = time, s tc

= contact time, s = coalescence time, s

ext ti ti T

= collision time for convection, s = collision time for bouncing, s = tank diameter, m

Ui

= particle velocity, m s"l

Ux Uj, Uk

= particle velocity in the loop, m s1 = individual velocities of coliding drops, m s1 = velocity difference over distance L, m s"~ = velocity difference over distance r, m s1 = drop volume, m 3 = position in the loop = RE + Rs minus the distance between drop centres, m

UL Ur V X Z

Greek letters = multifractal exponent

~(v, v ' ) = probability density of forming drops of volume v from drops of volume v' y E:

= coefficient of virtual mass, y = 0.75 = energy dissipation rate, m 2 s "3

(1"1)

= Kolmogorov scale, m

L(v, v ' ) = coalescence efficiency v(v) p

= number of daughter drops = density, kg m "3

p(a) cr q0

= prefactor in eq (2) = interfacial tension, N m "~ = dispersed phase volume fraction

REFERENCES Ba/dyga, J., Bourne, J. R., J. Chem. Eng. Japan 26, 738 (1993). Batdyga, J., Bourne, J. R., Chem. Eng. Sci. 50, 381 (1995). Batdyga, J., Bourne, J. R., Turbulent mixing and chemical reactions, Wiley & Sons, Chichester, 1999. Batdyga, J., Podg6rska, W., Canad. J. Chem. Eng. 76, 456 (1998). Chesters, A.K., Trans IChemE 69, Part A 259 (1991). Chhabra, A., Jensen, R., Phys. Rev. Lett. 62, 1327 (1989). Frisch, U., Parisi, G., "On the singularity structure of fully developed turbulence", in "Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics", North-Holland, Amsterdam, Holland (1985), pp. 84-88. Kolmogorov, A. N., Dokl. Akad. Nauk SSSR 30, 301 (1941). Konno, M., Muto, T. and Saito, S., J. Chem. Eng. Japan 21,335 (1988). Kuboi, R., Komasawa, I. and Otake, T.J., J. Chem. Eng. Japan 5, 423 (1972). Lagisetty, J. S., Das, P. K., Kumar, R., Chem. Eng. Sci. 41, 65 (1986). Meneveau, C., Sreenivasan, K. R., J. Fluid Mech. 224, 429 (1991). Obukhov, A. M., Dokl. Akad. Nauk SSSR 32, 22 (1941). Okamoto, Y., Nishikawa, M., Hashimoto, K., Int. Chem. Eng. 22, 88 (1981). Tsouris, C., Tavlarides, L.L., AIChEJ 40, 395 (1994). Wu, H., Patterson, G.K., Chem. Eng. Sci. 44, 2207 (1989). Acknowledgement.

T h e f i n a n c i a l s u p p o r t o f D S M R e s e a r c h is g r a t e f u l l y a c k n o w l e d g e d .

I 0th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

149

Measurement and Analysis of Drop Size in a Batch Rotor-Stator Mixer Richard V. Calabrese, Michael K. Francis*, Ved P. Mishra and Supathorn Phongikaroon Department of Chemical Engineering, University of Maryland, College Park, MD 20742-2111 USA ABSTRACT In order to evaluate the performance of high shear mixers, drop size has been measured in a batch rotor-stator mixer for dilute dispersions of low viscosity organics in water. Data were acquired, in-situ, using a novel video probe and automated image analysis software. Power draw was also measured. A class of mechanistic models, which account for drop interactions with turbulent velocity fluctuations and the shear field at various length scales, were developed to aid in data interpretation and provide a basis for correlation. Analysis of the data revealed that equilibrium drop sizes were of the order of the Kolmogorov microscale and that "mean shear" in the rotor-stator gap was not the predominant breakage mechanism. Furthermore, models that account for inertial drop-eddy interactions and submicroscale shear forces, suggesting a mixed breakage mechanism, equally correlated the data. 1. INTRODUCTION High shear mixers are broadly employed in chemical processes to produce emulsions and liquid-liquid dispersions. Despite their widespread use, there is almost no fundamental basis by which to theoretically predict or experimentally assess their performance. As a result, process development, scale-up and operation are often by trial and error, leading to higher processing costs, start-up problems and lost time to market. In order to develop a fundamental understanding, at least on a mechanistic basis, of how drop breakup affects liquid-liquid dispersion performance, it is necessary to develop mechanistic models and conduct definitive experiments to elucidate breakage mechanisms and provide a basis for data correlation. Fortunately, for turbulent conditions, there is a large body of literature that exists for stirred tanks that can guide this endeavor. However, high shear mixers are capable of producing smaller drops, so it is not certain that commonly accepted mechanistic theories and correlating equations for stirred vessels will apply. Furthermore, in-situ measurement of the drop size distribution (DSD) requires the use of novel high resolution imaging techniques. While our goal is to characterize dispersion and emulsification in continuous, in-line mixers, the work reported here is focused on understanding the breakup process and the development of techniques to measure the DSD. To this end we have monitored dilute dispersions produced, for turbulent conditions, in a Ross Model ME 100LC batch mixer with a 4 bladed rotor and various stator head geometry. Several low viscosity organics served as the dispersed phase so that internal viscous resistance to breakage could be ignored. A novel, high magnification video probe, with fully automated image analysis software, was used to * Current address: W. R. Grace & Co., Davison Division, Baltimore, MD 21226 USA

150 measure the DSD. A novel technique was also used to measure power draw. Systematic experiments were conducted to determine the dependency of drop size on power input, gap shear rate, stator head geometry and relevant physicochemical factors. A class of mechanistic models was developed that account for shear forces and drop-eddy interactions that could potentially cause breakup at various length scales. Fitting the mean drop size data to the various models provides physical insight into the breakage mechanism and a potential basis for data correlation. 2. E X P E R I M E N T A L P R O G R A M The experimental facility, which is comprised of a Ross Model ME 100LC batch mixer with a 4 bladed rotor, is shown in Figure 1. The rotor diameter is D = 3.4 cm and the standard stator inner diameter is 3.5 cm. The purpose built shaft and bearing assembly with the integrated vessel cover is not commercially available. The 2.5 liter stainless steel tank is fixed to the mixer with a Tri-clover clamp. It is completely filled with liquid during operation to prevent entrapment of air bubbles. Since the tank does not contain baffles, the mixing head is mounted off-center to promote top to bottom mixing. Two different stator heads were used. Six round turbulent jets emanate from the disintegrating head of Figure 1, while the slotted head of Figure 2 produces many finer slot jets. The standard rotor-stator

Figure 1. Batch rotor-stator mixing apparatus with close-up of "Disintegrating Stator Head"

!-i

_

......

I

Figure 2. "Slotted Stator Head"

gap width is 5 = 0.5mm. To test the effect of gap width and gap shear rate on drop size, a second slotted stator ring was machined to an inside diameter of 3.6 cm to produce a ~5 = 1 mm gap width. This geometry will subsequently be referred to as the "wide gap slotted mixing head".

151

2.1 High Magnification Video Probe The high magnification video probe, which is shown schematically in Figure 3, is capable of in-situ measurement of particle size distributions in the size range from 4 to 200 ~tm, with a resolution of 0.5 Ixm per pixel. As shown in Figure 1, the stainless steel probe, which has an immersion diameter of 1.5 cm, is inserted through the top of the mixing vessel. Fiber Optic

Light ~ Source

I

[

"~.~,~~~Camera "

I

Figure 3. Schematic of video probe and data acquisition system.

VCR [

I'

'1

'* ****

FrameC,rabber

The tip of the probe contains a 40X microscope objective which is used to image the particles and to focus light on the sampling volume. This is accomplished using internal relay lenses and a beam splitting mirror. Pulsed lighting, at the camera framing rate of 60 s"1, is used to freeze the motion of the particles in the flow field and present an image to the CCD video camera. The light source is a strobe with a flash duration of about 1 Its that is focused by an external lens and supplied to the probe via a fiber optic cable. A frame grabber controls the B&W camera and dark field lighting system, so that the images can be automatically digitized or output to a VCR. Automated image analysis routines were developed to condition the images, remove poorly focused particles and size the remaining drops using the characteristic ring signature of dark field illuminated objects. The video probe was designed and constructed by the authors. An extensive validation was performed using latex microspheres and soda glass particles to determine the maximum particle concentration, minimum measurable particle diameter (as a function of the index of refraction) and particle sampling volume. Corrections were made for sample volume size biases. Francis (1999) discusses the design, testing and validation of the video probe, as well as the image analysis software, in detail.

2.2 Liquid-Liquid Dispersion Experiments Three immiscible liquids, anisole, phenetole and chlorobenzene, were dispersed in water for the rotor speed range of 1,500 to 4,000 rpm. These substances were chosen since their viscosity was water-like and their refractive index was relatively large compared to water. Their physical properties are given in Table I. The system was maintained at 25 ~ by immersing the mixing vessel in a constant temperature bath. The dispersed phase concentration ranged from 0.08 to 0.24 % by volume. The purpose of this small variation was to test the assumption that the system was coalescence free. While the video probe was usually fully inserted into the vessel, the sampling location was also varied to test the assumption that drop size was spatially uniform. Each series of experiments were conducted

152 by filling the mixing vessel with water and then setting the rotor speed. The organic was then introduced with a syringe through a small sampling port in the vessel cover plate. At least 2 hours was allowed for the system to achieve an equilibrium DSD. After at least 800 counts of drop size were acquired, the rotor speed was increased and the system was again allowed to reach equilibrium. This procedure was repeated for the entire rotor speed range.

Table I. Physical Properties of Dispersed Phases Fluid:

Anisole

Phenetole

Chlorobenzene

Interfacial Tension with water (mN/m)

27.4

29.4

35.0

Density (kg/m3)

997

996

1,100

Viscosity (mPa.s)

1.1

1.3

0.8

2.3 Measurement of Power Draw The mixer motor input power was measured using a Load Controls, Inc. Model PH2A load meter. This was converted to power draw, P according to P

=

EM(n, PT)" PT - EM(n, PF)" PF

(1)

At each rotor speed, n, the total motor power, PT, and the power to overcome friction in the bearings and mounting hardware, PF, were measured. The later was obtained by running the system with the rotor removed from the shaft. To obtain fluid power draw it is necessary to know motor efficiency, EM, as a function of motor power and rotor speed. This was measured by removing the shaft and bearing assembly with its integrated cover from the mixer (see Figure 1) and replacing it with a free standing shaft containing a standard Rushton turbine. This turbine was used to agitate a baffled tank that was placed on a frictionless air bearing assembly. Measuring the torque required to prevent the tank from rotating yielded the actual power transmitted to the fluid. The motor efficiency was obtained by simultaneously monitoring motor power with the load cell. Efficiency experiments were conducted over a broad range of motor speed using water, corn syrup and their solutions. The procedure yielded a family of curves of efficiency versus motor power, at constant rotor speed, that could be used with Eq. 1. 3. MECHANISTIC MODELS FOR MEAN DROP SIZE In the absence of coalescence, mechanistic models for Sauter mean diameter, d32, can be developed by analogy to the approach for stirred tanks given by Chen and Middleman (1967), Calabrese et. al. (1986), Wang and Calabrese (1986) and others. If the drop is inviscid then only interfacial tension, (~, opposes drop deformation due to the disruptive force of the surrounding fluid. For a drop of diameter, d, the cohesive force (per unit area) due to interfacial resistance is given by xa -- aid. When the disruptive force is inertial or due to turbulent velocity fluctuations, the disruptive stress is given by: -

p

-

p

E(k)dk

(2)

153 p is the continuous phase density and v ' ( d ) 2 is the mean square velocity difference across the drop's surface. E(k) is the energy spectral density function that describes the energy contained in eddies of wave number k to k + dk. A maximum stable drop size, dmax exists when these disruptive and cohesive forces are in equilibrium so that XD(dmax) = xa(dmax). If it is further assumed that d32 0~ dmax, then a model for d32 follows from knowledge of the functional form of E(k), which depends upon the magnitude of dmax relative to the Kolmogorov microscale, rlK = (V3/e)TM. The kinematic viscosity, v = 11 / p is the ratio of the viscosity, rl, to density of the continuous phase, e is the local energy dissipation rate per unit mass and is characteristic of the maximum value in the system. It is often the case, as for stirred tanks, that dmax is small compared to the turbulent macroscale, which is of order of the stirrer diameter, D, and large compared to rlK SO that the ultimate drop size is determined by drop interactions with inertial subrange eddies. Then E(k) - e 2/3 k U 3 for D >> d >> rlK. This leads to a scaling law for d32 in terms of e and physical properties. For geometrically similar systems, it is argued that e is proportional to the power draw per unit mass of fluid. For constant Power No., the scaling law yields the well-known Weber No. correlation. The results are d32 "" G 3/5 13 -3/5 s -2/5

and

d32 / D - W e

3/5

[Inertial, D >> d >> rlK]

(3)

where the Weber No. is given by We = p n 2 D 3 / (~. As will be discussed below, rotor-stator mixers produce drops of order rlK and smaller. As a result, Eq.(3) may not be valid. If the stress on the drop is inertial, Shinnar (1961) suggests that v'(d) 2 - v -1 e d 2 for d < rlK. This leads to d32 -" G 1/3 p -2/3 11 1/3 F_,-1/3 and

d32 / D -

( W e R e ) 1/3

[Inertial, d < rlrd

(4)

where the Reynolds No. is given by Re = n D 2 / v. If the stress on the drop is inertial, Chen and Middleman (1967) suggest that E(k) = v -4 e 2 k -7 for d << rlK. This yields d32 -" G 1/7 p -5/7 11 4/7 E -2/7 and

d32 / D -

( W e R e 4)- 1/7 [Inertial, d << 11rd (5)

Shinnar (1961) further suggests that if the viscous stress for d < 11 is responsible for drop breakup, then XD - g 7' where the shear rate is given by y' -- (e / v) 1/2. This yields d32 -. i~ 1 P -1/2 11 -1/2 1~-1/2 and

d32 / D - ( W e R e - 1/2)- 1 [Viscous, d < rlK] (6)

To the authors' knowledge, the assertions made by Shinnar and by Chen and Middleman have not been tested due to lack of drop size data for d < riK. A final speculation is that the drops are broken by the high mean shear rate in the rotor-stator gap. Then XD - g y' where the shear rate is given by y' - Vtip / 15, with rotor tip speed, Vtip ~ n D. This yields d32 -, i~ 1 11 -1 5 Vtip"1

and

Ca - We / Re is the Capillary No.

d32 / 5 -- ( W e R e - 1 ) - 1

[Shear gap]

(7)

154 The mechanistic models represented by Eqs. (3) to (7) can be confirmed or rejected by comparison with data. Model discrimination will yield insight into the breakage mechanism and a basis for correlation. 4. RESULTS AND DISCUSSION Figure 4 is a plot of power draw versus rotor speed for the three stator head geometries of this study. It is seen that they all supply about the same amount of power to the fluid, with the slotted head drawing slightly more than the disintegrating (round) and wide gap slotted heads. When the Power No. is constant, P ~ n3. For these data P ~ nx, where X ~ 2.7, indicating that the Power No. is not constant. Therefore, while the first expression of Eqs. (3) to (6) are valid, the second expression, which is based on constant Power No., is not strictly valid. If these data are used to estimate the Kolmogorov microscale then 25 lxm > rlK > 15 gm for 1,500 RPM < n < 4,000 RPM. Local values of rig due to a local maximum in E could be somewhat smaller. Since rlK ~ E -1/4our estimates are reasonable. 100 t i"

Figure 4. Power draw versus rotor speed.

p_n x

~o

OSiotted, x = 2.69 A Disintegrating, x

=

2.67

@Wide gap slotted,x = 2.73 1 1000

10000 Rotor Speed, n (RPM)

Figure 5 shows Sauter mean diameter versus rotor speed for several experiments. At lower rotor speeds d32 is not much larger than riK, but at higher rotor speeds they are of the same order. Figure 5 shows several trends that are characteristic of the entire data set. 1. Over the limited range of dispersed phase concentration, t~, studied, drop size was independent of t~. Drop size was also independent of spatial location. 50 45 40 35

~, 30

Drop Fluid

Head

~%

9 Anisole Disintegrating 0.24

[]

o N 20 '~ 15 10 5 0 1000

!

n Anisole 90 3

t

o

t

I

'

2000

3000

4000

Slotted

0.19

Disintegrating 0.08

o CB

Slotted

0.08

A Phenetole

Slotted

0.18

5000

l~or s r ~ t ntm'M~

Figure 5. Sauter mean diameter versus rotor speed for several series of experiments. ~ is the volume % of dispersed phase.

155 2. Over the limited range of interfacial tension studied, no trends in d32 versus g could be discerned above the scatter in the data. 3. d32 for the slotted stator head was less than that for the round or disintegrating head. This is consistent with our observations of power draw. An additional observations is: 4. At constant rotor speed, the wide gap slotted stator head produced smaller drops than the standard gap slotted head, despite the fact that the nominal shear rate in the gap was reduced by a factor of two. This observation eliminates the model of Eq. (7) from consideration and somewhat invalidates the idea that drops are predominately broken up in the rotor-stator gap. It is more likely that breakup occurs in the turbulent jets emanating from the stator slots. Given the insensitivity of the data to liquid-liquid pair (via a), spatial location and t~, and the somewhat small variation with stator head geometry, it was decided to consider all of the data together for purposes of correlation and model validation. The data are fit by d32 = 0.44 dmax

Coefficient of variation R = 0.91

(8)

Chen and Middleman (1967) and Wang and Calabrese (1986) reported a ratio close to 0.6 for Rushton turbine stirred tanks. Figure 6 is a plot of d32 / D versus Weber No. The data are well fit by d32 / D = 0.40 W e -0.58

(9)

It therefore appears that Eq. (3) correctly scales the data and that breakage, like in turbine stirred tanks, is due to drop interactions with inertial subrang_goeseddies. For a Rushton turbine, Chen and Middleman (1967) reported d32 / D = 0.45 We " , a strikingly similar result. A careful examination of the data reveals that the correlations or second expressions of Eqs. (4) and (6), which apply for d < rlK, also provide a reasonable fit to the data. However, these two models exhibit a somewhat different dependence on e. Our results are for a single batch size and rotor diameter. Given the data of Figure 4, we can take E ~ P. Figure 7 presents a plot of d32 versus P. The data are well fit by d32 -- P -0.48, which compares more favorably with Eq. (6). It can therefore be concluded that the most likely mechanisms for breakup of drops whose size is near the Kolmogorov scale are due to interactions with inertial subrange eddies or due to sub-microscale viscous stresses.

Figure 6. Weber No. correlation.

156 There is an additional means to further discriminate among Eqs. (3), (4) and (6) based upon continuous phase viscosity. The inertial subrange model shows no dependency on viscosity. The viscous and inertial models for sub-Kolmogorov drops show opposite trends with viscosity. Experiments with different continuous phase viscosity are currently underway. 100 I

r

Figure 7. Correlation with power draw.

d~- p~.48

10

9Anisole oC h l ~

x Plz~etole , 1

I 10

100

Po~er Draw,P (Watts) 5. SUMMARY Analysis of the power draw data and drop size data for dilute dispersions in water reveals that for the range of experimental variables studied, the data are insensitive to physical properties, dispersed phase volume fraction and spatial location. Furthermore, there is only a small dependency on stator geometry. The drops are similar in size to the Kolmogorv length scale. It is unlikely that drops are predominately broken up in the rotorstator gap. It is more likely that breakup occurs in the stator slots or in the jets that they produce. Data correlation suggests a mixed turbulent breakup mechanism due to drop interaction with inertial subrange eddies and sub-Kolmogorov scale viscous stresses. The data are well correlated with power draw and with a Weber No. correlation that is close to that for a Rushton turbine stirred tank. 6. REFERENCES Calabrese, R. V., T. P. K. Chang, and P. T. Dang, "Drop Breakup in Turbulent Stirred-Tank Contactors - I: Effect of Dispersed Phase Viscosity," AIChE J, 32, 657 (1986). Chen, H. T. and S. Middleman, "Drop Size Distribution in Agitated Liquid-Liquid Systems," AIChE J., 13, 989 (1967). Francis, M. K., "The Development of a Novel Probe for the In Situ Measurement of Particle Size Distributions, and Application to the Measurement of Drop Size in Rotor-Stator Mixers", PhD Thesis, University of Maryland, College Park, MD, USA (1999). Shinnar, R., "On the Behavior of Liquid Dispersions in Mixing Vessels," J. Fluid Mech., 10, 259(1961). Wang, C. Y., and R. V. Calabrese, "Drop Breakup in Turbulent Stirred-Tank Contactors - II: Relative Influence of Viscosity and Interfacial Tension," AIChE Journal, 32, 667 (1986).

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

157

The impact of fine particles and their wettability on the coalescence of sunflower oil drops in water A W Nienow, A W Pacek, R Franklin and A J Nixon School of Chemical Engineering, The University of Birmingham, Edgbaston, Birmingham B 15 2TT, U.K. Coalescence rates of 5% sunflower oil drops in water have been measured under agitated conditions with and without fine particles being present. When using polymer particles of less than 10~tm in size, coalescence rates could be increased or reduced, the extent depending on their concentration and wetting characteristics. The greatest enhancement occurred with 1.4g/L PMMA particles wetted by oil and partially wetted by water. With monoglyceride fat crystals, similar results were found with very dramatic increases in coalescence with 1 to 2% fat with respect to sunflower oil while at 5%, coalescence was totally repressed. 1 INTRODUCTION

1.1 Reason for the Work The coalescence of sunflower oil (SFO) drops plays an important role in the production of low fat spreads. Typically, as part of the production process, a phase inversion step is required during cooling and agitation and the inversion is strongly dependent on coalescence rates. However, it is known that the phase inversion also depends significantly on the presence or otherwise of fine monoglyceride fat crystals which are formed during the cooling/mixing process. If they are present, phase inversion occurs; if they are absent, it does not [1 ]. On the other hand, in many cases, the addition of fine solids has been used as a means of preventing coalescence and thus acting as an emulsifier by stabilising dispersions [2]. Critical features in determining whether fine solids enhance or prevent coalescence is their wettability [3]. Other important aspects are the shape, concentration and size of the particles [4,5]. However, in many of these earlier studies, only stability was measured and coalescence rates were inferred. Thus, dispersions were produced and then left to stand under quiescent conditions. If they took a long time to settle/separate, then coalescence was considered to be slow; and if separation was rapid, coalescence was considered fast. It can be concluded firstly, that the above interactions are complex and not well understood. Secondly, since in the industrial process involving phase inversion, the impact of agitation on coalescence rates is important, it would be valuable to study coalescence under dynamic conditions, i.e., during agitation. Therefore, it was decided to study coalescence by the "step down in speed method" [6] in which the subsequent increase in drop sizes was followed in situ using the recently-developed video-technique [7].

1.2 Wettability/Coalescence Considerations The effect of wettability can be considered in terms of contact angles which affect how dispersed particles collect at interfaces. Fig.1 shows an example of how a particle of a

158 particular shape will sit at an interface [ 1] depending on the contact angle, ct. In general, it is postulated [ 1] that these particles, when they reside in the water phase (ct<45~ stabilise oil-inwater dispersion; whilst if they reside in the oil phase (135~176 they stabilise water-inoil dispersions especially if present in sufficient quantities as to cause steric hindrance [2]. Those particles with contact angles between the two can stabilise (large quantities-steric hindrance) or enhance coalescence. Enhancement is also related to the size and shape of the particles which has an effect on precisely where the particles sit at the interface. Thus, they may increase coalescence by bridging the film and encouraging drainage [5], especially if they are present in relatively small quantities. 2 EXPERIMENTAL

2.1 Choice of Mimic Particles Firstly, particles that could be considered to mimic the monoglyceride crystals were selected [8] as experiments using them would be easier to undertake and interpret. Three particulate materials were chosen; polyethylene (PE), polypropylene (PP) and polymethylmethacrylate (PMMA). The particle size, dispersed in SFO, was determined by a Malvern Sizer 2600. Table 1 gives the mean sizes and shape, and contact angles with respect to both SFO and to water, determined by either the Washburn method, the sensile drop technique or from the literature [8]. Thus, the PE and PP particles can be considered as fully wetted by the SFO and not wetted by water, and the PMMA preferentially wetted by SFO but also partially-wetted by water. 2.2 The Liquids The liquids were double distilled water and silica-treated SFO. The density of the SFO, Pd = 920 kg/m 3, the viscosity, 9d = 55 mPas and the interracial tension between water and sunflower oil, a = 26.7 + 0.2 mN/m at 20~ as measured by the pendant drop technique [8]. The fatty acid composition of the SFO does not change much from one batch to another but, as a natural product, it is not constant. It consists of 16 and 18 carbon chain acids, saturated and unsaturated [8] and its precise composition can also change on storage. Therefore, batches were replaced from time to time and even without the presence of particles, the results changed when the batch changed. Therefore, the conclusions are based on relative values within each SFO batch rather than absolute values compared across batches.

2.3 The Agitated Vessel The experiments were carried out in a baffled cylindrical glass vessel of 0.125 m internal diameter (T) equal to its height (H) with a six-bladed stainless steel Rushton turbine, diameter (D=0.5T) centred between the top and bottom. The vessel was placed inside a square waterfilled jacket and the temperature was controlled by a water-bath to 20~ + 0.1. The tank lid and the shaft beating were designed to prevent the entrapment of air. 2.4 The Experiment The particles were dispersed in SFO and the vessel was 95% filled with water. The SFO containing the particles was added to the vessel which was then closed with all air eliminated. The dispersion was stirred at a speed (N) of 480 rpm for 4 hours to reach steady state. The speed was then instantaneously reduced to 240 rpm, so that coalescence predominated and the drop size distribution (based on at least 500 drops) was determined with the video technique

159 [7] as a function of time, (t). The same procedure was repeated for SFO without particles. From the transients of Sauter mean sizes (d32), the coalescence rate (m) was calculated from: (dd32/dt) t=o c~ (2_ 22/3) (d32)t=~' (1) based on a model of Howarth [6].

2.5 Modification of Basic Method Using Monoglyceride Crystals The monoglyceride used was a mainly unsaturated one, Hymono 7804, and the melting point was 36~ [9]. 7804 up to 5% by weight with respect to SFO was added with stirring in a separate vessel and then heated to 60~ for 30 minutes and allowed to cool whilst standing at room temperature. At the point just prior to crystallisation of the 7804, the SFO/7804 mixture was added to the water in the vessel whilst preventing the entrapment of air. Two types of experiment were performed. In one set, the dispersion was stirred at 480 rpm following addition and further cooled using the water bath to 20~ + 0.1 ~ and held for 4 hours to allow full crystallisation. Then, a step change in impeller speed from 480 rpm to 240 rpm was used. In the other, the dispersion was reheated to 60~ at a constant impeller speed of 240 rpm and held for four hours (to ensure all 7804 was dissolved). The system was then slowly cooled to 20~ over a further four hours using the water jacket at 20~ During cooling, crystals of 7804 formed. Drop sizes were measured during each experiment. In addition, shear stress-shear rate data were determined using a Carrimed rheometer and interracial tension with the du Nouy technique (since the pendent drop technique was unreliable in the presence of crystals), coveting temperatures from 20 ~ to 60~ and concentrations from 0 to 5% w/w [9]. The data are summarised in Table 2. 3 EXPERIMENTAL RESULTS AND DISCUSSION

3.1 With Polymer Particles Figure 2 shows the impact of the concentration of polyethylene particles (PE) (starting from zero) on the variation of d32 with time of SFO (batch U2)drops after a step change from a steady state at 480 rpm to 240 rpm. The results for SFO (U2) are quite typical of the results found with other batches. There is a range of low concentrations where the addition of the PE has little or no impact on drop size or rate of coalescence (here at 2.3 g/L PE); then a narrow range of higher concentrations where the impact is very large (here at 4.7 g/L PE); and a range at higher concentrations where the addition of PE again has little or no effect, even possibly retarding coalescence (here at 9.2 g ~ PE). Observation of magnified drops using the video technique revealed PE particles on their surface especially when the drops were largest at 4.7 g/L PE. The coalescence rates calculated by the method of Howarth [6] are shown in Table 3 where it can be seen that the coalescence rate at 9.2 g/L had indeed dropped below that with SFO (U2) alone. At the same concentrations, the PE and PP particles produced a very similar impact on the coalescence of SFO (U2) (data not shown). If it is assumed that the extent of drop coverage is an essential feature that determines whether coalescence is enhanced or not as discussed earlier, it is important to estimate what the coverage is. The total surface area of drops, AD can be calculated from

160 AD = 6 VsFo (2) d32 where VsFo is the volume of SFO added. The ~ area covered, Ac, can be related to the number of particles, np, present and can be shown [8] to be Ac =100 Ap np AD

(3)

where the area of coverage (Ap) by a single polymer particle (diameter, dp) is n dp~/4. The coverage calculated in this way is also shown in Fig.2. The coverage gets greater as the drops get larger because the interfacial area of a fixed volume of drops, Ao, gets smaller. For these sizes of PE (and PP) particles, it appears that a coverage of around 5% is sufficient to cause a significant enhancement in coalescence rate. Lower values (<-3%) have a negligible effect and a coverage >-10% starts to produce steric hindrance. Similar results were found with the next batch ( U 3 ) o f SFO. However, even without particles, for the same step change with 5% SFO (U3), the coalescence rate was lower and at different concentrations of PE particles, this lowering continued with a maximum impact at 7 g/L (see Table 3b). SFO (U3) was then used to compare the impact of polymethylmethacrylate (PMMA) and PE particles. Fig.3 shows that with PMMA, the enhancement of coalescence is much greater and occurs at much lower concentrations in terms of mass added and % coverage compared to PE. The impact on coalescence rates can also be seen in Table 3c. In addition, the size distribution became much wider with PMMA with many small drops and a few very large. Table 4 shows dl0 (which is a good measure of the small drops) and d32 and d43 (both of which relate to the largest). It can be seen that for the PMMA addition to U3, the large drops have coalesced while many small ones have been left unaffected, with dl0 even lower than with SFO (U3) alone. On the other hand, for the PE particles, the large drops are not so affected but dx0 is about 50% greater than with SFO (U3) alone. Clearly, with PMMA particles, fully wetted by SFO and partially-wetted by water, the coalescence efficiency is greatly enhanced. This enhancement is postulated to be due to the particles being able to bridge the film between two oil drops. With PE and PP particles which are not wetted by the water, the film is destabilised but it is probably due to these particles causing the interface of the oil drops to be distorted (dimpled) as they collect there, thus thinning the film and requiring less drainage locally. When the concentration of PE and PP gets high enough, the dimpling effect is less pronounced, the interface becomes more rigid and steric hindrance becomes important. 3.2 With Hymono 7804 Fig.4 shows d32 for 5% SFO plus 7804 as a function of time for a step reduction in speed from 480 to 240 rpm at 20~ Again, there is very little difference in coalescence rates between SFO without 7804 and with 0.5% w/w 7804. At 1% w/w and 2% w/w, the rate is greatly increased and rather similar and at 5% w/w 7804, coalescence is essentially zero. These rates are quantified in Table 3d. It can be seen that this batch of SFO (H) had very low coalescence rates which were dramatically enhanced by the 7804. In addition, to these high rates of coalescence, large very complex, non-spherical drops were seen in the video pictures during coalescence at the 1% w/w and 2% w/w 7804 level. Perhaps the presence of these drops accounts for the fluctuation in size indicated in Fig.4.

161 Fig.5 shows the initial and final steady state values at 60~ and 20~ with and without 7804 when agitated at 240 rpm. The presence of 7804 significantly lowered the interracial tension particularly at 5% w/w but hardly altered the viscosity at 60~ (Table 2). At this temperature, the drops initially get smaller with increasing 7804 and then increase slightly,, perhaps because the density of the continuous phase increases [10] or because of interfacial affects associated with 7804 whilst still liquid [11]. Finally, at 5% they are notably smaller compared to SFO (H) without 7804. After the temperature is lowered over four hours (Fig.5), from zero up to 0.5% w/w 7804, the combination of changing viscosity, interfacial tension and the presence of crystals at temperatures <-36~ results, somewhat surprisingly, in a negligible change in drop size, i.e., no enhancement of coalescence rates. With 1% and 2% 7804, however, with very similar values of viscosity and interfacial tension, there is a dramatic increase in size again indicating very enhanced coalescence rates. However, this increase only occurs to a significant extent at temperatures below-30~ i.e., after crystals have started to form (data not shown). Thus, it can be concluded that in these much more complicated systems, the increase in drop size and coalescence rate is due to the formation of oil-wetted, water non-wetted crystals. At 5% w/w 7804, coalescence is inhibited to such an extent that drop sizes are below those found without 7804 and remain the same at all temperatures. It is suggested that the reason for inhibition of coalescence is again steric hindrance but also due to the significant increase in viscosity associated with the presence of crystals throughout the SFO, giving it structure and shear-thinning rheological properties (Table 2). Under these conditions, neither the reduction in speed (Fig.4) or temperature (Fig.5) has any affect on mean drop size. 4 CONCLUSIONS Polymer particles have an affect on the coalescence of SFO drops in water similar to those found with monoglyceride fat crystals and are therefore a reasonable mimic for the latter. In this work, following a step reduction in speed, the change in drop size of a 5% dispersion of SFO with or without such particles was followed. Non-water-wetted but fully oil-wetted particles of <10~m of PE and PP either have a minimal effect on coalescence rates (at low concentrations giving <-5% coverage of drops), markedly enhance it at concentrations giving between about 5 and 10% coverage and tend to reduce rates at >-10% coverage. Similar sized polymer particles of PMMA, partially water-wetted and fully oil-wetted, have a much more dramatic effect on coalescence rates but appear to particularly cause coalescence of larger drops. It is suggested that because of the difference in wettability of the PE and PP compared to the PMMA, the former particles enhance coalescence by encouraging drainage of the water film between oil drops whilst the latter does so by bridging the film between them. Monoglyceride fat crystals in the form of the predominantly fully-saturated monoglyceride Hymono 7804 produce a similar range of effects following a similar speed reduction. These effects are found over a concentration range up to 5% w/w relative to 5% SFO dispersion in water with a large enhancement of coalescence in the 1 to 2% range. Coalescence was also enhanced at the 1 to 2% w/w concentrations due to a lowering of temperature alone from 60 ~ to 20~ when agitating at 240 rpm just as is required in the production of low fat spreads; but at 5% w/w, coalescence was totally repressed.

162 5 ACKNOWLEDGEMENTS

A BBSRC CASE award (for AJN) is gratefully acknowledged as well as valuable discussions with Dr I P Norton, Unilever Research, Colworth. 6

NOMENCLATURE

Ac

d~o d32

d43 D N T t

VSFO cy (o

7804 PE PP PMMA SFO

% of area of SFO drop covered by polymer particles, % area of SFO drops covered by polymer particles, m 2 area of SFO drops, m 2 polymer particle diameter, m number mean drop size, m Sauter mean drop size, m mass mean drop size impeller diameter, m impeller speed, rev/s vessel diameter, m time, s volume of SFO drops, m 3 contact angle, deg interfacial tension, N/m coalescence rate, s~ Hymono 7804, a mainly unsaturated monoglyceride fat polyethylene particles polypropylene particles polymethylmethacrylate particles sunflower oil

REFERENCES

1. I.J. Campbell, I.T. Norton and W. Morley, Netherlands Milk Dairy Journal, 50 (1996) 167. D.E. Tambe and M.M. Sharma, Advances in Colloid and Interface Science, 52 (1994) 1-63. 3. J. Mizrahi and E. Bamea, British Chemical Engineering, 15 No. 4 (1970) 497-503. 4. I.J. Campbell, Food Colloids, Eds. R.D. Bee et al., RSC., 1989, p272-282. 5. K. Boode and P. Walstra, Colloids and Surfaces, 81 (1993) 121-137. 6. W.J. Howarth, AIChEJ, 13 (1967) 1007-1013. 7. A.W. Pacek, I.P.T. Moore, A.W. Nienow and R.V. Calabrese, AIChEJ., 40: 1940-1949. 8. R Franklin, PhD Thesis, The University of Birmingham, 1997. 9. A J Nixon, PhD Thesis, The University of Birmingham, 1999, to be published. 10. R.V. Calabrese, A.W. Pacek and A.W. Nienow, I.Chem.E. Res. Event, 1993, 642-644. 11. J.T. Davies and E. Rideal, Interfacial Phenomena, Academic Press, 1963. ,

Table I Data Characterising the Water/SFO/Polymer Particle Systems Polymer Shape Water ao Sunflower Oil a~ dl0* d32 * PE Irregular 100 + 1.7 24 + 1.4 3.9 6.1 PP Irregular 106 + 1.6 33 + 1.9 4.2 6.7 PMMA Spherical 71.4 0 to 20 6.9 7.0 PE & PP - oil wetted/not water wetted; PMMA-oil wetted/partially water wetted.

d43"

.

.

7.7 9.1 7.0 .

.

.

.

.

*l.tm

163 Table 2 Physical Properties of SFO wetted H ymono 7804 Interfacial Tension Concentration Temperature Viscosity mN/m % w 7804/w SFO(H) ~ Pa.s 25.1 0.0 20 0.072 a 60 0.016 a 25.5 0.5 20 0.086 b 12.1 60 0.019 a 16.4 1.0 20 0.071 b 8.2 60 0.015 a 11.1 5.0 20 0.715 b 2.8 60 0.014 a 4.0 Constant viscosity b Viscosity at a shear rate of 100 s~ (shear thinning)

Table 3 Coalescence rates of 5 vol % SFO water for a 480/240 step change : a) Batch U2/PE; b) Batch U3/PE; c) Batch U3/PMMA; d) Batch H/7804 a) U2 (d32 l.tm).o w [10.3 s"1] . . . . . b ) U3 ... (d32 ~m)t=o 137.2 0.8 U3 130.0 2.3 g/L 133.1 1.0 4.6 g/L 129.0 4.7 g/L 167.3 3.1 7 g/L 145.2 9.2 g/L 154.2 0.4 9.3 g/L 127.3

w [10 -3 S"l] 0.5 0.6 0.8 0.2

,

c) U3 U3 0.7 g/L 1.4 g/L

.

(d32 ~tm)t=0 130.0 155.3 173.5

w [10 -3 s"l] 0.5 3.0 4.2

d) H H 0.5% 1.0% 5.0%

(d32 btm).0 123.3 121.4 224.0 105.3

~ [10 -3 s-1] 0.14 0.17 2.4 0.062

Table 4 Mean Diameters for 5% SFO (U3) at 240 mins after a 480/240 step change with different PMMA concentrations and the PE for maximum coalescence rate. U3 0.7 g/L PMMA 1.4 g/L PMMA 7g/L PE dl0 [~tm] 95 77 81 147 d32 [btm] 300 560 670 371 d43 [~tm] 360 686 839 419 d32/d10 3.2 7.3 8.3 2.5 ,

,

PMMA

PE/PP

(

OIL

/ (a) 0~

~

(b) 45~

~

, ,

(c) 135~

) ~

Fig 1. Effect of shape and contact angle on the interfacial position of a solid particle at an oil/water interface [4]

]64 500

600] d32[pm]

d32[l~m] 1 1 1 1

~

/~13.4% /~U2 9.6%/ ..o.. 2.3g/IPE

9 400500 I

~.

400 300

/ 200

/~( /

~4.7g/I PE ---9- 9.2 g/I PE 3.2% .....

:.~'"'"

~T-,.~'~~

1.3%

~ 1 7 6

100

10

20

,

,

,

30

40

50

2.1%

1.8%~

T

/ ] 30C| 20C

0

~i,.,..,o_. - ~ . . . 2 4 % ..,

0

17.70 ~.'.~ 4%

.

22%[

/

/ /

.//.-"

/

.__

60

700

0

10

20

30 40 Time [min]

',

300

50

60

Fig 3. d32 versus time for 5% SFO (U3) with PMMA and PE for max. coalescence after 4801240 rpm step change.

9"

o

9

.9q

~ "....o..

0% w/w 7804 .. o.- 0.5% w/w 7804 ~ - 1.0% wlw 7804 ----9- 2.0% w/w 7804 5.0)~ w/w 7 ~

9

" ' l"~t

0.8%...'"'" o." .. o.. 0.7 g/I PMMA I ." ~1.4 g/I PMMA I o0.6% - ' ~ 7.0g/IPE~.~

d32[pm]

d32 [pm]

400

~

loc

Fig 2. d32 versus time for 5% SFO (U2) with different PE concentrations including coverage calculation after 480/240 step change.

500

.-"

~------

Time [min]

600

.__

t=600C t=200C

.

0

200

~1~,,...--,

,-.'-'v 9 ~

O

100

,o

o

o ~ 1 7 6 1 7 6

9 ~

~

9

~

~

41--------

0

60

120

180

240

Time [min] Fig 4. d32versus time after 480/240 rpm step change, effect of 7804.

0

!

|

!

|

1

2

3

4

5

Concentration of 7804 [% w/w] Fig 5. Steady state d32 versus concentration of 7804 at 240 rpm: 600C - 7804 liquid, 200C - 7804 crystals.

10nhEuropean Conference on Mixing H.E.A. van den Akker and i i Derksen (editors) 2000 Elsevier Science B. 17.

165

Influence of impeller type and agitation conditions on the drop size of immiscible liquid dispersions. M Musgrove a and S Ruszkowski b aBHR Group Limited, Cranfield, Bedford, MK43 0AJ, United Kingdom bProcter & Gamble Company, 8256 Union Centre Blvd, West Chester OH 45069, USA

1. BACKGROUND A common need in industrial liquid-liquid dispersion processes is to scale-up a drop size distribution from a small scale to the plant scale. For low-viscosity dispersions the correlations reported in the literature most commonly relate the Sauter mean diameter to impeller diameter ratio (d32/D) to We ~ with additional groups to account for effects of dispersed phase concentration and dispersed phase viscosity (see Zhou 1998 for a review of correlations). However until very recently almost all the work had been limited to Rushton disc turbines. Further, recent research (Zhou 1998, Pacek 1998) and industrial experience (as exemplified by the members of the FMP - Fluid Mixing Processes consortium at BHR Group) indicate that the We ~ correlations do not adequately describe the drop breakage mechanisms. This paper describes a study which was carried out between 1990 and 1996 as part of a larger research FMP programme on liquid-liquid dispersions (Portingell 1991, Calabrese 1992, Yung 1994, Musgrove 1996). The drop size distributions of immiscible liquid dispersions were measured in 0.17 and 0.29 m diameter tanks for a variety of impeller types: hydrofoils (Lightnin A310, Chemineer HE3), Rushton turbines, and pitched blade turbines. Silicone oil, chlorobenzene, xylene, cyclohexane and tri-butyl phosphate at a phase volume fraction of 0.13% were used as the dispersed phases, with distilled water as the continuous phase. The drop size distributions were measured using macro-photography and subsequent manual sizing to generate drop size distributions and Sauter mean diameters (d32) which correlated with the impeller geometry, operating conditions and continuous and dispersed phase physical properties.

2. EXPERIMENTAL The experiments were carried out in glass torispherical based vessels of 0.17 m and 0.29 m diameter. A standard configuration was used with four stainless steel baffles, H = T and T/3 off-bottom impeller clearance. A stainless steel lid was clamped onto the top of the vessel and all air removed. T/2 and T/3 Rushton turbines were used at both scales. At the larger scale PBTs and hydrofoil impellers were also used. Care was taken, before starting the experiment, to ensure that the vessel and vessel internals were clean and free of any contamination. The cleaning procedure included washing with a liquid detergent, soaking various parts of the apparatus in Decon 75 and then washing with distilled water. Most of the experiments were carried out using silicone oil (Dow Corning 200/5cS) as the dispersed phase and distilled water as the continuous phase. At the 0.29 m scale

166

additional experiments were carried out using xylene, chlorobenzene, cyclohexane and tributyl phosphate as the dispersed phase. Viscosities of the dispersed phases were between 0.5 and 5 mPa s and were low enough for them to be considered inviscid (Calabrese 1986). Interracial tension between the mutually saturated phases was measured using the pendant drop technique.

,,Dispersed phase

Silicone oil Xylen e ............ Chlorobenzene Cyclohexane Tri-butyl,, posphate .

.

.

.

.

.

In'terfacial tension (mN m)' 37 38 33

,,

'

.

.,

47 7

Experiments were carried out at speeds above the "just dispersion" speed, to ensure that the organic phase was completely dispersed. The organic phase was introduced into the vessel at 0.13% by volume using a pipette whose outlet was just above the tip of the impeller running at the desired speed. Macro-photography was used to record images of drops for subsequent analysis using a Magiscan image analyzer. Analysis was done manually by fitting circles to each drop image. For each experiment 500 to 800 drops were measured to generate the drop size distribution and calculate d32, the Sauter mean diameter. All measurements were made 2 hours after the injection of the dispersed phase. The low dispersed phase concentration was chosen to eliminate the effect of coalescence on the drop size distribution. Initial exploratory experiments showed that even at the low dispersed phase volume fraction used the dispersions were not "non-coalescing". Starting from an initial low impeller speed then increasing and decreasing speed resulted in d32 decreasing then increasing again. However the measured drop size distributions were independent of dispersed phase volume, implying that drop breakage rates were much greater than drop coalescence rates.

3. RESULTS AND DISCUSSION 3.1 Effect of local hydrodynamics

Zhou (1998) recently stressed the importance of relating drop sizes to the local impeller hydrodynamics. This was confirmed by a series of measurements for T/2 and T/3 Rushton disc turbines in the 0.17 and 0.29 m tanks. The results are plotted as d3= against impeller speed in figure 1. The same identical impeller (diameter 0.096 m) was used as a nominal T/2 impeller in the 0.17 m tank and a nominal T/3 impeller in the 0.29 m tank. Comparison of the two sets of data for this impeller (in the 0.27 and 0.29 m tanks respectively) shows that the two plots of d32 against speed lie almost exactly over one another. The same drop size distribution was obtained when the same impeller was operated at equal speed in different size tanks, despite large differences in the average power per unit volume. Clearly in this experiment

167

the drop size was controlled by the hydrodynamics close to the impeller, regardless of the size of tank it operated in.

Fig 1 Effect of RDT diameter on d32

Fig 2 Correlation of d32 with average dissipation rate in tank

At equal speed in the same size tank the large impeller produces smaller drops, as one might expect given its much higher power input.

3.2 Comparison of different impellers Virtually all the published mixing literature on liquid-liquid dispersions uses the Rushton disc turbine. A comparison of different impellers is of practical interest in helping to steer the choice of an appropriate impeller for a liquid-liquid dispersion process. It is also of importance in validating mechanistic models for drop breakage. Figure 2 presents d32 data as a function of average power per unit mass in the 0.29 m tank for a Rushton disc turbine (RDT), 4-bladed and 2-bladed 45 degree pitched blade turbines (PBT4-45 and PBT2-45), a 60 degree 4-bladed degree pitched blade turbine (PBT4-60), a Lightnin A310 and a Chemineer HE3, all of nominal diameter T/2. Unexpectedly the hydrofoil impellers, conventionally thought of as "low-shear" impellers, all produced smaller average drop sizes (d32) than the turbine impellers when compared at equal power per unit volume and impeller diameter. The average power per unit mass does not correlate the data well. However at least part of this could be attributed to variation in the impeller diameters (from 0.133 m to 0.155 m ) for what were nominally T/2 impellers. It has long been established that the turbulent kinetic energy close to the discharge of an impeller scales with tip speed squared as impeller diameter is changed (Nouri 1987). If one models the local energy dissipation rate by

168

k312

c~~

Eqn 1

L

Where k is the turbulent kinetic energy and L is the turbulence integral length scale. Then, if L is assumed to be proportional to D, for a given impeller geometry the local energy dissipation rate can be modeled by: Vt312

N3D 3

c ~---D-- oc~

oc

N3D 2

Eqn2

This expression can also be arrived at by dimensional analysis or by assuming that the volume of local energy dissipation varies in proportion to the impeller diameter cubed. Equation 2 can be used to compare different diameters of the same impeller geometry on the basis of local energy dissipation rate, even though it does not allow calculation of what this local value actually is. In a similar way differences in power number between impellers of similar geometry can be eliminated by using: s oc PoN3D 2

Eqn 3

This approach was supported by measurement of turbulent kinetic energy in the discharge of three 4-bladed pitched blade turbines. This was done using a Dantec 2 component LDA system, with shaft-encoding to allow the true turbulence component to be separated from the periodic fluctuations. The turbines had identical diameter (0.102 m) and identical projected blade widths and blade angles of 30, 45 and 60 degrees. The turbulent kinetic energies are normalized with tip speed squared and plotted against relative radial distance from the centre of the impeller (radial distance divided by impeller swept radius) in figure 3. As might be expected at equal tip speed the turbulent kinetic energy for the 60 degree PBT is greater than for the 45 degree PBT, which in turn is greater than for the 30 degree PBT. However when compared at equal power draw the average and peak turbulent kinetic energies are essentially the same for all three impellers. For example the averaged turbulent kinetic energy was 0.036, 0.030 and 0.032 m2s2 for the 60, 45 and 30 degree PBTs respectively. Since projected blade widths were the same, a reasonable assumption is that integral length scale and hence local dissipation rates are also the same for these three impellers. Further support for this approach to correlating local dissipation rates for impellers of similar geometry but different power numbers comes from a more extensive study conducted by Ruszkowski (1992). The d32 data are plotted as a function of PoN3D2 in figure 4. The quality of fit is not greatly improved compared with using average power per unit mass as in figure 2. Of course this approach neglects any differences between impellers in the discharge flow characteristics. Some impellers may generate high flow and relatively little turbulence close to the impeller, whereas others at the same power draw may generate more turbulence and less flow.

169

Fig 3 Effect of PBT blade angle on turbulent kinetic energy in discharge

Fig 4 Correlation of d32 with dissipation rate local to the impeller

A further correction is also needed for the number of impeller blades. When dealing with local hydrodynamics and local energy dissipation rates we must bear in mind that each blade acts as an individual flow-generating element, and locally a distinct flow field is produced by each blade. The total power drawn by each impeller is split across the number of blades it has. When comparing local energy dissipation rates the total impeller power draw must be divided by the number of blades on each impeller. Accordingly in figure 5 d32 data are plotted against: s oc P o N 3 D 2 1 n

Where n is the number of impeller blades. The d32 data are correlated well using this approach. The average slope is -0.45, close to the value of-0.4 which would be expected from the classic We -~ correlations reported in the literature (Calabrese 1986, for example). The gener'al good fit for all the impellers implies that a distinction between high flow/low turbulence against low flow/high turbulence impellers is not important in this context. The Rushton disc turbine (RDT) data are well-fitted by the correlation, which may seem surprising. The RDT has 2 vortex pairs per blade, and 2 regions of high energy dissipation compared with 1 per blade for the other impellers. So to correct for the number of local flow fields one ought to divide by the number of vortices, rather than the number of blades. However the RDT is a relatively poor flow generator (power number of 5 for a flow number of 0.75, Nouri (1987) compared with a PBT (power number of 1.4 for a typical flow number of 0.85 (Ranade 1989, Armstrong 1986). If the additional power drawn by the RDT is transformed to turbulence then one would expect higher energy dissipation rates in each vortex then for the PBT. This implies that the fit of the drop size data is somewhat fortuitous, with the required higher local energy dissipation rates being obtained by normalizing with 6 blades instead of 12 vortices.

170

Fig 5 Correlation of d32 with local perblade dissipation rate

Fig 6 Correlation of RDT d32 data with local per-blade dissipation rate

The data for the hydrofoil impellers (A310 and HE3) fall close to the PBTs. The implication is that in terms of the underlying hydrodynamics the behaviour of the hydrofoils is similar to that of the PBTs. The reasons for the unexpectedly small drop sizes produced by the hydrofoil impellers (figure 4) now becomes clear. If the hydrodynamics of the hydrofoils are similar to the PBTs then decreasing the number of blades from 4 to 3 produces a consequent increase in the local dissipation rate for each blade and hence a decrease in drop size. Use of 3 blades makes the hydrofoils act like "high-shear" impellers compared with PBTs of the same diameter as far as drop breakage is concerned. Hydrofoil impellers behave as "low-shear" impellers when compared with PBTs at equal power and equal torque. Under theses conditions the low power number (typically around 0.3) of a hydrofoil impellers allows a larger diameter to be used, with a consequent reduction in local dissipation rate and increase in drop size, compared with a PBT. The data for the hydrofoil impellers also appear to fall on a line of different slope than the PBTs. Both the A310 and HE3 show this difference. At first it was suspected this may be an experimental error, however the data were reproduced several times. The implication is that there may be different drop breakage mechanisms acting for the hydrofoil and PBT impellers, leading to the different slopes. Any possible change in breakage mechanisms could be due to the different geometry, or some other coupled factor. For example all the tests with hydrofoil impellers were done at much higher impeller speeds than the PBT impellers. Zhou (1998) has also reported a different trend for hydrofoil impellers and PBT impellers. The drop size data for T/2 and T/3 RDT impellers fromthe 0.17 and 0.29 m tanks are also poorly correlated(figure 6). The data fall into distinct groups for each impeller and tank combination. Baldyga (1995) and Blount (1995) have reported on possible changes in drop breakage mechanisms with changes in impeller diameter. It is possible that the poor correlation of data for hydrofoils and the larger range of RDT data stem from changes in the drop breakage mechanisms with scale and impeller type.

171

3.3 Effect of interracial tension The d32 data for theT/2 RDT in the 0.29 m tank using silicone oil, xylene, chlorobenzene, cyclohexane and tri-butyl phosphate are plotted in figure 7.

The following expression was used to correlate the data:

(PoN3D 2 1 n)-~ (o- / p)0.6 This can be re-arranged to give the classic We ~ correlation reported in the literature, with additional terms for Po and number of impeller blades. The data are well-correlated. Part of the reason is that it is difficult to obtain pure dispersed phases which give a wide range of interfacial tension with water. A small range for interfacial tension will inevitably result in well-correlated data. In this experiment the tri-butyl phosphate gave the poorest fit, and also had an interfacial tension which differed the most from the average. Fig 7 Effect of interfacial tension on d32 However the result is encouraging in showing that the effects of interfacial tension are reasonably well correlated by the classic We ~ correlation, in confirming that dispersed phase viscosity has no effect within the 0.5 to 5 mPa s range, and confirming that the dispersed phase density had no effect. Dispersed phase density is not normally considered as having an influence in turbulent drop breakage. However there may potentially be secondary effects, for example if a dispersed phase which is denser than water (chlorobenzene) is forced into a different part of the trailing vortices than a dispersed phase which is less dense than water. 4. CONCLUSIONS This study has demonstrated that: 1. Drop size data for liquid-liquid dispersions must be related to the local hydrodynamics where drop breakage takes place. For mixing impellers the local region is the impeller blade, rather the impeller. Each blade generates its own local flow field. 2. Drop size data can be successfully correlated for different impeller types on the basis of local hydrodynamics, albeit over a small range of scales. Based on local hydrodynamics the performance of hydrofoil and PBT impellers is similar. When compared at equal power input and diameter PBT impellers produce a larger d32 than hydrofoil impellers. 3. Overall d32 appeared well correlated with local hydrodynamics and interfacial tension for the nominal 0.15 m diameter impellers. However close examination of the data reveals

172

that hydrofoil impellers follow a different trend from PBT impellers, and the correlation could not be extended to impellers of 0.10 m diameter and smaller. The implication is that there are multiple mechanisms responsible for drop breakage, and further work is needed to elucidate these mechanisms. ACKNOWLEDMENTS This work was done within the framework of the FMP (Fluid Mixing Processes) consortium at BHR Group. The authors gratefully acknowledge the financial support and technical guidance form the FMP members. REFERENCES S Armstrong & S Ruszkowski, "Measurement and comparison of flows generated by different types of impeller types in stirred tanks", ppl.9-1.16, Proc Coil Agit Mecanique, Toulouse, France1986, J Baldyga & J Bourne, Chem. Eng. Sci. 50 (1995) 381 J Blount & R Calabrese, AIChE Annual Meeting, Miami Beach, FI, USA November 1995 R C Calabrese, C Y Wang & N P Bryner, AIChEJ, 32 (1986) 677 R. V. Calabrese "Analysis of dilute liquid/liquid dispersions: physical considerations, discriminating among mechanisms and modelling approaches", FMP report 054, July 1992 M. Musgrove, "Analysis of drop size in liquid/liquid dispersions at 0.29 m scale for various T/2 impellers", FMP report 1102, August 1996 J M Nouri, J H Whielaw & M Yianneskis, paper 35, Proceedings of the 2nd International Conference on Laser Anemometry - Advances and Applications. Strathclyde UK, September 1987 A W Pacek C C Man & A W Nienow, Chem Eng Sci, 53 (1998) 2005 L. Portingell "Liquid-liquid dispersions in a 0.17 m vessel: factors affecting mean drop size and drop size distribution", FMP report 1058, February 1991 V V Ranade & Joshi, Chem Eng Comm, 81 (1989) 197 S W Ruszkowski, Impeller discharge flow characteristics and their effect on mixing processes in stirred tanks. PhD Thesis, Cranfield University, 1992 B. Yung. "Discrimination of drop breakage mechanisms in relation to scale-up of liquid/liquid dispersions", FMP report 1080, June 1994 G Zhou and S M Kresta, Chem Eng Sci, 53 (1998) 2063

10th European Conference on Mixing H.E.A. van den Akker and =1.,I.Derksen (editors) 2000 Elsevier Science B. V.

173

Experimental findings on the scale-up behaviour of the drop size distribution of liquid/liquid dispersions in stirred vessels. G.W. Colenbrander Shell Research and Technology Centre, Amsterdam, P.O. Box 38000, 1030 BN Amsterdam, The Netherlands, E-mail: [email protected]

ABSTRACT Experiments were carried out in geometrically similar stirred vessels of three different diameters: 114 mm, 230 mm and 480 mm. The vessels contained oil-in-water dispersions with a dispersed phase volume fraction of 0.5. The stirrers were equipped with pitched blade turbines and the flow conditions were fully turbulent at all three vessel scales. Drop size distributions were measured with a Phase Doppler Anemometer system through a glass window in the perspex vessel walls. The results did not confirm the often applied scaling rule stating that drop sizes in turbulent flow are not dependent on equipment size if the turbulent energy dissipation rate is fixed. In addition to these steady-state experiments the response of the drop size distribution to a sudden increase or decrease in stirrer speed was determined in the two largest vessels. The time as well as the number of stirrer revolutions required to reach a new steady-state drop size distribution was found to increase with increasing vessel size.

1. INTRODUCTION When two immiscible liquids are mixed dispersions can be formed of drops of one of the liquids in a continuous phase, consisting of the other liquid. Such dispersions are frequently encountered in the process industries. Examples are reaction mixtures where reactants from one liquid phase have to be transferred to the other phase in order to react there with a second reactant. The drop size distribution determines the interracial surface area available for mass transfer and hence is one of the most important parameters determining the mass transfer rate. This is also important for wash and extraction operations frequently carried out in industry. Often the phases has to be separated again after an operation and if drops are too small the level of separation may be inadequate causing problems downstream in the process or affecting adversely product quality. A third example is suspension polymerisation where the polymerisation reaction takes place in the droplets, ultimately transforming into the polymer beads. Different product grades require different bead size distributions. In our work we limited ourselves to systems where the droplets undergo break-up as well as coalescence when subjected to a turbulent flow field. When this flow field is steady-state the break-up and coalescence processes may be in equilibrium such that a steady-state drop size distribution is established. Correlations exist to predict the drop size, but often the physical properties required as input for these correlations are unknown or small and

174 unknown quantities of surfactants are present which largely influence coalescence and breakup rates. For such applications it is useful to have validated scaling rules available which prescribe how experiments with the process fluids at a small scale have to be interpreted for application at larger (industrial) scale. In order to contribute to the development of scaling rules we carried out experiments in vessels of three different sizes, containing oil-in-water dispersions with a dispersed phase volume fraction of 0.5. In addition to experiments at constant stirrer speed we also determined the response of the drop size distribution to sudden changes in stirrer speed.

2. THE EXPERIMENTS The experiments were carried out in geometrically similar perspex vessels of three different sizes. Main dimensions are given in Table 1. The vessels had fiat bottoms and were equipped each with stirrers with three pitched blade impellers and four plate baffles. A lid was placed on top of the vessels and they were operated liquid-full in order to avoid air entrainment through the liquid surface.

Table 1 Main dimensions of the vessels Small vessel

Medium vessel

Large vessel

Internal diameter vessel

120 mm

230 mm

480 mm

Liquid height

184.2 mm

353.0 mm

737.0 mm

Liquid volume

2.1 litre

14.7 litre

133.4 litre

124.0 mm

258.5 mm

Diameter impellers

.

.

.

.

.

64.7 mm .

.

.

.

.

.

.

.

.

The dispersed phase consisted of Rh6ne-Poulenc 47V5 silicone oil with, according to specification, the following physical properties at 25~ density 920 kg/m 3, viscosity 5 mPas and surface tension 19.7 mN/m. The continuous phase in all experiments consisted of demineralised water, the dispersed phase volume fraction was 0.5. The experiments were carried out with a fixed concentration of Dobanol (91-10) added as a surfactant in order to obtain reproducible conditions and minimise the influence of accidentally present traces of surface-active agents in the vessels. Dobanol (91-10) is a Shell brand name for a compound with structure formula CH3(CH2)9-(CH2CH20)IoH. The end of the ethylene oxide group is polar (hydrophilic) whereas the other end of the molecule is apolar (oleophilic). We used a Dobanol concentration of 1.8 mg per litre oil/water mixture. Drop size distributions were measured with a Dantec Phase Doppler Anemometer (PDA) system through a glass window in the perspex vessel walls. Due to the high dispersed phase loading the measurement volume had to be located close to the wall in order to avoid too much scattering of the laser beams by drops present between the wall and the measurement volume. Rates at which drop sizes were measured during our experiments were in excess of 100 drops per second. With the system used drop diameters up to 1.3 mm could be measured.

175 In all 3 vessels experiments have been carried out where the stirrer speed was kept constant and the drop size distribution was not changing in time. The results of these experiments are presented in Section 3.1. In addition to these steady-state experiments the response of the drop size distribution to a sudden increase or decrease in stirrer speed was determined in the two largest vessels. In Section 3.2 these transient experiments are discussed. 3. RESULTS OF THE EXPERIMENTS

3.1 Steady-state conditions Figure 1 shows an example of the drop size distribution measured with the PDA system. From the drop size distributions obtained with the PDA system we determined the values of d84, ds0 and d16. The median drop size ds0 is defined such that 50% of the dispersed phase volume consists of drops larger than ds0. Similarly 84% of the dispersed phase volume is contained in drops larger than d84 and 16% in drops larger than d~6. In the small vessel 18 steady-state runs were carried out. For the medium size and large vessel the number of steady-state runs amounted to 29 and 11, respectively. A graphical representation of this data is given in Figure 2. The largest scatter in measured diameters for a given impeller tip speed is found for the medium size vessel at an impeller tip speed of approximately 2 m/s. For the 16 runs with steady-state time series at this impeller tip speed we calculated an average ds0 value of 563 micron and a standard deviation of 43 micron, 8% of the average value. For the other vessel/tip speed combinations a smaller number of experiments were done and the scatter in the droplet diameter values was smaller than for the medium size vessel at 2 m/s tip speed. Because of the small number of realisations we assume for each condition that the standard deviation is 8%.

Fig. 1: Drop size distribution determined from the measurement of 58159 drop diameters.

176

Fig. 2. Measured drop size diameters dl6 , ds0 and d84 as a function of impeller tip speed during steady-state conditions

177

800 700 600 2 500-

9 Small vessel

"~ 400.~

[] Medium vessel

300-

9 Large vessel

200 -

100 00.1

1.0

10.0

100.0

N3D2, m2/s3 Fig. 3. Average median droplet diameter measured in the 3 vessels as a function of N3D2, which is proportional to the stirring power density

Figure 3 contains the average ds0 values for silicone oil in the three vessels as a function of N3D2 with N the stirrer speed (revolutions per second) and D the impeller diameter. Stirring power density is given by Po N 3 Ds/VL with Po the stirrer power number and VL the liquid volume. For geometrically similar vessels D3/VL is constant and the group N3D2 is therefore proportional to the stirring power density. Figure 3 shows that when the same power density is applied to geometrically similar vessels of different size, the drop size produced increases with decreasing vessel size. And conversely, to produce droplet of a certain size, the required stirring power density decreases with increasing vessel size. This is in contrast with the often used correlation's where the power density required to produce a certain drop size is independent of vessel size. We characterised the shape of the droplet size distributions by the coefficients of variance (dso-ds4)/ds0 and (d16-dso)/dso. It was found that the droplet diameter does not exhibit a normal distribution: with few exceptions (d16-ds0)/dso is larger than (dso-ds4)/dso. Average values of these distribution parameters are given in Table 2 for the low and high impeller tip speed (approximately 2 and 3.5 m/s respectively). From this data we conclude that (ds0-d84)/ds0is independent of vessel size and stirrer speed. The (dl6-ds0)/ds0 data show more variation but we consider this to be statistical scatter, rather than a trend.

178 Table 2 Coefficients of variance of the drop size distributions (dso-d84)/dso Impeller tip speed

Small vessel

Medium vessel

Large vessel

2 m/s

0.25

0.26

0.27

3.5 rn/s

0.26

0.25

0.25

(d16-dso)/dso Impeller tip speed

Small vessel

Medium vessel

Large vessel

2 rn/s

0.35

0.35

0.47

3.5 m/s

0.40

0.40

0.37

3.2. Transient conditions

A total number of 12 experiments has been carried out where the response of the droplet size distribution to a sudden change in stirrer speed has been measured: 3 experiments in the large vessel and 9 in the medium size one. Figure 4 shows the stirrer speed response in the medium size vessel to a sudden change in speed setting from 313 to 537 rpm, controlled by computer. The stirrer speed response time is about 7 s. For the large vessel no computer recording of the stirrer speed could be made and the stirrer speed was controlled manually. For sudden changes in stirrer speed setting between 150 and 257 rpm in this vessel the stirrer response time amounted to some 10 s.

600 -

f

9

550 " 500

%"~

S**

450 " A0_ A

400 350 300 250 175

180

185

190

195

200

205

Time, s Fig. 4. Response of the stirrer in the medium size vessel to a sudden change in the stirrer speed settings from 313 to 537 rpm

179

Fig. 5. Response of the drop size distribution in the medium size vessel on changes in stirrer speed between 313 and 537 rpm

Figures 5 and 6 contain graphical representations of the droplet size response in the two largest vessels. The stirrer speed levels were chosen such that the ds0 steady-state levels were approximately 350 and 500 micron. As a measure for the response time of the drop size distribution we used the time required for the ds0 to reach the value do~d+ 0.9 (d.ew- dold)with do~dand ~ew the steady-state values of ds0 before and after the change in stirrer speed. The so defined drop size response time we called transient-up time when associated with an increase in stirrer speed and transient-down time when the stirrer speed was decreased. Figures 5 and 6 indicate that the response times of d16,dso and ds4 are similar. Table 3 contains the response times of ds0 to the change in stirrer speed. These transient-up and transient-down times are given in number of stirrer revolutions after the change in stirrer speed.

Fig. 6. Response of the drop size distribution in the large vessel on changes in stirrer speed between 150 and 257 rpm

180 Table 3 Response time (in number, of stirrer revolutions) of the drop size to changes in stirrer speed .

.

.

.

.

.

.

Medium vessel .

.

.

.

.

.

.

.

Large vessel .

.

.

.

Transient up

123 revs

257 revs

Transient down

140 revs

263 revs

The drop size response times in the medium size vessel were larger than the stirrer response time (7 s), but especially the transient-up times are not much larger than 7 s. It may therefore be that the response time to a stepwise change in stirrer speed would be faster than found in our experiments, especially for the transient-up. The drop size response times measured in the large vessel are an order of magnitude larger than the stirrer response time and will therefore not be significantly different from the response to an instantaneous step in stirrer speed. The large vessel data does not show a significant difference in number of stirrer revolutions required for the drop size to reach a new steady-state. The medium size vessel data show somewhat larger differences but in view of the large scatter in the data this seems hardly significant. Significant are the larger transient times in the large vessel compared to the drop size response in the medium size vessel, in clock time as well as in number of stirrer revolutions. Tentatively we conclude that increasing the vessel diameter by a factor two results in twice the number of stirrer revolutions required for the drop size distribution to reach the new steady-state and four times as much clock time. It will be challenging to find a physical explanation for this finding.

4. CONCLUSIONS 9 In geometrically similar vessels of different size equal specific stirring power input does NOT produce equal drop sizes. The drop size decreases with increasing vessel size. 9 The shape of the droplet size distribution as characterised by the coefficients of Variance is independent of vessel size and stirrer speed. 9 To reach a new steady-state drop size distribution after a change in stirrer speed requires the same number of stirrer revolutions for a transient-up (sudden increase in stirrer speed) as for a transient-down (decrease in stirrer speed). 9 To reach a new steady-state drop size distribution after a change in stirrer speed requires an increasing number of stirrer revolutions with increasing vessel size. Because of the small experimental data base we tentatively conclude that the number of stirrer revolutions required to reach a new steady-state is proportional to the vessel diameter.

10 th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

INVESTIGATIONS OF LOCAL DROP SIZE DISTRIBUTIONS S C A L E - U P IN S T I R R E D L I Q U I D - L I Q U I D D I S P E R S I O N S

181

AND

K. Schulze, J. Ritter and M. Kraume Technische Universitat Berlin, Institut fiir Verfahrenstechnik, StraBe des 17. Juni 136, 10632 Berlin, Germany Local drop size distributions in batch operated stirred vessels of different sizes were investigated using an endoscopic measurement technique which allowed the determination of drop sizes in coalescing liquid-liquid systems with volumetric dispersed phase fractions up to 0.5. It could be shown that most of the breakup as well as the coalescence processes determining the drop sizes take place near the stirrer region. Investigations with various dispersed phase fractions showed that the frequently used correlation d32-- We "~ (1 +C2 ~0) is no reliable description of the drop sizes at higher volume fractions of the dispersed phase. 1. INTRODUCTION Many chemical engineering processes involve mixing of two immiscible liquids such as solvent extraction or chemical reactions; sometimes eormected with heat transfer. The common objective of these operations is to increase the interfacial area for mass transfer by achieving small drop diameters. On the other hand the dispersed phase has to be separated from the continuous phase at'terwards so the entrainment of fine droplets has to be taken into consideration. Therefore, in order to optimise and control liquid/liquid dispersion processes it is essential to understand and to model the drop behaviour effeeted by variations of the physical properties and the mixing intensity. One important base for the mathematical modelling of these systems are reliable experimental data. Especially results from systems with technically relevant dispersed phase fractions up to 0.5 are insufficiently presented in literature. The objective of the present paper is to investigate the local dependence of the drop sizes in a liquid-liquid system with dispersed phase fractions of technical relevance. The results can be used to extend the understanding of the phenomena in a stirred coalescing system. In addition the two empirical scale-up criteria of equal power input per unit volume and equal tip speed are compared referring to the resulting drop size distributions. 2. FUNDAMENTALS The following equations represent the mathematical basis for the description of the interactions between turbulent processes in stirred liquid-liquid systems and the drop behaviour leading to a frequently used equation for the Sauter diameter [1- 4]. The mean square of the turbulent fluctuating velocity between two opposite points in a lokal homogenous isotropic turbulent field was calculated by Kolmogoroff [5]. In the case of drops these opposite points are supposed to be on the surface of a drop. The distance between the relevant points then equals the drop diameter dp.

182 2

2

w '2 = elok~ -dp7

(1)

The Weber number We @,max describes the ratio between kinetic energy of the drop E L and interracial energy Eo. This value marks a critical state where the drop is still stable.

Wea,~ '

=

E~,

=

p . w "~ . d p . , ~

o"

E~

(2)

Under the assumption of equal energy dissipation rates in various regions of the stirred tank Eq. (2) leads together with the stirrer We number to an equation for the maximum drop diameter. d.,.p,max = C I d

"n2d~--= -~Y C 1 9We -~

(6)

cr

According to several authors [6,7] this maximum drop diameter correlates linearly with the Sauter diameter d32. d32 = C 2 9We -0"6 d.

(7)

In most of the published correlations for d32 Eq. (4) is extended by the factor (1+ C3 q~) to consider the influence of the dispersed phase fraction in liquid-liquid dispersions. Several authors found exponents of the Weber number differently from -0.6. Shinnar [8] derived a dependency of d32 ~ We -~ for systems dominated by coalescence phenomena. Kipke [9] and Brooks [10] estimated an exponent of-0.3 whereas Hartland et al. [11] found -0.4 and Nienow et al. [12] -0.43. On the other hand experimental results of Baldyga [13] led to an exponent of -0.93. Relating to scale-up Eq. (4) leads to the well known scale-up rule of constant specific power input per unit mass for equal Sauter diameters. Some investigations came to a different conclusion as hey stated that the mean diameter remains the same in different Scales for constant tip speed [1, 14-16]. 3. EXPERIMENTAL 3.1 Equipment The liquid-liquid system used for the experiments consisted of toluene for the dispersed phase and distilled water for the continuous phase. The stirred tanks were glass vessels with a dished bottom, fitted with 4 equally spaced stainless-steel baffles, each one tenth of the tank diameter D and agitated with a 6-bladed Rushton turbine of d/D = 0.3 - 0.5. The vessel diameters D were 190, 300 and 400 mm (H/D=I). The coordinates of the endoscope position were r/D=~/4 in the altitude of the blades (stirrer region) and D/5 under the surface (bulk region). Size measurements were carried out after at least 2 hours of agitation. 3.2. Measurement technique An endoscope covered by a tube with a diameter of 7 mm and a glass window at its bottom allowed measurements of drop size distributions in a range of 30 to 1000 grn at any dispersed

183 phase fraction. The light supply was integrated into the endoscope and consisted of a flash light connected to the endoseope with fibre-optical light guides. A CCD-eamera mounted at the upper end of the endoscope took pictures of the drops in the vicinity of the glass window (Fig.l). In most eases drops appeared as well shaped spheres. So, a simple measure of the drops was possible. The size measurement was done manually supported by a semi-automatic determination of the drop diameters. The advantage of this inline microphotography Fig. 1 Drops of toluene in water resulted from the opportunity of local measurements in agitated vessels. Breakage and coalescence phenomena could also be observed (Fig 4). Further developments led to CCD-eameras with higher resolutions and flash lights with very short exposure times. A detailed analysis of drop size distributions with various numbers of drops showed that 200 to 400 drops are sufficient to determine the Sauter diameter with a deviation of less than 10 % typically even less than 5 %. A further increase of the number of counted drops led to insignificant improvements. 4. RESULTS Most of the drop size distributions measured in this investigation could be described as approximately normal distributions or logarithmic normal distributions. It is well known that pure breakage processes result in logarithmic normal distributions. In Fig. 2 drop size distributions were plotted in a probability net for various stirrer speeds. The measured distributions appeared as straight lines and were therefore logarithmic normal distributions. An increase in stirrer speed led to smaller drops, while the standard deviation remained nearly the same over the measured range as the slope of the straight lines was approximately constant. The relevant phenomena for the interpretation of the distributions shown in Fig. 2 were coalescence and breakage. In literature several approaches could be found that dealt with these basic processes in specific regions of mixing devices [17]. Shinnar's theory of turbulence-stabilised dispersions [8] was based on the assumption that drops could not be broken alter their creation because of their interracial energy. On the other hand coalescence through adhesive forces was assumed to be hindered for these drops due to turbulent fluctuating velocities. There was no discrimination between different locations in the stirred tank. Several authors divided the apparatus in two regions [18]. In the vicinity of the stirrer they mostly observed breakage processes whereas coalescence processes were found to take place on the circulation path in the bulk region [17]. In Fig. 3 drop size distributions were displayed which were measured at 5 different locations. The positions were chosen along the characteristic circulation path of the stirred liquid (here, for a Rushton turbine). A change in the distribution could be observed in the stirrer region whereas in the bulk of the liquid, only slight changes occurred. This was confirmed by the observation of deformed drops exclusively in the vicinity of the stirrer, as shown in Fig. 4 . At measurement point 2 the two relevant processes coalescence and breakage could be observed simultaneously. Consequently, in the physical system examined most of the drops

184

Fig. 2 Cumulative number distributions for various stirrer speeds [19]

Fig. 3 Local drop size distributions were broken in the narrow vicinity of the stirrer while coalescence can mainly be found in the effluent area of the stirrer. In vessels of different size the dependence of the drop size distributions measured in the stirrer and the bulk region on the dispersed phase fraction was investigated. An increase of the dispersed phase fraction led to enlarged Sauter diameters (see Fig. 5). This is due to the fact that if the dispersed phase fraction is doubled the frequency of drop collisions is increased by a factor of four. Based on the assumption of constant drop size a doubling of their number will double the probability of collision. Neglecting turbulence damping, i.e. constant breakage rates, this leads to a significant increase of the Sauter diameter.

185

Fig. 4 Drop coalescence and breakage in the stirrer region at measurement point 2 (see Fig. 3) With increased dispersed phase fractions discrepancies in the local drop size distribution and the corresponding Sauter diameter arose (Fig. 5). This effect got more pronounced the larger the tank diameter became. As the power input per unit volume was kept constant the average circulation time for the liquid in the stirred tank was increased with an enlarged tank diameter. Obviously this led to a higher number of coalescence processes resulting in larger Sauter diameters in the bulk region compared to the stirrer region.

tank diameter

Fig. 5 Local Sauter diameter at various dispersed phase volume fractions in tanks of different size Increased concentrations of the dispersed phase led to a lower influence of the Weber number on the Sauter diameter as shown by Fig. 6. An increase of d32 with higher phase fractions as given in Eq. (7) could not be confirmed. The parallel shitt predicted by the factor (1+C3 (0) was not measured in the experiments. Fig. 7 shows the corresponding drop size distributions for a mean power input of 510 W/m3. An increase of the dispersed volume fraction is followed by a shift of the drop sizes towards larger diameters. Furthermore, the drop size distributions are normally distributed. There is only a little decrease in the slope of the curves with increasing dispersed fractions.

186 1000

......

[pm] ---toluene/water ~-~Rushton turbine - D=150 mm _ld/D ffi 0.33 10

9=0.5

0.4 e........~=----'~'~...e

t__

d32 ~ We "~

0.3

o

E

o.2

el ,,m

0.1 L.....

10

0.05

e'~,,,e

......~'~~...~,

d32 ~ We -o.6

100

loo

[- ]

Weber number

1000

Fig. 6 Sauter diameter depending on the Weber number for various dispersed phase fractions.

t I I,I

e9

o

c

0.95

w

0.80

10 ~t

0.60

.Q ,am

E

mm

0

0

.>

_m =1

E

] t I I I

0.99 _~toluene/water ~Rushtonturbine H

O 3

D=150 mm

I q~= 0.05 0.1 "

~

-~d/D = 0.33

!

hill

I

~s = 510W / m 3 tl

f

..... -"~;

~"r(I

l/

0.40

o.osl

I

I

I

I/1/

1 .,fl

I/

/

~ ./" I

i J E --=v -~-,/ ~ /

I I'r [~""~1

~ I X j"

II ./1 / I

IJ~r

I Jlf .~'.f'l

I./~

V

0.2 - - 0 . 3 ~_ 0.4 L_0.5

I

f

.~'

I iI

1

1~J 2" b,"

I / J l _.~ .,~

~.,1, , f

l

1

I.

[.f..~'~

r

0

100

200

300

400

drop diameter

500

600 [pm] 700

dp

Fig. 7 Drop size distributions for various dispersed phase volume fractions. Many 5quids used in industrial processes consist of mixtures of several substances. Impurities 5ke suffactants and ions can be found regularly. These will form electrochemical double layers which also can have strong effects on coalescence. The formation of these layers can be quantified by the zeta-potential. Fig. 8 shows experimental results of the influence of Na + and CI" ions on the Sauter diameter. Although higher ion concentrations lead to a smaller thickness of the electrochemical double layer a decrease of the Sauter diameter was observed. Because of the manifold parameters influencing drop size distributions in liquid-liquid dispersions and the resulting difficulties for mathematical modelling of Sauter diameters the design of mixing devices is normally based on experiments conducted in a laboratory scale.

187

Fig. 8 Sauter diameter for various ionic concentrations For this purpose a reliable scale-up rule is necessary. Fig. 9 shows the application of two commonly used scale-up criteria. The first criterion of constant specific power input per unit

Fig. 9 Comparison of two commonly used scale-up criteria a) constant power input per unit volume b) constant stirrer tip speed

188 volume was derived from the dependence of the Sauter diameter on the Weber number (Eq. (7)). It was compared with a second criterion of constant stirrer tip speed. It can be seen from the cumulative volume distribution displayed in the figures that at lower dispersed phase fractions, constant stirrer speed appeared as a reliable scale-up criterion. This is in good agreement with results from breakage processes of solid particles. In stirred tanks the size reduction of solids correlates with the stirrer speed [20]. Obviously the size distribution is dominated by breakage processes in the case of low dispersed phase fractions. At higher dispersed phase fractions both criteria led to unsatisfactory results. 5. CONCLUSIONS With the developed inline measurement technique the determination of drop size distributions at different locations in the vessel and high dispersed phase fractions up to ~0=0.5 could be realised. For low dispersed phase concentrations significant differences of the local distributions could solely be observed in the stirrer region. With increased dispersed phase fractions (q~ > 0.1) discrepancies in the local drop size distribution and the corresponding Sauter diameter arose. This effect got more pronounced the larger the tank diameter became. The influence of the Weber number and the dispersed phase fraction on the Sauter diameter could not be described be the common relation d32-~ We "~ (1+C2 ~0). A comparison of scale-up criteria did not lead to clear results. In comrast to the standard scaleup for liquid-liquid systems the criterion of constant stirrer speed was the favourable one in the case of low dispersed phase fractions REFERENCES 1. Okufi, S.; Perez de Ortiz, Can. J. Chem. Eng., 68, S. 400-406, 1990 2. E.G. Chatzi, C.J. Boutris, C. Kipafissides, Ind. Eng. Chem. Res., vol. 30, pp. 536-543, 1991 3. R.V. Calabrese, C.Y. Wang, AIChE J., vol. 32, pp. 667-676, 1986 4. Zerfa, M.; Brooks, B.W., Chem. Eng. Sci., vol. 51, pp. 3223-3233, 1996 5. A. Kolmogoroff, Compt. Rend. Acad. Sci. USSR, vol.30, pp. 301-305, 1941 6. S. Jakubowsky; S. Sideman, Int. J. Multiphase Flow 3, pp. 171-180, 1967 7. D.I. Collias; R.K. Pmd'Homme; Chem. Eng. Sci.,vol.47, pp.1401-1410, 1992 8. R. Shinnar, Journal of Fluid Mechanics, vol. 10, pp. 259-275, 1961 9. K.-D. Kipke, verfahrenstechnik, vol. 15, pp. 563-566, 1981 10. B.W. Brooks, Tans IChemE, vol. 57, pp. 210-212, 1979. 11. S. Hartland, M. Laso, L. Steiner, Chem. Eng. Sci., vol. 42, pp. 2437-2445, 1987 12. A.W. Nienow, A.W. Pacek, C.C. Man, Chem. Eng. Sci., vol. 53, pp. 2005-2011, 1998 13. J. Baldyga, J. Chem. Engng. Japan, vol. 26, pp. 738-741, 1993 14. K. Ogawa, C. Kuroda, AIChE Symposium Series, 305, 91, pp. 95-101, 1995 15. M. Nishikawa, F. Mori, S. FujieAa, T. Kayama, J. Chem. Engng. Japan, vol. 20, pp. 454459, 1987 16. S. M. Kresta, G. Zhou, Chem. Eng. Sci., vol. 53, pp. 2063-2079, 1998 17. C.Tsouris; L.L. Tavlafides, AIChE J., vol. 40, pp. 395-406,1994 18. G.B. Tatterson, Scaleup and Design of Industrial Mixing Processes, Mc Graw-Hill (NewYork), 1994 19. J. Ritter, M. Kraume, Chem.-Ing.-Tech., vol. 71, pp. 717-720, 1999 20. K.Kipke, Chem. Ing. Tech., vol.52, pp. 658-659, 1980

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. K All rights reserved

189

Gas-liquid mass transfer in a vortex-ingesting, agitated draft tube reactor C. Leguaya'b, G. Ozcan-Taskin b & C.D. Rielly c a Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA b BHR Group, The Fluid Engineering Centre, Cranfield, Beds MK43 0AJ c Department of Chemical Engineering, Loughborough University, Loughborough, Leics. LE11 3TU

Gas-liquid mass transfer coefficients have been measured in a vortex-ingesting draft tube reactor, known as the Advanced Gas Reactor (AGR), over a range of specific power inputs and for various geometries, using the hydrogen peroxide reduction technique. Experimental results are presented for the effects of changing the liquid level and draft tube top clearance on kLa in both distilled and tap water. Correlations are proposed in terms of the specific power input, which allow prediction of the AGR mass transfer performance over a range of geometric conditions. These experimental data are also compared with empirical correlations from the literature. The AGR is shown to generate similar values of kLa to other self-inducing and gas sparged designs at the same specific power input, but with the advantage that the mass transfer performance is less susceptible to variations in the liquid level. 1. INTRODUCTION In many chemical and biochemical processes, one of the reactants is a gas, which undergoes reaction in a liquid solution. In conventional stirred reactors, the gas is continuously sparged into the vessel, but often a large fraction of the feed gas is not absorbed and disengages into the headspace. The headspace gas may be vented, or recycled using an external recompression loop, if the device is to be operated in a dead-end mode. However, when the reactant gas is hazardous to compress, for example in hydrogenations, chlorinations or oxidations, there are advantages in using a type of self-inducing gas-liquid reactor, which features internal recycling of the headspace gas. These self-inducing systems may operate by (i) surface aeration in a partially baffled, or unbaffled, vessel; (ii) self-induction through a hollow-bladed shaft and impeller; or, (iii) generation of surface vortices for gas ingestion. The advanced gas reactor (AGR) developed and patented by Praxair is an example of a vortex ingesting system and is the subject of this study (Leguay et al., 1998 a and b, 1999). The AGR (Fig. 1) comprises a helical screw impeller and concentric draft tube, mounted within the reactor vessel. The impeller generates a strong liquid down flow which impinges on the base of the vessel and is redirected into the annulus surrounding the draft tube; cruciform baffles at the exit from the draft tube remove the swirl generated by the impeller. At the top of the annulus, the upflow re-enters the draft tube, flowing around a further set of cruciform baffles, which may extend above the liquid level. These upper baffles promote the formation of surface vortices through which headspace gas is ingested. Gas bubbles, drawn into the suction side of the impeller, are dispersed and advected with the liquid flow. The liquid flow over the top of the draft tube causes a fraction of the bubbles to be recirculated, leading to longer gas residence times. The remaining fraction of bubbles disengages at the free surface, but subsequently this gas can then be recycled by vortex ingestion. Thus, the reactor can be operated in a "dead-end" mode with respect to the feed gas. Previous studies of the AGR by Leguay et al. (1998 a & b, 1999) have examined the two-phase hydrodynamics, flow patterns, liquid mixing, gas hold-up, gas entrainment rate and impeller discharge flow rate. In this paper, new results are reported on gas-liquid mass

190 transfer as a function of the specific power input and the geometry. This work also compares the mass transfer performance of the AGR with other self-inducing and gas sparged devices. 2. LITERATURE The volumetric gas-liquid mass transfer coefficient, kza, for gas-inducing reactors has received little attention in the literature, compared to conventional sparged systems, which have been extensively studied (Patwardhan and Joshi, 1999). Generally, in sparged reactors, mass transfer results are fitted to an empirical correlation of the form:

kLa = OC(F_,m )fl (Usg )r

(1)

where, em is the specific gassed power input, Usg is the gas superficial velocity and o; fl and ~' are empirical constants. For example, Smith et al. (1977) proposed the following correlation for a wide range of agitator designs and vessel diameters from 0.44 to 1.83 m:

kL a = 0.266(em)0.475 (Usg )0.4

(2)

Many correlations of the form of eq. (1) have been published, but even for a given set of test fluids, there are significant differences in the empirical constants (John, 1997). This variability is due partly to the various measuring methods (e.g. dynamic or steady-state) used by researchers, but also to the flow models assumed for the gas and liquid phases. In addition the values of the empirical constants depend on the fluid properties (specifically on the coalescence behaviour) and on the scale of operation. In the case of gas-inducing systems, the superficial gas velocity is no longer an independent variable, since it depends on the impeller speed; eq. (1) is often applied in a reduced form, i.e. kza ~ (em)/~,which empirically includes the relationship between Usg and era. Forrester et al. (1998) proposed such an empirical correlation to predict the mass transfer coefficient of a six-bladed concave self-inducing impeller, with a single orifice per blade

)0.80+0.06

kLa = (0.0191+ 0.0053)(e m

(3)

and Sawant et al. (1981) developed a similar correlation for a Denver froth flotation cell

kLa = 0.0195(era) 0'5

(4)

The controlling mechanisms of mass transfer are not identical in sparged reactors and the various gas-inducing systems. For example, in the AGR, the bubbles are ingested from the liquid surface, whereas in other systems, the bubbles grow and detach from a stationary sparger, or a moving impeller blade (Forrester, 1995). Furthermore, in the AGR, the gas superficial velocities are far greater in the draft tube than in the annulus, which could result in different mass transfer coefficients in these two regions. An aim of this work is to establish if some of these gas entrainment and dispersion mechanisms are more energy efficient for mass transfer than others.

3. EXPERIMENTAL The mixing equipment employed in this work has been described previously by Leguay et al. (1998a). In summary, experiments were carried out in a T = 0.29 m diameter flat bottomed AGR, as shown in Fig. 1. A concentric draft tube of diameter, Dd = 0.097 m was fitted in the vessel with cruciform baffles at the entry and exit. Two impellers were mounted on the shaft and operated in the draft tube: a downward pumping, four flight, helical screw of diameter

191 D = 0.088 m, with a six fiat bladed turbine underneath. The agitation speed was varied from 300 to 1300 rpm and all experiments were carried out under fully turbulent conditions. Liquid mixing studies (Leguay et al., 1998b) have shown that at this scale of operation, the liquid is well-mixed on the time scale required for mass transfer. Therefore, it was assumed that the entire mass transfer process could be adequately described by a single lumped volumetric mass transfer coefficient, kLa. On this basis, the overall volumetric mass transfer coefficient was measured using the steady-state hydrogen peroxide (H202) reduction technique (Hickman, 1988). Figure 1 shows a schematic diagram of the typical experimental set up required for these measurements. The method relies on the catalytic decomposition of hydrogen peroxide into oxygen and water according to the chemical reaction 2H202 ~ 2 H 2 0 + O 2 At the start of each experiment, the liquid phase is saturated with atmospheric oxygen using the vortexingestion capability of the AGR. Then, catalase is introduced into the liquid and H202 solution is pumped continuously into the reactor, Fig. 1 Schematic diagram of the experimental equipment where it decomposes (positions 1, 2 and 3 indicate dissolved oxygen probes) catalytically into oxygen and water. Initially the reaction product, dissolved oxygen, accumulates in the liquid phase, which then becomes supersaturated. The supersaturation acts as the driving force for desorption of oxygen into the ingested air bubbles. At steady-state the feed rate of H202 balances the reaction rate, r, and the oxygen production rate balances the oxygen mass transfer rate: r _ ,NH202

-~- 2VL

kLaAC

(5)

The concentration driving force, AC, may be determined from the dissolved oxygen concentration in the liquid bulk, CL, the gas phase oxygen concentration, Co, and the Henry's law constant, H, for oxygen in water. The dissolved oxygen (DO) concentration was measured at three locations in the vessel using fast response polarographic electrodes (see Fig. 1). Preliminary experiments showed the three DO probes gave very similar responses, confirming the validity of the well-mixed liquid assumption. Two assumptions were made regarding the gas-phase-mixing pattern:

(i)

well mixed

AC

=

(C L - CG(out) /

H)

(c,. - Co o., / , )- (c,. (ii) plugflow

(6)

/H )

AC= 1l~ l ; n ~ C L t" - ' c ( o u t ) l H)/~t" ' ' ' ' t - C a ( i n ) l H ' ])

(7)

The inlet gas phase oxygen concentration, Cc(i,), is the atmospheric air oxygen concentration, which is known; the exit gas oxygen concentration, Cc(ouo, is obtained from a

192 mass balance on the gas phase, knowing the gas entrainment rate, Qa (Leguay et al., 1999). Thus, the overall volumetric mass transfer coefficient, kLa, was determined over the range of geometries, shown in Table 1 and over a range of specific power inputs from 0.3 - 3.5 W/kg. Table 1. AGR geometries at which kLa was measured Liquid level, Ht/T Impeller top clearance,

1.2011.2711.44

|l

0.44

1.27 0.44

0.3810.44

0.09

0.06 0.08 0.09 0.11 0.13 0.14

0.09

Draft tube bottom clearance, Cb/T 0.2610.3310.50 0.37 0.35 0.33 0.32 0.30 0.28

0.33

Cimp/'T

Draft tube top clearance, C/T

...

1.27

The power dissipated by the impeller was measured using shaft-mounted strain gauges. Air was supplied by surface aeration only and the air volumetric flow rate, Qa, was measured using a bubble soap meter (Leguay et al., 1999). Over the range of experimental conditions studied, Qa was found to vary from to 1 to 12 1/min, corresponding to a range of superficial gas velocities of 0.3 to 3.4 mm s1 in the annulus, and 2 to 27mm s in the draft tube. The H202 feed rate was determined by recording the H202 reservoir weight loss in time (Fig. 1). Experiments were carried out using distilled and tap water as the test fluids. 4. RESULTS AND DISCUSSION 4.1 Measurements in distilled water

4.1.1 Effect of gas phase mixing model Figure 2 compares the mass transfer coefficients obtained in distilled water using the wellmixed and plug gas flow models, over a range of specific power inputs and for Ct/T = 0.09 and 0.14, keeping all other geometric ratios fixed; see Table 1. As expected, the well-mixed model gave about 10% higher kLa values than the plug flow model; the experimental error (from repeat runs) is estimated to be about _+15%. At this scale, the assumed gas phase models result in similar estimates of kLa, indicating that, during the bubble residence time, the oxygen partial pressure changes by a negligible amount by desorption. The data presented in the following sections have been calculated using the well-mixed gas model.

4.1.2 Effect of C f f Figure 2 also shows the kLa measurements, as a function of specific power input, for a range of draft tube top clearances, Ct/T. For a given specific power input below ~1 W/kg, kLa increases with decreasing draft-tube top clearance. This trend is consistent with the gas entrainment rate and hold-up results previously reported by Leguay et al. (1999). Lower draft tube top clearances enhance the intensity, size and frequency of the surface vortices responsible for gas ingestion. Consequently, more gas is entrained into the draft tube, resulting in higher gas superficial velocities and hold-ups and hence an increased interfacial area. Figure 2 shows, at specific power inputs greater than 1 W/kg and for C / / ' < 0.13, kLa becomes much less dependent on the draft tube immersion depth at constant era. This is an advantage over alternative self-inducing devices, whose mass transfer performance may be very sensitive to variations of the liquid height. An empirical correlation of the form of eq. (2), with the superficial gas velocity term lumped in the empirical constants, can be fitted to the experimental data obtained for C/T_< 0.09 (where, for a given era,kLa is a maximum):

kLa = 0.02 l(g m )0.69

(8)

193 0.1

0.1

Open symbols: plug flow Closed symbols: backmixed

Experimental Conditions: C tiT = 0.09 C impIT = 0.44 0.04 to 0.4 vvm

kLa = O . 0 2 1 ( e m ~ ~ ..d

0.01

x

Ill

Ct/T--0.06

9

kLa = 0.021(era)0'69

Experimental Conditions: x Ct/T--0.08 H LIT = 1.27 9Ct/T--0.09 Cimp/T =0.44 9Ct/T=0.11

x Ct/T=0.13 A Ct/T=0.14

0.04 to 0.5 vvm 0.001 0.1

I

I

I

I

'

' ' I I

[- HL/T=1.20I 9 HI./T=1.27I

I

I

I

i

i

,

A HI~=1.441

,i 0.01

1 10 Specific power input, trm / (W/kg)

Fig. 2 The effect of the gas phase mixing model on the calculated km at various C/rl" in distilled water.

""

0.1

I

I

,

,

,

,.Li

I

,

,

,

,

Specificpower haput, em / (W/kg)

,,,

10

Fig. 3 The effect of the total liquid height on kLa in distilled water.

4.1.3 Effect of Hz/T Measurements of kLa, carried out over a range of liquid levels keeping the top clearances constant (see Table 1), are shown in Fig. 3. Within experimental error, the volumetric mass transfer coefficient is independent of the liquid level. This is consistent with the power input, gas entrainment rate and hold-up measurements (Leguay et al., 1999) which were also found to be unaffected by changes in HtJT. These results indicate that increasing the length of the gas circulation pathway and the gas residence time, by increasing the liquid level, does not affect the mass transfer performance of the AGR, at this scale. This further demonstrates that there is negligible accumulation of oxygen in the bubbles during their residence time. 4.2 Measurements in tap water Figure 4 shows a comparison between tap water and distilled water results, for a range of C/T. Higher values of kLa were obtained in tap water than in distilled water, with a maximum difference of 40% at the highest specific power inputs. This is due to the presence of trace amounts of dissolved salts in tap water, which tend to decrease the frequency of bubble coalescence (Van't Riet, 1979). Consequently, in tap water, the average bubble size is smaller, resulting in a slightly higher interfacial area available for mass transfer. Comparison of the AGR mass transfer performance with other devices was made using the tap water results, since most of the researchers used this as a test fluid. As with distilled water, for low drafttube top clearances, km is a weak function of C/T, as shown in Fig. 4. For C/T <__0.09, the best mass transfer performance is obtained and kLa may be correlated by

kLa:O.O31(Em) 0"87

(9)

4.3 Comparison with other self-inducing devices Figure 5 compares the mass transfer coefficients measured in the AGR with the performance of (i) a concave-bladed self-inducing impeller with a single orifice per blade, given by eq.(3), and (ii) a Denver froth flotation cell, given by eq.(4), operated over the same range of specific

194 power inputs. The figures show four sets of results, corresponding to different draft tube top clearances, i.e. for different induced gas flow rates. The performance of the concave-bladed self-inducing impeller compares well with the AGR, at low draft tube immersion depths. For C/T < 0.11, at a given power input, the AGR can reach mass transfer coefficients up to twice the values obtained using Forrester's self-inducing impeller. This suggests that, for C//' < 0.11, the AGR ingests more gas than the self-inducing impeller, operated at the same specific power input. However, it is worth noting that eq.(3) was obtained using one orifice per blade and at one liquid level. Forrester (1995) reported that increasing the number of orifices per blade from one to four almost doubles the value of kLa at a given specific power input. With four orifices per blade, Forrester's devices would therefore compare well with the AGR performance at low draft tube immersion depths. Figure 5 also shows that, at low specific power inputs, the mass transfer performance of the AGR and the Denver cell are similar. However, as the specific power input increases, the AGR achieves mass transfer coefficients that are up to 240% greater than the Denver cell. .1 kt, a=O'O31(gm) 0"87

"e

9 A

% 0.01 t "~

" ~ 1 - 1 [] kLa = 0"021(era) 0"69

0.1 f [ e Ct/T=0.06 ~/ " Ct/T=0.09 , 0.08 ~/ * Ct/T=0.11 '~ ~[ I, Ct/T=0.14 ~/ ...... eq.(3) ~0.06 [ ~ ,.~ 0.04

Experimental Conditions: ~ 0 6 1 H l,/T=1.27 l Ct/T=0.09[ C imp/T =0.44 II' Ct/T=0.11 [ 0.04 to 0.5 vvm 9Ct/T=0.14I 0.001 9 0.1

................. 1 10 Specific power input, em / (W/kg)

Fig. 4 Comparison of kLa measurements at various C/T in tap (closed symbols) and distilled water (open symbols),

9 9 9 9

9 Ill . ..... 9 #,...... "" 9 ..

0.02

#

9 ....

ooO.-~176176 ,.,.- 9

0 0

1 2 3 4 Specific power input, e / (W/kg)

Fig. 5 Comparison of the A GR kLa values with the gas-inducing impeller, eq.(3) and the Denver cell eq.(4).

4.4 Comparison with conventional sparged gas-liquid reactors Comparisons were also made with existing correlations from the literature for conventional sp~ged systems, such as those reported by Smith et al. (1977). In this case, the superficial gas velocity is an independent variable, whereas in the AGR it depends on the impeller speed and tank geometry. Comparisons of the sparged reactor and the AGR mass transfer performance were made using the measured induced gas flow rates previously reported by Leguay et al (1999) and suitably averaged to reflect the differences in gas superficial velocities, U,gd and U~g~, in the draft tube and annulus, respectively

kLa : 0.266(~'m) 0"475(fl Us~4d+ (1- fl)UsO~4d) where fl = (Dd/T) 2 is the fractional volume of the draft tube region.

(10)

195 0.1

9

0.12

+20%/'"~

f

lj* ,""

'

9 ...... .... ---

Ct/T=0.09- T29 ..... ] Linek et al. (1987) - T29 [ Davies et al. (1985)- T31 [ Smith et al. (1977) - T44 to T183 Muskett (1987) - T61 to 267 I . . . . Van~ Riet (1979) - up to T150 .,'1"'"

0.10

/I ~0.08

i9 t

0.06

K," / , "

O

4 /"

f

0.04 l

[,~'~,"

,~"/,./"

I O Ct/T=0.06 ! [ 9 Ct/T--O.09

/ ,,'~"/ - " "

[. ctfr=O.ll

t ' ~ / ' /

0.01 0.01

"

,

,

""

0.02

~

i,~

"'"

.....

...""

9 ""

"'"

...,.-"

~--~'" '

[,A Ct/TT0:I ~

0.00 Predicted k La / s"1

0.1

Fig. 6 Comparison of the A GR kLa measurements at various C/T in tap water with the adapted Smith's correlation, eq.(lO).

.... 0

' ' ' ' ...... 1

2

Specificpower input, em / (W/kg)

3

Fig. 7 Comparison of the A GR kLa values in tap water with other literature data (T29 denotes that the experiments were conducted in a 29 cm diameter vessel)

Figure 6 shows the AGR experimental kLa values compared with those predicted from eq. (10), for a sparged system operated at the same gas supply rate. At low power inputs (and therefore low Usg) the data are within 20% of the values predicted by eq.(10). However, at higher em and Usg, eq.(10) underpredicts the AGR kLa values by about 40%. This result should be interpreted with caution, since the AGR experiments were carried out in a 0.29 m diameter vessel, whereas Smith's original correlation was restricted to vessel diameters of 0 . 4 4 - 1.83 m. To illustrate the effect of scale on kLa, the AGR mass transfer data obtained at C/T =0.09 are compared in Fig. 7 with predictions from literature correlations developed: (i) in sparged 0.30 m diameter tanks (Linek et al., 1987; Davies et al., 1985) and (ii) for larger diameter sparged tanks (Smith et al., 1977; Muskett, 1987a & b; Van't Riet, 1979). In Fig. 7, at a given era, the comparisons are made at the same gas supply rate as would be obtained by vortex ingestion in the AGR; consequently, the gas superficial velocities used in the sparged systems predictions were increased with increasing specific power input. Figure 7 shows significant discrepancies in the values of kLa for sparged systems at high specific power inputs, depending on the scale of operation. At a given specific power input and superficial gas velocity, higher kLa values are obtained in the smaller 0.3 m diameter tanks (i.e. the AGR scale) compared with the data for T > 0.44 m. The results of a few experiments carried out using the H202 reduction technique in a T = 0.29 m, conventional sparged system are shown in Table 2. These results, along with the literature data shown in Fig. 7, tend to suggest that for T < 0.44 m, the value of kLa is scale dependent. The AGR performance should therefore be compared to that of sparged vessels at the same scale of operation. In this respect, Fig. 7 shows reasonable agreement between the kLa values obtained in the AGR at C/F=0.09 and the performance predicted by Davies et al. (1985) for a sparged device of diameter 0.31 m, operated at the same gas supply rate as the AGR. The discrepancy between those two sets of data never exceeds 15%, which lies within the prediction error band given by Davies et al. (1985).

196 Table 2. Mass transfer measurements in a 0.3 m diameter sparged vessel; gm=0.86 W/kg, Us~=2.5x 10-3 ms -1 Conventional sparged vessel Davies et al. (1985) Muskett (1987) T29, T/3 45 ~ PBT, Tap water T31, RT, Tap water T61 to T267, RT and concave impeller, Tap water Mean kLa +'95% conf. int. Mean kLa + 20% Mean kLa + 10% 0.0344 + 0.00167 s 1 0.0296_+0.00592 s 1 0.0176+0.00176 s -1 5. CONCLUSIONS Previous characterisation of the liquid phase flow patterns, indicated that the AGR could be treated as a single well-mixed cell and therefore a lumped parameter approach is appropriate to model mass transfer at the 0.29 m diameter scale. The steady state H202 technique was used to measure kLa over a range of industrially realistic specific power inputs, between 0.3 and 3.5 W/kg and for various AGR geometries. In tap water and based on surface aeration alone, the AGR produces typical values of kLa ranging from 0.01 to 0.08 s 1. Increasing the draft tube top clearance gives a decrease in the mass transfer coefficient, in agreement with previous work which indicated that this change would decrease the volumetric flow rate of ingested gas. Overall, however, the AGR mass transfer performance is not very sensitive to Ct/T, which is an advantage over many other self-inducing designs. Changing the draft tube off-bottom clearance or the overall liquid height had a negligible effect on k~a. Compared to other gas-inducing and gas sparged devices, at a given specific power input, the AGR seems to perform slightly better than a concave bladed self-inducing impeller with only one orifice per blade. It was shown to perform considerably better that the Denver type froth flotation cell, indicating its ability to ingest a much higher flow rate of gas at a given specific power input. Finally, AGR mass transfer coefficients are similar to those reported in the literature for sparged reactors at the same scale, operating at the same specific power input and superficial gas velocity, but with the advantage of internal gas recycling. Acknowledgement: We gratefully acknowledge the financial and technical support provided by Praxair, Inc., Tarrytown NY, USA. CL gratefully acknowledges the support of an EPSRC studentship. We also thank Dr Ali A1-Khayat of B HRG for his input to our discussions.

REFERENCES Davies, S.N., Gibilaro, L.G.,Middleton, J.C., Cooke, M. and Lynch, P.M., (1985), 6th Eur. Conf. on Mixing, Wurzburg, West Germany, Paper 6, 27. Forrester S.E., (1995), PhD Thesis, University of Cambridge. Forrester, S.E., Rielly, C.D. and Carpenter, K.J., (1998) Chem. Eng. Sci., 53(4), 603. Hickman, A.D., (1988), 6th Eur. Conf. on Mixing, Pavia, Italy, 369. John, A. (1997), PhD thesis, University of Birmingham. Leguay, C.,Rielly, C.D. and 0zcan-Ta~kin,G., (1998a), I. Chem. E. Res. Event, paper 101. Leguay, C.,Rielly, C.D. and 0zcan-Ta~kin,G., (1998b), CHISA conference, Prague, paper G3.4, 19. Leguay, C.,Rielly, C.D. and 0zcan-Ta~kin,G., (1999), A. I. Ch.E. Ann. Meeting, Dallas, USA. Linek, V.Vacek, V. and Benes, P. (1997) Chem. Eng. J., 34, 11. Muskett, M.J., (1987a), PhD Thesis, Cranfield Institute of Technology. Muskett, M.J., (1987b), "Gas-liquid research review", FMP Report 030, BHR Group Ltd. Patwardhan, W.A. and Joshi, J.B., (1999), Ind.Eng.Chem.Res., 38, 49. S. B. Sawant, J. B. Joshi, V. G. Pangarkar and R. D. Mhaskar, Chem. Eng. J., 21, 11. Smith, J.M., Van't Riet, K. and Middleton, J.C. (1977), 2nd Eur. Conf. on Mixing, Cambridge, England, Paper F4, 51. Van't Riet, K., (1979), Ind. Eng. Chem. Processs Des. Dev., 18(3), 357.

I 0th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

197

Modelling of the Interaction between Gas and Liquid in Stirred Vessels G.L. Lane 1, M.P. Schwarz 1, and G.M. Evans 2 1 CSIRO Minerals, Box 312 Clayton Sth, Victoria 3169, Australia 2 Dept of Chemical Engineering, University of Newcastle, New South Wales 2308, Australia

Abstract Although existing literature demonstrates substantial progress in developing CFD methods for stirred tanks, most studies are limited to single-phase liquid flow. In modelling of multiphase mixtures, there are a range of additional complexities. Further development of CFD modelling is being investigated for gas-liquid contacting in a mixing vessel. This paper outlines the general method of simulation, and discusses modelling of the gas-liquid interaction. It is shown that predictions of gas distribution and holdup are sensitive to the specification of the drag force. This force is usually determined according to the drag force correlation of Ishii and Zuber [6]. There is evidence however that in a forced turbulent flow the drag coefficient on particles or bubbles is increased. Using the correlation of Ishii and Zuber [6], it is found that gas holdup is substantially underpredicted. Alternative methods of calculating drag coefficient in turbulent flow as proposed by Brucato et al. [7] and Bakker [10] are found to increase the predicted holdup but give an incorrect pattern of gas distribution. A modified correlation based on that of Brucato et al. [7] was found to give improved results, but the generality of the method is uncertain. To improve the accuracy of the CFD model, better knowledge of bubble drag coefficients is needed. 1. INTRODUCTION Computational fluid dynamics (CFD) modelling has considerable potential as a tool for design, process improvement and scale-up of stirred vessels. Existing literature has demonstrated substantial progress in developing CFD methods [ 1]. However, modelling has usually been limited to single-phase liquid flow, whereas multiphase mixtures are very common in industrial stirred tanks. For multiphase flow, the complexity of modelling increases considerably, and this remains an area for further development. CFD methods are presently being investigated for gas-liquid contacting in a stirred tank. The CFD method is applied to simulate a baffled tank stirred by a Rushton turbine. Progress in this work was reported recently [2]. This paper outlines the general method of simulation and modelling of the forces between gas and liquid. It is shown that the most important force is the drag force, which determines the gas slip velocity, and hence affects the distribution of gas in the vessel and the total gas holdup. There is evidence in the literature to indicate that the effective drag coefficient on bubbles is different in a turbulent liquid compared to that of a bubble rising in a stagnant fluid. In order to improve the accuracy of the CFD modelling method, several expressions for the modified drag coefficient have been tested.

198 Table 1. Nomenclature A B

b CA Co CL d DI2 F

g k L P Re S t T U Vd Vr X

added mass force (N m "a) body force (N m "3) factor in dispersion forces (-) added mass coefficient (-) drag coefficient (-) lift coefficent (-) particle or bubble diameter (m) diffusivity coefficient (m 2 s-1) drag force (N m -3) acceleration due to gravity (m S"2) turbulent kinetic energy (m 2 sE) lift force (N m 3) pressure (N m 2) Reynolds number (-) source or sink of mass (kg m -3 s-l) time (s) turbulent dispersion force (N m 3) velocity (m sl) drift velocity (m s-l) relative velocity (m sl ) position vector (m)

a E 7"/ A, ~t p

volume fraction (-) energy dissipation rate (m 2 S"3) particle-turbulence time ratio (-) Kolmogorov microscale (m) viscosity (N s m "E) density (kg m a) reference density (kg m "a) Po "rp12 particle relaxation time (s) q~tl2 integral turbulence timescale (s) particle-turbulence velocity ratio (-) ~r angular velocity (rad s -l) Subscripts br break-up co coalescence crit critical phase number i laminar L turbulent T reference value 0 1 liquid 2 gas

2. E Q U A T I O N S FOR T W O - P H A S E F L O W

For CFD modelling of gas-liquid flow in a stirred tank, the fluid flow may be described by the Eulerian two-phase equations, which are derived by averaging over an ensemble of a large number of bubbles. For turbulent flow, further time-averaging is also carried out. This introduces extra terms due to non-zero correlations of the fluctuating components, which must be modelled. The final form of the two-phase equations, as used in this study, is as follows:

t~(O~iPi~) + V .(o~iPiUi) = S i Ot t~(~iPiUi ) + V.((aiPiUi| Ot

(1) i + ].tT,i)(VU i + (Vui)T))

(2)

=-~ +cti(Pi - P0)g + Fi + T(1),i + T(2),i + Ai + Li + Bi + SiUi where i = 1 for the liquid and i = 2 for the gas (see Table 1 for symbols). Here, density-weighted Favre averaging is used, so that the averaged continuity equation, equation (1), is exact. Reynolds stresses in the continuous phase are modelled in terms of the eddy viscosity #T,, and mean velocity gradients, with the eddy viscosity determined from the standard k-e model. However, for the disperse phase, P+.2is taken as zero. The drag force is given by:

r 2 = - V 1 =-~o~2(alPl)

U 2-U1](U 2 - U 1 )

(3)

199

In CFD simulations of multiphase flow, the drag force is often the only interphase force considered [e.g. 3]. However, there are other forces which should be included to provide a more complete model, unless they can be shown to be negligible. Added mass and lift forces may be substantial if velocity gradients are large. These forces are given by [4]: A2 = - A 1 =~2Plfai \

DU2 Dt

L 2 = - L 1 =ot2PlCL(U2-U1)|174

D U a ]"

Dt / '

(4)

Extra turbulent correlation terms appear due to averaging of the interphase forces, and these correlations are modelled using a gradient transport hypothesis following the method of Simonin [5], which leads to turbulent dispersion forces. The first and second turbulent dispersion forces result from averaging of the time-varying pressure force and drag force respectively, and are given by: T(1),2 = -T(1),I = / 3

ll+ Or

(5)

PlkVCt2

T<2),2 --T~2)a - ~ a 2 ( a l P l )

(/

IVr[ D]2 V~ O~la2

b+rl r , and D12 -

1+'7, )

(6)

/91 ffl + Cfl~r 2

These expressions take into account the particle inertia and cross-trajectories effect due to particle slip. Hence, the expressions are functions of the ratio of particle relaxation time to large eddy time r/r, the ratio of slip velocity to turbulent velocity, ~r, and the effects of added mass given by b, where these expressions are evaluated as follows:

~r -- T~2

~r -- Ivrl '

-

b = 1+ C

Pa

A

.

(7)

+CA

The body force term Bi in equation (2) represents the centrifugal and Coriolis forces which apply in the rotating frame of reference only [2]. 3. DRAG F O R C E AND SLIP VELOCITY IN TURBULENT F L O W Of the several forces between the particle or bubble and the liquid, the drag force is usually the most important. Under steady conditions, a balance between drag and buoyancy forces results in the particle attaining a characteristic terminal velocity. This terminal velocity depends on the drag coefficient, which is usually determined from empirical correlations. For solid spherical particles, drag coefficients are determined by correlations describing the "standard" drag curve. For bubbles, the correlation of Ishii & Zuber [6], is often used. However these correlations refer to the steady motion of the particle or bubble through a stagnant liquid, and may not be adequate where there is a substantial level of externally forced turbulence in the liquid, as applies in a stirred tank. As summarised by Brucato et al. [7], there is a body of evidence in the literature which demonstrates that particle drag coefficients are modified by turbulence, often to a substantial degree. More recent experimental measurements include those of Magelli et al. [8], who measured the average settling velocity of solid particles in tall multiple impeller stirred tanks, and Brucato et al. [7], who carried out experiments in a Taylor-Couette apparatus. The data of these authors show fair agreement, with the settling velocity reduced to as low as 20% of the stagnant terminal velocity. The effect was found to increase with both particle size and mean

200 turbulent energy dissipation rate. Brucato et al. [7] proposed that the increase in drag coefficient may be related to the ratio of particle size to the Kolmogorov length scale, 2, according to:

CD -CD,o _ K(d ~3

CD,O

(8)

where the constant K was found to be 8.76 x 10-4. The decrease in particle slip velocity in a turbulent fluid may be explained in terms of the particle inertia, whereby the particle cannot follow the fluid motion exactly but is always under accelerating conditions as it responds to turbulent fluctuations. Particle motion may be modified by changes in the instantaneous drag coefficient, the effects of added mass, and the non-linearity of the drag force as the relative velocity varies [7]. No direct experimental data for the effect of turbulence on bubble rise velocity appears to be available. Nevertheless, bubbles are likely to be affected by turbulence in a similar manner to solid particles. Computer simulations of the motion of bubbles about 1 mm diameter have been carried out, by Maxey et al. [9] and by Spelt and Biesheuvel [10]. The bubble rise velocity was found to be decreased to as low as 4 0 - 50% of that in stagnant fluid. The effect increased with decreasing Kolmogorov timescale [9] or increasing turbulence intensity [ 10]. A method for calculating the modified bubble drag coefficient was suggested by Bakker [ 11], who proposed that the drag cofficient could be calculated from the standard drag curve using a Reynolds number based on a modified viscosity: Re* =

PlVrd

(9)

ILL + 2/~9~/T,1 However, the equation does not account for bubble size and apparently lacks an experimental basis. Nevertheless, Bakker applied the method in CFD modelling of a gassparged tank and obtained good agreement with experimental measurements. In other CFD simulations of gas-sparged stirred tanks [e.g. 12], the need for a modified drag coefficient is not recognised. Despite this, good agreement with experimental gas holdup measurements is still obtained in some cases. The explanation is not clear. In the simulations presented here, several possible equations are tested for the increase in bubble drag coefficient due to turbulence. The correlation of Brucato et al. [7], equation (8), has been tested since it has an experimental basis and the response of bubbles is likely to be similar to particles. The method of Bakker was also tested. Finally, a modified method based on the equation (8) was tested and found to give improved results. 4. SIMULATION METHOD The CFD method has been previously reported and more detailed information is available elsewhere [2]. Briefly, the CFD simulation models a 1.0 m diameter baffled tank with a standard Rushton turbine 0.333 m diameter. The geometry and operating conditions correspond to those of experimental measurements of bubble sizes and local gas holdup [ 13, 14]. Impeller speed is 180 rpm and air flow rate is 0.00164 m3/s. The tank including the impeller is modelled by a finite volume grid in cylindrical coordinates (see Figure 1). Impeller motion is accounted for using a Multiple Frames of Reference method, where the tank is divided into two zones, with a zone around the impeller in a rotating frame of reference where the impeller appears stationary.

201

The local Sauter mean bubble diameter was found experimentally to vary from 0.8 to 4.2 mm [ 13]. Although some progress has been made towards a model to predict local bubble sizes based on coalesence and break-up rates [2], in simulations presented here bubble sizes are specified in advance using an interpolation of the experimental measurements. This approach provides more certainty in studying the effect of different force terms. The equations are solved numerically using the commercial code CFX4.2, using additional user-supplied routines to implement the Multiple Frames of Reference method, to add and remove gas, and to specify interphase forces. Satisfactory completion of each simulation is based on several criteria, including sufficient reduction of the mass residuals, an accurate balance between rates of gas entering and leaving the tank, and a constant gas holdup. Several simulation cases have been run as follows, to test the use of different correlations for the bubble drag coefficient, while other model parameters remain constant: 9 C a s e (1): the drag coefficient was specified using the correlation of Ishii and Zuber [6]. 9 C a s e (2): the drag coefficient was modified by the correction factor according to equation (8), where ~ was calculated based on the local energy dissipation rate. 9 C a s e (3): the method of Bakker [ 11 ] was used in which the drag coefficient is based on a bubble Reynolds number according to equation (9). 9 C a s e (4): the drag coefficient is again calculated using equation (8), but K takes the lower value 6.5 x 10-6, and an overall averaged Kolmogorov length scale was calculated based on the global average dissipation rate. 5. RESULTS AND DISCUSSION Several simulation runs have been carried out, as described in Section 4 above, to investigate the specification of the drag coefficient. Accuracy of results can be assessed by comparison with experimental measurements [ 14] of total gas holdup and distribution of gas volume fraction, as shown in Figure 2. Case (1) applies the drag correlation for a bubble in stagnant liquid. Results indicate a pattern of gas distribution that is fairly similar to the experimental measurements (Figure 3). However, gas volume fractions are underpredicted, and the total gas holdup is only 1.0 % compared with an experimental measurement of 2.97 %. The magnitude of the various forces have been compared for case (1). The magnitude of the various forces can be compared with the buoyancy force, Apg, which is -9800 N/m 3. It can be seen (Figure 4) that added mass and lift forces are significant only in the immediate vicinity of the impeller. Dispersion forces were found to be significant mainly in the discharge stream. Therefore, in the bulk of the tank, bubble motion is dominated by drag and buoyancy forces, and consequently, the vertical slip velocity is found to be close to the terminal bubble rise velocity (-0.22 m/s in case (1)) throughout most of the tank. Thus, the equilibrium gas holdup results essentially from a balance between gas recirculation induced by liquid flow and the upward flux of gas as determined by the average vertical slip velocity. Various checks have been made which ifidicate that the prediction of mean velocities is quite accurate. Hence, the remaining uncertainty is the specification of drag coefficient. Further simulations have been run to test modifications to the drag coefficient. The distribution of gas volume fraction for case (2) is shown in Figure 5, where total gas holdup was overpredicted (4.9%). The pattern of distribution was much the same for case (3), where holdup was 3.2%. In both cases (2) and (3), although higher gas holdups are obtained, the pattern of distribution is wrong, due to a strong tendency for gas to collect in the lower recirculation loop. In these simulations, the modified drag coefficient is taken as a function of

202 the local energy dissipation rate, however these equations appear to be too sensitive to the high energy dissipation rate in the discharge stream, leading to very low slip velocities. Thus, the approaches of cases (2) and (3) lead to unsatisfactory results. In case (2), it was assumed that the value of & in equation (8) is a function of local energy dissipation rate. However, Micale et al. [3] applied equation (8) to modelling solids suspension using a global average value of the Kolmogorov scale. This approach was taken in case (4), which therefore smoothes over variations in e. With this approach, predictions for both the gas distribution pattern (Figure 6) and total gas holdup (2.6%) are closer to experimental data. However, the coefficient in equation (8) was reduced by about two orders of magnitude. While case (4) indicates improved results using a modified drag coefficient correlation, there is no certainty that the method is generally valid. It would appear that the equation of Brucato et al. [7] cannot be applied based on local e values and Kolmogorov scales, but only using an overall average for the Kolmogorov scale. This is a reflection of the experimental methods [7,8], where only averages were measured based on power input, although turbulent energy dissipation rates were very unevenly distributed, and local variations in e have thus been obscured. In a stirred tank, the value of near the impeller is at least 20 times higher than the average. In the Taylor-Couette apparatus, e is also unevenly distributed [15]. The use of a global average value of e is unsatisfactory since the correlation will not be generally valid for different tank configurations. Furthermore, in the Taylor-Couette device it is possible that observed settling velocities are affected by added mass and lift forces induced by Taylor vortices. Thus, there is further reason to doubt the accuracy of the correlation of Brucato et al. [7]. In summary, it would appear that a correlation is needed for the drag coefficient on bubbles in a turbulent flow, in order to improve CFD modelling of gas-stirred tanks. A generally applicable correlation does not appear to be available at present. Further investigation is recommended, in order to generate experimental data and provide such a correlation. 6. CONCLUSIONS For CFD modelling of gas dispersion in stirred tanks, it has been shown that the most important force is the drag force and the choice of model for drag coefficient is found to affect both the pattem and total holdup of gas. Using the equation of Ishii and Zuber [6], the total holdup is underpredicted, since the predicted gas slip velocity is too high. Altemative methods of calculating drag coefficient in turbulent flow according to Brucato et al. [7] and Bakker [ 11] increase predicted holdup but give an incorrect pattern of gas distribution. By using a modified correlation based on that of Brucato et al. accuracy of results was improved, however the generality of the method is uncertain. A generally applicable correlation for the drag coefficient of bubbles in turbulent flow does not appear to be available at present, and further investigation is recommended.

REFERENCES 1. Brucato, A., Ciofalo, M., Grisafi, F. and Micale, G., 1998, "Numerical prediction of flow fields in baffled stirred vessels: A comparison of altemative modelling approaches", Chem. Eng. Sci., Vol. 53, No. 21, pp. 3653-3684. 2. Lane, G.L., Schwarz, M.P., and Evans, G.M., 1999, "CFD Simulation of Gas-Liquid Flow in a Stirred Tank", Proc. 3rd Int. Symposium on Mixing in Industrial Processes, Osaka, Japan, Sept. 19-22 1999, pp. 21-28.

203 3. Micale, G., Montante G., Grisafi, F., Brucato, A., and Godfrey, J., 1999, "CFD Simulation of Particle Distribution in Stirred Vessels", Trans. L Chem. E., to be published. 4. Drew, D.A., and Lahey, R.T., 1987, "The Virtual Mass and Lift Force on a Sphere i n Rotating and Straining Inviscid Flow", Int. J. Multiphase Flow, Vol. 3, No.l, pp. 113-121. 5. Simonin, O., 1990, "Eulerian Formulation for Particle Dispersion in Turbulent TwoPhase Flows", Proc. 5th Workshop Two-Phase Flow Predictions, pp. 156-166. 6. Ishii, M. and Zuber, N., 1979, "Drag Coefficient and Relative Velocity in Bubbly, Droplet or Particulate Flows", AIChE Journal, Vol. 25, No. 5, pp. 843-855. 7. Brucato, A., Grisafi, F. and Montante. G., 1998, "Particle drag coefficients in turbulent fluids", Chem. Eng. Sci., Vol. 53, No. 18, pp. 3295-3314. 8. Magelli, F., Fajner, D., Nocentini, M. and Pasquali, G., 1990, "Solids distribution in vessels stirred with multiple impellers", Chem. Eng. Sci., Vol. 45, pp. 615-625. 9. Maxey, M.R., Chang, E.J., and Wang, L.-P., 1994, "Simulation of interactions between microbubbles and turbulent flows", Appl. Mech. Rev., Vol. 47, No. 6, part 2, pp. $70-$74. 10. Spelt, P.D.M. and Biesheuvel, A., 1997, "On the motion of gas bubbles in homogeneous isotropic turbulence", J. Fluid Mech., Vol. 336, pp. 221-244. 11. Bakker, A., 1992, Hydrodynamics of Stirred Gas-Liquid Dispersions, Ph.D. Thesis, Delft University of Technology, The Netherlands. 12. Jenne, M. and Reuss, M., 1997, "Fluid Dynamic Modelling and Simulation of GasLiquid Flow in Baffled Stirred Tank Reactors", Recents Progres en Genie des Procedes, Vol. 11, No. 52, pp. 201-208. 13. Barigou, M. and Greaves, M., 1992, "Bubble-size distributions in a mechanically agitated gas-liquid contactor", Chem. Eng. Sci., Vol. 47, No. 8, pp. 2009-2025. 14. Barigou, M. and Greaves. M., 1996, "Gas holdup and interfacial area distributions in a mechanically agitated gas-liquid contactor", Trans. I. Chem. E., Vol. 74, Part A, pp. 397--405. 15. Parker, J., and Merati, P., 1996, "An Investigation of Turbulent Taylor-Couette Flow Using Laser Doppler Velocimetry in a Refractive Index Matched Facility", J. Fluids Eng., Vol. 118, pp. 810-818.

Figure 1. Finite volume mesh for tank.

Figure 2. Experimental data for local gas holdup (volume fraction) [14].

204

Figure 3. Case 1 - gas volume fraction in a vertical plane half way between impeller blades.

Figure 4. Magnitude of the added mass and lift forces in a vertical plane calculated for Case 1.

Figure 5. Case 2 - gas volume fraction in a vertical plane half way between impeller blades.

Figure 6. Case 4 - gas volume fraction in a vertical plane half way between impeller blades.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

205

Experimental investigation of local bubble size distributions in stirred vessels using Phase Doppler Anemometry M. Sch/ifer, P. W/ichter and F. Durst Institute of Fluid Mechanics, Friedrich-Alexander-University Erlangen-Ntimberg, Cauerstrasse 4, D-91058 Erlangen, Germany http://www. 1stm. uni- erlangen, de The local bubble size distribution in gassed stirred tanks equipped with two different types of impellers, (1) a Rushton turbine and (2) a pitched blade impeller, was investigated by using phase Doppler anemometry (PDA). The accuracy of the PDA-system was checked on single bubbles of defined size that were produced at the tip of a syringe needle. The deviation of the mean diameter measured by the PDA-system from the actual size was less than 7%. The PDA-system was thereafter applied to analyse gas-liquid flows in stirred reactors. From the measurements the mean diameter d32 according to Sauter was calculated and it was found that the mean diameter varies locally between 0.65 mm and 1.5 mm for both types of impellers. The flow fields produced by the impellers have a considerable impact on the distribution of the mean diameter. Larger bubble sizes were detected in the ring vortices present in the large scale flow fields, with a significant increase occurring in the lower ring vortex produced by the Rushton turbine. In addition, break-up of bubbles in the discharge flow of the impellers was observed, but this does not necessarily lead to a drop in the mean diameter. Larger bubble sizes are also present more frequently in the discharge flow than in other regions so that other mechanisms have also to be taken into account, such as the low pressure region associated with the presence of trailing vortices and the recirculation of primary bubbles introduced to this region. The data obtained contribute to a better understanding of gas-liquid flows in stirred reactors. In addition, the development of numerical simulations of multiphase flows by computational fluid dynamics (CFD) requires better models than available to date and more detailed data for different types of flow configurations are required to validate the simulations. Therefore the data gained in the present study can significantly support the ongoing developments in CFD. 1.

INTRODUCTION

Many industrially relevant stirred vessel processes involve a gas-liquid dispersion. The challenge for an engineer is to select the best stirrer/vessel configuration from a wide range of possibilities to meet the requirements of the process, for example to ensure a sufficient masstransfer rate. It is common practice to investigate the integral quantities of gas-liquid stirred vessel flows, such as the gassed power input and the gas hold-up, and to present them in the form of diagrams of normalised quantities which permit the results of laboratory study to be transferred to large scale stirred vessel systems with linearly scaled-up geometries of the STR

206 [1]. Although the design rules resulting from such investigations are very useful for many practical applications, they permit very little information to be deduced on process and equipment improvements. Such improvements require local data on process related variables; for example, to prevent microorganism starvation, local gas hold-up and mass-transfer rates become important. In more recent times emphasis has been on the development of numerical computation techniques (CFD) for the purpose of establishing accurate and low-cost design and dimensioning methods to be applied to various processes in stirred vessels. Recently, CFD has also been applied to analyse gas-liquid flows in stirred vessels [2,3]. Numerical methods can be implemented for stirred tank reactors, however only in conjunction with precise, detailed experimental methods, which are essential for validation of the calculations and the models used in the numerical schemes. Especially for gas-liquid flows, an accurate validation requires local information on the mean velocities, mean diameters and the size distributions of the bubbles throughout the entire tank. To deliver local flow information new accurate measuring techniques have to be developed since conventional capillary techniques are not able to account for very small bubbles present in stirred tanks and thus can lead to errors [4]. Recently, video techniques have been developed and applied for measuring local bubble size distributions in stirred tanks [4]. However, the technique depends heavily on the accuracy of the post processing system, i.e. the image analysis. In this study Phase Doppler anemometry (PDA) was applied which offers the possibility to gain local information both on the mean and fluctuating velocities of particles, droplets or bubbles and on their size distribution simultaneously. Recently, PDA has been applied to study liquid-liquid dispersions and solid-liquid suspensions in stirred tanks [5,6]. PDA has the advantages of non-intrusive measuring techniques, but, since a maximum measurable bubble size and gas hold-up has to be taken into consideration, some limitations on the applicability of PDA exist. At the Institute of Fluid Mechanics in Erlangen a PDAsystem has been set up for measurement of local bubble size distributions in low hold-up gasliquid stirred vessel flows. Two different types of impellers were investigated, (1) a radial flow type Rushton turbine (RT) and (2) a downpumping pitched blade impeller (PB). For both impellers detailed information on the single-phase flow was available [7,8]. 2.

EXPERIMENTAL SET-UP

The test rig as shown in Figure 1 comprises three main parts, (1) the PDA-system, (2) the measuring section and (3) a traversing unit. The PDA-system consists of a HeNe-laser (632.8 nm), transmitting optics, receiving optics including a pair of photodetectors and further electronic facilities like PDA-filters and a PDA-processor to evaluate the signals. The lay-out of the optical set-up is based on light scattering calculations according to Lorenz-Mie theory and geometrical optics. The final optical set-up was optimised for the present gas-liquid system [9] and all relevant optical parameters and the characteristics of the measuring volume are given in table 1. The measuring section consists of a cylindrical baffled stirred vessel (T = 144 mm) equipped with (1) a standard Rushton turbine (D/T = 0.35) and (2) a four bladed pitched blade impeller (D/T = 0.35). The ungassed liquid height was equal to the vessel diameter (H = T). Four equally spaced baffles of width B = T/12 and thickness of 2 mm were mounted along the inner wall of the cylinder at a distance of 2.6 mm. Figure 2 shows the geometry of the mixing vessel and the coordinate system used. The details of the impellers are given in Figure 3. Both

207

Refractive index matching

Stirred Vessel

Laser

~

Transmitting Optics

/ ~7 " "

" "

/

/

MeasuringVolume

: : . ' / ~ -z r',~,.~.--

Receiving

/ Particles

Oscilloscope

Power

I

Supply

~

I: . . . . . - . J

Band-pa,~ Filters

PC Transient Recorder

Fig. 1. Set-up of PDA test rig for measurements in stirred tank reactors Table 1. Parameters of PDA-optics

Transmitting optics (air) Wavelength [nm] Focal length [mm] Beam waist diameter [mm] Beam intersection half angle [~ Shift frequency [MHz] Receiving optics (air) Focal length [mm] Off-axis/scattering angle [o] Elevation angle [~ Maximum bubble size [mm] Measuring volume (air) Number of fringes (without shift) Fringe spacing [pm] Diameter [pro] Length [mm]

632.8 400 0.65 3.58 1 310 90 0.54 2.4 97 5.07 496 7.384

impellers were mounted at a clearance of h = T/3. The blade thickness was 2 mm for the Rushton turbine (also disk thickness) and 0.9 mm for the pitched blade impeller. The complete measuring section was refractive index matched. For this purpose, the vessel walls, the baffles and the impeller blades were constructed from transparent Duran glass with the same refractive index as the working fluid (n = 1.468), a mixture of silicone oils. The fluid viscosity was 16.2 mPas and the density 1,021 kg/m 3. The surface tension of the mixture was well below that of water. In addition the tank was located in a hexagonal trough filled with the working fluid, in order to eliminate the distorting effect of the rounded surface of the tank on the path of the laser beams. The hexagonal trough was constructed with different angles between the vessel walls in order to be able to locate the receiving optics perpendicular to the surface of the trough for additional off-axis angles (see table 1), which might be required for measurements in other multiphase flow configurations.

208 A high-resolution measuring grid was realised in the entire flow field by automation of the data acquisition, whereby the PDA-optics were mounted on a 3-D traversing unit controlled by a PC via a controller. However, this allowed only for automated data acquisition along an axial profile at a constant radial position. Due to refraction at the vessel walls the receiving optics had to be traversed independently of the transmitting optics for every new location of the measuring volume in radial direction. The gas was introduced into the vessel by a syringe needle which was able to produce spherical bubbles of a defined size. This is very important for PDA-measurements, since the accuracy depends significantly on the sphericity of particles. The needle was located outside the impeller region at a radius of r = 26 mm (r/T = 0.181) at the bottom of the vessel (see Figure 2) so that the accuracy of the PDA-system could be checked by measurements of the bubble size produced above the tip of the syringe needle. ~.,

T= D = 50

144mm turn-

0.347 T

D = 5~mml= 0.35 T 3[/.5 m~n

q

!P

o

I

t~

r

I I

I

L iz.5

1

'~

I

i

I

D = 50 mm ~i 0.35 T :8 mrn ,

L', 1--, 14mm

Fig. 2. Vessel configuration 3.

12mm '1 I

rojected height

9

= 10mm

lJ ,-1 = 0.092T

Fig. 3.Geometry of Impellers

VALIDATION OF OPTICAL SET-UP

Although the PDA-technique does not need a calibration for bubble size measurements, it is recommendable to check the optical system, since it relies on exact positioning of the photo detectors. This was checked by locating the measuring volume above the tip of the syringe needle. The air introduced to the vessel was adjusted by a flow control system. In addition the frequency of the bubbles detaching from the tip of needle was monitored. For this the signals of the PDA-system and a stroboscope were used in parallel which led to the same frequencies. From the frequencies and the adjustable air flow rate the exact bubble size was determined and compared to those bubble sizes which resulted from the PDA-measurements. Figure 4 shows single bubbles typically produced at the tip of the syringe needle. The picture was taken by a CCD-camera with the help of a stroboscope. It is important to point out that the bubbles are of ideal spherical shape as indicated by the reference circle. The measuring volume of the PDA-system was placed at some distance from the tip of the syringe needle (at z/T = 0.472). 20,000 signals from single bubbles passing the measuring volume were detected and used for the evaluation of the bubble size distribution, which is depicted in Figure 5 in form of probability density function (pdf). The bubble size distribution

209

Fig. 4: Single Bubbles

Fig. 5. Bubble size and velocity distribution

is, as expected, narrow and a mean diameter of d~0 = 1.98 mm was determined. This is in good agreement to the bubble size that results from the bubble frequency and the air flow rate which revealed a value of 1.993 mm. Further tests were carried out at different bubble frequencies and air flow rates and the maximum deviation between the measured and actual bubble size was + 7%. In addition, Figure 5 contains the velocity distribution for the bubbles. Negative velocities indicate bubble rising towards the surface. A rising velocity of about 13 crn/s was obtained for the 2 mm bubbles which is in good agreement with theoretical and literature data [ 10]. 4.

RESULTS

The validated PDA-system was used to gain local information on gas-liquid characteristics in stirred tanks. Measurements were carried out in an r,z-plane located in the middle between two baffles (0 = 45 ~ with a radial and axial resolution of 6 - 8 mm throughout the entire tank. The stirrer speed was set to N = 800 r.p.m. (Vtip = nDN = 2.1 m/s) for the Rushton turbine (RT) and the pitched blade impeller (PBT) corresponding to a Reynoldsnumber of Re = NlY/v = 2,193. The power number for the RT was Po = 4.5 and for the PBT Po = 1.35 and this resulted in a specific power input of eT ----1.42 W/kg (RT) and eT = 0.43 W/kg (PBT). The experiments were conducted at a gas flow rate of 0.38 ml/s. The local bubble size distributions and the corresponding bubble velocities were recorded. Figures 6 and 7 show contour plots of the distribution of local bubble sizes (d32) throughout the entire vessel for the RT and the pitched blade impeller. The bubble sizes (d32) range from 0.65 mm to 1.5 mm. These size ranges are in agreement with findings in [4], where bubble mean diameters around 0.6 to 1.2 mm were reported for low surface tension fluids stirred by a RT for ev = 1 W/kg in a similar tank configuration. Regions of larger bubble sizes were detected for both types of impellers close to the tip of the needle, where the air is introduced to the vessel. For the RT the bubble size has already decreased after a short distance from the tip of the needle. It is important to point out that this is not due to break-up of bubbles, but increasingly more smaller bubbles from the bulk flow pass the measuring volume which dominate and decrease the mean diameter. For the PBT a

210

larger region of bigger bubble sizes was obtained around the air introduction, which extends up to the shaft. This can be explained by the recirculation zone which is generated by the PBT in this region [8]. A second region of larger bubbles was found for both types of impellers close to the shaft below the surface, where gas bubbles of relatively large size are sporadically sucked into the liquid via the free surface. In addition, larger bubbles are captured by the ring vortices present in the large scale flow fields. This is significant for the lower ring vortex produced by the RT, whereas only a minor increase in the bubble size becomes apparent for the upper vortex in the RT flow and the ring vortex produced by the PBT. This can be explained by the different flow fields produced by the impellers. The discharge flow of the RT leaves the impeller radially, splits at the vessel walls and forms two large scale vortices, one above and one below the centre line of the impeller [7]. The radial jet drives the large scale vortices and hinders also larger bubbles in escaping from the lower to the upper flow field. Therefore larger bubbles can be captured easily within the lower vortex. In contrast to this the upper ring vortex is of less intensity so that larger bubbles can not be kept continuously within the circulation of the liquid. The PBT generates only one large scale ring vortex that extends approximately over slightly more than half of the liquid height [8]. As a result the larger bubbles leave the circulation zone more frequently which is indicated by the smaller increase of the mean diameter within the large scale ring vortex. In addition, no circulation is present in the uppermost flow region of the vessel so that the bubbles, which leave the ring vortex, rise directly towards the free surface. That means that larger bubbles are not circulated back to the liquid volume and pass the measuring volume of the PDA-system in maximum once in their lifetime. Therefore, lower

211

mean diameters were obtained for the badly mixed uppermost flow regions, but it has to be taken into account, that this does not mean that local mass-transfer rates are better since also local gas hold-up rates are considerably lower in this region. In addition, the results point to an important fact for modelling gas-liquid flows in stirred reactors. In the discharge flow larger mean diameters were obtained than in other regions of the reactor, which is especially significant for the PBT. This might be in contradiction to the assumption, that in regions of high energy dissipation rates, as it is the case for the discharge of impellers, the lowest mean diameters should be obtained since here break-up of bubbles takes place. To clarify this, Figures 8 and 9 show the local bubble size distributions in form of a pdf at two different positions, from which the first one is located above the impeller and the second below within the discharge region. The peak value of both distributions is similar (appr. 500 lam). The distribution in the discharge flow is, however, considerably broader. That means on the one hand, that the probability of smaller bubbles to be obtained within the measuring volume is higher and therefore a break-up of bubbles is taking place due to the high energy dissipation rates in this region. On the other hand this does not result in a smaller value of the mean diameter since also larger bubbles are present in this region. The reason for this is not completely understood, since several phenomena have to be taken into account. Firstly, it is well known, that trailing vortices are present in the discharge of impellers, which generate an intense low pressure region and are able to capture larger bubbles [11]. Secondly, the influence of the air introduction has to be mentioned, from which larger primary bubbles are released to the recirculation flow in this region and as a result this can contribute to a larger mean diameter. It is interesting to note, that a similar observation was made in [4], where the lowest mean diameter was found at the uppermost measuring position (at z/T = 0.78). Here a further possible explanation could be derived from the video images, which revealed small bubbles detaching from the thinning end of elongated bubbles of size larger than 1 mm. This may lead to a further reduce of the mean bubble size.

Fig. 8. Pdf above impeller 5.

F i g . 9. Pdf below impeller

CONCLUDING REMARKS

Phase Doppler anemometry has been successfully applied to investigate gas-liquid flows in stirred tank reactors. The flow conditions within the reactor have a considerable impact on the

212 distribution of the local mean diameter throughout the entire reactor. The results show that the large scale vortices are able to capture larger bubbles, which is indicated by an increase in the mean diameter. This is significant for the lower ring vortex produced by a Rushton turbine, since the radial jet are hinders larger bubbles in escaping to the upper flow field. In addition, it was found that in the discharge flow of the impellers, especially for the pitched blade impeller, the mean diameter is higher than in other flow regions of the reactor. From this it can be concluded that care must be taken in modelling break-up phenomena in the discharge flow of impellers, since break-up due to high energy dissipation rates is not the only mechanism that determines the mean diameter. The low pressure region associated with the presence of trailing vortices and the recirculation of primary bubbles have also to be taken into account. This is of importance, for example, in population balances that are incorporated to CFD-codes for multiphase flow simulations [3]. In summary, detailed data sets for two different types of flow configurations in a stirred vessel have been produced. This data sets are now available for validation of numerical simulations and for further development of multiphase models incorporated into the CFDcodes. The data basis will be continuously extended by further measurements, in which parameters like impeller speed, gas flow rate and the viscosity of the fluid will be varied. Also further types of impellers will be considered in future investigations to extend the knowledge on gas-liquid flows in stirred tanks. In addition the investigations will have to aim at delivering information also on local gas hold-up rates. The present investigations were limited to relatively low overall gas hold-ups. These have to be increased to gain access to more technically relevant gas-liquid flows. It has to be established to what extent PDA can then be applied to such flow configurations. ACKNOWLEDGEMENTS The authors acknowledge financial support provided by the Deutsche Forschungsgesellschaft DFG (Gz: Du 101/45). REFERENCES 1. Tatterson, G.B., 1991, Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New York. 2. Bakker, A., Smith, J.M. and Myers, K.J., 1994, Chem. Eng., Vol. 101, No. 12, pp. 98-104. 3. Lo, Simon, 1998, paper submitted to the 4 th International Conference on Gas-Liquid and Gas-Liquid-Solid Reactor Engineering, Delft, August 23-25, 1999. 4. Machon, V., Pacek, A.W., Nienow, A.W., 1997, TranslChemE A, Vol. 75, pp. 339-348. 5. Zhou, G., 1997, PhD-Thesis, University of Alberta, Canada. 6. Yu, Z. and Rasmusson, A. C., 1999, Exp. in Fluids, Vol. 27, pp. 189-198. 7. Sch~ifer, M., H6fken, M. and Durst, F. 1997, Trans. I.Chem.E., vol. 75, Part A, pp. 729736. 8. Sch~ifer, M., Yianneskis M., W~ichter, P. and Durst, F., 1998, AIChE-Journal, Vol. 44, No. ~ 6, pp. 1233-1246. 9. Wachter, P., 2000, PhD-Thesis, Friedrich-Alexander-University Erlangen-Ntirnberg. 10. Bischof, F., 1994, PhD-Thesis, Friedrich-Alexander-University Erlangen-Ntirnberg. 11. Van't Riet, K. and Smith, J. M, 1972, Chem. Eng. Sci., Vol. 28, pp. 1031-1037.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

213

Void Fraction and Mixing in Sparged and Boiling Reactors Zhengming Gao, John M. Smith, Donglin Zhao and Hans Miiller-Steinhagen. Department of Chemical and Process Engineering, School of Engineering in the Environment University of Surrey, Guildford, GU2 5XH, UK The vertical void fraction'distributions with one or two Chemineer CD-6 impellers working in cold gassed, hot sparged and boiling systems are reported. The vertical void fraction in boiling systems is dramatically different from that in cold gassed or hot sparged systems whether in terms of value or distribution. The void fraction is much smaller in boiling systems than in cold gassed or hot sparging systems under the same gas phase output conditions. Hot sparged systems have similar vertical void fraction distributions, with maxima in similar locations and smaller void fraction, to those of cold gassed systems. The mixing times in ungassed, cold gassed, hot sparged and boiling systems agitated by a single CD-6 impeller are also reported. The results are of particular relevance to the design and operation of the reactors with hot sparging or boiling liquids. 1. INTRODUCTION Void fraction is an important performance variable in gas-liquid contacting and has been widely accepted for use in reactor diagnosis, selection and design. Being a sensitive reflection of the flow pattern formed in a vessel, local void fraction is potentially able to give much more information on the real hydrodynamics and mass transfer performance in a vessel. However, only a few workers [1-8] have investigated the void fraction distributions inan aerated vessel agitated by a single impeller. No studies at all have been reported on void fraction distributions when there are multiple impellers in a vessel, a configuration that is popular in industrial applications. The liquid phase mixing time is an important parameter for agitated reactor design and scale-up. There have been extensive studies with single-phase systems over the last 30 years. However there has been little work with aerated systems. Industrial mechanically agitated reactors frequently operate in regimes in which large volumes of vapour are generated. Typical examples include: batch organic syntheses carried out under reflux of a solvent, polymerisations in which water is produced and boiled off, and highly exothermic reactions in which temperature rise is controlled by the vaporisation of one of the reagents. Conditions when the liquid phase vapour pressure is significant we will term "hot sparged" or "boiling". These are radically different from the "cool" ambient environment usually used for most investigations of reactor hydrodynamics and mass transfer performance [9-11]. The vertical void faction distribution and liquid phase mixing time are two key parameters that can illustrate these differences. A start in the study of the dynamics of bubble saturation was made by Smith and Gao [11] who showed that small bubbles (in the order of 0.3mm diameter) are saturated within about

214 10ms. This implies that it is reasonable to generate very large sparging volumes by introducing inert gas into a heated, near boiling, liquid. The temperature of the liquid - which is at the equilibrium established by the balance between the heat supplied to the system and its removal as latent heat of evaporation- is used to calculate the vapour pressure of the liquid in the system and hence the total gas phase loading of the impeller. There is an implicit assumption that the introduced gas almost instantly becomes saturated with vapour at the operating temperature: we have been able to find no evidence, (e.g. from local temperature differences in the liquid), that this is not the case. 2. EXPERIMENTAL EQUIPMENT The void fraction experiments were carried out in an insulated vessel of 0.45m diameter and 1.1m high with four baffles and dished base, agitated by one or two Chemineer CD-6 hollow blade impellers of 0.177m diameter. With a single impeller in use the tank is filled with liquid to a depth equal to the tank diameter. With the dual impellers the fill height used is twice the tank diameter, as shown in Fig.1. The clearance between the bottom impeller and the vessel base is set to one third of the tank diameter. Four immersion heaters (each rated at 2.7 kW on full power) are mounted vertically in the base. Air is introduced into the vessel through a ring sparger (with twelve 3mm diameter holes). The vapour generated in the vessel is condensed in a coil heat exchanger and refluxed to the vessel. The drive motor is a specially wound AC motor driven by a variable frequency generator. The agitator shaft carries a Vibro Torque torsion transmitter giving outputs of torque, speed and shaft power. The liquid bulk temperature is measured using a high accuracy electronic thermometer.

Fig.1 Tank geometry and measurement points arrangement

215 With a single impeller and liquid depth equal to the tank diameter, local void fraction measurements were made with a point conductivity probe at 12 different positions in a vertical line, as shown in Fig.1. The measurement points were located midway between two baffles at a radial position 35mm away from the vessel wall. The spacing between the neighbouring points was 30mm. When the dual CD-6 impellers were used the liquid depth was twice the tank diameter: for this geometry the void distribution profile was measured at 27 locations in a vertical line, as shown in Fig.1. The rig used for the liquid phase mixing experiments is similar to that used by Smith and Katsanevakis [13] though now modified to allow much higher air sparge rates and with increased power ratings for the electric immersion heaters. The 80 litre vessel (0.45 m diameter and cylindrical side with a dished base and a flat lid just above the liquid surface) is heated by three candle form immersion heaters. Metered air is supplied through a ring sparger. The liquid height is 0.5m with a CD-6 impeller mounted at half depth of the liquid and driven by a motor with a variable frequency power supply. The mixing time has been successfully measured by the conductivity method using phosphoric acid as tracer. The pulse of tracer was added near the liquid surface at one side of the tank, while a specially designed conductivity probe was mounted on the other side of the vessel near the bottom of the tank. 3. RESULTS AND DISCUSSION

3.1. Void fraction in cold gassed systems The comparison of void fraction distributions for the single and dual impellers is shown in Fig.2. It is obvious that the distribution for single impeller (H--T) is similar to that for the lower impeller of dual impeller systems (H=2T) where the measuring positions are exactly the same for both single and dual impeller systems. This means that the hydrostatic and static pressure effects are of minor significance for cold gassed systems. The maxima in the void fraction distributions are just above the planes of the impellers. The Chemineer CD-6 is a typical hollow blade radial flow impeller, the main stream of bubbles is carried by the impeller discharge flow to the tank wall, while at the same time shifting upwards due to the buoyancy. This produces the maximum in the void fraction near the tank wall just above the impeller plane. 2 1.6 1.2 0.8 0.4 0 0

0.05

0.1 0.15 Void Fraction

0.2

0.25

Fig.2 Comparison of void distributions with single and dual CD-6 impellers

216

1.6 1.2 0.8 ~/=

0.4

t L

0

0.05

,f~~o.

-e- 2"CD-6, Cold gassed, N=4rps, Q=213.21/min -o- 2"CD-6, Cold gassed, N=4rps, Q=350.81/min

r

0.1 0.15 Void Fraction

!

0.2

0.25

Fig.3 Void fraction distributions in cold gassed systems, N=4rps

1.6 1.2 0.8 ! 0.4

0.05

O.1 O.15 Void Fraction

0.2

0.25

Fig.4 Void fraction distributions in cold gassed systems, N=5rps

The vertical void fraction distributions for dual CD-6 impellers in a cold gassed agitated vessel are shown in Fig. 3 and Fig. 4. Dual impellers have two maxima in the distribution, with the upper one showing a consistently higher gas fraction than the lower. As expected, local void fractions vary with both the stirrer speed and sparging rate: increasing with increases of either. Void fractions in the upper part of the tank are higher than hydrostatic pressure gradients would lead us to expect and imply that there is more gas recirculation above the upper impeller than above the lower one. The increase in gas fraction as the free surface is approached is probably an effect of secondary flow near the free surface.

217

1.6 1.2

t..

b.. N

0.8

L

04 0

,

~,

o

O.Ol

-o- 2"CD-6, Boiling,N=5rps, 4heaters, Q=4741/min -o- 2"CD-6, Boiling,N=5rps, 3heaters, Q=352.31/min _~_2"CD-6, Boiling,N=5rps, 2heaters, Q=230.51/min L_

0.02 0.03 Void Fraction

1

0.04

0.05

Fig. 5 Void distributions in boiling water

3.2. Void fraction in hot sparged and boiling systems The void fraction distributions in boiling liquid are shown in Fig. 5. The void fractions are almost zero below the level of the upper impeller. The void distribution in boiling systems is controlled by nucleation in the liquid phase and the response of vapour bubbles to flow and pressure fields. Although the superheated surface of heaters in dished bottom of the vessel generate some steam bubbles, hydrostatic pressure collapsed almost all of them once they left the heater surfaces. Hydrostatic forces also ensure that most of the nucleation and bubble development occurs near either the upper impeller or the free surface. Very few steam bubbles could be observed in the lower part of the vessel. Fig.5 also shows the effect of heat input on the void fraction distributions. It is quite clear that, as might be expected, the void fraction increases as the heat input increases. The effect of increasing the heating rate is therefore somewhat analogous to increasing the gassing rate in a sparged system.

1.6

1.2 N

0.8 0.4 9

0

0.05

0.1 0.15 Void Fraction

Cold gassed, Q=350.81/min Hot spaged, Q=3931/min Boiling, N=4rps, 3 Q=352.31/rnin

I

0.2

0.25

Fig.6 Void distributions, cold & hot sparged, boiling, N=4rps

218

Fig.7 Void distributions, cold & hot sparged, boiling, N=5rps The void fraction distribution comparisons between the sparged and boiling systems with dual CD-6 impellers are shown in Fig.6 and Fig.7. It is very clear that the hot sparged systems have similarly shaped void fraction distributions, with maxima in similar locations, to those of cold gassed systems. However, with the same total gas flow rate (i.e. including the contribution of the evaporation from the liquid phase), the local void fractions are always less in the hot sparged case than in cold conditions. In the hot sparged systems (the temperature is about 95~ water viscosity is about one third of that at room temperature. Calderbank [1] found that the bubble size increases with a decrease in liquid viscosity. This is consistent with our observation that the bubble size is larger in hot systems than that in cold ones, and also with the larger bubble rise velocities in hot systems [12]. These factors lead to bubble residence times being shorter in hot conditions and will correspondingly give lower void fractions at higher temperatures.

Fig.8 Mixing times in different operating systems

219 There is a very significant difference between the void distributions in boiling and sparged systems. The local void fraction for a boiling system is much lower than that for either cold o r hot sparged systems of similar gas phase throughput. As mentioned above, vapour bubbles are generated only in the upper part of the reactor, and almost all the recirculating bubbles collapse rapidly, leading to a very short residence time for the bubbles and hence very small void fraction.

3.3. Mixing time in sparged and boiling systems. Liquid phase mixing times in the ungassed, cold gassed, hot sparged and boiling systems are shown in Fig.8. It should be noted that the dissipation rate used here includes both shaft power and the contribution of the potential energy of the sparged gas bubbles. We see that the mixing times in ungassed, cold gassed and hot sparged systems are generally similar at the same power dissipation rate. However, the liquid phase mixing time in the boiling systems is approximately thirty percent lower than that in the other conditions with similar power dissipation rates. It is admitted that the contribution of vapour potential energy in the boiling case is impossible to estimate and has been neglected, though since vaporisation is occurring almost exclusively near the free surface, this energy input is likely to be small. Since small changes in local pressure or temperature will lead to evaporation or condensation with almost instantaneous expansion or collapse [9], this mechanism may provide extra energy that enhances the liquid mixing and lead to shorter mixing times. Fig.8 also shows that the mixing time is inversely proportional to the cube root of power dissipation rate, no matter what the operating regime is. This is consistent with other observations [13].

4. CONCLUSIONS In cold gassed systems, the distribution of void fraction is controlled mainly by the liquid flow field. There is a fairly uniform distribution of gas over the reactor height with distinct maxima in the void fraction just above the plane of these radial impellers. Hydrostatic and static pressure effects are generally of minor significance though with dual impellers of the two maxima in the distribution the upper one is consistently larger than the lower. Hot sparged systems have similar void fraction distributions, with maxima in similar locations, to those of cold gassed systems, though with the same total gas flow rate (i.e. including the contribution of the evaporation from the liquid phase), the local void fractions are always less in the hot sparged case than in cold conditions. Boiling systems have quite different void fraction distributions from the sparged systems whether in terms of value or distribution. The mean void fraction is much lower in a boiling system than in any sparged one with a similar gas phase output. Hydrostatic forces ensure that most of the nucleation and bubble development occurs near the upper impeller with only a few vapour bubbles in the lower part of the vessel, leading naturally to the maximum in the void fraction being near the free surface. The ungassed, cold gassed and hot sparged systems got the same liquid phase mixing time, but the boiling systems had a shorter mixing time under the same power dissipation rate. The mixing time is always inversely proportional to the cube root of power dissipation rate for different systems.

220 NOMENCLATURE H N Q T z

liquid height of the vessel, m impeller speed, s 1 gas flow rate, m 3 s 1 vessel diameter, m vertical distance from the base of vessel, m

REFERENCES 1. P.H. Calderbank, Trans IChemE, 36 (1958) 443. 2. A.W. Nienow, D.J. Wisdom and J.C. Middleton, Proc 2"4 European Conf on Mixing, (1977) FI-1. 3. Y. Nagase, H. Yasui, Chem Eng J, 27 (1983) 37. 4. M. Greaves and K.A. Kobbacy, Trans IChemE, 62 (1984) 3. 5. W.M. Lu, and S.J. Ju, 1986, Chem Eng J, 35 (1986) 9. 6. K. Takahashi, A.W. Nienow, Proc 7 th European Conf on Mixing, (1991) 285. 7. S.D. Vlaev, R. Mann and M. Valeva, Proc 8th European Conf on Mixing, (1994) 481. 8. A. Bombac, I. Zun, B. Filipic and M. Zumer, 1997, AIChE J, 43 (1997) 2921. 9. J.M. Smith and A.N. Katsanevakis, Trans IChemE, 71 (1993) 146. 10. J.M. Smith and C.A. Millington 1996, Trans IChemE, 74 (1996) 424. 11. J.M. Smith and Z. Gao, Proc 3rd international symposium on mixing in industrial processes, (1999) 189. 12. M. Jamialahmadi, C. Branch and H. Muller-Steinhagen, Trans IChemE, 72 (1994) 119. 13. S. Ruszkowski, Proc 8th European Conf on Mixing, (1994) 283. ACKNOWLEDGEMENT The authors are gratefully acknowledge the assistance provided by ICI Technology's Strategic Development Fund.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 .2000 Elsevier Science B. V. All rights reserved

221

A Numerical Investigation into the Influence of Mixing on Orthokinetic Agglomeration E.D. Hollander, J.J. Derksen a, O.S.L. Bruinsma b, G.M. van Rosmalen b, H.E.A. van den Akker ~ aKramers Laboratorium voor Fysische Technologic, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands bLaboratory for Process Equipment, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands

Abstract The scale-up behaviour of orthokinetic agglomeration in a 1, 10, 100 and 1000 liter vessel was investigated numerically, using constant Re-number, constant power input and constant impeller tip speed as scale rules. It was found that scale-up at constant ~ yielded the most constant results. At increasing volume, the influence of the macroscopic mixing time scale could be seen. In practice, laboratory scale experiments to determine agglomeration rate constants are carried out in stirred tanks of 1-10 liter. The results of this study indicate that this flow system exhibits too high turbulent intensities for agglomeration to occur readily. This is consistent with the experimental observation that small stirred tanks agglomerate badly, while large reactors do agglomerate. Laboratory scale stirred tanks will therefore be less suited for measuring these kinetic data.

1

Introduction

Orthokinetic agglomeration is a frequently encountered size enlargement mechanism in precipitating systems. Due to velocity gradients in the precipitating suspension, crystals may collide and subsequently grow together. Although the overall mass of the crystals is not altered by this mechanism, the total number of crystals is decreased. The Particle Size Distribution will therefore be altered drastically. Depending on the product properties required, agglomeration can both be beneficial and unwanted. In the manufacturing of pigments and dyes, for instance, the mono-dispersity of the material is essential. In this case, agglomeration should be avoided. In the production of aluminium hydroxide (a material with a very low growth rate), however, agglomeration can be used to increase the apparent growth rate of the material. The reactive nature of precipitation makes this form of crystallization sensitive for effects like micro- and macro-mixing. In addition to these effects, the agglomeration mechanism is also influenced by the distribution of velocity gradients in the reactor. If

222 one wants to understand orthokinetic agglomeration, all these effects have to be accounted for. Due to the non-linear nature of fluid flow, combined with the complex kinetics of orthokinetic agglomeration, scale-up behaviour of the agglomeration process will be very complex. For industrial purposes, however, predictions for reactor performance are essential. Usually, kinetic data are obtained in laboratory scale experiments. The standard scale-up rules for stirred tanks (e.g. scale-up at constant Reynolds number, constant power input or constant impeller tip speed), are then used to estimate industrial scale reactor performance. The simplifying assumptions introduced this way will inevitably lead to an unreliable designs for industrial scale reactors. In this research, a numerical study is presented on the scale-up behaviour of the agglomeration process. Reactor performances of a 1, 10, 100 and 1000 liter reactor scaled by the above-mentioned scale-up rules are simulated and compared. This way, better insight is gained on the coupling of mixing and agglomeration. In section 2, a theoretical description of the solution strategy is presented. Section 3 defines the numerical setup for this study. In section 4, the results of this study are given. Finally, section 5 is dedicated to some concluding remarks. 2

Theory

2.1 Lattice Boltzmann scheme and Large Eddy Simulation Usually, precipitation experiments are performed in baffled stirred tank reactors [1],[2]. The flow field in this type of reactors is notorious for it's complexity: the flow is 3dimensional, inherently time dependent and turbulent. To investigate the scale-up behaviour of agglomerators numerically, an efficient solver for fluid flow is needed. A computer code based on the lattice Boltzmann scheme is used for this purpose. Details on this code can be found in [3]. The scheme tracks the movement and collisions of particles that reside on a lattice. By defining appropriate collision rules and a sufficiently symmetric lattice, the particles will behave as an incompressible fluid [4],[5]. The continuity equation and the Navier-Stokes equations are solved this way. Industrial reactors are operated at very high Re-numbers (> 10~). A Direct Numerical Simulation of such a flow system is unfeasible. Therefore, some sort of turbulence model needs to be used to simulate the turbulent structures in the flow. In this research, Large Eddy Simulations (LES) were performed, using a Smagorinsky subgrid-scale model [6] to treat the impact of the unresolved flow scales on the resolved scales. A Smagorinsky constant cs of 0.12 was used. Although still computationally demanding, the LES turbulence model has a distinct advantage over Reynolds Averaged models like the k - c model. To understand the influence of fluid flow on particles, one has to be able to model the (time dependent) hydrodynamic environment of the particles as accurately as possible. While RANS models only give a description of the averaged flow, LES simulations do provide high resolution and time dependent information. 2.2 Agglomeration kinetics To be able to study the influence of fluid flow, flow depended agglomeration kinetics has to be provided. Experimental determination of these data has proved to be very

223

3" 10 -14 ~coll,O

.i"

112.10 -14 co

-

~ 1 . 1 0 -14

J 0

J !

I

5

10

15

20

Figure 1. Shear rate dependence of the agglomeration rate constant of Calcium Oxalate Monohydrate (solid line), according to Mumtaz [9]. The dashed line represents the collision rate, as formulated by von Smoluchowsky [11]. At increasing shear rate, it can be seen that the total number of collisions increases. However, the contact time between the primary particles decreases, while the viscous forces on the freshly formed agglomerate increase. These effects cause a decrease in the effective agglomeration rate constant at high shear rates.

difficult ([7], [8]). Mumtaz [9],[10] determined agglomeration kinetics for Calcium Oxalate Monohydrate numerically. Based on the model derived by Von Smoluchowski [11], agglomeration is coupled to shear rate. It is the authors' opinion that this model is the best available at the moment. The kinetics, depicted in fig.1 will therefore be used in this study.

2.3 Implementation of agglomeration Aside from the need to simulate fluid flow, a method for modelling the population of particles is needed. In [12], a Monte Carlo method is presented that can handle all events that occur in the precipitation process. Computational efforts for this method, however, are rather large. This is caused by the intrinsic statistical solution strategy, which implies the need for several Monte Carlo integrations every time step. On top of that, a fine grid for integrating population balances is needed ([13]). This makes the method difficult to handle at this point. In assuming that orthokinetic agglomeration is the only mechanism occurring, the integration can be simplified drastically since the analytical solution to this problem is known. For batch experiments, the solution is given by eq. 1.

0.%_ 0t

1

-- - - 2 ]~0m02 =:~ T/t0(t)--" 1

1

1

(1)

~ o t + mol,=o

with m0 the zeroth moment of the particle size distribution, and/30 the agglomeration rate constant. To link the shear rate needed for estimating/3o from fig.1 to the flow field

224 provided by the LES simulation, it is assumed that -~ can be set equal to V/~, with e the local energy dissipation rate as calculated in the LES simulations and v the kinematic viscosity of the fluid. This relation is similar to the one presented in [14]. Two strategies are adopted in integrating eq. 1 over the whole tank, being a particle tracking method, and a Eulerian technique.

2.4 Particle tracking strategy In precipitation, particle sizes of ~ 10 #m are encountered. The particle relaxation time for these sizes is in general much smaller than the smallest time scales of turbulence [15]. This implies that particles follow the fluid flow almost exactly. Particle tracking can be done without taking into account additional forces like drag or lift. In this research, particle trajectories are integrated using a 4th order adaptive stepsize Runge Kutta scheme [16]. A complication is, however, that the LES-computations (by definition) do not provide the subgrid scale velocities. Since particles are so small, they will feel these subgrid scale disturbances. To quantify the size of this effect, two particle tracking runs are done: one with no subgrid scale model and one based on a 'discrete eddy concept' model [17]. Since we are mainly interested in the dependence of agglomeration on fluid flow, gravity is excluded from particle tracking in this study. To incorporate agglomeration in particle tracking, an initial concentration is assigned to all particles. Based on the local deformation rate, ~0 is calculated. The concentration at the trajectory is calculated according to eq. 1, and subsequently transported according to the local fluid velocity. The model therefore assumes that a cluster of particles moves through the fluid like one particle would do. 2.5 Eulerian strategy For this method, a numerical grid was defined with an equal resolution to the fluid flow grid. To every cell, an initial concentration was assigned. Based on the local c, the agglomeration rate constant was computed. The new local concentration was computed with eq. 1. Based on the local resolved velocities, particle concentrations were transported. No subgrid scale model was incorporated in this method. 3

Numerical setup

Simulations are performed at a 120 • 123 • 123 grid for both the flow calculation and the integration of the population balances. Particle tracking is done with 5000 particles, initially distributed uniformly over the tank domain. This number was found to give adequate statistics for this flow problem. The physical dimensions of the various simulations done are given in table 1. As a base case (3=7=11), a 100 liter stirred tank with a standard Rushton turbine was used. The Re-number for this run was 2.104 . To cover the range from laboratory scale to industrial scale, reactors of 1, 10, 100 and 1000 liter are simulated. In runs 1-4, the Re-number is kept constant at 2 . 1 0 *, runs 5-8 are simulated at a constant power input of 2.39- 10 -~ m 2 s -3, and runs 9-12 at a constant impeller tip speed of 0.375ms -1. In tab. 1, it is assumed that the Po-number is equal to 5 for all simulations. For case 5, the Re-number is too low to justify this assumption. The error, however, is found to be small.

225 Table 1 Numerical setup. case volume

[m3]

1 2 3 4 5 6 7 8 9 I0 11 12

10 -3 10 -2 10 -1 1 10 -3 10 -2 10 -1 1 10 -3 10 -2 10 -I 1

Dimp

N

3.61.10 -2 7.78.10 -2 1.68.10 -1 3.61.10 -1 3.61.10 -2 7.78.10 -2 1.68.10 -1 3.61.10 -1 3.61.10 -2 7.78.10 - 2 1.68.10 -I 3.61.10 -1

1.53-101 3.30 7.11 .10 -1 1.53.10 -1 1.98 1.19 7.11 .10 -1 4.26.10 -1 3.30 1.53 7.11 .10 -I 3.3 -10 -1

[m]

Re

~

2.104 2.104 2.104 2.104 2.6.103 7.2.103 2.104 5.6.104 4.3.103 9.3.103 2.104 4.3-10 4

1.11 5.14-10 -2 2.39.10 -3 1.11 .10 -4 2.39.10 -3 2.39.10 -3 2.39 -10 -3 2.39.10 -3 1.11.10 -2 5.14-10 -3 2.39 -10 -3 1.11 .10 -3

[-]

8-3]

Ytip

[m

1.74 8.07.10 -1 3.75-10 -1 1.74.10 -1 2.25.10 -1 2.90.10 -1 3.75.10 -1 4.84.10 -1 3.75.10 -1 3.75 .10 -I 3.75 .10 -I 3.75.10 -1

All computations were started with a fully developed flow. The simulations were then run for 10 impeller revolutions. The simulation of a run typically took 2 days on two PIII/500 MHz processors.

4

4.1

Results Distribution in agglomeration rate constant

In fig. 2, a contour plot of/30 in a cross-section of base case 3 is shown. The right part of this figure shows that turbulence in the tank induces an erratic distribution of/30. The left side of Fig. 2 gives rio, averaged over several impeller revolutions. This figure shows that near the impeller region, agglomeration does not occur. Apparently, deformation rates are too high in this region. It can also be seen that, on average, the part of the reactor below the impeller has a higher/30 than the part above the reactor. This was found in several cases.

4.2

Inspection of predicted particle number decrease

In fig. 3, the total particle number decrease as a result of agglomeration for the three methods used is shown for the base case. It can be seen that the Eulerian technique and the particle tracking method without subgrid scale model yield virtually identical results, with an average difference of ~ 1% for all runs. For the integration of particle trajectories, a higher order scheme is used, than for particle displacement in the Eulerian method. This may cause the difference between these two methods. Comparing the Eulerian scheme and particle tracking with subgrid scale model shows a slightly higher average deviation of about 1.5%. This difference, caused by the fact that the latter does model unresolved turbulent fluctuations, is relatively small. Apparently, the subgrid scale motion does not have a large effect on the total particle displacement.

226

Figure 2. Typical average values of ~0 and an instantaneous distribution of ~o in an axial plane between baffles.

Figure 3. Tank averaged particle number concentration, normalized by the initial particle concentration, as a function of time, for the Eulerian method and the particle tracking (PT) methods used to compute agglomeration.

227 0.6

i

i

i

i

i"l

l r

'

i

I

'1 '

i

i

i

I

''" 1

0.5 I
.r,.4

0.4

i"~

"1

%

11/ / m

i

i

i

.

1/

-..~,

\%

\

/

\ %%

i I

a~

/

/

/

/

i

" - - .

%\%%

//

0.3

i

/

\ %

/

// o

0.2

/

/

Z 0.I 0

0.001

constant ~ I constant Re-numb_er _-.-~-_~ constant Vti p

-

s

s

!

!

|

|

0.01 Reactor Volume

|

|

lli

0.1

[m3]

.,

._i

|

|

i

i

||

1

Figure 4. Apparent reactor performance at various volumes. Different lines represent the ^ different scale-up rules used. /~0 is normalized with the maximum value of/30 giVen by fig. 1.

4.3 Scale-up rules Like in real-life experiments, the effective agglomeration constant ~0 can be estimated from the evolution of particle number concentration in time. This is achieved by fitting the tank averaged number concentration to the rate law in eq. 1. This ~0 represents the observed (reactor averaged) agglomeration rate constant, and is, similar to real-life experiments, a combination of the real kinetics and the influence of fluid flow. In fig. 4, ~0 is shown at various reactor volumes. From this graph, it can be seen that, depending on the scale rules used, reactor efficiency can alter drastically. The most constant reactor performance is found by scaling up at constant specific power input, ^ keeping the turbulent micro scales constant. There is, however, a slight decrease in ~0 at increasing reactor volume. A possible explanation can be found by inspecting fig. 2 more closely. For cases 5-8, the average ~0 was found to be higher in the bottom part of the reactor than in the top part. Consequently, the number decrease in this region will be higher. Due to the fact that agglomeration is second order in particle concentration, the reaction rate will decrease in the bottom part. This effect can be levelled by macroscopic transport of particles from the (more concentrated) top part of the reactor. This transport will become slower (the circulation time scales with N -1, which decreases in scaling-up) at increasing reactor volume resulting in a slight decrease of the observed ~0 at increasing reactor volume. Scale-up at constant impeller tip speed shows an optimum at the 100 liter reactor size. The spread in/~0 shows that this scaling rule is of little use for scale-up of agglomerators. Scale-up at constant Re-number shows a dramatic decrease in ~0 for small volumes.

228 The reason for this is the fact that the specific power input increases, which alters the (average) Kolmogorov scales of turbulence. At 100 liter, the Kolmogorov length scale s is 143 pm and the Kolmogorov 'shear rate' "Yk is 49 s -1. For 1 liter, s is 31 #m and "Yk is 1054 s -1. These conditions are too severe for agglomeration to occur. It is noteworthy to see that a 1 liter experiment at constant power input takes place at a Re-number of only 2600 in this study. In practise, experiments can never be done at such low Reynolds numbers, due to settling of the particles (recall that in the simulations, gravity is not incorporated). Usually, much higher Reynolds numbers are used at laboratory scale experiments. It can be seen from fig. 4, however, that at a Re-number of 20.104 (a typical value for laboratory scale experiments),/30 for a 1 liter reactor is almost zero. This is consistent with the well known observation that laboratory scale reactors usually do not agglomerate, while industrial scale reactors do. 5

C o n c l u s i o n s and r e c o m m e n d a t i o n s

Scale-up of orthokinetic agglomeration was investigated numerically, using LES based flow simulations. Using constant Re-number, constant specific power input and constant impeller tip speed as scaling rules, the apparent agglomeration rate constant /~0 was determined. It was found that, in the range of 1 to 1000 liter, scale-up at constant ~ gave the most constant/~0. At increasing volume, the agglomeration rate decreases, which is believed to be caused by the increase of the macroscopic mixing time. Scale-up at constant Re-number and constant impeller tip speed yielded a large spread in/30, and can therefore not be used as scaling rules for orthokinetic agglomeration. It was found that kinetic data at laboratory scale are usually performed at a too high Re-number. Results were consistent with the experimental observation that small scale experiments agglomerate less than industrial scale reactors. The main weakness of this research is the assumed agglomeration kinetics. The presented numerical technique can be used to construct an inverse technique to interpret kinetic data. By computing the flow field in the reactor and altering the kinetic model, a match can be found between numerical and experimental data. References

[1] M.L.J. van Leeuwen, C. Huizer, O.S.L. Bruinsma, M.J. Hounslow, and G.M. van Rosmalen. The influence of hydrodynamics on agglomeration of aluminum hydroxide: an experimental study. In International conference on mixing and crystallization, 1998. [2] A.S. Bramley, M.J. Hounslow, and R.L. Ryall. Aggregation during precipitation from solution, kinetics for calcium oxalate mono hydrate. Chem Eng Sci, 52:747-757, 1997. [3] J.J. Derksen and H.E.A. Van den Akker. Large eddy simulations on the flow driven by a rushton turbine. AIChE J., 45(2):209-221, 1999. [4] Daniel H. Rothman and St~phane Zaleski. Lattice-Gas Cellular Automata. Cambridge University Press, 1st. edition, 1997.

229 [5] J.G.M. Eggels and J.A. Somers. Numerical simulation of free convective flow using the lattice-boltzmann scheme. Int. J. Heat and Fluid Flow, 16:357-364, 1995. [6] J. Smagorinsky. General circulation experiments with the primitive equations: 1. the basic experiment. Mon. Weather Rev., 91:99-164, 1963. [7] H.S. Mumtaz, N.A. Seaton, and M.J. Hounslow. Towards a priori prediction of aggregation rates during crystallization. In Proceedings of the 1998 annual AIChE meeting, Miami, page ?, 1998. [8] E.D. Hollander, J.J. Derksen, O.S.L. Bruinsma, and H.E.A. Van den Akker. Measuring the effect of hydrodynamics on agglomeration. In Proceedings of the 1998 annual AIChE meeting, Miami, pages 156-161, 1998. [9] H.S. Mumtaz, M.J. Hounslow, N.A. Seaton, and W.R. Paterson. Orthokinetic aggregation during precipitation: A computational model for calcium oxalate monohydrate. Trans IChemE, 75:152-159, 1997. [10] M.J. et. al. Hounslow. A numerical study on the coupling of hydrodynamics and agglomeration. In Proceedings of the l~th International Symposium on Industrial Crystallization, Cambridge, page ?, 1999. [11] M. von Smoluchowski. Versuch einer mathematischen theorie der koagulationskinetik koloider 15sungen. Z. Phys. Chem., 92:156, 1917. [12] K. Rajamani, W.T. Pate, and D.J. Kinneberg. Time-driven and event-driven monte carlo simulations of liquid-liquid dispersions: A comparison. Ind. Eng. Chem. Fundam., 25:746-752, 1986. [13] E.D. Hollander, J.J. Derksen, O.S.L. Bruinsma, and H.E.A. Van den Akker. A numerical study on the coupling of hydrodynamics and agglomeration. In Proceedings of the l~th International Symposium on Industrial Crystallization, Cambridge, page ?, 1999. [14] P.G. Saffman and J.S. Turner. On the collision of drops in turbulent clouds. J. Fluid. Mech., 1:16-30, 1956. [15] R. Clift, J. R. Grace, and M. E. Weber. Bubbles, drops and particles. Academic Press, London, 1978. [16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipies in C. Cambridge University Press, 2nd. edition, 1992. [17] M. Sommerfeld, G. Kohnen, and M. Riiger. Some open questions and inconsistencies of lagrangian particle dispersion models. In Ninth symposium on 'Turbulent shear flow', Kyoto, Japan, pages 15.1.1-15.1.6, 1993.

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I0 th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

231

An experimental method for obtaining particle impact frequencies and velocities on impeller blades K.C. Kee a & C.D. Rielly b a Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA b Department of Chemical Engineering, Loughborough University, Loughborough, Leics. LE11 3TU

A method is proposed for the in-situ measurement of particle-blade impact velocities and frequencies in a stirred vessel, using a thin coating of plasticine to record craters formed by particle impacts. A model for the plastic deformation of the target material is described and an iterative method for calculating impact velocities from the dimensions of the craters is proposed. For a Rushton impeller blade, the impact velocities are close to the linear speed of the blade at the impact radius; the largest velocities therefore occur near the blade tip. The highest impact rates are also at the blade tip and below the disc of the Rushton turbine. In contrast, for a downward pumping 45 ~ pitched blade turbine, the most frequent impacts occur at small radii, on the lower half of the blade, where the lowest impact velocities are measured. 1. INTRODUCTION The impact of particles on impeller blades is a common feature of solid-liquid mixing processes. Fragile particles may undergo breakup or abrasion by collision with the blade; with hard particles, the blade itself may be eroded. In crystallization, secondary nucleation sites can be formed by attrition of asperities from parent crystals on impact with a blade. This hydrodynamic phenomenon has a significant effect on the nucleation rate and the crystal size distribution, but often simplistic assumptions are made about the particle-impeller collision dynamics, e.g. Synowiec et al. (1993) and Mazzarotta (1992) assumed the impact energy was proportional to the square of the impeller tip speed, regardless of crystal size and impact position. In solid-liquid abrasion, it has long been recognised that the flow field affects the particle impact velocities (Humphrey, 1990). Laitone (1979) and Kee & Rielly (1997) showed that the true impact velocity and angle always differs from the initial velocity and angle of a particle as it approaches a surface. This is an important phenomenon, since both the erosion rate and its mechanism depend on the impact energy and the impact angle (Hutchings, 1987). Despite the importance of this phenomenon, information on the distributions of impact position and velocity in mixing tanks is limited. Nienow (1976) examined the effect of crystal-impeller collisions on the rate of secondary nucleation by using an evenly covered grease coating which entrapped the particles. Takahashi (1992) and He et al. (1995) used a wax coating to record particle impact positions on Rushton and pitched blades. Contrary to Nienow (1976), they found the highest impact frequencies at the outer lower edges of the Rushton blades; for down-pumping turbines, the outer, upper edges of the blades suffered the highest frequency impacts. The differences were attributed to (i) transient effects; and (ii) differences in the hardness of the coated layers. In the present work, an improved coating technique has been used to obtain the distribution of impact positions on Rushton and downpumping, 45 ~ pitched blades in a mixing tank. The method is extended, so that the particle impact velocity may be estimated from an analysis of the crater size formed upon impact. 2.

EXPERIMENTAL APPARATUS AND METHODS

2.1

The Mixing Tank

The apparatus used was a flat-bottomed, cylindrical mixing tank made of acrylic perspex, with an internal diameter of T = 28.5 cm and equipped with 4 equally spaced baffles of width T/10;

232 the liquid height of H = 1.05T was constant in all experiments. The tank was fitted with a lid covering the water surface, to allow operation at impeller speeds in excess of the critical speed for surface aeration. Two single impeller types were used, namely a 6-bladed Rushton turbine and a 6-bladed 45~ downward-pumping turbine, positioned at a bottom clearance of C = 0.42T. The diameters of both the turbines were D = 0.70T. The width and length of the Rushton blades were W = 0.2D and L = 0.25D respectively and the diameter of the disc was 0.75D. The projected width and length of the 45~ downward pumping blades were W = 0.14D and L = 0.4D, respectively. The D/T ratio is larger than would normally be used, but was deliberately chosen to develop the crater analysis technique: large D / T ratios allow higher radial velocities of the flow to be obtained for both the Rushton (Cooper & Wolf, 1968) and the 45~ downward pumping blades (Mao et aI., 1997). Monosized glass ballotini particles were used with, diameters of 0.93, 1.75, 2.75 and 3.88 mm and a density of 2400 kg/m ~ A low particle mass fraction of 0.02 % was chosen to reduce the number of particle-particle interactions and to give a reasonable number of nonoverlapping impact craters. The ranges of impeller speeds for the Rushton turbine (150, 160 and 210 rpm) and the 45~ downward pumping turbine (275, 360 and 440 rpm) were chosen to be above the just-suspended speeds, Nj~s, for each system, based on Zwietering's (1958) criterion. The observed values of Nis were m good agreement with the correlations of Nienow (1985) and Raghava Rao et al. (1988).

2.2

The plasticine coating method

Initially, Takahashi et al.'s (1992) crayon method was used, but it was found that the hardness of the wax coating was not uniform and was very sensitive to the final solvent content. For the experiments reported here, plasticine was chosen as a suitable coating material. Removable sleeves were made from thin brass sheet, formed to fit tightly around individual impeller blades. Plasticine was sheeted out to a uniform thickness of 2 mm and was pressed onto the brass sleeve, to leave a visibly smooth surface. At the end of the experiments, the sleeves were removed from the blades and the impact distributions and crater profiles were analysed. The tank was operated at an elevated temperature to reduce the hardness of the plasticine and increase the sensitivity of the method for low energy impacts. An electrical immersion heater controlled the tank water temperature to within _+0.5~ Preliminary runs gave consistent numbers of impact marks for water temperatures above 40~ which was chosen as the operating temperature for subsequent experiments. Test showed that less energy was required to make an indentation on the heated plasticine, compared to the crayon coatings.

2.3

Experimental procedure

The contents of the tank were heated to 400(2 and the impeller was set to the required speed and turned off. The solid particles were then loaded into the tank and the agitator was restarted at the pre-set speed; the liquid flow starts up rapidly from rest and the particles become suspended. The impact distribution on the impeller blade for a specific particle size and impeller speed was obtained by averaging the distributions on 12 individual blades (three coated blades were obtained from 4 runs with the same experiment conditions). Similar to Takahashi et at.'s (1992) method, the impacts were analysed by dividing the blade area into 5x5 rectangular regions of equal size (for the Rushton blade, the 3 central regions are occupied by the impeller disc), as shown in Fig. 1. The numbers of impact marks in each region, ni, were obtained from an enlarged view of the image using the software NIH-Image. Takahashi et al. (1992) postulated that the start-up transient period has a significant effect on the impact distribution and in turn affects the impact rate. However, in the current study, similar local impact rates were obtained from two experiments, one of t = 30 s duration and another of t = 60 s duration, at each set of the current experimental conditions (Kee, 1999). As a result, the impact rate for each section of blade was simply defined as:

R~=n~/t

(1)

233 For the impact velocity analysis, the blades were also divided into a 5x5 array of rectangular regions; profiles of 4 selected craters in each rectangular region were measured using an enlarged view obtained from a CCD camera. Fig. 1 (a) Distribution of craters on a Rushton blade; Figure 1 shows a typical image of the plasticine (b) magnified view of a single crater coating from a Rushton blade and a magnified view of one crater. The crater shape resembles an ellipse. The impact direction can be determined from the direction of the major axis. In Figure 1(b), the particle travelled diagonally downward from left to right; the particle starts to rebound when the crater width, dc, is greatest. The crater length before, di, and after the rebound point, d r, were also measured and used to estimate the particle impact velocity and angle, as described below. _

Fig.2 Schematic diagram of a rigid particle impacting on a plastic surface: (a) initial impact stage; (b) final impact stage 3. NUMERICAL M O D E L USED TO OBTAIN IMPACT VELOCITIES Rickerby & MacMillan (1980) proposed a model to calculate the impact crater size formed by a particle with a known initial velocity vector. Here the method is to be used in reverse, i.e. to calculate the particle impact velocity, from measurements of the crater dimensions. The forward calculation is described first; full details are presented by Kee (1999). The model assumes that a rigid spherical particle impacts on a flat plastically deformable surface. Rotation of the particle prior to, and during, impact is ignored and the accumulation of deformed material ahead of the particle is neglected. As shown in Fig. 2, a particle with an initial velocity v0 = (v,~, Vyo)contacts the flat surface at point O, which serves as an origin for the Cartesian co-ordinate system. Initially, the particle is in contact with the entire surface of the crater, as shown in Fig. 2 (a). A normal force P and a shear force/1/' act on the projected area of contact AD, which is a circle of radius, b. , The friction coefficient/1 is assumed to remain constant. Thus, the instantaneous particle velocity, v = (vx, Vy), is given by

where

mdVx= dvy dt -/zP and m =P dt

(2)

P = ~b aH = rc(R 2 - y p2 )H ,

(3)

m is the mass of the impacting sphere and H is the dynamic hardness. The initial stage of

234 impact ends when the rear of the particle detaches from point D; see Fig. 2(b). In the final stage, the contact area between the particle and the crater is given by the arc AB in the twodimensional view of Fig. 2(b). The projection of the actual contact area onto the AB-plane coincides with the projection of area BC (A1) and area AC (A2) onto the same plane. It is relatively easy to determine these areas (Kee, 1999), so that the equations of motion become:

dvx

m~=-Psin

dt

where

dv y - / z P c o s y and m - y = P cos ?'- / ~ sin ?'

dt

P = (A1sin fl + A2 cos ~')H

(4) (5)

The trajectory of the centre of the particle of the particle is described by

dxp _

vx and

dyp

= Vy (6) dt dt In the forward calculation for a known initial velocity, eqs.(2) - (6) are integrated -

numerically through time, using a fifth-order Runge-Kutta routine with adaptive step size (Press et al., 1986). The calculation is stopped when: (i) the particle comes to rest within the material, or (ii) the particle has completed its rebound and has left the crater. The reverse problem, of finding the impact velocity of a particle from the size of the crater formed, was solved iteratively. As described previously, three measurements of the crater size (dj, dr and de) were obtained from image analysis, which could be compared with their predicted values (d/, dr and d/) from the model for a given initial velocity v0 = (v~0, Vy0). The convergence criterion for the iterative calculation of the impact velocity was chosen as

[(1-d-~c,/2+(1

dcJ

d! )2+ l (1- dr l211/2 0.05 7 2-rJ <

(7)

The error in the rebound distance had reduced weighting, because of the neglect of the accumulation of material at the rebound end of the crater. For each new iteration (n+l) the impact velocity components from the previous iterations were updated using

,(n+1)_._ (n)(3 xO

where

di -!-1dr l

Vx~ ~'4--~ii -4 ~r )

, (n+l) 9(n)(hmax

and

Vy0 =-Y0Lh~ax j

(8)

d'c =2~2Rhmax-h~a x (9) Takahashi et al. (1996) formulated a similar theory for normal impact of particles on

impeller blades, assuming plastic deformation. Their analytical solution is simply a special case of the current model and was used to verify the numerical solution method. However, as Laitone (1979) and Kee & Rielly (1997) demonstrated, particle impacts rarely occur normal to the plate. 4. CALIBRATION OF THE DYNAMIC HARDNESS In a dynamic impact process, the material exhibits a greater value of hardness than obtained from static tests (about twice the Vickers hardness value), due to the higher strain rates imposed. It is common to obtain the dynamic hardness of a material by fitting H to the results of impact experiments (Hutchings et al.,1976). The calibration method used here has been described by (Kee, 1999). At a water temperature of 40 ~ the calibrated dynamic hardness was 0.46+0.06 MPa for all of the impacts velocities. A coefficient of friction of 0.2 was also fitted to the experimental data from these calibration tests.

235

Figure 4. Percentage impact distribution on 45 ~ pitched blades (0.02 wt %) 5. RESULTS AND DISCUSSION 5.1 The distribution of impact positions Figures 3 and 4 show the percentage of impacts occurring on each of the 5x5 rectangular sections for the Rushton and 45 ~ pitched turbine blades, respectively, at different impeller speeds and for two particle sizes. The most darkly shaded areas represent the highest local impact frequencies for that blade; the left-hand side of each area is at the inner edge of the impeller blade, closest to the shaft. For the Rushton blades, the results shown in Figure 3 indicate that the lower regions, closest to the blade tip, were subject to the highest impact frequencies in all cases, which agrees with the findings of Takahashi et al. In contrast, however, Nienow (1976) found that most impacts were recorded at the inner edge of the Rushton blade, with more above the disc than below. This is surprising, since, unless N >> Njs there should be a lower solids concentration above the disc than below. Takahashi et al (1992) suspected that the grease coatings used in Nienow's experiments allowed the particles

236 near the outer regions to "skid", rather than being captured. Figure 3 also shows that the percentage of particle impacts in each region of the Rushton blades, became more uniform with increasing impeller speed, caused by the more homogeneous suspension of the particles in the tank; the percentage of impacts above the disc increased, whereas the percentage below decreased, with increasing impeller speed. Furthermore, the percentage of impacts in the radial direction also became more uniform, with increasing impeller speed. Takahashi et al (1992) noted that the impact distribution on the Rushton blades tended to be more uniform for smaller particles, as is also shown in Fig. 3. For the 45 ~ pitched blade turbine, the highest percentage impacts occur at the inner, lower regions of the blades, close to the shaft (see Fig. 4). In contrast, He et al. (1994, 1995) found the greatest number of impacts at the upper, outer regions of a down pumping 45 ~ pitched blade. The crayon method employed by Takahashi and his co-workers is claimed to detect particles with an impact energy greater than 6 x 10-7 J. Supposing that the maximum particle velocity is approximately equal to the linear speed of the impeller, then the maximum impact energy for a particle colliding with the blade at radius ri would be Emax = ~1 m (2/r.Nr/)2

(10) For the conditions used by He et al. (1994, 1995), eq.(10) indicates that no impacts would be detected on a crayon coating on the inner 25% of their down pumping blades. Later in this paper, it is show that the impact velocities at the lower inner regions are about half of those at the outer upper regions and so it is possible that the crayon method would not be able to record fully the particles hitting the inner regions of the pitched blade impeller.

5.2

The overall rate of particle impact

These experiments were conducted for a constant mass fraction of particles; consequently, the total number of particles in the tank, rip, depends on the particle diameter. To compensate for this effect, Figure 5 shows the total impact rates for each blade, normalised by the number of particles in the tank (i.e. Rp = ]~Ri / np), for each particle size. As would be expected, the overall impact rate increases as the impeller speed is increased. By comparison, the frequency with which fluid elements are pumped though the impeller 0.12 -*- 0.93 mm region is of order Fl N D O / V. '~ -z~- 1.75 mm • Assuming a flow number, ~ . 0.10 - I - 2.75 mm .-''''"'" Fl = 0.75 for a Rushton turbine d "6 0.08 ->~- 3.88 mm (Revill, 1982), then for this range of impeller speeds, the liquid passes through the "~ 0.06 -" impeller region with a frequency of between 0.7 and 1.1 s-~. Referring to Figure 5, the probability of a particle t~ .......... 41, impacting on the six Rushton blades during each pass is ,., 0.00 . . . . ' . . . . . . . . . ' ................... therefore between 10% and 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 60%, depending on the particle Impeller speed, N / rps size. Particles with greater inertia (larger Stokes numbers) Figure 5 The impact rate per particle for a single Rushton are more likely to hit the blades. blade for the four particle diameters. .

5.3

.

.

.

I

"

'

'1'

I

1'

"

'

'

I

.

.

.

.

I

.

.

.

.

I

.

.

.

.

I

.

.

.

.

Impact velocity

Figures 6 and 7 show the measured impact velocities of the particles on the blades, for both the Rushton and 45" pitched blade impellers, respectively; the number of arrows shown in each region is proportional to the frequency of impacts. The reference vectors above the

237

,

9

,

,,

v

N = 140 rpm, d = 2.8 mm T 4

9

~

N = 160 rpm, dp - 2.8 mm

N - 210 rpm, d - 2.8 mm

.~.

...

.,,,.

N = 140 rpm, d = 3.9 mm

..,,,.

N = 160 rpm, d = 3.9 mm P

N = 210 rpm, d = 3.9 mm P

Figure 6. Particle impact velocities on the Rushton impeller blades. , '--,,

j,

v

~

~

"X ~

----~

,,.

....

'~

i ~ ._.,..%

~

.~,,,

f

N = 275 rpm, d = 2.8 mm P

N = 360 rpm, d = 2.8 mm P

N = 440 rpm, d = 2.8 mm \

I

~

I

\

f ,

_

N = 275 rpm, d = 3.9 mm P

---.,--

---.--~-

..-..P

~ .

~

N = 360 rpm, d = 3.9 mm P

i

,,~

--..

9

'\

N=440rpm, d = 3.9mm P

Figure 7. Particle impact velocities on the 45 ~ pitched impeller blades. blades represent the linear velocity of the impeller at the centre of each region. For the Rushton blades the impact velocities generally increase by a factor of about 1.5, in the radial direction, from the inner edge of the blade to the outer tip. The particle impact directions are predominantly radially outward, with a downward axial component near the lower edge of the blade. This is consistent with the observed liquid velocity fields around the lower and outer blade edges of a Rushton turbine, measured in the reference frame of the rotating impeller (e.g. Stoots & Calabrese, 1995). The impact velocities at the inner edge of the blade are slightly higher than the blade speed (2~2Vri), but at the outer edge the particle impact and blade velocities are comparable. This indicates that, apart from the tangential component, other velocity components also have significant effects on the particle impact velocities near the blade inner edges. Furthermore, smaller particles are more easily accelerated due to their smaller Stokes number and therefore their impact velocities are generally greater than those of the larger particles. The 45 ~ pitched blades are longer than the Rushton blades, and the impact velocities at the outer tips are about twice those at the inner edge; see Fig. 7. The particle impact velocities on the pitched blades are significantly higher than the blade linear speed at the inner edge, but slightly lower than 21rJVri at the outer edge. Again, particles with lower Stokes numbers impact at higher velocities.

238 6. CONCLUSIONS A novel method has been developed to measure the distribution of particle impact positions on the impeller blades in a stirred tank. With the help of a theoretical analysis of the size of impact craters, the technique also allows the particle impact velocities to be obtained. The results indicate that the combined effects of a high impact frequency and a high impact velocity, subject the lower outer edge of the Rushton blades to high erosion rates. In contrast, although the impact concentration is highest at the lower, inner regions of 45 ~ pitched blades, the impact velocity is comparatively low; for the outer edge, the high impact velocity alone may cause the blades to undergo severe erosion. Indeed, as shown by various workers (Weetman, 1994; He et al., 1995), the outer edge of the pitched blade suffers the highest erosion r a t e - this implies that the impact velocity has a relatively more important role in erosion processes than other factors such as the impact frequency. REFERENCES Cooper, R.G. & Wolf, D. (1968), "Velocity profiles and pumping capacities for turbine type impellers", Can J Chem Eng, 46, 94-100. He, Y., Nomura, T., Takahashi, K. & Nakano, Y. (1994), "Effect of particle-impeller impact rate on crystal size distribution", Eighth Eur ConfMixing, Cambridge, IChemE Symp Ser 136, 383-390. He, Y., Takahashi, K. & Nomura,T. (1995), "Particle-impeller impact for a six-bladed 45 pitched turbine in an agitated vessel", J Chem Eng Jap, 28(6), 786-789. Humphrey, J.A.C. (1990), "Fundamentals of fluid motion in erosion by solid particle impact, lnt J Heat and Fluid Flow", 11(3), 170-195 Hutchings, I.M. (1987), "Wear by particulates", Chem Eng Sci, 42(4), 869-878. Hutchings, I.M., Winter, R.E. & Field, J.E., (1976), "Solid particle erosion of metals: the removal of surface material by spherical projectiles", Proc R Soc London, A 348, 379-392. Kee, K.C. & Rielly, C.D. (1997), "The impact of particles on flat blades", 9th Eur Conf Mixing, Paris, Rdcents Progrbs en Gdnie des Procddds, Vol 11, no 52 (1997), 57 - 64, ISBN 2-910239-25-X Kee, K.C. (1999), "Particle impacts on impeller blades", Ph.D. Thesis, Cambridge University. Laitone, J.A. (1979), "Aerodynamic effects in the erosion process", Wear, 56, 239-246. Mao, D., Feng, L., Wang, K. & Li, Y. (1997), "The mean flow field generated by a pitched blade turbine", Can J Chem Eng, 75, pp.307-316. Mazzarotta, B. (1992), "Abrasion and breakage phenomena in agitated crystal suspensions", Chem Eng Sci, 47(12), 3105-3111. Nienow, A.W. (1985), "The suspension of solid particles", (Ch. 16, Mixing in the process industries, ed., Harnby N., Edwards M.F. & Nienow, A.W.), Butterworths, London, 2nd Edition, pp.297-321. Nienow, A.W. (1976), "The effect of agitation and scale-up on crystal growth rates and on secondary nucleation", Trans IChemE, 54, 205-207. Press, W.H., Flannery, B.P., Teukolsky, S.A., & Vetterling, W.T. (1986), "Numerical recipes. The art of scientific computing", Cambridge University Press. Raghava Rao, K.S.M.S., Rewatkar, V.B. & Joshi, J.B., (1988), "Critical impeller speed for solid suspension in mechanically agitated contactors", AIChE J., 34, 1332-1340. Revill, (1982), Paper R1, Fourth Eur. Conf. Mixing, BHRA Fluid Eng, Cranfield, UK Rickerby, D.G. & MacMillan, N.H. (1980), "On the oblique impact of a rigid sphere against a rigidplastic solid", Int J Mech Sci, 22, 491-494. Stoots, C.M. & Calabrese, R.V. (1995), "Mean velocity relative to Rushton blade", AIChEJ, 41, 1-11. Synowiec, P., Jones, A.G. & Ayazi-Shamlou, P. (1993), "Crystal breakup in dilute turbulently agitated suspensions", Chem Eng Sci, 48(20), 3485-3495. Takahashi, K., Gidoh, Y., Yokota, T. & Nomura, T. (1992), "Particle-impeller impact in an agitated vessel equipped with a Rushton turbine", J Chem Eng Jap, 25(1), 73-77. Takahashi, K., Nakanko,Y. & Hokkirigaw, K. (1996), "Relationship between impact velocity of particle and size of crater formed on soft surface", J Chem Eng Jap, 29(4), 733-736. Weetman, R., (1994), "Development of an erosion resistant mixing impeller for large scale solid suspension application with CFD comparisons", IChemE Syrup Ser, 136, 49-56. Zwietering, Th. N. (1958), "Suspending solid particles in a liquid", Chem Eng Sci, 8, 244-253.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

239

Comparison between Direct Numerical Simulation and k-e Prediction of the Flow in a Vessel Stirred by a Rushton Turbine C. Bartels, M. Breuer, and F. Durst Institute of Fluid Mechanics, University of Erlangen-Nuremberg, Cauerstr. 4, 91058 Edangen, Germany The flow in a baffled tank reactor stirred by a Rushton turbine was simulated at a Reynolds number of 7275. Two different approaches were used for the turbulent flow predictions, a direct numerical simulation and a simulation with the standard k-e model. The flow patterns, the mean flow quantities, and the turbulent kinetic energy predicted by both methods were compared and showed a good overall agreement. Nevertheless, subtle differences were found in parts of the general flow field, e.g. the discharge flow of the impeller and in the distribution of turbulent kinetic energy. For most key features the DNS results are closer to experimental findings than the k-e predictions. 1 INTRODUCTION The k-e model is the most prevalent turbulence model in the numerical simulation of stirred vessel flows. Despite the fact that many other turbulence models (like the k-~ model, the RNGk-e model) have been introduced to improve the k-e model, it has remained the most frequently utilized model for routine investigations in industry. The reason for this predominance is that the k-e model has been quite successfully applied to many complex flow problems. Moreover, the weaknesses of the model are relatively well-known (e.g. swirling flows or recirculation zones) and can be accounted for. Unfortunately, both flow patterns mentioned above as shortcomings of the k-e model are present in the flow induced by a Rushton turbine. Swirling flow is present in the trailing vortices of the blades and recirculation zones exist behind the baffles and the blades. Therefore, it is not surprising, that flow predictions of turbulent flows in stirred vessels using the k-e model yield incorrect results. In comparison with experiments the following differences are often observed: a lower value of the turbulent kinetic energy in the discharge flow field of the impeller just outside of the impeller, the incorrect position of the two trailing vortices with respect to each other, and the radial position of the two large scale vortices. For these reasons the k-e model is often considered to be inadequate for the prediction of stirred vessel flows, despite of its widespread application. The above mentioned deficiencies of the k-e model have been attributed to several reasons, namely the assumption of local isotropy, the simplifications used in the derivation of the model transport equations for k and e, the determination of the model constants from simple flow situations (e.g. decaying grid turbulence) not comparable to stirred

240 vessel flows, and the use of wall functions in the formulation of boundary conditions. Up to date, no investigation has ever attempted to determine those of the before mentioned reasons that are responsible for the failure of the k-e model. In the present paper a simulation with a standard k-e model using the model constants by Launder and Sharma [1] is compared to a direct numerical simulation (DNS) at a Reynolds number of Re = N D 2 / v = 7275. The Reynolds number was chosen such that it was accessible to direct numerical simulation and at the same time the condition for the application of wall functions (y+ > 30) was not severely violated by the simulation with k-e model. The aim of this comparison is twofold. On one hand we would like to identify flow regions in a baffled vessel stirred by a Rushton turbine where a DNS simulation will improve the accuracy of a flow prediction. On the other hand such a comparison could indicate which terms of the transport equations for k and e are not correctly modeled by the standard k-e model. The material of the paper will be presented in the following order: First, we describe the geometry of the stirred vessel considered and give details of the used simulation procedures. Next, we compare the large scale flow field and the impeller flow field. Finally, we compare the distribution of the turbulent kinetic energy k for both kinds of simulation. 2 GEOMETRY AND SIMULATION PROCEDURE The geometry considered in the calculation, a baffled vessel stirred by a Rushton turbine, is described in the following. A closed, cylindrical vessel of diameter/height T = H = 0.152 m, equipped with four baffles of width b = 0.015 m ,,~ T / I O and a clearance between baffles and tank wall of s = 0.0026 m, is stirred by a Rushton turbine of the following dimensions: diameter/clearance D = h = 0.05 m ~ T/3, blade width/length w = D / 5 , W = D/4. The geometry is similar to the configuration experimentally investigated by Schiifer et al. [2] except that the disc and the blades were infinitely thin. Contrary to the experiments reported in [2], silicon oil was assumed as working fluid (density Q = 1039 kg/m 3 and dynamic viscosity 77= 0.01591Pa.s). The impeller rotated at n - 44.563 s -1 (= 2673.8 rpm), resulting in the above mentioned Reynolds number of Re = ~)nD 2/~7 = 7275 and a tip velocity U t i p = 7rnD = 7 mls. In the present paper, two different approaches have been utilized to simulate the effect of turbulent fluctuations in the working fluid: a direct numerical simulation resolving all turbulent length and time scales and a simulation using a standard k-e turbulence model. The basic transport equations describing the fluid flow in a stirred vessel are similar for both approaches. These are the conservation equations for mass and momentum of an incompressible and isothermal fluid (Navier-Stokes equations). In the DNS, the time-dependent Navier-Stokes equations are directly integrated using a very fine grid whereas for the simulations with turbulence model the Reynolds-Averaged form of the equations (RANS) including a standard k-e model are solved. In both approaches, the time-dependent geometry of the stirred vessel was simulated by a multiple frame of reference approach first proposed by Harvey et al. [3] which neglects the relative motion of impeller and baffles. Nevertheless, in a flow domain around the impeller the equations are solved in a rotating frame of reference whereas in the remaining domain the equations are integrated in a frame of reference at rest. Wechsler et al. [4] compared the multiple frame of reference approach with a fully time-dependent unsteady computation and found encouraging agreement for the flow induced by a pitched blade turbine in a simulation with turbulence model. For the DNS it is necessary in general to use a clicking/sliding mesh

241 technique for the coupling of rotating and fixed frame of reference. The multiple frame of reference approach can yield similar results to the most general approach only if the influence of the stirrer and the baffles in azimuthal direction is no longer significant at the interface. The computational domains in the DNS (and the RANS simulation) have been chosen to minimize this effect. The numerical procedure to solve the incompressible Navier-Stokes equations formulated in curvilinear coordinates is composed of the following components' second-order finite-volume discretization in time and space (central differences, deferred correction for the convective fluxes), collocated grid arrangement (momentum interpolation of Rhie and Chow [5]), pressure velocity coupling by means of the SIMPLE algorithm [6], the ILU approach of Stone as linear system solver, time integration by a second-order Crank-Nicolson scheme, structured multiblock grids. As mentioned above, in contrast to experiments no finite thickness of blades and disc was taken into account in the simulations. Moreover, only one half of the vessel was simulated exploiting the two-fold periodicity of the solution domain, thus significantly reducing the requirements on computational power and main memory. The underlying geometry is described by a block-structured grid of 15 blocks. The region computed in a rotating frame of reference is of cylindrical shape. It is confined by a radial distance of r = 0.046 m (~ 0.303 T) and is located between the axial positions z = 0.035 m (,,~ 0.23 T) and z = 0.065 m (~ 0.427 T). Although this region covers only 7.2% of the vessel volume, 33.4% (for DNS, for simulation with k-e model" 36.3 %) of the grid points were located in this region in order to resolve the trailing vortices which emerge from the blade tips of the Rushton turbine. On the grid used, each surface of the blade is covered by 24 • 32 control volumes. One half of the vessel is discretized by 136 (for DNS, for simulation with k-e model: 145) control volumes in the azimuthal direction. The total number of control volumes for the DNS was 1,901,824 and for the simulation with k-c model it was 2,082,816. 3 RESULTS The results of the DNS and the simulation with k-e model are depicted in Figs. 1 through 5. Each of the figures consists of three separate subfigures. The first (i.e. either the leftmost or the topmost) figure contains always the result of the prediction with k-e turbulence model. The next two figures depict the results of the DNS prediction of which the first shows time-averaged results and the second shows an instantaneous snapshot. The discussion of results will be started with the large scale flow field in the mid-baffle plane, see Fig. 1. In all three subfigures the two ring vortices driven by the radially propelled fluid are visible. It is also obvious that the lower vortex reaches the bottom of the vessel in all subfigures whereas the upper ring vortex does not extend up to the lid of the vessel. A closer comparison of the time-averaged DNS results and the simulation with k-e model reveals slight differences: The position of the two ring vortices is shifted in outward direction for the simulation with k-e model (RANS calculation). Whereas the centers for the DNS simulation are at R / T = 0.355/0.37 for the upper/lower vortex, respectively, the centers for the simulation with k-e model are at 0.38 and 0.395. In the DNS the upper ring vortex does not extend up to the same height as in the RANS calculation. In the DNS, the upper limit of the vortex is 0.71 and in the RANS calculation it is about 0.75.

242 The flow field in the vicinity of the agitator blades is depicted in Fig. 2 for a plane 5 ~ behind the blades and in Fig. 3 for a plane 15 ~ behind the blades. The flow pattern in the plane of the blades does not exhibit any significant difference of the two approaches and is therefore not shown. A difference in the flow pattern already present in the plane of the blades and also visible in magnifications of Figs. 2 and 3 are two small secondary vortices that develop in the DNS prediction near the junction of hub and disc. The RANS simulation does no provide any evidence for this flow feature. A comparison of the RANS and the DNS results in Fig. 2 reveals two important differences: The vortex roll-up behind the inner part of the blades, forming some kind of recirculation zone, has a much larger extent in the DNS results compared to the RANS results. The location of the two 'tip' vortices that form at about 5 ~ behind the blades is also closer to the axis in the DNS predictions. Both simulations show the distinct upward inclination of the flow field discharged from the impeller, The instantaneous flow-field of the DNS simulation shows a very interesting, wavy flow structure of the radial impeller flow, indicating a KelvinHelmholtz instability. A close comparison of the DNS and RANS results of the discharge flow field shows that the flow pattern of the RANS simulation is a little more wavy then that of the DNS results. The flow structure predicted by the DNS simulation receives strong support by experimental findings (cf. [7]). The flow field in a plane at an angle of 15~ behind the blades is characterized by the two 'tip' vortices that were already present in the plane 5~ behind the blades but meanwhile have been transported in radial direction by the large-scale flow field. It is interesting to note that the location of the vortices in the DNS results is now further away from the axis compared to the RANS results (especially for the lower vortex) indicating that the speed at which the vortices have been convected outwards is much higher for the DNS simulation. A second interesting difference between the two simulations is that the vortices in the DNS simulation are much closer to the plane of the disc (despite the same initial distance in the 5 ~ plane). The radial offset of the two vortices in the DNS simulation and the smaller distance of the vortices to the plane of the disc is in agreement with experimental findings. The two 'tip' vortices are dying out at an angle of about 35 ~ behind the blade. This result is identically produced by the DNS and the RANS simulation. As opposed to the above discussion of the time-averaged flow field, the discussion of the distribution of turbulent kinetic energy can only be considered as preliminary because the duration of averaging has not been long enough up to now (8.24 revolutions) so that the results are not yet completely stabilized. The results for the two simulation approaches have been depicted in Fig. 5 for the plane of the disc and in Fig. 4 for the mid-baffle plane. The comparison of the two simulation approaches for the plane of the disc reveals the following differences" Whereas the regions of increased turbulent kinetic energy k in the case of RANS simulations have a kidney like shape, these regions look more like a sickle with a detached head in the case of the DNS simulations. The shape found in the DNS simulations is much closer to experimental findings than that found in RANS simulations. Moreover, the distribution of turbulent kinetic energy in the outer flow region of the DNS simulations seems to be not as uniform as in the RANS calculations. The figure showing the contours of the instantaneous turbulent kinetic energy suggests that very small turbulent scales occur close to the impeller blades (where the grid is the finest) whereas in the outer regions of the flow field the length scales become larger and larger and finally dissipation occurs. The pictures shown in Fig. 4, depicts the contours of turbulent kinetic energy in the mid-

243 baffle plane. The most important differences between the two approaches is that the maximum value for the turbulent kinetic energy in the impeller discharge flow is much higher for the DNS simulation (k/U~ip = 0.118) than for the RANS simulation (k/U~i p = 0.079). This is in agreement with experiments where RANS simulations constantly underpredict the maximum value of the turbulent kinetic energy. Moreover, the turbulent kinetic energy is dissipated much faster close to the wall in the RANS simulation than in the DNS simulation. 4 CONCLUSIONS The comparison of the large scale flow field shows a good overall agreement of the observed flow features for both types of simulations. On the other hand, close comparison revealed differences between DNS and k-e simulations where the DNS predictions were almost always closer to experimental results. The results for the turbulent kinetic energy k also show some evidence of better agreement between experiment and DNS but it is not completely clear right now how reliable the information for turbulent kinetic energy is. This aspect of the current investigations will have to be reconsidered when the statistics of the results are improved. REFERENCES [1] B.E. Launder and B.I. Sharma, Application of the Energy Dissipation Model of Turbulence to Calculation of Flow near a Spinning Disc. Lett. Heat Mass Transfer, 15, 301-314, (1974). [2] M. Schafer, M. H6fken, and E Durst, Detailed LDV Measurements for Visualization of the Flow Field within a Stirred-Tank Reactor Equipped with a Rushton Turbine. Trans. IChemE, 75(A), 729-736, (1997). [3] A.D. Harvey, C.K. Lee, and S.E. Rogers, Steady-state Modeling and Experimental Measurement of a Baffled Impeller Stirred Tank. AIChE J. 41, 2177-2186, (1995). [4] K. Wechsler, M. Breuer, and E Durst, Steady and Unsteady Computations of Turbulent Flows Induced by a 4/45 ~ Pitched Blade Impeller. Journal of Fluids Engineering, 121, 318-329, (1999). [5] C. Rhie and W. Chow, Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation. AIAA J., 21, 1525-1532, (1983). [6] S. Patankar and D. Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows. Int. J. Heat Mass Transfer 15, 17871806, (1972). [7] M. Sch~fer, K. Wechsler, and E Durst, Advanced Methods for Investigation of SinglePhase Stirred Vessel Flows. Part I: Experimental Methods. Proc. of AIChE Annual Meeting, Miami Beach, FL, USA, November 1997, American Institute of Chemical Engineers 1998.

244

Fig. 1: Large-scale flow field, direction and magnitute of the velocity vectors projected onto the mid-baffle plane, k-e prediction (left), DNS time-averaged (middle), DNS instantaneous (fight).

Fig. 2: Impeller flow field, direction and magnitute of the velocity vectors in a plane 5 ~ behind the blades, k-e (top), DNS time-averaged (middle), DNS instantaneous (bottom).

245

Fig. 3" Impeller flow field, direction and magnitute of the velocity vectors in a plane 15~ behind the blades, k-e prediction (top), DNS time-averaged (middle), DNS instantaneous (bottom).

Fig. 4: Distribution of turbulent kinetic energy k/U.~i p in the mid-baffle plane, k-e prediction (left), DNS time-averaged (middle), DNS instantaneous (fight).

246

Fig. 5: Distribution of turbulent kinetic energy k/U~i p in the plane of the impeller disc. k-e prediction (top), DNS time-averaged (middle), DNS instantaneous (bottom).

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

247

The Use of Large Eddy Simulation to Study Stirred Vessel Hydrodynamics Andr6 Bakker, Lanre M. Oshinowo, and Elizabeth M. Marshall Fluent Inc., 10 Cavendish Court, Lebanon, NH, USA 03766 The application of large eddy simulation (LES) to the prediction of large-scale chaotic structures in stirred tanks is investigated. Flow regimes representing typical stirrer configurations were assessed: a single radial pumping impeller and a single axial pumping pitched blade turbine. The turbulent flow field in each configuration was calculated using LES turbulence models. The impellers were modeled using the sliding mesh model. The predicted flow patterns compared well with digital particle image velocimetry data reported in the literature, and exhibited the long time scale instabilities seen in the experiments. The results of these studies open the way to a renewed interpretation of many previously unexplained hydrodynamic phenomena that are observed in stirred vessels. 1.

INTRODUCTION

Prediction of mixing of multi-component fluids is important in many chemical process applications. Although laminar mixing is a complicated process per se, there is a far greater challenge in predicting the mixing in turbulent flows because of the intrinsic, chaotic nature of turbulent flows. In turbulent flows, large-scale eddies with coherent structures are mainly responsible for the mixing of passive scalars. The large-scale eddies embody themselves in the form of identifiable and organized distributions of vorticity. Recent experimental work suggests that large-scale, time-dependent structures, with periods much longer than the time of an impeller revolution, are involved in many of the fundamental hydrodynamic processes in stirred vessels. For example, local velocity data histograms may be bi-modal or tri-modal, even though they are usually analyzed as having only one mode in most laser-Doppler experiments (Bakker and Van den Akker, 1994a). The gas holdup distribution may be asymmetric and oscillating (Bakker and Van den Akker, 1994b). In solids suspension processes, solids can be swept from one side of the vessel to the other in a relatively slow oscillating pattern, even in dilute suspensions. Digital particle image velocimetry experiments have shown that large-scale asymmetries with periods up to several minutes exist in stirred vessels equipped with axial flow impellers (Myers et al. 1997). In this study, the application of LES to the prediction of these large-scale chaotic structures in stirred tanks is investigated. Several flow regimes representing typical stirrer configurations were assessed: a single radial pumping impeller, a single axial pumping pitched blade turbine, a single high-efficiency impeller, and a dual impeller combination. Because of space constraints, however, only the results for the former two systems will be discussed in detail in this article. Results include the blending of a tracer. Previous LES work reported in the literature considered radial flow impellers only (Eggels, 1996. Derksen and Van den Akker, 1999) and did not include blending studies. The impeller rotation was modeled using a sliding mesh model on unstructured grids (Mathur, 1994. Bakker et al. 1997). The theory behind LES, the model, and the results will be discussed.

248 2.

THEORETICAL

The LES Model The numerical modeling of these complicated mixing processes is a daunting task. Direct numerical simulation (DNS) provides the most exact approach in which the mechanism involved in turbulent mixing can be accurately represented. DNS requires resolving the smallest eddies which makes the approach prohibitively expensive even with the most powerful computers of the present day, and foreseeable future as well. On the other hand, the popular approaches based on the Reynolds-averaged Navier-Stokes (RANS) equations amount to averaging out the large eddies that are mainly responsible for mixing. One is left to model the effects of large eddies by relying on empirical data and phenomenological reasoning and hypotheses. Recently, the large eddy simulation (LES) approach has been used as an intermediate method between the extremes of DNS and RANS. In this approach, the governing equations are obtained by spatially filtering the Navier-Stokes equations. The large turbulent scales are computed explicitly, while the small scales are modeled using one of a number of available subgrid scale (SGS) models. The SGS models describe interactions between the resolved and unresolved scales. The LES approach is more general than the RANS approach, and avoids the RANS dependence on boundary conditions for the large scale eddies. Like DNS, the LES approach gives a three-dimensional, time dependent solution. The required computational resources for LES are between those of the DNS and RANS approaches. The LES model can be used at much higher Reynolds numbers than DNS because the computational effort is independent of the Reynolds number if the small scales obey the inertial range spectrum and the near wall effects are not important.

2.1

2.2

Governing Equations

The governing equations for LES are obtained by spatially filtering over small scales. Filtering eliminates the eddies whose scales are smaller than the filter width. In the current study, a top-hat filter of filter width to grid size ratio of two is used. Explicit filtering is not used. With this filter, differentiation and filtering operations commute only on uniform grids. The importance of commutation errors on non-uniform grids is a topic of current research. In the present work, it is assumed that the commutation error is a part of the error in the subgrid models. Applying the filtering operation to the momentum equation, we obtain:

~P-Ui 3t

+

~P-Ui U j ~xj

= -

~p ~x~

+

~ij ~

~O'ij

-)-~

~)xj

(I)

where xij is the filtered (subgrid scale) stress tensor. In the filtered equations, the terms represented by o'ij, called SGS stresses/fluxes, are of the form: O'ij "-- --(PUiUj -- p-'UiUj)

(2)

These SGS stresses/fluxes are unknown, and need to be modeled. Smagorinsky (1963) and Lilly (1966) developed the most basic subgrid scale model. In this model, the turbulent viscosity is modeled by:

/z, = pL~, I~1

(3)

249 where Ls is the mixing length for subgrid scales, and IS 1= 42~fi/~. The mixing length, ~ , is computed in FLUENT as lxLs = min (ted, C~V m) where K: and Cs are constants, d is the distance to the closest wall, and V is the volume of the cell. Yakhot et al. (1986) have obtained an RNG subgrid scale stress model by performing recursive elimination of infinitesimal bands of small scales. In the RNG SGS model, subgrid fluxes in the momentum equation are represented by: O'ij --~] a~aij = 2a.,,.,s, ij

(4)

This model differs from the Smagorinsky model in the way subgrid viscosity is calculated. In the RNG subgrid model, the effective viscosity,/leg =/1 +/it, is given by: /taf

=

[ 1+ H( /t2s /toll _ C)]m /t 3

(5)

where /tsgs = ~(CmgA) 2(2SijSij) ~r2and H(x) is the Heaviside ramp function. The coefficients, C s = 0.157 and C=100 are obtained from the theory. In highly turbulent regions, the filtering operation results in very high subgrid viscosity compared to the molecular viscosity, flsgs >>/t and /2ef~ ___-/t~gs. In this limit, the RNG theory based subgrid scale model returns to the Smagorinsky model with a different model constant. In weakly turbulent regions, the argument of the Heaviside function is negative, and the effective viscosity is equal to the molecular viscosity. The RNG SGS model in this limit correctly yields zero SGS viscosity in low Reynolds number flows without any ad-hoc modifications. 2.3

Numerical Method The numerical simulations are conducted using FLUENT 5. A detailed discussion of the numerical method and several validation studies of this code are given by Murthy and Mathur (1998). In this code, the domain is discretized into arbitrary unstructured polyhedra. The discretized form of the governing equations for each cell is obtained such that the conservation principles are obeyed on each polyhedron. In FLUENT, the linear equations are solved using an algebraic multigrid procedure. The results presented in this paper are obtained using central differencing for spatial discretization of the momentum equations, and timeadvancement via a second-order accurate implicit scheme. The transient impeller motion was modeled using the sliding mesh model for unstructured grids (Mathur, 1994). 3.

MODEL DESCRIPTION

The modeling results of the transient, turbulent hydrodynamics will be reported for two configurations. The first was a 45 ~ pitched-blade turbine (PBT) configuration consisting of a cylindrical, fiat-bottomed tank of internal diameter, T = 292mm, with four full-length baffles of width T/12. The free surface was at a height, H=T. The PBT had four blades, with a diameter D-0.35T, a blade width W=0.20D, and a blade thickness of 1 mm. The impeller center was positioned at a distance C----0.46T off the tank bottom. The impeller was mounted

250 on a 10mm diameter shaft rotating at 60rpm. This geometry was studied experimentally by Myers et al. (1997). The computational grid was defined by 527,000 unstructured, nonuniformly distributed, hexahedral cells. Approximately 180 seconds of actual time were simulated. All simulations were initiated from a zero-velocity flow field. The Rushton turbine configuration consists of a cylindrical, fiat-bottomed tank of internal diameter, T = 202mm, with four f-all-length baffles of width 22mm. The free surface was at a height, H=T. The impeller had a diameter D=T/3, a blade width W=0.20D and a blade thickness of one mm. The impeller center was positioned at a distance C=T/3 off the tank bottom. The impeller was mounted on a 5mm diameter shaft rotating at 290rpm. The computational grid was defined by 763,000 unstructured, non-uniformly-distributed, hexahedral cells. Approximately 3 seconds of actual time were simulated. The simulations were run on the parallel version of the FLUENT 5 code on dual processor Sun Ultra 60 machines. Each time step took an average of nine minutes of wall clock time. Implicit time steps of 0.01 to 0.05 seconds were used. 4.

RESULTS

4.1 Flow Field Results for the PBT Figure 1 shows vector plots after 162, 166, 170, and 174 seconds respectively for the simulations with the pitched blade turbine. These vector plots show the unsteadiness in the flowfield. Figure l a shows a relatively symmetric axial flow pattern. Figure l b shows an asymmetric flow pattern, with the flow on the left side of the vessel being axial, and the jet coming from the impeller attaching to the vessel bottom. On the right side of the vessel, the jet coming from the impeller bends radially and attaches to the vessel wall. Figure lc shows a flow pattern where the jet coming from the impeller attaches to the vessel wall on both sides and a secondary circulation loop has formed near the vessel bottom. Figure 4d shows a relatively symmetrical flow pattern where the axial jet from the impeller attaches to the vessel bottom. Qualitatively, these results compare strikingly well with the digital particle image velocimetry data reported by Myers et al. (1997). Their experimental data also showed the existence of these unsteady, asymmetric flow patterns. Note that these flow pattern oscillations have a time scale that is much longer than the time scale associated with the impeller blade passage frequency. Figure 2 shows the simulated flow patterns at three instances in time, spaced apart by half a blade passage period, or 0.125s. The overall flow pattern, which is asymmetric, does not change during that time. This also agrees with the experimental data reported by Myers et al. (1997). 4.2 Time Series Results for the PBT Figure 3 shows time series of the axial velocity in four locations. A Cartesian reference frame is used with the origin being on the axis at the liquid surface, and x being the downward axial direction. Significant fluctuations can be observed, with the axial velocity periodically changing direction in certain locations. These time series also show that the period of the fluctuations is not constant. Higher frequency variations are observed in locations (a) and (b), which are located just below the impeller blade tip, than in locations (c), close to the vessel bottom, and (d), which is close to the liquid surface. More advanced time series analysis, including spectral analysis, will be a topic of future research.

251

Figure 1 The pitched blade turbine flow pattern after 162, 166, 170, and 174 seconds.

Figure 2 Instantaneous velocity fields of the PBT taken during blade passage.

252 (a) x = O.185m y = -0.04m, z = -0.04m 0.2

.

.

.

.

.

.

0.1

-0.1

, 2500

2700

2900

3100

3300

3500

(b) x = 0.185m y = 0.04m z = 0.04m 0.2

o,

,. 0

V

-0.1

, 2500

,

2700

-

2900

9

v

,

,

3100

3300

3500

3300

3500

(c) x=0.25m y=0.05m z=0.05m 0.2 0.1 0

,~

-0.1 2500

2700

2900

3100

(d) x=O.O5m y=0.05m z=0.05m

1

0.2 0.1 0 -0.1

f

i

2500

2700

I

2900

i

3100

..

i

3300

3500

Figure 3 Time series plots of axial velocity for the PBT from 168.13306 (2500 time steps) to 178.12756s (3500 time steps) after start-up from a zero-velocity field.

4.3 Blending Results for the PBT Figure 4 shows the blending of a tracer injected just above the tip of the PBT. The results show that the tracer does not disperse symmetrically, indicating that such simulations need to be performed on full 360-degree grids, instead of periodic 90-degree sectors.

253

Figure 4 Blending of a tracer in the unsteady flow field created by a PBT. 4.4 Instantaneous Flow Field for the Rushton Turbine Figure 5 shows the unsteady flow field in the outflow of the Rushton turbine, at two instances in time separated by 0.024s, which corresponds to 0.7 of a blade passage period. Figure 5a shows the trailing vortex forming just behind the impeller blade on the left. The vortex core indicated on the fight in Figure 5a is the core of the trailing vortex behind the previous blade. Figure 5b shows how the vortex core coming from the blade has moved radially outwards. These results qualitatively agree with the results reported by Eggels (1996) and Derksen and Van den Akker (1999).

Figure 5 The unsteady flow created by a Rushtonturbine.

254 5. C O N C L U S I O N S In this study, the application of large eddy simulation (LES) to the prediction of largescale chaotic structures in stirred tanks was investigated. Flow regimes representing typical stirrer configurations were assessed. The predicted flow patterns for the PBT compared well with digital particle image velocimetry data reported in the literature, and exhibited the long time scale instabilities seen in the experiments. The results for the Rushton turbine compared well with LES simulations reported previously. The results of these studies open the way to a renewed interpretation of many previously unexplained hydrodynamic phenomena that are observed in stirred vessels, such as the time dependent settling of solids in dilute suspensions, bi-modal and tri-model velocity histograms observed in laser-Doppler experiments, large variations in measured blend times in seemingly identical experiments, and possibly even the observed asymmetric holdup distributions in low volume fraction gas dispersions. However, much additional research is needed to come to a full understanding of these phenomena. 6. R E F E R E N CES Bakker A., LaRoche R.D., Wang M.H., Calabrese R.V. (1997) Sliding Mesh Simulation of Laminar Flow in Stirred Reactors. TranslChemE, Vol. 75, Part A, page 42-44, January 1997. Bakker A., Van den Akker H.E.A. (1994a) Single-Phase Flow in Stirred Reactors. Chemical Engineering Research and Design, TranslChemE, Vol. 72, Number A4, July 1994, page 583-593. Bakker A., Van den Akker H.E.A. (1994b) Gas-Liquid Contacting with Axial Flow Impellers. Chemical Engineering Research and Design, TranslChemE, Vol. 72, Number A4, July 1994, page 573-582. Derksen, J., Van den Akker, H.E.A. (1999) Large Eddy Simulations on the Flow Driven by a Rushton Turbine. AICheJ, Vol. 45, No. 2, page 209-221. Eggels, J.G.M. (1996) Direct and Large-Eddy Simulations of Turbulent Fluid Flow Using the Lattice-Boltzmann Scheme. Int. J. Heat Fluid Flow, Vol. 17, page 307. Lilly, D. K. (1966) On the Application of the Eddy Viscosity Concept in the Inertial Subrange of Turbulence. NCAR Manuscript 123. Mathur (1994) Unsteady Flow Simulations Using Unstructured Sliding Meshes. AIAA 25th Fluid Dynamics Conference, June 20-23, 1994, Colorado Springs, CO. Murthy, J. Y., and S. R. Mathur (1998) A Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes. A A - 9 8 0860, January, 1998. Myers K.J., Ward R.W., Bakker A. (1997) A Digital Panicle Image Velocimetry Investigation of Flow Field Instabilities of Axial Flow Impellers. Journal of Fluids Engineering, Transactions of the ASME, Vol. 119, No. 3, page 623-632, September 1997. Smagorinsky, J. (1963) General Circulation Experiments with the Primitive Equations, I. The Basic Experiment. Month. Wea. Rev., Vol. 91, page 99- 164. Yakhot, A., S. A. Orszag, V. Yakhot, and M. Israeli (1986) Renormalization Group Formulation of Large-Eddy Simulation. Journal of Scientific Computing, Vol. 1, page 1-51.

10th European C6nference on Mixing H.E.A. van den Akker and,1.,1. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

Compartmental on Large

Eddy

modeling

255

of an 1100L DTB

crystallizer

based

flow simulation

Andreas ten Cate ~'b'c, Sean K. Bermingham a'b, Jos J. Derksen c, Herman M.J. Kramer ~ ~Laboratory for Process Equipment, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands bSection Process Systems Engineering, DelftChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands CKramers Laboratorium voor Fysische Technologie, Delft University of Technology, Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands Abstract

In this contribution, the development of a compartmental model for the dynamic simulation of an l l00L DTB crystallizer is presented. Design of the compartment structure was based on high resolution CFD simulation of the internal flow of the crystallizer. The CFD simulation of the turbulent flow field was based on a lattice-Boltzmann scheme with a Smagorinsky subgrid-scale turbulence model (cs was 0.11). The fully developed turbulent flow field was simulated at Re=240.O00 on 35.5.106 grid nodes. A detailed compartmental model with 21 compartments was derived. The model contained mass, heat and population balances for each compartment. From the CFD simulations, flow rates and local rates of energy dissipation for each compartment were determined. Explorative simulation results of the compartmental model are presented to demonstrate the influence of compartment structure, short circuiting flow and rate of energy dissipation on the evolving crystal size distribution. 1

Introduction

Evaporative crystallization is used in industry for the manufacturing of both bulk solid products (e.g. fertilizers and sugar) and fine chemicals. The hydrodynamic conditions in the crystallizer play an important role in the performance and behaviour of the crystallization process. The flow patterns in the equipment determine the crystal residence time in various sections of the crystallizer, while the turbulence characteristics are of influence on the mass transfer from the mother liquor to the crystal surface. In addition to this, crystal-impeller impacts can cause breakage and is therefore important for effects like secondary nucleation. All these effects influence product properties (e.g. crystal size distribution) and process behaviour. For scale-up of crystallizers models are needed that take into account the above described phenomena. Due to the complexity of the crystallization process, simplified mod-

256 els such as the Mixed Slurry Mixed Product Removal (MSMPR) model are used. In this model, the process variables (e.g. solute concentration, temperature, rate of energy dissipation) are assumed to be uniformly distributed over the entire crystallizer volume. Mass and energy balances for both solvent and solute are solved, combined with the population balance for the crystal distribution. Additional models can be incorporated to cover phenomena like mass transfer, crystal growth and dissolution, nucleation and attrition. One major drawback of this approach is that the influence of different phenomena (e.g. flow conditions and mass transfer on crystal growth) is lumped in effective model parameters. Compartmental modeling has been introduced to account for the interactions between kinetics and hydrodynamics on a meso scale. The importance of these interactions in scale-up and design is discussed by [1]. In compartmental models, process conditions may differ significantly from one compartment to another, but not within a compartment. Each compartment in such a model is described with the same set of model equations and model parameters. Differences between compartments with respect to nucleation, growth, dissolution, attrition, breakage and aggregation rates are therefore purely a result of process conditions which are not the same in each compartment. A compartmental model is characterized by its structure, i.e. the location and size of the various compartments; its connectivity, i.e. source and destination of interconnecting flow rates. Criteria for the design of a compartmental model can be based on the temperature, supersaturation, rate of energy dissipation and crystal size distribution([2]). The gradients in these conditions are influenced by the crystallization kinetics and material properties, the geometry of the crystallizer vessel and the operating conditions. An obvious barrier is that for a network of compartments the mass flows between the compartments need to be determined. These data are difficult to obtain experimentally. Therefore, in practice, rather crude compartmental models are used. In this study an attempt is made to derive a detailed compartmental model based on the Computational Fluid Dynamics (CFD) calculation of the flow pattern of a pilot scale 1100 L draft tube baffle (DTB) crystallizer. The objective of this study is to demonstrate the benefit of detailed computational fluid dynamics in the construction of compartmental models. Therefore, in section 2, the numerical method of the CFD simulations is presented as well as the obtained time-averaged CFD-data of the crystallizer. Section 3 discusses the construction of the compartmental model. In section 4, the results of both the CFD simulations and the different compartmental models are discussed.

2

CFD simulations

2.1 N u m e r i c a l s i m u l a t i o n o f t h e flow field The crystallizer studied in this contribution has a very complex geometry and is operated at highly turbulent conditions (Re _> 2. 105). Furthermore, the flow is inherently time dependent due to the rotating impeller. Resolving the flow pattern therefore requires a simulation technique that is capable of handling these features of the flow field at sufficient (temporal and spatial) resolution. In this intrinsically unsteady flow, a large eddy simulation (LES) approach to turbulence modeling is preferred. Due to its numerical efficiency, a lattice-Boltzmann discretization of the Navier-Stokes equations can provide

257 sufficiently high spatial resolution to do physically sound LES. The lattice-Boltzmann method applied in this research has proven to produce reliable results in simulations of theturbulent flow field of a stirred tank reported by [3] and [4]. The lattice-Boltzmann method has been developed in recent years and stems from the lattice gas cellular automata techniques that date back to the seventies and eighties. The concept of the lattice-Boltzmann method is based on the premise that the mesoscopic (continuum) behaviour of a fluid is determined by the behaviour of the individual molecules at the microscopic level. In the lattice-Boltzmann approach, the fluid is represented by fluid mass that is found on the nodes of an equidistant grid (lattice). The fluid mass mimics the behaviour of gas molecules. Collision rules that guarantee conservation of mass and momentum are applied. The elegance of the method is that although the collision rules are imposed completely local, the continuity equation and incompressible Navier-Stokes equations are recovered ([5, 6]). This characteristic makes the method extremely favourable for high-performance CFD on parallel computer platforms. Another major advantage of the lattice-Boltzmann method is that it is a time-dependent method. Thus it is suited to be combined with a subgrid-scale turbulence model for Large Eddy Simulation (LES). In LES, turbulent structures at two times the length of the grid spacing are effectively resolved while all turbulent fluctuations at smaller scales are filtered out. In this research, a standard Smagorinsky model ([7]) was applied, which treats subgrid-scale motion by means of an eddy-viscosity (ut) based on the local deformation rate. The Smagorinsky constant cs (the ratio between mixing length and grid spacing) was set to 0.11. Implementation of the geometry of the crystallizer in the lattice-Boltzmann scheme is based on a forcing algorithm presented by [4]. This algorithm is used to force the fluid to obtain prescribed velocities at the tank wall and baffles, the in- and outlets of the crystallizer and at the stirrer.

2.2

Crystallizer setup

The geometry of the crystallizer used in this research is given in figure 1. At the bottom of the crystallizer two inlets are positioned for the feed flow and external fines flow. The outer shell of the crystallizer is a settling zone with six withdrawal points for the removal of fine crystals. The outflow is passed through a heat exchanger for heat input and dissolution of fine crystals and then returned to the bottom section of the crystallizer. The inner section of the crystallizer contains the draft tube with four baffles. At the bottom side of the draft tube a marine type impeller is placed with a diameter of 48.5 cm. The slurry is circulated upwards through the draft-tube and downwards on

Table 1: Operating conditions of the 1100L DTB crystallizer Impeller speed Re Impeller Cvfeed Cv fines [r.p.m.] [-] [1/s] [i/s] Physical conditions 320 730.000 0.5 2.0 Impeller speed Re Impeller timestep gridspacing [r.p.timestep] [-] [#sits ] [mm/lu] Simulation conditions 1//3200 240.000 58 5.0

~

[m~ls] 2.4.10 -6 u [-] 1.4-10 -4

258

Figure 1: Crystallizer geometry and compartment structure. Side view and top view of the 1100L DTB crystallizer. Dimensions are given in cm.

the outside of the draft tube. The top section of the crystallizer is the boiling zone where supersaturation is generated. The conditions at which the crystallizer is operated are given in table 1. This table also contains the conditions and resolution at which the simulations were done. The inlet and outlet flows were included in the flow simulations. The Reynolds number that characterizes the flow is defined as Re = g D 2 / v where N is the rotational speed (rev/s), D the stirrer diameter and ~ the kinematic viscosity of the fluid. The Reynolds numbers of the physical system and the simulation differ a factor three. The difference in flow characteristics is expressed in the finest turbulent structures. For the design of a compartmental model, averaged flow data are only of interest and these will be practically identical due to the Reynolds similarity of the fully developed turbulent flow patterns. Although the fluid phase is a slurry, it is treated as a single phase with a uniform effective slurry viscosity.

2.3

Time averaged simulation results

The flow simulation was done on a cubic grid of 552 x 253 x 253 (~ 35.5.106) grid nodes. After start-up and development of the turbulent flow field, the simulation was carried out for 6.4 stirrer revolutions in order to capture averaged flow results. The whole computation was done on 8 nodes of a parallel cluster of pentiumIII 500 MHz processors. On this system calculation of one stirrer revolution took about 26 hours. For the crystallizer compartmental model, flow information is needed to determine the flows between compartments and the local kinetics. The choice of parameters will be discussed more extensively in section 3. In figure 2 various time averaged properties of the flow field are presented. These figures

259

Figure 2: Contour and Vector plots of the averaged flow field of the 1100L DTB crystallizer,

clearly demonstrate the distributed nature of the crystallizer. From the velocity contours one can clearly see that the fluid is accelerated in the stirrer section and at the boiling zone where the fluid is drawn back into the downcomer. The contour plot of the rate of energy dissipation shows that the baffles inside the draft tube cause a rate of energy dissipation that has the same order of magnitude as the stirrer. The contour plot of the turbulent kinetic energy clearly demonstrates the inhomogeneous flow conditions of the core of the draft tube and the outer zone. In table 2 the performance characteristics of the crystallizer marine type impeller as calculated from the CFD simulation are given. The literature values given were used in previous compartmental models by [8].

Table 2: Operating parameters estimated from the l l00L DTB crystallizer CFD([8]) From CFD From literature Power number [-] 0.47 0.40 Specific power input [W/kg] 1.73 1.96 Pumping number [-] 0.30 0.32

250 3

Compartmental modeling

3.1 S e t u p of t h e c o m p a r t m e n t a l m o d e l s The major properties that are produced with CFD and are relevant for the construction of a compartmental model are the fluid velocity and the rate of energy dissipation. These properties were used as criteria for the selection of the compartments. The compartmental model applied in this research is based on the framework and implementation of [9]. This model contains a mass- and heat-balance and a population balance to describe the evolution of the crystal population. The rate of secondary nucleation is calculated from a detailed model that describes the formation of attrition fragments as a result of crystal-impeller collisions ( [10] , [l l] ) . This mechanism is assumed to be the only source of nucleation and attrition. The growth of crystals is considered the result of a two-step process of (1) bulk to surface diffusion of the solute and (2) subsequent integration of the solute in the crystal lattice. In this mechanism, the rate of mass transfer is determined by the (turbulent) hydrodynamic conditions as characterized by the specific rate of energy dissipation e. Equation 1 is used to calculate the mass transfer coefficient; kd(L) : ~

ff~4 ~ 1/5 ~,3 /

V

1/3

§

2]

(1)

In this equation, kd is the mass transfer coefficient, DAB the diffusion coefficient of solute A in solvent B and L the particle length. The procedure for construction of the compartmental model was as follows: 1. Based on the velocity difference between the core and outer region of the draft tube, a choice was made to divide the draft tube in an inner section and an outer section. Because of a higher rate of energy dissipation at the baffles, the outer shell was divided in a baffle compartment and a compartment without baffle. The whole tube was divided in three horizontal sections to mimic plug flow behaviour in the axial direction. These compartments can be identified as compartments 1 to 11 in figure 1 2. At the top section, a strong outward velocity is found. Therefore, compartments 10, 11 and 12 were introduced to take into account the short circuiting effect of the slurry flow that is directly sent to the downcomer. 3. In compartment 14 all flows from compartments 10 to 13 are collected and are directed into the downcomer. The downcomer is again divided into three compartments to for plug flow behaviour. 4. Compartment 18 contains an outward flow to the settling zone (19) where a classification function is used that imposes the effect of settling on particle selection, based on a model proposed by [12]. 5. The bottom section contains a compartment where the feed and fines flow are introduced (20) and a stirrer section inside the draft tube where a high rate of energy dissipation is found (21). Based on this selection a compartment structure is made which is given in figure 3 (A). In this figure two examples, (B) and (C), of simpler model structures are given for comparison. (B) Represents the single MSMPR model as discussed in the introduction, extended with an external heat exchanger and stirrer compartment as attrition source.

261

Vapour

Product -i

~k

Vapour Product

27

"'126

(B)

Va

Product

I Fines loop

_sc

iio

,++!

Inlet/Stirrer section!

(A)

(C)

Figure 3: Compartment structures and connectivity; (A) a network of 21 compartments, (B) a single compartment with external fines loop and (C) a 5 compartment network. S is the stirrer compartment which functions as fines source, H T X is the external heat exchanger and fines dissolution loop.

262 A second compartment is added to include the volume of the fines-zone correctly. Model (C) represents a network of 5 connected compartments with a boiling zone and optional bypass (SC) to model the influence of a short-cut flow. With these three compartmental models simulations were done to investigate the influence of the model structure on the resulting crystal size. After selection of the compartments and setting up the compartment structure, two properties are determined from the CFD-data that determine the characteristic of each compartment. First, the ratio between the compartment specific rate of energy dissipation and the crystallizer volume-averaged specific power input is determined. This value is used in equation 1 to calculate the mass transfer coefficient. Second, the ratio between the volume flow of the main outlet and other outlets is calculated to determine the distribution of the outflow of each compartment to connected compartments. For example, the short circuit flow in compartment 1 of the compartment structure in figure 3 (C) is determined in this way to be approximately 60 % of the main circulation flow. The local turbulent condition also causes turbulent dispersion between compartments. The mass flux across a plane between two compartments is characterized by the average fluid velocity and the fluctuating part of the fluid velocity, u~, which can be estimated from the square root of the turbulent kinetic energy k;

U_l_ ~---

x

x

y

Turbulent dispersion can be implemented in the compartmental model by increasing each outlet flow with u•t and defining a back flow of the same size. Although the mechanism of turbulent transport can play an important role in the dynamic behaviour of the crystallizer, it has not been implement in the compartmental models at this stage. 3.2 S i m u l a t i o n r e s u l t s of t h e c o m p a r t m e n t a l m o d e l s To study the effect of non-uniform energy dissipation rates on the evolving crystal product, five explorative simulations were done. The simulations were done with growth parameters that were determined previously from experiments done with ammonium sulphate. In table 3 an overview of the simulation results is given. In this table, H refers to a simulation where the rate of energy dissipation was set homogeneous over the whole

Table 3: Steady-state crystal sizes and volume percentage of the product stream of 5 different model options. (H) indicates a homogeneous distribution of the rate of energy dissipation, NH indicates a non-homogeneous rate of energy dissipation and SC indicates that a shortcut stream was applied in the 5 compartment model. Simulation ~ of compartments Lower (L10) Median (Ls0) Upper (Lg0) and reference figure [#m] [#m] [#m] _ Run 1. 2 H 3B 291 657 1227 Run 2. 5 NH 3C 280 627 1192 Run 3. 5 NH, SC 3C 280 628 1194 Run 4. 21 NH 3A 274 638 1219 Run 5. 21 H 3A 285 670 1282 . . . . . .

263 crystallizer volume. Thus, for run 5 in each compartment the rate of energy dissipation is the same as the average power input of the whole crystallizer. 4

Discussion a n d conclusion

4.1 C F D r e s u l t s The time averaged results of the crystallizer CFD have produced new insight in the flow characteristics of the 1100 L DTB crystallizer. Figure 2 demonstrates a number of effects that were previously unknown or underestimated; 1. The flow inside the draft-tube appears to be much more inhomogeneous than was expected. The upward fluid velocity in the core of the draft tube is lower than the upward velocity found in the outer region of the draft tube. 2. The short circuit flow that is drawn from the draft-tube straight into the downcomer is estimated to be 60 percent of the main circulation flow. 3. The degree of turbulence (i. e. k and c) near the baffles on the inside of the draft tube is of the same order of magnitude as in the impeller region. Thus, the assumption that attrition is only caused by impeller collisions needs to be questioned. The operating parameters which characterize the impeller are in good agreement with literature values. This suggests that the CFD simulations can be considered reliable. 4.2 C o m p a r t m e n t a l m o d e l i n g The construction of a detailed compartmental model, as demonstrated in this paper, can be considered virtually impossible without the aid of reliable CFD-results. This is for instance demonstrated by the choice to divide the draft tube in core and outer section compartments. This choice is solely based on the observation that velocities encountered in the core of the crystallizer are lower than the velocities on the outside of the core and is not an obvious one. Nevertheless, the influence of the compartment structure on the evolving crystal product appears not to be pronounced, as can be seen in table 3. Currently, the effect of turbulence is only taken into account in the rate of mass transfer in equation 1 where it is taken to the power 1/5-th. Thus, the effect is expected to be very small which is confirmed by the simulation results. On the other hand, the difference between runs 4 and 5 indicate that although the influence of the rate of energy dissipation is small, it can still be observed. A surprising result is seen when comparing runs 2 and 3, where apparently the bypass is not of influence on the evolving crystal size distribution. Further development and study of the compartmental models is required to indicate whether an improved decoupling between the hydrodynamics and the kinetic processes has effectively been reached. References [1] Herman.J.M. Kramer, Sean K. Bermingham, and Gerda M. Van Rosmalen. Design of industrial crystallisers for a rquired product quality. J. Crystal Growth, 198/199:729737, 1999. [2] Sean K. Bermingham, H.J.M. Kramer, and Gerda M. Van Rosmalen. [3] J.G.M. Eggels. Direct and large-eddy simulations of turbulent fluid flow using the lattice-boltzmann scheme. Int. J. Heat Fluid Flow, 17:307, 1996.

254 [4] J.J. Derksen and H.E.A. Van den Akker. Large eddy simulations on the flow driven by a rushton turbine. AIChE J., 45(2):209-221, 1999. [5] Daniel H. Rothman and St~phane Zaleski. Lattice-Gas Cellular Automata. Cambridge University Press, 1st. edition, 1997. [6] S. Chen and G.D. Doolen. Lattice boltzmann method for fluid flows. Annu. Rev. Fluid Mech., 30:329-364, 1998. [7] J. Smagorinsky. General circulation experiments with the primitive equations: 1. the basic experiment. Mon. Weather Rev., 91:99-164, 1963. [8] Andreas M. Neumann, Sean K. Bermingham, Herman J.M. Kramer, and van Rosmalen, Gerda M. Modeling industrial crystallizers of different scale and type. Proceedings of the l~th International Symposium on Industrial Crystallization, 1999. [9] Sean K. Bermingham, Andreas M. Neumann, Peter J.T. Verheijen, and Herman J.M. Kramer. Measuring and modelling the classification and dissolution of fine crystals in a dtb crystalliser. Proceedings of the l~th International Symposium on Industrial Crystallization, 1999. [10] C. Gahn and A. Mersmann. Brittle fracture in crystallization processes part a: Attrition and abrasion of brittle solids. Chem. Eng. Sci., 54:1273-1282, 1999. [11] C. Gahn and A. Mersmann. Brittle fracture in crystallization processes part b: Growth of fragments and scale-up of suspension crystallizers. Chem. Eng. Sci., 54:1283-1292, 1999. [12] E. Barnea and J. Mizrahi. A generalised approach to fluid dynamics of particulate systems. Chem. Eng. J., 5:171, 1973.

10th European Conference on Mixing H.E.A. van den Akker and.I.d. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

265

Detailed CFD prediction of flow around a 45 ~ pitched blade turbine J.K. Syrjlinen and M.T. Manninen VTT Energy, Energy Systems, Process Simulation P.O. Box 1604, FIN-02044 VTT, Finland email: [email protected], [email protected] Numerical simulations of flow induced by a pitched blade turbine are reported. A high density grid was employed around the turbine blades in order to resolve the details of the flow field, including the trailing vortices. Results were compared to recent experimental data. The k-e turbulence model was used with either wall functions or a two-layer model in the nearwall region. Three different computational grids were used in the simulations. The trailing vortices were reproduced in the simulation in good agreement with the measured data. 1. INTRODUCTION Stirred vessels equipped with axial impellers and baffles have been investigated experimentally [1-8] and simulated numerically [9-18] extensively during the last decade. Using sliding mesh or multiple reference frame technique, the flow field can be predicted accurately enough for many practical applications without using experimental data [11-18]. However, the turbulence modeling has remained a problem. If the turbulence is not predicted accurately, the mixing processes in the tank are not described correctly either. RANS-based methods with wall functions are used in most simulations. The results are usually compared with experimental, time-averaged data. In most simulations, the k-e turbulence model seems to give the best results, although the predicted turbulent kinetic energy in the impeller outflow is clearly lower than measured. So far, the RNG k-e and Reynolds stress models have failed to improve the prediction of k [13-15]. With LES, promising results have been obtained in detailed modeling of a radial impeller [ 16]. If wall functions are used in the simulation, the overall density of the grid may not be adequate enough to capture all the relevant phenomena of the flow field. Usually, the main circulation loops are well reproduced in the simulations, but the flow near the impeller blade is not predicted very precisely. Consequently, this may lead to poor turbulence prediction in this region. On the other hand, turbulent kinetic energy may be overestimated by 50-400 % in ensemble-averaged experimental data if the periodic effect of the passing blade is not filtered out [7]. Therefore, comparing the measured and predicted angle-resolved values would be more useful. In this work, emphasis is put on detailed modeling of the flow in and near the impeller swept volume. EspeciaUy, the aim was to numerically recapture the tip vortex induced by the blade of a pitched blade turbine. Fluent 5 was used in the simulations [19]. The predicted

266

results are compared with the comprehensive measurements of Sch~er et al. [7,8]. Recently, Wechsler et al. [17,18] have simulated numerically the same geometry with a dense grid succesfully revealing the trailing vortex. They modeled the impeller blade and baffle as zerothickness surfaces and increased the Reynolds number in their calculation in order to ensure the validity of wall functions. In the current simulation, the geometry and Reynolds number were identical to the ones used in the experiment of Schafer et al. [7]. 2. SIMULATED GEOMETRY

The tank is cylindrical and it has a lid and four symmetrically placed baffles near the tank wall. The turbine is four-bladed and the pitch angle is 45 ~. The impeller shaft extends to the bottom of the tank. The dimensions are as follows: tank diameter T = 152 mm, liquid height H = T, impeller diameter D = 0.329T, blade width w = 0.264D, impeller mid-plane clearance from the bottom h = 0.33T, baffle width B = T/IO and clearance from baffle to the wall s = 2.6 mm. The blade and baffle thicknesses were 0.9 mm and 3 mm, respectively. The working fluid is silicone oil with densityp = 1039 kg/m 3 and dynamic viscosity # = 0.0159 Pas. In the simulated case, the impeller rotational speed was N = 2672 rpm. This resulted in impeller Reynolds number Re = 7280 and tip velocity Utip = 7 m/s. For symmetry reasons it was sufficient to model one fourth of the tank. Then the computational domain contained one impeller blade and one baffle. 3. NUMERICAL SOLUTION

3.1. Computational grids Three hexahedral, block structured grids were used in simulations. The size and structure of the first grid (Cmd 1) is typical with wall function approach. The total number of cells is 17 237 with 37, 27 and 18 cells in the axial, radial and tangential directions, respectively. The grid is slightly denser in the impeller swept volume. On the blade surface, a 9• cell grid was used. To seriously capture the trailing vortex, the second grid (Grid 2) was made considerably denser in the impeller swept volume. According to [7], the trailing vortex diameter varies from 5 to 15 ram. The grid on the blade surface was composed of 24 radial and 15 axial cells. Then the narrowest cross section of the trailing vortex is inside a 6• cell grid. 38 cells were used in tangential direction. The number of cells in axial and radial directions was 74 and 59, and the total size of the grid was 164 036 cells. The third grid (Grid 3) was made still denser in and near the impeller swept volume. 78 cells were used in circumferential direction, 109 in axial and 71 in radial direction. The grid on the blade surface consisted of 31><23 cells. Now the smallest vortex cross section would be covered by a 8• cell grid. The wall function approach ceased to be valid, and a near-wall turbulence model was employed. The grid was refined near the walls to have adequate resolution in the near-waU region. The total number of cells in this grid was 583 961.

3.2. Solution methods In the multiple reference frame approach [19] used in the Fluent 5 calculations, two fluid regions are allowed to rotate relative to each other in a non-moving grid. The flow in the

267 rotating region of the modeled geometry is solved in ~. rotating reference frame. A stationary reference frame is used in the non-rotating part .af the geometry. This results in an approximative steady state solution, because the relative position of the impeller and the baffle remain unchanged in the grid during the calculation. In case of a laboratory scale stirred tank equipped with an axial impeller, the sliding mesh technique can be replaced with the multiple reference frame approach without a loss of accuracy [13-15,17,18]. The great advantage of the multiple reference frame technique is that the simulation times are much shorter than with the sliding mesh method. Second-order accurate solution-dependent QUICK scheme was used for the space discretization of the convection terms in momentum and turbulence equations in all simulations. Also, a second-order accurate scheme was used to interpolate the cell face values of pressure in the momentum equations. The pressure-velocity coupling was solved using the SIMPLE algorithm. In addition to observing residuals, surface integrals of velocity components, turbulent kinetic energy and effective viscosity were monitored during the iteration. The solution was considered converged when these values remained constant. Typically, 5000-8000 iterations were sufficient for a converged solution. No serious convergence difficulties were encountered during the calculations.

3.3. Turbulence modeling The standard k-e turbulence model with wall functions was used in simulations with Grids 1 and 2. Grid 2 was not fully optimised, because the values of y+ were less than 20 on the blade surface. With Grid 3, the two layer zonal model of Fluent 5 was used [19]. In this model, the flow domain is divided into a fully turbulent region and a viscous-affected region near the wall based on turbulent Reynolds number Rey = p ~ y //l, where y is the distance to the nearest wall. In the fully turbulent zone (Rey > 200), the standard k-e model is used. In the viscous-affected region (Rey < 200), a fine grid is used to resolve the flow down to the viscous sublayer without wall functions. In the viscous-affected region, k is solved from the standard transport equation, but turbulent viscosity is calculated from /l t = p C ~ l ~ and dissipation from e----k 3/2 / [e. The length scales l/, and It are defined as -

ollo

n

oon

n

:

:

_

_ 7O

= 2Cl. 4. RESULTS

4.1. Main flow pattern The radial profiles of predicted axial velocity and turbulence kinetic energy are compared in a plane halfway between the baffles at three axial positions with experimental values in Figs. 1-2. In the simulation, the impeller blade is 15~ behind the plane between two baffles with Grids 1 and 2. In the simulation with Grid 3, the impeller blade is on the plane of the baffle. The experimental values in Figs. 1-2 are angle-resolved up to radial position 2r/7' = 0.395 and correspond to the positions of the blade in the simulations.

268

Figure 1. Radial profiles of scaled axial velocity Uax,/Uap at three axial positions in a plane between two baffles: (a) z/T= 0.145, (b) z/T= 0.46, (c) z/T= 0.67.

Figure 2. Radial profiles of scaled turbulent kinetic energy k / U~2~pat three axial positions in a plane between two baffles: (a) z/T = 0.145, (b) z/T = 0.46, (c) z / T = 0.67. Symbols as in Fig 1.

Figure 3. Velocity vectors on vertical planes near the blade. (a) ~ = 0 ~ (b) ~ = 30 ~ (c) t~ = 60 ~ Figures 1-2 show that the predicted main flow pattern is in good agreement with measured data. At z/T = 0.145, the axial flow from the impeller is somewhat stronger than the measurements suggest. In addition, the simulation predicts a secondary circulation near the axis (positive axial velocity), not seen in the experimental results at this position. In the upper part of the tank at z / T = 0.67, the axial velocity modeled with Grid 3 is close to the measured one. The predictions with Grids 1 and 2 differ qualitatively from Grid 3 results.

269 The predicted turbulent kinetic energy level is about 70% of the measured level below the impeller. Closer to the wall, the agreement is very good. Even the coarse grid simulations predict k fairly well in this location. Largest discrepancies are found at z/T = 0.67.

4.2. Angle-resolved velocity fields The velocity vectors predicted with Grid 3 near the impeller on a vertical surface on three tangential locations relative to the impeller blade are shown in Fig. 3. In Figure 3a the paper surface intersects the middle of the blade (~ = 0~ In Figs 3b and 3c, the blade is rotated from the paper surface towards the viewer by ~ = 30 ~ and t~ = 60 ~ respectively. The trailing vortex is clearly seen in Fig. 3. Similar result was obtained with Grid 2, whereas Grid 1 did not reveal the vortex. According to Schafer et al. [7], the trailing vortex stretches 130~ ~ behind the blade. In the simulations, the vortex is quite weak already at t~ = 90 ~ It is possible that the accuracy of the QUICK scheme is not adequate or the turbulent viscosity generated by the k-e model is too high thus damping the vortex too early. In wing tip vortex simulations, a fifth order accurate differencing of convection terms as well as a modification of the production term in the turbulence model were needed to supress numerical diffusion [20]. The scaled axial, radial and tangential velocity components (Uax/Utip, Urad//Utip and Ut,,,,s,/Ut~p)are shown as the function of the circumferential angle $ in Figs. 4-6. $ equals 0 in the middle of the blade and increases towards the following blade. In the axial mid-impeller plane (z/T = 0.329), the comparisons of circumferential profiles are made in two radial positions. Inside the impeller swept volume (r/T = 0.118), the predicted axial velocity (Fig. 4a) is in good agreement with the measured one for t~ > 20 ~ Close to the blade (~ < 20~ the downwards directed, predicted axial velocity is lower than measured. Weehsler et al. [17,18] reported higher axial velocities than measured here, caused by the infinitely thin blade. In the present simulation, the discrepancy may be due to a rather high value of k (Fig. 7a) which may cause overprediction of effective viscosity and underprediction of velocity. The prediction of the radial velocity shown in Fig. 5a agrees with measurements. In the mid-impeller blade, just outside the radial extent of the blade, r/T = 0.171, the predicted axial velocity follows closely the measured curve for ~ > 30 ~ (Fig. 4b). For 0 < 30 ~ the peak is lower and the flow is downwards in the vicinity of the blade. These deviations presumably reflect the slight difference in the location or velocities of the tip vortex. Predicted radial and tangential velocity profiles (Figs. 5b and 6b) have the same form as the experimental ones, but with an angular displacement by 5~ ~ Circumferential profiles of velocity components at the radial location of the trailing vortex outer edge (r/T = 0.171) in the axial plane z/T = 0.289 are compared in Figs. 4c, 5e and 6c. The predicted profiles show good qualitative agreement with measured data, with the exception of the missed tangential velocity peak in just below the impeller about 10~ behind the blade. The axial location of the trailing vortex is well predicted in the simulation. The predicted absolute values of axial and radial velocities are smaller than measured, which may be caused by a slight difference in the radial position of the predicted vortex. The damping effect caused by the discretization errors or turbulence modeling discussed earlier is another possible cause for the underprediction of velocities. It is evident that the near-wall approach provides the most accurate results in the vicinity of the impeller.

270

Figure 4. Circumferential profiles of UdUap at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z ~ = 0.329, r/T = 0.171, (c) z/T = 0.289, r/T = 0.171. Symbols as in Fig. 1.

Figure 5. Circumferential profiles of Ur~e'Ut~pat three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T = 0.329, r/T = 0.171, (c) z/T= 0.289, r/T = 0.171. Symbols as in Fig. 1.

Figure 6. Circumferential profiles of Utand'Unp at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T= 0.329, r / T = 0.171, (c) z ~ = 0.289, r/T= 0.171. Symbols as in Fig. 1.

Figure 7. Circumferential profiles of k / UT~p at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T = 0.329, r/T = 0.171, (c) z/T = 0.289, r/T = 0.171. Symbols as in Fig. 1.

271 4.3. Angle-resolved turbulent kinetic energy The circumferential profiles of the scaled turbulent kinetic energy, k/U,2~p, are compared

at the same locations as velocities in Fig. 7. In the mid-impeller plane, inside the impeller swept volume, k is clearly overpredicted, especially immediately behind the blade. The most significant effect caused by near-wall treatment of turbulence can be seen in front of the blade. There, the near-wall modeling suggests a relatively low level of k compared to the wall function method predictions. In the mid-impeller plane, outside the impeller swept volume, the agreement between calculated and measured k is relatively good 10~ ~ behind the blade. The predicted maximum value of scaled k is over 60% of the measured peak, agreeing with earlier simulations [17,18]. However, the predicted maximum location is about 10~ behind the blade. Immediately below the impeller, the level of k is fairly well predicted although all details of the experimental profile of k are not reproduced in the computation. As with the velocities, the best results in and near the impeller swept area are obtained with the near-wall model. 5. CONCLUSIONS Numerical simulation with a high density grid and k-e turbulence model showed that the tip vortex trailing the impeller blade could be captured with high precision. The predicted vortex has the same form and location as the measured one but dies out earlier. The average flow field outside the impeller region can be simulated with fair accuracy even with a lowdensity grid and wall functions. A dense mesh, preferably combined with near-wall turbulence modeling, is required for detailed prediction of the flow field in the vicinity of the impeller. The one-equation turbulence model used in the near-wall region in the current simulation provided the most accurate results both in the impeller swept area and in the bulk region of the tank. ACKNOWLEDGEMENTS We are grateful to M. Schafer for providing us the experimental data for comparison and wish to thank K. Wechsler and J. Majander for useful discussions. This work was financially supported by the National Technology Agency of Finland (Tekes). REFERENCES

1. H. Wu and G.K. Patterson, "Laser Doppler Measurements of Turbulent Flow Parameters in a Stirred Mixer", Chem. Eng. Sci., Vol. 44, pp. 2207-2221 (1989). 2. V.V. Ranade and J.B. Joshi, "Flow generated by pitched blade turbines I: Measurement using laser Doppler anemometry", Chem. Eng. Commun., Vol 81, pp. 225-248 (1989). 3. G.B. Tatterson, "Fluid Mixing and Gas Dispersion in Agitated Tanks", McGraw-Hill, New York (1991).

272 4.

L.M. Nouri and J.U. Whitelaw, "Particle Velocity Characteristics of Dilute to Moderate Dense Suspension Flow in Stirred Reactors", Int. J. Multiphase Flows, Vol. 18, pp. 2133 (1992). 5. K.J. Myers, R.W. Ward and A. Bakker, "A Digital Particle Image Velocimetry Investigation of Flow Field Instabilities of Axial-Flow Impellers", J. Fluids Eng., Vol. 119, pp. 623-632 (1997). 6. P. Mavros, C. Xuereb and J. Bertrand, "Determination of 3-d Flow Fields in Agitated Vessels by Laser-Doppler Velocimetry: Use and Interpredation of RSM Velocities", Trans IChemE, Vol. 76, Part A, pp. 223-233 (1998). 7. M. Schafer, M. Yianneskis, P. W~ichter and F. Durst, "Trailing Vortices around a 45 ~ Pitched-Blade Impeller", AIChe J., Vol. 44, pp.1233-1245 (1998). 8. M. Schafer, personal communication. 9. V.V. Ranade, J.B. Joshi and A.G. Marathe, "Flow generated by pitched blade turbines II: Simulation using k-emodel", Chem. Eng. Commun., Vol 81, pp. 225-248 (1989). 10. S.M. Kresta and P.E. Wood, "Prediction of the Three-Dimensional Turbulent Flow in Stirred Tanks", AIChE J., Vol. 37, pp. 448-460 (1991). 11. J.Y. Murthy, S.R. Mathur and D. Choudry, "CFD simulation of flows in stirred tank reactors using a sliding mesh technique", IChemE Symposium Series No 136, pp. 341348 (1994). 12. J.Y. Luo, R.I. Issa and A.D. Gosman, "Prediction of impeller induced flows in mixing vessels using multiple frames of reference", IChemE Symposium Series No. 136, pp. 549-556 (1994). 13. E.O.J. Majander and M.T. Manninen, "Numerical simulations of flow induced by a pitched blade turbine: Comparison of the sliding mesh technique and an averaged source term method", 3rd Colloquium on Process Simulation, 12-14 June 1996, Helsinki University of Technology, Espoo (1996). 14. E.O.J. Majander and M.T. Manninen, ''Numerical simulations of flow induced by a pitched blade turbine: The multiple reference frame technique", Technical Report LVT 3/98, VTT Energy, Espoo (1998). 15. J. Syrjanen and J. Majander, "Numerical modelling of flow induced by a propeller impeller: Fluent/UNS and CFX 4 simulation results", Technical Report LVT 5/98, VTT Energy, Espoo (1998). 16. J. Derksen and H.E.A. Van den Akker, "Large Eddy Simulations on the Flow Driven by a Rushton Turbine", AIChE J., Vol. 45, pp. 209-221 (1999). 17. K. Wechsler, M. Breuer and F. Durst, "Steady and Unsteady Computations of Turbulent Flows Induced by a 4/45 ~ Pitched-Blade Impeller", J. Fluids Eng., Vol. 121, pp. 318-329 (1999). 18. K. Wechsler, "Detailed Calculations in Singlephase Stirred Tank Reactors", Stirring and Mixing, International Seminar, 25-28 October 1999, LSTM-Erlangen, (1999). 19. FLUENT 5 User's Guide, Vol. 2, Fluent Inc., Lebanon, NH (1998). 20. J. Daeles-Mariani, D. Kwak and G. Zilliac, "On numerical errors and turbulence modeling in tip vortex flow prediction", Int. J. Num. Meth. Fluids, Vol. 30, pp. 65-82 (1999).

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

273

Comparison of CFD Methods for Modelling of Stirred Tanks G.L. Lane 1, M.P. Schwarz 1, and G.M. Evans 2 ~CSIRO Minerals, Box 312 Clayton Sth, Victoria 3169, Australia 2Dept of Chemical Engineering, University of Newcastle, New South Wales 2308, Australia

Abstract Simulation by computational fluid dynamics (CFD) is becoming an increasingly useful tool in analysis of the flow in mechanically stirred tanks. However, the development of accurate and efficient modelling methods is a continuing process. One significant complication in modelling of baffled stirred tanks is accounting for the motion of the impeller, since there is no single frame of reference for calculation. A number of approaches have been taken to this problem. In some cases an empirical model is provided for the impeller, whereas other methods are capable of predicting the effect of the impeller directly. In the latter category are the Sliding Mesh and Multiple Frames of Reference (MFR) methods. Results are presented for simulation of a standard configuration tank stirred by a Rushton turbine. Using the same geometry and finite volume grid, the fluid flow is simulated using both these methods. The Sliding Mesh and MFR methods are discussed and compared with respect to computation time and accuracy of prediction of mean velocities and turbulence parameters. It is found that the MFR method provides a saving in computation time of about an order of magnitude. Predicted mean velocities using both methods are compared with experimental data, and it is found that both methods provide good agreement with experimental data. Turbulence parameters are also compared with experimental data. It is found that both methods significantly underpredict the values of specific turbulent kinetic energy and rate of dissipation of turbulent energy. However, for the same grid density, the MFR method provides an improvement in the predictions. 1. INTRODUCTION Mechanically-stirred tanks are widely used in the process industries to carry out many different operations. These mixing tank operations exhibit complex three-dimensional fluid flow, leading to considerable uncertainty in design and scale-up, so that there is a need for more sophisticated methods for understanding the flow and contacting patterns. The progress in recent years in computational fluid dynamics (CFD) has led to increasing interest in the simulation of mixing tanks by this method, and CFD studies have been reported in the literature since the early 1980s. However, the development of modelling methods is a continuing process, and at each step of development it is necessary to assess the accuracy of CFD modelling by comparison against experimental data.

274 Table 1. Nomenclature

B g k P t U e

body force (N m-3)' acceleration due to gravity (m s 2) turbulent kinetic energy (m 2 s -2) pressure (N m -2) time (s) velocity (m s1) turbulent energy dissipation rate (m 2 S"3)

/2 p

viscosity (Pa s) fluid density (kg m -3) angular velocity (rad s l)

Subscripts L T

laminar turbulent

In most stirred tank configurations, baffles are fixed to the tank wall, which introduces a complication in the CFD modelling, since due to relative motion between the impeller and baffles, a single frame of reference is not available for carrying out computations. Various approaches have been taken to account for the impeller motion. These approaches vary in their range of applicability and the accuracy of the results. In some approaches the effect of the impeller is modelled empirically. For example, the impeller may be modelled by prescribing experimentally measured quantities as a boundary condition [1]. Such an approach limits the predictive capabilities of the model, and additionally, the method fails to capture the full details of the flow within the impeller. Details of the flow within the impeller region may be important for modelling multiphase flows such as gas-liquid flow, where there is a strong interaction between impeller and gas. Recent advances in CFD have led to the development of fully predictive methods for modelling baffled stirred tanks. Several approaches are available. One of the most frequently used methods at present is the Sliding Mesh method and a number of papers [e.g. 2, 3, 4] are available describing the method and discussing its accuracy. However, this is a timedependent method, which has the disadvantage of being highly computationally intensive. Therefore, several methods have been developed, in which the impeller can be modelled directly while making a steady-state approximation to the fluid flow. These methods include the Multiple Frames of Reference method [5], the Inner-Outer method [6] and the Snapshot method [7]. The present paper presents results for simulation of a baffled tank stirred by a Rushton turbine using two alternative methods, namely the Sliding Mesh and the Multiple Frames of Reference methods. In each simulation the same geometry is used and is represented by the same finite volume mesh, thus enabling comparison of the methods. Results are assessed in terms of computational efficiency and accuracy of the predictions as compared against available experimental data [8, 9, 10]. Often in CFD studies, accuracy of results is only assessed with respect to mean velocities. However, the turbulent parameters such as the turbulent kinetic energy, k, and the rate of dissipation of turbulent energy, e, are also of great importance in modelling of a range of mixing tank applications, since various mathematical sub-models are typically formulated as functions of k or e. Examples include modelling of micromixing, heat transfer to surfaces, interphase mass transfer, turbulent dispersion in multiphase flows, and break-up or coalescence of bubbles or droplets. Hence, as well as comparing against mean velocity measurements, it is also important to obtain accurate predictions of turbulence quantities.

275 Results are compared against turbulence data for k and e, obtained from several sources [8, 9, 10]. The CFD model matches the geometry of the mean velocity measurements obtained by Hockey [8]. Data from other sources is based on measurements using different tank sizes [9, 10], so it is assumed that the values can be scaled if non-dimensionalised by appropriate reference quantities. Comparison of several modelling methods has been previously presented by Brucato et al. [11]. Their work compares the impeller boundary condition method, the Sliding Mesh method and the inner-outer method. This paper provides additional new information specifically comparing the Sliding Mesh and MFR methods. 2. CFD M E T H O D The commercial code CFX4.2 was used to obtain numerical solutions to the equations for conservation of mass and momentum for an incompressible fluid using a finite difference mesh. Due to turbulent flow, the equations are solved in Reynolds-averaged form and may be given as: V.U-0 (1) o~(pU.__.__.~)+V. (pU|

3t

= - V P + V. ((#L + Pr )(VU + (VU) T ) + pg + B

(2)

Closure of the equations is obtained using the standard k-e. model to calculate the turbulent eddy viscosity BT. The tank geometry is the same as that used in the simulation by Luo et al. [3], for which Laser Doppler measurements of mean velocities are reported by Hockey [8]. The tank diameter, T, is 0.294 m, and the tank has a typical "standard" configuration, consisting of a six-bladed Rushton turbine with diameter D, of T/3, and four equally spaced baffles of width T/IO. The impeller clearance is T/3, blade height is D/5, blade length is D/4 and disc diameter is 0.75D. For the CFD simulation, this geometry is approximated (Figure 1) by a non-uniform finite difference grid, which has 48, 39 and 60 cell divisions in the axial, radial and azimuthal directions respectively. To reduce the amount of computational time required, a 180 ~ section of the tank is modelled assuming symmetry. The baffles, impeller disc, and impeller blades are treated as zero thickness walls and the impeller shaft is treated as a solid zone. The liquid is specified as water and the impeller rotational speed is 300 rpm, corresponding to a Reynolds number of 48,000. The Sliding Mesh method is a feature available in the commercial code CFX Version 4.2. In this method, the finite volume grid is divided into two domains, an inner domain which rotates with the impeller and an outer domain which is stationary. This is a transient method where at each time step the inner grid is rotated by a small incremental angle, and the flow field is recalculated taking into account the additional velocity due to the motion of the grid. Variables are interpolated between domains using an "unmatched grid" interface. The simulation was run from an initial condition of zero velocity until the developed flow pattern became periodically repeatable, indicating that a "steady-state" was reached. This was done in two stages: in the first stage coarse time steps were used corresponding to 30 ~ rotation of the impeller per step. After 10 full rotations of the impeller, steps were reduced to 12~ for a further 3 revolutions. The simulation was repeated using the Multiple Frames of Reference method. This feature is not currently available in CFX4 but was implemented using user-supplied Fortran routines.

276 The method has been previously reported by Luo et al. [5] and Tabor et al. [ 12], where further details are available. In this method, the impeller is also modelled directly and there are again two domains, a zone surrounding the impeller and a zone for the rest of the tank. The interface between rotating and stationary frames of reference was set at an axial distance +0.25D from the impeller centreline and radially at r = 0.75D. Two reference frames are used, so that in the impeller zone there is a rotating frame of reference in which the impeller appears stationary. In the rotating frame of reference, there is a body force term added to the momentum equation consisting of the Coriolis and centrifugal forces according to: B =-2pf~ | pf~ | (fl | X) (3) In this case a steady-state solution is calculated neglecting the time derivative term in equation (2). At the interface between the domains, the velocities are corrected by an implicit coupling method [5, 12] and spatial derivatives of the azimuthal velocity are also corrected. The simulation was run until the normalised sum of mass residuals was reduced to 103, which required approximately 600 iterations. 3. RESULTS AND DISCUSSION Simulation of the stirred tank has been carried out using both the Sliding Mesh and Multiple Frames of Reference methods. The results include predictions of mean velocities and the turbulence parameters k (the specific turbulent kinetic energy) and e (the rate of dissipation of k). An important finding is that the MFR method is much more computationally efficient due to the steady-state calculation, which requires only a fraction of the number of iterations, and achieves a solution with a computer processing time one tenth of that required for the Sliding Mesh method. This is in agreement with the findings of Luo et al. [5]. This may be an important advantage, particularly in modelling multiphase flows, where there are approximately twice as many equations to be solved, and convergence of the solution is substantially slowed due to the complexity of multiphase physics and the requirement to allow the distribution of the dispersed phase to evolve. In the authors' experience [13], the computational requirements of the Sliding Mesh can become overly excessive for simulating multiphase flows, and it has been shown [13] that gas-liquid- flow in a stirred tank can be modelled by the MFR method with much greater efficiency. To assess the suitability of the MFR method as an alternative to the Sliding Mesh method, the accuracy of these methods needs investigation. Results have been compared against data for radial profiles of the inean velocities at the level of the impeller mid-plane and several other positions in the tank using data obtained from Hockey [8] and also from Wu & Patterson [9]. In addition, turbulence quantities are compared against data of Wu & Patterson [9] and Deglon et al. [10]. Several of the comparisons are shown in Figures 2-5, although more extensive comparisons have been done which confirm the general findings stated here. For the mean velocity components, it is found that both methods give good agreement with experimental data, with only small discrepancies at certain positions. The same findings were obtained by Luo et al. [5]. Comparison with experimental k and e values shows that the Sliding Mesh method substantially underpredicts the values of these turbulence parameters in the impeller discharge stream. With the MFR method, the predicted values are significantly higher, yet the peak in the turbulence just off the blade tip is still underpredicted by about 50%. Closer to the wall, the two simulation methods give similar results and agreement with measurements improves. Values of k and e have been also been compared for several positions in the bulk of the tank,

277

and it is found that the Sliding Mesh and MFR methods tend to predict similar values. Values of k are actually in fair agreement with experiments for many positions, except near the centre below the impeller (Figure 5). Comparison with e in the bulk of the tank indicates substantial underprediction by the simulations. However, the accuracy of the experimental data might be called into question in this case, since the experimental method [10] does not predict e directly, and further confirmation might be needed. Ng et al. [14] have also assessed the accuracy of predictions of k in the impeller discharge stream for simulations using the Sliding Mesh method of a tank stirred by a Rushton turbine. They also found that the peak in the turbulence near the impeller tip was substantially underpredicted. By repeating the simulation with higher grid resolution and an embedded grid around the impeller blades, it was found that improved predictions of k were obtained, which was probably due to better prediction of velocity gradients around the impeller. However, at a grid density of 400 000 cells, the prediction of the peak k value was still only about 50% of the measured value. In the present results, it is found that the Multiple Frames of Reference method gives substantially improved predictions of k and e in the impeller stream compared with the Sliding Mesh method, for the same grid density. It is possible the MFR method could provide even better results using a higher grid resolution. The low predicted values of the turbulence quantities point to the need for caution when extending the CFD modelling to investigate phenomena in stirred tanks such as heat transfer or multiphase flow, where additional equations may need to be calculated as functions of the predicted k and e values. An interesting point is that if the total energy dissipation in the tank is calculated by integrating e over the tank volume, the power will be underpredicted due to low e values. Yet, it has been found [2] that if the pressure distribution over the impeller blades is used to calculate torque on the impeller, a good prediction of power input can be obtained. It might be possible for e values to be corrected before being used in other equations (e.g. for calculating bubble or drop breakage). If it could be assumed that the predicted pattern of distribution of e is roughly correct, then values could be scaled by a correction factor so that the integration of e values matches the power based on impeller torque. There is a need for more accurate modelling of the turbulence. One approach has been to increase the grid density, particularly around the impeller [ 14]. Measurements in the impeller stream have indicated that the turbulence is anisotropic [9], although in the bulk of the tank the turbulence may be isotropic. This implies that the assumption of isotropic turbulence in the k-e model may not be valid. To improve the prediction of turbulence quantities, it may be necessary to use a different turbulence model, for example a Reynolds stress model. However, such a model is more computationally demanding due to six additional equations which need to be solved, and for multiphase flows, the computation time may be excessive. In addition, the Reynolds stress model cannot be easily implemented in CFX4 when using the MFR method. 4. CONCLUSIONS Fluid flow in a baffled stirred tank has been simulated by computational fluid dynamics using both the Sliding Mesh and Multiple Frames of Reference methods. Comparison of mean velocities with experimental measurements indicates good agreement for both methods. The Sliding Mesh method is a time-dependent method where the time step is limited to a fairly small increment of rotation. For multiphase flows, this method may lead to excessive computation time due to the large number of time steps required. The MFR method provides a

278 means of reducing the computation time and so provides a more practical alternative, particularly for modelling gas-liquid contacting. The prediction of turbulence quantities is also important for modelling various aspects of stirred tank processes. Results show that both methods underpredict k and e, particularly in the impeller discharge stream, although the MFR method gives improved estimates. Caution should be taken when incorporating other mathematical models where equations are calculated as functions of k or e. CFD predictions of turbulence quantities in stirred tank simulations need further improvement.

REFERENCES 1. Bakker, A. & Van den Akker, H.E.A., 1994, "Single-phase flow in stirred reactors", Trans. L Chem.E., Vol.72, No. A4, pp. 583-593. 2. Lane, G. and Koh, P.T.L., 1997, "CFD Simulation of a Rushton Turbine in a Baffled Tank", Proc. Int. Conf. on Computational Fluid Dynamics in Mineral & Metal Processing and Power Generation, CSIRO, Melbourne, 3-4 July 1997, pp. 377-385. 3. Luo J.Y., Gosman, A.D., Issa, R.I., Middleton, J.C., & Fitzgerald, M.K., 1993, "Full Flow Field Computation of Mixing in Baffled Stirred Vessels", Trans. L Chem.E., Vol.71, Part A, pp. 342-344. 4. Lee, K.C., Ng, K. & Yianneskis, M., 1996, "Sliding Mesh Predictions of the Flows around Rushton Impellers", Fluid Mixing V, L Chem.E. Symposium Series, No. 140, pp. 47-58. 5. Luo, J.Y., Issa, R.I. & Gosman, A.D., 1994, "Prediction of Impeller Induced Flows in Mixing Vessels using Multiple Frames of Reference", L Chem.E. Symposium Series, No. 136, pp. 549-556. 6. Brucato, A., Ciofalo, M., Grisafi, F. & Micale, G., "Complete Numerical Simulation of Flow Fields in Baffled Stirred Vessels: the Inner-Outer Approach", L Chem.E. Symposium Series, No.136, pp. 155-162. 7. Ranade, V.V. & Dommeti, S.M.S., 1996, "Computational Snapshot of Flow Generated by Axial Impellers in Baffled Stirred Vessels", Trans. L Chem. E., Vol. 74, Part A, pp. 476-484. 8. Hockey, R.M., 1990, PhD Thesis, Dept. of Mech. Eng., Imperial College of Sci., Tech. & Medicine, London. 9. Wu, H., & Patterson, G.K., 1989, "Laser-Doppler Measurements of Turbulent-Flow Parameters in a Stirred Mixer", Chem. Eng. Sci., Vol. 44. No. 10, pp. 2207-2221. 10. Deglon, D.A., O' Connor, C.T., & Pandit, A.B., 1998, "Efficacy of a Spinning Disc as a Bubble Break-Up Device", Chem. Eng. Sci., Vol. 53, No. 1, pp. 59-70. 11. Brucato, A., Ciofalo, M., Grisafi, F. and Micale, G., 1998, "Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modelling approaches", Chem. Eng. Sci., Vol. 53, No. 21, pp. 3653-3684. 12. Tabor, G., Gosman, A.D., & Issa, R.I., 1996, "Numerical Simulation of the Flow in a Mixing Vessel Stirred by a Rushton Turbine", Fluid Mixing V, I. Chem.E. Symposium Series, No. 140, pp. 25-34. 13. Lane, G.L., Schwarz, M.P., & Evans, G.M., 1999, "CFD Simulation of Gas-Liquid Flow in a Stirred Tank", Proc. 3rd Int. Symposium on Mixing in Industrial Processes, Osaka, Japan, Sept. 19-22 1999, pp. 21-28. 14. Ng, K., Fentiman, N.J., Lee, K.C., & Yianneskis, M., 1998, "Assessment of Sliding Mesh CFD Predictions and LDA Measurements of the Flow in a Tank Stirred by a Rushton Impeller", Trans. I. Chem.E., Vol.76, Part A, pp. 737-747.

279

Figure 1. Finite volume mesh for stirred tank.

Figure 2. Comparison of axial velocity components obtained by each simulation method and experimental data (~ experimental [8], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial locations below and above the impeller.

Figure 3. Comparison of radial velocity components obtained by each simulation method and experimental data (~ experimental [8, 9], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial locations at impeller mid-plane and slightly above the impeller.

280 Normalised turbulent kinetic energy, x = 0.098 0.1 9,, 0.08 "

.==0.06

-

1

25-

~

.

Normalised energy dissipation rate, x = 0.098 41,

20"~

*

g15 -

~-

t~

0.04 ;,

0.02 '

_

~ , ~ ~

.~.,

O-

1.0

1.5

2.0

2.5

3.0

1.0

1.5

R/Ri

2.0

2.5

3.0

R/Ri

Figure 4. Comparison of k and E values obtained by each simulation method and experimental data ( , experimental [9], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial location at impeller mid-plane.

0.008 o~.~

Normalised turbulent kinetic energy, x = 0.049

0.006 0.004

9 _

Normalised energy dissipation rate, x = 0.049

0.6

"

I,

0.5>0.4 "

(I,

'

~.

.L.

~

'~

0.3-

0.002

0

O 0.0

0.5

1.0

1.5 R/Ri

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

R/Ri

Figure 5. Comparison of k and e values obtained by each simulation method and experimental data ( . experimental [8, 10], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial location below impeller in bulk flow, half-way between impeller and base.

10th European Conference on Mixing H,E.A. van den Akker and J.,l. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

281

Predicting the tangential v e l o c i t y f i e l d in s t i r r e d tanks using the Multiple Reference Frames (MRF) m o d e l w i t h v a l i d a t i o n by L D A m e a s u r e m e n t s Lanre Oshinowo a, Zdzislaw Jaworski b'c, Kate N. Dysterb, Elizabeth Marshall a and Alvin W Nienow b a Fluent Inc., 10 Cavendish Court, Lebanon, NH, USA 03766 b School of Chemical Engineering, University of Birmingham, Birmingham B 15 2TT, UK c also Chemical Eng. Faculty, Technical University of Szczecin, 70065 Szczecin, Poland Modeling the three-dimensional, transient motion of an impeller using the sliding mesh approach is the most rigorous and fully predictive method of CFD analysis of baffled stirred tanks. However, despite current advances in computational speed, this transient analysis takes a relatively long time. The multiple reference frame model, or MRF, is a steady state approximation of the impeller motion and is a viable alternative for such analysis. Recent work has shown numerical predictions of this flow field with the MRF model to give counterintuitive predictions where the tangential component of velocity is opposite to the impeller motion in large regions of the bulk liquid. Such predictions are also contrary to experimental observations. In the present work, the MRF model was used to correctly predict the tangential velocity distribution in a baffled, stirred tank containing a single-impeller (Rushton or pitched-blade turbine) under turbulent flow conditions. As a consequence of this work, a set of guidelines for applying the MRF model was determined and presented here. 1.

INTRODUCTION

In carrying out CFD optimization studies of impeller-tank geometry for macromixing, chemical reaction and precipitation, a fully predictive model for momentum transfer is required. Four such models for the impeller-tank interaction are reported in the literature: the multiple reference frames (MRF) model (Luo et al., 1993), the inner-outer model (Brucato et al., 1994), sliding mesh model (Murthy et al., 1994) and the force-field technique (Derksen and van den Akker, 1999). The latter approach was implemented within the lattice-Boltzman framework. The former methods were implemented within the control-volume approach. The sliding mesh model offers the advantages of modeling the transient behavior of the fluid motion in the tank but with a penalty of computational expense due to the time-dependent formulation. The inner-outer model divides the tank into two partially overlapping zones, one in a rotating frame, and the other in the stationary frame. An iterative matching of the solution obtained on the boundaries of the overlapping zones is required. The MRF model is the simplest being a steady-state approximation in which different cell zones move at different rotational speeds or on different axes of rotation. It requires least computational effort since the inner and outer frames are implicitly matched at the interface, requiting no additional iterative calculations. This facilitates rapid turnaround for design cycles and was chosen for this study. In addition, it does not require an a p r i o r i knowledge of the impeller discharge flow and turbulence. As an added benefit, the solution can be used as an initial condition for time-dependent sliding mesh simulations to avoid modeling the startup transients.

282 Often situations arise in which the swirl component of velocity in certain parts of the vessel, especially near baffles, is in a direction opposite to that of the impellers. CFD simulations of stirred tanks have predicted this reverse swirl in regions of the tank near the shaft, for example, where it is quite counter-intuitive. Furthermore, it has been observed in some studies that this reverse swirl persists throughout a significant part of the flow domain contrary to experimental observations (Harris et al, 1996, Venneker and Van den Akker, 1997). To date, there has been a general lack in the literature of validation studies comparing CFD predictions to measurements of the tangential velocity component in steady-state modeling of mixing tanks. Since an inert tracer or reactant is often added to the stirred liquid at the free surface, or close to it, the ability to correctly simulate the general direction (and magnitude which is typically small relative to the impeller tip speed) of the flow during the addition process is crucial. Macromixing, and subsequent stages in the mixing process, such as rni'cromixing, chemical reaction and precipitation can be significantly affected by the flow conditions at this initial process stage. The present paper models the three-dimensional turbulent flow in a baffled stirred tank, applying the MRF model to describe the motion of both radial Rushton and pitched-blade turbines. It will be shown by LDA measurements, that some reverse swirl does occur and this phenomenon can be predicted by CFD. It will also be shown that reverse swirl predicted in the CFD simulations, in locations where LDA does not show it, can be eliminated with improvements in the CFD problem definition and analysis. 2.

EXPERIMENTAL

The LDA system and signal processing used were essentially the same as those described elsewhere (Jaworski et al., 1996) except for the number of LDA channels, which was increased to two to measure the mean tangential and axial velocity components. A cylindrical, fiat-bottomed glass tank of internal diameter, T = 202mm, was provided with four full-length baffles of width 22mm. Distilled water was used at a height, H=T, with a square glass jacket placed around the tank to reduce the effect of wall curvature on the laser beams. This arrangement with the impeller drive was mounted on a traversing system, which could be moved in 3 dimensions with an accuracy of 0.01mm. Four, 6-bladed impellers were employed in the study, two Rushton and two pitched-blade turbines. Their diameters, D, were approximately T/4 and T/3, (51mm and 74mm for the Rushton turbines, and 48ram and 68ram for the pitched-blade turbines). All impellers had a blade width of 0.20D and thickness of 1 mm with the impeller centers positioned at a distance of H/3 off the tank bottom. Three horizontal planes were chosen for the location of measurement points, which had earlier indicated reverse swirl in CFD simulations. These were: 50, 90 and 120 mm above the impeller centers (or 117.3, 157.3 and 187.3 mm above the tank bottom). Three angular measurement positions, cr of 10, 45 and 80 degrees between adjacent baffles were used. The angle was measured in the direction of impeller rotation. The measurements were carried out at 18 radial distances from the tank axis, using 5 mm intervals and starting 10 mm from the axis. All data was collected in the ensemble average mode. Validation of the rms values was not considered since LDA measurements contain fluctuations due to blade passage frequency and large-scale instabilities as well as the "true" turbulence. RANS-based CFD aims to predict the latter but generally greatly underpredicts it (Brucato et al, 1998). Therefore, angle resolved measurements for comparisons in the discharge of the impeller to resolve the rms values were considered beyond the scope of this work since the primary interest is validating predictions of mean tangential flow far from the impeller. The average experimental error was estimated to be in the range of 1-2% of the impeller tip velocity.

283 3.

MULTIPLE REFERENCE FRAMES MODEL IMPLEMENTATION

The MRF model is used to model the impeller motion. The computational domain was divided into two zones, one in which the velocity is computed relative to the motion of the impeller and the second zone where the velocities are computed in a stationary frame relative to the walls and baffles. The velocity in the rotating frame is transformed to the stationary frame by V=Vr+~Xr (1) and the velocity gradient is Vv = Vv~ + V(~xr) (2) where v is the velocity in the stationary (absolute) frame and v, is the velocity in the relative (rotating) frame. At the boundary between the rotating and stationary zone, the diffusion and other terms in the governing equations obtain values for the velocities in the adjacent zone determined from Eqns. (1) and (2).

3.1

Azimuthal Averaging

The MRF model is best applied when the interaction between the impeller and the baffles is relatively weak. Hence, the optimal radial position of the interface between the stationary and rotating zones is roughly midway between the impeller blade tip and the inner radius of the baffles. However, when the impeller-baffle interaction is not weak (i.e. when D > T/2), the steady-state flow field can be dependent on the position of the impeller blades relative to the baffle position. Figure l(a) shows the impeller blades at three angular positions (0 ~ 15~ 30 ~ relative to the baffles. Figure l(b) shows the predicted tangential velocities at those angles and illustrates a periodic velocity profile repeating every 1/2 blade period with respect to the baffles. Therefore, for a 6 bladed impeller in a 4 baffle tank, an azimuthally-averaged mean flow field can be obtained over one-half of the blade period equivalent to 30 ~. In this work, the flow field was calculated with the impeller at two phase angles of 0 ~ and 15 ~ relative to the baffles and then averaged. 4.

NUMERICAL SIMULATION

The stirred tank with fluids of different viscosity was modeled in FLUENT 4.52. The three-dimensional, single-block, hexahedral element grids were automatically generated using MIXSIM 1.5, a mixing tank analysis tool. A 180 ~ sector was modeled due to the periodic symmetry in the flow. The condition of no slip was applied to all solid boundaries except for the liquid surface where a zero shear condition was applied. The simulations were considered conver~ed when the sum of the residuals was
4.1

Grid Sensitivity

The Sensitivity of the grid on certain integral quantities and the distribution of tangential velocity in the tank was investigated. Integral quantities: volume-averaged tangential velocity, w, flow number Fl, and power number Po, for different grid sizes, are shown in Table 1 for the laminar flow range. The solution appears more or less grid independent between 200K and 450K cells. However, the 197K grid solution is more similar to the 350K grid solution than the 255K grid. The reason is that the 197K grid was based on local refinement of the 68K grid in the impeller, baffle and tank wall regions, while the 255K grid was generated by doubling the 68K grid in the axial and radial directions.The mesh density in the impeller region has a significantinfluence on the integralquantitiesof flow and power draw. The refinement of the

284

Figure 1 (a) The dotted line represents the interface between rotating and stationary frames. (b) Normalized tangential velocity in the center of the discharge of a radial disk turbine at 3 different phase angles. Velocity profiles were obtained from a circular arc profile 1 mm outside the radius of the blade swept path. grid considers the regions where the root mean square deformation tensor is relatively high. The laminar study was performed as a precursor to future work beyond the scope of the present study. For turbulent flow (water; N = 290 rpm), the Power number approached the experimental value for the following grid sizes: 68K, Po = 4.16; 121K, Po --4.22; 350K, Po = 4.44; Po[expt= 4.69. The influence of the grid density on the distribution of negative swift in the turbulent flow field is illustrated in Figure 2. The maximum reverse swirl is predicted at the liquid surface. For the coarsest grid (Fig. 2(a)), the reverse swirl is predominant in the upper region of the tank. Above the grid size of 121K cells, the reverse swirl distribution is approximately invariant of the grid size. The Reynolds Stress Model was used.

4.2 Influence of the Turbulence Model The following turbulence models were compared: standard k-t; model, ReNormalization Group method (RNG) k-e model, and Reynolds Stress model (RSM). All turbulence simulations were performed first with the standard k-emodel before switching to the RNG k-e or RSM models. The effect of the turbulence models on the predictions of the tangential velocity distribution for different grid sizes is shown in Figure 3(a) to (c), plotted on the midbaffle plane. The k-e and RNG k-e models predict similar distributions of the reverse swirl in contrast to the RSM model, where nearly all the reverse swift near the shaft is eliminated. Table 1. Influence of grid size on the CFD predicted volume-averaged tangential velocity, Flow and Power numbers: Rushton turbine, D=T/3, N=317 rpm, ~t=39.5mPas; ]~residuals < lx10 5, Re=730. ||ll

i

Grid Size

Fl

Po

0.561 0.550 0.558 0.538 0.542

4.09 5.51 5.00 5.24 5.28

V

68,000 197,000 255,000 350,000 450,000

0.0518 0.0447 0.0518 0.0445 0.0455

i

285

Figure 2. Influence of grid density on the normalized negative tangential velocity wlvap distribution for the Rushton turbine (D=T/3) (a) 68K cells (b) 121K cells (c) 350K cells.

Figure 3. Normalized negative tangential velocity w/vap distribution for the Rushton turbine D=T/4, N=290rpm (a) k-e model; 467K cells, (b) RNG k-emodel, 121K cells, (c) RSM model; 121K cells, (d) RSM model; 121K cells, reduced MRF interface. Scale as in Figure 2. 4.3

Location of the MRF interface Analysis of the axial extents of the MRF interface revealed an influence on the tangential velocity distribution shown in Figure 3(c) and (d). The range of the tangential velocity affected by the change in the axial extents of the MRF boundary is between -0.02vap and 0. Round-off errors in the velocity transformation are likely to be responsible for this effect and the influence on the overall flow field is small. For the Rushton turbine impellers the MRF interface was set to +0.5D above and below the impeller while for the PBT impellers, the MRF interface was set from +0.5D to-2D.

286 5.

C O M P A R I S O N S W I T H E X P E R I M E N T A L DATA

The numerical predictions of the tangential and axial velocity are presented for the two Rushton and the two pitched blade turbines. Some of the predictions are compared with experimental results. All calculations were performed on a 160,000 cell grid using the RSM turbulence model with the axial extents of the MRF interface set to the values given in Section 4.3. Figure 4 shows the CFD predictions for the T/4 Rushton turbine to give the wellknown flow field in the mid-baffle plane, t~ = 45 ~ The discharge from the impeller is strongly radial with a negligible axial motion. There is a strong axial flow into the impeller from just above and below it whilst the velocity near the surface of the liquid is relatively low.

Figure 4. CFD flow field in the mid-baffle plane for the T/4 Rushton turbine. Figure 5 shows the normalized tangential velocity, w/yap, as a function of radial position, r, at three axial levels: 50, 90 and 120 mm as indicated in Fig. 4, and three angular positions 10~, 45 ~ and 80 ~. There is very good agreement between the numerical results and the experimental LDA measurements in the upper region of the tank where the velocity magnitude is low. The anisotropic nature of the turbulence in complex stirred tank flow influences the mean flow and the RSM model, accounting for the effects of curvature, swirl and rotation in the transport of Reynolds stresses, resolves the swirl correctly. Generally, the tangential velocity is positive. However, reverse swirl is measured near the liquid surface in the baffle region and behind the baffle in the t~ = 10 ~ plane and this is well predicted, except at +50mm and ~ - 10 ~ where the simulations overpredict the tangential velocity near the tank wall. To understand the flow pattern in the tank consider that as the flow discharges from the impeller and impinges on the tank wall, it spreads both upwards to the surface and downwards to the tank bottom (Fig.4), before returning axially to the impeller, describing the well-known double-loop pattern. However, the positive swirl interacting with the baffles needs to be conserved resulting in reverse swirl behind the baffles. The weaker predictions of tangential velocity occur at a location in close proximity to a transition between positive and reverse swirl requiring additional grid refinement to resolve the flow structure. Figure 6 shows the normalized axial velocity, u/vtip, as a function of the same radial, axial and the same angular positions as in Fig. 5. The positive axial flow near the wall and the

287

Figure 5. Radial profiles of normalized tangential velocity wlvtip at 50, 90 and 120 mm above the T/4 Rushton turbine. Symbols represent experimental data at 290 rpm in water.

Figure 6. Radial profiles of normalized axial velocity ulva,t, at 50, 90 and 120 mm above the T/4 Rushton turbine. Symbols represent experimental data at 290 rpm in water.

Figure 7. Radial profiles of normalized tangential velocity w/vap at 50 mm above the impeller and 10~ angular position. Symbols represent experimental data for a) the T/3 Rushton turbine at 290 rpm, b) the T/3 pitched blade turbine at 290 rpm and c) the T/4 pitched blade turbine at 290rpm.

288 negative axial flow in the center of the tank is predicted well. The positive axial flow is underpredicted in the baffle region due to the grid spatial discretization. The axial upward flow is strongest near the tank wall and baffle in Figure 6(c) for reasons discussed previously. The tangential velocity profiles for the three other impeller configurations are shown in Figure 7 (a-c) for the +50 mm and r = 10~ location behind the baffle. There is good agreement between the predictions and the measurements. The shape of the profiles are similar between similar impellers, but with different magnitude. For the Rushton turbine (Figure 7 (a)), the tangential velocity is quite uniform across the radius of the tank except in the baffle region close to the tank wall where reverse swirl can be observed. This flow feature was not captured by CFD for reasons of the grid. The pitched-blade turbine results (Figure 7 (b) and (c)) show a positive swift in the center of the tank and negative swift in the baffle

region. 6.

CONLUSIONS

The MRF model has been used to model the three-dimensional flow field in a stirred tank agitated with Rushton turbines and pitched-blade turbines. The tangential velocity distribution above the impeller has been correctly predicted in regions where the magnitude of the flow is low relative to the impeller tip speed. The occurrence of counter-intuitive reverse swirl in the simulation results has been identified as being caused by poor convergence, coarse grid density, the turbulence model of choice and the location of the MRF interface boundary. To eliminate reverse swirl and obtain accurate calculations of integral quantities, such as, power number, the grid must be sufficiently refined in the near impeller region, and adjacent to the walls experiencing direct impingement of the impeller discharge jet. Convergence of the solution, i.e., minimization of errors in variable equations, must be sufficiently deep. The k-ebased models were more likely to produce the reverse swirl than the RSM model. It can be concluded that steady-state modeling with the MRF is a valuable predictive tool in stirred tank analysis and design when single phase, turbulent flow occurs, provided the following are observed: 1. a refined grid near the impeller, and wall and baffles in the discharge stream is used; 2. a vertical interface boundary close to the mid-point between the tip of the impeller and the inner edge of the baffle and two horizontal ones which depend on the impeller type and should correspond to those used in this work. 7.

ACKNOWLEDGEMENTS

A part of the CFD work and all the experiments were undertaken at the University of Birmingham, UK, and EPSRC provided financial support for Z.J. 8.

REFERENCES

Brucato, A., Ciofalo, M., Grisafi, F., Micale, G. (1994). Inst. Chem. Eng. Symp. Ser., 136, 155 Brucato, A.,Ciofalo, M., Grisafi, F., Micale, G. (1998). Chem. Eng. Sci. 3653-3684. Derksen, J. and Van den Akker, H.E.A. (1999) AIChE J., 45 (2) 209-221. Harris, C.K., Roekaerts, D., Rosendal, F.J.J., Buitendijk, F.G.J, Daskopoulos, Ph., and Vreenegood, A.J.N. (1996) Chemical Engineering Science, 1569-1594. Jaworski Z., Nienow A.W. and Dyster K.N. (1996) Can. J. Chem. Eng., 74, 3-15. Luo, J.Y., Issa, R.I. and Gosman, A.D. (1994) IChemE Symposium Series, 136, 549-556. Murthy, J.Y., Mathur, S.R. and Choudhury, D. (1994) IChemE Symposium Series, 136, p. 341 Venneker, B.C.H. and Van den Akker, H.E.A. (1997) Recents Progres en Genie des Procedes, 11, no. 51,179-186.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

289

Numerical simulation of flow of Newtonian fluids in an agitated vessel equipped with a non standard helical ribbon impeller* G. Delaplace a, C. Torrezb, C. Andr6b, N. Belaubre a and P. Loisel ~'e alnstitut National de la Recherche Agronomique (I.N.R.A), Laboratoire de G6nie des Proc6d6s et Technologie Alimentaires, 369 rue Jules Guesde, B.P.39, 59651 Villeneuve d'Ascq C6dex, France bHautes Etudes Industrielles (H.E.I.), Laboratoire de G6nie des Proc6d6s, 13 rue de Toul, 59046 Lille Cedex, France eInstitut National de la Recherche Agronomique (I.N.R.A), Laboratoire de Biom6trie, Domaine de Vilvert, 78352 Jouy en Josas, France In this paper, a commercially available fluid dynamic (CFD) software package (FLUENT) was used to conduct numerical simulation for the flow field of Newtonian viscous fluids in a rounded bottom vessel equipped with an atypical helical ribbon impeller. The reliability of the numerical procedure was demonstrated on the basis of a comparison of the numerical results with experimental data in several ways. First for eight different Newtonian fluids, measured and numerically calculated power consumption were reported. Then, for different Newtonian fluids, measurements of axial circulation time data obtained by flow follower and conductivity method were compared to the predictions calculated by CFD approaches. Close agreement between the numerical computed CFD values and the experimental measurements were obtained. Finally, the flow velocity field computed under laminar regime for Newtonian liquids in the vessel were analyzed and discussed. 1. INTRODUCTION Mixing operation involving highly viscous Newtonian fluids are commonly encountered in the food and chemical industries. For such complex liquids, helical ribbon impellers are considered to be among the most efficient systems and therefore are frequently used in process industries (polymerisation reactors). In the last few years, the application of computational fluids dynamics (CFD) in stirred vessel gets more and more numerous, reflecting the growing maturity of commercial software. Although numerical analysis have been energetically attempted for a variety of phenomena in stirred vessels, there are very few reports concerning numerical simulations of the flow behaviour of Newtonian fluids in stirred vessel fitted with a helical ribbon impeller [1-4], because the shape of this agitator is not simple and therefore there are many complications arising from the complex design of the impeller/tank arrangement. In this paper, we firstly attempted numerical simulations of the flow behaviour of highly viscous Newtonian fluids in a rounded bottom vessel equipped with an atypical helical ribbon impeller using a commercially available fluid dynamic (CFD) software package (FLUENT). *This work was partiallysupportedby FEDERfounds.

290 Very few data are available for the helical ribbon impeller studied. Indeed this non standard helical impeller has two particularities: firstly, it is equipped with an anchor at the bottom in order to avoid the formation of a stagnant zone; secondly the size of the pitch ratio p/d ( p/d = 1.7 ) is greater than with a typical double helical ribbon mixer (0.5 < p/d <1.2). The reliability of the modelling approach was ascertained by verifying the numerically calculated results with experimental data in several ways. Both power consumption and axial circulation times data obtained experimentally were used for this purpose. 2. EXPERIMENTAL APPARATUS AND METHOD

2.1. Mixing system investigated A sketch of the mixing system investigated (atypical helical ribbon impeller and rounded bottom of the vessel) is shown in Figure 1. The dimensions of the mixing system are also reported in Figure 1. During all the experiments, the volume of liquid was kept constant and equal to 34.10 .3 m 3 which corresponds to a liquid depth, H of 0.409 m. The rotational speed of the impeller was measured by using a digital tachometer.

Fig. 1. Mesh, picture and geometrical parameters of the PARAVISC | mixing system studied (vessel diameter = 0.346 m; impeller diameter = 0.320 m; blades width = 0.032 m; impeller pitch = 0.560 m; impeller height = 0.340 m)

2.2. Fluids Seven Newtonian fluids covering a wide range of viscosities (1.7 Pa.s
2.3. Numerical simulation of the velocity distribution The numerical tools and solution procedure used by the commercial CFD finite volume software (FLUENT) to determine the fluid velocity profiles inside the vessel are well known [5] and will not be repeated in details here. In the following part, only details about the mixing system modeling options will be given. An unstructured mesh of 353573 tetrahedral cells has been built to represent the vessel. Note that the geometry of the mixing system used for simulation is very similar to the experimental equipment. The rounded bottom of the vessel has been accurately duplicated. Only additional parts of the impeller used to fix scrapers haven't been meshed (see Figure 1).

291 The boundary conditions at the impeller shaft and at the vessel walls were those derived assuming the no-slip condition. At the free surface, the boundary conditions were modeling by requiring that there is no normal velocity and zero normal gradient for all variables. A rotating reference frame was used to perform the simulation. 2.4. Data treatment used to compare experimental and numerical results 2.4.1. Power consumption A strain gauge torquemeter, in the range of 0 to 100 N.m was used to measure experimentally power consumption. For numerical approach, assuming that the power consumption supplied by the impeller is exclusively consumed as viscous dissipation of energy inside the vessel, we calculated the total power consumption P in the vessel by the summation of energy consumed in each volume control, throughout the vessel : p = # ~2

Vi

(1)

i

where /~ is the Newtonian viscosity of the tested fluids, Yi is the shear rate of cell i and Vi the volume of cell i. For each cell, ~ was deduced from the second invariant of rate of the deformation tensor A :A

9 =~/k:A

(2)

The results of power consumption measurements are presented in the form of the dimensionless relationship between the power number Po and the Reynolds number Re.

2.4.2. Axial circulation times Experimental axial circulation times data t e were obtained by two different methods: the flow follower technique and the conductivity method. Details of experimental set up are given elsewhere [6-7]. Both experiments were carried out and observed with the helix pumping upward (counter-clockwise direction of rotation) and downward (clockwise). When the passage of a freely suspended particle in a vessel is monitored (flow follower technique), t e is the average time taken by a particle to cross a given reference plane two times in the same direction. On the other hand, when the probe methods are used, the axial circulation time is defined as the time between two subsequent peaks on the response curve. For numerical approach, axial circulation times are obtained from the knowledge of both axial circulation flow rates Qa~ through an horizontal plane and the volume of liquid V in the tank: V

Axial circulation flow rate Q,~ is deduced from the numerical values of axial velocity u~x of each cell contained in the reference horizontal plane located at a height z above the bottom of the vessel:

292

aax - "1~ . [Uax,i[Ai

(4)

where Ai is the cross section area of cell i with the horizontal plane chosen. For this work, the two horizontal planes taken as reference to predict numerical values of circulation times were located at z = 0.230 and 0.295 m above the bottom of the tank (which corresponds respectively to z / H = 0.562 and 0. 721 ) 3. RESULTS AND DISCUSSION

3.1. Reliability of numerical method P o w e r consumption. Figure 2 shows for the laminar regime the comparison of the power curve obtained experimentally and numerically.

Fig. 2. Power consumption data obtained numerically (on the left) and experimentally (on the fight) for mixing system studied using Newtonian fluids. For both numerical and experimental methods, the Newtonian power curve obtained in the laminar regime are similar to each other and can be described by traditional relationships Po Re = K p. For O.1 < Re < 6 0 , by linear regression, we obtained Po Re = 312 for numerical results and Po Re = 315 for experimental results. By calculating the power consumption for a similar typical helical ribbon (not equipped with an anchor at the bottom) by the equation suggested by Delaplace and Leuliet [8], the product N p. Re = 249 obtained is 24% smaller. Thus we can assume that the excess of power consumption measured for our helical mixer is due to the anchor at the bottom of the impeller. Axial circulation times. Experimental results of axial circulation times for our mixing equipment suggest that the product n t c is constant in the laminar flow regime (Figure 3) and

independent of the pumping direction [6-7]. Using the least square method, we obtained for 10 <_Re <_40 n tc= 10.7 by flow follower method and we obtained for 10 <_Re <_100 n t c= 12.7 by conductivity method. Note that each point plotted on the axial circulation time

curve obtained by conductivity method is the result of a tracer injection and in this case t~ is deduced from the average period of the tracer response curve.

293

+ 25 Pa.s < ia < 33 Pa.s A 5.51 Pa.s < la < 6.29 Pa.s [3 3.21 Pa.s < kt < 3.46 Pa.s - 2.22 Pa.s < p < 2.41 Pa.s o 1.99 Pa.s < la < 2.09 Pa.s 9 Sol G; la = 1.72 Pa.s o 1.67 Pa.s < la < 1.69 Pa.s • 1.14 Pa.s < ~t < 1.21 Pa.s 9 0.46 Pa.s < la < 1.47 Pa.s AA

40

30

,

20

10

".8

0

i

i

i

i

....

i

40

30 | x.J

n tc = 1 2 . 7

o

. . . . . .

9 ,,i

O Sol D ; l.t = 4.3 Pa.s

20

o~_ ~ ~

10

i

. . . . . . .

10

100 Re, (-)

o

1 0

1

n tc = 10.7

A Sol F; la =2.8 Pa.s

~

J

1000 10

Re, (-)

i

i

i

I

,

,

i

100

Fig. 3. Axial circulation times data obtained experimentally by conductivity method (on the left) and by flow follower method (on the right) for mixing system investigated using Newtonian fluids. These experimental values of axial circulation number are qualitatively in agreement with those reported in the literature for typical double helical ribbon impellers (n t c between 5 and 25 [7]). It is difficult to make a more precise comparison with other helical mixers because the circulation number is dependent both on the mixing system and in a certain extent on the experimental technique used to measure it. Figure 4 shows numerical results obtained for dimensionless axial velocity vectors (u~ = Ua~/(ZC n D) in an r, 0 horizontal plane located at z / H = 0 . 5 6 2 when mixing Newtonian fluids (# = 2 . 8 2 P a . s , n = 3.66 rad.s -I , p = 1358 K g . m -3 , Re = 28.7 ) with the agitator rotate in counter-clockwise direction.

Fig. 4. Pumping action of the mixing system studied in an horizontal plane located at above the bottom of the vessel. O n the left: location of upward flows. O n the right: location of downward flows. z/H -0.562

294 Figure 4 shows clearly that upward flows U~x > 0)are generated by the impeller blade while downward flows (u~ < 0)are prominent in an area located between the helical blade and the stirring shaft. Theses results agree well with the experimental observations of tracer particles under same flow conditions. Indeed it has been observed that tracer particles descend around the stirring shaft and subsequently ascend in the vicinity of the vessel wall. Figure 5 shows for the laminar regime the results of axial circulation time curve obtained numerically. The product n t c attains a constant value again, n tc = 13.6. This value of circulation time is qualitatively in agreement with those obtained experimentally. Indeed the distribution of axial circulation time obtained by flow follower method is large, so the accuracy of this numerical result is quite acceptable (figure 5). 40-

= 25O

Sol F - ta = 2.8 Pa.s Re = 29 Counter -clockwise

o..

30,3 A 20-

20-

m~ Sol A; it= 60.0 Pa.s Sol B; la = 8.3 Pa.s * Sol C;/.t = 6.9 Pa.s o Sol D; la = 4.3 Pa.s - Sol F; la = 2.9 Pa.s + Sol O; ~t = 1.7 Pa.s

0

~10-

n tc =13.6

=

~ ~ - + O

10

-~

~ ~ - - ~

-~

5 -

0 V

Numerical approach -I

0.1

'

'

L''"'I

'

1

'

''""i

Re, (-)

'

10

'

'''"

1

V

~

~

~

v

~

V

~

~

v

A

v

= v

I=I v

= v

v ~

= v

l=l v

1=

o,,

d

,.4

,--'

~

~.~

~

v

v

1==,i

li,i

v

Fig. 5. On the left: Axial circulation time data obtained numerically with the mixing system investigated using Newtonian fluids. On the right: Histogram of axial circulation time distribution obtained experimentally for a Newtonian fluid by flow follower method. All these results regarding the power consumption and axial circulation times for different Newtonian fluids (1.7 Pa.s <# < 60 Pa.s) prove that numerical simulation method has sufficient reliability to predict flow behavior of highly viscous fluids in such mixing system, especially under laminar regime. 3.2. Analysis of numerical results obtained for the flow of Newtonian fluids Figure 6 shows respectively for various angular positions in the vessel (see bottom of Fig. 6) the numerical values of tangential, radial and axial velocities obtained at a given horizontal plane ( z / H = 0.5 62 ). In figure 6 radial locations of vertical arms (dashed lines) and helical

blades of the ribbon (dotted lines) have also been plotted. The results reported in Figure 6 show that the presence of the vertical cylinders which support the two blades generated significant local tangential and radial velocity components. This observation can be pointed out when comparing radius profiles of dimensionless tangential velocities generated close to the face of the blade (0 = 100 ~ and 0.83 < r* < 0.93 ) and along the vertical arms (0 = 2 ~ and 0.82 < r* < O.89 ).

295

Fig. 6. Dimensionless velocities obtained at an horizontal plane ( z / H = 0.562 ) when mixing Newtonian fluids ( # = 2.82 Pa.s , n = 3.66 rad.s -I, agitator rotate in counter-clockwise direction.

p =1358 Kg.m -3 ,

Re=28.7

).

The

Indeed in this case, the maximum dimensionless velocity is obtained at the outside face of vertical arms and is equal to 0.71 of the tip velocity zr n D. The maximum tangential dimensionless velocity along the blade (0 = 100 ~ and 0.83 < r* < 0.93 ) was only found to be equal to 0.45 of the tip velocity. Moreover, radial velocities components generated close to the face of the vertical arms (0 = 2 ~ and 0.82 < r* < 0 . 8 9 ) . was also found to be stronger than those generated along the

296 blades. Note that for angular position away from the blades and from the vertical arms of agitator (0 = 30 ~ 115 ~ and 170 ~ ) both radial and axial velocities were rapidly attenuated and take significantly smaller values Clu=land

lu:l <0J9

Results reported in Figure 6 show that axial flow is mainly generated by helical ribbon blades. Indeed values of axial velocities along the vertical arms ( 0 = 2 ~ and 0.82 <

< 0.89) are weak ( u ~
( Iu'~ l= 0. 25) generated by the helical blades of the agitator (0 = 100 ~ and 0.83 < r" < 0.93 ).

4. CONCLUSION In this article, numerical predictions based on CFD simulations of the flow of highly viscous Newtonian fluids in an agitated vessel with a non standard helical ribbon impeller were presented. The validity of the numerical method was ascertained by verifying the numerically calculated results with experimental data (Power consumption and axial circulation times). The analysis of numerical results for the flow of Newtonian fluids indicates that the impeller generates an important axial flow. The presence of vertical arms which support the blades produced a significant shift in the flow pattern, generating local radial flow in the vessel. REFERENCES 1. P.A. Tanguy, R. Lacroix, F. Bertrand, L. Choplin, E. Brito-de La Fuente, Finite element analysis of viscous mixing with a helical ribbon-screw impeller, AIChE J., 38 6 (1992) 939. 2. M. Kaminoyama, M. Kamiwano, Numerical analysis of reaction process of highly viscous liquids in a stirred vessel equipped with a double helical ribbon impeller, I. ChemE. Symposium Series 136 (1994) 541. 3. P.A. Tanguy, J. De la Villron, R. Labrie, J. Bousquet, D. Lebouvier, F. Bertrand, Evaluation of macromixing in a mechanically agitated vessel using chaos analysis, Rrcents Progr~s en Grnie des Procrdrs, 11 51 (1997), 259. 4. J. de la Vill6on, F. Bertrand, P.A. Tanguy, R. Labrie, J. Bousquet, D. Lebouvier, Numerical investigation of mixing efficiency of helical ribbons, AIChE J., 44 4 (1998) 972. 5. Creare.x, Inc., F L U E N T - Computational Fluid Dynamics Software User's Manual, Hannover, NH 1990. 6. G. Delaplace, J.Y. Dieulot, J.C. Leuliet, J.P. Brienne, Drtermination exprrimentale et prrdiction des temps de circulation et de mrlange pour un syst~me d'agitation hrlico'fdal, The Can. J. of Chem. Eng., 77 (1999) 447. 7. G. Delaplace, J.C. Leuliet, V. Relandeau, Circulation and mixing times for helical ribbon impellers - Review and Experiments, Experiments in Fluids, 28 (2000) 170. 8. G. Delaplace, J.C. Leuliet, Prrdiction des facteurs Kp et Ks pour des agitateurs ~ rubans hrlicoidaux traitant des fluides pseudoplastiques, Rrcents Progr~s en Grnie des Procrdrs, 11 53 (1997) 331.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

297

A contribution to simulation of mixing in screw extruders employing commercial C F D - s o f t w a r e M. Motzigemba, H.-C. Broecker, J. Priissa, D. Bothe, H.-J. Wamecke Chemical Technology and Engineering Department, University of Paderborn, D-33095 Paderborn, Germany aInstitute for Analysis, Martin Luther University Halle-Wittenberg, D-06099 Halle, Germany Single phase flow was simulated in a screw extruder. A translating reference frame technique was applied to be able to represent the screw rotation. The basic transformations and their implementation are presented together with details of the boundary conditions. In addition, the resuits of simulations using stationary, rotating and translating reference frames in an annular geometry of the same geometrical dimensions as the extruder geometry are compared with the analytical solution. In general, analytical and simulated data agree well. 1. I N T R O D U C T I O N The screw extruder is one of the most important processing machines for polymers, and it also takes over a growing number of tasks in the processing of chemical or pharmaceutical products and food. Examples are blending and compounding of polymers, mixing and compacting of detergents and processing of corn. Detailed knowledge of flow and mixing phenomena is important for selection and design of extruders, for optimisation of process control and scheduling and for control of mixed material properties. Computer simulations offer practical means to apply the full capabilities of material science and engineering to optimise extrusion processes. Extrusion is quite complex and involves different phenomena from thermodynamics, tribology and fluid dynamics. In addition, movement of the complex geometry is to be described. Even simulation of mixing in the subsystem dominated by continuous fluid flow is not easy due to the combination of multi-component models with adequate reference frames. In such extruders, mixing is taking place due to the interaction of rotating screw and stationary wall, in twin-screw systems also due to the strong screw-screw interaction, effecting convective flow and shear stresses. The flow patterns in single-screw systems can be viewed as a modified Couette flow; the complexity of flow in a twin-screw system is drastically magnified by the high stresses in the intermeshing region between the screws. The resulting flow is 3-dimensional due to the passage of the screw flights, periodic (in the laboratory frame) and, for the common application with highly viscous fluids, laminar. For the most interesting twin-screw extruder there exist numerous studies using 2Y2D-models not considering the intermeshing region, geometry with unwound channels, creeping flow with 2D-models, 3D-models with simple rotating boundary conditions or 3D-models with translating reference frame. 3D-simulations are done for several instantaneous positions of extruder ele-

298

Fig. 1. Screw Geometry

ments in [ 1]. Except for [2], all these studies have two things in common: they use specialised, self-programmed, single-purpose software code and do not properly represent the movement of the detailed extruder geometry. At present, there is demand for using commercial CFD-Software for this problem to benefit from the advantages of commercial code, e.g. shorter learning times, comprehensive environment for pre- and postprocessing, tested code. However, current multipurpose CFD-programs do not provide a clear-cut modelling or transformation option for the double screw extruder problem. This may change with the current version of POLYFLOW [3, 4], which uses a time-dependend mesh superposition technique, but is restricted to 1-phase flow. In this work, the simpler geometry of a single-screw extruder is considered in an attempt to test the described concept of a translating reference frame for the representation of the movement of the more complex geometry in a twin-screw extruder. A geometry setup preprocessor GEOMESH was employed to create the detailed geometry of the screw region in a singleflighted single-screw extruder (Fig. 1).

2. SIMULATION OF THE SINGLE-SCREW EXTRUDER There is a periodicity of geometry that makes it sufficient to analyse only one spatial rotation of the screw flight. For the single screw extruder it is even possible to reduce the domain to a thin slice by using a cylindrical coordinate system formulation for velocity; as this reduction is not practicable for the twin-screw extruder, it is also not used for this test system. The CFD-program FLUENT, using the finite volume method with a colocated variable arrangement, is employed for simulation of the 3D-domain. The total number of hexahedral cells for the extruder grid is 105,536 (Fig. 2). Because of the non-stationary geometry in the laboratory reference frame, one chooses a moving reference frame, in order to make the geometry stationary.

2.1. Translating reference frame Stationary geometry is achieved by using a reference frame, that translates with U =T.n

(1)

where n denotes the rotational speed, which is set to 200 rpm. The translating reference frame is implemented by means of the boundary conditions, since there is no option to do this directly in the employed CFD-program.

299

The field equations for the steady state, isothermal flow of a Newtonian, incompressible fluid are solved.

Fig. 2. Extruder grid

V.v=0

(2)

p ((v

(3)

where v denotes the flow velocity vector, p density, p pressure and/~ the dynamic viscosity. To be able to compare the translating reference frame with the rotating reference frame, the above mentioned periodicity has to be represented by boundary conditions, which can easily be implemented in both reference frames. Since the employed grid has two inflow areas, the builtin periodic boundary condition can not be used. Therefore, in a first step of computations, fixed velocities are set at inlets and Neumann boundary conditions are set at outlets. For comparability the set velocities are obtained from the rotating reference frame calculations with zero channel pressure drop by averaging over certain areas and transforming them into the translating reference frame. In the second step, the result of the first is used as an initial solution, and iterative periodic boundary conditions are established by writing the velocity data of outlet cells into corresponding inlet cells before each iteration. Non-slip boundary conditions for the barrel, u = - U , v = 0 , w = 0 , and screw surfaces, u = - U , v = 0, w = 2~r r. n , are employed, with u, v, w denoting axial, radial, circumferential velocity and r the local radius in a cylindrical coordinate system.

2.2. Rotating reference frame Stationary geometry is established by using a reference frame fixed to the screw, rotating with constant n. Selecting a built-in model, the field equations for the steady state, isothermal flow of a Newtonian, incompressible fluid are solved.

300

p.((v. V)v)=-Vp

+ flV2v + F ~er~a

(4)

with an inertial force vector V Inertia = p ( - 2 ~

x v-nx(n

x y))

(5)

where 12 denotes the angular velocity vector and y the position vector in the reference frame. The inflow and outflow boundary conditions are set in a three step procedure. In the first step a pressure of zero is employed at all inflow and outflow areas. At the inlets, that cover the channel, the flow vector direction is set in the direction of the channel. At the inlets, that cover the radial flight clearance, the flow vector is directed in the circumferential direction. The velocities at outflow boundaries are area weighted averaged over certain areas and employed in the second step as fixed velocities at inlets; Neumann boundary conditions are employed at outlets. In the third step iterative periodic boundary conditions are established by writing the velocity data of outlet cells into corresponding inlet cells before each iteration. Besides the simulation with one fluid, a two-species simulation is also done using the single phase model with identical properties of the substances. The simulations are performed on the basis of the second step by extending the inlet boundary conditions by a mass fraction of species 1, thereby defining the inflow of two unmixed polymer layers. 3. SIMULATION OF AN ANNULUS / CIRCULAR C O U E T T E F L O W Solving the flow field equations for axial, laminar flow in an annulus, containing an inner cylinder rotating with the constant angular velocity to, yields the steady state solution [5]:

u=u(r)= 2~

ln r/ R i r2 - R~ -(R~ - R~ ) lnRa/R i

R:a + R? -(R; - R? ) v=O w= w(r) = ~

Ri

lnRa/Ri

Ralr-r/Ro R,/Ri-Ri/Ra

where Ri and Ra denote the radius of the inner and outer cylinder, respectively.

Fig. 3. Geometry and grid of the annulus

(6)

301 The radii are chosen equal to the inner and outer radius of the extruder screw, respectively. The rotational speed is equal to the extruder simulation. The annulus can be seen as an idealized single screw extruder, for which an analytical solution of the flow equations exists. To assess the quality of different reference frames, the flow field of the annulus is simulated using a stationary, a rotating and a translating reference frame and compared with the analytical solution. Non-slip boundary conditions are employed for both cylinders. The inflow and outflow boundary conditions are, like those for the extruder, set in a two-step procedure. In the first step fixed velocity boundary conditions are employed at the annular inlet and Neumann boundary conditions at the annular outlet; the employed velocity was calculated from a simple flat channel approach [6]. In the second step the already described iterative periodic boundary conditions are employed at inand outflow boundaries. The geometry and the hexahedral O-grid used are shown in Fig. 3.

4. RESULTS FOR THE SCREW EXTRUDER For a comparison, the flow patterns are assessed from the common viewpoint of the rotating reference frame. The flow, calculated with the translating reference frame in Fig. 4, Fig. 5 and Fig. 6, consists of a Couette-flow - like main channel flow with superimposed circulation flow in a normal slice of the channel and an axial countercurrent motion taking place in the flight clearance. It agrees well with the flow calculated with the rotating reference frame, Fig. 7. The results are compared via the relative velocity differences between the two reference frames, with respect to the average absolute velocity in the rotating frame. Fig. 8 presents the results at each cell in a row of a transversal channel slice for the three components of velocity.

Fig. 4. Velocity vectors in the extruder

Fig. 5. Translating reference frame calculated circulation flow normal to the channel

302

Fig. 6. Velocity contours in the extruder (in m/s): axial with Au = 0.025 m/s, radial with Av = 0.004 m/s and circumferential with Aw = 0.05 m/s.

r

Fig. 7. Rotating reference frame calculated velocity contours, cf. Fig. 6.

Fig. 8. Relative velocity difference versus transverse channel coordinate In the rotating reference frame it is also possible to investigate computationally the mixing of two layers of fluid, illustrating the mixing effects of the circulating flow, Fig. 9. The results show visually that the redirection of flow at the screw flights is most important for the mixing and that after nearly one pitch the size of unmixed material volumes is smaller than the local cell size but that concentration is far from being uniform.

303

Fig. 9. Concentration distribution in a sequence of slices perpendicular to the channel direction

5. RESULTS FOR THE ANNULUS Calculations of the velocity are performed for each reference frame. Selected results of the axial, radial and circumferential velocity for the translating reference frame are shown in Fig. 10.

Fig. 10. Radial velocity prof'fles

To quantify the differences between the numerical and the analytical solution, the differences between the axial velocities u, yielding the maximum differences, are assessed. Fig. 11 shows normalised u-differences plotted against a radial coordinate increasing outwards. These small differences are achieved by decreasing the normalised residual below 104; for calculations terminated after residuals droppedbelow 10-3, the differences are only below 10%. Therefore, a termination criterion of 10-4 has also been applied in the extruder simulations.

304

Fig. 11. Relative axial velocity difference versus local radial coordinate

6. CONCLUSIONS As an initial step, the flow in a 3-D geometry of an extruder-analogue annulus has been investigated analytically and numerically by using multi-purpose CFD-software employing different moving reference frames. The computations showed quantitatively good agreement with the analytical solutions. It has been found that the employed O-Grid produces artefacts in the radial velocity field. The flow in a single-screw extruder has also been investigated numerically employing the translating reference frame and testing it against the rotating reference frame. The extruder simulation awaits further improvement of the grid to increase the agreement to the level of the annulus simulation. The next stage of the investigation will be to apply the translating reference frame to the twinscrew extruder geometry.

REFERENCES [ 1] HONGFEI CHENG,I. MANAS-ZLOCZOWER:Study of Mixing Efficiency in Kneading Discs of CoRotating Twin-Screw Extruders.Polymer Engineering and Science, Vol. 37, No. 6: 10821090, 1997. [2] o. W0NSCH: Simulation yon Mischvorgi~ngen in Schneckenmaschinen. Technische Mechanik, 18: 1-10, 1998. [3] POLYFI,OW S.A., POLYFLOW version 3.7 User's Guide, Louvain-la-Neuve, Belgium, 1999. [4] TH. AVALOSSE, V. RUBIN: Analysis of mixing in corotating twin screw extruders through nu-

merical simulation. Unpublished, Polyflow s.a., Louvain-la-Neuve, Belgium, 1999. [5] K.C. CHUNG,K.N. ASTILL:Hydrodynamic instability of viscous flow between rotating coaxial cylinders with fully developed axialflow. Journal of Fluid Mechanics, Vol. 8 part 4:641-655, 1977. [6] N.P. CHERF_MISINOFF:Polymer mixing and extrusion technology. Dekker, 1987.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

305

Experimental and CFD Characterization of Mixing in a Novel Sliding-Surface Mixing Device J-M. Rousseaux, Ch. Vial, H. Muhr and E. Plasari Laboratoire des Sciences du G6nie Chimique CNRS - Ecole Nationale Sup6rieure des Industries Chimiques INPL, 1 rue Grandville BP 451 54 001 Nancy Cedex, France

ABSTRACT The mixing process of two reagents fed continuously in the Confined Mixing Zone of the Sliding-Surface Mixing Device is studied, both experimentally and using a commercially available CFD package (FLUENT | 4.5). A 3-D description of the reactor geometry is employed. Solutions of the Navier-Stockes equation along with the standard two equations k-e turbulence model provide both the velocity and energy dissipation fields. The finite rate reaction model from FLUENT is used to simulate the mixing process. Experimental visualizations in a Plexiglas prototype have been performed. A good agreement between experiment and simulation is obtained. The turbulence quantities provided by the simulations are used to calculate a micromixing time. These values are compared to experimental values obtained in a pervious work. Excellent qualitative and quantitative agreements are obtained between simulation and experiment.

1. INTRODUCTION Rapid chemical processes such as precipitations and polymerizations are very sensitive to mixing conditions. For this reason, the design and development of novel mixing devices have an increasing importance in the modern chemical industry. In this work, a novel mixing device, namely the Sliding-Surface Mixing Device, is presented and its main mixing characteristics are evaluated. We believe that this apparatus can be successfully used in a wide range of operating conditions concerning single and multiphase processes, Newtonian and non-Newtonian media of various viscosities, semi-batch and continuous operating modes.

2. PRESENTATION OF THE SLIDING-SURFACE MIXING DEVICE The Sliding-Surface Mixing Device (SSMD) is composed of a tank equipped with a rotating disk whose diameter is equal to 80 % of the inner diameter of the tank.

306 This device separates the reactor in two mixing zones : a Confined Mixing Zone and a Moderately Mixed Zone. The Confined Mixing Zone (CMZ), is situated between the sliding disk and the bottom of the reactor, where the reactants are introduced. One central feed tube introduces reagent A and four lateral symmetrical feed tubes, all situated at equal distance r from the central feed tube, introduce reagent B. The gap height h between the sliding disk and the bottom of the reactor is very small and can be varied. The double interest of creating such a confined mixing zone inside the stirred tank reactor is to provoke a highly reproducible contact of the reagents in a perfectly defined zone and to create a very high localized shear stress at the very point where the reagents are fed, so as to achieve a rapid micromixing of the reagent feed streams. The Moderately Mixed Zone is situated above the rotating disk, where products can undergo slower processes.

3. OBJECTIVES OF THIS WORK The micromixing or mixing at the molecular scale in the CMZ is very sensitive to design (position of the four lateral feed tubes, gap height) and operating conditions (flow rate, mixing speed). Micromixing efficiency as a function of the operating parameters has already been studied experimentally with a stainless steel prototype [ 1]. In the range of process parameters investigated, it was found that micromixing times down to 10 ms can be achieved. The precise mechanism of the mixing process of the reagents was not clear however. Schlichting [2] already studied the flow of fluid near a rotating disk in a housing. However, the mixing process of two reagent feed streams near a rotating disk in a confined mixing zone has never been studied before. This problem should be carefully treated because of the existence of a boundary layer thickness, which may be of the same order of magnitude as the gap height, and because the flow regime can be either laminar or turbulent, depending on the radial position of the fluid in the CMZ [1,2]. The aim of this work is to study the mixing of two reagent streams fed continuously in the Confined Mixing Zone of the Sliding-Surface Mixing Device. A visualization technique is used to investigate the local mixing of the reagents. Numerical simulations are performed presenting the transport and the reaction of the 2 reagents fed continuously in the reactor. These simulations give values for turbulence dissipation rates and kinetic energy, which allows a comparison with experimentally found micromixing time values [1 ].

307 4. R E A C T O R P R O T O T Y P E AND METHODS USED The SSMD investigated in this work is a new Plexiglas (transparent) prototype having a volume of 2.5 L, an inner diameter of 150 mm, a disk diameter of 120 mm and a disk height of 10 mm. The gap height h is constant and equal to 2 mm for all experiments carried out. Two lateral feed pipe positions are investigated: r = 17.5 mm and r = 40.0 mm. The visualization technique adopted in this work is the conventional instantaneous decolourization system composed of an acid-base neutralization with phenolphtalein as indicator. The commercial code FLUENT 4.51 is used to perform numerical 3D simulations of the flow in the CMZ of the SSMD. A neutralization reaction is implemented using the Finite Rate Reaction Model option of FLUENT in order to investigate the local mixing process.

5. VISUALIZATION OF THE MIXING PLUMES The experiments are carried out in a continuous mode using the system H2SO4 + NaOH. The normality of the aqueous solution of NaOH is 0.040 N. 0.15 g/L of phenolphthalein is dissolved in some aqueous ethyl alcohol and is then added to the 0.040 N aqueous solution of NaOH. The normality of the aqueous acid solution is 0.042 N, in small stoichiometric excess (5 %). When the acid and alkali solutions are pumped continuously with identical flow rates (acid in the central feed tube and NaOH in the four lateral feed tubes). The mixing plume reveals itself as a red coloured zone. Once stability of the observed phenomena is reached, the reactor is photographed. The visualization technique enables to observe precisely the extent of the reaction zone without using any intrusive probe. These observations make it clear that the feed tube position plays a major role on the mixing process in the CMZ (see Figures 2 and 3).

Figure 2. Visualization sketch of the bottom of the SSMD with r= 17.5 mm and N=20 s1.

308

Figure 3. Visualization sketch of the bottom of the SSMD with r = 40.0 mm and N=20 s 1.

6. TURBULENCE MODEL The two equations k-e model is used in this study. This model requires one order of magnitude less computation time than higher order models and gives predictions of the mean velocities that are of comparable accuracy. The validity of the cell dimensions used in the CMZ is verified by using the wall distance parameter y+ criterion [3] : y+

0.207k2

(1)

VE

For a given cell dimension choice and once convergence of the flow is reached, it is possible to check, using the calculated values of k and c whether the value of y+ is not smaller than 12. If y§ < 12, the dimensions of the cells next to the walls need to be increased. The turbulence quantities k and e are used to calculate the micromixing times at the reagent feed points using the classical result of Corrsin for scalar mixing time [3] : tm

=

2--k

(2)

E

7. RESULTS OF THE SIMULATIONS AND DISCUSSION The CFD simulations are performed using the standard k-e model. The 3D grid used in this work is composed of 24• (i.e. 114 576) cells. The dimensions of the grid used are exactly the same as the dimensions of the lab reactor studied experimentally. Due to the four lateral (and symmetrical) feed tubes, only one fourth of the reactor should be simulated.

309 A compromise solution has to be found in order to use a maximum number of cells for a better description of the flow in the neighborhood of the disk (knowing that the gap height is only 2 mm) satisfying at the same time the y§ cell criterion (see eq. (7)). Figures 4 and 5 show the values Of the y§ criterion at the first slice situated below the disk (i.e. 0.20 mm above the bottom of the disk). It is clear that the y§ criterion is verified on approximately the whole domain. Hence, we may consider that our cell dimensions are acceptable. In addition, in order to substantiate our application of the k-e model and our cell dimensions, it is interesting to cite the work of Launder and Sharma [4]. Indeed, they carried out a theoretical study on the application of the energy-dissipation model of turbulence near a spinning disk and found that the k-e model of turbulence predicts accurately the flow, heat and mass transfer in the neighborhood of a rotating free disk, even when the flow is laminar. In the CMZ, depending on the rotating speed of the disk and the lateral position r where the flow is studied, the flow can be either laminar or turbulent. This point will be discussed later in this section. In this study, two feed pipe positions (r = 17.5 mm and r = 40.0 mm) and two rotating speeds (N = 20 s 1 and N = 50 s 1) are investigated and only one reagent feed rate is simulated (QA = QB = 240 mL/min).

Figure 4. Wall criterion y§ values for the slice situated at 0.2 mm from the bottom of the reactor (N=20sl).

Figure 5. Wall criterion y§ values for the slice situated at 0.2 mm from the bottom of the reactor (N = 50 s-l).

Simulations predict well the influence of the feed pipe location on the mixing process. The simulated mixing plumes are presented in Figures 6 and 7.

Figure 6. Simulated mixing plume. r = 17.5 mm ; N = 20 s 1.

310

Figure 7. Simulated mixing ~lume. r= 40.0 mm ; N = 20 s-.

The simulated mixing plumes (presented in Figures 6 and 7) are in good agreement with the experimental visualizations shown in Figures 2 and 3. Consequently, the finite rate reaction model from FLUENT, which has been criticized in the past [5], performs well in predicting the influence of operating conditions on reaction zone positions in the Confined Mixing Zone of the Sliding-Surface Mixing Device. The mean values of the turbulence quantities k and e obtained from simulation are used to calculate the micromixing time according to equation (8). The influence of the lateral feed pipe position on the micromixing time is now studied in more detail. The rotational Reynolds number (at a distance r from the axis) of a rotating disk is defined as : r Ea9

Re(r) - ~

r 2 2 ~rN

- ~

(3) v v For the flow around a disk rotating in a housing, two flow regimes have to be considered [1,2] : - for Re(r) < 105, the flow is laminar, for Re(r) > 2 105, the flow becomes turbulent. -

The experimentally found micromixing time values are modeled as follows [ 1] : 1 t m = .f~

(4)

with the following expressions of ~ for the operating conditions investigated in this work : -for laminar flow "

0 = 2250(rN) 2 +0.065

- f o r t u r b u l e n t f l o w " 0 = 8 7 5 ( r N ) 11'4 +

2.3510-2

(5) (6)

311

For N = 20 s1, the flow is laminar for all radial positions 17.5 mm < r < 40.0 mm, whereas for N = 50 s1, the flow is laminar for r < 25 mm and turbulent for r > 25 mm. Figure 8 compares the experimental micromixing time values modeled by equation (10) with mean simulated values.

Figure 8 : Comparison between experimental model (equation (10)) and simulated values.

From Figure 8, it can be concluded that a very good agreement between the experimentally obtained model and the simulations is obtained. At N = 20 s-1, the micromixing time decreases steadily when the feed pipe position r increases. At this rotating speed, the regime is laminar in the whole range of the investigated radial positions. In this case, only equation (11) is plotted. On the contrary, at N = 50 sl , the flow regime changes from laminar to turbulent, thus both equations (11) and (12) are used. Figure 8 shows that simulated micromixing times go through a maximum, predicting qualitatively well the change of regime when the radial distance is increased. As a conclusion, the lateral feed pipe positions should be chosen with great care in order to obtain optimum micromixing conditions.

8. CONCLUSIONS The comparison of simulation maps with photographs obtained from decolourization experiments show that the numerical simulation by CFD codes can well describe the process of mixing in the Confined Mixing Zone of the Sliding-Surface Mixing Device. In a previous experimental work [1], a generalized relation obtained from theoretical considerations allowed the calculation of micromixing time as a function of SSMD process parameters (r, h, Q, N). In this study, the k-e model is used to determine the micromixing time. The comparison of results obtained by CFD simulation with those obtained experimentally shows that micromixing time values under the same operating conditions are of the same order of magnitude. These facts prove that the k-e model applied with appropriate cell dimensions is well adapted to simulate mixing in this new device and can be successfully used to predict mixing effects in sliding-surface mixing devices of different geometric dimensions operating in a wide range of process conditions.

312

REFERENCES 1. Rousseaux, J.M., L. Falk, H. Muhr and E. Plasari, "Micromixing Efficiency of a Novel Sliding-Surface Mixing Device", AIChE J., Vol. 45, N ~ 10, pp. 2203-2213, Oct. 1999. 2. Schlichting, H., Boundary Layer Theory, McGRAW-HILL Series in Mechanical Engineering, New York, Sixth Edition, 1968. 3. Kruis, F.E. and L. Falk, "Mixing and Reaction in a Tubular Reactor : a Comparison of Experiments with a Model Based on a Prescribed PDF", Chem. Eng. Sci., Vol. 51, No. 10, pp. 2439-2448, 1996. 4. Launder, B.E. and B.I. Sharma, "Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow near a Spinning Disk", Letters in Heat and Mass Transfer, Vol. 1, pp. 131-138, 1974. 5. Hannon, J., S. Hearn, L. Marshall and W. Zhou, "Assessment of CFD Approaches to Predicting Fast Chemical Reactions", Paper 188, AIChE Annual Meeting, Miami Beach, FL, Nov. 15-20, 1998.

NOTATIONS

A, B Chemical species C~ Cel C~2 Parameters used in the k-e model K N

QA QB r t,,, u' U + y

Kinetic turbulent energy Rotation speed Feed rate of species A in the central feed tube Feed rate of species B in the lateral feed tubes Radial position Micromixing time Fluctuating velocity Velocity vector Dimensionless wall criterion

m 2 s2 -1 s mL min 1 mL min ~ m s m s1 m s~

Greek letters e vt v /9

to

Turbulent energy dissipation Kinematic turbulent viscosity Kinematic viscosity Fluid density Function depending on Reynolds number (see equations (4), (5) and (6)) Angular rotation speed

Abbreviations CFD Computational Fluid Dynamics CMZ Confined Mixing Zone SSMD Sliding-Surface Mixing Device

W kg 1 - m 2 s-3 m 2 s-1 m: s-1 kg m -3 S-2

rad s -1

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

313

AN I N V E S T I G A T I O N O F T H E F L O W F I E L D O F V I S C O E L A S T I C F L U I D IN A S T I R R E D V E S S E L Wentao Ju a, Xiongbin Huang', Yingchen Wang", Litian Shi' and Bin Zhangb, Jingxian Yuanb 'College of Chemical Engineering, Beijing University of Chemical Technology bDaqing EOR Equipment Cooperation By using the PIV (Particle Image Velocimetry), the flow fields of polyacrylamide (PAM) aqueous solutions with different concentration and with noticeable elasticity when the PAM concentration is relatively high were measured at different rotational speeds for two types of agitator. Radial flow is dominant in the flow field of high-concentration PAM solution and there are stagnant flow and secondary flow taking place at some areas in the vessel. The flow field of pseudoplastic fluid in the stirred vessel was numerically simulated using the "black-box" model. The simulation results are approximately consistent with the experimentally measured flow pictures. The velocity component values are also approximately in accordance with the experimental ones. I.

INTRODUCTION

Although there are few researches on determining the flow field of polymer mainly because of its opacity by using LDA (Laser Doppler Anemometry), there are still researches using some other experimental techniques. By using streak photography and a TV camera recorder, Kuboi et al. (1986)t~l obtained photographs of flow patterns for high viscosity Newtonian and shear-thinning non-Newtonian fluids. Elson et al. (1986)t21 described the use of X-ray-heavy metal tracer method for the study of mixing patterns in opaque fluids in agitated vessels. The numerical simulation of flow field in stirred vessels can be dated back to 1970's, but the non-Newtonian flow simulation is rather new in the field of CFD (Computational Fluid Dynamics) and flourishes in 1990's. These simulations are usually concentrated on certain limited geometry. Besides, the determination of turbulent model constants is also rarely performed for non-Newtonian fluids to the same extent as for Newtonian fluids. Hua Lu et al. (1995)t31 simulated the non-isothermal flow of the Carreau fluid in an agitated vessel induced by scraped-surface blades. Venneker et al. (1997)[41 presented results about the simulation of the turbulent flow of a pseudoplastic fluid in a stirred vessel. A qualitatively correct solution was obtained with the standard k-r on condition that at the impeller tip experimental data are used as boundary condition. However, there are differences between predictions and measurements. The study of Anne-Archard (1997)t51 "To whom correspondence should be addressed: Beijing University of Chemical Technology P.O.Box230, Beijing, 100029, P.R.China; E-mail: [email protected]

314 aimed at studying the influence of viscoelasticity and inertia on a mixing operation in laminar regime which is done in a numerical way by solving a mass transport equation in a vessel equipped with a paddle impeller. The result is useful to further the numerically treating of viscoelastic fluid mixing despite the simple geometry studied and its assumption of 2-D flow. 2. E X P E R I M E N T A L Six PAM aqueous solutions of different concentration were used in this study. The mean molecular weight of the PAM is 1.2 • 107~1.5 • 107. Solution rheology was investigated using a RV-II coeentric cylinder viseometer and a Carri-Med Rheometer CSL52oo(TA Instruments) with a cone and plate measuring system. A fiat bottom cylindrical 0.5m inner diameter stirred vessel made of polymethyl methacrylate with four symmetrically distributed baffles was used. Two types of axial-flow impellers of different diameters and blade angles were used, each having two (0.3m in diameter) or three blades (0.184m in diameter). The impeller clearance (distance between impeller and vessel) is 0.085m. By using the PIV system established by our own, sets of observations were made and recorded on films for different concentration solutions. Two types of light sources were used, namely, the continuous I-Ge lamp and three-pulse Xenon lamp to form light sheet. With the multi-pulse lamp, the velocity directions of tracing particles can be determined accurately and easily. Spherical fluorescent particles of about 300~500 ~tm in diameter were used as the tracers. An IBAS image analysis processor was used to digitize the images recorded on the 35mm films. 3.

RESULTS AND DISCUSSIONS

3.1 Rheological Experiments The steady shear equations for PAM solutions with different concentration are shown in table 1. Table 1 The power law equation for PAM solutions (13.8~ RVII) Concentration of PAM solution (ppm) ,1-r~"-' / Pa.s 500 K=0.00943, n=0.902 1000 K=0.0543, n=0.734 2000 K=0.594, n=0.641 3000 K=0.728, n=0.448 4000 K=1.47, n=0.406 5000 K=2.57, n=0.354 The flow behavior index of each PAM solution is smaller than 110, whicia' indicates the degree of the pseudoplasticity of the solution, i.e., the higher the solution's concentration the smaller the index, and the higher the pseudoplasticity. Using CSL]00, the dynamic storage shear modulus G'(w) and dynamical loss shear modulus G"(w) in the linear viscoelastic region for each PAM solution with different concentrations are measured. According to the Laun rule(Laun, 1986)t61,the primary normal stress coefficient v~(~)

315 can be estimated by equation (1):

v~(~)=2 ~ 2

(1)

1 ~,G"(o~)

The results of the estimated ~,~) are shown in Fig.1. As is well known, the angle of internal friction represents the relative magnitude of viscosity and elasticity of the fluid. The internal friction angles of each solution are shown in Fig.2. Because the lower the concentration of the solution the closer to 90* the internal friction angle, it can be concluded from Fig.2 that the elasticity of the low-concentration PAM solutions is less strong than that of the one with high concentration. It is justifiable to disregard the elasticity of low-concentration PAM solutions in the numerical simulation, i.e. the low-concentration PAM solutions can be considered to be purely pseudoplastic fluid. 1

11

v"v.~v,

.70 9

""-'-'-'-,-~.~

lo

"-,..'*'~.

"-

i~,..,..

.

-

~,t" ~

-I"

"v.

""'-'~'-~.*

"§

o Io,~

9

--~-- 500ppm .v

--v-- 3000ppm 4000~

o

"-.4.~.+_+_+s+f

:o,

........

o, ......... ; AnBular frequency r I ~'

,o

Fig.l The estimated primary normal stress coefficient value of PAM solutions with different concentration

....

!

0.01

. . . . . . . .

!

9

"',

......

i

"

. . . . . . . .

-

-

i

0.1 1 10 Ansular frequencym/ s"1 Fig.2 The internal l~iction angle of PAM solutions with

different concentration

3.2 Results of Flow Field Measurement

The Effects of PAM Concentratiort on the Flow Field a. The Characteristics of the Flow Field of Low-Concentration PAM Solutions Axial flow features can be observed in the whole field and there is one main axial flow circulation region. The value of the tangential velocity is approximate to that of the axial and radial velocity (see Pie.1 and Pie.2) The free surface is concave around the shaft and the surface is stable. The surface concave will be strengthened as the rotational speed increases when using the same type of agitator. The reverse taper flow region under the impeller not only expands but also evolves to a regular circulated flow (see Pie.3) which indicates a tendency to be radial flow. b. The Characteristics of the Flow Field of High-Concentration PAM Solutions Pica and Pie.5 are flow pictures of different concentration (1000 and 3000ppm respectively). The two pictures show that the radial flow is strengthened when the concentration of PAM is increased. This may be the result caused by the comprehensive effects of elastic force, viscous force and inertia force.

316

pic'1 sOOppmPAM exposure time 1/30s, three-blade agitator, rotational speed 250 rlmin, IGe lamp

Pic.2 500ppm PAM solution, exposure time 1/30s, three-blade agitator, speed 250 r/min, 0.085m from bottom (impeller region), rotational direction is clockwise, I-Ge lamp

No obvious concave of the free surface is observed. Rather apparent periodic Weissenberg effect has been observed at low rotational speed in numerous experiments and the fluid around the shaft rotates upward in a reverse direction of the rotating direction of the shaft. This effect is no longer obvious at high rotational speed because the inertia force plays a dominant role instead of the normal stress. The tangential, axial and radial velocities are all small except in the impeller region the tangential velocity is large. Pic.5 and Pic.6 are the orthographic view and bottom view respectively under the same experiment condition. The maximum value of the tangential velocity is 0 . 3 5 d s while the maximum value of both axial and radial velocity is about O.l6m/s.

Pic.3 SOOppm PAM solution, exposure time 1/3Os, two-long-bladed agitator, rotational speed 130r/min, I-Ge lamp

Pic.4 lOOOppm PAM solution, exposure time 1/3oS., two-long-blade agitator, rotational speed 1OOr/min, I-Ge lamp

317

Pic.5 3000ppm PAM solution, exposure time 15ms, two-long-blade agitator, rotational speed 130r/min, 3-pulse Xenon lamp

Pic.6 3000ppm PAM solution, exposure time 15ms, two-long-blade agitator, rotational speed 130r/rain, 3-pulse Xenon lamp

When using the same type of agitator, the tangential velocity of high-concentration solution is smaller than that of the low-concentration solution at the same rotational speed. We can deduce from Pic.6 and Pic.7 that the velocity difference between the impeller blade and the fluid is greater in high-concentration solution than that in low-concentration solution. This is caused by viscoelastic effect of the fluid and consequently results in the increase of power consumption. There are stagnant flows taking place in the areas where the wall and bottom intersect in the vessel (see Pie. 8). It can also be concluded that the stagnant flow region becomes larger in these areas as the solution concentration increases. Besides, there are also stagnant flows taking place near the baffles, which can be observed clearly in Pic.6. The Effect of Different TvDe of A~itator on the Flow Field When using different types of agitator the flow fields are different from each other for lowconcentration PAM solutions. The main axial flow region is larger when using three-blade agitator than that when using two-long-blade agitator. Pie.1 and Pica are flow field pictures of 500ppm PAM solution using three-blade and two-longblade agitator respectively at close impeller tip velocity (2.42m/s and 2.04m/s, respectively). The influence of agitator type on the flow field is noticeable for highconcentration PAM solutions. The stagnant flow region is larger when using three-blade agitator than that when using two-long-blade agitator. Pic.5 and Pie.9 Pic. 7 500ppm PAM solution, exposure time 33.3ms, are flow field pictures of 3000ppm PAM two-long-blade agitator, rotational speed 100r/min, Isolution using three-blade and two-long- Ge lamp.

318 blade agitator respectively at close impeller tip velocity (2.42m/s and 2.04m/s respectively). The reason for the expansion of the stagnant flow region is due to the velocity decrease caused by the strong pseudoplasticity, which enables the PAM solutions exhibit high viscosity because of the low shear rate in the distant areas from the impeller.

Pie.8 4000ppm PAM solution, exposure time 19 ms, two-long-blade agitator, rotational speed 100r/rain, 3-pulse Xenon lamp 4.

Pie.9 3000ppm PAM solution, exposure time 15 ms, three-blade agitator, rotational speed 250r/min, 3-pulse Xenon lamp

SIMULATION FOR PSEUDOPLASTIC

FLUID

4.1 Data Predisposing and Simulation Method It is indispensable to interpolate the uneven randomized distributed data to get further information from the flow pictures. The interpolated result of the impeller region is also used as boundary condition in the simulation using the "black-box" model. In order to simplify the calculation it is assumed that the fluid in the stirred vessel is incompressible pseudoplastic fluid disregarding its elasticity, and that the flow in the stirred vessel is steady-state symmetric turbulent flow. (When the PAM concentration is relatively low, from 500ppm to 2000ppm, the measured power number is constant in the measured range of impeller rotation speed. Therefore, the flow can be considered to be turbulent flow.) The shear viscosity is calculated using the power law model: n-1

1 Where, 77 is the shear viscosity of the pseudoplastic fluid, and ltr(;02 is the second 2 invariant of the rate-of-strain tensor ). The above method assumes the calculated velocity gradient equals the exact velocity gradient and the shear viscosity is determined by the velocity gradient. The simulation is performed with the standard k-6 model. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is adopted for solving the discretization equations.

4.2 Simulation Results The simulated flow field of 500ppm PAM solution when using three-blade agitator at

319 320r/min is shown in Fig.3, which is approximately the same as the flow field in Pie.3. The simulated flow also shows an obvious radial flow tendency as the experimental one. The value of the axial velocity is somewhat small, and there are stagnant flow taking place in the area near the wall and bottom. A very small vortex can be seen at the area where the wall and bottom intersect.

I~1

I

l

i

I

,/

1

1

-" "

,~ ~

~

I

/

'

/

I

,

!

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I .

~

/

# /

/

Fig.3 The small vortex in the area where the wall and bottom intersect 0.15 0.10

0.08,

":'~

"~

0.05

0.00

.~

0.00

"~

-0.08

>

-0.05

~.

-0.16

-9

-0.10,

0.05

"

0.~0 "

0.~5

o.io

numerical simulation value

.o.15.

numerical simulation value

.0.24 , 0.00

I ---0--weighted interpolation v a l u e 1

0,15

0.;0

o.~5

"~176

o.~o"

o.;~ " oJ,o " o.~s

Fig. 4a Axial velocity distribution, 500ppm PAM solution, 0.040m from bottom, rotational speed 100r/min, two-longblade agitator

o.~o

rlm

rim

Fig. 4b Radial velocity distribution, 500ppm PAM solution, 0.040m from bottom, rotational speed 100r/min, two-longblade agitator 0.20.1-

0.08.

~

0.00.

i o.

~ / 1

-0.24 ......... o.oo

o.o. -0.1-

v.,u.

numerical simulation value

--~

-02-

.~

-0.3-

I

-0.5

o.~,~

o.~o

o.~5 r/m

oDo

--o-- ~ighted interpolationvalue t numericalsimulationvalue

I

-0.4-

o.~

o.~o

Fig.5 Axial velocity distribution, 1000ppm PAM solution, 0.114m from bottom, rotational speed 100r/min, two-longblade agitator

ooo

o;~

o'1o

9

o:15 rim

.

~o

,

o~ ,

.

~6o

Fig,6 Axial velocity distribution, 2000ppm PAM solution, 0.114m from bottom, rotational speed 130r/rain, ~ l o n g blade agitator

Fig.4 to Fig.6 are some comparison examples of the velocity distribution between simulated value and experimental (interpolated) value. The key reason for the difference between the simulated and experimental values is the difference between the interpolated

320 and actual values caused by the extrapolation with the experimental data in some areas of the flow field where there are insufficient experimental data. 5.

CONCLUSION The PIV system used can efficiently get the flow field pictures. The aqueous solution of PAM is a viscoelastic fluid. The concentration of the solution plays a key role in influencing its flow field in a stirred vessel. Basically, the flow of the solution has a radial flow tendency while using the two axial flow agitators. The numerical simulation can obtain qualitatively correct solution. SYMBOLS G' dynamic storage shear modulus Pa v G" dynamical loss shear modulus Pa 6 k turbulence kinetic energy J/kg kg.m -~. ~, K consistency coefficient sn-2 n r u

flow behavior index radial distance axial velocity

I11 m . s -I

radial velocity m.s ~ turbulence energy dissipation rate W/kg shear rate s~ rate-of-strain tensor

r/a apparent shear viscosity Pa.s co angular frequency s"~ ~ul primary normal stress coefficient Pa.s 2

REFERENCES [1] Kubi, R., Nienow, A. W., "Intervortex Mixing Rates in High-viscosity Liquids Agitated by High-speed Dual Impellers", Chem. Eng. Sci., 123-133 (1986) [2] Elson,T. P., D. J. Cheesman and A. W. Nienow, "X-ray Studies of Cavern Sizes and Mixing Performance with Fluids Possessing a Yield Stress", Chem. Eng. Sci., 41, 25552562 (1986) [3] Hua Lu et al., J. "Non-isothermal Agitated Flow Fields of Non-Newtonian Fluids", Chem. Industry and Eng. (China), Vol. 46, No. 3 (1995) [4] Venneker, B. C. H., Van Den Akker H. E. A., "CFD Calculations of the Turbulent Flow of Shear-thining Fluids in Agitated Tanks", Proc. 9th. Eur. Conf. Mixing, 179-186 (1997) [5] Anne-Arehard, D., Boisson H. C., "Numerical Simulation of Newtonian and Viscoelastic 2-D Laminar Mixing in an Agitated Vessel", Proc. 9th. Eur. Conf. Mixing, 145-152(1997) [6] Laun H.M., J.Rheol., 30(30), 459-501 (1986)

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

321

F l o w of Newtonian and non-Newtonian fluids in an agitated vessel equipped

with a non-standard anchor impeller* G. Delaplace a, C. Torrezb, M. Gradeck c, J.-C. Leulieta and C. Andr6 b alnstitut National de la Recherche Agronomique (I.N.R.A), Laboratoire de G6nie des Proc6d6s et Technologie Alimentaires, 369 rue Jules Guesde, B.P. 39, 59651 Villeneuve d' Ascq C6dex, France bHautes Etudes Industrielles (H.E.I.), Laboratoire de G6nie des Proc6d6s, 13 rue de Toul, 59046 Lille Cedex, France CLEMTA-CNRS UMR 7563, 2 avenue de la forSt de H a y e - BP 160, F-54501 Vandoeuvre Cedex, France In this paper, a commercially available fluid dynamic (CFD) software package (FLUENT) was used to conduct numerical simulation for the flow field of Newtonian and NonNewtonian viscous fluids in a rounded bottom vessel equipped with an atypical anchor impeller. The reliability of the numerical procedure was demonstrated on the basis of a comparison of the numerical results with experimental data in several ways. First for different classes of fluids (Newtonian, shear-thinning and yield stress fluids) measured and numerically calculated power consumption were reported. For all the fluids investigated, standard deviation between the calculated and measured data was found inferior to 15%. Then, for a Newtonian fluid, measurements of tangential velocity fluctuation obtained by Laser-Doppler Velocimeter (LDV) in an area located near the shaft of the mixer were compared to the CFD values. Close agreement between the numerical computed CFD values and the LDV tangential velocity data were also obtained. Finally, the flow velocity field computed for nonNewtonian liquids in the vessel were analyzed and briefly compared to those obtained with Newtonian liquid at the same Reynolds number. 1. INTRODUCTION The agitation of highly viscous non-Newtonian fluids is a commonly encountered operation in many industrial processes. For such fluids, a detailed characterization of the three dimensional flow pattern inside the vessel [1] is of vital importance to gain a better understanding of the state of flow of the fluids and of the mechanisms responsible for homogenization and transport processes [2]. In the last few years, the application of computational fluids dynamics (CFD) in stirred vessels gets more and more numerous, reflecting the growing maturity of commercial software. Many of the mixing applications, however are restricted to the flow of Newtonian fluids in relatively simple geometry such as Rushton turbine [3] 'or Pitched Blade turbine, whilst flows in industrial mixing vessels have many complications arising both from the complex design of the impeller/tank arrangement (close clearance impellers and rounded bottom of the vessel) and from the complex rheological behavior of the agitated fluids. This work was partially supported by FEDER founds.

322 In this paper, a commercially available fluid dynamic (CFD) software package (FLUENT) was used to conduct numerical simulation for the flow field of viscous Newtonian and NonNewtonian liquids (shear-thinning and yield stress fluids) in a rounded bottom vessel equipped with an atypical anchor impeller. Although anchor agitators were commonly used in industry, very limited data are available for the anchor impeller studied. Indeed this non standard anchor impeller has one particularity, it is equipped with two horizontal blades at the bottom and the upper parts in order to avoid the formation of stagnant zone. The reliability of the numerical procedure was demonstrated on the basis of a comparison of the numerical results with experimental data in several ways. Global variable such as power consumption and local flow variables such as tangential velocities were used for this purpose. 2. EXPERIMENTAL APPARATUS AND METHOD

2.1. Mixing system investigated A sketch of the mixing system investigated (atypical anchor impeller and rounded bottom of the vessel ) is shown in Figure 1. The dimensions of the mixing system are also reported in Figure 1. During all the experiments, the volume of liquid was kept constant and corresponds to a liquid height, H of 0.361m. For each experiment, the agitator was rotated counterclockwise. The rotational speed of the impeller was measured by using a digital tachometer. A strain gauge torquemeter, in the range of 0 to 20 N.m, was used to obtain torque acting on the shaft.

Fig. 1. Picture Scheme, and geometrical parameters of the mixing system studied (vessel diameter=0.346 m; impeller diameter=-0.320 m; shaft diameter=0.025 m; horizontal and vertical blades width=0.030 m; impeller height=0.250 m).

2.2. Fluids 9 fluids (3 Newtonian, 4 shear-thinning and 2 yield stress fluids) covering a wide range of rheological behaviors were prepared to ascertain numerical results. The Newtonian fluids were aqueous solutions of glucose syrups of various water concentrations. The pseudoplastic fluids were aqueous solutions of Guar (SBI Viscogum HV 3000 A), a mixture of Saccharose and Carboxyl MethylCellulose CMC (CMC, Rh6ne Poulenc LTD), and an aqueous solution of Adragante gum (Powder 427, Alland & Robert). The two yield stress fluids were two aqueous neutralized solutions of Carbopol (Carbopol type 940, polyplastic S.A.). The flow curves of the tested fluids were obtained at the same temperature as that encountered in the mixing equipment. The viscous properties of the Newtonian and pseudoplastic fluids were measured in classical controlled rotational speed concentric cylinders (Contraves - Rheomat 30). Viscous properties of yield stress fluids were determined using a controlled shear stress viscosimeter (AR 1000 - T.A. Instruments) equipped with a cone and plate measuring system (Carri-Med CS 100).

323 To fit accurately all the flow curves of the tested fluid, "local" power laws model were used. This method consists to approximate each point of the flow curve by its tangency. The final relationship is: ~,,La --. k ( ~/ ) ~/m( ~, )-I

(1)

In equation (1), the parameters m ( ~ ) and k(~) are not constant as in Ostwald de Waele's model but varies with shear rate values (see Figure 2).

Fig. 2. Flow curves and rheological parameters of non-Newtonian fluids used Yield stresses of Carbopol solutions, z o , was indicated by fitting the Herschell-Bulkley model to the rheological data at low shear rates. Yield stresses determined in this way agrees well with stresses indicated by creep test on the controlled shear stress viscometer. As for shear-thinning fluids, "local" rheological law were used to fit flow curves (Figure 2).

2.3. Numerical simulation of the velocity distribution The numerical tools and the solution procedure used by the commercial CFD finite volume software (FLUENT) to determine the fluid velocity profiles inside the vessel are well known [4] and will not be repeated in details here. In the following part, only details about the mixing system modeling options will be given. A non uniform structured grid of 86592 cells (44x41x48 in the tangential, radial and axial directions respectively) has been built to represent a half of the vessel (the rest is deduced by symmetry). Note that the final geometry of the mixing system is quite similar to the experimental equipment. Even the rounded bottom of the vessel has been duplicated as shown in Figure 1. To obtain a good agreement of fluids flow along the clearance-wall, it was chosen to increase the density of the grid near the vessel wall. The boundary conditions at the impeller shaft and at the vessel wall were those derived assuming the no-slip condition. At the vessel axis, the values of the cells surrounding the point of consideration (on the axis) is assigned to the boundary cells bordering the axis. At the free surface, the boundary conditions were modeling by requiting that there are zero normal velocity and zero normal gradient of all variables. A rotating reference frame was used to perform the simulation. The apparent viscosity kta is constant for Newtonian liquids, while for non-Newtonian fluids txa is a function of shear rate ~. For each cell, ~, was deduced from the second invariant of rate of the deformation tensor A :Zi :

324

In its current stage of development FLUENT not allows us to use "local" laws to describe rheological behavior of fluids. To circumvent this difficulty, a subroutine has been developed in ours laboratories. This approach allows us to use local rheological models and fit accurately the flow curve of the tested fluids obtained by viscometers (see Figure 2). For numerical approach, assuming that the power supplied by the impeller is exclusively consumed as viscous dissipation of energy inside the vessel, the total power consumption, P, in the vessel was calculated by the summation of energy consumed in each volume control, throughout the vessel as shown in equation (4)

P= E~a]/2dV

(4)

I ,J ,K

where I, J, K indicate the number of each control volume divided into the r, 0 and z directions, respectively. Distributions of ~ and apparent viscosity g a in each discretized control volume were obtained from analysis mentioned above.

2.4. The LDV set-up A single component Laser Doppler Velocimeter with a 35mW He-Ne laser gun (DANTEC) was used to measure for a Newtonian fluid (Ix = 6.17 Pa.s) the velocity intensity profiles inside the mixing system. The optical components of the system comprised a beamsplitter, a, transmitting lens and receiving assembly. The electronic components of the LDV system included a photomultiplier, an oscilloscope, photodetectors and a microcomputer for data acquisition. Fluctuation of tangential velocities were measured in an horizontal plane located at Z / H = 0.692 (see Figure 5). Nine radial positions located between the shaft and the middle of the horizontal blade ( 0 . 1 9 < r * = 2 r / T < 0 . 4 0 ) were chosen as shown in Figure 4. The LDV apparatus was operated in the backscatter mode. When vertical blades of the anchor going by the laser beam, no Doppler shift could be detected. Acquisitions of this disconnecting time and rotationnal speed of the agitator allows us to obtain for each radial distance studied, the fluctuation of tangential velocity relative to the angular position of the blades (characterized by the angle a - see Figure 4) 3. RESULTS AND DISCUSSION

3.1. Reliability of numerical method Power consumption. Left part of Figure 3 shows, for Newtonian and non-Newtonian fluids tested, the comparison between experimental and numerical results of power consumption. Even for complex fluids possessing a yield stress, relative errors between experimental and numerical results have always been found inferior to 15 % For both numerical and experimental methods, the Newtonian power curve obtained in the laminar regime are similar to each other and can be described by traditional relationships between the Power number Po and the Reynolds number Re (fight part of Figure 3) eo Re = K p (5) Linear regressions on experimental and numerical power curve give (0.04 < Re < 3): Po Re = 183 for numerical approach and Po Re = 187 for experimental measurements.

325 Power consumption results obtained for our particular anchor impeller are qualitatively in agreement with those reported in the literature for others anchor agitators (Po Re vary between 100 and 370). A more accurate comparison is difficult to make because from our knowledge only [5] had investigated the flow of anchor impeller equipped with two horizontal blades as ours but these authors hadn't provided the value of the product Po Re in the laminar regime for their mixing system. Nevertheless calculating the power consumption for a similar anchor non equipped with horizontal blades by empirical correlation suggested by [6], we obtain in the laminar regime Po Re = 148 which is 26% smaller. So we can assume that excess of power consumption measured for our anchor mixer is due to the presence of horizontal blades.

Fig.3. On the left: Power consumption data obtained numerically and experimentally when mixing Newtonian, shear-thinning and yield stress fluids under laminar regime (0.04 < Re < 3 )at different rotational impeller speeds. On the right: Newtonian Power curve Tangential velocities: In order to ascertain the numerical simulation reliability to estimate under laminar regime local flow variables such as velocity components, values of tangential velocity obtained by LDV measurements have been compared to those calculated by CFD simulation. Figure 4 shows, for two radial distances r* =0.2 and 0.4, fluctuations of tangential velocities with the angular position of the blade obtained numerically. In this figure, experimental values of tangential velocities obtained experimentally when mixing Newtonian fluid under the same operating conditions have also been plotted. As shown in Figure 4, there is a good agreement between numerical and experimental results. Indeed variations of tangential velocity with blade angular position are quite similar and it can be seen that in the vicinity of the shaft, experimental and numerical tangential velocity reach their peak for the same angular blade position (ct = ~/2 rad). Even if small differences between experimental and numerical values of tangential velocity still exists, it is very difficult to obtain a more accurate agreement since the order of magnitude of remaining errors may be input to difficulties to localize the measurement point inside such industrial equipment. Note that measurements of tangential velocity have been carded out near the shaft and in this area numerical results are expected to be less accurate since both the tangential velocity intensity and the density of the grid are weak. Hence, the close agreement between experimental and simulated tangential velocities points out the validity of the numerical simulation method. All these results regarding the local tangential velocity, power consumption for both Newtonian and non-Newtonian fluids prove that numerical analysis method has sufficient reliability to predict flow behavior of highly viscous fluids in such mixing system, especially under laminar regime.

326

Fig. 4. On the left: Experimentally and numerically tangential velocities obtained for two radial positions when mixing Newtonian fluids (g=6.17 Pa.s and n=O.127s -1) with anchor impeller. All data are obtained in an horizontal plane located at z / H = 0.692 .On the right: Location of the velocity measurement points with LDA system angle a : angular position of the blade relative to angular position of measurement point. 3.2. Analysis of numerical results obtained for the flow of Newtonian and nonNewtonian fluids

Newtonian flow fields To analyze the flow pattern induced by anchor impeller studied, maximum values of fluctuating velocities obtained during one impeller rotation for various radial positions have been compared.

Fig. 5. CFD predictions of maximum velocities profiles obtained at different horizontal planes during one rotation of the impeller. Symbols used for various axial positions are given at the upper part of the scheme.

327 Figure 5 shows maximum values of velocity components (tangential, axial and radial) obtained for selected radial distances ( O < r * < l ) at different horizontal planes (0.21 < z/H < 0.97 ). Examination of Figure 5 reveals that tangential flow is dominant and becomes smaller with radial distances away from the impeller as reported by [1, 7]. Indeed, except at the upper parts of the vessel (at z/H = 0.91and 0.97 ), for all radial positions in the vessel, values of maximum tangential velocities is significantly higher than maximum values of axial and radial velocities (always inferior to 0.4). The results reported in figure 5 show clearly that close to the leading face of the upper and bottom horizontal blades (at z/H = 0.21 and z/H = 0.91 ), the rotation of the impeller generated a strong axial flow which was rapidly attenuated with both increasing axial (z/H = 0.34, 0.48, 0.59 and 0.8) and radial distances ( r* < 0.751 and r* > 0.925) away from the horizontal blades as mentionned by [1, 8]. Figure 5 also shows that maximum radial velocity was found at a radial position corresponding to the inner face of the vertical blade as found by [1, 7, 9, 10]. Note that our numerical value of maximum radial velocity at internal side of the blade is higher than those reported by [1, 7] but in agreement with numerical and experimental data of [9-10]. Moreover, this value was rapidly attenuated with increasing distance away from the radial location of the vertical blades. Non-Newtonian flow fields: To point out change in flow pattern which occurs in the vessel due to complex rheological properties of agitated fluids, velocities profiles obtained numerically at same Reynolds number for Newtonian and Non-Newtonian fluids have been compared.

Fig. 6. Velocities profiles obtained for an angular position normal to the vertical blades at z/H =0.59

328 Figure 6 shows an example of such velocities profiles obtained at z/H = 0.59 for an angular position normal to the vertical blades. This angular position have been chosen since it was observed from numerical results that change in flow are more significant with increasing distance away of the blades. Velocities profiles plotted in Figure 6 are quite interesting since they allows us to visualize main tendencies of change in flow observed when mixing nonNewtonian fluids and non-Newtonian fluids at same Reynolds number. Indeed, as shown in Figure 6, examination of numerical results reveals that: - for this agitator, few significant difference exist in flow when comparing flow patterns with weak shear-thinning fluids (CMC and 1.2% guar gum) to those obtained with Newtonian fluids. As it was already observed in pipe flow, it was also found throughout the vessel that the flow field of yield stress fluids (carbopol solutions) is quite similar to those obtained with high shear-thinning fluids (3% guar and 1.9% adragante gums). - tangential, axial and radial components of velocities away from the blades are slightly smaller than the corresponding values obtained for Newtonian fluids. 4. CONCLUSION Numerical predictions based on CFD simulations were obtained for the laminar flow fields of Newtonian and non-Newtonian fluids in a rounded bottom vessel stirred by a non-standard anchor impeller. The reliability of the method was ascertained by verifying the numerically calculated results with experimental data (power consumption and tangential velocities measurements). Flow patterns obtained numerically for Newtonian fluids shows us that the presence of the upper horizontal blade has a significant impact on the axial flow throughout the vessel. Finally the simulated flow velocity field computed at same Reynolds numbers for Newtonian-and non-Newtonian fluids allow us to observe change in flow pattern and is very helpful to relate flow behavior of complex fluids to their rheological properties. REFERENCES

1. M. Abid, C. Xuereb C., J. Bertrand, Hydrodynamics in Vessels Stirred with Anchors and Gate Agitators - Necessity of 3-D modelling, Trans IchemE, 70 Part A (1992) 377. 2. M. Ohta, M. Kuriyama, K. Arai, S. Saito, A two-dimensional model for the secondary flow in an agitated vessel with anchor impeller, J. of Chem. Eng. of Jap., 18 1 (I 985) 81. 3. C. Torrez, C. Andr6, Power consumption of a rushton turbine mixing viscous newtonian and shear-thinning fluids: comparison between experimental and numerical results, Chem. Eng. Technol., 21 7 (1998) 599. 4. FLUENT- User's Guide, (1996) 5. C.J. Hoogendoorn, A.P. Den Hartog, Model studies on mixers in the viscous flow region, Chem. Eng. Sci., 22 (1967) 1689. 6. K. Takahashi, K. Arai, S. Saito, Power correlation for anchor andhelical ribbon impellers in highly viscous liquids, J. of Chem. Eng. of Jap., 13 2 (1980) 147. 7. J. Bertrand, J.P. Couderc, Etude Numerique des 6coulements g6n&6s par une ancre dans le cas de fluides visqueux, Newtoniens ou pseudoplastiques, Entropie 125/126, (1985) 48. 8. Y. Murakami, K. Fujimoto, T. Shimada, A. Yamada, K. Asano, Evaluation of performance of mixing aparatus for high viscosity fluids, J. of Chem. Eng. of Jap., 5 3 (1972) 297. 9. M. Kuriyama, M. Inomata, K. Arai, S. Saito, Numerical solution for the flow of highly viscous fluid in agitated vessel with anchor impeller, AIChE J., 28 3 (1982) 385. 10. K. Arai, K. Takahashi, S. Saito, Correlation of velocity distributions in an anchor-agitated vessel using a bicubic B-spline function, J. of Chem. Eng. of Jap., 15 5 (1982) 383.

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 9 2000 Elsevier Science B. 1I. All rights reserved

329

Characterization of Convective Mixing in Industrial Precipitation Reactors by Real-time Processing of Trajectography Data B. Barillon and P.H. Jdzdquel Kodak European Research Laboratories, Kodak Industrie, Z.I. Nord, 71102 Chalon sur Sa6ne Cedex, France The hydrodynamic behaviour of silver halide precipitation reactors is characterized, at the laboratory and the pilot scales, by a particle tracking based method. Two CCD-cameras and a powerful image processing system record, in real-time at a frequency of 30 Hz, the trajectory of a neutrally buoyant particle. From the extracted data several configurations of reactors at both scales, including the positioning of the baffling system and the location of the mixer, are discussed. 1. BACKGROUND AND OBJECTIVES The silver halide microcrystals made for photographic purposes are, most of the time, obtained via double-jet precipitations between silver salt and halide salt solutions. This type of precipitation is performed in semi-batch reactors and requires an appropriate stirring whatever the volume of liquid in the reactor is. The reagents are allowed to contact each other in the vicinity of a mixer in order either to form solid silver halide or to contribute to grow the crystals already existing in the reactor. Because of the reaction kinetics, competition with mixing is very likely to happen: the characteristic mixing times are generally several orders of magnitude larger than the nucleation times, while they are comparable with the mean crystal growth times. If the reaction is as fast as or faster than mixing, the distribution of the products in the reactor will hence be mixing dependent and the resulting medium will be segregated. This can be critical when precipitation formulae are scaled up as any failure due to scaling issues may strongly impact the final photographic performances of the microcrystals. The goal of this study was to better describe the convective mixing (or macromixing) and compare it at different scales. Consequently, the characterization of the hydrodynamic behaviours was achieved for several reactors at the laboratory and pilot scales. The method we developed to characterize macromixing is an improved version of a particle tracking technique pioneered by Scofield 1'2. This technique had shown its capability to describe some of the typical features of the flows in model stirred reactors using, for instance, Rushton or Pitched-Blade turbines 3. However, it has not been practically used for industrial purposes because of very long and tedious work required to process the experimental data. In addition, the mathematical treatment of the trajectory to get practical and simple characteristics of the hydrodynamics inside the reactor is not as trivial as expected. The principle, based on a single particle tracking, consists in taking images, at a relatively high frequency, of a neutrally buoyant particle moving in a transparent reactor, by means of video cameras set around the tank. It is then possible to calculate, on each image, the

330 instantaneous position of this single particle, and to obtain, after a statistically significant lapse of time - typically one hour-, the whole particle trajectory: Eventually, information on the macroscopic hydrodynamics inside the reactor is extracted from the generated data. 2. EXPERIMENTAL DEVICE The major improvement brought here to the technique is a real-time processing of the images recorded at a frequency of 30 Hz, not only eliminating the need for the storage of a considerable amount of images in the view of a further processing, but also allowing the immediate storage of the -uncorrected- coordinates of the particle versus time. In concrete terms, the transparent reactor is immersed in a transparent square tank in order to minimize the distortion generated by the difference in the refraction index between water and air (the interface is a plane and not a portion of a cylinder). The reactor is equipped with transparent mixing devices as well. A 3 mm-diameter black plastic particle plays the role of a fluid tracer. By pouting some sodium salt in the reactor it is possible to adjust the density of the fluid to the density of the particle which can hence be considered as neutrally buoyant. Two 640x480 pixels CCD cameras cal~ture two synchronized images every 1/30 ~' second. Figure 1. Top view of the experimental device One of the two cameras has its optical axis perpendicular to the (0,X,Z) side of the tank whereas the other is perpendicular to the (0,Y,Z) side. The optical axes of the two cameras need to be orthogonal. This setup of the cameras allows us to obtain, on the one hand, the Xand Z- coordinates and on the other hand, the Y- and Z- coordinates of the particle (of. Fig. 1). Once the two images are captured they are transferred to a computerized modular vision system. Each image processing encompasses 2 steps which are the thresholding of the image light intensity and the detection of the particle position by determination of the luminous center of gravity of the image. Real-time treatment requires that this processing is shorter than the time between two consecutive image captures i.e. 1/30 th see. Let us point out the need for a very careful lighting of the reactor in order to avoid a wrong detection of the particle position. 3. DATA PROCESSING After correction of the optical bias and interpolation of the possible events that are not "seen" by the cameras, data processing leads to several useful characterization criteria such as instantaneous velocity, recirculation time distributions relatively to selected reactor cross section and intersections of the particle trajectory with these cross sections in order to obtain first qualitative information on the flow patterns and the associated characteristics. The way these characterization criteria are determined will be discussed in more details in the next section.

331 4. RESULTS AND DISCUSSION This technique was applied for the characterization and the improvement of both laboratory and pilot scale precipitation reactors. The first example refers to the pilot scale. Fig.2 represents the intersections between the trajectory and an horizontal reactor cross section. If the particle crosses the section in the upward direction or in the downward direction a circle, respectively a point, is drawn. By examining the cross section structure at different heights it is possible to get a precise idea of the flow patterns. Here, the flow pattern observed is due, on the one hand, to the pumping effect generated by the mixer, inducing a downward movement of the fluid in the central region of the reactor and, on the other hand, to the lateral discharge inducing an upward directed motion near the reactor walls. The projection of this pattern on the section is the external ring seen in figure 2. Figure 3 is a reproduction of the velocity map Figure 2 Flow pattern in horizontal section obtained after data processing. As the particle is (Z= 0.2m). Dimensions in cm. Pilot scale. assumed to be neutrally buoyant, the calculation of the velocity at each location also gives the value of the fluid velocity at this same location. The "velocity" of the particle is the magnitude of the velocity vector. Depending on the magnitude of the vector, a colored point is associated to each location and allows the visualization of the velocity field. The map is obtained by superimposition, by transparency, on one plane, of the velocity magnitudes associated to the different locations of the particle in the slice of the kettle delimited by the two planes y - y l and y -Y2. The image is then processed by means of a filter (Paint shop Pro| in order to emphasize the colored zones. This map witnesses the fact that there is limited velocity segregation in the reactor, even if the Figure 3 Velocitymap at pilot scale. Dimensions in cm. fluid is slightly accelerated in the vicinity of the shaft. Another fundamental criterion is the recirculation time distribution. The recirculation time is defined as the time needed by the particle to come back to a given horizontal cross section. A distinction was made between the time spent above the section (red distributions) and the time spent under the section (black distributions). Figure 4 illustrates the distribution obtained at the pilot scale. "Tmoy" corresponds to the mean time, " T m a x " to the longest event. There is

332 no "long events" in the reactor, the particle staying less than 15 seconds below the section. It means that no stagnant zone is to be found in the vessel. The distribution shows two peaks. The first one, near the y-axis, is probably due to turbulent eddies which make the particle come back very rapidly to the section. The second peak, at about 3 seconds, and the narrowness of the distribution point out the fact that the particle crosses the section at regular time intervals. It means that the particle follows more or less regularly a recirculation loop. In order to know whether the fluid motion is Figure 4 Recirculation time distribution below the predominantly vertical, one possibility is to cross section Z=0.2 m, pilot scale determine the intersections of the trajectory with a vertical cross section containing the shaft. In that case it is possible to obtain qualitative information on the radial and tangential components of the fluid. Another alternative is to draw in a pseudo 3-D representation the velocity vectors associated to the intersection points between the trajectory and a selected crosssection (cf. Fig.5). In the example given here, the radial and tangential components of the velocity vectors seem to be weak compared to the vertical component. This allows us to say that the Figure 5 Velocity vectors associated to the recirculation loop suggested by the recirculation intersection points between trajectory and cross time distribution consists in a vertical loop section (Z=0.12m), pilot scale. Dimensions in favoring exchanges between the bottom and the cm. surface of the reactor. Many other criteria could have been considered for data processing but the next part of this section will demonstrate that the four criteria listed so far (velocity maps, cross-sections, recirculation time distributions and velocity vectors) are sufficient to provide a reasonable basis for a comparison between the pilot and the lab scales. For the lab scale several configurations were studied. The most relevant parameters we played with were the baffling configuration and the type of mixer in the reactor. Figure 6 shows the velocity maps for two different baffling configurations, one with baffles distributed all around the mixer (conf.1), the other one with one vertical baffle set along the walls of the reactor (conf.2). The way the baffles are configured in the reactor changes the velocity map, namely the spatial location of the different ranges of velocity. In the circular baffle configuration the most agitated zones are the region near the mixer (high velocities) and the lower half of the reactor. In the wall baffle configuration, even if the location of the highest velocities is the same, the average velocities form a W-shaped zone consisting in the regions near the shaft and close to the walls. The differences in the velocities proves that the proximity baffles/mixer in conf,1

333

Figure 6 Velocitymaps at lab scale. Conf.1 with baffle around mixer(left) and conf.2 with wall baffles (right). Dimensions in cm. reduces macromixing in the reactor. Horizontal (fig.7) and vertical (fig.8) cross-sections for both configurations emphasize the differences suggested by the velocity maps. In conf. 1 the

Figure 7 Horizontal cross sections (Z=0.18m). Conf.1 (left), conf.2(right). Dimensions in cm. Lab scale. cross section reveals asymmetries in the flow pattern: the external ring is disrupted, suggesting a change in the direction of the flow, and its shape shows that the presence of the baffle around the mixer is not neutral. The baffling conf.2 is more similar to the one obtained at the pilot scale. The vertical section puts in evidence a different rotational movement around the shaft, the conf. 1 generating, near the reactor walls, a counter-clockwise movement of the fluid. This behavior is supposed to be due to a higher interaction between the baffle and the mixer in this configuration. Because there is a relationship between the number of intersections and the flowrate across the section, we can qualitatively conclude that conf.2 is more efficient in reducing the rotational movement than conf. 1. In fig.9 are represented the recirculation time distributions above (light) and below (dark) an horizontal section dividing the reactor in 2 equal volumes for both conf.1 and conf.2. This figure clearly shows that the distributions do not have the same shape. The one corresponding to conf. 1 presents a peak characteristic of a recirculation loop.

334

Figure 8 Vertical cross section containing the shaft. Conf.1 (left), conf.2 (right). Dimensions in cm. Lab scale. The distribution generated by the conf.2 baffle is much broader and shows no peak which confirms that there is no vertical well defined recirculation loop allowing the fluid to circulate between the top and the bottom of the reactor. The long times indicate that all the zones in the reactor are not equivalent in terms of residence time. This is consistent with the segregated velocity zones shown on the velocity map: the top of the reactor has a behavior which is close to a stagnant zone.

Figure 9 Recirculationtime distributions related to cross section Z=0.18m, conf.1 (left), conf.2 (right). Lab scale. In conclusion, the baffle around the mixer seems to generate significantly different flow patterns inside the reactor compared to pilot scale. If we assume that mixing is crucial for the precipitation process we must make sure that crystals see the same chemical environment (same supersaturations, for instance) and as a consequence have the same chemical and hydrodynamic history from one scale to another. This is the reason why conf.2 (and not conf. 1) could reasonably be considered as a scale down of the pilot scale described in the first part of the section. However one must keep in mind that macromixing cannot be dissociated from micromixing when a chemical reaction occurs and that the conclusions being drawn in this study solve only one part of the problem of scale up.

335

Experiments were also performed with another type of mixer. It consists in a vertical tube (several centimeters high) containing a pitched blade turbine. This mixer is supposed to provide an upward directed motion inside the tube and a well developed recirculation loop in the reactor. Horizontal baffles were attached to the tube to avoid air entrapment when the volume is low in the reactor. Several positions of the turbine inside the tube were studied. The two extremes correspond to the impeller at the bottom of the tube and at the top of the tube (cf. fig 10). Fig.11 represents the same horizontal cross-section for these two extremes. The position of the turbine inside the draft tube has an effect on the macromixing characteristics. When the turbine is located in the lower part of the tube, the flow field is upward directed. When the turbine is close to the top, the upward discharge shows severe disturbances: a downward directed motion appears along the shaft and the vertical Figure 10 Position of the turbine in discharge tunas into a more lateral one. In the horizontal the tube, bottom (left), top (right) cross sections the shape of the baffles appears clearly. This indicates that the baffles and the edges of the draft tube are not neutral. Not only the flow patterns but also the recirculation time distributions exhibit some differences, as illustrated in fig. 12. The distribution widens when the turbine is closer to the top. The velocity vectors (in

Figure 11 Horizontal cross-sections(Z=0.15m) with the pitched blade turbine at the bottom of the tube (left), at the top of the tube (right). Dimensions in cm. Lab scale.

Figure 12 Recirculationtime distributions relatedto cross section Z=0.15m, with the pitched blade turbine at the bottom of the tube (left), at the top of the tube (right). Lab scale.

336 fig.13) confirm the lateral discharge of the flow probably due to the presence of the baffles. The shift of the discharge decreases the efficiency of the macromixing. It appears that the efficiency of macromixing is directly related to the set up of the turbine inside the tube. Modifications in the positioning of the turbine in the tube can generate significantly different flow patterns inside the reactor.

Figure 13 Velocityvectors at cross section (Z=0.13 m) withthe pitched blade turbine at the bottom of the tube (let~), at the top of the tube (right). Dimensions in cm. Lab scale. 5. CONCLUSIONS Mixing being an important feature in precipitation processes, it was relevant to develop a tool able to characterize different configurations of reactors at different scales. The technique chosen is an improved particle tracking technique. The major improvement is the determination in real-time of the position of the particle in the reactor. With appropriately defined criteria it is then possible to extract the main features of the flow inside the reactor. The examples shown highlight the fact that with velocity maps, intersections of the trajectory with reactor cross sections, recirculation time distributions and determination of velocity vectors, it is possible to rebuild the flow patterns in the reactors and compare different configurations of baffling and mixers. Moreover, as this technique can be adapted at different scales, this tool becomes the tool of choice for solving scale up issues. In addition, it gives a good starting point if we want to model the hydrodynamic behavior of the fluid. This study is also a good example of application at an industrial stage of a technique developed in an academic framework.

REFERENCES 1. D.F. Scofield and C.J. Martin, "Mixing Time Distributions and Period Doubling in Stirred Tanks", Annual Meeting AIChe (Chicago,USA), Nov. 11-16 1990 2. D.F. Scofield, C.J. Martin and M.A. Pivovarov, "Underlying Geometry of Flows", Annual Meeting AIChe (Chicago,USA), Nov. 11-16 1990 3. S. Wittrner, "Caract~risation du Mrlange dans une Cuve Agitre par Trajectographie", PhD, INPL, Nancy, France, April 18 1996

10th European Conference on Mixing H.E.A. van den Akker and J.J. Derksen (editors) 2000 Elsevier Science B. V.

337

Characterisation of flow and mixing in an open system by a trajectography method P. Pitiot, L. Falk Laboratoire des Sciences du G6nie Chimique (LSGC-CNRS) 1, rue Grandville, BP. 451, 54001 Nancy CEDEX, France ABSTRACT This study deals with the description of the capacities of a Lagrangian approach -3Dtrajectography- to give some information about mixing in an open system. This original particle tracking leads us to a more detailed and local comprehension than with the classical Residence Time Distribution. Residence Time Distribution and Trajectory Length Distribution are compared, showing the high similarity of the two concepts for the studied tank. 1 INTRODUCTION Many techniques for flow and mixing characterisation in chemical reactors are available for chemical engineers. Among all these methods, one of the oldest but still in use method is the classical residence time distribution (RTD) method. Its main advantage is its ease of use and the ability to be directly employed to analyse reactors performance. The facility of such a technique is also to avoid sophisticated visualisation or in-situ measurements as it considers the system as a black-box, which response to specific excitation (as pulse tracer injection at the entry) is characteristic of the internal flow structure. Furthermore, the concept of RTD allows to compact the complex information in a single mathematical object, the distribution of residence time. If the RTD is classically used to detect the presence of dead flow or short-circuit zones in the system, it is however impossible to locate these particular zones. Additional methods, as Internal Age Distribution, are then necessary. On the other hand, with sophisticated investigation technique, as Laser Doppler Anemometry (1,2) or Particle Image Velocimetry (3,4), it is now possible to measure very precisely the turbulent flow field in complex system. Thanks to more sophisticated electronic systems, associated with convivial software, these techniques are increasingly easy to handle. But, because of the numerous data points, information is however not always easy to analyse, especially to link the mixing performance with the yield and the selectivity in chemical reactors. Between these two approaches, the concept of Lagrangian tracer is now being a developing alternative. For several years, we have developed such a method, named the trajectography method (5, 6, 7), to characterise local information in reactors. The purpose of the method is to track a small neutrally buoyant particle, for long observation time and to reconstruct the trajectory. From the analyse of this trajectory, it is possible to get information on the internal flow. The present study deals with the use of the 3D trajectory method to

338 investigate open flow system. In this article, we shortly describe the method and present the confrontation of the RTD obtained with the particle with a classical chemical tracer technique. 2

EXPERIMENTAL SETUP Following Scofield (8) idea, Wittmer (5) built a three-dimensional visualisation system. After several lightning improvements our tracer is a small particle (diameter 91.8 ram) and its density is the density of the fluid (difference less than 0.1%). This particle is made thanks to a micro-encapsulation process, using a solvent (water), a non-solvent (paraffin oil), a polymer (gelatine) and an adjusted amount of paraffin included to the polymer in order to optimize the density of the particle. Small molecules -like water- can go through the polymer, which is similar to a permeable membrane. Therefore, the density of our particle is auto-adjusted to the density of the fluid at ambient temperature (7). This particle tracking is applied to a 20 litres cylindrical stirred tank with a flat bottom (T =294 mm; H =281 mm). The plexiglas tank is equipped with four baffles and is placed in a glass cubic tank filled with water to reduce optical distortions. However, in this system, refraction phenomena are not negligible (7), as five different media can be distinguished (air, glass, water, plexiglas and fluid). A correction procedure taking into account the refraction and the real position of the cameras is applied in order to reconstruct the exact spatial position of the particle. The impeller used in this study is a classical Rushton turbine (D=T/3; C=T/3) at a rotation speed N = 120 rpm. For this open system (Fig. 1), the inlet is located near a baffle at the height of the impeller. The outlet is placed 270 ~ from the inlet (clockwise) at one centimetre from the free surface of the fluid. Pipes diameter equals to 8 mm and the circulation of the fluid is carried out by a peristaltic pump at a flow-rate of 1.8 litre per minute. Fig. 1. Sketch of the reactor. Two perpendicular and synchronised cameras (Fig. 2) track the displacements of the particle, in order to get the three coordinates of the particle as a function of time. At a frequency of 12.5 Hz, the two grey-level images are respectively transformed into a colour-component image (red-level and green-level). Thus, these two images do not loose their own identity and can be added together thanks to a PAL-RGB encoder to form a single colour image. This latter contains the two views of the particle at the same time. Thanks to this solution, a mechanical or an electronic synchronisation that would be expensive and unreliable, is avoided.

339

Fig. 2.3D-Trajectography; experimental setup. Colour images are digitised, compressed and stored on a special disk in real-time. With our acquisition-system, 30000 images can be stored, that allows a 40 minutes tracking. This time has to be linked to the mean residence-time (x) of a fluid aggregate (represented by the particle) in the tank. In our study, x equals to 11 minutes. If the behaviour of our tank is well represented by a CSTR-model, the Residence Time Distribution (RTD) should be represented by a decreasing exponential function, that means that 99% of the aggregates should go out of the tank after 5x and 98 % after 4x. As a result, with a 40 minutes (4x) acquisition-time, we can expect to get the major part of the Residence Time Distribution. After each experiment the colour images series are treated by image analysis to extract the position of the particle on each image. Then, for each record, we get image-coordinates of a few objects on the two planes and we have to reconstruct the 3D-trajectory in the tank coordinates-system. After bad objects elimination and the numerical intersection of optical axis (refraction), we have the trajectory of the particle, i.e. the three spatial coordinates as a function of time. HYDRODYNAMICS: TRAJECTORIES, RESIDENCE TIME DISTRIBUTION (RTD) AND TRAJECTORY LENGTH DISTRIBUTION (TLD)

3

3.1 Trajectories First, we represent side-views of characteristic trajectories (Fig. 3 to Fig. 5). On these figures (Fig. 3 to Fig. 5), the radial discharge flow and the subsequent recirculation loops induced by the Rushton turbine can be easily detected. Moreover, as the residence time (t~) increases, the trajectory looks more and more complex, but the structure of the flow remains identical.

340

The particle visits a larger volume of the tank when t~ is high. In Fig. 3, inlet and outlet can be distinguished; this is not possible on the two other figures, as the draw masks them. This also shows that inlet and outlet do not seem to disturb the global structure of the radial flow. At last, these three different trajectories illustrate the concept of RTD.

3.2 Residence Time Distribution (RTD) The results presented here concern 75 tracking-experiments carried out in the same experimental conditions as described bellow. In order to check that the particle is a good passive tracer of the fluid, we have confronted the RTD obtained by the tracking techni