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. H