10th European Conference on Mixing

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

Author: H.E.A. van den Akker (Editor) | J.J. Derksen (Editor)

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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

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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


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


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 DNaC1 + 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


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 (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 Otl0 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___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.%



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 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-- - 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.

> 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 > 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 2. 105). Furthermore, the flow is inherently time dependent due to the rotating impeller. Resolving the flow pattern therefore requires a simulation technique that is capable of handling these features of the flow field at sufficient (temporal and spatial) resolution. In this intrinsically unsteady flow, a large eddy simulation (LES) approach to turbulence modeling is preferred. Due to its numerical efficiency, a lattice-Boltzmann discretization of the Navier-Stokes equations can provide

257 sufficiently high spatial resolution to do physically sound LES. The lattice-Boltzmann method applied in this research has proven to produce reliable results in simulations of theturbulent flow field of a stirred tank reported by [3] and [4]. The lattice-Boltzmann method has been developed in recent years and stems from the lattice gas cellular automata techniques that date back to the seventies and eighties. The concept of the lattice-Boltzmann method is based on the premise that the mesoscopic (continuum) behaviour of a fluid is determined by the behaviour of the individual molecules at the microscopic level. In the lattice-Boltzmann approach, the fluid is represented by fluid mass that is found on the nodes of an equidistant grid (lattice). The fluid mass mimics the behaviour of gas molecules. Collision rules that guarantee conservation of mass and momentum are applied. The elegance of the method is that although the collision rules are imposed completely local, the continuity equation and incompressible Navier-Stokes equations are recovered ([5, 6]). This characteristic makes the method extremely favourable for high-performance CFD on parallel computer platforms. Another major advantage of the lattice-Boltzmann method is that it is a time-dependent method. Thus it is suited to be combined with a subgrid-scale turbulence model for Large Eddy Simulation (LES). In LES, turbulent structures at two times the length of the grid spacing are effectively resolved while all turbulent fluctuations at smaller scales are filtered out. In this research, a standard Smagorinsky model ([7]) was applied, which treats subgrid-scale motion by means of an eddy-viscosity (ut) based on the local deformation rate. The Smagorinsky constant cs (the ratio between mixing length and grid spacing) was set to 0.11. Implementation of the geometry of the crystallizer in the lattice-Boltzmann scheme is based on a forcing algorithm presented by [4]. This algorithm is used to force the fluid to obtain prescribed velocities at the tank wall and baffles, the in- and outlets of the crystallizer and at the stirrer.

2.2

Crystallizer setup

The geometry of the crystallizer used in this research is given in figure 1. At the bottom of the crystallizer two inlets are positioned for the feed flow and external fines flow. The outer shell of the crystallizer is a settling zone with six withdrawal points for the removal of fine crystals. The outflow is passed through a heat exchanger for heat input and dissolution of fine crystals and then returned to the bottom section of the crystallizer. The inner section of the crystallizer contains the draft tube with four baffles. At the bottom side of the draft tube a marine type impeller is placed with a diameter of 48.5 cm. The slurry is circulated upwards through the draft-tube and downwards on

Table 1: Operating conditions of the 1100L DTB crystallizer Impeller speed Re Impeller Cvfeed Cv fines [r.p.m.] [-] [1/s] [i/s] Physical conditions 320 730.000 0.5 2.0 Impeller speed Re Impeller timestep gridspacing [r.p.timestep] [-] [#sits ] [mm/lu] Simulation conditions 1//3200 240.000 58 5.0

~

[m~ls] 2.4.10 -6 u [-] 1.4-10 -4

258

Figure 1: Crystallizer geometry and compartment structure. Side view and top view of the 1100L DTB crystallizer. Dimensions are given in cm.

the outside of the draft tube. The top section of the crystallizer is the boiling zone where supersaturation is generated. The conditions at which the crystallizer is operated are given in table 1. This table also contains the conditions and resolution at which the simulations were done. The inlet and outlet flows were included in the flow simulations. The Reynolds number that characterizes the flow is defined as Re = g D 2 / v where N is the rotational speed (rev/s), D the stirrer diameter and ~ the kinematic viscosity of the fluid. The Reynolds numbers of the physical system and the simulation differ a factor three. The difference in flow characteristics is expressed in the finest turbulent structures. For the design of a compartmental model, averaged flow data are only of interest and these will be practically identical due to the Reynolds similarity of the fully developed turbulent flow patterns. Although the fluid phase is a slurry, it is treated as a single phase with a uniform effective slurry viscosity.

2.3

Time averaged simulation results

The flow simulation was done on a cubic grid of 552 x 253 x 253 (~ 35.5.106) grid nodes. After start-up and development of the turbulent flow field, the simulation was carried out for 6.4 stirrer revolutions in order to capture averaged flow results. The whole computation was done on 8 nodes of a parallel cluster of pentiumIII 500 MHz processors. On this system calculation of one stirrer revolution took about 26 hours. For the crystallizer compartmental model, flow information is needed to determine the flows between compartments and the local kinetics. The choice of parameters will be discussed more extensively in section 3. In figure 2 various time averaged properties of the flow field are presented. These figures

259

Figure 2: Contour and Vector plots of the averaged flow field of the 1100L DTB crystallizer,

clearly demonstrate the distributed nature of the crystallizer. From the velocity contours one can clearly see that the fluid is accelerated in the stirrer section and at the boiling zone where the fluid is drawn back into the downcomer. The contour plot of the rate of energy dissipation shows that the baffles inside the draft tube cause a rate of energy dissipation that has the same order of magnitude as the stirrer. The contour plot of the turbulent kinetic energy clearly demonstrates the inhomogeneous flow conditions of the core of the draft tube and the outer zone. In table 2 the performance characteristics of the crystallizer marine type impeller as calculated from the CFD simulation are given. The literature values given were used in previous compartmental models by [8].

Table 2: Operating parameters estimated from the l l00L DTB crystallizer CFD([8]) From CFD From literature Power number [-] 0.47 0.40 Specific power input [W/kg] 1.73 1.96 Pumping number [-] 0.30 0.32

250 3

Compartmental modeling

3.1 S e t u p of t h e c o m p a r t m e n t a l m o d e l s The major properties that are produced with CFD and are relevant for the construction of a compartmental model are the fluid velocity and the rate of energy dissipation. These properties were used as criteria for the selection of the compartments. The compartmental model applied in this research is based on the framework and implementation of [9]. This model contains a mass- and heat-balance and a population balance to describe the evolution of the crystal population. The rate of secondary nucleation is calculated from a detailed model that describes the formation of attrition fragments as a result of crystal-impeller collisions ( [10] , [l l] ) . This mechanism is assumed to be the only source of nucleation and attrition. The growth of crystals is considered the result of a two-step process of (1) bulk to surface diffusion of the solute and (2) subsequent integration of the solute in the crystal lattice. In this mechanism, the rate of mass transfer is determined by the (turbulent) hydrodynamic conditions as characterized by the specific rate of energy dissipation e. Equation 1 is used to calculate the mass transfer coefficient; kd(L) : ~

ff~4 ~ 1/5 ~,3 /

V

1/3

§

2]

(1)

In this equation, kd is the mass transfer coefficient, DAB the diffusion coefficient of solute A in solvent B and L the particle length. The procedure for construction of the compartmental model was as follows: 1. Based on the velocity difference between the core and outer region of the draft tube, a choice was made to divide the draft tube in an inner section and an outer section. Because of a higher rate of energy dissipation at the baffles, the outer shell was divided in a baffle compartment and a compartment without baffle. The whole tube was divided in three horizontal sections to mimic plug flow behaviour in the axial direction. These compartments can be identified as compartments 1 to 11 in figure 1 2. At the top section, a strong outward velocity is found. Therefore, compartments 10, 11 and 12 were introduced to take into account the short circuiting effect of the slurry flow that is directly sent to the downcomer. 3. In compartment 14 all flows from compartments 10 to 13 are collected and are directed into the downcomer. The downcomer is again divided into three compartments to for plug flow behaviour. 4. Compartment 18 contains an outward flow to the settling zone (19) where a classification function is used that imposes the effect of settling on particle selection, based on a model proposed by [12]. 5. The bottom section contains a compartment where the feed and fines flow are introduced (20) and a stirrer section inside the draft tube where a high rate of energy dissipation is found (21). Based on this selection a compartment structure is made which is given in figure 3 (A). In this figure two examples, (B) and (C), of simpler model structures are given for comparison. (B) Represents the single MSMPR model as discussed in the introduction, extended with an external heat exchanger and stirrer compartment as attrition source.

261

Vapour

Product -i

~k

Vapour Product

27

"'126

(B)

Va

Product

I Fines loop

_sc

iio

,++!

Inlet/Stirrer section!

(A)

(C)

Figure 3: Compartment structures and connectivity; (A) a network of 21 compartments, (B) a single compartment with external fines loop and (C) a 5 compartment network. S is the stirrer compartment which functions as fines source, H T X is the external heat exchanger and fines dissolution loop.

262 A second compartment is added to include the volume of the fines-zone correctly. Model (C) represents a network of 5 connected compartments with a boiling zone and optional bypass (SC) to model the influence of a short-cut flow. With these three compartmental models simulations were done to investigate the influence of the model structure on the resulting crystal size. After selection of the compartments and setting up the compartment structure, two properties are determined from the CFD-data that determine the characteristic of each compartment. First, the ratio between the compartment specific rate of energy dissipation and the crystallizer volume-averaged specific power input is determined. This value is used in equation 1 to calculate the mass transfer coefficient. Second, the ratio between the volume flow of the main outlet and other outlets is calculated to determine the distribution of the outflow of each compartment to connected compartments. For example, the short circuit flow in compartment 1 of the compartment structure in figure 3 (C) is determined in this way to be approximately 60 % of the main circulation flow. The local turbulent condition also causes turbulent dispersion between compartments. The mass flux across a plane between two compartments is characterized by the average fluid velocity and the fluctuating part of the fluid velocity, u~, which can be estimated from the square root of the turbulent kinetic energy k;

U_l_ ~---

x

x

y

Turbulent dispersion can be implemented in the compartmental model by increasing each outlet flow with u•t and defining a back flow of the same size. Although the mechanism of turbulent transport can play an important role in the dynamic behaviour of the crystallizer, it has not been implement in the compartmental models at this stage. 3.2 S i m u l a t i o n r e s u l t s of t h e c o m p a r t m e n t a l m o d e l s To study the effect of non-uniform energy dissipation rates on the evolving crystal product, five explorative simulations were done. The simulations were done with growth parameters that were determined previously from experiments done with ammonium sulphate. In table 3 an overview of the simulation results is given. In this table, H refers to a simulation where the rate of energy dissipation was set homogeneous over the whole

Table 3: Steady-state crystal sizes and volume percentage of the product stream of 5 different model options. (H) indicates a homogeneous distribution of the rate of energy dissipation, NH indicates a non-homogeneous rate of energy dissipation and SC indicates that a shortcut stream was applied in the 5 compartment model. Simulation ~ of compartments Lower (L10) Median (Ls0) Upper (Lg0) and reference figure [#m] [#m] [#m] _ Run 1. 2 H 3B 291 657 1227 Run 2. 5 NH 3C 280 627 1192 Run 3. 5 NH, SC 3C 280 628 1194 Run 4. 21 NH 3A 274 638 1219 Run 5. 21 H 3A 285 670 1282 . . . . . .

263 crystallizer volume. Thus, for run 5 in each compartment the rate of energy dissipation is the same as the average power input of the whole crystallizer. 4

Discussion a n d conclusion

4.1 C F D r e s u l t s The time averaged results of the crystallizer CFD have produced new insight in the flow characteristics of the 1100 L DTB crystallizer. Figure 2 demonstrates a number of effects that were previously unknown or underestimated; 1. The flow inside the draft-tube appears to be much more inhomogeneous than was expected. The upward fluid velocity in the core of the draft tube is lower than the upward velocity found in the outer region of the draft tube. 2. The short circuit flow that is drawn from the draft-tube straight into the downcomer is estimated to be 60 percent of the main circulation flow. 3. The degree of turbulence (i. e. k and c) near the baffles on the inside of the draft tube is of the same order of magnitude as in the impeller region. Thus, the assumption that attrition is only caused by impeller collisions needs to be questioned. The operating parameters which characterize the impeller are in good agreement with literature values. This suggests that the CFD simulations can be considered reliable. 4.2 C o m p a r t m e n t a l m o d e l i n g The construction of a detailed compartmental model, as demonstrated in this paper, can be considered virtually impossible without the aid of reliable CFD-results. This is for instance demonstrated by the choice to divide the draft tube in core and outer section compartments. This choice is solely based on the observation that velocities encountered in the core of the crystallizer are lower than the velocities on the outside of the core and is not an obvious one. Nevertheless, the influence of the compartment structure on the evolving crystal product appears not to be pronounced, as can be seen in table 3. Currently, the effect of turbulence is only taken into account in the rate of mass transfer in equation 1 where it is taken to the power 1/5-th. Thus, the effect is expected to be very small which is confirmed by the simulation results. On the other hand, the difference between runs 4 and 5 indicate that although the influence of the rate of energy dissipation is small, it can still be observed. A surprising result is seen when comparing runs 2 and 3, where apparently the bypass is not of influence on the evolving crystal size distribution. Further development and study of the compartmental models is required to indicate whether an improved decoupling between the hydrodynamics and the kinetic processes has effectively been reached. References [1] Herman.J.M. Kramer, Sean K. Bermingham, and Gerda M. Van Rosmalen. Design of industrial crystallisers for a rquired product quality. J. Crystal Growth, 198/199:729737, 1999. [2] Sean K. Bermingham, H.J.M. Kramer, and Gerda M. Van Rosmalen. [3] J.G.M. Eggels. Direct and large-eddy simulations of turbulent fluid flow using the lattice-boltzmann scheme. Int. J. Heat Fluid Flow, 17:307, 1996.

254 [4] J.J. Derksen and H.E.A. Van den Akker. Large eddy simulations on the flow driven by a rushton turbine. AIChE J., 45(2):209-221, 1999. [5] Daniel H. Rothman and St~phane Zaleski. Lattice-Gas Cellular Automata. Cambridge University Press, 1st. edition, 1997. [6] S. Chen and G.D. Doolen. Lattice boltzmann method for fluid flows. Annu. Rev. Fluid Mech., 30:329-364, 1998. [7] J. Smagorinsky. General circulation experiments with the primitive equations: 1. the basic experiment. Mon. Weather Rev., 91:99-164, 1963. [8] Andreas M. Neumann, Sean K. Bermingham, Herman J.M. Kramer, and van Rosmalen, Gerda M. Modeling industrial crystallizers of different scale and type. Proceedings of the l~th International Symposium on Industrial Crystallization, 1999. [9] Sean K. Bermingham, Andreas M. Neumann, Peter J.T. Verheijen, and Herman J.M. Kramer. Measuring and modelling the classification and dissolution of fine crystals in a dtb crystalliser. Proceedings of the l~th International Symposium on Industrial Crystallization, 1999. [10] C. Gahn and A. Mersmann. Brittle fracture in crystallization processes part a: Attrition and abrasion of brittle solids. Chem. Eng. Sci., 54:1273-1282, 1999. [11] C. Gahn and A. Mersmann. Brittle fracture in crystallization processes part b: Growth of fragments and scale-up of suspension crystallizers. Chem. Eng. Sci., 54:1283-1292, 1999. [12] E. Barnea and J. Mizrahi. A generalised approach to fluid dynamics of particulate systems. Chem. Eng. J., 5:171, 1973.

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

265

Detailed CFD prediction of flow around a 45 ~ pitched blade turbine J.K. Syrjlinen and M.T. Manninen VTT Energy, Energy Systems, Process Simulation P.O. Box 1604, FIN-02044 VTT, Finland email: [email protected], [email protected] Numerical simulations of flow induced by a pitched blade turbine are reported. A high density grid was employed around the turbine blades in order to resolve the details of the flow field, including the trailing vortices. Results were compared to recent experimental data. The k-e turbulence model was used with either wall functions or a two-layer model in the nearwall region. Three different computational grids were used in the simulations. The trailing vortices were reproduced in the simulation in good agreement with the measured data. 1. INTRODUCTION Stirred vessels equipped with axial impellers and baffles have been investigated experimentally [1-8] and simulated numerically [9-18] extensively during the last decade. Using sliding mesh or multiple reference frame technique, the flow field can be predicted accurately enough for many practical applications without using experimental data [11-18]. However, the turbulence modeling has remained a problem. If the turbulence is not predicted accurately, the mixing processes in the tank are not described correctly either. RANS-based methods with wall functions are used in most simulations. The results are usually compared with experimental, time-averaged data. In most simulations, the k-e turbulence model seems to give the best results, although the predicted turbulent kinetic energy in the impeller outflow is clearly lower than measured. So far, the RNG k-e and Reynolds stress models have failed to improve the prediction of k [13-15]. With LES, promising results have been obtained in detailed modeling of a radial impeller [ 16]. If wall functions are used in the simulation, the overall density of the grid may not be adequate enough to capture all the relevant phenomena of the flow field. Usually, the main circulation loops are well reproduced in the simulations, but the flow near the impeller blade is not predicted very precisely. Consequently, this may lead to poor turbulence prediction in this region. On the other hand, turbulent kinetic energy may be overestimated by 50-400 % in ensemble-averaged experimental data if the periodic effect of the passing blade is not filtered out [7]. Therefore, comparing the measured and predicted angle-resolved values would be more useful. In this work, emphasis is put on detailed modeling of the flow in and near the impeller swept volume. EspeciaUy, the aim was to numerically recapture the tip vortex induced by the blade of a pitched blade turbine. Fluent 5 was used in the simulations [19]. The predicted

266

results are compared with the comprehensive measurements of Sch~er et al. [7,8]. Recently, Wechsler et al. [17,18] have simulated numerically the same geometry with a dense grid succesfully revealing the trailing vortex. They modeled the impeller blade and baffle as zerothickness surfaces and increased the Reynolds number in their calculation in order to ensure the validity of wall functions. In the current simulation, the geometry and Reynolds number were identical to the ones used in the experiment of Schafer et al. [7]. 2. SIMULATED GEOMETRY

The tank is cylindrical and it has a lid and four symmetrically placed baffles near the tank wall. The turbine is four-bladed and the pitch angle is 45 ~. The impeller shaft extends to the bottom of the tank. The dimensions are as follows: tank diameter T = 152 mm, liquid height H = T, impeller diameter D = 0.329T, blade width w = 0.264D, impeller mid-plane clearance from the bottom h = 0.33T, baffle width B = T/IO and clearance from baffle to the wall s = 2.6 mm. The blade and baffle thicknesses were 0.9 mm and 3 mm, respectively. The working fluid is silicone oil with densityp = 1039 kg/m 3 and dynamic viscosity # = 0.0159 Pas. In the simulated case, the impeller rotational speed was N = 2672 rpm. This resulted in impeller Reynolds number Re = 7280 and tip velocity Utip = 7 m/s. For symmetry reasons it was sufficient to model one fourth of the tank. Then the computational domain contained one impeller blade and one baffle. 3. NUMERICAL SOLUTION

3.1. Computational grids Three hexahedral, block structured grids were used in simulations. The size and structure of the first grid (Cmd 1) is typical with wall function approach. The total number of cells is 17 237 with 37, 27 and 18 cells in the axial, radial and tangential directions, respectively. The grid is slightly denser in the impeller swept volume. On the blade surface, a 9• cell grid was used. To seriously capture the trailing vortex, the second grid (Grid 2) was made considerably denser in the impeller swept volume. According to [7], the trailing vortex diameter varies from 5 to 15 ram. The grid on the blade surface was composed of 24 radial and 15 axial cells. Then the narrowest cross section of the trailing vortex is inside a 6• cell grid. 38 cells were used in tangential direction. The number of cells in axial and radial directions was 74 and 59, and the total size of the grid was 164 036 cells. The third grid (Grid 3) was made still denser in and near the impeller swept volume. 78 cells were used in circumferential direction, 109 in axial and 71 in radial direction. The grid on the blade surface consisted of 31> 200), the standard k-e model is used. In the viscous-affected region (Rey < 200), a fine grid is used to resolve the flow down to the viscous sublayer without wall functions. In the viscous-affected region, k is solved from the standard transport equation, but turbulent viscosity is calculated from /l t = p C ~ l ~ and dissipation from e----k 3/2 / [e. The length scales l/, and It are defined as -

ollo

n

oon

n

:

:

_

_ 7O

= 2Cl. 4. RESULTS

4.1. Main flow pattern The radial profiles of predicted axial velocity and turbulence kinetic energy are compared in a plane halfway between the baffles at three axial positions with experimental values in Figs. 1-2. In the simulation, the impeller blade is 15~ behind the plane between two baffles with Grids 1 and 2. In the simulation with Grid 3, the impeller blade is on the plane of the baffle. The experimental values in Figs. 1-2 are angle-resolved up to radial position 2r/7' = 0.395 and correspond to the positions of the blade in the simulations.

268

Figure 1. Radial profiles of scaled axial velocity Uax,/Uap at three axial positions in a plane between two baffles: (a) z/T= 0.145, (b) z/T= 0.46, (c) z/T= 0.67.

Figure 2. Radial profiles of scaled turbulent kinetic energy k / U~2~pat three axial positions in a plane between two baffles: (a) z/T = 0.145, (b) z/T = 0.46, (c) z / T = 0.67. Symbols as in Fig 1.

Figure 3. Velocity vectors on vertical planes near the blade. (a) ~ = 0 ~ (b) ~ = 30 ~ (c) t~ = 60 ~ Figures 1-2 show that the predicted main flow pattern is in good agreement with measured data. At z/T = 0.145, the axial flow from the impeller is somewhat stronger than the measurements suggest. In addition, the simulation predicts a secondary circulation near the axis (positive axial velocity), not seen in the experimental results at this position. In the upper part of the tank at z / T = 0.67, the axial velocity modeled with Grid 3 is close to the measured one. The predictions with Grids 1 and 2 differ qualitatively from Grid 3 results.

269 The predicted turbulent kinetic energy level is about 70% of the measured level below the impeller. Closer to the wall, the agreement is very good. Even the coarse grid simulations predict k fairly well in this location. Largest discrepancies are found at z/T = 0.67.

4.2. Angle-resolved velocity fields The velocity vectors predicted with Grid 3 near the impeller on a vertical surface on three tangential locations relative to the impeller blade are shown in Fig. 3. In Figure 3a the paper surface intersects the middle of the blade (~ = 0~ In Figs 3b and 3c, the blade is rotated from the paper surface towards the viewer by ~ = 30 ~ and t~ = 60 ~ respectively. The trailing vortex is clearly seen in Fig. 3. Similar result was obtained with Grid 2, whereas Grid 1 did not reveal the vortex. According to Schafer et al. [7], the trailing vortex stretches 130~ ~ behind the blade. In the simulations, the vortex is quite weak already at t~ = 90 ~ It is possible that the accuracy of the QUICK scheme is not adequate or the turbulent viscosity generated by the k-e model is too high thus damping the vortex too early. In wing tip vortex simulations, a fifth order accurate differencing of convection terms as well as a modification of the production term in the turbulence model were needed to supress numerical diffusion [20]. The scaled axial, radial and tangential velocity components (Uax/Utip, Urad//Utip and Ut,,,,s,/Ut~p)are shown as the function of the circumferential angle $ in Figs. 4-6. $ equals 0 in the middle of the blade and increases towards the following blade. In the axial mid-impeller plane (z/T = 0.329), the comparisons of circumferential profiles are made in two radial positions. Inside the impeller swept volume (r/T = 0.118), the predicted axial velocity (Fig. 4a) is in good agreement with the measured one for t~ > 20 ~ Close to the blade (~ < 20~ the downwards directed, predicted axial velocity is lower than measured. Weehsler et al. [17,18] reported higher axial velocities than measured here, caused by the infinitely thin blade. In the present simulation, the discrepancy may be due to a rather high value of k (Fig. 7a) which may cause overprediction of effective viscosity and underprediction of velocity. The prediction of the radial velocity shown in Fig. 5a agrees with measurements. In the mid-impeller blade, just outside the radial extent of the blade, r/T = 0.171, the predicted axial velocity follows closely the measured curve for ~ > 30 ~ (Fig. 4b). For 0 < 30 ~ the peak is lower and the flow is downwards in the vicinity of the blade. These deviations presumably reflect the slight difference in the location or velocities of the tip vortex. Predicted radial and tangential velocity profiles (Figs. 5b and 6b) have the same form as the experimental ones, but with an angular displacement by 5~ ~ Circumferential profiles of velocity components at the radial location of the trailing vortex outer edge (r/T = 0.171) in the axial plane z/T = 0.289 are compared in Figs. 4c, 5e and 6c. The predicted profiles show good qualitative agreement with measured data, with the exception of the missed tangential velocity peak in just below the impeller about 10~ behind the blade. The axial location of the trailing vortex is well predicted in the simulation. The predicted absolute values of axial and radial velocities are smaller than measured, which may be caused by a slight difference in the radial position of the predicted vortex. The damping effect caused by the discretization errors or turbulence modeling discussed earlier is another possible cause for the underprediction of velocities. It is evident that the near-wall approach provides the most accurate results in the vicinity of the impeller.

270

Figure 4. Circumferential profiles of UdUap at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z ~ = 0.329, r/T = 0.171, (c) z/T = 0.289, r/T = 0.171. Symbols as in Fig. 1.

Figure 5. Circumferential profiles of Ur~e'Ut~pat three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T = 0.329, r/T = 0.171, (c) z/T= 0.289, r/T = 0.171. Symbols as in Fig. 1.

Figure 6. Circumferential profiles of Utand'Unp at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T= 0.329, r / T = 0.171, (c) z ~ = 0.289, r/T= 0.171. Symbols as in Fig. 1.

Figure 7. Circumferential profiles of k / UT~p at three positions: (a) z/T = 0.329, r/T = 0.118, (b) z/T = 0.329, r/T = 0.171, (c) z/T = 0.289, r/T = 0.171. Symbols as in Fig. 1.

271 4.3. Angle-resolved turbulent kinetic energy The circumferential profiles of the scaled turbulent kinetic energy, k/U,2~p, are compared

at the same locations as velocities in Fig. 7. In the mid-impeller plane, inside the impeller swept volume, k is clearly overpredicted, especially immediately behind the blade. The most significant effect caused by near-wall treatment of turbulence can be seen in front of the blade. There, the near-wall modeling suggests a relatively low level of k compared to the wall function method predictions. In the mid-impeller plane, outside the impeller swept volume, the agreement between calculated and measured k is relatively good 10~ ~ behind the blade. The predicted maximum value of scaled k is over 60% of the measured peak, agreeing with earlier simulations [17,18]. However, the predicted maximum location is about 10~ behind the blade. Immediately below the impeller, the level of k is fairly well predicted although all details of the experimental profile of k are not reproduced in the computation. As with the velocities, the best results in and near the impeller swept area are obtained with the near-wall model. 5. CONCLUSIONS Numerical simulation with a high density grid and k-e turbulence model showed that the tip vortex trailing the impeller blade could be captured with high precision. The predicted vortex has the same form and location as the measured one but dies out earlier. The average flow field outside the impeller region can be simulated with fair accuracy even with a lowdensity grid and wall functions. A dense mesh, preferably combined with near-wall turbulence modeling, is required for detailed prediction of the flow field in the vicinity of the impeller. The one-equation turbulence model used in the near-wall region in the current simulation provided the most accurate results both in the impeller swept area and in the bulk region of the tank. ACKNOWLEDGEMENTS We are grateful to M. Schafer for providing us the experimental data for comparison and wish to thank K. Wechsler and J. Majander for useful discussions. This work was financially supported by the National Technology Agency of Finland (Tekes). REFERENCES

1. H. Wu and G.K. Patterson, "Laser Doppler Measurements of Turbulent Flow Parameters in a Stirred Mixer", Chem. Eng. Sci., Vol. 44, pp. 2207-2221 (1989). 2. V.V. Ranade and J.B. Joshi, "Flow generated by pitched blade turbines I: Measurement using laser Doppler anemometry", Chem. Eng. Commun., Vol 81, pp. 225-248 (1989). 3. G.B. Tatterson, "Fluid Mixing and Gas Dispersion in Agitated Tanks", McGraw-Hill, New York (1991).

272 4.

L.M. Nouri and J.U. Whitelaw, "Particle Velocity Characteristics of Dilute to Moderate Dense Suspension Flow in Stirred Reactors", Int. J. Multiphase Flows, Vol. 18, pp. 2133 (1992). 5. K.J. Myers, R.W. Ward and A. Bakker, "A Digital Particle Image Velocimetry Investigation of Flow Field Instabilities of Axial-Flow Impellers", J. Fluids Eng., Vol. 119, pp. 623-632 (1997). 6. P. Mavros, C. Xuereb and J. Bertrand, "Determination of 3-d Flow Fields in Agitated Vessels by Laser-Doppler Velocimetry: Use and Interpredation of RSM Velocities", Trans IChemE, Vol. 76, Part A, pp. 223-233 (1998). 7. M. Schafer, M. Yianneskis, P. W~ichter and F. Durst, "Trailing Vortices around a 45 ~ Pitched-Blade Impeller", AIChe J., Vol. 44, pp.1233-1245 (1998). 8. M. Schafer, personal communication. 9. V.V. Ranade, J.B. Joshi and A.G. Marathe, "Flow generated by pitched blade turbines II: Simulation using k-emodel", Chem. Eng. Commun., Vol 81, pp. 225-248 (1989). 10. S.M. Kresta and P.E. Wood, "Prediction of the Three-Dimensional Turbulent Flow in Stirred Tanks", AIChE J., Vol. 37, pp. 448-460 (1991). 11. J.Y. Murthy, S.R. Mathur and D. Choudry, "CFD simulation of flows in stirred tank reactors using a sliding mesh technique", IChemE Symposium Series No 136, pp. 341348 (1994). 12. J.Y. Luo, R.I. Issa and A.D. Gosman, "Prediction of impeller induced flows in mixing vessels using multiple frames of reference", IChemE Symposium Series No. 136, pp. 549-556 (1994). 13. E.O.J. Majander and M.T. Manninen, "Numerical simulations of flow induced by a pitched blade turbine: Comparison of the sliding mesh technique and an averaged source term method", 3rd Colloquium on Process Simulation, 12-14 June 1996, Helsinki University of Technology, Espoo (1996). 14. E.O.J. Majander and M.T. Manninen, ''Numerical simulations of flow induced by a pitched blade turbine: The multiple reference frame technique", Technical Report LVT 3/98, VTT Energy, Espoo (1998). 15. J. Syrjanen and J. Majander, "Numerical modelling of flow induced by a propeller impeller: Fluent/UNS and CFX 4 simulation results", Technical Report LVT 5/98, VTT Energy, Espoo (1998). 16. J. Derksen and H.E.A. Van den Akker, "Large Eddy Simulations on the Flow Driven by a Rushton Turbine", AIChE J., Vol. 45, pp. 209-221 (1999). 17. K. Wechsler, M. Breuer and F. Durst, "Steady and Unsteady Computations of Turbulent Flows Induced by a 4/45 ~ Pitched-Blade Impeller", J. Fluids Eng., Vol. 121, pp. 318-329 (1999). 18. K. Wechsler, "Detailed Calculations in Singlephase Stirred Tank Reactors", Stirring and Mixing, International Seminar, 25-28 October 1999, LSTM-Erlangen, (1999). 19. FLUENT 5 User's Guide, Vol. 2, Fluent Inc., Lebanon, NH (1998). 20. J. Daeles-Mariani, D. Kwak and G. Zilliac, "On numerical errors and turbulence modeling in tip vortex flow prediction", Int. J. Num. Meth. Fluids, Vol. 30, pp. 65-82 (1999).


273

Comparison of CFD Methods for Modelling of Stirred Tanks G.L. Lane 1, M.P. Schwarz 1, and G.M. Evans 2 ~CSIRO Minerals, Box 312 Clayton Sth, Victoria 3169, Australia 2Dept of Chemical Engineering, University of Newcastle, New South Wales 2308, Australia

Abstract Simulation by computational fluid dynamics (CFD) is becoming an increasingly useful tool in analysis of the flow in mechanically stirred tanks. However, the development of accurate and efficient modelling methods is a continuing process. One significant complication in modelling of baffled stirred tanks is accounting for the motion of the impeller, since there is no single frame of reference for calculation. A number of approaches have been taken to this problem. In some cases an empirical model is provided for the impeller, whereas other methods are capable of predicting the effect of the impeller directly. In the latter category are the Sliding Mesh and Multiple Frames of Reference (MFR) methods. Results are presented for simulation of a standard configuration tank stirred by a Rushton turbine. Using the same geometry and finite volume grid, the fluid flow is simulated using both these methods. The Sliding Mesh and MFR methods are discussed and compared with respect to computation time and accuracy of prediction of mean velocities and turbulence parameters. It is found that the MFR method provides a saving in computation time of about an order of magnitude. Predicted mean velocities using both methods are compared with experimental data, and it is found that both methods provide good agreement with experimental data. Turbulence parameters are also compared with experimental data. It is found that both methods significantly underpredict the values of specific turbulent kinetic energy and rate of dissipation of turbulent energy. However, for the same grid density, the MFR method provides an improvement in the predictions. 1. INTRODUCTION Mechanically-stirred tanks are widely used in the process industries to carry out many different operations. These mixing tank operations exhibit complex three-dimensional fluid flow, leading to considerable uncertainty in design and scale-up, so that there is a need for more sophisticated methods for understanding the flow and contacting patterns. The progress in recent years in computational fluid dynamics (CFD) has led to increasing interest in the simulation of mixing tanks by this method, and CFD studies have been reported in the literature since the early 1980s. However, the development of modelling methods is a continuing process, and at each step of development it is necessary to assess the accuracy of CFD modelling by comparison against experimental data.

274 Table 1. Nomenclature

B g k P t U e

body force (N m-3)' acceleration due to gravity (m s 2) turbulent kinetic energy (m 2 s -2) pressure (N m -2) time (s) velocity (m s1) turbulent energy dissipation rate (m 2 S"3)

/2 p

viscosity (Pa s) fluid density (kg m -3) angular velocity (rad s l)

Subscripts L T

laminar turbulent

In most stirred tank configurations, baffles are fixed to the tank wall, which introduces a complication in the CFD modelling, since due to relative motion between the impeller and baffles, a single frame of reference is not available for carrying out computations. Various approaches have been taken to account for the impeller motion. These approaches vary in their range of applicability and the accuracy of the results. In some approaches the effect of the impeller is modelled empirically. For example, the impeller may be modelled by prescribing experimentally measured quantities as a boundary condition [1]. Such an approach limits the predictive capabilities of the model, and additionally, the method fails to capture the full details of the flow within the impeller. Details of the flow within the impeller region may be important for modelling multiphase flows such as gas-liquid flow, where there is a strong interaction between impeller and gas. Recent advances in CFD have led to the development of fully predictive methods for modelling baffled stirred tanks. Several approaches are available. One of the most frequently used methods at present is the Sliding Mesh method and a number of papers [e.g. 2, 3, 4] are available describing the method and discussing its accuracy. However, this is a timedependent method, which has the disadvantage of being highly computationally intensive. Therefore, several methods have been developed, in which the impeller can be modelled directly while making a steady-state approximation to the fluid flow. These methods include the Multiple Frames of Reference method [5], the Inner-Outer method [6] and the Snapshot method [7]. The present paper presents results for simulation of a baffled tank stirred by a Rushton turbine using two alternative methods, namely the Sliding Mesh and the Multiple Frames of Reference methods. In each simulation the same geometry is used and is represented by the same finite volume mesh, thus enabling comparison of the methods. Results are assessed in terms of computational efficiency and accuracy of the predictions as compared against available experimental data [8, 9, 10]. Often in CFD studies, accuracy of results is only assessed with respect to mean velocities. However, the turbulent parameters such as the turbulent kinetic energy, k, and the rate of dissipation of turbulent energy, e, are also of great importance in modelling of a range of mixing tank applications, since various mathematical sub-models are typically formulated as functions of k or e. Examples include modelling of micromixing, heat transfer to surfaces, interphase mass transfer, turbulent dispersion in multiphase flows, and break-up or coalescence of bubbles or droplets. Hence, as well as comparing against mean velocity measurements, it is also important to obtain accurate predictions of turbulence quantities.

275 Results are compared against turbulence data for k and e, obtained from several sources [8, 9, 10]. The CFD model matches the geometry of the mean velocity measurements obtained by Hockey [8]. Data from other sources is based on measurements using different tank sizes [9, 10], so it is assumed that the values can be scaled if non-dimensionalised by appropriate reference quantities. Comparison of several modelling methods has been previously presented by Brucato et al. [11]. Their work compares the impeller boundary condition method, the Sliding Mesh method and the inner-outer method. This paper provides additional new information specifically comparing the Sliding Mesh and MFR methods. 2. CFD M E T H O D The commercial code CFX4.2 was used to obtain numerical solutions to the equations for conservation of mass and momentum for an incompressible fluid using a finite difference mesh. Due to turbulent flow, the equations are solved in Reynolds-averaged form and may be given as: V.U-0 (1) o~(pU.__.__.~)+V. (pU|

3t

= - V P + V. ((#L + Pr )(VU + (VU) T ) + pg + B

(2)

Closure of the equations is obtained using the standard k-e. model to calculate the turbulent eddy viscosity BT. The tank geometry is the same as that used in the simulation by Luo et al. [3], for which Laser Doppler measurements of mean velocities are reported by Hockey [8]. The tank diameter, T, is 0.294 m, and the tank has a typical "standard" configuration, consisting of a six-bladed Rushton turbine with diameter D, of T/3, and four equally spaced baffles of width T/IO. The impeller clearance is T/3, blade height is D/5, blade length is D/4 and disc diameter is 0.75D. For the CFD simulation, this geometry is approximated (Figure 1) by a non-uniform finite difference grid, which has 48, 39 and 60 cell divisions in the axial, radial and azimuthal directions respectively. To reduce the amount of computational time required, a 180 ~ section of the tank is modelled assuming symmetry. The baffles, impeller disc, and impeller blades are treated as zero thickness walls and the impeller shaft is treated as a solid zone. The liquid is specified as water and the impeller rotational speed is 300 rpm, corresponding to a Reynolds number of 48,000. The Sliding Mesh method is a feature available in the commercial code CFX Version 4.2. In this method, the finite volume grid is divided into two domains, an inner domain which rotates with the impeller and an outer domain which is stationary. This is a transient method where at each time step the inner grid is rotated by a small incremental angle, and the flow field is recalculated taking into account the additional velocity due to the motion of the grid. Variables are interpolated between domains using an "unmatched grid" interface. The simulation was run from an initial condition of zero velocity until the developed flow pattern became periodically repeatable, indicating that a "steady-state" was reached. This was done in two stages: in the first stage coarse time steps were used corresponding to 30 ~ rotation of the impeller per step. After 10 full rotations of the impeller, steps were reduced to 12~ for a further 3 revolutions. The simulation was repeated using the Multiple Frames of Reference method. This feature is not currently available in CFX4 but was implemented using user-supplied Fortran routines.

276 The method has been previously reported by Luo et al. [5] and Tabor et al. [ 12], where further details are available. In this method, the impeller is also modelled directly and there are again two domains, a zone surrounding the impeller and a zone for the rest of the tank. The interface between rotating and stationary frames of reference was set at an axial distance +0.25D from the impeller centreline and radially at r = 0.75D. Two reference frames are used, so that in the impeller zone there is a rotating frame of reference in which the impeller appears stationary. In the rotating frame of reference, there is a body force term added to the momentum equation consisting of the Coriolis and centrifugal forces according to: B =-2pf~ | pf~ | (fl | X) (3) In this case a steady-state solution is calculated neglecting the time derivative term in equation (2). At the interface between the domains, the velocities are corrected by an implicit coupling method [5, 12] and spatial derivatives of the azimuthal velocity are also corrected. The simulation was run until the normalised sum of mass residuals was reduced to 103, which required approximately 600 iterations. 3. RESULTS AND DISCUSSION Simulation of the stirred tank has been carried out using both the Sliding Mesh and Multiple Frames of Reference methods. The results include predictions of mean velocities and the turbulence parameters k (the specific turbulent kinetic energy) and e (the rate of dissipation of k). An important finding is that the MFR method is much more computationally efficient due to the steady-state calculation, which requires only a fraction of the number of iterations, and achieves a solution with a computer processing time one tenth of that required for the Sliding Mesh method. This is in agreement with the findings of Luo et al. [5]. This may be an important advantage, particularly in modelling multiphase flows, where there are approximately twice as many equations to be solved, and convergence of the solution is substantially slowed due to the complexity of multiphase physics and the requirement to allow the distribution of the dispersed phase to evolve. In the authors' experience [13], the computational requirements of the Sliding Mesh can become overly excessive for simulating multiphase flows, and it has been shown [13] that gas-liquid- flow in a stirred tank can be modelled by the MFR method with much greater efficiency. To assess the suitability of the MFR method as an alternative to the Sliding Mesh method, the accuracy of these methods needs investigation. Results have been compared against data for radial profiles of the inean velocities at the level of the impeller mid-plane and several other positions in the tank using data obtained from Hockey [8] and also from Wu & Patterson [9]. In addition, turbulence quantities are compared against data of Wu & Patterson [9] and Deglon et al. [10]. Several of the comparisons are shown in Figures 2-5, although more extensive comparisons have been done which confirm the general findings stated here. For the mean velocity components, it is found that both methods give good agreement with experimental data, with only small discrepancies at certain positions. The same findings were obtained by Luo et al. [5]. Comparison with experimental k and e values shows that the Sliding Mesh method substantially underpredicts the values of these turbulence parameters in the impeller discharge stream. With the MFR method, the predicted values are significantly higher, yet the peak in the turbulence just off the blade tip is still underpredicted by about 50%. Closer to the wall, the two simulation methods give similar results and agreement with measurements improves. Values of k and e have been also been compared for several positions in the bulk of the tank,

277

and it is found that the Sliding Mesh and MFR methods tend to predict similar values. Values of k are actually in fair agreement with experiments for many positions, except near the centre below the impeller (Figure 5). Comparison with e in the bulk of the tank indicates substantial underprediction by the simulations. However, the accuracy of the experimental data might be called into question in this case, since the experimental method [10] does not predict e directly, and further confirmation might be needed. Ng et al. [14] have also assessed the accuracy of predictions of k in the impeller discharge stream for simulations using the Sliding Mesh method of a tank stirred by a Rushton turbine. They also found that the peak in the turbulence near the impeller tip was substantially underpredicted. By repeating the simulation with higher grid resolution and an embedded grid around the impeller blades, it was found that improved predictions of k were obtained, which was probably due to better prediction of velocity gradients around the impeller. However, at a grid density of 400 000 cells, the prediction of the peak k value was still only about 50% of the measured value. In the present results, it is found that the Multiple Frames of Reference method gives substantially improved predictions of k and e in the impeller stream compared with the Sliding Mesh method, for the same grid density. It is possible the MFR method could provide even better results using a higher grid resolution. The low predicted values of the turbulence quantities point to the need for caution when extending the CFD modelling to investigate phenomena in stirred tanks such as heat transfer or multiphase flow, where additional equations may need to be calculated as functions of the predicted k and e values. An interesting point is that if the total energy dissipation in the tank is calculated by integrating e over the tank volume, the power will be underpredicted due to low e values. Yet, it has been found [2] that if the pressure distribution over the impeller blades is used to calculate torque on the impeller, a good prediction of power input can be obtained. It might be possible for e values to be corrected before being used in other equations (e.g. for calculating bubble or drop breakage). If it could be assumed that the predicted pattern of distribution of e is roughly correct, then values could be scaled by a correction factor so that the integration of e values matches the power based on impeller torque. There is a need for more accurate modelling of the turbulence. One approach has been to increase the grid density, particularly around the impeller [ 14]. Measurements in the impeller stream have indicated that the turbulence is anisotropic [9], although in the bulk of the tank the turbulence may be isotropic. This implies that the assumption of isotropic turbulence in the k-e model may not be valid. To improve the prediction of turbulence quantities, it may be necessary to use a different turbulence model, for example a Reynolds stress model. However, such a model is more computationally demanding due to six additional equations which need to be solved, and for multiphase flows, the computation time may be excessive. In addition, the Reynolds stress model cannot be easily implemented in CFX4 when using the MFR method. 4. CONCLUSIONS Fluid flow in a baffled stirred tank has been simulated by computational fluid dynamics using both the Sliding Mesh and Multiple Frames of Reference methods. Comparison of mean velocities with experimental measurements indicates good agreement for both methods. The Sliding Mesh method is a time-dependent method where the time step is limited to a fairly small increment of rotation. For multiphase flows, this method may lead to excessive computation time due to the large number of time steps required. The MFR method provides a

278 means of reducing the computation time and so provides a more practical alternative, particularly for modelling gas-liquid contacting. The prediction of turbulence quantities is also important for modelling various aspects of stirred tank processes. Results show that both methods underpredict k and e, particularly in the impeller discharge stream, although the MFR method gives improved estimates. Caution should be taken when incorporating other mathematical models where equations are calculated as functions of k or e. CFD predictions of turbulence quantities in stirred tank simulations need further improvement.

REFERENCES 1. Bakker, A. & Van den Akker, H.E.A., 1994, "Single-phase flow in stirred reactors", Trans. L Chem.E., Vol.72, No. A4, pp. 583-593. 2. Lane, G. and Koh, P.T.L., 1997, "CFD Simulation of a Rushton Turbine in a Baffled Tank", Proc. Int. Conf. on Computational Fluid Dynamics in Mineral & Metal Processing and Power Generation, CSIRO, Melbourne, 3-4 July 1997, pp. 377-385. 3. Luo J.Y., Gosman, A.D., Issa, R.I., Middleton, J.C., & Fitzgerald, M.K., 1993, "Full Flow Field Computation of Mixing in Baffled Stirred Vessels", Trans. L Chem.E., Vol.71, Part A, pp. 342-344. 4. Lee, K.C., Ng, K. & Yianneskis, M., 1996, "Sliding Mesh Predictions of the Flows around Rushton Impellers", Fluid Mixing V, L Chem.E. Symposium Series, No. 140, pp. 47-58. 5. Luo, J.Y., Issa, R.I. & Gosman, A.D., 1994, "Prediction of Impeller Induced Flows in Mixing Vessels using Multiple Frames of Reference", L Chem.E. Symposium Series, No. 136, pp. 549-556. 6. Brucato, A., Ciofalo, M., Grisafi, F. & Micale, G., "Complete Numerical Simulation of Flow Fields in Baffled Stirred Vessels: the Inner-Outer Approach", L Chem.E. Symposium Series, No.136, pp. 155-162. 7. Ranade, V.V. & Dommeti, S.M.S., 1996, "Computational Snapshot of Flow Generated by Axial Impellers in Baffled Stirred Vessels", Trans. L Chem. E., Vol. 74, Part A, pp. 476-484. 8. Hockey, R.M., 1990, PhD Thesis, Dept. of Mech. Eng., Imperial College of Sci., Tech. & Medicine, London. 9. Wu, H., & Patterson, G.K., 1989, "Laser-Doppler Measurements of Turbulent-Flow Parameters in a Stirred Mixer", Chem. Eng. Sci., Vol. 44. No. 10, pp. 2207-2221. 10. Deglon, D.A., O' Connor, C.T., & Pandit, A.B., 1998, "Efficacy of a Spinning Disc as a Bubble Break-Up Device", Chem. Eng. Sci., Vol. 53, No. 1, pp. 59-70. 11. Brucato, A., Ciofalo, M., Grisafi, F. and Micale, G., 1998, "Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modelling approaches", Chem. Eng. Sci., Vol. 53, No. 21, pp. 3653-3684. 12. Tabor, G., Gosman, A.D., & Issa, R.I., 1996, "Numerical Simulation of the Flow in a Mixing Vessel Stirred by a Rushton Turbine", Fluid Mixing V, I. Chem.E. Symposium Series, No. 140, pp. 25-34. 13. Lane, G.L., Schwarz, M.P., & Evans, G.M., 1999, "CFD Simulation of Gas-Liquid Flow in a Stirred Tank", Proc. 3rd Int. Symposium on Mixing in Industrial Processes, Osaka, Japan, Sept. 19-22 1999, pp. 21-28. 14. Ng, K., Fentiman, N.J., Lee, K.C., & Yianneskis, M., 1998, "Assessment of Sliding Mesh CFD Predictions and LDA Measurements of the Flow in a Tank Stirred by a Rushton Impeller", Trans. I. Chem.E., Vol.76, Part A, pp. 737-747.

279

Figure 1. Finite volume mesh for stirred tank.

Figure 2. Comparison of axial velocity components obtained by each simulation method and experimental data (~ experimental [8], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial locations below and above the impeller.

Figure 3. Comparison of radial velocity components obtained by each simulation method and experimental data (~ experimental [8, 9], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial locations at impeller mid-plane and slightly above the impeller.

280 Normalised turbulent kinetic energy, x = 0.098 0.1 9,, 0.08 "

.==0.06

-

1

25-

~

.

Normalised energy dissipation rate, x = 0.098 41,

20"~

*

g15 -

~-

t~

0.04 ;,

0.02 '

_

~ , ~ ~

.~.,

O-

1.0

1.5

2.0

2.5

3.0

1.0

1.5

R/Ri

2.0

2.5

3.0

R/Ri

Figure 4. Comparison of k and E values obtained by each simulation method and experimental data ( , experimental [9], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial location at impeller mid-plane.

0.008 o~.~

Normalised turbulent kinetic energy, x = 0.049

0.006 0.004

9 _

Normalised energy dissipation rate, x = 0.049

0.6

"

I,

0.5>0.4 "

(I,

'

~.

.L.

~

'~

0.3-

0.002

0

O 0.0

0.5

1.0

1.5 R/Ri

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

R/Ri

Figure 5. Comparison of k and e values obtained by each simulation method and experimental data ( . experimental [8, 10], + Sliding Mesh simulation, 9 Multiple Frames of Reference simulation). Axial location below impeller in bulk flow, half-way between impeller and base.

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

281

Predicting the tangential v e l o c i t y f i e l d in s t i r r e d tanks using the Multiple Reference Frames (MRF) m o d e l w i t h v a l i d a t i o n by L D A m e a s u r e m e n t s Lanre Oshinowo a, Zdzislaw Jaworski b'c, Kate N. Dysterb, Elizabeth Marshall a and Alvin W Nienow b a Fluent Inc., 10 Cavendish Court, Lebanon, NH, USA 03766 b School of Chemical Engineering, University of Birmingham, Birmingham B 15 2TT, UK c also Chemical Eng. Faculty, Technical University of Szczecin, 70065 Szczecin, Poland Modeling the three-dimensional, transient motion of an impeller using the sliding mesh approach is the most rigorous and fully predictive method of CFD analysis of baffled stirred tanks. However, despite current advances in computational speed, this transient analysis takes a relatively long time. The multiple reference frame model, or MRF, is a steady state approximation of the impeller motion and is a viable alternative for such analysis. Recent work has shown numerical predictions of this flow field with the MRF model to give counterintuitive predictions where the tangential component of velocity is opposite to the impeller motion in large regions of the bulk liquid. Such predictions are also contrary to experimental observations. In the present work, the MRF model was used to correctly predict the tangential velocity distribution in a baffled, stirred tank containing a single-impeller (Rushton or pitched-blade turbine) under turbulent flow conditions. As a consequence of this work, a set of guidelines for applying the MRF model was determined and presented here. 1.

INTRODUCTION

In carrying out CFD optimization studies of impeller-tank geometry for macromixing, chemical reaction and precipitation, a fully predictive model for momentum transfer is required. Four such models for the impeller-tank interaction are reported in the literature: the multiple reference frames (MRF) model (Luo et al., 1993), the inner-outer model (Brucato et al., 1994), sliding mesh model (Murthy et al., 1994) and the force-field technique (Derksen and van den Akker, 1999). The latter approach was implemented within the lattice-Boltzman framework. The former methods were implemented within the control-volume approach. The sliding mesh model offers the advantages of modeling the transient behavior of the fluid motion in the tank but with a penalty of computational expense due to the time-dependent formulation. The inner-outer model divides the tank into two partially overlapping zones, one in a rotating frame, and the other in the stationary frame. An iterative matching of the solution obtained on the boundaries of the overlapping zones is required. The MRF model is the simplest being a steady-state approximation in which different cell zones move at different rotational speeds or on different axes of rotation. It requires least computational effort since the inner and outer frames are implicitly matched at the interface, requiting no additional iterative calculations. This facilitates rapid turnaround for design cycles and was chosen for this study. In addition, it does not require an a p r i o r i knowledge of the impeller discharge flow and turbulence. As an added benefit, the solution can be used as an initial condition for time-dependent sliding mesh simulations to avoid modeling the startup transients.

282 Often situations arise in which the swirl component of velocity in certain parts of the vessel, especially near baffles, is in a direction opposite to that of the impellers. CFD simulations of stirred tanks have predicted this reverse swirl in regions of the tank near the shaft, for example, where it is quite counter-intuitive. Furthermore, it has been observed in some studies that this reverse swirl persists throughout a significant part of the flow domain contrary to experimental observations (Harris et al, 1996, Venneker and Van den Akker, 1997). To date, there has been a general lack in the literature of validation studies comparing CFD predictions to measurements of the tangential velocity component in steady-state modeling of mixing tanks. Since an inert tracer or reactant is often added to the stirred liquid at the free surface, or close to it, the ability to correctly simulate the general direction (and magnitude which is typically small relative to the impeller tip speed) of the flow during the addition process is crucial. Macromixing, and subsequent stages in the mixing process, such as rni'cromixing, chemical reaction and precipitation can be significantly affected by the flow conditions at this initial process stage. The present paper models the three-dimensional turbulent flow in a baffled stirred tank, applying the MRF model to describe the motion of both radial Rushton and pitched-blade turbines. It will be shown by LDA measurements, that some reverse swirl does occur and this phenomenon can be predicted by CFD. It will also be shown that reverse swirl predicted in the CFD simulations, in locations where LDA does not show it, can be eliminated with improvements in the CFD problem definition and analysis. 2.

EXPERIMENTAL

The LDA system and signal processing used were essentially the same as those described elsewhere (Jaworski et al., 1996) except for the number of LDA channels, which was increased to two to measure the mean tangential and axial velocity components. A cylindrical, fiat-bottomed glass tank of internal diameter, T = 202mm, was provided with four full-length baffles of width 22mm. Distilled water was used at a height, H=T, with a square glass jacket placed around the tank to reduce the effect of wall curvature on the laser beams. This arrangement with the impeller drive was mounted on a traversing system, which could be moved in 3 dimensions with an accuracy of 0.01mm. Four, 6-bladed impellers were employed in the study, two Rushton and two pitched-blade turbines. Their diameters, D, were approximately T/4 and T/3, (51mm and 74mm for the Rushton turbines, and 48ram and 68ram for the pitched-blade turbines). All impellers had a blade width of 0.20D and thickness of 1 mm with the impeller centers positioned at a distance of H/3 off the tank bottom. Three horizontal planes were chosen for the location of measurement points, which had earlier indicated reverse swirl in CFD simulations. These were: 50, 90 and 120 mm above the impeller centers (or 117.3, 157.3 and 187.3 mm above the tank bottom). Three angular measurement positions, cr of 10, 45 and 80 degrees between adjacent baffles were used. The angle was measured in the direction of impeller rotation. The measurements were carried out at 18 radial distances from the tank axis, using 5 mm intervals and starting 10 mm from the axis. All data was collected in the ensemble average mode. Validation of the rms values was not considered since LDA measurements contain fluctuations due to blade passage frequency and large-scale instabilities as well as the "true" turbulence. RANS-based CFD aims to predict the latter but generally greatly underpredicts it (Brucato et al, 1998). Therefore, angle resolved measurements for comparisons in the discharge of the impeller to resolve the rms values were considered beyond the scope of this work since the primary interest is validating predictions of mean tangential flow far from the impeller. The average experimental error was estimated to be in the range of 1-2% of the impeller tip velocity.

283 3.

MULTIPLE REFERENCE FRAMES MODEL IMPLEMENTATION

The MRF model is used to model the impeller motion. The computational domain was divided into two zones, one in which the velocity is computed relative to the motion of the impeller and the second zone where the velocities are computed in a stationary frame relative to the walls and baffles. The velocity in the rotating frame is transformed to the stationary frame by V=Vr+~Xr (1) and the velocity gradient is Vv = Vv~ + V(~xr) (2) where v is the velocity in the stationary (absolute) frame and v, is the velocity in the relative (rotating) frame. At the boundary between the rotating and stationary zone, the diffusion and other terms in the governing equations obtain values for the velocities in the adjacent zone determined from Eqns. (1) and (2).

3.1

Azimuthal Averaging

The MRF model is best applied when the interaction between the impeller and the baffles is relatively weak. Hence, the optimal radial position of the interface between the stationary and rotating zones is roughly midway between the impeller blade tip and the inner radius of the baffles. However, when the impeller-baffle interaction is not weak (i.e. when D > T/2), the steady-state flow field can be dependent on the position of the impeller blades relative to the baffle position. Figure l(a) shows the impeller blades at three angular positions (0 ~ 15~ 30 ~ relative to the baffles. Figure l(b) shows the predicted tangential velocities at those angles and illustrates a periodic velocity profile repeating every 1/2 blade period with respect to the baffles. Therefore, for a 6 bladed impeller in a 4 baffle tank, an azimuthally-averaged mean flow field can be obtained over one-half of the blade period equivalent to 30 ~. In this work, the flow field was calculated with the impeller at two phase angles of 0 ~ and 15 ~ relative to the baffles and then averaged. 4.

NUMERICAL SIMULATION

The stirred tank with fluids of different viscosity was modeled in FLUENT 4.52. The three-dimensional, single-block, hexahedral element grids were automatically generated using MIXSIM 1.5, a mixing tank analysis tool. A 180 ~ sector was modeled due to the periodic symmetry in the flow. The condition of no slip was applied to all solid boundaries except for the liquid surface where a zero shear condition was applied. The simulations were considered conver~ed when the sum of the residuals was 25 mm. Figure 8 compares the experimental micromixing time values modeled by equation (10) with mean simulated values.

Figure 8 : Comparison between experimental model (equation (10)) and simulated values.

From Figure 8, it can be concluded that a very good agreement between the experimentally obtained model and the simulations is obtained. At N = 20 s-1, the micromixing time decreases steadily when the feed pipe position r increases. At this rotating speed, the regime is laminar in the whole range of the investigated radial positions. In this case, only equation (11) is plotted. On the contrary, at N = 50 sl , the flow regime changes from laminar to turbulent, thus both equations (11) and (12) are used. Figure 8 shows that simulated micromixing times go through a maximum, predicting qualitatively well the change of regime when the radial distance is increased. As a conclusion, the lateral feed pipe positions should be chosen with great care in order to obtain optimum micromixing conditions.

8. CONCLUSIONS The comparison of simulation maps with photographs obtained from decolourization experiments show that the numerical simulation by CFD codes can well describe the process of mixing in the Confined Mixing Zone of the Sliding-Surface Mixing Device. In a previous experimental work [1], a generalized relation obtained from theoretical considerations allowed the calculation of micromixing time as a function of SSMD process parameters (r, h, Q, N). In this study, the k-e model is used to determine the micromixing time. The comparison of results obtained by CFD simulation with those obtained experimentally shows that micromixing time values under the same operating conditions are of the same order of magnitude. These facts prove that the k-e model applied with appropriate cell dimensions is well adapted to simulate mixing in this new device and can be successfully used to predict mixing effects in sliding-surface mixing devices of different geometric dimensions operating in a wide range of process conditions.

312

REFERENCES 1. Rousseaux, J.M., L. Falk, H. Muhr and E. Plasari, "Micromixing Efficiency of a Novel Sliding-Surface Mixing Device", AIChE J., Vol. 45, N ~ 10, pp. 2203-2213, Oct. 1999. 2. Schlichting, H., Boundary Layer Theory, McGRAW-HILL Series in Mechanical Engineering, New York, Sixth Edition, 1968. 3. Kruis, F.E. and L. Falk, "Mixing and Reaction in a Tubular Reactor : a Comparison of Experiments with a Model Based on a Prescribed PDF", Chem. Eng. Sci., Vol. 51, No. 10, pp. 2439-2448, 1996. 4. Launder, B.E. and B.I. Sharma, "Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow near a Spinning Disk", Letters in Heat and Mass Transfer, Vol. 1, pp. 131-138, 1974. 5. Hannon, J., S. Hearn, L. Marshall and W. Zhou, "Assessment of CFD Approaches to Predicting Fast Chemical Reactions", Paper 188, AIChE Annual Meeting, Miami Beach, FL, Nov. 15-20, 1998.

NOTATIONS

A, B Chemical species C~ Cel C~2 Parameters used in the k-e model K N

QA QB r t,,, u' U + y

Kinetic turbulent energy Rotation speed Feed rate of species A in the central feed tube Feed rate of species B in the lateral feed tubes Radial position Micromixing time Fluctuating velocity Velocity vector Dimensionless wall criterion

m 2 s2 -1 s mL min 1 mL min ~ m s m s1 m s~

Greek letters e vt v /9

to

Turbulent energy dissipation Kinematic turbulent viscosity Kinematic viscosity Fluid density Function depending on Reynolds number (see equations (4), (5) and (6)) Angular rotation speed

Abbreviations CFD Computational Fluid Dynamics CMZ Confined Mixing Zone SSMD Sliding-Surface Mixing Device

W kg 1 - m 2 s-3 m 2 s-1 m: s-1 kg m -3 S-2

rad s -1


313

AN I N V E S T I G A T I O N O F T H E F L O W F I E L D O F V I S C O E L A S T I C F L U I D IN A S T I R R E D V E S S E L Wentao Ju a, Xiongbin Huang', Yingchen Wang", Litian Shi' and Bin Zhangb, Jingxian Yuanb 'College of Chemical Engineering, Beijing University of Chemical Technology bDaqing EOR Equipment Cooperation By using the PIV (Particle Image Velocimetry), the flow fields of polyacrylamide (PAM) aqueous solutions with different concentration and with noticeable elasticity when the PAM concentration is relatively high were measured at different rotational speeds for two types of agitator. Radial flow is dominant in the flow field of high-concentration PAM solution and there are stagnant flow and secondary flow taking place at some areas in the vessel. The flow field of pseudoplastic fluid in the stirred vessel was numerically simulated using the "black-box" model. The simulation results are approximately consistent with the experimentally measured flow pictures. The velocity component values are also approximately in accordance with the experimental ones. I.

INTRODUCTION

Although there are few researches on determining the flow field of polymer mainly because of its opacity by using LDA (Laser Doppler Anemometry), there are still researches using some other experimental techniques. By using streak photography and a TV camera recorder, Kuboi et al. (1986)t~l obtained photographs of flow patterns for high viscosity Newtonian and shear-thinning non-Newtonian fluids. Elson et al. (1986)t21 described the use of X-ray-heavy metal tracer method for the study of mixing patterns in opaque fluids in agitated vessels. The numerical simulation of flow field in stirred vessels can be dated back to 1970's, but the non-Newtonian flow simulation is rather new in the field of CFD (Computational Fluid Dynamics) and flourishes in 1990's. These simulations are usually concentrated on certain limited geometry. Besides, the determination of turbulent model constants is also rarely performed for non-Newtonian fluids to the same extent as for Newtonian fluids. Hua Lu et al. (1995)t31 simulated the non-isothermal flow of the Carreau fluid in an agitated vessel induced by scraped-surface blades. Venneker et al. (1997)[41 presented results about the simulation of the turbulent flow of a pseudoplastic fluid in a stirred vessel. A qualitatively correct solution was obtained with the standard k-r on condition that at the impeller tip experimental data are used as boundary condition. However, there are differences between predictions and measurements. The study of Anne-Archard (1997)t51 "To whom correspondence should be addressed: Beijing University of Chemical Technology P.O.Box230, Beijing, 100029, P.R.China; E-mail: [email protected]

314 aimed at studying the influence of viscoelasticity and inertia on a mixing operation in laminar regime which is done in a numerical way by solving a mass transport equation in a vessel equipped with a paddle impeller. The result is useful to further the numerically treating of viscoelastic fluid mixing despite the simple geometry studied and its assumption of 2-D flow. 2. E X P E R I M E N T A L Six PAM aqueous solutions of different concentration were used in this study. The mean molecular weight of the PAM is 1.2 • 107~1.5 • 107. Solution rheology was investigated using a RV-II coeentric cylinder viseometer and a Carri-Med Rheometer CSL52oo(TA Instruments) with a cone and plate measuring system. A fiat bottom cylindrical 0.5m inner diameter stirred vessel made of polymethyl methacrylate with four symmetrically distributed baffles was used. Two types of axial-flow impellers of different diameters and blade angles were used, each having two (0.3m in diameter) or three blades (0.184m in diameter). The impeller clearance (distance between impeller and vessel) is 0.085m. By using the PIV system established by our own, sets of observations were made and recorded on films for different concentration solutions. Two types of light sources were used, namely, the continuous I-Ge lamp and three-pulse Xenon lamp to form light sheet. With the multi-pulse lamp, the velocity directions of tracing particles can be determined accurately and easily. Spherical fluorescent particles of about 300~500 ~tm in diameter were used as the tracers. An IBAS image analysis processor was used to digitize the images recorded on the 35mm films. 3.

RESULTS AND DISCUSSIONS

3.1 Rheological Experiments The steady shear equations for PAM solutions with different concentration are shown in table 1. Table 1 The power law equation for PAM solutions (13.8~ RVII) Concentration of PAM solution (ppm) ,1-r~"-' / Pa.s 500 K=0.00943, n=0.902 1000 K=0.0543, n=0.734 2000 K=0.594, n=0.641 3000 K=0.728, n=0.448 4000 K=1.47, n=0.406 5000 K=2.57, n=0.354 The flow behavior index of each PAM solution is smaller than 110, whicia' indicates the degree of the pseudoplasticity of the solution, i.e., the higher the solution's concentration the smaller the index, and the higher the pseudoplasticity. Using CSL]00, the dynamic storage shear modulus G'(w) and dynamical loss shear modulus G"(w) in the linear viscoelastic region for each PAM solution with different concentrations are measured. According to the Laun rule(Laun, 1986)t61,the primary normal stress coefficient v~(~)

315 can be estimated by equation (1):

v~(~)=2 ~ 2

(1)

1 ~,G"(o~)

The results of the estimated ~,~) are shown in Fig.1. As is well known, the angle of internal friction represents the relative magnitude of viscosity and elasticity of the fluid. The internal friction angles of each solution are shown in Fig.2. Because the lower the concentration of the solution the closer to 90* the internal friction angle, it can be concluded from Fig.2 that the elasticity of the low-concentration PAM solutions is less strong than that of the one with high concentration. It is justifiable to disregard the elasticity of low-concentration PAM solutions in the numerical simulation, i.e. the low-concentration PAM solutions can be considered to be purely pseudoplastic fluid. 1

11

v"v.~v,

.70 9

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

lo

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"-

i~,..,..

.

-

~,t" ~

-I"

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"§

o Io,~

9

--~-- 500ppm .v

--v-- 3000ppm 4000~

o

"-.4.~.+_+_+s+f

:o,

........

o, ......... ; AnBular frequency r I ~'

,o

Fig.l The estimated primary normal stress coefficient value of PAM solutions with different concentration

....

!

0.01

. . . . . . . .

!

9

"',

......

i

"

. . . . . . . .

-

-

i

0.1 1 10 Ansular frequencym/ s"1 Fig.2 The internal l~iction angle of PAM solutions with

different concentration

3.2 Results of Flow Field Measurement

The Effects of PAM Concentratiort on the Flow Field a. The Characteristics of the Flow Field of Low-Concentration PAM Solutions Axial flow features can be observed in the whole field and there is one main axial flow circulation region. The value of the tangential velocity is approximate to that of the axial and radial velocity (see Pie.1 and Pie.2) The free surface is concave around the shaft and the surface is stable. The surface concave will be strengthened as the rotational speed increases when using the same type of agitator. The reverse taper flow region under the impeller not only expands but also evolves to a regular circulated flow (see Pie.3) which indicates a tendency to be radial flow. b. The Characteristics of the Flow Field of High-Concentration PAM Solutions Pica and Pie.5 are flow pictures of different concentration (1000 and 3000ppm respectively). The two pictures show that the radial flow is strengthened when the concentration of PAM is increased. This may be the result caused by the comprehensive effects of elastic force, viscous force and inertia force.

316

pic'1 sOOppmPAM exposure time 1/30s, three-blade agitator, rotational speed 250 rlmin, IGe lamp

Pic.2 500ppm PAM solution, exposure time 1/30s, three-blade agitator, speed 250 r/min, 0.085m from bottom (impeller region), rotational direction is clockwise, I-Ge lamp

No obvious concave of the free surface is observed. Rather apparent periodic Weissenberg effect has been observed at low rotational speed in numerous experiments and the fluid around the shaft rotates upward in a reverse direction of the rotating direction of the shaft. This effect is no longer obvious at high rotational speed because the inertia force plays a dominant role instead of the normal stress. The tangential, axial and radial velocities are all small except in the impeller region the tangential velocity is large. Pic.5 and Pic.6 are the orthographic view and bottom view respectively under the same experiment condition. The maximum value of the tangential velocity is 0 . 3 5 d s while the maximum value of both axial and radial velocity is about O.l6m/s.

Pic.3 SOOppm PAM solution, exposure time 1/3Os, two-long-bladed agitator, rotational speed 130r/min, I-Ge lamp

Pic.4 lOOOppm PAM solution, exposure time 1/3oS., two-long-blade agitator, rotational speed 1OOr/min, I-Ge lamp

317

Pic.5 3000ppm PAM solution, exposure time 15ms, two-long-blade agitator, rotational speed 130r/min, 3-pulse Xenon lamp

Pic.6 3000ppm PAM solution, exposure time 15ms, two-long-blade agitator, rotational speed 130r/rain, 3-pulse Xenon lamp

When using the same type of agitator, the tangential velocity of high-concentration solution is smaller than that of the low-concentration solution at the same rotational speed. We can deduce from Pic.6 and Pic.7 that the velocity difference between the impeller blade and the fluid is greater in high-concentration solution than that in low-concentration solution. This is caused by viscoelastic effect of the fluid and consequently results in the increase of power consumption. There are stagnant flows taking place in the areas where the wall and bottom intersect in the vessel (see Pie. 8). It can also be concluded that the stagnant flow region becomes larger in these areas as the solution concentration increases. Besides, there are also stagnant flows taking place near the baffles, which can be observed clearly in Pic.6. The Effect of Different TvDe of A~itator on the Flow Field When using different types of agitator the flow fields are different from each other for lowconcentration PAM solutions. The main axial flow region is larger when using three-blade agitator than that when using two-long-blade agitator. Pie.1 and Pica are flow field pictures of 500ppm PAM solution using three-blade and two-longblade agitator respectively at close impeller tip velocity (2.42m/s and 2.04m/s, respectively). The influence of agitator type on the flow field is noticeable for highconcentration PAM solutions. The stagnant flow region is larger when using three-blade agitator than that when using two-long-blade agitator. Pic.5 and Pie.9 Pic. 7 500ppm PAM solution, exposure time 33.3ms, are flow field pictures of 3000ppm PAM two-long-blade agitator, rotational speed 100r/min, Isolution using three-blade and two-long- Ge lamp.

318 blade agitator respectively at close impeller tip velocity (2.42m/s and 2.04m/s respectively). The reason for the expansion of the stagnant flow region is due to the velocity decrease caused by the strong pseudoplasticity, which enables the PAM solutions exhibit high viscosity because of the low shear rate in the distant areas from the impeller.

Pie.8 4000ppm PAM solution, exposure time 19 ms, two-long-blade agitator, rotational speed 100r/rain, 3-pulse Xenon lamp 4.

Pie.9 3000ppm PAM solution, exposure time 15 ms, three-blade agitator, rotational speed 250r/min, 3-pulse Xenon lamp

SIMULATION FOR PSEUDOPLASTIC

FLUID

4.1 Data Predisposing and Simulation Method It is indispensable to interpolate the uneven randomized distributed data to get further information from the flow pictures. The interpolated result of the impeller region is also used as boundary condition in the simulation using the "black-box" model. In order to simplify the calculation it is assumed that the fluid in the stirred vessel is incompressible pseudoplastic fluid disregarding its elasticity, and that the flow in the stirred vessel is steady-state symmetric turbulent flow. (When the PAM concentration is relatively low, from 500ppm to 2000ppm, the measured power number is constant in the measured range of impeller rotation speed. Therefore, the flow can be considered to be turbulent flow.) The shear viscosity is calculated using the power law model: n-1

1 Where, 77 is the shear viscosity of the pseudoplastic fluid, and ltr(;02 is the second 2 invariant of the rate-of-strain tensor ). The above method assumes the calculated velocity gradient equals the exact velocity gradient and the shear viscosity is determined by the velocity gradient. The simulation is performed with the standard k-6 model. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is adopted for solving the discretization equations.

4.2 Simulation Results The simulated flow field of 500ppm PAM solution when using three-blade agitator at

319 320r/min is shown in Fig.3, which is approximately the same as the flow field in Pie.3. The simulated flow also shows an obvious radial flow tendency as the experimental one. The value of the axial velocity is somewhat small, and there are stagnant flow taking place in the area near the wall and bottom. A very small vortex can be seen at the area where the wall and bottom intersect.

I~1

I

l

i

I

,/

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1

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,~ ~

~

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/

'

/

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,

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~

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# /

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Fig.3 The small vortex in the area where the wall and bottom intersect 0.15 0.10

0.08,

":'~

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0.05

0.00

.~

0.00

"~

-0.08

>

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~.

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-9

-0.10,

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0.~0 "

0.~5

o.io

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.o.15.


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I ---0--weighted interpolation v a l u e 1

0,15

0.;0

o.~5

"~176

o.~o"

o.;~ " oJ,o " o.~s

Fig. 4a Axial velocity distribution, 500ppm PAM solution, 0.040m from bottom, rotational speed 100r/min, two-longblade agitator

o.~o

rlm

rim

Fig. 4b Radial velocity distribution, 500ppm PAM solution, 0.040m from bottom, rotational speed 100r/min, two-longblade agitator 0.20.1-

0.08.

~

0.00.

i o.

~ / 1

-0.24 ......... o.oo

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-02-

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-0.5

o.~,~

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o.~5 r/m

oDo

--o-- ~ighted interpolationvalue t numericalsimulationvalue

I

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o.~

o.~o

Fig.5 Axial velocity distribution, 1000ppm PAM solution, 0.114m from bottom, rotational speed 100r/min, two-longblade agitator

ooo

o;~

o'1o

9

o:15 rim

.

~o

,

o~ ,

.

~6o

Fig,6 Axial velocity distribution, 2000ppm PAM solution, 0.114m from bottom, rotational speed 130r/rain, ~ l o n g blade agitator

Fig.4 to Fig.6 are some comparison examples of the velocity distribution between simulated value and experimental (interpolated) value. The key reason for the difference between the simulated and experimental values is the difference between the interpolated

320 and actual values caused by the extrapolation with the experimental data in some areas of the flow field where there are insufficient experimental data. 5.

CONCLUSION The PIV system used can efficiently get the flow field pictures. The aqueous solution of PAM is a viscoelastic fluid. The concentration of the solution plays a key role in influencing its flow field in a stirred vessel. Basically, the flow of the solution has a radial flow tendency while using the two axial flow agitators. The numerical simulation can obtain qualitatively correct solution. SYMBOLS G' dynamic storage shear modulus Pa v G" dynamical loss shear modulus Pa 6 k turbulence kinetic energy J/kg kg.m -~. ~, K consistency coefficient sn-2 n r u

flow behavior index radial distance axial velocity

I11 m . s -I

radial velocity m.s ~ turbulence energy dissipation rate W/kg shear rate s~ rate-of-strain tensor

r/a apparent shear viscosity Pa.s co angular frequency s"~ ~ul primary normal stress coefficient Pa.s 2

REFERENCES [1] Kubi, R., Nienow, A. W., "Intervortex Mixing Rates in High-viscosity Liquids Agitated by High-speed Dual Impellers", Chem. Eng. Sci., 123-133 (1986) [2] Elson,T. P., D. J. Cheesman and A. W. Nienow, "X-ray Studies of Cavern Sizes and Mixing Performance with Fluids Possessing a Yield Stress", Chem. Eng. Sci., 41, 25552562 (1986) [3] Hua Lu et al., J. "Non-isothermal Agitated Flow Fields of Non-Newtonian Fluids", Chem. Industry and Eng. (China), Vol. 46, No. 3 (1995) [4] Venneker, B. C. H., Van Den Akker H. E. A., "CFD Calculations of the Turbulent Flow of Shear-thining Fluids in Agitated Tanks", Proc. 9th. Eur. Conf. Mixing, 179-186 (1997) [5] Anne-Arehard, D., Boisson H. C., "Numerical Simulation of Newtonian and Viscoelastic 2-D Laminar Mixing in an Agitated Vessel", Proc. 9th. Eur. Conf. Mixing, 145-152(1997) [6] Laun H.M., J.Rheol., 30(30), 459-501 (1986)


321

F l o w of Newtonian and non-Newtonian fluids in an agitated vessel equipped

with a non-standard anchor impeller* G. Delaplace a, C. Torrezb, M. Gradeck c, J.-C. Leulieta and C. Andr6 b alnstitut National de la Recherche Agronomique (I.N.R.A), Laboratoire de G6nie des Proc6d6s et Technologie Alimentaires, 369 rue Jules Guesde, B.P. 39, 59651 Villeneuve d' Ascq C6dex, France bHautes Etudes Industrielles (H.E.I.), Laboratoire de G6nie des Proc6d6s, 13 rue de Toul, 59046 Lille Cedex, France CLEMTA-CNRS UMR 7563, 2 avenue de la forSt de H a y e - BP 160, F-54501 Vandoeuvre Cedex, France In this paper, a commercially available fluid dynamic (CFD) software package (FLUENT) was used to conduct numerical simulation for the flow field of Newtonian and NonNewtonian viscous fluids in a rounded bottom vessel equipped with an atypical anchor impeller. The reliability of the numerical procedure was demonstrated on the basis of a comparison of the numerical results with experimental data in several ways. First for different classes of fluids (Newtonian, shear-thinning and yield stress fluids) measured and numerically calculated power consumption were reported. For all the fluids investigated, standard deviation between the calculated and measured data was found inferior to 15%. Then, for a Newtonian fluid, measurements of tangential velocity fluctuation obtained by Laser-Doppler Velocimeter (LDV) in an area located near the shaft of the mixer were compared to the CFD values. Close agreement between the numerical computed CFD values and the LDV tangential velocity data were also obtained. Finally, the flow velocity field computed for nonNewtonian liquids in the vessel were analyzed and briefly compared to those obtained with Newtonian liquid at the same Reynolds number. 1. INTRODUCTION The agitation of highly viscous non-Newtonian fluids is a commonly encountered operation in many industrial processes. For such fluids, a detailed characterization of the three dimensional flow pattern inside the vessel [1] is of vital importance to gain a better understanding of the state of flow of the fluids and of the mechanisms responsible for homogenization and transport processes [2]. In the last few years, the application of computational fluids dynamics (CFD) in stirred vessels gets more and more numerous, reflecting the growing maturity of commercial software. Many of the mixing applications, however are restricted to the flow of Newtonian fluids in relatively simple geometry such as Rushton turbine [3] 'or Pitched Blade turbine, whilst flows in industrial mixing vessels have many complications arising both from the complex design of the impeller/tank arrangement (close clearance impellers and rounded bottom of the vessel) and from the complex rheological behavior of the agitated fluids. This work was partially supported by FEDER founds.

322 In this paper, a commercially available fluid dynamic (CFD) software package (FLUENT) was used to conduct numerical simulation for the flow field of viscous Newtonian and NonNewtonian liquids (shear-thinning and yield stress fluids) in a rounded bottom vessel equipped with an atypical anchor impeller. Although anchor agitators were commonly used in industry, very limited data are available for the anchor impeller studied. Indeed this non standard anchor impeller has one particularity, it is equipped with two horizontal blades at the bottom and the upper parts in order to avoid the formation of stagnant zone. The reliability of the numerical procedure was demonstrated on the basis of a comparison of the numerical results with experimental data in several ways. Global variable such as power consumption and local flow variables such as tangential velocities were used for this purpose. 2. EXPERIMENTAL APPARATUS AND METHOD

2.1. Mixing system investigated A sketch of the mixing system investigated (atypical anchor impeller and rounded bottom of the vessel ) is shown in Figure 1. The dimensions of the mixing system are also reported in Figure 1. During all the experiments, the volume of liquid was kept constant and corresponds to a liquid height, H of 0.361m. For each experiment, the agitator was rotated counterclockwise. The rotational speed of the impeller was measured by using a digital tachometer. A strain gauge torquemeter, in the range of 0 to 20 N.m, was used to obtain torque acting on the shaft.

Fig. 1. Picture Scheme, and geometrical parameters of the mixing system studied (vessel diameter=0.346 m; impeller diameter=-0.320 m; shaft diameter=0.025 m; horizontal and vertical blades width=0.030 m; impeller height=0.250 m).

2.2. Fluids 9 fluids (3 Newtonian, 4 shear-thinning and 2 yield stress fluids) covering a wide range of rheological behaviors were prepared to ascertain numerical results. The Newtonian fluids were aqueous solutions of glucose syrups of various water concentrations. The pseudoplastic fluids were aqueous solutions of Guar (SBI Viscogum HV 3000 A), a mixture of Saccharose and Carboxyl MethylCellulose CMC (CMC, Rh6ne Poulenc LTD), and an aqueous solution of Adragante gum (Powder 427, Alland & Robert). The two yield stress fluids were two aqueous neutralized solutions of Carbopol (Carbopol type 940, polyplastic S.A.). The flow curves of the tested fluids were obtained at the same temperature as that encountered in the mixing equipment. The viscous properties of the Newtonian and pseudoplastic fluids were measured in classical controlled rotational speed concentric cylinders (Contraves - Rheomat 30). Viscous properties of yield stress fluids were determined using a controlled shear stress viscosimeter (AR 1000 - T.A. Instruments) equipped with a cone and plate measuring system (Carri-Med CS 100).

323 To fit accurately all the flow curves of the tested fluid, "local" power laws model were used. This method consists to approximate each point of the flow curve by its tangency. The final relationship is: ~,,La --. k ( ~/ ) ~/m( ~, )-I

(1)

In equation (1), the parameters m ( ~ ) and k(~) are not constant as in Ostwald de Waele's model but varies with shear rate values (see Figure 2).

Fig. 2. Flow curves and rheological parameters of non-Newtonian fluids used Yield stresses of Carbopol solutions, z o , was indicated by fitting the Herschell-Bulkley model to the rheological data at low shear rates. Yield stresses determined in this way agrees well with stresses indicated by creep test on the controlled shear stress viscometer. As for shear-thinning fluids, "local" rheological law were used to fit flow curves (Figure 2).

2.3. Numerical simulation of the velocity distribution The numerical tools and the solution procedure used by the commercial CFD finite volume software (FLUENT) to determine the fluid velocity profiles inside the vessel are well known [4] and will not be repeated in details here. In the following part, only details about the mixing system modeling options will be given. A non uniform structured grid of 86592 cells (44x41x48 in the tangential, radial and axial directions respectively) has been built to represent a half of the vessel (the rest is deduced by symmetry). Note that the final geometry of the mixing system is quite similar to the experimental equipment. Even the rounded bottom of the vessel has been duplicated as shown in Figure 1. To obtain a good agreement of fluids flow along the clearance-wall, it was chosen to increase the density of the grid near the vessel wall. The boundary conditions at the impeller shaft and at the vessel wall were those derived assuming the no-slip condition. At the vessel axis, the values of the cells surrounding the point of consideration (on the axis) is assigned to the boundary cells bordering the axis. At the free surface, the boundary conditions were modeling by requiting that there are zero normal velocity and zero normal gradient of all variables. A rotating reference frame was used to perform the simulation. The apparent viscosity kta is constant for Newtonian liquids, while for non-Newtonian fluids txa is a function of shear rate ~. For each cell, ~, was deduced from the second invariant of rate of the deformation tensor A :Zi :

324

In its current stage of development FLUENT not allows us to use "local" laws to describe rheological behavior of fluids. To circumvent this difficulty, a subroutine has been developed in ours laboratories. This approach allows us to use local rheological models and fit accurately the flow curve of the tested fluids obtained by viscometers (see Figure 2). For numerical approach, assuming that the power supplied by the impeller is exclusively consumed as viscous dissipation of energy inside the vessel, the total power consumption, P, in the vessel was calculated by the summation of energy consumed in each volume control, throughout the vessel as shown in equation (4)

P= E~a]/2dV

(4)

I ,J ,K

where I, J, K indicate the number of each control volume divided into the r, 0 and z directions, respectively. Distributions of ~ and apparent viscosity g a in each discretized control volume were obtained from analysis mentioned above.

2.4. The LDV set-up A single component Laser Doppler Velocimeter with a 35mW He-Ne laser gun (DANTEC) was used to measure for a Newtonian fluid (Ix = 6.17 Pa.s) the velocity intensity profiles inside the mixing system. The optical components of the system comprised a beamsplitter, a, transmitting lens and receiving assembly. The electronic components of the LDV system included a photomultiplier, an oscilloscope, photodetectors and a microcomputer for data acquisition. Fluctuation of tangential velocities were measured in an horizontal plane located at Z / H = 0.692 (see Figure 5). Nine radial positions located between the shaft and the middle of the horizontal blade ( 0 . 1 9 < r * = 2 r / T < 0 . 4 0 ) were chosen as shown in Figure 4. The LDV apparatus was operated in the backscatter mode. When vertical blades of the anchor going by the laser beam, no Doppler shift could be detected. Acquisitions of this disconnecting time and rotationnal speed of the agitator allows us to obtain for each radial distance studied, the fluctuation of tangential velocity relative to the angular position of the blades (characterized by the angle a - see Figure 4) 3. RESULTS AND DISCUSSION

3.1. Reliability of numerical method Power consumption. Left part of Figure 3 shows, for Newtonian and non-Newtonian fluids tested, the comparison between experimental and numerical results of power consumption. Even for complex fluids possessing a yield stress, relative errors between experimental and numerical results have always been found inferior to 15 % For both numerical and experimental methods, the Newtonian power curve obtained in the laminar regime are similar to each other and can be described by traditional relationships between the Power number Po and the Reynolds number Re (fight part of Figure 3) eo Re = K p (5) Linear regressions on experimental and numerical power curve give (0.04 < Re < 3): Po Re = 183 for numerical approach and Po Re = 187 for experimental measurements.

325 Power consumption results obtained for our particular anchor impeller are qualitatively in agreement with those reported in the literature for others anchor agitators (Po Re vary between 100 and 370). A more accurate comparison is difficult to make because from our knowledge only [5] had investigated the flow of anchor impeller equipped with two horizontal blades as ours but these authors hadn't provided the value of the product Po Re in the laminar regime for their mixing system. Nevertheless calculating the power consumption for a similar anchor non equipped with horizontal blades by empirical correlation suggested by [6], we obtain in the laminar regime Po Re = 148 which is 26% smaller. So we can assume that excess of power consumption measured for our anchor mixer is due to the presence of horizontal blades.

Fig.3. On the left: Power consumption data obtained numerically and experimentally when mixing Newtonian, shear-thinning and yield stress fluids under laminar regime (0.04 < Re < 3 )at different rotational impeller speeds. On the right: Newtonian Power curve Tangential velocities: In order to ascertain the numerical simulation reliability to estimate under laminar regime local flow variables such as velocity components, values of tangential velocity obtained by LDV measurements have been compared to those calculated by CFD simulation. Figure 4 shows, for two radial distances r* =0.2 and 0.4, fluctuations of tangential velocities with the angular position of the blade obtained numerically. In this figure, experimental values of tangential velocities obtained experimentally when mixing Newtonian fluid under the same operating conditions have also been plotted. As shown in Figure 4, there is a good agreement between numerical and experimental results. Indeed variations of tangential velocity with blade angular position are quite similar and it can be seen that in the vicinity of the shaft, experimental and numerical tangential velocity reach their peak for the same angular blade position (ct = ~/2 rad). Even if small differences between experimental and numerical values of tangential velocity still exists, it is very difficult to obtain a more accurate agreement since the order of magnitude of remaining errors may be input to difficulties to localize the measurement point inside such industrial equipment. Note that measurements of tangential velocity have been carded out near the shaft and in this area numerical results are expected to be less accurate since both the tangential velocity intensity and the density of the grid are weak. Hence, the close agreement between experimental and simulated tangential velocities points out the validity of the numerical simulation method. All these results regarding the local tangential velocity, power consumption for both Newtonian and non-Newtonian fluids prove that numerical analysis method has sufficient reliability to predict flow behavior of highly viscous fluids in such mixing system, especially under laminar regime.

326

Fig. 4. On the left: Experimentally and numerically tangential velocities obtained for two radial positions when mixing Newtonian fluids (g=6.17 Pa.s and n=O.127s -1) with anchor impeller. All data are obtained in an horizontal plane located at z / H = 0.692 .On the right: Location of the velocity measurement points with LDA system angle a : angular position of the blade relative to angular position of measurement point. 3.2. Analysis of numerical results obtained for the flow of Newtonian and nonNewtonian fluids

Newtonian flow fields To analyze the flow pattern induced by anchor impeller studied, maximum values of fluctuating velocities obtained during one impeller rotation for various radial positions have been compared.

Fig. 5. CFD predictions of maximum velocities profiles obtained at different horizontal planes during one rotation of the impeller. Symbols used for various axial positions are given at the upper part of the scheme.

327 Figure 5 shows maximum values of velocity components (tangential, axial and radial) obtained for selected radial distances ( O < r * < l ) at different horizontal planes (0.21 < z/H < 0.97 ). Examination of Figure 5 reveals that tangential flow is dominant and becomes smaller with radial distances away from the impeller as reported by [1, 7]. Indeed, except at the upper parts of the vessel (at z/H = 0.91and 0.97 ), for all radial positions in the vessel, values of maximum tangential velocities is significantly higher than maximum values of axial and radial velocities (always inferior to 0.4). The results reported in figure 5 show clearly that close to the leading face of the upper and bottom horizontal blades (at z/H = 0.21 and z/H = 0.91 ), the rotation of the impeller generated a strong axial flow which was rapidly attenuated with both increasing axial (z/H = 0.34, 0.48, 0.59 and 0.8) and radial distances ( r* < 0.751 and r* > 0.925) away from the horizontal blades as mentionned by [1, 8]. Figure 5 also shows that maximum radial velocity was found at a radial position corresponding to the inner face of the vertical blade as found by [1, 7, 9, 10]. Note that our numerical value of maximum radial velocity at internal side of the blade is higher than those reported by [1, 7] but in agreement with numerical and experimental data of [9-10]. Moreover, this value was rapidly attenuated with increasing distance away from the radial location of the vertical blades. Non-Newtonian flow fields: To point out change in flow pattern which occurs in the vessel due to complex rheological properties of agitated fluids, velocities profiles obtained numerically at same Reynolds number for Newtonian and Non-Newtonian fluids have been compared.

Fig. 6. Velocities profiles obtained for an angular position normal to the vertical blades at z/H =0.59

328 Figure 6 shows an example of such velocities profiles obtained at z/H = 0.59 for an angular position normal to the vertical blades. This angular position have been chosen since it was observed from numerical results that change in flow are more significant with increasing distance away of the blades. Velocities profiles plotted in Figure 6 are quite interesting since they allows us to visualize main tendencies of change in flow observed when mixing nonNewtonian fluids and non-Newtonian fluids at same Reynolds number. Indeed, as shown in Figure 6, examination of numerical results reveals that: - for this agitator, few significant difference exist in flow when comparing flow patterns with weak shear-thinning fluids (CMC and 1.2% guar gum) to those obtained with Newtonian fluids. As it was already observed in pipe flow, it was also found throughout the vessel that the flow field of yield stress fluids (carbopol solutions) is quite similar to those obtained with high shear-thinning fluids (3% guar and 1.9% adragante gums). - tangential, axial and radial components of velocities away from the blades are slightly smaller than the corresponding values obtained for Newtonian fluids. 4. CONCLUSION Numerical predictions based on CFD simulations were obtained for the laminar flow fields of Newtonian and non-Newtonian fluids in a rounded bottom vessel stirred by a non-standard anchor impeller. The reliability of the method was ascertained by verifying the numerically calculated results with experimental data (power consumption and tangential velocities measurements). Flow patterns obtained numerically for Newtonian fluids shows us that the presence of the upper horizontal blade has a significant impact on the axial flow throughout the vessel. Finally the simulated flow velocity field computed at same Reynolds numbers for Newtonian-and non-Newtonian fluids allow us to observe change in flow pattern and is very helpful to relate flow behavior of complex fluids to their rheological properties. REFERENCES

1. M. Abid, C. Xuereb C., J. Bertrand, Hydrodynamics in Vessels Stirred with Anchors and Gate Agitators - Necessity of 3-D modelling, Trans IchemE, 70 Part A (1992) 377. 2. M. Ohta, M. Kuriyama, K. Arai, S. Saito, A two-dimensional model for the secondary flow in an agitated vessel with anchor impeller, J. of Chem. Eng. of Jap., 18 1 (I 985) 81. 3. C. Torrez, C. Andr6, Power consumption of a rushton turbine mixing viscous newtonian and shear-thinning fluids: comparison between experimental and numerical results, Chem. Eng. Technol., 21 7 (1998) 599. 4. FLUENT- User's Guide, (1996) 5. C.J. Hoogendoorn, A.P. Den Hartog, Model studies on mixers in the viscous flow region, Chem. Eng. Sci., 22 (1967) 1689. 6. K. Takahashi, K. Arai, S. Saito, Power correlation for anchor andhelical ribbon impellers in highly viscous liquids, J. of Chem. Eng. of Jap., 13 2 (1980) 147. 7. J. Bertrand, J.P. Couderc, Etude Numerique des 6coulements g6n&6s par une ancre dans le cas de fluides visqueux, Newtoniens ou pseudoplastiques, Entropie 125/126, (1985) 48. 8. Y. Murakami, K. Fujimoto, T. Shimada, A. Yamada, K. Asano, Evaluation of performance of mixing aparatus for high viscosity fluids, J. of Chem. Eng. of Jap., 5 3 (1972) 297. 9. M. Kuriyama, M. Inomata, K. Arai, S. Saito, Numerical solution for the flow of highly viscous fluid in agitated vessel with anchor impeller, AIChE J., 28 3 (1982) 385. 10. K. Arai, K. Takahashi, S. Saito, Correlation of velocity distributions in an anchor-agitated vessel using a bicubic B-spline function, J. of Chem. Eng. of Jap., 15 5 (1982) 383.

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

329

Characterization of Convective Mixing in Industrial Precipitation Reactors by Real-time Processing of Trajectography Data B. Barillon and P.H. Jdzdquel Kodak European Research Laboratories, Kodak Industrie, Z.I. Nord, 71102 Chalon sur Sa6ne Cedex, France The hydrodynamic behaviour of silver halide precipitation reactors is characterized, at the laboratory and the pilot scales, by a particle tracking based method. Two CCD-cameras and a powerful image processing system record, in real-time at a frequency of 30 Hz, the trajectory of a neutrally buoyant particle. From the extracted data several configurations of reactors at both scales, including the positioning of the baffling system and the location of the mixer, are discussed. 1. BACKGROUND AND OBJECTIVES The silver halide microcrystals made for photographic purposes are, most of the time, obtained via double-jet precipitations between silver salt and halide salt solutions. This type of precipitation is performed in semi-batch reactors and requires an appropriate stirring whatever the volume of liquid in the reactor is. The reagents are allowed to contact each other in the vicinity of a mixer in order either to form solid silver halide or to contribute to grow the crystals already existing in the reactor. Because of the reaction kinetics, competition with mixing is very likely to happen: the characteristic mixing times are generally several orders of magnitude larger than the nucleation times, while they are comparable with the mean crystal growth times. If the reaction is as fast as or faster than mixing, the distribution of the products in the reactor will hence be mixing dependent and the resulting medium will be segregated. This can be critical when precipitation formulae are scaled up as any failure due to scaling issues may strongly impact the final photographic performances of the microcrystals. The goal of this study was to better describe the convective mixing (or macromixing) and compare it at different scales. Consequently, the characterization of the hydrodynamic behaviours was achieved for several reactors at the laboratory and pilot scales. The method we developed to characterize macromixing is an improved version of a particle tracking technique pioneered by Scofield 1'2. This technique had shown its capability to describe some of the typical features of the flows in model stirred reactors using, for instance, Rushton or Pitched-Blade turbines 3. However, it has not been practically used for industrial purposes because of very long and tedious work required to process the experimental data. In addition, the mathematical treatment of the trajectory to get practical and simple characteristics of the hydrodynamics inside the reactor is not as trivial as expected. The principle, based on a single particle tracking, consists in taking images, at a relatively high frequency, of a neutrally buoyant particle moving in a transparent reactor, by means of video cameras set around the tank. It is then possible to calculate, on each image, the

330 instantaneous position of this single particle, and to obtain, after a statistically significant lapse of time - typically one hour-, the whole particle trajectory: Eventually, information on the macroscopic hydrodynamics inside the reactor is extracted from the generated data. 2. EXPERIMENTAL DEVICE The major improvement brought here to the technique is a real-time processing of the images recorded at a frequency of 30 Hz, not only eliminating the need for the storage of a considerable amount of images in the view of a further processing, but also allowing the immediate storage of the -uncorrected- coordinates of the particle versus time. In concrete terms, the transparent reactor is immersed in a transparent square tank in order to minimize the distortion generated by the difference in the refraction index between water and air (the interface is a plane and not a portion of a cylinder). The reactor is equipped with transparent mixing devices as well. A 3 mm-diameter black plastic particle plays the role of a fluid tracer. By pouting some sodium salt in the reactor it is possible to adjust the density of the fluid to the density of the particle which can hence be considered as neutrally buoyant. Two 640x480 pixels CCD cameras cal~ture two synchronized images every 1/30 ~' second. Figure 1. Top view of the experimental device One of the two cameras has its optical axis perpendicular to the (0,X,Z) side of the tank whereas the other is perpendicular to the (0,Y,Z) side. The optical axes of the two cameras need to be orthogonal. This setup of the cameras allows us to obtain, on the one hand, the Xand Z- coordinates and on the other hand, the Y- and Z- coordinates of the particle (of. Fig. 1). Once the two images are captured they are transferred to a computerized modular vision system. Each image processing encompasses 2 steps which are the thresholding of the image light intensity and the detection of the particle position by determination of the luminous center of gravity of the image. Real-time treatment requires that this processing is shorter than the time between two consecutive image captures i.e. 1/30 th see. Let us point out the need for a very careful lighting of the reactor in order to avoid a wrong detection of the particle position. 3. DATA PROCESSING After correction of the optical bias and interpolation of the possible events that are not "seen" by the cameras, data processing leads to several useful characterization criteria such as instantaneous velocity, recirculation time distributions relatively to selected reactor cross section and intersections of the particle trajectory with these cross sections in order to obtain first qualitative information on the flow patterns and the associated characteristics. The way these characterization criteria are determined will be discussed in more details in the next section.

331 4. RESULTS AND DISCUSSION This technique was applied for the characterization and the improvement of both laboratory and pilot scale precipitation reactors. The first example refers to the pilot scale. Fig.2 represents the intersections between the trajectory and an horizontal reactor cross section. If the particle crosses the section in the upward direction or in the downward direction a circle, respectively a point, is drawn. By examining the cross section structure at different heights it is possible to get a precise idea of the flow patterns. Here, the flow pattern observed is due, on the one hand, to the pumping effect generated by the mixer, inducing a downward movement of the fluid in the central region of the reactor and, on the other hand, to the lateral discharge inducing an upward directed motion near the reactor walls. The projection of this pattern on the section is the external ring seen in figure 2. Figure 3 is a reproduction of the velocity map Figure 2 Flow pattern in horizontal section obtained after data processing. As the particle is (Z= 0.2m). Dimensions in cm. Pilot scale. assumed to be neutrally buoyant, the calculation of the velocity at each location also gives the value of the fluid velocity at this same location. The "velocity" of the particle is the magnitude of the velocity vector. Depending on the magnitude of the vector, a colored point is associated to each location and allows the visualization of the velocity field. The map is obtained by superimposition, by transparency, on one plane, of the velocity magnitudes associated to the different locations of the particle in the slice of the kettle delimited by the two planes y - y l and y -Y2. The image is then processed by means of a filter (Paint shop Pro| in order to emphasize the colored zones. This map witnesses the fact that there is limited velocity segregation in the reactor, even if the Figure 3 Velocitymap at pilot scale. Dimensions in cm. fluid is slightly accelerated in the vicinity of the shaft. Another fundamental criterion is the recirculation time distribution. The recirculation time is defined as the time needed by the particle to come back to a given horizontal cross section. A distinction was made between the time spent above the section (red distributions) and the time spent under the section (black distributions). Figure 4 illustrates the distribution obtained at the pilot scale. "Tmoy" corresponds to the mean time, " T m a x " to the longest event. There is

332 no "long events" in the reactor, the particle staying less than 15 seconds below the section. It means that no stagnant zone is to be found in the vessel. The distribution shows two peaks. The first one, near the y-axis, is probably due to turbulent eddies which make the particle come back very rapidly to the section. The second peak, at about 3 seconds, and the narrowness of the distribution point out the fact that the particle crosses the section at regular time intervals. It means that the particle follows more or less regularly a recirculation loop. In order to know whether the fluid motion is Figure 4 Recirculation time distribution below the predominantly vertical, one possibility is to cross section Z=0.2 m, pilot scale determine the intersections of the trajectory with a vertical cross section containing the shaft. In that case it is possible to obtain qualitative information on the radial and tangential components of the fluid. Another alternative is to draw in a pseudo 3-D representation the velocity vectors associated to the intersection points between the trajectory and a selected crosssection (cf. Fig.5). In the example given here, the radial and tangential components of the velocity vectors seem to be weak compared to the vertical component. This allows us to say that the Figure 5 Velocity vectors associated to the recirculation loop suggested by the recirculation intersection points between trajectory and cross time distribution consists in a vertical loop section (Z=0.12m), pilot scale. Dimensions in favoring exchanges between the bottom and the cm. surface of the reactor. Many other criteria could have been considered for data processing but the next part of this section will demonstrate that the four criteria listed so far (velocity maps, cross-sections, recirculation time distributions and velocity vectors) are sufficient to provide a reasonable basis for a comparison between the pilot and the lab scales. For the lab scale several configurations were studied. The most relevant parameters we played with were the baffling configuration and the type of mixer in the reactor. Figure 6 shows the velocity maps for two different baffling configurations, one with baffles distributed all around the mixer (conf.1), the other one with one vertical baffle set along the walls of the reactor (conf.2). The way the baffles are configured in the reactor changes the velocity map, namely the spatial location of the different ranges of velocity. In the circular baffle configuration the most agitated zones are the region near the mixer (high velocities) and the lower half of the reactor. In the wall baffle configuration, even if the location of the highest velocities is the same, the average velocities form a W-shaped zone consisting in the regions near the shaft and close to the walls. The differences in the velocities proves that the proximity baffles/mixer in conf,1

333

Figure 6 Velocitymaps at lab scale. Conf.1 with baffle around mixer(left) and conf.2 with wall baffles (right). Dimensions in cm. reduces macromixing in the reactor. Horizontal (fig.7) and vertical (fig.8) cross-sections for both configurations emphasize the differences suggested by the velocity maps. In conf. 1 the

Figure 7 Horizontal cross sections (Z=0.18m). Conf.1 (left), conf.2(right). Dimensions in cm. Lab scale. cross section reveals asymmetries in the flow pattern: the external ring is disrupted, suggesting a change in the direction of the flow, and its shape shows that the presence of the baffle around the mixer is not neutral. The baffling conf.2 is more similar to the one obtained at the pilot scale. The vertical section puts in evidence a different rotational movement around the shaft, the conf. 1 generating, near the reactor walls, a counter-clockwise movement of the fluid. This behavior is supposed to be due to a higher interaction between the baffle and the mixer in this configuration. Because there is a relationship between the number of intersections and the flowrate across the section, we can qualitatively conclude that conf.2 is more efficient in reducing the rotational movement than conf. 1. In fig.9 are represented the recirculation time distributions above (light) and below (dark) an horizontal section dividing the reactor in 2 equal volumes for both conf.1 and conf.2. This figure clearly shows that the distributions do not have the same shape. The one corresponding to conf. 1 presents a peak characteristic of a recirculation loop.

334

Figure 8 Vertical cross section containing the shaft. Conf.1 (left), conf.2 (right). Dimensions in cm. Lab scale. The distribution generated by the conf.2 baffle is much broader and shows no peak which confirms that there is no vertical well defined recirculation loop allowing the fluid to circulate between the top and the bottom of the reactor. The long times indicate that all the zones in the reactor are not equivalent in terms of residence time. This is consistent with the segregated velocity zones shown on the velocity map: the top of the reactor has a behavior which is close to a stagnant zone.

Figure 9 Recirculationtime distributions related to cross section Z=0.18m, conf.1 (left), conf.2 (right). Lab scale. In conclusion, the baffle around the mixer seems to generate significantly different flow patterns inside the reactor compared to pilot scale. If we assume that mixing is crucial for the precipitation process we must make sure that crystals see the same chemical environment (same supersaturations, for instance) and as a consequence have the same chemical and hydrodynamic history from one scale to another. This is the reason why conf.2 (and not conf. 1) could reasonably be considered as a scale down of the pilot scale described in the first part of the section. However one must keep in mind that macromixing cannot be dissociated from micromixing when a chemical reaction occurs and that the conclusions being drawn in this study solve only one part of the problem of scale up.

335

Experiments were also performed with another type of mixer. It consists in a vertical tube (several centimeters high) containing a pitched blade turbine. This mixer is supposed to provide an upward directed motion inside the tube and a well developed recirculation loop in the reactor. Horizontal baffles were attached to the tube to avoid air entrapment when the volume is low in the reactor. Several positions of the turbine inside the tube were studied. The two extremes correspond to the impeller at the bottom of the tube and at the top of the tube (cf. fig 10). Fig.11 represents the same horizontal cross-section for these two extremes. The position of the turbine inside the draft tube has an effect on the macromixing characteristics. When the turbine is located in the lower part of the tube, the flow field is upward directed. When the turbine is close to the top, the upward discharge shows severe disturbances: a downward directed motion appears along the shaft and the vertical Figure 10 Position of the turbine in discharge tunas into a more lateral one. In the horizontal the tube, bottom (left), top (right) cross sections the shape of the baffles appears clearly. This indicates that the baffles and the edges of the draft tube are not neutral. Not only the flow patterns but also the recirculation time distributions exhibit some differences, as illustrated in fig. 12. The distribution widens when the turbine is closer to the top. The velocity vectors (in

Figure 11 Horizontal cross-sections(Z=0.15m) with the pitched blade turbine at the bottom of the tube (left), at the top of the tube (right). Dimensions in cm. Lab scale.

Figure 12 Recirculationtime distributions relatedto cross section Z=0.15m, with the pitched blade turbine at the bottom of the tube (left), at the top of the tube (right). Lab scale.

336 fig.13) confirm the lateral discharge of the flow probably due to the presence of the baffles. The shift of the discharge decreases the efficiency of the macromixing. It appears that the efficiency of macromixing is directly related to the set up of the turbine inside the tube. Modifications in the positioning of the turbine in the tube can generate significantly different flow patterns inside the reactor.

Figure 13 Velocityvectors at cross section (Z=0.13 m) withthe pitched blade turbine at the bottom of the tube (let~), at the top of the tube (right). Dimensions in cm. Lab scale. 5. CONCLUSIONS Mixing being an important feature in precipitation processes, it was relevant to develop a tool able to characterize different configurations of reactors at different scales. The technique chosen is an improved particle tracking technique. The major improvement is the determination in real-time of the position of the particle in the reactor. With appropriately defined criteria it is then possible to extract the main features of the flow inside the reactor. The examples shown highlight the fact that with velocity maps, intersections of the trajectory with reactor cross sections, recirculation time distributions and determination of velocity vectors, it is possible to rebuild the flow patterns in the reactors and compare different configurations of baffling and mixers. Moreover, as this technique can be adapted at different scales, this tool becomes the tool of choice for solving scale up issues. In addition, it gives a good starting point if we want to model the hydrodynamic behavior of the fluid. This study is also a good example of application at an industrial stage of a technique developed in an academic framework.

REFERENCES 1. D.F. Scofield and C.J. Martin, "Mixing Time Distributions and Period Doubling in Stirred Tanks", Annual Meeting AIChe (Chicago,USA), Nov. 11-16 1990 2. D.F. Scofield, C.J. Martin and M.A. Pivovarov, "Underlying Geometry of Flows", Annual Meeting AIChe (Chicago,USA), Nov. 11-16 1990 3. S. Wittrner, "Caract~risation du Mrlange dans une Cuve Agitre par Trajectographie", PhD, INPL, Nancy, France, April 18 1996


337

Characterisation of flow and mixing in an open system by a trajectography method P. Pitiot, L. Falk Laboratoire des Sciences du G6nie Chimique (LSGC-CNRS) 1, rue Grandville, BP. 451, 54001 Nancy CEDEX, France ABSTRACT This study deals with the description of the capacities of a Lagrangian approach -3Dtrajectography- to give some information about mixing in an open system. This original particle tracking leads us to a more detailed and local comprehension than with the classical Residence Time Distribution. Residence Time Distribution and Trajectory Length Distribution are compared, showing the high similarity of the two concepts for the studied tank. 1 INTRODUCTION Many techniques for flow and mixing characterisation in chemical reactors are available for chemical engineers. Among all these methods, one of the oldest but still in use method is the classical residence time distribution (RTD) method. Its main advantage is its ease of use and the ability to be directly employed to analyse reactors performance. The facility of such a technique is also to avoid sophisticated visualisation or in-situ measurements as it considers the system as a black-box, which response to specific excitation (as pulse tracer injection at the entry) is characteristic of the internal flow structure. Furthermore, the concept of RTD allows to compact the complex information in a single mathematical object, the distribution of residence time. If the RTD is classically used to detect the presence of dead flow or short-circuit zones in the system, it is however impossible to locate these particular zones. Additional methods, as Internal Age Distribution, are then necessary. On the other hand, with sophisticated investigation technique, as Laser Doppler Anemometry (1,2) or Particle Image Velocimetry (3,4), it is now possible to measure very precisely the turbulent flow field in complex system. Thanks to more sophisticated electronic systems, associated with convivial software, these techniques are increasingly easy to handle. But, because of the numerous data points, information is however not always easy to analyse, especially to link the mixing performance with the yield and the selectivity in chemical reactors. Between these two approaches, the concept of Lagrangian tracer is now being a developing alternative. For several years, we have developed such a method, named the trajectography method (5, 6, 7), to characterise local information in reactors. The purpose of the method is to track a small neutrally buoyant particle, for long observation time and to reconstruct the trajectory. From the analyse of this trajectory, it is possible to get information on the internal flow. The present study deals with the use of the 3D trajectory method to

338 investigate open flow system. In this article, we shortly describe the method and present the confrontation of the RTD obtained with the particle with a classical chemical tracer technique. 2

EXPERIMENTAL SETUP Following Scofield (8) idea, Wittmer (5) built a three-dimensional visualisation system. After several lightning improvements our tracer is a small particle (diameter 91.8 ram) and its density is the density of the fluid (difference less than 0.1%). This particle is made thanks to a micro-encapsulation process, using a solvent (water), a non-solvent (paraffin oil), a polymer (gelatine) and an adjusted amount of paraffin included to the polymer in order to optimize the density of the particle. Small molecules -like water- can go through the polymer, which is similar to a permeable membrane. Therefore, the density of our particle is auto-adjusted to the density of the fluid at ambient temperature (7). This particle tracking is applied to a 20 litres cylindrical stirred tank with a flat bottom (T =294 mm; H =281 mm). The plexiglas tank is equipped with four baffles and is placed in a glass cubic tank filled with water to reduce optical distortions. However, in this system, refraction phenomena are not negligible (7), as five different media can be distinguished (air, glass, water, plexiglas and fluid). A correction procedure taking into account the refraction and the real position of the cameras is applied in order to reconstruct the exact spatial position of the particle. The impeller used in this study is a classical Rushton turbine (D=T/3; C=T/3) at a rotation speed N = 120 rpm. For this open system (Fig. 1), the inlet is located near a baffle at the height of the impeller. The outlet is placed 270 ~ from the inlet (clockwise) at one centimetre from the free surface of the fluid. Pipes diameter equals to 8 mm and the circulation of the fluid is carried out by a peristaltic pump at a flow-rate of 1.8 litre per minute. Fig. 1. Sketch of the reactor. Two perpendicular and synchronised cameras (Fig. 2) track the displacements of the particle, in order to get the three coordinates of the particle as a function of time. At a frequency of 12.5 Hz, the two grey-level images are respectively transformed into a colour-component image (red-level and green-level). Thus, these two images do not loose their own identity and can be added together thanks to a PAL-RGB encoder to form a single colour image. This latter contains the two views of the particle at the same time. Thanks to this solution, a mechanical or an electronic synchronisation that would be expensive and unreliable, is avoided.

339

Fig. 2.3D-Trajectography; experimental setup. Colour images are digitised, compressed and stored on a special disk in real-time. With our acquisition-system, 30000 images can be stored, that allows a 40 minutes tracking. This time has to be linked to the mean residence-time (x) of a fluid aggregate (represented by the particle) in the tank. In our study, x equals to 11 minutes. If the behaviour of our tank is well represented by a CSTR-model, the Residence Time Distribution (RTD) should be represented by a decreasing exponential function, that means that 99% of the aggregates should go out of the tank after 5x and 98 % after 4x. As a result, with a 40 minutes (4x) acquisition-time, we can expect to get the major part of the Residence Time Distribution. After each experiment the colour images series are treated by image analysis to extract the position of the particle on each image. Then, for each record, we get image-coordinates of a few objects on the two planes and we have to reconstruct the 3D-trajectory in the tank coordinates-system. After bad objects elimination and the numerical intersection of optical axis (refraction), we have the trajectory of the particle, i.e. the three spatial coordinates as a function of time. HYDRODYNAMICS: TRAJECTORIES, RESIDENCE TIME DISTRIBUTION (RTD) AND TRAJECTORY LENGTH DISTRIBUTION (TLD)

3

3.1 Trajectories First, we represent side-views of characteristic trajectories (Fig. 3 to Fig. 5). On these figures (Fig. 3 to Fig. 5), the radial discharge flow and the subsequent recirculation loops induced by the Rushton turbine can be easily detected. Moreover, as the residence time (t~) increases, the trajectory looks more and more complex, but the structure of the flow remains identical.

340

The particle visits a larger volume of the tank when t~ is high. In Fig. 3, inlet and outlet can be distinguished; this is not possible on the two other figures, as the draw masks them. This also shows that inlet and outlet do not seem to disturb the global structure of the radial flow. At last, these three different trajectories illustrate the concept of RTD.

3.2 Residence Time Distribution (RTD) The results presented here concern 75 tracking-experiments carried out in the same experimental conditions as described bellow. In order to check that the particle is a good passive tracer of the fluid, we have confronted the RTD obtained by the tracking technique with the RTD obtained by a classical ionic tracer. This latter, determined by a classical Dirac-injection, leads to a mean residence time t s = 11.1 minutes. Just remind that x=V/Q for a CSTR, i.e. x~10.7 min. The molecular RTD is well represented (Fig. 6) by a CSTR-model' E ( t ) = exp ( - t / x ) . O n the same graph (Fig. -c

6), the RTD obtained with our particletracking experiments- is represented. No difference can be distinguished between this latter and the molecular RTD (statistical deviation less than 0.5 %). Fig. 6. Residence Time Distributions.

We can then conclude that our particle is a very good tracer of the fluid. 3.3 Trajectory Length Distribution (DLT) Villermaux (9) suggested that the residence time of a fluid tracer is not the only characteristic criterion of the flow; the length of its trajectory has also to be considered. This implies that the concept of Residence Time is not sufficient to determine the probability of meeting between two fluid elements (for a chemical reaction, for example): their trajectory must also be taken into account. For this purpose, Villermaux defined a novel concept: the Trajectory Length Distribution (TLD). For an open system, the idea is

341 the following: let us consider two different aggregates in an open system which RTD is known. These two fluid elements may have the same residence time, but may also move on very different trajectories, if they enter quiet or more turbulent zones. Thus, the lengths of their trajectory are different, even if their residence times in the tank are identical. Villermaux proposed a macromixing index (M) that is the ratio between the mean length of all trajectories encountered in the system and a characteristic dimension of this system (diameter of the tank (T), for example): M--lmean/Lcharacteristic. Two very different configurations can be distinguished: o:~ For a Plug-Flow Reactor: the length trajectory is the dimension of the reactor, so that M = I ~ For a CSTR: each aggregate moves on its own trajectory, which length is variable. So, the mean length may be high, leading to a high value of M. Experimentally, it is easy to get the RTD of one system as time is running. On the contrary, it is more difficult to get the TLD, since this concept implies that a lot of aggregates have to be tracked to give a good statistical representation of all lengths in the tank. The TLD concept is illustrated in Fig. 3 to Fig. 5 similarly as the RTD concept. Our experiments lead to a mean length of 70.6 metres. Fig. 7. Trajectory Length Distributions. If the diameter of the tank (T) is chosen as Lcharaeteristic, Villermaux macromixing index equals to 240, which is relatively high. In Fig. 7, the TLD is plotted and compared to the theoretical TLD (9) -exponential function as for the RTD with length instead of residence time: D(I) = (-1/~..__________~). exp

The relation between RTD and TLD in our stirred tank is obvious (Fig. 8). Fig. 8. Relationbetween time and length. This relation can be explained by the relative homogeneity of the velocity field, so that RTD and TLD are directly linked by the mean velocity in the tank; for our configuration, the mean velocity (slope of the line) equals to 0.108 rn/s (0.17 vtip).

342 4

T R A J E C T O G R A P H Y CONTRIBUTIONS

4.1 Reactor modelling The trajectography method may give a lot additional information on the structure of internal flow. Among the different results, one can mention the mean velocity field, the localisation of particular flow zone by Poincar6 sections, the determination of circulation time, etc... We illustrate here the capability of the method to validate internal flow structure model. The flow structure generated by the Rushton turbine is composed of two recirculation loops separated by the discharge stream plane of the impeller. The most simple flow model which can be proposed is based on two perfectly stirred tank (CSTR) of volume V1 and V2, with an exchange flow rate q = a Q . For ot-> oc, the RTD is equal to the RTD of a single CSTR. Let us define x l and x2, the instantaneous residence time of a fluid element in CSTR 1 and CSTR2 after a single passage, so that ti=Vi/(1 + a ) . Q . During its travel, the same fluid element can come back several times in the same zone before leaving the whole system. Therefore, one can define < x~ > and < x2 > as the mean residence time in reactor 1 and 2, considering the multiple passage of the fluid, so that: < x l > + <x2> = ~ = V / Q . One shows that <x~> and 5, Trans. IChemE.,Vol.71(A), ppl 1- (1993). 7. Tatterson G. B., Fluid mixing and gas dispersion in a agitated tank, McGraw-Hill, New York, (pp. 123) 8. Lee K. C., Yianneskis M., Turbulence properties of the impeller stream of a Rushton turbine, AICHE Journal; Vol.44, n~ pp. 12-24 (1998) 9. Cutter L. A., Flow and turbulence m a stirred tank, AICHE Journal; Vol.12, pp.35-45 (1966) 10. Bugay S., Analyse locale des dchelles caractdristiques du mdlange: application de la technique P.I. E aux cures agitdes, Thesis, Institut National des Sciences Appliqu~es de Toulouse (1998)


361

Analysis of Macro-Instabilities (MI) of the Flow Field in a Stirred Tank Agitated with Axial Impellers. Vesselina Roussinova and Suzanne M. Kresta Dept. of Chemical & Materials Engineering, University of Alberta Edmonton, Alberta, T6G 2G6 ABSTRACT In this study, the frequency distribution of velocity macro-instabilities (MI) in a stirred tank is reported. A fully baffled fiat-bottomed cylindrical stirred tank with diameter T=240mm and liquid height, H=T, was agitated with three different axial impellers (PBT, A310 and HE3). All of the velocity time series records were measured with a one-component Laser Doppler Velocimeter (LDV) on a rectangular grid of evenly spaced points upstream of the baffle at vertical distances z/T=0.3-0.75 from the bottom of the tank. The off-bottom clearance (C) and impeller diameter (D) were varied. The frequency content of the velocity time series was analyzed numerically using the Lomb periodogram algorithm, which is designed for unevenly spaced, Poisson-distributed data. The statistical space distributions of the frequencies show that in the case of a 45 ~ PBT with diameter D=T/2 and Re=48000 there is a resonant frequency that dominates, fui=0.62Hz. This frequency remains dominant when the off-bottom clearance (C) is changed, but two additional frequencies also appear and there is a distribution of minor frequencies around the dominant peaks. The off-bottom clearance (C) does not change the value of the dominant frequency, although the dominant frequency appears to resonate at one clearance and impeller diameter. Our frequency distribution analysis shows that for the A310 and HE3 there is no dominant frequency. The observed frequencies are scattered and broad banded. Once the resonant geometry was determined (45 ~ PBT, D=T/2, C/D=0.5), the study was extended to track the dependence of the dominant MI frequency on the impeller Reynolds number Re~, with variations in rotational speed N and fluid viscosity v. Our results clearly show that for Re> 104 the dimensionless frequency of the/V[I=fMl/q~ is constant. 1. INTRODUCTION Many publications document the large-scale, low-frequency motion in stirred tanks. Bruha and Fort [ 1] have shown that the frequency of occurrence of the macro instability (MI) is between 0.1 and ls q for the 6-bladed 45 ~PBT, T=0.3m and N=200-600rpm. They examine different geometries and track the dependence of MI on the impeller Reynolds number (Rex) from laminar regime up to the fully turbulent regime. The authors hypothesize that the MI

362 arises from transition of the flow field from single-loop to double loop circulation. While single loop circulation can be very stable, double loop circulation is inherently unstable due to the converging radial flow at the bottom of the tank [2, Fig.2c]. The strength of the converging radial flow will be at a maximum when the impeller discharge stream hits the tank wall exactly at the bottom corner of the tank. This condition coincides with the geometry of the resonant frequency discussed in this paper. The flow field contains a whole spectrum of frequencies due to unstable formations such as eddies and vortices. Close to the impeller blades, there are trailing vortices with dimensions of the order of D/10. The turbulence and the trailing vortices decay as the circulating flow spreads through the tank until in the upper third of the tank and close to the wall the dominant time varying component in the flow field is the MI. For solids distribution and surface feeds, this can be critical. The LDV used to probe MI measures the velocity of the seeding particle as it passes through the measuring volume. The process of particle arrival is random and therefore the time between the measured samples is non-uniform and follows a Poisson distribution. The non-uniform distribution of the time between the samples precludes the use of standard spectral estimators, which are based on equidistant times between samples. One of the most common standard spectral methods is the FFT (Fast Fourier Transform). The attractiveness of the FFT is the speed at which it can compute the spectral estimates. In order to calculate the spectrum with FFT the unevenly spaced time series could be converted to an evenly sampled sequence using one of various interpolation techniques. However; when uneven sampling is modeled as uniform sampling plus a stationary random deviation (the most reasonable scheme for this data), the resulting spectrum suffers from a low-pass filtering effect, which cuts off the upper frequency band. If the sampling dispersion is small compared with the mean sampling period the direct estimation of the spectrum is practically unbiased [3], but the sampling dispersion in our data is significant. The problems of accurate interpolation can be avoided by using the Lomb algorithm. The algorithm fits the time series using a least-square minimization procedure and it is suitable for direct analysis of unevenly sampled signals. 2. SPECTRAL ALGORITHM FOR UNEVENLY SAMPLED LDV DATA. The Lomb spectral method was originally proposed by Lomb [4] and has been used for astronomic time series analysis. It is based on general transformation theory [5] which shows that the projection of the signal x(t) onto the element of an orthonormal basis, b i (t) is the value, c that minimizes the mean square error energy E(c). E(c) is defined as an integral over the definition interval of the squared differences between x(t) and c. bi (t). The Lomb method implements this minimization procedure over the unevenly distributed sampled values of x(t) considering that the basis function is the Fourier kernel b i (t) = e j2,f,,. Suppose that x(t) is a continuous signal, where x(t) = x(ti), i= 1..N, and bi(t) is a orthogonal basis set which defines the transform, thus the coefficients c(i) that represent the x(t) in the transformation domain are:

363 4-0o

c(i) - ~x(t)bi (t)dt

(1)

-co

Calculated coefficients c(i) are those which minimize the squared error defined in [4] as 4-r

error - ~ (x(t) - c(i)bi (0) 2dt

(2)

-oo

In the case of evenly sampled data in the Fourier domain Eq.2 is well known as the discrete time Fourier transform (DTFT), its discretely evaluated version (DFT) and the associated fast algorithm used to compute it as (FFT). When the signal is available only at unevenly spaced time instants Lomb proposed to estimate the Fourier spectra by adjusting the model given as x(t n ) + sn - a cos(E~fit ~) + b sin(2~f it~)

(3)

in such a way that the mean square error s~ is minimized with the proper a and b parameters. It is easily proven that expression Eq.3 is a particular case for real signals from the more general formulation given as x ( t . ) + s~ = c(i)e j2"f't"

(4)

The x(t) and c(i) can be complex values. For any transform, not necessarily the Fourier transform, the expression will be x(t n) + s. = c(i)b~ (t n)

(5)

Minimization of s. variance (mean squared error) leads to minimization of N

~ l x ( t ) - c(i)b~(t~)l 2

(6)

n=l

The resulting value for c(i) should be 1

~

,

c(i) = ~-~X(tn)b~~.~(t~) lq

(7)

2

and k is defined by k = ~"[b~(t~)[ . rl=!

Now the Lomb normalized periodogram (spectral power as a function of angular frequency co = 2~f > 0) is defined by 1 Ps (~ - 2--~

{ [~j (x(tj) - x) cosc0(tj - x)~ ) ' j c o s 2 co(tj-x)

[~j(x(tj) - x) sin o(tj - "r } +

Ejsin 2 co(tj-x)

The mean, the variance of the signal X(tn) and the constant x are given by

(8)

364 1 r~ x = ~-i__~~x ( t , ) ,

-

1 N 0 2 = ~ - ~ _1 ~__~( x ( t i ) - X) 2

~ j sin 2o)t j tan(20)'l;) = ~-'jcos2c0tj

(9)

The constant x is a kind of offset that makes PN(co) completely independent of shifting all the t~' s by any constant. It makes Eq.8 identical to the equation obtained if one estimated the harmonic content of a data set, at given frequency co, by linear least-squares fitting. The Lomb algorithm requires N~N operations in order to calculate N~ frequencies from N data points. The program PERIOD implements the Lomb method (Press et al [6]). 3. E X P E R I M E N T A L SET UP In this study, a one-component LDV was used to probe the MI frequency. The LDV uses an Argon laser (k=514.5 nm) and a beam separation of0.0338m, which corresponds to a fringe spacing of7.6gm. Further details of the instrument configuration are given in Zhou and Kresta [7]. The signal processor operates in the frequency-domain burst detection mode and the analog signal was sampled at 2.5 MHz. The frequency of velocity measurements is determined by the particle arrival rate, which varied from 1500Hz to 700Hz in this work. Only particles which cross the measuring volume are detected, so the frequency of the velocity measurements is determined by the particle arrival rate or the seeding density. This

Figure 1. Sketch of the measuring grid leaves either the sample size (number of determinations) or the sample time (length of record in seconds) to be selected by the user. Experimental results show that both the sampling size and the sampling time are critical to velocity measurements. It was reported by Zhou [8] that the sample time must be long enough to cover at least 80 passages of the impeller blade. Calculations showed that for an impeller with4 blades rotating at 400 rpm, the sampling time should be no less than 3 seconds (i.e. 3 x 4 x 400 / 60 = 80 blade passages). If the sampling time is too short, the reproducibility of measurement is poor even with a sample size larger than 10,000 data points. For all of our measurements the number of determinations was set equal to 10,000, which means that the length of the record varies in different z/T planes from 10 to 20 seconds. This corresponds to

365 a sampling time of at least 100 rotations of the impeller and allows the capture of many MI events, which occur on the order of once every 5 rotations of the impeller. The stirred tank model used for this work had a tank diameter T= 0.240m equal to the liquid height T=H. To prevent air entrainment and surface vortexing, a lid was placed on the top of the tank and covered with 5 cm of water to seal the tank. There are four vertical Table 1. Experimental configurations Impeller type

Impeller diameter D, T/2 T/4

0.33,0.50,0.67 0.5, 1.0

A310

0.58T 0.35T

0.33, 0.5, 0.67 0.5, 1.0

HE3

T/3 T/4

0.4, 0.8, 1.0 0.5, 1.0

45 ~ P B T

C/D ratio

rectangular baffles of width W (W=T/10) spaced at an equal distance around the periphery of the tank at a small distance from the wall. The total number of grid points examined was 60, as shown in Figure 1" 6 points in the radial direction at 10 axial positions. The impeller diameters for the PBT were D=T/2 and T/4, for the HE3 D=T/3 and T/4, and for the A310 D=0.58T and 0.35T. The geometries examined are summarized in Table 1. 4. RESULTS The power spectra of the axial velocity time series were evaluated using the Lomb algorithm. The spectrum in Figure 2 shows the presence of both the BPF (Blade Passage Frequency) and the MI, but such spectra are typical only for locations close to the impeller. The BPF completely disappears from the spectrum by a radial position of 2r/D=l.33. For all grid points examined in this work the MI is the dominant frequency.

Figure 2. Frequency spectrum for the axial component at 2r/D=l.05 showing the presence of the BPF and MI

Once the frequency spectrum was calculated for all 60 locations, a histogram with the distribution of frequencies at the baffle was

366 constructed. Only the frequencies containing power higher than 75% of the maximum peak in an individual spectrum were included in the histograms. The selected bin size was kept the same for all impellers. 4.1 PBT

In the case of a PBT with D=T/2 and Rel=48000, the distribution of frequencies (Figures 3 a-c) shows a distinctive peak at fM~=0.62+ 0.02 Hz. This frequency remains dominant when the C/D ratios are varied. For C/D ratios 0.33 and 0.67, the histograms show the presence of additional frequencies f*=0.26 + 0.02Hz and t"*=0.76+ 0.01Hz and some scattered minor frequencies. The resonant geometry for the MI is D=T/2, C/D=0.5 as shown in Fig.3b. At this off bottom clearance, the dominant frequency is very coherent. Switching the impeller diameter to a small diameter (D=T/4) essentially eliminates the dominant frequency. In this case the impeller discharge stream either impinges of the bottom of the tank or decays significantly before reaching the tank wall. Once the dominant Nil frequency was determined, the experiments were extended by varying the fluid viscosity. All working fluids were Newtonian, with viscosities v, ran.~gingfrom lxl0 "6 m2/s (water) to 23xl 0 m2/s (water solutions of TEG ). The scaling of the dimensionless MI frequency, fMI/N, with Rel is shown in Figure 4. For Rel > 104 the fMI/N is constant at 0.18. Figure 3. Space distribution histogram of ~ frequency for 45~ with constant impeller diameter, D=T/2 and various C/D ratios: a) C/D=0.33, b) C/D=0.50 and e) C/D=0.67.

4.2. A310 and HE3

In Figures 5a-c the frequency space distribution histograms for the A310 are shown. The impeller diameter, D, in this

367 case is 0.58T. The impeller Reynolds number is Rex = 6.5xl 04, so the flow field is fully turbulent. The frequency of the MI is unstable, with fM~ranging from 0.3 to 0.44Hz. A single dominant frequency could not be identified. The maximum variability of the frequencies is observed in the case C/D=0.33. The flow pattern in this configuration is influenced by the proximity of the tank bottom to the impeller. This gives rise to a wide distribution of frequencies as there is a feed back from the impinging flow 0.30ta INMer v : l X l O 4 m21s at the bottom of the tank to the o h ~ o l v:3xlO 4 n~ls impeller discharge. The results • 'lEG v:SxlO 4 mZ/s 0.25for the HE3 are similar to those [] lEG v::23x104m21s for the A310. 0.20-

5. CONCLUSIONS Z

J

0.15-

0.10-

0.05.

0.Q0

.

' ;'

103

"

.

.

.

.

.

'lb

.

.

.

.

.

.

.

!

10~

LDV time series of axial velocities upstream of the baffle were successfully analyzed using the Lomb algorithm, an alternative to the FFT for the case of unevenly spaced data.

Re

Statistical analysis of the frequencies at 60 grid positions gives quantitative information about the dominant frequencies in the STR. This provides a way to search for a resonant or coherent frequency. For the case of a 45 ~ PBT with diameter D=T/2 and four baffles the dominant frequency of the MI was fMs=0.62I-Iz. This frequency remains dominant when the off-bottom clearance (C/D) is changed, but two additional frequencies appear (f*=0.26_+ 0.02Hz and f*=0.76 _+0.01Hz), and there is a distribution of minor frequencies around the dominant peaks. The frequency of the scales linearly with N for Rex greater than 104. A dominant frequency did not appear for the small PBT, D=T/4 at any C/D or for the A310 and HE3 impellers. Figure 4. Scaling of dimensionless f~a for the resonant geometry, 45~ PBT D=T/2, and C/D=0.5.

These results show that coherent MI's are extremely sensitive to both tank geometry and impeller design. ACNOWLEDGMENT The authors wish to acknowledge financial support of Lightnin and NSERC. REFERENCES 1. Bruha, O., I. Fort, P. Smolka, and M. Jahoda, 1996, Experimental study of turbulent macroinstabilities in an agitated system with axial high-speed impeller and with radial baffles, Coll. Czech. Chem. Comm., 61, 856-867.

368 2. Kresta, S.M. and P.E. Wood, 1993, The Mean Flow Field Produced by a 45 ~ Blade Turbine: Changes in the Circulation Pattern due to Offbottom Clearance, Can. J. Chem. Eng. 71, 42-53. 3. Lguna, P. and G. Moody, 1998, Power spectral density of unevenly sampled data by leastsquare analysis: Performance and application to heart rate signals, IEEE Transactions on Biomed. Eng., 45, 698-715. 4. Lomb N. R., 1976 Leastsquares frequency analysis of unequally spaced data, Astrophisieal J., 39, 447-462 5. Oppenheim, A.V. and A.S. Willsky, 1983, Signal and systems, Englewood Cliffs, NJ: Prentice-Hall 6. Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 1989, Numerical recipes, The art of scientific computing, Cambridge University Press, New York. 7. Zhou, G. and S.M. Kresta, 1996, Impact of the tank geometry on the maximum turbulence energy dissipation rate for impellers, A.I.Ch.E. Journal 42, 2476-2490.

Figure 5. Space distribution histogram of MI frequency for A310 with constant impeller diameter, D=0.58T and various C/D ratios: a) C/D=0.33, b) C/D=0.50 and e) C/D=0.67.

8. Zhou, G., 1996, Characteristics of turbulence energy dissipation and liquidliquid dispersions in an agitated tanks, Ph.D. thesis, University of Alberta, Edmonton, Alberta.

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

369

Local dynamic effect of mechanically agitated liquid on a radial baffle* J. Krat6na, I. Fo~t, O. Brf~a and J. Pavel Faculty of Mechanical Engineering, Czech Technical University, Technick~ 4, CZ-166 07 Prague 6, Czech Republic This paper presents experimental results of vertical (axial) distribution of a peripheral (tangential) component of dynamic pressure along the height of the radial baffle in a pilot plant cylindrical equipment with a six- or four-pitched blade impeller and four baffles. The measurements were obtained under a turbulent regime of flow using the trailing target with balancing springs at seven axial distances above the flat bottom. The experiments were carried out for both investigated impellers at five levels of impeller speed and at three levels of agitated liquid viscosity. The results of the measurement were interpreted in dimensionless form. The dimensionless mean dynamic pressure affecting the baffle exhibits maximum value at the bottom and very low values in the upper half of the baffle for all three investigated impeller off-bottom clearances h/T = 0.2, 0.35, 0.5. 1.

INTRODUCTION

A radial baffle in a mechanically agitated tank prevents rotation of a liquid resulting in the origin of central vortex and also increases the efficiency of the mixing [1]. An axially located pitched blade impeller in a cylindrical vessel with radial baffles exhibits the main force effects - axial and peripheral [2]. The distribution of the peripheral (tangential) component of dynamic pressure affecting a radial baffle at the wall of a cylindrical pilot plant reactor with axially located rotary impeller under turbulent regime of flow of agitated liquid was determined experimentally [3]. It follows from this study that the distribution of the dynamic pressure along the baffle depends significantly on the impeller type and its offbottom clearance. The aim of this study was to measure and analyse distribution of the dynamic peripheral pressure affecting the radial baffle of standard pilot plant mixing equipment with four baffles at the cylindrical vessel wall with an axially located pitched blade impeller under turbulent regime of flow of agitated liquid. 2.

EXPERIMENTAL

The experiments were carried out in a flat bottomed cylindrical pilot plant mixing vessel with four baffles (see Fig. 1) of diameter T = 0.3 m filled with water (~t= 1 mPa-s) or water-glycerol solution of dynamic viscosity ~t= 3 mPa.s and g= 6 mPa.s, respectively. The impeller was a standard pitched blade impeller (see Fig. 2) with six or four inclined plane blades. The direction of rotation was chosen to pump the liquid towards the vessel bottom. The range impeller frequency of revolution was chosen in the interval n = 3.33 s"l to 8.33 s1. For the originally developed measurement of the peripheral component of dynamic pressure affecting the baffle the system was modified as illustrated in Fig. 1. One of the This research was supported by grant No. OK 316/99 of the Czech Ministry of Education.

370 baffles was equipped with a trailing target of height hT and width B enabling it to be rotated along the axis parallel to the vessel axis with a small eccentricity and balanced by a couple of springs. Seven positions of the target Hx along the height of the baffle were tested. The angular displacement of target is directly proportional to the force F affecting the balancing springs (see Fig. 3). The flexibility of the springs was selected in such a way that the maximum target displacement was reasonably small compared with the vessel dimensions (no more than 5% of the vessel perimeter). A photo-electronic sensor scanned the angular displacement of the target and the output signal was treated, stored and analysed by the computer.

2.1.

Determination of trailing target loading

Let us consider the system with springs for determination of the torsional moment MT affecting during time t trailing target by oscillating agitated liquid. Then for the equation of dynamic model of the excited damped motion the following holds [4]: 1./3"(t)+ b. R23 9fl'(t)+ kp . R: . fl(t)= Mr(t ) ,

where I b k~ 13 R2, R3

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

(1)

moment of inertia [kg.m2], coefficient of damping [N.s.m'l], rigidity of the system [N.rad'l], angular target displacement [rad], radii of the axis of the trailing target from the spring clamping and the vessel wall, respectively [m] (see Fig. 5).

Fig. 1. Sketch of a flat bottomed agitated pilot plant mixing vessel with four radial baffles at the wall and an axially located pitched blade impeller and sketch of measurement of local peripheral force affecting the trailing target (H/T=I, h/T= 0.2, 0.35, 0.5, b/T = 0.1, hT= 10 mrn, B = 28 mm).

Fig. 2. Standard pitched blade impellers with a) four (z= 4) and b) six (z = 6) inclined plane blades (tx= 45 ~ D/T = 1/3, w/D = 0.2).

371 The rigidity of the system can be determined by mechanical calibration, i.e. from dependence between the force F affecting the balancing springs and angular displacement 13 (see Fig. 3). Characteristics of the oscillating model (inertia moment and damping coefficient) are determined from the measured period T of the oscillating movement (see Fig. 4). Ratio of the amplitude at the time t and the amplitude at the time t+T gives the so-called logarithmic decrement ln(fl,/fl,+r)= 6. T ,

(2)

where 8 is a parameter of reverberation. From quantities T and 8 the free angular frequency of the damped oscillations can be calculated: = 2,r/r ,

(3)

and, finally, the angular frequency of undamped oscillation a/"20 --" ~j.-('2 2 "~" a 2

(4)

9

Then moment of inertia and the coefficient of damping can be determined from the following relations: /

2

,

I = k a .R2/,(-2 o b = 2.6.

I/R~

(5) (6)

.

The first and second derivatives of the angular displacement [3 were calculated from the time course [~(t) numerically. Fig. 6 shows a time dependence of the torsional moment of the target with distorted results when the only linear term in Eq. (1) is taken into account, too.

2.2.

Radial prof'de of loading Let us assume the linear profile of the dynamic pressure distribution along the width of the baffle (see Fig. 7) Pk ( x ) = A x + C

(7)

,

with boundary conditions -0.206 -0.103 0.25

0

0.13

~ -

~'000

0.103 0.206 0.25

0.150

0.13 ~' 0.075 0.00 ~ 0.000 -0.13

-0.075

-0.25 -0.25 -500 -250 0 250 500 PC indicated displacement

-0.150

-0.13

Fig. 3. Results of mechanical calibration of balancing springs.

4

__..__L.__._._

5

6

t[s]

7

8

Fig. 4. Free angular frequency of system with trailing target.

372 X= 0" pk(0) = Pk,max, X= b" pk(b) = 0. Then the radial profile sought is

(8)

Pk (X) = Pk. .... O - x/b) , and the average value of the dynamic pressure over the width of radial baffle is 1b

1

1b

p:.o. = ~ 6j'p~ (x)dx = g :[P~'m'~(1-- x/b)dx = -2 Pk.max

(9)

"

Between the peripheral force affecting the trailing target and the average value of the dynamic pressure the following relation holds B

B

1

Fk = yPk (x)hrdx =hr ~Pk,m~x 0 -- x/B)dx = -~ Pk.max 0

" h r " B = Pk.av " h r " B

(10)

,

0

where hT is a target height of the and B is a target width. Centre of gravity of the force Fk lies in the centre of gravity of the linear profile of dynamic pressure (see Fig. 7) and its distance bc from the cylindrical vessel wall is (2/3)b. Value of the peripheral force Fk affecting the trailing target can be finally calculated from the torsional moment MT of target (see Eq. 1) MT MT Fk = R 3 - b----~= R 3 -(2/3)b 3.

"

(l 1)

RESULTS

All calculated components of the peripheral force, dynamic pressure and torsional moment can be expressed in dimensionless form of their mean (time averaged) values:

~=

F~ p.n2 .D 4 ,

(12)

P-~ = p . nP~ 2 .D 2 ,

(13)

M

9

=

M

Trailing

target

-~ 0.025

(7)

Pitched blade Turbine:

Again here the exponent over the modified Richardson number is lower for the disc turbine owing to high shear generation as compared to the pitched blade turbine. Combining the earlier work [4], where a extensive study was done with the respect to several impellers including the effect of D/T ratio, a generalized correlation has been proposed as'.

N•215215215 r

Np

V. 0 005)Bzx(AP) Ez t--:--.

(s)

The constants E1 and E2 are expressed in terms of the pumping effectiveness as follows; E1= 0.33 (NqfNp)~33 (9) and E2= 0.72 (Nq/Np) ~ (1 O) The typical values of Np and Nq for the disc turbine used in the experimentation are 4.9 and 1.03 respectively whereas for pitched blade turbine impeller, they are 1.6 and 0.944 respectively. The correlation given by equation 8 is independent of the type of the impeller used and also is valid under entire range of the operating parameters used in the present work excluding the viscosity effect, when Va/V is greater than 1.5%. 5. CONCLUSIONS: The mixing time is found to be dependent on the volume of the tracer pulse and also the density difference between the tracer pulse and the liquid bulk a~er a critical value of the Richardson number. Furthermore, the critical Richardson number is also dependent on the type of impeller used. In the lower range of tracer volumes, the exponent over the ratio A P/P is dependent on the operating speed. At lower speeds of rotation, the value of the exponent over the volume ratio is much higher as compared to the higher speeds of rotation (stirrer controlled regime) and hence the effect of volume ratio on the mixing time will be predominant at lower speeds. The mixing time is found to be independent of the viscosity difference over the range used in the expefimentatiorL It is also observed that when the buoyancy and viscous forces exerted by the tracer pocket are comparable with the momentum generated by the impeller, reduction in the mixing time takes place due to the fact that part oftracer pocket goes nearer to the impeller plane and is broken into smaller pockets. The situation is very similar to having multiple points of addition for the tracer pulse and it is recommended that when the quantities of the pulse to be mixed are substantially large, the pulse should be added at multiple points. A generalized correlation in terms of the pumping effectiveness of the impeller, volume of the tracer pocket and the density difference have been developed for the prediction of the mixing time.

393 6. NOMENCLATURE D Fa Fu g

H N Np Nq Ri T V. V

Diameter of Impeller (m) Buoyancy force exerted by tracer pocket (Kg m/s:) Upward lift force due to impeller momentum (kg m/s2) acceleration due to gravity (m/s2) Height of liquid in the vessel (m) agitator speed (rps) Impeller power number Flow number Richardson number Modified Richardson number Diameter of tank (m) Volume of the tracer pulse added (m3) Volume of the bulk (m3)

Greek:

Ap p ~. ~b A (N,)

Mixing time (s) Density difference between the added pulse and the bulk (kg/m3) Density of the liquid bulk (kg/m3) Viscosity of the added pulse (Pa-s) Viscosity of the liquid bulk (Pa-s) Difference between predicted and experimental dimensionless mixing time

REFERENCES

.

.

4.

5. 6.

C.D. Rielly andA.B. Pandit, Proc. of Sixth Eur. Conf. on Mixing (1988) 69-77, Pavia, Italy. I. Bouwmans, A. Bakker and H.E.A. Van den Akker, Trans. I. Chem. E., 75(A) (1997) 777-783. J.M. Smith and A.W. Schoenmakers, Chem. Eng. Res. Des. 66(1988) 16-21. P.R. Gogate and A.B. Pandit, Can. J. Cherm Engg., 77(5), (1999) 988-98 V.P. Mishra and J.B. Joshi, Chem.Eng. Res. Des. 73,5 (1994), 657-68. I. Bouwmam and H.E.A. Van den Akker, I. Cherm E. Symp. Series 121 (1990) 1-12.


I 0th European Conference on Mixing H.E.A. van den Akker and 3'..1. Derksen (editors) 9 2000 Elsevier Science B. V. All rights reserved

395

Mixing, Reaction and Precipitation: An Interplay in Continuous Crystallizers with Unpremixed Feeds N. S. Tavare Department of Chemical Engineering, University of Bradford West Yorkshire, BD7 1DP, UK A process involving elementary chemical reaction between two reactant species and subsequent precipitation of product in a continuous crystallizer with unpremixed feeds at the entry is modelled by the three- and four-environment models. The sensitivity of these models to Damk6hler number and dimensionless micromixing parameter is explored. The computationally efficient environment models appear to characterize satisfactorily the micromixing effects in a continuous reactive precipitator, while the IEM model description in a continuous MSMPR (mixed suspension mixed product removal) crystallizer is better suited to a nearly segregated system. Generally, the micromixing effects are much more important when the reactants enter separately than when they are initially premixed just before entry. Both reaction and crystallization performance are significantly influenced by the feed conditions.

Keywords: Micromixing, Reactive Precipitation, IEM micromixing model, Environment micromixing model, Continuous MSMPR Crystallizer 1.

INTRODUCTION

Although the general topic of mixing and mathematical analysis of continuous flow systems is of considerable interest, the bulk of the literature deals with configurations having single inlet and exit streams. When the reactants however enter separately the problem of unpremixed feeds evolves and one or both of the following situations arise; the feeds ports are apart and, as a result each feed stream experiences a different environment during part of its sojourn in the vessel; the rate of mixing of the different chemical species, when brought in close proximity is comparable to or slower than the other rate processes. Under these circumstances the micromixing effects have shown to be much more important than those for the case of premixed feeds [1-5]. Micromixing influences significantly the overall performance of a continuous crystallizer and its characterization for a reactive precipitation (or crystallization) system is important, the approaches developed in the field of chemical reaction engineering being extensively used. Tavare [6] extended the study of two types of micromixing models, (viz. environment and IEM (Interaction by Exchange with the Mean) models), developed for chemical reactors to a reactive precipitation system in a continuous crystallizer under the constraints of an MSMPR (mixed suspension mixed product removal) crystallizer with premixed feeds at the entry. These models are chosen because of their simplicity, versatility and computational economy. A variety of environment and IEM models have been proposed in the literature differing either in structure of environment interaction or in transfer

396 rates. Tavare [7] studied two-environment model for the case of premixed feed in the process of reactive precipitation configuration. The purpose of this article is to extend the same model for the case of unpremixed feeds using the same process configuration. Tavare [8] in his analysis of IEM (Interaction by Exchange with the Mean) micromixing model extended to a process of reactive precipitation showed that both reaction and crystallization performance characteristics are significantly different for the case of unpremixed feeds from those of the premixed feeds case. It would be desirable to compare the performance characteristics of these two models with unpremixed feeds. Although a variety of micromixing models for a chemical reactor have been proposed by many researchers in the field of chemical reaction engineering only a few previous studies appear to have addressed the problem of unpremixed feeds in the case of a reactive precipitation systems. A study of fast reactive precipitation of barium sulphate in a continuous stirred tank crystallizer for the case of unpremixed feeds was reported simulating the process with experimental verification by assuming that the vessel consists of two zones viz. complete segregation and molecular dissipation of concentration fluctuations [9]. The influence of intensity of mixing and mean residence on the rate of precipitation and mean product size were investigated. 2.

MICROMIXING MODELS

For the present analytical treatment, a continuous crystallizer with two feed streams each containing a single species is considered. These two species, A and B, react together homogeneously with first order reaction kinetics with respect to each of the reactants, the component A being assumed limiting. The archetype overall reaction considered is A + B--+C

rc = k cAc8

(1)

Precipitation of the solid product C resulting from this liquid phase reaction occurs simultaneously when the fluid phase becomes supersaturated with respect to component C. Conventional power law expressions of the form G = kg A c g

(2)

B = kb Aft'

(3)

and

are used to represent the growth and nucleation kinetics of the precipitation process respectively. The entire product in both the solid and liquid phases, together with unreacted material, leaves the crystallizer through a single exit. In general, the feed stream is described by its residence time distribution, flowrate and composition. The three-environment model as developed for a general chemical reactor having arbitrary separate feed streams [3] is extended to a reactive precipitator. This is essentially an outgrowth of the two-environment model developed from the original idea of Ng and Pippin [10] for a premixed feeds precipitator [7]. In this model two completely segregated entering

397 environments, one for each reactant feed stream in a two reactant species are assumed to transfer the material to a single maximum mixedness leaving environment at a rate proportional to their respective masses remaining in the entering environments. This model will accommodate separate flow streams having different flowrates and arbitrary stream residence time distributions (RTDs) in order to explore the sensitivity of the model to kinetic and micromixing parameters of a reactive precipitation configuration. A further refinement and extension of the three-environment, i.e., the four-environment model proposed by Mehta and Tarbell [5] is also extended to the foregoing reactive precipitation system. They developed this new model as they have shown inappropriateness of the three-environment model for complex reaction Idnetic systems because of intrinsic limitations associated with the structure of the leaving environment. Schematic representations of the three- and four-environment models of a crystallizer with two separate feed streams each of which contains a single reactant species are shown in Figures 1 and 2, respectively. Reactants A and B are admitted through their respective entering environments where they reside unreacted for their age V and subsequently transfer to the single leaving environment in the three-environment model and two separate but interacting leaving environments in the four- environment model, where they spend their residual lifetime ~ reacting. The entering environments are completely segregated while the leaving environments are in a state of maximum mixedness. The transfer of material from the entering to the leaving environment is first order in mass of the entering environment with transfer coefficient R and the reversible transfer between the leaving environment is also first order in the mass of each leaving environment with the same transfer coefficient as originally suggested by Mehta and Tarbell [5]. The detailed development of such three- and fourenvironment model to the case of unpremixed feed streams reactive precipitator will be presented elsewhere in future. 3.

RESULTS AND DISCUSSION

3.1 Effect of Damkiihler Number, ~ and Micromixing Parameters 1] and Using the physicochemical parameters (listed in Table 1 of [7]) calculations were performed to evaluate the crystallizer performance characteristics as predicted by the proposed models. Theresulting product size distributions from the crystallizer as illustrated by the conventional population density plots for a typical case of these models are shown in Figure 3 along with the population density plots under otherwise similar conditions for the extremes of micromixing viz. maximum mixedness (Model I), MM(I) and complete segregation (Model II), CS(II), from the previous studies [11 ]. Corresponding details of all these cases regarding dimensionless concentrations and product size distribution statistics are included in Table 1.

The dimensionless reaction group 7, i.e., the Damkrhler number, characterizes the dimensionless concentration of A at the vessel exit and hence determines the reaction performance. Keeping all other parameters in Table 1 constant, the sensitivity of both the reaction and crystallization performance characteristics to the Damkrhler number at typical values of micromixing parameters (1] = 10) and both the micromixing parameters at ), = 10 were explored for both these models by varying the parameters over the range 0.1-1000, covering a

398 104-fold range. The results of these calculations and from previous studies are reported in Figures 4 and 5.

Table 1" Performance characteristics of the IEM micromixing model (~/= ~1 = ~ = 10, 13 = 1.5; Figure 3)

Case

q/TI

x `4

Xc

L--,v,

CVw, %

NTXl 0.6, no./kg

tam MM(I)

~

0.136

0.567

930

50.0

0.24

CS(II)

0

0.096

0.540

353

22.5

1.20

IEM: Premixed

10.0

0.117

0.628

349

48.9

0.22

IEM: Unpremixed

10.0

0.166

0.601

348

48.8

0.19

2-EM: Premixed

10.0

0.111

0.545

958

57.9

0.40

3-EM

10.0

0.1072

0.2576

1233

49.8

0.21

4-EM

10.0

0.1089

0.2575

1219

48.3

0.21

The variations of the dimensionless concentrations of A and C, 2`4 and 2 c , in the solution phase m

and those of the weight mean size, L,v, coefficient of variation, CV and number concentration, N r, in the crystalline product phase, all at exit, with the Damk6hler number and micromixing parameters are shown in Figures 4 and 5, respectively. Also included in Figures 4 and 5 are these variations for the other three models (i.e., two-environment and IEM with premixed and unpremixed feeds) from previous studies. Low values of these parameters (3', rl ~) result in lower reaction rates and yield lower conversion. With an increase in 3', rlor ~,2A decreases and 2 c increases at lower values and then remains almost constant as a consequence of production of solid C. Both these models yield similar values, the significant difference between them being only in 24 at lower rl. They are however much different from those calculated in other model formulations. The micromixing parameters for the case of unpremixed feeds have significantly more influence on performance characteristics than in the case of premixed feeds. According to the environment model [7], when the micromixing parameter rl approaches zero, the transfer from the entering to the leaving environment reaches zero and the whole vessel tends to be in the completely segregated entering environment. For a large value of 1"1,most of the material from the entering environment is transferred to the leaving environment, occupying most of the vessel, and the whole reactor vessel approaches the state of maximum mixedness. Thus, with an increase in micromixing parameter rl, there should be a gradual movement of the performance characteristics from the completely segregated to the maximum mixedness case. Population density plots reported in Figure 3 and reaction and crystallization performance characteristics shown in Figures 4 and 5 appear to show this trend and approach these extreme limiting cases of micromixing over the range of the micromixing parameter rl. In the case of

399 the IEM model, both the reaction and crystallization performance characteristics for low ~ lie within those of the limiting cases of micromixing. The IEM model description used to characterize the micromixing effect for this case is akin to that of the completely segregated case and utilizes an additional step of first order mass exchange with the surrounding environment having the same residual lifetime. Consequently, with an increase in micromixing parameter ~ the performance characteristics should move away from those of completely segregated case and towards those of maximum mixedness case. The calculated performance characteristics for product C both in the solution and the solid phase appear not to reflect this trend. With an increase in ~, the relative contributions of terms in model equation change; they appear to influence the attributes of a clump and hence the average product characteristics at the outlet of a crystallizer. For high values of ~, the calculated average concentration of product C at the exit is higher than that for that for the case of maximum mixedness (Model I, MM(I)). Thus, the crystallizer operates at high average solution concentration and consequently low magma concentration of product C, yielding different product characteristics from those of the maximum mixedness (Model I, MM(I)). Since crystallization processes generally involve competitive and/or consecutive kinetic events, the states of extreme micromixing may not necessarily provide the bounds for crystallization performance characteristics (see, for example, [12]). Because of the small range of concentration between the two extreme micromixing levels, it appears that the changes in dimensionless concentrations are less sensitive to the micromixing parameters for the case of models with premixed feeds. The calculated dimensionless concentration profile for species A, YA, in both models lies within the small range of its values at the extreme micromixing levels while the calculated concentrations for C lie within the bounds for all 11 but, at high ~, exceed the bound of the extreme level. For the case of unpremixed feeds both reaction and micromixing parameters appear to exceed these bounds. Nevertheless the difference between the calculated values of species concentrations for these three- and four-environment models are small. The crystallization performance characteristics however are different and are influenced with the value of micromixing parameters. The analysis in the four-environment model probably yields a better representation than that in the three-environment model for the present simple reactive precipitation process. 3.2 Model Assessment

In the foregoing analysis, the results from the sensitivity analysis indicated that the environment models are computationally efficient and appear to characterize the micromixing effects over wide ranges of model parameters in a continuous reactive precipitator. The performance characteristics for unpremixed feeds are more sensitive to the variation of the micromixing parameter than those for premixed feeds. Several other parameters such as crystallization kinetics, residence time distribution, feed conditions, concentration and flowrate ratios and operating conditions may have significant influence. This analysis is concerned with a global characterization of the performance characteristics of a reactive precipitation system in a continuous crystallizer. Local characterization may perhaps provide valuable information regarding the micromixing process. Recently, attempts are being made to integrate computationally fluid dynamic and precipitation processes into commercial CFD codes in order to define both spatial distributions of instantaneous and local values of intensive properties of all species [13-18]. On many occasions, the detailed three dimensional, time dependent

400 numerical evaluations may not be feasible and other viable approaches may be advocated. A link between the systems approach of micromixing models and the fluid mechanics approach in the framework of the turbulence theory was attempted to simplify the analysis. (see for example, [5, 19-20]). Although a relationship between the mechanistic and direct turbulence models has been established the detailed numerical analysis might predict very different performance characteristics for complex reactions having different intrinsic timescales than the turbulent micromixing timescale [5, 21]. One of the difficulties associated with the application of the mathematical models in industrial practice was to estimate reliable micromixing parameters [22]. The basis of the general technique is the use of a known reactive crystallization system to evaluate the parameters of a model that represents by simulation the performance obtained experimentally from the crystallizer. A complete set of experimental data and a powerful parameter characterization technique are required to determine the model parameters and subsequently relate them to the crystallizer hydrodynamics. The analysis presented here lays no claim to providing a detailed physical insight into the interaction of mixing, reaction and subsequent crystallization, but rather provides a means of evaluating and subsequently predicting the performance for a given crystallizer configuration. NOTATION b B c c* Ac CS(II) CV E(u/) g G k

= nucleation order = nucleation rate, number/s.kg solvent = concentration, kmol/kg = saturation concentration, kmol/kg = concentration driving force, kmol/kg = complete segregation (Model II) = coefficient of variation on weight distribution, % = dimensionless residence time or age distribution = growth rate order = linear growth rate, m/s = reaction rate constant, kg/kmol.s 1% -- nucleation rate constant, number/[s.kg.(kmol/kg) b] kg = growth rate constant, m/[s.kg.(kmol/kg) g] L = crystal size, m MM(I) = maximum mixedness (Model I) = population density, number/m.kg n = crystal number concentration, number/kg NT = reaction rate, kmol/s.kg rc = time, age, s t = dimensionless concentration with respect to initial concentration of limiting reactant XA concentration of A, cA / ?A0 xC

= concentration of C, c c / C~o

Greek Symbols = dimensionless inlet concentration of B( = c~0 / c~0 )

401

Y n 0 L P~

= Damkrhler number (= kgA0r ) = dimensionless micromixing parameter for the IEM model = dimensionless micromixing parameter for the environment model (Rx) = dimensionless residence time (0 = t/x) = dimensionless residual lifetime (~ = ~/~ = 0 - q)) = crystal density, kg/m3 = parameter = mean residence time, s

Subscripts A,B,C CS(II) MM(I) w 1,2

= components = complete segregation (Model II) = maximum mixedness (Model I) = weight basis = inlet streams

Superscripts ^

= mean, outlet = dummy variable

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

C.P. Treleaven, A. H. Togby, Chem. Eng. Sci, 26 (1971) 1259. B.W. Ritchie and A. H. Togby, Chem. Eng. Commun., 2 (1978) 249. B.W. Pdtchie and A. H. Togby, Chem. Eng. J., 17 (1979) 173. B.W. Ritchie, Can. J. Chem. Eng., 58 (1980) 626. R.V. Mehta and J. M. Tarbell, AIChE J., 29 (1983) 320. N.S. Tavare, Chem. Eng. Sci., 49 (1994) 5193. N.S. Tavare, Computers Chem. Engng., 16 (1992) 923. N.S. Tavare, AIChE J., 41 (1995) 2537. J. Baldyga and R. Pohorecki, A reprint of the paper presented at the Conference on Mixing Colloque d' Agitation Mdcanique, ENSIGC, Toulouse (1986). 10. D. Y. C N g and D. W. T. Rippin, in Proc. Third European Symp. on Chemical Reaction Engineering, Amsterdam, September, 1964, pp 161-165, Pergamon Press, Oxford (1965). 11. J. Garside and N. S. Tavare, Chem. Eng. Sci., 40 (1985) 1485. 12. N. S. Tavare, Chem. Eng. Technology, 12 (1989) 1. 13. M. L. J. Van Leeuwen, O. S. L. Bruinsma, and G. M Van Rosmalen, Chem. Eng. Sci., 51 (1996) 2595. 14. M. L. J. Van Leeuwen, O. S. L. Bruinsma, and G. M Van Rosmalen, in B. Biscans and J. Garside (Eds.), Proc. 13th Symp. on Industrial Crystallization, Toulouse, France, (1996) pp 395-400. 15. M. L. J. Van Leeuwen, O. S. L. Bruinsma, and G. M Van Rosmalen, Proc. International Conference on Mixing and Crystallization, Tioman island, Malaysia, May (1998). 16. H. Wei and J. Garside, J., Acta Polytechnica Scandinavica, Chemical Technology Series No. 244 (1997) 9.

402 17. H. Wei and J. Garside, J., Trans IChemE, 75A (1997) 219. 18. J. Baldyga and W. Oricuch, Trans IChemE, 75A (1997) 160. 19. R. V. Mehta and J. M. Tarbell, AIChE J., 33 (1987) 1089. 20. J. Villermaux, ACS Symp. Ser. 226 (1983) 135. 21. L.-J. Chang, R. V. Mehta and J. M. Tarbell, Chem. Eng. Commun. 42 (1986) 139. 22. M. L. Call and R. H. Kadlec, Chem. Eng. Sci., 44 (1989) 1377.

403

Fig.I Schematic representation of the crystallizer and the three-environmental model

Fig.2 Schematic representation of the four-environmental model.

404

..... •.. 2'0 .......".........

....' ...................

Two-environment Three-environment Four-environment unpremixed, 1EM premixed, IEM

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

---____

""--.... ..ik

C

?? ---lo Mlvl(1)

. :-',--.,.

10

n

(no/m

kg)

",,., "-

- --~._L ,

v,'

..

.,

-~l

= 10, IEM

,.,, l

0

Figure

1000

3: P o p u l a t i o n

2000

L (~m)

de.~.si.t¥ p l o t s

3000

_- 1 0

405

Figure 4" Effect of the DamkOhler number

Figure 5: Effect of Micromixing parameters


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

407

Simulation of a tubular polymerisation reactor with mixing effects E. Foumier, L. Falk Laboratoire des Sciences du Grnie Chimique 1, rue Grandville 54000 NANCY FRANCE

ABSTRACT Manufacturing polymers with specific properties such as Molecular Weight Distribution (MWD) is a current issue for designing and driving industrial reactors. Actually, MWD is deeply dependant on mixing level at different scales in the reactor. The present paper aims at showing that a computational method, which is able to predict thermal and mixing effects on the MWD shape of a polymer, is now available. Practically, the studied case is anionic polymerisation of styrene which is carried out in a coaxial jet stirred tank without premixing system. The demonstration consists in comparing computational results of two completely different flow operating conditions driven on the same reactor. Actually, computational results allow to predict dispersion of MWD for both flows. Besides, results have brought to the fore undesired phenomena such as local hot point in recirculation loop, or the presence of very long chains next to the wall for less tormented flows. 1 INTRODUCTION Mixing effect on polymerisation reactions is known to have a strong influence on the Molecular Weight Distribution (MWD). For instance, continuous stirred tank reactors (CSTRs) give polymers of fairly broad MWD whereas ideal tubular reactors can give narrow MWD. In fact, flow characterisation in real tubular reactor are very complex and working conditions may be radically different from ideality. In order to investigate the effect of non ideal mixing conditions on polymer quality, modelling is an important tool for the comprehension of such complex phenomena and for the design and optimisation of polymerisation reactors. This paper presents some results obtained by high performance simulation technique on the effect of flow conditions. 2 COMPUTATION OF REACTIVE FLOWS First of all, it is necessary to deal with general considerations about mixing to understand the difficulties that arise while computing reactive flow. Generally, reactants enter the reactor without being pre-mixed. Streams tend to be perfectly mixed by several phenomena occurring at different scales. Theses phenomena condition scalar fields ~ according to the Reynoldsaveraged equation (1) : Convection 9 a,

Scalar fluxes ~ .

Molecular Diffusion

Meanreaction rate

408 Convection term describes macromixing achieved by fluid motion due to mean velocity. Scalar fluxes term represents fluid aggregates motion due to velocity fluctuations (turbulent diffusion) ensuring an efficient mixing at small scale. Mixing is achieved at ultimate scale by molecular diffusion. Finally, the mean reaction rate is a complex term dependant on the mean composition and composition fluctuations. When reactions are faster than any mixing process, chemical species are reacting while fluid is not yet perfectly mixed. In this case, the composition is not uniform at the scale where the competition takes place. Computational Fluid Dynamics (CFD) codes provide a description of all scales but sub-grid scales. When reactions compete with sub-grid mixing phenomena (micromixing), which is the case for polymerisation reaction, the concentration in each cell of the grid is actually heterogeneous. Therefore, the simple closure which consists in computing reaction rates with mean composition may lead to wrong results for fast reaction cases. A valuable solution (Pope 1985 [1]) allows to exactly compute reaction rates by considering the Probability Density Function (PDF) of scalars (composition and other transported quantities): the composition joint PDF.

2.1

Composition joint PDF transport equation

The composition joint PDF f~, defined by (2) estimates the probability that concentration vector, which components are concentrations and others transported quantities, equals ~ at time t and position x.

f~ (V;x,t)dv = P(o/ < ~(x,t) < ~ + d v )

(2)

Evolution of PDFJ~ by convection, turbulent diffusion, micromixing and reaction is governed by the transport equation of composition joint PDF (3). Convection

oz, +("')Ox, at

Reaction rate

+

.

Turbulent diffusion

M o l e c u l ~ diffusion

....

Ox i

Right hand side terms: the turbulent diffusion and molecular diffusion terms are not known and have to be closed. Turbulent diffusion term is typically closed by a gradient-diffusion model (4) which introduces the notion of turbulent diffusivity FT. (4) Ox~

Ox~

The molecular diffusion term may be closed by different micromixing models. In this study, we use the Interaction by Exchange with the Mean (IEM) model (5) proposed by Villermaux[2].

dt

t~ E

(5) micromixing time in turbulent flows

In order to solve the transport equation of the composition joint PDF, the turbulent flow field i.e. velocity, k and e are to be determined (where k is the turbulent energy and e is the

409 dissipation of k). This determination may be achieved through a CFD code (in our case, FLUENT CFD package), solving the Reynolds-averaged Navier Stokes equation (1). Then, a numerical solution of the PDF transport equation (2) is computed by a stochastic method (Monte Carlo algorithm). Such a method is particularly suitable when computing multidimensional problems since the CPU time increases linearly with the problem size while it increases exponentially when using finite differences or finite volumes techniques. It should be noticed that when polymerisation takes place, the viscosity of the reacting flow may increase significantly. Consequently, the Navier Stokes and the PDF transport equations are strongly interdependent. Since this study aims at establishing preliminary results, the viscosity of the reacting fluid is assumed constant. Additional work with more realistic assumption (variable viscosity) is in progress. 2.2 Computation by a stochastic method The stochastic solution algorithm (Pope 1985 [1], Roekaerts 1990 [3], Fox 1996 [4]) uses discrete PDF instead of continuous ones. The discrete PDF is a set of N notional particles with specific scalars which statistically represent the whole population of the continuous PDF. Calculation is performed in three steps. Firstly, notional particles are moved from a cell to a neighbour cell according to their convection and turbulent diffusion probability. Secondly, notional particles in each cell exchange scalar (matter and enthalpy) with each other particle according to the micromixing model. Finally, mass balance including reactions is integrated for each notional particle. PDF transport equation is integrated in time and space over the whole reactor. The final integration time corresponds to several mean residence times in the reactor.

3

APPLICATION

3.1 Anionic polymerisation of styrene initiated by sodium in THF Anionic polymerisation of styrene initiated by sodium in thetrahydrofuran (THF) is studied. This polyrnerisation can be considered as a non-terminating polymerisation ("living polymerisation") (Mtiller [5]), i.e. there is neither termination nor transfer processes. As a result, the MWD broadening cannot be due to chemical reaction but only to mixing effects. The polymerisation process can be divided into two reactions :

Initiation 9

I+M

k, >IM*

Propagation"

1M~

k,. >1M.,,§

- I denotes the initiator

where

- M denotes the monomer - IM~ denotes active chain of Degree of

Polymerisation : n The mechanism is modelled with the following assumption (Kim 1997 [6]) : 9 Initiation rate is infinite compared to propagation rate. Then, initiation and propagation can be assumed to be two consecutive reactions. 9 The propagation rate is independent of the Degree of Polymerisation (DP 9the number of molecules of monomer in a polymer chain). 9 The reactor is continuous and adiabatic. There is no heat transfer through the wall. However, convection, turbulent diffusion and micromixing allow transport of mass enthalpy. In order to save computational memory, the MWD is not calculated directly but by the use of lumped kinetic models and moments of the DP Distribution (DPD). This technique greatly spares the computational effort while the estimate of the concentrations and of the numberweight average Degree of Polymerisation is still possible. In many circumstances, these

410 parameters provide sufficient information about the effect of mixing on polymerisation. The ith moment of the DPD/at is defined by (6), noting that ~ is equal to the concentration of all active chains. Once the moments of the DPD are determined, the characteristic parameters of DPD given by equations (7) can be calculated. oo

lai = Z n i

(6)

[1M*,,]

n=l

Number Average Degree of Polymerisation Weight Average Degree of Polymerisation Polymolecularity Ratio

"x, = fl._.!.~ /a0 9xw = #2

(7)

: PR = Xw > 1 Xn

The reaction is calculated in each cell of the grid by solving the PDF transport equation with the scalar vector composed of the monomer concentration, the initiator concentration, the mass enthalpy, moments of order 0, 1 and 2 of the DPD. The consumption specific rates of monomer M, initiator I and mass enthalpy h are given by equations (8) while the generation terms for the moments are given by equations (9). r, = - k , [ M ] [ I ] rM = - k e [ M ] / a

0

A,H

(8)

rh = -

with A , H ~ 70 kd.mol -I P

(9)

,-., = k,[z]+ k~ t M l ~ 0 ru~ = k , [ I ] + k e [ M ] ( 2 / a , + /-to)

It should be underlined that the apparent kinetic constant kp, given by equation (10) (Miiller 1985 [5]) depends on the temperature and C*( =/ao), the total concentration of active chains. In fact, the value of kp is assumed to be independent of C*, calculating it with the limiting value C*~ reached for total consumption of the initiator.

=

105(210-71 ( 16.61031 C*

exp -

RT

m 3 mol-l.s -1

(10)

3.2 The coaxial mixing jet reactor The polymerisation is carried out in a coaxial mixing jet reactor as illustrated in figure 1. The central inflow (A) feeds the reactor within a mixture of initiator and solvent, whereas the annular inflow (B) is composed of a mixture of monomer and solvent. Mixing between the two streams is realised by the shear created by the high velocity difference. From dimensional analysis, it can be shown that the flow in such a coaxial mixing jet can be described by a dimensionless number, named the Craya-Curtet number (Becker et coll. 1963 [7]): C, = (uz/ul)~(D/d)2 -1

~/a-(u~/u,)

(11)

411

A critical value of the number C t c r i t = 0.976 (Barchillon and Curtet, 1964 [8]) has been experimentally determined. A recirculation loop appears when C t < Ct~ m , inducing a backmixing effect in the reactor.

I. : 20 cm

Inflow B Outflow

d:lcm

w.-

Inflow A

D:5cm

ul , r

Fig. 1 9Coaxial mixing jet reactor In order to illustrate the influence of mixing on the polymerisation reaction, two different flow configurations characterised by a different value of the Craya-Curtet number (cf. table 1) are compared: Configuration 1"

C t = 0.296

Configuration 2:

C t = 1.414

The reactor is designed for a specific production of polymer with a given mean molecular weight. For anionic polymerisation, the molar flow of polymer Fp can be calculated from the consumed molar flow rate of the initiator FI by:

F, = F~ The DP can be estimated by assuming perfect mixing and that all the chains have the same length. For total conversion, the degree of polymerisation is then equal to the ratio of the molar flow rate of monomer over the molar flow rate of initiator: D P ~ FM

F1 As our objective is to reach the same degree of polymerisation for both flow configurations, the inlet concentrations of monomer and initiator are adjusted to the inlet flow rates. Table 1 9Flow configuration parameters Inflow A (interior)

Inflow B (annular)

Velocity inlet : ul monomer concentration CM,1 Initiator concentration CI,I Inlet temperature T~ Velocity inlet : u2 monomer concentration CM,2 Initiator concentration CI,2 Inlet temperature T2

Configuration 1 4 m s~ 0 mol m -3 16 mol m "3 - 10~ 0.25 m.s ~ 4000 mol m "3 0 mol m "3 - 10~

Configuration 2 2ms" 0 mol m 3 32 mol m 3 -10~ 0.5 m s "~ 2000 mol m "3 0 mol m "3 -lO~

In the following example, the required production of polymer is fixed to 9Fp= 5 10 .3 mol.s 1, and the mean molecular weight is chosen at M = 39500 g mol 1. The degree of

412 polymerisation is nearly equal to d ~ = 380. As a result, FI = 5 10-3 mol.s 1 for the initiator and FM = 380 * 5 103 = 1.9 mol s1 for the monomer have to be introduced into the reactor. The characteristics of both flow are given in table 1.

3.3

Results

3.3.1 Mixing performance The obtained stream function field for both flow configurations are illustrated in figure 2. In agreement with the value of the Craya-Curtet number, a recirculation loop appears in flow configuration 1 which does not exist in flow configuration 2. The influence of the loop (backmixing) on mixing performance can be shown by calculating the radial mixture fractionf of a passive tracer: D/2 C r2dr

~

f=

0 D/2 r2 dr

(13)

~

0

By choosing for both configurations, the following inlet concentration (C1=1 and C2=0), the calculated values of the mixture fraction are : 9 at z=0 (inlet): fl,0 = f2,0 = 0.04 9 at infinite lengthfoo (complete mixing): fl,oo = 0 . 4 , f2,~o =0.143 In order to compare both configurations, a dimensionless mixture fraction is used:

(f -fo) Y-- (f~ - f 0 )

(14)

The longitudinal evolution of the dimensionless mixture fraction for the two configurations is drawn in figure 3. The mixing performances are very similar in both cases from the inlet to about 3 cm. For greater axial positions, the macromixing performance is significantly higher for configuration 1. More precisely, complete mixing is reached when axial position exceeds about 10 cm while for configuration 2 the maximal value of Y is about 0.9 at the outlet of the reactor. In conclusion, the back mixing occurring in configuration 1 strongly enhances the macromixing performance.

Fig. 2 StreamVelocityFunctionin the coaxial mixingjet reactor. Upper part: flow configuration 1. Lowerpart: flow configuration2. Fig. 3 Dimensionlessmixture fraction Y as function of axial position.

413

3.3.2

Computational results

Computational results, that are illustrated on figures 4, 5 make three distinct zones of the reactor stand out. Firstly, zone (1) is the area where initiator is encountered (figure 4) and where initiation takes place. Secondly, zone (2) is the area where monomer is encountered (figure 5) and where propagation takes place. Finally, zone(3), complementary to others, is the monomer exhausting zone where active chains are alone with solvent.

Fig. 4 Intiator concentration (mol m'3). Upper part: flow configuration 1. Lower part: configuration 2.

Fig. 5 Monomer concentration (mol m3). Upper part: flow configuration 1. Lower part: flow configuration 2.

First of all, initiation takes place in the first 5 cm of the reactor zone (1) where mixing state is very similar for both flow configurations (C.f. section 3.3.1, figure 3). The broadening MWD caused by competition between mixing and initiation must be very similar as well. Differences of the shape of M W D are actually due to competition between mixing and propagation. Let us compare phenomena occurring in zone (2) where propagation takes place. On one hand, for flow configuration 2, zone (2) can be divided into two sub-zones : 9 an intermediate zone situated between monomer exhausting zone and the area where monomer is abondam. The sub-zone is rich in monomer and in active chains as well, so propagation is quantitative. Propagation is stood out on figures 6 by a rise of temperature. Mean DP (figure 7) reaches low values close to 50 while PR (figure 8) reaches high values close to 12.

Fig. 6 Temperature (~ Upper part: flow configuration 1. Lower part: flow configuration 2.

Fig. 7 Mean Degree of Polymerisation. Upper part : flow configuration 1. Lower part: flow configuration 2.

Fig. 8 : Polymolecularity Ratio. Upper part: to flow configuration 1. Lower part: flow configuration 2. 9 by the wall, active chains concentration is very low because radial velocities are negligible. So active chains that are created in the central part of the reactor can only be

414 transported by turbulent difusion. Then, addition of a lot of monomer to scarce active chains provides very long chains (DP = 400-600) and PR reaches low values close to 2. Before considering the case of flow configuration 2, a noticing about PR has to be formulated: PR only qualifies a given dispersion of MWD when mean DP is also given. Indeed, for a difference of DP, dispersion of MWD is lower for long chains than for small ones. As a result, a global PR is estimated at the outflow and reaches a value of 5. As a conclusion, flow configuration 1 produce a polymer with a very large MWD because besides a the local dispersion, mean DP depends on its radial position. On the other hand, flow configuration 1 is less segregated than flow configuration 2. Active chains are forced back by the recirucation loop from central part where they are created to the periphery. In the recirculation loop, propagation is enhanced because the area is rich in monomer (figure 5) and in active chains. Besides, propagation is thermically auto-accelerated. Then, very long chains are created in the loop and temperature reaches 75~ (figure 6). This local temperature is greater than boiling temperature of THF and then physically not possible. In reality, termination reactions would have probably slowed down propagation and temperature would have been lower than predicted. Any way, even if temperature is not totally pertinent, computation has brought to the fore the existence of a hot point located in the loop. Besides, at the outflow, polymer goes out with a given PR equals to 3 (figure 8) with a mean DP equals to 300 (figure 7). Then, flow configuration 1, polymer has a rather large MWD but quality of polymer is the same at any radial position. 4 CONCLUSION Computing behaviour of a polymerisation reactor by solving composition joint PDF transport equation provides a good prediction of the MWD dispersion. Practically, computation allows to avoid some experimental test usually necessary to find reactor operating conditions which optimise the shape of MWD. Besides, computation manages to predict and localise undesired phenomena such as the presence of hot points in recirculation loop or long active chains by the wall. Although, the computational method proposed here provides interesting qualitative predictions, this method can be enhanced in the future by considering the evolution of viscosity with degree of polymerisation and temperature. REFERENCES

1. S.B. Pope, PDF method for turbulent reactive flows. Pro. Energy Combust. Sci., Vol.ll (1985) 119-192 2. J. Villermaux, Micromixing phenomena in stirred reactors. In Cheremisinoff, N.P. (ed.), Encyclopedia of Fluid Mechanics, vol. 2. Gulf Publishing Company, Houston; 707-771. 3. D. Roekaerts, Monte Carlo PDF method for turbulent reacting flow in a jet-stirred reactor. Twelfth Symposium on Turbulence, Rolla, MO U.S.A., (1990) 24-26 4. R.O. Fox, Computational methods for turbulent reacting flows in the chemical process industry. Revue de rInstitut Fran~ais du Prtrole, Vol.51, No. 2 (1996) 215-243 5 A.H.E. Mfiller, Carbanionic Polymerization: Kinetics and Thermodynamics. Encyclopedy of Polymers Engineering Science. Second Edition. N.Y. : John Wiley and sons. Chap. 26. (1985) 387-423. 6 D.M. Kim, E.B. Nauman, Nonterminating polymerisation in continuous flow systems. Ind. Eng. Chem. Res., 36 (1997) 1088-1094 7. H.A. Becker, H.C. Hottel, G.C. Williams, Mixing and flow in ducted turbulent jets. 9 th Symposium (International) on Combustion. Academic Press, London (1963) 7-20. 8. M. Barchilon, R. Curtet, Some details of structure of axisymetric confined jet with backflow. J. of Basic Engineering, Transaction ASME, 86 (1964) serie D,777-787.

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

415

Mixing equipment for particle suspension - generalized approach to designing F. Rieger and P. Ditl Czech Technical University in Prague Faculty of Mechanical Engineering, Department of Process Engineering Technick/t 4, 166 07 Praha 6, Czech Republic The proposed procedure for designing mixing devices for particle suspension is based on an experimentally proven experience that the plots of the energetic dimensionless criterion & expressing an impeller efficiency on the ratio dp/T are almost identical for many axial impellers within the relative vessel to impeller diameter T/D range from 2.5 to 4. This was confirmed experimentally for pitched three, four and six-blade turbines operating in flat and dished-bottom cylindrical baffled vessels at suspension mixing with solid particle concentration of 2.5 and 10% by volume, respectively. The effect of the bottom shape and the solid-phase concentration is discussed on the base of our extensive experimental program.

1. THEORETICAL BACKGROUND Numerous published studies dealing with particle suspension rather dim the situation than giving a reliable procedure for estimation of an impeller speed at which particles are just suspended. This is mainly caused by non-proper statistical treatment utilized by most authors treating their experimental data obtained for different hydrodynamic regions together. This leads to incorrect conclusions e.g. with respect to scale-up. A review of theories for particle suspension and related experimental techniques was published by Rieger and Ditl (1994). Several papers have been published since 1994 from which we would like to mentioned two papers, namely Corpstein, Fasano and Myers (1994) and Armenante and Nagamine (1998). Despite numerous published theories and designing equations still an industrial design often fails. We trust in a systematic experimental approach and theoretical backgrounds based on two different suspension mechanisms and that is why we are keeping since 1982 our own way.

Rieger and Ditl (1982) pointed out the existence of suspension mechanisms, which are different for fine and large particles. Based on this kmowledge the final correlation valid for the turbulent impeller region has been proposed by Rieger and Ditl (1994) in the form Fr'

Nf2Dp = f (1) Apg where the quantitative relationships for pitched six-blade turbine operating in a flat-bottom vessel are also given. A typical plot of Fr'vs. dyT is very similar to all plots u, vs. dp/T shown in this paper. The measured data have been originally correlated in a power form, separately for fine and large particles F~' = c ,

(2)

with different exponents a~ and ~ mad constants CI and C2 for fine and large particles, respectively.

416 Ditl and Nauman (1992) used a similar concept successfully for thin sheet particles. Dependencies F r ' vs. d / T have been experimentally determined for various impeller-vessel geometry and different concentrations of spherical particles. Originally proposed formulas reported by Rieger and Ditl (1994) described a particle suspension of large and fine particles separately. Rieger (1999a) substituted two separate relationships by a single one valid for the whole range of d;/T. C 4

-

F 1 -t = -

(3)

The constants Cu and C3 and exponents a and c can be calculated from lolown values of C/, C2, a/ and a2 as C4 = Ci, C 3 - Ci/C2, a = ai, c = a i - a2. The constant b reflects a transition regime of dfT and it can be adjusted at a value of b = 10. To evaluate energetic efficiency of an impeller for particle suspension the dimensionless criterion 7rs was proposed by Rieger (1993) in the form

re, = Po ~.l~;"3( % ) 7

(4)

where Po is a power number. Geometrical configuration having the lowest value of 7cs indicates energetically the most efficient system which means that particle suspension in the given system is achieved at the lowest impeller power input. As far as Po is constant in the turbulent region and Fr' =.f(d/T), then

r(d,,/r) as shown by Rieger (1999b) for the given solid-phase concentration and geometrical configuration of mixing equipment. Substituting into equation (4)for Fr "and Po from their definitions one obtains

]'ffs.--~

9

P

1

/9.,.

T7

(6)

The value of ~v, gives in the dimensionless form the impeller power input required for particle suspension of given physical properties in the tank of the diameter T having a particular vessel-impeller geometry. Substituting from equation (3) into equation (4) one obtains a quantitative representation of the dependency ~.,.= f(~/T) in the forrn 7

.Sa I dyll5

al

K 'T~s .- .-. . . . . . . . . . . .

+

where

1.5

dp

]b

(7) 1.5

417 K : po

(8) The new idea proposed in this paper is to base the designing procedure on the parameter 7c,. The definition of ~, overall sen:i-empirical correlation of Fr' = f(dp/T) for different vessel-impeller geometry have been recently published, unfortunately in hardly available proceedings or papers. 9.,"

c(-'

2. EXPERIMENTAL

The critical speed for particle suspension Nf was measured by a method described by Rieger and Ditl (1994). The value of Nf was evaluated from the measured course of sediment height at the vessel wall on the impeller speed. Extrapolating this plot to the zero sediment height, one obtains the value of N/which is very close to the value obtained visually according to Zwietering's definition. All data reported in this paper have been obtained in cylindrical tanks T = 0.2, 0.3, 0.4 and 1 m with four baffles of width 0.1 T. Both flat and dished-bottomed vessels were tested. A pitched three, four and six-blade turbine with the blades angle of 45 ~ having the different ratio of the vessel diameter to that of the agitator T/D = 2, 2.5, 3, 3.3 and 4 were tested. The impeller clearance was ~ = 0.5 D. The filling height in the vessel H was always equal to the vessel diameter T. As model liquids the aqueous solutions of glycerol and water were used and a solid-phase was modeled by glass ballotine of different mean diameters. Solid-phase volumetric concentrations c"= 2.5% and 10% were used in experiments. The results of these measurements have been previously published by Rieger and Ditl (1997) and Rieger (1997) and Rieger and Ditl (1994). 3. RESULTS The dependency ns vs. de/T has been plotted in groups each for different axial impellers and various relative vessel to impeller diameters T/D and a given solid-phase concentration. It was found that all plots within one group don't differ significantly. This allowed us to generalize the results. For each group the dependency ns vs. dp/T can be represented by one curve only within a confidence interval that covers all data measured. The approach described above was adopted for pitched six-blade turbine with the ratio of T/D ranging from 2 to 4 operated in a flat-bottomed vessel. These results have been published recently by Rieger and Ditl (1997) in the form of equation (2). Another group of results obtained for 45 pitched three, four and six-blade turbines by Rieger (1997) and dished bottom was also evaluated by the above described method. The dependency of rts vs. de/T for pitched six-blade turbine operated in flat-bottom tanks with different vessel to impeller diameters T/D at the volumetric solid-phase concentration 2.5 % by volume is shown in Fig.1 whereas the same for 10 % by volume is depicted in Fig.2. From Fig.1 it is clear that the relatively highest impeller (T/D - 2) has a lower suspension efficiency than the other T/D ratios. The week point is a bottom area below the shaft. For this reason this ratio T/D=2 cannot not be recommended for mixing in low concentrated suspensions. From Fig. 2 it follows that high speed mixers of relatively large diameter (T/D = 2) are quite efficient for mixing of concentrated suspensions of fine particles, however, they fail for relatively coarse particles dp/T > 0.01 frona the same reason described above.

418

0.1

71;s

9 T/D=2 0.01

T/D=2.5

-

T/D=3 ......

T/D=3.3 T/D=4

0.001

I

0.0001

,,

0.001

adr

J

0.01

....

0.1

Fig.l" Pitched six-blade turbine, flat-bottomed tank, c "= 2.5 vol.%.

T/D=2 T/D=2.5

0.1

T/D=3

71; s

...... 0.01

0.001 0.0001

[

.

,

0.001

T/D=3.3 T/D=4

I

0.01

0.1

d.rr Fig.2" 9Pitched six-blade turbine, fiat-bottomed tank, c"= 10 vol.%.

419

Fig.3" The dependency of z,, vs. dp/T for three, four and six-pitched blade turbine (P 3-BT, P 4-BT, P 6-BT), different T/D ratio, c"= 2.5 vol.%.

Fig.4: The dependency of 7csvs. dp/T for three, four and six-pitched blade turbine (P 3-BT, P 4-BT, P 6-BT), different T/D ratio, c"= 10 vol.%.

420 Similar results obtained in a dished-bottomed vessel are plotted in Figs. 3 and 4. This set of experiments is focussed on the effect of the number of the blades at the ratio T/D = 3.3. From both figures it is clearly seen that there is practically no difference between pitched three, four and six-blade turbine. It is also seen that for large particles 7cs =constant for both solid phase concentrations measured because of dished bottom eliminates "dead spaces" and improves circulation. To compare dished and flat bottomed tanks all data obtained with three axial impellers of different ratios T/D in the baffled flat bottom cylindrical vessel at solid-phase concentration c"= 2.5 vol.% are depicted in Fig.3 whereas Fig.4 shows the same data for solid-phase concentration c"= 10 vol.%. It follows from both figures that bottom shape affects preferably particle suspension of coarser particles since the lifting of particles from the corners behind the baffles and in the center below the impeller requires considerably higher power input. It is interesting that the power input needed for the suspension of the largest particles at cv= 10 vol.% approaches a power consumption reported for dished-bottom tanks. The data depicted in Figs.3 and 4 can be represented by equation (7) with the constants given in Table I. Constant K was calculated as an average value of constants K within one data group. For each plot ~., =f(dp/7) the constant K was calculated according to equation (8). Statistical acceptance of the average value of K was tested. All constants and exponents in equations (7) and (8) can be calculated from the constants and exponents appeared in the power form dependency Fr'vs. d//T given by equation (2) and power number, both obtained experimentally for a specific geometrical configuration. Table I: Constants of equation (7) cV[vol.%] K 2.45 2.5 10 302 10 31.6

Bottom shape Flat-bottom Dished bottom Flat-bottom Dished bottom

C3

al 0.6

2o.5

0.6

1.0 0.8

35.4 718 104

1.2 0.8

4. P R O P O S E D P R O C E D U R E

Based on the above outlined approach the following procedure can be recommended: 1. l~owing a suspension volume we can determine the vessel diameter T. For a given particle size, vessel diameter and particle concentration, the value of ~, can be determined either from the graph x.,.=f(d/T) or equation (7). 2. From the known value of ns one can calculate the power input needed for off-bottom particle suspension. 3. hnpeller speed required to lift particles can be determined by a common procedure if we kmow the power number of a chosen impeller type and T/D ratio.

Example. A suspension of voltunetric solid-phase concentration c" = 10 % composed from water and solid particles of I mm in dimvieter with solid-phase density 2500 kg/m 3 should be fhlly suspended by an axial impeller in the fiat bottomed tank of inner diameter 1 m. Critical impeller speed for particle suspension should be calculated.

421 Solution.

1. The value of 7c,. was determined from Fig.2 or equations (7) and (8) with constants given in Table I for the ratio dJT=O. 001. One obtains the value of ~r~=0. 01. 2. Power input required to lift particles in the state of suspension can be calculated from Equation (6) and one obtains the value of P= 649 W. 3. For pitched six-blade turbine with the ratio T/D=3 using the value of power number of P o = l . 7 one obtains the required impeller speed Nf=4.34 1 s -l. Alternatively, for pitched six-blade turbine with the ratio T/D=4 one obtains N f = 7. 01 s -I . 4. The results can be checked by equation (2) with constants from Rieger and Ditl (1997) so Fr"=0.397 for T/D=3 and F r ' = 0 . 8 9 7 for T/D=4. From these equations the values of critical impeller speed for suspension can be directly calculated as Nr=4.21 s -I for T/D=3 and Nr = 7. 06 s -~ for T/D=4. The difference in impeller speed calculated by both method is 3% for T/D=3 and 1% for T/D=4.

Acknowledgement: This research has been subsidized by the Research Project of Ministry of

Education of the Czech Republic J04/98:212200008.

NOMENCLATURE Cv

D dp Fr' g H H2 N Nr

P Po Re T l-t

volumetric concentration of solid particles in suspension, 1,% impeller diameter, m particle diameter, m 2 modified Froude number, Fr'=N Do/Apg acceleration due to gravity, m/s 2 height of the liquid level, m impeller clearance, m impeller speed, l/s, rpm critical impeller speed for particle suspension, l/s, rpm power input, W 3

5

P Ps

power number in mixed suspension, Po=P/psN D 2 Reynolds number, Re=ND p/l.t vessel diameter, m dynamic viscosity, Pa.s dimensionless number defined by equations (4) and (6) showing a suspension ability of the system from the energetical point of view liquid density, kg/m 3 suspension density, kg/m 3

Pt

solid phase density, kg/m 3

zr.,.

P

density difference,

p=pt-p, kg/m 3

422 REFERENCES

1. F. Rieger and P. Ditl, Suspension of Solid Particles, Chem. Eng. Sci. 49 (1994) 2219. 2. F. Rieger and P. Ditl, Suspension of Solid Particles in Agitated vessels, Proceedings of 4 th European Conference on Mixing, BHRA (1982) 263. 3. P. Ditl, E.B. Naunaan, Off-Bottom Suspension of Thin Sheets, AICHE Journal, 38, No.6 (1992) 959. 4. IF. Rieger, Calculation of Critical Agitator Speed for Complete Suspension, Reports of the Faculty of Chemical and Process Engineering at the Warsaw University of Teclmology, Vol. XXV, No 1-3 (1999a) 211. 5. F. Rieger, Research of Mixing at Mechanical Engineering Faculty of Czech Technical University at Prague, International Conference Mechanical Engineering'99, Bratislava (1999b) (in Czech). 6. F. Rieger, Efficiency of Agitators while Mixing of Suspensions, Proceedings of VI Polish Seminar on Mixing, TU Krakow (1993) 79. 7. F. Rieger and P. Ditl, Effect of Vessel to Impeller Diameter Ratio onto Particle Suspension, Scientific Bulletin of Lodge Technical University, No.21 (1997) 181. 8. F. Rieger, Influence of Impeller's Blade Number on Ability of Particle Suspension. Chem. pr m.72, No.3 (1997) 24 (in Czech). 9. P.M. Armenante and E.U. Nagamine, Effect of Low Off-Bottom Impeller Clearance on the Minimum Agitation Speed for Complete Suspension of Solids in Stirred Tanks, 53,No.9 (1.998) 1757-1775. 10. R. Corpstein, J.B. Fasano, K.J. Myers, The High-Efficiency Road to Liquid-Solid Agitation, Chemical Engineering, October (1994) 138-142.

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

423

Characterization and rotation symmetry of the impeller region in baffled agitated suspensions Ziyun Yu and ,~ke Rasmuson Department of Chemical Engineering and Technology, Royal Institute of Technology, SE- 100 44 Stockholm, Sweden The hydrodynamic conditions in the impeller region of an agitated suspension are investigated. The agitator is a 45 ~ pitched-blade turbine pumping downwards. A threedimensional phase-Doppler anemometer is used to measure local, instantaneous, threedimensional velocities of the fluid and of the suspended solids. A shaft-encoding technique is used to obtain angle-resolved data in order to resolve the turbulent velocity fluctuations from the impeller blade fluctuations. Measurements are performed in different vertical planes to investigate the influence of the baffles on the velocities, turbulent kinetic energy and Reynolds stresses, in the impeller region. It is found that this influence is not completely negligible. 1. INTRODUCTION Only a limited number of studies have been devoted to the hydrodynamics of pitched blade turbine agitation, a type of agitator often used in liquid blending and solids suspending (e.g. the studies by Ranade and Joshi [1 ], Kresta and Wood [2] and Pettersson and Rasmuson [3]). Investigations on actual distribution of solids in stirred suspensions are quite rare. In many processes the most important region in a mechanically agitated tank is the impeller region, where the flow is highly turbulent and anisotropic. In characterization of the turbulence the Reynolds stresses are important. However, they are rarely reported since velocity crosscorrelation measurements are required. Sometimes repeated 2-D measurements are used to obtain the complete Reynolds stress tensor. However, in this case there is usually not a perfect location match of the different measurements since the measurement volume is shaped like a strongly elongated spheroid. Three-dimensional phase-Doppler anemometry allows us to determine all three components of the local instantaneous velocity and the full set of Reynolds stresses in one single measurement. By using shaft-encoding technique, the angular location of each data value relative to the impeller blades can be identified. In this way the periodicity of the flow directly related to the impeller blades can be excluded in the determination of turbulence parameters. Shaft-encoding has been used to trace the behavior of the trailing vortex behind the impeller blades [4-6]. In the present work, the turbulence characteristics of the impeller region of a suspension agitated by a pitched-blade turbine in a cylindrical baffled tank are investigated. A threedimensional phase-Doppler anemometer, in conjunction with an optical shaft encoder, is used to measure the three components of time-resolved and angle-resolved velocities of the flow field. The three-dimensional velocity vectors of suspended particles are determined simultaneously. Measurements are carried out in the mid-plane between two baffles to characterize turbulence conditions around the impeller, and in different vertical planes to investigate the influence of the baffles on conditions in the impeller region.

424

2. Experimental 2.1. Agitated vessel configuration Experiments are performed in a cylindrical glass vessel with flat bottom and provided with four equally spaced baffles of stainless steel. The vessel has an inner diameter T = 210ram and a wall thickness of 4 mm. The working fluid is deionized water, which was filtrated twice to remove particles larger than 0.21xm. The liquid height in the vessel is equal to T, i.e. 210 mm. The vessel is installed inside a rectangular glass trough. The gap between the cylindrical tank and the trough is also filled with deionized water to minimize refraction effects at the surface of the cylindrical wall. The water in the trough is temperature-controlled to 20+0.1~ The impeller is a 45 ~ pitched, four-blade turbine pumping downwards with a diameter D of approximately equal to 0.4T. It is located with a clearance of T/3 from the bottom of the vessel. The baffles are connected to the lid of the tank. The rim of the lid is provided with a scale having 1~ resolution and an estimated accuracy of +0.5 ~ By turning the lid measurements in the different vertical planes relative to the baffles can be performed. The baffles width is about 0.1 T, with a clearance of 4mm from the wall and 4 mm from the bottom of the tank. The geometry of the tank and the agitator are shown in Figure 1. Dimensions are given in millimeters. The tank is mounted on a milling table which can be traversed in horizontal and vertical directions, with the positioning accuracy of 0.05 and 0.5 mm respectively. The impeller is driven by a variable speed electric motor and the maximum variation of the speed does not exceed +0.5% of the set rotation speed. An optical encoder is coupled to the shaft, which provides a marker pulse and a train of 360 pulses per revolution. The midpoint of one of the impeller blades is aligned with the marker pulse, thus each data point can be located relative to the impeller blade. Fig. 1. Geometry of the vessel and the impeller

2.2. Phase-Doppler anemometer The phase-Doppler system is a six-beam, fiber-based, three-component system (Dantec Measurement Technology), using the green (514.5 nm), blue (488.0 nm) and violet (476.5 nm) lines of an Ar-ion laser (Spectra Physics 2020). The two green laser beams intersect to form an interference pattern shaped like a strongly elongated spheroid, called probe volume.

425 From the light scattered by particles traversing the interference pattern, the local, instantaneous velocity vector perpendicular to the interference pattern can be determined. Two blue laser beams from the same probe (2-D) generate a second interference pattern that overlaps the green probe but is rotated 90 ~ This allows for determination of a second orthogonal velocity component. The two violet beams from a separate probe (l-D), at 90 ~ angle to the first, generate a third interference pattern perpendicular to the previous two. For particles traversing the cross-section region formed by all the three probe volumes, called the measurement volume, the complete 3-D velocity vector can be determined. The green light is used to determine the size of the spherical particles. Scattered green and blue light are collected by a receiver. The angle between the optical axis of the receiving optics relative to direct forward scattering of the green and the blue light is approximately 69 ~, assuring that the size estimation is predominantly based on reflected light. Violet light is collected by the 2-D probe. The measurement volume is about 0.003 mm 3 [7], which indicates the spatial resolution of the measurements. Data for the transmitting and the receiving optics and the detailed description of the system has been reported previously [8].

2.3. Particles Two types of particles are added to the fluid. Spherical metallic coated glass particles (TSI GmbH) with a specified density of 2.6 g/cm 3 and a number-mean- size of 4 / t m are used as fluid tracers. Glass beads with a high degree of sphericity (Duke Scientific Corporation) are used to simulate suspended particles. The glass beads are normal distributed in size, have a mean size of 321+9.6 #m and a standard deviation of 13.2 #m. The density is 2.42 g/cm 3. The two types of particles are denoted as seed particles and process particles, respectively. About 0.10 g of seed particles is added to the fluid, and the concentration of process particles is 0.06 per cent by volume. Pettersson and Rasmuson have examined the features of these particles

[3]. 2.4. Experiments All the measurements were carried out at a rotation speed of 450 rpm, which is just above the minimum suspending stirring rate for the tank configuration and the process particles used. The corresponding impeller tip velocity, Utip, is about 1.9 m/s. Three vertical planes relative to the baffles are studied. One is half'way between the baffles, i.e. 0 = 45~ another one is right behind the baffle, i.e. 0= 5 ~ and the last one is right in front of the baffle, i.e. 0 = 85 ~ In each vertical plane, 14 points of measurement were located around the impeller. Six of them were distributed on the horizontal plane 10 mm below the impeller centerline plane, 5 points on the plane 10 mm above the impeller centerline plane and 3 points were aligned on the vertical plane 4 mm away from the impeller tip. The measurement points and the cylindrical coordinate system used are shown in Figure 2. While 0 corresponds to tangential direction in the fixed frame of reference, ~ moves tangentially with the rotation of the impeller (see Figure 2). The former is taken as the tangential coordinate axis. Around 300,000 data values were collected in each measurement. About 55% were seeding particles, which means that there were more than 900 samples in each angular 2 ~ slot. Process particles occupy 1 to 2 % of the total number of data values.

426

Fig. 2. Location of measurement points

3.

RESULTS AND DISCUSSIONS

3.1. Mean velocity field The overall mean velocity components are calculated from ensemble averages over 360 ~. Velocity bias is corrected for by the magnitude of the instantaneous 3-dimensional velocity [9]. Figure 3 and Figure 4 present the general mean flow pattern measured in the planes of 0= 5 ~ 45 ~ and 85 ~ The flow has everywhere a strong axial component, at least in relative terms. Below the agitator also the tangential component is strong, while the radial contribution is quite small. Above the agitator there is a stronger influence of the radial component and the flow is towards the impeller axis. Fluid and particles have peak velocity below the impeller at about r = 33 mm (0.4D) and the maximum magnitude of the 3dimensional velocities is about 0.6 times the impeller tip speed. The observation of the fluid is in agreement with the results of others [1]. Close to the impeller tip, on the impeller centerline, the velocities are less than 0.1Utip. There is no big difference in the velocities of the fluid and the particles projected in radial-axial plane. In the projection in the radial-tangential plane, it can be observed that below the impeller the particles lag behind the fluid more and more when moving closer to the baffle. At 85 ~ just in front of the baffle, the lag is about 0.1Utip and a difference in flow

Fig.3. Mean velocity field over 360 ~ ensemble average in radial-axial plane: (a) 0 = 5~ (b) 0 = 45~ (c) 0 = 85 ~

427 -(a) . . . . . ,.~,..[85 o

0.5 Utip~ fluid

., ~ ]

~

85

o'

o., ~L ~

----------. particles ~1

'

fluid particle

45 ~

:l ,)/ i 0

10

20

30

r (mm) ~

40

0'

i0

20

r (mm)

..3;

40

..'~

Fig. 4. Mean velocity field over 360 ~ ensemble average in radial-tangential plane: (a) z = -10 mm; (b) z = 10mm. direction between the two phases can also be observed (Figure 4a). Above the impeller both fluid and particles tend to be more and more strongly forced towards the impeller when moving closer to the baffle plane (Figure 4b). Therefore the influence of the baffle on the conditions in the impeller region is not negligible. The influence of the baffle is clearer in the angle-resolved velocities of the fluid. The angle-resolved velocities are obtained from ensemble-averaging the instantaneous velocities over each angular slot (2~ Figure 5 shows radial-tangential velocity vectors at different vertical planes for three resolved angles (i.e. angles with respect to the impeller blades): = 10~ 30 ~ and 50 ~ It can be observed that the velocities vary with the baffle location in direction and in magnitude for each ~:. Below the impeller, just behind the impeller blade (~= 10~ the fluid is drawn towards the impeller shaft more strongly when close to the baffle plane. At ~= 50 ~ nearly in the mid-way of two blades, the maximum 3-D velocity is high just behind the baffle (0= 5~ It is lower in the plane mid-way between the baffles and becomes high again in the front of the baffle. The variation of the maximum velocity with respect to the baffle location is about 0.1Utip (see Figure 5c). Above the impeller, just behind the blade the influence of the baffle on the flow direction is significant, shown in Figure 5d. The flow direction changes with about 20 ~ depending on the baffle location. 3.2. Kinetic energy The results of kinetic energy are shown as the mean values over the whole impeller rotation. They are calculated from the turbulent fluctuating velocities by

k = !(U2z,turb 4" U2r,turb + bt20,turb) 2

(1)

where the turbulent fluctuating velocities are obtained from angle-resolved data as follows [ 10]"

428

(Uturb)2--< -~--~> (Uperi)2= 2

(2)

where the over-bar on the symbols denotes ensemble average of samples in one angular slot, and the bracket '< >' denotes ensemble average of slots in the revolution of the impeller (360~ In this work, we use 2 ~ as angular interval, therefore there are 180 slots in the whole revolution. It is not difficult to verify that the total fluctuating velocity, uwt, is consistent with the root-mean square of fluctuating velocities for the whole measurement. z=-10mm

z = 10mm

(a)

(d)

0.5Utip

0.5U~p

>

0 = 85 ~

0 = 85 ~

0 =45 ~

0=45 ~

~ = 10 ~

0=5 ~

,\

,

(b)

o.su.~._

0=5 ~ ....

t x...v....~.,v...

(e)

/

O.5Utip >

o:8,

t////

0 = 85 ~

0=45)

/ / / / ~

0 =45 ~

~=30 ~

0=5 ~ ,

,

~

,,

~

\\\',,

~\\\ ,,

( C)

0.5Utip

//~~ ~ " -

,/yl\

(f)

0.5Utip

>

0 = 85 ~

~=50 o

0 =45 ~

t?=5 ~ 10

20 30 40 r'----'~

50

1'0

\

\ \ \,,.

~. ~ \ \ \ \ 20

r

30

40

5'0

t"

v

Fig. 5. Angle-resolved velocity vectors projected in radial-tangential plane (a) z = - 1 0 , ~ = 10~ (b) z =-10, ~ = 30~ (c) z = - 1 0 , ~ = 50~ (d) z = 10, ~ = 10~ (e) z = 10, ~ = 30~ (f) z = 10, ~ = 50 ~

429

Figure 6 gives the kinetic energy distribution below the impeller at two vertical planes, 0 = 45 ~ and 0 = 85 ~ The overall pattem in the two planes is the same. The kinetic energy is low near the impeller shaft and decreases outside the impeller (at r = 45 mm). There is a maximum at the impeller tip (r = 41 mm). The kinetic energy increases in front of the baffle. At the impeller tip the value of the kinetic energy is about 15% higher at 0 = 85 ~ than that at 45 ~. This indicates that there is stronger intensity of the turbulence in the upstream of the baffle.

0.25

* 0=45~

0.20

~

~o:.~

~~

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

t-q

0.15 0.10

. .

iiiiii

0.05

.

0

10

.

.

20

.

30 r (ram)

40

i0

Fig. 6. Kinetic energy distribution along the radial distance below the impeller (z = -10 ram)

3.3. Reynolds stresses The Reynolds stresses are calculated from the total velocity fluctuation by the sample ensemble-averaging. All the Reynolds stresses together constitute the Reynolds stress tensor:

~.--

I

U z 9U z

U z 9U r

U z 9U 0

U r.u

z

U r "U r

U r "U 0

U0 . U z

UO 9l,t r

U 0 9U 0

(3)

where the cross-terms are the shear Reynolds stresses and the diagonal terms are the normal Reynolds stresses. The overall magnitude of the Reynolds stresses can be represented by the norm of the matrix, which is a non-negative scalar. In the Euclidean norm, we have

Ilrll

= ~/,Zm~(~T "V

(4)

where ~.r is the traverse matrix of "r, and 2max(~"r. ~') is the maximum eigenvalue of the matrix

(rr. r). Figure 7 shows the Euclidean norms of the Reynolds stress tensor, the normal stress tensor (taking the shear stresses as zero in the matrix) and the shear stress tensor (taking the normal stresses as zero in the matrix) below the impeller in the two vertical planes of 0 = 45 ~ and 0 = 85 ~. The normal stresses vary quite strongly with the radial location along the impeller with a clear peak value close to the tip. Just in front of the baffle the peak value is 15 % higher than the corresponding value in the mid-plane between the baffles. The shear stresses are clearly lower than the normal stresses. The shear stresses increase only weakly towards the tip of the impeller and the influence of the baffle is weaker.

430 0.25 0.20 -

fo:"

'

i[i:-

r

E

!!!!!

0.15 0.10-

--

0.05.

0.00 0

I

.

.

I

.

.

I

.

.

l

.

.

I

I

I

I

20 30 40 0 10 20 30 40 50 r (mm) r(mm) Fig.7. Norm of Reynolds stress tensor distributed along the radial distance below the impeller: ~ t o t a l Reynolds stresses" X normal stresses; ---o-- shear stresses. 4.

10

CONCLUSION

The baffles do exert an influence on the conditions in the impeller region. This influence is observed in the results over the fluid mean velocity field, the angle-resolved velocities, the kinetic energy and the Reynolds stresses, as well as in the behavior of larger process particles. The influence is not found to be very strong but is neither entirely negligible. Certain aspects deserve more attention. NOTATIONS D k T ~, U, u z, r, 0

r

diameter of impeller, mm turbulent kinetic energy, m2/s 2 inner diameter of tank, mm instantaneous, mean and fluctuating velocity, m/s axial (mm), radial (mm) and tangential (o) axis in cylindrical coordinate system angle of blade in the moving system of frame, o Reynolds stress tensor

il ll ax

norm of Reynolds stress tensor, m2/s 2

maximum eigenvalue of matrix

Subscripts peri tip tot turb

periodic impeller tip total turbulent z, r, 0 corresponding to cylindrical coordinate axes

REFERENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

V.V. Ranade and J. B. Joshi, Chem. Eng. Comm., 81 (1989), 197. S.M. Kresta and P. E. Wood, Chem. Eng. Sci., 48 (1993), 1761. M. Pettersson and A. C. Rasmuson, AIChE Journal, 44 (1998), 513. M. Yianneskis, Z. Popiolek and J. H. Whitelaw, J. Fluid Mech., 175 (1987), 537. C.M. Stoots and R. V. Calabrese, AIChE Journal, 41 (1995), 1. M. Sch~ifer, M. Yianneskis, P. Wachter and F. Durst, AIChE Journal, 44 (1998) 1233. Z. Yu and A. C. Rasmuson, Experiments in Fluids, 27 (1999), 189. M. Pettersson and A. C. Rasmuson, Trans. Inst. Chem. Eng., 75, Part A (1997), 132. D.K. McLaughlin and W.G. Tiedermann, Phys Fluids, 16 (1973), 2082 M. Fawcett, Personal communication, BHR group, UK (notes from M. Pettersson).


431

Solids suspension by the bottom shear stress approach M. Fahlgren, A. Hahn and L. Uby ITT Flygt, P. O. Box 1309, SE-171 25 Soina, Sweden

Synopsis. A local hydrodynamic condition for incipient motion of particles in a sediment bed is the achievement of a sufficient bottom shear stress. Once this is reached, near bottom turbulence characteristic determines whether entrainment occurs. Given the size and density of the solids - in the simple case of a unimodal distribution of spherical particles- and the density and viscosity of the liquid, it is argued that there is a critical value of the shear stress % for the lift force to overcome gravity. The critical value is found by experiment. A new experimental method is presented, where the lowest harmonic standing wave in a rectangular tank is used to generate a controlled shear stress pattern on the bottom. A universal reference curve is obtained, where the dimensionless bottom shear stress, which is a modified Froude number, is plotted against the local Reynolds number based on particle size. Knowledge of the hydrodynamic condition for incipient motion of particles enables solution of solids suspension problems in general geometries, and in liquids other than water. This is in contrast to some common engineering tools in use today. 1. INTRODUCTION

The condition for off bottom solids suspension has over the years been formulated in terms of system parameter correlations. The work most referred to is that by Zwietering [1]. In support of extensions and variations of Zwietering's correlation, local hydrodynamic conditions for particle suspension have been suggested, in terms of balances of various energies or forces. In the present work, the local condition only will be of interest, and no attempt is made at standard geometry correlations. Furthermore, the hydrodynamic condition for suspension will be discussed from the point of view of incipience of particle motion and entrainment into the flow, disregrading the various (global) criteria that are in use. However, a route to solids suspension design or prediction can be inferred from the method discussed below. The local hydrodynamic condition for a solid particle on a bed of similar particles to start to move under the action of liquid flow, is that the drag and lift forces supersede the gravity and bed friction (dragging motion) or liquid friction (rolling motion.) The statistical nature of the many-particle problem implies that drag and friction effects are not susceptible to deterministic modelling. However, the important balancing forces in suspending particles off bottom are lift and gravity, where the lift force depends on the (particle) Reynolds number Re, as well as to some extent on the local bottom topography.

432

In a set of experiments Shields examined the critical dimensionless shear stress (which is a densimetric Froude number) Frc = ~c / (ps - pl)gdp as a function of Re [2]. These two quantities are commonly called the Shields parameters. A large number of workers have repeated and extended this work [3, 4]. Due to the above mentioned statistical nature of the problem, the resulting reference curve has rather become a band. However, some of the width of this band is inevitably attributed to discrepancies in data collection and evaluation, which have been quite subjective. Most experiments have been carried out in flumes, although many other geometries have been considered. In this work, waves on the bottom of a closed tank [5] in combination with modern visualisation techniques, are shown to provide an experimental frame with less subjectivity. A common way of quantifying entrainment and transport makes use of the relation between the dimensionless bottom shear stress and Einstein's dimensionless transport parameter [6]. Such an approach to suspension in mixing tanks has been pursued by Gladki [7]. As has been shown by Cheng & Chiew [8], entrainment follows on incipient motion by a certain probability, given the turbulence characteristics close to the bottom. This probability can be increased by increasing the bottom shear stress, as shown in Fig. 3. 2. THE SEICHE GENERATOR The lowest harmonic (small amplitude) standing wave in a limited body of fluid is called seiche. The horizontal velocity field of the seiche in an inviscid fluid in a rectangular tank (Figures 1 and 2) is given by Massey [9] U(x, z,t) = -mcacosech[m(h + ~ a)]cosh[m(h + z ) ] c o s mx s i n tact

(1)

where a is the wave amplitude at the surface, m = m'L is the wave number, h is the equilibrium liquid depth and the celerity, c, is given by Eq. 2.

c =

+

tanh

.

(2)

In Eq. (2), ~ denotes surface tension, which will be neglected here, and g is the gravitational acceleration. The shear stress on the bottom, as mediated by a viscous boundary layer of width 5 1.8 a certain state of "saturation" occurs (Fig. 2 for 500 and 550 minl). The quantity of aspirated gas and the volumetric coefficient still increase although at a lower rate than previously. This is most probably due to the presence of a large space occupied by the gas phase in the tank and a decreasing turbulence caused by a large interfacial area. Since the mixing power still increases, the efficiency decreases. From all presented graphs it follows that the range of Fr' number, at which the impeller operates most efficiently, is quite broad and equals to 0.8 + 1.8. In this range the impeller efficiency is approximately constant.

454 5. CONCLUSIONS From the practical point of view, the self-aspirating disk impeller should operate in the range of the most efficient gas dispersion and at a maximum mixing intensity. This means that this type of impeller should be used at the modified Froude number Fr' --- 1.8 if there are no technological contraindications such as for example destruction of biomass structures. Naturally, the absolute energy consumption will be high but it will be used in the best way. If this value of the Fr' number is assumed, then it will be possible to determine on the basis of eq. (1) the required rotational speed of the impeller at a given diameter and liquid level over the impeller. It is worth to mention that under these conditions the self-aspirating disk impeller ensures the flow of up to a dozen kg O2/kWh, which is several times more than for surface aerators of much bigger dimensions [8]. SYMBOLS AND DIMENSIONLESS'NUMBERS In eqs. (2)-(6) the dimensionless numbers are defined as follows: 9 the power number Po = P/(N3.DS.9) or POG = PG/(Na'DS"o) 9 gas hold-up coefficient 9 = V~/(VL + VG) 9 gas flow number KG = V/(N.D 3) 9 the modified Sherwood number Sh = (kLa)/(g2/v) u3 where D - the impeller diameter, N - the impeller rotarional speed, H - level of liquid in the tank, h - distance of the impeller to the bottom, g - acceleration of gravity, k - consistency coefficient, n - flow behaviour index, P - power input, Pc - power input of aerated impeller, T - tank diameter, V - gas flow rate through the liquid being mixed, V c - liquid volume in the tank, V c - g a s volume in the two-phase mixture, kLa-Volumetric mass transfer coefficient, r l - dynamic viscosity, rlw- dynamic viscosity of water, v - kinematic viscosity, cr- liquid surface tension. REFERENCES 1. Kraslawski A., Rzyski E., Stelmach J., Selection of an Optimum Mixing-Aerating Device Based on Fuzzy Integral, Chem. Biochem. Eng. Q., 5 (1-2), 19-22, 1991 2. Zlokarnik M., Judat H., Rohr- und Schreibertthreh zwei lei stungsf'ahige Rtthreh zur FlUssgkeittsbegasung, Chemie Ing. Techn., 39, 1163, 1967 3. Heim A., Rzyski E., Stelmach J., Condition of Gas Aspiration by an Impeller, XI Celostatni Konferenci ,,Michani", Brno 1997 4. Patent RP nr 148476 5. Heim A., Rzyski E., Stelmach J., Mixing Power of Self-Aerating Disk-Type Agitators, 13th International Congress of Chemical and Process Engineering CHISA, Prague 1998 6. Heim A., Rzyski E., Stelmach J., Mieszadla samozasysajace. Zatrzymanie fazy gazowej, 16~h Polish Sciens Conference of Chemical and Process Engineering, Krak6w - Muszyna 1998 7. Zlokarnik M., Dimensional Analysis and Scale-up in Chemical Engineering, SpringerVerlag, Berlin 1991 8. SedzikowskiT., Uwagi w sprawie kryteriow wyboru systemow wg~ebnego lub powierzchniowego napowietrzania sciekow, Gaz, woda i technika sanitarna, 12 1990


455

A N O V E L G A S - I N D U C I N G A G I T A T O R SYSTEM FOR GAS-LIQUID R E A C T O R S FOR IMPROVED MASS T R A N F E R A N D MIXING E.A. Brouwera and C. Buurmanb aJONGIA NV, p.o.box 284, 8901 BB Leeuwarden, The Netherlands bMIXXIN consultancy, Nieuwstraat 55 B, Purmerend, The Netherlands Gas-inducing impeller systems are attractive for application in gas liquid reactors, in particular with "difficult" (or hazardous and expensive) gases, because they provide recirculation of the gas without a separate gas compressor. So far their commercial success has been rather limited due to disappointing gas induction rates and difficulties with the scale-up. novel gas-inducing agitator system has been designed, based on the combination of existing types of axial flow impellers with below it a gas-inducing device (Figure 1). The principle is that the impeller flow creates the necessary underpressure at the rear of the inducer elements, where the gas nozzles are positioned. The geometry of the impellerinducer combination has been optimised experimentally for maximum gas flow. Also the gas-liquid mass transfer was compared to that of a conventional (sparger) system. An accurate method was found for the prediction of the gas flow and a successful strategy developed Figure I Agitator- gas inducercombination for the scale-up problem. The result is a recirculation system which provides gas flows and mass transfer comparable to conventional systems at stirrer speeds well below the critical speed of the system. A

1. INTRODUCTION Gas inducing agitators for large chemical reactors suffer from two major problems - the prediction of the gas flow and mass transfer for the design is unreliable - for a large scale application the speed required for gas-inducing may easily exceed the critical speed (causing mechanical problems) 1.1 G a s f l o w p r e d i c t i o n

A major problem in the development of gas-inducing agitators is a correct prediction of the gas flow. Most of the literature give only reliable predictions for the operation condition where the gas induction starts. With the model proposed by Forrester et al [1,2] quantitative predictions of the induced gas flow appear to be possible. In their model the dynamic pressure balance accounts for the underpressure, the hydrostatic pressure, gas flow resistance, surface tension and the kinetic energy for bubble

456

formation. Applying this model to our experimentally observed gas flows we arrive at underpressure coefficients at the rear of the inducer elements which are well in line with literature data for flow around objects. For the conditions with the highest gas hold-ups we found a relatively small deviation and we incorporated a correction factor in the equations to account for this effect. 1.2. Mass transfer For the design of a large scale chemical reactor the mass transfer must be predictable. In order to verify this we carried out a number of mass transfer experiments in a model system. Special attention was paid to reactors with high aspect ratios. 1.3. Scale-up for subcritical speed The scale-up of gas inducing systems is a well-known problem. Geometrical scale-up leads to increased specific power consumption, because the increase in rotational speed otherwise cannot compensate sufficiently for the increase in liquid depth. As a consequence for large scale reactors we have chosen a design strategy where the gas inducer is positioned relatively closer to the liquid surface and where the mixing process for the whole of the reactor content is assured by axial flow impellers lower on the shaft. 2.

Figure 2 Importance of various contributions to the pressure balance

THE INDUCED GAS FLOW

2.1. Theory The work by Forrester et al [1,2] shows the importance of the kinetic energy for the bubble formation as a part of the pressure balance for gasflow. For a complete pressure balance one has to

take into account the pressure loss due to - flow resistance in the hollow shaft, the inducer arms, the nozzles - the surface energy , the hydrostatic pressure - the kinetic energy for bubble formation and compare these with the underpressure at the nozzles. Figure 2 shows the various contributions for a typical condition in our experiments.

The flow resistance for the hollow shaft and arms can be calculated simply on the basis of the well-known relations for friction in pipes. For the pressure drop in the nozzles the equation reads 1

2

Ap = C, 2PgV N and the resistance coefficient here is 0.5. The surface energy accounts for a pressure loss of

4.57

Ap=

4s db

The hydrostatic pressure is •P = Pdispersiongh' It should be noted that here the increase of the liquid height and the change in density due to the gas hold-up must be taken into account. For the kinetic energy Forrester et al [1,2] have found 2.25plqN ApKE = ~2d b(d~, - d~) They studied extensively the formation of bubbles and the size of bubbles emerging from the nozzles and found the following relation db _ 2~2nRN ~Cu-~ + l(d~, - d~) _dN 3qN For the underpressure at the nozzles aP=Cup

1

2

P,(2~RNn) 2

So when the underpressure coefficient CUPis known, the pressure balance can be solved (for each individual nozzle) and the resulting total gas flow can be found. 2.2.

Experimental

The concept of the combination of an axial impeller with a gas-inducing device underneath was tested in an UNDERPRESSURE COEFFICIENT experimental set-up of 200 1with water 1 9 ~ ~ and air. In these tests the stirrer speed and the immersion depth were varied. The geometry was varied with respect to the distance between the impeller and gas inducer, the angle between the impeller blades and the gas inducer arms and the angle of the nozzles on the arms. From the measured gas flow values of Cup were evaluated and the system could be optimised for maximum underpressure and gas flow. Figure 3 shows that the underpressure 0 60 120 180 NOZZLE ANGLE, DEG coefficients found (data points) are well in line with literature data (line) for the Figure 3 Underpressurecoefficients at nozzles flow around a cylinder in cross flow [3]. ,

/

.....

-1. /

9

9

9

9

i,

.

9

.

|

9

!

9

9

|

|

9

9

9

458 The differences are caused by the presence of gas bubbles and the less ideal cross flow. In figure 4 the calculated values for GASFLOW, m3/s the induced gas flow (on basis of these 0.0028 underpressure coefficients) are compared to the experimental values for various liquid levels and stirrer speeds. LOW LIQUI

0.002

3.

/ /

0.001

/HIGH / LIQUID

/ / L E V E L

'//6 "

"

'

'

'0"

STIRRERSPEED, rps

Figure 4 Measured gasflows compared to model calculations 3.2.

MASS T R A N S F E R

3.1. Theory On basis of gas flow and stirring power many design correlations for mass transfer have been proposed. One of the rare dimensionless correlations in this field was proposed by Henzler [4]. For a stirrer and sparger configuration he found for air and water k'a(v--~-2) 1'3 = 0.8-10 -4 (. ~M_M._)I/3 vg g gVg

Experiments

Mass transfer experiments have been performed in the 200 I test rig. This was done by first stripping the dissolved oxygen from the water with nitrogen. A "fast" oxygen probe was used to record the saturation process SATURATION WITH OXYGEN (REL.) after starting the mixer-gas inducer. The oxygen concentration increased as a 1 function of time exponentially with k~a as a 0.9 time constant. The extra delay in the 0.8 observed concentration caused by the membrane in oxygen probes however is 0.7 not negligible and one has to make a 0.6 correction for the time constant of the 0.5 D oxygen probe, resulting in ,

,,

0.4

k c p = 1 + _____E~pexp( - k ,a) c| k,a -kp

0.3 0.2 0.1 0

!

0

1 O0

200

300

400 TIME, s

kla -

k,a

-

kp

exp(-kp)

Figure 5 is a typical example of one of the experiments. For a given probe constant Figure 5 Evaluation of kla from saturation (kp = 0.22 s"1 in our experiments) the experiment value of k,a can be evaluated from the saturation curve. In figure 5 for example the line was fitted to the data for a k~a value of 0.01 s1.

459

Experimentally we found for aspect ratios of ca. 1 that the k~a values for the novel system are in line with those for stirred vessels with spargers (figure 6). However for high aspect ratios, with more than 1 impeller on the shaft, a different behaviour was observed. In our experimental set-up with H/T=3 (-600 I) (Figure 8) equipped with three axial flow impellers and one gas inducer we observed that the gas bubbles, introduced MASS TRANSFER COEFFICIENT (MEASURED), 1Is

0.06

0.8

SATURATIONWITH OXYGEN(REL) ""

0.05 0.6

0.04 0.03

.

~

0.4

0.02 0.01

0.2 II

!

9

I

!

0.02 o'.04 0.06 MASS TRANSFER COEFF,C,ENT (CALCULATED), 1Is Figure 6 Comparison of measured and calculated values for mass transfer (data points and model calculation)

In

!1

nl

GAS INDUCING MIXER

ADDITIONAL MIXERS

Figure 8 Example of geometry for reactors with high aspect ratio

0..

0

50

100

150 TIME,s

Figure 7 Mass transfer experiment in a reactor model w'dh H/T=3

under the upper impeller, are drawn down by the liquid stream only to a certain depth. Obviously in this upper zone the gas-liquid mass transfer takes place and the dissolved gas is distributed to the lower zones by mixing and turbulent exchange between the zones. It has been shown that a mathematical "three zone model", with realistic values for the mass transfer, interzone exchange rate and mixing, gives an accurate description of this behaviour. In figure 7 a typical oxygen saturation experiment is compared with the Calculated values for such a 3-zone model. The line of figure 7 represents the model with values k~a = 0.03 s"1 and an exchange velocity between the compartments of 0.3 nD. In terms of mass transfer and mixing behaviour this model can give a very satisfactory prediction for the observed experimental values.

460

4.

APPLICATION TO A CHEMICAL REACTOR

On the basis of this multizone model criteria can be set for a chemical reactor for the ratio between mass transfer, exchange rate and chemical reaction rate. In the reactor design the dimensions and stirrer speed are to be chosen so, that the induced gas flow, the specific power and the resulting mass transfer and exchange rate are sufficiently high to cope with the gas consumption by the chemical reaction. The novel system was applied in a 20 m3 hydrogenation reactor where it replaced a "surface aerator". It has demonstrated a better mass transfer, shorter batch times and a better flexibility in the operation of the process.

REFERENCES 1. S.E. Forrester, C.D. Rielly, K.J. Carpenter, Gas-inducing impeller design and performance characteristics, Chem Eng Sci 53 (1998) p603-615 2. S.E. Forrester, C.D. Rielly, Bubble formation from cylindrical, fiat and concave sections exposed to a strong cross-flow, Chem Eng Sci 53 (1998) p1517-1527 3. B. Eck, Technische StOmungslehre, Band 1, Springer Verlag, Berlin, 1988, p141 4. H.-J. Henzler, J. Kauling, Scale-up of mass transfer in highly viscous liquids, 5th European Conference on Mixing, W0rzburg 1985, p303-312 NOTATION Cp probe oxygen concentration co~ saturation concentration oxygen Cr resistance coefficient Cup underpressure coefficient db bubble diameter dN nozzle diameter D impeller diameter g acceleration of gravity h' immersion depth of gas inducer H total liquid height k~a volumetric mass transfer coefficient kp time constant probe n rotational speed z~p pressure drop qN gasflow through nozzle RN radius of nozzle position T tank diameter Vg superficial gas velocity VN velocity in nozzle F-.M specific mixing power v kinematic liquid viscosity pg density of gas p~ density of liquid interfaciai tension

mol/m 3 mol/m 3 m m m m/s 2 m m s1 s"1 s1 Pa m3/s m m m/s m/s W/kg m2/s kg/m 3 kg/m 3 N/m

10th European Conference on Mixing H.E.A. van den Akker and s 1 6 Derksen 3 (editors) 9 2000 Elsevier Science B. V. All rights reserved

461

Hold-up and gas-liquid mass transfer performance of modified Rushton turbine impellers Sandra C.P.Orvalho, Jorge M.T.Vasconeelos and Sebastiao S.Alves Center for Biologieal and Chemical Engineering, Chemical Engineering Department Instituto Superior Trenico, Av. Roviseo Pais, 1049-001 Lisboa, Portugal"

The effect of blade form on gas hold-up and interphase mass transfer rate generated by radialflow impellers was studied in a dual-impeller agitated tank. Five modified versions of the Rushton turbine were compared with the standard design. Within the precision of the experimental methods in this work, all the impellers provide the same gas hold-up e.v and specific mass transfer coefficient kLa at the same power input and supertieial gas velocity. I. INTRODUCTION Agitation is a very important factor in gas-liquid processes that need large gas handling capacity and effective gas dispersion, like fermentation and a variety of oxygenation and hydrogenation processes. Disk turbines are radial-flow impellers which are particularly suitable for gas-liquid dispersion through mechanical agitation. This is because the disk collects the gas underneath, forcing it into the high shear zone near the blades where bubble formation occurs, as well as because flow instabilities shown by open blade turbines are eliminated. The standard Rushton turbine, which is provided with six flat blades mounted radially on the disk, is one of the most usual impellers found in industry. However, two important drawbacks prevent the Rushton turbine from performing as an ideal gas disperser. Firstly, a significant fall in power demand is observed after gas is introduced that represents a loss of potential for heat and mass transfer. Secondly, flooding occurs at low gas flow numbers [1] thus restraining considerably the gas handling capacity of the impeller. Both disadvantages result from the formation of low pressure trailing vortices at the rear of the blade [2] which attract gas bubbles to form ventilated cavities [3]. Besides leading to an increase in local pressure which markedly reduces the power draw [4], these gas pockets command the hydrodynamic and dispersion characteristics of the turbine [5] Blade shape was found to affect both the ease of formation of the gas cavities and their size [5,6]. Using concave or similar blade geometries, it has therefore been possible to design disk turbines presenting flattened power characteristics and improved gas handling capacity before flooding [6,7]. In a different approach, the perforated blade design [8] was also claimed to present superior performance compared to the Rushton turbine. Regarding gas dispersion, not so many studies were published in the literature comparing the * This workwas supportedby JNICTProject PBIC/C/BIO/1988/95.

462 new designs to the standard one [5-14], still less focusing on gas-liquid mass transfer [6,9,10,14]. In some cases, the reported results were contradictory [6,9]. A comparison of six different blade profiles having the same vertical projected area has been previously carded out [15,23], showing that the streamlining of the blade leads to lower ungassed power number, higher gas flow number before flooding and increased insensitivity of power dissipation to gassing rate. Regarding mixing performance, different mixing time was found at the same rotational speed for the different impellers. However, at the same power input and superficial gas velocity, the mixing time obtained was the same. In this work, the comparison between the different blade shapes was extended to gas holdup (60 and volumetric mass transfer coefficient (kLa). 2. EXPERIMENTAL The experimental work was carried out in a fully baffled tank of diameter T = 0.4 m filled with water at an aspect ratio of 2 (Figure 1). Air was introduced at a flow rate QG equal to 0.125, 0.25, 0.5 and 1.0 vvm (volume per volume per minute), 1 vvm corresponding to a superficial velocity of 0.013 m/s. Agitation was provided by six different sets of dual six-bladed disk turbine stirrers with diameter D = T/3. For every turbine type, the rotational speed was set at constant values giving specific power inputs under ungassed conditions from 0.125 to 4.0 kW/m3. The stirrer off-bottom clearance was equal to T/2 and the impellers spacing was T (see Figure 1).

Fig. 1. Scheme of the experimental set-up.

Fig. 2. End view of the blade profiles (dash-point lines represent radial planes).

463 The impellers used were the standard Rushton turbine, the perforated flat bladed turbine (in accordance with Roman et al. [8]), the semi-cylindrical concave bladed turbine, the angled-90 ~ bladed turbine, the angled-60~ bladed turbine and the lancet bladed turbine. Figure 2 illustrates the blade shape of the impellers. The agitator power draw was calculated ~om measurements of the total torque applied to the shaft and the rotational speed. Torque was measured with a precision of + 1 % using inductive torquemeters mounted between motor and shaft and taking fi-iction into account. The stirring speed was measured using a photoelectric tachometer with an accuracy of_+0.008 s"l. The gas hold-up ev was determined by visual measurement of the liquid level before and after gassing. This method is acknowledgely inaccurate owing to the surface turbulence, but the repetitivity of measurements was verified to be within +10 %. The liquid-phase volumetric mass transfer coefficient kLa (taken as equal to the overall coefficient KLa) was measured using the steady state peroxide decomposition technique with manganese dioxide as catalyst [16]. The hydrogen peroxide solution was introduced into the tank in a well mixed region above the bottom (Figure 1). Measurement of the dissolved oxygen concentration was performed using duplicate oxygen meters WTW Oxi340 equipped with galvanic probes WTW CellOx 325. The probes were positioned at mid height of the tank with the head pointing upwards, in order to minimize errors due to bubble sticking to the probe membrane (Figure 1). Measurement at mid height of high aspect ratio vessels, when associated with well-mixed liquid and gas plug flow models, has been shown to give a representative value of the dissolved oxygen concentration for the evaluation of the steady-state kLa [17,18]. The kLa value was calculated from the steady state balance for oxygen in the liquid:

kLa =

Qperox 2VL 9A,o,C

(1)

where Q~,,,o~is the peroxide molar addition rate, VL the liquid volume and AlogCthe logarithmic mean driving force between the liquid bulk and the gas phase. The oxygen concentration in the outlet gas was calculated from the inlet value by assuming the simplest form of the steady state oxygen mass balance on the gas phase, where the volumetric gas flow rate QG is considered constant across the vessel. Under the experimental conditions in this work, this is an accurate assumption with an approximation of 5%. The kLa was determined at least twice under the same experimental conditions with a reproducib'flity within +_20%. 3. RESULTS

3.1. Gas hold-up Figures 3 and 4 show the hold-up results plotted versus the specific power input PC/VL at fixed values of the air flow rate Qo. Taking the experimental error into account, the hold-up is approximately independent of impeller type at given power input and gas flow rate values. Excluding experimental results apparently affected by systematic error (see Figure 3), a power regression of gas hold-up r in terms of the specific power consumption per unit volume Pe/VL (W/m3) and the superficial gas velocity vo (m/s) gives the following relationship with a correlation coefficient of 0.989:

464

(?G/

.(VG) 0"65

6 v = 0.10. [ , ~ )

(2)

Equation 2 is represented in Figures 3 and 4 by parallel full lines corresponding to different constant values of the gas flow rate QG. Deviations of Equation 2 relative to the experimental measurements fall within +30%, as shown in Table 1. Table 1 Correlations for hold-up compared to experimental data (Pc/VL in W/m3; vG in m/s) Errors (%) Correlation

Reference

6,, = 0 . 1 0 ( P c ~ L ) 0"37(VG) 0"65

This work

-2(pc~L) 0"375(VG) 0"62

~, = 8.35•

e.v = 0 . 2 6 8 (Pc~L)~

0"65

= 0 . 2 8 5 [(PG+VGpg)/VL] 0"303(VG)0"732

0.2

0.1

range

mean

+25

-30

7

Nocentini et al. [19] +27

-29

7

Bouaiti et al. [21 ]

+54

-40

13

Linek et al. [20]

+63

-36

19

-

"

/§

"7", 0.05 /

/

0.01

/

~"

Q~ = 0.25 vvm

Q6 = 0.125 vvm

!

i

!

|

i

I

i

!

i

i

0.25

0.5

1

2

4

0.25

0.5

1

2

4

PG/VL (kW/m3) o Concave

Angled-90 ~ Angled-60 ~

PG/VL (kW/m 3) Lancet Perforated Rushton

Fig. 3. Hold-up results. Full straight lines represent correlation 2 (encircled points for QG = 0.125 vvm and P ~ r = 4 kW/m 3 suggesting a systematic error were rejected). Dashed lines stand for 95% confidence interval.

465 0.2

/

/ /

/

0.1

/ / /

/

//...~//

'7", 0.05 / /

Qo = 0.5 vvm

0.01

Qo = 1.0 vvm

!

!

!

!

|

!

0.25

0.5

1

2

4

0.25

PG/VL (kW/m3) o Concave o Angled-90~ Angled-60~

-|

0.5

!

!

|

1

2

4

PG/VL (kW/m3)

Lancet Perforated Rushton

Fig. 4. Hold-up results (continued from Figure 3). Table 1 also shows that correlation 2 is very s'nnilar to the one proposed by Noeentini et al. [19] for multiple-Rushton turbines, both having an identical quality of firing to experimental data. Other correlations for air-water systems are compared in Table 1, like the one for the upper stages of quadruple-Rushton agitators by Linek et al. [20] or the one according to Bouaifi et al. [21 ] for different multiple impeller configurations. Results in this work are consistent with those by Saito et al. [7] stating that the parabolic blade shaped 6SRGT impeller and the Rushton impeller produce the same gas hold-up at the same power input and superficial gas velocity. Equal conclusion is found in the literature relative to the concave and Rushton turbines in single, dual and four-impeller systems [10,11], although in one of the works [10] differences were admitted that would not be detected by the precision of the measurements. This position is supported by results in a paper reporting that the convex disk turbine gives higher hold-up per unit power consumption than the concave disk turbine [12]. The situation is apparently clearer in highly viscous Newtonian media, where strong differences were found recently between the gas hold-up behavior of four disk type impellers, namely Rushton, concave, 6SRGT and parabolic bladed turbines [13]. From Equation 2, higher hold-up before flooding is expected with the impellers for which (i) the ungassed power fails less on gassing, and (ii) flooding occurs at higher gas flow rate. 3.2. Volumetric mass transfer coefficient kLa

kLa data were correlated separately for each impeller in terms of the specific power input PCC~Land the superficial gas velocity vo, according to the well known equation [22]"

466

(3) The optimized values of A, B and K are shown in Table 2. Results do not differ too much from impeller to impeller, showing partial overlapping of all the 95% confidence intervals associated with the individual correlations. Table 2 Parameters of kLa correlating Equation 3 (PcC~L in W/m3; vo in m/s) K Impeller

mean

A

B

Correlation error error error mean Mean coefficient (95% e.1.) (95% e.1.) (95% e.1.)

Rushton Perforated

0.0083 0.0032

0.0016 0.0007

0.62 0.66

0.02 0.02

0.49 0.41

0.02 0.03

0.997 0.995

Angled-90 ~

0.0054

0.0015

0.68

0.03

0.52

0.03

0.993

Concave

0.0113

0.0021

0.63

0.02

0.57

0.02

0.998

Angled-60 ~

0.0050

0.0011

0.64

0.02

0.46

0.03

0.997

Lancet

0.0080

0.0019

0.68

0.02

0.60

0.03

0.997

Overall correlation

0.0062

0.0010

0.66

0.01

0.51

0.02

0.989

The overall regression of all kLa, also presented in Table 2, is shown in Figure 5 by full straight lines, dashed lines standing for the error interval at 95% eoldidenee level. From Figure 5, the impellers appear to behave similarly in what concerns mass transfer, since differences are due to experimental and statistical errors. The Rushton and concave impellers are the only ones compared in the literature, but conclusions are contradictory [6,9,10]. Table 3 Correlations for kLa compared to experimental data Errors (%) Correlation

Reference

range

mean

kLa = 6.2xl 0"3( P c ~ L ) 0"66(VG) 0"51

This work

+41

-24

9

kLa = O.026 (P~L)~ (vo) ~

van't Met [22]

+42

-55

27

kLa = 0.015 (Pc~L) ~ (VG)~

Bakker et al. [10]

+59

-26

12

Linek et al. [20]

+66

-16

13

Nocentini et al. [19]

+85

-7

24

kLa = 8.61x10 "3[(PG+ vopg)/VL]~ 0 59

0 55

kLa = 1.5xl0"2(pc~L) " (vG)

~

467

0.2 /

0.1

/ /

//;/

/

"Trn 0.05 / ~'//, / 0.01

/ ~ Z/

Qa = 0.125 vvm

0.2

/

QG = 0.25 vvm

//'/

/ / . / /.-I-

0.1 -'-" 0.05

/ / / ~/

///~/////~// //~/// / /

0.01

Qo = 1.0 vvm

Qa = 0.5 vvm 0.25

0.5

1

2

4

PG/VL (kw/m3) o Concave Angled-90 ~ ,, Angled_60 ~

0.25

v [] +

0.5

1

2

4

PG/VL (kW/m3) Lancet Perforated Rushton

Fig. 5. kLa results (experiments under Qo = 1.0 vvm and PC~L= 0.25 kW/m3 were affected by systematic errors due to bubble/probe interference and thus were rejected). Results in this work allow the conclusion that kLa is not dependent on impeUer type, an outcome which has been suggested by van't Met [22] and was subsequently supported by several authors [7,9,14]. As shown in Table 3, where some eorrelatiom in the literature are compaired to kLa results, the generalized correlation obtained by van't Met in his survey of literature data for water [22] fits the data obtained in this work. The other correlatiom mentioned in Table 3 deal specifically with multiple-Rushton turbines [19,20] and concave blade disk impellers [10]. Despite the fact that kLa at the same Po/VL and vo does not depend on blade type, Equation 3 predicts that improved kLa is expected with those impellers (i) for which a higher proportion of ungassed power is still available as Pc/VL under gassed conditions, and (ii) for which higher values of vG can be handled without flooding.

468 4. CONCLUSIONS A comparison was done of six different blade profiles in disk turbines having the same diameter and blade projected dimemions, regarding gas hold-up and gas-liquid mass transfer performances. Gas volume fraction a, and volumetric oxygen transfer coefficient kt.a change in aceordanee with the energy dissipation rate achieved at the same rotational speed by the different radialflow impellers studied. However, at equal power and gassing rate, the different stirrers give the same e~ and kLa values, as long as experimental errors are taken into account. Better mass transfer performance and gas handling capacity may be expected after actual Rushton turbines are retrofitted with streamlined blade impellers of the same size, because lesser power fall on gassing and retarded impeller flooding may be obtained comparatively to the standard fiat bladed impeller. REFERENCES

1. Nienow, A.W., Warmoeskerken, M.M.C.G., Smith, J.M., Konno, M., Proc. 5ta Eur.Conf. Mixing (1985) 143. 2. van't Riet, K., Smith, J.M., Chew_ Eng. Sei., 30 (1975) 1093. 3. van't Riet, K., Smith, J.M., Chem. Eng. Sci., 28 (1973) 1031. 4. Bruijrg W., van't Riet, K., Smith, J.M., Chem. Eng. Res. Des., 52 (1974) 88. 5. van't Riet K., Boom, J.M., Smith, J.M., Chew_ Eng. Res. Des., 54 (1976) 124. 6. Warmoeskerken, M.M.C.G., Smith, J.M., Chem. Eng. Res. Des, 67 (1989) 193. 7. Saito, F., Nienow, A.W., Chatwin, S., Moore, I.P.T, J. Chew. Eng. Japan, 25 (1992) 281. 8. Roman, R.V., Tudose, Z.R., Gavrilescu, M., Cojocaru, M., Luca, S., Acta Biotechnol., 16 (1996) 43. 9. Smith, J., Dispersion of Gases in Liquids in Mixing of Liquids by Mechanical Agitation, Ulbrecht, J. J., Patterson, G. IC (Eds.), Gordon & Breach: New York, 1985, pp. 139-201. 10. Bakker, A., Smith, J.M., Myers, K.J., Chem. Eng., 101 (1994) 98. 11. Myers, K.J., Fasano, J.B., Bakker, A., I ChemE Symp. Series No. 136 (1994) 65. 12. Mehtras, M.B., Pandit, A.B., Joshi, J.B., I ChemE Symp. Series No. 136 (1994) 375. 13. Khare, A.S., Niranjan, K., Chem. Eng. Sei., 54 (1999) 1093. 14. Cooke, M., Middleton, J.C., Bush, J.R., Proc. 2~dInt. Conf. Bior. Fluid DynmrL (1988) 37. 15. Vasconcelos, J.M.T.,.Rodrigues, A.M., Alves, S.S., CHISA'98 (1998) Paper P1.39. 16. Vasconcelos, J.M.T., Nienow, A.W., Martin, T., Alves, S.S., McFarlane, C.M., Chem. Eng. Res. Des., Part A, 75 (1997) 467. 17. Noeentini, M., Chem. Eng. Res. Des., Part A, 68 (1990) 287. 18. Nocentini M., Pinelli, D., Magelli, F., Ind. Eng. Che~ Res., 37 (1998) 1528. 19. Noeentini M., Fajner, D., Pasquali, G., Magelli, F., Ind. Eng. Chem. Res., 32 (1993) 19. 20. Linek, V., Moucha, T., Sinkule, J., Chem. Eng. Sci., 51 (1996) 3203. 21. Bouaifi M., Roustan, M., Djebbar, R., Proc. 9t~ Eur. Conf. Mixing (1997) 137. 22. van't Riet, K., Ind. Eng. Chem. Process Des. Dev., 18 (1979) 357. 23. Vasconcelos, J.M.T., Orvalho S.C.P., Rodrigues, A.M.A.F., Alves, S.S., Ind. Eng. Chem. Res. (in press).


469

A SIMPLE METHOD FOR DETECTING INDIVIDUAL IMPELLER FLOODING OF DUAL-RUSHTON IMPELLERS A. Bomba6 and I. Zun Laboratory for Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, University ofLjubljana, Aw 6, 1000 Ljubljana, SLOVENIA An experiment was performed in a pilot size mixing vessel using single and dual Rushton impellers. The two-phase mixture was composed of air and deionized water. The structural function of the discharged two-phase flow was detected with a resistivity probe in the close vicinity of the outer edge of the impeller blade. In single-impeller stirring the local void fraction (or) was obtained by the signal processing method, which was previously calibrated. The ct values of the examined grid nodes of the discharge flow are presented for different air flow rates. Increasing the gas flow rate at constant impeller speed led to a sudden sharp decrease, approx. 80%, of the last ct value irrespective of the measured location. Such a hydrodynamic regime was recognized as flooding. Transformation of the time-domain structural function into frequency-domain enabled the identification of gas-filled cavity structures. Here, the flooding was recognized by the appearance of ragged cavities. Both methods present a loading-flooding transition (LFT) in the same hydrodynamic regime. Results were compared to those available from the literature and showed good agreement. 1. INTRODUCTION Recently some experimental studies of gas-filled cavity structures and local void fraction distribution in an aerated vessel have been presented (Lu and Ju 1987, Lu and Ju 1989, Bomba6 et al. 1997, Bomba6 and 2;un 2000). While Lu and Ju (1989) investigated gas-filled cavity configuration, flooding and pumping capacity using a constant temperature anemometer with single-impeller stirring, Bomba6 et al. (1997) and Bomba6 and 2;un (2000) mainly investigated gas-filled cavity structures and local void fraction based on resistivity probe response in an aerated vessel stirred with single and dual Rushton disk impellers, respectively. The local detection of the phases was based on the resistivity probe, which produced voltage responses. This was presented as a corresponding structural function Mpdefined as:

Mp(x,t) = ~

L

1, xis occupied by p 0, x is not occupied by p

P =

:L, G, s)

(1)

where in a two-phase flow field three states p are possible at a particular point x at any time t: the liquid phase L, gas phase G or phase interface S. Frequency analysis of the structural function Mp by discrete Fourier transformation enabled the presentation of the significant frequencies of an appearing gas phase. The Fourier coefficients Arkwere obtained from:

470 N-1

j2zik

X k = At ~ ] M p (t k)e ~r

(2)

k=0

where At denotes the time interval between successive instants ti. Among Fourier coefficient Ark that correspond to the frequency k/(NAt) only coefficients from k = 0 to k = N/2- 1 are meaningful. The criterion for gas-filled cavity structure recognition was described in detail (Bomba5 et al. 1997), as were the recognized structures: vortex-clinging structure (VC), structure with one large cavity (1L), structure with two large cavities (2L), small '3-3' structure ($33), large '3-3' structure (L33) and ragged cavities (RC). R-probe response, i.e. the corresponding structural function Mp, can be discriminated into a binary signal from which the local void fraction was calculated: l

where A Tdenotes total sampling time and Atoi gas-phase residence time. A signal processing technique developed in bubbly flow studies to discriminate the phases is described in detail in Zun et al. 1995. The simplified version with a single threshold at 6% of the signal amplitude was used as adequate phase discrimination procedure (Bomba~ 1994). The purpose of this paper is to present an experimental method based on interfacial structure characteristics, which enables the recognition of individual impeller flooding of a dual-impeller system. The method enabled simple estimation of individual impeller flooding. The results indicated the transition from a loading regime into flooding via different gas filled cavitystructures. The flooding transition of the lower turbine occurred at higher gas flow rates than that of single-impeller stirring, while the upper impeller remained (due to measuring restrictions) in a dispersing regime throughout all measured regimes. Good agreement with flooding prediction data from the literature was found for a single-impeller. 2. EXPERIMENTAL TECHNIQUE A cylindrical fiat-bottomed Perspex vessel of diameter 450 mm with four baffles mounted perpendicularly to the vessel wall was equipped with single and dual Rushton turbines. The geometry details for the single-impeller stirred vessel can be found in the work of Bomba6 et al. (1997), otherwise in Figure 1. Demineralized water and compressed air at room temperature were used in all experiments, a was measured in single-impeller stirring with a resistivity probe of tip diameter 11 ktm in the close vicinity of the outer edge of the impeller blade. The measuring grid consisted of 27 grid nodes as can be seen in the upper right corner of Figure 3. The radial distance between the two measuring lines (a, b, c) was set to 4mm while the axial distance (1,2,3,...9) was set to 3 mm. Gas-filled cavity detection was carried out in singleimpeller and dual-impeller stirring, in the last case with two probes at the same time. Tips were located at r = 75.5 mm and 22 mm upward from the lower blade edge in the area of the discharge flow, see Figure 2. The probes were powered by 5 V DC which produced a response magnitude of 2.3 V. The sampling rate was set to 500 Hz at 1 minute run for gas-filled cavity detection and 5 kHz at 3 minutes run for local void fraction measurements, respectively.

471

Figure 1. Geometrical parameters of the vessel and stirrers

Figure 2. Experimental setup

472 At the lowest impeller speed at least 600 cavities were detected for further statistical treatment. Measurements were always performed in the same manner, starting from low to high impeller speeds, with stepwise increasing of the gas flow rate at constant impeller speed. The reproducibility and directional sensibility was assumed to be equal to those evaluated in the work of Bomba~ et al. (1997).

3 RESULTS AND COMPARISON

3.1 Single impeller To determine the dependence of measuring locations on gas flow rate in loading-flooding transitions, ~ profiles of the impeller-discharge flow were measured in a standard vessel configuration with single impeller first. The dependence of cz on gas flow rate at given grid nodes is shown in Figure 3. It can clearly be seen that increasing the gas flow rate at constant impeller speed led at certain flow rates to a sudden sharp decrease of ot regardless of the measured location, ot was found to be approx. 20% of the previous value. Such a hydrodynamic regime (nF, qF) was recognized as a flooding point. LFT was also obtained comparatively by another criterion which was based on gas-filled cavity structure change. The appearance of ragged cavities was in correspondence with the sharp decrease of ot for the tested hydrodynamic regime. All further LFT were recognized on the bases of ragged cavities recognition by stepwise increasing of the gas flow rate at constant impeller speed. Such regimes were collected and depicted in a Fr-FI diagram, see Figure 4.

Figure 3: Dependence of c~ on gas flow rate at given grid nodes As can be seen, the LFT occurred via different gas filled cavity structure development. At lower values of Fr~_0.08 and Fl~_0.05 the transition appeared corresponding to structure change from VC to RC, which is in good agreement with the findings ofNienow et al. (1985). In the diagram area given by 0.08 and z/H e

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European Instructional Lectures: Volume 9, 2009; 10th EFORT Congress, Vienna, Austria (European Instructional Course Lectures)

Food Mixing: Principles and Applications

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