AN EXPERIMENTAL STUDY
ON TURBULENT MIXING
OF VISCO-ELASTIC FLUIDS
* ■: ■■■■■ 'iM ■ ■•-. m ... ■ -■ ■• = eI 1 •
:-'
■■ ; ; ! >PAUL
BARTELS
1988
AN EXPERIMENTAL STUDY
ON TURBULENT MIXING
OF VISCO-ELASTIC FLUIDS
Cover:
the interaction between an elastic polyacrylamide molecule
and two turbulent vortices
AN EXPERIMENTAL STUDY
ON TURBULENT MIXING
OF VISCO-ELASTIC FLUIDS
PROEFSCHRIFT
Ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus,
prof.dr. J.M. Dirken, in het openbaar
te verdedigen ten overstaan van een
commissie aangewezen door het College
van Dekanen op dinsdag 23 februari 1988
te 14.00 uur
door
Paul Vincent Bartels,
geboren te Rotterdam,
landbouwkundig ingenieur.
TR diss
1611
Dit proefschrift is goedgekeurd door de promotor
professor J.M. Smith D.Sc. C.Eng.
S T E L L I N G E N
behorende bij het proefschrift "An experimental study on
turbulent mixing of visco-elastlc fluids"
1 De axiale, de tangentiële en de radiale uitstroomsnelheden van een schuinbladige turblneroerder worden mede bepaald door de vlsco-elasticiteit van de gebruikte vloeistof en deze vloeistof kan daarbij ten opzichte van water in een tegengestelde
richting uitstromen. Dit proefschrift
2 De "BlaslusvergeliJking" kan worden toegepast om de
stromingsweer8tand van polyacrylamlde-oplossingen in een pijp te beschrijven, indien deze vergelijking uitgebreid wordt met een term die afhankelijk is van de elasticiteit van de
oplossing.
Dit proefschrift
3 Relaties tussen het meng- of stromingsgedrag In geroerde vaten en de zwichtspannlng van polymeeroplossingen, zoals deze
toegepast worden in de reologische modellen van Bingham, Casson of Herschel-Bulkley, zijn afhankelijk van de technische
mogelijkheden van de gebruikte reometers en hebben daarom geen werkelijke fysische betekenis.
dit proefschrift
ü Bij de interpretatie van de bepalingen van de rekviscosltelt van een vlsco-elastische oplossing met behulp van de "buisloze hevel methode" wordt ten onrechte niet vermeld dat
terugstroming van de vloeistof de meetmogellJkheden aanmerkelijk beperkt.
L. Nicodemia, B. De Cindio en L. Nicolals; Polymer Enging. and Sci. 1£ (1975) 679-683
5 Indien een oplossing een vlsco-elastisch gedrag vertoont, zal het vermogenskental voor turbulente menging met een radiaal uitstromende turbine lager zijn dan het geval is met water. Dit verschijnsel kan gebruikt worden om visco-elastische
oplossingen als zodanig te herkennen. Dit proefschrift
6 Het gecombineerde gebruik van fluorides in drinkwater. tabletten en tandpasta leidt bij de tandontwlkkeling van
kinderen in Nederland tot veel meer glazuurafwiJklngen dan tot nu toe verondersteld wordt. Een matiging In de toediening van fluoride aan zwangere vrouwen en kleine kinderen is daarom gewenst.
W.M.D. Niewland; Klinisch onderzoek naar het voorkomen van glazuurafwijkingen bij kinderen, met name fluorose, ACTA 1987
7 Het gedeelte van de genoten rente dat dient als dekking van inflatie en risico vormt geen reëel Inkomen en moet dan ook niet als zodanig belast worden.
8 Gezien de hiërarchische relatie tussen een hond en zijn baas dient de eigenaar van de hond strafrechtelijk vervolgd te worden voor mishandeling van derden door de hond, als ware de hond een handwapen.
9 Ook als het water in de R U n in de toekomst schoner wordt, blijft het onwaarschijnlijk dat de zalm zal terugkeren, aangezien de geschikte paalplaatsen In de afgelopen eeuw verdwenen zijn.
10 Voor de uitzendingen van de politieke partijen zouden de code-regels van de Reclameraad van toepassing dienen te zijn.
P.V. Bartels 23 februari 1988
"Ars longa, vita brevis"
A fluid, that's macromolecular
It's really quite weird - in particular The abnormal stresses
The fluid possesses
Give rise to effects quite spectacular
R.B. Bird
ACKNOWLEDGEMENTS
The enthusiasm and ideas of dr. ir. L.P.B.M. Janssen, now professor at the University of Groningen, stimulated this research at its start. Following his departure from the
Laboratory for Physical Technology my promotor prof. J. M. Smith D. So. has supervised the project. Most of the research described in this publication has been carried out in that latter period.
Thanks are due to the many students who contributed with their experimental work and interpretations of the results. Especially students, who have carried out research during their fourth or fifth year of the study, have made a large contribution to this thesis. They are Gerard van
Lookeren (chapters 3 and H), Arjen Markus (chapters 3 and 6)
Ruud van Beelen (chapters Ü and 5 ) . Tjia Liem (chapters 3 and 5 ) , Henk Kalsbeek (chapters 6 and 7) and Ruud Hogervorst
(chapter 8 ) .
I am indebted to dr. ir. J. Kunen for his collaboration in measuring the velocity profiles for polymer solutions in a
Ml mm diameter tube at Delft Hydraulics. I would like to thank
dr. ir. J. Blom (University of Twente, Enschede) and
dipl. lng. Z. Kolar (Interuniversity Nuclear Institute, Delft)
for the opportunities provided respectively to analyse the
dynamic viscosity of the experimental fluids and the diffusion in polymer solutions.
Further I would like to express my gratitude to all employees of the laboratory for Physical Technology for their help in the realization of the experimental equipment and for their assistance, while for the preparation of the graphics and photographs in this publication I like to thank
Ab Schinkel.
Finally, I extend my appreciation to TJia Liem and Ivo Bouwmans for the discussions and suggestions made, after reading the concept of this thesis, and to Els Kolff for the cover design.
Gerald Adang Ruud van Beelen Jo Bothmer Max Colon
Gerben van der Graaf Johan van Haastrecht Frank den Hartoi Paul Hendrikx John Hermans Ruud Hogervorst Henk Kalsbeek Arne Kool
Otto van der Lende Tjla Llem
Gerard van Lookeren Arjen Markus
Soerlw Peasar
Ineke van der Reijden - Stolk Henk Schaap
Peter Schippers Heroe Soedjak Alle Tigchelhoff Cees Versluijs
Quint van Voorst Vader
have made a contribution to the research project
"Mixing of visco-elastic fluids" as a part of their studies.
LIST OF CONTENTS page SAMENVATTING 13 SUMMARY 16 CHAPTER 1 1 GENERAL INTRODUCTION 19 1.1 Scope 19 1.2 Objective 20 1.3 Thesis structure 21 1.4 References 22 CHAPTER 2
2 THE VISCO-ELASTIC FLUIDS 23
2.1 Introduction 23
2.2 Chemistry of the experimental fluids 23
2.3 Degradation 25 2.4 Defining the viscosity of the solutions 26
2.4.1 The viscosity of
polyacrylamide solutions 26 2.4.2 Estimation of elastic properties 28
2.4.3 Carboxymethylcellulose 34 2.5 Molecular diffusion 34 2.5-1 Introduction 34 2.5-2 Diffusion experiments 35 2.5-3 Conclusion 38 2.6 Conclusions 38 2.7 Symbols 39 2.8 References 40 CHAPTER 3
3 HYDRODYNAMICAL DESCRIPTION OF VISCO-ELASTIC
FLUIDS IN TURBULENT PIPE LINE FLOW 43
3.1 Introduction 43 3.2 Drag reduction in literature 44
3.3 Literature about velocity profiles
of polymer solutions 49
3.4 Experimental 51 3.4.1 The tube and the fluids 51
3.4.2 The laser Doppler anemometer 52
3.4.3 The measuring volume 55 3.5 Drag reduction and time scales 57
3.6 Velocity profiles 67 3.6.1 Results with the 44 mm tube 67
3
3
3
78
9
CHAPTER <l Conclusions Symbols References75
76 78Ü HOMOGENIZATION IN TURBULENT PIPE LINE FLOW 83
Ü.1 Introduction 83 Ü.2 Modelling turbulent mixing 83
Ü.3 Measuring methods for the mixing experiments 85
Ü.Ü The acid - ba3e reaction 86 1.5 The iodine - thlosulphate reaction 87
1.6 Definition of homogenlzation 90
1.7 Experimental 93 1.7.1 The fluids 93 4.7-2 The set-up 93 1.8 Conversion along the tube 91
1.9 Comparison of the neutralization reaction
with the redox reaction 97 1.10 Influence on the mixing length
by polymer additives 98 1.11 Energy dissipation 102 1.12 The mixing energy required 111
a.13 Conclusions 112
1.11 Symbols 113
1.15 References 115
CHAPTER 5
5 MOTIONLESS MIXERS IN TURBULENT PIPE LINE FLOW 119
5.1 Introduction 119 5.2 Commercially available motionless mixers used 119
5.3 The energy requirement of the elements 122 5.1 The turbulence downstream of a mixing element 126
5.5 Homogenlzation by the mixing elements 132 5.5.1 An example of the effect of mixing
elements on the homogenlzation 132 5.5.2 The different types of mixing elements 133
5.5.3 The Influence of polymers added 135
5.6 Conclusions 136 5.7 Symbols 138 5.8 References 110
CHAPTER 6
6 HYDRODYNAMICS IN A STIRRED VESSEL 1Ü1
6.1 Introduction 1Ü1 6.2 Influence of the visco-elasticlty
on the large scale flow 1Ü2 6.3 The experimental set-up 1Ü5 6.3-1 The tank reactor 1Ü5 6.3-2 The laser Doppler anemometer 1Ü6
6.3.3 The viscosity used for the
velocity measurements 1Ü8
6.Ü The radial discharge of the impeller 1Ü9
6.Ü.1 The tangential Jet 1Ü9 6.Ü.2 Measurement of the tangential Jet 153
6.5 The pumping capacity 159 6.6 The trailing vortex 161 6.6.1 Literature 16Ü 6.6.2 Experiments 168 6.7 Turbulence in the vicinity of the impeller 170
6.8 Turbulence in the bulk of the vessel 17Ü
6.9 Conclusions 17" 6.10 Symbols 175 6.11 References 177 6.A Appendix: velocity distribution in the vessel
for water 181 6.B Appendix: velocity distribution in the vessel
for a solution of 1000 ppm PAAm 182
CHAPTER 7
7 ELASTICITY AND POWER CONSUMPTION
IN STIRRED VESSELS 183
7.1 Introduction 183 7.2 Experimental 183 7.2.1 The tank reactor 183
7.2.2 Power consumption l&U
7.2.3 Visualization 18Ü 7.2.Ü Experimental fluids 184
7.3 Influence of elasticity on the power number
for Rushton turbines 188 7.Ü Power consumption and the tangential jet 192
7.5 Visco-elastic fluids and inclined
blade turbines 19Ü 7.5-1 Inclined blade impellers 19"
7.5.2 Power number of inclined blade impellers
with Newtonian fluids 195 7.5-3 Effect of the fluid elasticity on the
flow field for axial Impellers 197 7.5.Ü Effect of the fluid elasticity on the
power consumption 199
7.6 Conclusions 202 7.7 Symbols 202 7.8 References 20Ü
CHAPTER 8
8 MIXING TIMES IN STIRRED TANK REACTORS 207
8.1 Introduction 207 8.2 Literature 207 8.3 The experimental fluids 210
8.Ü The measuring techniques used 211
8.U.1 Introduction 211 8.Ü.2 The light probe in the fluid 212
8.Ü.3 The light probe, measuring
over the full height 213
8.U.Ü The amylose indicator 215 8.Ü.5 The Injection system 216 8.Ü.6 Measuring procedure 217 8.5 Effect of the injection position 217
8.6 The homogenizatlon time for a Rushton turbine 219
8.7 Homogenizatlon and power consumption 225
8.8 Radially and axially discharging impellers 226
8.9 Conclusions 230 8.10 Symbols 231
8.11 References 232
CHAPTER 9
9 FINAL CONCLUSIONS 23Ü 9.1 Pipe line reactors and stirred tank reactors 235
9.2 Scale-up 237
EEN EXPERIMENTEEL ONDERZOEK NAAR
DE TURBULENTE MENGING VAN VISCO-ELASTISCHE VLOEISTOFFEN
SAMENVATTING
De hoeveelheid literatuur over het turbulent mengen van Newtonse vloeistoffen in geroerde vaten en in buisstromlngen is zeer groot. Veel minder aandacht is er besteed aan de betekenis van de visco-elasticlteit van vloeistoffen, zoals oplossingen van polymeren in water, op de turbulente menging, alhoewel vloeistoffen met dit soort reologisch gedrag vaak toegepast worden in de industrie. Het verschijnsel van weerstandsvermindering met visco-elastische vloeistoffen in turbulente pijpstroming is echter veelvuldig beschreven. Deze weerstandsvermindering wordt veroorzaakt door een
onderdrukking van de kleinste wervels, die de energie disslperen. Turbulente pijpstroming vormt, vanwege de vele literatuur, een goede basis, voor onderzoek naar de invloed die polymeertoevoegingen uitoefenen op de menging vanwege de
verminderde energiedissipatie.
Doel van dit onderzoek is het leggen van een verband tussen het visco-elastisch gedrag van een vloeistof en het stromingsgedrag, de energiedissipatie en de menging in turbulente buisstromlngen en geroerde vaten.
Het proefschrift begint met een karakterisering van de reologische eigenschappen van de waterige polyacrylamlde-oplossingen die als modelvloeistoffen gebruikt zijn.
Kenmerkend voor deze vloeistoffen is een karakteristieke tijd, die bepaald wordt met behulp van het Bird-Carreau
viscoslteitsmodel. De viscositeit is niet alleen belangrijk vanwege de visceuze krachten in de vloeistof, maar zal ook van invloed zijn op alle stromings-schalen via de elastische
krachten, die gerelateerd zijn aan een elasticlteitsgetal.
Dit kental wordt bepaald door de karakteristieke
vloeistoftijd, de viscositeit en de afmeting van de menger. Er kan een verdeling gemaakt worden tussen de
experimenten in pijpstromlngen. beschreven in hoofdstuk 3. " en 5. en in geroerde vaten, die verwerkt zijn in hoofdstuk 6, 7 en 8. Voor de beide processen zijn onderwerpen betreffende de hydrodynamica, de energiedlssipatie en de menging
onderzocht. Voor de bepaling van de vloeistofsnelheid is een "laser Doppler anemometer" gebruikt, terwijl de menging gemeten is met behulp van de ontkleurings-reactIe tussen natriumthiosulfaat en jood met amylose als indicator.
Voor beide mengsystemen is een duidelijke invloed van de visco-elasticitelt op de hydrodynamica gevonden voor de
plaatsen met een hoge afschuifsnelheid, zoals in de grenslaag bij pljpstroming en In de rolwervels achter de bladen van een Rushtonturbine. Dit resulteert in een aanzienlijke
vermindering van de energiedlssipatIe bij een zeer geringe toename van het elastlciteitsgetal. Bij de pljpstroming bestaat er tevens een drempelwaarde voor het Reynoldsgetal waarboven weerstandsvermindering optreedt voor een bepaalde polymeeroplossing. De frictlefaktor blijkt tevens een functie te zijn van het elastlciteitsgetal. Door de geringere
energieconsumptie verloopt het mengproces voor
polymeeroplossingen ook trager. De energie-dissipatle en
homogenisatie kunnen gerelateerd worden aan de karakteristieke vloeistof tljd en aan het elastlciteitsgetal.
In de buis is er een gedwongen grootschalige stroming, terwijl in een geroerd vat grootschalige circulatiestromingen bestaan, waarvan richting en groette kunnen veranderen. De elasticiteit vormt de bron van extra krachten, die andere stromingspatronen geven dan met water en stilstaande zones kunnen veroorzaken.
Voor roerders met een radiale of axiale uitstroming zijn de stromingspatronen bestudeerd. Bij schulnbladige roerders blijkt er een overheersende radiale uitstroming te kunnen ontstaan ten gevolge van de elastische krachten.
Met water wordt voor de schuinbladige roerders een lager vermogenskental dan voor een Rushtonturblne gevonden. Omdat de uitstroming meer radiaal gericht is, zal het vermogenskental voor deze roerders met visco-elastische vloeistoffen dan ook stijgen.
De radiale uitstrcming van een Rushtonturblne is met behulp van een model voor een tangentiële Jet
gekwantificeerd, en de samenhang met het vermogenskental is aangetoond.
In turbulente bulsstromingen en in geroerde vaten zal het mengrendement, gedefinieerd als de energie die nodig is om een zekere mate van menging te krijgen, niet duidelijk
veranderen voor een zelfde (tip)snelheid en voor lage waarden van het elasticiteitsgetal, hoewel er een minimum gevonden wordt voor zeer verdunde polymeeroplossingen. Voor de
geconcentreerdere polyacrylamlde-oplosslngen zal de benodigde mengenergie aanmerkelijk stijgen.
In de buis gemonteerde meng-elementen, zoals de Kenics of Sulzer SMX en SMV statische mengers, zijn eveneens
bestudeerd wat betreft het meng- en stromingsgedrag in en achter de elementen. De drukval over de mengers wordt slechts in geringe mate beïnvloed door de elasticiteit van
polymeeroplossingen, In tegenstelling tot de mengwerking, die afneemt bij een toenemende karakteristieke tijd van de
oplossingen. De opgewekte turbulentie achter een element blijkt dan nuttig te zijn om het energie-rendement bij het mengen met vlsco-elastische vloeistoffen te verbeteren. Indien de elementen op enige afstand van elkaar worden
gepositioneerd.
-SUMMARY
Extensive literature is available concerning the
turbulent mixing of Newtonian liquids in stirred tank reactors as well as in pipe line flow. Less attention has been paid to the Influence exerted by the visco-elastlc fluid behaviour, of aqueous polymer solutions for instance, on turbulent mixing,
though fluids with this rheologlcal behaviour are of frequent occurence in industry. The drag reduction phenomenon of
elastic liquids in turbulent pipe line flow is however well described. This drag reduction is caused by a damping of the smallest energy dissipating eddies. Because It has been
studied extensively, turbulent pipe line flow presents a good point of departure for research into the influence of polymer additions on the homogenization due to the diminished energy dissipation.
The objective of this thesis is to relate the
visco-elastic behaviour of fluids to the hydrodynamics, energy dissipation and homogenization for turbulent pipe line flow and stirred tank reactors.
This study starts with the characterization of the Theological behaviour of aqueous polyacrylamlde solutions. which were used as model fluids. An essential feature of these
fluids Is a characteristic fluid time, determined by using the Bird-Carreau model to describe the viscosity. Apart of the viscous forces, the viscosity will have an Influence on all
flow scales as a result of the elastic forces, which can be related to the elasticity number. This number is determined by the characteristic fluid time, the viscosity and the dimension of the mixer.
A division can be made between the experiments in tube flow, described in chapters 3, ü and 5, and in stirred tank reactors, dealt with in chapters 6, 7 and 8. For both the processes topics of hydrodynamics, energy dissipation and homogenization have been researched. A laser Doppler anemometer has been used for the determination of the
velocity, while the homogenization has been measured by means of the decoloration reaction of thiosulphate with iodine using
amylose as an indicator.
For both mixing systems a significant influence of visco-elasticity on the hydrodynamics is found in locations with a high shear rate, such as occur in the boundary layer of pipe line flow and in the trailing vortices of the blade of a Rushton turbine. This results in an dramatic drop of the overall energy dissipation for very low values of elasticity number. In tube flow there is a threshold for the drag
reduction phenomenon. The friction factor appears to be also a function of the elasticity number. The homogenization process of polymer solutions is also slower due to the lower energy dissipation, causing less dispersion. The energy dissipation and homogenization can be related to a characteristic fluid time and the elasticity number.
In the tube an imposed large scale flow occurs as a matter of course, while in a stirred tank large scale circulation flows exist, which are subject to change. The elasticity generates additional forces, giving different
patterns than in the case of water and giving rise to stagnant zones.
For impellers with an axial or a radial discharge the patterns have been studied. An inclined blade impeller may show a dominant radial discharge due to visco-elastic forces.
For water, a power number lower than that for a Rushton turbine has been established when inclined blade impellers are being used. Because the discharge is more radially directed, power consumption for visco-elastic fluids increases for this type of Impeller.
The radial outflow of an Rushton turbine has been quantified using a tangential jet model, and it has been related to the power consumption.
In turbulent pipe line flow and in stirred tank reactors the mixing efficiency, defined as the energy needed to obtain
a Bpeciflc rate of homogenlzation, will not change
significantly at a given (tip) velocity for low values of the elasticity number, although a minimum has been found; however, it increases considerably for concentrated solutions of
polyacrylamide.
The effect of inserted mixing elements, such as the Kenics or Sulzer SMX and SMV static mixers, on turbulent pipe flow has also been studied. The pressure drop of the mixers is almost independent of the concentration of polyacrylamlde, but mixing will be less when the characteristic time of the solutions Increases. In that case the turbulence downstream of an element appears to be useful for the Improvement of the energy efficiency for homogenlzatlon in visco-elastic fluids, if the elements are situated at some distance removed from one another.
C H A P T E R 1 GENERAL INTRODUCTION 1 . 1 SCOPE T h e m i x i n g o f f l u i d s i s c a r r i e d o u t in a l l p h a s e s o f t h e p r o c e s s i n d u s t r i e s . M o s t p r o c e s s f l u i d s p o s s e s s q u a l i t i e s o f c o m p l e x r h e o l o g l c a l b e h a v i o u r , e s p e c i a l l y t h o s e t h a t a r e
found in t h e f o o d - and b i o i n d u s t r i e s , p o l y m e r p r o c e s s i n g and c o a t i n g o p e r a t i o n s . S o m e e x a m p l e s of p r o d u c t s and p r o c e s s e s a r e a s f o l l o w s : - f e r m e n t a t i o n : * x a n t h a n g u m s and o t h e r e x t r a c e l l u l a i r p o l y s a c c h a r i d e s , used f o r e x a m p l e for e n h a n c e d o i l - r e c o v e r y * f e r m e n t a t i o n b r o t h s of m o u l d s - p a i n t i n d u s t r y : e m u l s i o n s - m a n u f a c t u r e of g l u e s and a d h e s i v e s - e m u l s i o n p o l y m e r i z a t i o n s ( l a t e x ) - s e p a r a t i o n p r o c e s s e s : f l o c e u l a t i o n T h e s e f l u i d s s h o w c o n s i d e r a b l e v a r i a t i o n in t h e i r m a n n e r o f d e v i a t i o n from i s o - v l s c o u s N e w t o n i a n b e h a v i o u r . A n u m b e r o f t h e m d i s p l a y e l a s t i c p r o p e r t i e s . T h i s time d e p e n d e n t r h e o l o g l c a l b e h a v i o u r , w h i c h i s c h a r a c t e r i z e d by a t e m p o r a r y s t o r a g e of e n e r g y , m a y be o b s e r v e d in p o l y m e r s , p o l y m e r s o l u t i o n s , e m u l s i o n s , d i s p e r s i o n s a n d m i c r o b i a l s u s p e n s i o n s ( O o l m a n et a l . \ 1 9 8 6 \ ) , but a l s o in g l y c e r i n e and o i l s s u c h a s h i g h m o l e c u l a r s i l i c o n e o i l , s o y b e a n o i l o r c o m m o n o i l s ( R i c c i u s and A r n e y \ 1 9 8 6 \ ) . V e r y l i t t l e i s k n o w n c o n c e r n i n g t h e e f f e c t s of f l u i d e l a s t i c i t y o n m i x i n g p e r f o r m a n c e i n e i t h e r l a m i n a r o r t u r b u l e n t f l o w r e g i m e s . For s t i r r e d v e s s e l s m o s t of t h e r e l e v a n t r e s e a r c h h a s b e e n c a r r i e d o u t In c o n n e c t i o n w i t h l a m i n a r m i x i n g , w h i l e t u r b u l e n t m i x i n g has b e e n r e s e a r c h e d t o 19
only a slight extent. In most cases elasticity is not perceived as a fluid property which can give rise to engineering problems in mixing. In a stirred reactor fluid elasticity may have a significant effect on the pumping
capacity of the impeller, the turbulent energy dissipation and its distribution throughout the vessel. As a result of this, there is a direct Influence on the circulation capacity, mass transfer and mixing performance. The elastic forces can also cause different flow patterns and stagnant zones.
A tank reactor is not very suitable for mixing concentrated polymer solution, because of the changes that occur in the flow-field and in the turbulence. Pipe line mixing can be better for homogenizatIon- and dispersion processes, as the external pumping capacity gives more
adequate control of the flow field. However, it has been known for forty years that even fluids which are practically non-elastic can cause a dramatic decrease in the pressure drop in
turbulent pipe line flow. This is called the Toms effect (Toms \ 1 9 " 9 \ ) . It may be assumed that this decrease in energy
dissipation will also result in less efficient mixing. More efficient mixing, at the expense of an additional pressure drop, can be affected with motionless mixing elements. For this purpose many different types of these elements which give different types of small-scale mixing flows are commercially available.
1.2 OBJECTIVE
In turbulent mixing operations the fluid viscosity is of minor importance. However, elasticity can cause significant changes in the flow when the related characteristic fluid time is of the same order of magnitude as that of the flows in the mixing process, whether on the macro or the micro scale.
The objective of the present work has been to observe the influences of the elasticity on the hydrodynamics, the power consumption and the turbulent diffusion of the
homogenization process, as compared to fluids with Newtonian properties. To quantify the elastic behaviour, a useful approach is necessarily based on a characteristic time and viscosity of the fluid. Water, being a Newtonian fluid, may be considered to have a very short characteristic time
-1Ü (water: 10 s ) .
For this purpose well known mixing processes have been selected: the tank reactor with a Rushton turbine and the pipe line reactor. Extensive literature is available for both types of mixers. Other mixer designs have also been studied in order to illustrate the special effects caused by elastic forces.
1.3 THESIS STRUCTURE
Different chapters will cover distinct aspects of the interaction between the mixing process and the rheology of the fluid. Following an introducing chapter two main sections may be distinguished: three chapters deal with the hydrodynamics, energy dissipation and mixing in tube flow, and the same topics will be discussed in connection with the stirred tank reactor in the three subsequent chapters .
The properties of the model fluids used, such as Theological behaviour, will be discussed in chapter two. In the third chapter, the hydrodynamics of aqueous polymer
solutions in pipe line flow will be discussed. A great deal of literature is available concerning the pressure drop and
related flow field for aqueous polymer solutions. The effects of elasticity are remarkable. These effects will be used to characterize the dilute polymer solutions. In subsequent chapters, lesser known areas will be researched. The homogenization of elastic fluids will be discussed in chapter Ü. In chapter 5. applications of industrial mixing devices to the polymer solutions will be shown. In the three subsequent chapters the influence of visco-elasticity on the hydrodynamics in stirred vessels, especially in the vicinity of the impeller, the related power consumption and
homogenizing will be discussed.
1.Ü REFERENCES
Oolman. T., E. Walitza and H. Chmiel 1986
Dynamic rheoloiical behavlor of mlcrobial suspensions
Food engineering and process applications; Vol. 1: Transport phenomena. M. Le Maguer and P. Jelen (edt. )
Elsevier Appli. Scl. Publ., London. 81-90 Riccius, D.D.J. and M. Arney 1986
Shear-wave speeds and elastic moduli for different liquids Part 2 Experiments
J. Fluid. Mech.. 171. 309-338 Toms, B.A. 19^9
Some Observations on the Flow of Linear Solutions through Straight tubes at Large Reynolds Numbers
Proc. first Int. Rheol. Congr. Holland (1918) Pt. II. 135 - 1«1
C H A P T E R 2
THE VISCO-ELASTIC FLUIDS
2.1 INTRODUCTION
The influence of visco-elasticity on the mixing process is difficult to measure. There are major problems in
controlling the behaviour of the polymer solutions during the time of the experiments, and also technical limitations in measuring the complex viscosity. This chapter will discuss the preparation of the experimental solutions and the
characterization of the rheological behaviour of the fluids used.
2.2 CHEMISTRY OF THE EXPERIMENTAL FLUIDS
In the experiments aqueous ionic solutions of polyacrylamide (PAAm) have been used as a typical
vlsco-elastic fluid. For the measurements of turbulent mixing concentrations up to 2000 weight ppm PAAm are used. These concentrations are used in Industry for separation processes such as, for example, floeculation and show the same
rheological behaviour as aome types of fermentation broth (Oolman et al. \ 1 9 8 6 \ ) . Relative to other polymers,
polyacrylamide (PAAm) has the best resistance against
degradation caused by light and shear or microbiological- and chemical reaction. This resistance, in combination with a relatively high elastic effect, has been the reason for
selecting this particular polymer. The polyacrylamide used in the experiments is a technical product of Dow Chemical
Company; Separan AP-30. Extensive literature is available concerning the behaviour of aqueous solutions of
-CH,-CH-C=0
NH,
POLYMER-CH, —
CH-C = 0
I
0' Na-CO-POLYMER Figure 2.iThe chemical structure of polyacrylamlde with co-polymer.
polyacrylamide, such as Sylvester and Tyler \1977\. Kuilcke et al. \1982\, Maehtle \1982\. and Wagner \198ü\. Separan
AP-30 is a partially hydrolyzed linear polymer with an anlonlc character and with a formula shown in figure 2.1. According to the manufacturer, the molecular weight is approximately 0.26 10 kg/mol. with a broad molecular weight distribution.
Because it is an electrolyte, the viscosity and elasticity are sensitive to counter ionic concentrations. In the experiments initial high concentrations of salt (KI) are frequently used. An example of a typical viscosity- concentration relationship
for these fluids is shown in figure 2.2. In this figure the relative viscosity ti has been used. This viscosity is the
r
ratio of the solution viscosity u to the solvent
app
viscosity u . The relative viscosity is measured, when
Separan A P - 3 0 (anionic character) 0.5% solution J U U U mPa s 1000 500 100 50 in "app
- 1 /
If. 1 ' /^ - - - '■-_
. pH 1 , 1 , 5"C 25°C 4 0 - C 200 4 0 0 . 1 0 -6k g / k g PAAm 10Fig. 2.2 The relation between the zero shear viscosity, denoted as the relative viscosity u , and the
concentration polyacrylamide (solutions also used in figure 3-10).
Fig. 2.3 Influence of pH on the viscosity of a polymer solution (Dow Chemical).
Ubbelhode viscosimeters are used. The viscosity also depends on the acidity (fisure 2.3)- A pH 8.0 gives a relatively high viscosity. At approximate pH 8.0 the dependence of viscosity on pH change is small.
2.3 DEGRADATION
A set of experimental measurements for one solution takes about a week. Because the polymer solutions are subject to some degree of degradation, precautions have been taken.
Separan AP-30 has been used because of its relatively low molecular weight for a drag reducing polymer. In the first moments of use. it is primarily the high molecular weight
constituents which are degraded. After about two days of shear some stabilization occurs. The final degree of degradation depends on the maximum stress present in the flow during the experiments. For example, a 1000 ppm solution retained
stability for one month in a stirred vessel, in which the impeller had a tip velocity of 1 m/s. Increasing this velocity to 1.5 m/s led to additional degradation.
In technical grade PAAm the residual traces of the initiator give an ageing effect, especially at lower polymer concentrations. These residues are sequestered by addition of 2-propanol. Used in a 2% solution this inhibits ageing nearly
entirely (Haase \ 1 9 7 2 \ ) , while the viscosity does not decrease significantly. For reproducibility the solutions for a series of measurements are dissolved from a master solution of
2 - 2.5X PAAm. This Initial solution is diluted with tap water for most experiments. It remains very stable after a retention time of 3 days, even though of diminished elasticity
potential, as a result of the shear history, effect of mixing, and also due to the presence of chemicals, such as the salt content of the tap water. The stability has been determined by the measurement of the viscosity and the power consumption. In figure 2.Ü the decline of the drag reduction in turbulent pipe flow is shown over several days of experiments. The drag
reducing effect is very sensitive to polymer degradation,
w h i l e t h e m o l e c u l a r w e i g h t f r a c t i o n g i v e s t h e g r e a t e s t e f f e c t a n d w i l l d e g r a d e f i r s t . 0 1 O 0 5 O 0 1 A f 0 0 0 1
>Tty-F i g . 2 . a T 1 1 1 1 t I I | 1 1 1 1—1—r—1—-Polyacrylamide solutions 25ppm 5 d a y s old 7 d a y s old 9 d a y s old 3 0 0 p p m 4 d a y s old 6 d a y s old 8 d a y s old andtl 0
,
Virk _ _i_L_X_L 5.10 J X T 5 K 5 'IO-Effect of the stress history on degradation of PAAm solutions, shown by an decrease of drag reduction. Prandtl: f- 0 - 5 = a.o log (Re f ° '5; - 0. Ü
virk f ~ ° "5 = 19 log (Re f0 , 5) - 32.5
2.Ü DEFINING THE VISCOSITY OF THE EXPERIMENTAL SOLUTIONS
2.Ü.1 The viscosity of polyacryamide solutions
Viscosity determinations were conducted on a Weissenberg Rheogonlometer R16, while for the low concentrations of polyacrylamide a Contraves Low Shear
Rheometer, owned by the Rheology Group (Department of Applied Physics, University of Twente) is used; in addition, a
Haake 500 and Ubbelhode viscosimeters were used. All solutions show a shear thinning behaviour, in which the shear stress zu
can be described by means of a power law model for the moderate shear rates j (figure 2.5). in scalar notation according to:
1 2 5 10 20 50 100 s-1
F i g . 2.5 T h e v i s c o s i t y p. and the first n o r m a l s t r e s s
d i f f e r e n c e zn - vn. p l o t t e d as p o w e r l a w f u n c t i o n s o f t h e s h e a r s t r e s s ( 0 . 9 X P A A m s o l u t i o n ) . ( 2 . 1 ) ( 2 . 2 ) * - k i " "1 app * W h e r e k Is the c o n s i s t e n c y and n t h e i n d e x , w i t h a v a l u e b e t w e e n 0 a n d 1 for s h e a r t h i n n i n g f l u i d s . At v e r y l o w s h e a r r a t e s r e a l f l u i d s tend t o h a v e a r e l a t i v e l y c o n s t a n t v i s c o s i t y . A z e r o s h e a r v i s c o s i t y p. can o be d e f i n e d as that w h i c h i s a p p r o a c h e d a s y n e a r s z e r o . At v e r y h i g h s h e a r r a t e s t h e d i l u t e s o l u t i o n s w i l l a p p r o x i m a t e t h e s o l v e n t v i s c o s i t y p. ( f i g u r e 2 . 6 ) . The P A A m s o l u t i o n s d o s n o t p o s s e s s a n y y i e l d s t r e s s a s d e t e r m i n e d by Van V l i e t \ 1 9 8 2 \ w i t h a p p a r a t u s c a p a b l e of m e a s u r i n g v e r y l o w s t r e s s e s in f l u i d s (Van V l i e t and H o o i j d o n k \ 1 9 8 U \ ) . T h e e x i s t e n c e of t h e y i e l d s t r e s s for p o l y m e r s o l u t i o n s i s d e b a t a b l e , as h a s b e e n s h o w n by B a r n e s a n d W a l t e r s \ 1 9 8 5 \- It is to b e e x p e c t e d that the u s e o f y i e l d s t r e s s e s in r h e o l o g i c a l m o d e l s w i l l d e c r e a s e w i t h t h e d e v e l o p m e n t o f m o r e s o p h i s t i c a t e d r h e o m e t e r s , and w i t h t h e a c c e p t a n c e o f t h e v i s c o - e l a s t i c b e h a v i o u r o f f l u i d s . 27
2.Ü.2 Estimation of elastic properties
Reliable techniques for measuring elastic phenomena in fluids are available to only a limited extent. For the
concentrated solutions of polyacrylamide the first r.ormai stress difference can be measured with the Weissenberg
Rheogoniometer and can be described, for moderate shear rates, according to a power law model (figure 2 . 5 ) :
(2.3) 'll ' ^22 = A ^
Using the Maxwell model
(2.Ü)
z * A = u if
d t
an estimation of the relaxation time A, a characteristic elastic property, can be given for lower shear rates, using the approach of White and Metzner \1965• and Cross \19V3\ by:
(2. 5)
Where G represents an elastic shear modulus.
These equations hold valid for low shear rates. Sometimes a factor 2 instead of 3 is used in the denominator
(Lodge \ 1 9 6 ü \ ) . This relation is useful in connection with measurements with the Weissenberg Rheogoniometer.
A different approach is provided ty quantifying the
elastic behaviour of the pseudo plastic polymer solutions from the data of the rheogram by a time constant (Cross \1973\)-A model has been presented by Bird and Carreau (Bird et al. \1977\, Carreau et. al. \ 1 9 6 8 \ ) . which uses a characteristic
time, applicable Instead of the relaxation time A, as an additional parameter to describe the apparent
viscosity JJ , including elasticity (figure 2 . 6 ) : app r! 2 G * * 1 1 3 " * 2 2 * 1 2 * 28
Fig. 2.6 The apparent viscosity u of a shear thinning solution as a function of the shear rate. The parameters of the Bird Carreau model are shown in the figure.
(2.6)
! i £ ^ L = ( i * (
V
) V
( n
-
1 1 / 2 1
In this equation t is the characteristic time of the fluid, ju0 the viscosity at zero shear rate and u the viscosity at
oo infinite shear rate.
The parameters for the solutions, according to
equation 2.6, have been estimated from data obtained with the Contraves Low Shear Rheometer using the method of Box \1965\. The use of the model is limited because measurements have only been possible for up to moderate shear rates. For n the
oo solvent viscosity has been used as a first aproximatlon.
The first time constant t can be estimated by the interception of the line for constant viscosity at low shear rates and the line for the power law relation, as shown In figure 2.6. The slope of the power law line equals (n-1).
A third m e t h o d i n v o l v e s the t h e o r e t i c a l b e a d - r o d - s p r l n g m o d e l w i t h i s o t r o p i c B r o w n i a n and h y d r o d y n a m l c f o r c e s for the p o l y m e r s in s o l v e n t s by R o u s e \ 1 9 5 3 \ and Z l m m \ 1 9 5 6 \ . M o s t o f t h e a p p l i c a t e d c o n c e n t r a t i o n s o f p o l y m e r m a y be c o n s i d e r e d to g i v e d i l u t e s o l u t i o n s a c c o r d i n g to R o u s e ' s t h e o r y . T h i s t h e o r y i s l i m i t e d to the r a n g e ( 2 . 8 ) 1 < c In) < 20 w h e r e c r e p r e s e n t s the w e i g h t c o n c e n t r a t i o n . T h e i n t r i n s i c v i s c o s i t y [y] Is d e f i n e d a s ( 2 . 9 ) ^ " l i mc to 0 — c
and the s p e c i f i c v i s c o s i t y u is d e n o t e d t>y
s p
( 2 . 1 0 )
U8 P =
A c c o r d i n g to R o u s e ' s t h e o r y the r e l a x a t i o n time for t h e p m o d e of v i b r a t i o n of the p o l y m e r s Is g i v e n by
6 [«] „
sM
( 2-
1 1 ) P ir c R T T h e g r e a t e s t e f f e c t i v e r e l a x a t i o n t i m e t (p = i ) o f t h e r e l a x a t i o n s p e c t r u m f o r s u c h a s o l u t i o n a s d e t e r m i n e d by R o u s e , can be e s t i m a t e d by ( B l o m et a l . \ 1 9 8 6 \ . H e r s h e y and Z a k i n \ 1 9 6 7 \ ) i A ( \ M ( 2 . 1 2 ) 6 (M - M „ ) M IT c R T 301000 500 mPas 100 50 10 5 -: — —i i — r - r ' i T T T i 1 i—i 1 i i M I 1 i t [ t r i l t : M - * O — w w w^ o ■PP * v 1 1 1 1 11 1 il 1 1 1 11 1 l i l 1 1 . 1 11 I I I Polyacrylamide solutions ( Oow Chemical, Separan AP-30 ) T = 25°C o 1000 ppm with salt Ö 2 0 0 0 ppm d i s t i l l e d w a t e r ? 2 0 0 0 ppm with salt T 5 0 0 ppm CMC solution A o o l i 1 1 l t i l l 1 1 i - l — 0.01 0 05 01 0.5 1 10 50 100 s-' 500 F i s . 2.7 R h e o g r a m s of d i l u t e d p o l y a c r y l a m i d e s o l u t i o n s . T h e t h e o r y o f Z i m m , w h i c h t a k e s i n t o a c c o u n t t h e h y d r o d y n a m l c a l l n t e r a c t o n s , r e s u l t s in t h e s a m e t y p e o f e q u a t i o n w i t h a r e l a x a t i o n t i m e w h i c h is a b o u t 1.5 t i m e s s m a l l e r . T h e c h a r a c t e r i s t i c t i m e t f o u n d a s a p a r a m e t e r in the Bird C a r r e a u m o d e l can b e a p p l i e d in e q u a t i o n 2 . 1 2 .
In f i g u r e 2.7 e x a m p l e s a r e g i v e n for s e v e r a l p o l y m e r s o l u t i o n s u s e d . T h e r e is an e v i d e n t d i f f e r e n c e in v i s c o s i t y b e t w e e n d i s t i l l e d w a t e r s o l u t i o n s and tap w a t e r s o l u t i o n s c o n t a i n i n g s a l t s . U s i n g a m o l a r v o l u m e o f 26 m / m o l for the S e p a r a n A P - 3 0 p o l y a c r y l a m i d e , a c h a r a c t e r i s t i c t i m e of 0.83 s can b e c a l c u l a t e d for t h e 1 0 0 0 p p m s o l u t i o n , w h i c h g i v e s a t of 0.8Ü s a c c o r d i n g t o i t s r h e o g r a m . D a t a f o m H i l l \ 1 9 7 2 \ , a s p r e s e n t e d in f i g u r e 2.8 (from G r e e n e e t a l . \ 1 9 8 2 \ ) . s h o w a v a l u e t w i c e a s h i g h . H o w e v e r , t h e s e v a l u e s a p p l y to f r e s h s o l u t i o n s . For a f r e s h s o l u t i o n o f 2 0 0 0 p p m 31
Figure 2.8
The first time constant t versus the
concentration PAAm, according to the data of Hill \1972\.
PAAm in distilled water, a fluid not used for experiments, a value of Ü s for t can be derived from figure 2.7. as well as from figure 2.8 and from the application of equation 2.12. Assuming that the volume occupied by the polymer molecules is too large to permit the application of Rouse's theory
(eq. 2.12). the relaxation time will be 1.5 times smaller. according to the theory of Zimm. This is true when c [M3 > 20. With c = 2 1 0 "3 and [M] in the order of 1 05 (figure 2.10),
this is the case. In spite of this difference, the example of the 2000 PAAm ppm solution shows the possibilities of the method to obtain a characteristic time for the fluid from an graphical Interpolation of its rheogram.
In figure 2.9 the ratio of the specific viscosity to the concentration PAAm is shown for solutions, used for mixing experiments described in chapters 3 and d. The data for this
figure are Identical to those indicated in figure 2.2. Figure 2.9 gives an indication of the high concentration of counter ions for the polymer used, while the hydrodynamical volume of the polymers (related to u /c) does not Increase at low
o
concentrations, because of the high concentrations of salts available to mask the polymer groups. The addition of salt3 will give a low viscosity (figure 2.7). but the degradation In time will also be less, due to the smaller hydrodynamical
I U
/
- I, 0.5 0.1' '-10 I ' I I . 1 1 I I » 50 100 500 1000.10-6 kg/kg 3?400.10-6 kg'kg
Fig. 2.9 T h e s p e c i f i c v i s c o s i t y as a f u n c t i o n o f t h e
concentration PAAm (solutions as shown in fig. 2.2)
0.2 10» 0.1 -• Msp 200 400.10-» kg/kg Figure 2.10 The specific viscosity as a function of the cone. PAAm (c) for several dilutions.
volume of the polymer. Figure 2.10 shows the same relation as In figure 2.9 for one solution after dilutions with distilled water. An Increase of c /c with lower concentrations of
sp
PAAm may be noted. Fewer polymer groups are masked by the salts, giving a larger hydrodynamical volume and a much longer relaxation time than in the case of the solutions used for figure 2.9, in accordance with equation 2.12.
The solutions used in the tube experiments have been degraded to a great extent and. due to the added salts and lso-propanol, can be aproximated by an overall lso-viscous behaviour for the shear rates involved. Apart from the apparent viscosity, with a value close to the zero shear viscosity, drag reduction is the only elastic phenomenon for these fluld6 that can be measured accurately.
2.Ü.3 Carboxymethylcellulose
In order to increase the viscosity, without creating a significant elastic effect, a high viscosity sodiumcarboxy-methylcellulose (CMC), has been used. Even though such
solutions are of a shear thinning and visco-elastic character at the concentrations used, the fluids, nevertheless, behave in an almost Newtonian manner (figure 2.7).
2.5 MOLECULAR DIFFUSION
2.5-1 Introduction
Molecular diffusion, within the smallest scale of turbulence, is the final stage in the mixing of chemical reactants (Harnby et al. \1985\) Though, generally, diffusion depends on the viscosity of the fluid, it has been
demonstrated that the macroscopic viscosity of polymer solutions does not dictate the molecular diffusion of ions (Astarlta \ 1 9 7 6 \ ) . A number of researchers, such as for
instance Dutta and Mashelkar \1985\. Belloni et al. \ 1 9 8 1 \ and Yoshida \1978\, have conducted experiments in several systems. Some enhancement in diffusion is even possible in shear rate dependent systems in some situations, due to slip and micro convection (Kllnger-Park and Hubbard \1985\)■
2.5.2 Diffusion experiments
The intradiffussion of ions in polyacrylamide solutions 22
has been studied, using radioactive tracers. Labeled Na and I have been used in an open capillary method.
developed from the set-up described by Brentel and Beronius \1978\. This method is based on the diffusion of tracer out of one end of a small capillary, filled with the polymer solution with tracer, into a fast flowing solution (figure 2.11). This solution is stirred in a buffer vessel and recycled through the tube with the capillary, which has been mounted
perpendicular to the wall of this tube, in such a way that the fluid flows along the open end, maintaining an almost zero concentration. The concentration of the labeled ions in the capillary Is measured with a scintillator. This experimental work has been carried out at the I.R.I., the Interunlversity Nuclear Institute in Delft. The solutions have been made of polyacrylamide dissolved in distilled water. Results are shown in table 2.1.
r
7ZZZZZZZZZZZZ
/1 tube with capillary 2 scintillator
3 single channel analyser 4 pump
5 vessel with magnetic stirrer.
Fig. 2.11 Schematic for the molecular intra diffusion measurements using radioactive tracers.
Table 2.1
tntradlffusion constants D in polyacrylamlde solutions for different types of ions
cone. PAAm labeled
X ion l o "9 m2/ s accuracy -9 2 10 in /s 0 0. 1 0.5 0. 1 0.5 0. 1 0. 1 ** + * Na* Na Na i~ i" i" I" 1.30 1.00 0. 8 0 0. 1 8 0. 12 0.03 O.iU 1 0 . 0 3 tO. 07 ♦ 0 . CU = 0. 06 ± 0 . 0 8 = 0. 0 3 = 0.02 2. 0 [ J a n s s e n and W a r m o e s K e r k e n M 9 8 2 \ ) * * S o l u t i o n c o n t a i n s 0.1 M N a C l ♦* S o l u t i o n c o n t a i n s 0.005 M 1 and 0.1 M Nal In s t e a d y - s t a t e e x p e r i m e n t s t h e i n t e r d l f f u s s l o n of KI ~ in a q u e o u s p o l y a c r y l a m l d e s o l u t i o n s has a l s o b e e n m e a s u r e d , u s i n g the p e n e t r a t i o n of I from a s a t u r a t e d c a r b o n t e t r a c h l o r l d e s o l u t i o n in an i o n i c a q u e o u s p o l y m e r s o l u t i o n w i t h a m y l o s e and i o d i d e a d d e d . T h i s t y p e of s o l u t i o n h a s b e e n used a l s o for the m i x i n g e x p e r i m e n t s in p i p e line f l o w , a l t h o u g h not d e g r a d e d ( f i g u r e 2 . 1 2 ) . T h e c h a n g i n g c o n c e n t r a t i o n g r a d i e n t a l o n g the tube has b e e n m e a s u r e d w i t h an a b s o r p t i o n m e t e r ( c h a p t e r 1 ) d u r i n g s e v e r a l d a y s . T h e r e s u l t s a r e p r e s e n t e d in t a b l e 2 . 2 . T h e trend ir. t h e d a t a a g r e e s r e a s o n a b l y w i t h t h e d a t a r e p o r t e d by M c C o n a g h y and
tube : lenglh 0.5m innerdiameter 16.4 mm
solution w i t i amylose
light absorption meter coloured solution with iodine and amylose
saturated
carbontetrachlonde iodine
F i g u r e 2 . 1 2
Schematic for the molecular diffusion measurements using colouration by iodine and amylose.
Table 2.2
diffusion constants D of I in polyacrylamide solution with amylose (measuring accuracy ±25 %)
% PAAm 0 o . 0 6 2 5 0. 1 2 5 0 . 25 2 - 1 m s l O "9 2 . 0 2 . 6 0 . 5 5 0 . 32 0 m Pa s 1 8 1 1 8 0
Hanratty \1977\. who used an electrochemical method. They arrived at a value of 0.86 10 m /s for the diffusion constant ID of Iodine (I complex) in a 100 ppm PAAm solution. Because it is practically impossible with this method to avoid convection for the solutions of low viscosity, the values for ID become more reliable at higher
concentrations. For the solution without polymer the value of JD, as presented in table 2.1, is almost twice the value
reported in literature for a similar concentration. For the solutions of about 0.1 X PAAm the diffusion constant ti is
nearly Identical for both methods, if I " ±s used as a tracer.
Both experiments show an effect on the diffusion of ions by the dissolved polyacrylamide. However, the decrease is not related to macroscopic fluid properties such as the
apparent viscosity. The change is small in comparison with the changes in the fluid behaviour or polyacrylamide
concentration. For the mixing experiments the co-ion I has been used. This ion is subject to an extra resistance in the polymer network caused by interaction of the electro negative charges. This effect Is relatively small at the concentrations I used, if I has been added.
2.5-3 Conclusion
It can be concluded that mixing times will not be influenced to a large extent by changes in the diffusion
constant, contrary to what might be expected from the increase in viscosity. In mixing experiments with concentrations up to 1000 ppm, the decrease of the diffusion rate at higher polymer concentrations will be of limited importance. This decrease is the result of a polymer network of greater density at a higher concentration and from the interactions between the polymer and the tracer Ion.
2.6 CONCLUSIONS
It is possible to characterize the polymer solutions by a fluid time, which Is the longest relaxation time. This time can be estimated from an Interpolation of the rheogram of the experimental solution, using the Bird Carreau model.
The diffusion constant for the iodine complex in the polymer solutions does not differ greatly from the values found in water.
2.7 S Y M B O L S A b c G k M n P R t. s t r e s s c o n s t a n t ( e q u a t i o n 2 . 2 ) s t r e s s i n d e x ( e q u a t i o n 2 . 2 ) c o n c e n t r a t i o n of p o l y m e r s h e a r m o d u l u s c o n s i s t e n c y m o l a r v o l u m e f l o w b e h a v i o u r i n d e x p m o d e o f v i b r a t i o n of p o l y m e r g a s c o n s t a n t c h a r a c t e r i s t i c f l u i d t i m e r e l a x a t i o n t i m e a b s o l u t e t e m p e r a t u r e N m "2 s " "1 ppm. kg k N m " N m s 3 , -1 m m o l J m o l_ 1K 1 a s K -1 G r e e k s y m b o l s : ï A
[«]
w
app ws Ms p Mo Ve
r s h e a r r a t e c h a r a c t e r i s t i c f l u i d t i m e i n t r i n s i c v i s c o s i t y d y n a m i c v i s c o s i t y a p p a r e n t v i s c o s i t y v i s c o s i t y of s o l v e n t s p e c i f i c v i s c o s i t y z e r o s h e a r v i s c o s i t y v i s c o s i t y at i n f i n i t e s h e a r k i n e m a t i c v i s c o s i t y d e n s i t y s t r e s s s h e a r s t r e s s f i r s t n o r m a l s t r e s s d i f f e r e n c e s -1 s P a s P a s P a s P a s P a s 2 -1 m s kg m N m ~Z - 3 N m N m - 2 -2 S p e c i a l s y m b o l : m o l e c u l a r d i f f u s i o n c o e f f i c i e n t m s 2 -1 39subscripts:
1 flow direction
2 direction of the velocity gradient
2.8 REFERENCES
Astarlta. G., 1976
Heat and Mass Transfer in Non-Newtonian Fluids,
Eur. Congr, on Chem. Ending. Working Party on Non - Newtonian Fluid Processing. Amsterdam, dune 30 - July 2 (1976). Dl-33 Barnes, H.A. and K. Walters 1985
The yield stress myth
Rheologlca Acta. 2ft, 323-326
Belloni, L.. M. Drifford and P. Turq 198Ü Counterion Diffusion in Polyelectrolyte Solutions Chem. Phys. . SjJ, IÜ7-15Ü
Bird. R.B.. R.C. Armstrong and O. Hassager 1977 Dynamics of polymeric liquids: Vol. 1 Fluid mechanics John Wiley 8. sons. New York
Blom. C., R.J.J. Jongschaap and J. Mellema 1986 (in Dutch) Inleiding in de Reologie
Technische Hogeschool Twente, Kluwer technische boeken, Deventer
Box, M.J. 1965
A new method of constrained optimization and a comparison with other methods
Computer J., 8. Ü2-52
Brentel, I., and P. Beronlus 1978
A new version of the continuous capillary method for determining tracer diffusion coefficients,
Radiochem Radioanal. Lett.. 3_3_, 7-12
Carreau, P.J.. I. Patterson and C.Y. Yap 1976
Mixing of Vlscoelastic Fluids with Helical-Ribbon Agitators I _ Mixing Time and Flow Patterns
Can. J. Chem. Englng. , 5_Ü, 135-IU2 Cross, M.M. 1979
Relation between vlsccelastlcity and shear-thinning behaviour In liquids
Rheologlca Acta, .18. 5. 609-6IÜ
Dutta, A. and R.A. Mashelkar 1985
Longitudinal Dispersion in Rectilinear Flow of Dilute Polymeric Liquids: Likely Role of Stress-Induced Migration Chem. Enging. Commun.. 3_2, 181-209
Dutta. A. and R.A. Mashelkar 1985
Longitudinal Dispersion in Rectilinear Flow of Dilute Polymeric Liquids: Likely Role of Stress-Induced Migration Chem. Englng. Commun.. 22, 181-209
Greene. H.L.. C. Carpenter and L. Casto 1982
Mixing characteristics of axial impellers with Newtonian and non-Newtonian fluids
Proc. Fourth Eur. Conf. on Mixing, NoordwiJkerhout, Netherlands. April 27-29, paper Dl, 109-126
Haase. H.C. and R.L. Mc Donald 1972
Dichotomies In the Viscosity Stability of Polyacrylamide Solutions I
J. Polym. Sci. Letters. Ed.. 1J3. Ü61-U67
Harnby, N., M.F. Edwards and Alvln W. Nienow (Edt.) 1985 Mixing in Process industries
(J.R. Bourne: Mixing in single-phase chemical reactors) Butterworth, London
Hershey, H.C. and J.L. Zakin 1967
A molecular approach to predicting the onset of drag reduction in the turbulent flow of dilute polymer solutions
Chem. Enging. Sci., £ 2 , 18-Ü7 Hill, C.T. 1972
Nearly Viscometric Flow of Viscoelastlc Fluids in the Disk and Cylinder System. II: Experimental
Trans. Soc. Rheology, 1^6, 2, 213-2Ü5
Janssen, L.P.B.M. and M.M.C.G. Warmoeskerken 1982 (in dutch) Fysisch technologisch bij-de-handboek
Delftse Uitgevers Maatschappij, Delft
Kulicke. W.M.. R. Knieswke and J. Klein 1982 Preparation, Characterization, Solution Properties and Rheologlcal Behaviour of Polyacrylamide
Prog. Polym. Sci., 8, 373-168
Kllnger-Park. P.U. and D.W. Hubbard 1985
Diffusion in Non-Ionic and Ionic Polymer Solutions: Effects of Shear Rate and Polymer Concentration
Chem. Enging. Commun. , 3.2. 171-201 Lodge, A.S. 196Ü
Elastic Liquids
Academic Press, London Machtle, W. 1982
Zur Alterung von wësserigen Polyacrylamid-Lösungen Makromol. Chem., 183, 2515-2525
Oolman, T., E. Walitza and H. Chmiel 1986
Dynamic Theological behaviour of microblol suspensions
Bouse. P.E.. 1953
A Theory of the Linear Viscolastic Properties of Dilute Solutions of Colling Polymers
The Jrnl. of Chem. Phys. . 2_1, 7, 1272-1280 Sylvester , N.D. and J.S. Tyler 1970
Dilute Solution Properties of Drag-Reducing Polymers Ind. Eng. Chem. Prod. Res. Dev., £• 518-553
Vliet, T. van 1982 Private communication
Depart, of Food Science, Agricultural University, Wagenlngen Wagner, P.J. 1981
Der Einfluss moleKularer Parameter auf das Verhalten turbulent strömender Polymerlösungen
Dissertation der Universitat Essen White J.C. and A.8. Metzner 1963
Development of Constitutive Equations for Polymeric Melts and Solu tions
J. Appl. Polymer Sci.. 7.. 1867-1889 Yoshida, N. 1978
Self-Diffusion of Small Ions in Polyelectrolyte Solutions J. Chem. Phys.. 6_2. U867-Ü871
Zimm. B.H. 1956
Dynamics of polymer molecules in dilute solutions: Viscoelastlclty, flow bifrengence and dielectric loss J. Chem. Phys., 2_a. 2, 269-278
C H A P T E R 3
HYDRODYNAMICAL DESCRIPTION OF VISCO-ELASTIC FLUIDS IN TURBULENT PIPE LINE FLOW
3.1 INTRODUCTION
The drag reducing effect in turbulent pipe line flow by polymer solutions with a visco-elaBtlc behaviour has been well known for over forty years. The effect is also called the Toms phenomenon, after the first investigator who has reported about it (Toms \19tt8\). Most of the subsequent papers deal with the relation between the diminished pressure drop and the type and concentration of the dissolved polymers. In this chapter the effect of the polymers on the hydrodynamics will be studied. In chapter k this effect will be related to the
mixing experiments. An extension for mixing devices will be made in chapter 5- A correlation is proposed to characterize
the rheologlcal behaviour of the fluids by the drag reduction data obtained.
In the last decade greater attention has been paid to the modification in the turbulent structure of the flow field. Especially after Introduction of Laser Doppler Anemometry (L.D.A.) reliable data concerning the changes in the flow field have been published. In this chapter laser Doppler velocity measurements In pipe flow have been used to Investigate the influence on the pressure drop and the turbulent flow field by aqueous polyacrylamlde solutions, also used for mixing
experiments in following chapters. Special attention is given to the velocity fluctuations which are important for the mixing in pipe line flow and the energy dissipation.
3.2 DRAG REDUCTION IN LITERATURE
In his review about the drag reduction phenomenon Hemming6 \ 1 9 7 6 \ gives more than a thousand references, Indicating the vast Interest researchers have taken in this subject. In the literature it has frequently been attempted to couple the properties of the solutions to the drag reduction. As shown in review papers by Hoyt \ 1 9 7 2 \ and Vlrk \ 1 9 7 5 \ this has never been entirely succesful. One reason is the diversity of the solutions, capable of interaction with the flow field, probably in several ways. Elata and Tiro \1965\ give a review concerning the different states of dissolution of the
polymers, with a changing effect on the Toms phenomenon. It is not even necessary for the polymer to be present In the entire flow field of the tube. Vleggaar \ 1 9 7 3 \ has shown the effect for concentrated polymer solutions, present in the centre of the tube only. This experiment has been extended by Frings \ 1 9 8 5 \ for polymer solutions, injected in the boundary layer. These injections give rise to a dramatical decrease in the pressure drop, provided that the diffusion of the polymers in the water is not significant. These papers show that the interaction between the additions to the solvent and the turbulent flow field are not of a straightforward nature. although it may be assumed that the related drag reduction is always the result of an alteration of perhaps local, turbulent structures.
The effect of geometries, other than the straight tube, on the pressure drop with polymer solutions suggests that the normal stress differences and the related elongational
viscosity of the vlsco-elastic polymer solutions are
Important. For Instance, Pisolkar \ 1 9 7 0 \ has shown that these solutions cause an drag enhancement when flowing through valves and fittings. In these cases there is a distortion of the wall layer or a decrease of the tube diameter, causing an elongation. Walters et al. \ 1 9 7 1 \ and Kelkar and Mashelkar \ 1 9 7 2 \ have studied the flow of polymer solutions in curved
pipes. They showed an enhancement in drag reduction in the transitional region but under turbulent conditions the
curvature of the tube had an adverse effect. This is caused by secondary flows in the curved pipe, due to the normal
stresses.
For the turbulent flow of homogeneous polymer solutions in straight tubes, most attempts to correlate the fluid
properties with the drag reduction data are based on the Deborah number:
characteristic fluid time A De =
-characteristic flow time L/U The characteristic fluid time A can be estimated by
determination of the stresses with a rheogonlometer, using equation 2.5- For the diluted solutions, the model of Rouse, as described in chapter 2, can be applied.
The estimation of the characteristic flow time depends upon the postulated mechanism. Astarlta et al. \1965.1969\ postulate that the energy dissipating eddies are made conservative by the vlsco-elastlc behaviour of the polymer solutions, causing a resistance to stretching flows, as found with vortices. Haas and Durst \ I Q 8 2 \ and Durst et al. \ 1 9 8 2 \ have given an extensive theory about this postulate, based on the flow through porous beds. The resistance will arise when the Deborah number on the scale of the energy dissipating vortices is in the order of 1. For eddies In the turbulent flow the characteristic flow time t is inversely
proportional to the vortex frequency u. Following the theoretical considerations of Tennekes and Lumley \ 1 9 7 2 \ (§ 1-5). the vortices of the Kolmogorov microscale, the smallest energy dissipating eddies, may be characterized by:
(3-2) t * l/w = (D/U) R e- 0'5 0
D is the diameter of the tube, which equals the diameter of the energy containing eddies, and U is a characteristic
velocity for the the macro scale vortices. Here, Tennekees and Lumley use the friction velocity u. for the flow near the
boundary layer. Astarlta and Marruccl \197U, § 7-5\ have defined the Deborah number as
(3-3a) De = A / tf
c o m b i n a t i o n w i t h e q u a t i o n 3 . 2 j c l v e s
( 3 . 3 b ) De = ( U / D ) A R e ° " 5
The theory, presented by Astarlta and Marruccl. also Includes an onset for the drag reduction at a certain Reynolds number, as has been shown by most experiments. Vlrk and
Merrill \ 1 9 6 9 \ have established that this onset point is the same Por different concentrations of a neutral polymer, for instance such as polyethyleneoxlde. although for weak
polyelectrolytes, such as polyacrylamide, the drag reducing behaviour in connection with the onset point depends on the concentration used.
A group of theories is based on changes in the wall shear layer. Kelkar and Mashelkar \ 1 9 7 2 \ have given a summary of these theories. Some are based on the mean burst interval period of the fluid elements from the wall, as used for instance by Achla and Thompson \ 1 9 7 7 \ and Wagner \198"\-However, this turbulent phenomenon Is difficult to quantify, and most theories do not give an exact detection criterion according to Kunen \198ü\ (see also Talmon et al. \ 1 9 S 6 \ ) . Other theories are based on the friction velocity u, which is an important parameter for the wall shear layer. Hershey and Zakin \ 1 9 6 7 \ have taken the reciprocal of the shear layer as
the characteristic flow time.
It is Interesting to note that all these theories come to the same formulation of the Deborah number:
(3.Ü) De « (u. /v ) A
Provided that the Blaslus type of relation
(üf = 0.316 Re"' 5) is valid, equations 3.3 and 3. ü can be
identified with one another (Kelkar and Mashelkar \ 1 9 7 2 \ ) . On the basis of the Deborah number, master curves have been made in most cases for relations describing the drag reduction.
A different approach, by Mizushlna and Usul \197U. 1977\. is based on the turbulent damping theory, giving good theoretical correlation with the data, however, these
relations are complex and hard to calculate. Walsh \ 1 9 6 7 \ proposed that polymer molecules slightly alter the energy balance of the turbulent fluctuations close to the wall and allow energy absorption by the polymers of disturbances, normally convected away from the wall. For very dilute solutions of different types of polymers, he found a
correlation for the drag reduction with the concentration of the polymer and the quadratic intrinsic viscosity of the solution used.
The drag reduction is limited, virk \ 1 9 7 0 \ has given an empirical correlation for the minimal friction factor at a specific Reynolds number
(3- 5) f " ° '5 = 19 los(Re f0 - 5) - 32.Ü
This equation can be approximated for 10 < Re < 10 by
(3.6) log(üf) = -0.015 - 0.50 log(Re)
Drag reduction may also be limited by the dimensions of the polymers, which cause a minimum scale for the energy
dissipating eddies (Walstra \ 1 9 7 Ü \ ) . This will only happen for very turbulent flows.
10 - U V i t h s / a s y m p t o t e " - - , '
/
• o A A , . '^
i w a t e r * n h p o l y m a r i n j e c t i o n Figure 3-1 Dimenslonless vele Ltj distributions inpipe line flow for polymer solutions:
Injection at the centre line of a concentrated Separan AP-30 solution
Cc ■ 0.1X) in water. Overall concentration: c = 20 ppm (Bewerdorf \ l ° 8 ü \ ) . 1 0 ' 1 0 ' 4 0 c ( p p m ) R e U . 0 ( w a l e r ) 1 2 5 0 0 2 9 5 m a * r e d a s y m p t o t e ( M i ' u s h i n a a n d UsuO V i r k ' s a s y m p t o t e u , . 11 7 in,,- 17 0 20 11500 2 22 50 12400 t 86 100 13200 ' 90 300 14400 3 22 10 38000 5 62 20 37400 5 36 50 34400 * 83 100 33800 4 21
rJt*
I' b: Homogeneous solutions of polyethylene oxide (D = 0.0253 m ) (Mizushlna and Usul \ 1 9 7 7 \ ) . . / o 8 40 76 too 190 2 1 41
-U . ( m « i 1 1 9 ! ' 1 6 2 3 6 17 V . f . ' s i s y m p t o t » / 'W .. 11 7 I n , . - 17 / / * ' . , )DB / . ' ' =D sr»* / - / O i < 4 '
/ * > <
A 5 ' / . V 4 " , » 67 / v a * ' » ' . , . ■ * - .„-r . • F t ^ ^ ' U,.2.5lny,-5 5 / ■ ' ' i -i c :Centre line injection of polyethylene oxide (Polvox WSR 301) at Re = 3-5 10 x/D gives the downstream distance to the injection point. NaCl concentration: 0.2X Polymer concentration injection fluid: 0.1 X Overall concentration of polymer: 50 t i
(McComb and Rabie 1982\)
3.3 LITERATURE ABOUT VELOCITY PROFILES OF POLYMER SOLUTIONS
In the studies of Seyer and Metzner \1°69\. Rudd
\1972\, Mizushina and Usui \1977\, Thielen \198l\, McComb and Rabie \ 1 9 8 2 \ and Bewersdorff \198ü\ it has been shown that in pipe line flow the boundary layer between the laminar sublayer and the inertial layer is thickened and the maximum turbulence intensity is located further from the wall with increasing concentrations of polymer. Whilst the absolute maximum of turbulence intensity is reduced, the intensity in the centre is not significantly affected. There is also a relative shift towards lower turbulence frequencies. The study of Bewersdorff \198Ü\ shows a different profile in the boundary layer than the experiments of Mizushina and Usui \ 1 9 7 7 \ as visible respectively in figures 3.1a and 3.1b. In these figures the axial velocity and the radial position have been made
dimenslonless by the friction velocity u. and by u/u.
respectively. Bewersdorff has injected a concentrated solution at the centre line of the tube. The cause for the differences can be found in the use of respectively a square duct or a
Ultimate profile U.= Aml n y . . Bm
* . . . = " 6
Newtonian wall law Ut = Ant n y * « B „
Fig. 3.2 Schematic of the mean velocity profile during drag reduction according to the three layer model of Virk \1975\. showing the elastic sublayer.
