An Experimental Study on Turbulent Mixing of Visco-elastic Fluids

AN EXPERIMENTAL STUDY ON TURBULENT MIXING OF VISCO-ELASTIC FLUIDS

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PAUL BARTELS 1988

Cover: the interaction between an elastic polyacrylamide molecule and two turbulent vortices


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 tangentile 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

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 reel Inkomen en moet dan ook niet als zodanig belast worden.

8 Gezien de hirarchische 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"

To Sophieke

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

6

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 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.

7

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.

a

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 3.6.2 Results with the 16.4 mm tube 72

3 3 3

7 8 9

CHAPTER

CHAPTER 6

6 HYDRODYNAMICS IN A STIRRED VESSEL 11 6.1 Introduction 11 6.2 Influence of the visco-elasticlty

on the large scale flow 12 6.3 The experimental set-up 15 6.3-1 The tank reactor 15 6.3-2 The laser Doppler anemometer 16 6.3.3 The viscosity used for the

velocity measurements 18 6. The radial discharge of the impeller 19 6..1 The tangential Jet 19 6..2 Measurement of the tangential Jet 153 6.5 The pumping capacity 159 6.6 The trailing vortex 161 6.6.1 Literature 16 6.6.2 Experiments 168 6.7 Turbulence in the vicinity of the impeller 170 6.8 Turbulence in the bulk of the vessel 17 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 18 7.2. Experimental fluids 184 7.3 Influence of elasticity on the power number

for Rushton turbines 188 7. Power consumption and the tangential jet 192 7.5 Visco-elastic fluids and inclined

blade turbines 19 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. Effect of the fluid elasticity on the

power consumption 199 7.6 Conclusions 202 7.7 Symbols 202 7.8 References 20

11

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. The measuring techniques used 211 8.U.1 Introduction 211 8..2 The light probe in the fluid 212 8..3 The light probe, measuring

over the full height 213 8.U. The amylose indicator 215 8..5 The Injection system 216 8..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 23 9.1 Pipe line reactors and stirred tank reactors 235 9.2 Scale-up 237

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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.

13

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.

m

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 tangentile 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 mengen stromingsgedrag in en achter de elementen. De drukval over de mengers wordt slechts in geringe mate benvloed 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.

ic-

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, 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

16

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

17

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.

ia

C H A P T E R 1

GENERAL INTRODUCTION

1 .1 SCOPE

The mixing of fluids is carried out in all phases of the process industries. Most process fluids possess qualities of complex rheologlcal behaviour, especially those that are found in the food- and bio industries, polymer processing and coating operations. Some examples of products and processes are as follows: -fermentation: *xanthan gums and other extra cellulair

polysaccharides, used for example for enhanced oil-recovery

*fermentation broths of moulds -paint industry: emulsions -manufacture of glues and adhesives -emulsion polymerizations (latex) -separation processes: floceulation

These fluids show considerable variation in their manner of deviation from iso-vlscous Newtonian behaviour. A number of them display elastic properties. This time dependent rheologlcal behaviour, which is characterized by a temporary storage of energy, may be observed in polymers, polymer solutions, emulsions, dispersions and microbial suspensions (Oolman et al. \ 1 9 8 6 \ ) , but also in glycerine and oils such as high molecular silicone oil, soybean oil or common oils (Riccius and Arney \1986\).

Very little is known concerning the effects of fluid elasticity on mixing performance in either laminar or turbulent flow regimes. For stirred vessels most of the relevant research has been carried out In connection with laminar mixing, while turbulent mixing has been researched to

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 \19"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

20

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

-1 (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 . 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.

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1. 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 - 11

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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. \1986\). 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

23

-CH,-CH-C=0 NH,

POLYMER

-CH, CH-C = 0 I 0' Na-

CO-POLYMER

Figure 2.i The chemical structure of polyacrylamlde with co-polymer.

polyacrylamide, such as Sylvester and Tyler \1977\. Kuilcke et al. \1982\, Maehtle \1982\. and Wagner \198\. 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 AP-30 (anionic character) 0.5% solution JUUU mPa s 1000 500

100 50

in

"app - 1 / If.

1 '

/ ^ - - - '

-

_

. pH 1 , 1 ,

5"C 25C 40-C

200 400.10-6kg/kg PAAm 10

Fig. 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).

24

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 \1972\), 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. 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,

25

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

and w i l l d e g r a d e f i r s t .

0 1

O 0 5

O 01

A f

0 0 0 1 >Tty-F i g . 2 . a

T 1 1 1 1 t I I | 1 1 1 11r1-Polyacrylamide solutions

25ppm 5 days old 7 days old 9 days old

300ppm 4 days old 6 days old 8 days old

andtl 0 ,

Virk _

_i_L_X_L 5.10 J XT 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.

virk f~"5 = 19 log (Re f 0 , 5) - 32.5

2. DEFINING THE VISCOSITY OF THE EXPERIMENTAL SOLUTIONS

2..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:

26

1 2 5 10 20 50 100 s-1

Fig. 2.5 The viscosity p. and the first normal stress difference zn - v n . plotted as power law functions of the shear stress (0.9X PAAm solution).

(2.1)

(2.2) * - k i"" 1 app *

Where k Is the consistency and n the index, with a value between 0 and 1 for shear thinning fluids.

At very low shear rates real fluids tend to have a relatively constant viscosity. A zero shear viscosity p. can

o be defined as that which is approached as y nears zero. At very high shear rates the dilute solutions will approximate the solvent viscosity p. (figure 2.6). The PAAm solutions do

s not possess any yield stress as determined by Van Vliet \1982\ with apparatus capable of measuring very low stresses in fluids (Van Vliet and Hooijdonk \ 1 9 8 U \ ) . The existence of the yield stress for polymer solutions is debatable, as has been shown by Barnes and Walters \1985\- It is to be expected that the use of yield stresses in rheological models will decrease with the development of more sophisticated rheometers, and with the acceptance of the visco-elastic behaviour of fluids.

27

2..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.) 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 \196\). 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. \1968\). 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

" *22

*12 *

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).

29

A third method involves the theoretical bead-rod-sprlng model with isotropic Brownian and hydrodynamlc forces for the polymers in solvents by Rouse \1 9 5 3 \ and Zlmm \1956\. Most of the applicated concentrations of polymer may be considered to give dilute solutions according to Rouse's theory. This theory is limited to the range

(2.8) 1 < c In) < 20

where c represents the weight concentration. The intrinsic viscosity [y] Is defined as

(2.9)

^ " l i m c to 0 c

and the specific viscosity u is denoted t>y sp

(2.10) U8P =

According to Rouse's theory the relaxation time for the p mode of vibration of the polymers Is given by

6 [] s M ( 2- 1 1 )

P ir c RT

The greatest effective relaxation time t (p = i) of the relaxation spectrum for such a solution as determined by Rouse, can be estimated by (Blom et al. \1986\. Hershey and Zakin \1967\)i

A ( \ M (2.12) 6 (M - M) M

IT c RT

30

1000

500 mPas

100 -

50 -

10 -

5 -

: i ir-r ' iTTTi 1 ii 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 i l 1 1 1 11 1 l i l 1 1 . 1 11 I I I

Polyacrylamide solutions

( Oow Chemical, Separan AP-30 ) T = 25C

o 1000 ppm with salt 2000 ppm dis t i l led water ? 2000 ppm with salt T 500 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

Fis. 2.7 Rheograms of diluted polyacrylamide solutions.

The theory of Zimm, which takes into account the hydrodynamlcal lnteractons, results in the same type of equation with a relaxation time which is about 1.5 times smaller.

The characteristic time t found as a parameter in the Bird Carreau model can be applied in equation 2.12.

In figure 2.7 examples are given for several polymer solutions used. There is an evident difference in viscosity between distilled water solutions and tap water solutions containing salts. Using a molar volume of 26 m /mol for the Separan AP-30 polyacrylamide, a characteristic time of 0.83 s can be calculated for the 1000 ppm solution, which gives a t of 0.8 s according to its rheogram. Data fom Hill \1972\, as presented in figure 2.8 (from Greene et al. \1 9 8 2 \ ) . show a value twice as high. However, these values apply to fresh solutions. For a fresh solution of 2000 ppm

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 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 10"3 and [M] in the order of 105 (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

IU

/ - 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 The specific viscosity as a function of the 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.

33

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..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 \1976\). A number of researchers, such as for instance Dutta and Mashelkar \1985\. Belloni et al. \1981\ 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\)

3

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.

r7ZZZZZZZZZZZZ/ 1 tube with capillary 2 scinti l lator 3 single channel analyser 4 pump 5 vessel with

magnetic stirrer.

Fig. 2.11 Schematic for the molecular intra diffusion measurements using radioactive tracers.

35

Table 2.1

tntradlffusion constants D in polyacrylamlde solutions for different types of ions

cone. PAAm labeled X ion lo"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. 80 0. 18 0. 12 0.03 O.iU

10.03 tO. 07 0. CU = 0. 06 0. 08 = 0. 03 = 0.02

2. 0 [Janssen and WarmoesKerken M 9 8 2 \ )

** Solution contains 0.1 M NaCl * Solution contains 0.005 M 1 and 0.1 M Nal

In steady-state experiments the interdlffusslon of KI ~ in aqueous polyacrylamlde solutions has also been measured, using the penetration of I from a saturated carbontetrachlorlde solution in an ionic aqueous polymer solution with amylose and iodide added. This type of solution has been used also for the mixing experiments in pipe line flow, although not degraded (figure 2.12). The changing concentration gradient along the tube has been measured with an absorption meter (chapter 1) during several days. The results are presented in table 2.2. The trend ir. the data agrees reasonably with the data reported by McConaghy and

3

tube : lenglh 0.5m innerdiameter 16.4 mm

solution wi 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 . 0625

0. 125

0 . 25

2 - 1 m s

l O " 9

2 . 0 2 . 6

0 . 55 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

37

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.

38

2.7 SYMBOLS

A b c G k M n P R t.

stress constant (equation 2.2) stress index (equation 2.2) concentration of polymer shear modulus consistency molar volume flow behaviour index p mode of vibration of polymer gas constant characteristic fluid time relaxation time absolute temperature

N m" 2 s"" 1

ppm. kg k N m" N m s 3 , -1 m mol

J m o l _ 1 K 1

a s K

-1

Greek symbols:

A []

w app w s Msp Mo

V

e r

shear rate characteristic fluid time intrinsic viscosity dynamic viscosity apparent viscosity viscosity of solvent specific viscosity zero shear viscosity viscosity at infinite shear kinematic viscosity density stress shear stress first normal stress difference

s- 1

s

Pa s Pa s Pa s

Pa s Pa s 2 -1

m s kg m N m~ Z

-3

N m N m

-2 -2

Special symbol:

molecular diffusion coefficient 2 -1 m s

39

subscripts: 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 198 Counterion Diffusion in Polyelectrolyte Solutions Chem. Phys. . SjJ, I7-15

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. 2-52 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_, 135-IU2

Cross, M.M. 1979 Relation between vlsccelastlcity and shear-thinning behaviour In liquids Rheologlca Acta, .18. 5. 609-6I 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

II0

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. 61-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, 18-7

Hill, C.T. 1972 Nearly Viscometric Flow of Viscoelastlc Fluids in the Disk and Cylinder System. II: Experimental Trans. Soc. Rheology, 1^6, 2, 213-25 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. 196 Elastic Liquids Academic Press, London

Machtle, W. 1982 Zur Alterung von wsserigen Polyacrylamid-Lsungen Makromol. Chem., 183, 2515-2525 Oolman, T., E. Walitza and H. Chmiel 1986 Dynamic Theological behaviour of microblol suspensions

ttl

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 strmender Polymerlsungen 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. U867-871

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

(12

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

3.2 DRAG REDUCTION IN LITERATURE

In his review about the drag reduction phenomenon Hemming6 \1976\ 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 \1972\ and Vlrk \1975\ 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 \1973\ has shown the effect for concentrated polymer solutions, present in the centre of the tube only. This experiment has been extended by Frings \1985\ 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 \1970\ 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. \1971\ and Kelkar and Mashelkar \1972\ 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

a u

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. \1982\ 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 \1972\ ( 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

45

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 lve s

( 3 . 3 b )

De = (U/D) A Re" 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 \1969\ 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 \1972\ 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 \1977\ and Wagner \198"\-However, this turbulent phenomenon Is difficult to quantify, and most theories do not give an exact detection criterion according to Kunen \198\ (see also Talmon et al. \19S6\). Other theories are based on the friction velocity u, which is an important parameter for the wall shear layer. Hershey and Zakin \1967\ 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.) De (u. /v ) A

Provided that the Blaslus type of relation (f = 0.316 Re"' 5 ) is valid, equations 3.3 and 3. can be

a 6

identified with one another (Kelkar and Mashelkar \1972\). 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 \1967\ 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 \1970\ 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.

This equation can be approximated for 10 < Re < 10 by

(3.6) log(f) = -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 \197\). This will only happen for very turbulent flows.

17

10 - U

Vi th 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 in pipe 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 \l8\).

1 0 ' 1 0 '

4 0 c ( p p m ) Re U .

0 ( w a l e r ) 1 2 5 0 0 2 95

m a * red a s y m p t o t e ( M i ' u s h i n a and 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 \1977\).

. /o 8

40 76

too 190 214

1 -

U . ( m i

1 1 9 ! ' 1

6 23 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 '

/ * >

3.3 LITERATURE ABOUT VELOCITY PROFILES OF POLYMER SOLUTIONS

In the studies of Seyer and Metzner \169\. Rudd \1972\, Mizushina and Usui \1977\, Thielen \198l\, McComb and Rabie \1982\ and Bewersdorff \198\ 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 \198\ shows a different profile in the boundary layer than the experiments of Mizushina and Usui \1977\ 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.= A m l n y . . B m

* . . .= " 6

Newtonian wall law U t = A n t 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.

9

round tube and in the state of dissolution of the polymer added. Polyacrylamide can be dissolved in a heterogeneous or a homogeneous state (Elato and Tiro \1965\). In the latter case the polymer molecules do not form clusters, like in the former case, and the concentration on micro scale is the same in every location. When the solution is heterogeneous the elasticity Is higher for the same concentration, giving a different modification of the flow field. Figure 3.1c presents the data of McComb and Rable \192\ for polymer injection, as has been done by Bewersdorff. In this case the same type of velocity distributions is found as by Mlzushlna and Usui. In this figure an additional logarithmic profile is present between the laminar layer and the core. This is called the elastic sublayer. With increasing concentration polymer the slope of the lines also increases up to an asymptote: the ultimate velocity profile of Vlrk \1975\. In all cases the data, measured for the maximum possible drag reduction, are in reasonable agreement with this ultimate velocity profile of Vlrk \1975\ which is formulated by (figure 3.2):

(3.7) U^ = 26.9 lQ(yJ - 17.0

According to the measurements of Vlrk this logarithmic velocity profile will be followed for all polymer solutions. This three layer model has been used by Vlrk to explain the exclstence of the asymptote as given by equation 3.5. For the more concentrated solutions only the thickness of this elastic sublayer will increase.

The diversity of velocity distributions found, shows that the relation between polymer additives and the turbulent flow field has not been solved, but it is clear that an elastic sublayer will develop, which will also cause a lower energy dissipation.

^o

3. EXPERIMENTAL

3..1 The tube and the fluids

The apparatus in which the laser Doppler experiments were done consists basically of a precision bore glass tube, 9 m lone with an inner diameter of 16.39 0.05 mm. The aqueous solutions are buffered in two 250 1 vessels as shown in figure 3.3- Two vessels are necessary for the mixing experiments with a redox reaction as will be shown In chapter 5. The solution is pumped through the tube by two Mono pumps with a variable rotational speed. Mixing in the two reservoirs is possible by means of bypasses, which are also used to control the exact flow velocity in the tube. In each pump line a pressure vessel and a small settling chamber with a contraction section at the beginning of the measuring pipe have been inserted. This set-up garantees a constant flow with a shear as small as possible. A collection vessel equiped with a thermostat (accuracy 0.5 K) and an Impeller has been mounted at ft m height. From this vessel the solution flows back to the

Fig. 3-3 Experimental set-up for measuring the mixing length by a colour reaction and the flow field by laser Doppler anemometry.

51

two 250 1 vessels. For the velocity measurements described in this chapter, the flow circulates continuously.

The pressure drop has been meassured over 0.5 and 5.0 m lengths of the tube with two Depex DP15 pressure difference transducers. The average velocity was determined by a electromagnetic induction flow meter.

The experimental fluids in this chapter are solutions of polyacrylamide. as described in 2.3- In table 3-1 ( 3-5) the fluid properties are presented, as determined by the viscosity and drag reduction measurements. The same type of solution has also been used for the experiments described in the chapters and 5-

3..2 The laser Doppler anemometer

Several velocity probes are commercially available. Visco-elastic liquids have special requirements because of the velocity-dependent normal stresses and the related fluid time scale. Therefore probes like the Pitot tube, the propeller and the hot wire anemometer, which are in direct contact with the fluid, are less suitable. Laser Doppler anemometry does not give any interference with the fluid, while the method is very direction sensitive, easy to use and to calibrate. The method is based on the measurement of the Doppler frequency shift of monochromatic light (in the present Investigation: X = 632.8 nm), scattered by very small dust particles in the fluid. By using two crossing monochromatic coherent beams of a He-Ne laser, a measuring volume in the fluid can be created (figure 3.). Only the scattered light of particles in this small volume will be detected. Although in reality the Interference occurs on the photodetector, the frequency shift f_ in the scattered light can be calculated by assuming an interference pattern in the measuring volume. The frequency shift is proportional to the velocity component u. perpendicular to this interference pattern, and inversely proportional to the distance s between maximum light intensities of the overall interference planes. This fringe

52

width of volume length of volume fringe distance

re 3. 1 measuring volume with ges created by the coherent chromatic laser s.

distance s depends on the angle e between the beams and the light source wavelength >,. So it is possible to write:

u = f s = 2 sin(e/2)

Limitations are given by the transparency of the fluid, the dimensions of the measuring volume, the complex electronics needed to process the signals with the hardware limitations and the related high cost of the apparatus. Beckwlth \1980\ summarizes the different types of anemometers which can be used. Watraslewlcz and Rudd \1976\ and Durst et al. \1976\ describe in detail the laser Doppler technique. For instance, Revill \1982\ and Costes and Couderc \198\ give an experimental comparison of anemometers for the measurement of the pumping capacity of turbine impellers, showing the accuracy of the laser Doppler system.

Turbulence profiles across the tube with the inner diameter of 16.tt mm have been measured with L.D.A. in all three directions separately at 7 m from the entrance of the tube described in the former paragraph. Measurements through the 2 mm thick glass wall are not possible because of the optical distortion. Therefore the method of Mishuzina and Usul \1977\ and Kunen \198tt\ has been used. A short thin wall section of tranparent plastic tube surrounded by a rectangular chamber filled with the experimental fluid served to minimize the optical distortion (figure 3.5). The plastic wall is about 0.12 mm thick. The chamber was equiped with optical glass windows. In this manner it was possible to

53

> ::"n O H

-U-^tr'-j-IJlri --""-'"-"-"--"-'

n -: - - , M - - ' 3 CP S T T T

overhead sheet 'sriei= 0.12 mm

dewaterlng (screw) perspex housing optical glass window glass tube seallngs de-aeration (valve)

Figure 3-5 The measuring chamber with a thin wall.

traverse close to the wall without significant optical distortion.

A reference beam method with forward scatter has been used with a TPD ttOO optical system, constructed by the 'Technisch Physische Dienst' at Delft, which is an institute related to the university. The system possesses a rotating grating for preshift and beam splitting (Oldengarm \1977\). The Doppler frequency is voltage converted by TPD trackers (Welling \1982\. Van der Molen and Van Maanen \1978\). A schematic of the optical system has been drawn in figure 3-6, while the electronical conversion of the diode signal is given in figure 3.7. The preshift was adjustable, and normally the choice was about 2 Mhz. High frequency trackers (TPD 1077) have been used for the Doppler frequency conversion, because of the very low mean velocity, theoretically zero, compared to the large velocity fluctuations in the radial direction. A further interpretation was accomplished by low and high pass filters, an average value voltmeter for the mean velocity and by HP 3721 A correlators. A HP 3720 A spectrum display was connected to one of the correlators. These correlators are capable of calculating the auto-correlation and the velocity

54

pholodiode 1

photodiode 2 *

Fig. 3-6 Schematic diagram of the set-up for the laser Doppler velocity measurements.

distribution. From both it is possible to obtain a value for the velocity fluctuations. Cross-correlation of the signals from photo detectors in each of the two beams (fiure 3.6) has been used for the measurement of the spectral density functions to diminish the ambiguity noise from random phase fluctuations (Van der Molen and Van Manen \1978\).

Complementary velocity field measurements have been carried out in a mm diameter tube with measuring chamber in collaboration with Kunen of the Laboratory for Aero- and Hydro dynamics in Delft at Delft Hydraulics (Kunen \198tt\, chapter 3). Simultaneous measurements in the radial and axial direction are possible, which have been analysed by computer. The system is almost identical to the anemometer described above. However, Bragg cells are used for the preshift and prisms are placed in the optical system for beam splitting.

3.1.3 The measuring volume

The anemometer in front of the 16.1 mm tube has been mounted on a table which can be traversed in the vertical direction (0.1 mm). The front end lens can be traversed in such a way that the laser Doppler measuring volume moves in a radial direction in the tube. The traversing has been calibrated by using the change in Doppler signal, when passing the walls. Because the rotating grating device can be rotated by 90 degrees, it was possible to measure in three directions. as shown in figure 3.8. The measuring volume was about 60 /am in width (b in figure 3.1). This is the width used for the

55

supply

medium freq. t racker

TPD 1077 2 MHz

ST photodiode

--.- vs. _--C^T^V^a,-

j ' ^ r ' . \

u: axial component

v: radial component

w: tangential component

Figure 3.8 Measuring method and definition of the velocity components in the pipe line flow.

photographs and a light absorption meter, presuming a Gaussian light distribution in the volume. This type of distribution is found for most light beams. The dimensions have been baaed on the volume given by an e reduction of the maximum light intensity. The dimensions of the volume determine the smallest scale of turbulence which can be measured. The spectral density function can be determined up to this scale.

It was not necessary to seed the water with particles, when using polymers. Occasionally latex particles of 1 m have been added to the water to obtain an optimum signal presence of 70 % of the measuring time.

3.5 DRAG REDUCTION AND TIME SCALES

In spite of the extensive literature which has appeared in the past years, there does not seem to be a convincing explanation for the mechanism of drag reduction. Perhaps several mechanisms are possible for different fluids, like solvents with polymer, micels or particles of a particular

57

0.05

0.02

i 1 1 r

41 = 0.3164 B o -

^

Re

/ / / x ' ' l i l * 20 V./ f * 30

V ' o 50 0

<

-L_

2.10' 5 10 '

f i s . 3 - 9 a 1 0 '

10-

-_

-

" \

-; _ 1

. |

\

1 1 1 I * ' " 1 1 ' ' 1 ' ' " | 1 ' | 1 1 ' I I

O

o solvent polymei solution o

o

\ %s~-^ 1 "<

i- Re v ^ \

10-' ' 10

-1 I I I I I I 11 I I I I 10 ' 1 0 ' 1 0 *

f i g . 3 - 9 b

10-

5 1 0 - ! ( -

10- '

- i 1 i l l

1 i

o

A

D

V

T

X \ '

c(ppm) N. 5

10

2 0

3 0

5 0

1 0 0

O(water) .\_

f i g . 3 - 9 c 5 10' 5 1 0 '

58

10-'

10-'

half open symbols: 100 ppm PAAm 2 h - 20 h used

closed symbols: after 2 h 0.01 X - 0.1 % polyacrylamide

fiS. 3-9 d

homoDoncout

2 . 1 0 - '

10-> 2

i _ _

\ \

"4 t

f

X O

X

64/Re

V

1 I 1

water Ptaeatoi 2850 c 50 ppm Priastof 2850 c 100 ppm

- 0 3*64^*'*

" - X -" - *

- R . * - . 1 1 1

5 .10 s 10* 2 10 4

f i e . 3 - 9 e f i g . 3 - 9 f

Fig. 3.9 Drag reduction data from literature: a: Wagner \198\, purified polyacrylamide solutions b: Virk \1975\, polyethylene oxide solution c: Mizushlna et al. \1971\. polyethylene oxide sol. d: Ogawa and Kuroda \1986\. polyacrylamide solutions e: Bewersdorf \198\. homogeneous and heterogeneous

polyacrylamide solutions Overall concentration: 50 ppm Separan AP-30 (Concentrations of injection solutions are given in per cent)

f: Gampert and Delgado \1985\, homogeneous polyacrylamide solutions.

5 9

shape. For dissolved polymers there seems to be an agreement that the phenomenon is a result of the interaction of the turbulence and vlsco-elastlcity as presented In paragraph 3.2.

In figure 3.9 the drag reduction is shown as found by several experimenters. In figure 3.9a the data are given for experiments of Wagner \198\ with polyacrylamlde solutions with a small molecular weight distribution. The onset for different concentrations polyacrylamlde is noticeable, as well as the parallel curves found for the relation between the friction factor and the Reynolds number under drag reducing conditions. The llneB for the friction factor of polymer solutions in this figure with logarithmic axis have a slope of approximately - 0.73. Also the measurements of Ogawa and Kuroda \1986\ (figure 3-9d) for polyacrylamlde with a correction of the viscosity for the pseudoplastic behaviour of the fluid (equation 3-9. using n = 0.97 - 1.0 and K = 1.3 - 1.8 for degraded 100 ppm PAAm solutions) give an almost constant slope of approximately -0.73- Our own experiments, presented in figure 3.10, show the same behaviour. In this figure, the Reynolds number Is based on the viscosity measured by Ubbelohde viscosimeters. For the concentrations of less than 100 ppm no deviation is visible for these straight lines. For higher concentrations, an additional pseudo drag reduction occurs at high velocities because of a Reynolds number that has been estimated too low. The polymer solutions have a lower apparent viscosity due to the pseudo plastic behaviour in that velocity region than has been measured by the Ubbelohde vlscoslmeter at low shear rates. For the same reason the 500 ppm solution gives a smaller friction factor than predicted by "Vlrks Law". In figure 3.11 the Reynolds number has been calculated for the 25 and 500 ppm PAAm solutions with an apparent viscosity based on a power law index n of 0.9 and 0.8, using the Reynolds number (Skelland: pp 16U-172 \1967\ and Dodge and Metzner \1959\)

6 0

0 1 0

0.05

0.01

0.005 -

Polyacrylamide (Separan AP-30)

O ppm o 25 ppm A 100 ppm V 200 ppm 300 ppm x 500 ppm

Blasius Virk

Flg. 3.10 Friction factor f versus the Reynolds number.

0.5

0 .1

0.05

0.01 - 41 t

0 0 0 5 b-

3.00 i " ^ J_

' M

10 ' 5 10 '

V \ V

I 1| i i -j | 1 1|in-i . Prandll : f - 0 . s = 4.0 log(Ref 5 ) -0 .4 . . Virk's asympt.: ( - = 19 log(Re10 5)-32.5 .

25 ppm *> 500 ppm

V ' \ ' V ^ . = l.0 0 9

v s

- Re 1.0 1 I I J -

' " V i r k 1 I 1 1 1 1

103 5.103 10 5.10* 1 0 '

Fis. 3.11 The influence of the power law index n, according to equation 3.9 for a 25 ppm (actual index n < 1.00) and a 500 ppm PAAm solution (actual n e 0.95).

61

R e _ g 2 -" D" ReNN -UQ 2 n " 3 (3 * l/n)"

Obviously the use of a zero shear viscosity AI0 introduces an uncertainty though for practical reasons cannot be avoided.

Presuming a linear relationship for the drag reduction with a slope of - 0.73. It should be possible to calculate the rheogram of the fluid, using equation 3-9-

Straight parallel lines for the drag reduction are often found. However, the slope can be different from the results presented above. Mlzushlna and Usul \197.1977\ give a theoretical basis to the drag reducing phenomenon showing a slope of approximately -1, also found for laminar flow, as shown in figure 3.9c. The theory used has been based on the damping of the turbulence by the polymers. The results of Gampert and Delgado \1985\ for the relation between the friction factor and the Reynolds number for the turbulent flow in a rectangular duct (figure 3-9f) are remarkable. They found a slope of about - 0.25 for the same type of PAAm, which has also been used by Wagner \19\. This effect can be related to both the use of a rectangular duct with a large aspect ratio for the measurements but perhaps mainly to a homogeneous state of dissolution as given by Bewersdorff XigS^X (figure 3-9e. discontinuous line), even though this state of dissolution has been assumed also for the results shown In figures 3.9a,b,c,d. On the other hand Darby \198\ has shown a decrease in the slope for fresh solutions. This can also be due to experimental reasons, such as degradation over the measuring traject for higher Reynolds numbers.

Taking the relation with a slope of - 0.73. which is an average for the results presented here for polyacrylamide solutions, the Reynolds number for the onset Re can be determined. With an increasing concentration of polyacrylamide the onset of drag reduction will occur at lower Reynolds numbers. Kor the very low Reynolds numbers the Blasius equation has been extrapolated. The effect of the geometry and the polymer additives can be calculated. An equation for the

62

friction factor versus the Reynolds number for the elastic fluids can be derived, postulating a slope of - 0.73 for this equation when the Deborah number on micro scale would become larger than 1. At the onset point, the Blasius equation (valid for U 1 0 3 < Re < 10 ), agrees with the relation used for the drag reducing fluids:

(3.10) log(Uf) = - 0.5 - 0.25 log(Re ) = - B - 0.73 log(Re )

This gives for the onset constant B (3.1D

B = - log(0.3l6 Re ' a ) = 0.5 - 0.U8 log(Re )

The time scale of the dissipation eddies is related to the overall Reynolds number by ( 3.2; eq. 3-2)

(3.12) t % D/U R e " ' 5 = e D 2 u _ 1 Re ~ 1 - 5 r o f

In paragraph 3.2 also a definition is given for the Deborah number, which will be 1 on micro scale for the onset point

(3.13) De = 1 a A/t s A (U/D) Re

The time scale of the fluid on micro scale in the onset point t will equal the shortest effective time of the fluid o relaxation spectrum, and can be calculated, using equations 3.10, 3-13 and the Blasius equation, assuming the friction velocity u. for the velocity scale of the interacting large scale vortices, according to

- (3.1U) __ 5.0 e 0 R e-l-375

D r "o

This time is related to the largest characteristic fluid time t , as defined by Rouse, by the relaxation spectrum. The time t identifies a polymer solution, and may be used for that reason (table 3-1). In chapter 2 an equation has been

63

given for the characteristic longest relaxation time t of the spectrum, which can be rewritten as

6 u. u. M sp s (3-15)

IT c R T

The time t . based on equation 3-l. is about a D

factor 30 smaller than the relaxation time, which can be found by extrapolating the data of Hill \1972\ for the PAAm solutions (figure 2.8). Reasons for this difference are the direct relation that has been assumed in equation 3-13t but most of all the different treatment of the solutions, giving less elasticity for the solutions in this work. Using a molar volume of 26 m /mol in equation 3-13 for t . a time of 0.03 s is found for the 300 ppm solution, while for t a

D value of 0.022 s has been extrapolated and calculated (table 3-1). Because of degradation the molar volume will in reality also be lower than that used above, especially for the low concentrations PAAm. It can be expected therefore that t forms an approximation of the characteristic fluid time found according to the Bird Carreau model. In figure 3-12 a direct relation between concentration PAAm and t_ can be

B seen as well, although the value of t has only been

B estimated. Because p. /c is proportional to the

sp concentration (figure 2.9). these relationships suggest that t can be identified with t., according to equation 3.13. B 1 S 0.04

0.02 -

-

B

"1 -

T

m^

'

i

""

1

1

tB = 69.4c

m,/1^

1

l

i

-

-

200 400 10-6 kg/kg

Figure 3-12 The characteristic time t versus the concentration of polyacrylamide.

6n

Using equations 3-13. 3-1 and 3-15 a relation can be given for the Reynolds number in terms of the onset point and the fluid properties

- 1 . 5 6 " M s p w s % t l " o Re % = = E l 2 2 2

T T R T C e D e D

The last term in this equation is the elasticity number (El). The number will also be used in chapter 6 and succeeding chapters to quantify the influence of the elasticity on the flow phenomena in stirred tank reactors.

Combination of equations 3-10 and 3-11 gives

(3.17) log(Hf) - - 0.5 + 0.8 log(Re ) - 0.73 log(Re)

Taking equations 3.11. 3-16 and 3.17 together gives a relation between the friction factor and the Reynolds number in the region for drag reduction:

log(f) = - k - 0.32 log(El) - 0.73 log(Re)

(3.18b) 1 M M H 6 M

- k. - 0.32 log( S P S ) - 0.73 log(Re) 2 2

D c ir e R T The relation is valid for Reynolds numbers larger than the onset Reynolds number and smaller than the Reynolds number at the Intersection of the ultimate drag reduction asymptote of Virk (equation 3-5) with the relation 3-18. The constant k in the equation 3-18 includes the interaction between the time spectrum in the turbulent flow and the fluid relaxation time spectrum. For the conditions used an estimation of k is 0.08. The second group in equation 3.18a and 3.18b is for practical signification. It gives the effect of the fluid elasticity and tube dimension on the drag. The part 1/D in equation 3.18b shows the Influence of the diameter on the drag

65

10-*

3.0

1.0

- 2 0

10-1

- 1 0

0 12 (Pas)'

Fig. 3-13 The characteristic time t and the onset Reynolds number Re versus the fluid dependent group of equations 3.16 and 3.18. showing the Influence of the elasticity on the drag reduction.

reduction. With an Increase of the tube diameter the drag reducing effect will decrease. The second part of this group, containing the fluid properties, depends on the concentration of polymer and the zero shear viscosity. The zero shear viscosity has been used as a approximation of the apparent viscosity in the experiments. In figure 3-13 the characteristic Reynolds number Re , determined by using equation 3.10 in figure 3.10, has been related to the concentration dependend group from equation 3-l8b. The value of the viscosity group will never be less than 2. This limit is set by the solution viscosity and the Intrinsic viscosity, with a value of 2.0 (figure 2.9). For purpose of convenience a straight line has been drawn, although an exponential increase can be seen. The experimental relation found according to figure 3-13 makes it acceptable to identify the fluid properties group with the elasticity number derived in

66

equation 3.18. The data for the drag reduction experiment are tabulated in table 3.1.

Within limits, the data show that the flow behaviour depends, in a simple way, on the polymer additives. For the experiments it is important that the fluid can be characterized by usins a characteristic fluid time and an elasticity number, found by a phenomenal interpretation of drag reduction in pipe line flow.

table 3.1

Effect of fluid properties on the dras reduction onset

concen c

ie"6

25 100 200 300 500

tration wo

10_3Pa s 1.05 1.25 1. 58 1.95 2.99

M M M o sp s c

(Pa s ) 2 2. 1 3. 1 k.6 6.2

12. 0

R 6f

10 3

17 5.6 3-0 1.9 1. 0

s 0.002 0. 008 O.Oltt 0. 022 0. 031

El

IQ"" c . o 8 0. 37 0.82 1. 60 3-78

3-6 VELOCITY PROFILES

3-6.1 Results with the UU mm tube

The velocity measurements in the larger tube give a good Impression of the effect of the visco-elastic fluids on the turbulent flow field. The instantaneous velocity, for instance , in the turbulent flow field can be split in a time averaged mean velocity U and the fluctuation velocity component u. The velocity fluctuations are characterized by u', which is defined by the root mean square (rms) value of u: u' = fu' . The turbulence intensity is defined by u'/U.

67

table 3-2

Parameters of the velocity measurements in the Ml mm tube. used for figures 3-1 to 3.17.

line no.

1 2 3 a 5 6

cone. PAAm c

ppm 0

0 80 80

An Experimental Study on Turbulent Mixing of Visco-elastic Fluids

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