design and scale-up of polycondensation reactors

DESIGN AND SCALE-UP OF POLYCONDENSATION REACTORS Hydrodynamics in horizontal stirred tanks and pervaporation membrane modules PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op dinsdag 10 december 2002 om 14.00 uur

door Gerardus Johannes Stefanus van der Gulik geboren te Uithuizen

Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. J.T.F. Keurentjes en prof.dr.ir. W.P.M. van Swaaij

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Gulik, Gerardus J.S. van der Design and scale-up of polycondensation reactors : hydrodynamics in horizontal stirred tanks and pervaporation membrane modules / by Gerardus J.S. van der Gulik. – Eindhoven : Technische Universiteit Eindhoven, 2002. Proefschrift. – ISBN 90-386-2704-1 NUR 913 Trefwoorden: chemische reactoren ; opschaling / hydrodynamica / membraantechnologie ; pervaporatie / menging / warmteconvectie / numerieke stromingsleer ; CFD / ultrasone computertomografie Subject headings: chemical reactors ; scale-up / hydrodynamics / membrane technology ; pervaporation / mixing / convective heat transfer / computational fluid dynamics ; CFD / ultrasonic computer tomography © Copyright 2002, G.J.S. van der Gulik Omslagontwerp: Paul Verspaget Druk: Universiteitsdrukkerij TU/e

Voor mijn ouders

Het in dit onderzoek beschreven onderzoek werd financieel gesteund door het Stan Ackermans Instituut, Akzo Nobel, Tejin Twaron en de NOVEM.

SUMMARY Condensation polymers are an important class of polymers, and have already been produced for a long time. For new processes, reactor design is often a time consuming process of trial and error, resulting in a reactor, for which little is known about the occurring physical and chemical conditions. Consequently, the conditions are difficult to reproduce at a different scale, making the quality of the produced polymer sensitive to the scale of operation. A proper insight into the hydrodynamics would enlarge the operating window and provides a route to a more straightforward scale-up. This includes the Residence Time Distribution, flow patterns, fluid velocities, and the mixing in the reactor under all flow conditions. The removal of water during polycondensations is also strongly influenced by the hydrodynamics. In this thesis, hydrodynamic aspects are studied for the design and scale-up of two types of reactor for the production of condensation polymers. The first type of reactor is a horizontal stirred tank reactor for which it is not straightforward to keep mixing conditions constant upon scale-up. In the second type of reactor, pervaporation membranes are used for the removal of water to favor product formation. Controlling hydrodynamics is important for increasing the heat and mass transfer between the membrane surface and the bulk liquid. Hydrodynamic aspects of the horizontal stirred tank reactor have been studied experimentally and numerically. The experimental characterisation has been performed using Planar Laser Induced Fluorescence, Pulse Response Measurements, Power measurements and Laser Doppler Anemometry. These techniques lead to information on overall circulation, poorly mixed zones, macro-mixing times, power consumption, velocities and turbulent quantities. Under laminar conditions the mixing in this reactor appears to be chaotic, which identifies the reactor as a good laminar mixer. Under turbulent conditions, however, the fluid mainly rotates like a solid body, which classifies the reactor as a moderate turbulent mixer. Macromixing times have been measured as a function of geometrical parameters and operation conditions. By correlation, equations are provided for scale-up of the reactor, while keeping the macro-mixing times constant. In more detail, the hydrodynamics in the horizontal reactor under turbulent conditions has been studied numerically using Computational Fluid Dynamics. For the turbulence modelling, the isotropic k-ε model and the anisotropic Differential Stress Model have been applied. A comparison of fluid velocities and turbulent quantities has been made with data from Laser Doppler Anemometry. The turbulence models proved to be able to describe the flow properties equally good. Passive scalar mixing could only be described properly using the anisotropic Differential Stress Model. Using these numerical tools, new routes come available for designing an improved reactor. Computational Fluid Dynamics has also been used for the design of a pervaporation membrane reactor. In this type of reactor it is important to reduce concentration and temperature polarization to obtain high water fluxes during operation. Polarization can effectively be reduced with secondary flow as induced by density differences. The secondary flow is found to be most effective in

horizontal set-ups increasing water fluxes up to 50%. The temperature segregation in the system, as induced by the secondary flow, has been measured using Ultrasonic Computer Tomography. With this experimental technique the propagation time of sound waves between several transducers is measured. The average temperature between the transducers can be calculated as the propagation time depends on temperature. With Computer Tomography, a 2D-distribution can be constructed from a number of average temperatures. The constructed 2Dtemperature distribution shows temperature segregation, which is a result of the secondary flow. This study is a compilation of numerical and experimental work performed on two reactors for polycondensation processes. From the work presented in this thesis it can be concluded that hydrodynamics has a major impact on the design and development of reactors for polycondensation reactions. CFD has been used as a numerical tool for studying hydrodynamics and designing reactors. The major challenge for the future will be to combine the insights in hydrodynamics with kinetic reaction schemes using CFD. This should lead to new reactor designs and modes of operation, providing reliable and sustainable processes for the future.

SAMENVATTING Condensatiepolymeren vormen een belangrijke klasse van polymeren die reeds gedurende lange tijd geproduceerd wordt. Bij het vergroten van de productieprocessen, moeten de reactoren veelal worden opgeschaald. Het reactorontwerp is dan echter vaak een tijdrovend proces van trial and error, resulterend in reactoren waarvan weinig bekend is over de heersende fysische en chemische condities. Bij schaalvergroting zijn de condities derhalve moeilijk te reproduceren, wat leidt tot ongewenste variaties in productkwaliteit. Goed inzicht in de hydrodynamica zou het werkgebied van de reactor kunnen vergroten en zou het opschalen kunnen vergemakkelijken. Onder de hydrodynamica wordt onder andere verstaan de verblijftijdspreiding, de mengpatronen, lokale vloeistofsnelheden en de menging in de reactor bij verschillende stromingsregimes. In dit proefschrift worden de resultaten beschreven van het onderzoek naar hydrodynamische aspecten van twee type reactoren waarin polycondensaties worden uitgevoerd. Het doel is de verkregen inzichten te gebruiken voor het opschalen en ontwerpen van de reactoren. Het eerste type reactor is een horizontale geroerde tankreactor waarbij de nadruk ligt op het voorspellen en controleren van de menging op verschillende schaalgroten. In het tweede type worden pervaporatie membranen toegepast voor het verwijderen van water uit het reactiemengsel waardoor productvorming wordt bevorderd. Het kunnen voorspellen en controleren van de hydrodynamica in dit type reactoren is belangrijk omdat daarmee de warmte- en stofoverdracht kunnen worden verbeterd tussen het membraanoppervlak en de vloeistofbulk. De hydrodynamische aspecten van de horizontaal geroerde reactor zijn zowel experimenteel als numeriek onderzocht. Experimenteel onderzoek is uitgevoerd met behulp de experimentele methoden Planar Laser Induced Fluorescence, pulsresponsie-metingen, Laser Doppler Anemometrie en vermogensmetingen. Deze methoden leveren inzicht in stromingspatronen, slecht gemengde zones, macromengtijden, vermogensverbruik, lokale vloeistofsnelheden en turbulente grootheden. Onder laminaire condities heeft de menging in de reactor een chaotisch karakter waardoor de menging relatief goed en snel verloopt. Onder turbulente omstandigheden draait de vloeistof onder invloed van de centrifugaalwerking voornamelijk rond zijn as als een star lichaam. De menging is dan niet erg effectief. Macromengijden zijn voor verscheidene condities bepaald als functie van geometrische variaties. Door correlatie van de data zijn empirische relaties afgeleid die kunnen worden toegepast om bij schaalvergroting de macromengtijden te kunnen voorspellen. Met behulp van de numerieke techniek Computational Fluid Dynamics, is de hydrodynamica in meer detail bestudeerd. Voor de turbulente modellering zijn het isotrope k-ε-model en het anisotrope Differential Stress Model gebruikt. Berekende vloeistofsnelheden en turbulente grootheden zijn vergeleken met data uit de Laser Doppler Anenometrie metingen. Beide turbulentiemodellen voorspellen de hoofdstromingen even goed. Opmenging van passieve scalairen, hetgeen wordt gebruikt om de mengsnelheid te kunnen kwantificeren, wordt alleen goed voorspeld met

behulp van het Differential Stress Model. Uitgebreidere toepassing van Computational Fluid Dynamics technieken zou het ontwerpen van dit type reactoren kunnen vergemakkelijken en versnellen. Compuational Fluid Dynamics technieken zijn ook toegepast bij het ontwerpen van het tweede type reactor; een pervaporatie membraanreactor. In dit type reactoren is het van belang de concentratie- en temperatuurpolarisatie te reduceren teneinde hoge waterfluxen te verkrijgen tijdens het productieproces. Polarisatie kan effectief worden gereduceerd door secundaire stroming die onstaat onder invloed verschillen in dichtheid in de vloeistof. Het is gebleken dat deze secundaire stroming het meest effectief is in horizontale opstellingen. Een toenamen van 50% in de waterflux wordt voorspeld. Aan temperatuurssegregatie in een horizontale modelopstelling zijn metingen verricht met behulp van Ultrasone Computer Tomografie. Met deze experimentele techniek wordt de voortplantingssnelheid gemeten van geluidsgolven tussen verschillende transducers. De gemiddelde temperatuur tussen twee transducers kan hiermee indirect worden vastgesteld omdat de voortplantingssnelheid afhankelijk is van de temperatuur. Met Computer Tomografie kan van een groot aantal gemiddelde temperaturen een 2-dimensionale temperatuursverdeling gereconstrueerd worden. Deze verdeling laat zien dat in de onderzochte horizontale membraanmodules de voorspelde temperatuurssegregatie inderdaad optreedt. Dit onderzoek is een compilatie van numeriek en experimenteel werk aan twee typen reactoren voor de productie van condensatiepolymeren. De hydrodynamica is van eminent belang voor het goed functioneren en kunnen ontwerpen van beide type reactoren. Grote vooruitgang kan worden geboekt door uitgebreidere toepassing van Computational Fluid Dynamics technieken. De grote uitdaging is te vinden in het combineren van de verkregen inzichten in de hydrodynamica en kinetische reactieschema’s in Computational Fluid Dynamics. Dit biedt perspectief voor nieuwe revolutionaire reactor ontwerpen en operationele condities, resulterend in betrouwbare en duurzame productieprocessen voor de toekomst.

CONTENTS

1. 2. 3. 4. 5.

THE NEED FOR CONTROLLING HYDRODYNAMICS IN POLYCONDENSATION REACTORS

1

HYDRODYNAMICS IN A HORIZONTAL STIRRED TANK REACTOR

15

HYDRODYNAMICS AND SCALE-UP OF HORIZONTAL STIRRED REACTORS

37

FLUID FLOW AND MIXING IN AN UNBAFFLED HORIZONTAL STIRRED TANK

59

HYDRODYNAMICS IN A CERAMIC PERVAPORATION MEMBRANE REACTOR FOR RESIN PRODUCTION

87

6.

MEASUREMENT OF 2D-TEMPERATURE DISTRIBUTIONS IN A PERVAPORATION MEMBRANE MODULE USING ULTRASONIC COMPUTER TOMOGRAPHY 107

7.

FUTURE PERSPECTIVES FOR PROCESS ENGINEERING OF POLYCONDENSATION REACTIONS DANKWOORD CURRICULUM VITAE

131

1 THE NEED FOR CONTROLLING HYDRODYNAMICS IN POLYCONDENSATION REACTORS

Chapter 1

2

1.1 Condensation polymers Condensation polymers are an important class of macromolecular products and include synthetic materials used as high-strength and/or high-toughness plastics and fibers (e.g., polyamides, polyesters and polycarbonates) as well as almost all hard resins (e.g., unsaturated polyesters, epoxy resins, urea-, melamine-, and phenol formaldehyde resins), thus covering a wide range of applications. Additionally, members of the same product family are used in relatively small but important sectors as sealants, elastomers, foams, and adhesive coatings (e.g., silicones, alkyd resins, and polyimides, Parodi et al., 1989). Condensation polymerization, also called step-growth polymerization, involves one or more reactants (monomers) possessing at least two reactive functional groups (Manaresi et al., 1989). In general, a polycondensation reaction can be described schematically in the following way: (n+1) A R1 A + n B R2 B

→ A R1 ( C R2 C R1 )n A + 2 n Q ←

(R.1)

End groups A and B react, forming a group C that becomes part of the polymer chain and a small molecule Q. The monomer molecules will disappear rapidly, and consequently, after a while only chains of monomers are coupling with other chains. In principle, the degree of polymerization is determined by a small excess of one of the monomers. When the relative excess equals 1/n, the average chain will contain n units1. n/(n+1) 1.0

1000

Pn [-]

0.995 0.99 100

0.975 0.95 0.925 0.9

10 0.9

0.92

0.94

0.96

0.98

1

Xc [-] Figure 1.1: Average degree of polymerization as a function of conversion Xc at different initial stoichiometric ratios n/(n+1). 1

The excess origins from the ratio of reactants in reaction R.1: (n+1)/n = 1 + 1/n

The need for controlling hydrodynamics in polycondensation reactors

3

The conversion, Xc, can be expressed as the degree of conversion of end groups, either based on A or B. In Figure 1.1, the number-averaged degree of polymerization 3n has been plotted against the degree of conversion Xc for several initial stoichiometric ratios n/(n+1). This figure shows that to obtain 100 units per chain with an initial stoichiometric ratio of 0.99, Xc has to be 0.995. Thus, to obtain long polymer chains both the stoichiometric ratio and Xc have to be close to unity. Often such long polymer chains are required to achieve the desired product quality.

4

Chapter 1

1.2 Reaction engineering of polycondensations The very high conversions that are required often lead to major technical problems (Thoenes, 1994; Vollbracht, 1989). For example, the small molecule Q (often water) may have to be removed to force the reaction to completion (Carothers, 1936; Sawada, 1976). This can be achieved by evaporation or chemical bonding. The removal of the small molecule from a viscous molten polycondensate is difficult as the surface tension and the viscosity of the liquid hinder the nucleation of vapor bubbles. Even when bubbles are formed, they have such a small volume that their rising velocity is extremely low. To obtain high conversions, the Residence Time Distribution (RTD) has to be very narrow (Thoenes, 1994; Westerterp et al., 1984; Biesenberger et al., 1983). Therefore, most often a batch reactor is used. A disadvantage of a batch reactor is that the viscosity of the reactor content increases dramatically during the process. This usually induces flow conditions to change from turbulent to laminar. A second disadvantage of a batch reactor is that at the beginning of the polycondensation process the reaction rate is high as the concentration of end groups is highest. Consequently, most heat of reaction has to be removed at the beginning of the process. It is difficult to remove all the heat within a short period of time from a large batch reactor because the surface to volume ratio is relatively low. Therefore, external cooling has to be applied or the scale of production has to be reduced. A number of reactors in series would sometimes be highly desirable, because different types of impeller can be used during different stages of the process and the heat production can be distributed over several reactors. However, a series of reactors can only be used when the RTD shows exact plug flow behavior or can be kept extremely narrow. For a polycondensation process it is difficult to find a suitable reactor, because the various requirements are often contradictory. The route to a successful polycondensation reactor is a time-consuming process of trial and error. From the resulting reactor, frequently little is known about the physical and chemical conditions. Consequently, these conditions are difficult to reproduce at a different scale, making the product quality sensitive to the scale of operation. However, successful operation upon scale-up can sometimes be maintained when several reactor-engineering aspects are known. Such aspects, for which generally the collective term ‘hydrodynamics’ is used, comprise the RTD of a reactor system, the flow patterns, the mixing under various flow conditions, and the mixing power. In this thesis, hydrodynamic aspects are studied for the design and scale-up of two types of reactor for the production of condensation polymers. The first type of reactor is a horizontal stirred tank reactor for which it is not straightforward to keep the mixing conditions constant upon scale-up. In the second type of reactor pervaporation membranes are used for the removal of the byproduct water.


5

1.3 Tools for studying hydrodynamics For the optimization of polymerization reactors, characterization of the occurring hydrodynamics is of prime importance. Two characterization methods are available: Experimental and Computational Fluid Dynamics (known as EFD and CFD, respectively). EFD is a collective term for several experimental techniques used to characterize the hydrodynamics. In this thesis the following experimental techniques have been used: •

•

•

•

•

Planar Laser Induced Fluorescence (PLIF) – With PLIF a fluorescent dye can be traced in a 2-dimensional laser sheet. Using a high-speed camera, a digital film can be made of the mixing pattern of the tracer in the reactor (Chapter 2; Miller, 1981; Owen, 1976). Pulse Response Measurements – With Pulse Response Measurements an inert tracer is injected in a batch reactor while at a different position the concentration of the tracer is monitored as a function of time. From these measurements macro-mixing times can be deduced (Chapter 3; Westerterp et al., 1984). Laser Doppler Anemometry (LDA) – LDA is an optical method for fluid flow research based on a combination of interference and Doppler effects. LDA allows the measurement of the local, instantaneous velocities of particles suspended in the flow. LDA has a high resolution power in time and is non-invasive (Chapter 4; Durst et al., 1981). Ultrasonic Computer Tomography (U-CT) – U-CT is based on the dependence of the propagation velocity of ultrasound on the temperature of a medium. Over a line between a speaker and a microphone the velocity of sound is measured and converted into an average line temperature. By measuring a large number of lines in a plane and using Computer Tomography, a 2-dimensional temperature distribution can be constructed. The technique is non-invasive (Chapter 6; Norton et al., 1984; Peyrin et al., 1983). Magnetic Resonance Imaging (MRI) – With MRI the proton density in a medium can be measured over a line. By measuring a large number of lines in a plane and using advanced computational techniques, a 2-dimensional proton density distribution can be constructed. This distribution can provide information on the location of different chemicals. The technique is non-invasive. Additionally, MRI can be used to measure local velocities by applying pulse field gradients (Chapter 7; Hornak).

CFD is a numerical tool for studying hydrodynamics. Fluid flow, mixing, heat and mass transfer in a prescribed geometry can be calculated on a computer. For this, several commercial packages are available, including CFX (AEA Technology, Harwell, UK), Fluent (Fluent Inc., Lebanon, USA), Star-CD (Computational Dynamics Ltd., London, UK). Only CFX has been used, in which both laminar and

Chapter 1

6

turbulent conditions can be handled. For turbulent flows, the Reynolds-Averaged Navier-Stokes equations can be solved with the use of appropriate turbulence models.

TDC H 2O

PPD

Figure 1.2: Flow state and initial location of the reactants in the Twaron® polymerization process.


7

1.4 Scale-up of a Horizontal Stirred Tank Reactor. The horizontal stirred tank reactor studied in this thesis is of the Drais type and is further referred to as the Drais reactor. The aromatic polyamide Twaron is produced in this type of reactor (Vollbracht, 1989; Banneberg-Wiggers et al., 1998). During the reaction, the flow regime changes from turbulent to laminar, due to a tremendous increase in viscosity. In both regimes mixing has to be sufficient as it has a large influence on the final product quality. The Twaron polyamide PPTA (Para-Phenylene TerphthalAmide) is produced via the condensation reaction of PPD (Para-Phenylene Diamide) with TDC (Terephthaloyl DiChloride). The fast propagation step is given as reaction (R.2): (n+1) H2N

NH2

+ n

O Cl C

O C Cl

H

H N

H O N C

O C

H N

NH2

+ n HCl

(R.2)

n

(PPD)

(PPTA or Twaron® polymer)

(TDC)

The degree of polymerization is controlled by adding a small amount of water that can terminate a reactive acylchloride group via the slow reaction (R.3): O R3 C

O Cl + H 2O

R3 C

O H + H Cl

(R.3)

Water is added at the start of the process because at the end of the process the viscosity is too high for mixing the water sufficiently with the reactor content. The initial presence of water, however, complicates the process, as it is able to terminate growing chains too early when mixing is insufficient, leading to short chains and a broad molecular weight distribution (MWD). The set of reactions (R.2 and R.3) is usually described as competitive-parallel. In production, the reactor is partially filled with the diamide component (containing the required amount of water), to which the diacylchloride component is added semi-batchwise. To allow for an exact stoichiometric ratio of the two monomers, only one injection point is used, as schematically depicted in Figure 1.2. The use of a single injection point implies that a good overall circulation is needed to allow the acylchloride molecules to react with all the amide-containing molecules throughout the reactor before termination with water occurs. Thus, a proper insight into the flow pattern, the mixing, and the mixing times are mandatory to guarantee constant product quality upon scale-up and to allow for quality improvements in existing equipment. These hydrodynamic aspects will be treated in the chapters 2 to 4 of this thesis.

Chapter 1

8

1.5 Design of the membrane reactor Alkyd resins are low molecular weight polyesters, formed by the equilibrium reaction between a di-alcolhol and a di-acid, as given in (R.4):

n R1 OH

+

n R2 COOH

Resin

+

H2O

(R.4)

Reaction conditions require elevated temperatures, typically above 200 °C. At these temperatures, the mixtures are moderately viscous (µ = 5-25 mPa·s). To obtain high yields, the reaction equilibrium has to be shifted to the right. This can be achieved by using a large excess of the alcohol (usually as solvent) and by removing the water. In current processes, water is removed afterwards or during the reaction by distillation (Figure 1.3A), which is not very efficient in the case of azeotrope formation. Additionally, due to the large reflux ratios required, energy consumption can be significant (Keurentjes et al., 1994). In this operation the energy input can be 10 times larger than the energy required for the removal of water.

H2O

Reactants Resin

Distillation unit

B Reactants Resin Reactor

A

Pervaporation membrane module

H2O

Reactor

C

Pervaporation membrane reactor

H22O

Reactants

Resin

Figure 1.3A-C: A) Current batch process for the production of resins with a distillation unit for the removal of water. B) Improved batch process in which the distillation unit has been replaced with a pervaporation membrane module. C) Continuous process in which the reactor and the distillation unit have been replaced with one pervaporation membrane reactor, running in a once-through continuous mode.


9

Membrane Water concentration Temperature

Water

Arbitrary units

Viscosity

Resin concentration

δT

δC

Figure 1.4: Concentrations, temperature and viscosity in arbitrary units near the membrane surface.

With ceramic pervaporation membranes, water can be removed selectively through the membrane by evaporation (Ho et al., 1992; Nunes et al., 2001; Rautenbach et al., 1984, 1985; Bakker et al., 1998; Koukou et al., 1999; Verkerk et al., 2001; Jafar et al., 2002). These membranes are selective to water as they are extremely hydrophilic and have pores that are about the kinetic diameter of water. Additionally, they are resistant to high temperatures. Use of these membranes allows for two alternative process schemes. In Figure 1.3B, a pervaporation membrane module is used instead of the evaporator. This bath-wise concept in Figure 1.3B is relatively easy to implement in current production processes, because only one unit operation has to be replaced. In Figure 1.3C, both the reactor and evaporator have been replaced by a membrane reactor. This concept is less easy to implement but has advantages, as it can be operated in continuous mode, which makes it easier to control the product quality. In both concepts, the reduction in energy consumption should be significant as, in principle, the only energy required is used for the evaporation of water. It can be anticipated that concentration and temperature polarization near the membrane represent a major problem. The polarization effects are schematically depicted in Figure 1.4. As water is transported through the membrane, the water concentration near the membrane surface will be low. As a result the resin concentration will be high, as the reaction will locally be in equilibrium. Also, the temperature will be low as water evaporates at the membrane surface adiabatically. Consequently, viscosity near the membrane will be relatively high. As a result, a relatively thick stagnant layer can be formed, severely reducing water transport from the bulk to the membrane surface. Optimizing the flow conditions near the membrane can reduce the occurring polarization effects. For the two process options given, different flow conditions are possible. For the process in Figure 1.3B, polarization can be reduced by applying

10

Chapter 1

high turbulence levels. For example, a powerful pump can recycle the liquid with high velocities over the membrane. However, this will reduce the energy savings. Using a powerful pump is not appropriate in the process depicted in Figure 1.3C, because velocities have to be low in order to limit equipment dimensions. Additionally, in Figure 1.3C nearly plug flow characteristics are required to obtain a narrow molecular weight distribution (MWD). It will be difficult to design a membrane reactor according to the process in Figure 1.3C with a flow pattern that reduces the polarization effects effectively, while fulfilling the requirements. In this thesis, the application of buoyancy forces is considered for obtaining a flow pattern that will effectively reduce polarization effects. Also, a relatively new experimental technique has been implemented for studying accompanying temperature effects.


11

1.6 Outline of this thesis This thesis describes the hydrodynamic aspects occurring in the two types of polycondensation reactor using both EFD and CFD. For the horizontal stirred tank reactor, fluid flow and mixing will be described both quantitatively and qualitatively. Insight into these hydrodynamic aspects will lead to efficient scale-up of the reactor in such a way that the product quality can be kept constant upon scale-up. For the membrane reactor the characterization of the flow and polarization effects should lead to the design of a once-through continuous reactor in which flow conditions are laminar. In Chapter 2, flow patterns of inert tracers under turbulent and laminar conditions are described. PLIF is used as the experimental technique. Experiments have been performed in three small-scale models of the Drais reactor. Mixing times have been determined, leading to guidelines for scale-up. In Chapter 3, the power input upon agitation has been measured in order to classify the configuration relative to other standard configurations. The energy input also provides the energy dissipation rate ε, which is crucial for defining well-mixed and poorly mixed regions. Mixing times have been measured using Pulse Response Measurements. Combined with the power measurements this leads to insight into the mixing efficiency as a function of scale. Chapter 4 is the final chapter on the Drais reactor describing LDA measurements under turbulent conditions. Mean and fluctuating velocities provide a general insight into the velocity scales and mixing processes occurring in the Drais reactor. LDA and PLIF measurements are compared with CFD calculations. To incorporate the turbulent properties, several turbulence models are available in CFD. Comparison shows that for a correct prediction of the PLIF experiments the selection of the appropriate turbulence models is the crucial parameter in CFD. Chapters 5 and 6 concern the pervaporation reactor. Chapter 5 contains a CFD study in which the effect of the orientation of a single membrane tube with respect to the gravity field has been studied. For these cases the relevance of buoyancy effects at laminar conditions has been examined. As an additional parameter the superficial velocity has been varied. Heat and mass transfer rates provide the essential parameters to distinguish the most optimal configuration. Chapter 6 describes the successful implementation of U-CT (Ultrasonic Computer Tomography) in a pervaporation membrane module. With this relatively new experimental technique, 2-dimensional temperature distributions can be measured. The measured temperature distributions are compared with CFD calculations.

12

Chapter 1

A review will be given in Chapter 7. Additionally, the implications for scaleup and future designs will be discussed. The set-up of this thesis is such that each chapter can be read separately. Consequently, some information will be repeated at more than one location. This approach has been chosen deliberately in order to enable the reader to go only through specific chapters of interest and to avoid a long list of references to other parts of this thesis.


13

References Bakker, W.J.W.; Bos, I.A.A.C.M.; Rutten, W.L.P.; Keurentjes, J.T.F.; Wessling, M.; “Application of ceramic pervaporation membranes in polycondensation reactions”, Int. Conf. Inorganic Membranes, Nagano, Japan, 1998, 448-451. Bannenberg-Wiggers, A.E.M.; Van Omme, J.A.; Surquin, J.M.; “Process for the batchwise preparation of poly-p-terephtalamide”, U.S. Pat., 5,726,275, 1998. Biesenberger, J.A.; Sebastian, D.H.; “Principles of Polymerization engineering”, John Wiley and Sons, New York, 1983. Carothers, W.H.; Trans. Faraday. Soc., 1936, 32, 39. Durst, F.; Melling, A.; Whitelaw, J.H.; “Principles and Practice of Laser Doppler Anemometry”, Academic Press, London, UK, 1981. Ho, W.S.W.; Sirkar, K.K. (Eds.); “Membrane Handbook”, Chapman & Hall, New York, 1992. Hornak, J.P.; “The basics of MRI”, http://www.cis.rit.edu/class/schp730/bmri/bmri.htm. Jafar, J.J.; Budd, P.M.; Hughes, R.; “Enhancement of esterification reaction yield using zeolite A vapour permeation membrane”, J. Membr. Sci., 2002, 199 (1-2), 117-123. Keurentjes, J.T.F.; Janssen, G.H.R.; Gorissen, J.J.; “The esterification of tartaric acid with ethanol: kinetics and shifting the equilibrium by means of pervaporation”, Chem. Eng. Sci., 1994, 49, 4681-4689. Koukou, M.K.; Papayannakos, H.; Markatos, N.C.; Bracht, M.; Van Veen, H.M.; Roskam, A.; “Performance of ceramic membranes at elevated pressure and temperature: effect of non-ideal flow conditions in a pilot scale membrane separator”, J. Membr. Sci., 1999, 155, 241-259. Manaresi, P.; Munari, A.; “Factors affecting rate of polymerization”, Comprehensive Polymer Science, Step Polymerization, 1989, 5, 35. Miller, J.N.; “Standard in fluorescence spectrometry”, Chapman & Hall, London, UK, 1981. Norton, S.J.; Testardi, L.R.; Wadley, H.N.G.; “Reconstruction internal temperature distributions from ultrasonic time-of-flight tomography and dimensional resonance measurements”, J. Res. Natl. Bur. Stand., 1984, 89(1), 65-74. Nunes, S.P.; Peinemann, K.-V. (Eds.); “Membrane Technology in the Chemical Industry”, Wiley-VCH, Weinheim, 2001. Owen, F.R.; “Simultaneously laser measurements of instantaneous velocity and concentration in turbulent mixing flows”, AGARD-CP193, Paper No. 27, 1976. Parodi, F.; Russo, S.; “Polycondensation and Related Reactions”, Comprehensive Polymer Science, Step Polymerization, 1989, 5, 1. Peyrin, F.; Odet, C.; Fleischmann, P.; Perdrix, M.; “Mapping of internal material temperature with ultrasonic computed tomography”, Ultrason. Imag., Conference Proceeding, July 1983, Halifax, 31-36. Rautenbach, R.; Albrecht, R.; “On the behavior of asymmetric membranes in pervaporation”, J. Membr. Sci., 1984, 19, 1-22. Rautenbach, R.; Albrecht, R.; “The separation potential of pervaporation. Part 2. Process design and economics”, J. Membr. Sci., 1985, 25, 25-54. Sawada, H.; “Thermodynamics in polymerization”, Chapter 6, Marcel Dekker, New York, 1976. Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. Verkerk, A.W.; Van Male, P.; Vorstman, M.A.G.; Keurentjes, J.T.F.; “Description of dehydration performance of amorphous silica pervaporation membranes”, J. Membr. Sci., 2001, 193(2), 227-238.

14

Chapter 1

Vollbracht, L.; “Aromatic Polyamides”, Comprehensive Polymer Science, Step Polymerization, 1989, 5, 374. Westerterp, K.R.; Van Swaaij, W.P.M.; Beenackers, A.A.C.M.; “Chemical Reactor Design and Operation”, John Wiley and Sons, New York, 1984.

2 HYDRODYNAMICS IN A HORIZONTAL STIRRED TANK REACTOR Abstract In this study, the hydrodynamics in a horizontal stirred tank reactor is investigated. This type of reactor is used in industry for fast polycondensation processes. Overall circulation, poorly mixed zones and macro-mixing times are determined in scale models under turbulent (Re > 105) and laminar (Re < 300) conditions using planar laser induced fluorescence. At both sets of conditions, the observed overall circulation is complex and changes when the length-todiameter ratio is varied. Under laminar conditions, the flow appears to be chaotic. The poorly mixed zones change in location, number, and life span at different length-to-diameter ratios. Dimensionless macromixing times under turbulent conditions are correlated with parameters variations and show nonlinear relationships in fill ratio, length-to-diameter ratio, and Reynolds number. Under laminar conditions, macro-mixing times could not be determined unambiguously, but they are only 2.5 times larger than under turbulent conditions.

This chapter is a slightly modified version of the publication: Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics in a Horizontal Stirred Tank Reactor”, Ind. Eng. Chem. Res., 2001, 40(3), 785-794.

Chapter 2

16

2.1 Introduction The design of a reactor for fast polycondensations is a major challenge for chemical engineers, as often several conflicting needs have to be fulfilled. Generally, two types of agitation are needed as the flow regime changes from turbulent to laminar, because of a tremendous increase in viscosity. In both regimes the mixing has to be sufficient as it has a large influence on the final product quality (Thoenes, 1994; Manaresi et al., 1989). This influence can easily be understood by considering the production process of Twaron, an aromatic polyamide produced via the polycondensation reaction of a diamide with a diacyl chloride (Gaymans et al., 1989; Vollbracht, 1989).The fast propagation step is given as reaction R.5: O (n+1) H2N R1

NH2

+ n Cl

O

C

R2

C

H

O

Cl

(R.5) H

H N R1 N

O H

C R2 C

N R1

H N

H + n H Cl

n

Theoretically, a small excess of the diamide determines the degree of polymerization. When the relative excess equals 1/n, the average chain will contain n+1 units. In practice, the degree of polymerization is controlled by adding water that can terminate a reactive acyl chloride group via the slow reaction R.6: O R3 C

O Cl + H 2O

R3 C

O H + H Cl

(R.6)

Water is added at the start of the process because, at the end of the process, the viscosity is too high to mix water sufficiently to molecular scale. The initial presence of water complicates the process, as water is able to terminate chains too early when mixing is insufficient, leading to short chains and a broad molecular weight distribution (MWD). Table 2.1: Order of magnitude of reaction time for reactions R.5 and R.6 (Jeurissen et al.; Borkent, 1976). Viscosity [Pa·s] Reaction time for propagation R.5 [s] Reaction time for termination R.6 [s] 10-3 10-4 102 10 10 102

In Table 2.1, the orders of magnitude of the reaction half-life times for both reactions are given for low and high viscosity levels. Going from low to high viscosity, the propagation rate slows by 5 orders of magnitude, while the termination rate remains unchanged. The propagation reaction is slowed down for two reasons. Firstly, RNH3+Cl--groups are formed, which are less reactive than

Hydrodynamics in a Horizontal Stirred Tank Reactor Injection L

Front C

D

17

Side

β h Hatch

Top

w

α

Figure 2.1: The Drais reactor, given in front, top and side views with length L, diameter D, blade angles α and β, blade width w, blade height h, and a hatch. The arrows point out the direction of rotation during operation.

NH2-groups (Gaymans et al., 1989). Secondly, the reactive end groups are more sterically hindered upon an increase in molecular weight. The chain stopper (water) will not be hindered as much, because it is a relatively small molecule. In production, the reactor is filled with the diamide-component (containing the required amount of water), to which the diacyl-component is added semi-batchwise. To allow for an exact stoichiometric ratio of the two monomers, only one single injection point is used. This implies that good overall circulation is needed to allow the acyl molecules to react with all the amide-containing molecules throughout the reactor before termination occurs. This implies that a fundamental insight into the hydrodynamic behavior of this type of reactor is mandatory to guarantee constant product quality upon scale-up and to allow for quality improvements in existing equipment. The polymerization described above is performed in a horizontal stirred tank reactor of the Drais type (Vollbracht, 1989). This multifunctional reactor, as depicted in Figures 2.1 and 2.2, can be used for powder mixing, turbulent fluid mixing and kneading at high viscosities with an energy dissipation up to 200 W/kg. Literature on the hydrodynamics in horizontal mixing vessels is very limited compared to the literature on vertical vessels. There is some literature on turbulent mixing, but literature on laminar mixing is absent. Ando et al. (1971a) studied power consumption and flow behavior under turbulent conditions in an unbaffled horizontal vessel with Rushton turbine impellers with Di/D = 0.9. They distinguish two flow states A and B. State A is obtained at a relatively low stirrer speed. The

18

Chapter 2

liquid is then pushed up by the impellers and sprayed, leading to the formation of fine liquid droplets and fine air bubbles. State B, the so-called hollow state, is obtained at a higher stirrer speed, providing a ring of fluid. As their research mainly focused on applications to gas-liquid absorption, mainly state A in baffled vessels was investigated (Ando et al., 1971b, 1974, 1981; Fukuda, 1990). This is due to a larger gas-liquid interface in the A state compared to state B. Ando et al. (1990) also studied turbulent mixing in an idealized horizontal vessel with baffles and multiple impellers. Macro-mixing times were measured, and a model was proposed for predicting them. It was established that the dimensionless macro-mixing time N⋅t is proportional to L/D. The information available in the literature on turbulent mixing in horizontal stirred tank reactors is not directly applicable to polycondensations in the Drais reactor for two reasons. First, the impeller geometry is completely different. Second, the fluid in the polycondensation process is in the hollow state (or the B state according to Ando et al., 1971a) because of a high stirrer speed. Therefore, we conducted an experimental study on the hydrodynamics in this type of reactor. For this purpose, mixing patterns, the life span of poorly mixed zones, and the macro-mixing time have been established experimentally. This is done for turbulent as well as laminar conditions for different reactor fill ratios. Subsequently, scaling rules will be defined based on these macro-mixing times.

Hydrodynamics in a Horizontal Stirred Tank Reactor

Figure 2.2: Stirrer geometry for reactor-15 (top) and reactor-20 (bottom). The white blades transport fluid to the right during rotation, the black stirrers transport fluid to the left. Thus, both have a pumping action toward the center of the reactor. The blades are evenly distributed over the shaft and are not drawn in perspective.

19

20

Chapter 2

2.2 Experimental Section The reactor used in this study was a horizontal stirred tank of the Drais type (Turbulent Schnellmischer, Drais Ltd, Mannheim, Germany), as depicted in Figures 2.1 and 2.2. Typical for the unbaffled cylindrical reactor is its horizontal position and the heavily designed impeller. This impeller can provide a high mixing power, needed to achieve sufficient mixing under highly viscous conditions. The reactor is characterized by length L and diameter D. Typical for the reactor is the clearance C, the distance between the blades and the reactor wall. For a small clearance, the blades perform a scraping action that keeps the reactor walls free from polymer material and also provides good heat exchange with the cooled walls. The blades have a pumping action towards the reactor center, providing an easy way to empty the reactor through the opened hatch. To determine macro-mixing times and the life span of poorly mixed zones, planar laser induced fluorescence (PLIF) was used. With PLIF, it is possible to make a digital film of the mixing of a tracer in a 2-dimensional plane in which the poorly mixed zones can easily be located. In contrast with other studies (Distelhoff et al., 1997; Kersting et al., 1995; Mayr et al., 1994; Moo-Young et al., 1972; Perona et al., 1998). PLIF is unobtrusive, has a small measurement volume, and is flexible in changing the monitoring point, as only the position of the laser sheet has to be changed. PLIF experiments were performed in three small-scale models of the Drais reactor. These reactors, as depicted in Figures 2.1 and 2.2, were all 0.18 m in diameter but differed in length, being 0.20, 0.27, and 0.36 m, providing L/D ratios of 1.1, 1.5, and 2.0, respectively. From this point on, these scale-models are referred to as reactor-11, reactor-15, and reactor-20, respectively. The blade width w was equal to 0.1 m, the blade height h was equal to 0.015 m, and the shaft had a diameter of 0.03 m. The blades were evenly distributed over the shaft for each reactor. The mutual angle of the blades was 180, 120, and 135° for reactor-11, reactor-15, and reactor-20, respectively. These angles correspond to industrial configurations and proved to provide the most stable fluid ring in partially filled reactors. The sidewalls, the shaft, and the impeller blades were made of stainless steel, and the cylindrical wall of glass. Under turbulent conditions, tap water was used as the reactor content for the three fill ratios 40, 60, and 100%. The rotational speed varied between 3.8 and 11.6 Hz, resulting in turbulent flow with Reynolds numbers ranging from 123,000 to 375,000, as defined by ρNDi2/µ. When the reactor was partially filled, the rotational speeds always resulted in the hollow state or the B state according to Ando et al. (1971a). For laminar conditions, glycerin (Heybroek, Amsterdam; purity >99.9%) was used at Reynolds numbers ranging from 90 to 270. Glycerin limits the use of PLIF to the examination of completely filled reactors because air bubbles lead to an untransparent fluid in partially filled reactors. Figure 2.3 shows the experimental arrangement for the PLIF experiments, which is comparable to the set-up used by Schoenmakers et al. (1997). The laser beam was generated by a 2-W Ar/Kr-laser (model Stabilite 2017-005, Spectra


21

Lens δ(t)

x Region of interest

Laser

.

Lasersheet

Camera PC

Figure 2.3: Sketch of the experimental set-up.

Physics) and had a wavelength of 488 nm. The beam was converted to a laser sheet with a thickness of 0.5 mm by a cylindrical lens (Dantec 9080XO.21). The position of the laser sheet in the vessel geometry was always parallel to the shaft. Therefore, the observed mixing process was always the mixing in the axial and radial directions. This is secondary mixing, superimposed on the mixing in the tangential direction. Disodium fluorescein (C20H10O5Na2) was used as the fluorescent dye (Merck, Darmstadt; purity >98wt%). This dye emits light with an intensity depending on the power of the laser light, the concentration, and the pH of the solvent. The power of the laser light was kept constant at 0.4 W. During every experiment, the final dye concentration was around 10-7 M. Therefore, the amount of injected solution, with dye concentration of 2·10-3 M, varied between 0.3 and 0.5 mL, depending on the reactor volume and fill ratio. The pH of both the injected solution and the reactor content was kept constant at a value of 10, as the intensity of the emitted light is independent of the pH at pH > 8. A high-speed camera (JAI CV-M30), connected to a PC, with an EISA compliant frame grabber (Magic) recorded the light that was emitted by the fluorescein molecules in the laser light plane. The commercial software package DMA-MAGIC was used for data acquisition. The value of the recorded gray scales ranged from 0 to 255, providing a resolution of 256 values. For complete mixing, a gray scale of around 150 was obtained. In the range from 0 to 255 the gray scale corresponds linearly with concentration. In Figure 2.3, the evaluated region is depicted as a dotted rectangle. This region was always set to the left part of the reactor. It was sufficient to monitor mixing in one half of the reactor because we observed that mixing was symmetrical with respect to the reactor center. The number of recorded images per second ranged between 30 and 120 and depended on impeller speed and expected mixing time. The number of pixels per image in the axial direction, i.e., from injection point to sidewall, ranged from 90 to 150 pixels

22

Chapter 2

for reactor-11 and reactor-20, respectively. In the radial direction, i.e., from shaft to cylindrical wall, the number was 68 for every image. This results in a spatial resolution of about 1×1×0.5 mm per pixel.


23

2.3 Results and Discussion This section starts with a brief description of the mixing pattern and the establishment of poorly mixed zones or islands, as observed in the PLIF images. Then, the technique for measuring concentrations and mixing times is shown. Finally, the mixing times are correlated with process parameters in order to formulate empirical correlations for scale-up.

2.3.1 Mixing Patterns and Chaotic Mixing The mixing of the injected dye in the axial and radial direction in reactor-11 is shown in Figures 2.4 and 2.5 for turbulent and laminar conditions, respectively. Although not the same, these mixing patterns show a resemblance. In Figures 2.4b and 2.5a, it can be seen that the dye is mainly transported in the axial direction as it flows from the central injection point towards the sidewall. Figures 2.4d and 2.5b show that the dye is subsequently transported in the radial direction along the sidewall, followed by transport to the bulk along the shaft in the axial direction. The overall circulation in Figures 2.4 and 2.5 is therefore turning counterclockwise. This is the opposite of what was expected, based on the center-oriented pumping action of the stirrer blades.

2.4a: 0.35 s.

2.4b: 0.55 s.

2.4c: 0.93 s.

2.4d: 1.65 s.

2.4e: 2.75 s.

2.4f: 5.6 s.

Figure 2.4: Mixing pattern in water in reactor-11 at N = 6 Hz.

Chapter 2

24

2.5a: 0.67 s.

2.5b: 1.0 s.

2.5c: 1.5 s.

2.5d: 2.37 s.

2.5e: 4.0 s.

2.5f: 7.4 s.

Figure 2.5: Mixing pattern in glycerin in reactor-11 at N = 8 Hz.

2.6a: 0.48 s.

2.6d: 2.50 s.

2.6b: 0.79 s.

2.6c: 1.50 s.

2.6f: 9 s. 2.6e: 4.01 s. Figure 2.6: Mixing pattern in water in reactor-20 at N = 6 Hz.

2.7a: 0.7 s.

2.7b: 1.5 s.

2.7c: 2.2 s.

2.7d: 3.1 s.

2.7e: 7.7 s.

2.7f: 8.0 s.

Figure 2.7: Mixing pattern in glycerin in reactor-20 at N = 6 Hz.


25

Figures 2.6 and 2.7 show how the dye is mixed in reactor-20. The PLIF images show that two regions are present that differ in overall circulation: one at the lefthand side of the PLIF image and one at the right-hand side. The circulation at the left-hand side is counterclockwise and thus shows resemblance with the circulation in reactor-11. The circulation at the right-hand side is clockwise. This circulation is visible in Panels a and b of Figures 2.7 in which the injected dye is transported first in the radial direction towards the shaft and subsequently in the axial direction along the shaft. By way of illustration, the generalized overall circulation in the radial and axial directions is depicted in Figure 2.8 for reactor-11, reactor-15 and reactor-20. The flow in Figures 2.5 and 2.7 can be considered as a 3-dimensional discontinuous periodic flow in which the impeller blades provide the discontinuous movement with a period equal to 1/N. This allows a comparison with the twodimensional flow in a cavity as studied by Leong and Ottino (1990) and Ottino (1991). Clearly, the laminar flow has a chaotic nature as it is capable of stretching and folding a region of fluid and returning it - stretched and folded - to its initial location after one period, i.e., one impeller revolution. Furthermore, the formed striations are reoriented when the impeller blades cross them. This event is visible in Figure 2.5d, in which the vertical impeller arm, as present in the right-hand side of the laser sheet, plows through the horizontal striations. These reorientations further enhance chaotic mixing.

Figure 2.8: Observed overall circulation in reactor-11, reactor-15, and reactor-20.

Chapter 2

26 3

100

Perimeter [cm]

80

-1

Liapunov-exponent σ [s ]

2.5 2

Reactor-20 N = 11 Hz

60 40 20 0 0

0.25

0.5

0.75

1

t [s]

1.25

1.5

1.75

2

1.5

Reactor-11 Reactor-15 Reactor-20

1

σ = 0.22N 0.5 0 0

2

4

6

N [Hz]

8

10

12

Figure 2.9: Liapunov exponents σ as a function of impeller frequency N. The inset provides the perimeter against time for reactor-20 at 11 Hz. The curve represents PE=PE0·exp(σ·t).

A quantitative indication for chaotic behavior is the Liapunov exponent σ in the formula PE=PE0·exp(σ·t). This formula (Leong and Ottino, 1989; Ottino, 1991) represents the stretching rate of the flow as it describes the perimeter of intermaterial area between the dye and the clear fluid as a function of time. A positive exponent σ implies exponential generation of intermaterial interface and, hence, implies chaotic flow. The inset in Figure 2.9 gives the perimeter against for reactor-20 at N=11 Hz. The initial exponential increase yields the Liapunov exponent by a fit procedure for which all the perimeter values after the maximum has been reached are ignored. The perimeter decreases after the maximum has been reached because the striations are lost when they become smaller than the pixel size. For all experiments under laminar conditions, the Liapunov exponent is given in Figure 2.9 as a function of impeller speed. The positive exponents increase linearly with impeller speed and appear to be independent of L/D. This indicates that chaotic mixing in all three reactors occurs in a similar fashion.

2.3.2 Poorly Mixed Zones and Islands Under turbulent conditions, poorly mixed zones are visible in Figures 2.4 and 2.6 as the areas in the reactors that remain dark the longest. These areas are visible in Figures 2.4e and 2.6e (reactor-11 and reactor-20), near the shaft between the two outer impeller blades (the same holds for reactor-15, although not shown here). Apparently, under turbulent conditions the location does not depend on L/D. Nevertheless, the poorly mixed zones are not very stagnant, as they disappear within seconds through turbulent dispersion.


27

Under laminar conditions, the islands (as a poorly mixed zone is usually referred to under laminar conditions) in reactor-15 and reactor-11 (Figure 2.5e) are located below the injection point. These islands are unstable and consequently disappear within seconds (compare e.g., panels e and f of Figure 2.5). This observation leads to the conclusion that reactor-11 and reactor-15 are globally chaotic. Leong and Ottino (1989) indicate that the existence of multiple folds along the island boundary is indicative for the instability of islands. The presence of a ‘rough’ island boundary in Figure 2.5e can be regarded as an indication for this. In reactor-20 (Figure 2.7e and f) four islands are visible throughout the reactor. These islands are very stable in the range of impeller frequencies applied. This leads to the conclusion that reactor-20 is not globally chaotic. The four islands, in fact, form four segregated torii, fluid elements that have often been observed in mixing vessels (Dong et al., 1994; Hoogendoorn et al., 1967; Lamberto et al., 1996; Lamberto et al., 1999; Nomura et al., 1997). These fluid elements act as barriers to mixing and are therefore highly undesirable in the polycondensation process as their existence allows early termination, hence leading to an undesired broad MWD. Probably, the segregated torii can be terminated by changing the mutual angle between the blades or by periodically changing the impeller speed (Harvey III et al., 1997; Lamberto et al., 1996; Leong et al., 1989; Unger et al., 1999).

2.3.3 Macro-Mixing Times under Turbulent Conditions For turbulent conditions, mixing is quantified by determining macro-mixing 1.1 1 0.9 0.8

G/Gn [-]

0.7 0.6 0.5 0.4 6

0.3

12 24

3.5

18

0.2 0.1 0 0

1

2

3

4

5

6

7

8

Time (in s) Figure 2.10: Normalized gray scale, plotted versus time, in reactor-11 at 6 Hz with a fill ratio of 40%. The dots represent the raw data, and the line represents the data after FFT filtering. The frequency plot from the Fourier transformation is depicted in the inset.

Chapter 2

28

times from response curves. These curves are created by plotting gray scales at a certain position in the PLIF images against the elapsed time. The chosen position is located as depicted in Figure 2.3 by an ‘x’ and is exactly two pixels from the sidewall and two pixels from the cylindrical wall. This position is selected to be representative for the mixing in the whole reactor as it is covered with fluid at every fill ratio. An example of a response curve is depicted in Figure 2.10 for the mixing in reactor-11, 40% filled with water. The dots represent the raw data after normalization on final gray scale. The maximum in gray scale is often obtained because of the appearance of the blades and air bubbles in the measurement point as a consequence of the presence of 60% air in the reactor. Because the stirrer had a constant speed, the disturbance of the blades could be removed by using fast Fourier transformation filtering (FFT filtering) from the commercial software package TablecurveTM by Jandel Scientific. A standard 40% smoothing level is used to zero 80% of the higher frequency components and all stirrer-related frequencies, resulting in the line in Figure 2.10. The accompanying frequency plot is also given in Figure 2.10 and is discussed in the Appendix. 1.6 1.4

G/Gn [-]

1.2 1 0.8 0.6

1.1

0.4

1.5

0.2

2

0 0

1

2

3

4

5

6

7

8

9 10 11 12 13

Time [s] Figure 2.11: Normalized gray scale plotted versus time at 6 Hz in reactor-11, reactor-15, and reactor-20 at a fill ratio of 100%.

Figure 2.11 shows normalized response curves after FFT filtering at 6 Hz in all three reactors that are completely filled. The profiles for reactor-11 and reactor-15 exceed unity, meaning that the dye is preferentially transported in the axial direction, toward the position where the gray scales are recorded. For reactor-20 the profile gradually rises to unity, without exceeding this limit. From Figure 2.11 it is concluded that when L/D is increased, it will take longer to mix the dye to the final concentration. Macro-mixing times are determined from the response curves after FFT filtering, like in Figure 2.11. As a representative value, the time that the normalized concentration differed by less than 10% from the final concentration was chosen. In Figure 2.12, these mixing times, hereafter referred to as t10, are represented as a function of the stirrer speed. The mixing times decrease with increasing impeller


29 Reactor-11, x=0.4

8

Reactor-11, x=0.6 Reactor-11, x=1.0

7


6


T10 [s]

5 4 3 2 1 0 3

4

5

6

7

8

9

10

11

12

N [Hz] Figure 2.12: Mixing time versus impeller frequency for the three reactors at three different fill ratios for turbulent conditions.

speed, whereas the macro-mixing times increase with increasing L/D. This is due to the fact that the distance between injection point and measuring position is larger with higher L/D. A fill ratio of 60% results in shorter mixing times than those for 100% or 40%. Mixing at a fill ratio of 60% can be shorter than mixing at 100%, because the slowest mixed zone near the shaft is absent at 60%. The difference from the mixing times at 40% can be a result of a lower overall circulation at 40%. From this it is suggested that, for a good circulation, a minimum amount of fluid is needed. These results also reveal that the mixing time is not linearly dependent on the fill ratio. It is supposed that the macro-mixing time will depend on the process and reactor variables as follows:

t m = t m ( ρ , µ , N , D, g , geometrical dimensions of the system)

(2.1)

Then, using dimensional analysis, the functional relationship can be arranged as

N D2 N 2 ⋅ D , geometrical dimensions as ratios N ⋅ t m = tm ρ ⋅ ⋅ , g µ

(

)

(2.2)

which applies to mixing vessels in general. The polycondensation process is operated in the hollow state in which the Froude-number N2·D/g is irrelevant. This leaves L/D as the only relevant geometrical dimension. When the obtained mixing times are correlated with the parameters varied by applying the relevant

Chapter 2

30

dimensionless numbers as given in equation 2.2, the following empirical correlation holds for D = 0.18m:

N ⋅ t10 = 16 ⋅ f ( x ) ⋅ Re

0.11

1.21

()

L ⋅ D

(2.3)

with

f ( x ) = 0.22 + ( x − 0.70) 2

(2.4)

A power series was chosen to describe the dependence of the mixing time on the fill ratio. The dimensionless mixing time N·t10 increases with the Reynolds number, although the contribution is less than 15%. The influence is small, as the mixing is turbulent over the entire range of applied impeller speeds. However, the positive power indicates that an increase in impeller frequency results in an increase in dimensionless mixing time. This suggests that mixing is less efficient at high stirrer speeds as the fluid tends to more solid-body rotation. The power of 1.2 in L/D indicates that the mixing mechanism is a combination of convection and dispersion, as 1 would indicate full convective flow and 2 full dispersive flow. As mentioned in the Introduction, Ando et al. (1971a) studied mixing in horizontal vessels with baffles in order to prevent the tendency to solid-body rotation. Their dimensionless mixing time correlated with L/D, indicating a larger convective contribution due to the baffles. Mixing in the reactor investigated here is more dispersive because of the absence of baffles, resulting in a dependency of (L/D)1.2 and, therefore, larger mixing times. Applying baffles can decrease mixing times under turbulent conditions. For high viscosity levels, however, baffles are not required, as viscous shear will damp out behind these baffles.

2.3.4 Macro-Mixing Times for Laminar Conditions Using the same black-box approach as used for turbulent conditions, macromixing times could be determined for laminar conditions. In the range of the parameters varied and with D = 0.18m, the empirical correlation N·t10 = 60·(L/D) can be obtained. This result is in good agreement with the correlations provided by Hoogendoorn and Hartog (1967) and Novak and Rieger (1975) for helical ribbon impellers in vertical vessels. According to this equation, mixing under laminar conditions is only 2.5 times slower than under turbulent conditions. However, establishing the mixing time in this manner is of course troublesome as the segregated torii in reactor-20 do not disappear within the determined macro-mixing times. A second method for quantifying mixing times is setting the macro-mixing time equal to the life span of the islands in reactor-11 and reactor-15. In the polycondensation process it is important to minimize this life span as the presence of islands implies a concentration ratio deviating from unity. This deviation results in an increased possibility of early termination, thereby leading to an undesired


31

40 35

Mixing time [s]

30

tm ∝

25

L/D=1.5

1 N 2 .6

L/D=1.1

20 15

tm ∝

10

1 N 1.3

5 0 4

5

6

7

8

9

10

11

12

N [Hz]

Figure 2.13: Life span of islands versus N, in reactor-11 and reactor-15 for laminar conditions.

broad MWD. In Figure 2.13, the life spans are plotted against impeller frequency. From Figure 2.13, it follows that the life span decreases with increasing impeller speed. It also follows that, at low impeller frequency, the life span in reactor-15 is larger than in reactor-11, whereas the two are comparable at high frequencies.

2.3.5 Mixing at Intermediate Viscosities In this study, we use only water and glycerin, both with Newtonian behavior. For the polycondensation process, one can imagine that viscosity and rheologic behavior will change continuously, and mixing behavior will go through a wide range of scenarios. Leong and Ottino (1990) investigated chaotic mixing in viscoelastic 2-dimensional flows at various viscosities. Upon an increase in viscosity, islands grow, and chaotic regions shrink. From this observation, it appears plausible to conclude, that, between our extreme cases (water and glycerin), nothing dramatic will occur. However, additional experiments by Leong and Ottino in which the shear rate was increased show that, in viscoelastic fluids, the number of islands increases, whereas in Newtonian fluids, this number is constant. Translating this observation to our practical situation indicates that, at higher impeller speeds (higher shear rate) more islands are formed. These observations show that studying hydrodynamics in the Drais reactor with viscoelastic fluids is mandatory for obtaining a complete picture of the mixing process.

Chapter 2

32

2.4 Conclusions This study provides information about flow patterns, the presence of poorly mixed zones, and macro-mixing times in three industrial configurations of the Drais reactor. Under turbulent conditions, the flow pattern shows flow circulation which is opposite to the pumping action of the impeller blades. The location of poorly mixed zones is the same in all three reactors. The dimensionless macro-mixing time N·t10 is correlated with L/D, stirrer frequency N, and fill ratio x. The obtained empirical correlation shows the following three relationships: • • •

N·t10 is at a minimum value at fill ratios around 60%. N·t10 increases more than linearly with increasing L/D ratio. The incorporated Reynolds number has a positive power, indicating that mixing at high stirrer speeds becomes less efficient.

Under laminar conditions, the flow patterns found indicate that the mixing is chaotic. Reactor-11 and reactor-15 are globally chaotic, whereas reactor-20 appears to have elements of order. This behavior shows that mixing in the Drais reactor is complex and complicates effective scale-up. The location, number, and life span of the islands, as well as the overall flow pattern in the Drais reactor, change when L/D is enlarged. Macro-mixing times could not be determined unambiguously, as the islands do not disappear within the measurement time. However, the macro mixing time seems to be only 2.5 times larger than under turbulent conditions.


33

Nomenclature C Di D G h L N PE PE0 Re t tm t10 V w x

clearance m impeller diameter vessel diameter gravitational constant blade height vessel length number of revolutions perimeter perimeter at t = 0 Reynolds number time mixing time time at which concentration only differs 10% from final concentration reactor volume blade width fill ratio

m m m/s2 m m Hz m m s s s m3 m -

angle between impeller blade and shaft angle between impeller blades in the tangential direction dynamic viscosity liquid density Liapunov exponent

° ° kg/(m·s) kg/m3 1/s

Greek α β µ ρ σ

References Ando, K.; Hara, H.; Endoh, K.; “Flow behavior and power consumption in horizontal stirred vessels”, Int. Chem. Eng., 1971a, 11, 735. Ando, K.; Hara, H.; Endoh, K.; “On mixing time in horizontal stirred vessel”, Kagaku Kogaku, 1971b, 35, 806. Ando, K.; Fukuda, T.; Endoh, K.; “On mixing characteristics of horizontal stirred vessel with baffle plates”, Kagaku Kogaku, 1974, 38, 460. Ando, K.; Shirahige, M.; Fukuda, T.; Endoh, K.; “Effects of perforated partition plate on mixing characteristics of horizontal stirred vessel”, AIChE J., 1981, 27(4), 599. Ando, K.; Obata, E.; Ikeda, K.; Fukuda, T.; “Mixing time of liquid in horizontal stirred vessels with multiple impellers”, Can. J. Chem. Eng., 1990, 68, 278. Borkent, G.; Tijssen, P.A.T.; Roos, J.P.; Van Aartsen, J.J.; “Kinetics of the reactions of aromatic amines and acid chlorides in hexamethylphosphoric triamide”, Recl. Trav. Chim. Pays-Bas, 1976, 95, 84. Distelhoff, M.F.W.; Marquis, A.J.; Nouri, J.M.; Whitelaw, J.H.; “Scalar mixing measurements in batch operated stirred tanks”, Can. J. Chem. Eng., 1997, 75, 641. Dong, L.; Johanson, S.T.; Engh, T.H.; “Flow induced by an impeller in an unbaffled tank”, Chem. Eng. Sci., 1994, 42, 549.

34

Chapter 2

Fukuda, T.; Idogawa, K.; Ikeda, K.; Ando, K.; Endoh, K.; “Volumetric Gas-phase mass transfer coefficient in baffled horizontal stirred vessel”, J. Chem. Eng. Jpn., 1990, 13(4), 298. Gaymans, R.J.; Sikkema, D.J.; “Aliphatic polyamides”, Comprehensive Polymer Science, Step Polymerization, 1989, 5, 357. Harvey III, A.D.; Wood, S.P.; Leng, D.E.; “Experimental and computational study of multiple impeller flows”, Chem. Eng. Sci., 1997, 52(9), 1479. Holmes, D.B.; Voncken, R.M.; Dekker, J.A.; “Fluid flow in turbine-stirred, baffled tanks - I -Circulation time”, Chem. Eng. Sci., 1964, 19, 201. Hoogendoorn, C.J.; den Hartog, A.P.; “Model studies on mixers in the viscous flow region”, Chem. Eng. Sci., 1967, 22, 1689. Jeurissen, F.T.H.; Surquin, J.; Private communication. Kersting, Ch.; Prüss, J.; Warnecke, H.J.; “Residence time distribution of a screw-loop reactor: experiments and modelling”, Chem. Eng. Sci., 1995, 50, 299. Lamberto, D.J.; Muzzio, F.J.; Swanson, P.D.; Tonkovich, A.l.; “Using time-dependent RPM to enhance mixing in stirred vessels”, Chem. Eng. Sci., 1996, 51(5), 733. Lamberto, D.J.; Alvarez, M.M.; Muzzio, F.J.; “Experimental and computational investigation of the laminar flow structure in a stirred tank”, Chem. Eng. Sci., 1999, 54, 919. Leong, C.W.; Ottino, J.M.; “Experiments on Mixing due to Chaotic Advection in a Cavity”, J. Fluid Mech., 1989, 209, 463-499. Leong, C.W.; Ottino, J.M.; “Increase in regularity by polymer addition during chaotic mixing in two dimensional flows”, Phys. Rev. Lett., 1990, 64(8), 874. Manaresi, P.; Munari, A.; “Factors affecting rate of polymerization”, Comprehensive Polymer Science, Step Polymerization, 1989, 5, 35. Mayr, B.; Nagy, E.; Horvat, P.; Moser, A.; “Scale-up on basis of structured mixing models: A new concept”, Biotechnol. Bioeng., 1994, 43, 195. Moo-young, M.; Tichar, K.; Takahashi, A.L.; “The blending efficiencies of some impellers in batch mixing”, AIChE. J., 1972, 18, 178. Nomura, T.; Uchida, T.; Takahashi, K.; “Enhancement of mixing by unsteady agitation of an impeller in an agitated vessel”, J. Chem. Eng. Jpn., 1997, 30(5), 875. Novák, V.; Rieger, F.; “Homogenization efficiency of helical ribbon and anchor agitators”, Chem. Eng. J., 1975, 9, 63. Ottino, J.M.; “Unity and diversity in mixing: Stretching, diffusion, breakup and aggregation in chaotic flows”, Physics of Fluids A3, 1991, 3(5), 1417. Perona, J.J.; Hylton, T.D.; Youngblood, E.L.; Cummins, R.L.; “Jet mixing of liquids in long horizontal cylindrical tanks”, Ind. Eng. Chem. Res., 1998, 37, 1478. Schoenmakers, J.H.A.; Wijers, J.G.; Thoenes, D.; “Determination of feed stream mixing rates in agitated vessels”, Proc. 8th European Mix. Conf. (Paris): Récents Progrès en Génie des Procédés, ed. Lavoisier, 1997, 11(52), 185. Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers: Dordrecht, 1994. Unger, D.R.; Muzzio, F.J.; “Laser-induced fluorescence technique for the quantification of mixing in impinging jets”, AIChE J., 1999, 45(12), 2477. Vollbracht, L.; Comprehensive Polymer Science, Step Polymerization, 1989, 5, 374.


35

Appendix

Radius [m]

In this appendix some remarks are made on the frequency plot in Figure 2.10. The plot shows spikes at frequencies that are characteristic for raw data like the impeller speed, i.e., 6 Hz and accompanying higher frequencies such as 12, 18 and 24 Hz. The peak around 3.5 Hz originates from differences in dye concentration in the tangential direction and is responsible for the large fluctuations in the response curve. In the response curve of reactor-11 in Figure 2.11, the fluctuations reappear with a time period of 1/3.5 s, the so-called circulation time tc (Holmes et al., 1964). As 10 periods can be distinguished, one can estimate a mixing time in the tangential direction of 3 s. Because the macro-mixing time t10 for this experiment was found to be 4.0 s, distributive mixing in the tangential direction is faster than in the axial and radial direction. The mixing in the tangential direction was also faster than that in the axial and radial direction in reactor-15 and reactor-20 because no large fluctuations were observed in the accompanying response curves in Figure 2.11. In these reactors, the dye is homogeneously distributed in the tangential direction before it has reached the position where gray scales are read. It is suggested that the clockwise circulation in the larger reactors, as depicted in Figure 2.8, enhances the tangential mixing. According to the frequency plot in Figure 2.10, the concentration in the tangential direction in reactor-11 fluctuates with a specific frequency. However, this frequency is not constant Injection throughout the reactor. Figure 0.18 2.14 shows the ratio of the frequency and impeller frequency in one-fourth of 0.15 reactor-11. The Figure shows that, near the wall, the fluid is 0.12 retained more than in the bulk as the ratio is smaller. The 0.09 largest ratio is 0.9, positioned between the two impeller 0.06 blades. The fluid in that area rotates almost as a solid-body and will therefore not be well0.03 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 -0.00 mixed, as is confirmed by the Axial distance [m] presence of the poorly mixed Figure 2.14: Ratio of fluid velocity to stirrer zone in Figure 2.4e at the speed in reactor-11. same position.

3 HYDRODYNAMICS AND SCALE-UP OF HORIZONTAL STIRRED REACTORS Abstract In this chapter, the hydrodynamics in horizontal stirred-tank reactors are investigated. The flow state, agitation power, and macromixing times have been determined experimentally. Two flow states, i.e., 'slosh' and 'ring', can be distinguished, with transition between the two states that shows hysteresis. The agitation power was determined by measuring the temperature increase upon mixing. The power number appears to be comparable to power numbers in unbaffled vertical vessels. A variation in fill ratio indicates that agitation energy dissipates uniformly throughout the reactor under laminar conditions. Under turbulent conditions, however, most energy is dissipated at the vessel wall. Using pulse-response measurements, macro-mixing times have been determined. The mixing times correlate with momentum input and liquid volume, thus indicating different hydrodynamics at large and small scales. A combination of mixing times and agitation power shows that, at small scale and intermediate fill ratios, the mixing is most energy-efficient.

This chapter is a slightly modified version of the publication: Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Hydrodynamics and Scale-Up of Horizontal Stirred Reactors”, Ind. Eng. Chem. Res., 2001, 40(22), 4731-4740.

38

Chapter 3

3.1 Introduction Horizontal stirred-tank reactors are widely used in industry. A commercial example is the unbaffled Drais reactor that can be used for multiple purposes such as powder mixing in catalyst preparation, liquid mixing and kneading during CMCproduction, and polycondensation processes (Vollbracht, 1989; BannenbergWiggers et al., 1998). In all applications, the reactor is only partially filled. A schematic representation of the Drais reactor is given in Figure 2.1. Typical for the unbaffled cylindrical reactor is its horizontal position and the heavily designed impeller. The impeller makes possible the application of high mixing power, which is needed to achieve sufficient mixing in viscous fluid processes. The reactor is characterized by its length L, diameter D, and clearance c, which is the distance between the blades and the reactor wall. Because the clearance is small, the blades perform a scraping action that keeps the reactor wall free from sticking material. The clearance also provides a region with high shear rates and good heat exchange with the cooled walls. The blades have a pumping action towards the reactor center, providing an easy way to discharge the reactor through the open hatch as represented in Figure 2.1. Despite the wide application of Drais reactors in liquid mixing, little is known about the hydrodynamics, mixing performance, and scale-up of such reactors. A literature survey shows that the literature on hydrodynamics in horizontal vessels is rather limited, in contrast with that on vertical stirred vessels. Some literature exists on the hydrodynamics under turbulent conditions. Ganz (1957) describes power measurements in horizontal stirred gas absorbers. Ando et al. (1971a) measured power input upon stirring in partially filled horizontal vessels in relation to flow behavior. In the vessel, two flow states could be distinguished. The 'slosh'-state is obtained at low stirring speed. The liquid is then pushed upward by the impeller and sprayed, which is ideal for use in gas absorption processes (Ganz, 1957; Ando et al., 1971b; 1974; 1981; Fukuda et al., 1990). The 'ring'-state, which is obtained at high stirrer speeds, results in a cylindrical liquid layer on the inside wall. Ando et al. (1990) also studied turbulent mixing in a horizontal vessel with baffles and multiple impellers. Macro-mixing times were measured, and a model was proposed for scale-up purposes. It has been established that the dimensionless macro-mixing time Ntm is proportional to L/D. Because literature on mixing in horizontal vessels under laminar conditions is virtually absent, we have to rely on mixing studies in vertical unbaffled tanks, for which many studies are available. The usual purpose of these studies has been to find geometries that provide good mixing performance. Data on mixing vary from author to author as a result of differences in geometry, definitions, experimental techniques, and fluid properties. However, a general consensus emerges concerning impeller designs. Judging from power input and mixing time experiments (Ando et al., 1990; Hoogendoorn et al., 1967; Novák et al., 1967), it can be concluded that the flow pattern of a good laminar mixer should include 1) axial flow, 2) all streamlines passing through the impeller region, 3) no closed streamlines occurring

Hydrodynamics and Scale-Up of Horizontal Stirred Reactors

39

outside the impeller region, and 4) frequent disruption of fluid along the wall. Figure 2.1 shows that conditions 2 and 4 will probably be achieved in the Drais reactor, because the impeller blades pass through the entire reactor volume. In Chapter 2, the flow patterns in a Drais reactor have been described, as investigated using planar laser induced fluorescence. It was found that axial flow in the reactor is rather effective. Also, closed streamlines can be present, acting as toroidal vortices. We have studied the macro-mixing in a Drais reactor filled with low- and high-viscosity liquids with the aim of obtaining a better understanding of the mixing during a polycondensation process in which the viscosity strongly increases as a result of the formation of polyaramid molecules (Bannenberg-Wiggers et al., 1998). To obtain a polymer product with the required quality in terms of MW (molecular weight) and MWD (molecular weight distribution), it is important to expose the polymerizing liquid to high shear rates (Agarwal et al., 1992). The high shear rates increase reaction rates through molecular orientation and rotational diffusion of the rods. This has been shown in an experimental study by Agarwal and Khakhar (1992; 1993), using two reactors in series in which the first reactor is a vertical reactor with a high-speed stirrer ensuring good overall mixing. The second reactor provides Couette-flow hydrodynamics, with nearly homogeneous high shear flow over the entire reactor but with little overall mixing. The authors were able to improve product quality considerably by increasing the shear rate in the second reactor. The Drais reactor investigated here, combines the important features of both reactors (Vollbracht, 1989; Bannenberg-Wiggers et al., 1998): a large impeller provides overall mixing while at the same time high shear rates occur in the small clearance between impeller and vessel wall. As the clearance of the Drais reactor is small, its volume is small compared to the total liquid volume: in completely filled reactors, this volume ratio is 2⋅10-4. Therefore, macro-mixing in the reactor has to be optimized so that all liquid in the bulk will pass the high-shear region in the clearance frequently. A second reason that emphasizes the importance of short macro-mixing times is that the polycondensation is performed in a semi-batch manner: one reactant is fed to the other and has to be mixed quickly throughout the reactor to prevent the occurrence of a premature termination reaction (Bannenberg-Wiggers et al., 1998; Chapter 2). Judging from the available literature, it is unclear what the macro-mixing time will be and how it will evolve in scale-up. Therefore, we have conducted an experimental study at different scales in which we established macro-mixing times by means of pulse-response measurements. The applied agitation power was also measured to link the mixing performance with power consumption. All measurements were performed in the 'ring'-state as the polycondensation process is performed at high stirrer frequencies, thus forcing the liquid into the 'ring'-state. As it is known that application of the 'ring'-state is required for a high MW to be obtained (Bannenberg-Wiggers et al., 1998), we determined the impeller frequency at which the 'ring'-state forms or disappears during operation. Also, flow behavior was studied as a function of vessel and stirrer geometry, impeller frequency, and fluid viscosity.

Chapter 3

40

3.2 Scale-up theory For scale-up of the polycondensation process, it is important to know how the mixing time evolves upon reaction. From the literature it is well-known that the mixing time will be a function of the process conditions and reactor configuration:

tm = ƒ(ρ,µ,N,DI,g,geometrical dimensions of the system)

[s]

(3.1)

Using dimensional analysis and omitting the Froude number (Fr), the functional relationship can be rearranged to:

Ntm = ƒ(Re, geometrical dimensions as ratios)

[-]

(3.2)

Ntm is the dimensionless mixing time and is expected to be independent of the Reynolds number (Re) under both turbulent and laminar conditions (Harnby et al., 1992). Under intermediate conditions, Ntm will be a power law in Re. In our specific case, Fr can be omitted as Fr is only important under conditions at which transition occurs between 'slosh'- and 'ring'-state. We are only interested in macromixing times in the 'ring'-state as the polycondensation process is performed in this state. To maintain tm constant during scale-up, a constant impeller frequency N is required according to equation 3.2. This will also result in a constant average shear rate γ& a , which is related to N by

γ& a = −

dv = k1 N dy

[1/s]

(3.3)

with k1 close to unity (Metzner et al., 1957; Thoenes, 1994). The highest shear rate, γ&max occurs in the clearance and can be estimated using

γ& max = −

dv dy

≈ c

πNDi πND − 0 = ∝ k2 N c D / 2 − Di / 2

[1/s]

(3.4)

Thus, γ&max is constant when N is kept constant, provided that the clearance is kept in a constant ratio with the vessel diameter. Keeping N constant in scale-up, however, is strongly reflected in the power requirement. Under turbulent conditions, the required power is given by

P = ρN P N 3 Di5

[W]

(3.5)


41

and the average energy dissipation rate per unit of mass is

ε=

N P f g N 3 Di2 P = x ρVl

[m2/s3] (3.6)

where ρ is the liquid density [kg/m3], NP is the power number [-], N is the agitation rate [1/s], Di is the impeller diameter [m], x is the fill ratio [-], Vl is the liquid volume [m3], and fg is the geometrical factor, which is 1.05 [-]. For geometrically similar systems, the power number NP can be rewritten as a functional relationship of dimensionless groups. Under turbulent conditions, only Re is relevant for the Drais reactor; thus

N p = k 3 Re a

[-]

(3.7)

[J/s]

(3.8)

Under laminar conditions, the following relation applies

P = k 4 µN 2 Di3 and consequently

ε=

P = k 5νN 2 ρVl

[J/(kg⋅ s)] (3.9)

with k4 and k5 as constants and ν the kinematic viscosity. For these equations, it is assumed that Np depends on Re-1 only, which is a good approximation when Re < 100. From equations 3.5 and 3.8, it can be seen that, for constant N, P increases with Di5 and Di3, respectively, which are both highly impracticable. Therefore, on larger scales, lower specific power input has to be applied which is usually obtained by reducing the impeller speed. This results in an increase in tm, which is undesirable in the polycondensation process. With this experimental study, we identify the limitations that can be faced during scale-up.

Chapter 3

42

3.3 Experimental Section 3.3.1 The Drais reactor To investigate the mixing process at different scales, four scale models of the Drais reactor were available. Typical geometrical data are given in Table 3.1. The small models are named reactor11, reactor15 and reactor20, where the numbers 11, 15, and 20 refer to the L/D-ratio. The large-scale model is reactor11-60, which refers to the L/D-ratio and the diameter. Reactor11 and reactor11-60 are geometrically similar. Reactor15 and reactor20 differ from reactor11 in angle β between blades (as shown in Figure 2.1) and in length. A detailed description has been given in Chapter 2. Table 3.1: Geometric data regarding the four reactors.

Parameter Reactor length L [m] Reactor diameter D [m] Reactor volume Vr [L] Impeller diameter Di [-] Blade width w [-] Blade height h [-] Blade thickness [-] Shaft diameter [-] β (°) Materials Shaft Cylindrical wall Side walls

Reactor11 0.198

5.0

180

Small-scale Reactor15 Reactor20 0.27 0.36 0.18 6.7 8.9 D⋅(29/30) D/2 D/12 D/90 D/6 135 120 Stainless steel Glass Stainless steel

Large-scale Reactor11-60 0.66 0.6 185 D⋅(29/30) D/2 D/12 D/54 D/10 180 Stainless steel Perspex Perspex

3.3.2 Transition in flow state The flow state was determined for different reactor geometries and fluid viscosities by observing the impeller frequency at which a fluid ring was formed or collapsed. The formation was complete when the inner gas/liquid surface was flat. Both transitions were clearly visible.

3.3.3 Power measurements Power measurements were performed in reactor11, reactor20, and reactor1160. Depending on the desired value of Re, tap water, glycerin (purity > 99.9% Heybroek, Amsterdam), or a mixture of the two was used as the working fluid. The


43

Figure 3.1: Schematic representation of the experimental set-up for the temperature measurements.

power input to the liquid by stirring was measured by determining the temperature increase of the liquid with time. The experimental set-up is depicted in Figure 3.1. Two Pt100-elements, denoted T1 and T2, measured the temperature in the reactors during agitation. There was no difference observed between T1 and T2 under turbulent conditions. Under laminar conditions, the difference never exceeded 0.2 °C. Pt100-element T3 measured the ambient temperature, Ta, during the experiments. All temperatures were measured with an accuracy of 0.1 °C. The power input followed from the energy balance over the mixing vessel, as given in equation 3.10.

ρC pVl

dT = P − h(T − Ta ) dt

[J/s]

(3.10)

where ρ is the liquid density [kg/m3]; Cp is the heat capacity [J/(kg⋅K)]; Vl is the liquid volume [m3]; T is the liquid temperature [K]; Ta is the ambient temperature [K]; P is the dissipated stirring energy [J/s]; and h is overall heat transfer coefficient [J/(K⋅s)]. The term on the left-hand side in equation 3.10 represents the accumulation term, P represents the dissipated stirring energy, and the last term represents the lumped losses to the environment. Assuming density, heat capacity, overall heat transfer coefficient and ambient temperature to be constant and temperatureindependent, differential equation 3.10 can be solved by considering the initial condition T(0) = T0, resulting in

 − ht T (t ) = Ta + (T0 − Ta ) exp  ρC V p l 

 P   + 1 − exp − ht  h  ρC V p l   

  [K]  

(3.11)

Chapter 3

44

3.3.4 Pulse-response experiments The experimental set-up for the pulse-response experiments is depicted in Figure 3.2. It was decided to perform measurements outside the vessel, because the clearance was too small for a probe to be placed inside. Therefore, the reactor contents were circulated through a spectrophotometer (A) placed in an external loop. The liquid was withdrawn from an outlet, placed at the same height as the clearance and fed back at the top in the reactor center. The flow rate in the loop was monitored by a flow meter (F). The measurement started when a small amount of a concentrated aqueous methylene blue solution was injected as a pulse, δ(t), in the reactor center. The concentration was measured by the spectrophotometer, which was connected to a PC for automatic monitoring.

δ F

A

t

A

PC

Figure 3.2: Schematic representation of the experimental set-up for the pulse-response measurements.

For reactor11-60, the volume of the piping was 450 mL, and the flow rate was 50 mL/s. Because the liquid volume was between 72 and 180 L, higher order mixing effects can be ignored. The time needed for the tracer to leave the reactor and reach the spectrophotometer was 0.9 s. This delay time was measured by injecting a small amount of tracer at the reactor outlet. Macro-mixing times were corrected for this delay. For the smaller reactors, the volume of the piping was 250 mL, the flow rate was 25 mL/s, the total liquid volume varied between 2 and 10 L, and the delay was 2.1 s. The total amount of injected solution was 3 mL. Using a high-speed camera, as described in chapter 2, the injection time was determined to be 0.20 ± 0.03 s. This time was negligible compared to the mixing time. An example of a tracer concentration response curve with time is given in Figure 3.3. The plotted response on the left y-axis was normalized using:

C n (t ) =

C (t ) − C (0) C∞ − C (0)

[-]

(3.12)


45 10 1

Cn [-]

2

1

σ [-]

1.2

0.8

0.1

0.6

0.01

0.4

0.001

0.2

0.0001 tm = 12.3

0 0

5

0.00001

10

15

20

25

Time [s]

Figure 3.3: Typical response curve with the corresponding variance for D = 0.6 m, L = 0.66 m, x = 100 %, and N = 4.8 Hz.

On the right y-axis of Figure 3.3 is plotted the variance of concentration, σ2, around the equilibrium value, as defined by

σ 2 (t ) = (1 − C n (t )) 2

[-]

(3.13)

The macro-mixing time was defined to be the time at which the variance was below 10-3. This resulted in tm = 12.3 s for the experiment in Figure 3.3. Every presented macro-mixing time (shown in Figures 3.11-14) is the average of at least three measurements. Table 3.2 provides the impeller frequencies used in the pulse-response experiments. Under turbulent conditions, the chosen frequency ensured that the liquid was in the 'ring'-state. Consequently, the values of Fr at large and small scale were similar. The values of Re were not similar, which was less important as it followed from literature that Ntm was independent of Re at the high applied values of Re (Coulson and Richardson, 1996). Using glycerin in reactor11-60 provides Re values up to 1730, which does not really justify the laminar classification. However, for convenience we have grouped all measurements using glycerin. Table 3.2: Range of impeller frequency and the corresponding shear rate Fr values as applied in the pulse-response measurements.

Reactor id.

D [m] Regime

N [Hz]

Laminar

3.0-11.2

Reactor11, -15, and -20

0.18

Reactor11-60

0.6

γ&

[1/s]

560-2100

Kinematic

γ& , tip speed, Re and Dynamic

Vtip[m/s]

Re [-]

Fr [-]

1.7-6.3

100-360

0.17-2.3

5

Turbulent

5.5-9.1

1040-1715

3.1-5.1

1.77⋅10 5 0.55-1.5 2.95⋅10

Laminar

0.6-4.8

110-900

1.1-9.0

210-1730 0.02-1.4

Turbulent

1.8-4.8

340-900

3.4-9.0

6.48⋅10 0.20-1.4 5 1.73⋅10

5

Chapter 3

46

3.4 Results and discussion 3.4.1 Flow state in water For the reactor filled with water, the Fr values at which fluid rings form or collapse, are plotted in Figure 3.4 against the fill ratio for reactor11, reactor20, and reactor11-60. The solid symbols mark the Fr value at which the fluid ring forms with increasing impeller speed, and the open symbols mark the Fr value at which the ring collapses with decreasing impeller speed. From Figure 3.4, it follows that, at all fill ratios hysteresis occurs between the values of Fr for ring formation and collapse. The values of Fr at which the transition occurs coincide for the ring collapse, however, they do not coincide for ring formation. At low as well as at high fill ratios, flow changes occur at higher values of Fr than is the case at intermediate fill ratios. This implies that, at intermediate fill ratios, weaker inertial forces, i.e., less power, are required to form a fluid ring. From Figure 3.4, the value of Fr can be determined at which a fluid ring is present. For the large-scale reactor, ring formation occurs around Fr = 1, which can practically be used as a criterion for ring formation. This information is important, as the fluid in the pulse-response measurements and power measurements has to be in the 'ring'-state for the polycondensation process (Bannenberg-Wiggers et al., 1998). Because of the hysteresis effect, however, it is possible to maintain the 'ring'state at stirrer frequencies lower than the frequency needed for ring formation. Ring formation:

Reactor11-60

Reactor11

Reactor20

Ring collapse:

Reactor11-60

Reactor11

Reactor20

2.5

2

Fr [-]

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

Fill ratio [-]

Figure 3.4: Flow state transition, expressed in Fr as function of the fill ratio for reactor11-60, reactor11, and reactor20.


47

3.4.2 Flow state in glycerin In reactors partially filled with glycerin, no fluid ring is formed at high stirrer speeds. For illustration, an image of the flow state is recorded with a high speed camera and is presented in Figure 3.5. Viscous forces are too high to obtain a fluid ring as in water. The gas phase is finely dispersed in the liquid, providing a milky fluid with small gas bubbles. Figure 3.5 also shows the presence of large holes in the fluid in the wake behind the blades. The holes do not completely extent to the cylindrical wall, indicating that the wall is entirely covered by a liquid film. The presence of the liquid film was proven to exist as small gas bubbles were found, using a stroboscope, that are transported tangentially by the impeller with a lower speed than the impeller speed. The presence of the liquid film probably is important, as this is the region with the highest shear rate.

Figure 3.5: Photograph of glycerin in reactor20 at a fill ratio of 40 % at 8 Hz.

3.4.3 Power measurements In Figure 3.6, the temperature rise is plotted as a function of time for five different situations. The time for measurement ranged from 5 minutes in reactor11 with glycerin to over 2 hours in reactor11-60 with water. By fitting the temperature profiles with equation 3.11, the applied agitation power P was obtained. The overall heat transfer coefficient h also follows from the fitting procedure. However, as the interpretation of these results appeared not to be very straightforward, these data have not been included. In Figure 3.7, the power number NP, as defined by P/ρN3D5, is given as a function of Re. The solid line represents measurements in a completely filled reactor11 and reactor11-60, which are geometrically similar but differ only in size. The measurements are in close agreement with data for a propeller mixer in an unbaffled vertical vessel as reported by Rushton et al. (1950). NP becomes independent of Re at high values of Re, i.e., under turbulent conditions. Accordingly, the power 'a' of Re in equation 3.7 approaches 0. The applied power is plotted against Re in Figure 3.8. From the slope of the lines at high values of Re, it follows that the applied power is proportional to the cubic root in impeller speed, which is in accordance with equation 3.5. Under laminar conditions, NP becomes inversely proportional to Re (the exponent 'a' in equation 3.7 approaches -1). From Figure 3.8, it follows that

Chapter 3

48 6.0

N = 12 Hz x = 40 % P = 154 W

5.0 4.0 T-T0 [°C]

Glycerin, small scale model

N = 8 Hz x = 80 % P = 87.4 W N = 12 Hz x = 100 % P = 48.6 W

3.0

Water, small scale model N = 8 Hz x = 100 % P = 17.5 W

2.0

Water, large scale model N = 2.4 Hz x = 100 % P = 202 W

1.0 0.0 0

1000

2000

3000

4000

5000

6000

7000

8000

Time [s]

Figure 3.6: Temperature rise versus elapsed time for the small- (D = 0.18 m) and large-scale (D = 0.6 m) models, filled with glycerin or water. Np(0.4,1.1)

Np(0.6,1.1)

Np(0.8,1.1)

Np(1.0,1.1)

Np(0.4,2.0)

Np(0.6,2.0)

Np(0.8,2.0)

Np(1.0,2.0)

1

Np [-]

0.1 100

1000

10000

100000

1000000

Re [-] Figure 3.7: Power number as a function of Re with the fill ratio and L/D ratio as parameters. 1000

D=0.18 P(0.001) P(0.003) P(0.014)

100 3.0

P [W]

P~N 2.6

P(0.099) P(0.297) P(0.527)

P~N

P(1.4) D=0.60

10 2.1

P~N

P(0.001) 1 10

100

1000

10000 100000 1E+06

1E+07

Re [-]

Figure 3.8: Power as a function of Re with viscosity and diameter as parameters.


49

10 Water Np (0.4) Np (0.6)

1 Np [-]

Np (0.8) Np (1.0) Glycerin Np (0.4)

0.1

Np (0.6) Np (0.8) Np (1.0) 0.01 0

0.5

1

1.5

2

2.5

3

3.5

4

Fr [-] Figure 3.9: Power number as a function of Fr with fill ratio and fluid type as parameters.

the applied power at low values of Re is proportional to the square root of the impeller speed, which is in accordance with equation 3.8. The power number NP is given as a function of Fr in Figure 3.9. Figure 3.9 shows that NP is virtually independent of Fr, apart from the case in which the reactor is completely filled with glycerin. For a horizontal reactor, Ando et al. (1971a) also found that NP is independent of Fr. In Figures 3.10A-D, the power P and dissipated energy per unit mass ε are given as functions of fill ratio for water and glycerin. With glycerin, the applied power was around 5 times higher than it was with water. From Figures 3.10A and 3.10B, it follows that, for water and glycerin, the applied power increases with increasing fill ratio. For water, the increase is less than proportional, as follows from Figure 3.10C in which ε decreases with increasing fill ratio. In water, most energy is dissipated at the impeller tips and at the cylindrical wall. At low fill ratios, this region represents a relatively larger volume than at high fill ratios. Therefore, ε will be higher at low fill ratios. Figure 3.10D shows that, in glycerin, ε is independent of the fill ratio, suggesting that energy is dissipated more uniformly throughout the fluid than in water. This also follows from Figure 3.10F, in which ε is plotted against Re. The values for ε at all fill ratios coincide and follow a power law of 2.1. As ε is homogeneous, the shear rate is also homogeneous. Consequently, the contribution of the high shear rate in the clearance to ε is limited, which can be explained by the small clearance volume.

Chapter 3

50 A) Water

70

8 , 1.1 10 , 1.1 12 , 1.1 14 , 1.1 8 , 2.0 12 , 2.0

60 P [W]

600

N L/D

50 40 30

N L/D

B) Glycerin

8 , 1.1 10 , 1.1 12 , 1.1 14 , 1.1 8 , 2.0

500

P [W]

80

400 300 200

20 100

10

0

0 0.4

0.5

0.6

0.7

0.8

0.9

0.4

1

0.6

0.7

0.8

0.9

1

90

25

C) Water

80

20

70

D) Glycerin

60 2 3

ε [m /s ]

15

2 3

ε [m /s ]

0.5

10

50 40 30 20

5

10 0

0 0.4

0.5

0.6 0.7 0.8 Fill ratio [-]

0.9

1

0.4

90

25 x L/D

2

3

ε [m /s ]

15 10 5

0.6 0.7 0.8 Fill ratio [-]

0.9

1

F) Glycerin

80 70

x L/D

60 2 3

0.4 , 1.1 0.6 , 1.1 0.8 , 1.1 1.0 , 1.1 0.4 , 2.0 0.6 , 2.0 0.8 , 2.0 1.0 , 2.0

20

ε [m /s ]

E) Water

0.5

0.4 , 1.1 0.6 , 1.1 0.8 , 1.1 1.0 , 1.1

50 40 30 20 10

0 100000

0 200000

300000 Re [-]

400000

500000

0

100

200 300 Re [-]

400

500

Figure 3.10: Results of the power measurements: (A) power in water against fill ratio,(B) power in glycerin against fill ratio, (C) ε in water against fill ratio, (D) ε in glycerin against fill ratio, (E) ε in water against Re, (F) ε in glycerin against Re. Parts (A)-(D) have N and L/D as parameters, and Parts (E) and (F) have fill ratio and L/D as parameters.

Using multivariable analysis, the relationship in equation 3.14 between NP and the varied parameters can be obtained for laminar conditions as

x 0.92  L  N P = 93 0.85   Re  D 

0.89

and for turbulent conditions as

[-]

(3.14)


N P = 0.15 x

0.36

L   D

51

0.44

[-]

(3.15)

The standard errors of the exponents are given in Table 3.3. For laminar conditions, NP proves to be nearly linear with liquid volume as the exponent for the fill ratio comes close to unity. Exponent 'a' approaches –1, which is in agreement with equation 3.7. Summarizing, it follows that, the value of NP in this case is similar to that of a propeller in an unbaffled tank. In water, ε is a function of the fill ratio and is highest near the vessel wall. In glycerin, ε is independent of the fill ratio and is homogeneous over the reactor. Table 3.3: Standard errors of the constants in the correlations for NP.

Conditions

Pre-exp. factor Exponent in x Exponent in (L/D) Exponent in Re

Lam. (Re < 400) 93 ± 9.6 5 Turb. (Re > 10 ) 0.15 ± 0.0055

0.92 ± 0.066 0.36 ± 0.064

0.89 ± 0.064 0.44 ± 0.072

-0.85 ± 0.02 -

3.4.4 Macro-mixing times in water under turbulent conditions The dimensionless mixing time Ntm in the large reactor11-60 is depicted in Figure 3.11A as a function of the fill ratio. It shows that, at every fill ratio, Ntm is independent of Re. Also, Ntm increases with increasing fill ratio, indicating that more impeller revolutions are required for a given degree of mixing with increasing liquid volume. From Figure 3.11B-D, in which Ntm is given for reactor11, reactor15, and reactor20, respectively, it follows that the lowest Ntm is found at a fill ratio of 0.7, which is in agreement with the results in chapter 2. This suggests that at low fill ratios (x < 0.4), the total amount of fluid is too low to provide good overall circulation in the fluid ring, whereas at high fill ratios (x ≈ 1), the poorly mixed zone near the shaft reduces the advantages of the better overall circulation. A comparison of Ntm for reactor11, reactor15, and reactor20 shows that Ntm increases with increasing L. Also, Ntm shows to be independent of Re, although Ntm increases slightly with increasing Re in reactor20. This increase indicates that the fluid tends toward solid-body rotation at higher stirrer speeds. Solid-body rotation can occur more easily in reactor20 than in reactor11 and reactor15, because of the relatively small effect of the sidewalls in reactor20. The above observations show that reactor11 and reactor11-60 differ significantly in hydrodynamic behavior despite their geometric similarity. In reactor11-60, the shortest mixing time is found at a fill ratio of 0.4, which is the lowest fill ratio applied, whereas in reactor11, it is found at a fill ratio of 0.7. The Ntm values at the two scales are also different. Apparently, turbulent and convective

Chapter 3

52 vtip [m/s]

Ntm [-]

2.6

5.2

2.6 90 80 70 60 50 40 30 20 10 0

90 80 70 60 50 40 30 20 10 0

1000000 3.0

3.4

3.8

1500000

2000000

4.2

5.0

4.6

200000

250000 Re [-]

300000

3.0

3.4

vtip [m/s] 3.8 4.2

4.6

2.6

200000 3.0

5.0 x [-] 1 0.7 0.6 0.5 0.4

B) Reactor11

150000

90 80 70 60 50 40 30 20 10 0

C) Reactor15

150000

2.6

10.4

90 A) Reactor11-60 80 70 60 50 40 No stable 30 fluid ring 20 10 0 500000

Ntm [-]

7.8

3.4

250000 3.8

4.2

4.6

300000 5.0

D) Reactor20

150000

200000

250000

300000

Re [-]

Figure 3.11: Dimensionless macro-mixing time against Re and impeller speed using water for (A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20.

mixing on both scales are different. An important difference between reactor11 and reactor11-60 is the wall-surface/liquid-volume ratio. This ratio is higher at small scale in reactor11 (22.2 m2/m3) than at large scale in reactor11-60 (6.66 m2/m3). As a result, the fluid tends more toward solid-body rotation at large scale and, therefore, shows increased mixing times. The observed trends indicate that the mixing time depends on scale and liquid volume in contrast to the conclusion of Harnby et al. (1992), who state that, for geometrically similar systems, Ntm is constant. Fox and Gex (1956), Middleton (1979), and Mersmann et al. (1976) have observed that, in vertical vessels, mixing times depend on liquid volume. Fox and Gex (1956) have correlated mixing times with liquid volume and momentum input according to: tm ∝

(N

Vl0.5 2

D4

)

0.42

[s]

(3.16)

Application of this approach has the advantage that the fill ratio can easily be implemented. When we use all presented mixing data, the following relationship is obtained:


(x π D L) = 66 ⋅ (N D ) 2

tm

53

0.68

4

[s]

4 0.42

2

(3.17)

Obviously, this is in good agreement with equation 3.16. This empirical relationship can be used for scale-up of the Drais reactor for processes under turbulent conditions (Re > 105).

3.4.5 Macro-mixing times in glycerin under laminar conditions When the reactor is filled with glycerin at fill ratios below 1, the gas phase is finely dispersed in the liquid. In the milky liquid thus obtained, as shown in the photograph in Figure 3.6, it is impossible to perform spectrophotometric pulseresponse measurements with the current set-up. Therefore, experiments have only been performed using completely filled reactors. In Figure 3.12A, Ntm is plotted against Re. It follows that the mixing time in glycerin is approximately 2.5-5 times longer than in water for the small and large reactors. Ntm is hardly dependent on Re on the different scales. Thus, in accordance with the observation of Harnby et al. (1992), Ntm is independent of Re for both laminar and turbulent conditions. However, this is only valid for a given scale, as the Ntm values for reactor11 and reactor11-60 differ despite their similar geometries. This emphasizes our previous observation that the hydrodynamics in the two geometrically similar systems are significantly different. Fox and Gex (1956) also provided a correlation for mixing times under laminar conditions:

tm ∝

(N

Vl 0.5 2

[s]

D4 )

1.25

(3.18)

500 400 300 100

Reactor11-60 Reactor20

100 90 80 70 60 50

Reactor15 Reactor11 100

Reactor11-60

tm [s]

Nt m [-]

200

Reactor20 Reactor15

A

10

1000 Re [-]

Reactor11

0.1

B 1

10

εε [m

2

3

/s ]

Figure 3.12: (A) Dimensionless macro-mixing time against Re and impeller speed, using glycerin. (B) tm as a function of ε using glycerin.

54

Chapter 3

Using the same approach, we obtain

( D L) = 530 × (N D ) 2

p 4

tm

0.98

[s]

4 0.44

2

(3.19)

The standard errors are given in Table 3.4. The differences between equations 3.18 and 3.19 can be a result of different geometries. Also, the results of Fox and Gex (1956) might be biased, as their experiments were performed in one single vessel in which only the liquid height was varied. Table 3.4: Standard errors of the constants in the correlations for t m

Conditions

Pre-exp. factor

Exponent in p/4 D2L

Exponent. in N2 D4

Laminar (Re < 1730) Turbulent (Re > 105)

519 ± 36.4 66 ± 3.1

0.98 ± 0.038 0.68 ± 0.022

-0.44 ± 0.014 -0.45 ± 0.022

3.4.6 Macro-mixing efficiencies The term Ntm is an efficiency parameter that represents the number of revolutions required to obtain the desired mixing at time tm. According to this term, the highest mixing efficiency is found in reactor11 at a fill ratio of 0.7. Less than 10 impeller revolutions are required to obtain the required mixing which is 5 times more efficient than in reactor11-60 or reactor20. 35

14

A) Reactor11-60

30

10

20

8

15

6

10

4

tm [s]

25

5

2

0

0 0

2

4

6

8

14

10

12

0.6 0.5 0.4 0

2

4

6

8

10

12

D) Reactor20

12

10

tm [s]

x [-] 1 0.7

14

C) Reactor15

12

B) Reactor11

12

10

8

8

6

6

4

4

2

2

0

0 0

2

4

6 e [m /s ] 2 3

8

10

12

0

2

4

Figure 3.13: tm as a function of e for (A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20.

6

8 2 3 e [m /s ]

10

12


55

In Figure 3.12B and 3.13A-D, the mixing times are plotted against the average energy dissipation ε for laminar and turbulent conditions, respectively. ε is calculated from equation 3.6, from which NP is calculated using the exponents in Table 3.3. From these figures, it can be concluded that, even though ε is the same, on average, tm in reactor11-60 is 5 times longer than in reactor11, although the length and diameter are only 3 times greater. In reactor11 at a fill ratio of 0.7, 1 m2/s3 is required to obtain mixing times of 1 s, whereas in reactor20, the same ε value provides a mixing time of 9 s. Therefore, an important conclusion is that scale-up of the Drais reactor based on energy consumption only, will lead to longer mixing times.

3.5 Concluding remarks In this study, we have examined the hydrodynamics in the Drais reactor at different scales. It has been concluded that the specific power consumption as a function of Reynolds number is the same on small and large scales. Over a wide range of Re values, the power number Np is comparable to Np of vertical unbaffled reactors with propellers. However, it has been shown that scale-up with constant power consumption will lead to longer mixing times at larger scale. This is reflected in Ntm, which depends linearly on the fill ratio at large scale and with the square root at small scale (Chapter 2). Thus, an important conclusion is that the hydrodynamics at different scale are different, making scale-up of the Drais reactor a difficult task. For scale-up purposes, however, the macro-mixing times have been correlated, using an approach as suggested by Fox and Gex (1956) that includes liquid volume and flux of momentum. Especially for turbulent conditions, the agreement in the obtained correlation is remarkable. Two flow states occur in this reactor, i.e., a 'slosh'-state at low vales of Fr and a 'ring'-state at high values of Fr. The transition between the two states as a result of changing impeller speed shows hysteresis. Using the criterion Fr=1 in scale up, one can determine the number of revolutions required to obtain the 'ring'-state. The 'ring'-state only applies for low-viscosity liquids. Using high-viscosity liquids, we observe a liquid phase with a dispersed gas phase. Nevertheless, still a liquid film exists on the cylindrical wall. The existence of this thin liquid film is important, because the highest shear rates are obtained in this film, which is crucial for obtaining products with high molecular weights in polymerization reactions (Agarwal et al., 1992). Mixing times under laminar conditions are approximately only 2.5 times higher than under turbulent conditions. In a previous paper, we showed that mixing under laminar conditions is very efficient, because of the chaotic nature. Striations were made visible using planar laser-induced fluorescence and shown to stretch, fold, and reorient throughout the whole reactor. The deformation process of the striations appears to occur homogeneously, which corresponds to the observed homogeneous energy dissipation as described in this paper. Because of this homogeneous dissipation, it can be concluded that the Drais reactor is a very efficient mixer, especially under laminar conditions.

Chapter 3

56

Nomenclature a b c C Cn C∞ Cp D Di fg Fr g h h k1-k5 L NP N P Re t tm T Ta T0 Vl Vr vtip w x y

constant constant clearance arbitrary concentration normalized concentration final concentration heat capacity vessel diameter impeller diameter geometrical factor Froude number acceleration of gravity overall heat transfer coefficient blade height constants length of the vessel power number agitation rate dissipated stirring energy Reynolds number time macro-mixing time liquid temperature ambient temperature temperature at t = 0 liquid volume reactor volume impeller speed blade width fill ratio distance

m J/(kg⋅K) m m m/s J/(K⋅s) m m 1/s J/s s s K K K m3 m3 m/s m m

blade angle mutual blade angle shear rate

°,degrees °,degrees 1/s

average power input per unit of mass dynamic liquid viscosity kinematic viscosity liquid density variance

m2/m3 kg/(m⋅s) m2/s kg/m3 -

Greek α β

γ& ε

µ ν ρ σ2


57

References Agarwal, U.S.; Khakhar, D.V.; “Enhancement of polymerization rates for rigid rod-like molecules by shearing”, Nature, 1992, 360, 53. Agarwal, U.S.; Khakhar, D.V.; “Shear flow induced orientation development during homogeneous solution polymerization of rigid rodlike molecules”, Macromolecules, 1993, 26, 3960. Ando, K.; Hara, H.; Endoh, K.; “Flow behavior and power consumption in horizontal stirred vessels”, Int. J. Chem. Eng., 1971a, 11, 735. Ando, K.; Hara, H.; Endoh, K.; “On mixing time in horizontal stirred vessel”, Kagaku Kogaku, 1971b, 35, 806. Ando, K.; Fukuda, T.; Endoh, K.; “On mixing characteristics of horizontal stirred vessel with baffle plates”, Kagaku Kogaku, 1974, 38, 460. Ando, K.; Shirahige, M.; Fukuda, T.; Endoh, K.; “Effects of perforated partition plate on mixing characteristics of horizontal stirred vessel”, AIChE J., 1981, 27(4), 599. Ando, K.; Obata, E.; Ikeda, K.; Fukuda, T.; “Mixing time of liquid in horizontal stirred vessels with multiple impellers”, Can. J. Chem. Eng., 1990, 68, 278. Bannenberg-Wiggers, A.E.M.; Van Omme, J.A.; Surquin, J.M.; “Process for the batchwise preparation of poly-p-terephtalamide”, U.S. Pat., 5,726,275, 1998. Coulson, J.M.; Richardson, J.F.; Backhurst, J.R.; Harker, J.H.; “Coulson and Richardson’s Chemical Engineering”, Vol. 1, 5th ed., Fluid flow, heat transfer and mass transfer; Butterworth-Heinemann Ltd: Oxford, England, 1996. Fox, E.A.; Gex, V.E.; “Single-phase blending of liquids”, AIChE J., 1956, 2(4), 539. Fukuda, T.; Idogawa, K.; Ikeda, K.; Ando, K.; Endoh, K.; “Volumetric Gas-phase mass transfer coefficient in baffled horizontal stirred vessel”, J. Chem. Eng. Jpn., 1990, 13(4), 298. Ganz, S.N.; Zh. Prikl.Khin., 1957, 30, 1311. Harnby, N.; Edwards, M.F.; Nienow, A.W. (Eds); “Mixing in the Process Industries”, 2nd ed., Butterworth-Heinemann Ltd: London, England, 1992. Hoogendoorn, C.J.; den Hartog, A. P.; “Model studies on mixers in the viscous flow region”, Chem. Eng. Sci., 1967, 22, 1689. Mersmann, A.; Einenkel, W.D.; Kappel, M.; “Design and scale up of mixing equipment”, Int. Chem. Eng., 1976, 16, 590. Metzner, A.B., Otto, R.E.; “Agitation of Non-Newtonian Fluids”, AIChE J., 1957, 3(1), 3. Middleton, J.C; “Measurement of circulation within large mixing vessels”, Proc. 3rd Eur. Conf. On Mixing, University of York, BRHA Fluid Eng. Cranfield, England, 1979, A2, 15. Novák, V.; Rieger, F.; “Homogenization efficiency of helical ribbon and anchor agitators”, Chem. Eng. J., 1975, 9, 63. Rushton, J.H.; Costisch, E.W.; Everett, H.J.; “Power characteristics of mixing impellers, Parts I and II”, Chem. Eng. Prog., 1950, 46, 395 & 467. Tatterson, G.B.; “Fluid mixing and gas dispersion in agitated tanks”, McGraw-Hill Inc: New York, United States of America, 1991. Thoenes, D.; “Chemical Reactor Development”, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. Vollbracht, L.; “Aromatic Polyamides”, Compr. Polym. Sci., Step Polym., 1989, 5, 374.

4 FLUID FLOW AND MIXING IN AN UNBAFFLED HORIZONTAL STIRRED TANK Abstract The turbulent flow field in a horizontal stirred tank reactor has been examined using Laser Doppler Anemometry and Computational Fluid Dynamics. The LDA experiments show that in the unbaffled reactor the mean velocities in tangential direction are at least one order of magnitude higher than in axial and radial direction. In these directions the turbulent and periodic fluctuations dominate over the mean velocities. Therefore, macro mixing in these directions is determined by turbulent dispersion. In the CFD calculations the isotropic k-ε model and the anisotropic Differential Stress Model have been applied to incorporate the turbulent properties of the flow. The mean flow properties are reasonably well predicted with both models. Both models underestimate the fluctuating properties. Scalar mixing experiments are simulated properly only when the anisotropic Differential Stress Model is used. This implies that when the choice for a turbulence model (isotropic k-ε model or anisotropic Differential Stress Model), is based on flow properties only, mixing simulations with passive and reacting scalars might fail completely for those cases where anisotropic turbulence is expected.

This chapter is a slightly modified version of: Van der Gulik, G.J.S.; Wijers, J.G.; Keurentjes, J.T.F.; “Fluid Flow and Mixing in an Unbaffled Horizontal Stirred Tank”, Submitted for publication in AIChE. J.

60

Chapter 4

4.1 Introduction Horizontal stirred tank reactors are widely used in industry. A commercial example is the unbaffled horizontal reactor of the Drais type (Turbulent Schnellmischer, Drais Ltd, Mannheim, Germany) as depicted in Figure 2.1. This reactor can be used for multiple purposes like powder mixing in catalyst preparation, liquid mixing and kneading during CMC production or polycondensation processes (Vollbracht, 1989; Bannenberg-Wiggers et al., 1998). In all applications the reactor is only partially filled (40% < x < 75%). The reactor is characterized by length L, diameter D and clearance B, which is the distance between the blades and the reactor wall. Since the clearance is small, the blades perform a scraping action that keeps the reactor wall free from sticking material. The small clearance also implies the presence of a region with high shear rates and good heat exchange. The blades have a pumping action towards the reactor center plane at xL = 0, providing an easy way to discharge the content of the reactor through the opened hatch as depicted in Figure 2.1. We have previously studied the mixing in the horizontal reactor (Van der Gulik et al., 2001a and 2001b) within the framework of the application in polycondensation processes in which viscosity increases several orders of magnitude during reaction. The choice for the Drais reactor has been made based on the good mixing performance when the reactor content is highly viscous. However, at the start of the polycondensation process, when the components are added and mixed, the reactor content has a low viscosity, resulting in turbulent conditions. Consequently, the mixing performance has also to be sufficient at turbulent conditions as this determines the local monomer ratio. We have characterized the mixing in the horizontal reactor at both low and high viscous conditions by determining power consumption (Van der Gulik et al., 2001a), mixing times (Van der Gulik et al., 2001a and 2001b) and the flow pattern (Van der Gulik et al., 2001b). Because the reactor is unbaffled and the impeller speed is usually high, it has been found that the fluid mainly flows in tangential direction. The flow in axial and radial direction can be interpreted as secondary flow superpositioned on the tangential flow. The secondary flow pattern has been studied by using Planar Laser Induced Fluorescence from which it follows that flow in axial direction dominates over the flow in radial direction (Van der Gulik et al., 2001b). Pulse response measurements have confirmed this observation (Van der Gulik et al., 2001a). For a quantitative description of the internal flow field computational fluid dynamics (CFD) calculations are becoming increasingly popular. CFD requires models that describe the turbulent properties of the flow. The k-ε model, as proposed by Launder and Spalding (1974), is based on an eddy viscosity hypothesis. One of the main assumptions is that the turbulence is isotropic. Although turbulence is in general anisotropic in mixing vessels (Kresta, 1998), the k-ε model is often used because of simplicity and computational convenience (Schoenmakers, 1998; Montante et al., 2001; Brucato et al., 2000; Togatorop et al., 1994; Read et al., 1997; Rousseaux et al., 2001; Brucato et al., 1998). Reynolds Stress Models (RSM)

Fluid flow and mixing in an unbaffled horizontal stirred tank

61

are computationally much more demanding (5 additional equations compared to the k-ε model) but are able to incorporate the anisotropic nature of turbulence. CFD studies on unbaffled vessels that compare the viability of both eddy viscosity models and RSM show different results. For example, Armenante et al. (1997) and Ciofalo et al. (1996) could only reproduce their Laser Doppler Anenometry data (LDA data) properly using RSM. Montante et al. (2001) used the k-ε model and a RSM to study the effect of the impeller clearance on the flow pattern in a vertical vessel. The k-ε model sufficed for all cases, except for the smallest clearance. In this paper we have examined CFD for describing the hydrodynamics in the Drais reactor. As a start we have checked the viability of the k-ε model and the Differential Stress Model (DSM), which is a RSM in differential form (Launder et al., 1975). For validation purposes, we have performed LDA measurements providing local velocities and turbulent quantities. Insight into these parameters should allow for effective scaling up and should allow definition of routes for improvement on the current reactor. Using CFD we have also performed scalar mixing simulations in the obtained flow fields for examining the mixing performance. The results of these simulations have been compared with previously reported PLIF data (Planar Laser Induced Fluorescence, Van der Gulik et al., 2001b) providing an additional validation on the turbulence modeling.

0.0

xL 0.17

0.33 0.05 0.17 = w⋅sin(30)

+

Ur -

- Ua + Figure 4.1: Quarter of the reactor in which the axial coordinate xL has been defined: xL = 0.0 in the reactor center and xL = 0.33 at the side wall. The direction of the axial velocity Ūa and the radial velocity Ūr have been exemplified.

62

Chapter 4

4.2 Experimental and numerical set-up 4.2.1 The horizontal vessel The vessel as depicted in Figure 2.1 was used in this study. Detailed geometrical data were given previously (Van der Gulik et al., 2001b). The diameter and the length were equal to 0.60 and 0.66 m, respectively, providing an L/D-ratio of 1.1. The clearance B was equal to 0.01 m. The impeller blades make an angle of 30° with the shaft. More detailed dimensions of the blades are given in Figure 4.1. The total blade length ‘w’, as defined in Figure 2.1, was 0.3 m. In Figure 4.1, the length, projected on the x-y-plane, was equal to 0.17 m. The blade height ‘h’ had a maximum of 0.05 m in the middle of the blade and was equal to 0 at the far ends of the blade.

4.2.2 LDA measurements LDA is an optical method for fluid flow research, based on a combination of interference and Doppler effects. LDA allows the measurement of the local, instantaneous velocities of particles suspended in the flow. LDA has a high resolution power in time and is non-invasive. A detailed description of the method has been given by Durst et al. (1981). Theoretical aspects of the turbulent flow occurring in the Drais reactor and how the turbulent flow has been investigated using LDA, are given in Appendix 4A. The LDA system used here has been described by Schoenmakers (1998) and Schoenmakers et al. (1997). A 2W Argon laser (Spectra Physics, Model Stabilite 2017-055) was used to provide two colored beams with wavelengths of 514.5 and 488 nm, respectively. Using a lens with a focal length of 400 mm both laser beams converged in a elliptical control volume of approximately 0.15 × 0.15 × 3.3 mm. An automated traverse system with the laser probe and the receiving optics was utilized in Backscatter acquisition mode. Two photo multipliers and two Burst Spectrum Analysers (Dantec, Denmark) were used for measuring two velocity components simultaneously. The LDA measurements were performed in a perspex model of the Drais reactor that was placed in a square perspex container. The impeller frequency was set to 3 Hz. All measurements were performed using water. The square perspex container was also filled with water to minimize lensing effects. Positioning of the measurement volume was realized using the procedures of Kehoe and Prateen (1987). Nevertheless, it was not possible to measure the tangential and radial components close to the cylindrical wall and the side wall. The measurement points are summarized in Table 1. To obtain the three velocity components, two separate measurements were taken for the same (r, Z) point: one in the plane at 0°, yielding the tangential and axial components, and one in the plane at -90°, yielding the axial and radial components. In axial direction steps were made of 0.01 m. Only half the reactor was examined since flow proved to be symmetrical over an axial plane at x


63

= 0. In each location 30,000 measurements were performed with an average data rate of 0.3 kHz. For seeding we used an aqueous dispersion of polystyrene particles with an average diameter of 4 µm. Table 4.1: Radial locations for performed measurements between axial positions 0 < xL< 0.29.

r 0.295 0.29 0.285 0.28 0.27 0.25 0.22 0.29 0.16 0.13

r/R 0.983 0.967 0.95 0.933 0.9 0.833 0.733 0.633 0.533 0.433

Tangential (0°)

√ √ √ √ √ √ √ √

Axial (0° and –90°) √ √ √ √ √ √ √ √ √ √

Radial (-90°)

√ √ √ √ √ √

The experimental error in mean velocity, periodic and turbulent fluctuations was less than 10%. The mean and fluctuating axial velocities in the 0° and -90° plane appeared to be the same within 10%, indicating that the average flow characteristics are rotation symmetric. Therefore, all experimental data will be presented as if measured in a single plane as presented in Figure 4.1. The direction of the axial and radial velocities has also been exemplified in Figure 4.1. A negative axial velocity is directed towards the reactor center whereas a positive velocity is directed towards the reactor side wall. A negative radial velocity is directed towards the shaft and a positive radial velocity is directed towards the cylindrical wall.

4.2.3 Computational Fluid Dynamics For the CFD calculations the Finite Volume Package CFX-4.2 was used (AEA Technology), installed on a Silicon Graphics Origin 200 workstation. The grid used is depicted in Figure 4.2. The grid was built up in Cartesian coordinates and comprised of 402 blocks and 374,820 cells. Because the angle between the impeller blades and the shaft is 30°, the grid strongly deviates from a computationally ideal orthogonal grid. The Block-Stone solver was required to solve the differential equations.

64

Chapter 4

Figure 4.2: The grid as used in the CFD Calculations, in front and side view.

The SIMPLE algorithm was adopted to couple the continuity and Navier–Stokes equations. Two turbulence models were used: the DSM and the standard k-ε model. Mathematical details are given in Appendix 4B. Because of the distorted grid, it was necessary to adopt the fully deferred correction for the ε equation in both turbulence models. On walls the conventional linear logarithmic “wall functions” were used (Launder and Spalding, 1974). The y+-values ranged from an average value of approximately of 300 on both the cylindrical wall and the impeller blade to over 1000 at the shaft. The latter region was thought to be less important. The 1storder accurate Hybrid differencing scheme was used for all advection terms. Additional calculations were performed using the 2nd-order CCCT-scheme (Curvature Compensated Convective Transport) for all equations. This scheme is a modification of the 3rd-order Quick-scheme in that it is bounded. Simulations were only carried out at the impeller speed at which the LDA measurements were carried out (i.e., 3 Hz). The rotating action of the impeller was implemented using the “rotating coordinates” approach. In this approach the coordinates of the complete grid rotated with 3 Hz while the vessel wall rotated back with 3 Hz. Consequently, the net velocity of the vessel wall equals 0 m/s. The Coriolis forces are implemented automatically in the “rotating coordinates” approach. The simulations were conducted as a transient event. The simulations with the k-ε model were started from still fluid conditions. Then 80 steps with a time step corresponding to 90° impeller rotation were applied using 25 iterations per time step. This corresponds to 20 complete impeller rotations of 6.66 seconds real time at 3 Hz. These large time steps were applied to obtain fluid motion in the tank. The attained solution was subsequently refined by allowing 12 rotations of the impeller with time steps covering an angular extent of 360/56 = 6.43°. The refined simulation thus consisted of 672 time steps and 25 iterations per time step.


65

For the simulations with DSM, the final solution obtained with the k-ε model was used as an initial guess. Subsequently, the geometry was allowed to rotate with time steps covering an angular extent of again 6.43° with a total of 672 time steps, i.e., 4 seconds of real time. Here also 25 iterations were used per time step. It was found that using the above-described time-marching procedure, a pseudo steady state was obtained in the vessel. This implies that at the end of the transient calculation the solution only differed from the solution in the previous time step in that the complete field was rotated with the required angular extent, indicating that sufficient convergence was obtained by the procedure adopted.

4.2.4 Scalar Mixing In the obtained velocity field, the mixing of an inert tracer was studied. In a cell next to the inlet (which is depicted in Figure 2.1) an initial concentration equal to 1 was defined. The progress of mixing was monitored at the position that is marked in Figure 2.3 with an ‘×’ (xL = 0.32, r = 0.295). The tracer was mixed in the previously mentioned pseudo steady state. Time steps in the tracer mixing simulations again covered an angular extent of 6.43°.

Chapter 4

66

4.3 Results and discussion LDA 4.3.1 Mean flow characteristics The time average tangential velocity Ūt proves to be almost independent of the axial position. Therefore, all the measurements performed at one radial coordinate have been averaged. These averages are presented as a function of the radius r in Figure 4.3 with a solid line. The location of the shaft, impeller arm, impeller blade and clearance are depicted on the x-axis. The impeller speed Uim as a function of the radius r is also depicted with a dotted line. The tip speed is equal to Uim(r = 0.29) = 2⋅π⋅r⋅N = 2⋅π⋅0.29⋅3 = 5.47 m/s. According to Figure 4.3, there appears to be a bulk region (0 < r < 0.22) where the tangential velocity is almost equal to the local impeller speed. The flow behavior in this bulk region can be qualified as ‘solid body rotation’. A second region can be distinguished near the cylindrical wall (0.22 < r < 0.3) where the tangential velocity is fairly constant (Ūt ≈ 3.8 m/s). In this region viscous forces reduce the tangential velocity. As the velocity of the cylindrical wall is equal to zero there will be a steep fall in tangential velocity very close to the wall. This could not be quantified as no measurements could be performed sufficiently close to the wall. 1 0.9

4.92

0.8

4.37

0.7

3.83

0.6

3.28

0.5

2.73

0.4

2.19

Ūt(r) [m/s]

Ūt (r)/Uim(0.29) [-]

5.47

Uim(0.29) = 5.47 [m/s]

0.3

1.64

Ūim(r)

0.2 0.1

1.09

Shaft

Clearance . 0.55 Blade .

Blade arm

0

0.00 0

0.05

0.1

0.15

0.2

r [m]

0.25

0.3

Figure 4.3: Tangential velocity Ūt as a function of the radial coordinate. On the left yaxis Ūt has been made dimensionless through division by the impeller tip speed. On the right y-axis the absolute velocity is given. On the x-axis, the location of the shaft, the blade arm, the blade and the clearance have been exemplified.

Fluid flow and mixing in an unbaffled horizontal stirred tank 0.2

r = R = 0.3 m 0.3

0.30

0.275

0.1

r [m]

0.25

0.05

0.29

0.225

0.28

0.2

0.27

0.175

0

0.25

-0.05 -0.1 -0.15

r [m]

Ua [m/s]

0.15

|

67

0.1

0.15

0.2

0.25

0.20 0.18

Impeller arm

0.15 0.13

0.19

0.1

0.10

B

0.075 0.05 0.025

0.05

0.23 Impeller blade

0.125

-0.25 0

0.25

0.22

A

-0.2

0.15

0.28 Clearance

-0.08 -0.06 -0.04 -0.02

xL [m]

0.05 0.03

0

0.3

0.08

Shaft

_0

0.00 0.02

Ua [m/s]

0.04

0.06

0.08

0.1

Figure 4.4: A) Time averaged axial velocity Ūa at several radii r, as a function of the axial coordinate xL. B) Averages of the velocities as presented in Figure 4.4A.

The time averaged axial velocities Ūa are depicted in Figure 4.4A as a function of the axial coordinate. The maximum absolute velocity is 0.2 m/s at (xL,r) = (0.1,0.29). At this position Ūt is equal to 3.8 m/s. Based on these measurements it can be concluded that the axial velocities are much smaller than the local tangential velocities. The axial velocity appears to be a function of the radius. There are regions with positive velocities (e.g., at r = 0.22) and region with negative velocities (e.g., at r = 0.28). As these regions cannot be determined easily from Figure 4.4A, the xL-averaged axial velocities are given as a function of the radius in Figure 4.4B. Near the cylindrical wall and near the impeller blade (0.3 < r < 0.27) the velocities are negative, meaning that these velocities are directed towards the reactor center. For 0.26 < r < 0.2 the axial velocities are a positive, thus directed towards the side wall. In Figure 4.5, the radial velocities are given. On average the velocity is negative, which violates with continuity. However, their order of magnitude is similar to the axial velocities. The highest velocities are measured near the side wall at high xL-values. 0.05

Ur [m/s]

0

|

-0.05

r [m]

-0.1

0.25

0.27 0.22

-0.15

0.19 0.16

-0.2 -0.25 -0.3 0

0.05

0.1

0.15

0.2

0.25

0.3

xL [m]

Figure 4.5: Time averaged radial velocity Ūr at several radii r, as a function of the axial coordinate xL.

Chapter 4

68 0.35

Average turbulent fluctuations

0.3

0.3

0.25

0.25

uP [m/s]

uT [m/s]

0.35

0.2 0.15 0.1

0.2 0.15 0.1

0.05

0.05

A

0 0

0.4

0.1

0.15

r [m]

0.2

0.25

0.3

0

0.4

0.3 0.25

uP [m/s]

0.3

0.2 0.15 Axial Tangential

0.1

Radial

C

0 0

0.1

0.15

0.2

0.25

xL [m]

0.3

0.1

0.15

r [m]

0.2

0.25

0.3

0.2 0.15 0.1 0.05

D

0 0.05

0.05

Periodic fluctuations on r = 0.27 m

0.35

0.25

0.05

B

0 0.05

Turbulent fluctuations on r = 0.27 m

0.35

uT [m/s]

Average periodic fluctuations

0

0.05

0.1

0.15

0.2

0.25

0.3

xL [m]

Figure 4.6: A) Time averaged turbulent fluctuation on r = 0.27 for axial, radial and tangential direction. B) Time averaged periodic fluctuation on r = 0.27 for axial, radial and tangential direction. C) xL- averages of the fluctuations presented in Figure 4.6A, but for all radii. D) xL-averages of the fluctuations presented in Figure 4.6B, but for all radii.

4.3.2 Turbulent quantities In Figure 4.6A, the time averaged turbulent fluctuations are given at a radial coordinate r = 0.27 m. The turbulent fluctuations for the three directions are of the same size, indicating that the turbulence shows isotropic behavior. The fluctuations in the vicinity of the side wall are higher than in the reactor center. Similar isotropic behavior in unbaffled vessels has also been observed by Montante et al. (2001). The periodic fluctuations at a radial coordinate r = 0.27 are presented in Figure 4.6B and show to be of the same order of magnitude as the turbulent fluctuations. Anisotropic behavior is observed near the side wall where the tangential fluctuations show a strong increase. In this region, the side wall scraper probably is responsible for the stronger periodic fluctuations. In Figure 4.6C and 4.6D, the xL-averaged values for the turbulent and periodic fluctuations, respectively, are plotted against the radial coordinate. The turbulent fluctuating velocities in Figure 4.6C indicate isotropy as the fluctuations in all three directions are of the same order of magnitude over the complete radius. The periodic fluctuations in Figure 4.6D show differences for the three directions, indicating the presence of an anisotropic turbulent mesostructure.


69

0.3

0.25

r [m]

0.2

0.15

0.1

ε [m2/s3]

0.05

0

0

0.05

0.1

0.15

0.2

0.25

0.3

xL [m]

Figure 4.7: Distribution of the energy dissipation rate ε for the measurable region.

In Figure 4.7, the energy dissipation rate ε is given in a 2-dimensional plot. This area represents 80% of the total reactor volume. Only the turbulent fluctuating part is included as the periodic fluctuations are supposed to represent pseudo turbulence. In the measured region, ε ranges from 2.5 m2/s3 near the cylindrical wall to 0.01 near the shaft. The maximum of 2.5 m2/s3 is located in the region where the impeller tips overlap. Integration of ε and multiplication by 2 to take the other half of the reactor into account, provides a dissipated energy of 45 W. On average, ε equals 0.31 m2/s3. Previously reported results (Van der Gulik et al., 2001a) provided a value of 375 W for the complete reactor so that almost all energy dissipates very close to the wall.

4.3.3 Overall remarks The detailed LDA information provides guidelines that can be useful in the use and scale-up of Drais reactors. For example, Figure 4.6B shows that the periodic fluctuations in the reactor center are lower than near the side wall. Consequently, mixing will be more efficient near the side wall. From this observation one can expect that the L/D-ratio should not be made too large, as the fluctuations in the vicinity of the reactor center lower upon an increase in L/D, thus decreasing the mixing performance. This is in agreement with results presented in a previous paper in which we have shown that mixing times increase more than linearly upon an increase in L/D (Van der Gulik et al., 2001a and 2001b). In Table 2 the average velocity and average turbulent components are given for r = 0.27 which are representative for the reactor. Note that the turbulent and periodic fluctuations in axial and radial direction are much higher than the averaged

Chapter 4

70

velocities. In axial direction, the difference is one order of magnitude. Therefore, the flow in axial and radial direction is exclusively determined by turbulent and periodic fluctuations. Consequently, macro mixing in these directions will be determined by turbulent dispersion. In tangential direction the convective flow is determining. The reactor can be considered as a moderate turbulent mixer as the liquid is mainly rotated in tangential direction and overall mixing is slow. Moreover, at radii below r = 0.22 the liquid rotates almost as a solid body. Figures 4.6C and 4.6D show that the fluctuations reduce at decreasing radii, meaning that the mixing performance reduces. Therefore, the reactor is usually used at fill ratios below 50%, so that at high impeller speeds a ring of fluid exists above r = 0.2 where at least some mixing occurs. This limitation in fill ratio up to 50% can be seen as a disadvantage of the Drais reactor. Table 4.2: RMS values in m/s of velocity and fluctuations scales at r = 0.27.

r = 0.27 Mean velocity Turbulent fluctuations Periodic fluctuations

Tangential 3.73 0.32 0.2

Axial 0.035 0.32 0.18

Radial 0.14 0.31 0.16


71

4.4 Results and Discussion CFD 4.4.1 Mean flow characteristics The calculated tangential velocities Ūt only depend on the radius which has also been observed in the LDA experiments. Therefore, all tangential velocities are xL-averaged analogous to the LDA velocities as represented in Figure 4.8. Overall, DSM combined with the Hybrid differencing scheme provides the best results as the data are the closest to the LDA data. Especially around the impeller blade at r = 0.27, the comparison is relatively good. The k-ε model with the Hybrid differencing scheme underestimates the tangential velocity around r = 0.27 whereas the DSM with the CCCT-differencing scheme underestimates the tangential velocity. The dissipative nature of the Hybrid differencing scheme seems to be effective in predicting the measured velocity. In the LDA data the highest tangential velocity is measured at r = 0.22m. All CFD solutions underestimate the velocity at this radius. As will be shown, axial velocities at this radius are also underestimated. Uim(0.29) = 5.47 m/s

1

5.47

0.9

4.92

k-e/hybrid

0.7

Ūt (r)/Uim(0.29) [-]

4.37

LDA-data

3.83

dsm/hybrid

0.6

3.28

dsm/ccct

0.5

2.73

0.4

2.19

0.3

1.64

0.2

1.09

Uim(r)

0.1

Tangential velocities .

0 0

0.05

0.1

Ūt (r) [m/s]

0.8

0.15

r [m]

0.2

0.25

0.55 0.00

0.3

Figure 4.8: xL-averaged tangential velocity Ūt as a function of the radial coordinate for the LDA measurements and the CFD calculations.

In Figure 4.9 simulated mean axial and radial velocities at r = 0.27 are compared with LDA data. The axial velocities in Figures 4.9A are all of the same order of magnitude and follow the same trend. In detail the velocities differ but regarding the quality of the numerical grid and the large periodic fluctuations that have been observed in the LDA measurements, the comparison is as good as expected. The simulated radial velocities in Figure 4.9B differ substantially from

Chapter 4

72 0.1

0.10

0

0.05

-0.1

Ūr [m/s]

Ūa [m/s]

0.15

0.00

-0.2

-0.05

-0.3

-0.10 -0.15

lda dsm k-e

r = 0.27

-0.4 -0.5 0

0.05

0.1

0.15 0.2 xL [m]

0.25

0.3

0

0.05

0.1

0.15 0.2 xL [m]

0.25

0.3

Figure 4.9: A) Comparison of the axial velocity Ūa, and B) the radial velocity Ūr. Both time averaged measured with LDA and calculated using the k-ε model and the DSM.

the measured velocities. Near the side wall a relatively strong radial flow (0.3 m/s) towards the shaft has been obtained. The effect of this flow phenomenon will be discussed in section 4.5. The strong radial velocity could not be confirmed using LDA data as no measurements could be performed near the side wall. In Figure 4.10A the xL-averaged axial velocities are compared with LDA measurements. The k-ε model and the DSM are well matched despite some minor differences near the shaft. The profile of the simulated velocities has the same nature as the LDA data, meaning that alternating negative (towards reactor center) and positive (towards side wall) velocities occur as a function of the radius r. The origin of the alternating direction can be explained from the vector plot in Figure 4.10B which has been obtained using the DSM with the Hybrid scheme. In this vector plot, the axial and radial components of the total velocity vector are 0.3 0.275 0.25 0.225 0.2

r [m]

0.175

DSM/Hybrid

0.15 0.125 0.1

A

k-e/hybrid

0.075

dsm/hybrid

0.05

C

dsm/ccct

0.025 0 -0.08 -0.06 -0.04 -0.02

B

LDA-data

Axial velocities

_0

0.02

0.04

0.06

0.08

0.1

Ua [m/s]

Figure 4.10: A) xL-averaged velocities Ūa as a function of the radius r. B) Vector plot representing the axial and the radial components. C) Exemplification of the point of view in the reactor for obtaining Figure 4.10B.

Plane with axial and radial velocity vectors


73

projected on a plane that crosses the shaft and the impeller blade near the side wall. The point of view is along the impeller blade as depicted in Figure 4.10C. The impeller blade (0.24 < r < 0.29) transports the fluid towards the reactor center while the flow reverses between the impeller blade and the shaft. Between impeller tip and cylindrical wall the flow appears to be directed towards the side wall which is not observed in the LDA measurements.

4.4.2 Turbulent quantities The periodic velocity fluctuations are not directly available from CFD data, but can be calculated using equation 4.1: kP =

1 1 2 N0

∑ [(U N0

− U 1 ) + (U 2 − U 2 ) + (U 3 − U 3 ) 2

1

2

2

]

[m2/s2] (4.1)

1

in which Ui are instantaneous local velocities and Ūi the time averaged velocities (Montante et al., 2001). N0 is the number of cells in the tangential direction and kP represents the energy content of the periodic fluctuations. The results are presented in Figure 4.11A. The data presented are the averages of all kP in tangential and axial direction. kP as calculated with both the k-ε model and DSM are of the same size as the measured kP. However, they differ in trend: the simulated kP remains more or less constant over the radius, the measured kP decreases with decreasing radius. The overestimated kP at low radii is probably due to the poor mesh quality near the shaft. Near the cylindrical wall, the differences in kP are approximately one order of magnitude. The turbulent fluctuations in all three directions are represented by the turbulent kinetic energy kT, which has been calculated using equation 4A.5 in Appendix 4A. Although the use of kT for DSM masks anisotropic properties, it makes comparison with kT from the k-ε model straightforward. The presented kT values in Figure 4.11B are xL- and time averaged. Clearly the k-ε model predicts the kT values better than DSM. However, both simulated kT are lower than the measured kT. Similar differences have been observed before and have been recognized as one of the major of discrepancies between LDA measurements and CFD calculations (Montante et al., 2001; Brucato et al., 1998; Armenante et al., 1997; Ng et al., 1998). Figure 4.11C shows the (xL-averaged and time-averaged) energy dissipation rate for the LDA measurements and the simulations. The simulated ε is significantly lower than the measured one. The simulated ε shows a strong increase near the cylindrical wall. The total dissipated energy using both turbulence models is given in Table 4.3. The predicted values appear to be about 30% of the measured value, for both the complete reactor and the part of the reactor in which LDA measurements could be performed (Van der Gulik et al., 2001a). The ratio between the values for the complete and for part of the reactor are the same for LDA and CFD data, meaning the distribution in dissipated energy appears to be similar.

Chapter 4

74

Table 4.3: Dissipated energy [W] in the complete reactor and in a part of the reactor using different experimental techniques and numerical models.

Experimental Temperature LDA measurements 375 (100%) -

Region Complete reactor Part of the reactor -0.29

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