Online monitoring of blending of non-Newtonian fluids in static mixer

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UNIVERSITY OF PISA Dipartimento DICI

Corso di Laurea Magistrale in Ingegneria Chimica

TESI DI LAUREA

ONLINE MONITORING OF BLENDING OF NON-NEWTONIAN FLUIDS IN STATIC MIXER

RELATORE

Prof. Ing. Elisabetta BRUNAZZI Prof. Ing. Federico ALBERINI

CONTRORELATORE Prof. Ing. Roberto MAURI

CANDIDATO Andrea ALBANO

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Introduction

1.1 Background

New Product Developments (NPD) is an ongoing activity involving frequent new and improved products being transferred from laboratory scale development to manufactur-ing at industrial scale. Traditional approaches to NPD focus the attention at laboratory scale with little or even no attention to a formulation’s ”manufacturability”. These is-sues are typically only addressed during scale up, at pilot scale and are often not fully resolved when the product goes to manufacturing scale. Such an approach to scale up also involves compromises. These result in not only longer and costlier scale up but also frequently increased production costs. These critical challenges also limit industry’s abil-ity to achieve more efficient and flexible processes. A way to deal with these challenges with a different approach is to improve the capabilities in terms of in situ measurements.

Online in situ monitoring can also help, with the help of the right instruments, to change a batch process to a continuous one gaining all the benefits of this kind of process with the compromise to improve process control measures to ensure consistent product quality. Trying to solve this challenges Electrical Resistance Tomography is used trying to monitor the mixing performance of static mixer in blending non-Newtonian fluids in laminar regime. Laminar mixing using static mixers has been the subject of interest in the last few years. Static mixers provide the opportunity to progress towards change from batch process to continuous with a reduction in inventory and plant space. Academics

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researches have focused studies in blending of fluids with complex rheology because of their wide use in industry. The aim of this researches are the development of knowledge of laminar mixing in static mixer is essential in order to achieve performance similar to traditional batch process.

Design the right geometry to achieve the desired degree of mixing is still a major challenge in pipe mixing with laminar flows, in fact in laminar flow there is a lack of an advective radial mixing mechanism since all fluid streamlines are in the axial direction flowing in parallel layers. Many works in literature deal with the comparison of different commercial static mixers, such as Kenics (KM) used in this thesis.

Laminar mixing is applicable in many different industrial applications including food ([Talansier et al.,2013]), personal care, household products, slurries, polymer manufac-ture and catalyst washcoats. All of these products have non-Newnotian rheology and this adds complication in understanding mixing performance.

1.2 Objectives and aims

The overall aim of this thesis is to investigate the ability of the Electrical Resistance Tomography (ERT) to evaluate mixing performance of a static mixer blending non-Newtonian fluids. To validate the results obtained with this technique Planar Laser In-ducted Fluorescence (PLIF) is used as it is the standard technique used widely in literature to evaluate mixing performance for both stirred vessels [Arratia and Muzzio,2004] and static mixers ([Lehwald et al.,2010], [Alberini et al.,2014a]). The ERT is an example of technique which allows online in situ measurements that could both decrease the cost of scale up and give the flexibility needed from the nowadays liquid products industries. The great advantages of ERT are:

• safe technique, no need of additional protections in the plant, • can be used directly in situ,

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• allows online quality control without stops the production, • post processing require less time,

• flexible if the geometry of the process is changed.

ERT has been used in a qualitative way in great number of works, this thesis wants to do a step forward for quantitive understanding thanks to its potentially future plant application. PLIF is not suitable for plan application for multiple reasons:

• since it uses laser is not a safe technique, there is a need of additional protections in the plant,

• to be used directly in situ require plant change in design, it requires a total black room to work reliably,

• requires to dope one of the fluid with a dye preventing the use of the resulting fluid as final product,

• requires transparent fluids, with opaque fluids the technique is not practicable, • the process has to be stopped to be used.

The comparison between the technique will be made using mixing performance param-eters such as intensity of segregation (CoV , LogV a); however those are statistical pa-rameter and they have been discussed at length and sometimes criticised in literature ([Kukukova et al.,2009]) in particular when a complex mixing pattern has to be charac-terised. These methods, if used in isolation can create misleading results. For this reason the comparison will be done with the areal intensity distribution ([Alberini et al.,2014b]) as well.

After the experimental part done in the laboratories, the processing and analysis of the data have required the use of the software MATLAB (MathWorks). The PLIF and ERT images processing has been found to be the crucial part of the analysis as further explained in the Chapter4. Once the codes have been implemented the difference in conductivity and in rheology have been related to the reliability of the ERT results.

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1.3 Thesis layout

• Chapter2

Contains a review of the literature on the fundamentals of mixing in terms of rheol-ogy of fluids and types of process used for the mixing. Relevant batch and contin-uous mixing equipment are rewired focussing on laminar flow and on the blending of non-Newtonian fluids. Moreover there will be a review of the literature on the fundamentals principles of ERT and its application.

• Chapter3

The material and methods used in this thesis work are described. A detailed de-scription of rig and apparatus for the different experiments is given focusing on PLIF and ERT experiments with a description of the v5r, the ERT instrument sup-plied by ITS. The characterisation of all the fluids, the greater problem of this the-sis, is presented including their rheology. Finally all the post process methods are detailed explained for both the mixing performance evaluation and for the injected area evaluation.

• Chapter4

The results obtained in post processing for both PLIF and ERT are shown. ERT limitation for the mixing performance and injected evaluation are investigated un-der the effect of different difference in conductivity and in rheology.

• Chapter5

Final overview of all the work done in this thesis is taken, with suggestions for the future works and ERT application in an industrial perspective.

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Chapter 2

Literature review

2.1 Introduction

A review is presented of current literature on the fundamentals of mixing of non-Newtonian fluids and on the various methodologies which can be applied to characterise mixing in equipment relevant to this thesis. In Section2.2 a review of the rheological behaviour of different fluids is presented focusing on non-Newtonian fluids. In Section2.3 litera-ture on batch and continuous mixing processes are compared. In Section2.4methods to quantify performance are reviewed focusing on alternative methods to determine scale and intensity. In Section2.5Planar laser inducted florescence is explained with some ap-plication in mixing performance evaluation. Finally in Section2.6Electrical Resistance Tomography and its application are presented.

2.2 Fluid mechanics fundamentals

2.2.1 Newtonian fluids

In continuum mechanics, a fluid is deformed as it flows due the application of external forces. Frictional effects are exhibited by the relative motion of molecules, this exhibits through the fluid’s dynamic viscosity which is a bulk fluid property. A Newtonian fluid is defined as a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional to the local strain rate. This means that for Newtonian fluid dynamic viscosity (µ) is constant hence there is a linear relationship between the applied

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shear stress (τ ) and the observed shear rate (˙γ). It can be shown that at the limit of small angular deformations, the shear rate is equivalent to the velocity gradient in the flow ([Byron R. Bird,2007]) this leads to Newton’s law of viscosity, written in 1-D flow in x−direction shown in Figure2.1.

F

A = τ = µ(− dVx

dy ) = µ ˙γ (2.1)

Figure 2.1: Schematic representation of unidirectional shearing flow

In general a fluid is Newtonian if, and only if, the tensor that describes the viscous stress and the strain are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow

τ = −p      1 0 0 0 1 0 0 0 1      + µ      0 ˙γ 0 ˙γ 0 0 0 0 0      (2.2)

If the fluid is not compressibile the tensor of the stress is simplified as2.1and the viscos-ity tensor reduces to two real coefficients, describing the fluid’s resistance to continuous shear deformation and continuous compression or expansion, respectively. Although no real fluid fits perfectly with the definition many common fluids such as water, glycerol and glucose syrup can be assumed to be Newtonian for practical calculations under ordi-nary conditions.

2.2.2 Non-Newtonian fluids

Fluids are called non Newtonian when presenting rheological behaviours that are more complicated than the simple linear relation between shear stresses and shear rates, char-acterising Newtonian fluids. Many important fluids in the chemical industry display

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non-Newtonian behaviour at moderate rates of strain. The range of fluids is wide and includes polymer solution, which have very large molecular weight and form long chains that give shear thinning or thickening behaviour, as well as emulsion and slurries containing sus-pended particles which may or may not be deformable. In polymer solutions this effect depends upon the history of the local strain rate experienced by the fluid due to elastic properties leading to a time-dependent effect; this manifests itself more strongly with in-creasing polymer concentration and lengthens the relaxation times of the polymer chains within the fluid. Other fluids do not experience time dependent behaviour and thus their flow rheology can be expressed simply in term of constitutive laws or non linear relation-ships between the shear stress and shear rate.

Time dependent

In general, materials with time-dependent rheological properties are called viscoelastic, as they exhibit both viscous and elastic characteristics when undergoing deformation. Many materials, in response to a step change in shear rate, take a finite time to reach their equilibrium state, while at a very short times they exhibit a solid-like behaviour. Such materials, where the viscosity decreases when the shear rate remains constant, are called thixotropic.

For some other fluids, called rheopexy or negative thixotropic ([Pradipasena and Rha, 1977]), applying a constant shear stress in time causes an increase in the viscosity, or even solidification. Those behaviour are shown in Figure2.2.

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Figure 2.2: Schematic of shear stress shear rate behaviour for time-dependent fluids [Wikipedia, 2017]

Examples of thixotropic fluids are clays, muds, honey and many kinds of paints. Exam-ples of negative thixotropic include gypsum pastes and printer inks.

Time independent

Non-Newtonian fluids present a time independent behaviour, where the apparent viscos-ity is not related and changes only as function of the applied shear rate. Generally the group si sub-divided in three categories: shear thinning, shear thickening and Bingham, or viscoplastic, fluids. In Figure2.3the constitutive properties of three of the most com-mon non-Newtonian fluids are represented.

• Shear thinning fluids

Shear-thinning fluids (also referred to as pseudo-plastic fluids) present an effective viscosity (that is the slope of the shear-stress curve of Figure 2.3 that decreases with increasing shear rate (and shear stress). A typical example is ketchup: when we squeeze it out of a bottle, it goes from being thick like honey to flow like a low viscosity fluid. The simplest model of shear-thinning behaviour is that of a suspension composed of micron-size, mutually attracting particles immersed in a

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Figure 2.3: Schematic of shear stress shear rate behaviour for time-dependent fluids [Mauri, 2015]

Newtonian fluid. When the suspension is quiescent, or at low shear rates, the sus-pended particles will tend to aggregate, forming clusters that oppose the shearing and therefore the fluid is viscous. On the other hand, as the shear rate increases, clusters will gradually dissolve, so that the effect of the attracting forces will de-crease, thus decreasing the viscosity. The most common costitutive equation used to describe the behaviour of a shear thinning fluid is the power law, which relates the shear stress to the shear rate as:

τ = K ˙γn (2.3)

where K and n are the consistency and flow indices, respectively (n < 1 for a shear-thinning fluid) (see Fig.2.4).

• Shear thickening fluids

Behave in the opposite manner, compared to shear-thinning fluids, i.e., their effec-tive viscosity increases at increasing shear rate (and shear stress). Their properties can often be approximated by a power law equation2.3with values of power law index, n >1. the term ”dilatant” has also been used for suspension where the par-ticles and the liquid of the suspension play a critical role on the overall rheology. • Visco-plastic fluids

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Figure 2.4: Schematic of shear stress shear rate behaviour for shear thinning fluids [Tagkey, 2008]

In this thesis, the fluids used are represented by this group of time independent non-Newtonian fluids. The most common fluid model for visco-plastic behaviour is the Bingham model. Bingham fluids are visco-plastic materials that behave as rigid bodies at low stresses but flow as viscous fluids at high stresses. A com-mon example is tooth- paste, which will not be extruded until a certain pressure is applied to the tube, then it flows out as a solid plug. As shown in Figure2.3, Bingham fluids do not exhibit any shear rate (and consequently no velocity and no flow), until a certain yield stress τ0is reached. Beyond this point, the shear stresses

increase linearly with increasing shear rates, and the slope of the line, µ0, is called

plastic viscosity. So, the constitutive relation for Bingham fluids reads:

     τ = τ0+ µ0˙γ for τ > τ0 ˙γ = 0 for τ < τ0 (2.4)

Another common model used to describe the rheological behaviour of shear thin-ning and viscoplastic fluids (due to the presence of yield stress) is the Herschel Buckley model. It is a simple generalisation of the Bingham plastic model, where the non linear flow curve si defined by an equation containing three constants:

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τ = τ0+ K ˙γn (2.5)

K and n are consistency and flow indices, and τ0 is the yield stress.

2.2.3 Rheological measurements

Shear viscosity measurement

A rheometer is a laboratory device which is used to measure the rheology behaviour of fluids. Different geometries can be used for the characterisation of different types of fluids. Cone and plate is the geometries used in this work, the difference among geometries is the gap between the plate and the sample platform. When the analysed fluid presents non-Newtonian rheological behaviour the cone and plate geometry has to be used because the gap changes as function of the angle α which allows a constant shear along the whole sample (see Fig.2.5).

Figure 2.5: Cone and plate rheometer geometries

To analyse the sample either a controlled shear stress or shear rate can be applied to obtain the behaviour under shear whereas some rheometer apply an extensional stress to determine the extensional viscosity of the fluid. The shear rate and stress factors are presented for cone and plate geometries:

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    

Fyω= _tan(α)1 ω Shear rate

FσM = _sπR33M Shear stress

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2.2.4 Mixing fundamentals

Two mechanisms are responsible for fluid mixing: diffusion and advection. Advection, transport fo matter by a flow, is required for mixing, The quality of the mixing is function of the energy input in the system which, in flow, increases with the gradient of pressure between two different points in the system. In fluid mechanics to identity which of the two mechanisms is predominant a non dimensional number called Schmidt (Sc) is used. Sc is a dimensionless number defined as the ratio of advection to diffusion and it is used to characterise fluid flows in which there are simultaneous momentum (advection) and mass diffusion convection processes. The higher Sc higher is the contribution of advection compared to molecular diffusion. In liquids molecular diffusion alone is not efficient for mixing since they possess comparatively low diffusivity values, thus in liquid mixing Sc >>1. The mixing of fluids is dependent upon the flow regime, i.e. whether the flow is laminar or turbulent, which can be determined according to the value of the Reynolds number Re= inertial forces viscous forces = ρuD µ (2.7) Laminar flow

The laminar regime prevails at low flow velocities where the pressure-velocity relation-ship is a function of the viscous properties fo the fluid. In this regime layers of fluid flow over one another at different speeds with virtually no mixing between layers. The flow in a straight pipe is a typical example of steady non-chaotic flows, in this chase the velocity v is a function of the radius of the pipe r:

v vmax

= 1 − (r R)

2 _(2.8)

called Poiseuille parabolic velocity profile, obtain with the no-slip condition at the wall. This mixing mechanism generate a poor mixing environment, because fluid motion is

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dominated by linear, viscous forces instead of non-linear inertial forces. Static mixers enable the limitation to be overcome by employing chaotic mixing mechanisms.

Chaotic flow

Chaotic flow involves the stretching and folding of fluid elements in laminar flow which enables the interfacial area between elements to grow at an exponential rate, as opposed to the linear growth rate which would normally be expected in laminar flow. Numerous experimental and computational examples have shown that real fluids flow can produce the type of stretching and folding that leads to chaos ([Ottino,1990]). Chaos is impossible in steady flow because it is completely characterised by time-invariant streamlines that coincide with path lines, and fluid elements lie within the same streamlines at all times. In fluid flows, a necessary but not sufficient condition of chaos is ”streamline crossing” of two streamline portraits taken at arbitrary times. The the crossing can create a special type of folding which is the preliminary step for mixing using a chaotic mechanism. The other condition created by the chaos is the stretching which can generate results in effective mixing within chaotic regions if it is accompanied by folding. Considering the Lyapunov theory ([Eden et al.,1991]), in a chaotic system the distance separating two fluid particles initially located very close to one another will diverge exponentially with time. Considering that the objective fo any mixing operation is to disperse cloisters of material, exponential divergence of clusters of material that are initially close to each other is extremely desirable of mixing applications.

Turbulent Flow

The turbulent flow regime prevail at high value of Reynolds number. Turbulence can be interpreted as some sort of critical phenomenon, like phase transition, with a net separa-tion between laminar and turbulent flows. In other words, there are no flows that are half laminar and half turbulent: as soon as the critical conditions are reached, laminar flow becomes turbulent. Turbulent flow is characterised by irregular movement with a non deterministic path which is intrinsically time-dependent. There is no observable patterns and no definite layers. The flow is governed primarily by inertial properties of the fluid in motion. In turbulent regime the fluid is subjected to random fluctuations in term of

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velocity and direction. Moreover, considering an empty pipe, there are three separate regions across the section of the pipe:

• Laminar flow next to the wall, where the velocity is below the critical value. • A central core of turbulent flow.

• Transition Zone between the two.

Most of the flow equations are empirical and they are used for the understanding and characterise such a complex phenomena.

2.3 Mixing equipment

The mixing of liquids is a key operation in which two of more miscible liquids are mixed together to reach a certain degree of homogeneity ([Suzanne M. Kresta,2004]). Mixing is always needed for the manufacture of a wide range of products such as food, personal care, home care and expanding to the production of catalysts. Blending may take place between high or low viscosity liquids, miscible and immiscible fluids. The mixing of flu-ids is generically achieved using either batch or continuous processes. In batch processes, stirred tanks and similar devices are used to blend fluids where the impeller generates the fluid motion. The amount of time required to have the desired degree of homogeneity is called blend time or residence time, time spent by the fluids inside the tank before being mixed. Static mixer of similar devices are used for continuous processes where fluids are pumped through mixing elements installed inside pipes. Whilst the flow regime of the system can be determined by using the Reynolds number. When non-Newtonian flu-ids are handled this approach is more complicated since the viscosity is not constant. Generally liquid with low viscosity are mixed in turbulent flow regime ([Nienow,1997], [Kumar and Narayan, 2009]), for highly viscous fluids a certain force may be needed in order to reach uniformity. Mixing of viscous liquids typically occurs in laminar flow ([Alvarez et al.,2002]) regime and stagnation points, known as islands of unmixedness, may form. Due to the complexity of the geometry usually the range of Re are different between continuous and batch process ([Chandra and Chopra,1992])

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2.3.1 Mixing of non-Newtonian fluids in batch processes

Most of the chemical production involves the use of mechanically agitated stirred vessel for manufacturing. Understanding of the behaviour of stirred vessels has received con-siderable research effort over the last few decades. Unfortunately, most of the fluids used in industry have non-Newtonian behaviour; research has generally focused on the blend-ing of sblend-ingle and multiphase low viscosity fluids in turbulent flow regime. As said before the rheology of non-Newtonian fluids complicates the study of fluid dynamics or the per-formance of any system where they are applied. The fundamental mixing mechanism for these vessels is via the transfer momentum to the material within the vessel via the physical movement of rotating impeller blades. Stirred tanks containing non-Newtonian fluids have been studied in gas-liquid systems ([Tecante and Choplin,1993]) or even in three phases (gas-liquid-solid) ([Kawase et al., 1997]). The huge variety of processes carried out in stirred vessels span a wide range of vessel sizes and geometries for opti-mal process efficiency. A standard nomenclature exists to describe the dimensions of a vertical cylindrical tank (see Fig.2.6).

Figure 2.6: Main dimensions of stirred vessel. T = tank diameter, D = impeller diameter, C= clearance, W= impeller withd; B= buffle witdh; H=liquid height

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clearance from the tank bottom (C) are the parameters used for standard configurations in terms of geometric rations such asD_T,C_T etc. In the last few years more studies have been focused to the fluid dynamics of non-Newtonian fluids. Different impeller types have been investigated to understand which one is more suitable for high performance. Usually the behaviour if Newtonian fluids is compared to non-Newtonian ones ([Aubin et al., 2000]) in order to elucidate the difference between the two systems. For fluids which present a non-Newtonian behaviour, research has been performed to obtain the shape of stagnant and moving regions within the vessels (due the formation fo caverns) which can occurs in the blending of viscoplatic and also pseudo-plastic fluids ([Hirata et al., 1994], [Galindo and Nienow,1992]). A further study investigates the cavern side using both experimental and computational approaches ([Adams et al.,2007]). The technique of particle image velocimetry (PIV) is largely used to investigate the fluids dynamic of stirred tank but also alternative techniques are applied in particular when the used fluids are opaque ([Fangary et al.,1999], [Simmons et al.,2009]).

2.3.2 Mixing of non-Newtonian fluid in continuous processes

Since in liquids the mixing obtained by diffusion is poor, it is necessary to design mixers which introduce chaos to the flow if the regime is laminar. This is the basis operation of static mixers which are used of this thesis. The mechanism in all static mixers is quite similar where a periodic forcing of the fluid stretches and folds the fluid streamlines. The most common application is the mixing of shear sensitive fluids such as polymers, where the fluids are mixed under very gentle process conditions. The flow inside the static mixer is characterised by an exponential rate of stretching and also reorientation due the repeated changes in flow direction (hence for this reason it is called chaotic mixing see Section2.2.4). In the literature the actual first patent on a static mixers dates from 1874, Sutherland describes a single element, multilayer, motionless mixer, used to mix air with a gaseous fuel ([Meijer et al.,2012]). A static mixer is a device with no moving parts and it relies on the motion of the fluid, due to external pumping, to move fluids through it. Static mixer is mostly ideal for continuous flow processes which require short resi-dence time. Unlike dynamic mixers, it could process a wide range of viscosities with a minimum space and maintenance requirements [Myers et al.,1997]. Since the shear

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force exerted on fluids is generally lower for static mixers, it is very ideal for industrial processes which are shear-sensitive [Junker et al.,1994].

During the last 10-15 years, many industries have moved from batch to continuous processes to reduce cost of utilities and space requirements for production. The flexibility fo production is also another important factor which addresses the development of inline mixing. The operation and design of static mixers are discussed in key texts, for exam-ple in Handbook of industrial mixing: science and practice [Suzanne M. Kresta,2004], unfortunately most of the correlation are described only for the mixing of Newtonian fluids. Two criteria are used to judge the efficiency of a static mixer: the first is en-ergy consumption and the second is its dimension. Most of the works about comparison of performance are concentrated on Newtonian fluids ([Pahl and Muschelknautz,1980], [Rauline et al.,1998], [Rauline et al.,2000], [Meijer et al.,2012]). Extensive blending data have been collected for the Sulzer SMV, KM and HEV mixers in the transitional and turbulent flow regimes using a laser induced fluorescence (LIF) technique ([Meijer et al., 2012], [Anderson and Meijer,2000]).

Industrial applications and commercial static mixers

In the Table2.1 are presented most of the commercial static mixers and the industrial applications in function of geometry and flow regime.

Table 2.1: Commercially available static mixer and industrial applications

Name Static mixer Area of application Mixer Geometry

Chemineer-Kenics [chemineer,2017]

Kenics Mixer (KM) Turbulent and Laminar blending of Liquid-Liquid and Gas-liquid dispersion. Polymer Extrusion.

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Table 2.1 – Continued from previous page

KMX-V Mixer Laminar blending and liquid dis-persion.

HEV Mixer Low-viscosity liquid-liquid blend-ing and gas-gas mixblend-ing.

UltraTab pH control, Chlorination and Chemical dosing/flash mixing. Disinfection and Polymer blend-ing in water treatment applica-tions. Turbulent flow mixers in desalination, Chemical Process-ing and any all low viscosity blending.

WVM Mixer Turbulent and Laminar Blending. Liquid-Liquid and Gas-liquid dis-persion. Gas-liquid dispersion applications and where mixing is required at very low flow condi-tions.

Koch-Sulzer [ Koch-Sulzer,2017]

SMX and SMX plus Mixing of Shear sensitive flu-ids such as polymers. Dispers-ing of gases and liquids in high-viscous fluids.

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SMV Dispersive mixing or mass transfer in the turbulent flow regime. Examples are oil/water or gas/liquid systems.

Wymbs Engi-neering Lightnin [Wymbs,2017]

Series 45 Inliner Mixing, Blending, Liquid and gas Dispersion and Emulsion For-mation. Direct steam heating. Plug flow inline chemical reac-tors. Laminar-flow heat transfer. Marbleising. Layer Formation.

Komax [Komax,

2017]

Triple action Any application of static mixer. Designed for use where addi-tives to the main pipeline flow have already been introduced upstream of the mixer

A Series Same as Triple action

M series Same as Triple action

CPS Uniform temperature and ve-locity profiles-from centerline to sidewall-produce continuous, predictable blending, dispersing, heat-exchanging and reaction time.

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Tube Originally designed for a reac-tive resin field, they are used in mixing particulate solids, liquids, and gases.

Brann and Lubbe

[Brann and

Luebbe,2017]

N-Form Static mixer Mixing hardeners, accelerators and colourants into resins. Mix-ing propellant and colour stock. Dispersing T iO2 suspensions

and acetic acid in caprolactam. Mixing water glass, catalysts and water. Continuous colour shading and dilution. Diluting retention media. Mixing glue. Continuous shading of printer’s ink. Food and drinks. Addition of fat to low-fat soft cheese and quark. Colouring glucose-sugar mixtures. Dispersing water in crude vegetable oil. Mixing hop extract and sugar solution into beer. Mixing sugar syrup, fruit concentrates and water cosmet-ics and detergents. Mixing sur-factants, preservatives, perfume and salt solutions with water. Neutralising sulphonic and fatty acids with sodium hydroxide so-lution.

Toray [Toray,2017] Static Mixer KOFLO Water treatment.

UET [UET,2017] Heliflo Series 1 Mixing of fluids with viscosities up to 100.000 cps as in the pro-duction of plastics.

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Heliflo Series 2 Low viscosity fluid mixing ap-plications such as oil disper-sion, color dilution, and coatings preparation.

Heliflo Series 3 Highly efficient in turbulent mix-ing applications such as wa-ter disinfection with chlorine and gas blending.

Heliflo PAC Shell and tube design for heat transfer in the production of plastics or a velocity increaser for dispersion. of fluids with vis-cosities up to 100.000 cps as in the production of plastics.

in this thesis one specific geometry is used for the experiments:

Figure 2.7: Kenics KM static mixers [chemineer,2017]

Kenics KM mixers (Fig.2.7) are equipped with helical mixing elements which direct the flow of material radially toward the pipe walls toward the pipe walls and back to the

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centre. Additional velocity reversal and flow division results from combining alternative right and left hand elements, thus increasing mixing efficiency [chemineer,2017].

2.4 Measure of mixing performance

2.4.1 Measures to quantify scale

The scale of segregation measures the thickness or dimension of the fluid striations in the lamellar structure generated by chaotic flow. Different approaches can be found in literature:

• Distribution of striation thicknesses

The distribution of striation thicknesses can be computed form the stretching dis-tribution ([Zalc et al.,2002a]) assuming that material filaments are stretched in one direction and simultaneously compressed in another direction at the same rate. • Determination of stretching field

Determination of stretching field is basically the characterisation of the elongations of fluid filaments in each region of a flow. The Lyapunov exponent measures the average stretching of fluid filaments after an infinite amount of time, the larger Lyapunov exponent is the more efficient a mixing process is.

• Determination of inter-material area density

An alternative approach is the inter-material area density which measures the amount of contact area between mixture components in each region of a flow.

2.4.2 Measures to quantify intensity

Generally the more commonly used approach for the classification of mixing is the study of intensity of segregation. The idea of using statistical tools for representing mixing performance was presented by [LaRosa and Manning, 1964]. In literature there are many approaches for the calculation of the intensity of segregation:

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The most popular method to describe the intensity fo segregation is the coefficient of variation, CoV , defined as

CoV = σ

C (2.9)

where σ is the standard deviation and C the average of the property (e.g. concen-tration) used to characterise the mixing through the device. A similarly derived quantity is the Log-variance which is defined as

logV a= Logσ2 = log " 1 N −1 n X i=1 (C − 1)2 # (2.10) where C is the normalised mixing quantity and N is the number of instantaneous measurement made on the mixing system.

• Segregation index

Alternative approach to the CoV is the segregation index [Men et al.,2007])which relates the degree of mixing to the standard deviation of mixing measure for dif-ferent sample points as

Is=

σ2 σ2

max

(2.11)

where σ_max2 is the variance for a completely segregate mixture and σ2is the vari-ance of the system. In particle cases this is measured by taking a sufficient number of random samples of a specific size and calculation their variance e.g. by mea-surement of the spatial distribution of the concentration of a dye.

• Poincare plot

Another approach to evaluate the efficiency of the blending is the evaluation of mixing patterns by particle tracking. This method applies the use of fluid tracer par-ticle which are injected in the flow and their location is tracked during the mixing process by computational or experimental visualisation methods. The efficiency of the mixing process is described by how rapidly the particle become dispersed in

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Figure 2.8: Example of Poincare plot [Suzanne M. Kresta,2004]

the system.

Equations and methods commonly used in the first two points are summarised in Ta-ble2.2.

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Scale of segregation

Striation thickness method Measure of striation dimensions from image analysis of CFD data Lyapunov exponent method Measure of stretching factor as: λ =ln

l0

= eΛt

Topological entropy method Measure of an average stretching factor of in a finite amount of time

Inter-material area method This measure is the amount of contact area between mixture components in each region of flow

Intensity of segregation

Coefficient of variaton CoV = σ C

Log-variance logV a = Logσ2_{= log}

1 N − 1 Pn i=1(C − 1) 2 Segregation index Is= σ2 σ2 max

Poincare plots Describe mixing patterns by particle tracking

Table 2.2: Summary of different approach to quantify the scale and intensity

Those different approaches for the measurement of mixing performance have been dis-cussed through the years, [Kukukova et al.,2009] suggested a new model to characterise mixing performance based on three key concepts: intensity of segregation, scale of segre-gation and exposure. In fact, if only a single variable is considered, the analysis of mixing performance can five misleading conclusions. This problem is illustrated in Figure2.9 that shows checkerboard patterns, which are organised from left to right by scale of the pattern which is equal to the number of the neighbouring cells on one side of each black square. In the three checkerboards the intensity described with the CoV is constant.

Figure 2.9: Three dimensions of mixing and segregation: intensity of segregation (CoV ), scale of segregation (striation thickness) and exposure (rate of change in segrega-tion) [Kukukova et al.,2009]

Other parameter is also defined, called exposure, which is the rate of reduction in segre-gation as:

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E ≈ Nt X i=1 Nb X i=1 1 2k 0_a ij(Ci− Cj) (2.12)

where Ntis the number of squares in check board, Nbis the number or the neighbouring

squares, k0 = 1 is the strength of the interaction, aij = 1 is the contact area per side, and

(Ci− Cj) is the concentration difference between two consecutive neighbours. A simple

system was considered in this work: the concentration of the black squares is defined as Ci = 1, and the white square as Ci = 0.

The Figure2.9explains how the scale and the intensity are different in terms of what they are suitable to measure. Considering only the scale moving to the right of the figure the mixing seems to improve, however considering only the intensity the mixing per-formance does not change. This suggests that all the variables play important roles in industrial mixing problems, and therefore they should all be considered to characterise different aspect of mixing performance. [Kukukov´a et al.,2008] investigates the appli-cation of spatial statistic methods in order to determine the effect of the sampling scale on the mixing performance. Two measures of mixing were used: Coefficient of variance CoV and the maximum striation thickness. For the evaluation of these parameters three sampling methods were tested: quadrants, probes and transects. Two CFD data sets were used as test cases: dispersion of floating particles in a turbulent stirred tank and laminar mixing of tracer particles in a micro-mixer. The objectives of the investigation were to explore the data resolution and sampling protocols needed to get accurate measures of CoV and striation thickness from ideal data sets, one turbulent and one laminar. The information collected form the results of data sets gave, sometimes conflicting answers in term of which was the better method. In the turbulence regime, the dominant mecha-nism of advection is also called macro-mixing, which is better identified by the scale of segregation which can be evaluated using the analysis of striation thickness. Recently, Alberini ([Alberini et al.,2014b], [Alberini et al.,2014a]) proposed an alternative method of examining the mixing performance, based on analysis of areas (striations) within the PLIF image which possess the same level of mixing, leading to an areal distribution of mixing intensity over the image. This method is described with reference to Figure2.10.

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Figure 2.10: Development of areal analysis method. (a) Identification of regions in the grayscale distribution with a given mixing intensity; (b) raw image; (c) example of image processing with regions of mixing intensity >60% in white [Alberini et al.,2014b]

Figure2.10(a) displays a typical distribution of grayscale values (which are proportional to dye concentration) which would be obtained from an image such as that shown in Figure2.10(b). Plug flow is assumed in the image analysis, so that each pixel is of the same importance. Thus, the mean value of grayscale in the image (corresponding to the fully mixed concentration, C∞), G , can be easily evaluated from the distribution. Using

the experimentally determined value of G, it is possible to calculate grayscale values corresponding to a given level of mixedness. Taking X% mixing as an example, this corresponds to grayscale values of either GX− = [1 + (1 − X)]G or GX+ = [1 + (1 −

X)]G. So for 95% mixing, GX+ = 0.95G and GX+ = 1.05G. Using MATLAB and

the freeware image analysis tool Image J, the pixels in the image are identified which correspond to GX− < G < GX+, thus corresponding to a mixing intensity of > X%:

this arbitrary region is shown in Figure2.10(a). These pixels are then set to white in the image, with the remaining out of range pixels being set to black (G= 0). An example of this procedure is shown in Figure2.10(c), where the fraction of the total cross-sectional area corresponding to this mixing intensity is then easily determined from the fraction of white pixels. By repeating this procedure over a range of values ofX, both discrete and cumulative areal distributions of mixing intensity are thus obtained (see Fig2.11).

2.5 Planar Laser Induced Fluorescence (PLIF)

PLIF is an optical technique which tracks the spatial concentration distribution of an added fluorescent dye at fixed intervals of time. A laser sheet is used to illuminate a plane in the flow and images are captured using an orthogonally mounted digital camera. The camera used contain either a charge-coupled device (CCD) or complementary

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metal-Figure 2.11: Bar graph showing an example of discrete areal intensity distributions ??

oxide-semiconductor (CMOS) based sensor, the latter are becoming increasing common due to reduced manufacturing cost and they are approaching parity with CCD sensors in terms of their quantum efficiency (sensitivity). The fluorescent dye illuminated by the laser sheet is excited by the laser light, it absorbs photons of laser light and re-emits photons of light at a lower frequency. For example a common set-up involves use of NanoPIV or diode lasers which emit at 532 nm (green) with Rhodamine 6G dye which produces fluorescent light 550-560 nm (yellow). Cut off filters can be used on the camera to prevent extraneous laser light from appearing in the image. This technique is able to detect the concentration maps of the system at different times of capture and has been used widely in stirred vessels investigating laminar flow, visualising the mixing patterns, studying the stretching of the fluid elements, and also characterising mixing by means of the gradient of concentrations (CoV) and segregation (striation thickness) in many differ-ent works ([Alvarez et al.,2002], [Ottino and Khakhar,2000],[Szalai et al.,2004], [Zalc et al.,2002b], [Arratia et al.,2006], [Zalc et al.,2001], [Alvarez et al.,2002]).

PLIF has also been demonstrated as a valuable method to quantify mixing times in agitated vessels both in the laminar and turbulent regime, by determination of the time required of the concentration of the dye to become uniform within a given confidence interval ([Simmons et al.,2009], [Hall et al.,2004]). [Ventresca et al.,2002] performed

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an investigation of dependence of laminar mixing efficiency of a motionless mixer upon viscosity ratio at low Reynolds number. The fluid used were aqueous solution of CMC which are transparent liquids with a non-Newtonian Rheology. The device used for the experiments consist of five elements of SMX static mixer. The mixing performance was evaluated using imaging analysis of the cross section of the pipe, detected at bottom of the static mixer. The images were detected using PLIF technique where resolved spatial variations of fluorescence intensity were recorded using a CCD camera. Ventresca in his work evaluated mixing performance considering both scale and intensity of segregation, for the first using correlograms and for the second CoV and intensity histograms are pre-sented. The conclusion of this work underlined the role of the viscosity ratio between the main flow and the injection on the mixing performance. For low viscosity ration, sta-tistical descriptions (CoV) were important indicators of mixing effectiveness ([Ventresca et al.,2002]). The scale described with correlograms is a viable statistic for indication of goodness of mixing, but following Ventresca’s work the information of the scale analysis can be misleading when diffusion occurs. However the main conclusion of this works was the intensity of segregation was the most valuable tool for the detection of level of mixing.

In another recents work ([Lehwald et al., 2012]) investigated the mixing using a pre-viously established method for the characterisation of micro-mixing and macro-mixing based on Two-tracer PLIF ([Lehwald et al.,2010]) carried out simultaneously with PIV for measuring the velocity and mixing fields induced by a static mixer element. Using the velocity map and an analysis of segregation index the mixing performance was de-termined. Du [Du et al.,2007] used PLIF to instantaneously visualise two-dimensional temperature distribution in a liquid/liquid cross-flow mixing process with flow channel where the excited fluorescence strength of Rhodamine B dye had linear relationship with the temperature finding that the mixing process of two liquids with different temperatures could then be characterised by the measured temperature fields. He also studied the effect of momentum ratio of two liquid flows on the mixing process, finding that larger momen-tum ratio between the jet and bulk flow benefited the mixing efficiency of the two liquids. Instead the heat transfer coefficient of water in the liquid sheets impingement process was compared with the turbulent heat transfer coefficient calculated by an empirical formula,

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results show that the impingement process can promote the heat transfer between the two liquids with different temperatures. Higher momentum ratio of two liquid flows leads to better mixing performance in terms of the transport phenomenon of temperature.

Alberini ([Alberini et al., 2014b], [Alberini et al.,2014a]) used PLIF images obtained by blending fluids with multiple static mixer to detect viscous stream filaments evident as spots when a fluid of higher viscosity was injected into a lower viscosity continuous phase, which is not predictable using conventional design approaches. This new method shows promise in unraveling the complexity of information-rich PLIF images, beyond a sole number-based mixing criterion. Zennels, [Zellner et al.,2015], used PLIF to en-able noninvasive in situ investigations of catalytic flow reactors: The method is based on the selective detection of two-dimensional absolute concentration maps of conversion-relevant species in the surrounding gas phase inside a catalytic channel. Exemplarily, the catalytic reduction of NO with hydrogen is investigated over a P t/Al2O3 coated diesel

oxidation catalyst by dye doped N O inside an optically accessible channel reactor, by quenching-corrected 2D concentration maps of the N O fluorescence above the catalytic surface are obtained under both, nonreactive and reactive conditions. The technique pre-sented has a high potential for a better understanding of interactions of mass transfer and surface kinetics in heterogeneously catalysed gas-phase reactions.

2.6 Electrical Resistance Tomography (ERT)

2.6.1 General information and applications

Electrical Resistance Tomography (ERT) is a measurement technique for obtaining in-formation about the contents of process vessels and pipelines. Multiple electrodes are arranged around the boundary of the vessels and pipeline at fixed locations in such a way that they make electrical contact with the fluid inside the vessel but do not affect the flow or movement of materials. A typical application is real time monitoring of multicom-ponent flow within process engineering units. Specific application where ERT has been successfully exploited.

[Williams et al.,1999] uses electrical resistance tomography for on-line auditing of an industrial hydrocyclone separation. The work demonstrates the retrofitting of electrodes

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into a commercially available separator and their use in laboratory, pilot plant scale in-vestigations of clay refinine. An accurate measurement of the air core size for different operational conditions were done; direct calculation of solid concentration profiles based on parametric reconstruction of conductivity data in three dimensions.

[Vlaev et al., 2000] used ERT to monitor solid/liquid filtration with a single inex-pensive 16-element ring sensor array, thanks to the high sensitivity of the UMIST Mark 1b-E data acquisition system Vlaev found that this tomography array can detect move-ment of the liquid level during filtration. This degree of sensitivity was also capable of detecting any tilt of the filter assembly, so that pathological behaviour due to malfunction or accidental displacement of the filter support plate could also be identified concluding that ERT has great potential for instrumenting and detecting ”pathological” behaviour of filtration processes in the pharmaceutical and fine chemicals industries. Moreover, illus-trative distortions of a filter cake formed by artificial surface depressions were readily observed in ERT tomograms.

[Hosseini et al.,2010] successfully determined the quality of solid/liquid mixing in an agitated tank measuring the degree of homogeneity using the concentration profile gen-erated from the ERT data. The ERT measurements were correlated to solid concentration profiles by which the degree of homogeneity was quantified. In his study, the effect of im-portant parameters such as impeller type, impeller speed , impeller off-bottom clearance , particle size, and solid concentration on the degree of homogeneity were explored. The results showed that the degree of homogeneity in the solid/liquid mixing was improved with increasing the impeller speed. However, after reaching the maximum achievable homogeneity, further increase in impeller speed was not beneficial but might be detri-mental. Hence, the measurement of the optimal impeller speed as a function of operating conditions and design parameters has vital role in achieving maximum homogeneity in a solid/liquid mixing system.

[Ismail et al.,2005] regarded the ERT as a successful method for visualising cross-sectional distribution and measuring multi-phase flows (MPFs) of oil and water thanks to its advantages over other tomography modalities, such as no radiation, rapid response,

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low cost, being non-intrusive and non-invasive, and the ability to withstand high tempera-ture and high pressure. It can deal with the complexity of MPF measurement by explicitly deriving the component distributions at two adjacent planes along a pipeline. Images of the component distributions can be cross-correlated to obtain the velocity profile of the flow. Multiplying the component concentration and velocity profiles yields a measure of volumetric flow rate for each phase accurately. Ismail stated that tomography methods could be applied to MPF measurement in two ways:

• as an instrument to determine the flow regime in order to correct or compensate the readings from currently available MPF meters, which are flow-regime dependent, • as a radically new flow regime-independent method of MPF measurement in its

own right, without having to resort to any other principles.

Furthermore it can potentially be used within a dual-modality system for simultaneously measuring the volume flow rate of oil, water and gas in oil well flows. Its capability of flow imaging would also be useful for process control because the cross-sectional image can give important additional information, such as phase distribution.

[Fransolet et al.,2005] presents an experimental analysis of the influence of the liq-uid rheology on the gas flow pattern in a bubble column reactor using aqueous solutions of xanthan as an example of non-Newtonian shear thinning fluid. Averaged gas holdup is determined by two experimental techniques: parietal pressure probes and electrical resistance tomography (ERT). ERT is also used to provide 2D images of the gas phase distribution in a column cross-section. Bubble size distributions are evaluated by a gas disengagement technique using the parietal pressure probes. All these techniques clearly show the gas flow pattern is different in Newtonian and non-Newtonian fluids. Homo-geneous flow regime, observed in water at low gas velocities, tends to disappear when viscosity increases. This evolution is visualised by a much less isotropic distribution of the gas phase within cross section of the column and by the appearance of much larger bubbles due to an increased coalescence phenomenon.

[Pakzad et al.,2008a] used ERT to visualise, in three dimensions, the concentration field inside a cylindrical mixing vessel equipped with a radial-flow impeller. The ability

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of ERT to work in opaque fluids makes this technique very attractive from an industrial perspective. An ERT system with a 4-plane assembly of peripheral sensing rings, each containing 16 electrodes, was used to measure the mixing time in agitation of xanthan gum solution which is a pseudoplastic fluid with yield stress. In this study, the effect of impeller speed, fluid rheology, power consumption, and Reynolds number on the mixing time was investigated. Leila stated that thanks to the 4-plane peripheral sensing rings the mixing time measured by the tomography system is more accurate than that measured using the conventional probe techniques in which only a few monitoring points inside the tank are employed for the mixing time measurements. The data obtained from the tomog-raphy system showed that the fluid rheology considerably affect the mixing time of the non-Newtonian fluids. As the yield stress of the solutions increases for a given impeller speed, low velocity regions exist in a larger portion of the fluid in the tank, resulting in an increase in the time required to reach homogeneity. The information yielded by ERT im-ages is extremely useful for determining levels of homogeneity within the mixing vessel, measuring cavern size, mapping flow paths, and identifying the dead zones. The ERT data are also useful for the validation of computational fluid dynamics (CFD) models used for predictions of non-Newtonian mixing behaviour.

[Pakzad et al.,2008b] demonstrated the ability of the ERT system to examine the flow pat-tern within the mixing vessel and monitor the mixing process by monitoring the dis-tribution of the tracer concentration as 2D tomograms. The formation and characteristics of the cavern formed in the mixing of opaque xanthan gum solutions were successfully monitored and measured. The images showed the presence of the well-agitated zones (cavern) around the impeller. Under the conditions where the cavern did not reach the tank wall, the cavern diameter was well correlated with Elson’s model.

2.6.2 Operating principles

An ERT system produces a cross-sectional image showing the distribution of electrical conductivity of the contents of a process vessel of pipeline from measurements taking at the boundary of the vessel. The system injects a current between a pair of electrodes and

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measures the resultant voltage difference between remaining electrode pairs according to a pre-defined measurement protocol. This interrogates an entire ‘slice’ through the measurement zone. A single measurement set consist of over 100 voltage measurements ([Dickin and Wang,1996]).

2.6.3 Electrode geometry and construction

The most common electrode geometry (see Fig.2.12) shows electrodes arranged at equal intervals around the boundary of a circular pipeline. The electrodes are connected to the data acquisition system (DAS) by co-axial cable which assists in reducing the effect of extraneous environmental noise and interference. The outer sheath of the co-axial cable is coupled to the feedback path of a voltage buffer to provide further noise immunity and the inner core is capacitively coupled to the input of the voltage buffer. The material for the electrode construction depends largely on the process application. The material should be more conductive than the fluids being imaged order to obtain reliable measure-ments. Typically the electrode material is stainless steel, silver, gold, platinum, silver palladium or any suitable material exhibiting a number of properties ([Dickin and Wang, 1996]). The dimensions of the electrodes are a function of the vessel diameter, range of conductivity to be measured, velocity of materials and the required imaging speed. A spare electrode, called ground electrode, positioned away from the measurement elec-trodes but in electrical contact with the internal fluid is required to ensure all voltage measurement are fixed against a common ground source.

2.6.4 Data Acquisition System

The Data Acquisition System (DAS) is responsible for obtaining the quantitative data describing the state of the conductivity distribution inside the pipeline. The data must be collected quickly and accurately in order to track small changes of conductivity in real-time thus allowing the image reconstruction algorithm to prove an accurate mea-surement of the true conductivity distribution. A sine-wave voltage output is fed into a voltage-to-current converter (referred to as a voltage controlled current source-VCCS). Current is used in preference to voltage as the electrical ‘probe’ due to the variation of contact impedance between electrodes and the fluid inside the sensor. The VCCS circuit

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Figure 2.12: Schematic diagram of electrode arrangement and placement [ITS,2017b]

maintain constants amplitude ove a wide range of resistance loads by employing two op-erational amplifiers and an analogue switch arrangement in the feedback path of one of the operational amplifier.

Measurement strategy

The measurement strategy or protocol for probing the conductivity distribution within the vessel via the electrodes arranged around the vessel boundary is of paramount im-portance. The different strategies have been develop such as Normal adjacent, Opposite strategy and Diagonal strategy [Breckon and Pidcock,1988]. In this work Normal Ad-jacent is the protocol used: in this strategy sensors with insulating boundaries with 16 electrodes arranged at equal intervals around the periphery of the sensor. As can be seen in Figure2.13, current is applied through two neighbouring electrodes (in this case elec-trodes 1 and 2) and the voltages are measured from the remaining pairs of neighbouring electrodes (in this case electrodes 3 and 4). Current is then applied through the next pair of electrodes and the voltage measurements are repeated. The procedure is repeated until all the independent measurements have been made. The adjacent measurement strategy

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yields N2 measurements, where N is the number of electrodes. However of these only

N (N −1)

2 are independent. Furthermore, to avoid electrode/electrolyte contact impedance

problems, the voltage is not measured at a current-injecting electrode and therefore the total number of independent measurements M is reduced to M = N (N −3)₂ . There-fore a 16-electrode sensor gives 104 independent measurements ([Seagar and Brown, 1987])([Gigu`ere et al.,2008]).

Figure 2.13: Normal adjacent protocol scheme [ITS,2017b]

In all cases, the voltage measurements pass through a multiplexer into a differential input amplifier which amplifies the potential difference between the two input voltage sig-nals. The amplifier has the ability to reject common-mode signals such as electrical noise. The sine-wave output of the differential amplifier is then fed into a programmable gain amplifier (PGA) to accommodate the wide dynamic range of voltage signals obtained from the many pairs of electrodes. A phase sensitive demodulator (PSD) is employed after the PGA to demodulate the voltage signals prior to low-pass filtering ([Dickin and Wang,1996]).

2.6.5 Image reconstruction

Following the acquisition of data from the boundary of the object to be imagers it s nec-essary to process this data using an appropriate image reconstruction algorithm. For an

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ERT system the reconstructed image will contain information on the cross-sectional dis-tribution of the electrical conductivity fo the contents within the measurement plane. A square gird of20 × 20 = 400 pixels represent the pipeline interior cross section, the cir-cular image is constructed using 316 pixels from the 400 pixel square grid (see Fig.2.14).

Figure 2.14: ERT grid for image reconstruction [ITS,2017b]

it is well known that, for electrostatic fields, when current lines are encountered by an interface of different conductivities, the current lines will deflect. Therefore image re-construction algorithms such as those for straight-ray transmission are inappropriate. In addition reconstruction algorithms developed for diffraction tomography such as ultra-sonic and optical are unsuitable since the propagation of electromagnetic field lines and the distribution of electrostatic field lines are governed by different differential equations. The choice of image reconstruction algorithm is a trade-off between accuracy of image and time required for reconstruction. There has been a demand for fast image recon-struction algorithms that can be used for the real time imaging of fast moving processes. Therefore much effort has been focused on the development of image reconstruction al-gorithms, both non-iterative and iterative for electrical tomography ([Yorkey et al.,1987]) ([Yang and Peng,2002]).

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The forward problem - sensitivity map

The image reconstruction process involves determining the electrical conductivity of each pixel within the image from the set of electrical measurements. This is known as the in-verse problem. However, these electrical measurements taken at the boundary of the pro-cess pipeline contain insufficient information to allow the inverse problem to be solved directly.

The model of a source free conducting inhomogeneous domainΓ0_{with a conductivity}

distribution σ(x, y), into which a steady-state current is injected and the corresponding voltage V(x, y) is measured is governed by Poisson’s equation as follows:

∇ · (σ(x, y)∇V (x, y)) = 0 in Γ0 _(2.13)

For a unique solution to exist, sufficient boundary conditions must be specified. [Saad et al.,2017] suggested the following condition:

             V = 0 At reference point R σ(δV δn) = +I on source electrode R σ(δV

δn) = −I on sink electrode

(2.14)

Where I denotes the current applied to the electrodes and n denotes the outward unit normal to the sensor.

The Finite Element Method (FEM) is used to solve Poisson’s equation for electrical resistance tomography by reducing it to a series of simultaneous equations describing the behaviour of each of the 316-pixels in Figure2.14. For a 16 electrode sensor there are 14 pair of electrodes for current injection when the adjacent measurement strategy is used. Therefore, for the ithcurrent injection, the use of the FEM converts the solution to Poisson’s equation to the following set of linear equations:

Avi = bi (2.15)

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within the finite-element mesh, vi is a vector representing the N unknown nodal

po-tentials and bi is a N x1 vector indicating the boundary conditions as described above.

Equation2.15is referred to as the ”forward problem” since all the elements resistivities are known and the node voltages are determined, can be solved using a variety of direct or iterative numerical methods.

The inverse problem

The idealised inverse conductivity problem is: Given all possible current densities j and their corresponding voltage distributions V find the conductivity σ in the interior of the body. In practice we cannot specify current densities, but only currents, which are applied through electrodes attached to the surface. Given the condition shown in equations2.13 and2.14plus the conservation of the charge condition

N

X

i

I = 0 (2.16)

the inverse problem become: Given all possible current patterns I = (I1, . . . , IN) and

their corresponding voltage patterns V = (V1, . . . , VN) find the conductivity σ inside

the body. Unfortunately this is impossible for the following reason: There are only N-1 linear independent current patterns. Any other current patterns must be a linear com-bination of L −1 basis patterns. Because the differential equation 2.13is linear, any voltage patterns must be linear combination fo the N −1 basis voltage patterns. There are only a finite number of linearly independent measurements. Form a finite number of measurements, we cannot obtain σ at every point in the interior of the body, we can only look for an approximation to σ that depends on a finite number of parameters. [Cheney et al.,1990] and his group develop an algorithm, called NOSER, for solving the inverse problem. NOSER (Newton’s One Step Error Reconstructor) is based on the method of least squares, its takes one step of a Newton’s method, using a constant conductivity as an initial guess. The code does not reproduce the conductivity accurately, unless it differs very little from a constant) but it yields useful images.

Another approach is the linearised back-projection between equipotential lines: That is, the potential difference, calculated by the forward solver, between two equipotentials

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on the boundary was back-projected to a resistivity value in the area enclosed by the two lines for all possible injection/measurement combinations. Kotre [Kotre,1994] de-veloped an enhanced back-projection algorithm utilising all the image pixel sensitivities, not only those on the diagonal, of the sensitivity matrix. This algorithm, used by the au-thors, is referred to as the sensitivity coefficient method since it is based on the sensitivity coefficient concept defined by Breckon and Pidcock [Breckon and Pidcock,1988]. The main advantage of this algorithm is that it can be performed in a single step using a pre-calculated pixel sensitivity matrix. The image is simply reconstructed via a matrix/vec-tor multiplication which can be performed rapidly on modern computers equipped with floating-point units, the sensitivity matrix is formed using the FEM-based forward solver.

[Yorkey et al.,1987] proposed an improved reconstruction algorithm, modified Newton-Raphson method varies a finite-element model of resistivities to fit a set of voltage mea-surements in a least-squared sense. Two procedures for calculating the Jacobian matrix are derived. One is standard, while the other is based on the compensation theorem. Results from two-dimensional computer simulations are compared to four other recon-struction algorithms, which are based on methods proposed by other authors. The mod-ified Newton-Raphson method provided significantly better reconstructions than any of the other methods in terms of perturbation, equipotential, iterative-equipotential, and the double-constraint methods.

An example of image reconstructed is shown in Figure2.15.

Note that the aim of this algorithms is to find the set of resistivity that produces voltages which minimise the difference between them and the voltage measured, thus the grid is not an exact representation but more an approximation.

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Chapter 3

Material and methods

3.1 Introduction

The ERT ability to evaluate mixing performance and injected area for a KM static mixer is presented, together with all the material and methods used in this thesis. In Section3.2the experimental rig is shown with the needed pumps calibration to ensure the flow condition. In Section3.3the fluid formulation of the fluids used in this thesis is done. In Section3.4 all the instrument and techniques required to visualise the flow for both ERT and PLIF are illustrated. Finally in Section3.5all the post process methods are explained.

3.2 Experimental Rig

Figure3.1 (a) shows an overall schematic of the experimental rig with Figure3.1 (b) giving a picture of the static mixer test section. KM static mixer of internal diameter (ID) of 25.4 mm (1”) with length of 0.22 m (_DL = 9) has been used. The flow to the mixer is delivered by an Albany rotary gear pump controlled using an inverter control WEG (model CF208). The secondary flow is introduced using a Cole-parmer Micropump (GB-P35) and is doped with fluorescence dye (Rhodamine 6G). In all the experience the injection of doped liquid is in the center of the pipe and placed as close as possible to the first static mixer element, the injector has a internal diameter of 8.9mm (see Fig.3.1c). After the static mixer there is a stainless steel pipe of internal diameter (ID) of 25.4 mm (1”) designed by ITS with two plane surrounded by 16 electrodes directly linked directly to V5R (see Section3.4.2) in order to enable flow measurement using ERT. A Tee piece

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(a)

(b)

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Figure 3.1: Schematics fo the static mixer rig

is placed at the end of the mixer section which ha glass window inserted on the corner of the Tee, normal to the axis of the main pipe, in order to enable flow measurements using PLIF, that requires optically transparent materials. A glass pipe section upstream of the Tee at the mixer section outlet provides optical access for the laser sheet to illuminate the transverse section. Two pressure transmitter were located both upstream (PR-35X/ 10 bar, Keller UK) and downstream (PR-35X/ 1 bar, Keller UK) of the static mixer section,

Online monitoring of blending of non-Newtonian fluids in static mixer

Contents

Chapter 1

Introduction

1.1

Background

1.2

Objectives and aims

1.3

Thesis layout

Chapter 2

Literature review

2.1

Introduction

2.2

Fluid mechanics fundamentals

2.3

Mixing equipment

2.4

Measure of mixing performance

2.5

Planar Laser Induced Fluorescence (PLIF)

2.6

Electrical Resistance Tomography (ERT)

Chapter 3

Material and methods

3.1

Introduction

3.2

Experimental Rig