Dipartimento di Ingegneria Civile e Industriale
Tesi di Laurea Magistrale in Ingegneria Chimica
Towards online rheological measurements in stirred
vessels
Candidato: Manuele Romano
Relatore: Prof. Ing. Elisabetta Brunazzi
Controrelatore: Prof. Ing. Roberto Mauri
(a) (b)
Master’s Thesis in Chemical Engineering
Towards online rheological measurements in stirred
vessels
Candidate:
Manuele Romano(a) Sign: ...
Supervisors:
Prof. Ing. Elisabetta Brunazzi(a) Sign: ...
Prof. Ing. Roberto Mauri(a) Sign: ...
Towards Online Rheological Measurements In Stirred Vessels
Abstract
Despite it may seem a simple operation, mixing is one of the least understood of the tasks in the process industry. While for some industrial applications well-established correlations for design and scale up are available, for other operations the mixing requirements evaluation is very critical. Problems related to poor mixing may affect product yield, quality, physical properties, and increase costs and time of manufacturing.
A very common form of mixing of liquids occurs in stirred vessels using different types of impeller geometries. In particular, blending of formulated liquids is a critical step in the manufacturing industry. Usually formulated liquids are viscous fluids with complex rheological properties, and they are mixed in stirred vessels in laminar regime to reduce power consumption. In these conditions, small clearance impellers like anchors are commonly used.
Research on mixing of non-Newtonian fluids in stirred vessels in laminar regime is quite poor. Most of the works refer to Metzner and Ottos’s model or its modifications. This model was meant to predict power consumption for shear thinning fluids, when all the rheological parameters are known. It assumes that in the region near the impeller the “effective” shear rate is proportional to the impeller rotational speed. Then, the effective viscosity is calculated from the effective shear rate, and so Reynolds number, Power number and power consumption. Many authors often use the effective viscosity as an overall average viscosity. Assuming that this can be done, if the rheology is known, at steady state the average viscosity can be evaluated directly from the impeller rotational speed.
However, in industrial applications what normally happens is that the fluid rheology evolves during the manufacturing process, conducted in batch configuration, and the critical objective is precisely monitoring its viscosity in real time for quality control.
The final objective of this work is to develop a new methodology for the online evaluation of a char-acteristic viscosity inside the vessel from torque measurements.
Experimental data of shear rate and viscosity distributions of fluids having different rheologies are collected using Particle Image Velocimetry, a flow visualization technique. These data and torque data are used together to build a simple statistical model, which takes in input the desired rheology and the torque measurements, and gives as output the distribution of the viscosity inside the vessel. The product will be finished when the average viscosity matches the desired value. This is a first step towards online rheological measurements in stirred vessels. Expanding the experimental database in the future work will improve the statistical model and allow its industrial application.
Contents
1 Introduction 1
1.1 Mixing in stirred vessels . . . 1
1.2 Aims and objectives of this work . . . 7
1.3 Thesis overview . . . 7
2 Fundamentals and Literature review 8 2.1 Fluid-dynamics fundamentals . . . 8
2.2 Rheologies . . . 15
2.3 Mixing in stirred vessels in laminar regime . . . 21
2.4 Particle Image Velocimetry . . . 29
3 Materials and methods 33 3.1 Fluids formulation . . . 33
3.2 Rheology measurements system . . . 35
3.3 Torque measurements system . . . 36
3.4 PIV system . . . 38
3.5 Post-processing . . . 40
4 Results and Discussion 47 4.1 Rheologies . . . 47
4.2 Kp evaluation . . . 49
4.3 Power input . . . 52
4.4 Flow fields . . . 57
5 Conclusions 70
5.1 Final remarks . . . 70
5.2 Publications . . . 72
5.3 Future work . . . 72
A Mathematics 75 A.1 Vector algebra in Cartesian coordinates . . . 75
A.2 Differential algebra in Cartesian coordinates . . . 78
A.3 Useful theorems . . . 81
A.4 Fourier Transform . . . 81
B Statistics: understanding the result of a regression analysis 83
C MATLAB codes 86
D Shear rate distributions: a, b and c of the Gaussian fitting 99
List of Figures
1.1 Conventional stirred tank layout . . . 2
1.2 Axial flow turbines . . . 3
1.3 Radial flow turbines . . . 4
1.4 Hydrofoil turbines . . . 4
1.5 High shear turbines . . . 5
1.6 Flow patterns with different impellers, impeller diameter and liquid viscosity . . . 5
1.7 Close-clearance impellers . . . 6
2.1 Shear flow of a fluid between two plates . . . 16
2.2 Typical rheograms of time-independent purely viscous fluids . . . 18
2.3 Phase diagram for elastic solid, purely viscous fluid and viscoelastic fluid . . . 19
2.4 Flow regimes in a stirred vessels . . . 22
2.5 2D PIV typical setup . . . 30
3.1 Rheology measurements system . . . 36
3.2 Mixing equipment . . . 37
3.3 Layout of the PIV system . . . 38
3.4 Planes investigated in the PIV experiments . . . 39
3.5 Example of flow field in a vertical plane . . . 42
3.6 Example of COV distribution . . . 43
3.7 Example of shear rate distributions . . . 44
4.1 Fitting models of rheograms of the fluids . . . 48
4.3 Rushton impeller . . . 51
4.4 Determination of the Rushton’s KP by linear interpolation . . . 51
4.5 Power number vs RPM . . . 52
4.6 Viscosity from torque meas. and from M&O models: CMC solutions . . . 53
4.7 Viscosity from torque meas. and from M&O models: Carbopol solutions . . . 54
4.8 Specific power vs RPM: silicone oil 350cSt and Boger . . . 56
4.9 Specific power vs RPM: silicone oil 1000cSt and Boger . . . 56
4.10 Specific power vs RPM: silicone oil 350cSt and silicone oil 1000cSt . . . 57
4.11 Flow fields on vertical planes: Silicone oil 350cSt. 80 RPM . . . 58
4.12 Flow fields on vertical planes: Boger fluid, 80 RPM . . . 58
4.13 Flow fields on horizontal planes: 5 mm from the bottom, 80 RPM . . . 59
4.14 Examples of COV field . . . 61
4.15 Shear rate distributions: CMC 1.0%, 80 RPM . . . 62
4.16 Shear rate distributions: CMC 1.0%, impeller at 0° . . . 63
4.17 Shear rate distributions: Boger vs Silicone oil 350cSt, impeller at 0° . . . 64
4.18 Average viscosity from PIV and effective viscosity from M&O: CMC 0.75% . . . 65
4.19 Average viscosity from PIV and effective viscosity from M&O: CMC 1.0% . . . 65
4.20 Average viscosity from PIV and effective viscosity from M&O: Carbopol 0.1% . . . 66
4.21 Average viscosity from PIV and effective viscosity from M&O: Carbopol 0.2% . . . 66
4.22 Sensitivity analysis of the logarithmic normalized mean shear rate . . . 68
5.1 Poster presented at the ChemEngDayUK 2017 conference in Birmingham . . . 74
List of Tables
2.1 Correlations for ksreported in literature . . . 28
3.1 Fluids formulations . . . 34
3.2 Timing between laser shots used in PIV experiments . . . 40
3.3 Structure of the matrix of PIV data before reshaping . . . 41
3.4 Structure of the matrices of PIV data after reshaping . . . 41
4.1 Rheological properties of the fluids . . . 49
4.2 Minimum and maximum Power numbers in the range 60 - 240 RPM . . . 53
4.3 Relative errors of some Metzner and Otto’s models on predicting the effective viscosity . 54 4.4 Relative standard deviations of some Metzner and Otto’s models on predicting the effective viscosity . . . 55
4.5 COV on x-velocity at 120 RPM: percentage of IA having COV < 5% . . . 60
4.6 COV on y-velocity at 120 RPM: percentage of IA having COV < 5% . . . 60
4.7 Fitted coefficients of the statistical model . . . 67
List of Symbols
Roman Symbols
g Gravitational acceleration on Earth [m/s2]
Jconv Convective flux [quantity/(m2· s)] Jdif f Diffusive flux [quantity/(m2· s)] u Velocity vector [m/s]
c Wall clearance of the impeller [m] D Diameter of the impeller [m] De Deborah number [−]
G Shear modulus [P a]
K Consistency index [kg/(m · s2−n)] KP Impeller power constant for laminar
regime [−]
KP∗ Modified impeller power constant for laminar regime, for power law fluids [−]
ks Metzner and Otto’s constant [−] L Characteristic length [m] N Impeller rotational speed [s−1] n Behaviour index [−]
P Power [W ]
p Mechanical pressure [P a] P o Power number [−]
qj Component of momentum vector along j [kg · m/s]
Re Reynolds number [−]
Re∗ Modified Reynolds number [−] T Diameter of the vessel [m] W i Weissenberg number [−]
t Time [s]
U Characteristic velocity [m/s]
Greek Symbols ˙γ Shear rate [s−1]
˙γef f Effective shear rate [s−1] γ Deformation [−]
Λ Torque [N · m]
λ Relaxation time of a viscoelastic fluid [s]
ν Kinematic viscosity [m2/s] ω Angular velocity [rad/s]
φ Phase displacement between stress and deformation [rad]
ρ Density [kg/m3] τ Shear stress [P a] τ0 Yield stress [P a] σ Stress tensor [P a]
τ Deviatoric stress tensor [P a] µ Dynamic viscosity [P a · s]
e
µ0 Real part of the complex viscosity [P a · s]
µ00 Imaginary part of the complex vis-cosity [P a · s]
µef f Effective viscosity [P a · s]
Abbreviations and Acronyms
2D-2C Two dimensions - two components HB Hershel-Bulkley fluid
IA Interrogation area(s) N Newtonian fluid
PIV Particle Image Velocimetry STn Shear-thinning fluid
V Viscoelastic fluid with constant vis-cosity
Mathematical Symbols and Notation
∂
∂t Partial (Eulerian) derivative D
Dt Material (Lagrangian) derivative d dt Total derivative a Scalar a a Vector a A Tensor A ∇ a Gradient of a ∇ · a Divergence of a ∇ × a Curl of a ∇2a Laplacian of a
“Deborah knew two things. First, that the mountains flow, as everything flows. But, secondly, that they flowed before the Lord, and not before man, for the simple reason that man in his short lifetime cannot see them flowing, while the time of observation of God is infinite.”
Chapter 1
Introduction
This Chapter is a brief introduction to the present work. In the first Section the rule of mixing in stirred vessels within the manufactury industry is introduced, describing conventional layout and industrial applications. In the second Section aims and objectives of this work are illustrated. The third Section is an useful overview of the dissertation.
1.1
Mixing in stirred vessels
Mixing is a unit operation which involves the reduction of inhomogeneities, in terms of composition, phase or temperature, in order to achieve a desired process result. Mass and heat transfer, chemical reactions, and product properties are usually the critical objectives. Practically all modern industrial processes involve some form of mixing. A very common one is the blending of liquids inside vessels. There many ways to perform mixing of fluids in vessels, and mechanical agitation is the most widely used. In this case, the main mechanism of mixing is the physical movement of material between various parts of the entire volume using a rotating system. Depending on the rotational speed, mixing may occur in laminar, turbulent or transitional flow conditions, as explained further. Also, the operation can be accomplished in continuous, batch, or fed-batch mode. Over 50% of the world’s chemical production involve mixing in stirred vessels [2].
An excellent source of information on various aspects of industrial mixing, including theory, applica-tions, scale-up procedures, cost management and experimental methods is Paul et al. [3].
1.1.1
Conventional layout
A conventional stirred tank consists of a vessel equipped with a rotating mixer. The vessel is generally a vertical cylindrical tank, while the mixer includes one or more impellers, shaft, seal, gearbox and a motor. Depending on the process objectives, different impeller geometries have been developed and
their performances are widely investigated in literature. Wall baffles are often installed in symmetric systems for transitional and turbulent mixing to prevent solid body rotation and cause axial mixing between the top and bottom of the tank. They consist in solid plates having width equal to 8% to 10% of the tank diameter and positioned in the path of tangential flow.
Figure 1.1: Conventional stirred tank layout; source: [3]
Generally tanks are designed with a height-diameter ratio close to unit, but higher aspect ratio tanks and horizontal vessels are sometimes used for particular applications. In tall tanks the mixer may be installed from the bottom to reduce the shaft length and avoid mechanical instability. In horizontal tanks and large product storages the main mixer, or additional ones, can be side entering. In small tanks angular top entering portable mixers are often used.
The conventional stirred vessel uses a flat or dished bottom, depending on the duty. For example, for solid suspensions dished bottoms are preferred because solids would tend accumulate in the corners of a flat bottom. Also, flow patterns below the impeller are influenced by the bottom shape, and so the mixing efficiency.
When heat addition to or removal from the process fluid is required, the vessel is equipped with appropriate heat transfer surfaces, such as jackets, baffle coils and helical coils.
The inlets and outlets are located based on the type of feed and the required rate of dispersion. Usually batch mixers can be fed from the top, while for processes requiring quick dispersion of the feed the inlet nozzle is located in a highly turbulent region.
1.1.2
Impellers
There are hundreds of impeller types in commercial use. The choice of the most effective impeller should be based on process requirements and knowledge of power consumption, pumping, flow patterns and
resulting shear rate levels. Impellers can be grouped as:
turbines, for low to medium viscosity fluids; – axial flow impellers;
– radial flow impellers;
– hydrofoil, with a particular blade profile to increase axial pumping at low shear; – high shear impellers;
close-clearance impellers, for high viscosity fluids; – anchors;
– helical ribbons;
specialty impeller designs for specific needs. Turbines
Turbines are used in transitional and turbulent flow applications with low to medium viscosity fluids. Axial flow impellers are used for blending, solids suspension, gas dispersion and heat transfer. The flow discharge has a strong axial component, as shown in Fig. 1.6 (c). Most applications require down-pumping of the flow towards the bottom of the tank, but sometimes up-down-pumping is more effective, such as gas dispersion and floating solids mixing. The most used axial impeller is the pitched blade turbine (PBT).
Figure 1.2: Axial flow turbines; source: [3]
Radial flow impellers discharge fluid radially outward to the vessel wall. With suitable baffles these flows are converted to strong top-to-bottom flows with two circulating loops, one below and one above
the impeller, as shown in Fig. 1.6 (a). Mixing occurs within each loop and less intensively between them. They can be used for any duty, but they are most effective for gas-liquid and liquid-liquid dispersion. The most used radial impeller is the 6-blades Rushton turbine. It is constructed with six vertical blades on a disk. Its standard relative dimensions consist of blade length of 1/4, blade width of 1/5, and the disk diameters of 2/3 and 3/4 with respect to tank diameter.
Figure 1.3: Radial flow turbines; source: [3]
Hydrofoil impellers are used when both axial component of the flow and low shear rate are desired. They produce about the same pumping but at lower shear and turbulence levels than a PBT. Geo-metries are quite complex and designed according to the desired flow patterns. An example is given in Fig. 1.6 (b).
Figure 1.4: Hydrofoil turbines; source: [3]
High shear rate turbines are used at high speeds in multi-phase mixing, such as gas dispersion, pigments addiction and making emulsions. They are low pumping and therefore are often used together with axial flow impellers.
Figure 1.5: High shear turbines; source: [3]
Figure 1.6: Flow patterns with different impellers, impeller diameter and liquid viscosity; source: [3]
Close-clearance impellers
Close-clearance impellers are used with viscous liquids and are designed to physically mix the fluid at low shear. These impellers are typically large in size, nearly the same size as the tank diameter. With anchors the main flow field is essentially a solid-body rotational flow around the impeller, similar to the ideal Couette flow of a viscous fluid confined in the gap between two rotating cylinders. Mixing and other transport phenomena are primarily due to secondary flows. Ohta et al. [4] have reported the existence of loop flows in both the upper and lower regions of the vertical cross section. The core region of the loop flow is a poorly mixed region because of the low shear and because, in addition, the fluid passing through the high-shear field, namely the clearance region between the blade and the wall, does not appreciably flow into this region. Sot this reason anchors are sometimes used together with secondary turbines to improve mixing in the inner region.
(a) anchor (b) helical ribbon
Figure 1.7: Close-clearance impellers
Helical ribbons provide top-to-bottom physical movement of the liquid. In addition to the outer helix, they can be designed with an inner helix pumping in the opposite direction. This is particularly needed for direct-action mixing for high viscosity materials such as high-concentrated polymer solutions, gums and pastes.
1.1.3
Typical applications
Stirred vessels are widely used in chemical and manufacturing industry for:
blending of homogeneous liquids such as lube oils, gasoline additives, dilution, and a variety of chemicals;
suspending solids in crystallizers, polymerization reactors, solvent extraction, etc.;
blending and emulsification of liquids for hydrolysis/neutralization reactions, extraction, suspen-sion polymerization, cosmetics, food products, etc.;
dispersing gas in liquid for absorption, stripping, oxidation, hydrogenation, ozonation, chlorina-tion, fermentachlorina-tion, etc.;
homogenizing viscous complex liquids for polymer blending, paints, solution polymerization, food products, etc.;
1.2
Aims and objectives of this work
The objectives of this Thesis are: use PIV for visualization of the flow fields of different fluids produced by an anchor impeller; review, comparison and validation of the models arising from the original M&O’s one; understand if M&O can be used to predict an average viscosity;
develop a new methodology for the evaluation of the average viscosity inside the vessel, suitable for quality control;
investigate the effects of viscoelasticity on the flow patterns and power consumption.
1.3
Thesis overview
This Thesis consists of five main chapters (e.g. Chap. 1), and each chapter is divided into sections (e.g. Sec. 1.1) and SubSections (e.g. SubSec. 1.1.1):
Chapter 1 is an introduction to this work. The research topics are briefly introduced from a scientific and industrial point of view, and the motivations and objectives are explained; Chapter 2 is an indispensable literature review to understand the next Chapters, how to collect
data and how to interpret the results. The fluid-dynamics fundamentals used in this work are reminded, pointing out the difference between convection and diffusion and their rule in mixing. The most common rheological behaviours and the corresponding constitutive equations are illustrated, with particular attention to the fluids used in this work. The present State of the Art on mixing in stirred vessels in the laminar regime is discussed, providing the theoretical background and showing the most important experimental results. Finally, the Particle Image Velocimetry technique is presented, explaining how it works and pointing out its strong and weak points;
Chapter 3 is the description of the materials and methods used in the experiments. The pre-paration of the fluids, the experimental systems setup, data collecting and post-processing are illustrated step by step;
Chapter 4 is dedicated to the presentation and explanation of the results. A comparison with existing literature is made, showing some interesting differences. Also, a new technique of indus-trial interest is provided, which allows the measurement of a characteristic viscosity from online torque measurements. This is a contribution towards online rheology measurements in stirred vessels;
Chapter 2
Fundamentals and Literature review
This Chapter illustrates the State of the Art in mixing in stirred vessels in laminar conditions and the necessary theoretical background for the next chapters. In the first Section some basic fluid-dynamics concepts are reminded. The second Section quickly illustrates, using a cause-effect approach, the relationship between the stress and the rate of deformation in different classes of fluids, focusing on time-independent viscous fluids and viscoelastic fluids which will be used in this work. The third Section is a literature review concerning mixing in laminar regime, and shows the effects of many parameters on power consumption. In the fourth Section the Particle Image Velocimetry technique is explained and references are given for further details.2.1
Fluid-dynamics fundamentals
Fluid-dynamics is the branch of physics which studies the behaviour of moving fluids.
Many natural phenomena and technological applications in any field of science relate to fluid flows. Some examples are sea currents, capillary flow in plant roots, blood circulation, wind turbines, For-mula1 car design and oil handling through pipelines. For this reason, scientists from different areas are interested in fluid-dynamics, and a huge research is made on it.
Most of the problems in fluid-dynamics require the resolution of differential equations, and they can be very complex. For example, the analytical resolution of Navier-Stokes equations is nowadays one of the seven most important unsolved problems of mathematics. The theoretical approach commonly uses assumptions to simplify the problems. For application of practical interest, experimental and/or numerical approach are sometimes preferred.
It would be practically impossible to exhaustively describe all the fluid-dynamics laws and assumptions, therefore only the ones useful for this work will be summarized. A more comprehensive argumentation may be found in Bird et al. [5] or in Mauri [6]. An useful reference for vector calculus in Fluid-mechanics is Kundu et al. [7].
2.1.1
Continuum assumption
Any kind of material is made of molecules and atoms. In a fluid, molecules collide with each other and slide over one another. The continuum assumption assumes that fluids are continuous rather then discrete, and it is used in many problems where the micro-scale nature of the material can be ignored because it has not significant effects on the solution. Under the continuum assumption the intensive properties such as density, temperature, velocity and pressure are well-defined at any point in space and vary continuously from one point to another, such that spatial derivatives can be calculated at any point.
2.1.2
Convection and diffusion
Physical quantities such as mass, momentum and energy can be transported from one point to another in two fundamentally different ways.
The first one is called convection, and it is related to the macroscopic movement of the medium. For example, convective transport of energy occurs when hot portions of material move towards a cold region: molecules carry a thermal energy depending on their temperature. Note that convection can only occur in fluids. In a solid only rigid motions are possible, with no effect on internal transport. The second mechanism is called diffusion, and it occurs without an effective movement of the medium. For example, a spoon dipped in hot water is heated because the molecules of water collide those of the spoon transferring thermal energy. Of course, molecules of water do not move inside the spoon. Diffusion can occur in fluids and solids, and between different materials put in contact as well. Actually, a third mechanism exists for energy. Radiative transfer occurs between two material points because of their temperature difference. Energy is transferred by electromagnetic radiation, even in absence of a material medium. Radiative transfer becomes very important in applications where great temperature differences exist, such as in boilers and furnaces. In this work its importance is negligible.
Convective fluxes
If f is an intensive physical quantity per unit of mass, the extensive quantity is F = ρ f V , where V is the volume. The amount of F which crosses the unit area per unit time is commonly called flux. Note that in mathematics the flux is generally defined instead as the integral of the rate of F over the surface. This might lead to disambiguation, but the intended meaning of the word flux is generally clear from the context.
The flux of a quantity can be convective if referred to the convection, or diffusive if referred to the diffusion.
Considering convection, if the fluid moves with velocity U , the mass crossing the surface S in the time dt is dM = ρ dV = ρ S U dt. The flux of mass is the ratio of the mass to the area per time, so JMconv= dM/(S dt) = ρ U .
If the fluid moves with a velocity field u, the convective flux becomes a vector with the same direction of the vector velocity:
JMconv= ρ u . (2.1)
The mass has a momentum dqj = dM uj along j, so the convective flux of the j-th component of momentum is
Jqj
conv
= ρ uju . (2.2)
The internal energy of the mass is dE = dM cV (T − T0), where cV is the thermal capacity per unit mass in J/(kg · K) and T0 is an arbitrary temperature reference, so the convective flux of energy is
JEconv= ρ cV (T − T0) u . (2.3)
Diffusive fluxes
Diffusive fluxes occur because of a gradient. By dimensional analysis diffusive fluxes can be expressed as the gradient multiplied by a diffusivity in m2/s, with a minus sign is because the transport occurs in the opposite direction of the gradient:
JMdif f ≡ 0 ; (2.4)
Jqj
dif f
= −ν ∇(ρ uj) ; (2.5)
JEdif f = −α ∇ (ρ cV T ) . (2.6)
Here ν is the momentum diffusivity also known as the kinematic viscosity and α is the thermal diffusivity. In the absence of concentration differences, the diffusive flux of mass is zero.
Note that if ρ cV = const we obtain JEdif f = −k ∇T , also known as the Fourier’s Law.
Reynolds number
A simple way to characterize the macroscopic properties of a fluid flow is to investigate the relative relevance of the convective momentum flux with respect to the diffusive momentum flux. If U is the characteristic velocity and L is the characteristic length, then the convective and diffusive fluxes are, respectively:
Jqconv≈ ρ U2; (2.7)
Jqdif f ≈ ν ρ U
L . (2.8)
Reynolds number is called after the English physician O. Reynolds and it is defined as
Re ≡ ρ U 2 νρ UL = U L ν = ρ U L µ . (2.9)
Here µ = ρ ν is the dynamic viscosity which will be better discussed in Sec. 2.2. Note that this definition includes only macroscopic quantities. For a micro-scale description of the flow other instruments are needed. For example, when a non-Newtonian fluid is mixed in a stirred vessel, viscosity is a function of the shear stress, which changes from point to point, so a characteristic viscosity needs to be defined to calculate Reynolds.
However, Reynolds number is used in the most part of experimental correlation in fluid-dynamics: transfer coefficients and overall mixing properties are often expressed as a function of Reynolds.
2.1.3
Laminar flow and turbulent flow
It is known, from the very beginning of the study of Fluid-mechanics, that there are two distinct ways in which a fluid flows into a pipe. At low flow rates pressure losses are proportional to the superficial velocity, defined as the ratio between the volumetric flow rate and the cross area. The fluid tends to flow without lateral mixing, and there are no eddies. At high flow rates pressure losses are about proportional to the superficial velocity squared. The flow is always tridimensional, even with symmetrical geometries, unsteady, with presence of eddies of different size interacting with each other, and characterized by chaotic changes in instantaneous pressure and velocity. Reynolds, in 1883, remarked that the critical velocity that separates these two flow regimes depends only on Reynolds number. Subsequent observations have demonstrated that other parameters, such as the roughness of the pipe are important. However, Reynolds number is generally sufficient to predict the flow regime in any application. For example, in the case of a fluid flowing into a pipe, laminar regime, corresponding to low flow rates, occurs when Reynolds number is below 2100, while turbulent regime occurs when Reynolds number is greater than 4000. The transitional regime occurs in the middle.
The difference between laminar and turbulent flows is critical for transport phenomena, because of the different transfer mechanisms. Laminar regime is characterized by high momentum diffusion and low momentum convection. The contrary is for turbulent regime. This reflects the physical meaning of Reynolds number.
2.1.4
Fluid mechanics laws in Eulerian formulation
In this SubSec. and in the next one the laws governing Fluid-mechanics are briefly explained. These laws are commonly called as the Conservation Laws, even if the term “conservation” might seem to be inappropriate. In fact, these equations express the rate of change of the physical quantities associated to a control volume at fixed position in the space or to a control element of fluid with fixed mass.
Conservation of mass
The rate of change of fluid mass inside a control volume Ω with fixed position is equal to the net convective flux of mass into the volume through the boundary surface ∂Ω. Physically, this means that mass is neither created nor destroyed in the control volume. Mathematically:
∂ ∂t ˆ Ω ρ dV = ˆ ∂Ω −ρ u · n dS . (2.10)
Here ρ is the density of the fluid in kg/m3,u is the velocity vector and n is the outward pointing unit vector from ∂Ω. This is the Eulerian-integral form of the conservation of mass equation. The differential form, also known as continuity equation, is obtained using the divergence theorem (see A.3):
∂ρ
∂t = −∇ · (ρ u) . (2.11)
Note that when the fluid is incompressible ρ = cost and the (2.11) reduces to ∇ · u = 0.
Conservation of momentum
The rate of change of fluid momentum inside a control volume Ω with fixed position is equal to the net convective flux of momentum into the volume through the control surface ∂Ω plus the external forces. Mathematically: ∂ ∂t ˆ Ω ρ u dV = ˆ ∂Ω −ρ u u · n dS + ˆ Ω f dV + ˆ ∂Ω −σ · n dS . (2.12)
Here f denotes the volumetric density of force in N/m3, σ is the stress tensor, and so σ · n is the superficial density of force along n in N/m2. This is the Eulerian-integral form of the conservation of momentum equation.
Stress tensor can be split up in two terms. The first is the effect of mechanical pressure and second is the deviatoric stress tensor:
σ = −p I + τ . (2.13)
p ≡ −1
3σii. (2.14)
The differential form, also known as Cauchy equation, is obtained using the divergence theorem and the continuity equation:
ρ∂u
∂t + ρ u · ∇u = −∇p + ∇ · τ + f . (2.15)
Note that (2.15) is a vectorial equation and it is equivalent to three scalar equations, one for each Cartesian axis.
Finally, if the deviatoric stress tensor is expressed in terms of the velocity field using the constitutive equation of the fluid, Navier-Stokes equations are obtained. For example, Navier-Stokes equations for incompressible fluids with constant viscosity are usually presented in the vectorial form as
ρ∂u
∂t + ρ u· ∇u = −∇p + µ∇
2u + ρ g . (2.16)
Here g is the gravitational acceleration constant in m/s2. It was assumed that gravity is the only volumetric force acting on the fluid.
Conservation of kinetic energy
The law of conservation of the kinetic energy in the Eulerian-differential form is obtained by dot-multiplying (2.15) for u : 1 2ρ ∂u2 ∂t + u · ∇ u2 2 = −∇p · u + ∇ · τ · u + f · u . (2.17)
Conservation of potential energy
The law of conservation of the potential energy in the Eulerian-differential form is obtained by dot-multiplying (2.11) for g h :
g h∂ρ
∂t = −g h ∇ · (ρ u) . (2.18)
Note that at fixed position g and h are constants.
Conservation of internal energy
The total energy of a system is the sum of the contributions of mechanical (kinetic and potential) energy and internal energy. The rate of change of fluid total energy inside a control volume Ω is equal
to the net convective flux of energy into the volume through the control surface ∂Ω plus the flux due to thermal conduction and the power of the external forces1. Mathematically:
∂ ∂t ˆ Ω ρ u 2 2 + g h + e dV = ˆ ∂Ω −ρ u 2 2 + g h + e u · n dS + ˆ ∂Ω −JEdif f · n dS + + ˆ Ω f · u dV + ˆ ∂Ω −σ · u · n dS . (2.19)
Using the divergence theorem, the continuity equation, the conservation of kinetic energy and the conservation of potential energy, after some mathematical manipulations the Eulerian-differential form of the conservation of internal energy is obtained:
ρ∂e
∂t + ρ u · ∇e = −p ∇ · u + τ : (∇ u) − ∇ · JE
dif f. (2.20)
The term σ : (∇ u) = −p ∇ · u + τ : (∇ u) represents the conversion of kinetic energy into internal energy by dissipation. It can be also observed that τ : (∇ u) = τ : S where S is the symmetric part of the tensor gradient of velocity. The asymmetric part, which represents a solid body rotation, does not contribute to the change of the energy. In fact, the double dot product of a symmetric tensor, here τ , for an asymmetric tensor is zero.
If the Fourier law assumptions are valid, the (2.20) becomes
ρ cV ∂T
∂t − k u · ∇e = −p ∇ · u + τ : (∇ u) + k ∇
2T . (2.21)
2.1.5
Conservation laws in Lagrangian formulation
The conservation laws described previously are been formulated from the point of view of a volume at fixed position in space (Eulerian point of view). Sometimes, it is useful to investigate the rate of change of a quantity from the point of view of a very small element of fluid with constant mass moving with the flow (Lagrangian point of view). This can be done using the material derivative formalism described in (A.2) and the Reynold’s transport theorem reported in (A.3). Eq.s (2.11), (2.15), (2.17) and (2.20) can be written in the Lagrangian form as follows (the proof is omitted for brevity):
Dρ Dt = −ρ ∇ · u ; (2.22) ρDu Dt = −∇p + ∇ · τ + ρ g ; (2.23) 1 2ρ Du2 Dt = −∇p · u + ∇ · τ · u + ρ g · u ; (2.24)
ρDe
Dt = −p ∇ · u + τ : (∇ u) − ∇ · JE
dif f. (2.25)
The (2.24) is the differential form of the famous Bernoulli equation.
Note that the contribution of convection does not appear in the the Lagrangian form of the equations. This is consistent with the definition of the fixed material element of fluid, which moves and changes its shape and its volume, but has a constant mass.
2.2
Rheologies
2.2.1
Fundamentals and Definitions
A stress is a surface density of force, and its SI units are P a = N/m2 (Pascal). The force can be normal or tangential to the surface on which it acts (any force with any direction is a sum of these two components), as is the stress.
A fluid is a substance capable of deforming in a continuous manner and indefinitely when it’s subjected to an external tangential (shear) stress. In other words, in quiet conditions a fluid can only oppose normal stresses, while if subjected to tangential stresses it necessarily starts moving. In quiet conditions normal stress is called pressure.
Note that this definition is phenomenological, because it only considers the fluid response to an external action, and it does not consider the micro-structure of the material.
Fig. 2.1 illustrates the concept. A fluid is bounded by two large parallel plates, of area A, separated by a distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U . The force is directly proportional to the area of the plate. The shear stress is
τ = dF
dA. (2.26)
In this simple case τ = F/A.
The fluid bounding the lower plate has zero velocity, while the fluid bounding the upper plate moves at the plate velocity U (no-slip conditions). The velocity gradient is
˙γ = du
dy (2.27)
and it is called the shear rate. In this simple case ˙γ = U/H. SI units for shear rate are s−1 (reciprocal seconds).
This situation is the simplest example of shear flow.
Figure 2.1: Shear flow of a fluid between two plates
µ = τ
˙γ. (2.28)
SI units for dynamic viscosity are P a · s (Pascal seconds).
The kinematic viscosity is the ratio of dynamic viscosity to density ν = µ/ρ. SI units for kinematic viscosity are m2/s (square meters per second), but cSt (centi-Stokes) are commonly used, because 1 cSt = 1 mm2/s is very close to kinematic viscosity of water at 20°C.
2.2.2
Rheologies and Constitutive equations
Rheology is the discipline of Fluid-mechanics which studies the relationship between stresses and shear rates, and finds applications in materials science, engineering, geophysics and physiology. It owes its name to the famous Heraclitus’ aphorism παντ α ρει, which means “everything flows”. The main objective of rheology is to provide the constitutive equation of the fluid, τ = τ ( ˙γ ; ...), which allows to calculate the stress from deformation rate. A rheogram is a plot of shear stress versus shear rate in simple shear flow, and it is widely used for classification of rheological behaviours. In practical application, fluid flows are actually more complex than the one described in Fig. 2.1, as is the relationship between stress and shear rate. For a complete description of the rheological behaviour of a fluid, 3-dimensional flows and time dependence of the physical quantities are required.
Fluids can be classified into:
purely viscous fluids, in which shear stress depends only on the rate of deformation and not on the extend of deformation;
– time-independent, in which the shear stress depends only on the instantaneous shear rate; – time-dependent, in which the shear stress depends on the past history of the shear rate; viscoelastic fluids, in which shear stress depends on the rate and on the extend of deformation. In
Time-independent purely viscous fluids
Time-independent purely viscous fluids are the largest class of engineering interest. They can be further divided in Newtonian and non-Newtonian fluids.
For Newtonian fluids the rheogram is a straight line passing through the origin. The slope is the viscosity and it is a constant (it depends only on temperature and perhaps pressure). Gases and low molecular weight liquids are generally Newtonian. The constitutive equation is written as
τ = µ ˙γ . (2.29)
For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, depends on shear rate. It is often called apparent viscosity to emphasize the distinction from Newtonian behaviour. Shear-thinning fluids have decreasing viscosity with increasing shear rate. Many polymer solutions and solids suspensions are shear-thinning fluids. Shear-thickening (dilatant) fluids show increasing viscosity with increasing shear rate. Dilatancy is quite rare, observed only in certain concentration ranges in some particle suspensions. For shear-thinning and shear-thickening fluids the constitutive equation can be written as
τ = K ˙γn. (2.30)
This relationship is commonly called power law. K and n are the consistency index and behaviour index, respectively. n is dimensionless, while K is in units of kg/(m · s2−n). The consistency index is a true constant, while the behaviour index can be considered a constant over wide ranges of shear rate. The fact that it is not a true constant is frequently irrelevant for engineering purposes [8]. For shear-thinning fluids n < 1. For shear-thickening fluids n < 1.
Non-Newtonian fluids include also yield-stress materials, for which a finite stress is required before deformation occurs. For Bingham plastic fluids the rheogram has a constant slope µ∞, called the infinite shear viscosity. Their constitutive equation is
τ = τ0+ µ∞ ˙γ . (2.31)
Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behaviour. Shear-thinning power law fluids with yield stress are called Herschel-Bulkley fluids. Their constitutive equation is
τ = τ0+ K ˙γn. (2.32)
Note that the last constitutive equation can be used as a generalization of all the different behaviours described. For example, Newtonian fluids may be meant as Hershel-Bulkley with τ0 = 0 and n = 1, while shear-thinning fluids have τ0= 0 and n < 1.
Figure 2.2: Typical rheograms of time-independent purely viscous fluids
Viscoelastic fluids
When a periodic stress is applied to an elastic body, at steady state the material reacts with a de-formation in phase with the stress. The maximum of dede-formation occurs when the stress is at its maximum, and the deformation is zero when the stress is zero.
With a purely viscous fluid, at steady state the material reacts with a deformation rate in phase with the stress. The maximum of deformation rate occurs when the stress is at its maximum, and the deformation rate is zero when the stress is zero (but not the deformation, because the fluid has changed its shape).
Viscoelastic fluids show an intermediate behaviour between elastic solids and purely viscous fluids. At steady state the stress is not in phase with the deformation, like in a elastic fluid, and not in phase with the deformation rate, like in a purely viscous fluid. Since deformation rate is the time derivative of deformation, they have a phase displacement equal to π/2. This means that a viscoelastic fluid is a substance in which the phase displacement between stress and deformation is φ with
0 < φ < π/2 . (2.33)
Many polymeric solutions and multi-component systems such as suspensions, emulsions, and mixtures show viscoelastic behaviour.
For a viscoelastic material the phase displacement φ depends on the frequency of the stress, while, at fixed frequency, it is a property of the substance. Actually, all the substances have viscoelastic behaviour, but some have φ ≈ 0 and they can be described with the elastic solid ideal model, while some have φ ≈ π/2 and they can be described with the purely viscous fluid ideal model.
Figure 2.3: Phase diagram for elastic solid, purely viscous fluid and viscoelastic fluid
˙γ(t) = 1 µτ (t) +
1
G˙τ (t) . (2.34)
The shear rate is equal to the sum of a viscous contribution and an elastic contribution. µ is the viscosity and in general it depends on ˙γ. G is the shear modulus, and in general it depends on γ. SI units for G are P a.
Assuming for semplicity constant µ and G and using the Fourier Transform (see A.4), we obtain
e˙γ(ω) = e τ (ω) µ + e˙τ (ω) G e˙γ(ω) = e τ (ω) µ + iω e τ (ω) G (2.35)
where ω is the angular velocity in rad/s. ω = 2π f where f is the frequency of the stress.
If we want to express the rheology of the fluid using the classic form of constitutive equations,eτ =µ · e˙e γ, we obtain e τ (ω) = 1 1 µ + iω G e˙γ(ω) (2.36) and so
e µ(ω) = 1 1 µ + iω G . (2.37) e
µ(ω) is called complex viscosity. The ratio µ/G ≡ λ has the units of a time, and it is the time of relaxation of the material. We can also express the complex viscosity putting in evidence its real and imaginary parts: e µ = µ 1 1 + iωλ = µ 1 1 + ω2λ2 − µ iωθ 1 + ω2λ2 = µ 0− i µ00. (2.38)
The real part of complex viscosity, µ0, represents the viscous contribution, as it keeps the stress and the deformation rate in phase. The imaginary part of complex viscosity, µ00, represents the elastic contribution, as it shifts the stress and the deformation rate by π/2 .
Phase displacement is φ = arctan µ 0 µ00 = arctan 1 ωλ . (2.39)
In general µ = µ ( ˙γ) and G = G (γ), so λ = λ (γ; ˙γ) and φ = φ (ω; γ; ˙γ). It is interesting to note that
lim ω→∞φ = 0 (2.40) and lim ω→0φ = π 2 . (2.41)
For high frequencies of the stress (short time scale) the material behaves like a solid, while for low frequencies of the stress (long time scale) it behaves like a pure viscous fluid.
Deborah number is called after the Prophetess Deborah who, in the Book of the Judges, proclaimed “The mountains flowed before the Lord” and it is defined as the ratio of the time of relaxation to the characteristic time of deformation process:
De ≡λ
t . (2.42)
Contained within the definition of the Deborah number is the idea that it is a dimensionless measure of the rate of change of flow conditions and is therefore related to flow unsteadiness (in a material or Lagrangian sense). In slowly changing or steady flows the time scale of deformation process is infinite and Deborah number is zero, regardless of the relaxation time. Thus, the Deborah number alone is insufficient to fully characterize effects due to viscoelasticity.
Weissenberg number is defined as the ratio of the elastic forces to the viscous forces acting on a fluid element at steady state flows (in a Lagrangian sense). Using the Upper-convected Maxwell model proposed by Oldroyd [9]: W i ≡ 2 λ µ ˙γ 2 µ ˙γ = 2 λ ˙γ ≈ λ U L, (2.43)
where the shear rate has been expressed as the ratio of the characteristic velocity U and the charac-teristic length L of the system, and the factor 2 can be neglected.
Deborah number and Weissenberg number are very similar in expression, so that they are frequently (but wrongly) used as synonym, and when the characteristic time of deformation process can be expressed as L/U they coincide. However, it is important to point out that they describe different effects. An useful article on the difference between Deborah and Weissenberg numbers is given by Poole [10].
Elastic properties of a viscoelastic fluid are related to shear-induced structure transformations, as pointed out by Malkin [11]. In polymer melts these transformations include formation of anisotropic structures and changes in the macromolecule entanglement topology, and they are related to the molecular weight distribution of the polymer.
2.3
Mixing in stirred vessels in laminar regime
2.3.1
Flow regimes in stirred vessels
Power number is a dimensionless parameter, useful to compare power data from different fluids mixed in stirred vessels. It is defined as
P o ≡ P
ρ D5N3, (2.44)
where P is the power input required to mix the fluid in W (Watt), D is the impeller diameter in m (meters) and N is the impeller rotational speed in s−1 (reciprocal seconds).
The Reynolds number is calculated using the impeller tip speed π D N as characteristic velocity and the impeller diameter as characteristic length, ignoring the constant π:
Re = ρ D 2N
µ . (2.45)
The Power number vs Reynolds number curve can be divided in three regions, which are directly analogous to the three regions of friction factor of flow in a circular pipe. Fig. 2.4 shows that in the first region, corresponding to low Reynolds and laminar conditions, Power number is inversely propor-tional to Reynolds number, in the third region, corresponding to high Reynolds and turbulent regime,
Power number is basically unaffected by Reynolds number, and the intermediate region corresponds to transitional regime.
Figure 2.4: Flow regimes in a stirred vessels; source: [13]
In turbulent regime the main mechanism of mixing is convection. Diffusion is still the predominant mechanism in the micro-scale. In laminar regime the main mechanism is diffusion. A gentle blending causes a continuous reorientation and redistribution of the fluid, increasing the interfacial area between different portions of the fluid and promoting diffusion.
While in a circular pipe the transitional region occurs in the narrow range of Reynolds number between 2000 and 4000 [8], in a stirred vessel it occurs at lower speeds and its extension is larger, depending on the impeller and on the fluid rheology. Metzner and Otto [8] have suggested a typical range between 10 and 1000. For anchor impellers Nagata [12] has reported a range from 10 to 300. Foucault, Ascanio and Tanguy [13] have found a range from 20 to 1000 for Rushton impellers and from 100 to values larger than 105for anchors.
It is quite intuitive that at fixed rotational speed the more viscous the fluid the higher is the Power number. It is also easy to see that in turbulent regime, where power number is constant, the power input is proportional to the impeller speed cubed. Therefore, if the fluid is viscous the required power can be very high. This is not only a cost, but it might also introduce technological problems, such as over-heating and excessive mechanical stress of the impeller and the shaft. For this reason, the mixing of high viscous fluids is typically performed in laminar regime, and sometimes in the transitional regime.
Furthermore, when turbine impellers are used highly viscous liquids velocity rapidly decay away from the impeller. This may result in formation of a cavern around the impeller. Mixing can be good inside the cavern and poor outside, where the fluid does not move, affecting the blending quality. Turbine
impellers are therefore not raccommended for use in the laminar regime, and close-clearance impellers are typically used [14]. This work focuses on mixing in laminar regime with anchors.
2.3.2
Power consumption with Newtonian fluids
In the laminar regime, with Newtonian fluids and absence of vortex on the free surface, Power number and Reynolds number are inversely proportional:
P o = KP
Re . (2.46)
KP is the impeller constant for laminar regime. It only depends on the geometry of the system, and not on the fluid.
Note that if Reynolds number is expressed, the (2.46) becomes
P o = KP µ
ρ D2N , (2.47)
which shows that Power number and so power input are proportional to the viscosity of the fluid. There are many works on the effects of geometry on KP. For example, Hemrajani and Tatterson [2] have reported that for a 2-blades anchor impeller Power number is a function of geometrical ratios such as KP ∝ c T −0.5 (2.48) and KP ∝ 0.89 h D + 0.11 , (2.49)
where c is the wall-blade clearance, T is the tank diameter, h is the impeller blade height and D is the impeller diameter. Espinosa-Solares et al. [15] have studied the combined effect of bottom clearance and wall clearance on the power consumption rate and proposed the correlation
KP = 41.1 " c T −0.23 + b T −0.27# , (2.50)
where b is the bottom clearance. They have observed that the power consumption decreases as the bottom and wall clearance increases, which is due to the change in the flow pattern.
Murthy Shekhar and Jayanti [16] have used CFD to simulate the flow around a 2-blades anchor impeller and have found
KP = 372 w D h D D − 0.5 w D c T −0.6 , (2.51)
where w is the width of the impeller. Again the power consumption decreases as the wall clearance increases.
2.3.3
Power consumption with non-Newtonian fluids: Metzner and Otto’s
model
Original model
When a non-Newtonian fluid is mixed in a stirred vessel, the apparent viscosity µa changes from point to point because of different values of the shear rate. A characteristic viscosity is needed to calculate Reynolds number and so the power input. The model proposed by Metzner and Otto [8] in 1957 is the basis of all the following research.
Imagine to have two identical sets of mixing equipment, one of which contains a Newtonian fluid and the other a non-Newtonian one. If these fluids are agitated in the laminar region at the same impeller speed, and the viscosity of the Newtonian is adjusted by diluting it or thickening it so that the power at each impeller is the same, then, because all variables are identical, the effective viscosities are the same in both the systems. Upon measuring the viscosity µ of the Newtonian fluid one knows the effective viscosity of the non-Newtonian µef f existing under those experimental conditions. This thought experiment defines the effective viscosity of the non-Newtonian fluid.
Note that while the viscosity µ of the Newtonian fluid and the apparent viscosity µaare local properties of the material and they do not depend on the geometry of the system, the effective viscosity µef f is not a local property of the material, but it’s an overall property of the system and it varies with the experimental conditions, namely the geometry of the impeller and its rotational speed. Since both these variables determine the shear rates in the mixer, the effective viscosity is shear-dependent. The model assumes that the region near the impeller can always be characterized by an average shear rate proportional to the impeller speed:
˙γef f = ksN , (2.52)
where ksis the Metzner and Otto’s constant and it is chosen as to have µef f = µa( ˙γef f) referring to the non-Newtonian rheological curve.
The procedure to determine ksis as follows:
1. Power input is measured and Power number is calculated; 2. the effective viscosity is calculated as
µef f =
P o ρ D2N KP
; (2.53)
3. the effective shear rate is calculated referring to the rheological curve of the fluid:
˙γef f = ˙γ (µef f) ; (2.54)
4. Finally, the proportionality constant ksis calculated as
ks= ˙γef f
N . (2.55)
Once ks is known, the required power input at any impeller speed can be calculated directly from the latter as follows:
1. knowing N , the effective shear rate is calculated by (2.52);
2. the effective viscosity is calculated from viscometric data as µef f = µa( ˙γef f); 3. a generalized Reynolds number is calculated as
Re = ρ D 2N µef f
; (2.56)
4. Power number is calculated as P o = KP/Re .
The model has been validated using shear-thinning and Herschel-Bulkley fluids with vessels having different clearance-diameter ratios. The proposed value for kswas equal to 13 and it was an average over all the experiments.
Note that, because of the definition of the effective viscosity, Reynolds number definition in (2.56) does not require any modification with respect to Newtonian fluids, and KP is the same for Newtonian and non-Newtonian fluids. In other words, if the effective viscosity and the resulting generalized Reynolds number are used, the Power number vs Reynolds number curves of any Newtonian or non-Newtonian fluid coincide.
The authors have not concluded anything about the dependence of ks on the fluid. In particular, ks might be expected to vary with the flow-behaviour index n. They have recommended that an extension of their work to shear thickening materials would be needed to prove conclusively whether or not ks depends on the behaviour index of the fluid and whether the generalized Reynolds number of (2.56) is applicable to correlation of power consumption data for all fluids.
Subsequent models
The classical approach proposed by Metzner and Otto consists in the use of the effective viscosity to calculate a generalized Reynolds number. In this way KP depends only on the impeller and the Power number vs Reynolds number curve is the same for any fluid. Murthy Shekhar and Jayanti [16] have used a theoretical constant value of 4π for ks, based on the approximation of the flow in an anchor-driven vessel to similar to the Couette flow between rotating concentric cylinders. It is interesting to point out that this value in very close to the 13 given by Metzner and Otto in their first work. Rieger and Novak [17] remarked that ks depends exclusively on the fluid and proposed the correlation
ks= n
2.21
n−1. (2.57)
Sestak et al. [18] have proposed another correlation in function of the rheology only:
ks= 35
n
1−n. (2.58)
Shamlou Ayazi and Edwards [19] have found a dependence on the clearance-diameter ratio, and have proposed ks= 33 − 172 c T . (2.59)
An alternative approach used with shear thinning fluids is to express Reynolds number in function of the fluid rheology and some geometrical factors such that the effective viscosity does not appear explicitly. ks is immediately defined by the expression of Reynolds. This implicitly leads to a dependence of ks on both the geometry and the behaviour index of the fluid. Note that, if the fluid is shear thinning, Metzner and Otto’s Reynolds number is
Re = ρ D 2N µef f = ρ D 2N K ˙γef fn−1 = ρ D2N K ksn−1Nn−1 =ρ D 2N2−n K ksn−1 . (2.60)
Beckner and Smith [20] have proposed
Re = ρ D
2N2−n
K [a (1 − n)]n−1 (2.61)
with a = 37 − 120 c/T , and so, by comparison with (2.60),
ks= a (1 − n) . (2.62)
Calderbrank and Moo Young [21] have proposed
Re = ρ D 2N2−n K " 9.5 + 9 T D 2 T D 2 − 1 #1−n 4 n 3 n + 1 n , (2.63)
and so, by comparison with (2.60), ks= " 9.5 + 9 T D 2 T D 2 − 1 # 3 n + 1 4 n n−1n . (2.64)
With this methods, KP is still the same for any fluid, but Power number can be evaluated directly from Reynolds number without having to evaluate a µef f first.
Finally, another approach is to define a modified Reynolds number which is not comparable with the one defined in (2.60). In this case, Kp becomes a function of the fluid. For example, Tanguy et al. [22] have used numerical simulation of power law fluids to investigate the effects of behaviour index on ks, proposing
Re∗= ρ D 2N2−n
K . (2.65)
With this definition, the KP∗ for a power law fluid is
KP∗= P o Re∗. (2.66)
The power input, which is the final result, must be the same regardless the definition used for Reynolds number, so P = K ∗ P Re∗ ρ N 3D5= KP Re ρ N 3D5, (2.67) which leads to ks= K ∗ P KP 1−n1 . (2.68)
It has be found that KP∗∝ exp(n) and
ks= 21.3 + 5.8 n . (2.69)
For shear thinning fluids the authors have considered the approximation ks = 25 ± 2 to be valid for practical purposes.
Issues of Metzner and Otto’s model
The Tab. 2.1 riassumes the correlations for ksreported in this work. A more extended list is given in [16].
Source c/T n ks
Metzner and Otto [8] = 0.24=1 13
Nagata [12] 0.05 0.2=0.8 25
Murthy Shekhar and Jayanti [16] 0.026=0.105 0.266=1 4π
Rieger and Novak [17] 0.05 0.50=0.97 nn−12.21
Sestak et al. [18] 0.05 0.07=1 351−nn
Shamlou Ayazi and Edwards [19] 0.021=0.133 0.2=0.75 33 − 172 Tc Beckner and Smith [20] 0.02=0.1057 0.249 − 0.73 [a (1 − n)]
n−1
a = 37 − 120 c/T
Calderbrank and Moo Young [21] 0.05 0.05=1.87 9.5 + 9( T D) 2 (T D) 2 −1 3 n+1 4 n n−1n Tanguy et al. [22] 0.05 0.3=1.7 25 ± 2
Table 2.1: Correlations for ks reported in literature
It can be seen that contradictory results have been reported in literature, considering that some authors have found ksto be a constant, some have found it to depend only on the geometry, some only on the rheology, and others on both the geometry and the rheology. These differences might arise from the different experimental conditions or from the use of a single geometry. Also, often the dependence of ks on the behaviour index and the geometrical factors is not strong from a numerical point of view. For example, observing (2.63), it may be noted that the second term f (n)n/n−1 has a value in the range 0.78=0.88 when the behaviour index is varied from 1 to 0.1, and ks shows a variation of only about 10% in this range.
Another problem with Metzner and Otto’s model is that it has been widely validated with power law fluids, but few works have been conducted using yield stress materials. With this kind of fluids the flow might be not fully shared because of the minimum shear needed to move the fluid. Anne-Archard et al. [23] have reported that the Metzner–Otto concept is valid for Herschel-Bulkley fluids insofar as the flow corresponds to a fully sheared regime, that is, when the fluid is used in its shear thinning domain, but ksmust include a dependence on Bingham number, defined as the ratio of the yield stress to a viscous stress. Viscoelastic fluids are not included at all.
The main use of Metzner and Otto’s approach is to predict power consumption of a stirred vessel operating at steady state, assuming that the rheology of the fluid is known. This can be very useful for the design of a new mixing equipment. However, the model uses only overall parameters and can not describe the flow field, nor the shear rate distribution inside the vessel, nor the viscosity distribution. It is important to emphasize that the effective viscosity is not an average viscosity inside the vessel. For
this reason, Metzner and Otto’s model can not be used to estimate the average viscosity. Moreover, if the process is unsteady, because, for example, the rheology changes during the manufacturing of the liquid, the model does not give the opportunity to monitor the rheology in real time, since, given the geometry, it basically takes in input only the impeller rotational speed, and not online measurements.
2.3.4
Effects of viscoelasticity on power consumption
It is known from literature that elasticity can strongly interfere with chemical and physical processes, including mixing, heat and mass transfers. Ramsay et al. [24] have shown that elasticity causes lower shear rates and axial mixing because of modifications of the flow patterns. Also, Ulbrect and Carreau [25] and Elson et al. [26] have pointed out that geometry strongly interacts with the elastic behaviour. The easiest way to compare power data from viscoelastic and purely viscous fluids is using Boger fluids (viscoelastic fluids whit constant viscosity) and Newtonian fluids. This allows to single out the contribution of elasticity.
The effect of viscoelasticity on the Power number vs Reynolds number curve appears to be somewhat contradictory in literature. However, most of the differences between papers arise from the choice of the dimensionless group accounting for the elastic component, from the choice of the time characterizing elasticity, or from the rheological method used to measure it, namely oscillatory, relaxation or normal stress difference measurements (see SubSec. 2.2.2, Viscoelastic fluids). For this reason, even when authors have worked with identical fluids and geometries, it is not always easy to quantitatively compare their results [27].
However, as elasticity acts as an energy storage, it would be expected that the power required to mix a viscoelastic fluid is greater than a purely viscous fluid with the same viscosity. Many studies have confirmed this hypothesis in the laminar [24, 27, 28] and the transitional regime [27, 29].
2.4
Particle Image Velocimetry
Particle Image Velocimetry (PIV) is a well-established optical method for flow visualization. Today it is the primary technique used in research.
A typical PIV apparatus consists of a laser and its optical arrangement, a camera, a synchronizer, the fluid under investigation and the seeding particles. Usually the final output is a 2D-2C (two dimensions - two components) flow filed, but more complex PIV setup, such as Stereoscopic PIV, can also measure components along the third axis, using two or more cameras at different positions.
Principles of PIV have been covered in many papers, such as Adrian [30]. Another excellent source of information on various aspects of PIV, including theoretical background and applications is Raffel [31].
2.4.1
2D PIV typical setup
The fluid is seeded with small particles, which are assumed to faithfully follow the flow of the fluid and not to influence it. A double pulsed laser illuminates a section of the fluid, so that particles are visible. A combination of cylindrical and spherical lens is used to reduce the laser beam into a thin sheet. A CCD camera is synchronized with the laser using a synchronizer, and it takes a pair of frames. The frames are split into a large number of interrogation areas (IA). Particle concentration is such that it is possible to identify individual particles in an image, but not to track them from the first frame to the second one, so a cross-correlation algorithm is used to calculate the most probable displacement vector in each IA. Then, the velocity vector is calculated in each IA using the physical length of the IA and the time between laser shots.
Figure 2.5: 2D PIV typical setup
2.4.2
Critical parameters
Some parameters are very critical for the goodness of a PIV experiment:
ideal particles should have the same density as the fluid and be spherical. Stokes number is defined as the ratio of the response time of the particle t0 to the time scale of the flow d/U , and so as
St = t0U
d , (2.70)
where U is the characteristic velocity of the fluid and d is the particle diameter. If St < 1 the particles follow fluid streamlines closely. If St < 0.1 error is below 1%. At low Reynolds number t0 is proportional to d2, so Stokes number is proportional to the particle diameter. Tipically the diameter is in the order of 10 to 100 micrometers. Refractive index of the particles should be different from the fluid one, so that the laser sheet incident on the fluid flow will reflect off of
the particles and be scattered towards the camera. Melling [32] have reviewed a wide variety of tracer materials that have been used in liquid and gas PIV experiments. Methods of generating seeding particles and introducing them into the flow have been discussed as well;
the size of the IA should be chosen to have at least 6 particles per IA on average;
the timing between laser shots needs to be chosen properly. If it is too short, it can be difficult to identify any displacement, because the limiting size is given by the physical length of a pixel. On the contrary, if the timing is too long, the ratio of the displacement to the time is not a good approximation of the instantaneous velocity. A first attempt value is given by 0.25 l/U , where l is the physical size of the IA and U is the characteristic velocity of the fluid;
the camera zoom and focus should be set such that the scattered light from each particle is from 2 to 4 pixels. With larger sizes optical disturbs might occur, with loss of precision.
.
2.4.3
Pros and Cons
The main advantages of 2D PIV are: the method is basically non-intrusive. If properly chosen, the seeding particles do not influence the fluid flow;
each displacement vector is a statistical average over many particle displacements inside the IA, so they can typically be accurate down to 10% of one pixel on the image;
high speed cameras allow to take a large number of image pairs in a short time. This is useful to computate average flow fields, reducing the noise and avoiding artefacts;
high speed cameras also allow the production of near-continuous information. The main cons are:
high power lasers are expensive and bring safety constraints; high speed CCD cameras are expensive;
the choice of the particles is not easy when the fluid is uncommon or it has extreme properties; image processing and post-processing requires high computational cost;
2D PIV is not be able to measure components along the z-axis (towards to/away from the cam-era), while the equations used in fluid-dynamics tipically include all three components. Moreover, z-axis components might also introduce interferences in the x and y components caused by par-allax;
displacements are average values. The resulting velocity field is a spatially averaged representa-tion of the actual velocity field. This affects the accuracy of spatial derivatives, which are often the most interesting output from PIV experiments.
Chapter 3
Materials and methods
Several types of experiments have been carried out to collect proper data and achieve the objectives of the Thesis. In this Chapter materials, equipments, layouts and procedures are described. This is useful for a complete interpretation of data and results, and allows reproducibility of the experiments. In the first Section the formulation of the fluids is illustrated step by step. In the second Section the equipment used for rheological measurements and the procedures are described. The third Section refers to torque measurements. The fourth Section shows the equipment used for PIV experiments, the experimental design, the procedures and the settings that have been used. Finally, the fifth Sections briefly explains the post-processing of data, which transforms the raw experimental measurements in the final results discussed in the next Chapter.
3.1
Fluids formulation
It is often not practical to use actual process fluids in a laboratory scale, as this can involve the use of expensive and obstructive safety precautions as well as inconvenient temperatures and pressures. To avoid these problems, suitable simulant fluids are usually used in research, that will behave in a manner representative of the process fluid.
Eight fluids with different rheological behaviours have been used in this work. They are reported in Tab. 3.1.
CMC solutions
CMC is provided as powder and its solutions are particularly useful for performing mixing experiments because they are inexpensive and the viscosity is relatively insensitive to small changes in temperature and dilution.
Two CMC solutions were prepared by gradually adding the desired quantity of CMC powder (Sigma-Aldrich, CAS no: 9004-32-4) to deionised water into a 1.5 liter vessel, while the mixture was mixed
