LDA measurements on Newtonian and non-Newtonian fluids in a stirred vessel

Bachelor Thesis

Simon de Koning

Delft, september 2009

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Abstract The hydrodynamics of a Newtonian fluid (water) and a non-Newtonian fluid (Blanose) were investigated using laser Doppler anemometry (LDA). Mean velocity profiles and RMS velocity profiles in water were compared to the data measured by Venneker (1999). The measured LDA data are also used to determine power spectral density functions (spectra) of the velocity fluctuations. These spectra showed a number of peaks, which were related to the blade frequency, or to fractions of this frequency. However, one peak was found at twice the blade frequency. The explanation for this high frequency peak is still uncertain. The spectra were also used to investigate the isotropic behaviour of the flow. It was found that at relative large distances from the impeller the flow was isotropic.

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Table of Contents Abstract..................................................................................................................................2

1. Introduction...................................................................................................................41.1 Background..........................................................................................................................41.2 Objective...............................................................................................................................41.3 Outline of the thesis............................................................................................................5

2. Theoretic outline...........................................................................................................62.1 Non-Newtonian fluid..........................................................................................................62.2 Turbulence ...........................................................................................................................62.3 Flow in stirred vessels........................................................................................................72.4 Laser Doppler Anemometry ............................................................................................8

3. Experimental setup and data processing ............................................................. 113.1 Stirred vessel .................................................................................................................... 113.1 LDA setup ......................................................................................................................... 123.3 Data processing ................................................................................................................ 13

3.3.1 Bias correction...........................................................................................................................143.3.2 Spectral analysis of LDA data .............................................................................................14

4. Experimental results................................................................................................. 174.1 Mean velocity profiles in water .................................................................................... 174.2 RMS velocities .................................................................................................................. 184.3 Power spectral density functions.................................................................................. 194.4 Local isotropy in the flow .............................................................................................. 27

5. Conclusions and recommendations ....................................................................... 305.1 Conclusions ....................................................................................................................... 305.2 Recommendations ........................................................................................................... 30

5.2.1 Experimental setup improvements .....................................................................................305.2.2 Future research ..........................................................................................................................31

Bibliography ...................................................................................................................... 32

Appendix............................................................................................................................. 34Power spectral density functions............................................................................................ 34

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1. Introduction 1.1 Background Non-Newtonian fluids are widely used in the process industry, and also in our daily life. Commonly used foods such as yoghurt, ketchup and soup, and emulsions like paint and latex all exhibit non-Newtonian characteristics. Although in daily life these liquids hardly flow in a turbulent way, in the food, fuel and paint industry the turbulent flow of these fluids is of great importance. Usually, manufacturing of these liquids starts with a Newtonian liquid which, because of the large scale of the equipment and the low viscosity, flows in a turbulent way. During the process, the rheological behaviour of the liquid changes gradually and can become more and more non-Newtonian, either because of reactions or because of additives. Usually, viscosity increases during the process but because of the large scale of the process equipment, the flow can still be in the turbulent regime. In order to have a sufficiently fast reaction and a reproducible product, it is important that reactants are intensively in contact with each other. The stirred vessel is one of the pieces of equipment that is used for this purpose. The fluid inside the vessel is mixed by an impeller so a large contact area between the reactants is created. Already in the 1960’s much research on mixing in stirred vessel was carried out, both in the laminar and the turbulent regime (Skelland 1967). However, research on the hydrodynamic behaviour of non-Newtonian fluids in stirred vessels focussed mostly on the laminar regime. The reason for this is the complex nature of turbulence itself combined with the wide variety of non-Newtonian fluids and their different behaviours. In the late seventies/early eighties an advanced measurement techniques, called laser Doppler anemometry (LDA), was developed. LDA made it easier to also investigate the turbulent flow of non-Newtonian fluids in stirred vessels (Venneker 1999). Strangely, most investigations since then still focus on Newtonian fluids in stirred vessels. Therefore, little is known of the hydrodynamic behaviour of non-Newtonian fluids compared to Newtonian fluids.

1.2 Objective A deeper insight in the hydrodynamics of non-Newtonian fluids may lead to more efficient manufacturing or better quality of non-Newtonian products. Therefore, further research on non-Newtonian fluids in stirred vessels is desirable. The focus of this Bachelor thesis is on the difference between Newtonian and non-Newtonian fluid motion in stirred vessels. The Newtonian fluid used in this investigation is tap water and the non-Newtonian fluid is Blanose, which is a pseudoplastic. Laser Doppler anemometry, will be used to measure velocities in a non-intrusive way. Mean velocities and RMS velocities measured in water will be compared to the data measured by Venneker (1999). The measured LDA data is also used to determine the power spectral density function (“spectrum”) of the velocity fluctuations. The spectrum gives a deeper insight into the flow field because it shows how the kinetic energy is distributed over different scales (frequencies). The fulfilment of this goal will be a small contribution in acquiring general knowledge on the hydrodynamic behaviour of non-Newtonian fluid motion in stirred vessels.

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1.3 Outline of the thesis The thesis consists of five chapters. In Chapter 2, a short review is given on (non-)Newtonian fluids, turbulence, flow in stirred vessels and LDA. Chapter 3 describes the experimental setup and explains which data processing techniques were used. In chapter 4, the acquired mean velocity profiles, RMS velocity profiles and the power spectra are presented and discussed. Finally, in Chapter 5 conclusions are drawn and recommendations for experimental setup improvements and future research are given.

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2. Theoretic outline 2.1 Non-Newtonian fluid In Newtonian fluids there is a linear relationship between the velocity gradients and stresses. For example, the shear stress τyx is given by (van den Akker and Mudde 1996)

�

τ yx = µduxdy, (2.1)

where µ is the dynamic viscosity and ux the velocity component in the x-direction. Representatives of fluids with Newtonian behaviour are water and air. Because of the importance of these fluids in many technological applications, most experimental (and numerical) research in fluid dynamics focuses on Newtonian fluids. However in the process-industry many fluids are non-Newtonian. Non-Newtonian fluids do not have a linear relation between the velocity gradients and stresses. Power-law fluids are a special type of non-Newtonian fluids. The relationship between the velocity gradient and the shear stress for Power-law fluids is (van den Akker and Mudde 1996)

�

τ yx = −Kduxdy

n−1

⋅duxdy, (2.2)

where K is the flow consistency index and n is the flow behaviour index. When the value of n is smaller than 1, the fluid is called a pseudoplastic. It follows from eq. (2.2) that for a pseudoplastic the shear stress decreases with increasing velocity gradient. In daily life there are many fluids with this behaviour. Examples are ketchup, paint and toothpaste. When you squeeze toothpaste out of the tube you are increasing the shear stress, therefore the toothpaste comes out with ease because of the lower viscosity. When the toothpaste lands on the brush the viscosity increases because there is no more shear stress and the toothpaste lies firmly on the brush.

2.2 Turbulence The flow of an incompressible, Newtonian fluid is described by the Navier-Stokes equations. In vector notation these equations read (Hanjalic et al. 2007)

�

ρ{∂ v ∂t

+ ( v ⋅∇) v } = −∇p + µ∇2 v , (2.3)

where

�

ρ is the density of the fluid,

�

v is the velocity of the flow and Δp is the pressure gradient in the flow. The dimensionless number that is associated with turbulence is the Reynolds number. The Reynolds number is the ratio of the inertial forces and the viscous forces in the flow

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�

Re =inertialviscous

=ρ( v ⋅∇) v

µ∇2 v =ρ

V 2

Lµ V

L2=ρVL

µ, (2.4)

where ρ is the fluid density, µ the dynamic viscosity, V is a characteristic velocity and L the characteristic length scale. When the Reynolds number is below a, geometry dependent, critical number the flow is laminar, which means that the flow has a regular, layered structure. When the Reynolds number is larger than the critical number the flow will be turbulent, which means that it is irregular with temporal and spatial fluctuations in velocity and pressure. A Fourier analysis of the fluctuating velocity (resulting in a so-called power spectral density function) shows the distribution of the kinetic energy in the velocity fluctuations over different scales (frequencies). In case of turbulent flow the power spectrum is continuous with a wide range of scales.

2.3 Flow in stirred vessels Stirred vessels are widely used as reactor vessels in the process industry. Fluids are mixed by the impeller. The stirred vessel is dimensioned to the standard geometry in which most dimensions depend on the tank diameter. This standard geometry makes it possible to compare the research done in vessels that differ in size. The flow inside stirred vessels is very complex, many experiments have already been done in the past. Most experimental and numerical research focused on turbulent flow of Newtonian fluids (e.g. Lee and Yianneskis 1998). But also the flow of non-Newtonian fluids has been investigated in the past (e.g. Metzner and Otto 1957). A deeper insight of the flow in stirred vessels is achieved by looking at flow patterns (fig. 2.1). These flow patterns where obtained by the help of tracer particles.

Figure 2.1: Gross flow patterns for radial flow in the standard stirred vessel geometry (Tatterson 1991). For flow in stirred vessels the characteristic velocity is the impeller speed, which scales with ND, where D is the impeller diameter and N is the number of impeller revolutions per second i.e., impeller frequency. Using the impeller diameter D as

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characteristic length scale the Reynolds number for Newtonian fluids in stirred vessels becomes (Venneker 1999)

�

Re =ρND2

µ. (2.5)

Because non-Newtonian fluids have a shear dependent viscosity, a characteristic viscosity or a shear rate must be defined. Metzner and Otto (1957) found that in the laminar regime the average shear rate

�

˙ γ av is linearly dependent on the impeller speed

�

˙ γ av = ksN, (2.6) where ks is the proportionality constant, which depends on the specific tank, and impeller geometry but is independent of the fluid, as long as it is inelastic (Ducla et al. 1983). The value of ks is determined by matching the power number Po for Newtonian and non-Newtonian fluids in the laminar regime. With this average shear rate the Reynolds number for Power-law fluids becomes (Venneker 1999)

�

Rea =ρN 2−nD2

Kksn−1 . (2.7)

Although the Metzner-Otto approach has been used in the transitional and turbulent regime, a much higher average shear rate can be expected in these cases (Van ‘t Riet 1975, Wichterle et al. 1984, Wichterle et al. 1985). Reynolds numbers in these experiments are however only used to indicate the flow regime, and not used for scale-up correlations. Therefore, even in the turbulent regime Rea is still calculated with eq. (2.7).

2.4 Laser Doppler Anemometry Laser Doppler anemometry (LDA) measures the velocity of small tracers that are distributed in a transparent fluid. If these tracers are sufficient small than it may be assumed that the velocity of the tracer is equal to the fluid velocity. The main advantage of LDA over conventional techniques, like a (glass fibre) probe or a filament, is that there is no physical object in the flow so that the flow is not disturbed during the measurement. The basics of LDA lie in the Doppler-shift. When light is scattered by a small moving particle a frequency shift will occur. This frequency shift is called the Doppler-shift and it is directly related to the particle velocity.

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Figure (2.2) shows a basic dual beam LDA system. A laser is used to produce a plane light wave with frequency f0. The single beam is split and the resulting beams are crossed (with crossing angle θ) to form a measurement volume at the intersection. A tracer particle moving through the measurement volume scatters light from both incident laser beams. The detector (usually a photomultiplier) receives the scattered light and its output signal oscillates with the Doppler frequency fD, which for the dual beam LDA system is given by

�

fD = v ( e i1 −

e i2)λ0

=2sin(θ 2)

λ0 v sinα, (2.8)

where

�

v is the velocity of the particle,

�

e i1 and

�

e i2 are the directional vectors of the two crossing laser beams, λ0 is the wavelength of light coming from the laser and

�

α is the angle between the velocity vector and the x-axis. A basic dual beam LDA thus measures the component of the velocity in the y-direction.

Figure 2.2: Schematic view of a dual beam LDA setup (Tummers 1999).

The detector used during the LDA-measurements in this investigation is a photomultiplier. The output signal of the photomultiplier is proportional to (Tummers 1999)

�

y(t) ~ cos(2πfD + Δϕ), (2.9) where Δϕ is the phase difference between the two light waves. This phase difference is assumed to be constant. This assumption is allowed because a laser is a coherent light source. From eq. (2.8) we see that the Doppler-shift can be positive or negative depending on the value of α. However the photomultiplier cannot distinguish between positive and negative values of the Doppler-shift because cos(-fD)=cos(fD). In its basic form LDA cannot determine the sign of the velocity. To make it possible to distinguish between positive and negative values a frequency pre-shift between the two incident waves is applied. With a Bragg cell one of the beams is shifted by a constant value fs. Assuming fs<<f0, then we have for the Doppler frequency (Tummers 1999)

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�

fD = f s +2sin(θ 2)

λ0 v sinα. (2.10)

In figure 2.3 this relationship had been visualized. If the shift frequency is chosen larger than the Doppler frequency that correspond to the smallest velocity in the flow (vmin), each value of |fd| is uniquely related to one velocity value. Due to this technique it is possible to determine the direction of the velocity.

Figure 2.3: The effect of frequency shift on the relationship between the particle velocity and the frequency of the photomultiplier output signal (Tummers 1999).

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3. Experimental setup and data processing 3.1 Stirred vessel The stirred vessel used in this investigation has an internal diameter T=273 mm and a height of 338 mm. The vessel has the standard configuration, which means that almost all dimensions are some constant fraction of the tank diameter T. The tank is stirred by a six-bladed disc impeller with diameter D=T/3 which is placed at a clearance height C=D. During the measurements the vessel was filled up to a height H=T. At this level, there was a free surface. Four baffles where placed on the inside of the wall of the vessel to create turbulent flow near the wall. All the dimensions of the vessel and their standard configuration are shown in table 3.1. Table 3.1: Dimensions of the vessel (in mm) and the standard configuration.

Vessel T 273

baffle width b=T/10 27

baffle thickness 5

Impeller D=T/3 90

blade height w=T/15 18

blade length lb=T/12 23

blade thickness db 2.5

disc diameter T/4 68

disc thickness 2.5

shaft diameter 15

Figure 3.1: Geometry of the stirred vessel with a six-bladed turbine. Side view (left) and top view (right). To minimize variations in optical distortion during traversing of the vessel with respect to the LDA setup, the cylindrical vessel was placed inside a square container

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with the space between the container and the vessel filled with tap water. Both the vessel and the container where made of Perspex.

3.1 LDA setup Figure 3.2 shows a schematic view of the experimental setup. A diode pumped solid-state (dpss) laser, manufactured by CNI, was used for all measurements. The laser emits 500mW of green light at a wavelength of 532 nm. The laser beam is reflected by two mirrors to a Bragg cell, which splits the beam and pre-shifts the frequency of one of the beams. It is possible to vary this pre-shift frequency. During the experiments a frequency shift of fs=0.8 MHz was used. After the Bragg cell, the beams pass through a lens and are then focused in the vessel to form a measurement volume.

Figure 3.2: A schematic view of the experimental setup. The laser beams were orientated in such a way that the radial velocity component is measured in all measurements. The dashed line in fig. 3.1 indicates at which locations the measurements were done. LDA measurements require particles to produce Doppler bursts. When measuring in tap water the use of seeding might not be necessary because pollution of small particles is large. However, when there were not sufficient Doppler bursts, seeding is used during this investigation. The seeding used during this experiment are hollow glass beads with a mean diameter between 8 µm and 12 µm. When a particle is passing through the measurement volume the scattered light is detected by a photomultiplier. The photomultiplier converts the light into a voltage and sends the signal to the LDA signal processor (IFA 750). For each particle the IFA digitizes the analog Doppler signal and determines the Doppler frequency (velocity), the arrival time and the transit time. The transit time is the duration of a Doppler burst and is used in the processing of the velocity data. A computer is used to control the IFA settings and to save the Doppler data. The signal coming from the

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photomultiplier is also visualized on an oscilloscope. With the oscilloscope it is possible to see whether the Doppler bursts are of high quality or noisy. Two working fluids are considered in this investigation. Besides tap water a pseudoplastic fluid is used during the experiments. The used fluid is made by dissolving Blanose in water. The mixture contains 0.1 weight percentage Blanose. Lower weight percentages showed no clear pseudoplasticity, while with weight percentages higher than 0.2% the apparent viscosities were too high to reach the turbulent regime (Venneker 1999). The precise rheological properties for the used mixture were not determined. Therefore only an approximation of the Reynolds number can be made, using the properties reporter by Venneker (1999). The used values are, for the flow consistency index K=31.6(x10-3[kgsn-2/m], the flow behaviour index n=0.77 and the proportionality constant ks=11.5. The impeller frequency during the measurements in tap water was 5 Hz. This impeller frequency results in a Reynolds number Re=40.5x103. The Reynolds number that Venneker (1999) used was Re=64.8x103. To achieve this Reynolds number in the available vessel an impeller frequency of 7.7 Hz was needed. However, an impeller frequency of 5 Hz turned out to be the upper limit, because otherwise the water would be splashed out of the vessel. Therefore, achieving the same Reynolds number as Venneker (1999) was not possible and the measurements were preformed at the highest possible impeller frequency of N=5 Hz. The impeller frequency during the measurements in Blanose was 2.43 Hz. This impeller frequency results in an approximate Reynolds number of Re=1.87x103. At this impeller frequency there was significant motion of the fluid without fluid being splashed out of the vessel. Therefore, this impeller frequency was chosen.

3.3 Data processing The acquired data from LDA can be processed in several ways. In this experiment the mean velocity, the root-mean-squared (RMS) velocity, the autocorrelation function (acf) and the spectral density function (sdf) were of interest.

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3.3.1 Bias correction There are a number of biasing effects in LDA measurements. The most important of these effects is velocity bias. This is the bias effect due to the fact that particles with a higher velocity have a (relatively) high probability to traverse the measurement volume. Therefore, more particles with a high velocity, with respect to particles with a low velocity will be detected. This effect has to be taken into account during the processing of the data. The unbiased mean velocity and the variance are calculated as weighted averages

�

u =uiω ii=1

N∑ω ii=1

N∑,

and

�

u'2 =u

i=1

N∑ 'i2ω i

ω ii=1

N∑,

where the subscript i denotes the ith velocity sample, N is the total number of velocity samples and ω is the weighting factor that takes the effects of the velocity bias into account. In this experiment the transit time of the particles is used as a weighting factor. The weighting factor is given by (Tummers 1999)

�

ω i = tri, where tri is the transit time of the ith particle.

3.3.2 Spectral analysis of LDA data A deeper and more quantitative insight in the velocity fluctuations is provided by the autocorrelation function (acf) and its Fourier transform, the spectral density function (sdf). The autocorrelation function is a measure how well a signal resembles a time-shifted version of itself. For a continuous signal the autocovariance function is defined as

�

R(τ) = u'(t)u'(t + τ ), where u’ is the velocity fluctuation (

�

u'(t) = u(t) − u ) and τ is the time-shift. Because the auto covariance is the mean of a product it is easy to see that the autocovariance function is an even function

�

R(τ) = R(−τ). The autocorrelation function is defined as

�

ρ(τ) =R(τ )R(0)

=u'(t)u'(t + τ)

u'2,

which is the autocovariance divided by the variance. Like the autocovariance function the autocorrelation function is also an even function and ρ(τ)=1 for τ=0. In this experiment the acfs are calculated for discrete signals. For an equidistant time series containing N samples, which are sampled with frequency, 1/Δt the auto covariance function can be calculated by (Priestley 1981)

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�

R(kΔτ) =1

N − ku(lΔt)u((l + k)Δt),

l=1

N −k

∑ with k = 0,1,...,N-1.

The discrete autocorrelation follows as

�

ρ(kΔt) =R(kΔt)R(0)

, with k = 0,1,...,N-1.

The estimation of the acf and sdf is normally done by equidistant sampling. However, in LDA the time between samples is random. This requires special spectral estimators for randomly sampled data. The approach which is used is called “slotting technique” (Tummers 1999) which computes a discretized version of the acf. The time lag axis is divided into equidistant intervals (the so-called slots) of width Δτ. A relatively small maximum time shift is considered when determining the autocorrelation function. The slot width is

�

Δτ =τmaxM,

where τmax is the maximum time shift and M the number of slots. Cross products of velocity samples uiuj with a difference in arrival times τ=ti-tj greater than τmax are ignored because these do not contribute to the acf for τ < τmax. The cross products with a difference in arrival time smaller than τmax are grouped in the kth lag time interval where k is determined from

�

k =ti − t jΔτ

+12

⎛ ⎝ ⎜

⎞ ⎠ ⎟ .

For every slot the variance is calculated based on the points lying in this specific slot. This process is called local scaling. The locally scaled version of the slotted autocorrelation function reads (Tummers 1999)

�

˜ ρ ={uiu j}(kΔτ)∑

{ui2}(kΔτ ) {u j

2}∑∑ (kΔτ ), with k = 0,1,...,M-1.

The numerator represents the slotted autocovariance function and in the denominator the function for the locally scaled variance is found.

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The normalized spectral density function is defined as the Fourier transform of the autocorrelation function. The discrete version of this transformation looks like (Tummers 1999)

�

S(ω) =Δτπ

12ρ(0) + ρ(kΔτ)W (kΔτ)cos(kωΔτ )

k=1

M −1

∑⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,

where W(t) is a variable window function. In this work a Tuckey-Hanning window was used

�

W (τ) = 1/2 +1/2cos(πττm),

with τ < τm. The width of this window can be conveniently varied with frequency by varying the value of τm with frequency as

�

τm (ω) = κ2πω,

where κ is a smoothing parameter.

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4. Experimental results This section first presents the mean velocities and root-mean-square (RMS) velocities as a function of the distance from the impeller for water and. Also, a comparison with the results of Venneker (1999) is made. Finally, the different power spectral density functions are presented for measurements in water and in Blanose.

4.1 Mean velocity profiles in water Figure 4.1 shows the mean velocities measured at different radial positions in the vessel. At each location approximately 60000 samples were acquired at a mean data rate of 200 Hz. The full line represents the (biased) mean velocity while the dashed line denotes the (unbiased) transit time weighted results. The measured mean velocity profiles in water can be compared to literature data. In this case data acquired by Venneker (1999). Figure 4.1 also shows a thick line which represents a fit to the mean velocity measured by Venneker (1999).

Figure 4.1: (biased) mean velocity profiles in water. It is clear that the data measured in the present investigation do not accurately reproduce the data of Venneker (1999). The reason for this is the different size of the vessels. The vessel that Venneker (1999) used was about 1.6 times larger in diameter than the one used in this experiment. Therefore, the impeller frequency must be (1.6)2=2.56 times larger to measure at the same Reynolds number. As mentioned in section 3.2, the required impeller frequency could not be achieved because water splashes out of the vessel. But a different Reynolds numbers alone, is not very likely to bring such large differences in the result. However, when the impeller is rotating at a frequency of 5 Hz the water stays in the vessel but there appear whirlpool like,

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vortices on the water surface. These vortices move down closely around the shaft. This phenomenon is called macro instability (MI), which is the topic of many investigations see, e.g. Nikiforaki et al. (2003). Marco instabilities were probably not present in the vessel Venneker (1999) because of the larger diameter of the vessel. Therefore, macro instabilities are most likely to create the deviation from the fit found by Venneker (1999).

4.2 RMS velocities The calculation of the RMS velocity is done using the same data which is used for the calculation of the mean velocity. Therefore, the results for the RMS velocity are also influenced by the MI. Figure 4.2 shows the results for the biased and biascorrected RMS velocity. A comparison with Venneker (1999) is useless because of the presence of MI. However, the figure gives some insight in the behavior of the fluid during the experiments. The turbulence kinetic energy k in fluid motion is defined by

�

k =12ρ(u'2 + v '2 + w'2), (4.1)

where ρ is the density of the fluid,

�

u'2 ,

�

v'2 and

�

w'2 are the variance of the radial, axial and tangential velocity component, respectively. In this investigation only the radial velocity component is measured and the axial and the tangential component are unknown. Thus, a precise calculation of the kinetic energy is impossible, but it is expected that the order of magnitude of the variance of the three velocity components do not differ much. Therefore, the region with the eddies of the highest energy is expected to be between r/R=1.25 and r/R=1.5, where the variance of the radial velocity component has its maximum.

Figure 4.2: RMS radial velocities in Water.

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4.3 Power spectral density functions In turbulent flow the velocity varies in space and time. The total energy in the velocity fluctuations is given by the turbulence kinetic energy (eq. 4.1). The power spectral density function describes how the energy of one of the components is distributed over different frequencies. In a turbulent flow the turbulence kinetic energy does not accumulate in a number of discrete frequencies. Instead there will be a wide range of active scales (frequencies) such that the sdf is a broad continuous function. When looking at the sdfs, certain effects (peaks) appear generally. These peaks appear at specific frequencies and have specific physical meaning.

Figure 4.3: sdf measured at position r/R=1.44 in water. The spectrum for water in fig. 4.3 shows four sharp peaks. Peak III corresponds to the blade frequency. The blade frequency (in rad/s) is the frequency of the impeller times the number of blades on the impeller. In the setup for water, where the impeller frequency is 5 Hz, this gives: 5x2xπx6=188.5 rad/s. Peak I occurs at a frequency of approximately 31 rad/s, which coincides with the impeller frequency (5x2xπ=31.4 rad/s). The appearance of this peak implies that one of the impeller blades is different from the others. A close inspection of the impeller revealed that one blade was longer than the others. Another peak (II) is found at twice the frequency of peak I. This implies that another blade is also different. Further inspection of the impeller showed that this is indeed the case. A fourth peak is found at a frequency of approximately 350 rad/s. This corresponds to twice the blade frequency. Measurements in Blanose also showed this high frequency peak (see fig. 4.4). The impeller speed during the measurements in Blanose was 2.43 Hz and the high frequency peak appeared at 185 rad/s (≈2.43x2xπx6x2=183).

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Figure 4.4: sdf for Blanose, measured at position r/R=1.11. Other investigations on stirred vessels (e.g. Nikiforaki et al. (2003) and Lee and Yianneskis (1998)) did not report such a high frequency peak. However, there are a few possible, physical explanations for this high frequency peak.The high frequency peak could be due to:

1. Macro instabilities, which are caused by the high impeller frequency. These MI rotate with a characteristic fundamental frequency. Investigation on MI by Nikiforaki et al. (2003) revealed a linear relationship (independent of the impeller design) between the impeller speed and the fundamental frequency of the rotating MI, f=aN where a is approximately 0.015. Therefore, it is very unlikely that MIs are the cause of appearance of the high frequency peak in the sdfs of water and Blanose. Also, the measurements on Blanose were done with a much lower impeller frequency of N=2.43 Hz. Macro instabilities where only visible in the vessel as from an impeller frequency of 3 Hz and higher. In spite of the absence of MI, the high frequency peak is observed for measurements on Blanose

2. Measurement error due to the use of an unstable solid-state laser. It is known that the solid-state laser sometimes produces artificial “velocity

data” at 50 Hz. However, when this happens the “artificial data” are easily detected in a velocity histogram as an unrealistic second peak. Such peaks did not occur in the present investigation suggesting that no “artificial data” were produced by the solid-state laser. Also, the high frequency peak does not occur at every distance from the impeller. Figure 4.5 shows that the peak does not occur in Blanose at r/R=1.44 but the peak does occur in Blanose at r/R=1.11 (see fig. 4.4). If the peak would be due to the unstable solid-state laser, then the peak has to appear at every distance from the impeller.

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Figure 4.5: sdf for Blanose, measured position r/R=1.44.

3. A vortex, which moves along with the impeller, creating a high frequency peak. To verify whether or not this third explanation makes sense an other measurement technique is needed. A technique, which can visualize the flow in stirred vessels such as Particle Image Velocimetry (PIV). Further investigation on this matter with PIV may reveal the origin of this high frequency peak.

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The next figures contain the sdfs of all the acquired data. In every plot also a line with a slope of -5/3 is drawn as a reference.

23

24

25

26

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4.4 Local isotropy in the flow Figure 4.6 shows the energy spectra of the radial velocity component in Blanose obtained at r/R= 1.89, 2.11 and 2.33. The spectra are plotted on double-log axes. A straight line with slope -5/3 is also drawn. This line represents the energy distribution predicted by Kolmogorov from a dimensional analysis for the inertial sub range. The -5/3 slope in the spectrum does not prove the existence of local isotropy, but it has been employed to lend confidence to the approximation of local isotropy (Kresta and Wood 1993). In figure 4.7 the energy spectrum of the radial velocity recordings in Blanose at r/R= 1.11, 1.22, 1.44 and 1.66 is plotted on double-log axes, and two straight lines with slopes -5/3 and -9/6 are also drawn. Clearly the slope is not -5/3, instead the slope is -9/6. Thus, isotropy is only observed for spectra recorded at a

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relative large distance from the impeller. This means that at a certain distance eddies in the flow receive more energy from the next larger eddies through the cascade process and less directly from the mean motion. The great advantage of this isotropy is that it allows neglecting of viscosity when using Computational Fluid Dynamics (CFD) for Blanose at sufficiently high Reynolds numbers, see Lee and Yianneskis (1998).

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Figure 4.6: Energy spectrum of the radial velocity in Blanose at r/R= 1.89, 2.11 and 2.33.

Figure 4.7: Energy spectrum of the radial velocity in Blanose at r/T= 1.11, 1.22, 1.44 and 1.66.

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5. Conclusions and recommendations

5.1 Conclusions A one-component LDA was used to measure the radial velocity component at various distances from the impeller in a stirred vessel filled with tap either water or a water/Blanose solution. The measured velocity time series were processed to mean velocities, RMS of the velocity and power spectra. From a comparison between the mean velocities (and RMS velocities) of the present investigation and those reported by Venneker (1999) it became clear that the vessel used in this investigation is not suitable for measurements at high Reynolds numbers because of the strong influence of macro instabilities (MIs). The power spectral density functions (sdfs) were found to be wide and continuous as in turbulent flow. The sdfs showed a number of peaks that could be related to the blade frequency, or to fractions of this frequency due to imperfection of the impeller. However, the measured spectrum also showed a peak at higher frequencies in some of the measurements. At this moment there is no clear explanation for the appearance of this high frequency peak so further research on this phenomena is desired. The sdfs also revealed the isotropic behavior of the flow at relative large distances from the impeller. This isotropic behavior makes it possible to neglect the viscosity when using Computational Fluid Dynamics (CFD) for Blanose at sufficiently high Reynolds numbers.

5.2 Recommendations There are recommendations regarding (1) the improvement of the experimental setup and (2) suggestions for further research.

5.2.1 Experimental setup improvements The experimental setup used during this Bachelor project is limited in its use. There are two major drawbacks of the setup.

• The size of the vessel. Because the Reynolds number (eq. 2.5) is proportional to the impeller frequency N and the square of the diameter of the vessel D, a high Reynolds number in a small vessel can only achieved with a high impeller frequency. However, this high impeller frequency produces macro instabilities, which influence the measurements. A solution to this problem is to use a larger vessel.

• The solid-state laser, which is used in this experiment, shows some stability issues. It is possible that these instabilities have influenced the measurements. Therefore it is recommend using a more stable laser (e.g. an Argon laser) for further investigations.

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5.2.2 Future research The results of this investigation raised more questions than they gave answers. Some topics for future research are listed below.

• New measurements of mean velocity, RMS velocity and power spectra have to be formed at a lower Reynolds numbers for which there are no macro instabilities. This way it is possible to see whether the data fits to results obtained from other researches, and whether the MI is responsible for the high frequency peak. Alternatively, measurements can be conducted in a new, larger vessel to allow a comparison with the results of Venneker (1999)

• Measurement of sdfs at a lower Reynolds number or a larger vessel to investigate whether the MIs are the origin of the high frequency peak in de sdfs.

• Measurement of sdfs with an other (i.e. more stabile) laser to investigate whether the laser is the cause of the high frequency peak in the sdfs.

• Research on stirred vessels with the use of time resolved PIV to see whether the high frequency peak is caused by some sort of tip vortex.

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Appendix

Power spectral density functions Table A.1: measuring time and number of samples of the measurements in water Position Measuring time Number of samples R 1h13min 3 359 888 R + 2.0 cm 1h9min 4 559 848 Table A.2: measuring time and number of samples of the measurements in Blanose Position Measuring time Number of samples R 1h17min 5 279 824 R + 0.5 cm 39min 3 179 894 R + 1.0 cm 1h30min 4 079 864 R + 1.5 cm 1h12min 2 999 900 R + 2.0 cm 1h6min 4 919 836 R + 2.5 cm 57min 1 319 956 R + 2.9 cm 1h30min 5 039 832 R + 3.5 cm 1h34min 5 939 802 R + 4.0 cm 48min 4 559 848 R + 4.5 cm 2h10min 5 099 830 R + 5.0 cm 1h22min 4 199 860 R + 5.5 cm 1h13min 2 999 900 R + 6.0 cm 1h35min 5 939 802