Chapter IVChapter IVChapter IVChapter IV 4.

(1)

Gas-liquid hydrodynamics in Taylor Flows with complex liquids Chapter IV - 62 -

Chapter IV

4. Results

Here follow the results obtained during the work's period are reported. The university of Birmingham made available different kinds of equipments that permitted a in depth study of the examined problem. Substantially this work starts with the characterization of fluids, particularly with the research of viscosity and superficial tension of complex liquids. That was useful mainly to have a more accurate idea of the unknown nature of the fluids considered. Besides in this chapter it is possible to find the results about the Mastersizer. In fact an analysis of particles was effectuated to clear out the dimension of the particles of alumina used in the PIV. After that the results obtained with HSV and PIV are reported in order to characterize the flow inside the capillary and the nature of fluid flow of particles used with the complex liquids.

The complex liquids used are:

• Water and polymer microspheres red fluorescing (3µm)

• Water and colloidal (ludox) 10%vol solution (ludox) and polymer microspheres red (3µm)

• Water and glycerol 50% wt solution and alumina (3µm)

(2)

- 63 -

4.1. Characterisation of fluids

In this chapter it’s analyzed particularly the characterization of fluids, showing the data obtained from the different used equipments.

4.1.1.Rheology results

4.1.1.1. Advanced rheometer

With this equipment is examined two kinds of solution. The first solution is water with 10% vol of ludox and the second one is water with 50% wt of glycerol.

4.1.1.1.1. Ludox solution

The peculiarity of this solution of water and ludox has not a non-Newtonian behavior also ludox has it. It showed below in the graphs. In appendix B there are more information about ludox.

Graph 1 viscosity vs time for ludox solution 10% per vol

The method used to find these graphs is explained in the Appendix A. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 20 40 60 80 v is co si ty ( P a s) time (s)

(3)

Gas-liquid hydrodynamics in Taylor Flows with complex liquids Chapter IV

- 64 - It’s showed the graph with the trend of shear stress vs shear rate.

Graph 2 shear stress vs shear rate of ludox solution 10%per vol

4.1.1.1.2. Glycerol solution

This kind of fluid is used to increase the viscosity of the fluid. This increase improve the relaxation time that permit a slow sedimentation of alumina particles inside the tank. The used fluid has a Newtonian behavior like it’s showed in figure.

Graph 3viscosity vs time for glycerol solution 50% per wt 0 2 4 6 8 10 12 0 50 100 150 200 250 300 sh e a r st re ss ( P a ) shear rate(1/s) 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48 0.52 0.56 0 10 20 30 40 50 60 V is co si ty ( P a s) Time (s)

(4)

- 65 -

Graph 4shear stress vs shear rate of ludox solution 10%per vol

0 1 2 3 4 5 6 0 100 200 300 400 500 600 700 800 S h e a r S tr e ss ( P a ) Shear Rate 1/s

(5)

- 66 -

4.1.2.Surface tension

In order to obtain a clear characterisation of the fluid the surface tension of the same ludox solution considered in the previous paragraphs has been analysed.

4.1.2.1. Torsion balance results

This graph represents the variation of surface tension for different solution of ludox with water.

Graph 5 surface tension vs % of ludox solution per vol

This graph represents the variation of surface tension for different solution of ludox with water.

Graph 6 surface tension vs % per vol of glycerol

0.058 0.059 0.06 0.061 0.062 0.063 0.064 0 2 4 6 8 10 12 14 16 su rf a ce t e n si o n ( N /m ) % vol of ludox 0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0 10 20 30 40 50 60 70 su rf a ce t e n si o n ( N /m ) % wt of glycerol

(6)

- 67 - The graphs are obtained with the torsion balance. Each value of a point is an averange between 10 values and the temperature of experiments is about 20°C. For the glycerol solution the surface tension is almost constant because the range is 0.0583 N/m and 0.063.

4.2. Malvern Mastersizer results

To confirm the information reported on the label of the product the tests are effectuated to verificate the real size of the alumina's particles. This test are made with the Mastersizer, the functioning was treated in the chapter on the Materials and Method. The alumina particles was bought from Sigma-Aldrich. They are considered two kinds of particles, 110 µm and 3 µm.

4.2.1. 110µm alumina particles

(7)

Gas-liquid hydrodynamics in Taylor Flows with complex liquids

The results obtained are:

Fig.44 The tables of the results obtained

liquid hydrodynamics in Taylor Flows with complex liquids Chapter IV

The tables of the results obtained

liquid hydrodynamics in Taylor Flows with complex liquids

(8)

- 69 - This graph shows the distribution of the diameter of particles of the examined sample.

Fig.45 Distribution of the diameter of particles

From the obtained data the percentage per volume mayor of the particles has size as expected from the label.

4.2.2. 3µm alumina particles

The input data inside the program of the Mastersizer are : Statistics Graph (1 measurements)

0.01 0.1 1 10 100 1000 10000 Particle Size (µm) 0 1 2 3 4 5 6 7 8 V o lu m e ( % )

(9)

The results obtained are:

Fig.46 The tables of the results obtained

The tables of the results obtained

(10)

- 71 -

Fig.47 Distribution of the diameter of particles

From the obtained data the percentage per volume major of the particles has not the size as expected from the label. This difference is originated from the agglomeration of the particles. the data obtained don't change the value of relaxation time also if the size of the particles are major.

Statistics Graph (1 measurements)

0.01 0.1 1 10 100 1000 10000 Particle Size (µm) 0 1 2 3 4 5 6 V o lu m e ( % )

(11)

4.3. HSV results

4.3.1.Flow pattern

During the first part of this work the HSV is used to discover the different flows inside the capillary. Substantially the liquids used are water

solution. Below the obtained pictures are showed Bubble Flow

Taylor Flow

uring the first part of this work the HSV is used to discover the different flows inside the Substantially the liquids used are water, water and ludox and glycerol 50% per weight solution. Below the obtained pictures are showed :

Fig.48

Fig.49

- 72 - uring the first part of this work the HSV is used to discover the different flows inside the and glycerol 50% per weight

(12)

Churn Flow

Film Flow

Fig.50

Fig.51

(13)

- 74 -

4.3.2. Flow conditions

With the HSV the flow conditions are evaluated:

the velocity of bubble is evaluated ,given that for each second the camera did 250 frames, and behind the capillary was put the meter. The high number of frames permitted to see the movement of the bubble during the ∆t:

where ∆t is ௡௨௠௕௘௥ ௢௙ ௙௥௔௠௘௦ ௧௢௢௞ ௜௡௧௢ ௖௢௡௦௜ௗ௘௥௔௧௜௢௡ ௡௨௠௕௘௥ ௢௙ ௙௥௔௠௘௦ ௣௘௥ ௦௘௖௢௡ௗ .

The liquid specific velocity is evaluated with the volumetric flow rate of liquid, then this flow was divided to the cross section of the capillary.

UL=

ொಽ ഏ ర஽

మ

With this values the different patterns are found and below the graph shows them.

(14)

This table summarizes the flow conditions for all samples: SAMPLES 1)WATER CONDITIONS GAS VELOCITY (m/s) 0.18 LIQUID VELOCITY (m/s) 0.02 VISCOSITY (Pas) 0.001 SURFACE TENSION (N/m) 0.07 DENSITY (g/l) 1

With this conditions the found flows bubble flow condition for water

This table summarizes the flow conditions for all samples:

)WATER 2)WATER WITH

LUDOX SOLUTION 3)WATER AND GLYCEROL 50%PER VOLUME WITH ALUMINA (3μm) 0.2 0.4 0.018 0.008 0.003 0.008 0.072 0.07 1 1.1

With this conditions the found flows are: bubble flow condition for water

Fig.52 - 75 - )WATER AND GLYCEROL 50%PER VOLUME WITH 4)WATER AND GLYCEROL 50%PER VOLUME WITH ALUMINA (110μm) 0.4 0.008 0.008 0.07 1.1

(15)

bubble flow condition for water and ludox

bubble flow condition for glycerol and water solution

bubble flow condition for water and ludox

Fig.53

bubble flow condition for glycerol and water solution

Fig.54

(16)

The distance between the bubbles have to be enough big( about 3 4 mm) to allow to have a big zone to study with the PIV. The shape of bubbles changes enough with different liquids, in fact with glycerol solution the shape is more circle than the water sol

of flow change specially the liquid velocity and the bubble velocity.

Samples Bond Number

1 2 3 4

The boundary between the regimes is determined properties and gas superficial velocity .

WeGs =ρGu 2

Gsd/ γ.

SAMPLES WEBBER NUMBER

1 2 3 4

The surface tension dominated regime was delimited by delimited by WeGs >20

A reasonable definition of the term capillary might be obtained by requiring the dominance of surface tension forces over buoyancy.

dimensionless bubble diameter in the diagonal direction can be number as

The distance between the bubbles have to be enough big( about 3 4 mm) to allow to have a big zone to study with the PIV. The shape of bubbles changes enough with different liquids, in fact with glycerol solution the shape is more circle than the water solution. With glycerol the conditions of flow change specially the liquid velocity and the bubble velocity.

Capillary Number Reynolds Number

1.26 0.00257

1.225 0.00833

1.386 0.04571

between the regimes is determined by the Weber number, which properties and gas superficial velocity .

WEBBER NUMBER Surface tension dominated regime

0.00166 YES 0.00200 YES 0.008228 YES 0.008228 YES

The surface tension dominated regime was delimited by WeGs <1 and the inertial regime was

A reasonable definition of the term capillary might be obtained by requiring the dominance of surface tension forces over buoyancy. Using the asymptotic values of 1.2 and 0.7, the bubble diameter in the diagonal direction can be correlated against the Capillary

- 77 - The distance between the bubbles have to be enough big( about 3 4 mm) to allow to have a big zone to study with the PIV. The shape of bubbles changes enough with different liquids, in fact ith glycerol the conditions

Reynolds Number 60 18 3.3 3.3

by the Weber number, which based on gas

Surface tension dominated regime

1 and the inertial regime was

A reasonable definition of the term capillary might be obtained by requiring the dominance of ymptotic values of 1.2 and 0.7, the correlated against the Capillary

(17)

- 78 - Graph 8 db square/dchannel vs Capillary Number

The method developed by Bretherton also gave an expression for the pressure drop over the bubble. In fact, the excess bubble velocity, film thickness and pressure drop are all related.

Graph 9 Δp vs Capillary Number

0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 0 0.01 0.02 0.03 0.04 0.05 db s q u a re /d ch a n n e l Capillary Number 0 5 10 15 20 25 0 0.01 0.02 0.03 0.04 0.05 Δ p ( N /m ^ 2 ) Capillary Number

(18)

4.3.3.Lenght

The gas length and the liquid length are showed below:

Graph 10 Lgas/D vs Re/Eo2

The Graph 10 shows the gas length over the diameter, as fuction of Re and Eo numbers, calculated according to the relation suggested by Laborie et al. (1998)

0 0.5 1 1.5 2 2.5 3 3.5 4 0 Lg a s/ D

The gas length and the liquid length are showed below:

shows the gas length over the diameter, as fuction of Re and Eo numbers, calculated according to the relation suggested by Laborie et al. (1998) where :

100 200 300

Re/Eo2

- 79 - shows the gas length over the diameter, as fuction of Re and Eo numbers, calculated

(19)

Graph 11 Lliq/D vs R 1/ReEo

Similarly Graph 11 shows the length of

The table shows the data obtain from the correlations: Sample 0 0.5 1 1.5 2 2.5 3 3.5 4 0 Ll iq /D

shows the length of liquid where:

The table shows the data obtain from the correlations:

Sample Lgas/D Lliq/D

1) 3.6480 0.879823 2) 1.77030 3.215383 3) 0.52036 3.508598 4) 0.52036 3.508598 0.001 0.002 0.003 0.004 1/ReEo

- 80 - 0.005

(20)

- 81 - These are the data obtained from the experiments:

Sample Lgas Lliq

1) 0.009 0.013

2) 0.006 0.015

3) 0.003 0.008

4) 0.003 0.008

The data obtained are compared with the correlation's data, below the graphs are showed:

Graph 12 gas length from correlation and gas lenght from experiments

0 0.002 0.004 0.006 0.008 0.01 0.012 0 1 2 3 4 5 Lg a s (m ) number of sample

gas length from correlation gas length from experiments

(21)

- 82 -

Graph 13 gas length from correlation and gas lenght from experiments

There are same difference between the values of correlation and that of experiments because the correlation is for circular capillary and the liquid is water.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1 2 3 4 5 Ll iq ( m ) number of sample

liquid length from correlation liquid length from experiment

(22)

- 83 -

4.4. PIV results

In this chapter are presented the results of PIV, in particular in the first part of this chapter are shown the pictures of PIV and after these are elaborated to determine the various greatness with Tecplot. The particle velocity in a still air when released from rest can be derived as (based on Newton’s 2nd law): p p p D p p p m g F m g d V dt dV m = − = −3

πµ

with

V

_p

(

t

=

0 )

=

0

Thus,

(

1 )

18

/ 2 p t p p p

e

g

d

V

τ

µ

ρ

−

=

where

µ

ρ

τ

18

2 p p p

d

=

= particle relaxation time.

Thus, particle relaxation time is the characteristic time for particle to transit from one state to another state. Because the particle relaxation time is generally very small, the particle dynamics are generally assumed to be in equilibrium. That is, the transition period is generally not taken into account for small particles. The horizontal distance particle traveled when injected horizontally into a still air is

dt dx d V d F dt x d m D p p p p p p 2 3

πµ

3

πµ

2 − = − = − =

with

x

_p

(

t

=

0 )

=

0

is the particle position, thus

)

1 (

)

1 (

18

/ / 2 p p t po p t po p p p

V

e

V

e

d

x

τ

µ

ρ

− −

−

=

−

=

where

V

_po is the initial particle velocity.

Therefore, the maximum traveling distance or the stop distance S is

S

=

τ

_p

V

_po. Note that the above equation is only valid for Stokes flow. Mercer (1973) derived the following approximate equation within 3% difference for initial Reynolds number up to 1500 as

1/ 3 1/ 3 0 0

Re

6 arctan

6 



ρ





=



−







ρ









d p g

S

(23)

- 84 - In this study the τ used to classify the different behaviours of the flow is:

given that the different density are considered, in fact during this work the complex liquids are used.

In particular :

• If τ < 10-3: 10-8 , the particles follow the liquid’s flow

(24)

- 85 -

4.4.1.Pictures of water and water with ludox

Here the pictures are shown, in particular the water and water with ludox because the behaviours are very similar. Substantially the particles follow the liquid flow and these particles are more visible than the alumina particles. These Fluorescent microspheres and particles emit bright and distinctive colours when illuminated by light of shorter wavelengths. This property improves their contrast and visibility relative to background materials. In addition to the features of conventional microspheres and particles, the fluorescent products offer extra sensitivity and detectability for analytical methods. The products can be stored at room temperature and can be dispersed in aqueous media or air without degrading their fluorescent properties.

a b

Fig.55 The PIV's results of the solution of water and polymer microspheres red fluorescing (3µm) a)PIV picture without vectors b) picture with vectors

(25)

- 86 -

c d

Fig.56 The PIV's results of the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm) a)PIV picture without vectors b) picture with vectors

(26)

- 87 -

4.4.2.Pictures of alumina solutions

Here the pictures are shown, in particular the alumina solutions because the behaviours are very similar. Substantially the particles don't follow the liquid flow and these particles are less visible than the others particles.

a b

Fig.57 The PIV's results of the solution of water and glycerol 50% wt solution and alumina (110µm) a)PIV picture without vectors b) picture with vectors

(27)

- 88 -

a b

Fig.58 The PIV's results of the solution of water and glycerol 50% wt solution and alumina (3µm) a)PIV picture without vectors b) picture with vectors

(28)

- 89 -

4.4.3.Magnitude velocity

Velocity vectors can be used to illustrate the magnitude and direction of the flow field throughout the solution domain. For 2D simulations, a plot of all velocity vectors gives an overall picture of the fluid behavior. Vectors need to be plotted on one or more planes or surfaces instead. Surfaces can be planar or non-planar, such as a surface of constant temperature or a surface of constant radius. The important point is that for vector plots to be meaningful, the vectors (with length and orientation) need to be clearly visible, so the surfaces or planes used to plot them need be chosen accordingly.

a b

c d

Fig. 59 a) Magnitude velocity for the solution of water and polymer microspheres red fluorescing (3µm) b) Magnitude velocity for the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm) c) Magnitude velocity for the solution of water and glycerol 50% wt solution and alumina (3µm) d) Magnitude velocity for the solution of water and glycerol 50% wt solution and alumina (110µm)

The volumetric liquid flow rate is between 10-4 m3/s -10-6 m3/s and the volumetric gas flow rate is about 10-3 m/s. In a and b the max velocity is about 0.3m/s whereas in c and d is 0.5m/s.

(29)

- 90 - Below the trend for all complex liquids used are shown, in particular every graph represents the trend ,fixed the Y about in the middle of space between the two bubbles. The red circle and the black line underline the zone in point.

Graph 14Trend of velocity for the solution of water and polymer microspheres red fluorescing (3µm)

Graph 15Trend of velocity for the solution of water and colloidal (ludox) 10%vol solution (ludox) and polymer microspheres red (3µm)

Each point of the graphs are an average of 10 images of PIV and the dark line is the polynomial that approximates the trend of the points.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 v e lo ci ty ( m /s ) Diameter capillary (mm) 0 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 v e lo ci ty ( m /s ) Diameter capillary (mm)

(30)

- 91 - Graph 16Trend of velocity for the solution of water and glycerol 50% wt solution and alumina (3µm) The plateau obtained during the experiments maybe there weren't particles in the specific space of the capillary.

T

Graph 17 Trend of velocity for the solution of water and glycerol 50% wt solution and alumina (110µm) Especially for the alumina solutions a lot of points are very strange maybe because the pump don't permit to give a good mixing of the particles.

0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 v e lo ci ty ( m /s ) Diameter capillary (mm) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 v e lo ci ty ( m /s ) Diameter capillary (mm)

(31)

- 92 - An no-dimensional velocity is found to compare the different trend:

U no-dimensional= ௎೑ ௎:

where Uf is the velocity of flow ;

and U=USL+USG where USL=

ொಽ ഏ ర஽ మ and USG= ொಸ ഏ ర஽ మ

The section is like circular capillary because in the corner of the square capillary there is not liquid, so it's better to approximate like circular section.

Graph 18 water solution vs alumina solution

The trend of the velocity is very important because the mass coefficient depends on the velocity (U):

KL ≈β Uα

Good mass transfer performance requires large interface area between gas and liquid (resulting directly from small bubble size and high gas fraction, given the fixed gas rate) and a high mass transfer coefficient (associated with local levels of turbulence). A high gas fraction is not always desirable since the profitability of a reactor is largely controlled by the quantity of liquid it contains. Excessive gas retention may also lead to overreaction. It is only necessary to allow enough time for the required mass transfer.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 1 1.5 2 2.5 3 3.5 U f/ U Diameter (mm)

trend of alumina solution trend of water solution

(32)

- 93 -

4.4.4.Vorticity

Vorticity, a vector quantity, is a measure of the rotation of the fluid. In terms of a fluid element, a nonzero vorticity implies that the element is rotating as it moves. The vorticity is defined as the curl of the velocity vector, U:

Vorticity can be defined in both 2D and 3D flows. In 2D flows, the direction is normal to the plane of the simulation. This means that for a 2D ax symmetric simulation of flow in a stirred tank, the vorticity is always in the circumferential direction:

In 2D simulations, positive values indicate counter clockwise rotations, while negative values indicate clockwise rotation. In a 3D simulation, vorticity can take on any direction, and plots of vorticity magnitude, rather than the individual components, are often the most helpful. The units of vorticity are s−1, the same as those used for shear rate. The fact that the vorticity is positive near the impeller and negative near the wall indicates simply that the direction of the curl is opposite in these two regions.

(33)

- 94 -

a b

c d

Fig.60 a)vorticity for the solution of water and polymer microspheres red fluorescing (3µm) b) vorticity for the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm) c) vorticity for the solution of water and glycerol 50% wt solution and alumina (3µm) d) vorticity for the solution of water and glycerol 50% wt solution and alumina (110µm)

The vorticity is evaluated like an shear rate (1/s), the red zone and blue zone are symmetrical an the range is, in a and b from -300 to 300 , in c and d from -200 to 200.

Below the trend for all complex liquids used are shown, in particular every graph represents the trend ,fixed the Y about in the middle of space between the two bubbles. The red circle and the black line underline the zone in point.

(34)

- 95 - Graph 18Trend of vorticity for the solution of water and polymer microspheres red fluorescing (3µm)

Graph 19Trend of vorticity for the solution of water and colloidal (ludox) 10%vol solution (ludox) and polymer microspheres red (3µm) -400 -300 -200 -100 0 100 200 300 400 0 1 2 3 4 v o rt ic it y ( 1 /s ) Diameter capillary (mm) -200 -150 -100 -50 0 50 100 150 200 0 1 2 3 4 v o rt ic it y ( 1 /s ) Diameter capillary (mm)

(35)

- 96 - Graph 20Trend of vorticity for the solution of water and glycerol 50% wt solution and alumina (3µm)

Graph 21Trend of vorticity for the solution of water and glycerol 50% wt solution and alumina (110µm)

For the alumina solution there are strange points for the same reason of magnitude velocity and noise. -250 -200 -150 -100 -50 0 50 100 150 200 250 0 1 2 3 4 v o rt ic it y ( 1 /s ) Diameter capillary (mm) -400 -300 -200 -100 0 100 200 300 0 1 2 3 4 v o rt ic it y ( 1 /s ) Diameter capillary (mm)

(36)

- 97 - Large eddy simulations are transient simulations designed to capture the fluctuations that are the result of turbulent eddies. For this reason, LES images and animations have the potential to capture small and large scale activity that would otherwise be averaged to zero with a RANS turbulence model. Some of the small scale activity includes the birth and death of eddies or small vortices. Some of the large scale activity includes low-frequency instabilities in stirred tanks. A common way to visualize the turbulent structure present in LES simulations of mixers is by animating vectors or isosurfaces of vorticity magnitude. Particle suspension from the base and drawdown from the surface are often required in gas–liquid agitated vessels and are influenced in a complex manner by gassing. There are no well-established correlations for the influence of gas. Particle suspension is probably controlled by the energy and frequency of turbulent bursts, and drawdown by details of local flow patterns and vorticity at the surface, both of which could be expected to be affected by the presence of gas bubbles. In the first two cases the vorticity is symmetric in the channel, whereas in the alumina and glycerol solution there isn't the symmetry.

(37)

- 98 -

4.4.5.Rate of strain

The rate of deformation or strain rate tensor is a collection of terms that together describe the complete deformation of a fluid element in motion. The deformation can be the result of linear strain, which gives rise to a linear deformation or stretching of the element, and shear strain, which gives rise to an angular deformation or change in shape of the element. The symmetric tensor has components of the generalized form

Although the tensor components themselves offer little insight into the behavior of the flow field, functions of the tensor components often do. In terms of the Cartesian coordinates x, y, and z, the diagonal terms are

Each of these terms represents a linear strain rate or rate of elongation of the fluid element in each of the three coordinate directions. The sum of these diagonal terms is the trace or first invariant of the tensor. For incompressible fluids, this quantity is always zero, since the volume of the fluid element must be conserved. In addition to the trace, another quantity, often referred to simply as the strain rate, is of interest. The strain rate, taken from the modulus of the tensor, is a positive-definite representation of all possible components of the strain rate tensor. It is used to determine the viscosity in strain-dependent non-Newtonian fluids and is also helpful as a reporting tool for mixing applications. In particular, regions with a high strain rate play an important role in liquid dispersion.

(38)

- 99 -

a b

c d

Fig.61 a)rate of strain for the solution of water and polymer microspheres red fluorescing (3µm) b) rate of strain for the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm) c) rate of strain for the solution of water and glycerol 50% wt solution and alumina (3µm) d) rate of strain for the solution of water and glycerol 50% wt solution and alumina (110µm)

The rate of strain is evaluated like an shear rate (1/s), the red zone and blue zone are symmetrical and the range is, in a and b from -200 to 200 , in c and d from -150 to 150.

(39)

- 100 - Below the trend for all complex liquids used are shown, in particular every graph represents the trend ,fixed the Y about in the middle of space between the two bubbles. The red circle and the black line underline the zone in point.

Graph 22Trend of rate of strain for the solution of water and polymer microspheres red fluorescing (3µm)

Graph 23Trend of rate of strain for the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm)

-200 -150 -100 -50 0 50 100 150 200 0 1 2 3 4 ra te o f st ra in Diameter capillary (mm) -250 -200 -150 -100 -50 0 50 100 150 200 250 0 1 2 3 4 ra te o f st ra in Diameter capillary (mm)

(40)

- 101 - Graph 24Trend of rate of strain for the solution of water and glycerol 50% wt solution and alumina (3µm)

Graph 25Trend of rate of strain for the solution of water and glycerol 50% wt solution and alumina (110µm)

For the alumina solution there are strange points for the same reason of magnitude velocity and noise. -150 -100 -50 0 50 100 150 0 1 2 3 4 ra te o f st ra in Diameter capillary (mm) -150 -100 -50 0 50 100 150 0 0.5 1 1.5 2 2.5 3 3.5 ra te o f st ra in Diameter capillary (mm)

(41)

- 102 - The rate of strain like the vorticity is very important because it changes the value of viscosity. The trend of the rate of strain should be important when the study of CFD will be done, in fact for example the Reynolds stress model (RSM) does not use the Boussinesq hypothesis. Rather than assume that the turbulent viscosity is isotropic, having one value as in the k–ε model, the Reynolds stress model computes the stresses. For 2D models, this amounts to four additional transport equations. Along with the transport equation for ε, which must also be solved in the RSM model, the effect of turbulence can be represented in the momentum equations with greater accuracy than can be obtained from the k–ε models. Flows for which the assumption of isotropic turbulent viscosity breaks down include those with high swirl, rapid changes in strain rate, or substantial streamline curvature. As computer power and speed have increased during the past several years, the use of the Reynolds stress turbulence model has become more widespread. In the first two cases the rate of strain is symmetric in the channel, whereas in the alumina and glycerol solution there isn't the symmetry.

(42)

- 103 -

4.4.6.Stream lines and vortex

In 2D simulations, a quantity called the stream function, ψ, is defined in terms of the density and gradients of the x- and y-components of the velocity, U and V. In terms of cylindrical coordinates, which are most appropriate for axisymmetric stirred tank models, the definition takes the form where U and V are the axial and radial components of velocity. The stream function is constant along a streamline, a line that is everywhere tangent to the velocity field. When defined in the manner above, ψ incorporates a statement of conservation of mass. The difference between the stream function defined on any two streamlines is equal to the mass flow rate between the streamlines. Thus when a pair of streamlines has close spacing, the implication is that the velocity is greater than when the same pair has wide spacing, since the same amount of mass must pass through the space between the lines. Streamlines therefore have the ability to convey not only the relative movement of the flow, but the relative speed as well .They are also close along the outer wall, but are more widely spaced elsewhere, where the flow recirculates in a larger area at a much slower speed. The vortex are found subtracting the bubble velocity. Below the pictures show the different vortex are generated inside the capillary. Below is related a graph where showed the obtained data from T. C. Thulasidas, M. A. Abraham and R. L. Cerro (1996) about the location of vortex center as a function of NCa.

(43)

- 104 - a

b

Fig.63 The streamlines(a) and the vortex (b) of the solution of water and polymer microspheres red fluorescing (3µm)

(44)

- 105 - a

b

Fig.64 the streamlines(a) and the vortex (b) of the solution of water and colloidal 10%vol solution (ludox) and polymer microspheres red (3µm)

The location of vortex for the water and water with ludox is in the center of the capillary. The more important difference is the size of vortex.

(45)

- 106 - a

b

Fig.65 The streamlines(a) and the vortex (b) of the solution of water and glycerol 50% wt solution and alumina (3µm)

(46)

- 107 - a

b

Fig.66 the streamlines(a) and the vortex (b) of the solution of water and glycerol 50% wt solution and alumina (110µm)

Whereas the location of vortex in the alumina solution is not defined because the flow is very chaotic.