Quantitative Parameters to Evaluate Mixing in a
Single Screw Extruder
By
Kiana Kalali
Degree of Master of Engineering
Department of Chemical Engineering
McGill University
Montréal, Québec, Canada
March 2011
A thesis submitted to the Graduate and Post-doctoral Studies Office in partial fulfillment of the requirements for the degree of Masters of Engineering
©Kiana Kalali, 2011 All rights reserved
2
Contents Abstract ......................................................................................................................................................... 1
ABRÉGÉ ......................................................................................................................................................... 2
Acknowledgments ......................................................................................................................................... 3
1 General Introduction ............................................................................................................................. 4
1.1 Main types of Mixing .................................................................................................................... 5
1.2 Distributive and Dispersive Mixing .............................................................................................. 5
1.3 Mixing measurement methods ...................................................................................................... 6
1.4 Interface growth and flow pattern ............................................................................................... 14
1.5 Mixing in a single screw extruder ............................................................................................... 15
1.6 References .................................................................................................................................. 18
2 Residence time distribution ................................................................................................................ 19
2.1 Introduction ................................................................................................................................ 19
2.2 Strain distribution in the extruder .............................................................................................. 20
2.3 Modeling ..................................................................................................................................... 23
2.4 Experimental section .................................................................................................................. 28
2.4.1 Material ............................................................................................................................... 29
2.4.2 Equipment ........................................................................................................................... 29
2.4.3 Experimental Procedure ..................................................................................................... 30
2.4.4 Calculation of experimental RTD functions ........................................................................ 31
2.5 Results and discussion ................................................................................................................ 31
2.5.1 Experimental verification of the models ............................................................................. 36
2.6 Conclusion ................................................................................................................................... 40
2.7 References .................................................................................................................................. 42
3 Image analysis ..................................................................................................................................... 44
3.1 Introduction ................................................................................................................................ 44
3.2 Theory ......................................................................................................................................... 46
3.3 Experimental section .................................................................................................................. 51
3.3.1 Material ............................................................................................................................... 51
3.3.2 Equipment ........................................................................................................................... 52
3.3.3 Experimental procedure ..................................................................................................... 54
3
3.4 Results and discussion ................................................................................................................ 56
3.5 Conclusion ................................................................................................................................... 61
3.6 References .................................................................................................................................. 63
4 General Conclusion ............................................................................................................................. 64
4.1 Summary ..................................................................................................................................... 64
4.2 Future Work ................................................................................................................................ 65
Appendix ..................................................................................................................................................... 66
Calculating viscosity and Power-law exponent (n) ................................................................................. 66
Matlab codes........................................................................................................................................... 71
Sample Images ........................................................................................................................................ 86
1
Abstract
Mixing is crucial in most polymer processing operations towards obtaining high-quality
products (e.g. tubing, tire treads and wire coverings). Material type, screw design, and processing
conditions all affect mixing profoundly. Different types of mixing elements have been developed
to improve mixing in the single screw extruder; however, the selection of these mixing elements
is not trivial. In this work, our purpose is to provide quantitative tools to select the best mixing
element.
Residence time distribution (RTD) and image analysis were used to compare mixing in
three different mixing elements: single flight, Maddox, and Saxton. Residence time distributions
were used to indirectly grasp an insight about the strain distribution inside the extruder.
Experimental RTD data were derived from silica tracer studies and compared to various mixer
models. A model based on a plug flow mixer in series with two continuous stirred tanks best fit
the experimental data in all three different mixing elements. For the image analysis method the
degree of mixing was determined. Mixtures of polyethylene resins with carbon black were
extruded and sliced. Subsequently, sliced samples were scanned to provide images showing the
distribution of the carbon black in the resin.
The RTD experiments showed that the mean residence time is highest in the Saxton
mixer and the lowest in the single flight element. Also, the RTD was broadest in the Saxton
mixer and narrowest in the single flight mixer. This means that the polymer in the Saxton mixer
experiences the widest range of strains and gets mixed more thoroughly. These results were
confirmed by image analysis, which showed that polymers mixed in the Saxton mixer were more
homogenously mixed compared to the two other mixing elements.
2
ABRÉGÉ
Les phénomènes de mélange dans les extrudeuses monovis ont été étudiés en détail depuis de
nombreuses années. Le défi principal est le choix des éléments de mélange les mieux appropriés
pour une tâche de mélange donnée. A ce jour, les fabricants d'équipements de mise en oeuvre des
matières plastiques s'appuient fortement sur des données expérimentales et leur expérience pour
opérer ce choix. Dans ce travail, notre objectif est de développer des critères d'évaluation
quantitatifs pour différents éléments de mélange dans les extrudeuses monovis. A cet effet, nous
comparons l'effat de mélange dans trois éléments de mélange différents (à savoir une zone de vis
à filet simple, une zone de mélange à filet barrière de type Maddock, et une troisième zone à
picots de type Saxton) en utilisant la distribution du temps de séjour et l'analyse d'images. Pour la
distribution du temps de séjour, une matière de traçage est injectée dans la machine, et sa
concentration dans l'extrudat déterminée par la pesée du résidu solide des échantillons. Pour
l'évaluation optique du mélange, des images de copeaux d'échantillons étaient analysées au
moyen d'un logiciel Matlab.
Donc, les différents éléments de mélange sont caractérisés, pour les matières utilisées dans cettte
étude (des polyoléfines), par la distribution du temps de séjour et la qualité de mélange obtenue
par analyse d'image. Mis ensemble, nos résultats confirment que la qualité de mélange obtenue
est directement liée à la distribution du temps de séjour; le meilleur résultat est obtenu avec le
mélangeur de type Saxton. Les deux paramètres peuvent être utilisés non seulement pour
l'évaluation, mais aussi la prédiction de l'effet de mélange dans d'autres conditions et
configurations.
3
Acknowledgments
I would like to thank my supervisor Prof. Milan Maric for his advice and guidance and
his support through the project. Without his help I would not be able to finish this project.
I also would like to thank Maillefer Company for sponsoring this project and for letting
me use all the laboratory equipments needed for this project. Special thank to Dr Schläfli from
R&D department I cannot thank him enough for his advice and guidance and for his great efforts
to support me through the project. I would have been lost without him.
I must also acknowledge Dr Rakhshanfar who has guided me through Matlab coding and
image analysis. I wish to thank all my group mates who have helped me with working in the
laboratory.
I wish to thank my family for helping me get through the difficult times, and for all the
financial and emotional supports and in particular, I must acknowledge my mother, Nayer
Moghimi, and my sister ,Katayoon Kalali, without whose love, encouragement, I would not have
finished this thesis.
I thank the technical staff of the Department of Chemical Engineering for their help in
understanding the workings of the building and the proper safety protocol to follow which helped
reduce the risk of accidents in the laboratory.
4
1 General Introduction
Mixing, defined as a process in which various components are subdivided and distributed
throughout the entire volume of a system, plays a vital role in almost every polymer processing
operation. Nowadays the need for developing new polymeric materials with improved properties
relies more on mixing existing polymers and additives together than on the synthesis of
chemically new polymers1. To develop improved materials, different additives and reinforcing
agents are mixed with polymers; the aim of mixing is to produce a uniform mixture of polymers
and these additives 2. Thus, efficient mixing in extrusion is essential in obtaining a consistent,
high-quality extruded product.
Basically, the reduction of the non-uniformity is accomplished by inducing physical
motion upon the ingredients in the mixture. There are three different mixing mechanisms
according to different types of motions: diffusive mixing, turbulence mixing, and convective
mixing. Diffusion and turbulence are limited to low viscosity materials. The high viscosity of
polymer melts suggests that convective mixing is the major mechanism for polymers. Diffusion
is the spontaneous spreading of particles without any external driving forces which is critical in
the mixing of gases and low viscosity liquids. In most polymer mixing operations the mixing
time is too short and therefore the contribution of diffusion is negligible3. Turbulent flow is the
motion of a fluid having local velocities and pressures that fluctuate randomly. The criterion to
achieve turbulent flow depends on the Reynolds number.
NReynolds = (Density × Velocity × Diameter)/ Viscosity
Generally if the Reynolds number exceeds 2000, the flow will be turbulent4. Although it
is an effective mechanism for mixing, the viscosities of polymer melts are too large (in range of
100 to 100,000 Pa.s)3 to achieve a sufficiently high Reynolds number. The exceedingly low
Reynolds number in polymer processing dictates the flow is laminar. Convective mixing, a
predominant mixing mechanism in polymer processing, is induced by imposing laminar shear,
elongation (stretching), and squeezing deformation on the mixture. Convective mixing is
5
generally achieved by pressure flow caused by a pressuregradient, or drag flow created by a
moving boundary, or a combination of both. Both of these processes occur in screw extruders,
the most ubiquitous piece of processing equipment used in the polymer industry.
1.1 Main types of Mixing
Mixing operations are often categorized as a function of the state of the materials which
are to be mixed together. If the materials to be mixed are both solids, solid-solid mixing occurs.
This type of mixing becomes important when both the polymer and the filler are in powder form,
such as the inlet to the extruder3. More frequently solid fillers are mixed with the molten
polymers; this type of mixing is termed solid-liquid mixing. Another important mixing type in
polymer processing is mixing of two or more polymer melts; this type of mixing is known as
liquid-liquid mixing. In some cases, the gas is mixed with the polymer melt, which is referred to
as liquid-gas mixing such as in a foaming operation.3
1.2 Distributive and Dispersive Mixing
It is quite common to classify a mixing operation by the physical properties of the
components involved. According to this classification two types of mixing, distributive and
dispersive, are distinguished. Distributive mixing occurs in systems with negligible cohesive
forces. In this type of mixing interfacial area between the components is increased by stretching
and components are spread throughout the volume. Therefore, in distributive mixing, shear
stresses are irrelevant to the mixing mechanism and it can be described by the extent of
deformation or strain that is experienced by the components. Distributive mixing can take place
in the solid as well as liquid state. When solid particles are mixed without any changes in
particles sizes, distributive solid-solid mixing has occurred. When a particulate solid is mixed
with molten polymer without any reduction in the particle size, distributive solid-liquid mixing
has occurred. Furthermore, mixing of two miscible polymer melts is an example of liquid-liquid
distributive mixing.
6
Dispersive or intensive mixing, causes the reduction of the size of a component which is
being held by cohesive forces. In dispersive mixing, the minor component needs to be broken
down, but this happens only when a sufficient degree of stress is applied to the components, such
as clusters of solid particles (agglomerates) or droplets suspended in a liquid. Similar to
distributive mixing, there are different types of dispersive mixing such as solid- solid dispersive
mixing, solid- liquid dispersive mixing, and liquid-liquid dispersive mixing. An example of
dispersive mixing is the mixing of carbon black agglomerates into a polymer by which the
agglomerate size has to be reduced below a certain minimum size to achieve good surface
quality of the final product. The main difference between distributive and dispersive mixing is
the role of the local stresses applied on the materials. Thus, the measurement of the stress, strain,
and interfacial area could be used as key indicators of the state of mixing5.
1.3 Mixing measurement methods
To achieve a complete characterization of the state of mixture, it is required to specify the
size, shape, orientation, and spatial location of every particle of the minor component. Generally,
mixtures are characterized by gross uniformity, texture, and local structure. Gross uniformity
means measuring quantitatively the distribution of the minor component throughout the system.
Perfect gross uniformity is indicated by identical concentrations formed in all samples taken
from a system. In a random mixing process, the maximum uniformity attained is given by the
binomial distribution5b
.5c
Consider a sample, randomly extracted from the mixture containing n particles, where n
is large enough for statistical treatment and small enough compared to the total number of
particles in the mixture. When the fraction of minor particles in the entire mixture is p, then b,
the probability that this randomly selected sample has exactly k minor particles, is given by the
binomial distribution5b
.
𝑏 𝑘; 𝑛. 𝑝 = 𝑛 !
𝑘! 𝑛−𝑘 ! 𝑝𝑘 (1 − 𝑝)𝑛−𝑘
(1)
As shown in equation (1) the distribution of the minor component depends on both the
average concentration of the minor component, p, and on the size of the sample, n. This is clear
by the definition of the variance of the binomial distribution given below5b
.
7
𝜎2 =𝑝 (1−𝑝)
𝑛 (2)
For a reducing in the non-uniformity in the mixture, the random distribution of minor
components is vital. This means that the probability of appearance of the minor component at
any place in the mixture must be constant. The best quality of the mixture occurs when the
probability of finding a particle of any component is the same at all positions in the mixture.
When the materials to be mixed have different physical properties, segregation occurs. In a
segregated mixture, the particles of one component preferentially appear in one part of the
mixture more than in other parts. The distance between the mixing state of each sample and that
of the statistically random one is measured by the degree of mixing5b
.
The degree of mixing, M, can be written as:
𝑀 = 𝜎2
𝑆2 (3)
where σ2 is the variance of the perfectly random sample (binomial distribution) and s
2 is the
experimentally measured variance of the samples expressed by Equation (4):
𝑆2 =1
𝑁−1 (𝑥𝑖 − 𝑥 )2𝑁
𝑖=1 (4)
In Equation (4), xi is the volume fraction of the minor component in the test sample i. It is
obvious that for a perfect random mixture M = 15b
.
The state of mixing should improve with time, so it is possible to monitor the rate of
mixing by measuring the mixing index at various times. The most important particle property
that affects the tendency to segregate usually is the particle size. Particles > 75µm in size have a
strong tendency to segregate; particles ˂ 75 µm in size usually show little tendency to segregate.
In the intermediate range, as the size of the particles decreases, the tendency toward segregation
will reduce3. As mentioned, difference in particle sizes is the main reason for getting segregation
and, therefore, making the particle sizes as uniform as possible can reduce the segregation. In
distributive solid-solid mixing making particles as small as possible is an appropriate approach to
8
reduce the segregation, while dispersive mixing involves the reduction in size of a cohesive
component.
Mixtures can also be characterized by texture as well as gross uniformity. The term“
texture” means that some compositional non-uniformity is reflected in patches, stripes, and
streaks5b
. There are two parameters that characterize the texture of samples: scale and intensity
of segregation. The texture of the mixture in relation to the quality of mixing depends on the
combination of both scale and intensity of segregation. However, mixing can only influence
scale of segregation. This is because the reduction of intensity of segregation occurs only by
diffusion, which is largely absent in mixing of polymers. Regarding scale of segregation, the size
of undistributed portions of the components can be reduced by supplying mechanical energy.
Turbulence, de-agglomeration and re-combination, fluid deformation by stretching or kneading,
and shear are the mechanisms used by mixing devices to reduce the scale of segregation5b, c
.
To measure the size of the undistributed portion of the components, the coefficient of
correlation, R(r), is used5b
. R(r), measures the degree of correlation between the concentrations at
two points separated by distance r. This is achieved by randomly “throwing” a dipole of length r,
and measuring the concentrations of the component in question at each point of the dipole. R(r)
is expressed as:
𝑅 𝑟 = 𝐶𝑖 𝑥 − 𝐶 (𝐶𝑖 𝑥+𝑟 −𝐶 )𝑁
𝑖=1
𝑁 𝑆2 (5)
where C (x) and C(x +r) are concentrations at the two points, 𝐶 is the mean concentration, N is
the total number of couples of concentrations, and S2 is the variance given by :
𝑆2 = 𝐶𝑖 𝑥 − 𝐶 2𝑁
𝑖=1
2
2𝑁−1 (6)
The values of the coefficient of correlation vary in the range from -1 to 1. It is -1 when
the origin and end of the vector r lie in different axes. The coefficient equals unity when both the
origin and the end of the r vector lie in the same axis. Finally, it is zero when the correlation is
random or, equivalently, when knowledge of the composition at the origin provides no
information about the composition at the end of the vector5c
.
9
The scale of segregation is the integral of the coefficient of correlation from distance r =
0 (R (0) = 1) to distance ζ at which there is no correlation (R (ζ) = 0):
𝑆 = 𝑅 𝑟 𝑑𝑟𝜁
0 (7)
One of the techniques to gauge the quality of the mixing is the measurement of the
variance. To measure the variance a number of small samples from the mixtures are extracted
and consequently the composition of each sample is determined. There are different techniques
to determine the composition of samples such as light transmittance, electrical conductivity,
titration, and particle counting5c
. Using the best technique for the particular material, a large
number of measurements easily can be taken and the concentration can be measured accurately.
Consider 𝐶𝑖 as the composition of individual samples and N as the total number of the samples.
Thus, the experimental average composition is:
𝐶 =1
𝑁 𝐶𝑖
𝑖=𝑁𝑖=1 (8)
The square of the standard deviation of the sample composition, S2, is the variance which is
given by
𝑆2 =1
𝑁−1 (𝐶𝑖 − 𝐶 )2𝑖=𝑁
𝑖=1 (9)
In homogenous mixtures, the compositions of all samples will be close to the average
composition and thus the variance will be close to zero. Tucker showed in his work that the
sample variance depends on not only the shape of the sample but also the size of the sample6. If
all the samples have equal size and shape, the sample variance is measured in a straightforward
fashion.
As mentioned before, the most significant mixing operations in polymer processing is
mixing of two or more polymer melts, a type of liquid –liquid mixing. In polymer processing,
due to the high viscosities of polymer melts, the flow is laminar. Interfacial surface area first was
used by Spencer and Wiley to measure liquid-liquid mixing in polymer processing.7 In mixing of
polymer melts, components are generally immiscible, so it is easy to identify the two
10
components and their interface. Thus, we can use the increase of interfacial area as a quantitative
measure for the extent of mixing. The striation thickness, r, is defined as the thickness of each
adjacent layer in a lamellar structure. The interfacial area per unit volume, Av, and the striation
thickness are related as follow5c
:
𝐴𝑣 = 1
𝑟 (10)
As mixing progresses, the interfacial area per unit volume increases and the striation
thickness decreases. Thus, it is possible to calculate the quality of mixing by determining the
interfacial area growth, or equivalently the striation thickness reduction.
The striation thickness is defined as the total volume divided by half of the total interfacial
surface:
𝑟 =𝑉
𝐴/2 (11)
If the minor component is initially introduced as randomly oriented cubes of height H and with a
volume fraction φ, the striation thickness can be expressed as:
𝑟 =2 𝐻
3𝜑𝛾 (12)
This equation indicates that striation thickness is inversely proportional to total strain
and volume fraction of the minor component. It is clear that for the large particles or in the case
of low volume fraction of the minor component, more strain is required to make any significant
reduction in striation thickness. As a result, it is more difficult to mix a small amount of a minor
component into a major component compared with a mixture of equal concentrations.
Consider an element subjected to simple shear flow which is determined by three
directional angles, ax, ay, and az. When the material is sheared with the primary flow in the „x‟
direction relative to the „y‟ direction, the ratio of the two areas at t0+∆t and t0 is obtained as
follows:
11
𝐴
𝐴0= 1 − 2 cos 𝛼𝑥 cos 𝛼𝑦𝛾 + 𝛾2𝑐𝑜𝑠2𝛼𝑥
1/2 (13)
where cos 𝛼𝑥 , and cos 𝛼𝑦 are directional cosines and, 𝛾, is the total strain undergone by material
mixed with each other.
Equation 13 shows that the interfacial area growth depends on its initial orientation and on the
magnitude of the shear strain imposed by the mixer.
For large values of 𝛾, Equation 13 reduces to:
𝐴0
𝐴= cos 𝛼𝑥 𝛾 (14)
This equation indicates that the interfacial area growth in a two-component material undergoing
large shear deformations is linear with shear strain5b, c
.
Due to the total strain being directly proportional to the interfacial growth function, it is
considered an important variable for the quantitative characterization of the mixing process.
Besides the increase in interfacial surface area, the reduction in striation thickness can be used to
evaluate mixing. Mohr, Saxton, and Jepson used striation thickness for evaluating mixing8. As
shown in Equation 11, the striation thickness is inversely proportional to interfacial surface area.
Therefore, Equation 14 can be easily re-written in terms of striation thickness instead of
interfacial surface area as shown below:
𝑆
𝑆0=
1
1+𝛾2 (15)
Equation 15 reveals the reduction in striation thickness is inversely proportional to the shear
strain; this is plotted in Figure 1-1.
12
Figure 1-1 striation thickness versus shear strain
Figure 1-1 shows that initially, striation thickness is reduced dramatically, however, the
striation thickness is reduced more slowly after the first rapid reduction. When the material
undergoes shear strains of 80 and greater, the reduction of the striation thickness will be equal to
zero. This indicates that the mixing is efficient at first, however, after about ten units of shear,
the mixing becomes inefficient. The mixing efficiency changes due to the change in orientation
of the element deformed in the shear field. The most favorable orientation, which is normal to
the flow direction is obtained initially. Later, as the element is sheared, its orientation will
change with shear strain. In the absence of the reorientation, the fluid particles orient just in the
direction of the flow which is not the favorable orientation for mixing. Therefore, in absence of
reorientation, the mixing efficiency reduces as the shear strain is increased. To avoid inefficient
mixing, reorientation is critical. Furthermore, according to Equation 13, at low strains, depending
on initial orientation, the interfacial area may increase or decrease. A decrease of interfacial area
occurs when the strain imposed on material de-mixes (which occurs when the fluid is sheared in
one direction a certain number of shear units, and equal and opposite shear will take the fluid
back to its original state) the components5a-c, 8
. As shown in Equation 14, initial orientation plays
an important role in interfacial area growth. Maximum increase in the interfacial area is achieved
when the initial surface is oriented perpendicular to the flow field. For maximum stretching of
the area, the optimal orientation can be calculated by differentiation of Equation 13.
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60
Str
iati
on
th
ick
nes
s ra
tio
(S
/S0)
strain(γ)
13
𝑑𝐴
𝑑𝛾= −𝐴0𝑐𝑜𝑠𝑎𝑥𝑐𝑜𝑠𝑎𝑦 (16)
The optimal orientation is achieved when the interfacial area element, along the z axis, is
perpendicular to the direction of shear, and the xy plane is at 450 to it.
𝐴
𝐴0 𝑚𝑎𝑥
= 𝑒𝛾/2 (17)
This important result indicates that if an interfacial area element is constantly maintained
at 450 to the direction of the shear, stretching becomes exponential with shear. This situation is
difficult to reach in extruders as there is always a possibility for slip to occur and not all of the
particles can face the mixing section in a desirable direction. Therefore, frequent reorientation is
desirable. As is shown below, in simple shear flow, reorientation reached by randomizing mixing
sections greatly improves the generation of interfacial area.
The interfacial area at the outlet of the short mixing section disrupting the simple shear
field is given by the following (mixing section produces randomly oriented minor component):
𝐴1 =1
2𝐴0𝛾1 (18)
where γ1is the total shear strain the fluid is exposed to before mixing section. It is assumed that
the shear strain in the mixing section itself is negligible. If the simple shear field is restored after
the mixing section, the total interfacial area after another exposure to shear strain γ1becomes:
𝐴1 =1
2𝐴0𝛾1 =
1
2
1
2𝐴0𝛾1 𝛾1 = 𝐴0(
1
2𝛾1)2 (19)
and n shear strain exposure of the same magnitude ᵞ1 , the total interfacial area will be:
𝐴𝑛 = 𝐴0 (1
2 𝛾1)𝑛
(20)
14
Thus, adding mixing sections that randomizes the minor component increases the
interfacial area. In a dynamic mixing device, such as an extruder, it is difficult to design mixing
sections with the most favorable orientation, but random orientation is more feasible5a
.
1.4 Interface growth and flow pattern
To determine the flow pattern most favorable in terms of interface growth, three strain
fields are considered: planar elongation, uniaxial elongation, and simple shear. These are
compared in Table 1-1.
Table 1-1 The average interface growth which is the integration of all the possible orientation and
maximum interface growth as a function of F(ɛ or ᵞ) where ɛ=𝒅𝒍
𝒍 is shown
Type of strain Interface growth function
A/A0
Maximum
Interface growth function A/A0
Average
Planar elongation Exp(ɛ) (1/2) exp(ɛ)
Uniaxial elongation Exp(ɛ /2) (4/5)exp(ɛ /2), for ᵞᵒ≥1
(4/5)exp(-ɛ), for ᵞᵒ≤1
Simple shear ᵞ ᵞ/2
According to Table 1-1, it is obvious that in terms of interfacial growth, elongational
flows are more efficient than simple shear. As for the identical reduction in striation thickness in
comparison with total elongation, higher total shear strain is needed. Therefore, to achieve
effective mixing, the interfacial area must not only be stretched, but also reoriented, as well as
randomized throughout the volume5a, 5c
.
15
To provide an efficient mixing, mixers must have complex geometries. Efficient mixing
is defined as the level of distribution of minor component into the matrix, which is a function of
both mechanical strain and the orientation of the interfacial area. When a mixer is irregular in
geometry, the entrance flows give extensional character. Also, the orientation given by shear
flow is disrupted and, consequently, mixing efficiency is increased5a, 9
.
1.5 Mixing in a single screw extruder
Single screw extruders are widely used in the polymer industry. This machine has been
extensively used for blending or compounding to produce uniform mixtures of polymers10
. A
regular single screw extruder is constructed of a rotating screw which is rotated inside the
extruder barrel (Figure 1-2). Extruders can be fed by either a polymer melt or with solid polymer
particles. Solid particles are fed into plasticating extruders throughout the hopper; the material is
conveyed, melted and pushed forward towards the end of the machine as the screw rotates.
Therefore, the most important functions in a single screw extruder are: solid conveying, melting,
melt conveying, and mixing. First, polymers are melted inside the extruder. As the solid particles
are conveyed in plug flow there is no relative motion between the solid polymer particles. Thus,
there is no or limited mixing while polymers are solid.
Figure 1-2: A single screw extruder11
16
To analyze mixing in a single screw extruder, the exact flow pattern is required. There
are many studies concerning mixing measurements proposing different methods such as
weighted average total strain (WATS)12
, the strain distribution by Moher et al8 , McKelvey
13,
and Pinto and Tadmor 12
, and the Weighted Average Deformation Characteristics ( WADC)14
.
The overall understanding about the flow pattern has been provided by monitoring the fluid
element through the extruder15
.
The mixing capacity of a single screw extruder is weak; therefore, to improve the mixing
capacity, mixing elements are designed to provide better mixing16
16a
.In single screw extruders,
the incorporation of mixing sections in the melt channel helps shear flow by increasing the
pressure drop in the extruder, and so the shear strain is increased. More significantly, the mixing
sections reorient the polymer interfaces, reducing the striation thickness in the material coming
out of the extruder9.
As mentioned previously, the extent of mixing directly relates to the total strain
undergone by fluid elements travelling down the screw channel. The strain history of each fluid
element depends not only on the shear rate but also on the time the element stays in the extruder,
so the residence time is a key parameter for characterizing the mixing performance of extruders.
The residence time is actually a distribution of residence times because the velocity field is not
uniform across the channel cross -section, and correspondingly, there is a distribution of strains
undergone by fluid elements5c, 10
.
Mixing phenomena in single screw extruders have been investigated for many years;
many studies, experimental and computational, textbooks and patents on this subject have been
published. Currently, for the designer of extrusion equipment, the key challenge is to select the
best mixing elements from the different existing mixing elements to accomplish satisfactory
mixing. Maillefer Company, a manufacturer of extrusion equipment, who sponsored this project,
uses various types of mixing sections in their screws. This company design many mixing
sections mainly based on experience and general consideration, but unfortunately the lack of
reliable quantitative parameter sometimes results in unsuccessful designs. To avoid this and for
providing quantitative values to characterize mixing phenomena in different mixing section, this
project aims at establishing a quantitative tool for evaluating different mixing sections. Thus, this
thesis proposes the comparison of the three most common mixing sections: one of the most basic
17
mixing sections (single flight), one of the most common distributive mixing sections (Saxton)
and one of the most common dispersive mixing sections (Maddox) to evaluate mixing in a single
screw extruder.
The residence time distribution indirectly provides insight about the total strain
experienced in the material and this was consequently measured here. Further, an image
analyzing techniques was used to evaluate the degree of mixing. This thesis will determine how
we can use residence time distribution and image analyzing methods to quantitatively compare
different mixing elements in a single screw extruder.
18
1.6 References
1. Manas-Zloczower, I., Analysis of mixing in polymer processing equipment. Case Western Reserve University: 1997; pp 5-8.
2. Rauwendaal, C., Mixing in polymer processing. M. Dekker: New York :, 1991. 3. Rauwendaal, C., Polymer mixing : a self-study guide. Hanser Publishers ; Hanser/Gardner Publishers:
Munich; Cincinnati, 1998. 4. Rauwendaal, C., Mixing in polymer processing. M. Dekker: New York, 1991. 5. (a) Rauwendaal, C., Polymer extrusion. Hanser Gardner Publications: Cincinnati, OH :, 2001; (b) Tadmor,
Z.; Gogos, C. G., Principles of polymer processing. Wiley-Interscience: Hoboken, N.J. :, 2006; (c) Baird, D. G.; Collias, D. I., Polymer Processing: Principles and Design. Wiley-Interscience: NY, 1998; (d) Manas-Zloczower, I.; Tadmor, Z., Mixing and compounding of polymers : theory and practice. Hanser Publishers ; Distributed in the USA and Canada by Hanser/Gardner Publications: Munich ; New York : Cincinnati :, 1994.
6. Tucker, C. L., Sample variance measurement of mixing. Chemical Engineering Science 1981, 36 (11), 1829-1839.
7. Galaktionov, O. S.; Anderson, P. D.; Peters, G. W. M.; Meijer, H. E. H., Mapping approach for 3D laminar mixing simulations: application to industrial flows. International Journal for Numerical Methods in Fluids 2002, 40 (3-4), 345-351.
8. Mohr, W. D.; Saxton, R. L.; Jepson, C. H., Mixing in Laminar-Flow Systems. Industrial & Engineering Chemistry 1957, 49 (11), 1855-1856.
9. D. I. Bigio, J. D. B. L. E. D. W. G., Mixing studies in the single screw extruder. Polymer Engineering & Science 1985, 25 (5), 305-310.
10. Kim, S. J.; Kwon, T. H., Measures of mixing for extrusion by averaging concepts. Polymer Engineering & Science 1996, 36 (11), 1466-1476.
11. Maric, M., single screw extruder
Chemical Engineering 584, p. P., Lecture7- Extrusion, Ed. 2009. 12. Pinto, G.; Tadmor, Z., Mixing and residence time distribution in melt screw extruders. Polymer
Engineering & Science 1970, 10 (5), 279-288. 13. J. M. McKelvey, Polymer Processing. John Wiley and Sons: New York, 1962. 14. Kwon, T. H.; Joo, J. W.; Kim, S. J., Kinematics and deformation characteristics as a mixing measure in the
screw extrusion process. Polymer Engineering & Science 1994, 34 (3), 174-189. 15. Amin, M. H. G.; et al., In situ quantitation of the index of mixing in a single-screw extruder by magnetic
resonance imaging. Measurement Science and Technology 2004, 15 (9), 1871. 16. (a) Han, C. D.; Lee, K. Y.; Wheeler, N. C., A study on the performance of the maddock mixing head in
plasticating single-screw extrusion. Polymer Engineering & Science 1991, 31 (11), 818-830; (b) Hu, G.-H.; Kadri, I.; Picot, C., One-line measurement of the residence time distribution in screw extruders. Polymer Engineering & Science 1999, 39 (5), 930-939.
19
2 Residence time distribution
2.1 Introduction
Large strains and frequent reorientation are key parameters for achieving good mixing.
The total strain experienced by a fluid particle is equal to the time integral of its strain rate, and
thus, the residence time distribution (RTD) indicates the total strain distribution imposed on the
fluid element. It is therefore highly useful to measure the RTD. Besides grasping insight about
mixing characteristics of the single screw extruder, the data obtained by the RTD function is of
interest for overall performance of the extruder, especially when time is an important element to
be considered (eg. reactions, degradation, etc.). RTD results can be used to investigate effects of
the different variables such as screw speed, temperature profile, pressure drop, and screw design
on the time it takes for the material to exit the extruder. Furthermore, it is possible to directly
measure the RTD in single screw extruders under almost normal operating conditions17
. RTD has
been extensively studied and some previous examinations of RTD in extruders include the
following: theoretical evaluation of RTD functions in melt extruders proposed by Pinto and
Tadmor12
; and the experimental evaluation of RTDs by Wolf and Resnick18
; and the non-
Newtonian power law model fluid derived by Bigg and Middleman19
from an isothermal
Newtonian flow model. RTD was measured experimentally in both plasticating and melt
extruders by Wolf and White20
. Influence of the leakage flow and radial temperature distribution
on the RTD in an extruder was reported by Sek21
, Roemer and Durbin22
, Kim and Skathkov23
,
and Tadmor and Klein24
.
Almost all analyses that seek information about mixing and strain distribution begin with
the definition of a RTD. The amount of time a fluid particle spends in the extruder to move from
the hopper (the entrance to the extruder) to the die( the exiting section from the screw) is known
as the RTD. The RTD was first introduced by Danckwerts (1953) as a function, f(t), so that the
exit age distribution, E(t)= f(t) dt defined as the fraction of exiting fluid which has spent a time
between t and t+dt in the system25
. If Q denotes the exiting volumetric flow rate and dQ the
fraction of it with residence time between t and t+dt, then the RTD function is given by:
20
𝐸 𝑡 = 𝑓 𝑡 𝑑𝑡 =𝑑𝑄
𝑄 (1)
A cumulative RTD function, F(t) can be defined as:
𝐹 𝑡 = 𝑓(𝑡)𝑑𝑡𝑡
𝑡0 (2)
where t0 is the minimum residence time. It is obvious that F(t) is the fraction of exiting fluid that
has spent a time less than or equal to t.
The mean residence time, 𝑡 , is then equal to:
𝑡 = 𝑡 𝑓(𝑡)𝑑𝑡∞
𝑡0 (3)
If the velocity profiles are known in a laminar flow system, the volumetric flow rate can
be calculated and consequently the RTD function can be determined. For cases where the RTD
cannot be calculated theoretically, experimental methods are used. These methods are based on
introducing tracers into the system and recording the concentration of the tracer at the exit of the
extruder5b, c, 26
.
2.2 Strain distribution in the extruder
As discussed earlier, the total strain plays a vital role in the quality of laminar mixing.
Because fluid elements follow different paths in the extruder and also have different residence
times, elements of fluid experience different strains. Strain distribution functions (SDF) are
introduced to quantitatively evaluate the fluid element's various strain histories and can be
related to RTDs.
The SDF, f(γ) dγ, is defined as the fraction of exiting flow rate that experienced a strain between
γ and dγ.The cumulative SDF,F(γ), is defined by the following expression:
𝐹 𝛾 = 𝑓 𝛾 𝑑𝛾𝛾
𝛾0 (4)
21
where γ₀ is the minimum strain.
F(γ) represents the fraction of exiting flow rate with strain less than or equal to γ.
The mean strain of the exiting stream is defined as5b, c
:
𝛾 = 𝛾𝑓 𝛾 𝑑𝛾∞
𝛾0 (5)
As the parallel plate geometry forms a simple model of melt extrusion, the strain distribution in
an extruder as exemplified by the SDF, and it can be examined in parallel plates with combined
pressure and drag flow5b
.
The derivation for the SDF in a single screw extruder often starts with the following
simplification. Assume a Newtonian fluid between parallel plates with a superimposed pressure
gradient and drag flow. The flow is fully developed, isothermal and laminar. The velocity profile
Vz is given as follows:
𝑉𝑧 = 𝜉 + 3𝜉 𝑞𝑝
𝑞𝑑(1 − 𝜉) 𝑉∘ (6)
where ξ = y/H (H = gap between the plates), 𝑉∘ is the velocity of the moving plate, and qp and qd
are the pressure and drag volumetric flow rates per unit width.
𝑞 = 𝑉𝑧𝐻
0𝑑𝑦 =
𝑉∘𝐻
2(1 +
𝑞𝑝
𝑞𝑑) (7)
𝑓 𝛾 𝑑𝛾 =𝑑𝑞
𝑞=
2
1+𝑞𝑝
𝑞𝑑
𝜉 + 3𝜉𝑞𝑝
𝑞𝑑(1 − 𝜉) 𝑑𝜉 (8)
𝛾 = 𝛾 𝑡 =𝐿
𝐻
1+3 𝑞𝑝
𝑞𝑑 (1−2𝜉)
𝜉+3𝜉 𝑞𝑝
𝑞𝑑 (1−𝜉)
(9)
22
The strain rate,𝛾 , is derived above and it is indicated by a time derivation of γ, (𝛾 = 𝑑𝛾/𝑑𝑡)
𝐹 𝜁 = 1 −𝜁2
1+𝑞𝑝
𝑞𝑑
1 +𝑞𝑝
𝑞𝑑(3 − 2𝜁) (10)
𝛾 𝜁 =1+ 3
𝑞𝑝
𝑞𝑑 1−2𝜁
𝜁[1+3𝑞𝑝
𝑞𝑑 1−𝜁 ]
(1+
𝑞𝑝
𝑞𝑑
2)𝛾 (11)
𝛾 = 2𝐿
𝐻(1+𝑞𝑝 𝑞𝑑 ) 𝐹𝑜𝑟 −
1
3 ≤ 𝑞𝑝 𝑞𝑑 ≤
1
3 (12)
It is clear that the ratio of the pressure to the drag volumetric flow rate plays a
dominant role on the SDF and also the mean strain. A positive pressure gradient via pressure
increase in the direction of flow (qp /qd˂0) will not only increase the mean strain, it will also
reduce the breadth of the distribution. Obviously, a negative pressure gradient has the opposite
effect. This result supports the experimental observation that an increase in positive pressure
gradient in the extruder improves mixing12
. The mean strain, as defined in Equation 12, is
proportional to L/H. This means that mean strain depends on geometry, and it is possible to
improve mixing by designing long and shallow conduits.
As shown in Equation 7, to calculate the SDF we need to know the velocity profile of all
fluid elements inside the extrusion channel. Calculating the velocity profile is very difficult;
therefore, instead of calculating SDF for analyzing mixing characteristics we used the RTD. This
is reasonable as the strain history of each fluid element depends not only on the shear rate but
also on the time the element stays in the extruder, so the residence time is a key parameter for
characterizing the mixing performance of extruders. Also it is possible to model the RTD
experimental results by using the cumulative residence time, F(t), and exit age residence time,
E(t), which will discuss later.
In order to enhance our understanding about the mixing characteristics of a single screw
extruder here in this work we compared mixing performance of three important mixing sections
23
(single flight, Maddock, Saxton). As discussed earlier, RTD is an easy and a direct way to study
the strain distribution function in a single screw extruder. Therefore, it is used as a method to
quantitatively evaluate mixing in these three different mixing sections. Also, we have chosen the
most general models among all different theoretical models developed for the RTD curves to
suggest the best model for each of the three different missing sections. The best model is defined
as the one that provides the statistical best to the experimentally measured RTD.
2.3 Modeling
Experimental RTDs are usually analyzed by E(t), and F(t) curves which determine the
variation in the concentration of tracer at the exit with time and the cumulative quantity of the
tracer at the exit with time, respectively27
. The screw geometry has significant effects on RTD
because different screw geometries have different hold-up volumes and mass flow patterns.28
Therefore, screws equipped with different mixing elements have different RTDs and
consequently different models can predict RTDs in different mixing elements. There are many
models developed to predict RTD in an extruder. The extruder geometry, shear and velocity
profiles were employed by Janssen et al (1979) and Chen and Pan (1993) 28
.With the assumption
of considering the melt material as a power-law fluid, a model for a two-dimensional flow field
in an extruder proposed by Bigg and Middleman (1974)29
. Levenspiel (1972)30
and Wolf and
Resnick (1963)18
used a series of ideal tank reactors to model the extrusion process. H. De Ruyck
developed a model through which the extruder was simulated as a complex reactor including
continuously stirred tanks as subreactors31
. The parallel plate and curved-channel flow models
was used by Pinto and Tadmor to derive the RTD functions for Newtonian flow in single-screw
extruders12
. Chen et al. developed an RTD model based on dividing the screw as a set of
individual turns connected in series with the assumption that each turn had a statistically
independent RTD32
.
Experimental RTD curves of an extrusion process typically consisted of an initial delay
time, then a rise in concentration of tracer particle and then followed by a gradual tail. The plug
flow behavior refers to the initial delay time while the tail characteristics are indicative of well-
mixed behavior. Thus, both plug-flow and well-mixed behaviour are provided by the RTD
curves of most extrusion processes. The plug-flow behaviour results in narrow residence time
24
distributions which represents good laminar mixing and also prevents the risk of thermal
degradation as there is no long-term “tail” for RTD5b
. The RTD tail represents the amount of
over-processing1 which determines the product quality
28. Here we used the most general models
to figure out which model is appropriate to describe the RTD experimental data of each mixing
section.
The model of plug flow in series with continuous stirred tank reactors (CSTRs) having
dead volume fraction proposed by Kumar, Ganjyal, Jones, and Hanna (2007) is used as a model
number 1 in this thesis 27
. Model number 2 is a plug flow in series with a cascade of CSTRs
without any dead volume. This model was first used by Jager et al. (1995) to characterize F(t)
curves and later by Singh and Rizvf (1997) to characterize E(t) curves28
. The model of an
infinite series of continuously stirred tank reactors which was first proposed by Levenspiel
(1972) and later by Davidson, Paton, Diosady, and Spratt (1983) is used as model number 3 here
30, 33. Furthermore, model number 4 is a plug flow in series with a perfect mixer and is known as
Wolf-White Model, which was first derived by Wolf-White (1976) 18
. Figure 2-1 shows the
schematic diagram of the three first models.
1 Over-processing occurs when materials stay in extrusion more than a time required to process them.
25
Figure 2-1 Schematic diagram of models 1, 2, and 3 used to describe mixing in a single screw extruder
26
To identify the parameter values of the above-mentioned RTD models, a least-squares
error-fit method was used. Through this model the optimum values are those that provide the
minimum S, which is given as:
𝑆𝐸𝑟𝑟𝑜𝑟 = (𝑦𝑖 − 𝑓 𝑥𝑖 )2𝑛𝑖=1 (13)
where yi represents experimental data and f (xi) refers to the given model data. Besides
computing each model parameters such as number of CSTRs,n, used in the model,and the dead
volume fraction, d, this method gives the best model as the one that has the minimum S when the
optimum parameter values are used.
The four mentioned models used to model the RTD are described theoretically below.
Model number 1: Plug flow in series with CSTRs having dead volume fraction. This
model consists of four parameters: mean residence t , fraction of plug flow p, number of CSTRs n
and dead volume fraction d. It includes plug flow as well as dead volume. The E (t) curve for this
model is derived as following:
𝐸 𝜃 =𝑏 𝑏 𝜃−𝑝 𝑛−1
𝑛−1 !𝑒𝑥𝑝 −𝑏 𝜃 − 𝑝 (14)
where b is calculated as:
𝑏 =𝑛
1−𝑝 1−𝑑 (15)
and the fraction of plug flow is:
𝑝 =𝑡𝑚𝑖𝑛
𝑡 (16)
where 𝑡𝑚𝑖𝑛 represents the first time the tracer starts appearing at the exit of the extruder. In this
model normalized time, 𝜃, was used as following:
27
𝜃 =𝑡
𝑡 (17)
Consequently the dimensionless RTD curves are:
𝐸 𝜃 =𝐸(𝑡)
𝑡 (18)
𝐹 𝜃 =𝐹(𝑡)
𝑡 (19)
where t is calculated by means of experimental data. (See equation 30)
Model number 2: Plug flow in series with cascade of CSTRs.
This model is similar to model number 1 expect that there is no dead volume for CSTRs. This
model hast three parameters: mean residence 𝑡 , fraction of plug flow p, and number of CSTRs n.
In this model, the dimensionless time is derived as following:
𝜃 =𝑡−𝑝𝑡
𝑡 −𝑝𝑡 (20)
The following equation is developed for this model to calculate E (t):
𝐸 𝑡 = 0 ; 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 𝑝 (21)
𝐸 𝑡 =1
𝑡 −𝑝𝑡 𝑛 (𝜃)𝑛−1
1
𝑛−1 !𝑒−𝑛(𝜃)
(22)
The two parameters n and p are estimated by using the least-square curve fitting model and 𝑡 is
calculated from experimental data.
Model number 3: Cascade of perfectly mixed reactors or continuously stirred tank
reactors (CSTRs).
This model includes two parameters: mean residence 𝑡 , and number of CSTRs, n. The E(t) is
calculated in this model as following:
28
𝐸 𝑡 =1
𝑡 𝑛(𝑛𝜃)𝑛−1 1
𝑛−1 !𝑒−𝑛𝜃
(23)
Model number 4: Plug flow in series with a perfect mixer.
This model is similar to model number two except this model consists of a plug flow mixer in
series with a perfect mixer. There are two parameters: mean residence 𝑡 , and fraction of plug
flow p in this model.The F(t) for this model is calculated as:
𝐹 𝑡 = 1 − 𝑒− (
1
1−𝑝)(
1−𝑝𝜃
𝜃); 𝐹 𝑡 ≥ 0 (24)
The E(t) curve for this model is plotted by using the following equation:
𝐸 𝑡 =1
(1−𝑝)𝑡 𝑒
𝜃−𝑝 (1
1−𝑝) ; 𝑓𝑜𝑟 𝜃 ≥ 𝑝 (25)
𝐸 𝑡 = 0; 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 𝑝 (26)
These four mentioned models were used in this work to fit the experimental data grasped out of
three different mixing elements: single flight, Maddock, and Saxton.
2.4 Experimental section
Different in-line and off-line methods have been proposed in the literature for RTD
measurement. An ash containing method based on thermal degradation was used for our
experiments. This method relies on weighing the ash residue and was used by Carneiro (2000)
with the same tracer, silica34
. Kim and coworkers (2004) used the same method but they used
aluminium flakes as a tracer35
.
29
2.4.1 Material
A polyethylene extrusion grade resin (Dowlex NG 5056D) was used as received; it was
kindly donated by Maillefer Company. A silica gel (Grade 7734, pore size 60 Å, 70-230 mesh)
was bought from Aldrich; it was used as received.
2.4.2 Equipment
The first part of the experimental work, collecting samples, was done at Maillefer
Company where a single-screw extruder NMC 45-24D (screw speed 215 rpm max, motor power
29 kW) was used. The three most important interchangeable mixing sections (single flight,
Maddock, and Saxton) were used separately. Figure 2-2 exhibits the three mixing elements used
in this experiment. The second part, burning samples, was done at McGill University where a
Fisher Scientific Isotemp Muffle Furnace, Model 550-58, was used to burn the samples and an
electronic lab scale was used to weigh the resulting ash residues.
Figure 2-2 Three Different Mixing Elements studied: a) single-flight b) Maddock c) Saxton (all with
Length=270mm, and diameter= 45mm)
30
2.4.3 Experimental Procedure
Fluid free of the tracer was extruded until steady output was achieved. After the extruder
reached steady state, and temperature was stabilized at 250° C, 100 gr of the tracer (silica gel)
was injected very quickly into the polyethylene feed stream and a stopwatch was started. The
extruder was operated at 20rpm screw speed and a feed rate of 9 kg/hr. Samples of extruded
strips (each weighing around 4 gr) were collected at each 30 sec intervals. This experiment was
performed for each of the three mentioned mixing sections. After collecting samples and
numbering them, each sample was weighed and replaced in a crucible noted with the sample
number. The crucibles were weighted before addition of the sample. Then the weight of crucibles
including samples was recorded. The crucibles filled with the samples were placed in the oven
and heated up to 550 °C for 3 hrs. After heating, the silica gel remained in the crucibles while all
the polyethylene parts had been degraded. The tracer concentration was determined by weighing
the silica gel residues of the samples. The experimental procedure is schematically shown in
Figure 2-3.
Figure 2-3: RTD experimental procedure, a plus injection of silica gel into the extruder operating
polyethylene, samples were cut numbered and RTD results grasped by weighing the residue of the silica
gel after PE were degraded.
31
2.4.4 Calculation of experimental RTD functions
By using experimental data and applying the following equations the residence time
distribution function, E(t) ,cumulative residence time distribution, F(t) , the mean residence time,
𝒕 , and the variance, 𝝈𝒕𝟐, were computed
36.
E t = C
C dt∞
0
=c
c∆t∞0
(27)
F t = E t dt = E t ∆t = c ∆tt
0
c∆t∞0
t
0
t
0
(28)
The concentration, C, represents the relative concentration of silica gel in each sample. In
our case different samples have different scales ranging from 3 gr up to 5 gr, thus, tracer
concentration, C, was determined with respect to the sample weight and the silica gel residue.
𝐶 =𝑊𝑅
𝑊𝑆 (29)
where,𝑊𝑅 , is the weight of silica gel residue in each sample, and 𝑊𝑆 is the sample weight. The
mean residence time, t , is defined as:
t = tcdt
∞
0
cdt∞
0
= tc∆t∞
0
c∆t∞0
(30)
And the variance σt2, is obtained from:
σt2 = t − t 2E t dt = t − t 2E t ∆t
∞
0
∞
0
(31)
The variance of the residence time was used to evaluate the spread of the distribution36
.
2.5 Results and discussion The experimental RTD results for three mixing sections: single flight, Maddock, and
Saxton are shown in Figure 2-4 .In this figure the operating conditions and screw speed were
kept constant for all three different mixing sections. The mean residence time and the variance
for these three different mixing sections are presented in Table 2-1. In addition, the mean
residence time and variance of the single flight mixing section and Saxton mixing section at two
other screw speeds are shown in Table 2-1. This was done since according to V. J. Davidson
32
(1983), operating variables such as screw speed have significant effects on RTDs33
. Figure 2-5,
2-6 and Table 2-1 present experimental RTD results at three different screw speeds: 20, 50, and
80 rpm in two different mixing sections: single flight and Saxton.
Figure 2-4 shows the residence time curves for three different mixing sections and each
have a similar shape, with the Saxton providing the broadest distribution and the single flight
providing the narrowest distribution. This is confirmed by comparing the variances for these
three different mixing sections (Table 2-1). As expected the highest variance occurs in the
Saxton mixing section and the lowest variance occurs in the single flight. The time at which the
tracer first emerges at 20 rpm in the Saxton mixing element is about twice that of the single flight
element at 20 rpm. The time at which the tracer first emerges in the Maddock mixing section is
more than one and a half times that of the single flight (Figure 2-4). This means that material was
exiting the extruder sooner in the single flight element than in either the Maddock or Saxton
screw configurations (Figure Error! Reference source not found.). It can be seen that there are
noticeable differences between the single flight and Saxton curves in terms of the initial
emergence time, height, and spread, while there are no large differences between Maddock and
Saxton curves. Only in the tail of the Saxton element is there is significant spread compared to
the Maddock element.
The mean residence time for the Saxton mixing section is the highest and that of the
single flight is the lowest. In the Saxton element, the mean residence time is about one and a half
times of that in the single flight. The mean residence time in the Maddock is about 1.3 times of
that in the single flight element.
Table 2-1 Values of 𝐭 and 𝛅𝟐 for three different mixing sections at different screw speeds at Q= 9 kg/hr
Screw Configuration Screw Speed(rpm) Mean residence time(min) Variance 𝛿2(s2)
single flight 20 7.2 1.5 x 105
single flight 50 3.4 3.4 x 104
single flight 80 3.1 2.8 x 104
Maddock 20 9.9 2.8 x 105
Saxton 20 11 3.8 x 105
Saxton 50 4.5 5.8 x 104
Saxton 80 3.5 3.6 x 104
33
Figure 2-4: Comparison of RTD for three different screw types (N= 20 rpm, Q= 9 kg/hr)
As shown in Figure 2-5 and 2-6, the screw speed influences the RTD. As the screw speed
increases, the RTD curve becomes narrower. For both the single flight and Saxton elements, the
mean residence time and variance decreased with increasing screw speed. In both the single
flight and Saxton elements, the reduction in mean residence time and variance is more significant
about 52% when the screw speed changes from 20 rpm to 50 rpm. Also there are no significant
changes in the mean residence times and variances for the single flight and Saxton elements, as
the screw speed change from 50 rpm to 80 rpm. In addition, the initial emergence time is
inversely proportional to the screw rotation speed (Figures 2-5, 2-6). For the single flight mixing
section, the E(t) curve at 20 rpm ended after 24 min, at 50 rpm, after 14 min and at 80 rpm, after
12 min. For the Saxton mixing section, the E(t) curve at 20 rpm ended after 31 min, at 50 rpm, it
ended after 13 min, and at 80 rpm, it ended after 11 min. By increasing the screw rotation speed,
more shears was applied to the material and consequently the shear heating increased and
materials were melted sooner. Therefore, more slip and more back-leakage occurred which
34
results in more axial mixing. The screw speed also affects the flow pattern by altering the hold-
up volume in each screw configuration, and thus, RTD will change by changing the screw speed.
Many researchers studied the effects of screw speed on RTD. Altomare and Ghossi (1986), and
Yeh and Hwang (1992) found that increasing screw speed shifts the RTD to the shorter times and
decreases the main residence time28
27
. Ziegler and Aguilar (2003) showed that screw speed
linearly affects RTD37
.
Figure 2-5: Effect of increasing screw speed on the RTD for single flight mixing section(Q= 9 kg/hr)
35
Figure 2-6: Effect of increasing screw speed on the RTD for Saxton mixing section(Q= 9 kg/hr)
.
Figure 2-7 shows the dimensionless plots of E(t) for three different mixing sections:
single flight, Maddock, and Saxton at 20 rpm screw speed and feed rate of 9 kg/hr. E(ɵ) is the
dimensionless RTD function and ɵ is the dimensionless residence time (ɵ = 𝑡𝑡 ). It can be seen
that the dimensionless RTD curves of E(ɵ) versus ɵ for the three different mixing sections
overlap with each other. This is in contrast with the results found by Xian-Ming Zhang (2008),
who showed that when the ratio of feed rate to screw speed is constant for a fixed screw
configuration, the dimensionless RTD function curves, E(t) curves, overlap with each other
regardless of screw speed and feed rate38. Although the ratio of feed rate to screw speed was
constant here, the screw configuration however was not fixed, the E(t) curve overlap with each
other.
36
Figure 2-7: The dimensionless RTD of the three different screw types (N= 20 rpm, Q= 9 kg/hr)
2.5.1 Experimental verification of the models
To determine which of the 4 different models is the best-fit model for the three different
mixing sections, Figures 2-8, 2-9 and 2-10 show the fits of experimental data to each of the
models.
The validity of the four different models for the single flight mixing section was
experimentally examined (Figure 2-8). As shown in Table 2-2, the best value for dead volume
fraction d in model number 1 according to the least squares method is zero, and therefore, there
is no difference between model number 1 and 2. According to the calculated errors for the
different models presented in Table 2-2, plug flow in series with cascade of CSTRs (model
number 2) is the best-fit model for the single flight mixing section. In terms of initial delay time,
a cascade of CSTRs (model 3) was in poor agreement with the experimental data. However,
37
model 3fits the data very well in the tail region. Plug flow in series with a perfect mixer (model
4) was the best model in terms of the RTD tail but it disagrees poorly with the initial delay time.
Figure 2-8: Experimental RTD curves and models for the single flight mixing section.(Screw speed =
20rpm, feed rate = 9 kg/hr)
Table 2-2 RTD models parameter values for the single flight mixing section by least square error
method, Model 1 has three parameters(n, d, and p), Model 2 has two parameters ( n, and p), Model 3 has
one parameter (n) and Model 4 has one parameter ( p) to be calculated by least square error method,
errors (S Error)are calculated by applying Equation 13
Models n D P Error (S Error)
1 2.1 0 0.2560 5.6%
2 2.1 - 0.2560 5.6%
3 2.7 - - 22.9%
4 - - 0.35 8.4%
38
Figure 2-9: Experimental RTD curves and models for the Maddock mixing section.(Screw speed =
20rpm, feed rate = 9 kg/h r(
Figure 2-9 shows the RTD models 1, 2, and 3 are in good agreement with experimental
data in terms of the first delay time for the Maddock mixing section. Here again there is no
difference between models 1 and 2 as d = 0 (Table 2-3). For the Maddock mixing section, the
best fit is also the plug flow in series with cascade of CSTRs with the lowest error (Table 2-3).
39
Table 2-3: RTD models parameter values for the Maddock mixing section by least square error method.
Model 1 has three parameters (n, d, and p), Model 2 has two parameters (n, and p), Model 3 has one
parameter (n) and Model 4 has one parameter (p) to be calculated by least square error method, errors
(S Error)are calculated by applying Equation 13
Models n d p Error (S Error)
1 2 0 0.37 5.5%
2 2 - 0.37 5.5%
3 4 - - 21.4%
4 - - 0.39 14.5%
Figure 2-10: Experimental RTD curves and models for the Saxton mixing section.(Screw speed = 20rpm,
feed rate = 9 kg/hr)
As shown in Figure 2-10, similar to the single flight and Maddock mixing sections, the
best fit model for the Saxton mixer was plug flow in series with cascade of CSTRs (model 2).
Compared to the two other mixing sections, the Saxton element has the greatest disagreement
with model 3 and in terms of fraction of plug flow, p, in model 4 it provides the largest fraction
(Table 2-4). This means that the Saxton mixer shows more plug flow or a perfect mixer behavior
compared to the two other mixing sections. This is reasonable as in terms of mean residence time
40
as the Saxton has the highest mean residence time. Also in terms of mixing degree, which will be
discussed more throughly in nexet chapter, it will be shown that the Saxton mixer has the highest
dgree of mixing. Taking this all together, the Saxton mixing section exhibited behaviour closer to
the plug flow than either the single flight or the Maddock mixers.
Table 2-3: RTD models parameter values for the Saxton mixing section by least square error method.
Model 1 has three parameters (n, d, and p), Model 2 has two parameters (n, and p), Model 3 has one
parameter (n) and Model 4 has one parameter (p) to be calculated by least square error method, errors
(S Error)are calculated by applying Equation 13
Models n d p Error (S Error)
1 2 0 0.3 7.3%
2 2 - 0.3 7.3%
3 2.6 - - 39.1%
4 - - 0.4 9.7%
2.6 Conclusion
The RTD studies were used to evaluate three different mixing sections: single flight,
Maddock, and Saxton. According to RTD curves, the Saxton mixer showed the widest RTD and
the narrowest RTD belonged to the single flight element. Also, the Saxton element exhibited the
highest mean residence time and the single flight mixing section showed the lowest mean
residence time. This means that material was exposed to a higher range of strains in the Saxton
mixer and was mixed more thoroughly. This finding is reasonable as the Saxton element has a
more complicated geometry with more flow restrictors, and thus, more back mixing occurs in
this screw configuration compared to the two other mixing sections. In addition, screw speed
affected RTD curves and the mean residence time decreased as the screw speed increased for
different mixing sections. This change was more significant for increasing screw speed from 20
rpm to 50 rpm than from changing screw speed from 50 rpm to 80 rpm.
Four different models were tested for three different mixing configurations: Model 1,
Model 2, Model 3, and Model 4. A plug flow in series with a cascade of CSTRs best fit the
experimental data in all three different mixing elements. For all three different mixing sections
41
the dead volume fraction d, which elongates the tail in the distribution was equal to zero. The
plug flow component of the RTD in model 4 for the Saxton mixer was the highest and for the
single flight was the lowest.
42
2.7 References
1. Manas-Zloczower, I., Analysis of mixing in polymer processing equipment. Case Western Reserve University: 1997; pp 5-8.
2. Rauwendaal, C., Mixing in polymer processing. M. Dekker: New York :, 1991. 3. Rauwendaal, C., Polymer mixing : a self-study guide. Hanser Publishers ; Hanser/Gardner Publishers:
Munich; Cincinnati, 1998. 4. Rauwendaal, C., Mixing in polymer processing. M. Dekker: New York, 1991. 5. (a) Rauwendaal, C., Polymer extrusion. Hanser Gardner Publications: Cincinnati, OH :, 2001; (b) Tadmor,
Z.; Gogos, C. G., Principles of polymer processing. Wiley-Interscience: Hoboken, N.J. :, 2006; (c) Baird, D. G.; Collias, D. I., Polymer Processing: Principles and Design. Wiley-Interscience: NY, 1998; (d) Manas-Zloczower, I.; Tadmor, Z., Mixing and compounding of polymers : theory and practice. Hanser Publishers ; Distributed in the USA and Canada by Hanser/Gardner Publications: Munich ; New York : Cincinnati :, 1994.
6. Tucker, C. L., Sample variance measurement of mixing. Chemical Engineering Science 1981, 36 (11), 1829-1839.
7. Galaktionov, O. S.; Anderson, P. D.; Peters, G. W. M.; Meijer, H. E. H., Mapping approach for 3D laminar mixing simulations: application to industrial flows. International Journal for Numerical Methods in Fluids 2002, 40 (3-4), 345-351.
8. Mohr, W. D.; Saxton, R. L.; Jepson, C. H., Mixing in Laminar-Flow Systems. Industrial & Engineering Chemistry 1957, 49 (11), 1855-1856.
9. D. I. Bigio, J. D. B. L. E. D. W. G., Mixing studies in the single screw extruder. Polymer Engineering & Science 1985, 25 (5), 305-310.
10. Kim, S. J.; Kwon, T. H., Measures of mixing for extrusion by averaging concepts. Polymer Engineering & Science 1996, 36 (11), 1466-1476.
11. Maric, M., single screw extruder
Chemical Engineering 584, p. P., Lecture7- Extrusion, Ed. 2009. 12. Pinto, G.; Tadmor, Z., Mixing and residence time distribution in melt screw extruders. Polymer
Engineering & Science 1970, 10 (5), 279-288. 13. J. M. McKelvey, Polymer Processing. John Wiley and Sons: New York, 1962. 14. Kwon, T. H.; Joo, J. W.; Kim, S. J., Kinematics and deformation characteristics as a mixing measure in the
screw extrusion process. Polymer Engineering & Science 1994, 34 (3), 174-189. 15. Amin, M. H. G.; et al., In situ quantitation of the index of mixing in a single-screw extruder by magnetic
resonance imaging. Measurement Science and Technology 2004, 15 (9), 1871. 16. (a) Han, C. D.; Lee, K. Y.; Wheeler, N. C., A study on the performance of the maddock mixing head in
plasticating single-screw extrusion. Polymer Engineering & Science 1991, 31 (11), 818-830; (b) Wong, A. C. Y.; Lam, Y.; Wong, A. C. M., Quantification of dynamic mixing performance of single screws of different configurations by visualization and image analysis. Advances in Polymer Technology 2009, 28 (1), 1-15.
17. Manas-Zloczower, I.; Tadmor, Z., Mixing and compounding of polymers : theory and practice. Hanser Publishers ; Distributed in the USA and Canada by Hanser/Gardner Publications: Munich; New York; Cincinnati, 1994.
43
18. Wolf, D.; Resnick, W., Residence Time Distribution in Real Systems. Industrial & Engineering Chemistry Fundamentals 1963, 2 (4), 287-293.
19. Bigg, D.; Middleman, S., Laminar Mixing of a Pair of Fluids in a Rectangular Cavity. Industrial & Engineering Chemistry Fundamentals 1974, 13 (3), 184-190.
20. Wolf, D.; White, D. H., Experimental study of the residence time distribution in plasticating screw extruders. AIChE Journal 1976, 22 (1), 122-131.
21. Sȩk, J. Technical University, Łódź, Poland 1979. 22. Roemer, M. H.; Durbin, L. D., Transient Response and Moments Analysis of Backflow Cell Model for Flow
Systems with Longitudinal Mixing. Industrial & Engineering Chemistry Fundamentals 1967, 6 (1), 120-129.
23. V. S. Kim, V. V. S., and Yu. V. Stungur, Plaste u. Kautschuk, 1978, 25, 352. 24. Tadmor, Z.; Klein, I., Engineering principles of plasticating extrusion. Van Nostrand Reinhold Co.: New
York, 1970. 25. Jaluria, Y., Fluid Flow Phenomena in Materials Processing---The 2000 Freeman Scholar Lecture. Journal
of Fluids Engineering 2001, 123 (2), 173-210. 26. Lidor, G.; Tadmor, Z., Theoretical analysis of residence time distribution functions and strain distribution
functions in plasticating screw extruders. Polymer Engineering & Science 1976, 16 (6), 450-462. 27. Kumar, A.; Ganjyal, G. M.; Jones, D. D.; Hanna, M. A., Modeling residence time distribution in a twin-
screw extruder as a series of ideal steady-state flow reactors. Journal of Food Engineering 2008, 84 (3), 441-448.
28. Singh, B.; Rizvi, S. S. H., RESIDENCE TIME DISTRIBUTION (RTD) AND GOODNESS OF MIXING (GM) DURING CO2-INJECTION IN TWIN-SCREW EXTRUSION PART II: GM STUDIES. Journal of Food Process Engineering 1998, 21 (2), 111-126.
29. Bigg, D.; Middleman, S., Mixing in a Screw Extruder. A Model for Residence Time Distribution and Strain. Industrial & Engineering Chemistry Fundamentals 1974, 13 (1), 66-71.
30. Levenspiel, O., Chemical reaction engineering. Wiley: New York, 1972. 31. De Ruyck, H., Modelling of the residence time distribution in a twin screw extruder. Journal of Food
Engineering 1997, 32 (4), 375-390. 32. Gao, J.; Walsh, G. C.; Bigio, D.; Briber, R. M.; Wetzel, M. D., Residence-time distribution model for twin-
screw extruders. AIChE Journal 1999, 45 (12), 2541-2549. 33. Davidson, V. J.; Paton, D.; Diosady, L. L.; Spratt, W. A., Residence Time Distributions for Wheat Starch in
a Single Screw Extruder. Journal of Food Science 1983, 48 (4), 1157-1161. 34. Carneiro, O. S.; Caldeira, G.; Covas, J. A., Flow patterns in twin-screw extruders. Journal of Materials
Processing Technology 1999, 92-93, 309-315. 35. Carneiro, O. S.; Covas, J. A.; Ferreira, J. A.; Cerqueira, M. F., On-line monitoring of the residence time
distribution along a kneading block of a twin-screw extruder. Polymer Testing 2004, 23 (8), 925-937. 36. Zhang, X.-M.; Xu, Z.-B.; Feng, L.-F.; Song, X.-B.; Hu, G.-H., Assessing local residence time distributions in
screw extruders through a new in-line measurement instrument. Polymer Engineering & Science 2006, 46 (4), 510-519.
37. Ziegler, G. R.; Aguilar, C. A., Residence time distribution in a co-rotating, twin-screw continuous mixer by the step change method. Journal of Food Engineering 2003, 59 (2-3), 161-167.
38. Zhang, X.-M.; Feng, L.-F.; Hoppe, S.; Hu, G.-H., Local residence time, residence revolution, and residence volume distributions in twin-screw extruders. Polymer Engineering & Science 2008, 48 (1), 19-28.
44
3 Image analysis
3.1 Introduction
Mixing of various ingredients is an important task in many industries such as food
engineering, pharmaceuticals, chemicals, cosmetics, cement, glass-making and polymer
processing1. The single screw extruder is one of the most important processing tools in the
polymer industry to compound and process plastics, rubbers, and fibers. However, the single
screw extruder fitted with a standard screw is an inefficient mixing device because it applies
insufficient low shear strain on the material2. To obtain the desired extent of mixing, the single
screw extruder may be equipped with specialized screws, modified feed zones, mixing sections,
and vents3. It is important to evaluate the extent of mixing in the extrusion process, as the
presence of poor regions of ingredient dispersion may cause products failure4. The mixing
quality of mixtures has been assessed by many methods such as measuring the total interfacial
area between the two mixture components as presented by Spencer and Wiley (1951)5,
calculating the weighted average total strain (WATS) used by Fenner (1979)6, determining the
scale and intensity of segregation as proposed by Danckwerts (1952)7, and evaluating striation
thickness as developed by Spencer and Wiley (1951)5. Besides these theoretical approaches for
evaluating mixing, there are some experimental techniques for evaluating mixing such as
residence time distribution (RTD), extrudate sectional cuts, flow visualization, and image
analysis. Shah (1979) And Gailus (1980) proposed using extrudate sectional cuts to count
stratiforms to evaluate mixing characteristics in a single screw extruder7. Mohr, Clapp, and Starr
(1961) used transparent barrel for flow visualization7. Heeschen (1995)
8 and Wightman (1996)
1
used image analysis method to quantitatively study mixing characteristics. Realpe (2003) used
image analysis to study concentration of powder mixtures9.
Image analysis methods are an automated method to analyze mixing when there are many
sections to be analyzed. For an example, there are dozens of stratiforms for each cross section
when an extrudate is sliced (Figure 3-1). Thus, the acquisition of mixing data for each specimen
45
could be quite tedious without using any automated form of stereology2 in which a computer
does the analysis7, 10
.
Figure 3-1 visualization of sliced extrudate 7
Image analysis is being used increasingly for mixing evaluation as data processing
equipment grows more powerful and less expensive. This method also has been used in the food
engineering field for determination of food quality. Surface color and surface texture of final
extrudates was measured by Brosnan and Sun (2004)11
and Tan, Gao, and Hsieh (1994)12
by
using an image processing method. Smolarz et al (1989), Alahakoon et al (1991), Barrett and
Peleg (1992) all used image analysis methods to evaluate structure of extruded food products12
.
There are three main steps for the image analysis method: 1) obtaining a section, 2)
acquiring a digital image of the section, and 3) analyzing the image10
. By using image analysis,
in-line, on-line and off-line measurements for assessing the mixtures quality are possible. The
differences of these measurements, in-line, on-line, and off-line, are based on the two first steps
of image analyzsis: obtaining a section and acquiring an image. In terms of analysis, there is no
difference whether pictures are taken in an in-line, on-line or off-line modes, as the analysis is
always performed on static images or frames. In the first group (in-line) the interior sections of
the mixture are analyzed by employing tomographic3 methods. Tomographic methods are based
on Maddock‟s “cold screw extrusion” technique, achieved by stopping the screw to cool the
barrel and screw by pushing or pulling the screw out of the barrel, and taking samples from the
screw channel 13
- 1, 13b, 14
. In on-line measurements, a transparent barrel provided visual
observation of the homogenization process4, 15,15b
. Using on-line methods, the dynamic mixing
characteristics of a single screw extruder was evaluated by Wong et al (1997)15b 16.
2 - the spatial interpretation of sections
3 - Tomography is imaging by sections or sectioning, through the use of any kind of penetrating wave.
46
For analyzing the image, different numerical simulations have been proposed by different
researchers. Alemaskin13a, 17
and Wang4 used Renyi/Shannon entropies to quantify distributive
mixing in polymer processing equipment.. Measuring the coefficient of variation was used by
Latif to evaluate the variation in the grey levels between black and white2. The Mean Gray
Values (MGV) was used by Obregon to develop a mathematical model for predicting calibration
curves18
and Realpe used gray image analysis9. Tan used statistical and spectral approaches
such as mean, standard deviation, and third moments to assess colour and surface texture
characteristics12
. Wightman was used image analysis to determine the mean, mode, starndard
deviation, variance, and skewness of local composition in the mixtures1. The variances of the
light intensity were used by Wong to quantify the mixing quality4.
The aim of this work is to develop a quantitative tool to assess the mixing characteristics
in a single screw extruder in three different kind of mixing sections. The degree of mixing of
carbon black with polyethylene in three important mixing sections: single flight (one of the most
basic mixing sections), Saxton (one of the most common distributive mixing sections), and
Maddock (one of the most common dispersive mixing sections) were measured by the image
analysis method. In this work, the off-line method in which the specimens were collected from
exiting extrudates exiting the extruder was used. After collecting samples and taking images
from them, images were analyzed using Matlab software version 7.1 (see appendix).
3.2 Theory Color homogeneity is one of the most important aspects of mixing which can be used to
determine the quality of the mixture. Due to the fact that the color homogeneity relies on the
mixing quality, evaluating the color homogeneity in a mixture can be a quantitative tool for
evaluating mixing efficiency in a single screw extruder. To analyze the color homogeneity in
each sample, the grey-level of each pixel is measured. In order to perform the analysis, we
should first determine the sample area that is a relevant region to process. To do this we should
recognize the boundary of each sample, so that the desired area to be analyzed is defined as the
area contained within the boundary.
To detect sample edge in the image, this work was done for examining derivatives in two
dimensions: horizontal and vertical. The pixel‟s gray-level changes sharply at the edges,
therefore, derivatives at edge points have large values. By taking the absolute value of these
47
derivatives and comparing them with a specified threshold, we can detect the edges. The gray
level values are ranged between 0 for black to 255 for white. To set the best value for the
threshold, according to the absolute values of the derivative (Figure 3-4) we can see that there
are no signals with changing in gray level more than 10 values before the first and after the last
signal magnitude jumps. Therefore, we set the threshold at 10. This means that the pixel gray
level changes at least 10 levels at the edge of sample image. Figure 3-2 shows the image of a
Maddock mixing section and a selected row as a sample. Figure 3-3 shows this row as a one-
dimensional signal and Figure 3-4 shows the absolute values of the derivative. The first and last
value above the threshold defines the starting and finishing points of images‟ edges in a row.
This algorithm was applied to all rows and columns in order to find the desired area. All the
pixels outside the desired area were recalled as removed pixels so that they were not used in the
analysis.
Figure 3-2 selected row in the image referring to the mixture of polyethylene (Dowlex NG 5056D) and
carbon black in the Maddock mixing section at 20rpm and feed rate of 10 kg/hr, operating tempreture at
250° C
Selected
row
48
Figure 3-3 One dimensional signals of selected row in Figure 3-2
Figure 3-4 The absolute values of the derivative of Figure 3-3
49
The next step is defining the mixing degree. In our experiment the best-mixed sample is a
homogenous sample that is all black and the worst case is a sample containing a white non-
mixed region. Figure 3-5 shows the hypothetically ideal figures of the best and the worst
mixtures.
Figure 3-5 a) the best mixture with mixing degree of 100% b) the worst mixture with mixing degree of
0%
The mixing degree for the best case is considered to be 100% and for the worst case it is
set to 0%. A simple criterion for measuring the mixing degree can be the ratio of black areas to
the whole sample. Therefore, the gray level pixels should be converted to a bi-level image i.e.
black and white pixels. For converting gray level pixels to bi-level pixels, the threshold method
was used. Using this method, values greater than the threshold are considered as white pixels and
values less than the threshold are considered as black pixels. Mixing degree is calculated as
follows:
𝐷 = 𝑃−𝑃𝑚𝑖𝑛
(1−𝑃𝑚𝑖𝑛 ) (1)
where mixed region ration, P, is:
𝑃 =𝐵
𝑇 (2)
and where B represents the number of black pixels. T represents the total number of pixels
including both black and white pixels. The minimum value of P,𝑃𝑚𝑖𝑛 , is a hypothetical value as
the mixing degree equal to zero, which is not feasible. Here 𝑃𝑚𝑖𝑛 is set at 50% which refers to
the condition where half of the mixture is white and the other half is black.
50
There is a major drawback with the above-mentioned algorithm. As shown in Figure 3-6 in some
cases, it is possible that two different samples have the same amount of degree of mixing, D,
while according to their images, it is obvious that one of them is the better mixture than the other
one and the mixing degree of one should be higher than the other. In Figure 3-6, there are two
samples which both have mixing degrees equal to 82%, but the “a” seems to be better mixed
compared to the” b “sample.
Figure 3-6 Two samples with mixing degree of 82%
An image filter was used to solve this problem. Through image filtering, the narrow lines
in the images can be removed. The length of the filter depends on the image resolution and the
extent of narrow lines that needed to be removed. Here we have used the Gaussian 5X5 filter to
filter the images shown in Figure 3-7. After filtering, the mixing degree for sample “a” is 84%
and for sample “b”, the mixing degree is 76%.
51
Figure 3-7 Filtered images a) filtered image with 84% mixing degree b) filtered image with 76% mixing
degree
Here in this work, we have calculated the mixing degree of images by using equation 1 and 2
after filtering them with Gaussian 5X5 filter.
3.3 Experimental section
The experimental procedure is divided into two sections: sample collecting and taking
images. The first section, sample collecting, was done at Maillefers‟ laboratory, while the other
section, taking images was done at McGill University.
3.3.1 Material
To assess the effects of different materials on mixing, here we used 2 different
polyethylene resins from which two different mixtures were prepared. All of the used materials
were kindly donated by Maillefer and they were used as received. The two different polyethylene
resins were: a linear low-density polyethylene extrusion grade resin (Dowlex NG 5056D), and a
52
medium density polyethylene extrusion grade resin (Eraclene FB 506). A linear low density
black polyethylene masterbatch (Plasblak LL4932) was used as a black pigment (i.e. tracer).
Two different mixtures of the afore-mentioned polyethylene resins were Mix 1 (25% weight
Dowlex, 73% weight Eraclene, 2% weight Plasblak) and Mix 2 (70% weight Dowlex, 28%
weight Eraclene, 2% weight Plasblak).
3.3.2 Equipment
For collecting samples, a single-screw extruder, (Maillefer NMC 45-24D, screw speed
215 rpm max, motor power = 29 kW) was used. To record rheological information such as
viscosity and power-law exponent (n) for the different materials and mixtures, the extruder was
attached to a slit die (Maillefer rheological die ENS Filiere rheologique). Figure 3-8 shows the
die used here. The die is constructed out of two sections: in one section, it is 7 mm in depth,
length=100 mm and a shallow section with 2.5 mm depth and length= 40 mm. There were four
pressure transducers measuring the pressure at the beginning and at the end of each section. The
most important part of the extruder was the interchangeable mixing sections all with length =
270 mm and diameter = 45 mm: single flight, Maddock, and Saxton, which were each used
separately.
Figure 3-8 Rheology die
First channel
Second channel
Pressure transducer
53
Figure 3-9 shows schematically the two sections of the die and the place and name of
each pressure transducer. The plastic cutter machine (Leitz Wetzlar) was used to slice the
samples (Figure 3-10). The optical microscope (Olympus BX60) was used to take pictures of the
sliced, compressed samples. A hydraulic compression press (Macklow-Smith S/N R33/92) was
used to make compressed samples out of sliced samples for taking images by microscope.
The second part, taking images from the sliced samples, was done at McGill. A scanner
(Canoscan 4400F) was used to take pictures from the sliced samples.
Figure 3-9 schematic picture of two sections of the die
Figure 3-10 Plastic cutter
« Rheo 1 » for
deep channel « Rheo 2 » for
deep channel
« Rheo 1 » for
shallow
channel
« Rheo 2 » for
shallow
channel
54
3.3.3 Experimental procedure
The materials (Dowlex, Eracelen, Mix1, and Mix2) with aconstant feed of the carbon
black masterbatch at 2% weight were extruded. To analyze the effects of screw speed on mixing,
the extruder was operated at two screw speeds: 20 rpm and 80 rpm. Feed rate at 20 rpm screw
speed was 10 kg/hr and at 80 rpm was 40 kg/hr rpm. After reaching to steady output and
stabilized temperature at 250° C, the samples were collected. Before collecting samples, extruder
operating conditions such as the average melt temperature, rate of output, and pressure profile in
the die were recorded by four pressure transducers located in the die. The extruder operating
conditions for each used material at different mixing sections and different same screw speeds
are shown in Table 3-1, 3-2, 3-3 and 3-4. The data presented in Table 3-1, 3-2, 3-3, and 3-4 was
used to automatically calculate the rheological properties (viscosity and power-law exponent (n))
of each material shown in Table 3-5(see Appendix).
Table 3-1 The operating conditions table for Eraclene FB506 material
Mixing
section
Screw
speed
Output
(kg/hr)
Ave melt
Temp °C
Height of the
channel (m)
Length of
channel (m)
Pressure at
Rheo1 (bar)
Pressure at
Rheo2 (bar)
Maddock 20 10.73 218 0.007 0.1 225.84 198.39
0.0025 0.04 123.84 58.13
Maddock 80 42.4 212 0.007 0.1 359.75 306.48
0.0025 0.04 190.28 91.43
Single
flight
20 10.8 218 0.007 0.1 216.60 189.56
0.0025 0.04 120.73 55.80
Single
flight
80 41.93 215 0.007 0.1 341.35 291.19
0.0025 0.04 184.71 88.52
Saxton 20 10.53 217 0.007 0.1 211.71 184.83
0.0025 0.04 118.97 55.8
Saxton 80 42.4 229 0.007 0.1 334.33 285.14
0.0025 0.04 180.62 86.31
55
Table 3-2 The operating conditions table for Dowlex NG5056G material
Mixing
section
Screw
speed
Output
(kg/hr)
Ave melt
Temp °C
Height of the
channel (m)
Length of
channel (m)
Pressure at
Rheo1 (bar)
Pressure at
Rheo2 (bar)
Single
flight
20 9.93 213 0.007 0.1 230.99 204.37
0.0025 0.04 129.25 55.43
Single
flight
80 38.73 228 0.007 0.1 400.45 343.13
0.0025 0.04 214.42 95.41
Maddock 20 9.8 215 0.007 0.1 232.74 208.59
0.0025 0.04 131.35 55.96
Maddock 80 38.66 240 0.007 0.1 387.26 333.32
0.0025 0.04 206.36 91.02
Saxton 20 9.73 216 0.007 0.1 236.58 211.72
0.0025 0.04 134.66 57.53
Saxton 80 38.53 218 0.007 0.1 427.11 356.65
0.0025 0.04 222.29 100.60
Table 3-3 The operating conditions table for Mix1 material
Mixing
section
Screw
speed
Output
(kg/hr)
Ave melt
Temp °C
Height of the
channel (m)
Length of
channel (m)
Pressure at
Rheo1 (bar)
Pressure at
Rheo2 (bar)
Single
flight
20 10.66 215 0.007 0.1 227.04 200.112
0.0025 0.04 127.11 58.053
Single
flight
80 41.6 222 0.007 0.1 374.92 318.709
0.0025 0.04 199.71 95.01
Maddock 20 10.6 220 0.007 0.1 217.07 190.577
0.0025 0.04 121.08 54.711
Maddock 80 41.33 229 0.007 0.1 349.7 298.468
0.0025 0.04 187.63 87.24
Saxton 20 10.4 219 0.007 0.1 221.9 194.797
0.0025 0.04 124.74 57.086
Saxton 80 41.33 224 0.007 0.1 360.22 307.416
0.0025 0.04 194.06 91.225
Table 3-4 The operating conditions table for Mix2 material
Mixing
section
Screw
speed
Output
(kg/hr)
Ave melt
Temp °C
Height of the
channel (m)
Length of
channel (m)
Pressure at
Rheo1 (bar)
Pressure at
Rheo2 (bar)
Single
flight
20 10.2 218 0.007 0.1 226.42 200.893
0.0025 0.04 127.23 55.194
Single
flight
80 39.66 235 0.007 0.1 377.95 323.711
0.0025 0.04 201.89 90.46
Maddock 20 9.93 218 0.007 0.1 232.36 206.833
0.0025 0.04 130.58 56.966
Maddock 80 39.33 226 0.007 0.1 390.83 334.223
0.0025 0.04 208.9 94.366
Saxton 20 9.93 218 0.007 0.1 230.61 204.488
0.0025 0.04 130.03 56.966
Saxton 80 39.26 222 0.007 0.1 405.07 345.829
0.0025 0.04 213.38 96.942
56
According to Table 3-1, 3-2, 3-3 and 3-4, we expect to have different mixing
characteristics for four different materials used here (Dowlex, Eracelen, Mix1, and Mix2).
After cooling the sample, it was sliced into 2.5 mm thick slices (5 slices for each sample).
For microscopy, the 2 slices of each sample were compressed at 200°C under pressure = 12
kg/cm2
for 20 min. After compressing samples, the microscope was used to take images with
2.5X magnification and 747.5 pixel/mm resolution. The sliced samples were scanned as a grey
scale positive film with 1200 dpi resolution and 37.15 pixel/mm resolution. The experimental
procedure is schematically shown in Figure 3-11.
Figure 3-11 Schematic experimental procedure, (a) the samples are compressed after being sliced and
the microscope is used to take images, (b) sliced samples were scanned
3.4 Results and discussion
Figure 3-13, 3-14, and 3-15 show zoomed images of different blends: Dowlex, Eraclene,
Mix1, and Mix2 in various mixing sections. These pictures were taken at 2X magnification and
57
747.5 pixels/mm resolution. As shown in Figure 3-13 the mixing quality becomes better as the
screw speed increases. Here we can see that a mixture of Dowlex/carbon black masterbatch has
fewer coarser striations at 20 rpm compared to the other mixtures. According to rheological data
shown in Table 3-5, Dowlex has the lowest viscosity of all the materials studied. This is
important in terms of mixing. First, lower viscosity fluids are more effective for distributive
mixing as there is no need to break any agglomerates down in distributive mixing. There low
viscosity permits more relative motion between the components in the mixture. This result is
confirmed with the
Calculated degree of mixing shown in Table 3-6 that were derived from scanned images
by using Equations 1 and 2. In Table 3-6, we can also see that the degree of mixing is increased
as the screw speed is increased. Comparing the degree of mixing for one screw configuration at a
particular screw speed, we can see that a mixture of Dowlex polyethylene resin with the carbon
black masterbatch has the highest degree of mixing, and the lowest degree of mixing belongs to
the Eraclene/carbon black mixture. This finding is confirmed by comparing viscosities provided
in Table 3-5, which shows that Dowlex has the lowest viscosity and Eraclene has the highest
viscosity.
Table 3-5 Viscosity and Power-law exponent calculated by rheology die for different material
Material Viscosity constant @ 200°C (Pa s-1) Power-law exponent (n)
Dowlex NG5065G 9822 0.55
Eraclene FB506 15396 0.41
Mix1 14653 0.46
Mix2 10590 0.49
Comparing degrees of mixing at a screw speed of 20 rpm screw speed for one type of
material but for different mixing sections (Table 3-6) we can see that mixtures in a single flight
mixing section had significantly lower degree of mixing compared to the Maddock and Saxton
elements. These results confirm the findings from the residence time distribution (RTD)
experiments discussed in Chapter 2. This finding is reasonable according to the geometries of
these three mixing elements (single flight, Maddock, and Saxton). As shown in Figure 3-12, we
can see that the single flight mixing element consists of a single primary helical flight but the
Maddock and Saxton have more complicated geometries. The Maddock mixer consists of a
series of grooves parallel to the screw length. The ends of each series are open which alternately
58
ends to the feed flow and discharge or exiting flow. There are two lands at each row: discharge
or mixing lands that are located between the entrance and discharge groove with diameter
slightly lower than the screw diameter and the wiping lands that are located between the
discharge and next entrance groove with the same diameter as the screw. The materials are
subjected to the high amount of shear at the mixing lands. The Saxton mixer consists of a
multi-flighted screw section with a reverse pitch channel cut into the flights. There are primary
and secondary flights in the Saxton mixing section, and the secondary flights have different helix
angle than the primary flights. The melt stream is sheared and recombined by facing the
interrupting secondary flights. There are no mixing regions in the geometry of the single flight
mixing section; there are high shear regions in the Maddock mixing section; in the Saxton mixer
the material are both sheared and reoriented, thus, the mixtures blended in the Saxton mixing
section have good qualities.
3-12: a) Maddock mixing section b) Saxton mixing section, c) single flight mixing section
59
Table 3-6 Degree of mixing calculated from scanned pictures
Mixing section Screw speed
(rpm)
Mixing degree %
Dowlex
Mixing degree %
Eraclene
Mixing degree %
Mix1
Mixing degree %
Mix2
Single flight 20 45 14 27 43
80 75 29 58 70
Maddock 20 76 65 65 68
80 97 81 85 89
Saxton 20 87 76 77 84
80 98 96 92 98
3-13: Zoomed images of different mixtures of polyethylene resins, each containing 2 w% carbon black
masterbatch in a single flight mixing sections at two different screw speeds 20 rpm and 80 rpm
(magnification = 2.5X, resolution.= 747.5 pixels/mm)
As shown in Figure 3-14 as speeds are increased, the mixing quality improves in all
cases. Mixing in the Dowlex/carbon black blends is easiest, even in the single flight element. In
the more viscous mixtures, the striations are quite noticeable and not fine. In the blends
compounded in the Maddock mixer (Figure 3-14) compared to those blended in the single flight
60
mixer (Figure 3-13) the striations become much finer in the Maddock mixer, even in the higher
viscosity mixtures. Figure 3-15 shows the zoomed images of the Saxton mixing section, where
we can see even at 20 rpm screw speed in the highest viscosity blend, the mixing is better
compared to the single flight or Maddock mixing element. This is quantified in Table 3-6. The
degree of mixing noticeably improves by increasing the screw speed in the single flight mixer,
the degree of mixing increased around 69% as the screw speed increases from 20 rpm into 80
rpm. This increase is much less noticeable for the Maddock (around 28%) and the Saxton
(around 18%), which means that mixing was sufficiently good at the lower screw speeds.
3-14 Zoomed images of different mixtures of polyethylene resins, each containing 2 w% carbon black
masterbatch in a Maddock mixing sections at two different screw speeds 20 rpm and 80 rpm
(magnification = 2.5X, resolution.= 747.5 pixels/mm)
61
Figure 3-15 Zoomed images of different mixtures of polyethylene resins, each containing 2 w% carbon
black masterbatch in a Saxton mixing sections at two different screw speeds 20 rpm and 80 rpm
(magnification = 2.5X, resolution.= 747.5 pixels/mm)
3.5 Conclusion
A new method for evaluating degree of mixing was developed as a means of comparing three
different elements in a single screw extruder. According to this method, the scanned images were
filtered by Gaussian 5X5 filter, and then the degree of mixing was calculated. As is sometimes
obvious from the visual inspection, the degree of mixing quantitatively provided a means of
comparison for the different elements. The Saxton mixing section provided mixtures with the
highest degree of mixing and mixtures compounded with the single flight mixing section had the
lowest degree of mixing. This finding is corroborated by the results of the RTD experiments in
Chapter 2.
Moreover, screw speed and material both affected mixing degree. The degree of mixing
increased as the screw speed increased in all elements. This change was more significant for
mixtures getting mixed by the single flight mixing section- it required the most energy to achieve
the mixing that was otherwise accomplished at lower screw speeds in more intensive mixing
elements such as the Saxton or Maddock elements. The effects of viscosity on material on
mixing quality were also investigated. The material with the lowest viscosity, Dowlex
62
NG5065G, provided the highest degrees of mixing in all screw speeds and configurations. The
mixtures of Eraclene FB506, the material with highest viscosity, had the lowest mixing degree in
all three different mixing sections.
63
3.6 References
1. Wightman, C.; Muzzio, F. J.; Wilder, J., A quantitative image analysis method for characterizing mixtures of granular materials. Powder Technology 1996, 89 (2), 165-176.
2. Latif, L.; Saidpour, S. H., Assessment of pigment distribution in molded samples using an image processing technique. Advances in Polymer Technology 1996, 15 (4), 337-344.
3. Rauwendaal, C., New directions for extrusion: compounding with single screw extruders. Plastics, Additives and Compounding 2002, 4 (6), 24-27.
4. Wong, A. C. Y.; Lam, Y.; Wong, A. C. M., Quantification of dynamic mixing performance of single screws of different configurations by visualization and image analysis. Advances in Polymer Technology 2009, 28 (1), 1-15.
5. Spencer, R. S.; Wiley, R. M., The mixing of very viscous liquids. Journal of Colloid Science 1951, 6 (2), 133-145.
6. Fenner, R. T., Principles of polymer processing. Chemical Pub.: New York, 1980. 7. Strasser, R. A.; Erwin, L., Experimental techniques for analysis distributive laminar mixing in
continuous flows. Advances in Polymer Technology 1984, 4 (1), 17-32. 8. Heeschen, W. A., A quantitative image analysis method for the determination of cocontinuity in
polymer blends. Polymer 1995, 36 (9), 1835-1841. 9. Realpe, A.; Velázquez, C., Image processing and analysis for determination of concentrations of
powder mixtures. Powder Technology 2003, 134 (3), 193-200. 10. Rauwendaal, C., Polymer mixing : a self-study guide. Hanser Publishers ; Hanser/Gardner
Publishers: Munich; Cincinnati, 1998. 11. Brosnan, T.; Sun, D.-W., Improving quality inspection of food products by computer vision--a
review. Journal of Food Engineering 2004, 61 (1), 3-16. 12. Tan, J.; Gao, X.; Hsieh, F., Extrudate Characterization by Image Processing. Journal of Food
Science 1994, 59 (6), 1247-1250. 13. (a) Alemaskin, K.; Manas-Zloczower, I.; Kaufman, M., Entropic analysis of color homogeneity.
Polymer Engineering & Science 2005, 45 (7), 1031-1038; (b) Dal Grande, F.; Santomaso, A.; Canu, P., Improving local composition measurements of binary mixtures by image analysis. Powder Technology 2008, 187 (3), 205-213.
14. Wightman, C.; Muzzio, F. J., Mixing of granular material in a drum mixer undergoing rotational and rocking motions I. Uniform particles. Powder Technology 1998, 98 (2), 113-124.
15. (a) Kim, S. J.; Kwon, T. H., Enhancement of mixing performance of single-screw extrusion processes via chaotic flows: Part II. Numerical study. Advances in Polymer Technology 1996, 15 (1), 55-69; (b) Wong, A.; Lam, Y., Visualization study on the dynamic mixing quality during single-screw extrusion. Journal of Polymer Research 2008, 15 (1), 11-19.
16. Muerza, S.; Berthiaux, H.; Massol-Chaudeur, S.; Thomas, G., A dynamic study of static mixing using on-line image analysis. Powder Technology 2002, 128 (2-3), 195-204.
17. Alemaskin, K.; Manas-Zloczower, I.; Kaufman, M., Color mixing in the metering zone of a single screw extruder: Numerical simulations and experimental validation. Polymer Engineering & Science 2005, 45 (7), 1011-1020.
18. Obregón, L.; Velázquez, C., Discrimination limit between mean gray values for the prediction of powder concentrations. Powder Technology 2007, 175 (1), 8-13.
64
4 General Conclusion
4.1 Summary
Mixing in single screw extruders has been studied extensively for many years. The key
challenge is to select the best mixing elements from among the different existing mixing
elements to accomplish satisfactory mixing. To develop quantitative parameters for evaluating
different mixing elements in single screw extruders, here we compared mixing in three different
mixing elements using residence time distribution and image analysis methods. For residence
time distributions, the tracer was injected and the concentration was detected by weighing the
ash residues of the sample. For image analyzing, the mixture of polyethylene based black master
batch (a mixture of carbon black dispersed in polyethylene) and polyethylene resins were
extruded, and the resulting samples for analysis were taken from sliced samples that were
scanned after collecting and slicing. Matlab software was used for analyzing the scanned images
and calculating the degree of mixing. The residence time distribution studies provided mean
residence times to evaluate different mixing elements. The image analyzing method provided
degree of mixing as a quantitative parameter to compare different mixing elements. Taken
together, our results confirmed the experimental data and demonstrated that for non-thermal
sensitive polymers, the higher the mean residence time, generally the better was the mixing. The
degree of mixing was higher in Saxton mixing section compared to single flight and Maddock
mixing sections. There were fewer striations in images of mixtures mixed by the Saxton element
than in mixtures taken from the Maddock and the single flight element. The single flight element
was clearly an inefficient mixing section. These two parameters, residence time distribution and
65
degree of mixing, not only can be used as quantitative parameters to evaluate different types of
mixing elements but also can be used as a data basis for predictive simulations.
4.2 Future Work
To further complete this study, more mixing sections must be tested. Here we compared
three most common mixing sections, but for making an application window for different mixing
sections, more mixing sections are needed to be studied. Also different mixing mechanisms,
dispersive and distributive, must be studied for these three mixing sections.
According to the fact that the axial shear rates in the center region of the screw channel
are very low, for improving axial mixing in extrusion processes, the mixing sections must be
used to redistribute the material from the center region to the outer region of the channel.
Observing the complex geometries of these two mixing sections (Figure 4-1) we can see that
there are some random slots between the flights in Saxton mixing section while in the Maddock
section, the flouted flights are lined perpendicular to each other and there are no slots between
each flight. So we can propose that the mixing characteristic in a Maddock mixing section can be
improved by creating some slots to vary the spacing between the flights. By doing this in fact,
streams of different axial velocities will be created.
4-1 a) Maddock mixing section b) Saxton mixing section
66
Appendix
Calculating viscosity and Power-law exponent (n)
The following equations are set in the control panel of the single-screw extruder (Maillefer NMC
45-24D) which are automatically calculated as the four pressure transducers are the die be
activated.
𝜂 = 𝑚0 ∗ exp(𝛽 ∗ 200)
𝑚0 = exp(ln( 𝑚0))
ln 𝑚0 = 𝑙𝑖𝑛𝑒𝑠𝑡 (𝑦𝑖 , 𝑇𝑖 , 𝑀𝑖)
𝑦𝑖 = ln(𝑑𝑝 ∗𝐻
2)
𝑀𝑖 = ln(𝑉) ∗ 2 +1
𝑛𝐹 ∗ (
2
𝑓𝑝 ∗ 𝑤 ∗ 𝐻2)
𝑉 =𝑜𝑢𝑡𝑝𝑢𝑡
𝑟ℎ𝑜/3600
𝛽 = 𝑙𝑖𝑛𝑒𝑠𝑡 (𝑦𝑖 , 𝑇𝑖 , 𝑀𝑖)
𝑛 = 𝑙𝑖𝑛𝑒𝑠𝑡 (𝑦𝑖 , 𝑇𝑖 , 𝑀𝑖)
Nomenclature in data sheets:
Output Mass throughput kg/h
Mass Temp(𝑇𝑖) . Melt temperature at exit °C
Rheo 1 Pressure reading of first transducer bar
Rheo2 Pressure reading of second transducer bar
H Height of the channel m
L Length of channel m
W Width of channel m
rho estimated melt density kg/m3
fp Estimated shape factor of rectangular channel -
𝑛𝐹 Power law exponent (fitted) beta temperature shift factor m0 viscosity constant at 0°C
67
y T M=LN(VRate*(2+1/n)*2/(fp*W*H*H)) dP meas dP calc Ratio
12.24217 222 5.846793 59.243 54.33585 0.917169
11.4001 0 4.472313 25.523 28.78309 1.127731
12.15388 235 5.856929 54.237 54.4666 1.004233
11.40025 218 4.498805 25.527 27.91369 1.093497
12.19667 226 5.84849 56.608 54.3378 0.959896
11.4233 218 4.472313 26.122 27.55025 1.054676
12.80451 222 7.293854 116.433 124.5112 1.069381
12.34601 0 5.919374 73.613 65.95675 0.895993
12.76055 235 7.303989 111.426 124.8108 1.120123
12.32436 218 5.88737 72.036 62.14 0.862624
12.78803 226 7.29555 114.53 124.5157 1.087188
12.33858 218 5.919374 73.068 63.1317 0.864013
n beta LN(m0) m0 R min 0.494715 -0.0002 9.307785 11023.51 0.862624
Mix2
Data set
rho 770
w 0.015
n 0.49 Starting value- Compare to value from LINEST
Output
Mass temp. Rheo 1 Rheo2 H L fp
39.26667 222 405.072 345.829 0.007 0.1 0.45
9.933333 232.356 206.833 0.007 0.1 0.45
39.66667 235 377.948 323.711 0.007 0.1 0.45
10.2 218 226.42 200.893 0.007 0.1 0.45
39.33333 226 390.831 334.223 0.007 0.1 0.45
9.933333 218 230.61 204.488 0.007 0.1 0.45
39.26667 222 213.375 96.942 0.0025 0.04 0.83 9.933333
130.579 56.966 0.0025 0.04 0.83
39.66667 235 201.886 90.46 0.0025 0.04 0.83 10.2 218 127.23 55.194 0.0025 0.04 0.88 39.33333 226 208.896 94.366 0.0025 0.04 0.83 9.933333 218 130.034 56.966 0.0025 0.04 0.83
68
Mix1
Data set
rho 770
w 0.015
n 0.46 Starting value- Compare to value from LINEST
Output
Mass temp. Rheo 1 Rheo2 H L fp
41.6 222 374.921 318.709 0.007 0.1 0.45
10.66667 215 227.04 200.112 0.007 0.1 0.45
41.33333 229 349.699 298.468 0.007 0.1 0.45
10.6 220 217.068 190.577 0.007 0.1 0.45
41.13333 224 360.215 307.416 0.007 0.1 0.45
10.4 219 221.88 194.797 0.007 0.1 0.45
41.6 222 199.705 95.01 0.0025 0.04 0.83 10.66667 215 127.113 58.053 0.0025 0.04 0.83 41.33333 229 187.632 87.24 0.0025 0.04 0.83 10.6 220 121.077 54.711 0.0025 0.04 0.88 41.13333 224 194.058 91.225 0.0025 0.04 0.83 10.4 219 124.738 57.086 0.0025 0.04 0.83
y T LN(VRate*(2+1/n)*2/(fp*W*H*H)) dP meas dP calc Ratio
12.18965 222 5.936925 56.212 52.0885 0.926644
11.45368 215 4.575948 26.928 29.78156 1.10597
12.09686 229 5.930494 51.231 48.54461 0.947563
11.43732 220 4.569679 26.491 28.29792 1.068209
12.12701 224 5.925643 52.799 50.82888 0.962686
11.45942 219 4.55063 27.083 28.32265 1.045772
12.69824 222 7.383985 104.695 113.5745 1.084813
12.28217 215 6.023009 69.06 64.93614 0.940286
12.65627 229 7.377555 100.392 105.8474 1.054341
12.24237 220 5.958243 66.366 60.06175 0.905008
12.6803 224 7.372704 102.833 110.828 1.077748
12.26157 219 5.997691 67.652 61.75512 0.912835
n beta LN(m0) m0 R min
0.460372 -
0.00964 11.521 100810.8 0.905008
69
Dowlex NG5056G
Data set
rho 770
w 0.015
n 0.55 Starting value- Compare to value from LINEST
Output Mass temp. Rheo 1 Rheo2 H L fp 9.93333333 213 230.998 204.371 0.007 0.1 0.45
9.13333333 210 234.18 206.677 0.007 0.1 0.45
38.5333333 218 427.112 356.653 0.007 0.1 0.45
9.8 215 232.745 208.591 0.007 0.1 0.45
38.6666667 240 387.261 333.324 0.007 0.1 0.45
38.7333333 228 400.454 343.132 0.007 0.1 0.45
9.73333333 216 236.586 211.717 0.007 0.1 0.45
9.93333333 213 129.255 55.436 0.0025 0.04 0.83 9.13333333 210 130.618 56.402 0.0025 0.04 0.83 38.5333333 218 222.293 100.606 0.0025 0.04 0.83 9.8 215 131.358 55.959 0.0025 0.04 0.88 38.6666667 240 206.365 91.024 0.0025 0.04 0.83 38.7333333 228 214.426 95.412 0.0025 0.04 0.83 9.73333333 216 134.669 57.529 0.0025 0.04 0.83
y T LN(VRate*(2+1/n)*2/(fp*W*H*H)) dP meas dP calc Ratio
11.44244 213 4.415641 26.627 28.64995 1.075974
11.47481 210 4.331676 27.503 27.98273 1.017443
12.41555 218 5.771269 70.459 58.0234 0.823506
11.34497 215 4.402127 24.154 28.01576 1.159881
12.14833 240 5.774723 53.937 49.30369 0.914098
12.2092 228 5.776445 57.322 53.99017 0.941875
11.37414 216 4.395301 24.869 27.7029 1.113953
12.34881 213 5.862702 73.819 70.9332 0.960907
12.35417 210 5.778736 74.216 69.28127 0.933509
12.84864 218 7.218329 121.687 143.6577 1.180551
12.36998 215 5.790692 75.399 67.17407 0.890915
12.79508 240 7.221784 115.341 122.0689 1.058331
12.82643 228 7.223506 119.014 133.672 1.123162
12.39281 216 5.842362 77.14 68.58844 0.889142
n beta LN(m0) m0 R min
0.548186 -
0.00749 10.69007 43917.59 0.823506
70
Eraclene FB506
Data set
rho 770
w 0.015
n 0.414 Starting value- Compare to value from LINEST
Output
Mass temp. Rheo1 Rheo2 H L fp
10.7333333 218 225.837 198.392 0.007 0.1 0.45
42.4 212 359.749 306.478 0.007 0.1 0.45
10.8 218 216.602 189.561 0.007 0.1 0.45
42.4 229 334.333 285.143 0.007 0.1 0.45
41.9333333 215 341.356 291.199 0.007 0.1 0.45
10.5333333 217 211.713 184.833 0.007 0.1 0.45
10.7333333 218 123.842 58.133 0.0025 0.04 0.83
42.4 212 190.281 91.427 0.0025 0.04 0.83
10.8 218 120.727 55.798 0.0025 0.04 0.83
42.4 229 180.623 86.314 0.0025 0.04 0.88
41.9333333 215 184.712 88.528 0.0025 0.04 0.83
10.5333333 217 118.974 55.798 0.0025 0.04 0.83
y T LN(VRate*(2+1/n)*2/(fp*W*H*H)) dP meas dP calc Ratio
11.4727 218 4.638437 27.445 28.63181 1.043243
12.13591 212 6.012231 53.271 51.42135 0.965278
11.45787 218 4.644628 27.041 28.70537 1.06155
12.05621 229 6.012231 49.19 49.10316 0.998235
12.07568 215 6.001163 50.157 50.77108 1.012243
11.4519 217 4.619627 26.88 28.48672 1.059774
12.23243 218 6.085497 65.709 58.40842 0.888895
12.64083 212 7.459291 98.854 104.8987 1.061148
12.22048 218 6.091689 64.929 58.55847 0.901885
12.59377 229 7.400795 94.309 97.77082 1.036707
12.61345 215 7.448224 96.184 103.5722 1.076813
12.19311 217 6.066688 63.176 58.11243 0.91985
n beta LN(m0) m0 R min
0.414366 -
0.00271 10.18458 26491.41 0.888895
71
Matlab codes
%% RTD modeling program
clc clear close all load processed_data
%% tmin_single20 = 12; t=deltat*(0:length(Et_single20)-1); p_single20 = t(tmin_single20)/tbar_single20; theta_single20 = t/tbar_single20;
tmin_maddox20 = 23; t=deltat*(0:length(Et_maddox20)-1); p_maddox20 = t(tmin_maddox20)/tbar_maddox20; theta_maddox20 = t/tbar_maddox20;
tmin_sax20 = 24; t=deltat*(0:length(Et_sax20)-1); p_sax20 = t(tmin_sax20)/tbar_sax20; theta_sax20 = t/tbar_sax20;
%% %singleflight 20 disp ('-----------Table 2 Singleflight--------------------'); disp ('data n d p error');
Et_single20 = tbar_single20*Et_single20; Et_maddox20 = tbar_maddox20*Et_maddox20; Et_sax20 = tbar_sax20*Et_sax20; Et_single20(1:8) = 0; Et_maddox20(1:18) = 0; Et_sax20(1:18) = 0;
hold on plot(theta_single20, Et_single20,'ok','LineWidth',3);
[n,d,p] = find_flow_params(1,Et_single20',theta_single20,p_single20); e_single20_1 = flowmodel(1,theta_single20,n,p_single20,d); e_single20_1 = max(Et_single20)*e_single20_1/max(e_single20_1); err_single20_1 = sum((e_single20_1-Et_single20').*(e_single20_1-
Et_single20')); plot(theta_single20, e_single20_1,':r','LineWidth',2); disp (['Model1 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_single20_1)]);
[n,d,p] = find_flow_params(2,Et_single20',theta_single20,p_single20); e_single20_2 = flowmodel(2,theta_single20,n,p_single20,d); e_single20_2 = max(Et_single20)*e_single20_2/max(e_single20_2);
72
err_single20_2 = sum((e_single20_2-Et_single20').*(e_single20_2-
Et_single20')); plot(theta_single20, e_single20_2,'+b','LineWidth',2,'MarkerSize',6); disp (['Model2 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_single20_2)]);
[n,d,p] = find_flow_params(3,Et_single20',theta_single20,p_single20); e_single20_3 = flowmodel(3,theta_single20,n,p_single20,d); e_single20_3 = max(Et_single20)*e_single20_3/max(e_single20_3); err_single20_3 = sum((e_single20_3-Et_single20').*(e_single20_3-
Et_single20')); plot(theta_single20, e_single20_3,'--g','LineWidth',3); disp (['Model3 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_single20_3)]);
[n,d,p] = find_flow_params(4,Et_single20',theta_single20,p_single20); e_single20_4 = flowmodel(4,theta_single20,n,p,d); e_single20_4 = max(Et_single20)*e_single20_4/max(e_single20_4); err_single20_4 = sum((e_single20_4-Et_single20').*(e_single20_4-
Et_single20')); h = plot(theta_single20, e_single20_4,'-.m','LineWidth',3); disp (['Model4 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_single20_4)]);
xlabel('Normalized time'); ylabel('Residence time distribution function E(t)'); title ('Singleflight'); legend('Experimental','Model 1','Model 2','Model 3','Model 4'); xlim([0 4.2]); saveas(h,'sigleflight_20','tif') ;
%% %maddox disp (' '); disp ('-----------Table 3 Maddox--------------------'); disp ('data n d p error'); figure hold on plot(theta_maddox20, Et_maddox20,'ok','LineWidth',3);
[n,d,p] = find_flow_params(1,Et_maddox20',theta_maddox20,p_maddox20); e_maddox20_1 = flowmodel(1,theta_maddox20,n,p_maddox20,d); e_maddox20_1 = max(Et_maddox20)*e_maddox20_1/max(e_maddox20_1); err_maddox20_1 = sum((e_maddox20_1-Et_maddox20').*(e_maddox20_1-
Et_maddox20')); plot(theta_maddox20, e_maddox20_1,':r','LineWidth',2); disp (['Model1 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_maddox20_1)]);
[n,d,p] = find_flow_params(2,Et_maddox20',theta_maddox20,p_maddox20); e_maddox20_2 = flowmodel(2,theta_maddox20,n,p_maddox20,d); e_maddox20_2 = max(Et_maddox20)*e_maddox20_2/max(e_maddox20_2); err_maddox20_2 = sum((e_maddox20_2-Et_maddox20').*(e_maddox20_2-
Et_maddox20'));
73
plot(theta_maddox20, e_maddox20_2,'+b','LineWidth',2,'MarkerSize',6); disp (['Model2 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_maddox20_2)]);
[n,d,p] = find_flow_params(3,Et_maddox20',theta_maddox20,p_maddox20); e_maddox20_3 = flowmodel(3,theta_maddox20,n,p_maddox20,d); e_maddox20_3 = max(Et_maddox20)*e_maddox20_3/max(e_maddox20_3); err_maddox20_3 = sum((e_maddox20_3-Et_maddox20').*(e_maddox20_3-
Et_maddox20')); plot(theta_maddox20, e_maddox20_3,'--g','LineWidth',3); disp (['Model3 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_maddox20_3)]);
[n,d,p] = find_flow_params(4,Et_maddox20',theta_maddox20,p_maddox20); e_maddox20_4 = flowmodel(4,theta_maddox20,n,p,d); e_maddox20_4 = max(Et_maddox20)*e_maddox20_4/max(e_maddox20_4); err_maddox20_4 = sum((e_maddox20_4-Et_maddox20').*(e_maddox20_4-
Et_maddox20')); h = plot(theta_maddox20, e_maddox20_4,'-.m','LineWidth',3); disp (['Model4 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_maddox20_4)]);
xlabel('Normalized time'); ylabel('Residence time distribution function E(t)'); xlim([0 3.2]); title ('Maddox'); legend('Experimental','Model 1','Model 2','Model 3','Model 4'); saveas(h,'maddox_20','tif') ;
%% %saxton 20 disp (' '); disp ('-----------Table 4 Saxton--------------------'); disp ('data n d p error'); figure hold on plot(theta_sax20, Et_sax20,'ok','LineWidth',3);
[n,d,p] = find_flow_params(1,Et_sax20',theta_sax20,p_sax20); e_sax20_1 = flowmodel(1,theta_sax20,n,p_sax20,d); e_sax20_1 = max(Et_sax20)*e_sax20_1/max(e_sax20_1); err_sax20_1 = sum((e_sax20_1-Et_sax20').*(e_sax20_1-Et_sax20')); plot(theta_sax20, e_sax20_1,':r','LineWidth',3); disp (['Model1 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_sax20_1)]);
[n,d,p] = find_flow_params(2,Et_sax20',theta_sax20,p_sax20); e_sax20_2 = flowmodel(2,theta_sax20,n,p_sax20,d); e_sax20_2 = max(Et_sax20)*e_sax20_2/max(e_sax20_2); err_sax20_2 = sum((e_sax20_2-Et_sax20').*(e_sax20_2-Et_sax20')); plot(theta_sax20, e_sax20_2,'+b','LineWidth',2,'MarkerSize',6); disp (['Model2 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_sax20_2)]);
74
[n,d,p] = find_flow_params(3,Et_sax20',theta_sax20,p_sax20); e_sax20_3 = flowmodel(3,theta_sax20,n,p_sax20,d); e_sax20_3 = max(Et_sax20)*e_sax20_3/max(e_sax20_3); err_sax20_3 = sum((e_sax20_3-Et_sax20').*(e_sax20_3-Et_sax20')); plot(theta_sax20, e_sax20_3,'--g','LineWidth',3); disp (['Model3 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_sax20_3)]);
[n,d,p] = find_flow_params(4,Et_sax20',theta_sax20,p_sax20); e_sax20_4 = flowmodel(4,theta_sax20,n,p,d); e_sax20_4 = max(Et_sax20)*e_sax20_4/max(e_sax20_4); err_sax20_4 = sum((e_sax20_4-Et_sax20').*(e_sax20_4-Et_sax20')); h = plot(theta_sax20, e_sax20_4,'-.m','LineWidth',3); disp (['Model4 ' num2str(n) ' ' num2str(d) ' '
num2str(p) ' ' num2str(err_sax20_4)]);
xlabel('Normalized time'); ylabel('Residence time distribution function E(t)'); title ('Saxton'); xlim([0 3.1]); legend('Experimental','Model 1','Model 2','Model 3','Model 4'); saveas(h,'saxton_20','tif') ;
function [e] = flowmodel(model_num,theta,n,p,d)
switch model_num
case 1
b = n/(1-p)*(1-d); theta(theta<p) = p; x1 = b*(theta-p); f1 = floor(n)-1; f2 = n-1; fac = f2*factorial(f1)/f1; x2 = b*(x1.^(n-1))/fac; e = x2.*exp(-x1);
case 2
theta(theta<p) = p; theta = (theta-p)/(1-p); x1 = n*(n*theta).^(n-1); f1 = floor(n)-1; f2 = n-1; fac = f2*factorial(f1)/f1; x2 = x1/fac; e = x2.*exp(-n*theta);
case 3
x1 = n*(n*theta).^(n-1); f1 = floor(n)-1;
75
f2 = n-1; fac = f2*factorial(f1)/f1; x2 = x1/fac; e = x2.*exp(-n*theta);
case 4 e = (1/(1-p))*exp(-(theta-p)/(1-p)); e(theta<p) = 0; otherwise disp('error in model number'); end
function [n,d,p] = find_flow_params(model_num,Et,theta,p)
switch model_num case 1 N_V = 2:0.1:4; D_V = 0:0.01:.3; lse = zeros(length(N_V),length(D_V)); r=1; for n = N_V c=1; for d = D_V e = flowmodel(model_num,theta,n,p,d); etheta = max(e)*Et/max(Et); lse(r,c) = sum((e-etheta).*(e-etheta)); c=c+1; end r= r+1; end
[val di] = min(min(lse)); [val ni] = min(min(lse'));
d = D_V(di); n = N_V(ni); p = [];
case 2
N_V = 2:0.1:4; r=1; lse = zeros(1,length(N_V)); for n = N_V e = flowmodel(model_num,theta,n,p,[]); etheta = max(e)*Et/max(Et); lse(r) = sum((e-etheta).*(e-etheta)); r= r+1; end [val ni] = min(lse); n = N_V(ni); d = []; p = []; case 3 N_V = 2:0.1:4;
76
r=1; lse = zeros(1,length(N_V)); for n = N_V e = flowmodel(model_num,theta,n,p,[]); etheta = max(e)*Et/max(Et); lse(r) = sum((e-etheta).*(e-etheta)); r= r+1; end [val ni] = min(lse); n = N_V(ni); d = []; p = []; case 4 P_V = 0:.01:.9; r=1; lse = zeros(1,length(P_V)); for p = P_V e = flowmodel(model_num,theta,[],p,[]); etheta = max(e)*Et/max(Et); lse(r) = sum((e-etheta).*(e-etheta)); r= r+1; end [val pi] = min(lse); p = P_V(pi); n = []; d = [];
otherwise disp('error in model number'); end
% Experimental compare
clc clear close all
load experimental_results
deltat = 10.0;
smooth_single20 = relative_weight_single20; [maxd maxi] = max(relative_weight_single20); smooth_single20(maxi:end) = smooth(relative_weight_single20(maxi:end),3); us_single20 = resample(smooth_single20,3,1); us_single20(us_single20<0) = 0; s_single20 = sum(us_single20*deltat); Et_single20 = us_single20/s_single20; Ft_single20 = cumsum(us_single20*deltat); t=deltat*(0:length(us_single20)-1); tbar_single20 = sum(t'.*us_single20*deltat)/s_single20; t_single20 = t/tbar_single20;
smooth_single50 = relative_weight_single50; [maxd maxi] = max(relative_weight_single50); smooth_single50(maxi:end) = smooth(relative_weight_single50(maxi:end),3); us_single50 = resample(smooth_single50,3,1);
77
us_single50(us_single50<0) = 0; s_single50 = sum(us_single50*deltat); Et_single50 = us_single50/s_single50; Ft_single50 = cumsum(us_single50*deltat); t=deltat*(0:length(us_single50)-1); tbar_single50 = sum(t'.*us_single50*deltat)/s_single50; t_single50 = t/tbar_single50;
smooth_single80 = relative_weight_single80; [maxd maxi] = max(relative_weight_single80); smooth_single80(maxi:end) = smooth(relative_weight_single80(maxi:end),3); us_single80 = resample(smooth_single80,3,1); us_single80(us_single80<0) = 0; s_single80 = sum(us_single80*deltat); Et_single80 = us_single80/s_single80; Ft_single80 = cumsum(us_single80*deltat); t=deltat*(0:length(us_single80)-1); tbar_single80 = sum(t'.*us_single80*deltat)/s_single80; t_single80 = t/tbar_single80;
smooth_sax20 = relative_weight_sax20; [maxd maxi] = max(relative_weight_sax20); smooth_sax20(maxi-7:end) = smooth(relative_weight_sax20(maxi-7:end),3); us_sax20 = resample(smooth_sax20,3,1); us_sax20(us_sax20<0) = 0; s_sax20 = sum(us_sax20*deltat); Et_sax20 = us_sax20/s_sax20; Ft_sax20 = cumsum(us_sax20*deltat); t=deltat*(0:length(us_sax20)-1); tbar_sax20 = sum(t'.*us_sax20*deltat)/s_sax20; t_sax20 = t/tbar_sax20;
smooth_sax50 = relative_weight_sax50; [maxd maxi] = max(relative_weight_sax50); smooth_sax50(maxi:end) = smooth(relative_weight_sax50(maxi:end),3); us_sax50 = resample(smooth_sax50,3,1); us_sax50(us_sax50<0) = 0; s_sax50 = sum(us_sax50*deltat); Et_sax50 = us_sax50/s_sax50; Ft_sax50 = cumsum(us_sax50*deltat); t=deltat*(0:length(us_sax50)-1); tbar_sax50 = sum(t'.*us_sax50*deltat)/s_sax50; t_sax50 = t/tbar_sax50;
smooth_sax80 = relative_weight_sax80; [maxd maxi] = max(relative_weight_sax80); smooth_sax80(maxi:end) = smooth(relative_weight_sax80(maxi:end),3); us_sax80 = resample(smooth_sax80,3,1); us_sax80(us_sax80<0) = 0; s_sax80 = sum(us_sax80*deltat); Et_sax80 = us_sax80/s_sax80; Ft_sax80 = cumsum(us_sax80*deltat); t=deltat*(0:length(us_sax80)-1); tbar_sax80 = sum(t'.*us_sax80*deltat)/s_sax80; t_sax80 = t/tbar_sax80;
78
smooth_maddox20= relative_weight_maddox20; [maxd maxi] = max(relative_weight_maddox20); smooth_maddox20(maxi-7:end) = smooth(relative_weight_maddox20(maxi-7:end),3); us_maddox20 = resample(smooth_maddox20,3,1); us_maddox20(us_maddox20<0) = 0; s_maddox20 = sum(us_maddox20*deltat); Et_maddox20 = us_maddox20/s_maddox20; Ft_maddox20 = cumsum(us_maddox20*deltat); t=deltat*(0:length(us_maddox20)-1); tbar_maddox20 = sum(t'.*us_maddox20*deltat)/s_maddox20; t_maddox20 = t/tbar_maddox20;
%% hold on %plot(t_single20,tbar_single20*Et_single20,'--ks','LineWidth',2,
'MarkerSize',8); %plot(t_single50,tbar_single50*Et_single50,':*k','LineWidth',2,
'MarkerSize',8); %h_single = plot(t_single80,tbar_single80*Et_single80,'-.ok','LineWidth',2,
'MarkerSize',8);
plot((0:length(Ft_single20)-1)*deltat,tbar_single20*Et_single20,'--
sr','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','r'); plot((0:length(Ft_single50)-
1)*deltat,tbar_single50*Et_single50,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_single = plot((0:length(Ft_single80)-1)*deltat,tbar_single80*Et_single80,'-
.ok','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','k');
title('Singleflight'); ylabel('E(t)'); xlabel('Residence time'); legend('20 rpm ','50rpm','80 rpm'); xlim([0 1800]);
figure hold on %plot(t_sax20,tbar_sax20*Et_sax20,'--ks','LineWidth',2, 'MarkerSize',8); %plot(t_sax50,tbar_sax50*Et_sax50,':*k','LineWidth',2, 'MarkerSize',8); %h_sax = plot(t_sax80,tbar_sax80*Et_sax80,'-.ok','LineWidth',2,
'MarkerSize',8); plot((0:length(Ft_sax20)-1)*deltat,tbar_sax20*Et_sax20,'--sr','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','r'); plot((0:length(Ft_sax50)-1)*deltat,tbar_sax50*Et_sax50,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_sax = plot((0:length(Ft_sax80)-1)*deltat,tbar_sax80*Et_sax80,'-
.ok','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','k'); title('Saxton'); ylabel('E(t)'); xlabel('Residence time'); legend('20 rpm ','50rpm','80 rpm'); xlim([0 2100]);
figure hold on
79
plot(t_single20,tbar_single20*Et_single20,'--rs','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','r','MarkerEdgeColor','r'); plot(t_maddox20,tbar_maddox20*Et_maddox20,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b','MarkerEdgeColor','b'); h_all_dl = plot(t_sax20,tbar_sax20*Et_sax20,'-.ok','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','k','MarkerEdgeColor','k'); title('All at 20 rpm'); ylabel('E(t)'); xlabel('Dimentionless residence time'); legend('Singleflight ','Maddox','Saxton'); xlim([0 4.2]);
figure hold on plot([0:length(Et_single20)-1]*deltat,tbar_single20*Et_single20,'--
rs','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','r'); plot([0:length(Et_maddox20)-
1]*deltat,tbar_maddox20*Et_maddox20,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_all_d = plot([0:length(Et_sax20)-1]*deltat,tbar_sax20*Et_sax20,'-
.ok','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','k'); title('All at 20 rpm'); ylabel('E(t)'); xlabel('Residence time(Sec)'); legend('Singleflight ','Maddox','Saxton'); xlim([0 2100]);
%% figure hold on plot((0:length(Ft_single20)-1)*deltat,tbar_single20*Ft_single20,'--
rs','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','r'); plot((0:length(Ft_single50)-
1)*deltat,tbar_single50*Ft_single50,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_single_ft = plot((0:length(Ft_single80)-
1)*deltat,tbar_single80*Ft_single80,'-.og','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','k'); title('Singleflight'); ylabel('F(t)'); xlabel('Cumulative residence time'); legend('20 rpm ','50rpm','80 rpm'); xlim([0 2100]);
figure hold on plot((0:length(Ft_sax20)-1)*deltat,tbar_sax20*Ft_sax20,'--rs','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','r'); plot((0:length(Ft_sax50)-1)*deltat,tbar_sax50*Ft_sax50,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_sax_ft = plot((0:length(Ft_sax80)-1)*deltat,tbar_sax80*Ft_sax80,'-
.ok','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','k'); title('Saxton'); ylabel('F(t)'); xlabel('Cumulative residence time'); legend('20 rpm ','50rpm','80 rpm');
80
xlim([0 2100]);
figure hold on plot(t_single20,tbar_single20*Ft_single20,'--rs','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','r'); plot(t_maddox20,tbar_maddox20*Ft_maddox20,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_all_dl_ft = plot(t_sax20,tbar_sax20*Ft_sax20,'-.ok','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','k'); title('All at 20 rpm'); ylabel('F(t)'); xlabel('Dimentionless Cumulative residence time'); legend('Singleflight ','Maddox','Saxton'); xlim([0 4.2]);
figure hold on plot((0:length(Ft_single20)-1)*deltat,tbar_single20*Ft_single20,'--
rs','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','r'); plot((0:length(Ft_maddox20)-
1)*deltat,tbar_maddox20*Ft_maddox20,':*b','LineWidth',2,
'MarkerSize',8,'MarkerEdgeColor','b'); h_all_d_ft = plot((0:length(Ft_sax20)-1)*deltat,tbar_sax20*Ft_sax20,'-
.ok','LineWidth',2, 'MarkerSize',8,'MarkerEdgeColor','k'); title('All at 20 rpm'); ylabel('F(t)'); xlabel('Cumulative Residence time(Sec)'); legend('Singleflight ','Maddox','Saxton'); xlim([0 2100]);
%% saveas(h_single,'single','tif') saveas(h_sax,'saxton','tif') saveas(h_all_dl,'all_dimless','tif') saveas(h_all_d,'all_dim','tif') saveas(h_single_ft,'single_ft','tif') saveas(h_sax_ft,'saxton_ft','tif') saveas(h_all_dl_ft,'all_dimless_ft','tif') saveas(h_all_d_ft,'all_dim_ft','tif')
variance_single20 =sum((((0:length(Et_single20)-1)-
tbar_single20).^2).*Et_single20'*deltat); variance_single50 =sum((((0:length(Et_single50)-1)-
tbar_single50).^2).*Et_single50'*deltat); variance_single80 =sum((((0:length(Et_single80)-1)-
tbar_single80).^2).*Et_single80'*deltat); variance_maddox20 =sum((((0:length(Et_maddox20)-1)-
tbar_maddox20).^2).*Et_maddox20'*deltat); variance_sax20 =sum((((0:length(Et_sax20)-1)-
tbar_sax20).^2).*Et_sax20'*deltat); variance_sax50 =sum((((0:length(Et_sax50)-1)-
tbar_sax20).^2).*Et_sax50'*deltat); variance_sax80 =sum((((0:length(Et_sax80)-1)-
tbar_sax20).^2).*Et_sax80'*deltat);
81
disp('---------------------Table 1---------------------------') disp('Screw Configuration Screw Speed Tbar(min) Variance') disp([' Single flight 20 ' num2str(tbar_single20/60,2) ' '
num2str(variance_single20,2)]); disp([' Single flight 50 ' num2str(tbar_single50/60,2) ' '
num2str(variance_single50,2)]); disp([' Single flight 80 ' num2str(tbar_single80/60,2) ' '
num2str(variance_single80,2)]); disp([' Maddox 20 ' num2str(tbar_maddox20/60,2) ' '
num2str(variance_maddox20,2)]); disp([' Saxton 20 ' num2str(tbar_sax20/60,2) ' '
num2str(variance_sax20,2)]); disp([' Saxton 50 ' num2str(tbar_sax50/60,2) ' '
num2str(variance_sax50,2)]); disp([' Saxton 80 ' num2str(tbar_sax80/60,2) ' '
num2str(variance_sax80,2)]);
save processed_data
82
Sample Images
Mixing
section
Screw speed
(rpm)
Mixing degree %
Dowlex
Mixing degree %
Eraclene
Mixing degree %
Mix1
Mixing degree %
Mix2
Single
flight
20
80
Maddock
20
80
Saxton
20
80
83
