WALL SLIP IN PIPE RHEOMETRY OF
MULTIPHASE FLUIDS
A thesis submitted to The University of Manchester
for the degree of Doctor of Philosophy
in the School of Chemical Engineering and Analytical Science
2013
NUR ‘ALIAA ABD RAHMAN
School of Chemical Engineering and Analytical Science
LIST OF CONTENTS
LIST OF CONTENTS
2
LIST OF FIGURES
6
LIST OF TABLES
10
NOMENCLATURE
11
ABSTRACT
16
DECLARATION
17
ACKNOWLEDGEMENTS
18
PREFACE
19
1
INTRODUCTION
20
1.1
Non-Newtonian fluids and rheology
20
1.2
Multiphase fluid systems
22
1.3
Pipe rheometry
23
1.4
Wall slip – mechanisms, effects and quantification
24
1.5
Problem statements and scope of the research
29
1.6
Objectives of the research
31
2
LITERATURE REVIEW
34
2.1
Rheology – Definition, history and concept
34
2.2
2.3
2.1.1
Measurement of rheological properties
36
2.1.2
Viscosity and the classical extremes of elastic and viscosity
39
Non-Newtonian fluids
45
2.2.1
Yield stress
46
2.2.2
Constitutive equations of rheological models
48
2.2.3
Viscoelasticity and viscoplasticity
52
Shear-thinning fluids
54
2.3.1
Ice cream
55
2.3.1.1
Ice cream formulation/ingredients
57
2.3.1.2
Manufacturing of ice cream and microstructure creation
61
2.3.1.3
Research on ice cream rheology
69
2
2.3.2
Citrus dietary fibre (CDF)
71
2.3.3
Magnesium silicate (talc powder)
73
2.4
Rheometry
73
2.5
Flow of fluids in pipes
76
2.6
2.5.1
Correction for entrance effect
78
2.5.2
Rabinowitsch correction for nonparabolic velocity profile
80
Wall slip effects
81
2.6.1
Factors influencing wall slip
83
2.6.2
Mechanism of wall slip
85
2.6.3
Correction for wall slip
92
2.6.3.1
Mooney method
92
2.6.3.2
Modified-Mooney methods
97
2.6.3.3
Tikhonov Regularisation-based Mooney method
99
2.7
Analysis on wall slip effects in the flow of multiphase systems
101
2.8
Viscous heating
112
2.9
Conclusions
115
3
DESIGN, BUILD AND COMMISSIONING OF PIPE RHEOMETRY RIG
119
3.1
Introduction
120
3.2
Selection of equipment
120
3.3
3.2.1
Selection of piping system
121
3.2.2
Selection of continuous industrial ice cream freezer
127
Instrumentation of the rig
131
3.3.1
Temperature transducer
131
3.3.2
Pressure transducer
132
3.4
Other equipment
133
3.5
Experimental arrangement of pipe rheometry
134
3.6
Installation of the rig
134
3.7
Commissioning of the rig
136
3.8
Conclusions
140
4
WALL SLIP AND VISCOUS DISSIPATION IN ICE CREAM PIPE RHEOMETRY
142
4.1
Introduction
142
3
4.2
Materials and Methods – Ice cream production
144
4.3
Results and Discussion
146
4.3.1
Temperature gradient of ice cream at the wall
146
4.3.2
Flow curves and wall slip analysis
151
4.3.3
Energy balances of viscous dissipation
158
4.3.4
Fat droplet size distribution
161
4.4
Conclusions
163
5
WALL SLIP IN PIPE RHEOMETRY OF CITRUS DIETARY FIBRE SUSPENSIONS
166
5.1
Introduction
166
5.2
Materials and Methods
168
5.3
Results and Discussion
169
5.3.1
Flow curves and rheological behaviour
169
5.3.2
Wall slip analysis
173
5.4
Conclusions
181
6
WALL SLIP IN PIPE RHEOMETRY OF MAGNESIUM SILICATE SLURRIES
183
6.1
Introduction
183
6.2
Materials and Methods
184
6.3
Results and Discussion
185
6.3.1
Flow curves and rheological behaviour
185
6.3.2
Wall slip analysis
189
6.4
Conclusions
195
7
DISCUSSIONS ON WALL SLIP ANALYSIS
197
7.1
Mooney equation – dimensionless approach
197
7.2
Wall slip in ice cream flow
200
7.3
Wall slip in multiphase suspensions
202
7.4
Influence of non-homogeneous shear flow
203
7.5
Conclusions
208
8
CONCLUSIONS AND FUTURE WORK
210
8.1
Conclusions
210
8.2
Future Work
215
4
8.2.1
Application of combined ultrasonic pulsed Doppler velocimetry and pressure drop
(UPDV-PD)
217
8.2.2
Application of electrical resistance tomography
219
REFERENCES
220
APPENDIX A
238
APPENDIX B
241
5
LIST OF FIGURES
Fig. 1.1: Slip velocity at the interface between the apparent slip layer and the bulk of
the fluid
27
Fig. 1.2: Scope of the project
32
Fig. 2.1: Linear extension of a rectangular bar (after Steffe, 1996)
37
Fig. 2.2: Shear deformation of a rectangular bar (after Steffe, 1996)
38
Fig. 2.3: Pure shear (after Dutch, 1999b)
39
Fig. 2.4: Concept of flow resistance in steady simple shearing flow
41
Fig. 2.5: Deformation of a Hookean solid on the application of stress. The material
section ABCD becomes A'B'C'D'
43
Fig. 2.6: Typical non-Newtonian behaviour curves as comparison to the curve of
Newtonian liquid
46
Fig. 2.7: Rheogram of shear-thinning behaviour
55
Fig. 2.8: Schematic diagram of the physical structure of ice cream showing air
bubbles, ice crystals and fat globules
58
Fig. 2.9: Ice cream displaying yield stress (credit to iStockphoto)
69
Fig. 2.10: Parabolic velocity profile for Newtonian fluid
77
Fig. 2.11: Schematic diagrams for plug flow, slip flow and shear flow in a straight
pipe
82
Fig. 2.12: Wall slip/depleted layer formed at the inner surface of a pipe during
multiphase fluid flow
87
Fig. 2.13: Pressure decreases linearly along the pipe length where f is constant and δ
is assumed to be constant too
89
Fig. 2.14: Pressure is assumed to decrease non-linearly along the pipe length if the
depleted layer thickness increases
89
Fig. 2.15: Force balance for pressure driven flow between two flat stationary surfaces 90
Fig. 2.16: Example of successful Mooney plots
96
Fig. 2.17: Unviable 4Q/πR3 against 4/R plots with negative –y axis intercept
96
Fig. 3.1: The first section of the pipeline
125
Fig. 3.2: The test section pipes
127
Fig. 3.3: CS200 continuous ice cream freezer
129
Fig. 3.4: Front and side views of CS200 continuous ice cream freezer
129
6
Fig. 3.5: Pressure transducer design no.2
133
Fig. 3.6: Schematic of ice cream production rig
135
Fig. 3.7: Detail of pipe rheometer test section for ice cream
135
Fig. 3.8: Detail of pipe rheometer test section for CDF suspensions and magnesium
silicate dispersions
136
Fig. 3.9: Results from two different arrangements of temperature transducers obtained
by swapping the positions of the transducers on 33.9 mm pipe diameter.
The measured temperatures were independent of transducer positioning
139
Fig. 3.10: Bagley plots constructed to determine the entrance pressure loss for 33.9
mm pipe diameter. The fitted lines intercept the y-axis at approximately
zero values, hence the loss is assumed to be negligible.
139
Fig. 4.1: Measured temperature gradient of ice cream at the wall against apparent
shear rate for different pipe diameters
147
Fig. 4.2: Ratio of viscous dissipation to calculated heat transfer per unit length against
apparent shear rate for different pipe diameters
150
Fig. 4.3: Wall shear stress against apparent shear rate data for ice cream flow in four
different pipes at -5oC. Fitted curves and the respective flow indices are
shown in the figure.
152
Fig. 4.4: Wall shear stress against apparent shear rate for different pipe diameters
154
Fig. 4.5: Mooney plot of apparent shear rate against 4/R at values of constant wall
shear stress
154
Fig. 4.6: Tikhonov regularisation fitted shear stress against shear rate for different
regularisation parameters
156
Fig. 4.7: Tikhonov regularisation fitted slip velocity against shear rate for different
regularisation parameters and comparative points from Mooney plots
156
Fig. 4.8: Tikhonov regularisation fitted fraction of flow due to slip against wall shear
stress for different pipe diameters
157
Fig. 4.9: Slip layer thickness calculated from measured temperature gradients at the
wall against wall shear stress
160
Fig. 4.10: Apparent viscosity against shear rate for Tikhonov regularisation fitted
bulk flow of ice cream, ice cream wall slip region and matrix with no ice
previously measured by Martin et al. (2008)
7
161
Fig. 4.11: Fat aggregate size distributions of ice cream mix and ice cream samples
flowing out from different pipes. There is a significant difference in the
distribution of fat globule size in ice cream mix and ice cream samples.
Fig. 5.1: SEM image of Herbacel AQ Plus Citrus Fibres
162
169
Fig. 5.2: Wall shear stress against apparent shear rate data for citrus fibre suspensions
- (a) 2%, (b) 3% and (c) 4% (w/w) flow in four different pipes. Power law
fitted curves are shown in the figure.
171
Fig. 5.3: Mooney plot of apparent shear rate against 4/R for (a) 2% (b) 3% and (c)
4% (w/w) concentrations of CDF suspensions. Non-linear relationship was
obtained and the straight lines fitted intercept the ordinate axis at negative
values for all cases.
174
Fig. 5.4: Apparent shear rate against 4/R2 for (a) 2% (b) 3% and (c) 4% (w/w)
concentrations of CDF suspensions. The plots appear to be quite linear.
However, a degree of non-linearity is still significant.
177
Fig. 5.5: ESEM images of (a) dry citrus fibre powder and (b) wet citrus fibre powder.
A significant increase in particle size was observed after water was
introduced.
180
Fig. 6.1: SEM image of Micro-Talc AT Extra (Martin et al., 2004)
184
Fig. 6.2: Wall shear stress against apparent shear rate data for magnesium silicate
slurries - (a) 10%, (b) 16%, (c) 20%, (d) 24% and (e) 28% (w/w) flow in
four different pipes. Fitted curves are shown in the figure.
187
Fig. 6.3: Mooney plot of apparent shear rate against 4/R for (a) 10%, (b) 16%, (c)
20%, (d) 24% and (e) 28% (w/w) concentrations of magnesium silicate
slurries. Non-linear relationship was obtained and the straight lines fitted
intercept the ordinate axis at negative values for all cases.
191
Fig. 6.4: Apparent shear rate against 4/R2 for (a) 10%, (b) 16%, (c) 20%, (d) 24% and
(e) 28% (w/w) concentrations of magnesium silicate slurries. Linear least
squares fits were applied to these data and appear plausible.
194
Fig. 8.1: Experiment conducted to observe the effect of the condition of wall surface
on slip: (a) a line was drawn before the experiment started; (b)
discontinuities of the line was observed when using normal smooth wall
surface and (c) the marker line on the fluid was continuous to the marker
line on the rotor surface when a sandpaper was attached to the surface of
the rotor
217
8
Fig. 8.2: Proposed UPDV-PD rheometry for ice cream study
9
218
LIST OF TABLES
Table 2.1: Roles of microstructures in ice cream (Hyde and Rothwell, 1973; Marshall
and Arbuckle, 2000; Clarke, 2004; Goff, 2010d; Goff, 2010e)
62
Table 2.2: Comparisons between the classical Mooney and modified-Mooney
methods
98
Table 2.3: Summary of all capillary and pipe flow wall slip analyses on various
systems
103
Table 3.1: Examples of pipe diameters and length used in previous studies
122
Table 3.2: Standard features for continuous ice cream freezer (model CS200)
130
Table 3.3: Technical features of CS200 continuous ice cream freezer
131
Table 3.4: Specifications of PT-100 sensor
132
Table 3.5: Specifications of S model transducer
133
Table 3.6: Sample raw data: pressure vs. length; pressure vs. time and temperature
vs. time for ice cream flow in 33.9 and 8.7 mm pipe diameters
Table 4.1: Ice cream formulation
138
145
Table 4.2: Comparison of the energy dissipated in different pipe sizes at similar ice
cream flow rate
151
Table 4.3: Consistency index, M and flow behaviour index, n for ice cream flowing
in different pipes
153
Table 5.1: Rheological data for citrus powder suspensions at different concentrations 172
Table 6.1: Rheological data for magnesium silicate slurries at different
concentrations
188
10
NOMENCLATURE
Roman
a
acceleration (m s-1)
A
area (m2)
b
slope of curve
cp
specific heat capacity (J kg-1 K-1)
cp,air
specific heat capacity of air (J kg-1 K-1)
d50
median particle aggregate size (μm)
d90
maximum particle aggregate size (μm)
D
pipe diameter (m)
e
material constant related to the properties of the binder and particles
Ed
rate of work for deforming a fluid in the flow through pipe (W or J s-1)
Es
rate of work for slip along the wall (W or J s-1)
ET
rate of work required to deliver a flow of fluid/total energy dissipation (W or
J s-1)
f
friction factor
F
force (N)
Fn
normal force (N)
g
gravity acceleration (m s-2)
G
rigidity modulus (Pa)
Gr
Grashof number
h
height (m)
i
dimensionless constant (Cross and Carreau models)
I
constant parameter with the dimension of time (Cross and Carreau models)
j
gradient of the log τw against log sh curve
11
k
thermal conductivity (W m-1 K-1)
kair
thermal conductivity of air (W m-1 K-1)
kinsulation thermal conductivity of insulation layer (W m-1 K-1)
L
length (m)
Lo
initial length (m)
m
mass (kg)
M
consistency coefficient (Pa sn)
n
flow index
NJ
number of wall shear stress divisions
NK
number of apparent shear rate divisions
Na
Nahme number
Nu
Nusselt number
p
index parameter
P
pressure (Pa)
Pent
entrance pressure (Pa)
Pcorrected corrected pressure (Pa)
Pr
Prandtl number
Q
volumetric flow rate (m3 s-1)
Qshear volumetric flow rate due to bulk shear (m3 s-1)
Qslip
volumetric flow rate due to apparent slip (m3 s-1)
H
rate of heat transfer (W or J s-1)
r
radial distance (m)
R
inside pipe radius (m)
Rmin
minimum pipe radius (m)
RT
total of inside pipe radius and insulation thickness (m)
12
Re
Reynolds number
ReMR
Metzner-Reed Reynolds number
S1
Tikhonov regularisation error term
S2
Tikhonov regularisation smoothness term
t
time (s)
T
mean ice cream temperature over the radial direction (oC)
Tair
air temperature (oC)
Ticecream ice cream temperature (oC)
Tsurface surface temperature (oC)
Tw
ice cream temperature at the wall (oC)
ΔT
temperature rise (K)
u
convective heat transfer coefficient (W m-2 K-1)
U(r)
velocity profile term
V
velocity (m s-1)
VB
bulk velocity (m s-1)
Vmean mean velocity (m s-1)
Vshear shear velocity (m s-1)
Vslip
apparent slip velocity (m s-1)
Vslip
vector of fitted slip velocities
Vslip

dimensionless apparent slip velocity
Vx
local velocity (m s-1)
WCD
resistance to heat transfer by conduction
WCV
resistance to heat transfer by convection
x
axial distance (m)
Δx
change in length (m)
13
XD
entrance length (m)
Y
Young’s modulus (Pa)
z
slip exponent dependent on the fluid
Greek
α
Navier’s slip coefficient
β
temperature sensitivity (K-1)
βair
volumetric thermal expansion coefficient (1/Tair) (K-1)
φ
slip coefficient
ѱ
time constant
σ
normal stress (Pa)
ε
normal strain (Pa)
δ
slip layer thickness (μm)
γ
shear strain (%)

shear rate (s-1)
γ
vector of shear rates
app
apparent shear rate (s-1)
sh
bulk shear rate (s-1)
slip
slip layer shear rate (s-1)
 w
true wall shear rate (s-1)
 
dimensionless shear rate
λ
regularisation parameter
μ
viscosity (Pa s)
μair
viscosity of air (Pa s)
14
μapp
apparent shear viscosity (Pa s)
μo
asymptotic values of viscosity at very low shear rates
μmatrix matrix viscosity (Pa s)
μp
plastic viscosity (Pa s)
μslip
slip layer apparent viscosity (Pa s)

dimensionless viscosity

asymptotic values of viscosity at very high shear rates
θ
dimensionless temperature
ρ
density (kg m-3)
ρair
density of air (kg m-3)
τ
shear stress (Pa)
τw
wall shear stress (Pa)
τwc
critical wall shear stress (Pa)
τy
yield stress (Pa)

dimensionless shear stress
Φ
particle volume fraction
Prefixes
d
infinitesimal change
∂
finite change
15
ABSTRACT
The University of Manchester
Nur ‘Aliaa Abd Rahman
Doctor of Philosophy
Wall Slip in Pipe Rheometry of Multiphase Fluids
2013
Multiphase fluids are widely available in our everyday life. Many of the materials
we use and eat every day are classed as multiphase and characterisation of their
properties is required to improve quality and manufacture. Being typically nonNewtonian fluids, rheological characterisation of multiphase fluid systems is indeed
a complex procedure. Apparent wall slip, or more precisely wall depletion effect
near the wall, is an important phenomenon which often occurs in the flow of
multiphase fluids in pipes. Wall slip has its own advantages and disadvantages in the
processing and pipe flow of multiphase fluids. One of the main problems it causes is
the underestimation of the viscosity and the true flow curve of the fluids reported
during experimental measurements. Standard correction methods often account for
this effect, but there have been many instances reported where the data does not
comply with the technique. This study aimed to present a wall slip analysis of a
selection of non-Newtonian multiphase fluids during flow in pipes. A pipe
rheometry rig was specially designed and built for the purpose of the research which
includes four interchangeable pipes with different diameters; three sets of pressure
and temperature transducers located at three different points along the test section;
an electronic mass balance; and a PC data logger for control, monitoring and data
collection purposes. Three distinct non-compliant multiphase fluids were chosen for
study i.e. ice cream, citrus dietary fibre (CDF) suspensions and magnesium silicate
slurries. The experiments were carried out in pressure driven shear flow. The flow
data were analysed using the classical Mooney method along with a Tikhonov
regularisation-Mooney method. For ice cream flow data, the analysis indicated that
significant apparent wall slip occurred in all flows and there was a small but
significant increase in the temperature near the wall which indicated the occurrence
of viscous heating phenomenon. Energy balances indicated that the apparent wall
slip effect was not due to the existence of a thin slip layer of rarefied low viscosity
fluid next to the wall. It was found that the results were better understood as being
the result of a moderately thick layer of slightly heated ice cream next to the wall.
Mooney and Tikhonov regularisation-Mooney methods were confirmed to be
incompatible with the wall slip behaviour of CDF suspensions and magnesium
silicate slurries. The incompatibility of the method to analyse wall slip is attributed
to the inconsistent ratio of Vslip/τw and δ/μslip at constant wall shear stress. It is
concluded that this was due to the microstructure changes and shear-induced reorientation of the particles in CDF suspensions and magnesium silicate slurries
during flows which resulted in the inconsistency of the slip layer thickness and
consequently affected the wall slip characterisation process. The principal
contribution of this research work is to present a comprehensive wall slip analysis in
pipe rheometry of multiphase non-Newtonian fluids, which are of particular interest
in engineering process design.
16
DECLARATION
No portion of the work referred to in the thesis has been submitted in support of an
application for another degree or qualification of this or any other university or other
institute of learning.
Copyright Statement
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owns certain copyright or related rights in it (the “Copyright”) and she has given The
University of Manchester certain rights to use such Copyright, including for
administrative purposes.
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This page must form part of any such copies made.
iii. The ownership of certain Copyright, patents, designs, trademarks and other
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described in this thesis, may not be owned by the author and may be owned by third
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The
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Library’s
regulations
(see
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policy on presentation of Theses.
17
ACKNOWLEDGEMENTS
‘Without these special people, my thesis work would never be completed’.
My heartfelt appreciation goes to my supervisor, Dr. Peter Martin for his endless
guidance, sincerity and tolerance throughout my studies and difficult times. Thank
you for helping me to understand the fundamental and application of rheology, fluid
mechanics and heat transfer in pipeline flow of fluids.
Special appreciation to the internal examiner, Dr. Alastair Martin (The University of
Manchester) and external examiner, Dr. Ian Wilson (University of Cambridge) for
all the constructive comments towards the thesis and research work.
My biggest gratitude to the Government of Malaysia and Universiti Putra Malaysia
(UPM) for the financial support for this three and a half-year research programme. I
would also like to acknowledge The Royal Society (Research Grant RG090409) for
the funding to build up the rig.
Special thank you to the laboratory staff at the School of Chemical Engineering and
Analytical Science, The University of Manchester especially Mr. Alan Fowler for
the help and support in building up the pipe rheometry rig and made this research
study possible. Also, thank you to my colleagues at the university for the wonderful
times we have had together. To my wonderful friends in Malaysia and in the UK,
thank you for lending me your ears and shoulders to cry.
My eternal gratitude goes to my beloved parents, Hj. Abd. Rahman Said and Wan
Zaharah Megat Hashim; and my parents-in-law, Hj. Ismail Othman and Hjh. Umi
Othman for their endless prayer, support and encouragement that have enabled me to
pursue my education to this degree and help me to sail through the winding journey.
They are the ‘wind beneath my wings’. Special thanks to my siblings, siblings-inlaw and relatives for their ongoing motivation.
To my dearest husband and best friend, Mohd. Hafizul Ismail, thank you for your
love, support and encouragement. Your patience and understanding are greatly
appreciated. Specially for my darling son, Adam Danial, you are my smile, laughter
and joy; easing all my worries in life.
‘So, verily, with every hardship there is ease. Verily, with every hardship there is
ease.’ (Inshirah, 94: 5-6).
18
PREFACE
The author, Nur ‘Aliaa Abd Rahman has a bachelor’s degree in Process and Food
Engineering (Second Class Upper) and a Master’s degree in Food Engineering, both
awarded by Universiti Putra Malaysia in 2007 and 2009, respectively. The author’s
area of interest is in food process engineering and she has published her research
work on the processing of fruit juice previously in various peer-reviewed research
journals. Her passion in studying and exploring more on food process engineering
has brought her to the School of Chemical Engineering and Analytical Science, The
University of Manchester to pursue her PhD degree focusing on the wall slip
phenomenon in pipe rheometry of multiphase fluids especially food. The three and a
half-year of the research study has increased her knowledge and understanding on
the engineering of multiphase fluids flow especially ice cream under the supervision
of Dr. Peter Martin. She has been awarded a scholarship by the Government of
Malaysia and Universiti Putra Malaysia to undergo the PhD research programme.
Upon completion of the degree, the author will serve Universiti Putra Malaysia as a
lecturer cum researcher. A part of this work based on the ice cream study was
presented at the XVIth International Congress on Rheology in Lisbon, Portugal (510 August 2012). The author was also awarded a travel award by IChemE Food and
Drink Special Interest Group for the congress. A research paper based on this PhD
work has been published recently in Journal of Food Engineering:
Rahman, N.A.A., Fowler, A. and Martin, P.J. (2013). Wall slip and viscous
dissipation in ice cream pipe rheometry. Journal of Food Engineering, 119 (4): 731737.
19
1
INTRODUCTION
The Introduction starts with the definition and background of nonNewtonian fluids and rheology. Then the topic grows with the description of
multiphase systems. Wall slip phenomenon which occur during the flow of
complex non-Newtonian fluids are introduced in the subsequent section that
describes the mechanisms, effects and quantification of wall slippage. The
basis of this research is explained in section 1.5, ‘Problem statements and
scope of the research’. This chapter ends with the objectives of the research
and overview of the upcoming chapters.
1.1
Non-Newtonian fluids and rheology
Flow of fluids or materials is an interesting and challenging field to
explore. It became of interest to mankind thousands of years ago. The
intuitive concepts of fluid ‘thinness’ and ‘thickness’ has developed into a
more technical and scientific understanding of the concepts. They are related
to how fluids or materials deform and flow when force or strain is applied.
The study of the manner in which materials flow when subjected to applied
stress or strain is termed ‘rheology’ (Steffe, 1996). Rheology is a field which
focuses on describing the flow behaviour of fluids. Examples of complex
fluids we encounter every day are tomato ketchup, ice cream, honey,
20
chocolate, toothpaste and shampoo. They are called complex due to their
physical nature which is directly related to their microstructure. From
everyday use, we can observe or feel the difference between these types of
fluids and simple fluids such as tap water and cooking oil. Tap water and
cooking oil can easily flow when we pour them from their containers. This is
not the case when we try to pour mayonnaise from the jar or press toothpaste
from the tube. They will only flow after a certain amount of force is applied
on them. This force is required to overcome the fluid’s yield stress i.e. the
stress at which fluids/materials starts to flow. Complex fluids can also be
found in pharmaceutical, fuel and ceramic industries. Typically, complex
fluids fit into the category of non-Newtonian fluids, while simple fluids such
as water and cooking oil fit into the category of Newtonian fluids. This
classification of fluids will be described in depth in Chapter 2.
Non-Newtonian fluids can often be classified as having either shearthinning or shear-thickening behaviour. Shear-thinning fluids will decrease in
viscosity when the shear rate increases while in the case of shear-thickening,
the viscosity will increase when the shear rate increases. The increase or
decrease in viscosity of these kinds of fluids with shear rate is often nonlinear. This is in contrast to fluids which have no yield stress and where
viscosity is constant, which is known as Newtonian behaviour. The
complexity of non-Newtonian fluids is usually attributed to their molecular
properties or microstructure. Microstructure of fluids plays an important role
either the fluids exhibit shear-thinning or shear-thickening behaviour. The
fluids are either classified as single phase or multiphase. Most polymers are
single phase systems which are formed by linear or entangled molecular
21
chains. Fluids such as ice cream, starch suspension, yogurt, mayonnaise and
shampoo are multiphase systems which are formed by two or more
substances of different phases. This study focuses on the rheology of
multiphase non-Newtonian fluids with shear-thinning characteristics.
Generally, a rheometer is utilised to characterise the flow behaviour of nonNewtonian fluids.
1.2
Multiphase fluid systems
Multiphase fluids are formed by two or more different phases. Many
of them are formed by two phases such as mayonnaise (oil droplets and
water), shampoo (oil droplets and water), ceramic paste (solid ceramic
particles and water) and starch/fibre suspensions (solid fibre particles and
water). Some fluids have more complex structure such as sorbet (semi-frozen
sugar solution, air bubbles and ice crystals) and ice cream (semi-frozen sugar
solution, air bubbles, ice crystals and fat droplets). Complex microstructure
usually leads to complex flow behaviour. It would be a great challenge to
study the rheological behaviour of multiphase fluids systems. The existence
of different phases in a system will cause microstructural interaction that
complicates flow and its measurement. The interaction of the fluids with the
solid wall of the equipment further complicates the process.
During flow of multiphase fluids in rheometers or pipelines in
factory, the fluids are being sheared by the action of force imparted by the
rheometry tube or pipe wall. The shearing effect becomes more pronounced
when the flow rate increases or the size of the tubes or pipes decreases. The
particles in the fluids will move past each other, between liquid phase and
22
past the wall. As multiphase fluid is flowing, the serum phase is ‘squeezed
out’ from the fluids and forms a thin layer adjacent to the solid wall surface
(Higgs, 1974). Shear stress is the highest at the wall and zero at the centre of
the tube/pipe. The difference in shear stress in the radial direction provides a
driving force for the migration of the particles towards the region of lower
stresses at the centre (Khan et al., 2001). This phenomenon is termed wall
slip or wall depletion effect. Wall slip phenomenon in pipes is the main focus
of this research study and the exploration of this phenomenon in the flow of
multiphase fluids will be explained throughout this thesis.
1.3
Pipe rheometry
Tube rheometers are very useful in collecting rheological data and
can be divided into three categories: glass capillaries, high pressure
capillaries and pipe rheometers. In this study, pipe rheometry was adopted in
the development of the measurement system. Pressure gradients are required
to create flow along pipes. The raw data sets for tube rheometers are pressure
drop and volumetric flow rate (Steffe, 1996). The pressure drop is easily
measured using pressure transducers while volumetric flow rate may be
readily determined by dividing the mass flow rate by the density. The mass
flow rate in pipe system is measured using a mass flow meter.
Two of the important phenomena that need to be considered in the
pipe rheometry are viscous dissipation/heating and wall slip. Fluid transport
through a channel inevitably requires a driving force. Multiplying that
driving force by the average flow velocity gives the power required to drive
the fluid (Hardt and Schonfeld, 2007). The work input is converted into
23
thermal energy, i.e. the fluid is heated up while being transported along a
channel, an effect being referred to as viscous heating (Hardt and Schonfeld,
2007). This process is irreversible transfer of mechanical energy to heat by
the flow working against the viscous stresses. Viscous heating/dissipation is
an undesired phenomenon in most cases.
Viscous dissipation becomes an important factor especially for highly
viscous and high shear flows since viscous dissipation changes the
temperature distribution by playing a role like an energy source (Pinarbasi et
al., 2009). The effect of viscous dissipation has been reported to be
significant in the capillary flow of a variety of polymer melts, including those
which exhibit wall slip (Kamal and Nyun, 1980; Rosenbaum and
Hatzikiriakos, 1997). The highest shear rates are obtained near the wall
which results in higher wall temperature and lower fluid viscosity. Viscous
dissipation is a mechanism which can cause phenomenon analogous to ‘wall
depletion effects in flow’ or wall slip (Elhweg et al., 2009).
1.4
Wall slip – mechanisms, effects and quantification
Wall slip in multiphase fluid flow occurs due to the displacement of
the dispersed phase(s) away from the solid boundaries leaving a lower
viscosity, depleted layer of liquid (Barnes, 1995). It is termed ‘apparent slip’
in the case of multiphase systems because the wall slippage is due to phase
separation. This kind of slippage is different from the slippage occurs due to
loss of adhesion between the fluid and the wall i.e. true slip. True slip
phenomenon normally occurs during flow of single phase polymer solutions
where there is no phase separation near the wall. Apparent slip occurs when
24
the rheological behaviour of the fluid found adjacent to the wall is different
from that of the bulk of the fluid flowing in the channel (Cohen and Metzner,
1985; Kalyon, 2005). Unlike true slip, for which the fluid slips in the
immediate vicinity of the wall due to adhesive or cohesive failure, apparent
slip occurs at the interface between the slip layer and the bulk fluid flowing
in the channel (Kalyon, 2005).
Wall slip in multiphase system such as in suspensions is closely
related to the migration effects encountered in liquids containing very little or
moderate amounts of particulates (Yilmazer and Kalyon, 1989). There may
also be physicochemical interactions present between the particulates and the
wall of the capillary rheometer tubes causing depletion of the particles close
to the wall (Khan et al., 2001). From the study by Yilmazer and Kalyon
(1989) on the flow of very dilute suspensions in a capillary, they have found
that particles move away from the wall and the centre, to a distance of
approximately 0.6 radii from the centre. The movement of the particulates
away from their original streamlines occurs due to instabilities in flow. Flow
instabilities in flows of concentrated suspensions typically occur in the
relatively low wall shear stress range (Yilmazer et al., 1989). The critical
value of wall shear stress, τwc below which the flow instabilities occur, is
dependent on the capillary/pipe diameter where τwc decreases with increasing
diameter, D.
In the apparent slip mechanism of concentrated suspensions, the
migration of the solid particles away from the wall generates an essentially
particle-free ‘slip layer’ adjacent to the capillary wall with a thickness, δ
(Yilmazer and Kalyon, 1989). Slip layer thickness is dependent on
25
deformation rate (Kalyon et al., 1993). The slip layer thickness will be
consistent along the pipe in steady state flow. It will decrease as the shear
stress increases i.e. flow rate increases in the sample. When multiphase fluids
are sheared, large velocity gradients are produced in slip layer, resulting in
apparent slippage of the bulk slurries. The creation of this slip layer makes
any flow of fluids easier due to the lubrication effects (Chen et al., 2010).
The thickness of the lubricating slip layer can be determined as (Yilmazer
and Kalyon, 1989):

Vslip  slip
w
Eq. (1.1)
on the basis of the dynamics of fully developed, incompressible, multiphase
and isothermal flow where Vslip is the slip velocity, μslip is the viscosity of the
slip layer and τw is the wall shear stress. The slip layer is assumed to be
Newtonian.
The classical technique to quantify the slip velocity was introduced
by Mooney (1931). Mooney method for determining the slip velocity of
incompressible fluids when isothermal, stationary, and laminar flow is based
on the assumption that the wall slip velocity (under constant wall conditions)
depends only on the shear stress at the wall (Graczyk et al., 2001). In the case
of capillary dies, the use of dies of different diameters but with the same
length-to-diameter ratio is required to obtain the relationship between
apparent shear rate and reciprocal die diameter at constant values of wall
shear stress. Mooney method will be described further in Chapter 2.
Alternatively, the slip velocity may be obtained by calculating the difference
between apparent shear rates at constant wall shear stress for slipping and
26
non-slipping flow (Mourniac et al., 1992). For apparent slip mechanism, wall
slip velocity is taken as the velocity at the interface between the fluid
comprising of the apparent slip layer and the bulk of the fluid flowing in the
channel (Kalyon, 2005) as illustrated in Fig. 1.1.
Vslip
Vslip
Fig. 1.1: Slip velocity at the interface between the apparent slip layer and the
bulk of the fluid
Wall slip velocity is usually linked to shear stress by a power law
function with a temperature- and pressure-dependent slip coefficient
(Yilmazer and Kalyon, 1989; Hatzikiriakos and Dealy, 1992). Wall slip
velocity, Vs is defined as the difference between the velocity of the fluid
adjacent to the wall and the velocity of the wall and is considered to be a
non-linear function of the absolute value of the wall shear stress, τw
(Yilmazer and Kalyon, 1989; Zhang et al., 1995; Kalyon, 2005):
Vslip   w
z
Eq. (1.2)
where φ is the slip coefficient and z is the slip exponent dependent on the
fluid, the materials of construction and the surface characteristic of the flow
channel (Ramamurthy, 1986).
Slip is often neglected in viscosity measurement but it is important in
mathematical modelling of processes where the condition of the boundary
27
between the material being processed and the wall of equipment which
contains it needs to be specified (Halliday and Smith, 1997). The apparent
shear viscosity, μapp for rough dies was higher than for their smooth
counterparts indicating that slip is occurring in the smooth dies (Mourniac et
al., 1992). Apparent shear viscosity can be calculated using the following
equation:
 app 
w
 app
Eq. (1.3)
Wall slip will cause the underestimation of viscosity of bulk fluid due
to the existence of lower viscosity fluid layer near the wall. This complicates
the measurement process and leads to the inaccuracy of the experimental
data. Hence, correction for wall slip is needed in order to obtain the accurate
viscosity value. The contribution of slip at the wall to the volumetric flow
rate in capillary flow was found to increase with decreasing shear stress,
giving rise to plug flow at sufficiently low shear stress values (Kalyon et al.,
1993).
Slip is a complex mechanism and its behaviour might be different
from one fluid system to another. However, slip has its own advantages
during flow and processing of multiphase fluids. It lowers the viscosity near
the wall region which helps in fluid flow and subsequently lowers the process
energy requirements in the system i.e. reducing the pumping power required.
Other advantages of wall slip will be described in the following chapter.
Despite of the advantages, slip removal or correction is necessary during
rheological measurement to determine the fluid true flow properties. As
stated before, wall slip lowers the viscosity of the fluid measured in any
28
rheometry system and the effects can be seen when measurement is taken
using different tubes/pipes diameter – the measured viscosity is lower in
smaller diameter.
1.5
Problem statements and scope of the research
Knowing that wall slip is significant in pipe flow of multiphase fluids,
the author planned to investigate this phenomenon. Characterising wall slip
of multiphase fluids is a challenging process which requires a suitable
arrangement of equipment and instrumentation to enable the data collection
process. A pipe rheometry rig capable of measuring pressure, temperature
and velocity profile was to be designed and built to enable the rheological
studies and wall slip analysis. Three multiphase fluids of different
characteristics were chosen to be studied i.e. ice cream, citrus dietary fibre
(CDF) suspensions and magnesium silicate slurries. Careful work planning
was vital to bring this target into reality. The instrumentation of pressure and
temperature transducers was important to monitor and track changes of the
fluid properties when flowing in pipes. This arrangement also enabled
quantification of wall slip and viscous dissipation phenomena in pipe flow.
This was important especially in the flow of ice cream due to its temperature
sensitive nature. The design and development of the pipe rheometry system
will be described further in the upcoming chapter. The work plan is presented
in the flow chart shown in Fig. 1.2. This flow chart illustrates the stages of
the whole experimental work plan performed which explains the scope of this
research study.
29
The first stage was the design and development of pipe rheometry rig.
The design of the system was developed and sent for fabrication. Once the
piping system was fabricated, the instrumentation of pressure and
temperature transducers; electronic mass balance and PC data logger were
performed. After everything was set up, experimental work started with the
pipe rheometry of ice cream. Ice cream was produced using an industrial
freezer (scraped surface heat exchanger) which was located next to the rig.
Pressure, temperature and flow measurements were conducted and the data
was collected using the data logger. Analysis of experimental data was done
to obtain the important plots and consequently the associated analysis. When
the experimental work on ice cream has been completed, the experiment was
repeated using CDF suspensions and magnesium silicate slurries. CDF
suspensions were produced by mixing and homogenization processes while it
was just mixing process for the magnesium silicate slurries. Pressure and
flow rate measurements were conducted to obtain the data to be used in the
analysis. For both CDF suspensions and magnesium silicate slurries,
experiments were repeated using different solid concentrations in order to
study the effects of different concentrations on the flow behaviour. The
accomplishment of this experimental procedure helps in characterising wall
slip phenomenon in complex non-Newtonian multiphase fluids. It is also to
prove the reliability of the pipe rheometry system developed to study fluid
flow behaviour and most importantly in quantifying wall slip. The method
developed to measure pressure, flow rate and especially temperature would
be useful for the industrial factories.
30
1.6
Objectives of the research
The general objective of this study was to present a comprehensive
study on wall slip analysis of several important non-Newtonian multiphase
fluids i.e. ice cream, CDF suspensions and magnesium silicate slurries. To be
specific, this study aimed:

to design and build an integrated rig for pressure drop, temperature
and velocity profile measurements during pipe flow of multiphase
fluids with ice cream as the main focus

to study wall slip and viscous dissipation phenomena during ice
cream flow in pipes

to study wall slip phenomenon in the flow of CDF suspensions and
magnesium silicate slurries of different concentrations during flow in
pipes
31
Conceptual design
Design and
development of
pipe rheometry rig
Selection of equipments
Fabrication and
instrumentation
Rheological studies
Production of ice
cream
Ice cream
Pressure and flow
measurement
Data analysis
Measure particle
size distributions
Ice cream mix
Production of CDF
suspensions
Production of magnesium
silicate slurries
CDF suspensions
Magnesium silicate
slurries
Pressure and flow
measurement
Pressure and flow
measurement
Frozen ice cream
τw vs. 
Wall slip analysis
τw vs. 
Data analysis
Wall slip analysis
Viscous heating
analysis
Completion of the
research
Fig. 1.2: Scope of the project
32
Data analysis
τw vs. 
Wall slip analysis
This thesis consists of eight chapters. Chapter 1 has introduced the
background of non-Newtonian fluids, multiphase fluids, pipe rheometry, wall
slip and explained the scope of the research. Chapter 2 explains in depth all
the backgrounds, theories and studies associated with this research project.
Chapter 3 covers the design and development of pipe rheometry rig to study
flow of fluids. Chapter 4 describes the pipe rheometry of ice cream, and the
effects of wall slip and viscous heating on the flow of ice cream. Chapter 5
explains the pipe rheometry and wall slip analysis of citrus dietary fibre
suspensions at different solid concentrations. Chapter 6 discusses on the pipe
rheometry and wall slip analysis of magnesium silicate slurries also at
different solid concentrations. Chapter 7 discusses on the wall slip analysis in
all of the multiphase systems studied. Finally Chapter 8 concludes the
research work and recommends possible future studies.
33
2 LITERATURE REVIEW
This chapter discusses the background and theory associated with the
presented research. The chapter starts with a brief introduction of the
definition, history and concept of rheology. More technical terms in the field
of rheology such as Newtonian, non-Newtonian, shear-thinning as well as the
constitutive model equations to show the flow behaviours are described as
the chapter progresses. The rheometry of non-Newtonian fluid, the associated
parameter corrections required for the determination of the true rheological
properties of fluids and the effect of viscous heating are discussed. The key
literature that forms the background to this research is detailed carefully to
serve as a foundation for the subsequent chapters.
2.1
Rheology – Definition, history and concept
The term ‘rheology’ dates back to 1929 when it was coined by
Professor Bingham of Lafayette College, Indiana, on the advice of a
colleague, Professor Markus Reiner (Barnes et al., 1989; Tanner and Walters,
1998). It is broadly defined as ‘study of deformation and flow of matter’ and
the definition was accepted when the American Society of Rheology was
founded in 1929 (Barnes et al., 1989). It is the study of the manner in which
materials flow when subjected to stress or strain (Steffe, 1996).
34
The science which we now know as rheology is in one sense very old;
and it certainly predates the formal introduction of the term in 1929 by
centuries if not millennia (Tanner and Walters, 1998). The basic concept of
‘thinness’ and ‘thickness’ relate to what we understand as viscosity and play
an important role in the understanding of flow. The most prominent figures
who thought about flow in philosophical contexts are the philosophers
Confucius (traditional dates 551 - 479 BC) and Heraclitus (traditional dates
540 – 475 BC) (Tanner and Walters, 1998). The phrase ‘panta rhei’
(everything flows) of Heraclitus and the statement ‘The master stood by a
river and said “Everything flows like this without ceasing, day and night”’ in
the Analects of Confucius [new translation by Simon Leys, 1996] (Tanner
and Walters, 1998) are two similar philosophical quotes about ‘everything
flows’. There is the phrase: ‘The mountains flowed before the Lord’ in the
famous song of the Jewish prophetess Deborah (Book of Judges Chapter 5
v5) which influenced Markus Reiner, a prominent figure in modern rheology
to propose the definition of the popular dimensionless Deborah number on
the strength of reference, the idea that eventually ‘everything flows, if you
wait long enough, even the mountains’ (Reiner, 1949; Tanner and Walters,
1998).
Rheology deals with three primary theoretical concepts: kinematics,
conservation laws and constitutive relations (Tanner and Walters, 1998).
Kinematics is the science of motion and describes how bodies deform with
time. Conservation laws deal with forces, stresses and various energy
interchanges arising from motion. Constitutive relations are more specific to
a particular or, class of bodies and serve to link motion and forces, thus
35
completing the description of the deformation (Tanner and Walters, 1998).
Focusing on the key words of the definition in isolation - deformation, flow
and matter – suggest that rheology is, by nature, multidisciplinary and it is
rooted in mathematics, physics, mechanics, but it is also rooted in life, and
therefore everybody has fundamental experience with the basic concept of
rheology (Young, 2011).
2.1.1
Measurement of rheological properties
Rheological properties are determined by measuring force and
deformation as a function of time (Tabilo-Munizaga and Barbosa-Cánovas,
2005). The fundamental concepts of stress and strain are keys to all
rheological measurements. Stress is defined as a force per unit area and is
expressed in Pascal (N m2). It is a vector quantity so has a direction; normal
stresses act on planes perpendicular to this direction and shear stresses act on
planes parallel to this direction.
Normal stresses are termed tensile when an element is being stretched
and compressive when it is being squashed. Normal stress can be expressed
by the equation:

Fn
A
Eq. (2.1)
where σ is the normal stress, Fn is the normal force and A is the area on
which the force acts.
Shear stress occurs when the force acts in parallel to a surface and
causes one element of fluid to slide over the adjacent element. The most
general definition is that shear acts to change the angles in an object (Dutch,
1999a). Shear stress can be expressed by the equation:
36

Fs
A
Eq. (2.2)
where τ is the shear stress, Fs is the shear force and A is the area.
Strain is a dimensionless quantity of relative deformation of a
material. It is the amount of deformation an object experiences compared to
its original size and shape (Dutch, 1999a). The type of strain (normal or
shear) can also be determined by the direction of the applied stress with
respect to the material surface.
Normal strain (ε) occurs when the stress is normal to the sample
surface. It is also known as longitudinal or linear strain. It is the strain that
changes the length of a line without changing its direction and can be either
compressional or tensional (Dutch, 1999a). Fig. 2.1 shows an example of
normal strain.

L
Lo
Eq. (2.3)
where ε is the normal strain, δL is the change in length and Lo is the initial
length.
Fig. 2.1: Linear extension of a rectangular bar (after Steffe, 1996)
37
Shear strain, γ is the strain that changes the angles of an object and
causes lines to rotate (Dutch, 1999a). It can be categorized into two which are
simple shear and pure shear. Simple shear involves rotation about a point in
one parallel direction with no change in area. In longitudinal strain, the
concern is about the line that changes in length but in shear strain, it is about
the change in angles. Fig. 2.2 illustrates simple shear.
Fig. 2.2: Shear deformation of a rectangular bar (after Steffe, 1996)
When the upper plate of a rectangular bar of height, h is linearly
displaced by an amount of δL while the lower surface remains stationary, an
angle called angle of shear will form. The ratio of deformation to original
height is tan γ, where γ is the angle the sheared line makes with its original
orientation.
Shear strain, γ = Deformation / Original height
Eq. (2.4)
= tan γ
Pure shear is harder to see than simple shear because there is no
stationary frame of reference. It is the deformation resulting in no change of
area. There is no rotation, only compression and extension. It can be
explained in Fig. 2.3 (Dutch, 1999b).
38
Fig. 2.3: Pure shear (after Dutch, 1999b)
Consider a block deformed without changing the area as shown in the
top row of Fig. 2.3. It looks like the only deformation involved is
compression and extension. From the diagonals of the block as shown in the
bottom row of Fig. 2.3, it can be seen that there is indeed shear because the
angle between the diagonals changes. Neither of the situations described in
simple and pure shear is likely to occur exactly in nature as most
deformations do involve area changes (Dutch, 1999b). However, they are
useful ideal concepts, and deformation does closely relate to these two types
of deformation.
2.1.2
Viscosity and the classical extremes of elastic and viscosity
The wide definition of rheology would allow a study of all materials
that flow. However, the emphasis is often on the materials that exist between
the classical extremes of a Hookean elastic solid and a Newtonian viscous
liquid. Before exploring these two extremes, it is helpful to consider the
definition of solid and liquid. Solid and liquid are two of the four
fundamental states of matter, the others being gas and plasma. Solid is
39
termed as matter which does not flow to take on the shape of its container
and does not expand to fill the entire volume available to it. Hence, it is
characterised by structural rigidity and resistance to changes of shape or
volume. It consists of atoms which are tightly bound to each other. Liquid is
defined as matter which is able to flow and take the shape of the container. It
has a definite volume but no fixed shape. Liquid is made up of tiny vibrating
particles of matter, such as atoms and molecules, which held together by
intramolecular bonds. Now, let us explore what are classical extremes of
Hookean elastic solids and Newtonian viscous liquids are about. In 1678,
Robert Hooke developed his ‘True Theory of Elasticity’ which proposed that
‘the power of any spring is in the same proportion with the extension
thereof’, i.e. if you double the tension, you double the extension (Barnes et
al., 1989). This forms the basic argument behind the theory of classical
(extremely small- or infinitesimal-strain) elasticity. At the other end of the
spectrum, Isaac Newton (1687) addressed the problem of steady shear flow
in a fluid and the ‘Principia’ published contains the famous hypothesis: ‘the
resistance which arises from the lack of slipperiness of the parts of the liquid,
all other things being equal, is proportional to the viscosity with which the
parts of the liquid are separated from one another (Tanner and Walters,
1998). This ‘lack of slipperiness’ is what is now called viscosity, and it is
subconsciously measured when rubbing a fluid between thumb and index
finger.
Viscosity is synonymous with ‘internal friction’ and is a measure of
‘resistance to flow’ (Barnes et al., 1989). All fluids offer resistance to any
force tending to cause one layer to move over another. The application of
40
shearing force (forces parallel to the surface over which they act) is required
in the relative motion between layers. The resisting forces must be in the
opposite direction to the applied shear forces. Fig. 2.4 illustrates the concept
of ‘resistance to flow’, where two parallel planes, each of area A, at y = 0 and
y = h, with the intervening space filled with liquid. The upper plane moves
with relative velocity, V while the lower plane remains stationary. The
stationary fluid between the two planes is now being sheared i.e. flow and the
lengths of the arrows between the planes are proportional to the local
velocity, Vx in the liquid.
V, F
A
y
x
h
F
Fig. 2.4: Concept of flow resistance in steady simple shearing flow
The force per unit area, F/A acting on the liquid to produce the motion is
denoted by τ and is proportional to the velocity gradient (or shear rate,  )
V/h, i.e. if you double the force, you double the velocity gradient (Barnes et
al., 1989). The flow described above is steady simple shear. This relationship
can be expressed as:
 
or can be written as:
41
V
h
Eq. (2.5)
  
Eq. (2.6)
where τ is the shear stress,  is the shear rate, and the constant coefficient of
proportionality, μ is called viscosity or shear viscosity.
Fluids for which the rate of deformation is proportional to the shear
stress are called Newtonian fluids after Sir Isaac Newton (Çengel and
Cimbala, 2006). Fluidity of Newtonian fluids is independent of shear rate.
Examples of these fluids are water, air, gasoline and oils. Newtonian liquid
will continue to flow as long as stress is applied and the liquid will be
sheared at the same rate as the shear stress applied. These properties can be
expressed by the linear relationship as shown in Eq. (2.6).
In the nineteenth century, Navier and Stokes independently developed
a consistent three-dimensional theory for Newtonian viscous liquid. The
governing equations for such a fluid are called the Navier-Stokes equations
(Barnes et al., 1989). It describes how the velocity, pressure and density of a
moving fluid are related:




DV

 P  g   2V
Dt
Eq. (2.7)
Navier-Stokes equations describe the motion of fluid substances. These
equations arise from applying Newton’s second law to fluid motion which
postulates that the acceleration, a of a body is parallel and directly
proportional to the net force, F acting on the body, is in the direction of the
net force, and is inversely proportional to the mass, m of the body i.e., F =
ma.
In the case of a Hookean solid, a shear stress, τ applied to the surface
y = h based on Fig. 2.2 results in an instantaneous deformation as shown in
42
Fig. 2.5. Once the deformed state is reached, there is no further movement,
but the deformed state continues as long as the stress is applied (Barnes et al.,
1989).
y
y

D
D’
C
C’

A
A’
B
x
B’

x
Fig. 2.5: Deformation of a Hookean solid on the application of stress. The
material section ABCD becomes A'B'C'D'
The angle, γ is called the shear strain and the relevant constitutive equation
to relate all the parameters is
  G
Eq. (2.8)
where G is referred to as the rigidity modulus. In a solid, shear stress is a
function of strain, but in a fluid, shear stress is a function of strain rate.
For two centuries, Hooke’s Law for solids and Newton’s Law for
liquids appeared to be satisfactory. However, in the nineteenth century,
scientists began to have doubts. In 1835, Wilhem Weber carried out
experiments on silk threads and found out that they were not perfectly elastic.
It is a solid-like material, however, its behaviour cannot be described by
Hooke’s Law alone because there are elements of flow in the deformation
pattern, which are clearly associated more with a liquid-like response (Barnes
et al., 1989). In 1867/68, James Clerk Maxwell made an influential
contribution from a paper entitled ‘On the dynamical theory of gases’ which
43
proposed a mathematical model for a fluid possessing some elastic properties
by putting forward the idea that ‘viscosity in all bodies may be described
independently by hypothesis’ by the equation (Barnes et al., 1989; Tanner
and Walters, 1998):
d
d 
Y

dt
dt 
Eq. (2.9)
where σ is the stress and ε the strain. Y is Young’s modulus and ѱ is a time
constant. Maxwell used this equation to calculate gas viscosity and no real
explanation was given for the equation. It is perhaps ironic that the concepts
of the rivals Hooke and Newton were united forever by Maxwell in his
equation (Tanner and Walters, 1998).
However, these two classical extremes of Hookean elastic solids and
Newtonian viscous liquids are outside the scope of rheology. Newtonian fluid
mechanics based on the Navier-Stokes equations is not regarded as a branch
of rheology and neither is classical elasticity theory (Barnes et al., 1989). The
main emphasis is with materials between these classical extremes, just like
Weber’s silk threads and Maxwell’s elastic fluids, which can be classified as
non-classical behaviour. The kind of fluids that have this kind of behaviour is
called non-Newtonian fluids. Fluidity of non-Newtonian fluids may vary
with shear rate, time, frequency or amplitude of periodic strain and other
variables.
This study focuses more on the rheological behaviour of fluid food
i.e. non-Newtonian fluids in particular. The elastic behaviour of many fluid
food is small or can be neglected (material such as dough is an exception)
leaving the viscosity function as the main area of interest.
44
2.2
Non-Newtonian fluids
Rheology is the study of non-Newtonian liquids (Massey, 2006). As
soon as viscometers became available to investigate the influence of shear
rate on viscosity, researchers found departure from Newtonian behaviour of
many materials such as dispersions, emulsions and polymer solutions (Barnes
et al., 1989). Fluids that have a reduced viscosity when the rate of shear
increases are said to have shear-thinning behaviour, although the terms
temporary viscosity loss and pseudoplasticity have also been employed. The
examples of materials with shear-thinning behaviour are margarine, ice
cream, milk, clay, gelatine, blood, liquid cement and fluids with suspended
particles. Fluids that exhibit an increase in apparent viscosity with the rate of
shear are said to have shear-thickening or dilatancy behaviour (solution with
suspended starch, sand and concentrated solutions of sugar in water).
Newtonian, shear-thinning and shear-thickening fluids are categorized as
time-independent materials.
Fig. 2.6 illustrates the curves of shear stress against shear rate for
non-Newtonian time-independent fluids and we can see how the curves are
different from the curve of Newtonian liquid. Newtonian liquids, by
definition, have a straight line relationship between the shear stress and the
shear rate with zero intercept (Steffe, 1996). Hence, that is why all fluids
which do not exhibit this behaviour are called non-Newtonian.
45
Shear
Stress
(Pa)
Herschel-Bulkley
Bingham
Shear-thinning
Newtonian
Shear-thickening
Apparent shear
rate (1/s)
Fig. 2.6: Typical non-Newtonian behaviour curves as comparison to the
curve of Newtonian liquid
From the curves in Fig. 2.6, we can see there are fluids with so-called
Bingham and Herschel-Bulkley which start to deform at certain shear stress
value. Bingham plastic material can resist a finite shear stress and thus
behave as a solid, but deform continuously when the shear stress exceeds the
yield stress and thus behave as a fluid (Çengel and Cimbala, 2006). The
stress at which the materials start to deform is called yield stress. Examples
of Bingham fluids are toothpaste and mayonnaise. The function involves
relating shear stress and shear rate calculated from experimental data. The
behaviour of the fluid is visualized as a plot of shear stress versus shear rate,
and the resulting curve is mathematically modelled using various functional
relationships which will be discussed in the upcoming subsection.
2.2.1
Yield stress
Yield stress is the minimum shear stress required to initiate flow
(Steffe, 1996). It is related to the level of internal structure in the material,
which must be destroyed before flow can occur (Tabilo-Munizaga and
Barbosa-Cánovas, 2005). Yielding corresponds to the transition from the
46
solid regime to the liquid regime, associated with the beginning of
irreversible changes in the element configuration (Coussot, 2005). When the
applied stress is less than a certain critical stress which is the yield stress,
such fluids do not flow but deform plastically like a solid with definite strain
recovery upon the removal of stress; and when the yield stress is exceeded,
the fluid flows like a truly viscous material with finite viscosity (Nguyen and
Boger, 1992). There are various methods for determining yield stress both
directly and indirectly, each requires different applications. One common
indirect method is to extrapolate the shear stress versus shear rate curve back
to the shear stress intercept at zero shear rate (Steffe, 1996). Direct
measurements generally rely on some independent assessment of the yield
stress as the critical shear stress at which the fluid yields or starts to flow i.e.
from experiments such as stress relaxation, creep/recovery, the vane methods
and cone penetration. Yield stress can be categorized into two groups. Static
yield stress is the yield stress measured in an undisturbed sample, while the
dynamic yield stress is often determined from extrapolation of the
equilibrium curve (Steffe, 1996).
Many fluids of industrial significance have been shown to exhibit
flow properties intermediate between those of a solid and a liquid. This
important class of fluids, known for their viscoplastic behaviour,
encompasses quite a wide range of concentrated suspensions, pastes,
foodstuff, emulsions, foams and composites (Bird et al., 1983; Nguyen and
Boger, 1992). The finite spread of a liquid-like food when poured out which
seems to come to a constant thickness is a clear everyday example of yield
stress to the non-expert (Barnes, 1999). There are various kinds of liquids
47
that appear to have yield stress. Food that exhibit yield stress behaviour are
ketchups, molten chocolate, mayonnaise, yoghurts and purees. For other
materials, this behaviour can be seen in clay, oil paint, toothpaste, printing
inks, ceramic pastes, surface-scouring liquids, bloods and foams.
Several empirical and theoretical models have been proposed and
used to describe the rheological behaviour of time-independent viscoplastic
fluids i.e. Bingham, Herschel-Bulkley and Casson models (Nguyen and
Boger, 1992). The practical usefulness of yield stress is important in
engineering design and operation of processes where handling and transport
of industrial suspensions are involved i.e. the minimum pump pressure
required to start a slurry pipeline, the levelling and holding ability of paint
and the entrapment of air in thick pastes (Nguyen and Boger, 1992).
2.2.2
Constitutive equations of rheological models
Constitutive equations that predict the shape of the general flow curve
need at least four parameters (Barnes et al., 1989). One of the equations is
called the Cross model, named after Malcolm Cross, an ICI rheologist who
worked on dye stuff and pigment dispersions (Barnes, 2000). He found that
the viscosity of many suspensions could be described by the equation of the
form
  
1

 o    1  ( I ) i
Eq. (2.10)
o  
i
 I 
  
Eq. (2.11)
or, when rearranged,
where μo and μ∞ refer to the asymptotic values of viscosity at very low and
very high shear rates, respectively, I is a constant parameter with the
48
dimension of time and i is a dimensionless constant. When this model is used
to describe non-Newtonian liquids, the degree of shear-thinning is dictated by
the value of i, with i tending to zero describes more Newtonian liquids, while
the most shear-thinning liquids have a value of i tending to unity. The Cross
equation can be reduced to Sisko, power law and Newtonian behaviour if
various simplifying assumptions are made (Barnes, 2000). An alternative to
the Cross model in Eq. (2.10) is the Carreau model:
  
1

 o    (1  ( I ) 2 ) i / 2
Eq. (2.12)
Both Cross and Carreau models are the same at very low and very high shear
rates, and only differ slightly at I  ~ 1 (Barnes, 2000).
Certain approximations made to the Cross model, have introduced a
number of other popular and widely used viscosity models (Barnes et al.,
1989). For μ<<μo and μ>>μ∞, the Cross model reduces to:

o
( I ) i
Eq. (2.13)
which, with a simple change of the variables I and i, can be written
  M n
Eq. (2.14)
This is the well-known power law (or Ostwald-de-Waele) model where M is
called the consistency index and n is the power law or flow behaviour index.
The power law model is a general relationship used to describe the behaviour
of non-Newtonian fluids. It is used extensively to describe the nonNewtonian flow properties of liquids in theoretical analysis as well as in
practical engineering applications.
49
For most structured liquids at high shear rates, μ<<μo and the Cross
model reduces to (Barnes et al., 1989; Barnes, 2000):
   
o
( I ) i
Eq. (2.15)
Many real flows take place for structured liquids at shear rates where the
viscosity is just coming out of the power law region of the flow curve and
flattening off towards μ∞. This situation is dealt by simply adding a
Newtonian contribution to the power law description of the viscosity giving:
     M n1
Eq. (2.16)
     M n1
Eq. (2.17)
or in terms of shear stress,
this is called the Sisko equation, and it is very good at describing the flow
behaviour of most emulsions and suspensions in the practical everyday shear
rate range of 0.1 to 1000 s-1 (Barnes, 2000). If n in the Sisko model is set to
zero (Barnes et al., 1989):
   
M

Eq. (2.18)
which, with a simple redefinition of parameters can be written as
   y   p 
Eq. (2.19)
where τy is the yield stress and μp the plastic viscosity (both constant). This is
the Bingham model equation.
Over a reasonable range (normally only over a one-decade range
approximately) of shear rates, the shear stress seemed to be a linear function
of shear rate, but displaced upwards by a constant value, which is called the
yield stress which was found by extrapolation to where the shear rate was
50
zero (Barnes, 2000). Bingham investigated systems where the plot of shear
stress against shear rate showed a simple straight-line-with-intercept-type
behaviours, and such liquids are called Bingham plastic as described
previously. However, power law model of Eq. 2.14 fits the experimental
results for many materials over two or three decades of shear rate, making it
more versatile than the Bingham model (Barnes et al., 1989). However, care
should be taken in the use of the model when employed outside the range of
data used to define it.
Another general relationship used to described the behaviour of nonNewtonian fluid is the Herschel-Bulkley model (Steffe, 1996):
   y  M n
Eq. (2.20)
This model is appropriate for many fluid foods. It combines the power law
characteristic with yield stress. Herschel-Bulkley model is very convenient
because Newtonian, power law (shear-thinning when 0<n<1 or shear
thickening when 1<n<∞) and Bingham plastic behaviour may be considered
as special cases of the Herschel-Bulkley model (Steffe, 1996). M is
commonly referred to viscosity (μ) and plastic viscosity (μp) for Newtonian
and Bingham plastic model, respectively.
Up to this point, the discussion has been about how a given shear rate
results in a corresponding shear stress, whose value does not change as long
as the value of the shear rate is maintained. However, this is often not the
case in practical situations. The measured shear stress, and hence the
viscosity, can either increase or decrease with time of shearing and such
changes can be reversible or irreversible. This can be termed as timedependent materials. Time-dependent materials have a viscosity function
51
which depends on time (Steffe, 1996). They are the additional types of nonNewtonian behaviour that arise when the apparent viscosity changes with the
time for which the shearing force is applied (Massey, 2006). Based on the
accepted definition, a gradual decrease of the viscosity under shear stress
followed by a gradual recovery of structure when the stress is removed is
called thixotropy (Barnes et al., 1989) such as starch-thickened baby food or
yoghurt. The opposite type of behaviour, which involves a gradual increase
in viscosity under stress, followed by recovery is called rheopexy or antithixotropy such as latex dispersions or plastisol pastes. This study only
focuses on the time-independent behaviour, i.e. shear-thinning, and this brief
description on time-dependent behaviour is just for the readers to understand
this kind of behaviour.
The class of so-called non-Newtonian fluids is also well described by
the term viscoelastic, which describes the dual nature of their behaviour
(Boger and Walters, 1993). It is not unreasonable to assume that all real
materials are viscoelastic, i.e. in all materials, both viscous and elastic
properties coexist (Barnes et al., 1989). The particular response of a sample
in a given experiment depends on the time-scale of the experiment in relation
to a natural time of the material which means, if the experiment is relatively
slow, the sample will appear to be viscous rather than elastic, whereas, if the
experiment is relatively fast, it will appear to be elastic rather than viscous
(Barnes et al., 1989).
2.2.3
Viscoelasticity and viscoplasticity
Viscous materials resist shear flow and strain linearly with time when
a stress is applied. Materials with the ability to change temporarily when
52
stress is applied and revert back to original form when stress is removed is
called elastic. Viscoelastic materials possess both viscous and elastic
properties when undergoing deformation. Examples of viscoelastic materials
are bitumen, nylon, flour dough, polymers and metal at very high
temperature. Viscoelastic materials exhibit time-dependent strain where the
relationship between stress and strain depends on time. Viscoelastic materials
are both non-Newtonian and thixotropic materials. The energy of the
materials is lost when load is applied and then removed. This is attributed to
the thixotropic characteristic of the materials. Hysteresis is observed in the
stress-strain curve, with the area of the loops being equal to the energy lost
during the loading cycle (Meyers and Chawla, 2008). The materials are also
characterised by the stress relaxation (step constant strain causes decreasing
stress) and creep (step constant stress causes increasing strain) properties.
Viscoelasticity is actually a molecular rearrangement of materials. When a
stress is applied on a viscoelastic material, such as polymer, parts of the
polymer chain move and change position. The rearrangement is termed
creep. The creep behaviour gives the prefix visco- and the ability of the
material to fully recover when stress is removed gives the suffix –elasticity
(McCrum et al., 2003).
For inelastic behaviour of solids, the rate-dependent behaviour can be
described by viscoplasticity. The inelastic behaviour is plastic deformation
which means that the material undergoes unrecoverable deformations when a
load level is reached i.e. yield stress. The theory of viscoplasticity is required
to describe the behaviour beyond the limit of elasticity or viscoelasticity.
Viscoplasticity is characterised by a yield stress, below which the materials
53
will not deform, and above which they will deform and flow according to
different constitutive relations (Mitsoulis, 2007). Viscoplastic behaviour can
be described by Bingham plastic, Herschel-Bulkley and Casson models. Food
like margarine, mayonnaise and ketchup are also examples of viscoplastic
materials.
2.3
Shear-thinning fluids
Shear-thinning behaviour is very common in fruit and vegetable
products, polymer melts, as well as cosmetic and toiletry products (Steffe,
1996). When flowing, these materials may exhibit three distinct regions when
the experimental data are plotted into rheogram of shear stress versus shear
rate: a lower Newtonian region where the apparent viscosity (μo), called the
limiting viscosity at zero shear rate, is constant with changing shear rates; a
middle region where the apparent viscosity (μ) is changing (decreasing for
shear-thinning fluids) with shear rate and the power law equation is a suitable
model for the phenomenon; and an upper Newtonian region where the slope
of the curve (μ∞), called the limiting viscosity at infinite shear rate, is
constant with changing shear rates (Steffe, 1996). Fig. 2.7 shows the
rheogram of fluids with shear-thinning behaviour.
54
Shear
Stress
(Pa)
Slope = μo
Slope = μ∞
Upper
region
Middle
region
Lower
region
Shear
Rate
(1/s)
Fig. 2.7: Rheogram of shear-thinning behaviour
The middle region is most often examined when considering the
performance of food processing equipment, while the lower Newtonian
region may be relevant in problems involving low shear rates such as those
related to the sedimentation of fine particles in fluids (Steffe, 1996).
Rheological models used to describe flow behaviour of a shear-thinning, nonNewtonian fluid can be described by one or more rheological models.
Various
materials
with
non-Newtonian
characteristics
are
encountered in everyday life. Most of them display shear-thinning behaviour
rather than shear-thickening. This study aimed to investigate the rheological
behaviour of three non-Newtonian shear-thinning fluids which were ice
cream, citrus fibre suspension and talc powder (magnesium silicate) slurry.
The process of characterising the flow behaviour of these materials was
challenging and exciting due to their unique characteristics. These specific
materials are introduced in the following subsections.
2.3.1
Ice cream
Ice cream is a popular food and known for its soft texture,
temperature sensitivity and creamy mouthfeel. It is an example of complex
55
microstructured material consisting of ice crystals, air bubbles and fat
globules contained within a viscous liquid matrix. Ice cream is made and
eaten in almost every country in the world. Ice cream was first enjoyed a long
time ago; it is believed to be first created 3000 years ago when the emperors
of China enjoyed snow ice cream (Mallare, 2010). The Roman Emperor is
said to have sent slaves to the mountains to bring snow and ice to cool and
freeze the fruit drinks (Goff, 2010a). These were followed by a story that
Marco Polo introduced a new snow ice cream to Italy after his return from
China in year 1295. It was believed that in 1533, an Italian chef took the ice
cream recipe to France. The story continues with Charles I took the ice cream
recipe to England in 1600s (Mallare, 2010). The exact history is not
documented, but there is some consensus about the story of ice cream. It was
believed that the first improvement in the manufacture of ice cream was from
the invention of hand-cranked ice cream freezer by Nancy Johnston in 1846
(Goff, 2010a). In 1926, the first continuous process freezer was perfected by
Clarence Vogt and continually improved by other manufacturers. According
to Clarke (2004), in reality, the history of ice cream is closely associated with
the development of refrigeration techniques which can be traced in several
stages related to this – a) cooling food and drinks by mixing it with snow or
ice, b) the discovery that dissolving salts in water produces cooling, c) the
discovery (and spread of knowledge) that mixing salts and snow or ice cools
even further, d) the invention of ice cream maker in the mid-19th century and
e) the development of mechanical refrigeration in the late 19th and early 20th
centuries. Unilever and Nestlé are the largest individual worldwide
producers, with about 17 and 12% of the market respectively (Clarke, 2004).
56
They are still the key global ice cream manufacturers to date. The global ice
cream production is large and developing tremendously.
A typical ice cream consists of about 30% ice, 50% air, 5% fat and
15% matrix sugar solution by volume (Clarke, 2004). It contains four main
constituent phases of the ice cream microstructure; which are solid ice and
fat, liquid sugar solution and gas. In the United Kingdom, it is defined as a
frozen food product containing a minimum of 5% fat, 7.5% milk solids other
than fat (i.e. protein, sugars and minerals) while in the United States, ice
cream must contain at least 10% milk fat and 20% total milk solids and must
weigh a minimum of 0.54 kg l-1 (Clarke, 2004). Ice cream is obtained by
heat-treating and subsequently freezing an emulsion of fat, milk solids and
sugar (or sweeteners), with or without other substances. Ice cream can be
categorized into three different classes depending on the composition of
ingredients used. They are premium, standard and economy. Premium ice
cream has relatively high amount of dairy fat and low amount of air while the
economy ice cream uses vegetable fat and contains more air. However, these
terms have no legal standing in the United Kingdom (Clarke, 2004).
2.3.1.1 Ice cream formulation/ingredients
The science of ice cream consists of understanding its ingredients,
processing, microstructure and texture and crucially, the links between them
(Clarke, 2004). It has been called ‘just about the most complex food colloid
of all’ by Dickinson (1992) due to its extremely complex, intricate and
delicate substance. The ingredients and processing create the microstructure.
Fig. 2.8 shows a schematic diagram of the physical structure of ice cream.
57
This structure is drawn based on author own understanding after referring to
several references (Marshall and Arbuckle, 2000; Clarke, 2004).
Fig. 2.8: Schematic diagram of the physical structure of ice cream showing
air bubbles, ice crystals and fat globules
Ice cream consists of ice crystals, air bubbles and fat droplets in the
size range of 1 μm to 0.1 mm dispersed in a viscous solution of sugars,
polysaccharides and milk proteins, known as the matrix. Overrun is defined
as percentage of increase in volume of ice cream greater than the amount of
mix used to produce that ice cream (Goff, 2010b). Ice cream has the
following composition (Goff, 2010c):

greater than 10% milk fat by legal definition, and usually between 10%
and as high as 16% fat in some premium ice creams.

9 to 12% milk solids-non-fat: this component, also known as the serum
solids, contains the proteins (caseins and whey proteins) and
carbohydrates (lactose) found in milk.

12 to 16% sweeteners: usually a combination of sucrose and glucosebased corn syrup sweeteners.

0.2 to 0.5% stabilizers and emulsifiers.
58

55 to 64% water which comes from the milk or other ingredients.
Fat performs several functions in ice cream: it helps to stabilize the
foam, it is largely responsible for the creamy texture, it slows down the rate
at which ice cream melts and it is necessary to deliver flavour molecules that
are soluble in fat but not water (Clarke, 2004). Fat globules of 0.5 – 1.5 µm
diameter were evident at the gas – liquid interfaces and protruded into the
bubble cavity (Martin et al., 2008). Sources of fat used in industrial ice cream
productions are butterfat, cream and vegetable fat.
The serum solids or milk solids-non-fat (MSNF) are an important
ingredient in ice cream because they improve the texture of ice cream (due to
protein functionality), help to give body and chew resistance to the finished
product, are capable of allowing a higher overrun without the characteristics
snowy or flaky textures associated with high overrun (also due to protein
functionality), and may be a cheap source of total solids, especially whey
powder (Goff, 2010c). Milk proteins for ice cream manufacture are obtained
from several different raw materials: milk (concentrated, skimmed or whole),
skimmed milk powder, whey powders and buttermilk or buttermilk powder.
Sweetening agents improve the texture and palatability of the ice
cream, enhance flavours and are usually the cheapest source of total solids
(Goff, 2010c). Sugars can also influence the texture of ice cream in another
way because they affect the viscosity of the matrix. Sucrose is the sugar most
commonly used in ice cream. However, it has become common in the
industry to substitute all or a portion of the sucrose content with sweeteners
derived from corn syrup because it contributes a firmer and more chewy
59
body to the ice cream, is an economical source of solids and improve the
shelf life of the finished product (Goff, 2010c).
The main purpose of the stabilizer in an ice cream mix is to absorb
any ‘free’ water and so to prevent the formation and growth of large ice
crystals during the processing and storage of the ice cream (Hyde and
Rothwell, 1973). The stabilizers in use today include locust bean gum, guar
gum, carboxymethylcellulose (CMC), Xanthan gum, sodium alginate and
carrageenan. Stabilizers help to prevent heat shock that usually happen
during the final product distribution which causes some of the ice to partially
melt when the temperature rises and then refreeze as the temperature is once
again lowered. Every time the heat shock happens, the ice cream becomes
more granular to the mouthfeel. Each of the stabilizers has its own
characteristics and often, two or more of these stabilizers are used in
combination to lend synergistic properties to each other and improve their
overall effectiveness (Goff, 2010c).
Emulsifiers are substances that concentrate and orient themselves in
the interface between the fat and the concentrated sugar solution and reduce
the interfacial tension of an immiscible system. Emulsifiers aid in developing
the appropriate fat structure and air distribution necessary for the smooth
eating and good meltdown characteristics desired in ice cream (Goff, 2010c).
The reduction of interfacial tension due to the displacement of proteins on the
fat surface by emulsifiers promotes a destabilization of the fat emulsion (due
to a weaker membrane). Egg yolk was used as ice cream emulsifier in most
original recipes. Today, mono- and di-glycerides and also polysorbate 80 are
the emulsifiers that pre-dominate most ice cream formulations.
60
Water is the medium in which all of the ingredients are either
dissolved or dispersed and it forms a high proportion of ice cream (typically
60 – 72% w/w). During freezing and hardening, the majority of the water is
converted into ice. Table 2.1 summarizes the roles of each microstructure in
ice cream.
2.3.1.2 Manufacturing of ice cream and microstructure creation
Simply mixing together the ingredients and freezing them does not
make good quality ice cream because this does not produce the
microstructure of small ice crystals, air bubbles and fat droplets held together
by the matrix (Clarke, 2004). The basic factory processes for making this
microstructure are mix preparation (which consists of dosing and mixing of
the ingredients, homogenization and pasteurization), ageing, freezing, and
hardening. The preparation of the mix involves moving the ingredients from
the storage areas to the mix preparation area, weighing, measuring or
metering them, and mixing or blending them (Marshall and Arbuckle, 2000).
i)
Mixing of ingredients
The mixing process is designed to blend together, disperse and
hydrate (dissolve) the ingredients in the minimum time with optimal energy
usage (Clarke, 2004). The ingredients are selected based on the desired
formulation and the calculation of the recipe from the formulation and
ingredients chosen, then the ingredients are weighed and blended together to
produce what is known as the ‘ice cream mix’ (Goff, 2010d).
61
Table 2.1: Roles of microstructures in ice cream (Hyde and Rothwell, 1973; Marshall and Arbuckle, 2000; Clarke, 2004; Goff, 2010d;
Goff, 2010e)
Microstructure
Fat droplets
Ice crystals
Ingredients
 Fat
 Skimmed
milk powder
 Emulsifiers
Process Involved




Mechanisms of Formation
 help to stabilize the foam
 largely possible for the creamy
texture
 slow down the rate of melting
 help improve the whipping
qualities of mix and body and
texture of ice cream
 help to stabilize the air bubbles in
ice cream as they come into
contact with the air-matrix
interface during freezing
 Emulsifiers aid in developing the
appropriate fat structure and air
distribution necessary for the
smooth
eating
and
good
meltdown characteristics desired
in ice cream
Water
(comes Freezing (heat is  Immediately formed when the Smaller ice crystals, smoother
from the milk or removed from the
textures
mix touches the cold barrel walls
other ingredients) viscous liquid)
 Rapidly scraped off by the
rotating scraper blades of the
factory freezer and all the small
ice crystals are dispersed into the
mix by the beating of the dasher
Pasteurization
Homogenization
Ageing
Freezing
 Pasteurization melts the fat
 Homogenization
reduces
fat
globule diameters
 Ageing
causes
protein
displacement on the fat globule
surface
and
promotes
fat
crystallization
 Shear during freezing also causes
some of the fat droplets to collide
and coalesce
Function
62
Air bubbles
Air
Freezing
injected
barrel
Liquid Matrix






-
Sugars
Water
Milk proteins
Stabilizer
Flavours
Colours
- air is The beating of the dasher shears the  gives the desired texture to the
into the large air bubbles entered and breaks
finished ice cream
them down into many smaller ones
 results in a considerable increase
in volume which is commonly
referred as overrun
 dispersion of small air bubbles is
vitally important to obtain good
quality ice cream
a) Sweetening agents
 improve
the
texture
and
palatability of the ice cream
 enhance flavours
 affect the viscosity of the matrix
b) Milk proteins:
 improve the texture of ice cream
 stabilise oil-in-water emulsions
against coalescence by providing
a membrane around fat droplets
 help to give body and chew
resistance to the finished product
 allow higher overrun without
snowy or flaky textures
c) Stabilizers:
 absorb any ‘free’ water and so to
prevent the formation and growth
of large ice crystals
 help to prevent heat shock
63
ii)
Pasteurization
Pasteurization of ice cream mix is required because this process
destroys all pathogenic microorganisms, thereby safeguarding the health of
consumers (Marshall and Arbuckle, 2000). It is the biological control point in
the system. It also reduces the number of spoilage organisms such as
psychotrophs and helps to hydrate some of the components (proteins,
stabilizers).
Pasteurization - (1) renders the mix substantially free of vegetative
microorganisms, killing all of the pathogens likely to be in the ingredients,
(2) brings solids into solution, (3) aids in blending by melting the fat and
decreasing the viscosity, (4) improve flavour of most mixes, (5) extends
keeping quality to a few weeks, and (6) increases the uniformity of product
(Marshall and Arbuckle, 2000).
There are two basic methods which are batch or low-temperature
long-time (LTLT) and continuous or high-temperature-short-time (HTST).
Batch pasteurization is done by heating the ice cream mix in large jacketed
vats equipped with some means of heating, usually steam or hot water to at
least 69oC (155 F) and held for 30 minutes to satisfy legal requirements for
pasteurization. Continuous pasteurization is usually performed in a heat
exchanger at 80oC (175 F) for 25 seconds. The HTST system is equipped
with a heating section, cooling section, and a regeneration section (Goff,
2010d).
iii)
Homogenization
The mix is also homogenized which forms the fat emulsion by
breaking down or reducing the size of the fat globules found in milk or cream
64
to less than 1μm (Goff, 2010d). In the homogenizer, the hot mix (>70oC) is
forced through a small valve under high pressure (typically up to about 150
atm) where large fat droplets are elongated and broken up into a fine
emulsion of much smaller droplets (about 1μm in diameter) which greatly
increasing the surface area of the fat (Clarke, 2004).
Two-stage homogenization is sometimes used with lower pressure
(about 3-5 atm) to reduce clustering of the small fat droplets after the first
stage to produce a thinner, more rapidly whipped mix and reduces the rate of
meltdown. Homogenization helps to form the fat structure which makes a
smoother ice cream, gives a greater apparent richness and palatability, better
air stability and increases resistance to melting (Goff, 2010d).
After homogenization, the milk proteins readily adsorbed to the bare
surface of the fat droplets (Clarke, 2004). Proteins stabilize oil-in-water
emulsions against coalescence by providing a strong, thick membrane around
the fat droplet. It is hard for the droplets to come into close contact due to the
interactions between the proteins on the outside of the droplets, known as
steric stabilization.
iv)
Ageing
The mix is then aged for at least four hours at between 0 and 4oC and
is gently stirred from time to time. This allows time for the fat to cool down
and crystallize, and for the proteins and polysaccharides to fully hydrate
(Goff, 2010d). Two important processes take place during ageing - 1)
adsorption of the emulsifiers to the surface of the fat droplets which replace
some of the milk protein, and 2) crystallization of the fat inside the fat
droplets due to nucleation. These actions help improve the whipping qualities
65
of mix and also the body and texture of ice cream. It is essential that ageing is
long enough for crystallization to occur and for emulsifiers to displace some
of the protein since both of these processes are important precursors to the
next stage in ice cream production (Clarke, 2004). The displacement of some
of the protein by emulsifiers produces a weaker membrane which is able to
stabilize the emulsion under the static conditions in the ageing tank, but
makes it unstable under shear.
v)
Freezing
So far, only one part of the microstructure has been formed – the fat
droplets. Pasteurization melts all of the fat and homogenization reduces fat
globule diameters (Marshall and Arbuckle, 2000). Ageing causes protein
displacement on the fat globule surface and promotes fat crystallization. The
other parts of ice cream are created in the freezing process which is the core
of the manufacturing process (Clarke, 2004). The mix at approximately 4oC
is pumped from the ageing tank into the factory freezer. The simultaneous
aeration, freezing and beating of the factory ice cream freezer converts mix
into ice cream which generates the ice crystals, air bubbles and matrix.
The modern factory ice cream freezer is known as scraped surface
heat exchanger in food industry. It is designed to remove heat from (or add
heat to) viscous liquids. It is a cylindrical barrel equipped with rotating
stainless steel dasher which has scraper blades that fit very closely inside the
barrel. The dasher has two functions: to subject the mix to high shear and to
scrape off the layer of ice crystals that forms on the very cold barrel wall
(hence the term ‘scraped surface heat exchanger’) (Clarke, 2004). Freezing is
also required to break down air cells incorporated during mixing, since
66
whipping alone did not lead to small air bubbles (Chang and Hartel, 2002).
The air bubbles formation, fat coalescence and ice crystal formation
processes occur in the freezing process are described in the following
sections:
a) Air bubbles formation
Incorporation of air into the mix is necessary to give the desired
texture to the finished ice cream (Hyde and Rothwell, 1973). The
manufacture of high quality ice cream requires careful control of both
overrun and air cell size distribution (Sofjan and Hartel, 2004). Air is injected
into the barrel through a system of filters to ensure that it is clean, dry and
free from microbiological contamination (Clarke, 2004). The incorporation
of air during freezing results in a considerable increase in volume which is
commonly referred as overrun (Hyde and Rothwell, 1973). The creation of a
dispersion of small air bubbles of average diameter of 70 µm (Goff, 2010e) is
vitally important to obtain good quality ice cream. The beating of the dasher
shears the large air bubbles entered and breaks them down into many smaller
ones: the larger the applied shear stress, the smaller the air bubbles (Clarke,
2004). Structural changes as overrun increased resulted in ice cream that
were slightly softer (higher penetration depth) and slightly more resistant to
melt down (slower melting rate) (Sofjan and Hartel, 2004).
b) Fat coalescence
The shear also causes some of the fat droplets to collide and coalesce
because the mixed protein-emulsifier layer makes the emulsion unstable
under shear as described previously. The function of emulsifiers in ice cream
is actually to de-emulsify the fat. The choice of fat type and the ageing
67
process ensure that some of the fat in ice cream mix is solid so the droplets
can partially coalesce, i.e. they form a cluster that retains some of their
individual nature (Clarke, 2004). These kinds of fats are also known as deemulsified or destabilized fats. They help to stabilize the air bubbles in ice
cream as they come into contact with the air-matrix interface during freezing.
The balance between fat, protein and emulsifier are critical for the
manufacture of ice cream because it controls the stability of the emulsion and
hence the ease of aeration and the stability of the air bubbles (Clarke, 2004).
c) Ice crystals formation
A layer of frozen mix is immediately formed when the mix touches
the cold barrel wall. It is rapidly scraped off by the rotating scraper blades of
the factory freezer and all the small ice crystals are dispersed into the mix by
the beating of the dasher. There are large temperature gradients inside the
barrel, both radial (from colder at the wall to warmer at the centre) and axial
(from warmer near the inlet and colder towards the outlet) (Clarke, 2004). The
crystals are melted after being dispersed into the warmer mix at the centre of
the barrel and subsequently cool the mix down. Near the inlet the crystals all
melt, but about one-third of the way through the barrel the mix becomes cold
enough for the ice crystals to survive (Clarke, 2004). Hence, the increase in
viscosity due to ice formation begins about one-third of the distance along the
barrel. The increase in viscosity affects the whipping characteristics of the
mix; therefore most of the small air bubbles are formed in the final two-thirds
of the barrel.
68
2.3.1.3 Research on ice cream rheology
Studies on various rheological behaviour of ice cream have been
conducted by researchers. Briggs et al. (1996) investigated the viability of
testing the yield stress of ice cream by the vane method. The ability of ice
cream to be dipped or scooped is a direct consequence of yield stress. From
the study, they found out that yield stress decreased as temperature increased.
Fig. 2.9 illustrates the image of ice cream as an example of soft solid that is
readily deformable but displaying yield stress and supporting own weight. Ice
cream flow displayed a yield stress under certain conditions, and a power-law
relationship under others; these suggest that a temperature dependent
Herschel-Bulkley model would be appropriate, although further study of
these is required to obtain a complete characterisation (Martin et al., 2008).
Fig. 2.9: Ice cream displaying yield stress (credit to iStockphoto)
Research has tended to focus on the use of ingredients to influence
the microstructure and the constituents of ice cream. More attention is now
being given to the development of microstructure during processing
69
(Wildmoser et al., 2004). The role of air bubble size in stabilizing the product
and creating a creamier mouthfeel has become appreciated (Chang and
Hartel, 2002; Eisner et al., 2005). The rheology of ice cream is very complex:
it depends on the number, size and shape of the suspended ice, fat and air
particles, the concentration of the sugars, proteins and polysaccharides and
the temperature (Clarke, 2004). An appealing texture and rheology are
critical aspects of ice cream product quality. Success is achieved by careful
manipulation of the four main constituent phases of the ice cream
microstructure: ice crystals, air bubbles and fat globules contained within a
viscous liquid matrix (Martin et al., 2008).
Wildmoser et al. (2004) investigated the impact of ice cream
microstructure on the rheological behaviour and the quality characteristics of
ice cream using oscillatory thermo-rheometry (OTR). They observed that in
the low temperature range, the ice crystal microstructure governs the
rheological behaviour of ice cream. This is because at temperature below
10oC, the degree of connectivity of ice crystals was higher and caused lower
flowability. They also observed that at temperature below 10oC, the higher
the overrun and the more finely dispersed the air bubbles, the higher the ice
cream flowability in the OTR test. This is due to the interruption of ice
crystals microstructure by the air bubbles resulting in a better scoopability of
ice cream. For the temperature range above 0oC, the ice crystals in ice cream
are completely melted and therefore air and fat phases play a major role in
the rheological behaviour which has caused an increased level of creaminess.
According to the study done by Eisner et al. (2005), the air cell structure has
proven to be one of the main factors influencing melting rate, shape retention
70
during meltdown and the rheological properties in the molten state, which are
correlated to creaminess. This section has described the science of ice cream
in depth. The next section will discuss on citrus dietary fibre (CDF).
2.3.2
Citrus dietary fibre (CDF)
Dietary fibre is defined as the edible parts of plants or analogous
carbohydrates that are resistant to digestion and absorption in the human
small intestine with complete or partial fermentation in the large intestine
(Tungland and Meyer, 2002). This definition was proposed by the American
Association of Cereal Chemists in 2000 as the most preferred definition of
dietary fibre. Dietary fibre was one of the first ingredients to be associated
with the health trend in the 1980s, particularly in bakery and cereal products;
and now dietary powder is used in the production of ice cream, margarine
and yogurt. Food developers use fibre ingredients for their functional
properties such as solubility, viscosity and gelation forming ability, and
water, oil, mineral and organic molecule-binding capacities (Tungland and
Meyer, 2002). Dietary fibres are also used in food products to increase the
viscosity, providing ‘body’, formation/stabilization of emulsions and
formation/stabilization of foams (Diepenmaat-Wolters, 1993). Citrus dietary
fibre (CDF) is among the well-known fruits dietary fibres used. Orange,
grapefruit, lemon, lime and mandarin are the example of citrus fruits. CDF is
the product obtained from harvest-fresh dejuiced, de-oiled and carefully dried
citrus fruits. As other fruit dietary fibres, CDF has very high dietary fibre
content and water binding capacity. The fibres do not dissolve in water due to
their high content of cellulose and hemi-cellulose but they show excellent
71
swelling properties (Anonymous, 2002). The high swelling properties is
responsible for the rapid increase of product viscosity of aqueous systems.
The use of CDF has benefited the food processing industry. It
enhances stability of the food system in margarine, contributes to the
improvement of body in ice cream, improves succulence in liver spread and
frankfurter-style sausage and contributes to the pseudoplastic flow behaviour
of low fat oil-in-water emulsions (Fischer, 2007). Fibrous structure of CDF
can be reduced by imposing high shear stress e.g. with a high pressure
homogenization (Anonymous, 2002). The product does not build a threedimensional gel structure after homogenization so they are easily pumpable
and can homogeneously be mixed with liquids. Fruit fibres make an ideal
ingredient for consistency adjustment of aqueous systems due to the
characteristics of smooth structure, stable water binding of the fibres and
good swelling properties without the risk of lump formation.
Fruit purees and pulps are generally characterised as non-Newtonian
fluids due to complex interactions amongst the components. Different
equations were used for the description of semi-liquid fruit products that
represent the most suitable rheogram fit. The most widely used empirical
rheological models are the power law (Grigelmo-Miguel et al., 1999;
Haminiuk et al., 2006; Dak et al., 2007) and Herschel-Bulkley model
(Jimenez et al., 1989; Bhattacharya et al., 1991; Bhattacharya et al., 1992;
Ahmed and Ramaswamy, 2004; Dutta et al., 2006; Tonon et al., 2009). In
this research, citrus dietary powders of different concentrations were mixed
with water to form suspensions and their behaviour when flowing in pipes
was investigated.
72
2.3.3
Magnesium silicate (talc powder)
Magnesium silicate (talc powder) is a chemical compound consisting
of magnesium, silicon and oxygen. The talc powder particles are nonspherical in shape. Magnesium silicate is used as talcum powder to make
baby powder, chalk and paint. It is also widely used in food, ceramics,
pharmaceuticals and cosmetics industries. In food industry, synthetic
magnesium silicate is added in table salt to prevent caking and added to filter
media to absorb impurities from used oil. In ceramic industry, powdered talc
is used in many extruded products as filler. It may be added to bulk out a
material, to modify its flow properties, or to modify the final product
properties (Martin et al., 2004). In our study, talc powders of different
concentrations were dispersed in water and the flow behaviour was
characterised. In order to measure the rheological properties of fluid,
experimental work need to be performed using various equipment developed
specifically to study rheology. The next section, Rheometry, will discuss on
the experimental methods of measurement in rheology.
2.4
Rheometry
There are various available methods for evaluating the rheological
properties of materials. In evaluating solids, one is typically looking at a
stress-strain relationship as opposed to a fluid where a shear stress-shear rate
relationship is studied (Steffe, 1996). Rheometry is a general term used to
define the experimental methods of measurement of rheological properties,
while a more narrowly defined term, viscometry, is typically used in
measurements of viscosity (Malkin and Isayev, 2006). Techniques for
73
measuring rheological properties of polymeric materials have been well
described previously by Macosko (1994). The rheology of food has been
described extensively by Steffe (1996) and Rao (2006).
The study of Newtonian and non-Newtonian fluids requires
considerable care and adequate instrumentation because poorly designed
instruments can provide data that can be misleading and of little value. The
methods of viscosity measurement are based on direct utilization of the main
equation, Eq. (2.6), which defines the concept of viscosity. A rheometer must
be capable of providing readings that can be converted to shear stress (τ) and
shear rate (  ) in the proper unit Pa and s-1, respectively (Rao, 2006). Both
values are termed as local values, i.e. they are referred to at some points in
space occupied by liquid. Thus, in determining the shear stress and shear rate,
the solutions of the problem of hydrodynamics are utilized to provide
relationship between measured macroparameters and dynamic (stress) and
kinematic (shear rate) characteristics of stream at a point (Malkin and Isayev,
2006). A well designed instrument should allow for the recording of the
readings so that time-dependent flow behaviour can be studied.
For viscosity measurement, the flow in the selected geometry should
ideally be steady, laminar and fully developed, and the temperature of the test
fluid should be maintained uniform (Rao, 2006). Both of the concepts of
laminar and fully developed flow are usually used in studies on flow and heat
transfer and are best explained using flow in a straight pipe. In steady fully
developed flow, the radial velocity profile does not change along the length
of the tube. Laminar flow means that all fluid elements are flowing parallel to
each other and that there is no bulk mixing between adjacent elements.
74
Reynolds number, named after Osborne Reynolds is used as a criterion for
laminar flow of a Newtonian fluid in a smooth tube with density, ρ and
viscosity, μ which can be represented by:
VD
 2000

Eq. (2.21)
In general, because non-Newtonian fluids especially fluid food are highly
viscous, laminar flows are usually encountered (Rao, 2006).
For fluids that exhibit Newtonian behaviour, viscometers that operate
at a single shear rate (e.g. glass capillary) are acceptable. For fluids that
exhibit non-Newtonian behaviour, data should be obtained at several shear
rates and the commonly used rheometers are concentric cylinder, plate and
cone (cone-plate), parallel disk, capillary/tube/pipe and slit flow. The
constitutive equations for shear stress in capillary/tube/pipe and slit
geometries can be simply deduced from the pressure drop over a fixed length
after fully developed flow has been achieved. In rotational rheometer
geometries i.e. concentric cylinder, cone-plate and parallel disk, shear stress
can be calculated from the measured torque and the dimensions of the test
geometry being used. However, in contrast to shear stress, the derivation of
expressions for the shear rate requires solution of the continuity and
momentum equations with the applicable boundary conditions (Rao, 2006).
In the measurement using cone-plate geometry, the shear rate depends only
on the rotational speed and not on the geometrical characteristics. In all other
flow geometries (capillary/tube/pipe, concentric cylinder and parallel disk),
the dimensions of the measuring geometry play important roles.
75
In this study, a pipe rheometry system was specially developed to
investigate the flow behaviour of ice cream, citrus dietary fibre suspensions
and talc powder (magnesium silicate) slurries.
2.5
Flow of fluids in pipes
Several researchers have developed pipe rheometry rigs to study the
rheological behaviour of non-Newtonian fluids in pipelines. Gratão et al.
(2007) conducted a work to study the rheological behaviour of soursop juice
using experimental apparatus developed for pressure loss measurement.
Soursop
juice
exhibits
non-Newtonian
power
law
behaviour
and
experimental friction factors in the fully developed, laminar, pipe flow were
reported. Various rheological characterizations have been done on other nonNewtonian materials flowing in pipes. An example of multiphase material is
coal-water paste which is a complex and highly concentrated suspension of
water and coal powders. It behaves as a non-Newtonian Herschel-Bulkley
fluid in the experimental ranges (Lu and Zhang, 2005). In the work done by
Slatter (2008) to study the phenomena associated with the pipe flow of highly
concentrated sludge in different pipe sizes, the sludge flowed in the laminar
regime and showed shear thinning characteristic typical of homogeneous
non-Newtonian flow. The flow systems developed in these studies aimed to
obtain two important parameters from the experimental work carried out i.e.
the pressure drop ΔP across a specified length, L of pipe and volumetric flow
rate, Q. There are some studies done using capillary rheometry in order to
imitate flow of fluids in pipes and during extrusion process. The velocity
profile of pressure-driven flow of a Newtonian fluid in a pipe/capillary is a
parabola (Morrison, 2001).
76
2
PR 2   r  
U (r ) 
1    
4L   R  
Eq. (2.22)
where r is the radius at which the velocity distribution need to be determined,
ΔP is the pressure difference, R is the pipe radius and L is the length of pipe.
Fig. 2.10 shows a parabolic velocity profile of Newtonian fluid.
Fig. 2.10: Parabolic velocity profile for Newtonian fluid
Integration of this velocity profile results in the Hagen-Poiseuille law, which
is the pressure drop/flow rate relationship for fluids flowing through a long
cylindrical pipe:
Q
PR 4
8L
Eq. (2.23)
where Q is the volumetric flow rate obtained by dividing mass flow rate over
density. This equation explains that for a specified flow rate, the pressure
drop and thus the required pumping power is proportional to the length of
the pipe and the viscosity of the fluid, but it is inversely proportional to the
fourth power of the radius (or diameter) of the pipe (Çengel and Cimbala,
2006). The fluid is assumed to be viscous and incompressible, while the flow
is assumed to be laminar through a channel of constant circular cross-section
that is substantially longer than its diameter, and there is no acceleration of
fluid in the channel. This equation is also applied for blood flow in capillaries
77
or veins, for air flow in lung alveoli and for the flow through a drinking
straw. For Newtonian fluids, the pressure drop and flow rate data generated
were transformed into wall shear stress ΔPR/2L versus shear rate at the wall
for Newtonian fluid 4Q/πR3, which is a straight line of slope equal the
viscosity.
PR
4Q
 3
2L
R
Eq. (2.24)
The quantity 4Q/πR3 is called the apparent shear rate,  app, which means that
it is the shear rate one could deduce from the flow rate presuming that the
flow is Newtonian and that there is no wall slip.
The notations τw and  app will be used to describe wall shear stress
and apparent shear rate in this thesis:
w 
PR
2L
Eq. (2.25)
4Q
R 3
Eq. (2.26)
app 
Eq. (2.26) for apparent shear rate is valid only for Newtonian fluids in the
absence of wall slip. If there is no slip at the wall of capillary/pipe, the
apparent shear flow,  app for non-Newtonian fluid can be corrected to be  w
by the Rabinowitsch correction as described in Section 2.52.
2.5.1
Correction for entrance effect
A correction to the measured pressure is often necessary due to losses
that occur during the reduction in radius between the barrel and the
capillary/pipe. Contraction flow at the capillary entrance region in the
pipe/capillary causes an extra pressure drop due to stretching of fluid
elements (Ahò and Syrjälä, 2006). The raw pressure drop is calculated as the
78
pressure difference between the inlet and outlet pipe/capillary. There is a loss
of pressure at the entrance to the pipe/capillary, thus the true pressure drop
across the length is smaller than the raw pressure drop. The entrance pressure
losses must be subtracted from the raw pressure drop in order to calculate the
correct pressure drop across the pipe/capillary. This correction is known as
Bagley correction (Bagley, 1957) and can be determined from Bagley plot.
Eq. (2.25) can be rearranged as
P  2 w
L
R
Eq. (2.27)
To perform this correction, data are taken on a variety of
pipes/capillaries; a plot of ΔP versus L/R at constant wall shear stress (at
constant apparent shear rate at steady state) should go through the origin and
have a slope equal to 2τw. If the line does not go through the origin, this is a
reflection of combined entrance and exit pressure losses (Morrison, 2001).
These losses can be corrected by subtracting the y-intercept, ΔPent of such
plot from the raw pressure drop, ΔP. The Eq. (2.27) now becomes:
P  2 w
L
 Pent
R
Eq. (2.28)
and
ΔPcorrected = ΔP - ΔPent
Eq. (2.29)
and the true wall shear stress value is
w 
Pcorrected R
2L
Eq. (2.30)
The entrance/exit loss correction is thus a correction to the pressure
measurement. In principle, the significance of the entrance pressure drop
compared to the pressure drop across the capillary/pipe decreases with
79
increasing L/R. The large L/R however can lead to other errors; the longer the
pipe/capillary is, the greater the pressure effect becomes (Ahò and Syrjälä,
2006). Increase in pressure can have a significant effect on viscosity of
material for example some polymers and therefore too large L/R should be
avoided. Moreover, the role of viscous heating gets more pronounced with
the longer pipe/capillary, hence it is preferable to perform the viscosity
measurements using capillary/pipe with relatively small L/R to avoid the
effects of pressure and viscous heating.
2.5.2
Rabinowitsch correction for nonparabolic velocity profile
For non-Newtonian fluids, the shear rate will be different than the
shear rate obtained in Eq. (2.26). The correct shear rate at the wall for a nonNewtonian fluid may be calculated from the following equation (Morrison,
2001),
1
4
 w  app (3 
d ln app
d ln  w
)
Eq. (2.31)
The quantity in the brackets is the Rabinowitsch correction (Rabinowitsch,
1929). To calculate the Rabinowitsch correction from data, a plot of ln  app
versus ln τw is made. The slope of the straight line is the term needed in the
Rabinoswitsch correction. Once the apparent shear rate is corrected to the
true shear rate, the viscosity may be calculated as the ratio of the shear stress
at the wall to the true shear rate at the wall of the pipe/capillary,

w
 w
Eq. (2.32)
However, if slippage takes place at the wall, the Rabinowitsch correction
cannot be applied to obtain the corrected shear rate for non-Newtonian fluid.
Therefore a procedure for correction for slip needs to be done to obtain the
80
correct viscosity of the fluid. In the next section, the wall slip phenomenon
will be described in detail and the correction methods to remove wall slip
from the calculation will be presented.
2.6
Wall slip effects
As described previously, wall slip or better defined as ‘wall depletion
effects’ in suspension flows, occurs in the flow of multiphase fluids in
viscometers (and rheometers) because of the displacement of the dispersed
phase away from solid boundaries, leaving a lower-viscosity, depleted layer
of liquid near the wall. The term ‘apparent slip’ was suggested by Yoshimura
and Prud’homme (1988) to describe the phenomenon in foods and other twophase materials because the slip is due to phase separation while in the case
of molten polymers and polymer solutions, there is no phase separation near
the wall (true slip). In the case of true slip in very viscous liquids such as
polymer melts, they do lose complete adhesion with respect to the wall and
slide along them; and the frictional force at the wall can be a function of the
local pressure (Barnes, 1995).
Fig. 2.11 illustrates the velocity profiles for plug flow (fully wall
slip), slip flow and shear flow (no wall slip) in laminar flow regime in a
straight pipe.
81
Plug flow
(total wall slip)
Slip flow
(wall slip)
Shear flow
(no wall slip)
Fig. 2.11: Schematic diagrams for plug flow, slip flow and shear flow in a
straight pipe
Wall slip can be beneficial in various ways in terms of material
processing and product structure. In a process such as extrusion and
moulding, wall slip can be beneficial by reducing process energy
requirements and pressures. In the extrusion of composites, wall slip is
expected to play an important role in the reduction of the surface tearing that
is responsible to ‘sharkskin’ phenomenon (Hristov et al., 2006). In the study
by Fatimi et al., (2012) on the strategies to improve injectability of calcium
phosphate (CaP) biomaterials that are used in bone and dental surgery, they
suggested that favouring a slight particle depletion from the wall or
eventually wall slip using surface treatment of the inner part of the needles
can contribute to the improvement of practical injection conditions. From the
observations done by Higgs (1974) on flow of many foodstuff in glass tubes,
it is obvious that many of them exhibit a wall effect or wall slip which could
be used to advantage.
However, the occurrence of slip results in complications in both the
measurement and modelling of the flow behaviour of materials (Rides et al.,
2008). Wall slip or wall depletion effects are proved to contribute to the
discrepancies observed in the measured viscosity data in the low shear rate
82
range (Gregory and Mayers, 1993; Barnes, 1995). Slip is well known to
occur during the flow of two-phase systems between smooth solid boundaries
(Pal, 2000).
2.6.1
Factors influencing wall slip
Among the factors influencing wall depletion phenomenon are size of
the dispersed particles, concentration of bulk suspension, dimensions of the
measuring geometries, temperature of the process and the magnitude of shear
stress applied. The effects of wall depletion increase with the increase in
particle size. Particles will move away from the wall at a distance of the
same size as the particle size, hence if the size of the particle is large, the
distance will be larger, thus increasing the depleted layer thickness. In the
study by Pal (2000) on the oil-in-water emulsion, he observed that at low
shear stresses, the flocs are large in size. As the ratio of the floc size to gap
width is large at low shear stresses, the slip effects are expected to be
important. With the increase in shear stress, the flocs of dispersed particles
undergo breakdown and their size decreases. Eventually at high shear
stresses, the flocs disappear and the emulsion consists of only primary
particles. As the primary particles are very small compared to the gap width,
slip effects become negligible.
Slip effects are found to be more severe when a disperse system is
flowing inside a small diameter channel. Several researchers have observed
the dependence of slip on the measuring geometry (Cohen and Metzner,
1985; Mourniac et al., 1992; Corfield et al., 1999). The effect of wall slip
becomes more pronounced in smaller pipe diameters which have higher
perimeter to cross-sectional area ratios. With decreasing local geometry
83
(diameter), the surface-to-volume ratio increases and thus, the effect of the
lubricated layer is more significant, hence the dependence of slip on the
measuring geometry (Sofou et al., 2008).
Varying the concentration of disperse particles in the multiphase
system also influences the wall depletion effects. In the study by Chen et al.
(2010) on the slip flow of coal-water slurries in pipelines, they observed that
at low concentration and at low shear stress, slip is contributed by the
formation of lubricant layer which was always present even without shear.
The formation of this layer due to static wall depletion effects causes the
material to flow in plug flow under the yield stress value. At higher
concentration, the liquid rich layers arising from static wall depletion only is
not thick enough for wall slip to develop. When the critical wall shear stress
for slip was larger than the yield stress, wall shear stress is large enough to
induce a particle migration from near the pipe wall towards the bulk. Increase
in concentration causes the decrease in slip layer thickness; decrease in wall
slip velocity due to increase in flow resistance; and rapid increase in yield
stress. For further higher concentrations, a slip layer would develop only if
significant shear deformation occurs.
Egger and McGrath (2006) studied the depletion layer thickness in
oil-in-water emulsion. At low volume fraction of the disperse phase, the
system has a greater capacity for enabling a concentration gradient to form
from the wall to the bulk solution, i.e. the droplet wall repulsions dominate.
However, as the amount of droplets increases, the repulsion between the
droplets due to packing restrictions starts to dominate over the droplets wall
repulsion which makes the depletion layer thickness decreases. When a
84
minimum volume fraction has been reached, the formation of a depletion
layer is no longer possible.
The temperature applied during processing also contributes to wall
depletion effects behaviour. Chen et al. (2010) observed that increase in
temperature causes the slip velocity to be higher due to the decrease in
viscosity but it does not affect the thickness layer or particle migration effect.
They also observed that increase in particle size and temperature causes the
formation of slip layer due to the decrease in flow resistance. According to
Nguyen and Boger (1992), at low shear stress (below the yield stress),
depleted layer thickness increases as shear rate increases and the fluid flows
as a plug (plug flow). After the shear stress reaches the yield stress value, the
thickness will decrease gradually indicating the shear deformation of the
material. This is closely related to the wall slip effect influenced by particle
sizes as explained previously where at low shear stress, the particles are
larger, and hence the slip layer is thicker. Under shear deformation, the large
particles undergo breakdown and become smaller which leads to the decrease
in slip layer thickness and reducing the effect of wall slip.
2.6.2
Mechanism of wall slip
During the flow of dispersed systems such as concentrated
suspensions and emulsions, they are brought into contact with a smooth solid
surface, and the displacement of the dispersed phase away from the solid
boundary occurs. Dispersed particles migrate from the region of high shear
near the wall towards the bulk fluid, the region of less intense shear rates.
The layer of fluid adjacent to the wall will be of lower viscosity than the bulk
fluid suspension. Due to the significantly smaller thickness of the slip layer
85
than the channel gap, the formation of the slip layer would give the
appearance of wall slip; hence why it is called ‘apparent slip’ at the wall
(Cohen and Metzner, 1985; Jiang et al., 1986). According to Barnes (1995),
this phenomenon arises from steric, hydrodynamic, viscoelastic, chemical
and gravitational forces acting on the dispersed phase immediately adjacent
to solid boundary. According to Kalyon (2005), during the flow of a
suspension of rigid particles, the particles cannot physically occupy the space
adjacent to a wall as efficiently as they can away from the wall. This leads to
the formation of generally relatively thin, but always present, layer of fluid
adjacent to the wall, i.e. the ‘apparent slip layer’ and has been observed in
different types of suspensions and gels (Bertola et al., 2003; Meeker et al.,
2004; Kalyon, 2005). This layer was detected as early as 1920 by Green
(1920) in the flow of paint suspensions under a microscope.
According to Bingham (1922), slip comes from a lack of adhesion
between the material and the shearing surface, and the result is that there is a
layer of liquid between the shearing surface and the main body of the
suspension. During the flow of a dispersed system, the formation of an
apparent slip layer free of particles with a thickness, δ, which adheres to the
wall generates a significant variation of the shear viscosity over the flow
cross-sectional area (Kalyon, 2005). The shear viscosity of the depleted layer
is significantly smaller than the shear viscosity of the bulk suspension away
from the wall, which gives rise to a step change in the slope of the velocity
distribution. Since the viscosity of continuous phase is usually quite low, the
creation of such a layer results in a lubrication effect and hence lower
viscosities were observed than true viscosities (Pal, 2000).
86
Reiner (1960) assumed that the apparent slip layer is Newtonian, and
due to that, Kalyon (2005) in his analysis assumed that the thickness of the
apparent slip layer is sufficiently small that the separation of it from the bulk
suspension to form the slip layer does not affect the shear viscosity of the
suspension. He also assumed that the thickness of the apparent slip layer is
constant under steady flow conditions and is defined by the properties of the
concentrated suspension.
The slip/depleted layer formed near the wall of a horizontal pipe is
illustrated in Fig. 2.12. Apparent shear rate is termed as the strain rate applied
on a fluid.
Homogeneous
fluid

Vslip

Liquid
layer
Fig. 2.12: Wall slip/depleted layer formed at the inner surface of a pipe
during multiphase fluid flow
Based on Fig. 2.12, shear rate in the slip layer can be written as:
 
V
Eq. (2.33)

Wall shear stress is written as:
 w   slipslip
87
Eq. (2.34)
By combining Eq. (2.33) and Eq. (2.34) yields:
 w   slip
Vslip

Eq. (2.35)
Based on Eq. (2.35), depleted slip layer thickness is inversely proportional
with wall shear stress. Thus, it can be deduced that at low wall shear stress,
slip is apparent due to the thicker depleted layer. Wall shear stress is also
expressed as  w 
PR
.Substituting this expression into Eq. (2.35),
2L
PR  slipVslip

2L

Eq. (2.36)
As a fluid flows in a pipe, the fluid pressure will drop to overcome the
resistance or friction exerts by the pipe wall. Based on Fanning friction factor
formula which relates the friction to the wall shear stress in fluid flow
calculations yields:
w 
fVslip
2
2
Eq. (2.37)
Combining Eq. (2.35) with (2.37) yields:
 slipVslip fVslip 2


2
Eq. (2.38)
Thus it can be deduced that as the slip layer thickness increases, the
friction or resistance exerts by the wall decreases. Generally, during flow in
pipe at fixed shear rate, friction factor f is normally constant along the pipe,
hence, pressure decreases linearly (Fig. 2.13) along the pipe and it was
assumed that δ is constant too.
88
P
f constant
 constant
L
Fig. 2.13: Pressure decreases linearly along the pipe length where f is
constant and δ is assumed to be constant too
However, if the slip layer thickness is not constant along the pipe, let
us assume that it increases; resistance/friction along the pipe will decrease
and affect the pressure drop. Pressure in the pipe will decrease non-linearly
with distance (Fig. 2.14). As δ →R, ΔP will become smaller and the pressure
will not decrease much.
P
 increases
f decreases
L
Fig. 2.14: Pressure is assumed to decrease non-linearly along the pipe length
if the depleted layer thickness increases
Fig. 2.15 shows the force balance for pressure driven flow between
two flat stationary surfaces.
89
τ + ∂τ
P + ∂P
Fig. 2.15: Force balance for pressure driven flow between two flat stationary
surfaces
Pressure is constant in radial direction if streamlines are parallel, therefore no
flow in radial direction assuming homogeneous material. Force balance over
unit length of pipe yields:
w  
R dP
2 dx
Eq. (2.39)
Force balance over radial element of pipe yields:

r

w R
Eq. (2.40)
Assuming that there is wall slip with a Newtonian slip layer and its thickness
is negligible compared to pipe radius (δ << R), the slip layer thickness is as
shown in Eq. (1.1) i.e.:

Vslip  slip
w
Eq. (2.41)
The apparent slip layer thickness is not affected by channel gap or the
volumetric flow rate and the fluid forming the apparent slip layer and the
bulk suspension are incompressible. According to Barnes (1995), the typical
thickness of the slip layer is in the order of magnitude of 0.1 – 10 μm in
90
geometries hundreds or thousands of times bigger. Kalyon (2005) has
analysed the structure of the concentrated suspension of potassium chloride
(KCl) particles and the thickness values of the slip layer were determined to
be in the range of 2 to 30 μm.
There are a number of studies on shear-induced particle migration in
nonhomogeneous flow fields (Koh et al., 1994; Lukner and Bonnecaze,
1999). Their measurements confirmed that the velocity profile in a
concentrated suspension was blunted due to the migration of particles from a
high shear area to low shear area and different from that of a Newtonian
fluid. In nonhomogeneous flow, the shear rate and shear force are
nonhomogeneous in a section perpendicular to the direction of flow, which
causes the migration of particles (Wang et al., 2007). Particle migration
could be described based on theory which assumes that there are two primary
causes for particle migration such as particle interaction and local variations
of concentration-dependent viscosity (Leighton and Acrivos, 1987; Phillips et
al., 1992; Miller and Morris, 2006).
Wang et al. (2010) observed that in the flow of concentrated
suspension in a capillary flow, particles migrate towards the centre of the
capillary tube causing nonhomogeneous concentration profile. If the effect of
particle migration is significant, the pressure gradient in the capillary tube
decreases with increasing axial distance of the capillary. With an increase in
flow length, the particles will migrate more and thus reduce the concentration
gradually over the wall. Hatzikiriakos and Dealy (1992) reported that the slip
velocity is not uniform along the flow length if pressure-dependent slip
velocity prevails. This statement was confirmed by Wang et al. (2010) in
91
their proposed slip model which showed that the slip velocity increases with
decreasing hydraulic pressure in a capillary, with the smallest slip at the inlet
where the hydraulic pressure is the largest.
2.6.3
Correction for wall slip
If wall slip effects were not taken into account in the measurement
and modelling of the flow behaviour of materials, the viscosity of the
materials or the pressure gradients in pipe/capillary would be underestimated.
As shear rheology is a material property, it should be independent of the test
geometry. The most obvious manifestation of slip is that one obtains different
answers in different-sized geometries when calculating viscosities from
formulae that assume no slip in the flowing liquid, in particular the apparent
viscosity calculated in this way always decreases with decrease in geometry
size, e.g. tube radius, gap size and cone angle (Barnes, 1995). The
characterisation of wall slip includes the Mooney method (Mooney, 1931),
modified-Mooney methods (Jastrzebski, 1967; Wiegreffe, 1991; Crawford et
al., 2005) and Tikhonov Regularisation-based Mooney method (Yeow et al.,
2000; Yeow et al., 2003). All of these methods will be discussed in the
following subsections.
2.6.3.1 Mooney method
Mooney method (Mooney, 1931) is a graphical inverse problem
solution technique used to analyse laminar flow with apparent wall slip along
capillaries and pipes. Mooney performed a macroscopic analysis of the
slipping phenomenon for fully developed, incompressible, isothermal and
laminar flow in circular tubes. It does not differentiate between the true slip
92
(due to loss of adhesion of material and solid wall) and slip due to wall
depletion, but it is an analysis of the observed macroscopic behaviour.
Mooney (1931) presented a method for determining the apparent wall
slip velocity from a set of flow curves obtained from pipes of different
diameters. The analysis assumes that:

The fluid is homogeneous.

The streamlines are parallel and thus pressure is constant in the radial
direction.

The fluid may be described by a constitutive equation relating shear
stress, τ, to shear rate,  , of a general form, τ = f(  ).

The wall slip velocity, Vslip, is a discontinuity and is a function only of
the wall shear stress, τw.

The flow is fully developed.
Again, a force balance on a fluid element of radius r yields the equations:

r

1 dP


2 dx 
r


R w
Eq. (2.42)
Eq. (2.43)
where dP/dx is the pressure gradient in the axial x direction and R is the
inside pipe radius. For a non-Newtonian fluid which only changes with shear
rate and nothing else, the integration of the velocity of a radial element of
fluid yields the equation:
4Q
4
4
 Vslip  3
3
R
R
w
  w


0
93
2
f
1
 d
Eq. (2.44)
where Q is the volumetric flow rate. The left hand term is known as the
apparent shear rate, app ; the first term on the right accounts for the
contribution of flow due to slip; the second term on the right accounts for the
contribution of flow due to shear in the bulk of the fluid. This second term is
only a function of wall shear stress. Therefore, if wall shear stress is constant,
this term is constant too. Mooney proposed reading off values of apparent
shear rate at constant wall shear stress from flow curves from different pipe
radii. A plot of apparent shear rate against 4/R then yields a linear trend with
gradient equal to the slip velocity and ordinate axis intercept related to the
bulk shear term. This plot will be a straight line if the slip velocity is
constant. Conversely, if the plot yields a straight line, the slip velocity must
be constant. If the slip velocity and wall shear stress are constant, the ratio of
slip layer thickness to slip layer viscosity must be constant too. Therefore, if
the slip layer is Newtonian (i.e. of constant viscosity) then the slip layer
thickness is constant.
Apparent slip is more often attributed to the shear of a thin layer of
low viscosity fluid (e.g. a rarefied suspension or pure binder) between the
wall and the bulk fluid. The analysis requires data for the variation of the
apparent shear rate against the radii, at a constant wall shear stress. The
derivation of the equation is detailed in Appendix A. If this layer is of
thickness δ and apparent viscosity μslip, then the slip layer shear rate, slip , is:
slip 
V slip
Eq. (2.45)

94
Eq. (2.44) simply states that the apparent shear rate is the sum of the slipping
and shearing components (the terms on the right hand side). Using the
following definition for the apparent shear rate,  app ,
app 
4Q
R 3
Eq. (2.46)
one may express Eq. (2.44) as follows:
app 
4
4
Vslip  3
R
w
  w



2
f
1
 d
Eq. (2.47)
0
The last expression on the right hand side is the pure shearing component of
the apparent shear rate,  sh, thus
app 
4
Vslip   sh
R
Eq. (2.48)
As explained previously, the slopes in  app against 4/R plots which determine
the coefficients of slip are straight lines. Each line, being straight, has a
constant slope; and therefore the coefficient of slip is constant for all points
through which the line is drawn. But all these points refer to the same
shearing stress. Points referring to a different stress lie on a different line,
having different slope and indicating therefore a different coefficient of slip
(Mooney, 1931). Mooney assumes that the slip is a function of the shearing
stress at the wall, but it may be any function. The slip velocity may be written
as:
V slip

w
 slip
Eq. (2.49)
Therefore, if the Mooney method is used to analyse such apparent slip
phenomenon, there is an implicit assumption that the ratio of slip layer
thickness to apparent viscosity is constant.
95
Mooney analysis is said to be successful when the plot of 4Q/πR3
against 4/R yields a straight line with Vslip as the gradient of the slope and γ̇
as the –y intercept (true shear rate due to shear flow). The plots obtained shall
look like in Fig. 2.16. Fluids which are able to be analysed by this method are
assumed to be homogeneous (as this is the basis of this method). Any
observed changes in 4Q/πR3 are due entirely to slippage and are independent
of fluidity (Mooney, 1931).
4Q/πR3
τw4
τw3
τw2
τw1
4/R
Fig. 2.16: Example of successful Mooney plots
However, if the plots of 4Q/πR3 against 4/R intercept the negative –y axis, it
indicates that the data do not comply with the Mooney method (Fig. 2.17).
4Q/πR3
τw4
τw3
τw2
τw1
4/R
Fig. 2.17: Unviable 4Q/πR3 against 4/R plots with negative –y axis intercept
96
If the plots intercept the negative –y axis, the true shear rate value is
negative and flow rate due to wall slip has larger value than total flow rate.
This is definitely unviable. This phenomenon indicates that there is a
difference in behaviour between the fluid successfully analysed by the
Mooney method and the unsuccessful ones. This is believed due to
inhomogeneity in the flow system which makes it difficult to determine the
slip velocity and true shear rate.
2.6.3.2 Modified-Mooney methods
In most studies, Mooney method is suitable for polymer melts, but
cannot be applied to a non-Newtonian suspension fluid with significant
particle migration (Wang et al., 2010). Several researchers have proposed,
modified-Mooney methods for a generalized-Newtonian fluid (Jastrzebski,
1967; Wiegreffe, 1991; Crawford et al., 2005). The modifications are listed
in Table 2.2. In those modified methods, they have introduced added
parameter in the plot of 4Q/πR3 against 1/R by Mooney (1931) to obtain
positive intercepts of the lines due to negative intercepts obtained when
using the classical Mooney analysis. As mentioned previously, the negative
intercepts means negative bulk shear value and thus is unviable.
The pure shear rate,  sh obtained upon the removal of slip in Mooney
and modified-Mooney methods can be used in Rabinowitsch correction to
correct the bulk shear rate,  sh for the non-Newtonian velocity profile, and
obtain the true wall shear rate,  w as described previously. The Rabinowitsch
correction is expressed in terms of  sh , as follows (Crawford et al., 2005):
1
4
 w   sh (3 
97
d ln  sh
)
d ln  w
Eq. (2.50)
or
1
4
1
j
 w   sh (3  )
Eq. (2.51)
where j is the gradient of the τw versus  sh log-log curve. The true viscosity
of the sample is then calculated using Eq. (2.32). Finally, as the slip velocity
is known at various stress levels, the relationship between the slip velocity
and the wall shear stress can be generalized in Navier slip law of the form:
 w  Vslip p
Eq. (2.52)
where α is the Navier’s slip coefficient and p is an index parameter.
Table 2.2: Comparisons between the classical Mooney and modifiedMooney methods
Paper
Mooney
(1931)
Analysis
procedure
4Q/πR3
against 1/R
Vslip
calculation
Vslip = b/4
Remarks
-
Jastrzebski
(1967)
4Q/πR3
against 1/R2
Vslip = b/4
-
Wiegreffe
(1991)
4Q/πR3
against 1/R2
4φτw = 4Vslip R

b = 4φτw
Crawford et
al. (2005)
4Q/πR3
against 1/Rp+1
4φτw = 4Vslip Rp

Proposed a new slip
analysis which assumes
that at a given stress
level, the wall slip
velocity, Vs is a function
wall shear stress, τw and
capillary radius, R
b = 4φτw
p is determined using
optimization algorithm
and varied so as to
minimize the overall
error between the data
points and the best-fit
lines on the modifiedMooney plots


98
Lam et al.
(2007)
Q/πR3τwe
against 1/R
b
-
Wang et al.
(2010)
[4n/(3n+1)]
[z+3] against
1/R
b

Includes Rabinowitsch
correction for nonNewtonian fluids and
the non-linear effects of
slip velocity on wall
shear stress
b = slope of the curve
2.6.3.3 Tikhonov Regularisation-based Mooney method
The Mooney analysis can also be implemented using Tikhonov
regularisation. This method is called Tikhonov Regularisation-based Mooney
(TRM) analysis and is used to process capillary viscometry data for materials
with a bulk yield shear stress and in the presence of wall slip. It is a least
squares method which utilised linear algebra to calculate the least squares
best fit slip velocity and also calculates the least squares best fit true flow
curve of the bulk fluid. It addresses certain limitations of the graphical
Mooney analysis by using all the available experimental data, taking into
account experimental errors and not introducing any new errors (Martin and
Wilson, 2005). In this method, the Rabinowitsch correction is incorporated in
the analysis and a successful analysis yields the optimal material bulk shear
stress-shear rate dependence, yield stress of the bulk material and interface
shear stress-apparent slip velocity dependence from the capillary/pipe flow
data without assuming any form of constitutive equation.
This Tikhonov regularisation-based method is applied to find an
approximate solution to Eq. (2.44) using measured experimental data (Yeow
et al., 2000; Yeow et al., 2003). In this method, the interval between the
minimum and maximum τw in a data set is divided into NJ uniformly spaced
99
points and the unknown slip velocities at these points are represented by
vector Vslip. Similarly the maximum integration interval in Eq. (2.44) is
divided into NK uniformly spaced points and the unknown shear rates at these
points are represented by  . The precision of the approximate solution is
quantified using the sums of the squares of the deviation between the
approximate solution and the experimental data, S1. The smoothness of the
approximate solution is judged using the sums of the squares of the second
derivatives of the approximate solution at the internal discretization points,
S2. Tikhonov regularisation minimizes a linear combination of these two
quantities S1+λS2 to yield a solution, where λ is an adjustable numerical
factor known as the regularisation parameter. The condition   y   0 must be
satisfied and was solved iteratively using the approach of Yeow et al. (2000).
Yeow et al. (2003) illustrated the successful application of this TR Mooney
analysis to previously published data where classic Mooney analysis had
already been successful.
Martin and Wilson (2005) summarised all the capillary flow wall slip
for pastes by reapplying the TRM analysis to some cases in the literature
where the classic Mooney analysis was reported to have failed, and where the
experimental data were available. They concluded that for the paste
materials, it was only moderately successful at best, yielding viable results in
only three of eleven reported sets of data, where the use on the data of
Jastrzebski was successful. For the foams and polymers, the analysis was
much more successful, yielding viable results in all six cases where data were
reported. They also suggested that the Jastrzebski interface condition should
no longer be used and if the microstructure of the bulk material changes with
100
the capillary radius, then the rheological properties are likely to change as
well and thus Eq. (2.44) is no longer valid and any form of Mooney analysis
will be invalid. The application of TRM analysis was adopted by several
researchers and the method yielded successful results (Chen et al., 2010; Ma
et al., 2012).
2.7
Analysis on wall slip effects in the flow of multiphase systems
Wall slip behaviour in various types of non-Newtonian systems has
been analysed by researchers since a long time ago using the classical
Mooney, modified Mooney and until recently, Tikhonov RegularisationMooney methods. Those studies cover a very wide range of materials
including polymers, concentrated suspensions and pastes. Table 2.3 details
the summary of all capillary and pipe flow wall slip analyses for various
systems. Most polymers such as polylactide, polypropylene, polyethylene
and polystyrene are single phase fluids which have either straight chain or
entangled structure.
From the summary in Table 2.3, it can be seen that Mooney analysis
is compatible with the data for single phase polymers with linear structure
(Zhao et al., 2011; Ansari et al., 2012; Othman et al., 2012). For polymers
with entangled structure, classic Mooney analysis was found to give
unphysical slip velocity data (Crawford et al., 2005; Zhao et al., 2011). Most
of the multiphase fluids analysed using the Mooney method gave negative
bulk shear results. From the observation, it is concluded that there is a link
between the microstructure of the fluids and the incompatibility of the fluid
rheological data with Mooney slip analysis. Fluids which contain particles
101
with irregular shape such as kaolinite platelets, talc platelets, silica powder
and calcium phosphate leads to difficulty in rheological characterization
(Jastrzebski, 1967; Lanteri et al., 1996; Martin et al., 2004; Fatimi et al.,
2012).
Difficulty in characterising the rheological behaviour of multiphase
systems was also reported in dense systems with a very viscous deformable
phase, consisting of hydrated dispersed phase such as potato starch paste
(Halliday and Smith, 1997; Cheyne et al., 2005) and doughlike material
(Singh and Smith, 1999; Sofou et al., 2008). A similar challenge was also
found in aqueous foam dispersion which generates rheological data that is
incompatible with Mooney analysis (Herzhaft et al., 2005). In a more
concentrated complex system such as coal-water slurry used in coal
combustion/gasification processes, increase in particle concentration causes
the formation of a slip layer that is mainly affected by shear-induced particle
migration effects rather than static wall depletion effects (Chen et al., 2010).
Shear-induced particle migration effects causes some dissociation taking
place in the fluid which leads to phase separation i.e. particle migration or
changes in local solids concentration, particle packing density or structure
during flow (Cheng, 1984; Lanteri et al., 1996; Chevalier et al., 1997). This
phenomenon indicates the inappropriateness of homogeneous model.
102
Table 2.3: Summary of all capillary and pipe flow wall slip analyses on various systems
Paper
System
Structure/behaviour
N
Rmin
Classic
Modified
(mm)
Mooney
Mooney
Analysis
Analysis
TR
Mooney (1931)
Concentrated suspension
- Kaolin/water
 Multiphase system
 Wall depletion effect
 Complex system – at very high shear rate, the flow
behaviour will change after reaching the
maximum shear rate value which might be due to
the turbulence in the water between the particles
of clay, eventhough the particles themselves are
moving in essentially parallel lines with the
turbulence
5
0.4

-
-
Chung and
Cohen (1985)
Paste (polymer)
- Glass fibres (16 x 400
μm)/thermoplastic melts
 Multiphase system
 Wall depletion effect
 Breakage of glass fibres at high shear rates change
the rheological properties of material
4
0.25

-
-
Graczyk et al.
(2001)
Paste
- Alumina powder/silicon
oil
 Multiphase system
 Wall depletion effect
 Complex structure
2
1.5

-
-
Tsao et al.
(1993)
Paste (composite)
- Si powder + SiC
whiskers
(5μm)/thermoplastic
melts
 Multiphase system
 Wall depletion effect
 Suspension viscosity is affected significantly by
particles size distribution and particle surface
roughness – broader particle size distributions and
round, smooth particles result in lower viscosities
3
0.4

-
-
103
Dubus and
Burlet (1997)
Suspension
- Alumina powder
(0.6μm)/thermoplastic
melts
 Multiphase system
 Wall depletion effect
3
1.0

-
-
Yilmazer and
Kalyon (1989)
Highly filled suspension
(60% v/v)
- (NH4)2SO4 powder
(23μm)/polymer
 Multiphase system
 Wall depletion effect
 Migration effects increase with increasing shear
stress due to the reduction in the concentration of
spheres
 Flowing as plug flow at high shear as opposed to
low concentration system
3
0.66

-
-
Kalyon et al.
(1993)
Concentrated suspension
- Al + (NH4)2SO4 powder
(23μm)/polymer




Multiphase system
Wall depletion effect
Complex structure
Flow behaviour changes depending on the
concentration of the suspension
 Shear-thinning at very high concentration (76.5
vol% solids)
 Shear-thinning followed by shear-thickening with
increasing wall shear stress at medium
concentration (60%)
3
0.65

-
-
Corfield et al.
(1999)
Concentrated paste
- Potato granules/water
 Multiphase system
 Wall depletion effect
 Complex physical structure and response to
mechanical forces
 Particles absorb water and consequently swell
4
1.15

-
-
104
Halliday and
Smith (1997)
Concentrated paste
- Potato granules/water
 Multiphase system
 Wall depletion effect
 Complexity of time-dependant temperature and
water response to granules
 Undergoes changes at elevated temperatures when
it changes in combination with water loss
 At ambient temperature, particularly at low water
content, it could approach the situation where they
exhibit solid rather than fluid properties
3
0.5

(successful
-
-

-
-
but
sometimes
gives -ve
bulk shear)
Higgs (1974)
Suspension
- French mustard/tomato
puree
 Multiphase system
 Wall depletion effect
Ansari et al.
(2012)
Polymer melt
- High density
polyethylene




Single phase system
Linear structure
True slip
Flow instabilities at flow rates greater than a
critical value causing surface defects on the
surface of extrudates i.e. melt fracture phenomena
(sharkskin or surface melt fracture; slip-stick or
oscillating melt fracture; and gross melt fracture)
6
0.26

-
-
Ardakani et al.
(2011)
Paste
- Toothpaste
 Multiphase system
 Wall depletion effect
 Has yield stress, thixotropic and Bingham plastic
behaviour
 Homogeneous – liquid migration is assumed to be
negligible
4
0.43

-
-
105
Othman et al.
(2012)
Zhao et al.
(2011)
Polylactide
- PLA 7001D
- PLA 2002D
- PLA 3051D
- PLA 3251D
Polymer melts
-
Polypropylene
High-density
polyethylene
Polystyrene
 Flexible polymer chain expended linear random
coils
 Single phase polymer
 True slip (loss of adhesion) after critical slip stress
3
0.43
 Single phase polymer
 True slip
3
0.5
 Straight chain stucture
 Entangled structure
-
Polymethylmethacrylate
Kalyon (2005)
Concentrated suspension
- KCl particles/elastomer
binder
 Two-phase system
 Wall depletion effect slip
3
0.5
Hristov et al.
(2006)
Polymer
- Wood flour/high-density
polyethylene
 Stick-slip
 Yields rough and smooth structure (sharkskin)
- Rough structure when material stick at the
wall with momentarily increasing pressure
- Smooth structure when material slip at the
wall with low pressure
3
1.0
106

-
-


-
-
x (neg. bulk
shear)
x (neg. bulk
shear)

-
-
-
-
-
-

-
-
Jastrzebski
(1967)
Concentrated suspension
- Kaolinite platelets
(0.55μm)/water
 Multiphase system
 Wall depletion effect
 When hydrated, various degrees of association due
to the interaction between the particles and
between the particles and the water molecules
 At high concentrations, they form flocculated
structure showing a 3-dimensional arrangement
resembling gel
3
1.65
x (non linear)
Singh and Smith
(1999)
Suspension
- Wheat meal/water
 Multiphase system
 Wall depletion
 Viscosity decreased with the addition of water and
with the increase in temperature
 Starch absorbs water and swell
3
1.0
x (neg. bulk
shear)
Cheng (1984)
Suspension
- Polystyrene spheres
(17μm)/aqueous glycerol
solutions




Multiphase system
Wall depletion effect
Cannot be treated as homogeneous system
Packing structure is central to the behaviour of
dense suspensions
 Phase separation i.e. particle migration or changes
in local solids concentration or particle packing
density or structure occurs during flow
2-3
1.25
x (non-linear)

Jastrzebski
method
(successful)
-
Martin et al.
(2004)
Paste
- Talc platelets
(8μm)/water +surfactant
 Multiphase system
 Wall depletion effect
 Complex structure (irregular shape) which leads to
difficulty in rheological characterization
 Shear-induced re-orientation of the talc platelets
during flow
3
0.5
x (neg. bulk
shear)

Jastrzebski
method
(successful)
x
107

Proposed
Jastrzebski
method
-*
x
Jastrzebski method
(failed)
-
Lanteri (1996)
Paste
- Silica powder (1100μm)/LDPE+fluidizer




Chevalier et al.
(1997)
Paste
TiO2 powder/acid+water
Harrison et al.
(1987)
Khan et al.
(2001)
3
0.85
x (non-linear)
 Multiphase system
 Wall depletion effect
 Some dissociation taking place in the material
(indicating the inappropriateness of homogeneous
model) – confirms the validity of a biphasic model
2
0.78
x (neg. bulk
shear) Qs>Q
Suspension
- MCC powder/aqueous
lactose solution
 Multiphase system
 Wall depletion effect
 Complex deformation of the material when the
material is forced to flow into the die
3
0.5
x (non-linear)
Suspension
- Alumina
(0.4μm)/aqueous
polymer solution
 Multiphase system
 Wall depletion effect
 Material flow response is highly dependent upon
the interfacial characteristics of the boundary,
therefore, the nature of the boundary is very
important in the paste processing operation
 During flow, the interfacial resistance naturally
induces inhomogeneities within the flow which
produce complex stress and strain (or shear) rate
fields within the bulk of the flowing paste material
 Reaction or processing forces involved during the
deformation are greatly influenced by the induced
inhomogeneities flow conditions
3
0.5
x (Qs>Q)
Multiphase flow
Wall depletion effect
Irregular shaped particles
He found that the flow behaviour of paste is
dependent on the radius
108
Proposed a
heterogeneous model

-*
Wiegreffe
method
(successful)
-*
x
Jastrzebski method
(failed)
-*

Jastrzebski
method
(successful)
-*
Herzhaft et al.
(2005)
Dispersion
- Foam/hydroxy-propylguar + biocide + KCl
 Multiphase system
 Wall depletion effect
 Unstable system where drainage, coalescence and
Ostwald ripening leads to an alteration of the
bubbles size distribution - must be handled with
great care
2
3.85
(pipe)
x (neg. bulk
shear)
Cheyne et al.
(2005)
Suspension
- Potato starch paste
 Wall depletion effect slip
 Dense suspension with a very viscous deformable
phase, consisting of a hydrated starch gel
4
1.0
x (neg. bulk
shear)
-
-
Crawford et al.
(2005)
Polymeric material
- Polydimethylsiloxane
gum
 True slip
 Highly entangled, linear structure
3
0.25
x (neg. bulk
shear)
 Wiegreffe method
(bad correlation)
 Crawford method
(successful)
-
Sofou et al.
(2008)
Bread dough
- Flour/water system
 Wall depletion effect
 Possess both viscoelastic and viscoplastic
properties
3
0.43
x (neg. bulk
shear)
 Geiger (1989)
method
(successful)
-
Fatimi et al.
(2012)
Concentrated suspension
- 40% biphasic calcium
phosphate/
hydroxypropylmethylcell
ulose




3
0.42
x (neg. bulk
shear)
Two-phase system
Wall depletion effect slip
Particles are not spherical
Shape and size vary from one particle to another
109

Jastrzebski
method
(successful)
-
-
-
Chakrabandhu
and Singh
(2005)
Suspension
- Green peas/aqueous
CMC
 Multiphase system
 Wall depletion effect
 Complex due to large size of solid
1
11
(pipe)
-
Delgado et al.
(2005)
Suspensions
- Air bubbles/Lithium
complex soap in
mineral/synthetic oil
 Multiphase system
 Wall depletion effect
 Very high viscosity and very low flow rates
8
7.92
(pipe)
-
Zhou and Li
(2005)
Paste
- PVA fibers, silica sands
Methocel
powders/cement and slag
 Multiphase system
 Wall depletion effect
4
-
 Jastrzebski method
(successful)
-
Hicks and See
(2010)
Dough
- Hard wheat flour/water
 Multiphase system
 Wall depletion effect
 Complex material consisting of a polymeric
network of gluten proteins with around 60% v/v
starch filler particles
2
1.0
-
 Geiger method
(successful)
-
Meng et al.
(2000)
Suspension
- Coal powder
(450μm)/water
 Multiphase system
 Wall depletion effect
 Complex highly concentrated system
2
10
-

Jastrzebski
method
(successful)
-
Lu and Zhang
(2002)
Suspension
- Coal powder
(<6000μm)/water
 Multiphase system
 Wall depletion effect
 Complex highly concentrated system
4
12.5
-

Jastrzebski
method
(successful)
-*
3
110

Proposed an
alternative
method and
modified
Mooney
equation so that
τw = f(r,Φ)n

Proposed a
modified
equation
-
-
*
s
s
x
Chen et al.
(2009)
Suspension
- Coal/water slurries
 Multiphase system
 Wall depletion effect
 Complex non-Newtonian behaviour which vary
exquisitely with slight increase in concentration
3
25
(pipe)
x (neg. bulk
shear)
 Jastrzebski method
(bad correlation)
 Crawford method
(successful)
Chen et al.
(2010)
Suspension
- Coal/water slurries
Same as previous
4
25
(pipe)
-
-
Wang et al.
(2010)
Concentrated suspension
melts
- Spherical glass
particles/polymer
ethylenevinyl acetate
(EVA)
 Wall depletion effect
 Complex structure that leads to many complex
flow properties
4
0.5
-
4
10
(pipe)
-
Suspension
 Wall depletion effect
- Petroleum-coke/water
 Complex structure
slurries
- Petroleum-coke/sludge
slurries
*Successful TRM from the analysis of Martin and Wilson (2005)
Ma et al. (2012)
111

Introduced
modified
method
-
-

-

2.8
Viscous heating
It has been discussed on how the wall slip effects can contribute to
measurement errors. However, another issue other than wall slip effects
which can disrupt the measurement of fluid flow is the phenomenon of
viscous heating/dissipation. This phenomenon normally affects the flow of
highly viscous fluids. The portion of the power driving the flow which
overcomes frictional resistance is converted into thermal energy, i.e. the fluid
is heated up while being transported through a channel (Hardt and Schonfeld,
2007). This is typically an undesired phenomenon. The interaction between
viscous dissipation and fluid flow is of great importance in a variety of
applications that involves the flow of viscous fluids with temperature and
shear rate dependent properties (Pinarbasi and Imal, 2005). The highest shear
rates are obtained near the wall which results in increased fluid temperature,
thus possibly reduced fluid viscosity, in this region. The dissipation is zero at
the centreline and maximum at the wall.
This phenomenon can be demonstrated by performing a simple
experiment with a metal paper clip as described by Winter (1987): bend the
clip wide open and close it repeatedly until the clip breaks. Then, touch the
metal near the region of the break and feel the high temperature. The
mechanical energy for bending the metal has been converted into internal
energy. Viscous heating phenomenon can be observed in many applications
such as polymer processing flows where the injection moulding or extrusion
at high rates causes significant temperature rise. In aircraft engineering,
aerodynamic heating in the thin boundary layer around high speed aircraft
112
raises the temperature of the skin. In food processing, viscous heating has a
significant effect in the flow of ice cream (Elhweg et al., 2009).
Expenditure of energy is required in order to overcome resistance
forces in the deformation and flow of materials. This energy is dissipated, i.e.
during the flow it is converted into internal energy (heat) in the material. The
increase in internal energy expresses itself in a temperature rise (Winter,
1987). The total rate of work required to deliver a flow of fluid, ET is given
by the equation:
ET  Q( P1  P2 )
Eq. (2.53)
where Q is the total flow rate (m3s-1) and P1-P2 is the pressure difference.
In a horizontal pipe of uniform cross-section where there is no slip at
the boundary, the energy for the pipe flow is completely dissipated into the
fluid. However, when the fluid slips at the wall, one part of the energy for
flow through the pipe is dissipated at the slip surface and the remaining part
is dissipated in the volume of the deforming fluid (Winter, 1987). The rate of
work for slip along the wall, Es becomes
Es 
( P1  P2 )VslipD 2
4
Eq. (2.54)
where Vslip is the slip velocity near the wall (m s-1) and D is the pipe diameter
(m). The rate of work for deforming the fluid in the flow through the pipe
becomes
Ed  Q( P1  P2 )  Es
Eq. (2.55)
In order to determine the ratio of viscous heating to heat conduction,
Nahme number, Na (Macosko, 1994) is used to evaluate the viscous heating
effect in material flow:
113
Na 
R 2
4k
Eq. (2.56)
where β is the temperature sensitivity of viscosity (K-1) defined as
(1/μ)(∂μ/∂T) and k is the thermal conductivity (W m-1 K-1) of the fluid.
Expressing Na in terms of the apparent shear rate for the flow of power law
fluids (Elhweg et al., 2009) gives:
Mappn 1 R 2
Na 
4k
Eq. (2.57)
where M and n are the consistency and flow behaviour indices of the fluid,
respectively. The dimensionless number indicates how much temperature rise
will affect the viscosity. The higher the number, the more pronounced the
effect of viscous heating. The effect becomes more significant when Na ≥ 1
(Winter, 1987; Macosko, 1994). Therefore, the viscous heating might locally
influence the temperature even if the number is smaller than 1 and a safe
value for neglecting the effects of viscous heating seems to be when the
number is ≤ 0.1 (Winter, 1987). In nearly isothermal processes, the Nahme
number is a measure of how much viscous heating affects the temperature
dependent viscosity. Large values of Na indicate that isothermal conditions
cannot be maintained.
The average wall temperature increase for adiabatic pipe flow is
calculated as (Winter, 1987):
Tw 
P
c p
Eq. (2.58)
By incorporating Eq. (2.25) into Eq. (2.58) and solving for axial temperature
gradient in the slip layer near the wall, a balance of viscous dissipation with
mean fluid temperature increase yields a simple general equation:
114
2 w
dT

d x c p 
Eq. (2.59)
where T is the mean fluid temperature over the radial direction, x is the
distance in the axial direction along the pipe, cp is the specific heat capacity
and ρ is the density of the fluid. The derivation of the equation is detailed in
Appendix B.
Elhweg et al. (2009) reported a numerical simulation for temperature
dependent viscosity which illustrated that the velocity profile of ice cream
flow in pipes becomes increasingly steep near the wall due to the decrease in
viscosity associated with local heating in this region. This effect has also
been illustrated for other highly viscous fluids where the viscosity is sensitive
to temperature, such as magma flows (Costa and Macedonio, 2003). The net
effect of this could resemble a highly sheared slip layer, although no previous
study has clarified how this might be manifested or how it should be
analysed.
2.9
Conclusions
This chapter has summarised the theories and background associated
with this research study in depth. The definition, history and concept of
rheology were detailed at the beginning of the chapter to introduce to the
readers the basic of rheology. The topic expanded to the classification of
fluids into Newtonian and non-Newtonian behaviours. A section was
developed to explain the behaviour of non-Newtonian fluids and the
constitutive equations that have been used to describe the fluids such as
Cross, Sisko, power law, Herschel-Bulkley and Bingham models. More
115
important technical terms such as shear-thinning (pseudoplastic), shearthickening (dilatant), thixotropy, rheopexy, viscoelasticity and yield stress
were introduced. The explanation became more focused towards fluids which
exhibit shear-thinning behaviour. This study has investigated the rheological
behaviour of three different shear-thinning fluids which were ice cream,
citrus dietary fibre suspensions and magnesium silicate (talc powder) slurries.
Three sections were developed to explain the characteristics of the materials
in terms of their microstructure, production, usage and consumption.
The chapter
continued to
explain the flow of fluids
in
pipes/capillaries. In order to obtain the correct measurements of the
rheological behaviour, several corrections analyses need to be performed
such as correction for entrance effect (Bagley plots), shear rate
(Rabinowitsch method) and wall slip effects. The chapter became more
focused by describing wall slip effects phenomenon which occurs in the flow
of non-Newtonian fluids in pipes/capillaries. The factors and mechanisms
influencing wall slip were described in detail with the support of previous
literatures on that particular topic. The methods of correction for wall slip
were explained to describe various developed methods originated from the
classic Mooney analysis. Various modified-Mooney analyses were
performed by several researchers to analyse the slip effects when the Mooney
method failed to analyse their data. An interesting analysis method of
removing wall slip was introduced by incorporating mathematical Tikhonov
regularisation analysis and Mooney method to analyse the flow data. This
method was adopted in our study to characterise the fluid studied. A
summary of all capillary and pipe flow wall slip analyses on various systems
116
based on previous literatures was presented and correlation can be made
between the microstructure of the material used with the success/failure of
the Mooney method to analyse wall slip. Wall slip in the flow of most
multiphase fluids and highly entangled single phase material could not be
analysed using the classical method.
Apart from wall slip effects, another phenomenon which interrupts
the measurement of rheological behaviour of fluid is viscous heating. The
effect of viscous heating is more pronounced in the flow of highly viscous
fluids. A method to calculate the dissipated energy in the system was outlined
and a formula to obtain a dimensionless parameter called Nahme number
used in various studies to determine the ratio of viscous heating to heat
conduction was introduced. The topics discussed in this chapter provide a
background to this research study.
As stated previously in the Introduction chapter, the objective of the
research was to present a comprehensive study on wall slip analysis of ice
cream, CDF suspensions and magnesium silicate slurries. In order to perform
the study, a pipe rheometry rig was designed and built to enable the
rheological measurement. The data obtained from the measurement was used
to investigate the wall slip phenomenon in the pipe flow of the fluid tested.
Viscous heating analysis was also incorporated in the study of ice cream pipe
rheometry due to its very high viscosity and temperature sensitivity
properties. The rheological study of ice cream in situ on manufacturing line
by Martin et al. (2008) has quantified the effect of viscous heating which was
assumed to be contributed to the melting of the ice crystal phase. The effect
of apparent wall slip was also observed based on the measured pressure drop
117
dependency on pipe radius. A simulation analysis by Elhweg et al. (2009)
confirmed that viscous heating is significant in ice cream flow. However, up
to now, there are no other studies which further look into the relation
between the effect of viscous heating in pipe flow to the observed apparent
slip effect. The pipe rheometry rig built with integrated temperature and
pressure transducers would be able to bring this objective into reality and
increase the knowledge of ice cream rheology to the next level.
There are no studies have been reported on the pipe rheometry of
citrus fibre suspension. The mostly used suspension in the rheology literature
is wood pulp. For fluid food, most of the rheological studies were performed
using offline rheological measurement. In this research, the flow behaviour
and wall slip phenomenon of CDF suspensions of different concentrations
were investigated to gain more understanding on suspension flow. To the
author’s knowledge, there are no other studies in the literature which reported
on the flow behaviour of low concentration magnesium silicate (talc powder)
slurries. Hence, to further strengthen this research work, the flow behaviour
of magnesium silicate slurries was also studied using the pipe rheometry rig
developed and the wall slip phenomenon was investigated too.
The next three chapters embark on pipe rheometry studies of three
different shear-thinning fluids: Chapter 3 will describe the design and
development of pipe rheometry rig to study the flow; Chapter 4 will describe
the experimental and data analysis on ice cream; Chapter 5 will be on citrus
dietary fibre suspensions; and Chapter 6 will be on magnesium silicate (talc
powder) slurries.
118
3
DESIGN, BUILD AND COMMISSIONING OF PIPE
RHEOMETRY RIG
The initial plan was to design and build an integrated rig to measure
pressure drop, temperature and velocity profile during pipe flow of
multiphase fluids. The main focus of the design was on ice cream. However,
after asking for the opinions and advice from an expert on ultrasonic velocity
profiling system from Sweden, it was understood that the measurement of
velocity profile of ice cream flow requires a longer period of time for the
system to be established and mastered apart from the high cost needed in
setting it up. We decided to remove the velocity profiler system from the
design
and
continued
with
only pressure
drop
and
temperature
measurements.
In this chapter, the design, build and commissioning of a pipe
rheometry rig and its application are described. The chapter starts with the
introduction of the design work. This is followed by the selection of
equipment section describing the continuous scraped surface ice cream
freezer and piping system in detail. The instrumentation of the rig is
explained afterwards and a process and instrumentation diagram (P&ID) is
illustrated.
119
3.1
Introduction
One of the aims of this research project was to design and build a
controlled system to investigate wall slip and viscous dissipation phenomena
during flow in pipeline. This pipe rheometry was used to measure pressure
drop, temperature and volumetric flow rate. Several pressure and temperature
transducers were positioned at different points in the system to enable the
measurement process. These instruments together with an electronic mass
balance were connected to a PC data logger for monitoring and control
purposes. Pipes of different diameters were installed to enable the
quantification of different phenomena that occur during flow such as viscous
heating and wall slip. This system was built mainly to study the flow of
multiphase slurries that exist in various forms such as in food. The first food
material investigated in this system was ice cream. The system was also
utilised to investigate citrus dietary fibre suspensions and magnesium silicate
slurries. In order to enable the study of ice cream flow, the initial design step
was to plan a suitable equipment arrangement for ice cream flow.
3.2
Selection of equipment
In ice cream processing, the most important equipment is the scraped
surface heat exchanger (continuous industrial ice cream freezer) which
performs the core operation in ice cream processing – freezing. Its function is
to aerate, freeze and beat the mix to convert it into ice cream and generates
the ice crystals, air bubbles and matrix to obtain the desirable ice cream
structure. The description of the freezer chosen is described in upcoming
subsection.
120
Selection of the test pipe section is very crucial in this work and there
are a lot of important aspects to be considered. From the studies conducted
by Martin et al. (2008), they found out that viscous heating occurs during ice
cream flow in pipe especially at very low temperatures. Ice cream gets
warmer due to internal friction when flowing inside pipes. As the
temperature lowers, the viscosity increases and makes the ice cream more
resistant to flow. The friction is higher near the wall and causes the
increasing effect of viscous heating. The increased dissipation of heat warms
the ice cream even more. This enhances the wall slip effect. The design of the
pipe rheometry aimed to enable the observation of fluid flow inside pipes of
different diameters and to quantify the pressure and temperature gradients
along the length of pipes. The selection of size, length and type of pipes is
described in the following subsection.
3.2.1
Selection of piping system
It is important to have a range of pipe diameters as well as range of
flow rates to be studied. Various parameters from previous studies and the
parameters on the new design are listed in Table 3.1. The length listed in the
table are of three different types: 1) entry length to the measuring point
(Takeda, 1986); 2) tube length (Dogan et al., 2003; Dogan et al., 2005;
Martin et al., 2008; Elhweg et al., 2009); and 3) distance between two
pressure sensors (Ouriev and Windhab, 2003; Ouriev et al., 2003; Wiklund et
al., 2007; Birkhofer et al., 2008; Wiklund and Stading, 2008).
121
Table 3.1: Examples of pipe diameters and length used in previous studies
Author,
Year
Diameter
(Inch, m)
Length (m)
Flow
rate
(m3/s)
Takeda
(1986)
0.5”,
0.012 m
1.80
8.28 
10-5
Water
This study was done to
measure velocity profiles
using ultrasonic Doppler
shift detection device.
Dogan et
al. (2003)
2”,
0.0532 m
12.00
2.11 
10-4
Tomato
concentrates
This study developed an inline
rheometer
using
ultrasonics to measure
velocity
profiles
and
pressure drops to determine
shear rate and the stress
distributions in a pipe.
Ouriev et
al. (2003)
1¼”,
0.032 m
1.00
7.13 
10-5
Chocolate
suspension
This study developed an inline ultrasonic method for
investigation of the flow
behaviour of concentrated
suspensions.
Ouriev
and
Windhab
(2003)
0.9”,
0.023 m
1.27
5.32 
10-4
Highly
concentrated
shear
thinning and
shear
thickening
suspensions
This study investigated the
transient pressure driven
shear flow of highly
concentrated suspensions
using a novel Dopplerbased ultrasound velocity
profiler – pressure drop
(UVP-PD)
methodology
and compared with those in
steady flow.
Dogan et
al. (2005)
0.8”,
0.0204 m
0.85
9.35 
10-5
Acid-thinned
starch
This work was done to
study the application of an
in-line ultrasonic-based
rheological
characterization method for
measuring the rheological
properties of acid-thinned
and native cornstarch
suspensions and gels.
Wiklund
et al.
(2007)
1.4”,
0.0355 m
2.52
1.22 
10-3
Industrial
nonNewtonian
fluids
This work developed a
methodology for measuring
rheological flow properties
in-line, in real-time, based
on simultaneous
measurements of velocity
profiles using a UVP
technique with pressure
difference technology.
1.8”,
0.0455 m
122
Materials
Description
Wiklund
and
Stading
(2008)
0.89”,
0.0225 m
2.52
1.22 
10-3
Industrial
suspension
This study evaluated the
application
of
in-line
ultrasound Doppler-based
UVP-PD
rheometry
method for non-invasive,
real-time
rheological
characterization
of
complex
model
and
industrial suspensions.
1.4”,
0.0355 m
1.9”,
0.0485 m
Martin et
al. (2008)
0.2”,
0.0049 m
to
1.0”,
0.0254 m
1.5
1.6 
10-6
Ice cream
This study reported that
viscous heating effects,
were observed at relatively
low shear rates for the
commercial ice cream, but
not the model ice cream
foam. This was attributed
to the melting of the ice
crystal phase in the
commercial ice cream and
hence, absence from the
model ice cream foam.
Birkhofer
et al.
(2008)
0.6”,
0.016 m
1.0
8.04 
10-6
Cocoa butter
This study investigated the
dynamic response of the
cocoa
butter
shear
crystallization process to a
step
reduction
in
temperature of a two stages
shear
crystallizer
by
measuring the UVP-PD in
a pipe section. It was
observed that the cocoa
butter suspension is shear
thinning for which the
value of the power law
exponent decreased with
increase
in
the
concentration of cocoa
butter crystals.
Elhweg et
al. (2009)
0.4”,
0.0098 m
to
2.0”,
0.051 m
1.9
1.6 
10-6
Ice cream
They
confirmed
that
viscous heating has a
significant effect on the
observed
rheological
behaviour of ice cream.
New
Design
½” – ,
0.0127 m
¾” –
0.0191 m
1” ,
0.0254 m
1½” ,
0.0381 m
1.2
~5.6 
10-5
Wall slip and viscous
 Ice cream
 Citrus fibre heating effect are to be
suspension observed.
 Magnesium
silicate
slurry
123
A 3 m 1.5” (outer diameter) stainless steel flexible tube was attached
to the freezer outlet for ice cream to flow to the rheometry arrangement. It
was planned to locate three sets of pressure and temperature transducers at
different points in the test section to enable the measurement process. Four
interchangeable stainless steel pipes of different diameters were installed and
the studies were planned to be done in one diameter pipe at a time. The
surface roughness of the pipes was 0.13 μm (obtained from the supplier) and
the internal diameters of the pipes were 8.7 mm, 15.2 mm, 21.5 mm and 33.9
mm. Each pipe had a 0.5 m entry section for flow to develop followed by
transducers spaced at 0.10 m and 0.50 m apart followed by a further 2 m
length of pipe with an open exit. All pipework up to and including the test
section was lagged in 25 mm thick glass fibre insulation. The entrance length
required for the flow to be fully developed was determined by using the
equation developed by Poole and Ridley (2007):
XD
 [(0.24n 2  0.675n  1.03) 6  (0.0567 Re MR )1.6 ]1 / 1.6
D
Eq. (3.1)
where XD is the entrance length, D is the pipe diameter, n is the power law
index and ReMR is the Metzner-Reed Reynolds number (Metzner and Reed,
1955):
ReMR =
VB 2n D n
M
8(
n n
)
6n  2
Eq. (3.2)
where ρ is the bulk density, VB is the bulk velocity and M is the consistency
index. This equation is applicable for flow conditions that are truly laminar
which valid in the range 0.4<n<1.5 and 0<ReMR<1000.
There are two sections in the piping system. The first section is a
straight pipe completed with three different valves. Fig. 3.1 illustrates the
124
first section of the piping system. The product that comes out from the
freezer is connected to this section by a flexible hose. This section functions
to transfer the product from the freezer to the second section which is the test
section. A pressure relief valve, 3-way valve (plug cock valve) and a control
tap (butterfly valve) are positioned in the first section. Relief valve is used to
control the pressure in the system which can be built up by a process upset or
equipment failure. If the pressure is too high in the pipe, this valve will let the
ice cream to flow through its other outlet to relief the high pressure in the
pipe. Plug valve can be fully open or fully closed. It can be used to direct the
ice cream away from entering the test section and let the ice cream flows
through its other outlet into the side barrel. Butterfly valve is used for
regulating flow and allows for quick shut off in the case of process upset. As
a whole, the purpose of these valves is to control the flow of ice cream
entering the test section.
Fig. 3.1: The first section of the pipeline
125
The size of the pipe in this first section is 1½” and is about 1 meter in length.
This section is made from dairy pipe and is completed with five ring joint
type (RJT) union size 1½”, one tri-clamp ferrule size 1½” and one APV
ferrule size 1½”.
For the second section (test section), there are four sizes of pipes that
had been chosen which are ½” (0.0127 m), ¾” (0.0191 m) , 1” (0.0254 m)
and 1½” (0.0381 m) (Fig. 3.2). The pipes were supplied by Swagelok,
Manchester, United Kingdom. The length of each pipe is 1.2 meter. The pipe
diameters stated previously are the outer diameters of the pipes. However, for
measurement and calculation procedures, the internal diameter for each pipe
is used throughout the thesis. All the tubes have the same inlet size which is
1½” and comes with a ferrule to reinforce the pipe joint with the first
section’s pipe and to prevent splitting. A reducer is used for ½”, ¾” and 1”
pipes to join the larger 1½” inlet to the pipe. For each pipe, there are three
sets of ¼” NPT female boss (labelled A in Fig. 3.2) and ¼” x 0.035” tubes
(labelled B in Fig. 3.2) for the attachment of pressure and temperature
transducers respectively.
126
Fig. 3.2: The test section pipes
3.2.2
Selection of continuous industrial ice cream freezer
In this study, the aim was to prepare 200 litres (0.2 m3) batch of ice
cream premix for each experimental run. The production rates used in several
studies were in the range of 1.6 x 10-6 m3 s1 to 1.20 x 10-3 m3 s1. In this
study, the maximum flow rate was estimated to be around 200 L hr1 or
5.610-5 m3 s1. Hence, the flow rate is still in the same range as the flow
rates in other studies which author had been referring to. An industrial ice
cream freezer that has the production rate of 200 L hr1 is needed. Another
aspect that is important is the weight of the machine. It was required to
comply with the allowed single equipment item mass in the laboratory which
is 1000 kg for the ease of transportation of the machine using the elevator.
127
After doing some research on the available continuous ice cream
freezers in the market, it was decided to choose APV Soren CS200 (Soren
SRL, Corsico, Italy). The CS200 has the maximum capacity of 200 L hr1 at
100% overrun. The weight of this machine is 790 kg which is below the
maximum mass allowed in the laboratory. It can be used to produce ice
cream of different overruns and formulations.
It has a long barrel and special design cooling system. The dasher
used is the open type dasher which is important to help whip the mix and
incorporate air. It features rugged stainless steel scraper blades that kept the
ice scraped off the surface of the freezer. Every aspect of the freezer can be
controlled including overrun, mix inlet and outlet flow rate, temperature of
ice cream and dasher speed. Fig. 3.3 illustrates the CS200 continuous ice
cream freezer and Table 3.2 lists the standard features for this CS200
continuous ice cream freezer.
Fig. 3.4 shows the front and side views of the freezer and the
technical features are listed in Table 3.3.
128
Fig. 3.3: CS200 continuous ice cream freezer
Fig. 3.4: Front and side views of CS200 continuous ice cream freezer
129
Table 3.2: Standard features for continuous ice cream freezer (model CS200)
Features
Details
Frame
ANSI 304 stainless steel frame with sliding doors
Cylinder
Chrome-plated nickel
Dasher
 Type 30 (open)
 ANSI 316 stainless steel
Scraper blades
Stainless steel (ANSI 316)
Mix pump


Rotary type
Powered with variable speed motor drive
Self-contained
refrigeration unit


Compressor
Condenser with pressure controlled water
valve
Full flooded refrigeration system
Back pressure valve for evaporation pressure
control
Evaporator made of stainless steel



Product back pressure For manual cylinder pressure control
valve
Overrun control system  Manual
 Visual airflow indicator
Control
panel
Other
and
power Contains:
 Main switch
 Start/stop push button
 Motor starters for all motors
 Motor overload protections
 Load indicator for dasher motor with two
level freeze-up protection
 Emergency switch



Companion sanitary ferrules and clamps
Manual bypass pump for Cleaning-in-Place
(CIP)
Piston type compressor
130
Table 3.3: Technical features of CS200 continuous ice cream freezer
Features
Refrigerant
Dimensions (m)
A
B
C
D
E
F
Electrical power unit to be connected to central refrigeration plant (kW)
Electrical power unit with self-contained refrigeration plant (kW)
Cooling water (l/h)
Net weight unit to be connected to central refrigeration plant (kg)
Net weight unit with self-contained refrigeration plant (kg)
Volume (m3)
3.3
Details
R404
A/NH3
1.32
0.75
1.87
0.76
0.31
0.23
6.2
11.7
3,300
680
790
4.7
Instrumentation of the rig
Three sets of pressure and temperature transducers were attached to
the pipe in the test section. These instruments were linked up with a PC data
logger used for monitoring, control and data collection purposes. The
following subsections describe the temperature and pressure transducers
integrated into the system.
3.3.1
Temperature transducer
In the study on ice cream rheology by Martin et al. (2008), for the
pipe length of 1.5 m, the wall temperature gradient dTw/dx was equal to
0.61C m1. Hence, it was assumed that in this study, the wall temperature
gradient will be in the same range due to the similar range of flow rates and
pipe sizes chosen. It was important to select an accurate transducer model for
temperature measurement. The temperature transducers chosen to be used in
the system were PT-100 sensor (Pico Technology, Cambridgeshire, United
Kingdom). They have an accuracy of ±0.03 oC and can read as low as -50oC
131
and a probe diameter that could be positioned flush with the internal pipe
surface. The specifications are as listed in Table 3.4.
Table 3.4: Specifications of PT-100 sensor
Features
Accuracy
Cable length
Operating temperature range
Probe diameter
Probe length
Probe material
3.3.2
Details
±0.03oC at 0oC
1 meter
-50→250oC
4mm
150mm
Stainless steel
Pressure transducer
In the previous study by Martin et al. (2008), the pressure in the
pipeline was 8 bar. In this study pressure transducers which can read up to 16
bar were selected to enable study of flow of different materials in 0–16 bar
range apart from ice cream. Pressure transducers available in the market
come in two designs – 1) the probe is in contact with the fluid (more
accurate) and 2) the probe is not in contact with the fluid (less accurate).
Design no. 2 (Fig. 3.5) where the probe is not in contact with the fluid
was chosen due to the cost factor. For this type of transducer, fluid (ice
cream) may get into the small channel between the probe and liquid surface.
Air might be trapped between the probe and fluid. The air will take the
pressure from the fluid and hence, the transducer will measure the pressure of
air formed instead of the fluid directly.
132
Fig. 3.5: Pressure transducer design no.2
There is a possibility of using more accurate transducers in the future.
The pressure transducer chosen were Swagelok® S model transducers
(Swagelok, Manchester, United Kingdom) which are able to withstand
pressure up to 16 bar. They have an accuracy of 0.5% which was estimated to
be sufficient for this project. The specifications are listed in Table 3.5.
Table 3.5: Specifications of S model transducer
Features
Accuracy
Details
0.5% limit point calibration
Operating pressure range 0 – 16 bar
Output signal (2 wire)
3.4
4 – 20 mA
Other equipment
Other equipment that were important in this system were a Silverson
Verson in-line processor (Silverson L5M-A, Chesham, UK) as a
homogenizer to break the particles to submicron diameters; a 200 L wheeled
feed barrel complete with an agitator (Model: NS-1, Lightning Mixers Ltd.,
133
Ponyton, England) to keep stirring the premix to be fed into the freezer; a 210
L product drum (Model: N0210, Dormex Containers Ltd., Cheshire,
England) to collect the ice cream after being transported in the pipeline; and
flexible tubes (DairyBits, Leicestershire, England) to transfer premix from
the feed barrel to the freezer, from the freezer to the pipeline and from the
pipeline to the product drum.
3.5
Experimental arrangement of pipe rheometry
An experimental arrangement for pilot scale production and
rheometry of ice cream was constructed as illustrated in Fig. 3.6 while the
associated test section is illustrated in Fig. 3.7. This consisted of a feed barrel
supplying pre-mix to the continuous scraped surface heat exchanger with a
maximum capacity of 200 L hr-1. Fig. 3.8 shows the detail of the pipe
rheometer test section for CDF suspensions and magnesium silicate slurries.
3.6
Installation of the rig
The ice cream rig was installed in the Morton Laboratory pilot plant,
The University of Manchester. The pipe work was installed on the support
clamps and the pipes were connected using tri-clamp sanitary fittings. The
freezer was connected to compressed air line and also to a cooling water
system at junction immediately above freezer.
Due to the closure of the Morton Laboratory in early 2012, the pipe
rheometry rig was moved to the pilot plant in the newly built James
Chadwick Building i.e. the new Chemical Engineering building. Further
134
experimental work, i.e. the pipe rheometry of CDF suspensions and
magnesium silicate slurries, was conducted in this new facility.
Fig. 3.6: Schematic of ice cream production rig
Pressure transducers
Insulation layer
0.5 m
0.1 m
Flow out
Flow in
Pipe
Temperature transducers
Fig. 3.7: Detail of pipe rheometer test section for ice cream
135
Fig. 3.8: Detail of pipe rheometer test section for CDF suspensions and
magnesium silicate dispersions
3.7
Commissioning of the rig
Some preliminary works were conducted in the commissioning
process of the rig. As mentioned before, the rig was initially designed and
built to study ice cream flow. Hence, all the preliminary work done were
related to ice cream. Initially, several formulations were tested to produce a
model ice cream with non-milk formulation that has similar properties with
the real ice cream when deformed and is able to last longer in comparison to
real ice cream which can perish easily due to the inclusion of dairy material.
However, the exclusion of milk from the formulation made the structure
harder and more sorbet-like. Therefore, we decided to include milk powder in
the final formulation to get a product with good structure and resemble real
ice cream. The formulation used to produce ice cream will be shown in
Chapter 4.
Several tests were conducted using the pipe rheometry rig to collect
some raw data to check on the reliability of the system. Ice cream was
pumped to the pipe system and measured the pressures and temperatures at
136
three different points over the 0.6 m test section. Table 3.6 shows the
examples of pressure vs. length, pressure vs. time and temperature vs. time
data for selected flow rates for the largest and smallest pipe diameters.
Pressure at the wall decreased linearly over the length of pipe. These trends
are consistent with invariant flow conditions over the test length i.e. steady
state condition. Pressure and temperature were also constant with time. The
data were slightly unstable at the beginning of the flow and started to be
constant after a moment. Similar trends were also observed for the flow in
15.2 and 21.5 mm pipe diameters.
In order to verify the temperature data obtained, we swapped the
positions of the temperature transducers along the pipe length and monitored
for any change in measured temperature. The data are shown in Fig. 3.9 for
trial using pipe with 33.9 mm diameter. The measured temperature gradients
were found to be independent of transducer position. Therefore, the
temperature transducers with the accuracy of ±0.03oC used were reliable to
measure the temperature near the wall. In the following chapters, all the
works presented were conducted in triplicate in order to show a high degree
of repeatability.
Entrance effect correction was performed to determine the entrance
pressure loss that occurs when ice cream flows from the freezer to the test
pipe. Fig. 3.10 shows the Bagley plots constructed at different apparent shear
rates for 33.9 mm pipe diameter. The straight lines fitted intercept the y-axis
at approximately zero values and the slopes for each line were approximately
equal to 2τw. Hence, the pressure loss is assumed to be negligible. The same
trends were obtained using pipes of 21.5, 15.2 and 8.7 mm internal diameters.
137
Table 3.6: Sample raw data: pressure vs. length; pressure vs. time and temperature vs. time for ice cream flow in 33.9 and 8.7 mm pipe
diameters
Pressure vs. length
Pressure vs. time
Wall temperature vs. time
0.5
Pressure (bar)
Pressure (bar)
0.6
0.4
0.3
R² = 0.9999
0.2
0.1
0
0
0.5
Length (m)
1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
L=0 m
L=0.1 m
L=0.6 m
0
500
Time (s)
Wall temperature (oC)
Pipe size: 33.9 mm
1000
-4.93
-4.94 0
-4.95
-4.96
-4.97
-4.98
-4.99
-5.00
500
1000
L=0 m
L=0.1 m
L=0.6 m
Time (s)
Pipe size: 8.7 mm
2
1.5
R² = 0.9998
1
0.5
0
0
0.5
Length (m)
1
L=0.1 m
2
L=0.6 m
1.5
1
0.5
0
0
200
400
Time (s)
138
600
Wall temperature (oC)
L=0 m
2.5
Pressure (bar)
Pressure (bar)
2.5
-4.8
-4.82 0
-4.84
-4.86
-4.88
-4.9
-4.92
-4.94
-4.96
500
1000
L=0 m
L=0.1 m
L=0.6 m
Time (s)
Arrangement 1
Arrangement 2
-4.92
-4.94 0
500
1000
-4.95
L=0 m
-4.96
L=0.1 m
-4.97
L=0.6 m
-4.98
-4.99
-5.00
Wall temperature (oC)
Wall temperature (oC)
-4.93
-4.93 0
1000
-4.94
-4.95
L=0 m
-4.96
L=0.1 m
-4.97
L=0.6 m
-4.98
-4.99
-5.00
Time (s)
500
Time (s)
Fig. 3.9: Results from two different arrangements of temperature transducers
obtained by swapping the positions of the transducers on 33.9 mm pipe
diameter. The measured temperatures were independent of transducer
positioning
0.500
0.450
Pressure drop (bar)
0.400
0.350
0.300
Apparent shear rate
(s-1)
0.250
0.200
17
0.150
21
0.100
28
0.050
34
0.000
0
10
20
30
40
L/R
Fig. 3.10: Bagley plots constructed to determine the entrance pressure loss
for 33.9 mm pipe diameter. The fitted lines intercept the y-axis at
approximately zero values, hence the loss is assumed to be negligible.
To track the changes of ice cream condition near the wall, the
temperature transducers were installed in the opposite direction from the
pressure transducers. This enables us to monitor changes in both pressure and
139
temperature at a particular point accurately. They are very useful and reliable
to help tracking the changes in ice cream structure when flowing in pipe.
Pressure reading is important in determining whether there is a change in the
structure of ice cream near the wall. It is related to the wall shear stress and
wall slip velocity in the pipe. The instability/change in the pressure reading
over time indicates the inconsistency of wall slip with time. Temperature
sensors are really important in this study as an indicator of the
microstructural changes near the pipe wall. Increase in temperature indicates
the melting of ice crystals and the highly concentrated matrix phase is locally
diluted with water and lowers its viscosity. This contributes to the
enhancement of wall slip phenomenon. Knowing that ice cream also has
thixotropic behaviour (time-dependent non-Newtonian fluid), a new batch of
ice cream mix is required for each experimental run in order to get reliable
and repeatable set of data. The accomplishment of this experimental
procedure helps in characterising the wall slippage in complex nonNewtonian fluids such as ice cream.
3.8
Conclusions
This chapter presented the design, build and commissioning of the
pipe rheometry rig. The pipe rheometry rig was successfully built and
instrumented with pressure and temperature transducers to enable the data to
be recorded in real time. Several work were conducted during the
commissioning process such as developing the model ice cream formulation
and preliminary test run using the pipe rheometry rig. Some examples of raw
data were presented to demonstrate the reliability of the rig to perform flow
140
measurement. The upcoming chapters shall discuss the use of this pipe
rheometry rig throughout the study for ice cream, CDF suspensions and
magnesium silicate slurries flow measurements.
141
4
WALL SLIP AND VISCOUS DISSIPATION IN ICE
CREAM PIPE RHEOMETRY
The first multiphase fluid examined in this research work was ice
cream. In this chapter, wall slip and viscous dissipation of ice cream were
studied using the pipe rheometry rig previously described. The chapter starts
with the introduction of the experimental work. Then it is followed by ice
cream production, measurement of temperature near the wall, pressure drop
and flow rate. This chapter ends with the results of the experimental test and
also the conclusions of the work.
4.1
Introduction
The application of rheology is important in characterising the
behaviour of complex soft solids like ice cream. During flow, the material
microstructure, and therefore quality, will change. Processing within the
freezer is designed to achieve the desired structure, but flow along pipes can
be detrimental (Eisner et al., 2005). This chapter presents new results and
analysis to characterise the effect of pipeline flow on ice cream with
particular emphasis on wall slip and viscous dissipation i.e. the two important
phenomena that need careful attention in ice cream pipe rheometry.
142
A recent attempt to measure the steady-shear rheology of a
commercial ice cream in situ on manufacturing lines, and the steady-shear
rheology of ice-free model ice cream foam using a laboratory Multi-Pass
Rheometer (MPR) was done by K. N. Odic from Unilever R&D (United
Kingdom), and the results were analysed by Martin et al. (2008). This
analysis has quantified for the first time the significant effects of viscous
heating in ice cream flow along industrial process pipes. This was attributed
to the melting of the ice crystal phase in the commercial ice cream and
suggested that further investigation of this would require consideration of the
ratio of viscous dissipation against heat transfer by conduction. They also
observed that the measured pressure drop dependency on pipe radius was
consistent with the effect of apparent wall slip. Pinarbasi and Imal (2005)
concluded similar findings that viscous heating effects should not be
neglected in the analysis of the flow of polymeric solutions and melts,
although this is at shear rates several orders of magnitude greater than for ice
cream.
As mentioned previously, a simulation analysis conducted by Elhweg
et al. (2009) confirmed that viscous dissipation has a significant effect on the
observed rheological behaviour of ice cream. The effect of both viscous
dissipation and wall slip mechanisms is to lower the apparent viscosity of the
material. Elhweg et al. (2009) also observed that the velocity profile of ice
cream flow in pipes becomes increasingly steep at the wall due to the
decrease in viscosity associated with local heating in this region. The effect
on wall temperature is noticeable under certain conditions and indicates that
143
viscous dissipation should be considered in the design of piping runs
(Elhweg et al., 2009).
Therefore, viscous dissipation and wall slip are indeed two significant
features in ice cream flow due to high viscosity of ice cream, the temperature
sensitivity of the matrix and the microstructure sensitivity to local
temperature (via the ice fraction). To the authors’ knowledge, no studies on
wall slip and viscous dissipation in ice cream pipe rheometry have been
reported. This chapter presents a comprehensive study on wall slip and
viscous dissipation in ice cream pipe rheometry and provides new results and
analysis to interpret both phenomena.
4.2
Materials and Methods – Ice cream production
An 8% fat ice cream premix was prepared with formulation given in
Table 4.1 for use in all experiments. Batches of 200 L were mixed by an axial
flow impeller and then passed through a Silverson Verson in-line processor
(Silverson L5M-A, Chesham, UK) at 10230 rpm to break the oil droplets up
to sub 20 μm diameters. The ice cream mix was pumped to a continuous
scraped surface heat exchanger (APV Soren CS200, Soren SRL, Corciso,
Italy). This was operated to produce ice cream at 5.0oC and with an overrun
of 90% (i.e. a gas volume fraction of 0.47).
The formulation used was similar to that reported by Martin et al.
(2008) and so the continuous liquid matrix volume fraction was
approximately 0.42 and ice crystal fraction was approximately 0.11. The
measured density of the ice cream was 523 kg m3. The specific heat capacity
144
of the ice cream at 5.0C was estimated to be 17600 J kg1 K1 by
differentiating the enthalpy-temperature data reported by Cogne et al. (2003).
Table 4.1: Ice cream formulation
Ingredients
Mass (%)
Water
64.3
Oils/Fats (Vegetable oil)
8.0
Protein (skimmed milk powder)
11.5
Sugar (sucrose)
16.0
Stabilizer (Xanthan gum)
0.1
Emulsifier (Polysorbate)
0.1
The pipe rheometry system was pre-cooled by flowing ice cream
through it until temperature measurements were stable as monitored by the
two temperature transducers located at a distance of 0.6 m. A jig was used to
align the surface of the temperature transducer probes to the pipe wall
surface. Mass flow rate was measured continuously by logging the weight of
the discharge over time with an electronic balance connected to the PC. One
batch of premix was prepared each day and a series of experiments
conducted at four different flow rates through one pipe section. A pipe
section of different diameter would be used the next day, all experiments
were performed in triplicate and data was averaged before subsequent
analysis. The entrance length required for the flow to be fully developed was
estimated to be 0.03 m using the equation developed by Poole and Ridley
(2007) as described in Chapter 3, thus entry effects were assumed to be
negligible.
Pressure drop and flow rate data were generated and transformed into
wall shear stress, τw and apparent shear rate,  app using
w 
PR
2L
145
Eq. (4.1)
app 
4Q
R 3
Eq. (4.2)
where ΔP is the pressure drop along the pipe, R is the internal pipe radius, L
is the length of the stainless steel pipe and Q is the volumetric flow rate
(obtained by dividing mass flow rate by ice cream density).
Fat droplet size distributions were measured by laser diffraction
method using Malvern Mastersizer (Malvern Instruments, Worcestershire,
United Kingdom). The measurements were carried out at room temperature.
Both samples of mix and ice creams were dissolved in deionized water,
stirred at a rate of 1785 L min1 for 2 min to degas and break loose
aggregates before the measurement process.
4.3
Results and Discussion
This section will describe all the results obtained from the
experimental work done on ice cream using the pipe rheometry system
developed.
4.3.1
Temperature gradient of ice cream at the wall
The three temperature readings along the test section lengths showed
a small but significant and repeatable temperature gradient, which appeared
linear over the three points. The calculated temperature gradients are
presented in Fig. 4.1. It is apparent that the temperature gradient increased
with apparent shear rate for each pipe and that the gradient increased as the
pipe radius decreased. Over all of the data from all of the pipes, there appears
to be an approximately proportional relationship between the temperature
gradient and the apparent shear rate.
146
Fig. 4.1: Measured temperature gradient of ice cream at the wall against
apparent shear rate for different pipe diameters
The experiments were conducted in triplicate and showed a high degree of
reproducibility. The standard deviation of calculated wall shear stresses and
temperature gradients were less than 0.2% of the mean value. For clarity only
the mean of the triplicate values is presented here.
There are two heat sources in the experiment (since both kinetic and
potential energy are constant): viscous dissipation from within the flow and
heat transfer from the atmosphere through the insulation and pipe into the ice
cream. The effects of these on the average radial ice cream temperature were
detailed in Eq. (2.59). There will be a radial temperature distribution in both
cases, but each involves most heating occurring in the near wall region and
thus trends from these equations are indicative of the ice cream wall
temperature gradients. Increasing pipe radius tends to reduce the temperature
gradient in both dissipation and conduction cases and this trend are evident in
Fig. 4.1. Increasing apparent shear rate results in increasing wall shear stress.
147
Thus viscous dissipation tends to result in increasing wall temperature
gradient with apparent shear rate, and this trend was observed in the data.
However, wall temperature gradient due to heat transfer tends to decrease
with apparent shear rate due to the increase in mass flow. Heat conduction
would give rise to more heating and higher temperature gradients at slower
flow rates (Martin et al., 2008). Therefore, on balance the results indicate that
viscous dissipation is both a significant and the dominant heat source in the
experiments. These results are used in section 4.3.3 to equate the measured
wall temperature increase with apparent wall slip.
This was also borne out by calculated estimates of the rate of heat
transfer from the air into the pipe. The dominant resistances to heat transfer
in this case are convection between air and the outer surface of the insulation,
and radial conduction through the insulation. Rate of heat transfer, H (W)
was calculated from:
T  Ticecream
H  air
WCV  WCD
Eq. (4.3)
where
WCV 
1
u  2RT L
Eq. (4.4)
and
WCD 
ln( RT / R)
2Lkinsulation
Eq. (4.5)
where Tair and Ticecream are the air and ice cream temperatures, respectively. R
is the inside pipe radius while RT is the total of the inside pipe radius and
insulation thickness. WCV is the resistance to heat transfer by convection while
WCD is the resistance to heat transfer by conduction. The pipe wall thickness
is ignored in this case. L is the length of the test section, u is the convective
heat transfer coefficient to the outer surface of the insulation and k is the
148
thermal conductivity of the insulation layer i.e. 0.034 W m-1 K-1. Convective
heat transfer coefficient was obtained by using the following equation,
u
Nu  k air
2 RT
Eq. (4.6)
Nu is the Nusselt number which can be calculated using this equation,
Nu = 0.53 Pr Gr0.25
Eq. (4.7)
where Pr and Gr are the Prandtl and Grashof numbers, respectively. They
can be obtained via the following formulas,
Pr =
Gr 
c p ,air  air
Eq. (4.8)
k air
 air g air 2 (2 RT ) 3 (Tair  Tsurface)
 air 2
Eq. (4.9)
where cp,air, μair, kair and ρair are the specific heat capacity, viscosity, thermal
conductivity and density of air, respectively. βair is the volumetric thermal
expansion coefficient which equals to 1/Tair. Rate of heat transfer per unit
length (W m-1) can be calculated by dividing H with L.
The ratio of energy dissipation to heat conduction increases with
increase in volumetric flow rate. Increasing flow rate results in increasing
wall shear stress and consequently increasing viscous dissipation. Total
energy dissipation, ET (W) can be obtained as follows:
ET   wVbulk 2RL
Eq. (4.10)
where Vbulk is the velocity of the bulk ice cream. Total energy dissipation per
unit length (W m-1) can be calculated by dividing EdTotal with L. Fig. 4.2
illustrates the ratio of energy dissipation to heat conduction per unit length
against apparent shear rate.
149
Fig. 4.2: Ratio of viscous dissipation to calculated heat transfer per unit
length against apparent shear rate for different pipe diameters
It is apparent that as the flow rate increases, the heating effect caused
by viscous dissipation increases in all pipes. Radial heat conduction gives
rise to more heating at slower flow rates especially in the smallest pipe while
viscous dissipation gives rise to more energy dissipation at faster flow rates.
To reduce the heat transfer effect into the pipe, more insulation layer should
be added in the future. However, it is clear that viscous dissipation is the
dominant heat source in the experiments.
The amount of dissipated energy in the slip region is the combination
of the effect between pressure differences, slip velocity and pipe size while
the amount of energy dissipated in the bulk (fluid) is a function of pressure
difference and volumetric flow rate. As stated previously, the increases in
bulk flow rate causes the pressure drop, wall shear stress and viscous
dissipation to increase. At constant pipe flow rate, the pressure difference
150
will increase with the decrease in pipe diameter. The comparison was done
when ice cream flowed at nearly similar flow rate inside two different pipes
i.e. D = 33.9 mm and D = 21.5 mm as shown in Table 4.2.
Table 4.2: Comparison of the energy dissipated in different pipe sizes at
similar ice cream flow rate
33.9
21.5
Pipe size (mm)
1
0.03
0.03
Flow rate (kg s )
1
17.04
63.04
Apparent shear rate (s )
4
ΔP (Pa)
3.8410 8.19104
2.08
3.87
Energy dissipated in slip region (J s1)
1
0.05
0.10
dT/dx (K m )
1.31
2.57
Nahme number, Na
The dissipated energy and the consequent temperature gradient at the
wall was higher in 21.5 mm pipe compared to 33.9 mm pipe. The ratio of
viscous heating to heat conduction was determined using Nahme number, Na
as described in section 2.8 (Chapter 2). Eq. (2.59) was used to calculate Na in
terms of apparent shear rate for the flow of power law fluids. The value of
the temperature sensitivity to viscosity, β for ice cream is 0.7 K-1 (Elhweg et
al., 2009). Na was found to be higher in 21.5 mm pipe. The values of Na
number larger than 1 indicates that viscous heating is significant and
dominant in this flow. This number shows how much temperature rise will
affect the viscosity. The higher the number, the more pronounced the effect
of viscous heating. Therefore, it can be concluded that dissipation of energy
is dependent on the pipe size at similar ice cream flow rate in this case.
4.3.2
Flow curves and wall slip analysis
The experimental work generated flow data set and flow curves for
8.7, 15.2, 21.5 and 33.9 mm pipe diameters which are presented in Fig. 4.3.
These appear to approximately obey power law behaviour. The fitted curves
151
are shown in the figure. The standard deviation of calculated wall shear
stresses and apparent shear rates were very small. For clarity only the mean
of the triplicate values is presented here. A strong pipe radius dependence is
evident, such that at constant wall shear stress the apparent shear rate
increases in smaller radius pipes. Such an effect is frequently indicative of
apparent wall slip effects in suspension flows as represented by Eq. (2.44).
The respective consistency and flow behaviour indices for the ice
cream flow are shown in Table 4.3. At the largest pipe diameter, ice cream
displays a strong shear thinning behaviour indicated by the low n value.
However, as the pipe diameter decreases, n increases which indicate that the
ice cream became less shear thinning.
Fig. 4.3: Wall shear stress against apparent shear rate data for ice cream flow
in four different pipes at -5oC. Fitted curves and the respective flow indices
are shown in the figure.
152
Table 4.3: Consistency index, M and flow behaviour index, n for ice cream
flowing in different pipes
Pipe size (mm) Consistency index, M Flow behaviour index, n
33.9
21.5
15.2
8.7
257
105
65
26
0.26
0.46
0.55
0.69
The Tikhonov regularisation (TR) method analysis was conducted on
the data. An in-house program, written in Mathematica, was used to
implement the method, detailed further by Martin and Wilson (2005). Fig.
4.4 shows the fitted flow curves for the four pipe diameters. The fitted true
flow curve represents the no-slip flow of the ice cream in all the pipes. The
deviation between the true flow curve and the measured flow curve towards
the x-axis is an indication of the amount of wall slip. Interpolations between
data points were made for values of constant wall shear stress and used to
create the Mooney plots shown in Fig. 4.5. These data are reasonably linear:
the gradients of the fits shown give the respective slip velocities and, if the
fits were extrapolated, the ordinate axis intercepts indicate that bulk shear
also occurs.
153
Fig. 4.4: Wall shear stress against apparent shear rate for different pipe
diameters
Fig. 4.5: Mooney plot of apparent shear rate against 4/R at values of constant
wall shear stress
The Mooney method was also implemented using Tikhonov
regularisation with both NJ and NK = 101. A range of regularisation
parameters were tested to find a suitable balance between goodness of fit and
curve
smoothness.
An
advantage
154
of
the
Tikhonov
regularisation
implementation over the graphical Mooney plot is that it yields a true flow
curve for the fluid without requiring any assumed form for the constitutive
equation.
Fig. 4.6 shows the true flow curves calculated with three contrasting
values of regularisation parameter. Fig. 4.7 shows the corresponding
calculated wall slip characterisation, along with the points found from the
gradients of linear fits in Fig. 4.5. Both the flow and slip characterisations
show erratic and unrealistic behaviour for λ = 0.1 and an overly smoothed
characterisation for λ = 1000. The results for the intermediate value of λ = 10
retain a degree of non-linearity in the rate dependence whilst also
representing a physically plausible characterisation – consequently the results
calculated with λ = 10 were used for the rest of this study. The use of a
regularisation parameter allows errors in the data to be appropriately
accommodated.
The ice cream characterisation shows a bulk fluid which has a yield
stress τy = 545 Pa and appears to obey a Bingham plastic model where shear
stress varies linearly with shear rate. A linear fit to the curve yields the flow
model:
  545  4.40
Eq. (4.11)
The wall slip characterisation shows the slip velocity increasing with wall
shear stress, although this does not appear to be proportional and non-linear.
155
Fig. 4.6: Tikhonov regularisation fitted shear stress against shear rate for
different regularisation parameters
Fig. 4.7: Tikhonov regularisation fitted slip velocity against shear rate for
different regularisation parameters and comparative points from Mooney
plots
The overall fit of this ice cream characterisation is presented along
with the measured flow data in Fig. 4.4, where a reasonably good overall fit
is evident. It is evident that the fitted true flow curve displays some shear
156
thinning behaviour. However, a shear thickening effect is evident as the
apparent slip becomes more significant at smaller diameters and this
corresponds to the fitted shape of the slip velocity in Fig. 4.7. Similar shear
thickening effect was also observed with λ = 1000, while λ = 0.1 generated
curves with unrealistic behaviour as stated before.
Fig. 4.8 shows the calculated fraction of flow due to wall slip against
wall shear stress. It is apparent that all of the flows in the different pipe
diameters experienced complete wall slip at the lowest flow rates studied.
Wall slip was the dominant contributor to flow over the whole data set, but as
the wall shear stress increased up to a third of the flow, the flow is shown to
have been contributed by bulk shear in the pipe. As indicated by Eq. (2.44),
wall slip contributes a greater proportion of flow at small pipe radii.
Fig. 4.8: Tikhonov regularisation fitted fraction of flow due to slip against
wall shear stress for different pipe diameters
157
4.3.3
Energy balances of viscous dissipation
The results presented in sections 4.3.1 and 4.3.2 are indicative of two
well-known effects: viscous dissipation in highly viscous, temperature
sensitive fluids, and apparent wall slip in multiphase suspensions. This
section presents an energy balance analysis to clarify the nature of the
apparent slip indicated by the analysis and relate this to the observed viscous
dissipation effects.
Section 2.6.3.1 introduced Eq. (2.45) which approximates the shear
within a thin slip layer where curvature can be neglected and the slip layer
fluid is Newtonian. Mooney’s method relies on the ratio δ/μslip being constant
for there to be linear trends on the Mooney plot. One of the clearest reported
instances of this was reported by Kalyon (2005) which found that the
apparent slip layer consisted purely of the liquid binder, and thus had
constant viscosity, and the slip layer thickness was constant for a given
suspension. Further, he reported that the slip layer thickness was correlated
with the particle size, solids volume fraction and maximum solids packing
fraction – and was thus independent of shear rate and channel geometry. The
slip layer thickness in such a scenario is typically of the same order or
magnitude as the solid particle diameter.
Martin et al. (2008) detailed experiments where the rheology of an
ice-free ice cream matrix was measured at 5C by both a parallel plate
rheometer and a Multi Pass Rheometer. The formulation was similar to that
used in this study and thus a similar matrix rheology would be expected. The
Carreau model represented the data well and was fitted for the matrix
viscosity, μmatrix, as:
158

 matrix  1.18  395 1  690 2

0.299
Eq. (4.12)
An iterative method was used to solve for the slip layer thickness
based on Eq. (2.41) for the slip layer viscosity and the calculated slip
velocities. The resultant slip layer thicknesses ranged from 0.208 mm to
0.239 mm and the slip layer viscosities ranged from 1.36 Pa s to 1.40 Pa s.
Ice crystals are typically less than 50 μm in diameter (Martin et al., 2008).
Therefore, slip layers of this thickness would be several factors thicker than
the particle size which is not consistent with Kalyon’s (2005) model of the
slip layer.
An energy balance was conducted to predict the axial temperature
gradient in the slip layer which yielded (refer the derivation in Appendix B):
d Tw
2 w

dx
cp 
Eq. (4.13)
This analysis, combined with thicknesses calculated assuming a matrix slip
layer, predicts an axial temperature gradient of ice cream at the wall ranging
from 0.557 K m1 to 0.752 K m1. These predictions are typically a factor of
20 greater than the measured values and so this does not appear to be a
reasonable representation of the apparent slip in the ice cream flow.
An alternative approach was adopted to account for the measured
axial temperature gradient. An energy balance based on Eq. (4.13) was used
to calculate the slip layer thickness required to match the measured axial
temperature gradient at the wall. The corresponding slip layer viscosity was
then calculated using Eq. (2.41). Fig. 4.9 shows the calculated slip layer
thicknesses. The thickness is generally constant over the range of wall shear
stresses, but a significant pipe radius effect is apparent. The slip layer
159
thicknesses are significant ranging from 10% to 17% of the pipe radius. Slip
layer thickness decreases as pipe size decreases. The analysis assumes that
curvature can be neglected in the slip layer, so there are some errors
introduced. For the sake of simplicity and clarity of presentation, with the
aim of elucidating the measured results, this simplifying assumption is kept.
Fig. 4.9: Slip layer thickness calculated from measured temperature gradients
at the wall against wall shear stress
Fig. 4.10 compares the apparent viscosity calculated for the bulk ice
cream, calculated for the wall slip layer, and previously measured by Martin
et al. (2008) for the matrix with no ice. The calculated viscosity for the wall
slip layer appears to follow on closely from that calculated for the bulk
viscosity, and is considerably greater than the matrix viscosity.
It appears to be coincidence that the ratio δ/μslip was relatively
constant in these experiments, despite changes in both terms. This allowed
approximate interpretation of the results using the Mooney method. Despite
the apparent success of the analysis, the picture that emerges from the
160
analysis of the temperature measurements is significantly different. This
leads to important differences in the interpretation and characterisation of the
ice cream flow. The flow does not exhibit apparent slip through the
occurrence of a thin slip layer at the wall. Instead, apparent slip is due to a
much thicker layer of ice cream near the wall which shears disproportionately
compared to the rest of the ice cream. This picture is consistent with the
runaway viscous dissipation model, where heating reduces viscosity and thus
perpetuates further shear and so on.
Fig. 4.10: Apparent viscosity against shear rate for Tikhonov regularisation
fitted bulk flow of ice cream, ice cream wall slip region and matrix with no
ice previously measured by Martin et al. (2008)
4.3.4
Fat droplet size distribution
Particle size is important as it influences many properties of
particulate materials and is a valuable aspect of product quality. The shear
forces during freezing process could cause both coalescence and breakup of
the droplets in the mix, however it is normally reported that coalescence is
the dominant process. The degree of fat coalescence can be estimated by the
161
volume fraction of the primary particles (Goff and Spagnuolo, 2001). The
comparison of the fat droplet size distributions between ice cream mix and
ice cream samples flowing out from different pipes is shown in Fig. 4.11.
Homogenized ice cream mix has very small particle size which can be seen
in the distribution plot. There was a significant difference between the fat
globules distributions in the ice cream mix and the ice cream samples
withdrawn from different pipes. Particle size distributions become more
bimodal in ice cream because the applied shear during the freezing process
causes fat globules in ice cream to undergo partial coalescence or fat
destabilization. As a result, the clumps and clusters of fat globules form and
build an internal fat structure network (Goff, 1997).
Fig. 4.11: Fat aggregate size distributions of ice cream mix and ice cream
samples flowing out from different pipes. There is a significant difference in
the distribution of fat globule size in ice cream mix and ice cream samples.
The highest median fat aggregate size (d50 = 2.77 µm) and the highest
maximum size (d90 = 18.16 µm) were found in ice cream samples flowing out
from the smallest pipe. The lowest median fat aggregate size (d50 = 1.79 µm)
162
and the lowest maximum size (d90 = 14.39 µm) were found in the ice cream
samples flowing out from the biggest pipe. Droplets exceeding a critical size
of 30 – 40 µm are undesirable because they would result in a buttery
mouthfeel (Eisner et al., 2005). The higher shear stress at the wall in smaller
pipe i.e. higher apparent shear rate contributed to fat aggregation in ice cream
sample. The fat globule aggregates formed during the partial coalescence
process are responsible for surrounding and stabilizing the air cells and
creating a semi-continuous network of fat throughout the product resulting in
smooth-eating texture and resistance to melt down. The purpose of droplet
size distribution measurement in this study was to observe the difference in
fat droplet sizes between the ice cream mix and ice cream flowing from
different pipes.
4.4
Conclusions
This chapter presented a study of wall slip in ice cream pipe
rheometry using both Mooney plots and Tikhonov regularisation. This was
complemented with measurement of ice cream temperature at the wall which
allowed for analysis of energy balances in the near wall region. Pipe radius
dependence was evident in the flow curves, indicative of wall slip effects.
This apparent slip was amenable to analysis by the Mooney method and
indicated the contribution of slip to flow ranged from 70% to 100%. A
significant increase in ice cream temperature next to the wall along the length
of the pipe was measured in all cases and was attributed to viscous
dissipation. Energy balances indicated that the apparent wall slip effect was
not due to the existence of a thin slip layer of matrix fluid next to the wall.
163
Instead, it was found that the results were better understood as being the
result of a moderately thick layer of slightly heated ice cream next to the
wall. The increased temperature and shear thinning nature of the ice cream
led to runaway shear near the wall which tended to dominate the overall
flow. Whilst these flows may be interpreted as wall slip, the origin of the
phenomenon is different from that in most suspension flows and significantly
alters interpretation of results. Measurement of ice cream temperature in pipe
flow is straightforward and inexpensive. This chapter has shown its utility in
yielding insight into viscous dissipation and flow conditions, the authors
therefore recommend it for both industrial plant and test equipment. Fat
droplet size distribution was also obtained. Fat droplet size was higher in the
ice cream samples compared to the ice cream mix. Fat droplet size was more
pronounced to aggregation during the freezing process due to the higher
shear stress that affects the fat structure. The aggregation of fat alongside
with the melting of ice crystals affects the viscosity of ice cream which is
lower at higher apparent shear rate.
Characterising slippage of ice cream flowing in pipe is indeed a very
challenging process. The whole operation needs to be carefully controlled
due to the highly sensitive nature of ice cream. A good arrangement of
equipment and instrumentation is important to enable the data collection
process. To characterise slippage, a very good set of pressure drop data is
required to calculate the wall shear stress and subsequently the information is
used to calculate slip velocity. Experiments need to be done using at least
two pipe diameter sizes (four were used in this study) to see the effect of
slippage in different pipe diameter. The information on the occurrence of
164
viscous dissipation is vital in this case as viscous dissipation causes the ice
phase to melt and subsequently enhances slippage. The flow behaviour of
complex materials is different from one to another. The behaviour observed
in ice cream is different from the behaviour observed in other complex fluid
system such as fibre suspensions and magnesium silicate slurries. Further
discussions on wall slip analysis will be presented in Chapter 7.
165
5
WALL SLIP IN PIPE RHEOMETRY
DIETARY FIBRE SUSPENSIONS
OF
CITRUS
In this chapter, the flow behaviour of citrus dietary fibre (CDF)
suspensions was determined using the same pipe rheometry rig as previously
described. This suspension is expected to exhibit wall slip when flowing in
pipelines. The chapter starts with the introduction of the experimental work
to determine the rheological behaviour of CDF suspensions. This chapter
expanded with the data analysis of the fluid to determine wall slip and critical
discussion on the analysis method used are presented.
5.1
Introduction
Flow of fibre suspensions is a very important process in the
manufacturing of wide range of products such as pulp and paper, foodstuffs,
textile and fibre-reinforced composites. The manufacture of pulp and paper is
the largest among these industries. Pulp and paper studies are the most
reported research in fibre suspensions scientific literature which focused on
some particular aspect of pulping and papermaking as well as measurement
of the rheological properties of pulp suspensions as reviewed by
Derakhshandeh et al. (2011). The rheology of fibre suspensions is complex
due to the complexity in the microstructure and behaviour. Fibre orientation
166
and migration away from solid boundaries create a depletion layer near the
wall, which enhances apparent wall slip effect that complicates rheological
measurements (Nguyen and Boger, 1992; Barnes, 1995; Swerin, 1998).
Wiklund et al. (2005) reported a study of noninvasive measurements in pulp
suspensions at consistencies ranging from 0.74% (w/w) up to 7.8% (w/w)
using ultrasound velocity profiling (UVP) and laser Doppler anemometry in
an experimental flow loop. The result from UVP technique strongly indicated
the existence of a shear layer close to the pipe wall. The study was also able
to measure the thickness of the layer directly. More recently, Derakhshandeh
et al. (2010) studied the flow behaviour of 0.5 – 5% pulp suspensions using
both conventional and coupled ultrasonic Doppler velocimetry-rheometry
techniques. They observed shear thinning behaviour, and Newtonian
behaviour beyond a critical level of shear stress. Pipe flow of pulp
suspensions have been studied over the years especially to discuss the friction
loss in the flow regimes (Duffy, 2003; Duffy, 2006).
Another industry in which the flow of fibre suspensions is important
is food manufacturing. Most of the studies done to date on the flow
properties of fluid food (Bhattacharya and Bhattacharya, 1994; Ma and
Barbosa-Canovas, 1994; Bhattacharya and Bhat, 1997; Grigelmo-Miguel et
al., 1999) were performed using offline rheological measurement.
Rheological data of fluid food particularly on fruit dietary fibre suspension
when flowing in pipelines are not available. Most of the data previously
reported for CDF were on the effect of CDF on the physical, chemical and
sensory properties of food (Anonymous, 2002; Dervisoglu and Yazici, 2006;
Fischer, 2007). In this study, the rheological properties of CDF suspension
167
were determined. The purpose of the research presented here was to
investigate the effect of different concentrations of CDF on the flow
properties of the suspension and on wall slip phenomenon. The
understanding of the rheological behaviour of CDF suspensions, in terms of
shear rate, shear stress and wall slip is of paramount importance in the design
of flow systems, selection of pumps and for scale-up and mechanization of
the process.
5.2
Materials and Methods
Herbacel AQ Plus Citrus Fibre (Herbafood Ingredients GmbH,
Werder (Havel), Germany) was used in this investigation. Fig. 5.1 shows a
scanning electron microscope (SEM, acquired on FEI QuantaTM 200 ESEM,
FEI Europe, Netherlands) image of the dry CDF powder. The particles are
non-spherical and different in shapes and sizes which indicate different
cellulose structures due to different citrus fruits used to produce the powder.
It is difficult to obtain precise particle size distribution data for these particles
since most available methods contain some uncertainty over which
dimensions of the particles are measured. This is especially so for the long
fibrous structure which the size is difficult to measure. However, a particle
size analysis was still performed and a particle size distribution obtained by
laser
diffraction
using
Malvern
Mastersizer
(Malvern
Instruments,
Worcestershire, United Kingdom) measured all particles having diameters
between 38 and 334 μm, with a mean of 171 μm.
168
Fig. 5.1: SEM image of Herbacel AQ Plus Citrus Fibres
CDF powder was mixed with water at room temperature to make 2, 3
and 4% (w/w) suspensions i.e. volume fraction of 0.07, 0.10 and 0.14,
respectively. The powder was added gradually into water and stirred using
agitators, to break the lumps and to form a smooth suspension. The
suspension was then pumped through a Silverson in-line processor (Silverson
L5M-A, Chesham, UK) at 10230 rpm for 30 mins to homogenize it.
5.3
Results and Discussion
This section describes the results obtained from the experimental
work done on CDF suspensions using the pipe rheometry system.
5.3.1
Flow curves and rheological behaviour
Figs. 5.2 (a) to (c) show the measured flow curves for CDF
suspensions. The data were fitted to power law model. It can be seen that the
flows show strong dependency on pipe radius, such that at constant wall
169
shear stress the apparent shear rate increases in smaller pipes. Such an effect
is frequently indicative of apparent wall slip effects in suspension flows as
represented by Eq. (2.44).
(a)
(b)
170
(c)
Fig. 5.2: Wall shear stress against apparent shear rate data for citrus fibre
suspensions - (a) 2%, (b) 3% and (c) 4% (w/w) flow in four different pipes.
Power law fitted curves are shown in the figure.
(Error bars bound regions of possible values)
Table 5.1 shows the rheological data of CDF suspensions when
flowing in different pipes such as flow behaviour index and consistency
index at various levels of concentrations. Consistency index, M is a measure
of the consistency of the fluid, the higher the M the more ‘viscous’ the fluid;
flow behaviour index, n is a measure of the degree of non-Newtonian
behaviour, the greater the departure from unity the more pronounced are the
non-Newtonian properties of the fluid (Tanner, 1985).
For all the cases, the flow behaviour index, n of less than 1 observed
in each pipe indicates that CDF suspensions are non-Newtonian
pseudoplastic fluids. The highest and the lowest values of n were observed at
the lowest and highest concentrations of solids for all cases which indicates
171
that non-Newtonian behaviour increases with increased level of solids in the
fluids.
Table 5.1: Rheological data for citrus powder suspensions at different
concentrations
Pipe size, D
(mm)
Citrus fibre powder
concentration in the
suspension
(% w/w)
Consistency
index, M
(Pa sn)
Flow
behaviour
index, n
33.9
2
3
4
2
3
4
2
3
4
2
3
4
12.8
30.6
36.4
7.70
17.3
21.4
4.70
11.4
13.9
0.85
2.27
3.18
0.29
0.22
0.20
0.33
0.32
0.31
0.37
0.35
0.34
0.57
0.55
0.51
21.5
15.2
8.7
Increasing the concentration of the powder also increased the
consistency index of the fluids. The lowest and highest values were observed
at the lowest and highest concentrations of solids. The same behaviour was
observed in the rheological studies of maize flour suspension (Bhattacharya
and Bhattacharya, 1994), rice-blackgram suspensions (Bhattacharya and
Bhat, 1997), and peach dietary fibre suspensions (Grigelmo-Miguel et al.,
1999).
At the largest pipe diameter, the suspension displays a strong shear
thinning behaviour at every solids concentration indicated by the low n value.
However, as the pipe diameter decreases, n increases which indicate that the
suspensions became less shear thinning. n increases and moves towards
172
Newtonian behaviour i.e. n = 1. However, the suspensions are still in shear
thinning region.
5.3.2
Wall slip analysis
The interpolations between data points made for values of constant
wall shear stress to generate Mooney plots show an apparently non-linear
relationship between the apparent shear rate and 4/R and they are presented
in Figs. 5.3 (a) to (c). Although a straight line could be fitted to these data
with a gradient of the slip velocity, the line would intercept the ordinate axis
at a negative value which, considering Eq. (2.44), is not consistent with the
physical model presented. Thus no conclusions about the nature of the flow
can be drawn from these Mooney plots.
Tikhonov Regularisation-Mooney analysis performed on the data was
unsuccessful to generate any output for all the CDF suspensions. Hence, no
results from TRM analysis are available to be discussed here.
(a)
173
(b)
(c)
Fig. 5.3: Mooney plot of apparent shear rate against 4/R for (a) 2% (b) 3%
and (c) 4% (w/w) concentrations of CDF suspensions. Non-linear relationship
was obtained and the straight lines fitted intercept the ordinate axis at negative
values for all cases.
(Error bars bound regions of possible values)
174
Successful Mooney analysis have been previously reported for pastes
such as potato starch, french mustard, tomato puree, toothpaste, glass
fibres/thermoplastic melts, and alumina/silicon oil (Higgs, 1974; Chung and
Cohen, 1985; Corfield et al., 1999; Graczyk et al., 2001) as detailed in Table
2.3 in Chapter 2. However, other physically unrealistic cases have been
reported, most importantly by Jastrzebski (1967) on kaolinite pastes. He
reported the empirical result that wall shear stress against wall slip velocity
data over a range of capillary diameters aligned on the same curve when the
wall slip velocity was divided by the diameter, but offered no physical
justification of this feature (Martin et al., 2004).
As shown in Table 2.2 in Chapter 2, for Jastrzebski method, apparent
shear rate data is plotted against 4/R2 as opposed to the Mooney method,
where apparent shear rate data is plotted against 4/R. The Jastrzebski method
is usually called Jastrzebski-Mooney method. Jastrzebski-Mooney plots for
the CDF suspensions are presented in Fig. 5.4. Linear least squares fits were
applied to these data. Even though the plots appear to be quite linear,
however, a degree of non-linearity is still significant which indicates that the
method is also unable to analyse wall slip in CDF suspensions flow correctly.
Previous studies have reported unsuccessful attempts to obtain slip velocity
using both Mooney (negative bulk shear and non-linear plots) and
Jastrzebski-Mooney methods on microcrystalline cellulose/aqueous lactose
suspension (Harrison et al., 1987) and wheat suspension (Singh and Smith,
2009).
175
(a)
(b)
176
(c)
Fig. 5.4: Apparent shear rate against 4/R2 for (a) 2% (b) 3% and (c) 4% (w/w)
concentrations of CDF suspensions. The plots appear to be quite linear.
However, a degree of non-linearity is still significant.
(Error bars bound regions of possible values)
Where materials are thought to be slipping and shearing, but which
give unsuccessful Mooney diagram, the materials could be considered as
‘badly behaved’. Where pipe flow experiments yield successful Mooney
diagrams, the fluid could be considered as ‘well-behaved’. Since both
Mooney and TRM methods do not comply with the data of CDF suspensions,
wall slip effect cannot be analysed using this method. Due to that, the bulk
yield stress, slip velocity and slip layer thickness values cannot be obtained.
However, this material must still be characterised in order to make process
prediction.
Mooney plot which yields a straight line at constant wall shear stress
indicates constant slip velocity. Section 2.6 introduced Eq. (2.44) which
177
approximates the shear within a thin slip layer where curvature can be
neglected and the slip layer fluid is Newtonian. As mentioned previously,
Mooney method relies on the ratio δ/μslip being constant for there to be linear
trends on the Mooney plot. The constant ratio of slip velocity to wall shear
stress indicates constant slip layer thickness to slip viscosity ratio. The
unsuccessful Mooney plots generated as shown in Fig. 5.3 illustrate that the
slip velocity (gradient of the graph) was not constant at constant wall shear
stress, thus the ratio δ/μslip was not constant. The slip layer in CDF
suspensions is assumed to consist of water and a very low concentration of
citrus fibre which does not have effect on the viscosity. Hence the slip layer
viscosity is constant. Therefore, it was deduced that the slip layer thickness, δ
was the factor affecting wall slip in pipe rheometry in CDF suspensions.
As a suspension or slurry flows, the microstructure will change.
Fibres are known for their swelling properties when water is added to them.
We attempted to look into the microstructure of dry and wet CDF powders. A
pair of environmental scanning electron microscope (ESEM, acquired on FEI
QuantaTM 200 ESEM, FEI Europe, Netherlands) images (Fig. 5.5) illustrates
the changes of CDF powder particles when water was introduced to them.
The approximate size of a fibre particle measured based on the scale on the
images showed that the particle size increased from 39.6 to 47.9 μm after
water was introduced. We also attempted to measure particle size in
suspension using laser diffraction method. However, the measurement does
not represent the actual particle size in the system as a whole. This was due
to experimental limitations where only small amount of liquid sample is
added to the dispersing unit.
178
The particles swelled greatly in water and the orientation of the
particles and the microstructure of the suspension changed during flow. The
different shapes and sizes of the fibre particles as shown in Figs. 5.1 and 5.5
caused the slip layer to have inconsistent thickness near the wall hence why
the ratio δ/μslip was not constant. The changing orientations of the particles
during flow change the microstructure of multiphase fluids with lower solids
concentrations, and quite possibly across the cross section of the pipe as well.
If this is so, the constitutive parameters will vary correspondingly over the
diameter of the pipe. Thus, the second term on the right side of Eq. (2.44)
may not be constant over the 4/R range, and the Mooney analysis will fall
down. Simulation studies done by Pozrikidis (2002) on wall layers in less
dense suspensions have clearly indicated the reorientation of non-spherical
particles under shear near the wall.
Increase in solid concentration causes the wall shear stress to increase
at constant apparent shear rate due to higher pressure gradient required for
the thicker suspension to flow. Based on Eq. (2.49), increase in wall shear
stress will cause the slip velocity to increase. Apart from the increased
difficulty in the particles movement in the system, higher particle content
causes more water to be absorbed by the fibre particles hence generally
reduces the slip layer thickness compared to the system with lower solid
concentration.
179
~ 47.9 μm
~ 39.6 μm
(a)
(b)
Fig. 5.5: ESEM images of (a) dry citrus fibre powder and (b) wet citrus fibre powder. A significant increase in particle size was observed
after water was introduced.
180
In order to investigate the rheological properties of CDF suspensions,
other approaches that could be done to measure the rheology and avoid wall
slip are by using pipes with higher surface roughness or using rheometer with
roughened wall surface or vane spindles. The true rheological properties
could be obtained from this kind of measurement methods. The wall slip
velocity could be determined by calculating the difference between the
apparent shear rates at constant wall shear stress for slipping and nonslipping flow as shown in Eq. (2.48). The most effective method to
investigate the rheological properties of fibre suspensions is by visualising
the pipe flow using direct measurement method as described by Wiklund et
al. (2005) and Derakhshandeh et al. (2010).
5.4
Conclusions
This chapter presented a study of wall slip in CDF suspensions pipe
rheometry. The rheological behaviour of CDF suspensions when flowing in
pipelines was experimentally determined and they behaved like nonNewtonian, pseudoplastic fluids. An increase in the powder concentration
causes a decrease in the flow behaviour index (indicating the increase in nonNewtonian behaviour) and increase in the consistency index. Standard
approaches for determining wall slip velocity using Mooney analysis yielded
non-physical results. Another approach based on Tikhonov regularisationMooney method was also unsuccessful to characterise CDF suspensions. The
incompatibility of the method to analyse wall slip is attributed by the
inconsistent ratio of Vslip/τw and δ/μslip at constant wall shear stress. The same
phenomenon applies to the suspension with higher concentration. This is
181
attributed to the microstructure changes and shear-induced re-orientation of
the particles in CDF suspension during flows which caused the inconsistency
of the slip layer thickness and consequently affect the wall slip
characterisation process. An attempt was made to plot Jastrzebski-Mooney
graphs. Even though the plots appear to be quite linear, however, a degree of
non-linearity is still significant which indicates that the method is also unable
to analyse wall slip in CDF suspensions flow correctly. Chapter 6 presents
the wall slip study in the pipe rheometry of another multiphase fluid used in
this research i.e. magnesium silicate slurries. Further discussions on the wall
slip analysis in this study are presented in Chapter 7.
182
6
WALL SLIP IN PIPE RHEOMETRY OF MAGNESIUM
SILICATE SLURRIES
In this chapter, the flow behaviour of magnesium silicate slurries was
determined using the same pipe rheometry rig as previously described.
Magnesium silicate slurries were also expected to exhibit wall slip when
flowing in pipelines. The chapter starts with the introduction of the
experimental work to determine the rheological behaviour of magnesium
silicate slurries. This chapter expanded with the data analysis of this fluid to
determine wall slip and critical discussion on the analysis method used are
presented.
6.1
Introduction
Previous studies have reported on the rheology and wall effects in the
flow of kaolinite platelets suspension (Jastrzebski, 1967) and talc platelets
paste (Martin et al., 2004). Most of the research previously reported on talc
platelets i.e. magnesium silicate relates to its rheological behaviour when
used in extruded ceramics as filler. Talc powder is also used in baby powder,
paint and other materials as explained in Chapter 2. In this study, magnesium
silicate slurries of different concentrations were studied to characterise the
flow behaviour and wall slip phenomenon. Apart from that, this added
183
experimental procedure was aimed to test the reliability of the pipe rheometry
system developed to characterise the rheological behaviour as well as to
enable the analysis of wall slip of various types of multiphase fluids.
6.2
Materials and Methods
Micro-Talc AT Extra i.e. magnesium silicate (Norwegian Talc (UK)
Ltd., UK) was used in this investigation. Fig. 6.1 shows an SEM image of the
magnesium silicate powder obtained from the previous study of Martin et al.
(2004) with particles having diameters between 1 and 20 μm in diameter,
with a mean of 7 μm. The particles have non-spherical shape and are soft
platelets, as indicated by the worn edges. Magnesium silicate powder was
mixed with water at room temperature to make 10, 16, 20, 24 and 28% (w/w)
slurries i.e. volume fraction of 0.04, 0.06, 0.07, 0.09, and 0.11, respectively.
The powder was added gradually into water and stirred using agitator. The
slurries were kept stirred in the feed tank during the experimental process to
avoid sedimentation.
Fig. 6.1: SEM image of Micro-Talc AT Extra (Martin et al., 2004)
184
6.3
Results and Discussion
This section describes the results obtained from the experimental
work done on magnesium silicate slurries using the pipe rheometry system.
6.3.1
Flow curves and rheological behaviour
Figs. 6.2 (a) to (c) show the measured flow curves for magnesium
silicate slurries. It can be seen that the flows show strong dependency on pipe
radius, such that at constant wall shear stress the apparent shear rate increases
in smaller pipes. Such an effect is frequently indicative of apparent wall slip
effects in suspension flows as represented by Eq. (2.44).
(a)
185
(b)
(c)
186
(d)
(e)
Fig. 6.2: Wall shear stress against apparent shear rate data for magnesium silicate
slurries - (a) 10%, (b) 16%, (c) 20%, (d) 24% and (e) 28% (w/w) flow in four
different pipes. Fitted curves are shown in the figure.
(Error bars bound regions of possible values)
187
Table 6.1 shows the rheological data of magnesium silicate slurries
when flowing in different pipes.
Table 6.1: Rheological data for magnesium silicate slurries at different
concentrations
Pipe size, D
(mm)
Magnesium silicate
concentration in the
suspension
(%)
Consistency
index, M
(Pa sn)
Flow
behaviour
index, n
33.9
10
16
20
24
28
10
16
20
24
28
10
16
20
24
28
10
16
20
24
28
0.87
1.02
1.21
1.33
1.68
0.89
1.07
1.24
1.43
2.02
0.93
1.13
1.27
1.65
2.32
0.94
1.14
1.31
1.73
2.44
0.96
0.95
0.94
0.93
0.87
0.63
0.62
0.60
0.58
0.54
0.49
0.48
0.47
0.43
0.40
0.37
0.36
0.35
0.32
0.30
21.5
15.2
8.7
For all the cases, the flow behaviour index, n of less than 1 observed
in each pipe indicates that magnesium silicate slurries are non-Newtonian
fluids at low concentration. The highest and the lowest values of n were
observed at the lowest and highest concentrations of solids for all cases
which indicates that non-Newtonian behaviour increases with increased level
of solids in the fluids. Increasing the concentration of the powder also
increased the consistency index of the fluids. The lowest and highest values
were observed at the lowest and highest concentrations of solids.
188
6.3.2
Wall slip analysis
The interpolations between data points made for values of constant
wall shear stress to generate Mooney plots show an apparently non-linear
relationship between the apparent shear rate and 4/R and they are presented
in Figs. 6.3 (a) to (c) for CDF suspensions. Similar to CDF suspensions flow,
the fitted line intercepts the ordinate axis at a negative value which again, is
not consistent with the physical model presented. Thus no conclusions about
the nature of the flow can be drawn from these Mooney plots. A non-linear
relationship was also reported for higher solids concentration of magnesium
silicate in paste for extrusion (Martin et al., 2004), potato starch paste
(Cheyne et al., 2005) and bread dough (Sofou et al., 2008).
Tikhonov Regularisation-Mooney analysis performed on the data was
unsuccessful to generate any output for all the CDF suspensions. Hence, no
results from TRM analysis are available to be discussed here.
(a)
189
(b)
(c)
190
(d)
(e)
Fig. 6.3: Mooney plot of apparent shear rate against 4/R for (a) 10%, (b)
16%, (c) 20%, (d) 24% and (e) 28% (w/w) concentrations of magnesium
silicate slurries. Non-linear relationship was obtained and the straight lines
fitted intercept the ordinate axis at negative values for all cases.
(Error bars bound regions of possible values)
191
Jastrzebski-Mooney plots for the magnesium silicate slurries are
presented in Fig. 6.4. Linear least squares fits were applied to these data and
appear plausible. Previous studies known to the author which have also
reported successful attempts using Jastrzebski-Mooney methods when the
analysis using Mooney method generated non-linear plots or gave negative
bulk shear intercept. The successful Jastrzebski-Mooney plots were obtained
from the studies on polystyrene spheres/aqueous glycerol solution
suspensions (Cheng, 1984); coal powder suspension (Meng et al., 2000; Lu
and Zhang, 2002), alumina/aqueous polymer solution suspension (Khan et
al., 2001); talc paste (Martin et al., 2004); foam/hydroxyl-propyl-guar
suspension (Herzhaft et al., 2005), and fibres/cement paste (Zhou and Li,
2005).
(a)
192
(b)
(c)
193
(d)
(e)
Fig. 6.4: Apparent shear rate against 4/R2 for (a) 10%, (b) 16%, (c) 20%, (d)
24% and (e) 28% (w/w) concentrations of magnesium silicate slurries. Linear
least squares fits were applied to these data and appear plausible.
(Error bars bound regions of possible values)
194
Most of the previous studies which have discussed similar problems
with the Mooney analysis either suggest that the experimental error was
possible, or resort to using the Jastrzebski-Mooney method (Martin et al.,
2004). A simple understanding of the problem can be gained by considering
microstructural changes in the slurry during pipe flow. It is possible that the
microstructure of many slurry materials will change during flow, and quite
possibly across the section of the pipe. The low solids fraction of the slurry
enhanced shear-induced re-orientation of the talc platelets. If this is so, the
constitutive parameters will vary correspondingly over the diameter of the
pipe. Thus, the second term on the right hand side of Eq. (2.44) may not be
constant over the 4/R range, and the Mooney analysis will fall down. Shearinduced re-orientation of the talc platelets may cause the slip to have
inconsistent thickness near the wall and made the ratio of δ/μslip to be
inconsistent.
6.4
Conclusions
This chapter presented a study of wall slip in magnesium silicate
slurries pipe rheometry. Magnesium silicate slurries showed non-Newtonian
behaviour during flow in pipes. As observed for CDF suspensions, an
increase in the powder concentration causes a decrease in the flow behaviour
index (indicating the increase in non-Newtonian behaviour) and increase in
the consistency index. Both Mooney and Tikhonov regularisation-Mooney
methods were also unable to characterise the fluid flow. The incompatibility
of the method to analyse wall slip is attributed by the inconsistent ratio of
Vslip/τw and δ/μslip at constant wall shear stress. The same phenomenon applies
195
to the slurries with higher concentration. Shear-induced re-orientation of the
particles in magnesium silicate slurry during flows which caused the
inconsistency of the slip layer thickness which affect the wall slip
characterisation process. An attempt made to plot Jastrzebski-Mooney graphs
yields plausible results with the plots appear more linear than the graphs
obtained from Mooney analysis. Further discussions on the wall slip analysis
are presented in Chapter 7.
196
7
DISCUSSIONS ON WALL SLIP ANALYSIS
Chapter 4, 5 and 6 have discussed on the experimental work that were
successfully performed using the pipe rheometry rig. In this chapter, further
general discussion on wall slip phenomenon is presented. Mooney equation
as described in Chapter 2 was developed into dimensionless form based on
radial-dependent rheology. This approach illustrates a better picture of the
analysis done in the previous two chapters. This chapter also discusses on
phase migration phenomenon which is important especially during the pipe
flow of CDF suspensions and magnesium silicate slurries.
7.1
Mooney equation – dimensionless approach
In this section, Mooney equation (Eq. 2.44) was developed into
dimensionless form. The dimensionless shear stress and shear rate terms are:
 
  

w
Eq. (7.1)

Eq. (7.2)
app
From Eq. (2.44), the Mooney equation can be made dimensionless as
follows:
197
4Q
4
4
 Vslip  3
3
R
R
w
  w


2
f
1
 d
Eq. (7.3)
0
 
1
 R 3 
4
4  R 3  w 2
Vslip 


  f
3 
R
 4Q   w  4Q   0
1
 d
Eq. (7.4)
Knowing that Q/πR2 is equal to Vmean, and 4Q/πR3 is equal to  app ,
1
Vslip
Vmean
 
1  4  w2
 3
  f
 w  app   0
1
 d
Eq. (7.5)
By non-dimensionalising all the terms, the final dimensionless form of
Mooney equation for non-Newtonian fluid becomes:
  1

1  Vslip  4    2 f
1
 d 
Eq. (7.6)
 0
Dimensionless wall shear stress is a function of dimensionless apparent shear
rate and temperature while apparent shear rate is a function of dimensionless
wall shear stress and temperature as shown in Eqs. (7.7) and (7.8)
respectively:
   f  , 
Eq. (7.7)
   g  , 
Eq. (7.8)
Dimensionless radial distance can be written as:
r 
r
R
Eq. (7.9)
r   
Eq. (7.10)
Dimensionless temperature is a function of dimensionless wall shear stress as
well as dimensionless radial and axial distances:
  hr , x
Eq. (7.11)
  h , x
Eq. (7.12)
198
Hence, dimensionless apparent shear rate is a function of dimensionless wall
shear stress, temperature and axial distance as shown in Eq. (7.13).
   g  ,   i , x
Eq. (7.13)
Therefore, the Mooney equation for a non-Newtonian fluid can be written as
Eqs. (6.14) and (6.15):
 1

1  Vslip  4    2 g  ,  d  
Eq. (7.14)
 0
  1

1  Vslip  4    2 i , x  d  
Eq. (7.15)
 0
Based on these equations, if the dimensionless axial position is
constant, then the second term on the right is fixed and thus the
dimensionless slip velocity is known. Any change in velocity profile due to
viscous dissipation will be accounted for within the second term on the right,
and thus only slip effects in addition to this will be represented by the
dimensionless slip velocity term. However, if the dimensionless axial
position is not constant, then the dimensionless bulk flow term will not be
constant. This is the case in this study, thus whilst the wall shear stress is
constant across the data set, there is a change in the velocity profile which is
not just a function of r/R, i.e. τ/τw, and thus there is an effect which may
appear as wall slip. It is not inevitable that the Mooney plot will yield a
straight line, but it appeared to be the case in the data reported here for ice
cream. If the plot was not linear the same effect might be taking place, just
not with a constant thickness to viscosity ratio.
In the case of ice cream, the ratio δ/μslip is known but the temperature
increase of the ice cream near the wall is also desired. Calculating the energy
199
dissipation rate for different thicknesses at given shear stresses as described
in Chapter 4, enables knowledge of slip layer thickness and thus slip layer
viscosity can be calculated. Therefore, viscous heating actually does not
cause any term in Mooney equation which corresponds to wall slip, but this
is at a point along the length of the pipe short enough for axial changes to be
insignificant. The flows in the different pipes will not be at an equivalent
dimensionless axial position; therefore they will display different flow
characteristics. This may appear as a shift of flow curves for each pipe radius.
This shift may follow the Mooney pattern, but only if the slip velocity is only
a function of the wall shear stress. This is true when the thickness to viscosity
ratio remains constant. Therefore, viscous heating effects are likely to appear
as slip effects in ice cream flow, and these may or may not obey Mooney’s
equation. It is possible that they occur in addition to other slip phenomena at
the wall.
7.2
Wall slip in ice cream flow
Ice cream flows under plug flow condition below the yield stress.
Under low shear, ice crystals start to melt due to rising temperature caused by
heat dissipation from the friction between ice cream and the pipe wall. The
highly concentrated matrix phase is diluted with water and its viscosity
lowers. This makes the ice cream flow like a solid plug with an envelope of
liquid film acting as lubricant. Ice cream will continue to flow in this manner
until yield stress is reached. After yield stress, deformation of ice cream
structure near the wall takes place. As described previously in Chapter 4, the
apparent wall slip effect found in ice cream flow was not due to the existence
200
of a thin slip layer of matrix fluid next to the wall. Instead, it is the result of a
moderately thick layer of slightly heated ice cream next to the wall. The low
viscosity of the heated ice cream near the wall causes high shear rates at the
boundary layer which subsequently amplified a shear rate-induced effect i.e.
shear thinning. The increased temperature and shear thinning nature of the
ice cream led to runaway shear near the wall which tended to dominate the
overall flow. Whilst these flows may be interpreted as wall slip, the origin of
the phenomenon is different from that in most suspension flows and
significantly alters interpretation of results.
The complex structure of ice cream should be taken into account
during transportation in pipelines, handling to the distributors and even
during viscometry/rheometry measurements as slight changes in one
structure will affect other structures in the complex system. The major
contributor to slippage during ice cream flow is the melting of ice phase.
Thus it can be seen that the adverse effect of viscous dissipation in the flow
and shear induced structural changes do affect the wall slip phenomenon near
the wall. Ice cream transportation in pipelines should not be done in very
small pipes as this will affect the structure greatly. Bigger pipes should be
used to limit the complex interactions to only the structure near the wall in
order to prevent further structural damage initiated near the solid boundaries.
These findings open more opportunities for further research to be done in this
area especially on the effect of viscous dissipation that can lead to structural
changes if it is not controlled.
201
7.3
Wall slip in multiphase suspensions
Based on the rheogram fit of both CDF suspensions and magnesium
silicate slurries both in Chapter 5 and 6, they also exhibit wall slip effects due
to the dependency of flow on pipe diameter. However, the classic method for
analysing wall slip based on Mooney analysis failed to characterise the data
of both fluids where the plots yielded non-linear relationship between the
apparent shear rate and 4/R. Although a straight line could be fitted to these
data with a gradient of the slip velocity, the line would intercept the ordinate
axis at a negative value which is unviable. As explained previously, the
unsuccessful Mooney analysis shows that the Vslip was not constant at
constant τw and thus the ratio δ/μslip was not constant too. The irregular shapes
and sizes of the fibre and talc particles caused the slip layer to have
inconsistent thickness near the pipe wall.
In the flow of CDF suspension, the dispersed phase forms flocs when
first pumped into a pipe. At low shear stress/rate, the sizes of the flocs
formed by a group of individual particles are larger than the size of the
particles themselves. The steric hindrance between the flocs and the pipe wall
at rest or at very low stress causes the flocs to be displaced at a distance of
the same size as the flocs. The slip layer thickness will continue to increase
until a critical shear stress is reached just before the fluid starts to deform.
Before the critical point, the fluid will flow in the form of plug due to total
slip. Deformation of the fluid structure takes place after that critical point.
When the stress near the wall increases due to increase in the fluid flow rate,
the flocs near the wall will move past each other rapidly and may cause the
breakdown of the flocs into individual particles. Reduction of the dispersed
202
phase size from large flocs to smaller particles reduces the steric hindrance
with the wall and subsequently causing the slip layer thickness to decrease.
This behaviour was also observed by other researchers (Nguyen and Boger,
1992; Egger and McGrath, 2006; Mallik et al., 2009).
Fibre and talc particles are subjected to intense mechanical
stimulation from both fluid flow and pipe wall. Forces are acted upon the
suspensions by pumping and shear forces from the boundaries. Talc powder
particles are soft platelets and have irregular structure. The low solids
fraction in the suspension contributes towards the shear-induced reorientation of the talc platelets during flow i.e. across the cross section of the
pipe. This also happens in citrus fibre suspension where the fibre particles
orientation leads to nonhomogeneous shear flow. Furthermore, the fibre
suspension becomes denser as the particles absorb water and expand in size.
This consequently makes the suspension becomes thicker.
7.4
Influence of non-homogeneous shear flow
In the study done by Jastrzebski (1967) on concentrated suspensions,
he stated that the values of slip coefficients are functions not only of the
shear stress but also of the tube radius which leads to the determination of the
values of the corrected slip coefficients which are independent of the tube
radius. Jastrzebski introduced the slip coefficient φ = Vs/τw in the original
derivation of Mooney equation which slightly modified it. The empirical
equation development leads to a plot of Q/πR3τw vs. 1/R2 at constant τw where
the slip coefficient, φ is the slope of the straight line. He concluded that the
changes of the corrected slip coefficients are closely related to the structural
203
characteristics of the suspension. Nguyen and Boger (1992) have provided
similar explanations as to how the slip behaviour changes with increasing
shear stress due to the changes in the structure of the material. The
microstructural changes described appear plausible, although no evidence has
been presented to support the explanation (Martin and Wilson, 2005).
To date, there are various modified-Mooney correlations developed to
suit particular experimental data when Mooney’s method failed to analyse
them as described earlier in Chapter 2. However, there are again purely
empirical and no physical relations have been done to explain what is
occurring in the system that makes the classic analysis failed. It is understood
(based on Table 2.3 in Chapter 2) that Mooney analysis is more successful in
analysing the wall slip behaviour of single phase fluid such as polymer melts
which have homogeneous structure in the system. Most of the studies which
dealt with complex multiphase fluids such as concentrated suspension and
paste were unable to characterise the slip behaviour via the classic method.
Being a multiphase fluid system with solid particles moving in
Newtonian/non-Newtonian fluid, makes the characterisation process difficult.
So far, relatively little attention have been given to relating the
incompatibility of the classic wall slip analysis method with the behaviour of
particle movement due to the gradients in collision frequency and in
concentration of particles in the system. This nonhomogeneous distribution
of solid particles in the system is contributed by the flow-induced particle
migration phenomenon. The migration from the highly sheared and/or highly
concentrated zones is due to particle collisions in that area which results in
the local increase in the suspension velocity. According to Lanteri et al.
204
(1996), most of the authors whose results are not compatible with Mooney’s
analysis adopted Wiegreffe’s analysis, in which he writes:
Vslip ( w , R) 
 w
4R
Eq. (7.16)
where φ is called the corrected slip coefficient. Such expressions failed to
represent an intrinsic behaviour law for the material which is independent of
the geometry of the considered measuring apparatus.
This present study has proven that the wall slip behaviour of CDF
suspensions and magnesium silicate slurries cannot be described by
considering them as homogeneous materials with slipping conditions as
assumed in the classic Mooney analysis. Lanteri et al. (1996) have proposed
a method to analyse their data by developing a heterogeneous model which
takes into account the separate contributions of solid and liquid phases to the
global behaviour of the bulk material. They found out that the apparent shear
rate depends on 1/R2 at a given τw and this dependency is not the consequence
of a discontinuity in the velocity at the interface between the material and the
mould, but is due to a relative motion of the two phases. Jastrzebski (1967)
concluded that the slip caused by the wall effect is due to the non-uniform
distribution of the dispersed solid particles which are at lower concentration
at the wall surface than any other points farther away from the wall tube.
Even though the model proposed by Lanteri et al. (1996) provides a good
agreement between numerical simulations and experimental data, however, a
mismatch observed between the prediction of the model and the experimental
data suggested that the proposed method is not sophisticated enough. It is
thought that the shear-induced particle migration phenomenon in
205
nonhomogeneous shear flow should be taken into account in the failure of the
classic analysis.
Dynamic segregation caused by shear-induced particle migration
occurred as a result of the competition between gradients in particle collision
frequency and gradients in viscosity of the suspension. The simulation study
of wall slip in simple shear flow by Ahuja and Singh (2009) demonstrated
that apparent slip is present even in simple shear flow where the shearinduced
particle
migration
is
absent.
Under
the
conditions
of
nonhomogeneous shear flow, there can be migration near the wall which can
increase the depletion of particles resulting into enhanced slip (Medhi et al.,
2011). In multiphase system, the particle deficient layer i.e. slip layer can be
observed even if there was no flow as a result of static geometric depletion
effect. As explained before, this effect could result from steric,
hydrodynamic, viscoelastic, chemical and gravitational forces acting on the
solid particles adjacent to the wall (Barnes, 1995). The existence of shearinduced migration during bulk flow contributes to the creation of depleted
layer and further enhances the wall slip apart from other factors such as large
dispersed particles, smooth walls, low speeds or flow rates as well as wall
and particles carrying electrostatic charges. Shear-induced particle migration
reaches equilibrium because of the increase in viscosity of the zones where
particles are migrating to and can be expected to stop before the local particle
volume fraction is high enough to reach the so-called random loose packing
i.e. the maximum volume fraction of solid particles obtained when they are
packed randomly. In nonhomogeneous shear flows of concentrated
206
suspensions, the particle migration is irreversible i.e. an individual particle is
not moving back to its original streamline.
Leighton and Acrivos (1987) proposed expressions for particle
migration in nonhomogeneous shear flows based on the effect of a spatially
varying interparticle interaction frequency and a spatially varying effective
viscosity. Their mechanism detailed the expressions for the scaling of the
particle flux with concentration, shear rate and particle dimension. Particle
migration has been much studied since then. Phillips et al. (1992) used this
flux expression to develop a diffusion equation that describes the evolution of
particle concentration profiles over time. The stress in the fluid is modelled
as a Newtonian fluid with an empirical relationship for the dependence of the
viscosity on the local volume fraction of particles. The particle distribution in
the flow is determined from a conservation equation that accounts for flowinduced particle migration. Even though the model is successful in describing
the time-dependent development of concentration variations in flow between
concentric cylinders, the resulting model is not based directly on fundamental
principles that may be generalized to mixed or nonviscometric flows (Yapici
et al., 2009). Several modelling approaches have been proposed by
researchers to predict particle migration in concentrated suspensions and its
relation to velocity and stress fields. This method can be adopted to analyse
the particle migration in a fluid system which contributes to wall slip that is
failed to be analysed by Mooney equation. As explained before, if the shearinduced re-orientation of particles is really occurring in the fluid, the
constitutive parameters will vary correspondingly over the diameter of the
pipe. This will cause the second term on the right hand side of Eq. (2.44) may
207
not be constant over 4/R range, and the Mooney analysis will fall down. In
such cases, it may be possible to find some interface relationship which
yields apparent viable Mooney plots such as Jastrzebski and Wiegreffe
methods but there would be no grounds to believe that the plots provide true
values of wall slip.
From this overview, it is again believed that the physical structure of
the particles in the fluid system as well as the shear-induced particle
migration/re-orientation at higher shear stress/rates contributes to the
complexity in the wall slip analysis of multiphase flow and makes the
Mooney’s slip analysis incompatible. A physical model that can describe the
counterbalance reaction between shear-induced particle migration as well as
particle concentration and viscosity gradients in the flow field would be able
to help interpret the complex behaviour in detail. Furthermore, the physical
structure of the particles in fluids is believed to have direct consequence with
the occurrence of shear-induced particle migration which develops at very
high shear stress/rates due to breakdown of irregular particles which clump
together (in the case of paste) and the rupture of gel-like flocs formed by
hydrated water-absorbent particles (in the case of fibre suspension).
7.5
Conclusions
In this chapter, Mooney equation was presented in dimensionless
form to provide a clearer picture of the wall slip analysis done on the data of
the fluids studied especially ice cream. The incompatibility of the classic
Mooney method to analyse wall slip in CDF suspensions and magnesium
silicate slurries pipe flow is believed due to the changes of physical
208
microstructure of both the citrus fibres and talc platelets. Citrus fibre particles
vary in shapes and sizes and they swelled when mixed with water and formed
quite a viscous deformable phase at low concentration. Talc platelets have
irregularities in their shapes and formed a loose and randomly orientated
arrangement in dispersion. This inhomogeneity of the microstructures leads
to the inconsistency of the slip layer thickness near the wall and consequently
affecting the analysis. It is also strongly believed that shear-induced particle
migration phenomenon is also responsible for the incompatibility of Mooney
method to analyse the data. When the microstructure changes with radius or
when there is a relative motion of the different phases in the bulk material,
the assumption used in the Mooney analysis that the material is homogeneous
is invalidated, and thus no form of Mooney or Mooney-based analysis will be
valid.
209
8
CONCLUSIONS AND FUTURE WORK
This three and a half-year research project has been dedicated to
characterising wall slip in pipe rheometry of multiphase fluids i.e. ice cream,
citrus dietary fibre suspensions and magnesium silicate slurries. This chapter
concludes the outcomes of the research work and recommends future
research in this field.
8.1
Conclusions
The initial objective of this study was to design and build an
integrated rig for pressure drop, temperature and velocity profile
measurements during pipe flow. The main focus was on the flow of ice
cream. However, due to limited cost and time, it was decided to remove the
velocity profile measurement system from the design. Therefore, a pipe
rheometry rig was designed and built to enable the measurement of pressure
drop and temperature during pipe flow of multiphase fluids and the initial
setup was for ice cream. Viscous dissipation and wall slip during ice cream
flow in pipes have been successfully studied with an interesting and
important finding reported on the origin of wall slip that differ from other
multiphase fluids. Wall slip phenomenon was further investigated for pipe
flow of other multiphase fluids i.e. CDF suspensions and magnesium silicate
210
slurries. We successfully collected all the required data for analysis.
However, we did not manage to fully characterise wall slip in pipe flow of
both fluids. We further looked into the microstructure of both materials used
and have made some assumptions on wall slip analysis. The following
paragraphs shall conclude each chapter of this thesis briefly.
Chapter 3 describes the design and development of the pipe
rheometry rig used in this study. The system was carefully designed to make
this study possible. The pipe rheometry rig is complete with temperature and
pressure sensors as well as electronic mass balance. All of these were
connected to a PC data logger for monitoring, control and data collection
purposes. There are four interchangeable pipes of different diameters
fabricated to be installed on the pipe rheometry rig. This is for the purpose of
quantifying wall slip phenomenon in different pipes. The system will be used
by the student/researcher to investigate wall slip behaviour in the pipeline
flow of multiphase fluids. The equipment arrangement was made to enable
the pipe rheometry study of ice cream, citrus fibre suspensions and
magnesium silicate slurries.
Chapter 4 was dedicated solely on the study of wall slip in ice cream
pipe rheometry using both Mooney plots and Tikhonov regularisation. An
industrial scraped surface heat exchanger was installed next to the pipe
rheometry rig to enable the continuous ice cream production and data
collection process. This was complemented with measurement of ice cream
temperature at the wall which allowed for analysis of energy balances in the
near wall region. Pipe radius dependence was evident in the flow curves,
indicative of wall slip effects. This apparent slip was amenable to analysis by
211
the Mooney method and indicated the contribution of slip to flow ranged
from 70% to 100%. A significant increase in ice cream temperature next to
the wall along the length of the pipe was measured in all cases and was
attributed to viscous dissipation. Energy balances indicated that the apparent
wall slip effect was not due to the existence of a thin slip layer of matrix fluid
next to the wall. Instead, it was found that the results were better understood
as being the result of a moderately thick layer of slightly heated ice cream
next to the wall. The increased temperature and shear thinning nature of the
ice cream led to runaway shear near the wall which tended to dominate the
overall flow. Whilst these flows may be interpreted as wall slip, the origin of
the phenomenon is different from that in most suspension flows and
significantly alters interpretation of results.
Chapter 5 presented a study of wall slip in CDF suspensions pipe
rheometry. The rheological behaviour of the fluid when flowing in pipelines
was experimentally determined and CDF suspensions display nonNewtonian,
pseudoplastic
behaviour.
An
increase
in
the
powder
concentration causes a decrease in the flow behaviour index (indicating the
increase in non-Newtonian behaviour) and increase in consistency index.
Standard approaches for determining wall slip velocity using Mooney
analysis yielded non-physical results. Another approach based on Tikhonov
regularisation-Mooney method was also unsuccessful to characterise both of
the fluids. The incompatibility of the method to analyse wall slip is attributed
to the inconsistent ratio of Vslip/τw and δ/μslip at constant wall shear stress. The
same phenomenon applies to the suspensions with higher concentration. This
is believed to be attributed to the microstructure changes and shear-induced
212
re-orientation of the particles in CDF suspension during flows which caused
the inconsistency of the slip layer thickness and consequently affect the wall
slip characterisation process. An attempt was made to plot JastrzebskiMooney graphs. Even though the plots appear to be quite linear, however, a
degree of non-linearity is still significant which indicates that the method is
also unable to analyse wall slip in CDF suspensions flow correctly.
Chapter 6 presented a study of wall slip in magnesium silicate slurries
pipe rheometry. The slurries also display non-Newtonian, pseudoplastic
behaviour. Both Mooney and Tikhonov regularisation-Mooney methods were
also unable to characterise the fluid flow. Similar to the CDF suspensions
study, the incompatibility of the method to analyse wall slip is attributed to
the inconsistent ratio of Vslip/τw and δ/μslip at constant wall shear stress. The
same phenomenon applies to the slurries with higher concentration. Shearinduced re-orientation of the particles in magnesium silicate slurry during
flows caused the inconsistency of the slip layer thickness which affect the
wall slip characterisation process. An attempt made to plot JastrzebskiMooney graphs yields plausible results with the plots appear more linear than
the graphs obtained from Mooney analysis.
Chapter 7 has further discussed on wall slip analysis. Mooney
equation was presented in dimensionless form to illustrate a better picture of
wall slip analysis done in the studies especially on ice cream. The
incompatibility of the data from the pipe flow of CDF suspensions and
magnesium silicate slurries with the classic Mooney method was also
described and related to the nonhomogeneous nature of the fluids. Shear-
213
induced particle migration is also believed to be responsible for the
incompatibility of the analysis.
The pipe rheometry rig developed was demonstrated to be suitable for
fluid flow experimental work. It is capable of providing the pressure,
temperature and mass data in real-time. The data obtained are utilised to
determine the rheological properties of the fluids. The rig designed offers the
flexibility to interchange the pipes according to the preferred size. This
enables wall slip characterisation process where data obtained from different
pipe diameters are analysed using the available mathematical method. The
availability of temperature measurement using the sensors enables
temperature monitoring and the data can be further utilised to analyse the
viscous heating phenomenon during complex flow that is believed to enhance
the wall slip effect especially in ice cream. The development of the rig was
important for the completion of this research study. It enabled the author to
perform extensive experimental work dedicated to study and understand the
effect of wall slip in pipe flow of multiphase fluids.
Characterising wall slip is indeed a very challenging task. A good
arrangement and instrumentation is important to enable the data collection
process. As mentioned by Sochi (2011) on his review on the slip at fluidsolid interface, the literature of wall slip phenomenon is full of contradicting
views and results, experimental as well as theoretical, and hence many issues
will remain unresolved for a long time to come.
214
8.2
Future Work
The flow behaviour of complex materials is different from one to
another. Wall slip is a complex phenomenon and its quantification is
challenging. The complexity in the characterisation of wall slip in the flow of
multiphase fluids can be reduced by having an improved system to
exclusively study the phenomenon. As explained in previous chapters, it is
recommended that a physical model should be developed and computational
simulation on multiphase flow should be performed to explain the
counterbalance reaction between shear-induced particle migration, particle
concentration and viscosity gradients in the flow field. This physical model
coupled with direct observation on the fluid system enables full
characterisation of the rheological behaviour and interpretation of the
complex behaviour in detail. Electrical tomography or laser Doppler
velocimetry can be designed to purposely fit the test section in the pipe
rheometry system developed to make this further study possible. Classic
Mooney technique can also be adapted to calculate slip velocity and
observation can be done if there is a correlation between the compatibility of
the technique on the data of particular complex fluids. The approach
proposed is intended to provide a new insight in wall slip characterisation
process and improve the understanding of this complex phenomenon in
depth.
Roughened wall surfaces could be used in an attempt to eliminate
wall slip. Other than that, slip can also be prevented by attaching a rough
material such as sandpaper or serrated surfaces. A simple experiment was
conducted to see the effect of wall surface condition on wall slip
215
phenomenon of CDF suspension. The experiment was conducted using a
cylinder-type viscometer with a rotational inner cylinder and a stationary
outer cup. 2% CDF suspension was filled in the annular space between the
stationary cup and the cylindrical rotor driven through a torque measuring
head by a motor. The torque generated on the moving cylinder at 25 oC
provides the shear stress at the outer wall of the rotor. The experiment was
conducted in two conditions of the surface of the rotor: i) smooth wall
surface and ii) rough wall surface by attaching a sandpaper of 22 μm
roughness. In the steady torsional flow experiment, a line was drawn to cover
the free surface of the suspension and the edges of the container and rotor to
provide guidance on the wall slip behaviour of the CDF suspension as shown
in Fig. 7.1(a). For case (i) i.e. for smooth wall surface, there are
discontinuities in the marker line at both surfaces of the container and rotor
as shown in Fig. 7.1(b) indicating that there is significant wall slip during
simple shear flow of the CDF suspension.
For case (ii) where sandpaper was used to make the surface rough, the
marker line at the surface of suspension was continuous and connected to the
marker line at the edges of the moving surface as shown in Fig. 7.1(c) which
indicates no-slip condition. Hence, it is proven that wall slip can be
eliminated by attaching a rough material to the wall or by roughening the
surface of the wall where high shear is imposed.
However, it would have been a great challenge to apply this technique
to alter the surface of the inner wall of our pipes or even to fabricate a new
set of pipes during the course of the study. The technique may involve an
increased degree of experimental complexity and has its own difficulties.
216
(a)
(b)
(c)
Fig. 8.1: Experiment conducted to observe the effect of the condition of wall
surface on slip: (a) a line was drawn before the experiment started; (b)
discontinuities of the line was observed when using normal smooth wall
surface and (c) the marker line on the fluid was continuous to the marker line
on the rotor surface when a sandpaper was attached to the surface of the rotor
Direct observation method using imaging technique could be a better
and sound application to further exploring wall slip in depth. The following
subsections propose the incorporation of the technique used in this study with
direct observation method.
8.2.1
Application of combined ultrasonic pulsed Doppler velocimetry and
pressure drop (UPDV-PD)
Due to complex microstructure of multiphase fluids especially ice
cream, it is rather difficult to monitor the rheological behaviour especially
when flowing inside pipes. The combination of ultrasonic pulsed Doppler
velocimetry with flow curves backed out from pressure gradient
measurements is important to establish a robust analysis of this flow. Fig. 7.2
illustrates the proposed UPDV-PD rheometry system for ice cream study
(based on Fig. 3.6).
217
Water
Compressed
air
Test section
Multiplexer
unit
I-5
9
I-3
UVP
adaptor
I-6
E-4
PC
Water return
V-2
I-1
8
I-4
I-2
7
V-3
6
V-1
4
3
5
2
Flexible
hose
Oscilloscope
1
E-2
Flexible
hose
E-1
E-3
Fig. 8.2: Proposed UPDV-PD rheometry for ice cream study
Several flow adapter cells for housing the ultrasound transducers need
to be installed in the flow loop. In order to do this, the pipes available need to
be cut at two points to enable the installation of the UVP adaptor. Ultrasound
transducers are used together with a UVP instrument with a multiplexer unit.
The multiplexer instrument allows direct access to the demodulated echo
amplitude (DMEA). A four channel digital oscilloscope is used as an integral
part of the data acquisition scheme, enabling simultaneous measurements of
the flow velocity profiles and monitoring of the acoustic properties, sound
velocity and attenuation, directly in-line. The multiplexer instrument and the
other hardware devices were connected to a master PC. Flow curves will be
backed out of the directly measured velocity profile from UPDV, enabling
clear interpretation of apparent wall slip effects and a robust material
characterisation. UPDV is capable to generate velocity profiles of the flow
inside pipe and can be used to directly determine the slip layer thickness as
well as slip velocity
218
8.2.2
Application of electrical resistance tomography
Electrical resistance tomography (ERT) is used to determine the
distribution of electrical conductivity from the measurement of voltage
around the periphery of pipelines. It is applied to image the distribution of
materials in some region of interests by obtaining a set of measurements
using sensors that are distributed around the periphery (York, 2001). Until
recently, there is no research has been done on the application of electrical
resistance tomography to monitor the flow of ice cream in industrial pipes.
ERT system has been used to measure the multiphase flow. However, it is
only limited to gas – liquid system and is not available for gas-liquid-solid
system. The most relevant publication relates to the measurement of phase
hold-ups for gas-liquid-solid three phase system (Jin et al., 2010). Hence, it is
useful to combine ERT with pressure drop measurement which is an
established traditional method to measure the rheological behaviour of gasliquid-solid three phases system in ice cream. ERT is used to generate
velocity profiles of the flow inside pipe and can be used to directly determine
the slip layer thickness as well as slip velocity. Apart from that, ERT can be
utilised to monitor the shear-induced migration behaviour of other
suspensions/slurries to further understand the wall slip behaviour. The
incorporation of pressure drop technique together with the ERT system is
expected to be a valuable tool for the development of robust measurement of
ice cream rheological behaviour in pipes as well as other multiphase systems.
219
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APPENDIX A
Derivation of Mooney Equation
For fully developed, laminar flow in a horizontal pipe,
i) A force balance on a differential volume element of radius, r and length, ∂x in the
flow direction gives:
τ
R
r
P + ∂P
τ
x
Vmean
Q
P
τ
P + ∂P
∂x
The volume element involves only pressure and viscous forces. Thus, pressure and
shear forces balance each other:
Pr 2  P  P r 2   2rx
 Pr 2   2rx

P  2r

x
r 2
P
2

x
r
Hence, we obtain the shear stress in the radial element:
r  P 
 --------------------------- Eq. (i)
2  x 
  
Shear stress at the wall becomes:
w 
R  dP 

 ------------------------- Eq. (ii)
2  dx 
238
ii) Fluid might slip at the wall:
Total flowrate, Q  Qslip  Qshear

Qslip  VslipR 2

For Qshear,
Vshear
r
Flow through the annulus
dr
Area = 2πr dr
r R
Qshear 
V
shear
2πr
dr
2rdr
r 0
Hence,
r R
Q  VslipR   Vshear 2rdr
2
r 0
r R
Q  VslipR 2  2  Vshearrdr
r 0
 r2 
 
d
r R
2
2
Q  VslipR  2  Vshear   dr
dr
r 0
r R
r R 2

r2 
r dVshear 

Q  VslipR  2 Vshear   
dr


2  r 0 r 0 2 dr


2
At
r = R, Vshear = 0
r = 0, Vshear = 0
r R

r2 
Hence, the term Vshear 
2  r 0

is zero at r = R and r = 0. The equation now
becomes:
239
Q  VslipR 2  2
r R
r 2 dVshear
 2 dr dr
r 0
r R
r
Q  VslipR  
2
dVshear
dr
dr
2
r 0
Q  VslipR 2  
r R
r
2
dr
2
dr ------------------------ Eq. (iii)
r 0
Q  VslipR  
r R
r
2
r 0
-   f  
   f
At
1
  -------------------- Eq. (iv)
r = R, τ = τw
r = 0, τ = 0
By inserting Eqs. (i), (ii) and (iv) into Eq. (iii), the equation becomes:
2
  w
 R 
Q  VslipR      f
 0   w 
2
R 3
 Q  VslipR  3
w
2
  w


2
1
  R d
w
f 1  d ------------------ Eq. (v)
0
Eq. (v) is total fluid flow rate. To convert this into apparent shear rate,  i.e. 4Q/πR3
term, multiply Eq. (v) with 4/πR3. The final equation becomes:
4Q
4
4
 Vslip  3
3
R
R
w
  w


2
f
1
 d
0
240
APPENDIX B
Derivation of axial temperature gradient in the slip layer
The average wall temperature increase for adiabatic pipe flow is calculated as
(Winter, 1987):
Tw 
P
------------------------------ Eq. (i)
c p
Based on Eq. (2.25) in Chapter 2,
P 
 w 2L
R
------------------------------ Eq. (ii)
By inserting (ii) into (i), the average wall temperature increase becomes:
Tw 
 w 2L
Rc p
R is the pipe radius and L is the length of the pipe. For the slip layer near the wall,
thickness of the layer is δ, and a change in the length of the volume element in the
flow direction is termed Δx. By substituting the terms, the equation now becomes:
Tw 
 w 2x
 c p
Tw
2 w

x  c p
By changing into partial derivative form, the final equation to estimate the axial
temperature gradient in the slip layer near the wall becomes:
dTw
2 w

dx  c p
241