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Review
Rheology of pulp fibre suspensions: A critical review
B. Derakhshandeh, R.J. Kerekes, S.G. Hatzikiriakos n, C.P.J. Bennington 1
Department of Chemical and Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3
a r t i c l e i n f o
Article history:
Received 8 February 2011
Received in revised form
1 April 2011
Accepted 16 April 2011
Available online 22 April 2011
Keywords:
Pulp fibre suspension
Biomass
Rheology
Complex fluids
Viscoelasticity
Multiphase flow
a b s t r a c t
This paper reviews past studies on the measurement of rheological properties of pulp fibre suspensions.
Such suspensions are complex fluids important in the manufacture of many pulp-fibre based products,
such as communication papers, hygiene products, packaging, as well as other fibre-based materials.
Pulp suspensions play a role in other biomass conversion processes as well. This review focuses on key
properties of fibre suspensions, such as regimes of behaviour based on inter-fibre contact, apparent
yield stress, apparent viscosity, and viscoelasticity. Difficulties encountered in measurement of these
properties due to flow regime changes, heterogeneous mass distribution, and formation of depletion
layers at solid boundaries is discussed and methods to overcome them are reviewed.
& 2011 Elsevier Ltd. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.1. Fibre properties and consistency ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.2. Fibre contacts and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.3. Forces on fibres and flocculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.4. Rheology of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3. Apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.2. Apparent yield stress of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.3. Modified rheometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3463
3.4. Vaned-geometry devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3463
3.5. Findings of various approaches to measure apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3464
3.6. Modelling apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4. Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.1. Suspensions of synthetic fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.2. Shear viscosity of pulp suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.3. Extensional viscosity of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3466
5. Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3466
6. Fluidisation of pulp suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467
7. Applications in pipe flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467
8. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3469
Contents lists available at ScienceDirect
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Chemical Engineering Science
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2011.04.017
n Corresponding author.
E-mail address: [email protected] (S.G. Hatzikiriakos).
1 Posthumous
Chemical Engineering Science 66 (2011) 3460–3470
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tip. Local velocities were measured in the gap between R1 and Ry.
The results showed good agreement with apparent yield stress
values obtained using the linear shear stress ramp method, i.e. the
shear stress at which the instantaneous viscosity was a max-
imum, thereby verifying this simpler technique as a reliable
measurement of apparent yield stress (Derakhshandeh et al.,
2010a).
3.5. Findings of various approaches to measure apparent yield stress
All workers found the apparent yield stress to depend on a
power of the consistency. At low consistency, the power depen-
dence is for the difference between the consistency being tested
and a threshold consistency at which networks form. The premise
here is that consistencies below the threshold do not contribute
to mechanical strength. Thalen and Wahren (1964a) defined the
threshold by a sediment concentration, Cs, as shown in Eq. (4)
below. Later, Martinez et al. (2001) defined it by the gel crowding
number NG as shown in Eq. (5).
sy ¼ aðCm�CsÞb ð4Þ
or
sy ¼ aðN�NGÞb where NG ¼ 16 ð5Þ
These equations apply for low consistency pulp suspensions,
typically in the range in which paper is formed, typically Cm¼
0.5–1%. However, most processing of fibre suspensions takes
place at larger consistencies, typically are 3% or more. In this
range, the apparent yield stress equations can be simplified to:
sy ¼ aCbm ð6Þ
where Cm is consistency in % and a has units Pa.
Values for a and b measured in earlier studies were sum-
marised by Kerekes et al. (1985). The studies examined yield
strength in various ways, including shear strength of pulp plug
surfaces in flowing suspensions, and defined these strengths in
differing ways. In addition, many other factors differed among the
studies, such as wood species and pulping method. Not surpris-
ingly, results varied considerably. Values of a were in the range
1.8oao24.5 (Pa) and b in the range 1.69obo3.02. Many key
variables contributing to these differences were not measured or
reported. For example, Dalpke and Kerekes (2005) found fibre
length to be very important, with longer fibres causing larger
apparent yield stress.
At consistencies Cm48%, fibre suspensions generally contain
substantial air. Bennington et al. (1995) measured the apparent
yield stress in ranges of consistency having up to 90% air content by
volume, obtaining the following expression for apparent yield stress
sy ¼ 7:7� 105C3:2m ð1�jgÞ
3:4A0:6 ð7Þ
0:004rCmr0:5 and 0rjg r0:9
where jg is the fractional gas content, A is the fibre axis ratio with
the pre-factor numerical constant in Pa. This equation is valid for
both mechanical and chemical pulps. At high gas contents, it was
found that apparent yield stress could be well described by fibre
volume fraction Cv (Bennington et al., 1990):
sy ¼ aCbv ð8Þ
All the measurements of apparent yield stress have consider-
able scatter, often as much as 100%. To determine an average
value, Bennington et al. (1990) defined a relative apparent yield
stress to be that measured in a single test divided by the average
apparent yield stress for all tests performed under the same
experimental conditions. Using the relative apparent yield stress,
experimental data were compared on a normalised basis and
approximated by a Gaussian distribution. The coefficient of
variation for the apparent yield stress of pulp suspensions found
to be 20% and for synthetic fibres to be 45%.
Scatter in the data is due to many factors, including how
apparent yield stress is defined and the method of measurement.
For example, apparent yield stresses obtained by quasi-static
methods were all larger than those measured using dynamic
methods. Other difference are illustrated in the methods and
definitions employed in studies of Gullichsen and Harkonen
(1981), Bennington et al. (1990), Swerin et al. (1992), Wikström
and Rasmuson (1998), Dalpke and Kerekes (2005), and
Derakhshandeh et al. (2010a). An example is given in Table 1
for apparent yield stress of a bleached softwood kraft pulp.
Fig. 6. Velocity profiles across the gap for SBK pulp suspensions at several mass
concentrations. Solid lines are the best fits to the data by the Herschel–Bulkley
model with constants listed in the insert (Derakhshandeh et al., 2010a).
Table 1
Apparent yield stress of SBK suspension obtained using different methods.
Reference Measurement method sy (Pa)
Cm¼3%
sy (Pa)
Cm¼6%
Bennington et al. (1990) Baffled concentric-cylinder, ultimate shear strength 176 1220
Swerin et al. (1992) Couette cell, oscillatory experiments 19.3 117
Damani et al. (1993) Parallel plate geometry, oscillatory experiments – 60
Wikström and Rasmuson (1998) Baffled concentric-cylinder, ultimate shear strength 131 1100
Ein-Mozaffari et al. (2005) Concentric-cylinder, ultimate shear strength 350 –
Dalpke and Kerekes (2005) Vane in large cup geometry, ultimate shear strength 130 –
Derakhshandeh et al. (2010a) Vane in large cup geometry, ultimate shear strength 248 –
Derakhshandeh et al. (2010a) Vane in large cup geometry, maximum viscosity 154 –
Derakhshandeh et al. (2010a) Velocimetry–rheometry 137 –
B. Derakhshandeh et al. / Chemical Engineering Science 66 (2011) 3460–34703464
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The values range from 19.3 to 350 Pa for a consistency of 3%, and
from 60 to 1220 Pa for a consistency of 6%.
3.6. Modelling apparent yield stress
Several workers developed mathematical models of fibre net-
works to predict apparent yield stress. Bennington et al. (1990)
derived an equation based on network theory that included fibre
aspect ratio and Young’s modulus as follows:
sy ¼ cEA2Cv3 ð9Þ
where A is the fibre aspect ratio, E is the fibre’s Young’s modulus,
c is a constant and Cv is the volume concentration of the pulp
suspension. All fibres were assumed to be rod-like with a common
area moment of inertia. In later work, Wikström and Rasmuson
(1998) considered differing area moments of inertia for pulp fibres
and modified Eq. (9) by using the fibre stiffness (the product of
elastic modulus and area moment of inertia). This modification led
to the following equation proposed for the apparent yield stress
values:
sy ¼ cEA2 1�
d4
D4
� �
Cv
3
ð10Þ
where d and D are the inner and outer diameters of the fibres.
4. Shear viscosity
4.1. Suspensions of synthetic fibres
Relatively few studies have been devoted to measuring the
viscosity of pulp suspensions, although there have been a sub-
stantial number for the simpler case of synthetic (plastic, glass)
fibres of uniform size and shape. Although this review is focused
on pulp fibres, there are sufficient similarities to synthetic fibres
to warrant a brief discussion here.
References to many of the major studies of synthetic fibres can
be found in review papers and literature surveys in papers on the
subject, for example Ganani and Powell (1985), Bennington and
Kerekes (1996), Petrie (1999), Kerekes (2006), and Eberle et al.
(2008). Some studies are particularly relevant to pulp fibre
suspensions. Nawab and Mason (1958) measured the viscosity
of dilute suspensions of thread-like rayon fibres in castor oil. They
employed a bob and cup viscometer equipped with a microscope
to check for wall slippage. Fibre aspect ratio had a strong effect on
suspension viscosity and its shear dependence. In other work,
Blakeney (1966) examined the effect of fibre concentration on the
relative viscosity of suspensions of straight, rigid nylon fibres
with aspect ratio of about 20. He found that at Cv40.0042,
viscosity increased dramatically with concentration. Interestingly,
this condition is NE1.
Ziegel (1970) measured the viscosity of suspensions of glass
rods, glass plates and asbestos fibres in high viscosity polymer
fluids and compared them with those of spherical particles. Horie
and Pinder (1979) measured the viscosity of suspension of nylon
fibres over a wide range of consistency and shear rates; they
found thixotropy and that thickness of the shearing layer in the
viscometer to depend on time of shearing.
Kitano and Kataoka (1981) employed a cone-plate rheometer
to study the steady shear flow properties of suspensions of
vinylon fibres in silicone oil up to Cm¼7%. Ganani and Powell
(1986) studied the rheological behaviour of monodisperse glass
fibres both in Newtonian and non-Newtonian suspending media
at fibre volume fractions of 0.02, 0.05, and 0.08. Milliken et al.
(1989) employed falling-ball rheometry to measure viscosity of
monodisperse randomly oriented rods in a Newtonian fluid.
Suspensions exhibited Newtonian behaviour at Cvo0.125 and
a sharp transition at Cv40.125 at which viscosity depended on
third power of concentration. This corresponds to N¼33.
Petrich et al. (2000) studied the relationship between the fibre
orientation distribution, fibre aspect ratio, and the rheology of
fibre suspensions. They measured both specific viscosity and
normal stress differences. Chaouche and Koch (2001) examined
the effect of shear stress and fibre concentration on the shear-
thinning behaviour of rigid fibre suspensions. They showed that
fibre bending and a non-Newtonian suspending liquid played
a major role in shear-thinning behaviour of suspension at high
shear rate values. Switzer and Klingenberg (2003) modelled
the viscosity of fibre suspensions. They showed viscosity to be
strongly influenced by fibre equilibrium shape, inter-fibre friction,
and fibre stiffness.
4.2. Shear viscosity of pulp suspensions
Steenberg and Johansson (1958) studied flow behaviour of
suspensions of unbleached sulphite pulp in a custom-made
parallel-plate viscometer. They measured shear stress-shear rate
relationships over a wide range of flow velocity and consistencies
up to 2.5%. They observed two transition points: a maximum at
low shear rates and a minimum at high shear rates. These
correspond to similar observations in pipe flow (discussed later).
The transition points shifted towards higher shear rates as the gap
clearance decreased. They measured viscosities at shear rates
above the second transition point where pulp was considered to
be a fully sheared medium. At these high shear rates, they found
Newtonian behaviour up to about NE100, with viscosity slightly
larger that water. This work showed that various flow regimes
may exist in rheometers.
In another early study, Guthrie (1959) measured the apparent
viscosities of pulp suspensions at Cmo2% to calculate Reynolds
numbers in a pipe. He found viscosity increased dramatically with
consistency above a critical value of Cm¼1.4%. Above this, pulp
suspension viscosity showed no significant dependence on the
fibre length over the length range of 0.2–0.67 mm. The likely
explanation for this observation is that consistency 1.4% at fibre
length 0.67 mm give about N¼20, which is near the gel crowding
number. Below this the suspension behaves as dilute and length
would not be expected to be important. An abrupt change in
suspension behaviour at this point is to be expected.
Chase et al. (1989) surveyed the variation of torque versus
rotational rate of hardwood and softwood pulp suspensions to
study the effects of fibre concentration and freeness on the
viscosity parameter. For both suspensions, viscosities increased
linearly with consistency. The viscosity of hardwoods decreased
linearly with freeness, while the viscosity of softwoods increased
initially and then decreased with a decrease in freeness. They also
concluded that pulp behaves as a Bingham plastic fluid on the
basis that pulp exhibits an apparent yield stress.
Chen et al. (2003) studied the flow behaviour of pulp suspen-
sions in a modified parallel-plate rheometer. The lower plate was
replaced with a Petri dish to prevent the suspension overflow, but
no modification was made to minimise wall slippage. Softwood
and hardwood bleached kraft pulps were mixed in different ratios
but the total mass concentration kept at 0.05%, giving a very
dilute suspension i.e. NE5. They measured shear stress as
a function of shear rate and performed stress relaxation experi-
ments. Using a CCD camera, they identified three flow regimes as
illustrated in Fig. 7. In the first regime, Newtonian flow was
observed at low shear rates. In the second regime, they observed
unstable flow, with jumps in the shear stress dependant on shear
rate. The stress jumps were attributed to the flocculation of pulp
fibre suspensions. They measured a mixture of softwood and
B. Derakhshandeh et al. / Chemical Engineering Science 66 (2011) 3460–3470 3465
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Rheology of pulp fibre suspensions A critical review
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rheology of pulp suspensions have been invaluable. Financial
assistance from the Natural Sciences and Engineering Research
Council of Canada (NSERC) grant number CRDPJ 379851 is greatly
appreciated.
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Review
Rheology of pulp fibre suspensions: A critical review
B. Derakhshandeh, R.J. Kerekes, S.G. Hatzikiriakos n, C.P.J. Bennington 1
Department of Chemical and Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3
a r t i c l e i n f o
Article history:
Received 8 February 2011
Received in revised form
1 April 2011
Accepted 16 April 2011
Available online 22 April 2011
Keywords:
Pulp fibre suspension
Biomass
Rheology
Complex fluids
Viscoelasticity
Multiphase flow
a b s t r a c t
This paper reviews past studies on the measurement of rheological properties of pulp fibre suspensions.
Such suspensions are complex fluids important in the manufacture of many pulp-fibre based products,
such as communication papers, hygiene products, packaging, as well as other fibre-based materials.
Pulp suspensions play a role in other biomass conversion processes as well. This review focuses on key
properties of fibre suspensions, such as regimes of behaviour based on inter-fibre contact, apparent
yield stress, apparent viscosity, and viscoelasticity. Difficulties encountered in measurement of these
properties due to flow regime changes, heterogeneous mass distribution, and formation of depletion
layers at solid boundaries is discussed and methods to overcome them are reviewed.
& 2011 Elsevier Ltd. All rights reserved.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.1. Fibre properties and consistency ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.2. Fibre contacts and forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.3. Forces on fibres and flocculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3461
2.4. Rheology of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3. Apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.2. Apparent yield stress of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3462
3.3. Modified rheometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3463
3.4. Vaned-geometry devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3463
3.5. Findings of various approaches to measure apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3464
3.6. Modelling apparent yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4. Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.1. Suspensions of synthetic fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.2. Shear viscosity of pulp suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3465
4.3. Extensional viscosity of fibre suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3466
5. Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3466
6. Fluidisation of pulp suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467
7. Applications in pipe flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467
8. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3469
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ces
Chemical Engineering Science
0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2011.04.017
n Corresponding author.
E-mail address: [email protected] (S.G. Hatzikiriakos).
1 Posthumous
Chemical Engineering Science 66 (2011) 3460–3470
Page 6
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tip. Local velocities were measured in the gap between R1 and Ry.
The results showed good agreement with apparent yield stress
values obtained using the linear shear stress ramp method, i.e. the
shear stress at which the instantaneous viscosity was a max-
imum, thereby verifying this simpler technique as a reliable
measurement of apparent yield stress (Derakhshandeh et al.,
2010a).
3.5. Findings of various approaches to measure apparent yield stress
All workers found the apparent yield stress to depend on a
power of the consistency. At low consistency, the power depen-
dence is for the difference between the consistency being tested
and a threshold consistency at which networks form. The premise
here is that consistencies below the threshold do not contribute
to mechanical strength. Thalen and Wahren (1964a) defined the
threshold by a sediment concentration, Cs, as shown in Eq. (4)
below. Later, Martinez et al. (2001) defined it by the gel crowding
number NG as shown in Eq. (5).
sy ¼ aðCm�CsÞb ð4Þ
or
sy ¼ aðN�NGÞb where NG ¼ 16 ð5Þ
These equations apply for low consistency pulp suspensions,
typically in the range in which paper is formed, typically Cm¼
0.5–1%. However, most processing of fibre suspensions takes
place at larger consistencies, typically are 3% or more. In this
range, the apparent yield stress equations can be simplified to:
sy ¼ aCbm ð6Þ
where Cm is consistency in % and a has units Pa.
Values for a and b measured in earlier studies were sum-
marised by Kerekes et al. (1985). The studies examined yield
strength in various ways, including shear strength of pulp plug
surfaces in flowing suspensions, and defined these strengths in
differing ways. In addition, many other factors differed among the
studies, such as wood species and pulping method. Not surpris-
ingly, results varied considerably. Values of a were in the range
1.8oao24.5 (Pa) and b in the range 1.69obo3.02. Many key
variables contributing to these differences were not measured or
reported. For example, Dalpke and Kerekes (2005) found fibre
length to be very important, with longer fibres causing larger
apparent yield stress.
At consistencies Cm48%, fibre suspensions generally contain
substantial air. Bennington et al. (1995) measured the apparent
yield stress in ranges of consistency having up to 90% air content by
volume, obtaining the following expression for apparent yield stress
sy ¼ 7:7� 105C3:2m ð1�jgÞ
3:4A0:6 ð7Þ
0:004rCmr0:5 and 0rjg r0:9
where jg is the fractional gas content, A is the fibre axis ratio with
the pre-factor numerical constant in Pa. This equation is valid for
both mechanical and chemical pulps. At high gas contents, it was
found that apparent yield stress could be well described by fibre
volume fraction Cv (Bennington et al., 1990):
sy ¼ aCbv ð8Þ
All the measurements of apparent yield stress have consider-
able scatter, often as much as 100%. To determine an average
value, Bennington et al. (1990) defined a relative apparent yield
stress to be that measured in a single test divided by the average
apparent yield stress for all tests performed under the same
experimental conditions. Using the relative apparent yield stress,
experimental data were compared on a normalised basis and
approximated by a Gaussian distribution. The coefficient of
variation for the apparent yield stress of pulp suspensions found
to be 20% and for synthetic fibres to be 45%.
Scatter in the data is due to many factors, including how
apparent yield stress is defined and the method of measurement.
For example, apparent yield stresses obtained by quasi-static
methods were all larger than those measured using dynamic
methods. Other difference are illustrated in the methods and
definitions employed in studies of Gullichsen and Harkonen
(1981), Bennington et al. (1990), Swerin et al. (1992), Wikström
and Rasmuson (1998), Dalpke and Kerekes (2005), and
Derakhshandeh et al. (2010a). An example is given in Table 1
for apparent yield stress of a bleached softwood kraft pulp.
Fig. 6. Velocity profiles across the gap for SBK pulp suspensions at several mass
concentrations. Solid lines are the best fits to the data by the Herschel–Bulkley
model with constants listed in the insert (Derakhshandeh et al., 2010a).
Table 1
Apparent yield stress of SBK suspension obtained using different methods.
Reference Measurement method sy (Pa)
Cm¼3%
sy (Pa)
Cm¼6%
Bennington et al. (1990) Baffled concentric-cylinder, ultimate shear strength 176 1220
Swerin et al. (1992) Couette cell, oscillatory experiments 19.3 117
Damani et al. (1993) Parallel plate geometry, oscillatory experiments – 60
Wikström and Rasmuson (1998) Baffled concentric-cylinder, ultimate shear strength 131 1100
Ein-Mozaffari et al. (2005) Concentric-cylinder, ultimate shear strength 350 –
Dalpke and Kerekes (2005) Vane in large cup geometry, ultimate shear strength 130 –
Derakhshandeh et al. (2010a) Vane in large cup geometry, ultimate shear strength 248 –
Derakhshandeh et al. (2010a) Vane in large cup geometry, maximum viscosity 154 –
Derakhshandeh et al. (2010a) Velocimetry–rheometry 137 –
B. Derakhshandeh et al. / Chemical Engineering Science 66 (2011) 3460–34703464
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The values range from 19.3 to 350 Pa for a consistency of 3%, and
from 60 to 1220 Pa for a consistency of 6%.
3.6. Modelling apparent yield stress
Several workers developed mathematical models of fibre net-
works to predict apparent yield stress. Bennington et al. (1990)
derived an equation based on network theory that included fibre
aspect ratio and Young’s modulus as follows:
sy ¼ cEA2Cv3 ð9Þ
where A is the fibre aspect ratio, E is the fibre’s Young’s modulus,
c is a constant and Cv is the volume concentration of the pulp
suspension. All fibres were assumed to be rod-like with a common
area moment of inertia. In later work, Wikström and Rasmuson
(1998) considered differing area moments of inertia for pulp fibres
and modified Eq. (9) by using the fibre stiffness (the product of
elastic modulus and area moment of inertia). This modification led
to the following equation proposed for the apparent yield stress
values:
sy ¼ cEA2 1�
d4
D4
� �
Cv
3
ð10Þ
where d and D are the inner and outer diameters of the fibres.
4. Shear viscosity
4.1. Suspensions of synthetic fibres
Relatively few studies have been devoted to measuring the
viscosity of pulp suspensions, although there have been a sub-
stantial number for the simpler case of synthetic (plastic, glass)
fibres of uniform size and shape. Although this review is focused
on pulp fibres, there are sufficient similarities to synthetic fibres
to warrant a brief discussion here.
References to many of the major studies of synthetic fibres can
be found in review papers and literature surveys in papers on the
subject, for example Ganani and Powell (1985), Bennington and
Kerekes (1996), Petrie (1999), Kerekes (2006), and Eberle et al.
(2008). Some studies are particularly relevant to pulp fibre
suspensions. Nawab and Mason (1958) measured the viscosity
of dilute suspensions of thread-like rayon fibres in castor oil. They
employed a bob and cup viscometer equipped with a microscope
to check for wall slippage. Fibre aspect ratio had a strong effect on
suspension viscosity and its shear dependence. In other work,
Blakeney (1966) examined the effect of fibre concentration on the
relative viscosity of suspensions of straight, rigid nylon fibres
with aspect ratio of about 20. He found that at Cv40.0042,
viscosity increased dramatically with concentration. Interestingly,
this condition is NE1.
Ziegel (1970) measured the viscosity of suspensions of glass
rods, glass plates and asbestos fibres in high viscosity polymer
fluids and compared them with those of spherical particles. Horie
and Pinder (1979) measured the viscosity of suspension of nylon
fibres over a wide range of consistency and shear rates; they
found thixotropy and that thickness of the shearing layer in the
viscometer to depend on time of shearing.
Kitano and Kataoka (1981) employed a cone-plate rheometer
to study the steady shear flow properties of suspensions of
vinylon fibres in silicone oil up to Cm¼7%. Ganani and Powell
(1986) studied the rheological behaviour of monodisperse glass
fibres both in Newtonian and non-Newtonian suspending media
at fibre volume fractions of 0.02, 0.05, and 0.08. Milliken et al.
(1989) employed falling-ball rheometry to measure viscosity of
monodisperse randomly oriented rods in a Newtonian fluid.
Suspensions exhibited Newtonian behaviour at Cvo0.125 and
a sharp transition at Cv40.125 at which viscosity depended on
third power of concentration. This corresponds to N¼33.
Petrich et al. (2000) studied the relationship between the fibre
orientation distribution, fibre aspect ratio, and the rheology of
fibre suspensions. They measured both specific viscosity and
normal stress differences. Chaouche and Koch (2001) examined
the effect of shear stress and fibre concentration on the shear-
thinning behaviour of rigid fibre suspensions. They showed that
fibre bending and a non-Newtonian suspending liquid played
a major role in shear-thinning behaviour of suspension at high
shear rate values. Switzer and Klingenberg (2003) modelled
the viscosity of fibre suspensions. They showed viscosity to be
strongly influenced by fibre equilibrium shape, inter-fibre friction,
and fibre stiffness.
4.2. Shear viscosity of pulp suspensions
Steenberg and Johansson (1958) studied flow behaviour of
suspensions of unbleached sulphite pulp in a custom-made
parallel-plate viscometer. They measured shear stress-shear rate
relationships over a wide range of flow velocity and consistencies
up to 2.5%. They observed two transition points: a maximum at
low shear rates and a minimum at high shear rates. These
correspond to similar observations in pipe flow (discussed later).
The transition points shifted towards higher shear rates as the gap
clearance decreased. They measured viscosities at shear rates
above the second transition point where pulp was considered to
be a fully sheared medium. At these high shear rates, they found
Newtonian behaviour up to about NE100, with viscosity slightly
larger that water. This work showed that various flow regimes
may exist in rheometers.
In another early study, Guthrie (1959) measured the apparent
viscosities of pulp suspensions at Cmo2% to calculate Reynolds
numbers in a pipe. He found viscosity increased dramatically with
consistency above a critical value of Cm¼1.4%. Above this, pulp
suspension viscosity showed no significant dependence on the
fibre length over the length range of 0.2–0.67 mm. The likely
explanation for this observation is that consistency 1.4% at fibre
length 0.67 mm give about N¼20, which is near the gel crowding
number. Below this the suspension behaves as dilute and length
would not be expected to be important. An abrupt change in
suspension behaviour at this point is to be expected.
Chase et al. (1989) surveyed the variation of torque versus
rotational rate of hardwood and softwood pulp suspensions to
study the effects of fibre concentration and freeness on the
viscosity parameter. For both suspensions, viscosities increased
linearly with consistency. The viscosity of hardwoods decreased
linearly with freeness, while the viscosity of softwoods increased
initially and then decreased with a decrease in freeness. They also
concluded that pulp behaves as a Bingham plastic fluid on the
basis that pulp exhibits an apparent yield stress.
Chen et al. (2003) studied the flow behaviour of pulp suspen-
sions in a modified parallel-plate rheometer. The lower plate was
replaced with a Petri dish to prevent the suspension overflow, but
no modification was made to minimise wall slippage. Softwood
and hardwood bleached kraft pulps were mixed in different ratios
but the total mass concentration kept at 0.05%, giving a very
dilute suspension i.e. NE5. They measured shear stress as
a function of shear rate and performed stress relaxation experi-
ments. Using a CCD camera, they identified three flow regimes as
illustrated in Fig. 7. In the first regime, Newtonian flow was
observed at low shear rates. In the second regime, they observed
unstable flow, with jumps in the shear stress dependant on shear
rate. The stress jumps were attributed to the flocculation of pulp
fibre suspensions. They measured a mixture of softwood and
B. Derakhshandeh et al. / Chemical Engineering Science 66 (2011) 3460–3470 3465
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Rheology of pulp fibre suspensions A critical review
Author's personal copy
rheology of pulp suspensions have been invaluable. Financial
assistance from the Natural Sciences and Engineering Research
Council of Canada (NSERC) grant number CRDPJ 379851 is greatly
appreciated.
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