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InvitedKeynotePaperonFundamentalsofrheologicalclassificationandmeasurementofhighsolidsconcentrationslurriesandpastes.

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NigelHeywood

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Fundamentals of rheological classification and measurement

of high solids concentration slurries and pastes

N I Heywood and N J Alderman Aspen Technology, Harwell, UK

Proc Hydrotransport 16 Conference, Santiago, Chile, pp 675 to 699, 26 to 28 April 2004.

SYNOPSIS

Recently interest in the design of handling and transfer systems involving highly concentrated,

fine particle non-Newtonian slurries and pastes has increased substantially. This is to some

extent the result of increased interest in the disposal of high concentration tailings in the

mining industry, either on land or through mine backfilling, and partly facilitated through

recent improvements in thickener technology. Coupled with the application of suitable types

and concentrations of flocculants, these thickener improvements allow much higher

concentrations of thickened tailings to be created. This results in many benefits such a

reduction in the required land area for tailings disposal because smaller volumes of tailings are

involved as the solids concentrations increase, and because storage slopes can be increased as

paste viscosity and yield stress levels are increased.

However, high concentration pastes can also lead to problems in the rheological

characterisation and control of their rheological properties. Relatively small changes in solids

content, particle size distribution, pH and other slurry parameters can have very large effects

on rheological properties, so it is even more important than previously to control all these

slurry parameters over a narrower degree of variability than for lower concentration slurries, if

relatively constant rheological properties of the high solids content material are to be

maintained. Sampling and preconditioning for rheological measurements also become more

critical, and time-dependent and as well as time-independent rheological behaviour may be

exhibited. This time-dependency could be fully reversible, partially reversible or irreversible

thixotropy, or, more unusually antithixotropy.

This paper describes the main categories of rheological behaviour that high concentration

slurries can exhibit. Some of the behaviour conforms to conventional rheological property

while other behaviour can cause significant problems when trying to characterise the slurries

and apply the information in process engineering design. The paper also describes approaches

to make both relevant and accurate rheological measurements by making measurements only

over relevant shear rate and/or shear stress ranges, and identifying and correcting for four of

the most important errors in viscometry.

NOTATION

Cv solids volume concentration -

d sphere diameter m

dr diameter of rotor in progressive cavity pump m

D diameter of tube m

Da agitator diameter m

e rotor eccentricity in progressive cavity pump m

ks proportionality constant between torque-averaged shear rate and agitator speed -

K consistency coefficient in power law or Herschel-Bulkley models N sn m

-2

L tube or bob length m

m exponent in Eqn (3) -

n power law exponent in power law or Herschel-Bulkley models -

n local power law exponent -

N rotational speed rps

Np rotor speed in progressive cavity pump rps

p pitch of rotor in progressive cavity pump m

Rb bob radius m

Rc cup radius m

S constant defined by either Eqn (2) or (3)

Ta mixing tank diameter m

To error-free torque N m

Tac critical value of Taylor number -

V mean velocity in pipe m s-1

VS wall-slip velocity m s-1

Vt terminal settling velocity of a sphere under gravity m s-1

ratio of bob diameter to cup diameter -

γ shear rate s-1

avγ average shear rate s-1

γmax

maximum shear rate s-1

γp

average shear rate in a progressive cavity pump s-1

γw

wall shear rate s-1

difference between cup and bob diameters m

ηB plastic viscosity Pa s

ηC Casson viscosity Pa s

ηN Newtonian viscosity Pa s

ηr relative viscosity Pa s

density kg/m3

shear stress Pa

e equilibrium shear stress Pa

w wall shear stress Pa

τy yield stress Pa

τyB Bingham yield stress Pa

τyC Casson yield stress Pa

τyHB Herschel-Bulkley yield stress Pa

cylinder or agitator angular velocity rad.s-1

c critical cylinder angular velocity at which primary laminar flow breaks down rad.s-1

1 INTRODUCTION

The purpose of this paper is to summarise the types of rheological properties that “high solids

content” slurries or pastes can exhibit. The paper also outlines the types of rheological

measurements that can be undertaken with both the identification, and, where possible,

correction of associated viscometric errors, using either tube or rotational viscometers. The

aim is to characterise the slurry rheological properties for the purposes of system design.

High concentration slurries occur in the process industries in a wide range of physical forms

depending largely on the moisture level in the solids and on the particle size and particle

distribution. In most solids handling plants, it is usual to find intermediate and final products

in at least one of these physical states. Various terms are applied to wet solids depending on

their appearance, their handling properties and consequently the type of handling equipment

used either to store or convey the material. Starting with dry bulk solids, a progressively

increasing mixture level would cause the material to move through various classifications

such as a damp powder, cake, paste, high concentration slurry, low concentration slurry, and

finally, dirty liquid. This paper focuses on high concentration slurries and pastes only.

The possibility of some viscometric errors, such as the “wall-slip” effect which can occur in

both tube and rotational viscometry, increases in high solids content pastes. It is important to

be aware of the existence of these errors, and either to minimise their significance or to carry

out additional rheological measurements to characterize and correct for these errors. Only in

this way can serious errors in system design be avoided. Sometimes conventional viscometry

is not applicable because anomalous rheological behaviour is exhibited (1). This can

sometimes be explained through microscopic scale considerations, but cannot readily be

quantified using conventional rheological data reduction techniques. In these situations, full-

scale flow processes need to be simulated or “mimicked” on the small-scale, and scale-up

rules developed to predict behaviour at full-scale.

2 CLASSES OF RHEOLOGICAL BEHAVIOUR

Fluids, in general, may be classified by their phase condition as homogeneous,

pseudohomogeneous (that is, able to be treated as homogeneous) or heterogeneous. A number

of factors affect the phase condition, one of which is sedimentation under gravity. Slurries

may be either settling or non-settling. For instance, coarse-particle, granular slurries tend to

settle whereas stabilised or concentrated, flocculated fine-particle slurries have a low settling

tendency. However, the distinction between non-settling and settling slurries is not clear as

judgement has to be made with regard to the time frame over which settling is deemed to be

important. This can be done by considering the specific engineering application. For example,

for pipeline design the slurry settling rate needs to be related to the residence time of the slurry

in horizontal, inclined or vertical pipework. Three different procedures allow the distinction

between non-settling and settling slurries to be made with respect to the application (2).

If solid particles in a suspending liquid medium settle sufficiently slowly under gravity, the

slurry can be characterised rheologically. These ‘non-settling’ slurries are considered to be

pseudohomogeneous (that is, the particles are uniformly distributed throughout the liquid).

The usual viscometric methods for measuring the flow curve of homogeneous fluids can be

used for these slurries, with appropriate modifications to account for specific slurry attributes

(e.g., fibrous particles, large maximum particle size, very high viscosity, etc).

If, however, the solid particles in a suspending medium undergo rapid settling, the slurry can

no longer be treated as pseudohomogeneous (that is, the slurry is heterogeneous). Here, flow

curve measurements using rotational or tube viscometry are difficult and often meaningless.

Sometimes, it is possible to use the Metzner-Otto method (3) to measure the flow curve. This

method maintains the particles in suspension whilst carrying out the flow curve measurement.

Pseudohomogeneous slurries may be further classified into various rheological types on the

basis of their rheological behaviour when subjected to shear. Slurries and pastes can exhibit

either time-independent or time-dependent flow properties. In either case, the slurry is

characterised by a viscosity (i.e. the ratio of shear stress to shear rate) which varies with

changes in shear rate or shear stress. For a time-independent slurry, the viscosity is constant

with time at a given shear rate or shear stress level. The slurry can be Newtonian (constant

viscosity with varying shear rate), shear-thinning (reduction in viscosity with increasing shear

rate), viscoplastic (reduction in viscosity with increasing shear rate, once a yield stress has

been exceeded), or shear-thickening / dilatant (increase in viscosity with increasing shear

rate).

Time-dependent property occurs when slurry viscosity varies with time under constant shear

stress or shear rate conditions. This behaviour is sometimes also referred to in the literature as

“shear-sensitive”, but the use of this term is to be discouraged owing to its imprecision. Slurry

viscosity has also been observed to either increase or decrease with time as a result of changes

in one or more of the slurry physical properties which combine to determine the slurry

rheological properties. For instance, slurry viscosity has been observed to increase with shear

owing to particle attrition effects, and the corresponding progressive increase in the

percentage of “fines” in the slurry which exhibit Brownian motion.

The most frequently met time-dependent property is thixotropy which is characterised by

shear rate-dependent and time-dependent viscosity. For a well-rested slurry sample (that is,

one that has been allowed to rest for a time to recover from the effects of previous shear and

so rebuild its internal structure), the application of shear results in a decrease in viscosity with

the time of shear. However, for a slurry sample that has undergone significant shear, reducing

the shear stress level from a previous higher level, or removing the shear entirely, will lead to

a progressive viscosity increase with time arising from the rebuilding of structure in the slurry.

Quite frequently some people make the error of referring to a time-independent, shear-

thinning slurry as “thixotropic”. This is a confusion in terminology. In fact, most industrially-

relevant thixotropic slurries and pastes are also shear-thinning, but by no means all shear-

thinning slurries are also thixotropic.

2.1 Time-independent behaviour

It is important to know whether the flow properties are time-dependent. The simplest case is

where the flow curve does not depend on the time period over which the slurry is sheared. A

constant shear rate (or conversely shear stress) results in a constant shear stress (or conversely

shear rate) being observed instantaneously. This flow curve is said to be time-independent.

For many slurries, the variation of viscosity with shear rate as a function of solids content is

complex when measured over a wide shear rate range (Figure 1). At low solids concentrations,

the viscosity curve indicates shear-thinning behaviour (viscosity decreases with shear rate)

with well-defined zero-shear and infinite shear viscosities. At higher solids concentrations,

shear-thickening behaviour

Figure 1 Typical slurry viscosity curves (Cheng, 1980)

(viscosity increases with shear rate) is observed at high shear rates. This becomes more

pronounced with increasing solids concentration. The characteristic shear rates for either the

onset of shear-thinning or shear-thickening behaviour decrease with increasing concentration.

The slope of the viscosity curve at low shear rates within the shear-thinning region also

increases, until at very high solids concentrations it attains a value of -1. If replotted as a flow

curve, this slope of -1 shows a yield stress and the flow behaviour would be viscoplastic (4).

A consequence of this behaviour is that, strictly speaking, there are no such materials as

“Newtonian” slurries and “non-Newtonian” slurries. Instead there may be slurries which

exhibit Newtonian behaviour over a wide (and perhaps the only relevant) shear rate range, and

there are slurries which may exhibit non-Newtonian behaviour over the relevant shear rate

range, but which may also exhibit Newtonian behaviour outside this shear rate window.

Furthermore, there are slurries which may exhibit both Newtonian and non-Newtonian

property within the relevant shear rate window.

It follows from Figure 1 that a single point viscosity measurement is often inadequate to

describe the flow behaviour of slurries. Here, for any end-use of the viscosity data, the

relevant shear rate (or shear stress) range must be assessed and used in determining the

measurement conditions for the viscometer. A common error is to allow the shear rate (or

shear stress) range of the one and only available viscometer to dictate the test conditions for

measurement rather than to define the relevant range upfront. Another common error is to use

the full shear rate or shear stress range of an instrument when a much narrower window is

often all that is required. Making more rheological measurements than is necessary is a waste

of time, but more than one viscometer may be needed to cover the required shear rate or shear

stress range. Once the shear rate or shear stress window is estimated, a check on the

viscometer suitability to measure viscosity levels over the specified shear rate or shear stress

range must be made. Sometimes the shear stress or shear rate and the viscosity may be too low

or too high for the viscometer, so one or more alternative viscometers must be used.

2.2 Time-dependent behaviour

If a time-dependent slurry is held under constant shear for a sufficiently long time (typically

minutes but sometimes hours for thixotropic materials), the viscosity will attain a steady-state

value. Plotting this viscosity as a function of shear rate in the form of a shear stress versus

shear rate plot will provide the equilibrium flow curve for thixotropic materials. This means

that under steady-state operation in a process such as pumping slurry through a pipeline, it

may be possible to relate the frictional pressure loss for the flow of thixotropic slurries to the

equilibrium flow curve as if they were time-independent.

It is important to know whether the flow properties of the slurry are time-independent or time-

dependent in a number of applications. In steady pipe flow, a time-independent slurry will

have a constant pressure gradient along the pipeline whereas a time-dependent slurry will

develop a variable pressure gradient along the pipeline until the slurry structural state has

reached an equilibrium level. If pipe flow is interrupted, the structure of a time-dependent

slurry will develop causing the viscosity to increase with time. On start-up, the pump must be

capable of moving a pipeline of slurry at the higher viscosity associated with the static slurry.

The opposite of thixotropic property, i.e., the situation where the viscosity of a previously

unsheared slurry increases progressively with time under constant shear conditions, is very

rare in industrial environments provided there is no change to the fundamental slurry physical

properties, such as total solids content, particle size distribution, pH, temperature, etc. This

rare behaviour has been referred to as “rheopexy”. However, Cheng (5) pointed out that the

term “rheopexy” should really be used as a subset of the thixotropy definition, and the

alternative terms negative thixotropy, or antithixotropy, should be employed when describing

the opposite behaviour to thixotropy. The full complexity of time-dependent behaviour is

described using several terms: rheopexy, negative thixotropy (antithixotropy), and irreversible

thixotropy (rheomalaxis or rheodestruction), and partial thixotropy (see Table 1).

Table 1 Comparison of different classes of time-dependent slurry flow behaviour

At rest when

freshly

prepared

Under shear At rest after

shear

Rate of structural build

up at rest after shear

Thixotropy Gel Breaks down Fully recovers +ve, the rate of build up at

rest is faster than at higher

shear rates

Rheopexy Gel Breaks down

under moderate

and high shear

rates

Fully recovers +ve, the rate of build up at

rest is slower than at

moderately low shear rates

Negative

Thixotropy

(Antithixotropy)

Fluid Builds up Breaks down -ve

Irreversible

Thixotropy

(Rheomalaxis,

Rheodestruction)

Gel Breaks down Does not recover

at all

zero

Partial Thixotropy Gel Breaks down Partially recovers +ve

NB Rheopexy as stated in this table is the original definition. However, since the 1950’s rheopexy has become

synonymous with the negative thixotropy definition (see BS 5168:1975 - Glossary of rheological terms)

3 FLOW CURVE MEASUREMENT

3.1 Good sampling practice

When high concentration slurries and pastes are sampled for laboratory analysis, good

sampling practice is crucial to ensure that the samples are indeed representative of the bulk

properties of the slurry or paste. Ideally, fine particle slurry samples should not be taken in

flowing situations, and certainly not close to shearing wall surfaces, where various

hydrodynamic effects can lead to variations in both solids content and particle size

distribution. Normally several samples should be taken from a well-mixed vessel. Moreover,

the use of syringes or pipettes for taking slurry samples is strongly discouraged since this will

affect the shear history of the sample prior to the flow curve measurement. In addition, their

use will increase the variability of solids concentration and particle size distribution between

samples. Placement of the slurry sample in the viscometer should always be done carefully

with a spatula or pouring from an agitated bottle.

As the slurry solids content increases above about 40% by volume, both solids concentration

and particle size distribution (psd) can have a progressively bigger effect on slurry viscosity

and flow curve shape, and therefore classification. Figure 2 shows how rapidly slurry viscosity

increases with solids content. Relatively small variations in the solids content of high

concentration slurries can lead to disproportionately large variations in slurry viscosity. The

plot shows the typical relationship between the slurry relative viscosity (ratio of slurry

viscosity to viscosity of suspending liquid) and the solids volume concentration. By way of

example, at a solids concentration of 60% by volume, the relative viscosity, according to the

Eilers equation (6), will be about 24, but varies from 17 to 34 if the solids content were to

vary from 57% to 63%. Such sensitivity to solids content variations increases at progressively

higher solids contents, as indicated by the increasing slope of the curve in Figure 2.

1

10

100

1000

0 10 20 30 40 50 60 70

Solids concentration, CV

Re

lativ

e v

isco

sity

,

63%57%

39

24

17

r

2

v

vr

C1.351

C1.251η

Eilers Equation

1

10

100

1000

0 10 20 30 40 50 60 70

Solids concentration, CV

Re

lativ

e v

isco

sity

,

63%57%

39

24

17

r

2

v

vr

C1.351

C1.251η

Eilers Equation

Figure 2 Typical plot of slurry relative viscosity versus

solids concentration by volume

A second parameter which can affect slurry viscosity is the slurry psd. Farris (7) has shown

theoretically how psd variations can have very large effects on slurry viscosity for fully-

dispersed slurries at solids concentrations above about 40% by volume. Figure 3 gives a

simple example of a bimodal slurry (7), and how variations in the proportions of the fine and

coarse solids can lead to very large variations in slurry viscosity. Multimodal slurries exhibit

similar behaviour at high concentrations, but again, not at lower concentrations. As in the case

of the effect of overall solids concentration on slurry viscosity, the greater the solids content

of the slurry the greater the sensitivity of slurry viscosity to variations in psd. It follows that

the greater the solids content, the more care should be taken in slurry sampling, and probably

the larger the number of slurry samples required for rheological testing.

Figure 3 Predictions of relative viscosity of a slurry as function of coarse

fraction in a bimodal psd slurry (After Farris, 1968)

Other parameters which affect slurry viscosity include pH, particle shape and temperature. All

these parameters also need controlling during sampling and testwork.

3.2 Shear rate and shear stress ranges

With the range of viscometers commercially available, it is possible to measure viscosity over

12 orders of magnitude of shear rates from 10-6

to 106s

-1. However, most process applications

require data only over two to three orders of magnitude of shear rate. Table 2 lists the shear

rates typical of various slurry flow processes of interest to engineers (8-13).

When a material passes through the process plant, it invariably encounters situations where

different shear rate ranges apply. For instance, slurry can be agitated in a storage vessel and

can be subjected to a range of shear rates, depending on the design and speed of the agitator. It

will then experience different shear rate ranges in a discharge pump and pipe system.

Table 2 Shear rates typical of some slurry flow processes

Flow Process Shear rate, γ (or shear stress, )

expression Ref

Typical shear

rate range, s-1

Sedimentation of solids

in a suspending liquid

d

V3γ t

max

7n9d

Vγ t

av for power law slurry

14,15 10-6 to 10-3

Pipeline flow D

8V

4n'

3n'1γw

16 1 to 103

Mixing and stirring -

Proximity agitator aa

a

maxDT

ΩDγ

11

10 to 103

Mixing and stirring -

Non-Proximity agitator

(Metzner-Otto method)

Nkγ sav where ks ranges between 9 and 15 4,17

Pumping - Progressive

cavity pump

e

)p)d((1.5πNγ

0.522

rp

p

or

e

)p)d(4e(π0.8Nγ

0.522

r

2

p

p

8 100 to 103

3.2.1 Shear rate and shear stress estimation

Methods exist for defining the shear rate range for many processing applications (18). The

minimum shear rate often approaches zero whereas the maximum shear rate can be estimated

according to the application. Equations which can be used to estimate shear rate for several

flow situations are summarised in Table 2.

If a method for shear rate estimation in a particular processing application is not available,

define the flow region of interest, determine the differences between fluid velocities at two

points across the region (one point will often be the surface of some moving element such as a

pump impeller or mixer agitator, and the other will be a stationary equipment surface) and

divide this velocity difference by the separation distance between the two points. The full

shear rate range for measurement is estimated by considering the minimum shear rate as the

combination of the lowest difference in fluid velocities across the largest dimensional

separation, and the maximum shear rate using the highest fluid velocity difference across the

smallest dimensional separation.

Rather than ensuring the viscosity/flow curve measurement is carried out over an appropriate

shear rate window, it is sometimes more appropriate to carry out the measurement of the shear

stress appropriate to the application. Here, a controlled stress rheometer should be used in

preference to a controlled shear rate rheometer. However, the newer models of controlled

shear rate rheometers have a controlled stress option, and this capability should be used.

3.3 Recommended viscometer geometries for slurries and pastes

Flow curve measurements for engineering design should be made using either a tube or

coaxial cylinder. The use of cone-and-plate or parallel plate geometries is also feasible in

some circumstances, provided problems arising from particle-jamming (a gap separation of at

least 30 to 50 times the mean particle diameter is required to avoid any particle-jamming

effects), sample expulsion at high shear rates, and setting the gap are all appropriately dealt

with (2). Use of a rotating disc viscometer should also be avoided because of the problem of

defining the shear rate (which varies across the disc) and its somewhat restrictive shear rate

range (typically 1 to 100 s-1

), although the geometry is appropriate for quality control.

3.3.1 Tube viscometry

Tube viscometers are generally once-through batch devices consisting of either a horizontal or

vertical length of precision-bored, straight tube through which the test slurry is passed at

varying rates from a reservoir. However, recirculating pilot-scale pipeline viscometers are also

commonly used. Tube diameters typically range from 0.3 mm to 5 mm. In the controlled rate

tube viscometer, a piston or ram forces the fluid through a horizontal or vertical tube at a

constant flowrate and the resultant pressure drop is measured. In the controlled pressure tube

viscometer, compressed air (or nitrogen) is applied to drive the slurry through a horizontal or

vertical tube and the resultant volumetric flowrate is measured. Measurement of the pressure

drop and volumetric flow rate allows the shear stress and shear rate to be calculated (18, 19).

3.3.2 Coaxial cylinder viscometry

The coaxial cylinder viscometer consists of a bob (the inner cylinder) located in a cup (the

outer cylinder) with the test sample contained in the annular gap between the bob and cup.

This viscometer can be operated either in the controlled-rate or controlled-stress mode.

There are two types of controlled-rate instruments, the Couette and the Searle types. In a

Couette-type, the cup is rotated and the torque exerted on the bob by the test sample is

measured. In the Searle-type, the cup is stationary and the bob is both the rotating element and

the torque driver. In controlled-rate instruments, the bob or cup is rotated at a constant speed

that can be sequentially stepped by the operator, or by using a steadily-changing speed ramp.

The resultant torque on the bob is measured by a torsion spring. In controlled-stress

instruments, constant torques that can be sequentially changed or a torque ramp is applied to

the bob, and the resultant speeds are measured. Expressions for shear stress and shear rate on

the cylindrical surface of the inner cylinder are available (18, 20).

3.4 Flow curve measurement for slurries and pastes

Flow curve measurements should be repeated at least twice. This will allow assessment of the

variability of viscosity with solids concentration and particle size distribution. With controlled

rate instruments, the flow curve is usually generated using at least three cycles of ramping the

speed, either by a sequence of intermittent steps or a continuous ramp. This is done up to a

maximum set speed in a given time period and down again over a similar period. The

resulting torque is measured. In controlled stress instruments, the torque is ramped in a similar

manner, and the resultant speed is measured. The time over which the shear ramp up or down

is to be carried out is left to the operator to decide. However, a good starting point would be

the time that gives 30 s per step, but this will depend on the slurry.

Repeated shear cycles carried out on the test sample will show if the sample exhibits time-

dependent flow behaviour such as thixotropy. If the up and down curves for the first and

successive cycles coincide the sample is said to exhibit time-independent flow behaviour.

However, if hysteresis loops between the up and down curves are observed for each

successive cycle, then the sample is said to exhibit time-dependent flow behaviour, in which

case repeat the experiment where the speed (or torque) is held constant until the torque (or

speed) attains a steady value before the ramping the speed (or torque) to the next value. This

will yield an equilibrium flow curve where the up and down curves coincide.

The solids concentration above which anomalous rheological behaviour starts to occur (1) is

somewhat relative. For 1 m particles, the solids volume concentration is about 60%, whereas

for 100 m particles, it occurs at about 40%. When carrying out flow curve measurements on

these materials, a check for reproducibility should be made. This can be done using four test

methods:

1. testing again on the same sample;

2. testing a new sample for the same bulk sample using the same viscometric geometry;

3. testing a new sample from the same bulk sample with the same viscometric geometry but

with different dimensions;

4. testing a new sample from the same bulk sample with a different viscometric geometry.

Even after eliminating or correcting for major viscometric errors (see Section 3.5), two or

more of these tests will tend to give a flow curve band rather than a unique flow curve. The

existence of a flow curve band arises from either the lack of reproducibility in the test sample

of the inhomogeneity of the bulk sample (as in the case for test 2) or the solids moving

separately from the liquid phase and taking up different packing structures depending on the

shearing conditions (as is the case for tests 1, 3 and 4). If the flow curve band can be attributed

to the irreproducibility in composition of the small volume of the test sample, then the mean

of the flow curve data should be taken. However, if the flow curve band arises from the

rearrangement of packing structures during shear, then it is advisable to construct an upper

bound to the flow curve data for a conservative assessment.

3.5 Errors in flow curve measurement

There are four common error sources for both tube and co-axial cylinder viscometers which

should normally be properly accounted for.

3.5.1 Secondary flow (co-axial) and transitional / turbulent flow (tube)

Flow curve measurements are made for (primary) laminar flow conditions only. The data

collected must be checked to ensure they are not subject to secondary laminar flow conditions

in a co-axial cylinder viscometer, or transitional / turbulent flow conditions in a tube

viscometer. This is achieved by calculating the laminar flow limit for the slurry under test and

rejecting data subject to secondary or transitional / turbulent flow. Alternatively, an

experimental method using different sizes of the same viscometric geometry can be used in

which the flow curve data affected by secondary or transitional / turbulent flow (assuming the

end effect and wall slip effects have been corrected for) show up as deviations from the main

curve (21).

3.5.2 End-effect

This error arises because the assumption that the tubes or cylinders are infinitely long in the

derivation of shear stress and shear rate equations cannot, of course, be met in practice. End-

effect can be dealt with by minimisation, experimental determination or prediction (19).

3.5.3 Wall-slip

Wall-slip may occur when a slurry or paste is sheared. This effect gives a resultant wall shear

stress at a given wall shear rate lower than expected owing to the formation of a thin layer of

fluid (caused by the depletion of the dispersed phase at or near the shearing surface) having a

viscosity lower than the bulk of the fluid. Conversely, for a given wall shear stress, the

measured shear rate is greater than the true shear rate. In tube viscometry, for example, wall-

slip is present when curves of wall shear stress versus nominal wall shear rate obtained with

different tube diameters do not superimpose after all other corrections to the data have been

made. This effect can be quantified through a wall slip velocity, Vs. Different methods are

used to estimate Vs values to correct flow curve data for wall-slip error.

3.5.4 Viscous Heating

Viscometric tests should be designed to avoid significant temperature increases due to viscous

heat dissipation, which occurs when any fluid is sheared. However, serious experimental

errors are generally found only in high viscous fluids at high shear rates. The data collected

must be checked to ensure that they are not affected by viscous heating. This is done by

calculating the viscous heating limit given for the appropriate geometry. Alternatively, an

experimental method using different sizes of the same viscometric geometry can be used in

which the flow curve data affected by viscous heating, assuming the end effect and wall slip

effects have been corrected for, show up as deviations from the main curve (21).

Figure 4 Errors in flow curve measurement

Figure 4 summarises the way in which these four errors affect the flow curve. Higher-than-

expected viscosity values will be obtained if the torque-speed data are affected by end-effect

in coaxial cylinder and tube viscometers. Data affected by secondary or turbulent flow will

also give rise to apparently erroneously large viscosity values. Lower-than-expected viscosity

values will be obtained if the torque-speed data are affected by the wall-slip effect and viscous

heating.

3.6 Correcting for errors

3.6.1 Correcting for errors in tube viscometry

Transitional/turbulent flow - The expression for wall shear rate is valid only for laminar flow

conditions and does not apply for transitional or turbulent flow. Hence any transitional or

turbulent flow data collected during the course of an experiment must be rejected. This

requires that the laminar flow limit for the slurry under test should be calculated. The criterion

for the laminar flow limit for power law non-Newtonian fluids is given by Eqn (1)

D

V8

)2 +(n

n4

n3 + 1

404

n

8

D ρ > τ

2

2

2

w )1/()2( nn

(1)

This equation can be plotted as a double logarithmic plot of w against 8V/D to give the

laminar limit line (Figure 5); data to the left of the line (which has a slope of 2) correspond to

laminar flow, while data lying on it or to the right of the line must be rejected. The laminar

flow limit with n = 1 is the worst case, leading to a conservative approach to data elimination.

Laminar flow limit

with n = 1

Wall s

Laminar flow limit

with n = 1

Wall s

Figure 5 Plots of τW versus 8V/D for digested sewage sludge passing

through a 18.85mm tube with the laminar limit

End effect - This arises in the form of additional pressure losses at the entrance to and from

the tube. These losses result from:

(a) energy losses arising from viscous or elastic behaviour as the fluid converges at the

entrance to the tube or diverges at the end of the tube,

(b) losses in the kinetic energy of the fluid brought about by rearrangement of the fluid

streamlines on entry and exit,

(c) time-dependent properties of the fluid which result in an additional variation of wall shear

stress along the tube entrance or exit length for steady flow, and

(d) energy losses, arising from either elastic deformation of viscoelastic fluids or structural

change of thixotropic fluids, not recovered during flow in the tube.

It is usual to treat the losses at the exit as being negligible compared with those at the

entrance. Equations are available for estimating the entrance length to obtain 98% of fully-

developed flow of Newtonian, power law and Bingham plastic fluids (10). However, it is

preferable to correct for end effect experimentally using a number of tubes of the same

diameter but of different lengths (16, 20).

Wall slip - To correct for wall slip, the wall slip velocity, Vs, is assumed to be given by (19)

ws SτV (2)

and this assumption is validated when the plots of 8V/D versus 1/D for various constant

values of w are straight lines. Otherwise, Vs can be expressed as (22)

m

ws

D

SτV

(3)

where m takes a value such that plots of 8V/D versus 1/Dm + 1

for various constant values of

w give straight lines; this condition is usually found by trial-and-error, assigning different

values of m (such as 1.25, 1.5, 2.0, 2.5 and 3.0).

As an example, plots of wall shear stress versus nominal wall shear rate (8V/D) were obtained

for a fine particle slurry using a controlled-pressure tube viscometer fitted with three tube

diameters of 0.408, 0.602 and 0.981 mm, Figure 6. The three curves did not superpose, and so

wall-slip was shown to be present. To correct for wall-slip, wall-slip velocities were estimated

using the procedure outlined above. Wall-slip velocities ranged from 44 mm/s at a wall shear

stress of 350 Pa to 68 mm/s at 600 Pa. Values of 8(V-VS)/D were then used in place of 8V/D

for the three curves to give a single wall-slip corrected curve as shown by the uppermost solid

line in Figure 6. This shows that if the 0.408 mm tube alone had been used, the slurry

viscosity would be underestimated by about 10%.

It should be noted that, in general, the relationship between slip velocity and wall shear stress

using the tube geometry will not usually apply to other viscometric geometries. Thus, if wall-

slip is present and the application for the viscometric data is pipeline design, the tube

viscometric geometry should be selected, if possible.

0

200

400

600

800

0 2500 5000 7500 10000 12500 15000 17500

8V/D (s-1

)

Sh

ear

str

ess (

Pa)

0.981 mm

0.602 mm

0.408 mm

Figure 6 Shear stress versus nominal wall shear rate for a fine particle slurry

using three capillary tube sizes

3.6.2 Correcting for errors in coaxial cylinder viscometry

Secondary Flow- For experimental data not to be affected by secondary flow when the bob is

rotating (Searle type), the following criterion developed for Newtonian fluids (used as an

indicator for non-Newtonian fluids) must be satisfied:

Ta

Ω

)β - (1

4β δ ρ > τ

1/2c

2

3/222

(4)

where = Rb/Rc , = Rc - Rb, is rotational speed in rad/s and Tac is the Taylor number at

the critical speed at which primary laminar flow breaks down, c, given by Alderman (2) as

1.8cβ

10502350Ta

(5)

This equation can be plotted as a double logarithmic plot of against to give the laminar

limit line (straight line with slope of 2); data to the left of the line correspond to laminar flow,

while data lying on it or to the right of the line must be rejected for further analysis.

If the cup is rotating (Couette type), the higher velocity at the cup radius has a stabilising

effect on fluid flow and secondary laminar flow occurs at a speed about 10 times higher than

that for the case where the bob is rotating (23).

End effect – This error arises because of the viscous drag on the ends of finite length cylinders

and can account for up to 30% of the measured torque. End effect can be minimised with

limited success by

• using a bob with a large L/Rb ratio,

• trapping an air bubble in the fluid using a bob with a recessed bottom, or

• using a Mooney-Ewart bob whose bottom is conically shaped to give the same

Newtonian shear rate as that in the annulus.

Alternatively, end effect can be determined experimentally so that the data can be corrected.

This involves the use of either a series of bobs of same radius but of different lengths or a

single bob partially immersed to different depths (16, 20).

Wall slip - This can be identified when flow curve data obtained with a series of bobs of

different radii do not coincide after the data have been corrected for end effect. This effect is

normally dealt with either by using bobs and cups with walls roughened to eliminate the wall

slip or by experimental methods which allow the wall slip velocity to be determined and

corrected for. The assumption that roughened wall surfaces eliminate wall slip should be

verified using bobs and cups of varying degrees of roughness (20, 24).

3.7 Rheological assessment of the yield stress property

The yield stress (y) of a slurry is defined as either the minimum stress required to maintain

steady shear flow in a material when using a controlled rate (CR) viscometer or the maximum

stress that the material will sustain without developing steady shear flow when using a

controlled stress (CS) viscometer. For stresses below y, the slurry behaves as an elastic solid

whereas for stresses above y, the slurry flows with a shear rate dependent on the excess stress (

- y). The relationship between the shear rate and the excess stress is linear for slurries exhibiting

Bingham plastic behaviour and non-linear for slurries exhibiting viscoplastic behaviour. As the

yield stress is often linked to the material structure, it will depend on the slurry’s previous shear

history. Thus, a low y will occur when the material structure is broken down by shear whereas a

higher y will result when the material is allowed to rest causing the structure to develop.

Many methods are used for yield stress measurement of high concentration slurries (11, 25),

including vane rheometry (26). A vane rheometer has several advantages over conventional

measuring geometries including the insertion of a vane causes much less structural damage to

the sample and the absence of wall-slippage since the yield surface is within the material itself.

The vane immersed in the cup containing the slurry sample is either made to rotate at a fixed rate

and the resulting torque is measured as a function of time, or small step increases in torque are

applied until the resulting strain is no longer constant with time. From the maximum torque, the

yield stress can be evaluated. Provided care is taken to ensure no wall slip, it has been found for

many materials that the yield stress results agree well with those obtained from extrapolation of

the flow curve obtained from CS rotational viscometry beyond the lowest measurable shear rate

to the zero shear rate.

The slump test is also available for yield stress measurement of high concentration pastes (27).

This test is carried out by first filling the slump cone in three layers, each layer being tamped 25

times. After smoothing the surface, it is placed upside-down on a surface. The cone is raised

vertically causing the material to slump. From the slump height, the yield stress is evaluated.

There is much controversy in the literature over the existence of a yield stress (25). However, the

requirement for the concept of a yield stress depends on the application. If a slurry is

characterised by its property at zero shear rate, a yield stress cannot be measured because the

slurry property attained is always unknown and cannot be obtained by extrapolation. One would

perpetually be developing more sophisticated viscometers for measurement of ever-decreasing

shear rates. So, no-one will ever measure a true yield stress! However, from an engineering

perspective, such measurements are expensive and unnecessary because a part description of the

flow curve over the relevant shear rate range using a yield stress is sufficient. The magnitude of

the yield stress was shown (28) to be dependent either on the length of the observation time used

to determine whether flow is established or has ceased in a CS experiment, or on the lowest

shear rate used in a CR experiment. As the yield stress is essentially then a time-dependent

property, it becomes important to ensure the choice of the observation time or shear rate to use is

matched to the characteristic time or shear rate to which the results are to be applied.

3.8 Rheological assessment of time-dependent property

3.8.1 Hysteresis loop tests

The presence of time-dependent flow behaviour in slurries/pastes can be assessed by carrying

out repeated shear cycles on a test sample in a rotational viscometer (5). If hysteresis loops

between the up and down curves are observed for each successive cycle, the sample is said to

exhibit time-dependent flow behaviour. Figure 7 shows positive hysteresis loops

(“clockwise”) for slurries whose internal structure is breaking down under cyclic shearing, but

negative loops (“anticlockwise”) are also obtained for slurries whose internal structure is

increasing under cyclic shearing. With positive loops, the first loop may show the catastrophic

breakdown of a yield stress but the following few loops will not generally coincide.

However, as the shearing cycles are repeated, two successive loops will eventually

superimpose, giving the equilibrium loop. This is invariant with respect to the number of

shearing cycles involved. The area between the up and down curves of any one loop test is

indicative of the thixotropic nature of the slurry. This area is zero for slurries exhibiting time-

independent flow behaviour. In other words, the up and down curves should superimpose,

provided that there are no errors arising from measurement, as discussed in Section 3.5. As

there is always some error in the measurement involved, this superposition will always be

approximate.

The hysteresis flow loop measurement is useful as a quality control tool to compare two or

more samples of slurry or paste from the same or different batches, provided that the

equilibrium flow curve is used as the basis for comparison in order to eliminate any differing

previous shear history effects. However, the loop test suffers unfortunately from two

disadvantages, namely:

(i) it is often carried out too quickly and inertia effects from the measuring head are

introduced but not always recognised.

(ii) the response cannot be resolved separately into shear rate and time (29).

Figure 7 Typical behaviour of a thixotropic slurry when subjected to

repeated cyclic shearing

3.8.2 Step shear rate experiments

If the slurry appears to exhibit time-dependent flow behaviour, the experiment where the

speed (or torque) is held constant should be repeated until the torque (or speed) attains a

steady value before the ramping the speed (or torque) to the next value. This will yield an

equilibrium flow curve in which the up and down curves coincide. Typical results are depicted

in Figure 8.

Alternatively, a different type of step-shear rate experiment is carried out to determine “lines

of constant structure” (5). Instead of waiting for the shear stress to attain an equilibrium value

after a change in shear rate has been imposed upon the sample, the new shear rate is held for

only as long as necessary to obtain the initial shear stress and the initial slope of the shear

stress versus time plot, before the shear rate is returned to a reference shear rate.

Figure 9 summarises how this type of step-shear rate experiment is carried out (5). The sample

is first sheared at a reference shear rate γ=γER

until equilibrium is attained and = eE. The

shear rate is then suddenly changed to a test shear rate γ=γT A

and the shear stress is recorded

as a function of time. The shear rate is returned to γE

and the sample brought again to

equilibrium. A new test shear rate γ=γBT is now applied and the shear stress recorded. This

sequence of sample conditioning at γE

and tested at γT

is repeated for other values

of ,γ=γCT etc.

Figure 8 Thixotropic behaviour observed from step changes in shear rate

Figure 9 Step-shear rate experiments to obtain constant-structure curves

From the shear stress versus time traces, the initial shear stresses 0 immediately after the step

change to γT

are read and plotted versus ; γT and also the reference point .)τ ,γ( eET

It is assumed

that the structure has not had time to change just before and just after the step change and so

all the data points refer to the same structure, that in equilibrium at . γE

The flow curve plotted

(Figure 10) is therefore a “line of constant structure”.

By repeating the test using a range of reference shear rates ,γ ,γ ,γ = γHGFR a family of constant

structure curves can be mapped out (Figure 10). Constant structure curves combined with the

measurement of curves illustrating the rate of change of structure with time (4) provide a full

characterization of a time-dependent slurry. However, the practical application of this method

has still to be developed adequately for many slurry engineering applications.

Figure 10 Plotting constant-structure curves

4 FLOW CURVE MODELLING AND INTERPRETATION

The main classes of flow curve exhibited by slurries under steady-state shear are given in

Table 3. These are idealised representations as most slurries, as we have seen, show more than

one flow curve classification over the measurable shear rate range of 10-6

to 106 s

-1. As a

result, sometimes more complex models are required, such as the Cross (30) or Sisko (31)

models. These models can be particularly useful in product formulation, but for most process

engineering applications the simpler models involving just two or three model parameters will

generally suffice.

Having completed the calculation procedure for the corrected flow curve, the data may be

amenable to a single curve fit. Sometimes, because of considerable scatter in the data, it may

be more appropriate to construct at least two curves: a mean curve obtained from regression

analysis using all the data and an upper bound curve obtained from regression analysis using

(, γ ) data selected from the curve that was initially drawn by eye. The upper bound curve

would normally represent the worst case for many engineering applications. This curve would

normally account for any possible variations in solids concentration, particle size distribution,

particle shape and pH.

Table 3 Flow curve models

Flow Curve Description Flow Curve Model

Newtonian Newtonian model: γ η = τN

Shear-thinning or

pseudoplastic Power law model: γK = τ

n n < 1

Shear-thickening

(dilatant) Power law model: γK = τ

n n > 1

Bingham plastic Bingham plastic model: γη + τ = τByB

Viscoplastic

Casson model: 0.5

C

0.5

yC

0.5 γηττ

Generalised Bingham plastic or Herschel-Bulkley model:

γK + τ = τn

yHB

Further factors can cause difficulties in attempting to draw a single flow curve through the

data. These factors include the use of two or more different viscometric geometries which may

give differing degrees of phase separation during shear, sample variability taken from the

same batch, and uncorrected errors associated with the use of any viscometric geometry.

5 FLOW CURVE MODEL SELECTION AND MODEL PARAMETER

ESTIMATION

It is often not immediately obvious from the data which of the flow models summarised in

Table 3 should be selected for further design usage. The following approach is suggested:

(i) Plot all the (, γ ) data on linear axes and separately, on double logarithmic axes. This

is to assess the suitability of the Newtonian, Bingham plastic and power law models.

(ii) If there is considerable scatter in the data, decide by eye or from the correlation

coefficient obtained by linear regression analysis whether a straight line through the

linear or the log-log plot gives the better representation. Similarly decide for the upper

bound curve. If one of these alternatives is acceptable, the use of the Herschel-Bulkley

model is probably not warranted.

(iii) If neither of these alternatives appears satisfactory because there is significant

curvature of the data on both linear and log-log plots, the following can arise:

(a) If there is data curvature on the log-log plot with the slope of the curve

increasing with shear rate axis and if the linear plot does not produce a straight

line then the Herschel-Bulkley model should adequately describe the data.

(b) If there is data curvature on the log-log plot and the slope of the curve

decreases with shear rate axis, the use of the Herschel-Bulkley model is

inappropriate as this implies a negative yield stress parameter. However, a

curve fit is possible and would result in a negative value for the yield stress

parameter. Force-fit either a Bingham plastic or power law model to the data.

Estimates need to be made for the parameters defined in the flow models given in Table 3. As

the Herschel-Bulkley model can be reduced to the Newtonian, power law and Bingham plastic

models, a least squares regression analysis can first be performed on the (, γ ) data to obtain

yHB, K and n. It may then be possible to simplify the model by setting the yHB to zero if the

estimate is close to zero and/or setting n to 1 if the estimate is close to unity. Two methods are

commonly used when carrying a regression analysis on the (, γ ) data:

• a non-linear least squares regression on unweighted data,

• a non-linear least squares regression on weighted data.

Standard non-linear regression software packages can be used. Alternatively non-linear

regression can be performed using Microsoft Excel via the ‘Solver’ tool (32). Both methods

above provide sets of yHB, K and n estimates which give viscometric data predictions to ± 2%

of the original data within the original shear rate range. Outside this shear rate range,

agreement can be poor and extrapolations to shear rates above the highest experimental value

should be avoided. Extrapolation to shear rates below the lowest experimental value is

unavoidable when predicting frictional pressure loss for laminar pipeflow (16).

6 CONCLUDING REMARKS

Slurry rheological properties need to be adequately measured for many process engineering

applications, including equipment selection and sizing, and specification of operating

conditions. These measurements may need to take into account both time-dependent flow

properties, as well as time-independent properties. Unfortunately, the application of time-

dependent flow property data to process engineering is not nearly so well-developed as it is

for time-independent property, and further work is required in this area. However, additional

testwork which assesses any time-dependent property can be very useful in design.

The flow properties of relatively low concentration fine particle slurries and pastes can usually

be readily measured. However, at higher solids concentrations, unusual responses from slurry

samples to conventional measurement methods should be expected (1). These responses

include time-dependency due to particle packing rearrangements under shear, the “stick-slip”

phenomenon, particle holdup in tube viscometers, and the influence of pressure when the

solids concentration starts to approach the maximum packing fraction. These effects are

documented in the rheological literature and may need to be considered during rheological

data interpretation for high concentration slurries.

By making both relevant and accurate rheological measurements on slurry samples that have

been correctly taken, and, if necessary, pre-conditioned, measurement effort can be minimised

and focused on the engineering requirement more precisely. The need for accuracy may

introduce additional laboratory experiments, particularly if end-effect and/or wall-slip errors

are identified. However, ensuring that only viscometric data relevant to the process

engineering applications of interest are collected will minimise the effort. Figure 11 gives the

steps to take to ensure the relevance and accuracy of slurry rheological measurements.

Determine shear rate / shear

stress over which flow curve

should be measured

Consider the impact of slurry

properties on viscometric

geometry selection, including

1. Maximum particle size

2. Concentration

3. Temperature

4. Pressure

5. Corrosive?

6. Volatile?

7. Abrasive?

Select viscometric geometry

Obtain flow curve using loop tests

Undertake additional tests to

determine time-dependent

behaviour

Yes Minimise effects of error, or

correct flow curve data

Obtain corrected flow curve

Select flow model

Obtain model parameters

for design calculations

AP

PLIC

AT

ION

SLU

RR

Y

CO

NS

IDE

RA

TIO

NS

VIS

CO

ME

TE

R

SE

LE

CT

ION

FL

OW

CU

RV

E

ME

AS

UR

ME

NT

FL

OW

CU

RV

E

CO

RR

EC

TIO

NS

FL

OW

CU

RV

E

MO

DE

LLIN

G

Are there significant time-

dependent effects?

Yes

Are the data subject to

secondary/turbulent flow or

viscous heating effect?

Are the data subject to

end-effect error?

Are the data subject to

wall-slip error?

Minimise effects of error, or

correct flow curve data

Yes

Eliminate data affected by

secondary/turbulent and/or

viscous heating effects

Yes

No

No

No

No

Figure 11 Summary flowchart for making slurry flow curve measurements

7 REFERENCES

1. Cheng, D C-H (1984) “Further observations on the rheological behaviour of dense

suspensions”, Powder Technology, 37, 255-273.

2. Alderman, N J (1996) “Non-Newtonian Fluids: Obtaining viscometric data for frictional

pressure loss estimation for pipeflow”, ESDU 95012, ESDU International plc, London,

UK.

3. Metzner, A B & Otto, R E (1957) “Agitation of non-Newtonian fluids”. AIChEJ, 3, 3-10.

4. Cheng, D C-H (1980) “Viscosity-concentration equations and flow curves for

suspensions”, Chem and Ind, 17 May, 403-406.

5. Cheng, D C-H (1987) “Thixotropy”. Int J of Cosmetic Sci., 9, 151-191.

6. Eilers, V H (1941) Kolloid-Z, 97, 313.

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viscosity data”. Trans. Soc. Rheol., 12, 281-301.

8. Alderman, N J (2004) “Shear rate estimation for viscosity/flow curve measurement”,

Aspen Technology report.

9. Barnes, H A, Hutton, J F & Walters, K (1989) An Introduction to Rheology, Elsevier,

Amsterdam.

10. Macosko, C W (1994) Rheology – Principles, Measurements and Applications, Wiley-

VCH, New York.

11. Steffe, J F (1996) Rheological Methods in Food Process Engineering, 2nd ed., Freeman

Press, Michigan.

12. Barnes, H A (2000) A Handbook of Elementary Rheology, Institute of Non-Newtonian

Fluid Mechanics, University of Wales, Aberystwyth.

13. Mezger, T G (2002) The Rheology Handbook, Vincentz Verlag, Hannover, Germany.

14. Patton, T C (1979) Paint Flow and Pigment Dispersion, 2nd

ed, John Wiley & Sons Inc,

New York.

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of Wales, Aberystwyth, UK.

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680p. Published by Elsevier Applied Science, Barking, UK. Now distributed by Kluwer

Publications, Dordrecht, the Netherlands (1991).

17. Chhabra, R P & Richardson, J F (1999) Non-Newtonian flow in the Process Industries:

Fundamentals and Engineering Applications, Butterworth-Heinemann, Oxford, pp 338-

339.

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slurry flow curves”. Chemical Engineering Progress, Part 1 : April and Part 2 : May.

19. Alderman, N J & Kruszewski, A P (2004) “Viscosity measurement : tube viscometry”,

Aspen Technology report.

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Technology report.

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Rheology, Vol 1, pp 250-253, Sydney, Australia.

22. Jastrzebski, Z D (1967) “Entrance effects and wall effects in an extrusion rheometer

during flow of concentrated suspensions”, Ind Eng Chem, 6, 445-454.

23. White, J L (1990) Polymer Engineering Rheology, John Wiley & Sons Inc, New York.

24. Cheng, D C-H & Parker, B R (1976) “The determination of wall-slip velocity in the

coaxial cylinder viscometry”. In Proc 7th Int Cong Rheol., Gothenberg, Sweden, 518-519.

25. Barnes, H.A. (1999) “The yield stress – a review or ‘ ’ – everything flows?” J Non

Newt. Fluid Mech., 81, 133-178.

26. Barnes, H.A. and Nguyen, Q-D. (2001) “Rotating vane rheometry – a review”, J Non Newt.

Fluid Mech., 89, 1-14.

27. Paterson, A.J.C. (2002), “Is slump a valid measure of the rheological properties of high

concentration paste slurries”, Proc of Hydrotransport 15, BHR Group, Banff, Canada, 3-5

June, 361-374.

28. Cheng, D.C-H. (1986) “Yield stress: A time dependent property and how to measure it”.

Rheol. Acta., 25, 542-554.

29. Barnes, H A (1997), “Thixotropy – a review”. J Non-Newtonian Fluid Mech., 70, 1-33.

30. Cross, M M (1965) “Rheology of non-Newtonian flow: equation for pseudoplastic

systems”. J Colloid Sci, 20, 417-437.

31. Sisko, A W (1958) “The flow of lubricating greases”. Ind. Eng. Chem., 50, 1789-1792.

32. Roberts, G P, Barnes, H A & Mackie, C (2001) “Using the Microsoft Excel ‘Solver’ tool

to perform non-linear curve fitting, using a range of non-Newtonian flow curves as

examples”, Applied Rheology, Sept/Oct, 271-276.