Viscometric data interpretation and correlation for non-Newtonian fluids

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-FLO-302

Interpretation and Correlation of
Viscometric Data

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CONTENTS
0

INTRODUCTION/PURPOSE

4

1

SCOPE

4

2

FIELD OF APPLICATION

4

3

DEFINITIONS

4

4

NON-NEWTONIAN FLUID BEHAVIOR

4

4.1
4.2
4.3

Introduction
Classification of Non-Newtonian Fluids
Caution

4
6
14

5

VISCOMETER MEASUREMENTS FOR
TIME-INDEPENDENT FLUIDS

16

5.1
5.2
5.3
5.4
5.5
5.6

Concentric Cylinder Viscometers
Cone and Plate Viscometers
Parallel Plate Viscometer
Tube or Capillary Viscometer
Checks for Consistency of Data and Interpretation
Estimate of Process Shear Rate

16
17
18
18
20
23

6

MODEL FITTING TO FLOW CURVES

24

6.1
6.2
6.3
6.4

Power Law
Bingham Plastic
Direct use of Numerical Data
Rheological Models Involving Temperature Dependence

24
25
26
26


7

CHARACTERIZATION OF TIME-DEPENDENT LIQUIDS

29

7.1
7.2
7.3
7.4
7.4

Sample Loading
Tests at Constant Shear Rate
Dynamic Response Measurement
Changes in Shear Rate
Concluding Remarks

29
29
30
31
33

8

TECHNIQUES FOR CHARACTERIZATION OF
VISCOELASTIC LIQUIDS

33

8.1
8.2
8.3
8.4

Stress Relaxation
Oscillatory Shear Measurements
Normal Force Measurement
Elongational Viscosity Measurement

33
33
33
34

9

NOMENCLATURE

34

10

BIBLIOGRAPHY

35

APPENDICES

A

EQUATIONS FOR VISCOMETERS

36

A.1

EQUATIONS FOR CONCENTRIC CYLINDER
VISCOMETERS

36

A.2

EQUATIONS FOR CONE AND PLATE VISCOMETERS

38

A.3

EQUATIONS FOR PARALLEL PLATE VISCOMETER

39

A.4

EQUATIONS FOR TUBE OR CAPILLARY VISCOMETER

39


TABLES
1


5

2

SOME EXAMPLES OF NON-NEWTONIAN MATERIALS

15

3

SUMMARY OF CHARACTERISTICS OF CONCENTRIC
CYLINDER VISCOMETERS

16

SUMMARY OF CHARACTERISTICS OF CONE AND PLATE
VISCOMETERS

18

SUMMARY OF CHARACTERISTICS OF TUBE OR CAPILLARY
VISCOMETERS

18

SUMMARY OF CONSISTENCY CHECKS

20

4

5

6

FIGURES

1

SHEARING BETWEEN PARALLEL PLANES

5

2

SIMPLE TESTS OF A QUALITATIVE NATURE WHICH INDICATE
WHETHER OR NOT A LIQUID DEPARTS FROM NEWTONIAN
BEHAVIOR
7

3

VISCOELASTIC MATERIAL BEHAVIOR

8

4

PSEUDOPLASTIC BEHAVIOR

10

5


10

6

LIQUIDS WITH ZERO AND INFINITE SHEAR VISCOSITIES

11

7

DILATANT BEHAVIOR

11

8

LIQUID HAVING A YIELD STRESS

12

9

TIME-DEPENDENT LIQUID BEHAVIOR

13

10

CO-AXIAL OR CONCENTRIC CYLINDER VISCOMETER

17


11

CONE AND PLATE GEOMETRY

17

12

TUBE OR CAPILLARY VISCOMETER

19

13

DETERMINATION OF END EFFECTS IN TUBE VISCOMETERS

21

14

DERIVATION OF END EFFECTS FOR PRESSURE DRIVEN TUBE
VISCOMETERS
22

15

TYPICAL DATA FOR THIXOTROPIC FLUIDS

22

16

APPLICATION OF POWER LAW

24

17

TYPICAL IDEAL BINGHAM PLASTIC BEHAVIOR

25

18

NON-LINEAR BINGHAM PLASTIC BEHAVIOR

26

19

EFFECT OF TEMPERATURE

27

20

TYPICAL CONSTANT SHEAR RATE DATA

30

21

DYNAMIC RESPONSE DATA

31

22

CHANGE IN SHEAR RATE DATA

31

23

OTHER FORMS OF RESPONSE TO CHANGES IN SHEAR RATE 32

24

TORQUE SPEED PLOTS

37

25

TORQUE vs SPEED PLOT FOR BINGHAM PLASTICS

38

DOCUMENTS REFERRED TO IN THIS PROCESS
ENGINEERING GUIDE

40


0

INTRODUCTION/PURPOSE

This guide is one of a series of guides on non-Newtonian flow prepared for GBH
Enterprises.

1

SCOPE

This document provides guidance on the interpretation of viscometric data
obtained from different measuring devices and their correlation into forms usable
for design purposes. It does not cover the design of pipes or equipment handling
non-Newtonian fluids.

2

FIELD OF APPLICATION

This Guide applies to Process Engineers in GBH Enterprises.

3

DEFINITIONS

For the purpose of this Guide, the following definitions apply:
Viscometer

An apparatus used to measure the viscosity of a liquid.

Rheometer

An apparatus used to measure other flow behavior
properties (e.g.elastic properties) in addition to the viscous
properties.

With the exception of terms used as proper nouns or titles, those terms with initial
capital letters which appear in this document and are not defined above are
defined in the Glossary of Engineering Terms.

4


4.1

Introduction

All gases, pure liquids and dilute inorganic solutions are Newtonian in their flow
behavior. By definition, such materials obey Newton's law of viscosity, i.e. when
sheared in laminar flow, the shear stress (see Note 1) is directly proportional to
the shear rate (see Note 2 and Figure 1).

Notes:
(1)

Shear stress:

shear force per unit area causing the
shearing motion. (N/m2)

(2)

Shear rate:

equivalent to the velocity gradient in
one-dimensional shearing motion. (s -1)

(3)

Viscosity:

the ratio of shear stress / shear rate (Ns/m2)
1 Ns/m2 = 1 Pa.s = 1 kg/s.m = 1000 cP = 10 P

FIGURE 1

SHEARING BETWEEN PARALLEL PLANES

The proportionality constant, µ1 is termed the shear velocity or simply viscosity
and for a Newtonian fluid it is constant and independent of shear rate,

.

The viscosity, however, varies strongly with temperature and to a lesser extent
with pressure.

Thus for a Newtonian liquid (such as ethanol or petrol) the laminar viscosity, µ, is
a constant which does not vary with the processing conditions (provided the
temperature and pressure remain constant). Such materials may be pumped at
any flowrate, stirred at any speed and the viscosity will have a unique value
provided laminar flow is maintained. In turbulent flow, even for a Newtonian
liquid, the shear stress is no longer directly proportional to the shear rate. It is
possible to define a 'turbulent viscosity' as the ratio of shear stress to shear rate,
but its value will depend on the shear rate. Note that the viscosity used in the
definition of Reynolds number is the laminar viscosity, even under turbulent
conditions.
However, many industrial liquids exhibit 'non-Newtonian' characteristics, i.e. in
laminar flows, Newton's law of viscosity is inadequate to describe their flow
behavior. Such materials are shown in Table 1.
TABLE 1


Some Fluids Which Exhibit Non-Newtonian
Fluid Behavior
Suspensions
Slurries
Pastes
Creams
Gels
Greases

Waxes
Plastics
Rubbers
Paints
Foodstuffs
Polymer Solutions

The listing in Table 1, is by no means exhaustive, but if any liquid is two-phase
(e.g. emulsions, pastes, slurries) or if the liquid contains polymeric materials
(plastics, rubbers, polymer solutions), then non-Newtonian behavior should be
suspected.
The only sure way to determine whether or not a liquid is non-Newtonian is to
take measurements in a Viscometer or Rheometer of the shear stress-shear rate
behavior of the material. However, some simple tests of a qualitative nature are
given in Figure 2 to indicate whether or not the liquid departs from Newtonian
behavior. It must be stressed that these checks only lead one to suspect unusual
flow behavior (they are not a replacement for detailed measurements in a
Viscometer).


In order to describe this behavior quantitatively and to generate data suitable for
engineering design, it is necessary to take measurements in a Viscometer (to
measure the viscous properties of the liquid) or a Rheometer (to measure the full
flow properties or rheology, e.g. elastic properties). The measurement of the
rheology or flow properties in various Viscometer and Rheometers is dealt with
later in this Guide. However, we will initially consider some of the types of nonNewtonian fluid behavior.
4.2

Classification of Non-Newtonian Fluids

In the case of a Newtonian liquid (which is just a special case of the more
general non-Newtonian behavior) it has already been seen that the viscous
stress, Ԏ is directly proportional to the shear rate (velocity gradient),
the viscosity, µ, is the constant of proportionality viz.,

, and

Thus the shear stress is a unique function of the shear rate, given viscosity µ
(which is a material property to be measured in a Viscometer).
There is a class of non-Newtonian liquids called 'time-independent liquids' and
for these materials the shear stress also depends on only the shear rate (but not
in the simple linear form of Equation! 2). That is,

For such materials it is possible to define an apparent viscosity, µa, which is now
dependent upon shear rate


Thus for time-independent materials, the apparent viscosity changes with shear
rate (it will also vary with temperature and to a lesser extent with pressure). The
material responds instantaneously (for practical purposes) to imposed changes in
shear rate and there is no time-dependency.
In contrast to the above time-independent (i.e. instantaneously responsive) nonNewtonian materials, some liquids exhibit time-dependency and the viscous
properties depend upon the shear rate and the time of shearing, i.e.

Such materials are termed 'time-dependent'.
Finally, there are some industrial liquids which, in addition to viscous properties,
also possess the characteristic of 'elastic solids'. These materials exhibit 'elastic
recovery' after deformation, they possess large 'elongational viscosities' when
stretched and also exert 'normal forces' in steady shearing flows (see Figure 3).


FIGURE 2 SIMPLE TESTS OF A QUALITATIVE NATURE WHICH INDICATE
WHETHER OR NOT A LIQUID DEPARTS FROM NEWTONIAN BEHAVIOR


With a viscoelastic liquid, the shear rate at any instant depends upon the
previous history of the liquid in addition to the current conditions and the
relationship between shear stress and shear rate is very complex.
Non-Newtonian liquids can therefore be classified according to:
(a)

time-independent behavior;

(b)

time-dependent behavior;

(c)

viscoelastic behavior.

In the following sections a more detailed consideration is given to timeindependent liquids since, for such materials, the techniques of quantitatively
describing their viscous flow properties in Viscometers are well established.
Furthermore, the use of such properties in engineering design of pump/pipeline
systems, heat exchangers, mixing vessels, etc. is reasonably well understood.
A briefer discussion is devoted to time-dependent and viscoelastic liquids
because such liquids are difficult to characterize fully in Viscometers and
Rheometers. Furthermore, the incorporation of such data into design procedures
is difficult (and in some cases impossible) with our present state of knowledge.
4.2.1 Time-independent Liquids
These materials can be divided into three types:
Pseudoplastic
liquids
.
Dilatant liquids

-

for which the apparent viscosity, µa, reduces as the
shear rate,
increases

-

for which the apparent viscosity, µa, increases
as the shear rate,
increases.

Bingham plastic
liquids

-

these materials possess a yield stress Ԏy and unless
the shear stresses exceed this value
flow will not commence.

Each type is now considered separately below.
(a)

Pseudoplastic liquids

A typical flow curve, plot of shear stress, Ԏ, vs shear rate,
from a Viscometer for a pseudoplastic liquid is shown in

obtained
Figure 4.


In this Figure it is seen that the apparent viscosity, µa, (defined on the ratio of the
shear stress to shear rate) reduces as the shear rate increases. Such behavior is
also termed 'shear-thinning' and is found in many non-Newtonian liquids.
FIGURE 4


The data can be replotted as apparent viscosity, oa, vs shear rate ˙ and this
clearly shows the reduction in apparent viscosity as the fluid is subjected to
greater and greater velocity gradients (see Figure 5).


FIGURE 5


In some cases it is found that materials which are pseudoplastic at moderate
shear rates tend to Newtonian behavior (i.e. viscosity becomes independent of
shear rate) at low and high values of the shear rate. These limiting viscosities are
often called the zero-shear viscosity, µoo, and the infinite shear viscosity, µo .
This type of behavior is illustrated in Figure 6.


FIGURE 6

(b)

LIQUIDS WITH ZERO AND INFINITE SHEAR VISCOSITIES

Dilatant liquids

The term dilatant indicates that the material increases in apparent viscosity as
the shear rate increases. Such behaviour is often termed shear-thickening. A
typical flow curve is shown in Figure 7 as a plot of shear stress vs shear rate and
also apparent viscosity vs shear rate.


FIGURE 7

DILATANT BEHAVIOR

This type of behavior is less common than shear-thinning (pseudoplastic)
behavior and such materials will present processing difficulties since the more we
try to shear the material, the greater is the resistance offered by the fluid.
(c)

Bingham plastic liquids

In the case of such materials it is not possible to set up velocity gradients (shear
rates) in the fluid until a finite stress, the yield stress, Ԏy, is exceeded. A typical
flow curve for such materials is shown in Figure 8.
FIGURE 8

LIQUID HAVING A YIELD STRESS


The precise value of the yield stress, Ԏy, is often difficult to obtain because its
evaluation requires the extrapolation of viscometric data down to a shear rate of
zero. Indeed, there is much controversy over the existence of 'yield stresses'.
However, the need for an accurate value of the yield stress, Ԏy, is only important
in very low shear rate processes. At moderate and high shear rates the precise
value of the yield stress becomes less important.
All the above materials which are time-independent should respond essentially
instantaneously to imposed changes of shear rate or shear stress and this should
be apparent in obtaining Viscometer data at a variety of shear rates. If the
response to changes in shear rate is finite and significant the material should be
treated as being time-dependent.
4.2.2 Time-dependent Liquids
In the case of liquids which are classed as being time-dependent, the apparent
viscosity depends not only upon the current value of the shear rate but upon the
previous shear rate history to which the liquid has been subjected. If such a liquid
is rested for a long time and then subjected to a steady constant shear rate, two
modes of behaviour are possible. In the case of thixotropic materials the
apparent viscosity reduces with time whereas anti-thixotropic (sometimes called
rheopectic) material exhibits an increase in viscosity with time (see Figure 9).
FIGURE 9

TIME-DEPENDENT LIQUID BEHAVIOR


In both cases, i.e. thixotropic and rheopectic, after shearing at constant rate for a
long time, the apparent viscosity approaches an equilibrium viscosity.
If a time-dependent material is sheared to an equilibrium apparent viscosity at a
given shear rate and then the stress is removed, the apparent viscosity will
gradually increase in the case of the thixotropic fluid and reduce for the antithixotropic fluid. The rates at which viscosities are reduced or increased are not
necessarily the same. For example, for thixotropic liquids, the rate at which the
viscosity is reduced by shearing is often much faster than the recovery under
zero or low shear rate conditions.
Time-dependent materials are difficult to characterise in a Viscometer because
merely putting the material into a Viscometer imposes a shear history upon the
fluid. Furthermore the necessity to obtain dynamic data (i.e. variation in apparent
viscosity with time) requires an instrument with rapid response characteristics.
However, with care relevant Viscometer data can be collected and used in
design and further details are given in Clause 6. At least, Viscometer data will
indicate the worst possible state of the material to be handled and a
conservative design will always result.
It should be noted that thixotropy is the most common type of time-dependent
behavior.
4.2.3 Viscoelastic Fluids
In steady flows in channels of constant cross section the viscoelastic fluid exerts
normal forces, see Figure 3 (c), perpendicular to the shearing motion and
information on such forces can be obtained from Viscometers with normal force
measuring facilities. In other respects in such steady flows, the effects of
elasticity are not detected.
However, in steady flows through channels of varying cross-sections, e.g.
expansions, contractions, valves, orifice plates and in unsteady flows e.g. startup of a pipe-line or in a pulsating flow, the elastic properties of the material can
become very important. In order to obtain relevant data, Viscometers with
oscillatory shearing features can be used. Also extensional Rheometers, jet
thrust Rheometers can be used. More details are given in Clause 7.
Unfortunately, at the present time, data on the rheology of viscoelastic liquids can
only be obtained from Viscometers and Rheometers in very idealised flow
situations. The use of such data for more realistic flows of engineering
significance is still at the forefront of research.

Reliable, general design techniques to handle such fluids have not yet been
developed.
4.3

Caution

(a)

In the preceding treatment of the general characteristics of non-Newtonian
flow behavior, the major characteristics have been isolated and discussed
separately. However, one fluid may exhibit different aspects of nonNewtonian behaviour at the same time or at different shear rates. Thus
one type of behaviour does not preclude other types. For such complex
fluids a knowledge of the processing conditions may enable one to
determine the particular characteristics relevant to that process and often
only one type of non-Newtonian behaviour will determine the flow.

(b)

Much of the literature concerned with rheology is aimed at the prediction
of non-Newtonian characteristics from molecular structure or material
morphology. This is not a reliable approach and rheological data for
engineering design must be obtained by carefully planned Viscometer
experiments. These tests should be carried out over a range of shear
rates relevant to the equipment to be designed (and at appropriate
operating temperatures).
Table 2 gives a rough guide of the type of behavior expected from
different industrial materials. This should be treated with caution. There is
no replacement for careful viscometry to establish rheological
characteristics. It is dangerous to have too many preconceived ideas.

(c)

The selection of the most appropriate type of Viscometer or rheometer is
important. The operating characteristics and the suitability of the most
common types are discussed in Clause 5, 7 and 8.

(d)

Viscometer measurements should always be taken under laminar flow
conditions when the flow is dominated by viscous forces. This is
appropriate since in many cases the high apparent viscosities of nonNewtonian liquids means that they are processed under laminar
conditions. However, there are some instances, e.g. in slurry
transportation in large pipelines, when turbulent flow can be achieved. In
such cases the use of laminar Viscometer data is open to doubt and it
may be that some scale-up from a small pilot plant, operating in turbulent
flow, is the best way to proceed.


TABLE 2

SOME EXAMPLES OF NON-NEWTONIAN MATERIALS


5

VISCOMETER MEASUREMENTS FOR TIME-INDEPENDENT FLUIDS

Design methods are generally only available for time-independent fluids. It is
possible to carry out some design calculations for thixotropic or viscoelastic fluids
but the procedures are usually very complex and require specialist knowledge.
For time-independent fluids the measurement and data handling procedures are
quite straightforward. Even the pitfalls of the subject are quite well documented.
This section describes the measurement procedures for several Viscometer
geometries together with the standard data processing techniques. The
characteristics of each type of Viscometer are also listed. It concludes with
advice on checking for consistency of the data and their interpretation and on
how to estimate the process shear rate for typical process conditions.
The equations applicable to each type of Viscometer are given in Appendix A.
5.1

Concentric Cylinder Viscometers

A concentric cylinder Viscometer is shown diagrammatically in Figure10. Its main
characteristics are summarized in Table 3. Concentric cylinder Viscometers are
thus suitable for low to moderate viscosity measurements. High viscosities may
be determined in small diameter concentric cylinder Viscometers. Temperature
control is good.
Errors may arise from possible end effects or wall slip, see 5.5.1.
TABLE 3

SUMMARY OF CHARACTERISTICS OF CONCENTRIC
CYLINDER VISCOMETERS

Viscosity Range

-

Low to Moderate

Shear Rate

-

Low to Moderate

Temperature Control

-

Good

Difficulties

-

End Effects
Wall Slip
Turbulence

High Viscosities

-

Require Small Cylinders

High Shear Rates

-

Require (i) Narrow Gap
and/or (ii) Annular Gap.


FIGURE 10 CO-AXIAL OR CONCENTRIC CYLINDER VISCOMETER

5.2

Cone and Plate Viscometers

A cone and plate Viscometer is shown diagrammatically in Figure! 11. The main
characteristics are summarised in Table 4 and are therefore suitable for
moderate to high viscosity measurements. Low viscosities can be measured if
the Viscometer has large cones and small angles. Good temperature control can
be difficult.
FIGURE 11

CONE AND PLATE GEOMETRY


TABLE 4

SUMMARY OF CHARACTERISTICS OF CONE AND PLATE
VISCOMETERS

Viscosity Range

-

Moderate to High

Shear Rate Range

-

Moderate to High

Temperature Control

-

Can be Difficult

Difficulties

-

Relative Cone-Plate Location

Restrictions

-

Gap Angle must be less than 4°

Low Viscosities

-

Require

(i) Small Gap Angles

and/or
(ii) Large Cones
Low Shear Rates

5.3

-

Require Large Gap Angles.

Parallel Plate Viscometer

The parallel plate Viscometer is similar to the cone and plate geometry but does
not have uniform shear rates. Its main application is in determining normal force
data in studies of viscoelastic fluids.
5.4

Tube or Capillary Viscometer

A tube (or capillary) Viscometer is shown diagrammatically in Figure 12. Its main
characteristics are summarized in Table 5. Tube Viscometers are thus suitable
for moderate to high viscosities although low viscosities can be measured using
small diameter tubes. Temperature control is good, but relatively large samples
are required.
Errors may arise from wall slip and end effects.


TABLE 5

SUMMARY OF CHARACTERISTICS OF TUBE OR CAPILLARY
VISCOMETERS

Viscosity Range

-

Moderate to High

Temperature Control

–

Good

Difficulties

-

(i) Relatively Large Sample Required

Shear Rate Range

(ii) End Effects
(iii) Wall Slip
Low Viscosities

-

Require Small Diameter Tubes.

FIGURE 12 TUBE OR CAPILLARY VISCOMETER

5.5

Checks for Consistency of Data and Interpretation

Errors can arise from shortcomings either in the data themselves or in the way
they have been interpreted. Table 6 summarizes consistency checks for different
Viscometer data and their likely causes.

5.5.1 Coaxial Cylinder Viscometers
Equation (23) provides for appropriate data interpretation for non-Newtonian flow
properties. Consequently data for two cylinder combinations of different Ri /Ro
ratio should give the same
Ԏ vs ˙

relationship. If this is not the case there may be a problem of wall

slip (see Note). If this is the case considerable extra data will be needed to allow
for a proper interpretation. At least three inner cylinder diameters should be used
to collect shear stress-speed data. This data can be processed by the procedure
of Oldroyd [Ref. 1] to determine slip velocity and bulk rheology values.

Note:
True wall slip, i.e. a finite wall velocity, is a rare phenomenon e.g. it can occur in
some flows of polymer melts. More usually, wall slip refers to the formation of a
low viscosity layer of liquid next to a solid surface. For example, when slurries
are sheared, particles migrate away from the solid surface leaving a low
viscosity, particle-free layer.
This is not uncommon, especially for non-Newtonian slurries, and care must be
taken to check for unusually low results from such measurements. Standard
measuring systems are sometimes unable to cope with difficult solid dispersions
due either to 'jamming' of the relatively small gaps, excessive wall slip caused by
migration of the dispersed phase away from the sensing member or rapid
sedimentation of the dispersed phase. One practical solution is to replace the
standard sensing member by a miniature agitator which minimises these
problems. After characterising the agitator it can be used to measure torque,
shear stress and viscosity in the usual manner. [Ref 6]
5.5.2 Tubular Viscometers
Equation (6) is necessary for the processing of non-Newtonian flow data to give a
value of shear rate at the tube wall.


If data are available for more than one tube diameter the Ԏ(R) vs
relationship should be the same in all cases. If this is not so the data should be
reprocessed by the procedure of Mooney [Ref. 2] to determine the slip velocity
and then bulk properties.
End Effects in Tube Viscometers
Tube Viscometers are subject to additional pressure losses as the test fluid
enters the tube. This effect can be eliminated by using several tubes of different
length but the same diameter and then determining values of ΔP/L for various
values of Q from plots of ΔP vs L (see Figure 13).
FIGURE 13 DETERMINATION OF END EFFECTS IN TUBE VISCOMETERS

Data of this form are conveniently collected with mechanically driven (Instron,
etc.) tube Viscometers. With pressure driven Viscometers, a cross plot of ΔP vs
Q data may be necessary (see Figure 14).


FIGURE 14 DERIVATION OF END EFFECTS FOR PRESSURE DRIVEN
TUBE VISCOMETERS

In attempting to eliminate end effects, problems may be encountered if the fluid
has thixotropic properties. If the data processing procedure above does not seem
to be satisfactory a check should be made using a rotational Viscometer (either
cone and plate or coaxial cylinder) to check for stress variation with time at
constant rotational speed:
FIGURE 15 TYPICAL DATA FOR THIXOTROPIC FLUIDS


In the early stages of a project it is more convenient to study thixotropic
properties in a rotational Viscometer. Further details are given in Clause 6.
5.5.3 Viscoelasticity
The viscoelastic character of the test fluid may make itself apparent by 'die swell'
in the extrudate from the tube Viscometer. There is no particular procedure which
can be used to improve data processing to allow for this characteristic but it is
wise to record the observation as viscoelasticity since it may be important in
understanding any potential flow problems. It is not a simple matter to collect and
use viscoelastic data for process design, and quality control is about the most
ambitious objective than can be secured by measurements of viscoelasticity.
Further details on appropriate Viscometer techniques for the assessment of
viscoelasticity are given in Clause 7.
5.6

Estimate of Process Shear Rate

Experimental data, from Viscometers cover only a finite range of shear rates. It is
therefore important that this range of shear rates should cover the shear rates
encountered in process for which the data are to be used. Extrapolation or
Viscometer data outside its range of validity is extremely dangerous. Methods for
estimating the process shear rate for typical processconditions are given below.
5.6.1 Steady Laminar Pipe Flow
In steady laminar pipe flow the nominal wall shear rate (i.e.wall shear rate for a
Newtonian fluid in the same pipe at the seam mean velocity) is 8V/D where V is
mean velocity, and D is the pipe diameter.
It is reasonable therefore to obtain Viscometer data over a range of shear rates
around this nominal value.
Although the shear rate at the pipe centre is zero, Sheffield and Metzner [Ref. 3]
have shown that data much below the nominal shear rate do not greatly affect
the flow.


5.6.2 Turbulent Pipe Flow
In turbulent pipe flow the wall shear rate is high and it is the apparent viscosity of
the fluid at this shear rate which is important. The calculation of the wall shear
rates is one of trial and error [Ref. 4].
5.6.3 Laminar Flow in Mixing Vessel
In a mixing vessel operating under laminar conditions the average shear rate ẏA
in the vessel can be estimated from:

ẏA = ҜsN

………….

(7)

where Ҝs = shear rate constant for impeller. 'Best' current values are given
below:
for

propellers

10

disc or flat-bladded turbines

11.5

angled blade turbines

13

anchors [Ref. 7]

33 - 172(C/dT)

helical ribbon-screw [Ref. 8]

60 (S/dR)0.65

Thus knowing the impeller speed N, the appropriate Viscometer range should be
around the value of ẏA given above.
5.6.4 Turbulent Flow in Mixing Vessel
In a mixing vessel in which the flow is turbulent, the flow is no longer dominated
by viscous forces and the precise shear rate is not too important. It is
Recommended that the previous equations (for laminar mixing) are used here.


6

MODEL FITTING TO FLOW CURVES

Many design calculations can be conducted using rheological data expressed in
the form of a simple mathematical model - usually shear stress related to shear
rate. Obviously the model can be no more accurate than the data to which it is
fitted and common sense must be used in selecting appropriate models. There
are a variety of simple models to use which relate specific circumstances.
6.1

Power Law

The power law model is frequently applied to the description of shear-thinning or
pseudoplastic flow properties. The form of the equation is:

where K is referred to as the consistency index and n as the flow behaviour
index. The consistency index gives some indication of the viscous resistance
offered to flow and the flow behaviour index gives an indication of the degree of
non-Newtonian behaviour of the fluid. The value of K often changes with
temperature or solids concentration; n is virtually independent of temperature but
may be affected by changes in particle size or form. The value of n lies
between 0 and 1 for shear-thinning fluids. The further away from a value of 1 the
more non-Newtonian is the fluid character.
A log-log plot of shear stress-shear rate data will show if the power law model is
applicable (see Figure 16). In this plot many fluids show an extensive range of
shear rate over which the power law will apply. However in the low shear rate
range it is likely that a degree of Newtonian behaviour will be observed and there
will be differences in slope observed in various regions of the plot.


FIGURE 16 APPLICATION OF POWER LAW

Similar slope changes may be observed at high shear rate. The power law
description may be used in design calculations if the shear rates are known to lie
within the power law region or may subsequently be shown to lie in this region.
The power law parameters may be determined from the slope (n) and the
extrapolated intercept, K at t = 1.
The power law model may also be used to describe the less common property of
dilatancy or shear-thickening. For these fluids the value of the flow behaviour
index (n) will be greater than 1.0, once again the departure from a value of unity
is an indication of the degree of non-Newtonian behavior. The value of K is again
an indication of resistance to flow. However, the comparison of values of K only
has real meaning where fluids have similar values of n. (This restriction applies to
both shear-thinning and shear-thickening fluids.)
6.2

Bingham Plastic

The Bingham Plastic model is used to describe shear-thinning fluids which have
a yield stress. The form of the equation is:


where the yield stress, ty, is the stress value which must be exceeded before
shear commences. The term, µp, plastic viscosity, is a parameter and not to be
Confused with the usual meaning of viscosity except when µp > ty.
A linear plot of shear stress/shear rate data as illustrated in Figure 17 will show if
the Bingham plastic model is applicable.
FIGURE 17 TYPICAL IDEAL BINGHAM PLASTIC BEHAVIOR

Many fluids give a plot of this kind, particularly for narrow ranges of shear rate. It
is important that linearity in such a case should not be regarded as justification
for an assumed yield stress by extrapolation, particularly if that extrapolation is
over a significant range of shear rate. If an understanding of the yield stress is
important to a particular exercise, it is desirable that data at low shear rate should
be available to improve the accuracy of the extrapolation.
It is frequently found that such plots are not entirely linear, both at low shear
rates and elsewhere. Figure 18 is such an example.


FIGURE 18 NON-LINEAR BINGHAM PLASTIC BEHAVIOR

In such cases extrapolation is more difficult but a modified Bingham model,
Herschel-Bulkley or generalised Bingham, can be used. This is expressed as:

The parameters Kp and n can be determined in a manner analogous to that used
for power law fluids by plotting log (tx - ty) vs log ẏ The intercept at ẏ = 1 gives a
value of Kp and the slope gives the value of n (usually n < 1). It may be
necessary to establish the best relationship bytrial and error, estimating various
rates of ty then Kp and ẏ to lead to a plot of t vs ẏ giving best least squares fit.
6.3

Direct use of Numerical Data

The use of mathematical models to describe the shear stress-shear rate
behaviour of non-Newtonian liquids has appeal, particularly if the form of
equation is amenable to subsequent mathematical manipulation, e.g. to evaluate
flow rate-pressure drop relationships. Also the use of models permits
extrapolation of the data, but this can be very dangerous when taken too far
and the use of a mathematical model (once adopted) can hide this extrapolation.


In many cases it is found that the viscometric data do not fit conveniently any of
the simple models often used, e.g. power law or Bingham plastic. However, this
is often not serious because the numerical data together with interpolation
methods can often be incorporated directly into design procedures thus
eliminating the model fitting stage. This procedure has the added advantage that
checks can continuously be made to examine whether or not the data are
to be extrapolated.
6.4

Rheological Models Involving Temperature Dependence

Most non-Newtonian liquids exhibit viscous properties which are highly
dependent upon temperature; as the temperature increases the viscosity usually
decreases (although a few complex materials act in the opposite way). It should
be remembered at all times that these variations in viscosity due to temperature
changes can often be more dramatic than the non-Newtonian changes in
apparent viscosity due to changes in shear rate.
This has two major implications:
(a)

Careful temperature control should be exercised in all Viscometer tests
and the temperature should be matched accurately to the process
temperature.

(b)

In heat transfer operations the variation in viscosity with temperature in the
equipment will be of significance in design and rheological data should be
collected over the appropriate range of temperatures.

In order to collect rheological data over a range of temperatures for timeindependent materials, measurements are taken of shear stress against shear
rate at a variety of temperatures (see!Figure! 19).


FIGURE 19 EFFECT OF TEMPERATURE

In many cases this data can be fed into a computer store and used directly in
design procedures, the appropriate process viscosity being obtained from the
test data by means of an interpolation method for the process shear rate and
temperature.
In some instances it is helpful to use a model of the temperature and shear rate
dependence of apparent viscosity and this can then be used to evaluate oa
under the appropriate processing conditions. This has the advantage that if the
model is physically realistic, extrapolation of the data outside the test range of
shear rate and/or temperature is feasible.
Two approaches are commonly used. The first is based on the Arrhenius
equation developed for Newtonian fluids, i.e.

where A and a are constants independent of temperature.


The equivalent form based on the power law equation is:

And this is rewritten as:

all of which can be found from viscometric tests over a range of temperatures.
If the material is found to have a significant yield stress, ty, the above equation
can be modified to give:

A second approach is based upon an alternative to the Arrhenius equation, i.e.

where µ1 is the viscosity at reference temperature T1 and β1 is a parameter,
independent of temperature. The power law modification to the form is:


and again these parameters can be obtained from Viscometer data covering an
appropriate range of shear rates and temperatures. As before, if the material has
a significant yield stress the equation becomes:

Equations (14) and (17) are the form:

A third approach convenient for numerical analysis evaluates the function F(T) as
a polynomial in T, e.g.


7

CHARACTERIZATION OF TIME-DEPENDENT LIQUIDS

In this section a brief description is given of some of the ways of determining
useful rheological data for time-dependent non-Newtonian liquids. Emphasis is
placed upon relatively simple procedures which will generate quantitative data
which can be used for engineering design purposes. Also the measurements will
give some idea as to the way in which such materials should be efficiently
processed e.g. it is advisable with thixotropic liquids which build up a
structure (and hence a high apparent viscosity) at very low shear rates or on
standing, to keep the material subjected to high shear rates at all times. Stopping
the process will produce a high viscosity liquid which could give severe start-up
problems.
The tests described below for time-dependency are best carried out in rotational
Viscometers, preferably in a co-axial cylinder device. Severe problems arise in
interpreting data from capillary Viscometers for time-dependent liquids.
In many cases it is often not necessary to know the complete rheological
characterisation of the liquid - indeed this is a research project in its own right for
many materials. However, the procedures outlined below will give useful data on
the material in its worst state and under normal steady state operating conditions.
7.1

Sample Loading

The viscous properties of a time-dependent liquid depend upon the previous
shearing to which the material has been subjected. This creates an immediate
problem for characterization in a Viscometer because the act of loading the
sample into the instrument creates a shearing action.
In order to load the Viscometer with a sample, and to achieve a reproducible test
procedure, it is recommended that the liquid is put into the Viscometer and then
sheared in the device at a constant shear rate until the shear stress reading it
becomes essentially constant (i.e. it has reached its equilibrium value, see 4.2).
Care must be taken in choosing the shear rate, since it is possible to destroy a
very large part of the rheological structure. This effect can be used to advantage
as a measurement technique by breaking the structure initially and then reducing
the shear rate; provided that the material recovers in a practicable time. The
material can then be used directly for characterisation at other shear rates or
alternatively the sample can be allowed to rest for a fixed time before the start of
the characterisation. This latter procedure permits good thermostatting of the
sample for cases where the test temperature is important.


7.2

Tests at Constant Shear Rate

After the sample has been loaded into the Viscometer, sheared to equilibrium
and then rested, as described in the previous Clause, the extent of the material's
time-dependency can be assessed. This is done by setting the Viscometer at a
constant shear rate,
and observing the change in the measured shear
apparent viscosity, µa), see Figure 20.
stress, Ԏ, (or
FIGURE 20 TYPICAL CONSTANT SHEAR RATE DATA

This process can be repeated (including loading each time) at a variety of shear
rates relevant to the actual processing conditions and the extent of the timedependency can be assessed by comparing the initial and equilibrium stresses or
apparent viscosities. Also data presented in this way give a measure of the rate
at which structure is either broken down or built up under shearing conditions.
In deciding whether or not the changes from Ԏ1 to Ԏeq (or µ1 to µeq) are
significant, the importance of the viscous properties in the actual process and the
accuracy of the design equations to be used must be assessed. However, as a
rule of thumb, changes of less than 10% in stress or viscosity with time under
constant shear rate conditions indicate that timedependency can be ignored for
design purposes. Changes of about 30% or more should certainly be allowed for.
If the actual process is one of approximately constant shear rate, then the curves
above indicate the extremes of behaviour of the liquid which can be encountered.
For example, for a thixotropic liquid the apparent viscosity of the rested sample is
the highest likely value and its use in design will be conservative.

7.3

Dynamic Response Measurement

When interpreting the significance of the measurements of time-dependency
obtained from rotational Viscometers at constant shear rate, it should be
remembered at all times that the material is also shear rate dependent. For
example, if we take a sample which has been subjected to shear rate ẏ1 for time
t1 in the Viscometer, at the time Ԏ1 the material is shear dependent and sudden
changes in shear rate will bring instantaneous responses in apparent viscosity,
as well as the longer term time-dependent effect shown above.
Consider the case of a thixotropic liquid which is sheared at ẏ1 for time Ԏ1, see
Figure 21.
FIGURE 21 DYNAMIC RESPONSE DATA

If the Viscometer is quickly changed to a series of new shear rates ẏa, ẏb, ẏc, etc.
the non-Newtonian (in this case, shear-thinning response) will be revealed, i.e.
we observe that the apparent viscosity is dependent upon shear rate. This test
needs to be conducted rapidly so that the longer term time-dependent effects of
shearing are eliminated. Of course, this calls for an instrument with a good
dynamic response, otherwise the measurements reflect the mechanical
responses of the instrument to rapidly imposed speed changes.


7.4

Changes in Shear Rate

In some cases it is important to know how the material will behave if it is
subjected to changes in shear rate. In such instances relevant Viscometer data
can be obtained by subjecting a sample to one shear rate, ẏ1 for a period of time
Ԏ1 and then changing the shear rate instantaneously to a new value ẏ2, see
Figure 22.
FIGURE 22 CHANGE IN SHEAR RATE DATA

Here the case is illustrated of a thixotropic liquid which is subjected to a decrease
in shear rate from ẏ1 to ẏ2 at time t1. At time t1 the instantaneous change in
shear rate brings about the initial dynamic response giving a lower stress and
higher apparent viscosity because the shear rate is lowered and it is assumed
that the material is shear-thinning. After time t1 a recovery of viscosity is
observed. It is worthy of note that the rate of breakdown of viscosity under shear
is not necessarily the same as the rate of recovery. Often it is found that
breakdown is a much more rapid process than recovery but this is not
necessarily so in all cases.
Other possible forms of response to imposed shear rate changes in the
Viscometer are shown in Figure 23. In these Figures the rapid initial dynamic
response to changes in shear rate has been neglected and the curves drawn are
typical of the responses obtained from a Viscometer with a poor dynamic
response to imposed speed changes.

Using the Viscometer in this way the response of time-dependent materials to
changes in the process shear rate can be modelled.
FIGURE 23 OTHER FORMS OF RESPONSE TO CHANGES IN SHEAR RATE


7.5

Concluding Remarks

The detailed study of the rheology of time-dependent non-Newtonian liquids is a
complex topic. However, in this Guide relatively simple tests have been outlined
which enable measurements to be taken which are useful in design.
(a)

A procedure for loading the Viscometer to obtain reproducible results is
presented.

(b)

Simple steady shear rate tests allow the reader to assess whether or not
the material is significantly time-dependent. If it is, such experiments allow
an assessment to be made of the best and worst possible states of the
material during processing.

(c)

Dynamic response tests are outlined which indicate the shear rate
dependency of the apparent viscosity. Such tests should be carried out
rapidly and require an instrument with a good dynamic response to speed
changes.

(d)

Tests are outlined which permit Viscometer data to be obtained in a way
which will model shear rate changes experienced during processing.

8

TECHNIQUES FOR CHARACTERISATION OF VISCOELASTIC
LIQUIDS

There are a number of commercial Rheometers which allow some
measurements to be made of viscoelastic properties. In almost all cases it is
difficult to either interpret or apply the measurements made. The most immediate
use that can be made of such measurements is in quality control problems.
There are a number of different types of measurement that can be made and
these are:
8.1

Stress Relaxation

This is probably the least demanding experiment in terms of the instrumentation
required and so is relatively inexpensive. After a known shear history, shearing is
stopped and the stress-time (decay) process recorded.
Measurements of strain relaxation can also be made with some instruments and
it may be possible to convert some stress relaxation measurements to strain.

8.2

Oscillatory Shear Measurements

This test facility is available on a number of instruments. Measurements may be
carried out at a range of frequencies and amplitudes, usually in rotational
Viscometers using cone and plate geometry. The data collected are usually more
tractable if small amplitudes are used (usually referred to as the 'linear' range).
The data collected are usually in the form of amplitude ratio (output
displacement, input displacement) and phase angle (output relative to input
displacement). If the fluid tested is inelastic the input and output displacement will
have a phase angle of 90°. For fluids having elastic properties the phase angle
will be less than 90° and viscosity and elastic moduli can be calculated from the
components of the output displacement in phase and out of phase with the input
amplitude. These two moduli are usually plotted as functions of frequency.
8.3

Normal Force Measurement

Facilities for this measurement are available to varying degrees of sophistication
in several rotational Viscometers. In this experiment, the elasticity of the test fluid
manifests itself as a force tending to separate cone and plate or plate and plate
in steady shear experiments. The data from these two types of experiment can
be used to determine first and second normal force differences respectively, both
as functions of shear rate.
8.4

Elongational Viscosity Measurement

Experiments to determine elongational viscosity can be conducted in a number of
ways. One of the most convenient in terms of experimental accuracy is the
suspended siphon test. For many viscoelastic fluids it is possible to establish an
open siphon in which a stream of liquid is lifted from a reservoir to a tube by
virtue of its axial tensile strength. By using measurements of the weight of the
suspended thread of liquid, the liquid flow rate and the shape of the thread
(photographs) elongational viscosity may be determined as a function of
extension rate.


10 BIBLIOGRAPHY
Ref

Source

[1]

W L Wilkinson, 'Non-Newtonian Fluids', Pergamon, New York,
1960.

[2]

K Walters, 'Rheometry', Chapman & Hall, 1975.

[3]

G Astarita et al (eds), 'Rheology' Vol. 1, 2, Plenum Press, 1980.

[4]

R I Tanner, 'Engineering Rheology', Clarendon Press, Oxford,
1985.

[5]

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

[6]

K J Carpenter, T E Heald, 'Derivation of Rheological Data from
Non-Standard Sensors on the Haake Rotovisco Viscometer'

[7]

P A Shamlov, M F Edwards, 'Power Consumption of Anchor
Impellers in Newtonian and Non-Newtonian Liquids' Chem. Eng.
Res. Des., Vol 67, Sept 1989.

[8]

L Choplin, T Merquinol, 'Mixing of Viscoelastic Fermentation Broth
with Helical Ribbon-Screw (HRS) Impeller' 6th European
Conference on Mixing, Pavia, Italy 1988.


APPENDIX A
A.1

EQUATIONS FOR VISCOMETERS

EQUATIONS FOR CONCENTRIC CYLINDER VISCOMETERS

A.1.1 Equations for Newtonian Fluids
Newtonian Fluids
The level of viscous drag at any time is described in terms of force per unit area,
the shear stress, Ԏ.

Viscous drag is a function of the number of intermolecular collisions occurring per
unit time, this being described in terms of velocity gradient or shear rate, ẏ.

For Newtonian fluids the shear stress-shear rate relationship is adequately
described by the viscosity which is constant at constant temperature (and
pressure).

Viscosity = Ԏ /ẏ

=µ

Note that the calculation of the shear rate is only valid for Newtonian fluids.
No allowance for the influence of end effects is included. For discussion of this,
refer to 5.5.2


A.1.2 Standard Equations for non-Newtonian Fluids
The shear stress, Ԏ, may be calculated as for Newtonian fluids, but the shear
rate requires a calculation procedure based on the data collected over a range of
speeds.
(a)

Plot the torque-speed data on log-log coordinates as shown in Figure 24
(where the data result in a linear plot) or Figure 24 (b) (where a non-linear
plot results).

FIGURE 24 TORQUE SPEED PLOTS

(b) Calculate the shear rate ẏ as follows:
case (i) for linear plot

ẏ


where np is the slope of the linear log-log plot
case (ii) for non-linear plots

ẏ

where np is the slope of the log T vs log N plot at each value of N. As np varies
with N, the appropriate value must be applied for each value of N used in the
calculation.
The procedure is only valid if (ln s)/np < 0.5, which is generally the case for
commercial Viscometers. A more cumbersome calculation procedure is required
if this is not the case.
This procedure can also be applied to the frequently encountered pseudoplastic
or shearthinning fluids and the less common dilatant or shear-thickening fluids.
A.1.3 Calculation for Bingham Plastic Fluids
For Bingham plastic fluids which give a linear plot of torque against speed, as
illustrated in Figure 25, the Newtonian relationship for shear rate:

may be used.
However, if the plot is not linear the data should be re-examined as for A.1.2(b),
case (i) and (ii) using a log T vs log N plot. It is most likely that case (ii) will apply
(non constant value of np) and shear rate may be calculated in the same way,
remembering the restriction


FIGURE 25 TORQUE vs SPEED PLOT FOR BINGHAM PLASTICS

A.2 EQUATIONS FOR CONE AND PLATE VISCOMETERS
Provided the gap angle (θc) is less than 4°, the shear rate can be calculated from
the angle and speed alone and the equations are valid for both Newtonian and
non-Newtonian fluids.


A.3

EQUATIONS FOR PARALLEL PLATE VISCOMETER

The shear stress at radius R is given by:


A.4

EQUATIONS FOR TUBE OR CAPILLARY VISCOMETERS

For Newtonian and non-Newtonian fluids, the shear stress and shear rate vary
with radial position in the tube. The values are therefore usually calculated at the
tube wall as follows:
Shear stress at the wall

The bracketed term in Equation 31(b) is called the Rabinowitsch correction.
Since m can be as low as 0.2, this correction can be highly significant in
determining the true wall shear rate.
If this plot is not linear, a value of m must be taken for each value of Q.


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