Non-Newtonian flow and applied rheology

523 Pages·2008·5.49 MB·English

by  Chhabra R.P.

Preface to First Edition
Non-Newtonian fl ow and rheology are subjects which are essentially interdisciplinary in
their nature and which are also wide in their areas of application. Indeed non-Newtonian
fl uid behaviour is encountered in almost all the chemical and allied processing indus-
tries. The factors which determine the rheological characteristics of a material are highly
complex, and their full understanding necessitates a contribution from physicists, chem-
ists and applied mathematicians, amongst others, few of whom may have regarded the
subject as central to their disciplines. Furthermore, the areas of application are also
extremely broad and diverse, and require an important input from engineers with a wide
range of backgrounds, though chemical and process engineers, by virtue of their role in
the handling and processing of complex materials (such as foams, slurries, emulsions,
polymer melts and solutions, etc.), have a dominant interest. Furthermore, the subject
is of interest both to highly theoretical mathematicians and scientists and to practising
engineers with very different cultural backgrounds.
O wing to this inter-disciplinary nature of the subject, communication across subject
boundaries has been poor and continues to pose diffi culties, and therefore, much of the
literature, including books, is directed to a relatively narrow readership with the result
that the engineer faced with the problem of processing such rheological complex fl uids,
or of designing a material with rheological properties appropriate to its end use, is not
well served by the available literature. Nor does he have access to information presented
in a form which is readily intelligible to the non-specialist. This book is intended to
bridge this gap but, at the same time, is written in such a way as to provide an entrée to
the specialist literature for the benefi t of scientists and engineers with a wide range of
backgrounds. Non-Newtonian fl ow and rheology is an area with many pitfalls for the
unwary, and it is hoped that this book will not only forewarn readers but also equip them
to avoid some of the hazards.
C overage of topics is extensive and this book offers a unique selection of material.
There are eight chapters in all.
The introductory material, Chapter 1, introduces the reader to the range of non-
Newtonian characteristics displayed by materials encountered in every day life as well
as in technology. A selection of simple fl uid models which are used extensively in proc-
ess design calculations is included here.
Chapter 2 deals with the characterization of materials and the measurement of their
rheological properties using a range of commercially available instruments. The impor-
tance of adequate rheological characterization of a material under conditions as close as
possible to that in the envisaged application cannot be overemphasized here. Stress is
laid on the dangers of extrapolation beyond the range of variables covered in the experi-
mental characterization. Dr. P.R. Williams (Reader, Department of Chemical Biological
Process Engineering, Swansea, University of Wales, UK) who has contributed this chap-
ter is in the forefront of the development of novel instrumentations in the fi eld.
The fl ow of non-Newtonian fl uids in circular and non-circular ducts encompass-
ing both laminar and turbulent regimes is presented in C hapter 3 . Issues relating to the
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xiv Preface to First Edition
transition from laminar to turbulent fl ow, minor losses in fi ttings and fl ow in pumps, as
well as metering of fl ow, are also discussed in this chapter.
Chapter 4 deals with the highly complex but industrially important topic of mul-
tiphase systems – gas/non-Newtonian liquid and solid/non-Newtonian liquids – in pipes.
A thorough treatment of particulate systems ranging from the behaviour of particles
and drops in non-Newtonian liquids to the fl ow in packed and fl uidized beds is presented
in Chapter 5 .
The heating or cooling of process streams is frequently required. C hapter 6 discusses
the fundamentals of convective heat transfer to non-Newtonian fl uids in circular and non-
circular tubes under a range of boundary and fl ow conditions. Limited information on
heat transfer from variously shaped objects – plates, cylinders and spheres – immersed
in non-Newtonian fl uids is also included here.
The basics of the boundary layer fl ow are introduced in C hapter 7 . Heat and mass
transfer in boundary layers and practical correlations for the estimation of transfer coef-
fi cients are included.
The fi nal Chapter 8 deals with the mixing of highly viscous and/or non-Newtonian
substances, with particular emphasis on the estimation of power consumption and mix-
ing time, and on equipment selection.
A t each stage, considerable effort has been made to present the most reliable and
generally accepted methods for calculations, as the contemporary literature is inundated
with confl icting information. This applies especially in regard to the estimation of pres-
sure gradients for turbulent fl ow in pipes. In addition, a list of specialist and/or advanced
sources of information has been provided in each chapter as “ Further reading ” .
In each chapter a number of worked examples have been presented, which, we
believe, are essential to a proper understanding of the methods of treatment given in
the text. It is desirable for both a student and a practising engineer to understand an
appropriate illustrative example before tackling fresh practical problems themself.
Engineering problems require a numerical answer and it is thus essential for the reader
to become familiar with the various techniques so that the most appropriate answer can
be obtained by systematic methods rather than by intuition. Further exercises which the
reader may wish to tackle are given at the end of the book .
Incompressibility of the fl uid has generally been assumed throughout the book,
albeit this is not always stated explicitly. This is a satisfactory approximation for most
non-Newtonian substances, notable exceptions being the cases of foams and froths.
Likewise, the assumption of isotropy is also reasonable in most cases except perhaps for
liquid crystals and for fi bre-fi lled polymer matrices. Finally, although the slip effects are
known to be important in some multiphase systems (suspensions, emulsions, etc.) and in
narrow channels, the usual no-slip boundary condition is regarded as a good approxima-
tion in the type of engineering fl ow situations dealt with in this book.
In part, the writing of this book was inspired by the work of W.L. Wilkinson N on-
Newtonian Fluids, published by Pergamon Press in 1960, and now long out-of-print, and
it is hoped that readers will fi nd it to be a welcome successor.
R.P. Chhabra
J.F. Richardson

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Preface to Second Edition
In presenting this new edition, we would like to thank the many individuals from all over
the world who have pointed out errors, and, more importantly, have made suggestions
for improvement in the text. Bearing in mind these ideas together with the points raised
in independent reviews of the fi rst edition, the entire text has been reviewed. Therefore
where the need was recognized, the presentation has been improved by re-organizing, or
by expanding the existing material, or by adding new text and/or illustrative examples
throughout the whole book to facilitate comprehension of the material covered here.
Apart from the overall general updating to include state-of-the-art information, the spe-
cifi c changes made to the fi rst edition are summarized here. The discussion in Chapter 1
has been sharpened to highlight the wide occurrence of, and the effects arising from,
non-Newtonian fl ow behaviour in diverse industrial settings including food, pharma-
ceutical, personal care and household products, as well as in biomedical and biologi-
cal process engineering applications. Included here is also a new section to emphasize
the importance of the intimate link between the micro-structure and rheology of a fl uid
which is used widely to formulate new products, especially in food and personal-care
product sectors to meet ever increasing expectations of the consumers. Chapter 2 has
been completely re-vamped to highlight the importance of proper rheological charac-
terization of complex fl uids, together with a discussion on the relative merits and de-
merits of various rheological techniques in current use. Conversely, online rheological
measurements are increasingly used to monitor product quality continuously. This topic
is addressed in a new section on online viscometry in this chapter. While very little new
material has been added to C hapter 3 , the existing coverage has been expanded signifi -
cantly by providing detailed derivations of important relationships in order to improve
clarity and several new illustrative examples have been introduced here. The role of
non-Newtonian characteristics in continuous thermal treatment of foods is highlighted
in a new section on the residence time distributions of the two phases in Chapter 4 .
In addition, this chapter provides a short introduction to the two-phase fl ow of gas and
drag reducing polymer solutions in pipes, as practised in the oil industry. In Chapter 5 ,
the existing discussion on the estimation of drag or the terminal falling velocity of non-
spherical (isometric) particles settling in non-Newtonian media has been expanded to
refl ect the recent developments in this fi eld. Chapters 6 and 7, which have been accepted
in their existing form, have remain largely unchanged. Finally, the discussion on liquid
mixing in Chapter 8 has been strengthened in general by adding several new examples
and by adding more detailed discussions on static mixers, novel impeller designs and the
prediction of mixing times in particular. Lastly, several new exercises have been added
at the end of the book for the benefi t of the student.
W ith the rapid advances occurring in this vast interdisciplinary fi eld, both the selec-
tion of material and its arrangement are becoming increasingly diffi cult, and must be to a
great extent a matter of personal choice, but we hope that this new edition will continue
to provide a sound basis for a study of the fundamentals of the subject and will also be
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xii Preface to Second Edition
of some value to practising professionals who must deal with such diffi cult materials on
a day-to-day basis. In closing, we very much hope that our readers will continue to make
suggestions for further improvements in this work.
R.P. Chhabra
J.F. Richardson
March 2008
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Acknowledgements (First Edition)
The inspiration for this book originated in two works which have long been out-of-print
and which have been of great value to those working and studying in the fi eld of
non-newtonian technology. They are W.L. Wilkinson’ s excellent introductory book,
Non-Newtonian Flow (Pergamon Press, 1959), and J.M. Smith’ s chapter in the fi rst two
ed itions of Coulson and Richardson ’ s Chemical Engineering, Volume 3 (Pergamon Press,
1970 and 1978). The original intention was that R.P. Chhabra would join with the above
two authors in the preparation of a successor but, unfortunately, neither of them had the
necessary time available to devote to the task, and Raj Chhabra agreed to proceed on his
own with my assistance. We would like to thank Bill Wilkinson and John Smith for their
encouragement and support.
The chapter on Rheological Measurements has been prepared by Dr. P.R. Williams,
Reader in the Department of Chemical and Biochemical Process Engineering
at the University of Wales, Swansea – an expert in the fi eld. Thanks are due also to
Dr. D.G. Peacock, formerly of the School of Phamacy, University of London, for work
on the compilation and processing of the Index.
J.F. Richardson
January 1999

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Chapter 1
Non-Newtonian fl uid behaviour
1.1 Introduction
One may classify fl uids in two different ways; either according to their response to the exter-
nally applied pressure or according to the effects produced under the action of a shear stress.
The fi rst scheme of classifi cation leads to the so called ‘ compressible ’ and ‘ incompressible ’
fl uids, depending upon whether or not the volume of an element of fl uid is dependent on its
pressure. While compressibility infl uences the fl ow characteristics of gases, liquids can nor-
mally be regarded as incompressible and it is their response to shearing which is of greater
importance. In this chapter, the fl ow characteristics of single-phase liquids, solutions and
pseudo-homogeneous mixtures (such as slurries, emulsions, gas–liquid dispersions) which
may be treated as a continuum if they are stable in the absence of turbulent eddies are con-
sidered depending upon their response to externally imposed shearing action.
1.2 Classifi cation of fl uid behaviour
1.2.1 Defi nition of a Newtonian fl uid
Consider a thin layer of a fl uid contained between two parallel planes a distance d y
apart, as shown in F igure 1.1. Now, if under steady state conditions, the fl uid is subjected
to a shear by the application of a force F as shown, this will be balanced by an equal and
opposite internal frictional force in the fl uid. For an incompressible Newtonian fl uid in
laminar fl ow, the resulting shear stress is equal to the product of the shear rate and the
viscosity of the fl uid medium. In this simple case, the shear rate may be expressed as the
velocity gradient in the direction perpendicular to that of the shear force, i.e.,
⎛ ⎞
FA (cid:2) τyx (cid:2) μ⎝⎜⎜⎜⎜(cid:3)ddVyx⎠⎟⎟⎟⎟(cid:2) μγ(cid:2)yx ( 1.1)

Surface area A
F dVx
y
dy
x
Figure 1.1 Schematic representation of unidirectional shearing fl ow
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2 Non-Newtonian Flow and Applied Rheology: Engineering Applications
Note that the fi rst subscript on both τ and γ(cid:2) indicates the direction normal to that
of shearing surface, while the second subscript refers to the direction of the force and
the fl ow. By considering the equilibrium of a fl uid layer, it can readily be seen that at
any shear plane there are two equal and opposite shear stresses – a positive one on the
slower moving fl uid and a negative one on the faster moving fl uid layer. The negative
sign on the right hand side of equation (1.1) indicates that τ is a measure of the resist-
yx
ance to motion. One can also view the situation from a different standpoint as: for an
incompressible fl uid of density ρ , equation (1.1) can be written as:
μ d
τ (cid:2)(cid:3) (ρV ) ( 1.2)
yx ρ dy x

The quantity ‘ ρ V ’ is the linear momentum in the x -direction per unit volume of the
x
fl uid and hence τ represents the momentum fl ux in the y- direction and the negative
yx
sign indicates that the momentum transfer occurs in the direction of decreasing velocity
which is also in line with the Fourier’s law of heat transfer and Fick’s law of diffusive
mass transfer.
The constant of proportionality, μ (or the ratio of the shear stress to the rate of shear)
γ(cid:2)
which is called the Newtonian viscosity is, by defi nition, independent of shear rate ( yx )
or shear stress ( τ ) and depends only on the material and its temperature and pressure.
The plot of shear ysxtress ( τ yx) against shear rate ( γ(cid:2)yx) for a Newtonian fl uid, the so-called
‘ fl ow curve ’ or ‘ r heogram’ , is therefore a straight line of slope, μ , and passing through
the origin; the single constant, μ , thus completely characterizes the fl ow behaviour of a
Newtonian fl uid at a fi xed temperature and pressure. Gases, simple organic liquids, solu-
tions of low molecular weight inorganic salts, molten metals and salts are all Newtonian
fl uids. The shear stress–shear rate data shown in Figure 1.2 demonstrate the Newtonian
fl uid behaviour of a cooking oil and a corn syrup; the values of the viscosity for some
substances encountered in everyday life are given in Table 1.1 .
Figure 1.1 and equation (1.1) represent the simplest case wherein the velocity vector
which has only one component, in the x- direction and it varies only in the y -direction.
Such a fl ow confi guration is known as simple shear fl ow. For the more complex case of
three-dimensional fl ow, it is necessary to set up the appropriate partial differential equa-
tions. For instance, the more general case of an incompressible Newtonian fl uid may be
expressed – for the x -plane (area oriented normal to the x -direction) – as follows ( Bird
et al ., 1987, 2002 ):
τxx (cid:2)(cid:3)2μ∂∂Vxx (cid:4) 23μ⎛⎝⎜⎜⎜⎜∂∂Vxx (cid:4) ∂∂Vyy (cid:4) ∂∂Vzz⎞⎠⎟⎟⎟⎟⎟ ( 1.3)

τxy (cid:2)(cid:3)μ⎛⎝⎜⎜⎜⎜∂∂Vyx (cid:4) ∂∂Vxy⎞⎠⎟⎟⎟⎟⎟ ( 1.4)

τxz (cid:2)(cid:3)μ⎛⎝⎜⎜⎜⎜∂∂Vzx (cid:4) ∂∂Vxz⎞⎠⎟⎟⎟⎟ ( 1.5)

S imilar sets of equations can be drawn up for the forces acting on the y- and z -planes;
in each case, there are two (in-plane) shearing components and a normal component.
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Non-Newtonian fl uid behaviour 3
0 10 20 30 40 50 60 70 80
550
Cooking oil (T = 294 K)
100 Corn syrup (T = 297 K) 500
90 450
80 400
Slope = m = 11.6 Pa s
70 350
a)
P
s ( 60 300
s
e
str
ar 50 250
e
h
S
40 200
30 150
20 Slope= m = 0.064 Pa s 100
10 50
0 0
0 400 800 1200
Shear rate (s−1)
Figure 1.2 Typical shear stress–shear rate data for a cooking oil and a corn syrup
Figure 1.3 shows the nine stress components schematically in an element of fl uid. By
considering the equilibrium of a fl uid element, it can be shown that τ (cid:2) τ ; τ (cid:2) τ
yx xy xz zx
and τ (cid:2) τ . The normal stresses can be visualized as being made up of two compo-
yz zy
nents: isotropic pressure and a contribution due to fl ow, i.e.,
P (cid:2)(cid:3)p(cid:4)τ ( 1.6a)
xx xx
P (cid:2)(cid:3)p(cid:4)τ ( 1.6b)
yy yy
P (cid:2)(cid:3)p(cid:4)τ ( 1.6c)
zz zz
where τ , τ , τ contributions arising from fl ow are known as deviatoric normal
xx yy zz
stresses for Newtonian fl uids and as extra stresses for non-Newtonian fl uids. For an
incompressible Newtonian fl uid, the isotropic pressure is given by:

p (cid:2)(cid:3)1(P (cid:4)P (cid:4)P ) (1.7)
3 xx yy zz
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4 Non-Newtonian Flow and Applied Rheology: Engineering Applications
Table 1.1 Typical viscosity values at room temperature
Substance μ (mPa s)
Air 10 (cid:3) 2
Benzene 0.65
Water 1
Molten sodium chloride (1173 K) 1.01
Ethyl alcohol 1.20
Mercury (293 K) 1.55
Molten lead (673 K) 2.33
Ethylene glycol 20
Olive oil 100
Castor oil 600
100% Glycerine (293 K) 1500
Honey 104
Corn syrup 10 5
Bitumen 10 11
Molten glass 10 15
y
P
yy
τ τ
yz yxτ
xy
τ
zy
P Flow
τ xx
τ xz
zx
P
zz x
z
Figure 1.3 Stress components in three-dimensional fl ow
From equations (1.6) and (1.7) it follows that:
τ (cid:4)τ (cid:4)τ (cid:2) 0 (1.8)
xx yy zz
For a Newtonian fl uid in simple shearing motion, the deviatoric normal stress compo-
nents are identically zero, i.e.,
τ (cid:2) τ (cid:2) τ (cid:2) 0 (1.9)
xx yy zz
Thus, the complete defi nition of a Newtonian fl uid is that it not only possesses a con-
stant viscosity but it also satisfi es the condition of equation (1.9), or simply that it satisfi es
the complete Navier–Stokes equations. Thus, for instance, the so-called constant viscosity
Boger fl uids ( Boger, 1976 ; Prilutski et al., 1983) which display constant shear viscosity
but do not conform to equation (1.9) must be classed as non-Newtonian fl uids. A cursory
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