transport properties of fluids their correlation prediction & estimation

Transport Properties of Fluids
Transport Properties of Fluids
Their Correlation, Prediction and Estimation
Edited by
Jtirgen Millat
NORDUMInstitutfiir Umwelt undAnalytik
GmbH,
Kessin/Rostock, Germany
J. H. Dymond
The University, Glasgow, UK
C. A. Nieto de Castro
University of Lisbon, Portugal
IUPAC
CAMBRIDGE
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© The International Union of Pure and Applied Chemistry 1996
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Library of Congress Cataloguing in Publication data
Transport properties of fluids
:
their correlation, prediction and estimation / edited by Jurgen Millat,
J.H. Dymond, C.A. Nieto de Castro.
p.
cm.
ISBN 0-521-46178-2 (hardcover)
1.
Fluid dynamics. 2. Transport theory. 3. Fluids—Thermal properties.
I. Millat, Jurgen. II. Dymond, J. H. (John H.) III. Castro, C.A. Nieto de
QC151.T73 1996
530.4'25-dc20 95-12766
ISBN-13 978-0-521-46178-8 hardback
ISBN-10 0-521-46178-2 hardback
ISBN-13 978-0-521-02290-3 paperback
ISBN-10 0-521-02290-8 paperback
This book is dedicated by its editors and authors to the memory of Professors
Joseph Kestin (1913-1993)
and
Edward A. Mason (1926-1994).
Their outstanding research and inspiration contributed greatly to the concept and
content of this volume.
Contents
List of contributors page ix
Foreword xiii
Part one:
GENERAL
1 Introduction

7.
Millat, J. H. Dymond and
C.
A. Nieto de Castro 3
2 Technological Importance
W.
A.
Wakeham
and
C.
A. Nieto de Castro 6
3 Methodology
C. A. Nieto de
Castro
and W A. Wakeham 17
Part two:
THEORY
4 Transport Properties of Dilute Gases and Gaseous Mixtures
J. Millat, V
Vesovic
and
W.
A. Wakeham 29
5 Dense Fluids
J. H.
Dymond,
E. Bich, E.
Vogel,
W A. Wakeham,
V

Vesovic
and
M.
J. Assael 66
6 The Critical Enhancements
/ V Sengers and
J.
Luettmer-Strathmann 113
Part three:
DATA REPRESENTATION
7 Correlation Techniques
D.
G. Friend and
R.
A. Perkins 141
8 Equations of State
K. M. de
Reuck,
R. J, B.
Craven
and
A.
E. Elhassan 165
vn
viii Contents
Part four:
APPLICATION OF SELECTED METHODS
9 Computer Calculation
C.Hoheisel
189

H. / M.
Hartley
and
D.
J. Evans 210
10 Modified Hard-Spheres Scheme
J. H. Dymond and
M.
J. Assael 226
11 The Corresponding-States Principle: Dilute Gases
E. A. Mason and
F.
J. Uribe 250
12 The Corresponding-States Principle: Dense Fluids
M. L Huber and
H.
J. M. Hanley 283
13 Empirical Estimation
B.
E. Poling 296
Part five:
APPLICATION TO SELECTED SUBSTANCES
14 Pure Fluids
E.
P.
Sakonidou, H. /?. van den
Berg,
J. V Sengers,
J. Millat, V
Vesovic,

A. Nagashima, J. H.
Dymond,
R.
Krauss and
K.
Stephan 311
15 Binary Mixtures: Carbon Dioxide-Ethane
W.
A.
Wakeham,
V
Vesovic,
E.
Vogel
and
S.
Hendl 388
16 Reacting Mixtures at Low Density - Alkali Metal Vapors
P.
S. Fialho, M. L V
Ramires,
C.
A. Nieto de Castro
and
J.
M. N. A. Fareleira 400
Part six:
DATA BANKS AND PREDICTION PACKAGES
17 Data Collection and Dissemination Systems
R.

Krauss, K. Stephan, A. I. Johns, J.
T.
R. Watson,
K. M. de
Reuck,
R. J. B.
Craven,
A. E. Elhassan,
K. N. Marsh, R. C.
Wilhoit
and
A.
A. Vasserman 423
Index 478
Contributors
M. J. Assael
Faculty
of Chemical
Engineering,
Aristotle
University,
Univ.
Box
453,
GR-54006Thessaloniki,
Greece
E. Bich
Universitdt Rostock, Fachbereich Chemie, D-18055 Rostock, Germany
R. J. B. Craven
IUPAC Thermodynamic Tables Project Centre, Department of Chemical Engineering and

Chemical
Technology,
Imperial
College,
London SW7
2BY,
UK
K. M. de Reuck
IUPAC Thermodynamic Tables Project Centre, Department of Chemical Engineering and
Chemical
Technology,
Imperial
College,
London SW7
2BY,
UK
J. H. Dymond
Chemistry Department, The University, Glasgow G12 8QQ, UK
A.
E. Elhassan
IUPAC Thermodynamic Tables Project Centre, Department of Chemical Engineering and
Chemical
Technology,
Imperial
College,
London SW7
2BY,
UK
D.
J. Evans

Research School of Chemistry, The Australian National University, Canberra ACT 0200,
Australia
J. M. N. A. Fareleira
Technical University of Lisbon,
Av.
Rovisco
Pais,
P-1096
Lisbon,
Portugal
P.
S. Fialho
University of Azores, P-9702 Angra do Heroismo
Codex,
Portugal
D.
G. Friend
National Institute of Standards and
Technology,
Thermophysics Division,
Boulder,
CO 80303,
USA
H. J. M. Hanley
National Institute of Standards and
Technology,
Thermophysics
Division,
Boulder,
CO 80303,

USA
IX
x Contributors
C. Hoheisel
Ruhr-Universitdt Bochum, Theoretische Chemie, D-44799 Bochum, Germany
M. L. Huber
National Institute of Standards and
Technology,
Thermophysics Division,
Boulder,
CO 80303,
USA
A. I. Johns
National Engineering Laboratory, East
Kilbride,
Glasgow G75
OQU,
UK
R. Krauss
Universitdt
Stuttgart,
Institut fur Technische Thermodynamik und Thermische Verfahrens-
technik,
D-70550
Stuttgart,
Germany
J. Luettmer-Strathmann
Institute for Physical Science &
Technology,
University

of
Maryland,
College
Park,
MD 20742,
USA
K. N. Marsh
Thermodynamics Research
Center,
Texas
A&M
University
System,
College
Station,
TX 77843-
3111,
USA
E. A. Mason*
Department of Chemistry, Brown
University,
Providence, R.I., 02912, USA
J. Millat
NORDUM Institut fur Umwelt und Analytik
GmbH,
Gewerbepark Am
Weidenbruch,
D-18196
Kessin/Rostock,
Germany

A. Nagashima
Faculty of Science and
Technology,
Department of Mechanical Engineering, Keio University,
3-14-1
Hiyoshi,
Yokohama
223, Japan
C. A. Nieto de Castro
Chemistry Department, Faculty of Sciences, University of Lisbon, Campo Grande, Ed. Cl -
Piso 5, P-1700 Lisbon, Portugal
R. A. Perkins
National Institute of Standards and
Technology,
Thermophysics Division,
Boulder,
CO 80303,
USA
B.
E. Poling
College of Engineering, The University of Toledo, Toledo, Ohio 43606-3390, USA
M. L. V. Ramires
Chemistry Department, Faculty of Sciences, University of Lisbon, Campo Grande, Ed. Cl -
Piso 5, P-1700 Lisbon, Portugal
E. P. Sakonidou
Van der Waals-Zeeman Laboratory, University of Amsterdam, Valckenierstraat
65-67,
NL-1018 XE Amsterdam, The Netherlands
J. V. Sengers
Institute for Physical Science

&
Technology,
University
of
Maryland,
College
Park,
MD 20742,
USA
* Deceased November 1994.
Contributors xi
K. Stephan
Universitdt Stuttgart, Institut fiir Technische Thermodynamik und Thermische Verfahrens-
technik,
D-70550 Stuttgart, Germany
F.
J. Uribe
Department of Physics, Universidad Autonoma
Metropolitana,
Av.
Michoacdcan y Calz. de la
Purisima, 09340 Mexico,
D.F.,
Mexico
H. R. van den Berg
Van der Waals-Zeeman Laboratory, University of Amsterdam, Valckenierstraat 65-67,
NL-1018 XE Amsterdam, The Netherlands
A. A. Vasserman
Odessa Institute of Marine Engineers, Mechnikova Street
34,

Odessa, Ukraine
V. Vesovic
Department of Mineral Resources Engineering, Imperial
College,
London SW7
2BP,
UK
E. Vogel
Universitdt Rostock, Fachbereich Chemie, D-I8055 Rostock, Germany
W. A. Wakeham
Department of Chemical Engineering and Chemical
Technology,
Imperial College, London
SW7
2BY,
UK
J. T. R. Watson
National Engineering Laboratory, East
Kilbride,
Glasgow G75
OQU,
UK
R. C. Wilhoit
Thermodynamics Research
Center,
Texas
A&M
University
System,
College

Station,
TX 77843-
3111,
USA
Foreword
The Commission on Thermodynamics of
the
Physical Chemistry Division of the Inter-
national Union of Pure and Applied Chemistry is charged by the Union with the duty to
define and maintain standards in the general
field
of thermodynamics. This duty encom-
passes matters such as the establishment and monitoring of international pressure and
temperature scales, recommendations for calorimetric procedures, the selection and
evaluation of reference standards for thermodynamic measurements of all types and
the standardization of nomenclature and symbols in chemical thermodynamics. One
particular aspect of the commission's work from among this set is carried forward by
two subcommittees: one on thermodynamic data and the other on transport properties.
These two subcommittees are responsible for the critical evaluation of experimental
data for the properties of fluids that lie in their respective areas and for the subsequent
preparation and dissemination of internationally approved thermodynamic tables of the
fluid state and representations of transport properties.
The Subcommittee
on
Transport Properties
has
discharged
its
responsibilities through

the work of groups of research workers active in the
field
drawn from all over the world.
These groups have collaborated in the preparation of representations of the viscosity,
thermal conductivity and diffusion coefficients of pure fluids and their mixtures over
wide ranges of thermodynamic states. The representations have almost always been
based upon an extensive body of experimental data for the property in question accu-
mulated over many years by the efforts of laboratories worldwide. The results of this
work have been published under the auspices of the subcommittee, with international
endorsement, in Journal of Physical and Chemical Reference Data and International
Journal of
Thermophysics.
The series of papers produced provides equations that de-
scribe the properties as a function of temperature and density that can be readily coded
to yield transport properties at any prescribed thermodynamic state with a defined un-
certainty.
In 1991, in collaboration with the Commission on Thermodynamics, the Subcom-
mittee on Transport Properties brought together a wider group of experts to con-
tribute to Volume III in the series Experimental
Thermodynamics,
which was edited by
Xlll
xiv Foreword
W. A. Wakeham, J. V. Sengers and A. Nagashima under the title Measurement of
the
Transport Properties of Fluids (Blackwell Scientific, Oxford). The volume describes
the state of
the
art with respect to the instrumentation for the determination of the trans-
port properties of fluids which has been employed in the acquisition of the data upon

which many of the representations of the properties have been based. However, the
representations of the transport properties of fluids have also relied heavily on the use
of kinetic theory to fill gaps for thermodynamic states where no experimental results
exist or for properties where measurements have not been performed. The theory is then
employed as a secure means of interpolation or even extrapolation from a smaller set
of high-quality information.
The present volume was conceived by the Subcommittee on Transport Properties
and the Commission on Thermodynamics to be a complement to the description of
experimental techniques. Its purpose is therefore to outline the principles that underlie
the statistical mechanical theories of transport processes in
fluids
and
fluid
mixtures in a
way that leads to results that can be used in practice for their prediction or representation
and to give practical examples of how this has been implemented. The brief to the
editors of
this
book from the subcommittee has been admirably fulfilled by the team of
authors that they have assembled. The coverage of the theory of transport properties is
concise yet comprehensive and is developed in a fashion that leads to useful
results.
The
sections on applications work their way through increasingly complicated archetypal
systems from the simplest monatomic species to dense mixtures of polyatomic fluids of
industrial significance and always with
the
emphasis on practical
utility.
This approach is

concluded with examples of practical realizations of
the
representations of
the
properties
incorporated in computer packages.
The book is intended to be useful for engineers who have to make use of represen-
tations of transport properties in order that they should understand the methodology
that lies behind published correlations as well as the limitations of the development.
It is also intended to be a summary of the status of the field at a particular moment
for practitioners of the subject. It is thus a book which is intended to bring the latest
state of knowledge to bear on problems of practical importance through international
collaboration and thus fulfill one of the main objectives of IUPAC.
W. A. Wakeham
Chairman
Commission on Thermodynamics
International Union of Pure and Applied Chemistry
Part
one
GENERAL
Introduction
J. MILLAT
NORDUM Institut fur Umwelt und
Analytik,
Kessin/Rostock,
Germany
J. H. DYMOND
The University, Glasgow, UK
C. A. NIETO DE CASTRO

University of Lisbon, Portugal
Accurate knowledge of transport properties of pure gases and liquids, and of their mix-
tures,
is essential for the optimum design of the different items of chemical process
plants, for determination of intermolecular potential energy functions and for devel-
opment of accurate theories of transport in dense fluids. A previous IUPAC volume,
edited by Wakeham et al. (1991), also produced by Commission 1.2 through its Sub-
committee on Transport Properties, has described experimental methods for the accu-
rate determination of transport properties. However, it is impossible to measure these
properties for all industrially important fluids, and their mixtures, at all the thermody-
namic states of interest. Measurements therefore need
to
be supplemented
by
theoretical
calculations.
This present volume, which is complementary to the previous publication, discusses
the present state of theory with regard to the dilute-gas state, the initial density depen-
dence, the critical region and the very dense gas and liquid states for pure components
and mixtures. In all cases, the intention is
to
present the theory in usable form and exam-
ples are given of
its
application to nonelectrolyte systems. This will be of particular use
to chemical and mechanical engineers. The subtitle of this volume Their correlation,
prediction and estimation' reflects the preferred order of application to obtain accurate
values of transport properties. Careful correlation of accurate experimental data gives
reliable values at interpolated temperatures and pressures (densities), and at different
compositions when the measurements are for mixtures. Unfortunately, there are only a

limited number of systems where data of such accuracy are available. In other cases,
sound theoretical methods are necessary to predict the required
values.
Where informa-
tion is lacking - for intermolecular forces, for example - estimation methods have to
be used. These are of lower accuracy, but usually have more general applicability.
In view of the outstanding need for accurate theoretical prediction, this volume gives a
clear presentation of current theory as applicable to
fluids
and
fluid
mixtures in different
density ranges. As a result of the substantial advances made in recent years it is now
4 J. Millat, J. H. Dymond and
C.
A. Nieto de Castro
possible to describe exactly the low-density transport properties as well as the critical
enhancements.
The dilute-gas theory is presented here for the first time in terms of effective col-
lision cross sections in a comprehensive readily usable form which applies to both
polyatomic fluids and monatomic fluids. This description should now be used exclu-
sively but, because it is relatively new, expressions are given for the macroscopic quan-
tities in terms of these effective cross sections, and certain simple relationships be-
tween these effective cross sections and the previously used collision integrals are also
described.
It is possible to account for the initial density dependence (at least for viscosity) and,
although the description is not rigorous, it is sufficient for this often relatively small
contribution
to
the transport properties. For higher densities, the modified Enskog theory

can be used in a consistent manner, although this does have limitations. This becomes
obvious from the fact that different empirical modifications have been proposed and
applied to different regions of the transport property surface. Therefore, for certain
ranges of thermodynamic states, an empirical estimation scheme based on the density
dependence of the excess transport property
is
frequently
to be
preferred. For liquids and
dense gases under conditions where the critical enhancements are negligible, methods
based on hard-sphere theory give the best representation of experimental data.
Although experimental transport properties are measured at different temperatures
and pressures, it is the density, or molar volume, which is the theoretically important
variable. So, for the prediction of transport properties, it is necessary to convert data
at a given temperature and pressure to the corresponding temperature and density, or
vice versa, by use of a reliable equation of state. Accordingly, an account is given in
this volume of the most useful equations of state to express these relationships for gases
and liquids. For dense fluids, it is possible to calculate transport properties directly
by molecular simulation techniques under specified conditions when the molecular
interactions can be adequately represented.
A
description
is
included in this book of these
methods, which are significant also for the results which have aided the development
of transport theory.
When the above methods fail, estimation methods become important. Schemes based
on the Corresponding-States Principle which are particularly important in this respect
are described. In order to demonstrate clearly just when the methods of correlation, the
theoretical expressions and estimation techniques are applicable, examples are given

of transport-property data representation for systems of different complexity: simple
monatomic fluids, diatomic fluids, polyatomic fluids (specifically, water and refriger-
ant R134a), nonreacting mixtures and (dilute) alkali-metal vapors as an example of a
reacting mixture.
Rapid access to transport property data is essential for the efficient use of proposed
correlation and prediction schemes. As a result, experimental data have been stored in
many data banks worldwide and the final section of this volume describes a number of
Introduction 5
the major data banks, which also incorporate methods for the calculation of transport
properties.
The authors are aware of the fact that, although this book demonstrates the significant
progress that has been made in this field in the past decade, there is still a need for
additional experimental and theoretical work in many parts of the transport-property
surface. In the individual chapters, an attempt is made to specify the relevant needs in
each density domain. It should be noted that this volume is restricted to a discussion
of nonelectrolytes. In spite of their technological importance, ionic systems, including
ionized gases and plasmas, molten metals and aqueous electrolyte solutions are not
included because of the different nature of the interaction forces.
A
complete description
of the transport properties of these fluids and fluid mixtures would occupy another
volume.
The editors acknowledge with thanks the contributions which have been made by all
the authors. They have attempted to produce a reasonable uniformity of style and apol-
ogize for any gross inconsistencies which remain. It is appreciated that not all theories
of transport and estimation methods have been covered. For these omissions, and for
all errors, the editors accept full responsibility. Finally, it is with the greatest pleasure
that the editors acknowledge the support of members and corresponding members of the
IUPAC Subcommittee on Transport Properties of Commission 1.2 on Thermodynamics.
Particular thanks are extended to its chairman, Professor

W.A.
Wakeham, for his many
constructive comments and his unending enthusiasm and encouragement throughout.
Reference
W.A. Wakeham, A. Nagashima & J.V. Sengers, eds. (1991). Experimental Thermodynamics,
Vol.
Ill: Measurement of the Transport Properties of Fluids. Oxford: Blackwell
Scientific Publications.
2
Technological Importance
W. A. WAKEHAM
Imperial College, London, UK
C. A. NIETO DE CASTRO
University of Lisbon, Portugal
2.1 Introduction
Fluids, that is gases and liquids, are self-evidently prerequisites for normal life. They
also play a major role in the production of many artefacts and in the operation of much
of the equipment upon which modern life depends. Occasionally, a fluid is the ultimate
result of a technological process, such as a liquid or gaseous fuel, so that its existence
impinges directly on the public consciousness. More often, fluids are intermediates in
processes yielding solid materials or objects, and are then contained within solid objects
so that their public image is very much less and their significance not fully appreciated.
Nevertheless, every single component of modern life relies upon a fluid at some point
and therefore upon our understanding of the fluid state.
The gross behavioral features of a fluid are well understood in the sense that it is
easy to grasp that a gas has the property to completely fill any container and that a
liquid can be made to flow by the imposition of a very small force. However, beyond
these qualitative features lie a wide range of thermophysical and thermochemical
properties of fluids that determine their response to external stimuli. This analysis
concentrates exclusively on

thermophysical
properties and will not consider any pro-
cess that involves a change to the molecular entities that comprise the fluid. The most
familiar thermophysical properties are those that determine the change in state of a
fluid that results from an external stimulus, for example the change of temperature
of a mass of fluid that results from the input of a quantity of heat to it. Such prop-
erties,
which relate to differences between two states of thermodynamic equilibrium,
are known as thermodynamic properties. On the other hand, those properties which
are concerned with the rate of change of the state of a fluid as a result of a change
in external conditions, or with the transport of mass, momentum or energy between
different parts of a fluid which is not in a uniform state, are known as
transport
prop-
erties and form the subject of this volume. The purpose of this chapter is to illustrate
the importance of the transport properties of fluids in science and technology.
Technological
Importance 7
2.2 Areas of technological interest
It will be shown later in the book (see Chapters 4 and 5) that the transport of mass,
momentum and energy through a fluid is the consequence of molecular motion and
molecular interaction. In the low-density gas phase the mean free path of the molecules
is very much greater than a molecular diameter. It is then the free molecular motion that
contributes mostly to the transport, and molecular collisions are relatively rare events
involving only two molecules at any one time. Such molecular collisions modify the
transport process by deflecting molecules from their original course. Thus the nature of
the collision, which is determined by the forces exerted between a pair of molecules,
necessarily determines the magnitude of the
flux,
of

mass,
momentum or energy induced
by a gradient of molecular concentration, flow velocity or temperature in the gas. The
fluxes, J, of the transported quantities and the imposed gradients, V7, are normally
related via simple, phenomenological, linear laws such as those of Fick, Newton and
Fourier (Bird et al 1960)
(2.1)
Here, X is the transport property associated with the particular process under consid-
eration. It follows that the transport coefficient, which itself may be a function of the
temperature and density of
the
fluid, will reflect the interactions between the molecules
of the dilute gas. For that reason there has been, for approximately 150 years, a purely
scientific interest in the transport properties of fluids as a means of probing the forces
between pairs of
molecules.
Within the last twenty years, at least for the interactions of
the monatomic, spherically symmetric inert gases, the transport properties have played
a significant role in the elucidation of these forces.
As the density of the fluid is increased the free motion of molecules is increasingly
dominated in the transport process by the interactions among the molecules and espe-
cially groups of
them.
The mean free path becomes smaller and of the order of several
molecular diameters. The details of the interactions between the molecules therefore
become less important compared to the fact that so many interactions take place. Thus,
when the dense liquid state is attained, it seems that quite simple models of
the
interac-
tion between molecules are adequate for a description of the behavior of the transport

properties (see Chapters 5 and 10). In the extreme case of a fluid near its critical point
the specific intermolecular interaction becomes totally irrelevant, since the transport
properties of the fluid are determined by the behavior of clusters and their size rather
than anything else (see Chapter 6). Thus, the scientific importance of transport prop-
erties under these conditions becomes one of seeking to describe the behavior of the
property itself through appropriate statistical mechanical theory rather than as a tool to
reveal other fundamental information.
The importance of
the
transport properties of
fluids
in
technology
is
maintained across
the entire spectrum of densities. Almost all chemical-process plants make use of fluids
either in process streams or as a means to heat and cool those streams. The process of
8
W.
A.
Wakeham
and
C.
A. Nieto de Castro
heat exchange between two
fluid
streams is conducted in
a
heat exchanger whose design
must be such

as to
permit
the
requisite heating or cooling of a process stream
to be
carried
out within prescribed limits of temperature. The rate of heat exchange and, therefore,
the design of the heat exchanger is dependent on the physical properties of the fluids
involved. A knowledge of these properties is evidently
a
prerequisite for the design. The
design of chemical reactors, particularly those that make use of porous solid
catalysts,
or
of separation equipment, requires a knowledge of the diffusion coefficients for various
species in
a
mixture of
fluids
in addition to the viscosity and thermal conductivity. Errors
in the values of the properties used
to
design
a
given item of
a
chemical plant can produce
a significant effect on the capital cost of that
item,
as well as unexpected increases in the

operating
costs.
Errors of this kind
have
effects that can propagate throughout
the
design
of the entire plant, sometimes becoming amplified and threatening its operability.
Similarly, the design of refrigeration or air conditioning equipment requires
a
knowl-
edge of
the
viscosity and thermal conductivity of
the
working
fluid
in the
thermodynamic
cycle in order to determine the size of the heat-transfer equipment and fluid pumps re-
quired
to
meet
a
specified
duty.
Moreover, the viscosity and thermal conductivity of fluid
lubricants is of great significance to the process of lubrication. Considerable efforts are
expended to select and synthesize fluids with particular characteristics for the viscosity
of lubricants as

a
function of temperature to ensure proper operation of lubricated equip-
ment under a variety of operating conditions. Indeed, it is particularly in this area that
the need for some accurate standard reference values for the viscosity of fluids is most
acute because of the need to provide meaningful intercomparisons of data obtained by
different manufacturers. In addition, most of
the
equipment used in industry to measure
or to control properties of the process streams needs to be calibrated with respect to
standard reference data, which, sometimes, require international validation.
Transport coefficients occur in all forms of continuum, hydrodynamic equations
concerned with mass, momentum and energy conservation once constitutive equations
for the fluids of interest are introduced. Such equations are frequently encountered in
trying to model mathematically technological processes with a view to their refinement.
Attempts to model such processes mathematically (usually numerically) are frequently
limited by a lack of knowledge of the physical properties of the materials involved
including the transport coefficients of the fluids.
Increasingly, there is a demand for improved safety of technological processes. The
term 'safety' may include environmental damage of various kinds as well as a direct
threat to life and property. Here, too, transport properties of fluids have a significant
impact. For example, the description of the process of pollutant dispersion contains
diffusive and convective components into which the transport coefficients of the gas
or liquid medium enter. There is a growing requirement for a demonstration of the
pedigree of every number that is employed in a calculation intended to demonstrate
a safety case for industrial plant so as to satisfy regulatory or legislative bodies both
nationally and internationally. Such requirements dictate that there should be a body
Technological
Importance 9
of approved, preferably internationally approved, data which can be used as a standard
source in all calculations of this kind.

It is clear therefore that the transport properties of fluids are of significance in many
areas of technology. At least in some of those areas, the accuracy of the data may be of
considerable economic and practical significance. The following sections are intended
to provide quantitative examples of this assertion.
2.3 Examples
2.3.1 Intermolecular forces
As was remarked earlier, all of the transport coefficients, as well as other properties,
of a dilute gas depend upon the intermolecular forces that exist between the molecules
in the gas. Thus, on the one hand a knowledge of the intermolecular forces enables all
the dilute-gas properties to be evaluated at an arbitrary temperature even if they have
not all been measured. Equally, it might be expected that accurate measurements of the
transport properties of the gas might be used to determine the forces between pairs of
molecules. It was not until 1970 that such a process was shown to be feasible and then
only with the aid of input from other sources of information, but it is interesting to note
that one of the factors that contributed to the slow development of this process was the
inconsistency of the available data for the viscosity of a gas with independent sources of
information. The inconsistency was
finally
traced to errors in early measurements of the
viscosity of gases that were finally eliminated by improved experimental design. The
final importance of the transport properties of gases in elucidating the forces between
some molecules is best illustrated by means of
the
example of
argon.
Argon has always
been an archetypal system for this study because of the spherical nature of the molecule
and the consequent spherical symmetry of the pair potential and simplicity of the kinetic
theory.
The viscosity of

a
dilute gas composed of spherically symmetric, structureless mole-
cules is related to the pair potential through the equation
(2 2)
n
L f
n
* " 4(k
B
T/7tm)V
2
6(2000)
h
where
&B
is the Boltzmann constant, T the absolute temperature and m the molecular
mass.
In addition, 6(2000) is an effective cross section that contains all of the dynamical
information related to the intermolecular potential that acts between the molecules. It
is explicitly related to the pair potential for this interaction in later chapters of this
book (see Chapter
4).
Finally, /^ is a factor near unity that accounts for kinetic theory
approximations beyond the first and is extremely weakly dependent on the nature of the
intermolecular interaction.
By
1972
the viscosity of argon had been determined over a range of temperatures from
120 K to 2000 K with an accuracy of better than 2%. At around that time, independent
10

W.
A.
Wakeham
and
C.
A. Nieto de Castro
1000
0-3
Fig.
2.1.
A modern version of the intermolecular pair potential for argon. Solid
line:
a repre-
sentation
of the full potential; symbols: A inversion of gas viscosity; • inversion of second
virial
coefficient.
measurements of the spectrum of bound argon dimers and molecular-beam-scattering
data became available for the first time. When all of the available data for argon were
combined it was possible to determine the intermolecular pair potential for argon for
the first time (Maitland et al 1987).
Subsequently, and of greater significance in the context of
this
volume, it was shown
that it was possible to determine the pair potential of monatomic species directly from
measurements of the viscosity of the dilute gas, by a process of iterative inversion
(Maitland et al 1987). As an illustration of the success that can be achieved, Figure 2.1
compares the pair potential that is obtained by application of the inversion process
to the viscosity data for argon with that currently thought to be the best available
pair potential for argon which is consistent with a wide variety of experimental and

theoretical information.
The same techniques
have
been employed
to
determine the pair potential for other like
and unlike interactions among the monatomic gases (Maitland et
al.
1987). Attempts
have also been made to apply the same sort of techniques to polyatomic gases (Vesovic
& Wakeham 1987). However, because of the nonspherical nature of the pair potential
and the sheer magnitude of the computational effort required to evaluate the effective
Technological
Importance
11
cross sections in such a case progress has been limited until recently. The advent of very
much faster computers now holds out the hope that such systems may become more
amenable to study.
2.3.2 Process-plant design
As an example of the importance of the transport properties of fluids in the design of
chemical process plant the catalytic reactor for
the
synthesis of methanol from hydrogen
and carbon monoxide shown in Figure 2.2 is considered. The feed gases consist of a
mixture of hydrogen and carbon monoxide which enter the reactor through a gas-
gas exchanger, which is an integral part of the pressure vessel for the reactor. In the
heat exchanger the incoming gases at 30 MPa are heated from ambient temperature to
610 K by the product gases from the two catalyst beds in the reactor. In the particular
design shown, a second heat exchanger
is

used between the two catalyst beds to provide
interstage cooling. Because the two heat exchangers are incorporated into the reactor
their design must be specified with only a small safety margin if the size of the entire
reactor is to remain within acceptable limits. Furthermore, for the preheater, there is
little
flexibility
in operation by which design deficiencies may be overcome in operation
because many of the variables are determined by the requirements of the catalytic
reaction zones. In order to study the effect of the transport properties of the gases
upon the design of this equipment a standard design methodology has been applied
(Armstrong et
al.
1982) in which a set of realistic, but arbitrary, values for the physical
properties of the gas streams has been adopted as a reference case to yield a reference
PRODUCT
PREHEATER
INTERSTAGE
UNIT
WATER
CATALYST
BED
CATALYST
BED
FEED
Fig. 2.2. A catalytic reactor for the synthesis of methanol.