Table Of ContentKinetic and
Thermodynamic Lumping
of Multicomponent
Mixtures
Proceedings of an ACS Symposium on Kinetic and Thermodynamic
Lumping of Multicomponent Mixtures, Atlanta, GA, April 15,1991
edited by
Gianni Astarita
Department of Chemical Engineering, University of Naples, Italy and
Department of Chemical Engineering, University of Delaware, Newark, DE, U.S.A.
Stanley I. Sandler
Department of Chemical Engineering, University of Delaware, Newark, DE, U.S.A.
ELSEVIER
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PREFACE
As the information which, in principle, could be available for any scientific or
engineering problem is so vast, one always needs to lump information together into
a more manageable subset in order to proceed. For example, every field theory in
which matter is regarded as a continuum is implicitly based on a lumping
procedure since we know that matter is composed of molecules. But, of course,
molecules are composed of atoms, and those of subatomic particles, and so on.
Thus continuum, molecular and atomic descriptions of matter represent different
degrees of lumping. The idea of lumping is one which is used, more or less
consciously, in a large variety of fields.
This book is the proceedings of a Symposium on Kinetic and
Thermodynamic Lumping of Multicomponent Mixtures held at the American
Chemical Society Meeting in Atlanta, GA in April 1991. The thermodynamic and
kinetic behavior of multicomponent mixtures is an area where the requirements of
lumping have been clearly identified, and the techniques and results of lumping
have been analyzed in considerable detail. Papers presented at the Symposium
consisted of both invited and contributed papers. Each invited paper was a review
of a subfield within the landscape of the symposium while the contributed papers
contain detailed analyses of specific problems. The invited papers by R. Aris, Κ. B.
Bischoff, F. J. Krambeck, G. Froment, R Cotterman and J. M. Prausnitz, review the
lumping of multicomponent mixtures in the context of different problems, while J.
Wei's paper deals with applications of the lumping concept to other problem areas.
The contributed papers deal with a large variety of kinetic and thermodynamic
applications.
One of our goals in organizing this symposium was to bring together active
researchers in this field to report on and discuss the progress which has been
made in the lumping of mixtures of very many components for a number of
different applications, and to identify the important problem areas which remain. A
second goal was to prepare this proceedings volume which could serve both as an
introduction for people entering this field, and as a reference book for more
experienced researchers. We hope that after this book is studied, the reader will
feel we have accomplished this latter goal.
We wish to express our appreciation to all the authors for their time, effort
and their valuable contributions. They lucidly presented their work at the
Symposium, and then prepared the final versions of the manuscripts in a camera-
ready form in a timely fashion that has allowed rapid publication of this volume. We
also wish to thank the Industrial Chemistry Division of the American Chemical
Society for having included our Symposium as part of the April 1991 National
Meeting. Most importantly, we wish to thank Lorraine E. Holton for all her help
with the organization of the Symposium and these Proceedings.
Gianni Astarita and Stanley I. Sandler
Naples (Italy) and Newark, DE (USA)
Kinetic and Thermodynamic Lumping of Multicomponent Mixtures, 1
edited by G. Astarita and S.L Sandler
Elsevier Science Publishers B.V., Amsterdam, 1991 — Printed in The Netherlands
LUMPING REVISITED: GLOBAL ENVIRONMENT CHANGES
JAMES WEI
Department of Chemical Engineering
Massachusetts Institute of Technology
Rm. 66-540, Cambridge, MA 02139
Abstract
Good lumping schemes are needed for the understanding
and management of global environmental changes. The
Global Circulation Models of today, used to predict the
global climate in the next century, employ about eight
thousand cells to model the atmosphere and the ocean.
Higher resolution models are contemplated to provide
needed regional information, but would require an
increase of computation speed by 104 to 106. The Carbon
Emission Budget used to maintain statistics on gaseous
emissions, and to predict the impact of policy
decisions, requires a commonly accepted lumping scheme.
INTRODUCTION
More than two decades ago, my colleague Jim Kuo and I were
dealing with the problem of catalytic reforming of naphtha to make
high octane gasoline, where there are too many molecular species
to deal with individually but not very much information about
their properties. We wrote in a paper published in I&EC
Fundamentals in 1969: "One may, however, partition the species
into a few equivalence classes (or lumped classes), and then
consider each class as an independent entity... Such lumping also
gave petroleum processing the PONA analysis, in which all species
are divided into four classes: paraffins, olefins, naphthenes, and
aromatics" (Wei and Kuo, 1969). We took the word "lumping" from
"Finite Markov Chains" by Kemeny and Snell, (Kemeny and Snell,
1960). They wrote "Lumping reduces a (Markov) process with very
2
large number of states to a process with a more manageable smaller
number of states, at the sacrifice of obtaining less precise
information." They also described the opposite step of
"Expanding" a Markov chain into a larger chain which gives more
detailed information. The priiiciples of the Exactly Lumpable
Systems are easy to describe and the system has very nice
properties, but most of the time we have to deal with systems
where the lumping is not exact.
I did not revisit the lumping problems in the next two
decades, and watched from a distance the many excellent papers
published on the kinetics and thermodynamics of lumping. The use
of the four-letter word "Lump" was called into question,
especially since the economists use the more elegant word of
"Aggregation". The Merriam-Webster dictionary describes "Lump" as
a word of Swedish or Norwegian origin, meaning a stump or a piece
cut off from a log. The following definitions are relevant:
(v.t.) to speak of collectively; also, to group together
indiscriminately; (n.) the whole aggregation, collection, lot; as
taken in the lump; (v.t. Colloq.) to put up with (something
distasteful) ; as, if you don't like it, you must lump it. In any
case, the usage of "Lumping" in the chemical engineering
literature is so entrenched now that if you don't like it, you
must. ...
One of my current research problems is Global Warming, and
what engineers can do about it. There are many challenging
problems that involve lumping schemes. Let us discuss two of them
here: The Global Circulation Model and the Carbon Emission
Budget. We derive two relations which will be used later.
Let us consider a system with η lumps, each occupied by an
extensive variable a. Thus we have a space A in n-dimensions,
{
and a particular state can be described by an η-vector a:
a = ( a, , a , ... a)
2 n
We are interested in a property Ρ(a). Lumping is an operation
that combines states to produce a lower dimensional system
L(a) - a=(£S, ...S)
lt 2 a
and a property function defined at a lower dimension
UP)
An exact lumping is obtained when
A good approximate lumping is obtained when
P(£)-P(a)
kl (1)
1 Ρ (a)
Now let us consider Ρ to be a linear property so that
And let us lump a and a together so that
4 2
^=a+a
1 1 2
which leads to the value of
Thus the value of ρ depends on the value of a, so we do not
have exact lumping. If we were to take p as the average of p, and
1
p, then the ratio of the maximum error to the property described
2
in equation (1) would be
i5^-(pa^pa) _ (p-p) (a-a)
1 1 1 1 2 2 1 2 2 x
Piai+P2a2 Piai+PA2
2
4
This is the Principle of Homogeneity, so that if there is a
small difference between p and p., the error would be small. If
2
a state is usually present in a small value so that a is much
1
greater than a, and that p.a. is much greater than pa, then the
2 2 2
maximum error ratio is
Pi (*i+*> ~Piai-Pa2 ^ P1-P2 ) ( az ) ()
2 2 ( 3
Piai+Pa2 Pi ai
2
This is the Principle of Materiality, where errors are
negligible as long as a^ is very small.
GLOBAL CIRCULATION MODELS
The prediction of weather (short term) and climate (long
term) are presently conducted on Global Circulation Models (GCM).
The scale and complexity of these models stretches computer
technology to its limits and much beyond. A current model divides
the earth atmosphere into cells that are about ten levels in the
vertical direction for the atmosphere and the ocean, 22 grids from
South Pole to North Pole (8 degrees in latitude) and 36 grids
around the equator (10 degrees in longitude), which means almost
8,000 cells. The width of a cell is 1117 km at the Equator, and
194 km at latitude 80. The time step is about 40 minute
increments. The entire surface of the United States can be
covered by a dozen cells. This is much too coarse for an accurate
prediction of rain fall and agriculture at the San Joaquin Valley,
which has a width of 100 km. But to make an "equilibrium"
simulation run, assuming that atmospheric carbon dioxide suddenly
doubled in concentration, 400 computational hours are needed on
the Cray-1, (Washington, 1990).
The climate community is talking about higher resolution
5
models by making the grid sizes 10 times smaller to 100 km width,
which means that the time step would also have to be 10 times
smaller, so that the increase in computation intensity would go up
by a factor of 10,000. This would require a Ten-Teraflop
computer. We certainly want to know about "transient" simulation
runs, to see how long it would take the atmosphere to reach
equilibrium, and preliminary estimations are of the order of 50
years! Furthermore, we want to know the consequences of
intervention policies such as a US tax of $100/ton of carbon, or
of the relaxation of the population control efforts in China, so
we need to run many "scenarios" about the consequences of policy
changes. Now we are talking about a Petaflop computer!
Is there any way to devise a smarter lumping scheme, so we
would not need 8 million cells to do justice to the earth
atmosphere? Is it pretty dumb to put so many grid points in the
Polar region, so Antarctica has more than 100 cells and Greenland
has 15 cells compared to 12 cells for USA? Should the cells
follow earth contours, such as the Great Lakes and the Appalachian
Mountains, rather than the latitude-longitude grids? The
principle of Homogeneity in equation (2) should play a role in the
lumping scheme. This is a truly challenging problem lumping.
ANTHROPOGENIC EMISSION OF CARBON DIOXIDE
The earth atmosphere contains 740 billion tons of carbon
dioxide, which is growing at the rate of 3 billion tons per year,
and the ocean is absorbing 3 billion tons per year as well,
(Schneider, 1989). It is generally agreed that if the atmospheric
greenhouse gases doubles in concentration, the average temperature
6
on earth would increase by 1.5 to 3.5 ° C, which may lead to a
general sea level rise, severe storms, and changing rainfall
patterns. The understanding and control of the rates of
anthropogenic greenhouse gases emission, over the course of the
next century, is a very important challenge.
Let us consider the global kinetic problem of the growing
emission of carbon dioxide through burning fuels. Currently,
there are 5 billion people on earth, each contributing his or her
share of the 5 billion tons of carbon emitted. In addition, de
forestation and changes in land use may add another 2 billion tons
of carbon. It is anticipated that the earth population will grow
to 10 billion or more into the next century, so there would be
added growth pressure on the emission of carbon. Furthermore,
global economic growth through industrialization and trade would
mean more wealth and energy use, adding more carbon emission,
(Bolin, 1986).
Some of the questions that we want to resolve include:
(a) What are the largest sources of emission, and what are the
major factors behind them?
(b) What are the fastest growing sources of emission, and what
are the major factors behind them?
(c) What are the major targets of opportunity for reduction by
changing social habits and ways of life? What are the
associated costs?
(d) What are the major targets of opportunity for reduction by
changing the technologies for economic activities and energy
use? What are the associated costs?
(e) How do we set an overall target of reduction, distribute
equitably to the various sectors of emission, and achieve our
objectives at the least cost?
(f) What instruments of policy would be easy to enforce, and
effective in meeting our targets?
7
SCHEMES OF LUMPING CARBON EMISSION
How should we aggregate or lump the more than a hundred
nations on earth? One scheme was given by the World Resources
Institute (World Resources Institute, 1990) , together with the
United Nations Environmental Programme, which divides the world
into 7 regions:
Region No. of nations Population, millions
Africa 49 648
N. and C. America 15 427
S. America 12 297
Asia 37 3109
Europe 27 498
USSR 1 288
Oceania 5 27
Total 146 5292 million
This is not a particularly good lumping scheme for keeping track
of the emission of carbon dioxide, for instance within the Asia
lump, Japan behaves very differently from India. It does not
satisfy the Principle of Homogeneity of equation (2).
A different scheme was used by the Environmental Protection
Agency (U.S. Environmental Protection Agency, 1989) in their
famous "Stabilization" report of 1989.
Region Population, million
United States 239
OECD Western Europe, Canada 430
OECD Pacific 144
USSR/Eastern Europe 416
China/CP Asia 1140
Middle East 111
Africa 570
Latin America 402
South/southeast Asia 1417
Total 4870
Notice that the total population of the two surveys are not the
same. There is a good deal more uniformity within each lump.
