Kinetic 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 Amsterdam — Oxford — New York — Tokyo 1991 ELSEVIER SCIENCE PUBLISHERS Β. V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 655, Avenue of the Americas New York, NY 10010, U.S.A. ISBN 0-444-89032-7 © Elsevier Science Publishers Β.V., 1991 All rights reserved. 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No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any meth­ ods, products, instructions or ideas contained in the material herein. Although all advertising material is expected to conform to ethical (medical) standards, inclusion in this publication does not constitute a guarantee or endorsement of the quality or value of such product or of the claims made of it by its manufacturer. This book is printed on acid-free paper. Printed in The Netherlands ν 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.