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Acknowledgment Steven Greenwald assisted in the digital calculations. Nomenclature Ca = capillary number, pu,/u Ca*, CaO = lower speed limit on Ca, Equations 6 and 7 F = F function, Equation 3 g = acceleration of gravity Go = Goucher number, R(pg/2u)W 2 = film thickness, cm. = flow thickness, Equation A-4, cm. r = radial coordinate, cm. R = radius of cylinder, cm. T = dimensionless flow thickness, k(pg/pu,)’/* u4 = surface velocity at liquid-gas interface, cm./sec. UtC = wire withdrawal velocity, cm./sec. V = volume flow rate, cc./sec. W = dimensionless velocity, u,(p/R2pg) W* = lower speed limit on W Y = Y function, Equation 2 U = local velocity in film, cm./sec. GREEK LETTERS Fc = viscosity, poises P = density, gram/cc. U = surface tension, dynes/cm. literature Cited Chien, S.-F., J. Appl. Mech. 33, 222 (1966); Deryagin, B. V., Levi, L. N., “Film Coating Theory,” Chaps. 2 to 4, Deryagin, B. V., Titiyevskaya, A. S., Dokl. Akad. Nauk USSR 50, Focal Press, New York, 1964. 307 (1945). Gutfinger, Chaim, Tallmadge, J. A., Znd. Eng. Chem. 59, No. 11, Jeffreys, H., Proc. Cambridge Phil. SOC. 26, 204 (1930). Matsuhisa, Seikichi, Bird, R. B., A.Z.Ch.E. J. 11, 588 (1965). Rajani, G. R., M. S. thesis (chemical engineering), Drexel In- 18 (1967); 60, No. 2, 74 (1968). stitute of Technolow. June 1968. Tallmadge, J.A., A.fCh.E. J. 12,810 (1966). Tallmadge, J. A,, Labine, R. A., Wood, B. H., IND. ENG. CHEM. Tallmadge, J. A., Soroka, .4. J., Chem. Eng. Sci., in press. Tallmadge, J. A., Tlrhite, D. A,, Znd. Eng. Chem. Process Design White, D. A., Ph.D. dissertation, Yale University, April 1965. White,D. A,, Tallmadge, J. A., A.I.Ch.E. J. 12, 333 (1966). White, D. A., Tallmadge, J. A , , A.Z.Ch.E. J. 13, 745 (1967) FUNDAMENTALS 4,400 (1965). Develop. 7, 503 (1 968). RECEIVED for review March 25, 1968 ACCEPTED October 24, 1968 Work supported by the National Science Foundation Grant GK- 1206. A LUMPING ANALYSIS IN MONOMOLECULAR REACTION SYSTEMS AnalJysis of the ExactlJy Lumpable &stem J A M E S W E 1 A N D J A M E S C . W . K U O Research Department, Central Research Division, Mobil Research and Development Corp., Princeton, N. J. 08540 Because of the enormous number of chemical species encountered in many chemical reaction systems (especi- ally those related to the petroleum industry), it is often expedient to lump all the species into a few groups for practical purposes. This paper presents a theoretical study on the exact lumping of a monomolecular reac- tion system. One can obtain the necessary and sufficient conditions under which the kinetics of the lumped classes can be exactly described by a complex first-order reaction scheme. Many implications of these lumpability conditions are investigated. The lumpability of a monomolecular reaction system coupled with diffusion has also been studied, The analysis for the system with diffusion is similar to analysis without diffusion. N MANY chemical processes, particularly those related to the I petroleum industry, the number of molecular species in- volved often runs into the thousands. For instance, in a naphtha-reforming process, one would like to investigate the reforming kinetics, the feedstock characterization, and the product characterization. In these investigations, one is unable to deal with each chemical species separately. One may, however, partition the species into a few equivalence classes (or lumped classes), and then consider each class as an independent entity. Such a lumping process is by no means strange to us. We often consider, for example, all oxygen molecules as “oxygen” even though the kinetic energies of the individual oxygen molecules are different. Such lumping also gave petroleum processing the PONA analysis, in which all species are divided into four classes: paraffins, olefins, naphthenes, and aromatics. There are many reasons to lump-for example, in the simula- tion of a petroleum reformer, we may lump in the chemical analysis of the feedstock, such as a PONA analysis, in describing the kinetics of the system and in determining the quality of the products, such as the octane number. A reformer system is “perfectly lumpable” if one can make a single lumping scheme that is good at all of these stages. Here, however, we concentrate entirely upon describing the kinetics of the system, and search only for the kinetic lumpability conditions. As we proceed, we must realize that lumping always leads to a loss of information, and that the information lost in the kinetic lumping may be important, for example, in the characteriza- tion of the product. Among all the conceivable chemical reaction systems, we particularly consider the monomolecular reaction system (complex first-order kinetic system). For this system, we 114 I & E C F U N D A M E N T A L S