HigginsWhitaker_AICHEJ_2012

Aris r introduction to the analysis of chemical

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Unformatted text preview: agues. The students who participated in this effort are too numerous to mention, but their contributions are an essential part of this article. Notation A A* Ae Acj B cA I MWA M M ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ N NJA P RA ¼ ¼ ¼ ¼ RA ¼ R¼ Rs ¼ RM ¼ RNP ¼ RP ¼ atomic matrix row reduced echelon form of the atomic matrix atomic/electronic matrix interfacial area contained within the volume, V (m2) Bodenstein matrix qA/MWA, molar concentration of species A (mol/m2) unit matrix molecular mass of species A (g/mol) mechanistic matrix total number of stable molecular species and Bodenstein products (reactive intermediates) number of stable molecular species in a multicomponent system number of J-type atoms associated with molecular species A pivot matrix net molar rate of production of species A per unit volume (mol/ m3s) net global molar rate of production of species A (mol/s) column matrix of net rates of reaction for stable molecular species (mol/m3s) column matrix of heterogeneous net rates of reaction for stable molecular species (mol/m2s) column matrix of the total net rates of production for stable molecular species and Bodenstein products (reactive intermediates) (mol/m3s) column matrix of net rates of production of the nonpivot species (mol/m3s) column matrix of net rates of production of the pivot species (mol/m3s) AIChE Journal February 2012 Vol. 58, No. 2 RB ¼ column matrix of net rates of production of Bodenstein products (mol/m3s) RNP ¼ column matrix of global net rates of production for the nonpivot species (mol/s) RP ¼ column matrix of global net rates of production for the pivot species (mol/s) r ¼ column matrix of reference reaction rates (mol/m3s) rA ¼ net mass rate of production of species A per unit volume (kg/ m3s) rI ¼ reference reaction rate for elementary step I (mol/m3s) S ¼ stoichiometric matrix T ¼ number of atomic species t ¼ time (s) vA ¼ species A velocity (m/s) V ¼ volume of a fixed control volume (m3) Greek letters qA ¼ mass density of species A, kg/m3 Literature Cited 1. Aris R. Introduction to the Analysis of Chemical Reactors. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1965. 2. Amundson NR. Mathematical Methods in Chemical Engineering: Matrices and Their Application. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1966. 3. Cerro RL, Higgins BG, Whitaker S. Material balances for chemical engineers. Available at: http://www.ekayasolutions.com/MatBalance/ Contents.php, in press. 4. Truesdell C, Toupin R. The classical field theories. In: Flugge S, editor. Handbuch der Physik, Vol. III, Part 1. New York: Springer Verlag, 1960:226–858. 5. Noble B. Applied Linear Algebra: Englewood Cliffs. New Jersey: Prentice-Hall, Inc., 1969. 6. Sankaranarayanan TM, Ingle RH, Gaikwad TB, Lokhande SK, Raja T, Devi RN, Ramaswany V, Manikandan P. Selective oxidation of ethane over Mo-V-Al-O oxide catalysts: insight to the factors affecting the selectivity of ethylene and acetic acid and structure-activity correlation studies. Catal Lett. 2008;121:39–51. 7. Frank-Kamenetsky DA. Conditions for the applicability of Bodenstein’s method in chemical kinetics. J Phys Chem. (USSR) 1940;14: 695–702. ¨ 8. Bjornbom PH. The relation between the reaction mechanism and the stoichiometric behavior of chemical reactions. AIChE J. 1977;23: 285–288. 9. Smith WR, Missen RW. Mass conservation implications of a reaction mechanism. J Chem Educ. 2003;80:833–838. ¨ 10. Bjornbom PH. The relation between the reaction mechanism and the stoichiometric behavior of chemical reactions. AIChE J. 1977;23: 285–288. 11. Horn F, Jackson R. General mass action kinetics. Arch Rational Mech Anal. 1972;47:81–116. 12. Bodenstein M, Lind SC. Geschwindigkeit der Bildung der Bromwasserstoffes aus sienen Elementen. Z Physik Chem. 1907;57:168– 192. 13. Wood BD, Quintard M, Whitaker S. Jump conditions at non-uniform boundaries: the catalytic surface. Chem Eng Sci. 2000;55:5231– 5245. 14. Whitaker S. The species mass jump condition at a singular surface. Chem Eng Sci. 1992;47:1677–1685. 15. Arce PE, Quintard M, Whitaker S. The art and science of upscaling. ´ In: Galan MA, de Valle EM, editors. Chemical Engineering: Trends and Developments. West Sussex, England: Wiley, 2005:1–40. ´ 16. Ochoa-Tapia JA, del Rıo JA, Whitaker S. Bulk and surface diffusion in porous media: An application of the surface averaging theorem. Chem Eng Sci. 1993;48:2061–2082. 17. Whitaker S. The Method of Volume Averaging. Dordrecht: Kluwer Academic Publishers, 1999. 18. Feynman RP, Leighton RB, Sands M. The Feynman Lectures on Physics, Vol. I. New York: Addison-Wesley Pub. Co., 1963. 19. Porter SK. How should equation balancing be taught? J Chem Educ. 1985;62:507–508. Published on behalf of the AIChE DOI 10.1002/aic 547 Z Appendix A: Homogeneous and Heterogeneous Reactions Our analysis of the stoichiometry of heterogeneous reactions is based on conservation of atomic species expressed as A¼N X Axiom II: (A1) Here, we adopt the clas...
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