HigginsWhitaker_AICHEJ_2012

HigginsWhitaker_AICHEJ_2012 - Local Global and Elementary...

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Local, Global, and Elementary Stoichiometry Brian G. Higgins and Stephen Whitaker Dept. of Chemical Engineering and Material Science, University of California at Davis, Davis, CA 95616 DOI 10.1002/aic.12593 Published online May 6, 2011 in Wiley Online Library (wileyonlinelibrary.com). Stoichiometry refers to conservation of atomic species. In this article, local refers to a point at the continuum level, global refers to the macroscopic balance level, and elementary refers to conservation of atomic species associated with distinct kinetic steps. The role of stoichiometry in the determination of the pivot matrix and the mechanistic matrix is presented. The elements of both these matrices are referred to as stoichiometric coefficients; however, both sets of coefficients are different and both play different roles in the analysis of chemical reactors. V V C 2011 American Institute of Chemical Engineers AIChE J, 58: 538–552, 2012 Keywords: reactor analysis, reaction kinetics Introduction Aris 1 describes stoichiometry as the bookkeeping of atomic species. This bookkeeping goes hand-in-hand with the axiom for conservation of mass that we state as Axiom I: @ q A @ t þr±ð q A v A Þ¼ r A ; A ¼ 1 ; 2 ; ::::; N ð 1 Þ Here q A is the mass density of species A , while v A is the velocity of species A and r A is the net mass rate of produc- tion of species A owing to chemical reaction. The fact that mass is neither created nor destroyed by chemical reaction is expressed as Axiom II: X A ¼ N A ¼ 1 r A ¼ 0 (2) In this representation, we have used N to signify the num- ber of stable, identifiable species. In any chemical reaction there are molecular fragments and active species that exist only in small amounts, and the precise nature and amount of these fragments are generally unknown. Hidden behind the statement made by Eq. 2 is the assumption that what is unknown and undetectable does not play a significant role in the concept of conservation of mass. When dealing with chemical reactions, the molar forms of Axioms I and II are preferred and the first of these is given by Axiom I: @ c A @ t þr±ð c A v A Þ¼ R A ; A ¼ 1 ; 2 ; ::::; N ð 3 Þ Here, we have used c A ¼ q A = MW A ; R A ¼ r A = MW A (4) in which MW A represents the molecular mass of species A . The fact that mass is neither created nor destroyed by chemical reaction can also be expressed as Axiom II: X A ¼ N A ¼ 1 MW A R A ¼ 0 (5) Here, it is important to recognize that R A has the meaning R A ¼ net molar rate of production per unit volume of species A owing to chemical reactions 8 < : 9 = ; (6) which can also be interpreted as Correspondence concerning this article should be addressed to S. Whitaker at [email protected] V C 2011 American Institute of Chemical Engineers 538 AIChE Journal February 2012 Vol. 58, No. 2
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R A ¼ molar rate of creation of species A per unit volume owing to chemical reactions 8 > < > : 9 > = > ; ² molar rate of consumption of species A per unit volume owing to chemical reactions 8 > < > : 9 > = > ; ð 7 Þ Clearly Eqs. 6 and 7 are equivalent descriptions of R A , thus, the reader is free to chose which ever set of words is most appealing. In this article, we will use the phrase ‘‘net
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