HigginsWhitaker_AICHEJ_2012

With a little intuitive thought one can construct

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Unformatted text preview: he row reduced echelon form with 3 In any two successive rows with non-zero elements, the leading 1 of the lower row occurs one column to the right of the leading 1 of the higher row. 540 At this point, we return to Eq. 14, ignore the rows of zeros, and make use of a row/column partition to obtain (24) From this picture one must deduce the equations that are necessary to solve problems. With a little intuitive thought one can construct ‘‘constraints’’ on the net rates of production given by Published on behalf of the AIChE 1 RC2 H6 ¼ À RCO2 ; 2 7 RO2 ¼ À RCO2 ; 4 3 RH2 O ¼ þ RCO2 2 (25) February 2012 Vol. 58, No. 2 AIChE Journal 7 C2 H6 þ O2 ! 2 CO2 þ 3H2 O; 2 (28d) and a rigorous method of constructing these types of schemata is described elsewhere (See Appendix C3 of Ref. 3). Rather than become involved in the lengthy analysis needed to develop Eqs. 28, it is much easier to follow the route outlined by Eqs. 13 through 23. We begin this process with a visual representation of the atomic matrix given by Molecular Species ! C2 H6 2 carbon 2 6 hydrogen4 6 oxygen 0 O2 CO2 CO H2 O C2 H4 C2 H3 OOH 3 01 10 2 2 7 00 02 4 45 22 11 0 2 ð29Þ Figure 1. Complete homogeneous combustion of ethane. in which C2H6, O2, and CO2 represent the nonpivot species. Use of this form of the atomic matrix in Eq. 13 leads to However, one intuitive solution may not be the same as another intuitive solution, and sometimes the coefficients in this set of equations get reversed and a significant error occurs. Use of Eq. 23 to obtain Eqs. 25 avoids this possibility. We now move beyond the single independent ‘‘homogeneous reaction’’ suggested by Figure 1, and we consider the catalytic oxidation of ethane to produce ethylene (C2H4) and acetic acid (CH3COOH) along with carbon dioxide (CO2), carbon monoxide (CO), and water (H2O). This situation is illustrated in Figure 2 where we have suggested that a ‘‘heterogeneous reaction’’ occurs at the cÀj interface. In reality the reaction mechanism will be much more complex than suggested in Figure 2 where we wish only to emphasize that the reaction occurs at a fluid–solid interface. In Appendix A, we show that both homogeneous and heterogeneous reactions can be treated within a single framework. Given the two reactants and the five products illustrated in Figure 2, one might count atoms to obtain; 3C2 H6 þ 5O2 ! CO2 þ CO þ 5H2 O þ C2 H4 þ CH3 COOH ð26Þ 2 2 2 Axiom II : 4 6 0 01 00 22 10 02 11 RC2 H6 RO 2 RCO2 RCO RH2 O RC2 H4 6 36 226 6 4 45 6 6 026 6 4 3 7 7 23 7 0 7 7 ¼ 405 7 7 0 7 5 RC2 H3 COOH (30) in which all the atomic species (C, H, and O) are present in at least one nonpivot species. Using several elementary row operations, we can express the atomic matrix in row reduced echelon form so that Eq. 30 takes the form 2 61 6 6 60 6 4 0 0 0 1 0 0 1 1 2 0 3 3 1 7 2 À À 2 6 3 2 2 1À 3 3 2 3 1 3 2 3 3 2 6 76 76 76 76 76 56 6 4 RC2 H6 RO2 RCO2 RCO RH 2 O RC2 H4 3 7 7 23 7 0 7 7 ¼ 405 7 7 0 7 5 RC2 H3 COOH (31) however, one could also count atoms to develop a different result given by 4C2 H6 þ 8O2 ! 2CO2 þ 2 CO þ 8H2 Oþ C2 H4 þ CH3 COOH ð27Þ Here, it should be clear that counting atoms does not work and the development of pictures representing the partial oxidation of ethane requires a significant effort. Sankaranarayanan et al.6 have studied the catalytic oxidation of ethane based on the following set of pictures 1 C2 H6 þ O2 ! C2 H4 þ H2 O 2 (28a) 3 C2 H6 þ O2 ! CH3 COOH þ H2 O 2 (28b) 5 C2 H6 þ O2 ! 2CO þ 3H2 O 2 (28c) AIChE Journal February 2012 Vol. 58, No. 2 Figure 2. Partial oxidation of ethane. Published on behalf of the AIChE DOI 10.1002/aic 541 Application of the obvious row/column partition leads to 2 10 6 40 1 00 2 1 6 3 RC 2 H 6 0 6 7 761 76 0 5 4 RO2 5 þ 6 À 62 6 4 RCO2 1 2 1À 3 2 32 3 0 6 6 Â6 4 2 3 2 À 3 2 3 RCO RH 2 O RC2 H4 RC2 H3 COOH 3 2 3 1 3 2 3 3 7 7 7 7 7 5 23 0 7 7 67 7 ¼ 4 0 5; 5 0 Figure 3. Undetermined products. ð32Þ of Eq. 11 that applies to the control volume illustrated in Figure 4. There we have illustrated the local rate of production for species A, designated by RA, and the global rate of production for species A, designated by RA. The latter is defined by and this immediately provides an example of the pivot theorem given by Z RA ¼ 2 1 2 23 0À À À 2 3 2 36 3 3 37 RCO 6 7 RC2 H6 6 2 1 76 R H 2 O 7 7 4 RO 2 5 ¼ 6 1 À 7 À 76 62 6 3 3 74 RC2 H4 5 6 7 RCO2 4 5 RC H COOH 23 2 2 2 À1 À À 3 3 3 V (33) Equating the elements of the left- and right-hand sides provides the net rates of production of the nonpivot species in terms of the net rates of production of the pivot species. 1 2 2 RC2 H6 ¼ 0 À RH2 O À RC2 H4 À RCH3 COOH 3 3 3 and we often use an abbreviated description given by 8 9 <global rate of = R A ¼ production of ; A ¼ 1; 2; :::; N : ; species A Z V (34b) 2 2 2 RCO2 ¼ ÀRCO þ RH2 O À RC2 H4 À RCH3 COOH 3 3 3 A¼N X NJA RA dV ¼ 0; J ¼ 1; 2; :::; T (37) A¼1 (34c) A¼N X A¼1 Z RA dV ¼ 0; NJA J ¼ 1; 2; :::; T (38) V Use of the definition of the global rate of production for species A giv...
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