HigginsWhitaker_AICHEJ_2012

In appendix b we show that there is an equivalent

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Unformatted text preview: one chooses a ‘‘reference reaction rate’’ designated by rI, rII,…,rK. These reference reaction rates are used to express the elementary rate of production of species A as AIChE Journal (43) in which MAk is a component of the mechanistic matrix.8 This representation of the elementary rate of production can be used with Eq. 42 to express the net rate of production of species A as (40) This result is essential for the analysis of macroscopic systems because it specifies the global net rates of production that must be inferred from measurements of the composition, selectivity, conversion, and yield. In Appendix B, we show that there is an equivalent pivot theorem for atomic species balances as opposed to the molecular species balances associated with Eq. 40. For atomic species, this same pivot theorem appears in the transient analysis of batch reactors and the steady-state analysis of continuous stirred tank reactors (CSTR). Elementary Stoichiometry: R k ¼ MA k rk ; A Published on behalf of the AIChE (47) DOI 10.1002/aic 543 This partition leads to the following two matrix equations 2 32 3 R1 M1 I M1 II M1 III : M1 K 6 R2 7 6 M2 I M2 II M2 III : M2 K 7 676 7 6 R3 7 6 M3 I M3 II : : :7 6 7¼6 7 6:7 6 : : : : :7 676 7 4:5 4 : : : : :5 RN MN I MN II MN K |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 3 rI 6 rII 7 67 6 rIII 7 (48a) 67 4:5 rK stoichiometric matrix 2 3 rI MNþ1 I MNþ1 II : : MNþ1 K 6 rII 7 RNþ1 67 4 : 5¼4 : : :: : 5 6 rIII 7 67 4:5 : : : MM K RM MM I |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} rK Bodenstein matrix 2 3 Elementary kinetic schema IV: 2 3 kIV H þ HBr À H2 þ Br ! (52d) Elementary kinetic schema V: kV 2Br À Br2 ! (52e) It is important to note that the schemata listed in Eqs. 52 are not linearly independent. In fact, there is no physical requirement that this be the case, thus, the stoichiometric matrix S that appears in Eq. 49 may have a rank less than N. This list of schemata represents a set of pictures and what we need for analysis are equations. We indicate how to develop these equations in the following paragraphs. When the rules of mass action kinetics11 are applied to the picture given by Eq. 52a, we obtain the following equation (48b) In terms of compact notation, we express the first of these two results as R ¼ Sr (49) in which R is the same column matrix of net rates of production that appears in Eq. 13. We refer to S as the stoichiometric matrix9; however, we must be careful to note that the pivot matrix appearing in Eq. 23 is also composed of stoichiometric coefficients. Because of this, there is a possibility of confusion when referring to ‘‘stoichiometric coefficients.’’ The second of Eqs. 48 can be expressed as RB ¼ Br I Elementary chemical reaction rate equation I: RBr2 ¼ ÀkI cBr2 ; (53) and on the basis of this expression, we choose the ‘‘reference reaction rate’’ to be rI  kI cBr2 Elementary reference reaction rate I: (54) Next we need to make use of elementary stoichiometry, thus, we recall Eq. 41 with k ¼ I to obtain A¼M X A¼1 (50) NJA RI ¼ 0; A J ¼ Br (55) in which RB is the column matrix of net rates of production for the Bodenstein products, and B is the Bodenstein matrix. The classic resolution of this set of equations is to impose the condition of local reaction equilibrium expressed as As this process involves only Br2 and Br, we see that Eq. 55 provides Local reaction equilibrium : and this immediately leads to RB ¼ 0 (51) and use the result from Eq. 50 to eliminate the concentration of the Bodenstein products from the column matrix of reference reaction rates in Eq. 49. In this manner, Eq. 49 becomes a key element in the design of a chemical reactor. At this point, we are ready to return to Eq. 43 and indicate how one applies elementary stoichiometry in the process of determining the elements of the mechanistic matrix. To accomplish this, we consider the classic example10 of the reaction of hydrogen with bromine to form hydrogen bromide. The elementary chemical kinetic steps associated with the reaction of H2 with Br2 to form HBr are illustrated by: kI ! Elementary chemical kinetic schema I : Br2 À 2Br (52a) 2 RI 2 þ RI ¼ 0 Br Br Elementary stoichiometry I: Br þ H2 À HBr þ H ! H þ Br2 À HBr þ Br ! 544 DOI 10.1002/aic RI Br 2 (57) SCHEMA I ð52bÞ Elementary chemical kinetic schema III: kIII I RBr2 ¼ À This represents a mathematical statement that the atoms of bromine are conserved during the process illustrated by Eq. 52a; however, this result is usually considered to be intuitively obvious and thus is not identified as elementary stoichiometry. One can think of Eq. 57 as an example of ‘‘counting atoms’’ in the same manner that led from the picture given by Eq. 24 to the equation given by Eq. 25. Howeve...
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