HigginsWhitaker_AICHEJ_2012

If those other species are present in very small

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Unformatted text preview: en by Eq. 35 leads to the following global form of Axiom II: Global Stoichiometry Statements concerning the products of a reaction are generally made on the basis of macroscopic observation. Thus, the products associated with the catalytic combustion of ethane are determined by the macroscopic observation of a reactor such as we have illustrated in Figures 1 and 2. In many cases, there may be other species in the product stream that are difficult to detect, and we have suggested this possibility in Figure 3. If those other species are present in very small amounts, the prediction given by Axiom II will be acceptable. Here, it should be clear that the implementation of Axiom II depends on the judgment that one makes concerning the molecular species that are present in the system. Up to this point we have discussed the local form of Axiom II, that is, the form that applies at a point in space. However, when Axiom II is used to analyze the reactor shown in Figure 3, we will make use of an integrated form DOI 10.1002/aic (36) The integral can be taken inside the summation operation, and we can make use of the fact that the elements of NJA are independent of space to obtain These three results are consistent with the pictures given by Eqs. 28; however, the development of Eqs. 34 is easier, and the reliability of the result is greater. 542 (35) When dealing with a problem that involves the global rate of production, we need to form the volume integral of Eq. 11 to obtain (34a) 1 7 2 1 RO2 ¼ RCO À RH2 O þ RC2 H4 À RCH3 COOH 2 6 3 3 8 9 < net macroscopic molar rate = RA dV ¼ of production of species A : ; owing to chemical reactions Axiom II ðglobal formÞ: Published on behalf of the AIChE A¼N X NJA RA ¼ 0; J ¼ 1; 2; :::; T A¼1 (39) Figure 4. Local and global rates of production. February 2012 Vol. 58, No. 2 AIChE Journal Here, one must remember that RA has units of moles per unit time while RA has units of moles per unit time per unit volume, and thus, the physical interpretation of these two quantities is different as illustrated in Figure 4. The analysis at the macroscopic scale follows that given by Eqs. 13 through 23 and it leads to the pivot theorem for the global rates of production given by RNP ¼ P RP Pivot Theorem: Elementary Stoichiometry Knowledge of the global net rates of production indicated by Eq. 35 allows us to analyze reactors from the macroscopic point of view, but it does not allow us to design reactors. For design we need to be able to predict the net rates of production in terms of the concentration of the species involved in the reaction, that is, cA, cB, and so forth. To accomplish this, we need to represent the net rates of production, RA, RB, RC, and so forth in terms of chemical reaction rates or kinetic models. One approach is to develop models based on a series of elementary rates of production identified as RI , RII , RIII ,…. RI , RII , …, RI , RII , RIII , and so forth. To A A A B B M M M illustrate how this is done, we begin with a set of K elementary rates of production that are constrained by elementary stoichiometric conditions given by A¼M X A¼1 NJA Rk ¼ 0; J ¼ 1; 2; :::; T ; A k ¼ I; II;:::; K ð41Þ Here, we note that there are M species under consideration. The first N of these species are stable molecular species, such as we have illustrated in Figures 1 and 2, whereas species N þ 1 through M represent the unstable species that are identified in Figure 3 as the ‘‘other species.’’ We will refer to these species as ‘‘reactive intermediates’’ or as ‘‘Bodenstein products.’’7 The sum of the elementary rates of production provide us with the net rate of production as indicated by RA ¼ k¼K X k¼I Rk ; A A ¼ 1 ; 2 ; :::; N ; N þ 1 ; :::; M A ¼ 1; 2; :::; M; k ¼ I; II;:::;K RA ¼ k¼K X MA k rk ; A ¼ 1 ; 2 ; :::; N ; N þ 1 ; :::; M February 2012 Vol. 58, No. 2 (44) k¼I In matrix form we express this result as 2 32 R1 M1 I 6 R 2 7 6 M2 I 6 76 6 R 3 7 6 M3 I 6 76 6:76: 6 76 6 : 7¼6 : 6 76 6 RN 7 6 M N I 6 76 6 RNþ1 7 6 MNþ1 I 6 76 4:54: RM MM I M1 II M2 II M3 II : : MN II MNþ1 II : : M1 III M2 III : : : : : : : : : : : : : : : : 3 M1 K M2 K 7 7 :7 7 :7 7 :7 7 :7 7 :7 7 :5 MM K 2 3 rI 6 rII 7 67 6 rIII 7 67 4:5 rK (45) while in compact matrix notation we have RM ¼ Mr (46) Here, M is the mechanistic matrix, RM is the column matrix of all the net rates of production, and r is the column matrix of reference reaction rates. The Bodenstein products (N þ 1 through M) are often subjected to the approximation of local reaction equilibrium that is also referred to as the steady-state assumption or the steady-state hypothesis or the pseudo steady-state hypothesis. These are appropriate phrases when kinetic mechanisms are being studied by means of a batch reactor; however, the phrase local reaction equilibrium is preferred because it is not process-dependent. To apply the approximation of local reaction equilibrium, it is convenient to construct a row/row partition of Eq. 45 as indicated by (42) Associated with each elementary kinetic model, k ¼ I, II,…,K,...
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