This preview shows page 1. Sign up to view the full content.
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 steadystate
assumption or the steadystate hypothesis or the pseudo
steadystate 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 processdependent. 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,...
View
Full
Document
This document was uploaded on 01/26/2014.
 Winter '14

Click to edit the document details