This preview shows page 1. Sign up to view the full content.
Unformatted text preview: he row reduced echelon form with
3 In any two successive rows with nonzero 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 righthand 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...
View
Full
Document
This document was uploaded on 01/26/2014.
 Winter '14

Click to edit the document details