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Unformatted text preview: sic continuum point of view and
assume that this result is valid everywhere. That is to say
that Axiom II is valid in homogeneous regions where RA
changes slowly and it is valid in interfacial regions where
RA changes rapidly. We follow the work of Wood et al.13
and consider the cÀj interface illustrated in Figure A1.
The volume V encloses the cÀj interface and extends into
the homogeneous regions of both the cphase and the jphase. The total net rate of production of species A in the
volume V is represented by
Z
RA dV ¼
V Z
RAc dV þ Vc Z
RAj dV þ Vj V NJA RA dV ¼ A¼1 Z
þ
Acj Z A¼N
X
Vc A¼N
X A¼1 Vc Z
þ Vj
A¼N
X RAs dA NJA RAc dV þ A¼1 Z A¼N
X
Vj NJA RAs dA; NJA RAj dV A¼1 NJA RAs dA; J ¼ 1; 2; :::; T A¼N
X NJA RAc ¼ 0; A¼1 A¼N
X NJA RAj ¼ 0; J ¼ 1; 2; :::; T J ¼ 1; 2; :::; T A¼1 (A5)
This leads to the obvious constraint on the heterogeneous
rates of production, RAs, and we summarize our results associated with Axiom II as
A¼N
X NJA RA ¼ 0; J ¼ 1; 2; :::; T (A6) A¼1 Axiom II ðhomogeneous; cphaseÞ A¼N
X NJA RAc ¼ 0; A¼1 J ¼ 1; 2; :::; T ðA7Þ
Axiom II ðhomogeneous; jphaseÞ A¼N
X NJA RAj ¼ 0: A¼1 J ¼ 1; 2; :::; T ðA8Þ NJA RAj dV A¼1 ðA4Þ At this point we require that the homogeneous net rates of
production satisfy the two constraints given by (A2) Acj A¼N
X A¼1 Axiom II (general) Here, the dividing surface that separates the cphase from
the jphase is represented by Acj, and the heterogeneous
rate of production of species A is identiﬁed by RAs. This
quantity is also referred to as the surface excess reaction
rate.14 Multiplying Eq. A2 by the atomic species indicator
and summing over all molecular species leads to
Z A¼N
X Z
NJA RAc dV þ Acj NJA RA ¼ 0 A¼1 Z A¼N
X 0¼ Axiom II ðheterogeneous; cj interface A¼N
X NJA RAs ¼ 0: A¼1 J ¼ 1; 2; :::; T ðA9Þ (A3) A¼1 From Eq. A1, we see that the lefthand side of this result
is zero, and we have These results can be obtained for a more general system
(see Ref. 14) in which the cÀj interface represents a moving
and deforming surface. For a reactor in which only homogeneous reactions occur (see Figure 1), we make use of
Eq. A6 in the form
2 Axiom II: A R ¼ 0; 3
RA
6 RB 7
67
R¼6 : 7
67
4:5
RN (A10) For a catalytic reactor in which only heterogeneous reactions occur at the cÀj interface (see Figure 2), we make use
of Eq. A9 in the form
2 Axiom II:
Figure A1. Catalytic surface.
548 DOI 10.1002/aic Published on behalf of the AIChE A Rs ¼ 0; 3
RAs
6 RBs 7
6
7
Rs ¼ 6 : 7
6
7
4:5
RNs February 2012 Vol. 58, No. 2 (A11) AIChE Journal The pivot theorem associated with homogeneous reactions
is obtained from Eq. A10, and the analysis leads to Eq. 23,
which is repeated here as
Pivot Theorem (homogeneous reactions): RNP ¼ P RP (A12)
The pivot theorem associated with heterogeneous reactions
is obtained from Eq. A11 and is given here as
Pivot Theorem (heterogeneous reactions): ðRs ÞNP ¼ P ðRs ÞP
(A13)
Here we note that the axiom for the conservation of
atomic species takes exactly the same form for homogeneous reactions (Eq. A10) and for heterogeneous reactions
(Eq. A11). In addition, the application of the principle of
conservation of atomic species takes exactly the same form
for homogeneous reactions (Eq. A12) and for heterogeneous reactions (Eq. A13). The fact that both the axiom and
the application take exactly the same form for homogeneous and heterogeneous reactions has led many to ignore the
difference between these two distinct forms of chemical
reaction.
In general, measurement of the net rate of production is
carried out at the macroscopic level, thus, we normally
obtain experimental information for the global net rate of
production. For a homogeneous reaction, this takes the form
Z
RA ¼
RA dA; A ¼ 1 ; 2 ; ::: ; N
(A14)
V whereas the global net rate of production for a heterogeneous reaction is given by
Z
RA ¼ RAs dA; A ¼ 1 ; 2 ; ::: ; N (A15) A cj Here, we note that the global net rates of production for
both homogeneous reactions and heterogeneous reactions
have exactly the same physical signiﬁcance, thus, it is not
unreasonable to use the same symbol for both quantities.
Given this simpliﬁcation, the global version of the pivot theorem represented by Eq. 40 applies to both homogeneous
and heterogeneous reactions. Elementary Stoichiometry and Upscaling
Moving on to the applications of elementary stoichiometry
indicated by Eq. 41, we remark that upscaling of the results
from elementary stoichiometry is routine and follows the
procedure outlined by Eqs. A14 and A15. However, the key
quantity of interest is the representation given by Eq. 46 that
we repeat here as
RM ¼ M r Figure A2. Upscaling the reference reaction rates. hRM i ¼ February 2012 Vol. 58, No. 2 RM dV (A17) V h R M i ¼ M h ri (A18) Here, the simple upscaling results given for Axiom II are
lost, and the upscaling process becomes quite complex. For
heterogeneous reactions, the reference reaction rates, rI, rII,
rIII, and so forth, will depend on surface con...
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