Unformatted text preview: centrations, thus
surface area averaging16 of the nonuniform catalytic surface
is the ﬁrst step in the sequence suggested in Figure A2.
Given the spatially smoothed surface conditions associated
with the reference reaction rates, one can upscale those
quantities through the hierarchy of length scales indicated in
Figure A2 using the method of volume averaging.17 Appendix B: Atomic Species Balances
The atomic species balance has some advantages when
carrying out calculations by hand because the number of
atomic species balance equations is almost always less than
the number of molecular species balance equations. We
begin our development of atomic species balance equations
with Axioms I and II given by
Z
Z
d
Axiom I :
cA dV þ
cA vA Á n dA
dt
V
A
Z
¼
RA dV ; A ¼ 1; 2; …; N ðB1Þ
V Axiom II : AIChE Journal Z Given that the mechanistic matrix is independent of any
upscaling process, we use Eq. A16 to obtain (A16) This result is used in the design of chemical reactors, thus,
it is the local volume average form that is needed.15 In terms
of the volume V indicated in Figure A2, the quantity of
interest is hRMi deﬁned by 1
V A¼N
X NJA RA ¼ 0; J ¼ 1; 2; … T (B2) A¼1 To develop an atomic species balance, we multiply Eq.
B1 by NJA and sum over all molecular species to obtain Published on behalf of the AIChE DOI 10.1002/aic 549 A¼N
X NJA A¼1 d
dt Z
cA dV þ A¼N
X Z
Z A¼N
X
NJA cA vA Á ndA ¼
NJA RA dV A¼1 V A¼1 V Ae (B3)
Here, we have made use of the fact that vA Á n is zero
everywhere except at the entrances and exits that we have
denoted by Ae. On the basis of Axiom II, we see that the
last term in this result is zero, and our atomic species balance takes the form
A¼N
X d
NJA
dt
A¼1 Z
cA dV þ A¼N
X Z V cA v Á ndA ¼ 0; J ¼ 1; 2; …; T NJA A¼1 Figure B1. Batch and steadystate reactors. Ae (B4)
in which we have imposed the very reasonable approximation that vA Á n ¼ v Á n at the entrances and exits. In Eq.
B4, we have indicated explicitly that there are T atomic species balance equations instead of the N molecular species
balance equations given by Eq. B1. When T
N it may be
convenient to solve material balance problems using atomic
species balances.
One application of the atomic species balance given by
Eq. B4 is based on the total molar concentration of the
Jtype atoms that is given by
(
cJ ¼ A¼N
X )
NJA cA A¼1 8
< 9
total molar =
¼
concentration ;
:
;
of J type atoms J ¼ 1; 2; … ; T
(B5) Use of this result in Eq. B4 leads to the atomic species
macroscopic balance given by
d
Atomic Species Balance :
dt Z Z
cJ dV þ V cJ v Á ndA ¼ 0;
ðB6Þ This form of the atomic species balance is sometimes useful for solving macroscopic mole balance problems with
chemical reactions.
The form represented by Eq. B4 is also useful for the
analysis of batch reactors and steadystate reactors such as
those illustrated in Figure B1. For the batch reactor, we can
express Eq. B4 as A¼1 NJA dnA
¼ 0;
dt J ¼ 1; 2; ::; T (B7) in which the total number of moles of species A in the batch
reactor is given explicitly by
Z
nA ¼ cA dV
V Integration of Eq. B7 from t ¼ 0 to t ¼ tf yields
550 DOI 10.1002/aic t
Z¼tf NJA DnA ¼ 0; D nA ¼ A¼1 ðdnA =dtÞdt J ¼ 1; 2; …; T
t¼ 0 (B9)
and in matrix form the ﬁrst of these becomes
AðDnÞ¼ 0 (B10) This result is analogous to Eq. 13 and one can follow the
analysis from Eq. 13 to Eq. 23 to obtain the pivot theorem
for a batch reaction
Pivot Theorem ðbatch reactorÞ : ðDnÞNP ¼ PðDnÞP (B8) (B11) Turning our attention to the continuous ﬂow reactor
(CSTR) illustrated in Figure B1b, we express the ﬂux term
in Eq. B4 as
Z
Z
Z
cA v Á ndA ¼ À
cA jv Á njdA þ
cA jv Á njdA ¼ DnA
Ae Aentrance Aexit (B12) A J ¼ 1; 2; …; T A¼N
X A¼N
X Use of this expression in the steadystate form of Eq. B4
leads to
A¼N
X NJA DnA ¼ 0 J ¼ 1; 2; …; T (B13) A¼1 and in terms of the nomenclature used in Eq. 13, we have
AðD n Þ ¼ 0 (B14) This provides another application of the pivot theorem
that we express as
Pivot Theorem ðCSTRÞ: ðD nÞNP ¼ PðD n ÞP (B15) Both the Eqs. B11 and B15 provide useful tools for the
treatment of batch reactors of the type illustrated in Figure
B1. The derivation of these results is based on Axiom I and
Axiom II along with the uniqueness proof of the row
reduced echelon form of the atomic matrix. Published on behalf of the AIChE February 2012 Vol. 58, No. 2 AIChE Journal Appendix C: Conservation of Charge Ne1 ¼ 0; The axiom associated with conservation of atomic species
was originally stated as Ne2 ¼ À2; ionic species such as SO2À
4 Ne3 ¼ þ1; ionic species such as Naþ A¼N
X Axiom II : NJA RA ¼ 0; J ¼ 1; 2; :::; T (C1) A¼1 in which N represents the total number of identiﬁable stable
species such as ethane (C2H6), butadiene (C4H6), water
(H2O), sulfuric acid (H2SO4), ans so forth. In addition, if we
have a solution of sulfuric acid, the species would include
ions such as Hþ and SO2À . The matrix form of Eq. C1 is
4
given by
2 2 N11
6 N21
6
6
6 N31
Axiom II : 6
6:
6
6
4:
NT 1 N12
N22 N13 ::::::
:::::: N1;NÀ1;
N2;N...
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