Environmental Engineering Reference
In-Depth Information
is that of the Lagrange multipliers. The constraint of this problem is the elemental
balance, i.e.,
X
N
a
ij
n
i
=
A
j
,
j
=1,2,3,
…
:
,
k
ð
Eq
:
5
:
32
Þ
i
=1
where
a
ij
is the number of atoms of element
j
in the species
i
.
A
j
is defined as the total
number of atoms of the element
j
in the reaction mixture. To form the Lagrangian
function (
L
), the Lagrange multipliers
λ
k
, multiplied with the elemental
balance constraint, and the resulting terms are added to G
t
as follows:
λ
j
=
λ
1
,
…
,
!
L
=G
t
+
X
j
=1
λ
j
X
k
N
a
ij
n
i
−
A
j
ð
Eq
:
5
:
33
Þ
i
=1
The partial derivatives of Equation (5.33) are set equal to zero in order to find the limit:
=
∂
+
X
G
t
∂
∂
L
j
λ
j
a
ij
=0
:
:
ð
Eq
5
34
Þ
∂
n
i
n
i
Equation (5.34) can be written as a matrix that has
i
rows, and those are solved
simultaneously with the constraints as defined in Equation (5.32). Because the first
term on the right is the definition of the chemical potential, Equation (5.34) can be
written as
μ
i
+
X
j
λ
j
a
ij
=0
ð
Eq
:
5
:
35
Þ
Combination of Equation (5.30) and Equation (5.35) gives
+
X
G
f
,
i
+R
u
Tln
y
ðÞ
Δ
j
λ
j
a
ij
=0
ð
Eq
:
5
:
36
Þ
Example 5.6 Calculation of the equilibrium composition based on
minimization of the Gibbs free energy
Ethane and steamare fed to a steamcracker at 1000K and a total pressure of 1 bar at a
ratio of 4 mol H
2
O to 1 mol ethane. Estimate the equilibrium distribution of the
products (CH
4
,C
2
H
4
,C
2
H
2
,CO
2
, CO, O
2
,H
2
,H
2
O, and C
2
H
6
). Values of
Δ
G
f
i
at 1000 K are:
G
f ,
CH
4
=4
mol
−1
G
f
,
CO
2
=
mol
−1
G
f
,
CO
Δ
:
61 kcal
Δ
−
94
:
61 kcal
Δ
mol
−1
=
−
47
:
942 kcal
G
f ,
C
2
H
4
=28
mol
−1
G
f ,
O
2
= 0 kcal
mol
−1
G
f ,
H
2
O
=
mol
−1
Δ
:
249 kcal
Δ
Δ
−
46
:
03 kcal
G
f ,
C
2
H
2
=40
mol
−1
G
f ,
H
2
= 0 kcal
mol
−1
G
f
,
C
2
H
6
=26
mol
−1
Δ
:
604 kcal
Δ
Δ
:
13 kcal
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