Biomedical Engineering Reference
In-Depth Information
This is a greatly simplified example of the stoichiometric modeling of a
gasification reaction. The complexity increases with the number of equations
considered. For a known reaction mechanism, the stoichiometric equilibrium
model predicts the maximum achievable yield of a desired product or the
possible limiting behavior of a reacting system.
7.5.2.3 Nonstoichiometric Equilibrium Models
In nonstoichiometric modeling, no knowledge of a particular reaction mecha-
nism is required to solve the problem. In a reacting system, a stable equilibrium
condition is reached when the Gibbs free energy of the system is at the mini-
mum. So, this method is based on minimizing the total Gibbs free energy. The
only input needed is the elemental composition of the feed, which is known
from its ultimate analysis. This method is particularly suitable for fuels like bio-
mass, the exact chemical formula of which is not clearly known.
The Gibbs free energy, G
total
for the gasification product comprising
N species (i
5
1,
...
, N) is given by:
X
X
N
N
n
i
P
n
i
G
f
;
i
1
n
i
Δ
G
total
5
n
i
RT ln
(7.73)
i
5
1
i
5
1
G
f
;
i
is the Gibbs free energy of formation of species i at standard
pressure of 1 bar.
Equation (7.73)
is to be solved for unknown values of n
i
to minimize
G
total
, bearing in mind that it is subject to the overall mass balance of indi-
vidual elements. For example, irrespective of the reaction path, type, or
chemical formula of the fuel, the amount of carbon determined by ultimate
analysis must be equal to the sum total of all carbon in the gas mixture pro-
duced. Thus, for each jth element we can write:
X
where
Δ
N
a
i
;
j
n
i
5
A
j
(7.74)
i
5
1
where a
i,j
is the number of atoms of the jth element in the ith species, and A
j
is the total number of atoms of element j entering the reactor. The value of
n
i
should be found such that G
total
will be minimum. We can use the
Lagrange multiplier methods to solve these equations.
The Lagrange function (L) is defined as:
!
kJ
X
1
λ
j
X
K
N
L
G
total
2
a
ij
n
i
2
A
j
=
mol
(7.75)
5
j
i
1
5
5
where
λ
ϕ
is the Lagrangian multiplier for the jth element.
To find the extreme point, we divide
Eq. (7.75)
by RT and take the
derivative:
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