Biomedical Engineering Reference
In-Depth Information
it is straightforward to use the number of moles extent n
ext,i
for each of the N
R
reactions. Thus,
we can define the change in the number of moles of species j through the relation
X
N
R
n
j
¼
n
j0
þ
n
ij
n
ext
;i
(3.65)
i
¼1
To summarize, we can always find a single concentration variable that describes the change in
all species for a single reaction, and for N
R
simultaneous reactions, there must be N
R
indepen-
dent variables to describe concentration changes. For a single reaction, this problem is simple
(use either C
A
or f
A
), but for a multiple reaction system, one must set up the notation quite
carefully in terms of a suitably chosen set of N
R
concentrations or conversions ( f
i
's) or n
ext,i
's.
Finally, the rate of reaction for a given species j in a set of N
R
reactions can be computed via
X
N
R
r
j
¼
n
ij
r
i
(3.66)
i
¼1
The rate of reaction for any given species is the summation of its reaction rates in all the
reactions.
Example 3-3. Elemental balance. Consider the fermentation of carbon monoxide in
aqueous solution using the bacterium Methylotrophicum to produce acetic acid, butyric
acid, and additional cells. Based on the reported yields, stoichiometry is given by
þ
v
O
2
O
2
þ
v
N
2
N
2
þ
v
H
2
O
H
2
O
CO
/
v
X
X
where X represents the Methylotrophicum cells. Assuming that cells produced in this reaction
can be represented by the Roel's average biomass (that is, 1 C-mol cells
0:52
CO
2
þ 0:21
CH
COOH
þ 0:006
CH
CH
CH
COOH
þ
3
3
2
2
¼
CH
1.8
O
0.5
N
0.2
),
determine all the remaining stoichiometric coefficients.
Solution: Let X
CH
1.8
O
0.5
N
0.2
. To find the remaining stoichiometric coefficients, we need
to perform atom balances individually. In this reaction, we have four atoms involved: C, O,
N, and H. Four balance equations can be written independently.
C-mole balance:
¼
1 0
v
O
2
0
v
N
2
0
v
H
2
O
¼ 0:52 þ 0:21 2 þ 0:006 4 þ
v
X
(E3-3.1)
O-mole balance:
1 20
v
O
2
0
v
N
2
v
H
2
O
¼ 0:52 2 þ 0:21 2 þ 0:006 2 þ 0:5
v
X
(E3-3.2)
N-mole balance:
0 0
v
O
2
2
v
N
2
0
v
H
2
O
¼ 0:52 0 þ 0:21 0 þ 0:006 0 þ 0:2
v
X
(E3-3.3)
H-mole balance:
0 0
v
O
2
0
v
N
2
2
v
H
2
O
¼ 0:52 0 þ 0:21 4 þ 0:006 8 þ 1:8
v
X
(E3-3.4)
Equations
(E3-3.1)
through
(E3-3.4)
need to be solved simultaneously to find all the stoi-
chiometric coefficients.
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