Biochemical Reactions - A First Look
81
and, for the third term
E (S 6 T)
* £[(F R ,r p "l£,) ^ r ] = 4 ( FR,P *'R r^
r ]=
FKT
r
P~'RrE (SST) =
F R fP _1R rF
(5)
while the fourth term is given by Eq. (3.43). By combination of these results we find that the contributions
from the first and second term cancel and
F = F - F R r p - ‘R rF
(6)
Since the last term is positive, the variance-covariance matrix for the estimated rates is always smaller than
that for the measured rates.___________________________________________________________________
Example ЗЛО Calculation of best estimates for measured rates
We return to the experimental data of von Meyenburg (1969), which were presented in Example 3,5 (see
Fig. 3.5), and we want to find better estimates for the measured variables. We first consider the data for low
dilution rates where no ethanol is formed and in Example 3.13 we will consider the data for higher dilution
rates.
At low dilution rates the measured rates are <},,
,
qc
and
q
x. There are two non-measured rates -
ammonia utilization <y„and formation of water
qw.
Thus with the biomass composition specified in Eq. (1) of
Example 3.5 we have
'1
0
1
1
Ï
'0
0^
2
0
0
1.83
;
E c =
3
2
1
2
2
0.56
0
1
0
0
0.17
j
,1
0;
With a total of six components and four elemental balances, the number of degrees of freedom F = 2, and
since four volumetric rates are measured
{K
= 4) the system is over-determined. From Eq. (3.39) the
redundancy matrix is found to be
1
0
1
1
0
-0.286
-0 .2 8 6
0.014
0
0.572
0.572
-0.028
0
0.858
0.858
-0 .0 4 2
and rank(R) = 2. It is easily seen that the last two rows are proportional to the second row, and we therefore
delete these two rows and find the reduced redundancy matrix
R. =
0
-0 .2 8 6
-0 .2 8 6
0.014
(3)
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