Biochemical Reactions - A First Look
83
8
= F R , P" R (.qm
5
= 10
(
2.2
}
-
2.8
-
1.0
^
0.8
and the estimate for the measured rates is therefore
<L = q
m- s =
( -
0.142^1
-0 .0 6 0
0.064
0.078
j
(
8
)
(9)
It is concluded that the measurements are very good, since the estimated rates differ only slightly from the
raw data. It is, however, justified to make the small corrections, and we can next calculate the variance
covariance matrix for the estimated rates by using Eq. (6) of Note 3.3.
f
0.209
0.112
0.116
-0.092
0.112
0.154
-0.152
-0.041
-0 .1 1 6
-0.152
0.151
-0.034
0.092
0.041
-0.034
0.127
It is observed that the diagonal elements in F are smaller than the diagonal elements in F, and the variance
of the estimates is therefore smaller than for the raw measurements. The estimates q m
are therefore likely to
be more accurate and reliable. Whereas the errors of the raw measurements are uncorrelated, it is observed
from Eq. (10) that the errors of the estimated rates are correlated, since the rates are correlated through the
constraints in Eq. (3.38).
In the analysis we have used the volumetric rates, whereas yield coefficients were used in Example 3.5. The
present error analysis could, however, just as well be carried out using the yield coefficients (simply replace
the volumetric rate vector with a vector containing the yields).______________________________________
Normally the variance-covariance matrix is assumed to be diagonal, i.e., the measurements are
uncorrelated. However, the volumetric rates are seldom measured directly, but they are based on
measurements of so-called primary variables, which may influence more than one of the measured
volumetric rates. An example is measurement of the oxygen uptake rate and the carbon dioxide
production rate, which are based on measurement of the gas flow rate through the bioreactor
together with measurement of the partial pressure of the two gases in the exhaust. If there is an error
in the measured gas-flow rate, this influences both of the above-mentioned rates, and errors in the
measured rates are therefore indirectly correlated. The same objection holds for measurements of
many other volumetric rates, which are in reality obtained by combination of a concentration and a
flow-rate measurement. In all these cases of indirect error correction it is difficult to specify the true