Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 303

Letting,

d C C
d t

y ¼ ln z ¼ ln (7.50)

and

C A0 1
2

x ¼ ln C C (7.51)

one can transform the initial differential regression model to a linear regression model:

y ¼ ax þ b (7.52)

Therefore, the final regression is very easy to perform and visually pleasing. Here, the
discrepancy between the linear model, Eqn (7.52) , and the initial differential model, Eqn
(7.35) , can be reduced by good practice in experimental design.
However, because of the range of data in Table 7.3 , the simplified approach outlined above
is not suitable as new experimentation is not feasible. To illustrate the linear regression from
a differential method, we let

C A0 1
2

n 13

29

n

x n ¼ C C C C (7.53)

the Eqn (7.46) becomes

z ¼ k f x n (7.54)

Table 7.5 shows the experimental data ( Table 7.3 ) as converted to the new coordinates ( x n , z )
for n ¼ 1.5 and 2. The derivatives, z , is evaluated by a central difference scheme, i.e.

C A 0 1
2

n 13

29

n C A 0 1

2

n 13

29

n

C C;i C C;i þ C C;iþ1 C C;iþ1

x n;i ¼ (7.55)

2

and

C C; i þ1 C C; i
t iþ1 t i

z i ¼ (7.56)

where i is the index for the data points, in incremental time.
Figure 7.8 shows the correlation on the ( x n , z ) plane. The correlation for n ¼ 1.5 is shown

in Fig. 7.8 a, whereas the correlation using n ¼ 2 is shown in Fig. 7.8 b. The best-fit lines are

given by

z ¼ 0:003024x 1:5 (7.57)

and

z ¼ 0:001344x 2 (7.58)
The corresponding correlation coefficients are given by 0.9165 and 0.9124, respectively.