Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 302

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 100 200 300 400

t , min

FIGURE 7.7 Kinetic parametric estimation using an integral method.

7.8.2. Differential Methods
With the integral methods of approaches, one often has to assume a value for the order of
reaction. Trial-and-error may not be everyone's favorite. To avoid the trial-and-error nature
of the integral method, the experimental data in Tab l e 3 can be differentiated to obtain

C A 0 1 n

k f 13

n

d C C

z ¼

d t ¼ k f

C C C C (7.46)

2 29

The values of z at selected values of C C 's can be obtained either through direct numerical
differentiation of the data ( C C , t ) series or by first regressing the data series according to
a polynomial form

C C ¼ a 0 þ a 1 t þ a 2 t þ a 2 t 2 þ a 3 t 3 . (7.47)

and then differentiating to obtain the derivative or z .
By so doing, one effectively reduces the differential model to an algebraic regression
model. However, Eqn (7.46) is still nonlinear. One will need to use a nonlinear regression
program to perform the parametric estimation. Therefore, this is still not favorable in the
Physical Chemistry and Reaction Engineering texts. Steps are often taken during the plan-
ning of experiments to avoid this situation. For this problem, one can notice that if C C is small,
Eqn (7.46) may be approximated by

z z k f C A 0 1

n

C C (7.48)

2

Equation (7.48) can be rearranged to give

C A 0 1
2

ln z z ln k f þ n ln C C (7.49)