Page 206 - Modeling of Chemical Kinetics and Reactor Design

176 Modeling of Chemical Kinetics and Reactor Design

1 1 1 1 k C
= + • + 2 • B (3-244)
r k k k C kk C
O O 1 A O 1 A
which is of the linear form

Y = C + C X + C X 2 (3-245)
O
1
2
1
where X = 1/C and X = C /C .
1
2
B
A
A
The practice of transforming a nonlinear model into linear form can
result in invalid estimates of the coefficients and consequent mis-
interpretation of the fitted model. The problem is associated with
the assumption of a constant error variance (Assumption 3). If the
measured rates r have a constant error variance, then the errors
associated with ln r (Equation 3-241) or 1/r (Equation 3-244) will not
have a constant error variance. The degree of violation will depend
on the range of the measured reaction rate values. If the data contain
both very small rates and very large rates, then the error variance will
change appreciably over the data set and ordinary least squares esti-
mates of the coefficients will give poor estimates. However, if the rate
values are not spread over a large range, then model transformations
of Equations 3-241 and 3-244 will not significantly distort the esti-
mated coefficients. An illustration of the effects of linearizing model
transformation of Equation 3-240 is given in Example 3-1.
Generally, a model form is first proposed (i.e., a hypothesis that
must be tested) using plots of the data. A simple plot can show obvious
inadequacies in the model form and can suggest a better form. Alter-
natively, it can show excessive scatter in the data and warn against
overconfidence in an adequate fit. The least squares estimates of the
parameters are determined using a weighted or an optimization (i.e.,
trial-and-error) search procedure if required. This gives the best
estimates if the assumptions are valid. Further testing the adequacy
of the fitted model requires using both plots of the residuals and the
sum of squares of the residuals. Finally, an estimate should be made
of the precision of each parameter estimate by statistical analysis (e.g.,
95% confidence intervals for the parameters).
Often, one or more model forms in chemical reaction kinetics may
fit the data. Although it is tempting to want to justify a specific model
as the mechanism of the reaction, it is preferable to only infer that
the model could be the mechanism. It is also desirable that the reaction
mechanism taking place be understood in order to solve a problem in