Page 77 - Introduction to chemical reaction engineering and kinetics

3.5 Notes on Methodology for Parameter Estimation 59

(3) The function y = a + bx + cx2 + dx3 is linear with respect to the parameters a,
b, c, d (which may be determined by linear regression), but not with respect to
the variable x .
The reaction orders obtained from nonlinear analysis are usually nonintegers. It is
customary to round the values to nearest integers, half-integers, tenths of integers, etc.
as may be appropriate. The regression is then repeated with order(s) specified to obtain
a revised value of the rate constant, or revised values of the Arrhenius parameters.
A number of statistics and spreadsheet software packages are available for linear re-
v
gression, and also for nonlinear regression of algebraic expressions (e.g., the Arrhenius
“OF
0 equation). However, few software packages are designed for parameter estimation in-
volving numerical integration of a differential equation containing the parameters (e.g.,
equation 3.4-8). The E-Z Solve software is one package that can carry out this more dif-
ficult type of nonlinear regression.

Estimate the rate constant for the reaction A + products, given the following data for
reaction in a constant-volume BR:

tlarb. units 0 1 2 3 4 6 8
c,/arb. units 1 0.95 0.91 0.87 0.83 0.76 0.72

Assume that the reaction follows either first-order or second-order kinetics.

SOLUTION This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9

v
“O-v
0 (with n = 2), which correspond to the relationships developed for first-order and second-
order kinetics, respectively. However, here we illustrate the use of nonlinear regression
applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized
integrated forms. The method employs user-defined functions within the E-Z Solve soft-
ware. The rate constants estimated for the first-order and second-order cases are 0.0441
and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in
E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental
data and the predictions from either the first- or second-order rate expression. This lack of
sensitivity to reaction order is common when fA < 0.5 (here, fA = 0.28).

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain
some insight from a residual plot, which depicts the difference between the predicted
and experimental values of cA using the rate constants calculated from the regression
analysis. Figure 3.10 shows a random distribution of residuals for a second-order re-
action, but a nonrandom distribution of residuals for a first-order reaction (consistent
overprediction of concentration for the first five datapoints). Consequently, based upon
this analysis, it is apparent that the reaction is second-order rather than first-order,
and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is
much smaller for second-order kinetics than for first-order kinetics (1.28 X 10V4 versus
5.39 x 10-4).
We summarize some guidelines for choice of regression method in the chart in Figure
3.11. The initial focus is on the type of reactor used to generate the experimental data