Page 282 - Elements of Chemical Reaction Engineering 3rd Edition

254 Collection and Analysis of Rate Data Chap. 5

This procedure is continued by further varying a and k until we find their
best values, that is, those values that minimize the sum of the squares. Many
well-known searching techniques are available to obtain the minimum value
u&.I1 Figure 5-10 shows a hypothetical plot of the sum of the squares as a
function of the parameters a and k:

u2 = f(k, a) (5-35)

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"J

Figure 5-10 Minimum sum of squares.

In searching to find the parameter values that give the minimum of the
sum of squares u2, one can use a number of optimization techniques or soft-
ware packages. The procedure begins by guessing parameter values [e.g., Table
5-2 (a = 1, k = 1 s-l)] and then calculating r, and then u2 for these values
(see, e.g., the sixth column in Table 5-2). Next a few sets of parameters are
chosen around the initial guess, and u2 is calculated for these sets as well. The
search technique looks for the smallest value of u2 in the vicinity of the initial
guess and then proceeds along a trajectory in the direction of decreasing u2 to
choose different parameter values and determine the corresponding u2. The
trajectory is continually adjusted so as always to proceed in the direction of
decreasing u2 until the minimum value of u2 is reached. A schematic of this
Vary the initial procedure is shown in Figure 5-1 1, where the parameter values at the mini-
guesses Of mum are a = 2 and k = 5 s-], If the equations are highly nonlinear, the initial
parameters to make
sure you find the guess is extremely important. In some cases it is useful to try different initial
true minimum guesses of the parameter to make sure that the software program converges on

"(a) B. Carnahan and J. 0. Wilkes, Digital Computing and Numerical Methods (New
York: Wiley, 1973), p. 405. (b) D. J. Wilde and C. S. Beightler, Foundations of Opti-
mization (Upper Saddle River, N.J.: Prentice Hall, 1967). (c) D. Miller and M. Fren-
klach, Int. J. Chem. Kinet., 15, 677 (1983).
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