Page 289 - Elements of Chemical Reaction Engineering 3rd Edition

Sec. !XI Least-Squares Analysis 26 1

N N

we want the values of OL and k that will make s2 a minimum.
If POLYIMATH is used, one should use the absolute value for C,,, which
is the: term in brackets in Equation (5-37), that is,
n
-
= C [CAmi- (abs[CAia- (1 - a)kti])1’(1-a)]2 (5 3 8)
i=l
Another way to solve for the parameter values is to use time rather than
concentrations:

(5-39)

That is, we find the values of k and a that minimize
N N
s2 = 1 (tmi - tCi)2 = (5-40)
i= 1
Discussion of weighted least squares as applied to a first-order reaction i:; pro-
vided on the CD-ROM.

5.5.3 Weighted Least-Squares Analysis

Both the linear and nonlinear least-squares analyses presented above
assume that the variance is constant throughout the range of the measured vari-
ables. If this is not the case, a weighted least-squares analysis must be used to
obtain better estimates of the rate law parameters. If the error in measurement
is at it fixed level, the relative error in the dependent variable will increase as
the independent variable increases (decreases). For example, in a first-order
decay reaction (C, = CAo@), if the error in concentration measurement is
0.01 CAo, the relative error in the concentration measurement [o.olcA()/cA(t)]
will increase with time. When this error condition occurs, the sum to be iini-
rnized for N measurements is
N
W,
u~ =: 1 [y, (exptl) - y, (calc)l2
I= 1
where Wi tr weighting factor.
is
]For parameter estimation involving exponents, it has been shown hat a
weighted least-,squares analysis is usually necessary. l4 Further discussion on
weighted least squares as applied to a first-order reactnon is given on the CD-ROM.

14A C. Noms, Computational Chemistry: An Introduction to Numerical Solution (New
York: Wiley, 19Sl>, and D. M. Himmelblau, Process Analysis by Statistical Meithods
(New York: Why, 19701, p. 195.