Page 204 - Modeling of Chemical Kinetics and Reactor Design

174 Modeling of Chemical Kinetics and Reactor Design

the relative error in the concentration measurement [0.01C AO /C (t)]
A
increases with time. It is possible to minimize the sum of N measure-
ments by:

N
σ = ∑ W i i ( [ y exp tl ) − (calc )] 2 (3-236)
2
y
i
= i 1
where W is the weighing factor.
i
The weighted least-squares analysis is important for estimating para-
meter involving exponents. Examples are the concentration time data
–kt
for an irreversible first order reaction expressed by C = C AO e , and
A
the reaction rate-temperature data expressed by − ( r A ) = k C e −E a RT .
A
O
These equations are of the form
Y = Ae –BX (3-237)
where Y = C or (–r ) and X = t or 1/T, respectively.
A A
Linearizing Equation 3-237 gives
ln Y = ln A – BX (3-238)
It is also possible to determine A and B that minimize the weighted
sum of squares. The weighting function is the square of the inde-
pendent variable, and the function to be minimized is

N
σ = ∑ i 2 [ yln i (exp tl )−ln y i (calc )] 2 (3-239)
2
y
= i 1
VALIDITY OF LEAST SQUARES FITTING

The validity of least squares model fitting is dependent on four
principal assumptions concerning the random error term ε, which is
inherent in the use of least squares. The assumptions as illustrated by
Bacon and Downie [6] are as follows:

Assumption 1: The values of the operating variables are known
exactly. In practice, this is interpreted to mean that any uncertainty
associated with a value of an operating variable has much less effect
on the response value than the uncertainty associated with a measured
value of the response itself.
Assumption 2: The form of the model is appropriate. Statistically, this
is expressed as E(ε) = 0 for all data, where ε is the random error.