Page 205 - Modeling of Chemical Kinetics and Reactor Design

Reaction Rate Expression 175

Assumption 3: The variance of the random error term is constant
over the ranges of the operating variables used to collect the data.
When the variance of the random error term varies over the operat-
ing range, then either weighted least squares must be used or a
transformation of the data must be made. However, this may be
violated by certain transformations of the model.
Assumption 4: There is no systematic association of the random
error for any one data point with the random error for any other
data point. Statistically this is expressed as: Correlation (ε , ε ) =
v
u
0. For u, v = 1, 2, . . . n, u ≠ v,
where ε = random error for experimental run number u
u
ε = random error for experimental run number v
v
PROBLEMS AND ERRORS
IN FITTING RATE MODELS

Several methods are used to fit rate models, the two most common
of which often give erroneous results. The first is the transformation
of a proposed rate model to achieve a model form that is linear in
the parameters. An example is the nonlinear model:

r = kC C b (3-240)
a
A B
which can be transformed into linear form by taking logarithm of
both sides

ln r = ln k + a ln C + b ln C B (3-241)
A
Equation 3-241 is a linear model of the form

Y = C + C X + C X 2 (3-242)
2
1
1
O
where X = ln C and X = ln C .
A
1
B
2
Another example of model transformation to achieve linearity is the
change from the nonlinear rate equation
r = kk C A (3-243)
O
1
1 + kC + k C B
2
A
1
to the equation