Page 203 - Modeling of Chemical Kinetics and Reactor Design

Reaction Rate Expression 173

Equations 3-232, 3-233, and 3-234 can be solved simultaneously
to determine the unknowns C , C , and C , which are the rate constant
O
2
1
k, and the orders (a and b) of the reaction respectively. This type of
problem is best expedited using a computer program. Appendix A
reviews the multiple regression that is being employed to determine
the rate constant k and the orders (a and b) of the reaction. The
computer program (PROG3) on the CD-ROM determines the rate
constant k and the orders (a and b) of the reaction. Other parameters
such as the reaction order, frequency factor k and activation energy
O,
E can be determined using regression analysis.
a
NONLINEAR ANALYSIS

Another method for determining rate law parameters is to employ
a search for those parameter values that minimize the sum of the
squared difference of measured reaction rate and the calculated reac-
tion rate. In performing N experiments, the parameter values can be
determined (e.g., E , C , C , and C ) that minimize the quantity:
O
2
1
a
N
r
σ = s 2 = ∑ (r im − ) 2 (3-235)
ic
2
−
NK NK − 1
–
= i 1
2
where σ = variance
2
s = Σ (r – r ) 2
im ic
N = number of runs
K = number of parameters to be determined
r = measured reaction rate for run i
im
r = calculated reaction rate for run i
ic
Nonlinear least squares curve fitting using the Microsoft Solver is
reviewed in Appendix B.
WEIGHTED LEAST SQUARES ANALYSIS
A weighted least-squares analysis is used for a better estimate of
rate law parameters where the variance is not constant throughout the
range of measured variables. If the error in measurement is corrected,
then the relative error in the dependent variable will increase as the
independent variable increases or decreases.
Consider a first order reaction with the final concentration expressed
–kt
by C = C e . If the error in concentration measurement is 0.01C ,
A AO AO