5.68J/10.652J Spring 2003 Lecture Notes Tuesday April 15, 2003
Kinetic Model Completeness
We say a chemical kinetic model is
complete
for a particular reaction condition when it
contains all the species and reactions needed to describe the chemical processes at that
reaction condition to some specified level of accuracy. In other words, the kinetic model
is complete when all the reactions whose rate constants it sets equal to zero really are
negligible for the specified reaction conditions and error tolerance.
Note that this definition of
complete
does NOT assume that the rate constants or any
other parameters in the model are correct, just that the reactions excluded from the model
are indeed negligible.
If you believe you can list all possible reactions of the species in the model, a necessary
condition for a model to be complete is that the rates of these reactions to form species
not included in the model must be smaller than some error tolerance; this approach was
introduced by R.G. Susnow et al. (1997). However, this condition is not sufficient: e.g.
one of the minor neglected species could be a catalyst or catalyst poison with a very big
influence on the kinetics at very low concentrations. If the kinetics are strictly linear you
can show that this necessary condition is sufficient (Matheu et al. 2002, 2003); an
important example of strictly linear kinetics are the ordinary master equations used to
compute pressuredependent reaction rates.
Of course, you usually do not know all the possible reactions. But for purposes of kinetic
model reduction one usually assumes that the initial “full” model is complete; you are
satisfied if the reduced model reproduces the full model to within some error tolerances
under some reaction conditions.
Sensitivity Analyses
A. Definitions of Sensitivities
Suppose you have a model you think is complete at some reaction condition, and you
have some estimates for
k
and
Y
o
so you can numerically solve
d
Y
/dt =
F
(
Y
,
k
)
Y
(t
o
) =
Y
o
to get your best prediction for the trajectory
Y
(t). (Y is normally made up of many mass
fractions y
i
(t) and a few other timevarying state variables e.g. T(t).) You never know the
rate (and other) parameters
k
and the initial concentrations
Y
o
exactly, so you are always
interested in how much the predicted trajectory would vary if these values were a little
different than the values you assumed, i.e. how sensitive is your prediction to the values
of these parameters? A common way to express this is to write
Y
(t) as a Taylor expansion
in the parameters
k
and
Y
o
:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documenty
i
(t) = y
i
(t; k,Y
o
) +
Σ
(
∂
y
i
(t)/
∂
k
n
)(k
n
’– k
n
)
o
+
Σ
(
∂
y
i
(t)/
∂
y
m
)( y
o
m
’ y
o
m
) + …
The rate constants k usually depend on T and P, so if T and/or P vary with time the first
term gets to be a mess. The most popular approach (used in the latest versions of
CHEMKIN, but not consistently in earlier versions) is to imagine that all of the rate
constants k
n
(T,P) are multiplied by scaling factors D
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '03
 RogerD.Kamm
 Reaction, rate constants, Sensitivities

Click to edit the document details