On the Formation of Nitrogen Oxides During the Combustion of ...

A.2 Concepts of Kinetics Reduction
power, the net reaction rates of the latter can be set to zero, leading to algebraic
relations. The steady-state approximation searches for species whose
production and consumption cancel out immediately. By doing so, many intermediate
species can be eliminated from a detailed mechanism if their net
production rates tend towards zero. In practice, species are regarded as quasisteady
if their net production rate is small compared to their respective production
and consumption rates [11, 43, 336]. This process of analyzing the
timescales was successfully automated, resulting in the so-called computer
assisted reduction method (CARM). It evaluates reaction rates and automatically
eliminates species at quasi-steady state on the basis of results obtained
from perfectly stirred reactor models, employing more or less detailed mechanisms
[67, 90, 299].
The computational singular perturbation (CSP) method investigates the dynamics
of the source term vector and tries to find the directions in which it
will rapidly reach steady-state. Eigenvalues and eigenvectors of the Jacobian
matrix of the source term represent chemical timescales and reaction groups,
respectively [11, 43, 256].
The intrinsic low-dimensional manifold (ILDM) method was presented by
Maas and Pope [262] in 1992. Similar to the CSP method, the ILDM method
tries to find directions in state space in which the source term rapidly reaches
steady-state. The method is based on the assumption that a combustion system
follows certain trajectories in state space during the combustion process.
However, these trajectories are not associated with any particular species or
reaction. Since the steady-state assumption is applied locally, different reaction
paths can be captured. The common part of all trajectories can be described
as a parameterized curve of only a single progress variable. Assuming
all but two processes to be in steady-state results in a surface parameterized
by two variables and so on. This process generally leads to intrinsic lowdimensional
manifolds, in which the number of dimensions represents the
number of slow processes. However, the higher the number of dimensions,
the more rapidly the trajectories through state space will approach the manifold.
As time approaches infinity and independently of the initial conditions,
all solution trajectories of stable systems will approach the same equilibrium
point. Thus, movement along the ILDM corresponds to the evolution of the
predefined slow processes, and the equilibrium point effectively would be a
zero-dimensional manifold [191, 262, 399, 400].
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