The Directed Relation Graph (DRG) method [6], [7], [8] identifies unimportant species in a reaction mechanism by resolving species coupling without any a priori knowledge of the system. It requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. It has a distinct advantage over the Principal Component Analysis in that it does not require the inclusion of sensitivity analysis results in the solution on which it operates. DRG does require local temperature, pressure, and species composition data to calculate the production rate of a species, which can be accomplished using calls to Ansys Chemkin core routines. From this it constructs a directed relation graph with all of the species in the mechanism and uses a user-specified error tolerance to remove unimportant species from the mechanism.
Direct species coupling can be defined by the immediate error to the production rate of
a species A, introduced by the removal of another species B from the mechanism. Such
immediate error, noted as 
, can be expressed as
 | (3–20) |
where A and B indicate the species, 
indicates the 
th reaction of the mechanism, 
indicates the stoichiometric coefficient of species A in the
th reaction, and 
is the reaction rate of the 
th reaction that can be readily obtained from Ansys Chemkin. The denominator
represents the total absolute contributions to the production rate of species A by all
reactions in the mechanism involving species A. The numerator represents those contributions
by reactions involving species B. When the immediate error 
is larger than a pre-defined error-tolerance level, it means that the
removal of species B from the mechanism will introduce an error to the production rate of
species A that is beyond that which has been deemed acceptable to the user. Therefore, if
species A is to be kept in the skeletal mechanism, species B must also be kept to avoid
non-negligible error in the concentration of species A. In addition, if the removal of
species C will introduce a significant error in species B, retaining species A will also
require species C to be retained in the skeletal mechanism, because of the indirect species
coupling between species A and C. On the other hand, if a species is not required either
directly or indirectly by any species to be retained in the skeletal mechanism, this species
can be safely removed because it will not introduce any significant error.
To identify all of the direct and indirect species coupling effects in a reaction mechanism, a directed relation (virtual) graph is used to map all of the species in the mechanism and all of the dependencies between the species. Such a graph can be constructed using the following rules:
Each vertex in the graph is uniquely mapped to a species in the reaction mechanism.
The starting vertices of the graph correspond to the major species (that is, fuel and oxidizer) in the reaction mechanism.
A directed edge A
B exists in the graph if and only if the corresponding error
is larger than a pre-defined error tolerance.
The directed relation graph in Reaction Workbench is constructed using the following steps according to the rules listed above:
Calculate all of the immediate error values
, between all species in the reaction mechanism.Eliminate all of the immediate error values
that are less than the pre-defined error tolerance.Sort all the remaining directed edges in decreasing order of
.Initialize an empty graph with only vertices, corresponding to all the species in the reaction mechanism.
Mark the vertices corresponding to any reactants, fuel molecules, or oxidizer components, with 1 and the rest with -1.
Insert directed edge A
B into the graph in decreasing order of 
. If A is marked with a positive value and B is marked with -1,
depth-first search with B as root and mark every newly discovered vertex with the value
.Eliminate all species with the marked value -1 from the reaction mechanism.
Eliminate all reactions involving the eliminated species from the mechanism.