Other projects:
P-Dependence & Model Construction
Model Reduction and Numerical Toolkit
Adaptive Chemistry for Reacting-Flow Simulations
Kinetic Model Reduction and Numerical Toolkit for Kinetics
B.Bhattacharjee,
J.Tolsma, W.H.Green,
Prof. Paul
Barton
Supported by Alstom Power; EPA Center for Airborne Organics
Portions of the work done in collaboration with A.Singer
Motivation for Performing Kinetic Model Reduction
In recent years, increasingly stringent environmental regulations and
competitive markets have highlighted the need for detailed modeling of
product and pollutant formation in chemical processes. As a result, large-scale
mechanisms comprising thousands of well-parameterized reactions and hundreds
of species have been developed. Detailed chemistry can now be simulated
on the order of seconds in closed, spatially homogeneous (batch or steady-state)
reactors. However, most industrial processes are spatially-inhomogeneous.
In such cases, the chemical activity is strongly coupled to heat and mass
transfer within the reacting-flow system. The computational cost of the
partial differential equation system that must be solved increases rapidly
with the complexity of the
chemistry and the system geometry. Furthermore, the inherent stiffness
of many chemical systems (e.g. combustion) accounts for upto 90% of this
cost. Multi-dimensional, turbulent simulations of reacting flows using
large-scale mechanisms are currently impossible. Despite rapid escalation
in hardware capacity, CPU and memory loads will remain impractial in the
forseeable future. In practice, detailed mechanisms are pruned by inspection,
and a single reduced kinetic model is used in the reacting-flow simulation.
Not only is this process tedious and error-prone, but the single, simplified
model is rarely valid over the entire composition space of the reacting-flow
field. As a result, the predictive value of the model is compromised.
Ideally, detailed chemistry would be incorporated into reacting-flow simulations,
without incurring the full computational cost of integrating large-scale
mechanisms. The adaptive chemistry approach addresses this problem by
replacing the full mechanism with a number of locally-accurate reduced
kinetic models (rather than a single, global reduced model). In a limited
region of the flowfield, the chemical activity is often reproduced accurately
by a small subset of the original reactions and species. Thus, by selecting
an appropriate reduced model to replace the full chemistry, a detailed
reacting-flow simulation can be performed with reasonable computational
load. Clearly, the success of any adaptive chemistry implementation depends
on the availability of a library of reduced models. Herein lies the importance
of a robust, efficient and system-independent methodology for kinetic
model reduction.
A Linear, Integer Programming Framework for Model Reduction
In the course of this project we have developed and implemented an optimization-based
framework for kinetic model reduction. We assume that the user has a structurally-complete,
well-parameterized kinetic mechanism which accurately models the system
of interest. We refer to this as the 'full' mechanism. In order to generate
a reduced model (at a specified point in composition space) an integer
optimization problem is solved. The reduced model thus obtained has the
same
number of species, but fewer reactions than the full mechanism. The linearity
of our formulation guarantees that the optimization problem is solved
to global optimality- i.e. the smallest possible reaction subset is always
selected to satisfy the specified error tolerance. When the number of
reactions significantly exceeds the number of species in the reduced mechanism,
our method typically yields a linear decrease in integration CPU cost
with the number of reactions.
Furthermore, the linear integer program and can be solved much more robustly
and efficiently than (nonconvex, isoperimetrically-constrained) optimization
formulations previously described in the literature. The algorithm has
been tested on several combustion mechanisms, the largest being the Lawrence
Livermore n-heptane combustion mechanism (2446 reactions). A model library
comprising ~600 reduced models for a 2-D laminar methane flame was generated
from
the GRImech 3.0 mechanism (325 reactions). For more details, please refer
to Optimally
Reduced Kinetic Models: Reaction Elimination in Large-Scale Kinetic Mechanisms.
Ranges of Validity and Parametric/Operating Uncertainty
Reduced kinetic models are derived using point constraints in our
optimization problem. Thus, they are strictly valid only at the nominal
composition and temperature at which they were reduced. To be of practical
value, a reduced model must be associated with a range of validity (in
composition space) over which it is guaranteed to predict the chemistry
accurately. One heuristic approach to assigning ranges of validity is
to apply multiple point constraints during model reduction. One can then
assume that the reduced model obtained is valid everywhere within the
convex hull of those points. We are currently developing a methodology
to identify rigorous ranges of validity for reduced models. An initial
(usually optimistic!) guess is provided for the range of validity of a
reduced model. The interval error associated with model reduction is then
evaluated over this range of compositions. The type of interval extension
used (e.g. natural or Taylor) determines the tightness and computational
cost of the error estimation. If a composition interval is found to be
infeasible (i.e. it exceeds the user's tolerable range of error) the composition
interval is reduced (e.g. by bisection), and the interval error is re-evaluated.
This process is repeated until the user's error criteria are satisfied,
and the final interval is assigned as the range validity for the reduced
model. . Since the reduced model is known to be valid at the nominal reduction
point, this 'brute force' approach guarantees finite termination. However,
this approach is not very efficient. Neither does it guarantee that the
largest possible feasible region is correctly identified.
In the case of the model reduction problem, we are interested in locating
a range of compostion and temperature over which the model reduction error
is tolerable. This is a special case of the general feasibility problem
where one is interested in identify a valid parameter or operating space
over which certain constraints are known to hold, e.g. in modeling a CSTR
an engineer may be interested in the range of operating temperatures over
which a certain selectivity is achieved. In the literature, such problems
are usually framed as semi-infinite programming problems, having a finite
number of decision variables, and an infinite number of constraints. We
propose to reformulate the feasibility problem using interval-valued constraints
as described above. We are currently developing an
algorithmic framework to improve the efficiency of the search and identify
the largest possible feasible region.
Software
We have developed a software package called RIOT (Range Identification
and Optimization Tool for Kinetic Modeling) to perform the model reduction
and range validation tasks described above. The full mechanism, and relevant
thermodynamic data are assumed to be available in CHEMKIN-compatible format.
The user is required to specify the compositions point(s) at which a reduced
model is to be constructed. This may be done directly, or by specifying
mesh points in a PREMIX/OPPDIFF output file. The CHEMKIN II library is
then used to perform the rate and thermodynamic calculations necesary
to formulate the problem constraints. The resulting linear, integer program
is then solved using the CPLEX MIP callable library. Finally, a range
of validity is calculated using Taylor interval extensions generated by
DAEPACK. RIOT generates two output files: a CHEMKIN mechanism file describing
the reduced model, and a summary file detailing the reactions retained
and the range of validity for the reduced model. Since the quality (tightness)
of the interval extensions obtained is strongly influenced by the functional
form of the rate calculations, only irreversible, Arrhenius-type reactions
should be used in the RIOT code. General (i.e. mechanisms containing reversible
and fall-off reactions) should be preprocessed using MECHMOD and FITARR.
A graphical description of the software is shown below: