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An Intelligent System for Reaction Kinetic Modeling and
CatalystDesign
Santhoji Katare,† James M. Caruthers, W. Nicholas Delgass,
andVenkat Venkatasubramanian*
Center for Catalyst Design, School of Chemical Engineering,
Purdue University,West Lafayette, Indiana 47907
The continuing development of high throughput experiments (HTEs)
in catalysis has dramaticallyincreased the amount of data that can
be collected in relatively short periods of time. Evenwhen HTEs can
afford “Edisonian” discovery, how can the increasing amounts of
data beconverted to knowledge to guide the next search in the vast
design space of catalytic materials?To address this question, we
recently proposed a catalyst design architecture that uses
detailedkinetic models. In this paper, we describe Reaction
Modeling Suitesa rational, automated, andintelligent environment,
based on systems, artificial intelligence, and optimization
techniquesthat aid the development of kinetic models. We
demonstrate its utility in developing a kineticmodel for propane
aromatization on zeolite. We also show the proof-of-concept of how
a geneticalgorithm-based search strategy can be used to search for
kinetic parameters that correspondto an improved catalyst.
1. Introduction
The design of new materials possessing desiredmacroscopic
properties or performance characteristicsis an important, although
difficult, problem. Materialsdesign finds applications in the
development of diversematerials such as polymers, polymeric
composites,blends, paints and varnishes, refrigerants,
solvents,drugs, pesticides, and so on. The traditional
approachrequires the designer to hypothesize a molecule ormaterial,
synthesize it, and experimentally evaluate itto see if it meets the
desired properties or performancecriteria and to reformulate the
design if the targets arenot met. This Edisonian guess-and-test
method is time-consuming, expensive, cumbersome, and
complicatedstime-consuming and expensive because of the nature
ofthe experiments and cumbersome and complicatedbecause of the
underlying huge, nonlinear search space.
In the area of catalyst design, experimentally tuningcatalyst
structure to improve performance is well known.1Advances in surface
science techniques2 that enablemanipulation of individual atoms on
the catalyst surfacein real time have3 immensely contributed to
improvedunderstanding of the catalysts. With the advent of
highthroughput and combinatorial methods, experimentalguidance
techniques such as hierarchical screening,4evolutionary ideas,5 and
those based on statistics6 havebecome relevant. Despite these
efforts, the nonlinearityand the size of the underlying search
space still pose astrong challenge to systematic design. Also,
designtechniques that are mainly driven by experiments willonly
enable in the collection of information, and unlessthere is a
method to convert that information intoknowledge and insight, a
general catalyst design meth-odology would remain an unsolved
problem.
Theory and model based catalyst design strategies arewell known
in the literature. These include the idea of
using qualitative reasoning and knowledge-based sys-tems,7,8
efforts toward using computational models andcalculations to guide
the search for new materials,9,10and those that use detailed
microkinetic models to studycatalytic systems.11 A more
comprehensive review ofcatalyst design techniques is available
elsewhere.12
Computer-aided materials design13 offers an attrac-tive
alternative to the above approaches, whereby thedesign problem
involves the use of computer-basedprocedures to systematically
identify appropriate mo-lecular structures that satisfy a set of
desired properties.In general, the overall task requires the
solution of twosubproblems as shown in Figure 1: the forward
prob-lem, which involves the computation of performancemeasures or
physical, chemical, and/or biological prop-erties from the product
structure and formulation/composition; and the inverse problem,
which entails theidentification of the appropriate molecular
structure orcomposition given the desired macroscopic properties.To
solve the inverse problem, which is the true designproblem, a
robust forward model is essential. Thisforward model development is
complicated because theunderlying system is often complex. The main
chal-lenges include identification of the key descriptors
thatcharacterize the system under study and developmentof a
methodology to link the material descriptors toperformance. We
recently proposed a methodology14 forbuilding forward models for
designing catalysts. This
* To whom correspondence should be addressed.
E-mail:[email protected]. Phone: 765 494 0734. Fax: 765
4940805.
† Current address: Department of Chemical Engineering,University
of Houston, Houston, TX 77204.
Figure 1. Schematic of the forward and inverse problems
incomputer-aided materials design.
3484 Ind. Eng. Chem. Res. 2004, 43, 3484-3512
10.1021/ie034067h CCC: $27.50 © 2004 American Chemical
SocietyPublished on Web 02/11/2004
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involves a systematic, rational, and iterative techniquefor
knowledge extraction (KE) from high throughputexperimentation data.
The KE procedure facilitatesconvergence to a quality predictive
model from an initialapproximate model by systematically
incorporating anynew information about the system as and when it
isavailable. In this paper, we describe the ReactionModeling Suite
(RMS)sa collection of systems, optimi-zation, and artificial
intelligence based tools that enableKE by aiding the expert in
building robust kineticmodels.
The main objective of this work is the design anddevelopment of
systems tools for integrating largeamounts of diverse sets of data
with the hypothesisgeneration and screening process, at a pace
com-mensurate with the rate of data production. In particu-lar,
emphasis would be on developing user-drivensystems tools to offer a
systematic, less error-prone,automated environment for an expert to
postulate,evaluate/optimize, and refine reaction mechanisms.
Thetools would allow rigorous analysis of multiple
reactionmechanisms in the light of data, with minimum
humanintervention. This would make the whole process userfriendly
and quick. The rest of this paper is organizedas follows. RMS will
be described in the next sectionalong with a brief review about the
requirements of anautomated kinetic model building system and the
state-of-the-art in this area. In section 3, the various
capabili-ties of the RMS tools in hypothesis screening,
reactionnetwork analysis, model refinement, model discrimina-tion,
and experimental formulation will be demon-strated by developing a
kinetic model for propanearomatization on H-ZSM-5 zeolite catalyst.
This sectionwill also include a proof-of-concept of the
inverseproblem that involves the search for an improvedcatalyst for
paraffin aromatization. The main contribu-tions will be summarized
and general conclusions willbe drawn in the final section.
2. Reaction Modeling Suite
The key requirement of any model building procedureis the rapid
screening of an expert-postulated reactionmechanistic hypothesis
that could explain the data. Thisprocess should be fast enough to
keep pace with the rateof data generation from combinatorial and
high through-put experiments. Another challenge is to develop
user-driven tools that naturally relate to the domain expert.Toward
this end, we have developed the RMS thatenables rational, automated
reaction kinetic modelingand thus facilitates knowledge archiving
and retrieval.The software in RMS allows easy encoding of
reactionchemistry knowledge in terms of pseudo-English lan-guage
rules and enables automated and fast hypothesistesting by screening
through multiple hypothesis in asystematic manner.
Traditionally, the reaction-modeling problem has beenan art
tackled by chemists and chemical engineers orother domain experts.
On the basis of their experienceor knowledge about the system at
hand, the experts firstformulate a set of heuristics or rules that
appear togovern the process. These rules directly translate to
areaction mechanism, and a mathematical model is thenconstructed
from it. Depending on the discrepancies inthe predictions of the
model and experimental results,the experts go back to the initial
stage of rule formula-tion and consider alternative or additional
rules. Whenthe reaction network consists of a large number of
reactions and chemical species, the development of
themathematical model becomes cumbersome. The overallprocess as
shown in Figure 2 is therefore “Edisonian”and is often protracted,
cumbersome, and expensive. Itis protracted because even a slight
change in one of thereactions in the mechanism leads to multiple
changesin the mathematical equations that represent thesereactions.
Since building a feasible mechanism startingfrom a plausible set of
steps is often iterative, the wholeprocess becomes time-consuming
and highly prone toerrors. Any efforts to automate the same using
computer-assisted methods can lead to considerable savings intime
and money.
The importance of developing robust kinetic modelsfor
understanding the underlying chemical system iswell-known in the
literature. For example, concepts suchas stoichiometric network
analysis,15 mathematicallycontrolled comparison and canonical
representation ofdifferential equation models,16 development of
largescale reaction networks based on elementary reac-tions,11,17
deduction of reaction mechanisms given a setof elementary steps,18
and reverse engineering of reac-tion mechanisms19,20 have attracted
widespread atten-tion. Software tools for automating the process
ofreaction model building have also been available. Table1 shows a
list of software tools available in the literaturethat aid the
process of modeling chemical reactionnetworks. The tools have been
categorized based ontheir ability to (1) formulate a reaction
network fromhigh level chemistry rules, (2) visualize the
reactionnetwork, (3) parse the reaction network to get a
math-ematical model, (4) solve the model and optimize
theparameters, and (5) analyze the results statistically.
Fordetailed reviews of software tools for reaction kineticmodeling,
the reader is referred to Arkin21 and Katare.12
Drawing ideas from the traditional modeling meth-odologies, we
propose a set of tools with a systemsviewpoint for effectively and
efficiently implementingthe ideas of chemical reaction modeling
within theperspective of computer-aided materials design.
Specif-ically, the current work deals with a framework
fordeveloping forward prediction models for surface reac-tions from
a catalyst design (Figure 1) perspective. Thekey steps involved in
the process of model building areas follows: generation of the
simplest plausible reactionmechanism; translation of the mechanism
to a compu-tationally tractable mathematical model; solving
themodel to estimate the parameters in light of highthroughput
and/or insufficient experimental data; refin-
Figure 2. Traditional model building process.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3485
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ing the model to better fit the data by altering themechanism;
suggesting new experiments that could helpdiscriminate among
multiple models.
The main challenges involved are as follows:Mechanism Generation
from Reaction Rules.1. Unambiguous representation of the large
number
of reactions and species.2. A compiler that understands the
generic reaction
rules and a network generator that applies these
rulesrecursively to all possible reactant species.
3. Pruning the reaction mechanism to get the simplestpossible
model that can explain the data consistently.
4. Assimilating thermokinetic data and experimentalinformation
involving the various reactions/species tominimize the number of
thermodynamic and/or kineticparameters to be optimized and to aid
the process ofparameter estimation.
Parameter Estimation.5. Robust solvers that handle the large
number of
differential-algebraic equations and parameter estima-tion
techniques that will evaluate the validity of theproposed mechanism
to model data.
6. Evaluation of the multiple solutions for the param-eters that
explain the data equally well.
Feature Extraction.7. Feature extraction techniques to identify
the dis-
crepancies between the key features of the modelpredictions and
that of the data so that they can be usedfor MR and experimental
formulation.
8. Mapping the feature discrepancies to the mecha-nistic rules
that generated the reaction mechanism.
Statistical Analysis.9. Estimation of the robustness of the
model.
We now describe RMS (Figure 3), the tools thatprovide solutions
to most of the above challenges.Specifically, in this section, we
present our implementa-tion of an English language
rules-to-reaction networkcompiler that translates pseudo-English
language rulesinto a chemical reaction network. Then we describe
ahybrid algorithm for parameter estimation that affordsa thorough
and efficient search of the nonlinear param-eter space. This is
then followed by a feature extractionprocedure that enables a
natural way for evaluating thevalidity of a model in light of the
data. Finally, weexplain the statistical analysis tools that have
beendeveloped to analyze the robustness of a kinetic model.
2.1. English Language Rules-to-Reaction Net-work Compiler.
Building a kinetic model is initiatedby an expert who proposes an
initial set of reaction rulesthat is most likely to explain the
experimentally ob-served product distribution. For example, for a
solid acidcatalyst system, adsorption, desorption, protolysis,
beta-scission, oligomerization, dehydrogenation, aromatiza-tion,
etc., form a plausible rule set which gives rise to alarge number
of elementary reactions. The first step ofpostulating a hypothesis
as rules is the most critical oneas it drives the subsequent
process of model buildingand evaluation. Moreover, the task of
model refinementbased on the model-data mismatch is typically
aimedat altering one or more of these basic reaction rulesrather
than independently changing the elementaryreaction steps. This is
because changing a single ruleaffects several chemically similar
elementary steps.Thus, the process of hypothesis screening is
largelydependent on how well the expert is able to postulateand
iteratively manipulate these reaction rules. There-
Table 1. List of Software Tools That Aid in the Process of
Modeling Chemical Reaction Networksa
no. software descriptors reference
1 Reaction Description Language F Pricket and Mavrovouniotis,
1997302 DBsolve VPOS Goryanin et al., 1999993 E-cell VPOb Tomita et
al., 19991004 Gepasi POS Mendes, 19931015 CRNT POS
ftp://ftp.che.rochester.edu/pub/feinberg/6 Dynetica VPO You et al.,
20031027 XMG FPOS Green et al., 2001398 NetGen FPOS Broadbelt et
al., 1994249 IBM CKS VPOS
www.almaden.ibm.com/st/msim/ckspage.html
10 MKM POS http://www.aue.auc.dk/∼stoltze/mkm/main.html11
Mitsubishi FPOS Hostrup and Balakrishna, 20017512 Chemkin POS Kee
et al., 198910313 KINAL A POS Turanyi, 199038
a The descriptors show the ability of the tool to formulate a
reaction network from higher level rules (F), visualize the network
(V),parse the network into mathematical model (P), solve the model
and optimize the parameters (O) and analyze the results
statistically (S).b Solves but does not optimize the
parameters.
Figure 3. Reaction Modeling Suite.
3486 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
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fore, any effort to automate the model building processshould
aid the expert as much as possible in his/hernatural working
language.
2.1.1. Representation of Molecules and Reac-tions. Software that
enables hypothesis screening andmodel building should provide as
much flexibility aspossible to the expert in postulating and
manipulatingthe various reaction rules. It should be possible
toinclude new reaction rules readily. The molecules andreaction
rules should be encoded in the natural languagethat is used by an
expert while formulating them.Previous works on automated model
generation haveused several representations including extended
SMILESnotation22 and bond-electron (BE) matrices23 to repre-sent
molecules and BE matrices,24 functional groupvectors,25-27 and
pseudo-English language rules28 torepresent reaction rules. SMILES
and BE matrices canbecome cumbersome29 as the complexity of the
reactiveintermediates increases. Extending the functional
groupvectors representation to non-hydrocarbon chemistrymay not be
straightforward.
Prickett and Mavrovouniotis30 have tried to overcomethe above
shortcomings by introducing a pseudo-Englishlanguage to describe
reaction rules along with a com-piler to translate these
instructions into a reactionnetwork. Their Reaction Description
Language (RDL)uses a sequence of commands to locate the reaction
site,to manipulate the reactant to form the product, and toprune
the reaction network with a syntax that mimicsthe way reactions are
typically described by the chem-ists.
RMS has been designed to facilitate hypothesis gen-eration and
screening, and one of its key requirementsis that it should enable
the initiation of this processusing a natural language interface.
RDL31 satisfies thiscondition, and so we have designed and
developed theReaction Description Language Plus Plus (RDL++) asa
system that extends RDL. RDL++ can be used tomodel reaction
mechanisms on solid acid-based cata-lysts, like zeolites, and is
designed to be more user-driven and extendible. Also RDL++ has been
integratedwith other tools that are geared toward building
robustreaction kinetic models.
2.1.2. The Reaction Description Language PlusPlus (RDL++). RDL++
is a compiler that translateschemistry rules in pseudo-English
language to a reac-tion network. Molecules and reaction networks
inRDL++ are represented in an object oriented fashionalong the
lines of RDL.28 Molecules are graphs whosenodes represent the atoms
and the edges denote theconnectivity between the atoms. The various
attributesof a molecule are shown in Table 2. The molecule
ischaracterized by its atoms, bonds, fragments (rings andchains),
and its role as a reactant or a product in aparticular reaction. An
atom’s attributes include itsneighboring atoms, the list of bonds,
its charge, elementtype, and the molecule or the fragment to which
itbelongs. A bond has its list of atoms and the list of
itsneighboring atoms, order, and the identity of themolecule or
fragment to which it belongs. Finally, the
fragment is represented by its type (ring or chain), alist of
atoms and bonds, and the molecule to which itbelongs.
A schematic of the RDL++ is shown in Figure 4.RDL++ consists of
(1) a compiler and a networkgenerator that transforms reaction
rules and globalpruning rules to a reaction network and (2) a
modelgenerator that generates a model from the reactionnetwork and
a set of grouping rules. The compilertranslates the English
language rules to intermediatecode or patterns which are then
recursively applied bythe network generator to all the species in
the reactionmixture to generate the reaction network.
Reaction rules in RDL++ are similar to that of RDL28and consist
of three important blocks of statements withspecific roles: (1)
identification of the reaction site(s)among the reactive species,
(2) transformation of thereactive sites to products, and (3) local
pruning of thereactions based on the reactive sites or on the
productsformed. The pruning rules restrict the type of
reactantsthat can enter a reaction and the products that can
beformed. A typical reaction rule that describes adsorptionof a
paraffin to form a carbonium ion (eq R.1) is shownin Table 3. Every
statement is in the form of a produc-tion rule and is enclosed in
parantheses. Comments arepreceded by a pair of forward slashes. The
statementat the beginning of every rule describes the name of
thereaction. This is followed by the identification of the
rateconstant of the reactions generated by this rule. The rulethen
identifies the reaction sitesa neutral carbon in thereactant. The
local pruning rules constrain the variouspossibilities, and in the
current version of paraffinadsorption (Table 3), it is required
that the reactant bea paraffin and that any cyclic species be
forbidden. Also,
Table 2. Attributes of a Molecule, Atom, Bond, and a Fragment in
RDL++
attributes
molecule atom, bond, fragment, reactant/product, network to
which it belongsatom neighboring atoms, list of bonds, charge,
element type, molecule or fragment to which it belongsbond list of
atoms, list of neighboring atoms, order, molecule or fragment to
which it belongsfragment type, list of atoms and bonds, molecule to
which it belongs
Figure 4. Schematic of the Reaction Description Language
PlusPlus.
Table 3. A Typical Rule in RDL++: Adsorption of aParaffin To
Give a Carbonium Ion
{// description of the rule(reaction-name “adsorption of
paraffin”)
// rate constant definition(rate-constant kaa)
// reaction site identifier(label-site m1 reactant)(label-site
c1 (find neutral-carbon))
// local pruning statements(require (paraffin m1))(forbid
(cyclic m1))(forbid (less-than (size-of m1) 2))
// transformation statements(add-charge c1)(connect c1
neutral-hydrogen)}
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3487
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the size or the number of carbon atoms in the reactanthas been
constrained to be less than 8. So, the pruningrules enforce the
condition that paraffin adsorption canonly take place on an acyclic
paraffin up to C7. Note thatany of these pruning rules can be
relaxed as per therequirement of the system under
consideration.12
2.1.3. Network Generator. The reaction rules areconverted by the
compiler to an intermediate code thatcontains the information about
the generic reactions.These patterns are now applied to all the
species in thereaction mixture to create the actual reaction
network.For example, if there are four rules and two
initialreactants, after the application of the four rules to thetwo
initial species, the rules are again applied to theproducts formed
from in the first pass. This process isrepeated until each of the
rules is applied to all thespecies in the reaction mixture.
2.1.4. Model ReductionsPruning of ReactionNetworks. Since the
analysis of complex reactionnetworks typically requires more data
than are oftenavailable, the mathematical model of the
reactionnetwork formed from the reaction rules is often pruned.The
area of simplification of mathematical models thatdescribe reaction
mechanisms has been reviewed byTomlin and co-workers32 and
Mavrovouniotis.33 Themethods of model reduction can be widely
divided intotwo partssreduction based on the time scale analysisand
the techniques that are not based on the timeevolution of the
species. Pseudo-steady-state analysisthat converts the differential
equations into algebraicequations, computational singular
perturbation basedon reaction rates, and the inertial low
dimensionalmanifold technique of Mass and Pope34 that use
thespecies rate trajectory to distinguish between the slowand fast
rates are examples of the former class. Sensi-tivity analysis,
which studies the importance of reac-tions and species to identify
the redundancy in thereaction network, lumping of a group of
species such asisomers or chemically similar groups, forms the
basisof time-independent techniques for model reduction.
In summary, the reaction network is often prunedusing a variety
of techniques,32,33,35 including sensitivityanalysis,36-38
math-programming methods,39-42 andmanifold techniques.34,43
However, these methods de-pend on the elimination of species and/or
kinetic stepsthat are not important for a particular data set,
wherethere is no assurance that this species and/or
reactionmechanism will not become important for other
reactionconditions. In contrast, Mavrouvouniotis and Prickett31have
suggested model reduction methods where knownreactivity
relationships between different species areused to eliminate
unimportant reactions; alternatively,reaction rate based techniques
have been used to controlthe size of the network.44
RDL++ consists of two types of pruning rulesslocalpruning rules
that are restricted to a particular reactionrule and global pruning
rules that are applied to all thereaction rules. The local pruning
rules include (1)forbidding a particular reactant from undergoing
areaction or a product from being formed, (2) limiting the
number of carbons in the reactants and/or the products,(3)
requiring or forbidding a particular pattern in thereactant and/or
product. For example, adsorption ofparaffin is restricted; paraffin
with fewer than twocarbonssmethaneswill not adsorb (Table 3). The
globalpruning rules apply to all the reaction rules and hencecan
restrict the formation of certain types of productsby any reaction
rule. As shown in Table 4, the globalpruning rules, for example,
forbid the formation ofspecies with two adjacent double bonds,
triple bondsamong the productssdefined as a “trifin”
product,species that have a double bond and a positive chargeand
species with charges on two different atoms.Another powerful global
pruning rule is forbidding theformation of any isomers. This
reduces the size of thenetwork to a great extent and is
particularly usefulwhen building models with data that cannot
distinguishbetween different isomers. Although the word
“pruning”implies that it happens after the actual
transformationtakes place, pruning rules defined in terms of
thereactants forbid the concerned reaction from beingexecuted for
unqualified reactants.
2.1.5. Examples of Network Generation withRDL++. We illustrate
the utility and versatility of theRDL++ chemistry compiler in
translating the pseudo-English language rules into a reaction
network usingan example of a set of paraffin reactions on a
zeolitecatalyst. This reaction mechanism consisting of
paraffinadsorption, desorption, dehydrogenation, and protolysisis a
critical subset of the reactions leading to paraffinaromatization,
a commercially important reaction forthe transformation of paraffin
to gasoline. The RDL++rules corresponding to these reactions are
shown inTable 3 and Tables 5-7 and their representative reac-tions
in R.1 through R.4, respectively. Specifically,reaction set S1
consists of (1) adsorption of a paraffin toform a carbonium ion
(Table 3), (2) desorption of thecarbonium ion to give back the
paraffin (Table 5), (3)carbonium ion dehydrogenation to give a
carbenium ion
Table 4. Global Pruning Rules
{(forbid (adjacent double-bond))(forbid (trifin product))(forbid
positive-carbon attached-to double-bond)(forbid (double
charge))(forbid (isomer product))}
Table 5. Carbonium Ion Desorption To Give a Paraffin
{(reaction-name “desorption of carbonium”)(rate-constant
kad)(label-site c1+ (find positive-carbonium))(label-site h1 (find
neutral-hydrogen attached-to c1+))(disconnect c1+
h1)(subtract-charge c1+)}
Table 6. Dehydrogenation of a Carbonium Ion GivesRise to a
Carbenium Ion and H2
{(reaction-name “dehydrogenation of carboniums”)(rate-constant
kcd)(label-site c1+ (find positive-carbonium))(forbid (quaternary
c1+))(label-site h1 (find neutral-hydrogen attached-to
c1+))(label-site h2 (find neutral-hydrogen attached-to
c1+))(disconnect h1 c1+)(disconnect h2 c1+)(connect h1 h2)}
3488 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
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and H2 (Table 6), and (4) protolysis of carbonium ion togive a
paraffin and a carbenium ion (Table 7).
These reactions describe the adsorption of paraffin toform
carbonium ions which subsequently protolyze anddehydrogenate to
give carbenium ions or desorb to giveback the paraffin. To
eliminate infeasible products, thefirst four global pruning rules
as shown in Table 4,which forbid adjacent double bonds and triple
bonds, acarbon with a charge attached to a double bond, andthe same
molecule with two charges, are used ingenerating the reaction
network.
To study the effect of the reaction rules on the inputspecies,
different reactantsspropane (C3), butane (C4),isobutane (2m-C3),
pentane (C5), 2-methylbutane (2m-C4), and 2,2-dimethylpropane
(2,2m-C3)shave beenused to generate the reaction networks. The
choice ofthe input species was based on the fact that we wantedto
examine the effect of the size of the reaction networkand the type
of species formed depending on the variousisomers of small alkanes
as input reactants. Also, theeffect of the symmetric nature of the
reactants on theresultant reaction network will be studied. The
numberof paraffin, carbonium, and carbenium ions and the
totalnumber of species including H2 in the product mixturefor
reaction mechanism S1 with different input speciesare tabulated in
Table 8.
Propane adsorbs to give a three-carbon carbonium ionwhich can
then protolyze to give methane and ethylcarbenium ion, or ethane
and methyl carbenium ion.Ethane subsequently can adsorb to form a
two-carboncarbonium ion, which can then protolyze to give amethyl
carbenium ion and methane. In the final reac-tion mixture methane,
ethane, and propane along withtheir adsorbed carbonium species are
present. The two-and three-carbon carbenium ions are mainly formed
bythe dehydrogenation of the respective carbonium ionsand a methyl
carbenium ion is formed when ethyl
carbonium ion undergoes protolysis. The total numberof
elementary reactions that result from each of the fourrules is as
shown in Table 9. Due to the restriction inthe paraffin adsorption
rule (Table 3), species with fewerthan two carbons cannot undergo
adsorption; hence,methane does not adsorb to give a single carbon
car-bonium ion. The increase in the number of isomers
withincreasing carbon numbers in the reactant molecule,and hence
the increase in the number of possible validreaction sites, is
responsible for the increase in thenumber of reactions in the
various reaction networks.
The symmetry of the molecules also affects thenumber of isomers
and hence affects the number ofreactions due to each of the
reaction rules and the totalof species that are present in the
reaction network. Forexample, 2-methylbutane, which has the most
numberof isomers as compared to that of the other input
species,gives rise to the maximum number of ions. Also thenumber of
reactions due to protolysis, which involvesbreaking of a bond
between two carbons of a carboniumion, increases for reactants that
are asymmetric sinceasymmetric species have more isomers.
Similarly, since2,2-dimethylpropane is the most symmetric
speciesamong all the five carbon reactants considered in thisstudy,
it gives rise to the lowest number of reactionsand total number of
species among all the five carbonreactants. All the above
computations took only a fewseconds on an Intel Xeon dual processor
machine with1 GHz processors, 512K cache, and 2 GB RAM runningunder
the RedHat Linux 7.3 operating system. Thecomputer program
approximately consists of 14K linesof C++ code.
Reaction set S1 primarily consisted of paraffin activa-tion
reactions in which the paraffin was adsorbed andthe resulting
carbonium ion was transformed to paraffinand carbenium ions.
Besides the chemistry of carboniumions, the process of paraffin
aromatization also involvesthe transformation of carbenium ions.
The carbeniumions formed due to the dehydrogenation and
protolysisof carbonium ions or by the adsorption of an olefin
(R.5)can desorb to give olefins (R.6), break into smallercarbenium
ions (R.7), combine with an olefin to form alarger carbenium ion
(R.8), swap its positive charge with
Table 7. Protolysis of a Carbonium Ion To Form aCarbenium Ion
and a Paraffin
{(reaction-name “protolysis of carbonium ions”)(rate-constant
kp)(label-site c1+ (find positive-carbonium))(label-site c2 (find
neutral-carbon attached-to c1+))(label-site h1 (find
neutral-hydrogen attached-to c1+))(disconnect c1+ c2)(disconnect
c1+ h1)(connect c2 h1)}
Table 8. Product Distribution as a Result of ReactionSet S1 with
Different Paraffins as Inputa
input paraffincarbonium
ionscarbenium
ions
total no. ofspecies
including H2
C3 3 3 4 11C4 4 5 6 162m-C3 4 5 6 16C5 5 8 9 232m-C4 6 11 12
302,2m-C3 5 7 7 20
a All isomers of all the species are generated.
Table 9. Number of Reactions as a Result of ReactionSet S1a
reaction typeinput I II III IV
total no. ofreactions
C3 3 3 3 3 12C4 5 5 5 6 212m-C3 5 5 5 5 20C5 8 8 8 10 342m-C4 11
11 11 14 472,2m-C3 7 7 6 7 27a All isomers of all the species are
generated.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3489
-
a paraffin/monoene/diene (R.9), or combine with aparaffin to
give a larger paraffin (R.10).
The above reactions are only representative of thevarious
reaction rules. These reactions can be easilymanipulated by
changing only a few words in therules.12 In summary, reaction set
S2 consists of thefollowing paraffin and olefin reactions: (1)
adsorptionof paraffin to form a carbonium ion (Table 3);
(2)desorption of the carbonium ion to give back the paraffin(Table
5); (3) carbonium ion dehydrogenation thatresults in a carbenium
ion and H2 (Table 6); (4)protolysis of carbonium ion to give a
paraffin and acarbenium ion (Table 7); (5) adsorption of olefin to
forma carbenium ion (Table 10); (6) Desorption of thecarbenium ion
to give back the olefin (Table 11); (7)â-scission of a carbenium
ion to a smaller carbeniumion and an olefin (Table 12); (8)
oligomerization of acarbenium ion and an olefin to form a larger
carbeniumion (Table 13); (9) hydride transfer between a
carbeniumion and a paraffin/monoene/diene (Table 14); (10)
alky-lation of a carbenium ion with a paraffin to give rise toa
larger paraffin (Table 15). It is important to recall thatthe
network generator operates recursively on all the
new species formed during the course of generation ofthe network
starting from the initial set of reactantsspecified by the user.
When the generation of isomersis forbidden by the global rule
(Table 4), all isomers areconsidered to be the same. For example,
when a five-carbon carbenium ion desorbs to give an olefin,
thedouble bond could be placed on either side of the carbonatom
that originally had the charge. Specifically
However, when the formation of isomers is forbidden,2-pentene is
considered to be the same species as1-pentene, and so the second
reaction (R.12) does notbecome part of the reaction network.
Consequently, thenumber of species and the total number of
reactions inthe reaction network reduce to a great extent when
theglobal rule that forbids the generation of isomers isused. This
could be very useful in generating compactreaction networks
especially when the kineticist doesnot have access to analytical
data that can distinguishamong the various isomers. Although the
current imple-mentation of RDL++ retains the first isomer formedand
rejects the subsequent ones, one could use thermo-kinetic
information to make this procedure more chemi-cally consistent.
Table 10. Adsorption of Olefin To Form a Carbenium Ion
{(reaction-name “adsorption of olefin”)(rate-constant
koa)(label-site b1 (find double-bond))(label-site c1 (find
neutral-carbon attached-to b1))(label-site c2 (find neutral-carbon
attached-to b1))(forbid (diene m1))(decrease-order-of
b1)(add-charge c1)(connect c2 neutral-hydrogen)}
Table 11. Desorption of the Carbenium Ion To Give Backthe
Olefin
{(reaction-name “desorption of adsorbed olefins”)(rate-constant
kod)(label-site c1+ (find positive-carbon))(label-site c2 (find
neutral-carbon attached-to c1+))(label-site b1 (find single-bond
connecting c1+ c2))(label-site h1 (find neutral-hydrogen
attached-to c2))(disconnect c2 h1)(increase-order-of
b1)(subtract-charge c1+)}
Table 12. â-Scission of a Carbenium Ion in to a SmallerCarbenium
Ion and an Olefin
{(reaction-name “Beta-scission”)(rate-constant kb)(label-site m1
reactant)(label-site c1+ (find positive-carbon))(label-site c2
(find neutral-carbon attached-to c1+))(label-site c3 (find
neutral-carbon attached-to c2))(label-site c4 (find neutral-carbon
attached-to c3))(label-site b1 (find single-bond connecting c1+
c2))(label-site b2 (find single-bond connecting c3 c2))(label-site
b3 (find single-bond connecting c3 c4))(forbid (less-than (size-of
m1) 4))(disconnect c2 c3)(add-charge c3)(subtract-charge
c1+)(increase-order-of b1)}
Table 13. Oligomerization of a Carbenium Ion and anOlefin To
Form a Larger Carbenium Ion
{(reaction-name “Oligomerization”)(rate-constant
kolig)(label-site m1 reactant)(label-site b1 (find
double-bond))(label-site c1 (find neutral-carbon attached-to
b1))(label-site c2 (find neutral-carbon attached-to b1))(forbid
(diene m1))(search-network-for
(label-site m2 reactant)(label-site c3+ (find
positive-carbon))
)(require (less-than (plus (size-of m1) (size-of m2))
8))(decrease-order-of b1)(connect c1 c3+)(add-charge
c2)(subtract-charge c3+)}
3490 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
-
To demonstrate the effect of forbidding the generationof isomers
on the size of the reaction network, we willanalyze the reaction
networks that result from reactionset S2 with and without the
global rule that restrictsisomer generation. The product
distribution as a resultof the reaction network from reaction set
S2 whenisomers are not generated is shown in Tables 16-19.The
number of reactions as a result of the variousreaction rules is
shown in Table 20. The notations inthe tables follow the IUPAC
convention. The abbrevia-tions m and e stand for methyl and ethyl,
respectively.Also the superscript ) denotes the double bond and
theprefix to the carbon (C) denotes the location of thedouble bond
or that of the positive charge as the casemay be. The total number
of paraffin, olefin, carbonium,and carbenium ions is shown in Table
16. It is interest-ing to note that the number of all the species
generatedremains the same irrespective of the input reactantsexcept
for the number of olefins when 2,2-dimethylpro-pane (2,2m-C3) is
the input. However, as shown in
Tables 17-19, the paraffins, olefins, and carbonium ionsformed
from each of these species as inputs are quitedifferent from each
other. For example, the five-, six-,and seven-carbon paraffins
generated when pentane isthe input reactant are linear as against
the branchedproducts obtained when 2-methylbutane is the input.All
the paraffins except methane, as shown in Table 17,adsorb to give
the corresponding carbonium ions shownin Table 19. This is because
the paraffin adsorption rule(Table 3) forbids adsorption of methane
that leads tothe formation of the highly unstable
single-carboncarbonium ion. Similar to the paraffins and the
car-bonium ions, the distribution of carbenium ions (notshown as
they are identical to that of carbonium ionsshown in Table 19) is
similar to the distribution ofolefins (Table 18). This is intuitive
because olefinsadsorb to give the corresponding carbenium ions.
Forexample, for the case when propane is the input
species,2-ethyl-1-butene (2e-1C4)) adsorbs to give rise to 3m-3C5+
as follows:
Table 20 shows the number of reactions as a resultof each of the
reaction rules and the total number ofreactions in the reaction
network. The alkylation reac-tion (Table 15) that involves the
fusion of a carbeniumion with an alkane to give rise to a larger
alkane is theleast restricted rule and so accounts for the
mostnumber of reactions. This is because any of the paraffinand
carebenium ion species in the reaction network canundergo this
reaction provided that the resultant paraf-fin species has fewer
than eight carbon atoms. Theâ-scission reaction involves the
fragmentation of acarbenium ion to a smaller carbenium ion and an
olefin.Also this rule requires the presence of a three-carbonlinear
chain attached to positive carbon. When propane(C3),
2-methylpropane (2m-C3), or 2,2-dimethylpropane(2,2m-C3) are the
input reactants, carbenium ions thatsatisfy this constraint are not
created, and henceâ-scission reactions do not occur.
The statistics of the reaction network that resultsfrom the
seaction set S2 when the generation of allisomers of all species is
allowed is shown in Table 21and Table 22. From Table 23, it is
evident that thenumber of species and the number of reactions in
thereaction network increase when isomers are generated.The
reaction network generated is the smallest whenthe most symmetric
molecules2,2-dimethyl propane (2,-2m-C3)sis used as the input
reactant. This is becausethis molecule results in products that are
highly sym-metric and hence have fewer isomers. The vast changein
the size and type of the reaction networks that resultbecause of
just one change in the reaction rules (forbid-
Table 14. Hydride Transfer That Transfers a Chargefrom a
Carbenium Ion to a Paraffin
{(reaction-name “Hydride transfer”)(rate-constant kh)(label-site
m1 reactant)(label-site c1+ (find positive-carbon))(forbid (allylic
m1))(search-network-for
(label-site m2 reactant)(label-site c2 (find
neutral-carbon))(label-site h1 (find neutral-hydrogen attached-to
c2))(require (less-than (plus (size-of m1) (size-of m2))
8))(require (or (and (paraffin m2) (at-least (size-of m2)
2))))(forbid (allylic m2))
)(disconnect c2 h1)(add-charge c2)(connect c1+
h1)(subtract-charge c1+)}
Table 15. Alkylation of a Carbenium Ion with a ParaffinTo Give
Rise to a Larger Paraffin
{(reaction-name “Alkylation”)(rate-constant kalk)(label-site m1
reactant)(label-site c1+ (find
positive-carbon))(search-network-for
(label-site m2 reactant)(label-site c2 (find
neutral-carbon))(label-site h1 (find neutral-hydrogen attached-to
c2))(require (paraffin m2))
)(require (less-than (plus (size-of m1) (size-of m2))
8))(connect c1+ c2)(disconnect h1 c2)(subtract-charge c1+)}
Table 16. Product Distribution as a Result of Reaction Set S2
with Different Paraffins as Inputa
inputno. of
paraffinno. ofolefin
total no. ofgas-phase
species
no. ofcarbonium
ions
no. ofcarbenium
ionstotal no.of ions
C3 7 6 14 6 7 13C4 7 6 14 6 7 132m-C3 7 6 14 6 7 13C5 7 6 14 6 7
132m-C4 7 6 14 6 7 132,2m-C3 7 5 13 6 7 13
a Isomers of different species are ignored. Total number of gas
phase species includes the H2 molecule.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3491
-
ding isomers) demonstrates the power and utility of theRDL++
compiler for translating English language rulesto elementary
reactions.
The computation time for the various runs of reactionset S2 both
with and without taking into considerationthe various isomers is
given in Table 23. This table alsoshows the number of gas phase
(paraffin + olefin + H2)and surface species (carbonium and
carbenium ions)reactions and the total number of reactions for
thevarious input reactants. Clearly, the time taken togenerate a
reaction network increases with its size.Accounting for isomers
takes longer, 38-42 s, as
compared to under 1 s when the isomer information isexcluded.
All the computations were performed on theIntel Xeon dual processor
machine with 1 GHz proces-sors, 512K cache, and 2 GB RAM. The
effectiveness ofthe compiler is evident as it can be used to
generatemultiple reaction networks by minimal change in
highlyintuitive rules rapidly.
2.1.6. Automatic Reaction-Network-to-Model Gen-erator. The first
phase of the RDL++ compiler gener-ates the reaction network that is
consistent with thechemistry rules as specified by the user.
However, totest the validity of these reaction networks
againstexperimental data, a mathematical model has to begenerated.
Typically in reaction engineering examples,this corresponds to
formulating an ordinary differentialequation model to explain
transient data or an algebraicequation model to fit steady-state
data. Law of massaction kinetics is used to translate a reaction
networkto such a mathematical model. This process could bevery
tedious and error prone when dealing with reactionnetworks with
more than 10-20 elementary steps. Theautomatic
reaction-network-to-model generator (Figure4) automates this step
by transforming the reactionnetwork into a set of differential or
algebraic equationsdepending on the data available. Each of the
elementarysteps generated by RDL++ is scanned for every speciesand
the law-of-mass-action terms contributing to theconsumption, and/or
production of these species isconstructed. As an example, consider
the followingreactions:
Reaction terms that affect the concentration of speciesB (CB)
are -k1CACB and k2CCCB2. Corresponding termsfor the species A, B,
and C are then used to constructthe mathematical model:
The reactions in the network have parameters thatcan be grouped
according to the concept of “similarspecies undergo similar
reactions under similar rates”.Empirical grouping rules such as
Polanyi relations tocorrelate activation energies to adsorption
energies,reactivity relationships, variation of activation
energieswith carbon numbers etc. based on experiments,
com-putations, and theory can be used to group the various
Table 17. Various Paraffin Species Formed as a Resultof Reaction
Set S2a
C3 C4 2m-C3 C5 2m-C4 2,2m-C3
C1 C1 C1 C1 C1 C1C2 C2 C2 C2 C2 C2C3 C3 C3 C3 C3 C3C4 C4 2m-C3
C4 C4 2m-C3C5 C5 2m-C4 C5 2m-C4 2,2m-C33m-C5 C6 2m-C5 C6 3m-C5
2,3m-C43e-C5 3m-C6 2,3m-C5 C7 3m-C6 2,2,3m-C4
a Isomer information is ignored.
Table 18. Olefins Formed as a Result of Reaction Set S2a
C3 C4 2m-C3 C5 2m-C4 2,2m-C3
C2) C2) C2) C2) C2) C2)1C3) 1C3) 1C3) 1C3) 1C3) 1C3)1C4) 1C4)
2m-1C3) 1C4) 2C4) 2m-1C3)2C5) 2m-1C4) 3m-1C4) 1C5) 2m-1C4)
3,3m-1C4)2e-1C4) 3C6) 1m-2C5) 1C6) 3m-1C5) 2,3,3m-1C4)3e-2C5)
2e-1C5) 2(1m-C2)1C4) 1C7) 4m-1C6) -
a Isomer information is ignored.
Table 19. Carbonium Ions Formed as a Result ofReaction Set
S2a
C3 C4 2m-C3 C5 2m-C4 2,2m-C3
C2+ C2+ C2+ C2+ C2+ C2+1C3+ 1C3+ 2C3+ 1C3+ 1C3+ 2C3+2C4+ 1C4+
2m-1C3+ 1C4+ 2C4+ 2m-2C3+2C5+ 2C5+ 3m-2C4+ 1C5+ 2m-1C4+
2,2m-1C3+3m-3C5+ 2C6+ 4m-2C5+ 2C6+ 3m-2C5+ 2,3m-2C4+3e-3C5+ 3m-3C6+
2,3m-3C5+ 2C7+ 4m-2C6) 2,3,3m-2C4+
a Isomer information is ignored.
Table 20. Number of Reactions as a Result of ReactionSet S2.
Isomers of Different Species Are Ignored
reaction typeinput I II III IV V VI VII VIII IX X
total no. ofreactions
C3 6 6 6 9 6 6 0 15 15 21 90C4 6 6 6 10 6 6 4 15 15 21 952m-C3 6
6 6 10 6 6 0 15 15 21 91C5 6 6 6 8 6 6 4 15 15 21 932m-C4 6 6 6 9 6
6 3 15 15 21 932,2m-C3 6 6 6 8 5 5 0 13 15 21 85
Table 21. Product Distribution as a Result of Reaction Set S2
with Different Paraffins as Input Taking into Account theVarious
Isomers of Different Speciesa
inputno. of
paraffinno. ofolefin
total no.of gas-phase
species
no. ofcarbonium
ions
no. ofcarbenium
ionstotal no.of ions
C3 24 55 80 76 79 155C4 23 56 80 82 81 1632m-C3 23 53 77 83 78
161C5 22 51 74 77 76 1532m-C4 22 56 79 78 79 1572,2m-C3 22 51 74 76
73 149
a Total number of gas phase species includes the H2
molecule.
A + B 98k1
C
C + 2B 98k2
D (R.14)
dCA/dt ) -k1CACB
dCB/dt ) -k1CACB - k2CCCB2
dCC/dt ) -k1CACB - k2CCCB2
3492 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
-
reactions so that the number of model parameters canbe
reduced.45 These grouping rules also impose con-straints to make
sure that the model parameters arecorrelated such that they do not
vary independentlyleading to nonphysical values. For example, the
rateparameters of a reversible reaction cannot vary inde-pendent of
the equilibrium constant of that reaction.
2.2. Genetic Algorithm Based Hybrid Pseudoglo-bal Parameter
Estimator. The development of pre-dictive models is a
time-consuming, knowledge inten-sive, iterative process where an
approximate model isproposed to explain experimental data, the
modelparameters that best fit the data are determined, andthe model
is subsequently refined to improve its predic-tive capabilities.
Ascertaining the validity of the pro-posed model is based upon how
thoroughly the param-eter search has been conducted in the
allowable range.The determination of the optimal model parameters
iscomplicated by the complexity/nonlinearity of the
model,potentially large number of equations and parameters,poor
quality of the data, and lack of tight bounds forthe parameter
ranges. Thorough search of the param-eters is necessary to obviate
the wrong conclusionsabout the effectiveness of a proposed
mechanism.
Recently, we critically evaluated a hybrid searchprocedure12,46
that employs a genetic algorithm foridentifying promising regions
of the solution spacefollowed by the use of an optimizer to search
locally inthe identified regions. We also reported that
thisalgorithm is capable of finding global minima for testcase
problems47 as determined by a deterministic globaloptimizer,48 but
with significant savings in time. Theperformance of this hybrid
method in the presence ofnoise was found to be satisfactory. Also,
this hybridtechnique has been able to locate multiple solutions
thatare nearly as good with respect to the “sum of squares”error
criterion but imply significantly different physicalsituations. In
this section, we will compare this meth-odology with another
stochastic techniquesadaptiverandom search.49 In section 3, we will
propose a 13-parameter model that results in 60 differential
algebraicequations for propane aromatization on a zeolite cata-lyst
as a more challenging test case to validate thisalgorithm.
We will now compare the hybrid procedure with apopular parameter
estimation method available in theliteraturesthe direct search
optimization techniquebased on use of randomly chosen sample points
andadaptive reduction of the search space.50 Belohlav
andco-workers51 have used this method for estimating theparameters
of a model for toluene dehydrogenationbased on the following
reaction scheme
where A, B, and C represent toluene, methylcyclohex-ene, and
methylcyclohexane, respectively. The model51consists of a set of
three ordinary differential equationsto describe the time evolution
of the concentration ofspecies A, B, and C and 14 data points for
each of thespecies is used for estimating the five parameters
(k1-k5) in the model. To minimize the correlation among
theestimated parameters, the authors have used the de-terminant of
the multiresponse data as the criterion forestimating the
parameters as suggested by Box andDraper.52 The first two rows of
Table 24 show the bestset of parameters as reported by Belohlav and
co-workers51 and those obtained by our hybrid searchprocedure,
respectively. The corresponding predictionsare shown by the solid
and dashed curves, respectively,in Figure 5. It is interesting to
note that although thepredictions are nearly indistinguishable, the
parametersare slightly different and the objective function
valueobtained by the hybrid procedure is marginally lower
Table 22. Number of Reactions as a Result of ReactionSet S2
Taking into Account the Various Isomers ofDifferent Species
reaction typeinput I II III IV V VI VII VIII IX X
total no. ofreactions
C3 89 76 71 114 96 96 54 96 81 114 887C4 80 80 75 119 98 98 56
96 82 116 9002m-C3 88 83 78 123 98 98 57 98 84 118 925C5 77 77 72
115 96 97 55 96 81 115 8812m-C4 78 78 73 117 96 96 54 96 81 114
8832,2m-C3 76 76 71 114 98 96 54 96 81 114 876
Table 23. Comparison of Reaction Networks Generated by Reaction
Set S2 with and without Isomer Generation
S2 without isomers S2 with isomers
input
no. ofsurfacespecies
no. ofgas-phase
speciestotal no. ofreactions
time(s)
no. ofsurfacespecies
no. ofgas-phase
speciestotal no. ofreactions
time(s)
C3 13 14 90 0.86 155 80 887 39C4 13 14 95 0.91 163 80 900
402m-C3 13 14 91 0.86 161 77 925 42C5 13 14 93 0.86 153 74 881
382m-C4 13 14 93 0.92 157 79 883 382,2m-C3 13 14 85 0.83 149 74 876
37
Figure 5. Concentration-time curves for species A, B, and C
inthe toluene hydrogenation model.51 Solid lines correspond to
theparameters reported by Belohlav and co-workers,51 dashed
linesrepresent the best solution obtained by the hybrid procedure,
andthe dotted lines show the predictions for the parameter set
whoseobjective function value is at most five times that of the
bestsolution of the hybrid procedure.
A a B f C
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3493
-
than that reported by Belohlav et al.51 This may bebecause of
the errors in the integration routines used.
A crucial point commonly associated with most non-linear
parameter estimation problems is the multiplicityof the solutions.
This becomes especially importantwhen there is error associated
with the data availablefor estimating the parameters. To further
investigatethis issue, we examined all the solutions of the
hybridprocedure with at most five times the objective functionvalue
(SSE) corresponding to that of the best solution.The dotted lines
in Figure 5 show the predictions of theworst solution of this set.
The parameter values corre-sponding to this worst solution are
shown in Table 24.With the assumption that a typical kinetic
experimenthas 15-20% error in data, the predictions from thisworst
solution cannot be distinguished from the predic-tions
corresponding to the best solution as shown in thedashed lines in
Figure 5. Figure 6 shows the relativevariation of the 41 solutions
whose objective functionvalue is at most five times that of the
best solution. Therelative variation in a parameter is determined
as theabsolute value of the difference between the parameterand the
average value of the parameter scaled by theaverage value;
specifically, kiscale ) |ki - kiavg|/kiavg. It isinteresting to
note that parameters k2 and k5 can varyup to almost 100% of their
average value. This meansthat these parameters could be twice as
much as theiraverage values among all the solutions. Multiple
solu-tions and large parameter variation among them couldbe an
indication that we have insufficient data toeffectively estimate
the parameters or that the proposedmechanism does not explain the
data completely. Thesesolutions could be of potential interest in
planningfurther experiments for discriminating among competi-tive
models for this problem.
As discussed in the previous paragraph, the hybridsearch
procedure, using GA for identifying promisinginitial guess values
followed by the application of atraditional local optimizer, is
able to find the globaloptimum. It is important to note that local
optimizersalone can be very successful for relatively small
refinedproblems with well-defined and small parameter
bounds;however, for initial screening of large amounts of data,the
above procedure would be natural choice. Also, thismethod is useful
from a design perspective when theexpert is interested in multiple
solutions. In section 3,we will examine how this hybrid method
works for muchlarger problems that are of particular interest
indetermining optimal reaction networks for real systems.
2.3. Feature Extractor. The main aim of developingan automated,
user-driven tool kit such as ReactionModeling Suite is to aid an
expert in building large-scale kinetic models. One of the key
postulates in thiseffort is that any system to aid an expert should
followthe thought process of the expert. In our opinion, thehuman
expert does not primarily think in terms of thedetailed
mathematical formulation of a model; ratherhe or she thinks in
terms of the “rules” that lead to thatmodel and the features that
result from the model.Accordingly the RDL++ compiler acts as an
informationgathering tool from the user through which the expertcan
key in the rules and it also translates the input rulesinto a
mathematical model automatically. The modelparameters are then
robustly estimated using the GA-based hybrid parameter estimation
technique as ex-plained in section 2.2. The expert is now
interested inanalyzing the predictions that resulted from the
modelthat was based on the rules.
The analysis of predictions vs data, at least duringthe early
stages of model development, is not primarilyvia a least-squares
fit but rather through a comparisonof the “features” of the data vs
those of the model. Asshown in Figure 7a, the expert would vote for
model 2that captures the features of the true performance(dotted
line) even though model 1 has better quantita-tive fit to the data.
Clearly, the expert does not thinkin terms of the squared errors at
individual data pointsor in terms of the sum of the squared errors
or in otherstatistical lack-of-fit measures that quantitatively
ad-dress the difference between the model predictions andthe data.
For example, in the simple catalytic reactionA goes to B, the
“rule” that A must be adsorbedreversibly leads to the “feature”
that the rate of produc-tion of B will show a maximum with
increasing tem-perature. Similarly, the features could be the
initialslope of the rate curve, the kink at the top of the
curve(Figure 7b), or the time at which the selectivity
curvesaturatessessentially the key landmarks that the ex-pert is
interested in explaining through the model.
Information about the mismatch between the featuresof the data
and the model predictions is used to
Table 24. The Performance of the Hybrid Procedure onthe Toluene
Hydrogenation Model 51 Where k1 to k5 Arethe Parametersa
model k1 k2 k3 k4 k5 objective
Belohlav et al. 0.023 0.005 0.011 1.9 1.8 3.088 × 10-8hybrid,
best 0.0234 0.0039 0.0106 1.6958 1.6953 2.883 × 10-8hybrid, worst
0.0226 0.0034 0.0071 1.2139 0.8438 1.424 × 10-7
a The best solution from the hybrid method and the solutionwith
at most five times the objective function value of this
bestsolution are reported. The corresponding
concentration-timecurves are as in Figure 5.
Figure 6. Multiple solutions for the toluene
hydrogenationproblem51 whose objective function value is up to five
times thatof the best solution but whose parameter values are
widelydifferent. The scaled parameter values have been calculated
asthe absolute difference between the actual value and the
averagevalue and then scaled by the average value.
Figure 7. Need for feature extraction: (a) rate vs time;
(b)concentration vs time.
3494 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
-
postulate new rules or to modify the existing rules toimprove
the modelsan iterative process that we callmodel refinement.
Evidently, the ability to automati-cally extract the features
facilitates model refinement,and in this section, we explain the
process of automati-cally and robustly extracting the features from
a curvewhich could be either experimental or based on themodel
predictions.
To avoid the scenarios shown in Figure 7 and to aidthe process
of model refinement, we propose a feature-based model evaluation to
screen and compare models.These models could be different
mathematical realiza-tions of a process that are derived based on
differentassumptions or could be because of the multiple param-eter
values for the same underlying model.
The objective is to come up with a criterion forestimating the
goodness-of-fit of multiple models in theirability to explain data
using an objective function basedupon the critical features of the
model predictions andthe data. For example, consider a model with
twoparameters. If nf is the number of features identified tobe
critical in the data, Mfi(x1,x2) corresponds to the ithfeature as
predicted by the model, and Dfi(x1,x2) corre-sponds to the
corresponding feature in the data, thenthe following objective
function would suit our purposes
where wi is the weighing factor for the feature i thatdepends on
the importance of the feature and ∑i)1
nf wi )1. For the purpose of illustration, let us choose
twoslopes s1 and s2 as the critical features and let m1 andm2 be
the corresponding model predicted slopes. Thenthe above objective
function would reduce to
An automatic feature extraction procedure would fa-cilitate the
evaluation of the above criterion simple. Thisbecomes more
important especially when the data areavailable at a higher rate
and accuracy and automatedmodel discriminations strategies are
required to buildrobust kinetic models.
In the chemical engineering literature, feature extrac-tion
techniques have been used for the purposes of trendanalysis of
process data in order to exploit the temporalinformation and to
reason about the process state. Themain activities involved are (i)
identification of qualita-tive trends and (ii) mapping from trends
to operationalconditions. To deal effectively with a multitude
ofprocess data and extract the underlying importanttrends and
events in the process, Janusz and Venkata-subramanian53 proposed a
framework for the automaticgeneration of such qualitative process
trend descriptionsdirectly from sensor data. A trend is represented
as asequence of seven primitives that are piecewise unimo-dal or
quadratic segments. These primitives form thealphabets of their
trend description language. Thisqualitative filter provided a
meaningful compaction oflarge amounts of numerical data without
losing theessential information about the trends.
Other examples of qualitative process trend analysisinclude use
of an expandable “composite” shape libraryto approximate a noisy
process signal by a polynomial,54
a knowledge-based interpretation of sensor patterns,55a
technique for data compression and trending calledpiecewise linear
online trending that adapts to processvariability and noisy data,56
pattern matching betweenthe observed fault trends and the ones in a
knowledgebase,57 application of trend based temporal techniquesfor
medical diagnosis,58 a B-Spline based technique fordata compression
and automatic trend extraction,59combination of the primitives
based trend descriptionlanguage53 with a fuzzy-logic-based
multivariate infer-ence framework for temporal-reasoning,60 and
auto-mated identification of process trends based on
aninterval-halving procedure.61 A more recent review ofthe
qualitative methods used in the process trendanalysis and fault
diagnosis is available elsewhere.62,63
There are significant differences between the sensornetwork
process data and kinetic data that are the focusof this paper.
Unlike process data from plants, kineticdata are typically
available in small amounts. Althoughthe volume of data from high
throughput experimentshas been increasingly available at a higher
rate andaccuracy in the recent years, this is still not a match
tothe historical process data available from chemicalplants.
Kinetic data generally do not show abnormaldeviations or
unexplainable trends. If there are ir-regularities in curves from
good experimental setups,they will mostly be systematic and
repeatable. The datafrom experiments is noisy, but the noise is far
less ascompared to that in process data.
Qualitative process trend extraction algorithms do notuse a
priori information about the processes as this istypically not
available in sensor network data; however,in the case of kinetic
data, one typically knows the curvesignatures for faulty
experiments such as malfunctionin reactor setup, catalyst
deactivation, temperaturespike, etc., and this information can be
used to rejectcertain features in the curves that are not
interesting.Also, the expert typically has some information
aboutwhich features are important and which are not.
Thisinformation can be used so that the feature extractionalgorithm
does not have to look for the unimportantfeatures. Considering the
above differences between thekinetic data and the data from process
plants, we cannotuse the automatic trend extraction algorithms such
asthe interval-halving procedures61 that have been devel-oped
mainly to deal with large volumes of noisy datawithout user
intervention and with minimal a prioriinformation.
Any feature extraction algorithm devised for charac-terizing
kinetic data should be able to overcome adifferent set of
challenges. First, it is highly likely thatin the context of
kinetic data, some features that occuronly for very short periods
of time, and hence treatedas noise by automatic noise rejection
algorithms, couldactually be the most important features. So, a
progres-sive, detailed-to-coarse feature identification with
up-dates from the user is required. Second, not all thefeatures are
equally important for a kineticist trying tomodel data. For
example, the initial lag in a rate curvemay be more important than
the relatively large devia-tions at saturation at later times.
Also, features suchas the slope of the curve, offset, etc., at
lower space timesin a plug flow reactor may be more important as
theynonlinearly affect the features at the end of the
reactor.Similarly the concentrations of species with fewercarbon
numbers may be more important than that ofthe long-chain
hydrocarbons in a polymerization reactor,
minx1x2
∑i)1
nf
wi||Mfi(x1,x2) - Dfi(x1,x2)|| (1)
minx1,x2
∑i)1
2
wi||(mi - si) (2)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3495
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as the smaller species act as the seeds for the longerones. So,
a mechanistic rank ordering of features thatare important for the
problem at hand is required so asto facilitate the understanding of
which features wouldneed to be fixed first and which are relatively
unimpor-tant in the process of MR. Third, in the case of
kineticdata we are interested in not only the qualitative trendsin
terms of the slopes of the curves but also the absolutevalues of at
least some of the features. Finally, data maybe sparse in certain
time ranges. Unlike process datathat has almost equal density over
large periods of time,kinetic data may not be available over the
entire designparameter space. So any feature extraction
algorithmfor kinetic data should not rely on the density of datato
extract meaningful features.
We propose the following feature extraction algorithmthat
systematically interprets the kinetic data curvesby identifying key
features of the curve, e.g., increasingrate, decreasing rate,
change of slope, inflection point,etc. realized through the
generation of a “feature vector”.
(1) With the help of a human expert, draw a smoothcurve passing
through the experimental data. Thisensures that the features
generated are not affected bythe noise and irregularities in the
data.
(2) As shown in Figure 8, identify the critical pointswhere
there are abrupt changes in the value, first orsecond derivatives
of the curves. Report any unimodalor quadratic primitives that
match this section of thecurve. Populate the feature vector with
the primitive,slope, curvature, intercept at end points, and the
rangeof the independent variable (typically time) that corre-sponds
to this section of the curve.
(3) Interacting with the user, rank order features bothin terms
of the important species and time regimesaccording to their
importance. If the user flags certainfeatures to be unimportant
according to the user, redostep 2 by merging time ranges of the
identified featureswith the adjacent ones. If the user specifies a
particulartime range to be of greater importance, calculate
thefeature vector in that range.
(4) Repeat step 2 for the curves generated by themodel
simulations and calculate the sum of the squareddeviations between
the model and the data features.Populate the feature vector of the
model curve with thismetric. Optimize on the model parameters using
thisobjective function criterion with any parameter estima-tion
routine such as the GA-based hybrid algorithmexplained in section
2.2.
The example shown in Figure 9 demonstrates thefeature extraction
technique to compute the similaritybetween two curves. The expert
identifies the significantfeatures in the data (Figure 9a) by
analyzing the abruptchanges in the value, slope, and the curvature.
The datacurve is partitioned into five intervals within which
thecurve is characterized by one feature. For example, inthe first
time interval, the expert is interested in thelag period and in the
second interval between t1 and t2,the slope of the curve (at t2) is
considered to beimportant. The points at which the curve saturates
att4 and at t5 are considered to be the critical features inthe
last two intervals. The primitives that closely matchthe curves in
each of the time intervals is extracted bycomputing the slope and
curvature and is as shown inTable 25.
Now consider a model that results in the curve asshown in Figure
9b. This curve is analyzed for thecritical points, and the various
time intervals arecalculated. The primitives and the important
featuresof this curve are also shown in Table 25. It is
interestingto note that the first time point at which
significant
Figure 8. Methodology for automated feature extraction.
Figure 9. Example to illustrate the feature extraction
algorithmto compute the similarity between two curves. Data and the
expertpostulated curve through data are shown in the top figure (a)
andthe hypothetical model curve is in the bottom figure (b).
Table 25. Critical Features of the Data and the ModelCurves
Shown in Figure 9
data model
time interval primitive feature primitive feature
t1-0 E Erange t1 t1′t2-t1 B Bslope at t2 s st3-t2 D D- - -t4-t3
A Aordinate at t4 b b′t5-t4 E Fordinate at t5 a a
3496 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
-
change in slope occurs is different in the data (t1) andthe
model (t1′) curves. Similarly, the saturation pointat time t4 is
different between the two curves, b and b′.Also, the basic shape as
characterized by the primitivebetween t4 and t5 in the data curve
is E whereas thatin the model curve is F. The objective function in
eq 1is used to account for the differences in the two curvesas
and w1 + w2 + w3 ) 1. To quantify the differences inthe
qualitative aspects of the primitives E and F, thefuzzy similarity
matching indices60 to quantify thedifferences between the various
primitives. For example,primitives A and C are not completely
different and sothey are assigned a similarity index of 0.25. This
meansthat primitive A is 25% similar to primitive C.
Similarly,primitives E and F are closer to each other by 75%.
The above algorithm for feature extraction is simpleand
extensively uses domain knowledge about thesystem available from
the user regarding the featuresand their relative importance.
Unlike interval-halving-based fully automated algorithms,60 it does
not rely onthe density of the data. This algorithm also allows
foriterative correction of features and so if it misses animportant
feature, it can go back and locate it with thehelp of the user. The
user-defined features on theexperimental data are used to guide the
automaticextraction of the features from the model curves usingthe
critical points. The feature-based objective function(eq 2) can
thus be computed for estimating the param-eters and screening
through multiple models.
2.4. Statistical Analyzer. Statistical analysis of themodels is
necessary in order to screen, compare, andimprove them. The most
common statistical methodused for analyzing the quality of the
model is based onthe sensitivity of the model output with respect
to themodel parameters defined as a partial derivative intro-duced
via a Taylor series expansion
where the partial derivatives ∂ci/∂kj known as the first-order
local concentration sensitivity coefficients areevaluated by the
parameters kj, one at a time at time tand the effect on
concentrations measured at time t +∆t. Kinetic models generally
involve ordinary differen-tial equations such as
where c is a dimensional concentration vector. Theabove ODEs can
be differentiated with respect to themodel parameters kj to
give
where J(t) ) ∂f/∂c and the initial condition for ∂c/∂kj isa zero
vector. Equations 5 and 4.6 are coupled throughthe matrices ∂f/∂c
and ∂f/∂k; hence eq 6 can only besolved if the concentration values
calculated in eq 6 areavailable at times where these matrices are
calculated
during the numerical solution of eq 6. This is achievedby
solving the (m + 1)n equations in eqs 5 and 6simultaneously. A more
efficient algorithm for thesolution of the sensitivity differential
equations is thedecoupled direct method64 which uses the fact that
eqs5 and 6 have the same Jacobian. The result of the abovesolution
procedure is a set of sensitivity coefficients ∂ci/∂kj. Since the
parameters and the various outputquantities of the model may have
different units espe-cially when the reactions are of different
reaction orders,normalized sensitivity matrix defined as the
fractionalchange in concentration ci caused by a fractional
changeof parameter kj
is typically used for further analysis. The variance onthe
parameter estimates can be computed using
where s2 is an unbiased estimate of the model predictionerror
expressed as the difference between the modelpredictions (ci) and
the experimental data (ĉi) as
for a model with p parameters and in the case wherethere are n
experimental data points.
The local sensitivity coefficients and the other metricsdefined
above have been used for identifying redundantspecies and redundant
reactions35,42 and hence to reducea kinetic model. Other techniques
such as concentrationsensitivity analysis,32 reaction rate
analysis,36 principalcomponent analysis,65 and lumping analysis66
have beenused for the investigation and reduction of
reactionmechanisms especially for the description of
combustionreactions. For a more comprehensive review of
thestatistical methods for the analysis of reaction mecha-nisms,
refer to Tomlin et al.32 Computer softwarepackages such as SENKIN67
and KINALC38 implementone or more of these methods. An alternative
to theselocal methods would be the global sensitivity
analysisproceduresstudy of the effect of the parameters on themodel
output without the assumption of any individualsolution. For
example, methods68 such as the Fourieramplitude sensitivity test
simultaneously perturb allrate parameters by sine functions with
different fre-quencies and analyzes its effect on the
concentrations.These methods are computationally expensive
especiallyfor models with a large number of parameters.
To address the above concerns, the Statistical Ana-lyzer in RMS
uses techniques from both local and globalsensitivity analysis for
ascertaining the robustness ofa kinetic model. A kinetic model is
defined to be robustif it is accurate in explaining the data even
when themodel parameters have not been estimated with suf-ficient
accuracy. Typically kinetic model parameterssuch as rate and
equilibrium constants cannot beestimated accurately because of the
errors in the modeland the data and the errors in the estimation of
theparameters. It is useful to see how these errors arepropagated
through to the model predictions. We pos-tulate that the various
errors are localized as errors in
S )kjci
∂ci∂kj
)∂ ln ci∂ ln kj
(7)
Var(k) ) JTJ-1 s2 (8)
s2 ) ∑i)1
n
(ci - ĉi)2n - p (9)
min w1(t1 - t1′)2 + w2(b - b′)
2 + w3(E - F)2 (3)
ci(t,k + ∆k) ) ci(t,k) + ∑j)1
m ∂ci
∂kj+ ... (4)
dc/dt ) f(c,k),c(0) ) c0 (5)
dc
dt
∂c
∂kj) J(t)
∂c
∂kj+
∂f(t)
∂kj(6)
j ) 1, ..., m
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3497
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the estimated parameters, and we use these errors tocompute the
error in the model predictions.
Local sensitivity analysis deals with sensitivity coef-ficients
computed by local perturbations around the bestminimum and is
expressed as the standard deviations,confidence bounds, and
correlation matrix69,70 of theparameters. To illustrate the
concepts of the localsensitivity analysis, we consider a model for
CO oxida-tion on a supported metal catalyst as described inAppendix
A. Specifically, we show as to how the errorin the model, in the
data, and in the parameterestimation procedure can be propagated to
the modelpredictions. This model consists of three steps:
molec-ular adsorption of CO in a quasi-equilibrated mannerwhere the
forward step of adsorption and the reversestep of desorption are of
almost the same rate; dissocia-tive irreversible adsorption of
oxygen; the surface reac-tion between the two adsorbed species to
give CO2. Thethree parameters that describe this process are
theequilibrium constant of the CO adsorption process (K1),the rate
constant for oxygen adsorption (k2), and the rateconstant for the
surface reaction (k3). Table 26 showsthe values of the estimated
parameters, and the corre-sponding predictions of the rate of CO2
production withrespect to the variation in the partial pressures of
COand O2 are as in Figure 10. The large standard devia-tions and
the confidence intervals in parameter k3 showthat for a reasonably
good prediction of the data, thevalue of k3 has not been estimated
accurately.
Figure 11 pictorially represents the inference re-gions70 of the
parameters for 90 and 95% confidencelimits. The large variation in
the ordinate of the firstplot again shows that the parameter k3 has
not beenestimated accurately. Also, the plots show how
theparameters are correlated to each other. Parameters k2and k3 are
correlated negativelyswhen k2 increases k3decreasessand parameters
K1 and k2 are positively
correlated. This information is also available from
thecorrelation matrix shown in Table 26. We claim thaterrors in the
model, in the data, and in the estimationprocedure have been cast
as the errors in the modelparameters, and we now propagate this
error to themodel predictions in the following manner. Assumingthat
the parameters follow a Gaussian distribution withmean as the best
estimates and standard deviationgiven by the errors in the
estimates (Figure 12), werandomly sample 1000 parameter sets from
this distri-bution and simulate the model with these parametersets.
The resulting predictions are shown in the bottomtwo plots in
Figure 12. The thick lines show the 3σdeviation of the model
predictions and the dots showthe experimental data.71 The error
bars on the modelpredictions show clearly that even though the
parameterk3 has not been accurately estimated, the model
isrobuststhe predictions are accurate. Also, we can seethat the
accuracy of parameter k3 affects the predictionsof the variation of
CO2 rate with the partial pressure ofCO more than that of the
predictions of the variationof CO2 rate with the partial pressure
of O2. In the caseof a large reaction network, inferences of this
kind canbe used to ascertain as to which part of the
reactionnetwork is sensitive to which parameters.
Information available by locally perturbing the pa-rameters
around the best solution may not be sufficientto analyze the model
especially when a large numberof parameter sets explain the data
equally well. Typi-cally the ranges in which parameters of large
reactionnetworks lie are not known accurately. Also the models
Figure 10. Predictions of CO oxidation model where
carbon-monoxide adsorption is quasi-equilibrated and adsorption
ofoxygen is irreversible. The parameters are as in Table 26.
Table 26. Values of the Rate Constants, StandardDeviations, 80%
Confidence Interval, and the CorrelationMatrix for the CO Oxidation
Model Where CarbonMonoxide Adsorption Is Quasi-Equilibrated
andAdsorption of Oxygen Is Irreversible
80% confidenceinterval correlation matrixparameter mean std
dev
log K1 -3.75 0.24 -4.09 -3.42 1.000 0.967 -0.753log k2 -3.87
0.11 -4.03 -3.72 1.000 -0.849log k3 3.57 168.53 -222.18 229.30
1.000
Figure 11. Inference regions based on the confidence intervalsof
the parameters for the CO oxidation model where carbonmonoxide
adsorption is quasi-equilibrated and adsorption ofoxygen is
irreversible.
Figure 12. Parameter correlations assuming a Gaussian
distri-bution and the µ ( 3σ error bars on the model predictions
for theCO oxidation model where carbon monoxide adsorption is
quasi-equilibrated and adsorption of oxygen is irreversible. The
dots inthe bottom plots show the data from Cant, Hicks, and
Lennon.71
3498 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
-
that describe these networks may not have all therequired
components either stoichiometrically or interms of constraints
among the model parameters atleast in the beginning of the
model-building procedure.In such situations, the data can be
predicted by a largenumber of parameter sets equally well. It is
importantto estimate these multiple minima as they may cor-respond
to physically different situations of the underly-ing system. This
information is especially useful whendesigning a catalyst. For
example a set of parameterscould correspond to the high coverage of
CO on thecatalyst surface and yet another set of parameters
couldcorrespond to the high coverage of O2. It is likely thatfor
the range of partial pressures of CO (higher thanthat of O2) used
to collect the data, high coverage of O2is physically infeasible.
Hence, it is important to identifyand analyze the multiple
minima.
To address this concern, the Statistical Analyzer inReaction
Modeling Suite uses techniques to identifymultiple minima and
analyzes them. The GA-basedhybrid parameter estimation technique
discussed earlierin this section was one such technique. The
effectivenessof this method in identifying multiple minima wasshown
in section 2.2 and Figure 6. Another methodknown as the
trough-walking algorithm that tracks theminima in the local
neighborhood of any given minimais described in Appendix B. This
method starts withmultiple initial guess values, and once a local
optimumis found, the local neighborhood is analyzed to find
anyother close minima. This ensures that we find most ofthe local
minima around any given minima. The numberof local minima that we
are able to find depends on thevalue of the parameter that controls
the size of the localneighborhood searched at any step (EDBOUND )
0.05)and also on the ruggedness of the fitness landscape.
The vertical bars in Figure 13 show the parametervalue ranges of
all the minima found at the varioustemperatures for the CO
oxidation model described inAppendix A. Each of these minima
predicts the datawithin a sum of squared errors between the
modelpredictions and the data of 0.1. It is interesting to notethat
the O2 adsorption rate constant (k2) has been foundwith great
accuracy unlike the other parameters (K1 andk3) that show large
variations. This information is alsoavailable from the standard
deviations of the param-eters from the local sensitivity analysis
(Table 26). Moreimportantly, the information about the multiple
minimahelps us in validating the model predictions. For ex-ample in
Figure 13, the equilibrium constant K1 for theexothermic reaction
of CO adsorption has a positiveslope and the rate constants have
negative slopes. Thisis in accordance to the physical realities of
this system.
2.5. DiscussionsReaction Modeling Suite. In thissection, we
describe a user-driven, automated set oftoolssReaction Modeling
Suite (RMS) that aids theexpert in constructing robust kinetic
models. Specifi-cally, RMS is designed to allow the expert to
initiatethe kinetic modeling sequence in a simple reactionchemistry
language, converts the reaction network intoa mathematical model,
optimizes the model parametersusing a hybrid algorithm, extracts
the features of thedata and model prediction curves, and
statisticallyanalyzes the robustness of the model.
The RDL++ compiler that translates the English-language rules to
a reaction network is generic, extend-able, and intuitive, thereby
affording an easy-to-useinterface for a practitioner. The
English-language rulesinput is more user friendly as compared to
that of thebond order-bond electron matrices24 and the
structureoriented lumping vectors25 as the rules are in thenatural
language used by a chemist to describe thereactions. The human
expert can readily create multiplehypotheses and change the size of
the reaction networksfrom a few species and reactions to several
hundreds ofspecies and reactions by manipulating a few steps inthe
reaction rules. Any new rule can be easily added,and the existing
rules can be changed with little effort.The capability of RDL++ to
track down all the isomersand generate all reaction steps that
involve all theisomers of any species is very useful for
describingreaction networks whose characteristics change with
thethree-dimensional structure of the species involved.Also, during
the initial stages of modeling a network,the expert can simply turn
off the isomer generationglobal rule and, with the limited amount
of analyticaldata, try to explain the reaction. The expert can
alsomanipulate the size of the reaction network by changingthe
number of carbon atoms present in a reactant or aproduct in any of
the reaction rules. The use of globalrules to prevent the formation
of chemically infeasiblespeciessallylics, species with a positive
charge and adouble bond, trifins, species with triple bonds,
specieswith more than two double bonds, species with apositive
carbon attached to a double bond, etc.senablesthe expert to keep
the reaction network feasible.
RDL++ has been designed and implemented alongthe lines of
Reaction Description Language,31 but RDL++forms a part of RMS which
handles all the operationsof building a kinetic model starting from
the formulationof chemistry rules to the analysis of the
performance ofa kinetic model. With this in mind, RDL++ has
beendesigned to be more extendable, user-driven, and ef-ficient
than RDL. Specifically, new rules such as de-sorption, cyclization,
and hydride transfer have been
Figure 13. Multiple values of rate constants for the CO
oxidation model where carbon monoxide adsorption is
quasi-equilibrated andadsorption of oxygen is irreversible.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3499
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developed based on the language for solid acid chemistryand
reactions on catalytic surfaces. New keywords andsyntax for
carbonium ions, trifin (species with triplebonds), allylic (species
with a double bond and a positivecharge), and monoene, diene, and
triene (species withthree double bonds), have been included in
RDL++ soas to enrich the palette for the user. New model
pruningconcepts to reduce the size of the reaction network havebeen
introduced. The concept of global rules preventsduplication of
pruning steps in individual reaction rules.Also the user now has
the powerful ability to forbid theformation of isomers. The size of
the resulting model iscontrolled by the size of the carbon chain in
thehydrocarbon reactants or products rather than the less-intuitive
“generation count”72 that is based on the depthof the reaction
network. We have also shown as to howa tool such as RDL++ can be
integrated with other toolsfor automated hypothesis generation and
testing inorder to build robust kinetic models. Finally, RDL++has
been developed in a C++ environment which ismore structured and
user friendly compared to that ofLISP.
Possible improvements to RDL++ include an
XML(http://www.w3c.org) based interface to interactivelydefine new
keywords and to extend existing keywords.An intelligent
backtracking mechanism that enablescausal reasoning of the
individual terms in the math-ematical model to the elementary
reaction steps of thenetwork and/or directly to the section of the
rules wouldmake the whole compiling process more transparent
andcould have potential implications in model refinement.With this
added feature to perform qualitative sensitiv-ity analysis, the
user will be able to selectively modifya set of rules to manipulate
the terms in the modelwhich would result in different features in
the perfor-mance curves.
The Feature Extractor module in RMS is used to aidthe expert in
extracting the features in the data curvesand then use this
information to develop an objectivefunction to compare the features
in the model and thedata. An expert-system-like framework that can
becontinually updated with user-supplied informationabout the
different features and their relative impor-tance can make this
process more efficient. Also, theprimitives that have been
currently used in RMS arebased on the description of the first- and
second-orderderivatives of the curves. New definitions of
primitivesthat use the domain knowledge would be more attrac-tive.
For example, simple kinetic reaction mechanismsgive rise to
standard rate laws73 which in turn give riseto specific features in
the performance curves. Forexample, a second-order surface reaction
gives rise to asquare in the denominator of the
Langmuir-Hinshel-wood rate expression and a saturation curve.
Similartrends can be encoded as primitives and more complexrate
expressions c