An Intelligent System for Reaction Kinetic Modeling and...

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

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

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

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)}


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)}


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.


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+)}


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.


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) -


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+


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


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


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


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


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.


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)


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


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


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


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.


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

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