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theesignan nod labe-arte au-atedssion.ppli-ationmationdictions emis-s of
G. N. Pontikakis
G. S. Konstantas
A. M. Stamatelos1
e-mail: [email protected]
Mechanical and Industrial EngineeringDepartment,
University of Thessaly,383 34 Volos, Greece
Three-Way Catalytic ConverterModeling as a ModernEngineering Design ToolThe competition to deliver ultra low emitting vehicles at a reasonable cost is drivingautomotive industry to invest significant manpower and test lab resources in the doptimization of increasingly complex exhaust aftertreatment systems. Optimization clonger be based on traditional approaches, which are intensive in hardware use antesting. This paper discusses the extents and limitations of applicability of state-of-thmathematical models of catalytic converter performance. In-house software from ththors’ lab, already in use during the last decade in design optimization studies, updwith recent, important model improvements, is employed as a reference in this discuEmphasis is on the engineering methodology of the computational tools and their acation, which covers quality assurance of input data, advanced parameter estimprocedures, and a suggested performance measure that drives the parameter esticode to optimum results and also allows a less subjective assessment of model preaccuracy. Extensive comparisons between measured and computed instantaneousions over full cycles are presented, aiming to give a good picture of the capabilitiestate of the art engineering models of automotive catalytic converter systems.@DOI: 10.1115/1.1787506#
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IntroductionThe catalytic converter has been in use for the past 30 yea
an efficient and economic solution for the reduction of pollutaemitted by the internal combustion engine, the latter beingpowertrain for almost all vehicles in use today. The widespreuse of the catalytic converter was the response of the automoindustry to the legislation of developed countries, which polimits to the most important gaseous pollutants emitted by bgasoline and diesel engines.
Since the concern about the environmental impact of the emsions of the vehicles fleet is steadily growing—especially in urbareas, where air pollution has become a major issue—emislegislation becomes gradually stricter. Accordingly, this has ledcontinuous efforts of the automotive industry to improve the eciency of the catalytic converter@1#. Today’s emission standardhave been lowered so much that the catalytic converter techogy has been pushed to its limits and it became apparent thaorder to build vehicles that comply with the legislation, automtive engineers should tune the whole system of engine, piping,catalytic converter@2#. Thus there emerged the need to view tcatalytic converter as a component of an integratedexhaust after-treatment systemthat should be designed very accurately.
In this context, the role of modeling of the components of ehaust aftertreatment systems is becoming increasingly imporespecially as regards the catalytic converter, which is the mcrucial device of such systems. Since the introduction of catalconverters in production vehicles, catalytic converter models hbeen appearing in the literature in parallel with the developmof new catalytic converter technologies. Nevertheless, the aracy, reliability, and application range of catalytic converter moels is still questioned. The number of modeling applications inautomotive industry remains limited, especially when contrasto the plethora of models that appear in the literature@3–15#. It
1Author to whom correspondence should be addressed.Contributed by the Internal Combustion Engine Division of THE AMERICAN SO-
CIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OFENGINEERING FORGAS TURBINES AND POWER. Manuscript received by the ICEDivision, December 1, 2002; final revision received September 1, 2003. AssoEditor: D. Assanis.
906 Õ Vol. 126, OCTOBER 2004 Copyright ©
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seems that a complicated landscape of approaches and methogies has been created, causing an uncertainty as regards thelidity and applicability. Most probably, this adversely affects theapplication in everyday practice, although modern modelmethodologies have been greatly improved and, in many cathey have been successfully incorporated in the process of exhaftertreatment systems design@16–20#.
In what follows, we attempt to inverse this situation, first bsketching an overview of modeling approaches that could helpnavigation through the complicated landscape of this field ofsearch. Subsequently, we present our choice of modelingproaches along with some supporting tools, in order to compicomplete methodology that provides the required high accurlevels for the current state-of-the-art exhaust aftertreatmenttems design. This methodology combines a significantly updaversion of theCATRAN @21# modeling code with optimization toolstailored to the computer-aided estimation of the model’s chemkinetics parameters. The first steps of the developed optimizamethodology have already been presented elsewhere@22#. Herewe update and enhance it by incorporating recent improvemof the catalytic converter model and better integration withsupporting tools.
Navigation in the Modeling LandscapeA great number of models have been presented until tod
featuring a multitude of approaches and levels of modeling deThe diversity of published works on the field indicates thatdefinite answers have been given to the catalytic converter meling problem@23#. There are several reasons for this situation
• Modeling objectives and application range. Not all publishedworks share common objectives and application range, varyfrom fast, approximate models to very detailed, computationaintensive models. Fundamental research models formulationally attempts to describe phenomena as accurately as possrequire a lot of input data and usually can be tested only intremely simplified catalyst behavior scenarios. For applicatioriented models, formulation depends on the system or dewhere modeling is applied as well as the design parameters uinvestigation. In this case, accuracy may be sacrificed becausconstraints such as simplicity or flexibility.
iate
2004 by ASME Transactions of the ASME
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• Problem complexity. Catalytic converter operation involveheterogeneous catalytic chemical reactions, which are couwith simultaneous heat and mass transfer and take place uhighly transient conditions. Under such conditions, it is difficultdescribe chemical phenomena quantitatively or even qualitatiEach set of inevitable approximations and simplifying assumtions that are included essentially defines a different modeapproach, and no one can a priori be considered better or wthan any other one.
• Rapidly changing washcoat technology. Catalytic convertermanufacturers continuously improve the chemical characterisof the catalytic converter washcoats, in an effort to produce ecient as well as low-cost designs. Thus chemical kinetics resehas to keep track of most modern washcoat developments, winhibits the acquisition of in-depth knowledge about the wacoats chemical behavior and may lead to system-specific consions and results.
• Performance assessment difficulty. Finally, there is no consensus on how to assess model performance, so that modelssimilar scope can be compared. The introduction of such a modological tool should help towards identifying the weaknesand advantages of different approaches and provide with a qtitative criterion for model comparison.
The variety of modeling approaches that have been propoand tested until today can be largely attributed to the above poNevertheless, similarities may be noticed among many differmodels and the majority of them share a common structure, wis dictated by the structure and operating concept of the cataconverter itself.
Specifically, the catalytic converter is essentially a batch of pallel channels, which have been covered in their interior bchemically active washcoat layer. The structure of most publismodels follows the structure of the converter itself so that emodel can be divided into three distinct levels: The washclevel, the channel level, and the reactor level. Below, we atteto clarify what is modeled at each level and discuss which aremost common choices that one has to make when developicatalytic converter model.
Washcoat Level. Washcoat modeling is local in nature. Athis level, local phenomena at each point of the washcoat athe channel axis are considered. Two dominant phenomena shbe modeled: diffusion and simultaneous reaction within the wacoat.
Heat and mass transport through the boundary layer of theis normally accounted for by employing mass and heat-trancoefficients. Heat transfer through the washcoat is normally omted, since the washcoat is approximately isothermal@24#. Finally,several approaches exist for the modeling of mass transthrough the washcoat: from completely neglecting washcoatfusion to detailed calculation of species profiles diffusinreacting in the washcoat solving the corresponding balance etions. The former can be viewed as a zero-dimensional approwhile the latter is one- or two-dimensional and implies significaadded computational cost.
For the reaction modeling, the mission of the model is twofo~i! To identify the prevailing physical and chemical phenome
and formulate an appropriate reaction scheme.~ii ! To assign a rate expression to each reaction.When building the reaction scheme, the primary choice is
tween elementary or overall reactions. Elementary reactionsscribe in detail the real steps with which heterogeneous cataproceeds. On the other hand, overall reactions view heterogencatalysis phenomenologically as a one-step reaction and no imediate steps are considered.
Elementary reactions usually employ simple Arrhenius-tyrate expressions@25,26#. Overall reactions use more complicaterate expressions, which are either totally empirical or theybased on the Langmuir-Hinshelwood formalism and containsome empirical terms@9,12,27#. Essentially, the overall reactio
Journal of Engineering for Gas Turbines and Power
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approach favours the simplification of the reaction scheme, atexpense of using more complicated and highly empirical ratepressions; in a sense, the complexity of the reaction schemetails is hidden into the mathematical formulation of the rate epressions. An interesting fact that is not yet fully understoodfundamental researchers in chemistry and chemical kinetics isthe above-mentioned engineering approach has succeeded inducing unexpectedly high accuracy in matching the performaof catalytic converters in real cycles@7,23#.
Channel Level. At the channel level, the local informationprovided by the washcoat model is exploited. The objectivesthe model are the following:
~i! to determine the mass and heat transfer between the exhgas and the solid phase~substrate and washcoat! of the converter;
~ii ! to determine the exhaust gas characteristics~temperatureand species concentrations! along the channel.
At this level, chemical and physical phenomena in the washcare viewed as heat or mass sinks/sources. Profiles of gas temture and species concentration between the channel wall andbulk flow are computed along the channel axis.
The exhaust gas flow through the channel is laminar andusually approximated with plug flow. Thus one-dimensional hand mass balance equations for the exhaust gas are formulatthis level of modeling. The option here is between a transienta quasisteady approach. If transient~time-dependent! terms of theequations are omitted, steady-state balances remain. The qsteady approach implies that these steady-state balancessolved for each time step as solution proceeds in time for differboundary conditions. The boundary conditions are imposed bytransient reactor model.
By omitting transient terms, the steady-state approach estially assumes that the there is no accumulation of heat or masthe gas flow, which is a realistic assumption@28#. The objective ofthis approximation is to simplify the balance equations and redthe computational cost that is involved in their solution.
Reactor Level. At the reactor level, channel-level information is exploited. Specifically, channels interact with each otonly via heat transfer. Thus channels are viewed as heat sisources and only one problem is tackled: Heat transfer in the sphase, i.e., conductive heat transfer in the monolith aconvection–radiation to the surrounding air. At this level, the hsources computed for each channel at the kinetics and chalevel calculations are used to estimate the temperature field omonolith.
Heat-transfer calculations may be one, two, or three dimsional, depending on the desired accuracy. One-dimensional~1D!reactor level models treat all the channels of the monolith idecally ~i.e., subject to identical boundary conditions!. On the otherhand, 2D computations divide the monolith into sectors~clustersof channels! and the channel level computations are done for eone of the distinct sectors@10#. Finite-volume or finite elementapproaches may be employed in the 3D computation@6,11,15#.
Evidently, each catalytic converter model incorporates a lotassumptions and approximations. At each level of catalytic cverter modeling, choices have to be made regarding modecomplexity and detail, which affect the accuracy and applicabiof the resulting model. For the engineer, modeling detail is alwdictated by the engineered object. Thus before deciding for acific model formulation, the model’s purpose should be clarifieTherefore we give below a brief discussion about the role of calytic converter modeling within the process of modern exhaaftertreatment systems design and operation.
Role of Modeling in the Design of Ultra Low EmissionExhaust Aftertreatment Systems
According to emission legislations for passenger cars and vemissions are measured over an engine or vehicle test cwhich is an important part of every emission standard. These
OCTOBER 2004, Vol. 126 Õ 907
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cycles are supposed to create repeatable emission measurconditions and, at the same time, simulate a real driving condiof a given application.
Consequently, the primary objective of the automotive indusis to build cost-effective exhaust lines that will enable the vehito succeed in the legislated driving cycle tests. Because ofincreasingly stringent emission standards, we observe a trenwards more complicated exhaust lines. For successful designengineer must consider not only the catalytic converter but,stead, the exhaust line~engine settings and control–exhaupiping–catalytic converter! as a system—an exhaust aftertrement system. This progressively makes the experimental testinthe exhaust line more difficult and more expensive, requirmore time and experimental data to tune it appropriately.
In order to decrease cost and design time for developmmodeling of the whole exhaust aftertreatment system wouldextremely helpful. If fast, reliable models were available to tindustry, new exhaust line configurations could be tested rapand at a reasonable cost; additionally, optimization of the exhline components could be aided by numerical optimization produres to achieve improved configurations and lower overall emsions. The underlying notion is the incorporation of computaided engineering~CAE! practices in the design of exhaust linewhich is currently under way in the automotive industry.
Consequently, the targets for the exhaust line modeling areby the industry. Currently, an indicative wish list for an idemodeling tool of the kind is the following:
• Reliability. It should be cross checked and validated thoughly in real-world case studies before conclusions and dedecisions can be drawn using it.
• Speed. It should run reasonably fast~that is, faster than reatime!, with common computer equipment so that it can be tesadjusted, and used within the time and cost constraints ofautomotive industry.
• Versatility. It should be easy to modify and apply to differesystem configurations, in order to enable their assessmenttuning.
• Ease of use. It should be easy to validate and use by automtive engineers who are not modeling experts.
• Minimum input data. It should require input data that may bacquired by routine experiments, in order to keep cost low anprevent input data uncertainty and errors.
• Simplicity. It should follow the fundamental engineering ruof thumb to keep complexity low. In this way, engineers–uscan easily have control over their modeling tools and easily ginsight to model behavior and results.
Traditionally, modeling tools and CAE procedures in geneare used in many areas of automobile design. The modeling ocatalytic converter proved to be a very complex problem, thouand this area resisted the extensive application of modeling. Atoday, such modeling tools are not fully accepted in the fieldcatalytic converter optimization, let alone the CAE-driven desof the whole exhaust line.
Nevertheless, progress on this field of research is fast. Thetomotive industry clearly identifies the importance of modelitools and recent improvements of catalytic converter models aresponse to this trend. Below, we present some work in this dition. We employ an updated version of a catalytic converter meling code, which has been continuously developed since 1@10# and has been employed on various design projects. Theware is combined with the development of a performance meafor the assessment of modeling quality, which is utilized withingenetic algorithm optimization procedure for the computer-aidestimation of reaction kinetics parameters. Our purpose is tovide a demonstration of the role and applicability of current moeling software in the design process of modern exhaust aftertment systems.
908 Õ Vol. 126, OCTOBER 2004
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Model DescriptionBelow, we present an updated version of theCATRAN catalytic
converter model. The model’s design concept is the minimizatof degrees of freedom and the elimination of any superflucomplexity in general. The main features of the model arefollowing:
• transient, one-dimensional temperature profile for the sophase of the converter~reactor level modeling!;
• quasisteady, 1D computation of temperature and concentraaxial distributions for the gaseous phase~channel level modeling!;
• simplified reaction scheme featuring a minimum set of redreactions and an oxygen storage submodel~washcoat level mod-eling!.
Below, the detailed description of model formulation is givefor each modeling level.
Washcoat Level Modeling. The first task of washcoat modeling is to define how the simultaneous phenomena of diffusand reaction in the washcoat will be taken into account. What wbe adopted in this work is the ‘‘film model’’ approach, whichthe simplest and most widely used one~e.g., Refs.@9#, @11#!. Thefilm model approximates the washcoat with a solid–gas interfawhere it is assumed that all reactions occur. This approximaessentially neglects diffusion effects completely, and assumesall catalytically active cites are directly available to gaseous-phspecies at this solid–gas interface.
This has been questioned by Zygourakis and Aris@24# andHayes and Kolaczkowski@29#. They provide evidence that concentration gradients in the washcoat are present and may sigcantly affect the operation of the monolithic converter, especiain high temperatures. Nevertheless, significant complexity istroduced in the models in order to explicitly consider diffusionthe washcoat. Therefore washcoat diffusion is not implemenhere and its effect is lumped into the kinetic parameters ofmodel.
The approximation for the solid–gas interface states thatspecies that diffuse to it through the boundary layer are remofrom the gas phase due to reactions:
rg
Mgkm, jS~cj2cj ,s!5Rj . (1)
The left-hand side of the above equation describes mass trathrough the boundary layer of the gas flow. Parameterkm is themass transfer coefficient in the boundary layer,cj denotes speciesconcentration at the gaseous phase, andcs, j is corresponding con-centration at the gas–solid interface.S is the geometrical surfacearea of the washcoat, i.e., channel wall area per channel voluFor a channel with hydraulic diameterdh , we readily find:S54/dh .
On the right-hand side of Eq.~1!, the rateRj refers to theproduction or consumption of each species at the solid–gas inface. ForNR reactions, each taking place with a rater k , the rateof consumption or production of a speciesj is
Rrea,j5dgS(k51
NR
~aj ,kr k!, (2)
whereaj ,k is the stoichiometric coefficient of speciesj in reactionk, d is the washcoat thickness, andg is the specific catalyst areai.e., catalytically active area per washcoat volume.
The second task of washcoat modeling is to choose a reacscheme and the appropriate expressions for the reaction rater k .We opt for overall reactions, because they provide the user wimore compact and comprehensible reaction scheme and thecomputationally less expensive. Here, the three-way catalytic cverter ~3WCC! will be considered, which is designed for sparignition engines exhaust. Below, we briefly discuss whatchoose to implement in this case.
Transactions of the ASME
Table 1 Reaction scheme and rate expressions of the model
Reaction Rate expression
Oxidation reactions
1 CO11/2O2→CO2
r15A1e
2E1 /RgTcCOcO2
G2 H211/2O2→H2O
r25A2e
2E2 /RgTcH2cO2
G3, 4 CaHb1(a10.25b)O2→aCO210.5bH2O
rk5Ake
2Ek /RgTcCaHbcO2
G, k53,4
NO reduction
5 2CO12NO→2CO21N2 r 55A5e2E5 /RgTcCOcNO
Oxygen storage
6 2CeO21CO→Ce2O31CO2 r 65A6e2E6 /RgTcCOcCcap7, 8 CaHb1(2a1b)CeO2→ r k5Ake
2Ek /RgTcCaHbcCcap, k57, 8
→(a10.5b)Ce2O31aCO10.5bH2O9 Ce2O311/2O2→2CeO2 r 95A9e2E9 /RgTcO2
(12c)Ccap10 Ce2O31NO→2CeO211/2N2 r 105A10e
2E10 /RgTcNO(12c)Ccap
Inhibition term
G5T~11K1cCO1K2cCxHy!2~11K3cCO
2 cCxHy2 !~11K4cNO
0.7!, Ki5ki exp~2Ei /RgT!
K1565.5 K252080 K353.98 K454.793105
E1527990 E2523000 E35296,534 E4531,036
Auxiliary quantities
c523moles CeO2
23moles CeO21moles Ce2O3,
dc
dt52
r 91r 10
Ccap1
r 61r 71r 8
Ccap
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In the present model, the oxidation reactions rates of COHC are based on the expressions by Voltz et al.@27#. It is inter-esting to note that the expressions developed by Voltz et al. a30 years ago for a Pt oxidation catalyst continue to be succeswith little variation, in describing the performance of Pt:Rh, PPd:Rh, and even tri-metal catalysts. For the HC oxidation,have to note the real exhaust gas contains a very complex mixof several hundreds of different hydrocarbon species with vability in composition depending on the driving conditions@30#.In practice, the diversity of the HC mixture is usually taken inaccount by considering two categories of hydrocarbons, eaching oxidized in different temperature: an easily oxidizing H~‘‘fast’’ HC !, and a less-easily oxidizing HC~‘‘slow’’ HC !. Here,the ‘‘fast’’ HC is represented by propene (C3H6) and the ‘‘slow’’HC is represented by propane (C3H8). In practice, only the totalhydrocarbon content of the exhaust is measured. Throughoutwork, it is assumed that the exhaust HC consisted of 85% ‘‘faHC and 15% ‘‘slow’’ HC. This assumption was done for modelinpurposes due to the lack of more accurate data and, accordinour experience, it gives satisfactory results.
For the reaction between CO and NO we use an empiricaaction rate adopted from Pattas et al.@31#, which predicts a vari-able order of reaction for the CO oxidation from NO, dependon CO concentration. The expression is qualitatively consiswith the results of Koberstein and Wannemacher@32#, which pre-dict that the reaction order for CO in the CO–NO reaction tendunity as the reactants’ concentrations tend to vanish. Finally,drogen oxidation is also included in the model.
All reduction–oxidation reactions employed in the modalong with their rate expressions, are given in Table 1.
Apart from the redox reactions, oxygen storage phenomplay a principal role in the efficiency of the 3WCC. Oxygen stoage occurs on the ceria~Ce!, which is contained in large quantitiein the catalyst’s washcoat~at the order of 30% wt!. Under netoxidizing conditions, 3-valent Ce oxide (Ce2O3) may react with
Journal of Engineering for Gas Turbines and Power
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O2 , NO, or H2O and oxidize to its four-valent state (CeO2).Under net reducing conditions, CeO2 may function as an oxidiz-ing agent for CO, HC, and H2 .
Oxygen storage is taken into account by an updated reacscheme. The new scheme consists of five reactions which accfor ~i! Ce2O3 oxidation by O2 and NO, and~ii ! CeO2 reduction byCO and fast/slow hydrocarbons The model uses the auxilquantity c to express the fractional extent of oxidation of thoxygen storage component. It is defined as
c523moles CeO2
23moles CeO21moles Ce2O3. (3)
The extent of oxidationc is continuously changing during transient converter operation. Its value is affected by the relativeaction rates of reaction nos. 6–10. The rates of reactions,given in Table 1, are expected to be linear functions ofc. Specifi-cally, the oxidation rate of the oxygen storage component issumed proportional to the active sites of Ce2O3 , i.e., to Ccap(12c). On the other hand, the oxidation rate of CO and HCCeO2 is assumed proportional to (Ccapc). Moreover, the rates ofthese reactions should be linearly dependent on the local contration of the corresponding gaseous phase reactant.
The rate of variation ofc is the difference between the rate thCe2O3 is oxidized and reduced:
dc
dt52
r 91r 10
Ccap1
r 61r 71r 8
Ccap. (4)
The updated model includes an analytical solution for Eq.~4! forc at each node along the catalyst channels.
Channel Level Modeling. For the formulation of thechannel-level model, two usual simplifications are employ@26,33#, namely:
OCTOBER 2004, Vol. 126 Õ 909
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• The axial diffusion of mass and heat in the gas phase is nligible.
• The mass and heat accumulation in the gas phase is neglig~This comprises the assumption for the quasi-steady-state nof the problem.!
The first assumption is generally accepted and is employemost models, e.g., those of Chen et al.@11# and Siemund et al.@9#. Only the boundary layer effect on mass transfer is accounfor, using a mass transfer coefficientkm , which is a function ofthe Sherwood number. Bulk flow is approximated with plug flowith uniform temperatureTg and species concentrationscj . Thevelocity of the flow readily results from the mass flow rate:uz
54m/(rpdh2).
The second assumption is more controversial and bothquasi-steady and the transient approach have been tested iliterature. Shamim et al.@15# have presented a model that incoporates transient terms at channel level modeling. On the ohand, Young and Finlayson@8# and Oh and Cavendish@3# argueabout the validity of the quasi-steady approximation. They pvide justification on the basis of the large ratio of thermal to mtime constants of the problem. Following this line of thought, tquasi-steady approach is used here as well, for simplicity andcomputational cost.
In order to write the mass balance for the exhaust gas, a mbulk valuecj is employed for the gas–phase concentration of especies. Likewise, a valuecs, j is considered for the concentratioof each species at the solid–gas interface. Using the quasi-stestate approximation and neglecting diffusion and accumulaterms, the mass balance for the gas phase becomes
rguz
]cj~z!
]z5rgkm, jS@cj~z!2cs, j~z!#. (5)
Similarly to the above, a mean bulk valueTg is used for theexhaust gas temperature, and a solid phase temperatureTs is in-troduced for the monolith and the solid–gas interface. Energtransferred to and from the exhaust gas only due to convecwith the channel walls. Thus the energy balance for the gas pbecomes
rscpuz
]Tg~z!
]z5hS@Ts~z!2Tg~z!#. (6)
Parameterh is the heat-transfer coefficient and is calculated afunction of the Nusselt dimensionless number.
Finally, the boundary conditions for the temperature, mass flrate, and concentrations are given from measurement at theverter’s inlet:
cj~ t,z50!5cj , in~ t !,
Tg~ t,z50!5Tg, in~ t !, (7)
m~ t,z50!5min~ t !.
The species that are considered in the exhaust gas flow arefollowing: j 5CO, O2 , H2 , HCfast, HCslow, NOx , N2 .
Reactor Level Modeling. At the reactor level modeling, heatransfer between channels and between the reactor and itsrounding are modeled. The principal issue here is to decidone-, two-, or three-dimensional modeling of the heat transshould be employed.
The reactor model presented in this work is a one-dimensioheat-transfer model for the transient heat conduction in the molith. Heat losses to the environment via convection and radiaare also taken into account. Its primary assumptions are thelowing:
• Heat losses from the front and the rear face of the monoare neglected.~To our knowledge, this is the case with all modethat have appeared in the literature.!
910 Õ Vol. 126, OCTOBER 2004
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• Since the catalytic converter is always insulated, simpler mels approximate the convert as adiabatic. Heat losses to therounding are taken into account but, owing to the model’snature, they are inevitably distributed uniformly in each monlith’s cross section.
• Flow rate and temperature profiles of the exhaust gas atinlet of the filter are considered uniform. An average valueflow rate and temperature is measured, and gas flow is distribuniformly to each channel.
The temperature field in the converter is described by the eqtion of transient heat conduction in one dimension, with hsources being convection from the exhaust gas, the enthalpyleased from the reactions, and convection to ambient air,
rscp,s
]Ts
]t5ks,z
]2Ts
]z21hS~Tg2Ts!1(
k51
NR
~2DHk!r k1Qamb.
(8)
Finally, the boundary condition needed for the solution of the hconduction equation refers to the heat losses to ambient air:
Qamb5Smon@hamb~Ts2Tamb!1«s~Ts42Tamb
4 !#. (9)
Two- and three-dimensional reactor models have also appearthe literature, e.g., the models of Heck et al.@34#, Chen et al.@11#,Zygourakis@35#, and Jahn et al.@36#. These models are indispensable if the exhaust gas at the converter inlet exhibits a sevenonuniform flow profile but also require mass flow rate and teperature profiles at the inlet of the catalytic converter. Such dare not usually available in routine engine-bench or driving cyconverter tests, which are the main application field for our modTherefore the 1D approach is preferred, since its modeling dematches routine input data quality, provides sufficient accuraand has low computational power requirements.
Tuning Procedure
General. The kinetics submodel introduces into the catalyconverter model a set of parameters that have to be estimatedreference to a set of experimental data. The introduction of table parameters is inevitable regardless of the formulation ofreaction scheme. Tunable parameters take into account thetivity of the specific washcoat formulation as well as any othaspect of catalytic converter operation that is not included inmodel explicitly.
In the present model, the tunable parameters are the activaenergyEk and the preexponential factorAk that are included in thereaction rater k of each reactionk. In total, there is a maximumnumber of 20 tunable parameters, however, the values of action energies are more or less known from Arrhenius plots athus only the pre-exponential factors are tuned~ten tunable pa-rameters!. Apart from reaction activity, tunable parameters valuinclude the approximations that have been done during moformulation. Among them, the most important are:
• effect of reaction scheme and rate expressions simplificatio• effect of exhaust gas input data uncertainty;• effect of neglection of diffusion in the washcoat;• effect of 1D reactor model approximation.Since~i! the rate expressions of the model are empirical and~ii !
effects of model approximations are lumped into the tunablerameters, the latter do not correspond to real kinetic parameRather, they should be viewed as fitting parameters of the mo
The traditional method to tune a model was to manually adthe parameters by a trial-and-error procedure, starting from aof realistic values~known from previous experience! and modify-ing them gradually, so that the model results compare well wthe measured ones. The usual method for computation vemeasurement comparison is inspection of plotted results. Tmanual tuning introduces human intuition and experience in orto ~i! assess model performance, and~ii ! fit the model to the givenexperimental data. This is a questionable practice because
Transactions of the ASME
t
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• inspection is dependent on the scale that results are vieand it may therefore be misleading;
• there is considerable difficulty to compare the performancedifferent models that are presented in the literature;
• it does not provide any confidence about the quality oftuning.
The above drawbacks of manual performance assessmenparameter tuning have led to efforts for the development ocorresponding computer-aided procedure. Since model tuninessentially a parameter-fitting problem, the underlying idea ofattempts has been to express the problem mathematically aoptimization problem. This involves the introduction of an itertive optimization method, which maximizes or minimizes an ojective function that indicates goodness of fit. The objective fution required by the optimization procedure is thus aperformancemeasureof the model, i.e., a mathematical measure that assesmodel’s performance in a quantitative manner.
A few attempts for the development of a complete compuaided tuning methodology have been presented in the past. Mtreuil et al.@12# were the first to present a systematic attemptthe tuning of the parameters of their steady-state three-way clytic converter model. Dubien and Schweich@37# also published amethodology to determine the pre-exponential factor and thetivation energy of simple rate expressions from light-off expements. Pontikakis and Stamatelos@7,38# introduced a computeraided tuning procedure for the determination of kinetic parameof a three-way catalytic converter model from driving cycle tebased on the conjugate gradients method. Glielmo and Sa@39# presented a simplified three-way catalytic converter mooriented to the design and test of warm-up control strategiestuned it using a genetic algorithm.
All of the above efforts used a performance measure basethe least-squares error@40# between measured and computedsults. Except for Glielmo and Santini, all other works are basedgradient-based methods for the optimization of the performameasure@41#. Gradient methods are faster, more accurate,may be used as black-box methods, but assume that the optimtion space is unimodal~i.e., it contains a single extremum!. Ge-netic algorithms, used by Glielmo and Santini, are better suitemulti-modal optimization but they are slower, less accurate,have to be appropriately adapted to the target problem.
In the present work, we present recent progress that hasmade in the field of computer-aided parameter estimation. Itcontinuation of the work of Pontikakis and Stamatelos@7# andupdates both performance measure definition and optimizamethodology. In the following sections the requirements thaperformance measure should comply with are presented, aperformance measure that satisfies these requirements is deFinally, a genetic algorithm is applied as an optimization methology @23#.
Formulation of the Performance Measure. The perfor-mance measure that is formulated below exploits the informaof species concentrations measurements at the inlet and the oof the catalytic converter. Specifically, it is based on the convsion efficiencyEj for a pollutantj. Herein, we take into accounthe three legislated pollutants, thusj 5CO, HC, NOx .
To account for the goodness of computation results compawith a measurement that spans over a certain time period, ane for each time instance must be defined. The latter should gthe deviation between computation and measurement for theversion efficiencyE. Summation over time should then be peformed to calculate an overall error value for the whole extenthe measurement. Here, the error is defined as
ueu5uE2Eu. (10)
Absolute values are taken to ensure error positiveness. Thisdefinition also ensures that 0<ueu<1, since it is based on convesion efficiency.
Journal of Engineering for Gas Turbines and Power
wed
of
he
andf ag isalls ana-b-c-
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The error between computation and measurement is a funcof time and the tunable paremeter vector:e5e(t;q), whereq isthe formed by the pre-exponential factor and activation energyeach reaction of the model:
q5@A1 ,E1 ,A2 ,E2 , . . . ,ANP,ENP
#T. (11)
We nameperformance function f(t;q)5 f „e(t;q)… a function ofthe errore, which is subsequently summed over some time pert to give the performance measureF. Here, the performance function is defined as
f ~ tn ;q!5ue~ tn ;q!uemax~ tn!
. (12)
Time t take discrete values,tn5nDt, with Dt being the discreti-zation interval which corresponds to the frequency data measuThe quantityemax is the maximum error between computation ameasurement, and it is defined as
emax~ tn!5max$E~ tn!,12E~ tn!%. (13)
The performance measure can be subsequently formed usingfunction of the sum of the performance function over time:
F~q!5FS (n50
N
f ~ tn ;q!D , N5t/Dt. (14)
In this work, we define the performance measureF as the meanvalue of the performance function over the time period of intere
F~q!51
N (n50
N
f ~ tn ;q!51
N (n50
Nue~ tn ;q!uemax~ tn!
. (15)
The performance measure defined in Eq.~15! is used for the as-sessment of the performance of each of the three pollutantsHC, NOx . The total performance measure is computed asmean of these three values:
F5FCO1FHC1FNOx
3. (16)
The above performance measure presents advantageous fecompared to the classical least-squares performance measur
• It ranges between two, previously known, finite extreme vues. Extremes correspond to zero and maximum deviationtween calculation and experiment.
• The extrema of the performance measure are the same fophysical quantities that may be used and all different measments where the performance measure may be applied. Thathe performance measure is normalized so that its extrema dodepend on the either the measured quantities or the experimprotocol.
Furthermore, out of the different versions of performance msure tested within the specific genetic algorithm optimization pcedure, the specific one gave us the best convergence, withouneed to rely on empirical weight factors@23#.
Optimization Procedure. Having defined the performancmeasure for the model, the problem of tunable parameter esttion reduces in finding a tunable parameter vectorq that maxi-mizesF.
The parameter vectorq is not used directly in the optimizationprocedure. Instead, we perform parameter reparametrizationthe pre-exponential factorA, defined as
A510A⇔A5 log A (17)
and the transformed tunable parameters vector becomes
q5@A1 ,E1 ,A2 ,E2 , . . . ,ANP,ENP
#T. (18)
Then, the problem of tunable parameter estimation is expresse
OCTOBER 2004, Vol. 126 Õ 911
.-
s
nr
i
h
a
etn
o
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t
e
p-rac-
bedsignncy,n is2.
adelstssful
ure-ter
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Maximize F8~q !512F~q !
51
3N (j 5CO,HC,NOx
(n50
Nuej~ tn ;q !uej ,max~ tn!
, N5t/Dt.
(19)
This is a constraint maximization problem, since the componeof vectorq are allowed to vary between two extreme values, iq i ,min<qi<qi,max. Previous experience@7# has shown that the parameter space that results from this problem formulation is mumodal. Thus an appropriate optimization procedure shouldused. Here, a genetic algorithm has been employed for the mmization of Eq.~19!. A brief description of the genetic algorithmoperation concept and the main features of the implementaused herein are presented below.
A genetic algorithm is a kind of artificial evolution. Whaevolves is a population of solutions to a problem; each solutioan individual of the population. Individuals are born, mate, repduce, are mutated, and die analogously to nature’s paradigm.cornerstone of this evolution process is that more fit individuare given more advantage to live longer and propagate theirnetic material to the next generations.
The main steps taken by the algorithm are the following:~i! Initialization. A set of points in the optimization space
chosen at random. This is the initial population of the genealgorithm, with each point corresponding to an individual of tpopulation.
~ii ! Encoding. Each individual is encoded to a finite binarstring ~chromosome!. In brief, each real intervalbq i ,min ,qi,maxc ismapped to the integer interval@0,2,# and subsequently transformfrom integer to binary. Then, the chromosome is built by concenating the binary strings that correspond to the value of eparameterq i . This encoding is called ‘‘concatenated, multiprameter, mapped, fixed-point coding’’@42#.
~iii ! Fitness calculation. The fitness of each individual in thpopulation is computed using Eq.~19!. It should be noticed thafitness calculation requires that the model be called for each ividual, i.e., as many times as the population size.
~iv! Selection. Random pairs of individuals are subject to tounament, that is, mutual comparison of their fitnesses@43#. Tour-nament winners are promoted for recombination.
~v! Recombination (mating). The one-point crossover operato@42# is applied to the couples of individuals that are selectedrecombination~parents!. One-point crossover works on the chrmosomes~binary encodings! of the individuals. The resultingchromosomes~children! are inserted to the population replacintheir parents and they are decoded to produce their corresponreal parameter vector.
~vi! Mutation. A small part of the population is randomly mutated, i.e., random bits of the chromosomes change value.
~vii ! Steps~iii !–~vii ! are repeated for a fixed number of geerations or until acceptably fit individual has been produced.
The randomized nature of the genetic algorithm enables iavoid local extrema of the parameter space and converge towthe optimum or a near-optimum solution. It should be notthough, that this feature does not guarantee convergence to
Table 2 Parameters of the genetic algorithm
Encoding type binaryCrossover operator one-point crossoverMutation operator binary mutationPopulation size 100Crossover probability 0.6Mutation probability 0.03Encoding resolution 10 bit~,510!Limits 1007,Ai,1023
912 Õ Vol. 126, OCTOBER 2004
ntse.,
lti-be
axi-
tion
tis
o-Thealsge-
stice
y
at-ach-
di-
r-
rfor-
gding
-
-
toardsd,the
global optimum. This behavior is common to all multimodal otimization techniques and not a specific genetic algorithm chateristic.
The implementation of the genetic algorithm that was descriabove is not the only one possible. There are a number of dedecisions and parameters that influence the operation, efficieand speed of the genetic algorithm. The present implementatioclassical, though. Its characteristics are summarized in Table
Application ExamplesThe model’s predictive ability is going to be demonstrated in
number of typical applications. The first step is to tune the mofor a specific catalyst configuration in typical driving cycle teand the estimate the kinetic parameters of the model. Succetuning in this phase implies that~i! the model incorporates theappropriate degrees of freedom in order to match the measment,~ii ! the tuning methodology is able to tackle the parameestimation optimization problem successfully, and~iii ! experimen-tal data are in a certain level of accuracy and has been procethrough a quality assurance procedure@44#.
First, a MVEG test is considered,~European test cycle!. Athree-way catalytic with a 2.4-l volume, two beds, 400 cpsi, 6mil wall thickness, underfloor converter with 50 g/ft3 Pt:Rh 7:1precious metal loading is installed on a passenger car withengine displacement. The measured catalyst’s performance oncar is presented in Fig. 1, by means of the measured instantanCO, HC, and NOx emissions at converter inlet and exit, over th1180-sec duration of the cycle.
Obviously, the specific converter attains a significant oveefficiency: The emissions at catalyst’s exit are diminished aftercold start phase. However, the emissions standards themselvequite low: Thus the model should not only accurately predict calyst light-off, but it should also be capable of matching the calyst’s breakthrough during accelerations, decelerations and ecially in the extra-urban, high-speed part of the cycle.
In order to match this catalyst’s behavior, the model was tunusing the genetic algorithm. The measured emissions duringfull NEDC were employed as a reference period for parameestimation~t51180 s in Eq.~19!!. CPU time of the order of 48 hwas required for the computation of 135 generations~100 indi-viduals each! on a Pentium IV 2.4-GHz PC. However, if onneeds faster execution time, a part of the cycle containing thestart and some hot operation could equally do with just one-thof the time.
The tuning process resulted to the kinetics parameters of T3. The evolution of the most important kinetics parameters valof the individuals as determined by the genetic algorithm durthe 135 generations is presented in Fig. 2. It may be observedthe genetic algorithm converges to specific values for theserameters. This behavior is not observed for all kinetics parameand this is taken into account for improvements in the kinetscheme@22#. The computed results which are produced whenmodel is fed with the frequency factors determined by the 13generation of the genetic algorithm, are summarized in the foof cumulative CO, HC, and NOx emissions at catalyst’s exit, compared to the corresponding measured curves in Fig. 3. The errthe prediction of cumulative emissions does not generally exc5% during the cycle for any species.
Also, it is useful to mention that although the specific genealgorithm matures after about 120 generations, the convergenclear even from the 20th generation: This means that we cocompute good kinetics parameters even with 8 h CPU time. Thusthe genetic algorithm approach can no more be considereforbidding due to excessive computation time. For comparispurposes, it should be mentioned that the tuning process ismost critical part of the modeling job, and it could take seveworking days from experienced engineers. Moreover, its rewas not guaranteed. With the new developments, whenevergenetic algorithm fails to converge, is an indication that the qu
Transactions of the ASME
Fig. 1 Measured instantaneous CO, HC, and NO x emissions at converter inlet and exit, over the 1180-sec duration of the cycle:2-l-engined passenger car equipped with a 2.4-l underfloor converter with 50-g Õft 3 Pt:Rh catalyst
c
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ity of the test data employed in the tuning process is questionaThus the tuning effort with the specific data is stopped andquality assurance team must examine and possibly reject thedata@44#.
In addition, computed and measured temperatures at theverter’s exit are compared in Fig. 4. Apparently, the modelcapable of matching the catalyst’s behavior with a remarkaaccuracy, allowing the performance of design optimization sties.
Since one of the objectives of this paper is to quantifyattainable accuracy, we proceed to a more detailed comparisomodel predictions and measurements in the form of instantanCO, HC, and NOx emissions. Prediction of instantaneous emsions over the full extent of a real legislative cycle is a significchallenge to any catalytic converter model.
Figure 5 presents the computed and measured instantaneouemissions at converter inlet and exit during the first 600 secNEDC: Apparently, the model successfully matches light-off bhavior of the catalyst, as well as subsequent breakthrough duacceleration. The role of oxygen storage and release reactiomatching the CO breakthough behavior is better assessed bcluding in the graph the computed degree of filling of the to
Table 3 Kinetics parameters tuned to the Pt:Rh catalyst
Reaction A E
1 CO11/2O2→CO2 2.000E120 90,0002 H211/2O2→H2O 2.000E119 90,0003, 4 CaHb1(a10.25b)O2→aCO210.5bH2O 1.800E120 95,000
2.710E119 120,000NO reduction
5 2CO12NO→2CO21N2 3.403E109 90,000Oxygen storage
6 2CeO21CO→Ce2O31CO2 7.832E109 85,0007, 8 CaHb1(2a1b)CeO2→ 1.283E110 85,000
→(a10.5b)Ce2O31aCO10.5bH2O 3.631E113 85,0009 Ce2O311/2O2→2CeO2 2.553E109 90,00010 Ce2O31NO→2CeO211/2N2 4.118E110 90,000
Journal of Engineering for Gas Turbines and Power
ble.thetest
on-is
bled-
hen ofous
is-nt
s COof
e-rings inin-
tal
washcoat’s oxygen storage capacity.~See, for example, the smabreakthroughs, which are of the order of 200 ppm, with maximpeaks of the order of 1000 ppm, that are accurately matchedthe computation!.
A comparison of computed and measured CO emissions forcycle part from 600 to 1180 s is given in Fig. 6. In the extra-urbpart of the cycle, the efficiency of the converter is reduced duehigh flowrates. Again, the breakthroughs are observed whenoxygen stored in the washcoat is gradually depleted. The pretion of the model is remarkably good, especially when the ordemagnitude of the breakthroughs is considered. This succesprediction indicates that the oxygen storage reactions thatimplemented in the model are capable of modeling the phenenon with high accuracy. An exception to this good behaviorobserved only in the interval between 930 and 980 sec. Here texists room for further improvement of the storage submodelreaction scheme.
Figures 7 and 8 present the computed and measured instneous NOx emissions at converter inlet and exit during the twhalves of the NEDC: Apparently, the model successfully matclight-off behavior of the catalyst, as well as subsequent Nxbreakthrough behavior during accelerations. Oxygen storagcritical also here. The prediction accuracy is remarkable whennotes that the breakthroughs are of the order of a few ppmweak point of the model is spotted in the comparison of compuand measured behavior between 980 and 1100 sec. Again, ththe subject for future improvements in the oxygen storage smodel.
The situation appears equally good in Figs. 9 and 10 wherecomputed and measured instantaneous HC emissions at convinlet and exit during the first 600 s and the rest of NEDC agiven. The connection between HC breakthroughs and oxystorage phenomena in the washcoat is apparent also in thisThe model results are of comparable quality as the previousures, since the model predicts the events~HC breakthroughs! ofthe extra-urban part of the cycle, not only qualitatively, but aquantitatively, in a certain extent. HC light-off behavior is al
OCTOBER 2004, Vol. 126 Õ 913
914 Õ V
Fig. 2 Evolution of the values of three selected kinetics parameters in the 100 individuals of the populationduring 135 generations. The convergence to the final values is apparent even from the 40th generation.
o theyticess-ntt’sis, iffirst
matched with a very good accuracy, taking into account the cplexity of the hydrocarbons composition that is modeled onlytwo representative components.
The overall performance of the specific model prediction’s cbe quantitatively assessed by means of evaluating the performmeasure which is defined in Eqs.~15! and~16!. The performancemeasure takes the value:F850.959.
ol. 126, OCTOBER 2004
m-by
anance
The above results indicate that the model formulation hascapability to match typical measurements of a three-way catalconverter, and that the tuning methodology may be used succfully to fit the model to the measured data. Another importafinding is that the model is capable of predicting the catalysperformance outside the region where it has been tuned. Thatwe tested carrying out the tuning process based only on the
Fig. 3 Computed and measured cumulative CO, HC, and NO x emissions at converter exit during NEDC:2-l-engined passenger car equipped with a 2.4-l underfloor converter with 50-g Õft 3 Pt:Rh catalyst
Transactions of the ASME
Journal of Engi
Fig. 4 Measured converter inlet temperatures, computed, and measured converter exit tem-peratures during NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converterwith 50-g Õft 3 Pt:Rh catalyst
irt
tic
ac-re-rter.
400 s~see above discussion on genetic algorithm tuning!, and themodel demonstrated an equally satisfactory predictive abilitythe catalyst performance throughout the full NEDC cycle.
Additional evidence is provided below about the model’s abilto predict the operating behavior of a different catalytic conveconfiguration. This is achieved by using the model to predict
neering for Gas Turbines and Power
for
tyterhe
behavior of an alternative configuration of a three-way catalyconverter with no further kinetic parameter adjustments.
Therefore, as a next step in the assessment of the model’scuracy and predictive ability, the model is employed in the pdiction of the performance of an alternative underfloor conveof the same washcoat type, which is1
4 the size of the original one
Fig. 5 Computed and measured instantaneous CO emissions at converter inlet and exit during thefirst 600 sec of NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converter with50-gÕft 3 Pt:Rh catalyst
OCTOBER 2004, Vol. 126 Õ 915
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Fig. 6 Computed and measured instantaneous CO emissions at converter inlet and exit during thesecond half of the NEDC „600–1180 sec of NEDC …: 2-l-engined passenger car equipped with a 2.4-lunderfloor converter with 50-g Õft 3 Pt:Rh catalyst
rd
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Since the same catalyst formulation and precious metal loadinemployed, modeling of this case is performed using the sakinetics parameters of Table 3 which where estimated fororiginal converter. Only the external dimensions of the conveare changed to the ones of the reduced size converter an
26, OCTOBER 2004
g ismetheter
no
further tuning of the kinetic parameters is performed. The resof the model are then compared to the measured results forconverter in Fig. 11, in the form of cumulative CO, HC, and NOxemissions. Apparently, the model is capable of predicting thenificant change in all three pollutants emissions that is caused
Fig. 7 Computed and measured instantaneous NO x emissions at converter inlet and exit duringthe first 600 sec of NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converterwith 50-g Õft 3 Pt:Rh catalyst
Transactions of the ASME
Journal of En
Fig. 8 Computed and measured instantaneous NO x emissions at converter inlet and exit during thesecond half of NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converter with50-gÕft 3 Pt:Rh catalyst
i
tionex-
eous
the reduction of the converter’s volume, without changes inkinetic parameters. The performance measure now takesvalue: F850.921, which indicates a sufficiently accurate predtion of measured performance.
gineering for Gas Turbines and Power
itsthe
c-
A better insight on the model’s performance, also in associawith oxygen storage and release behavior, can be made, forample, by a comparison of computed and measured instantanHC emissions at the converter’s exit, during the full cycle~Fig.
Fig. 9 Computed and measured instantaneous HC emissions at converter inlet and exit during thefirst 600 sec of NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converter with50-gÕft 3 Pt:Rh catalyst
OCTOBER 2004, Vol. 126 Õ 917
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Fig. 10 Computed and measured instantaneous HC emissions at converter inlet and exit during thesecond half of NEDC: 2-l-engined passenger car equipped with a 2.4-l underfloor converter with50-gÕft 3 Pt:Rh catalyst
ia
a
ionpsi,
12!. Here, the model demonstrates the capacity of takingaccount the change in total oxygen storage capacity, to accurpredict the effect of size reduction. The same good behaviodemonstrated with CO in Fig. 13, and NOx ~not shown here!.
26, OCTOBER 2004
ntotely
r is
As a next step, we check the model’s predictive ability withdifferent catalyst type, namely, a Pd:Rh 14:1, 150 gr/ft3, 0.8-lvolume converter, which is now fitted in a close-coupled positon a 2.2-l engined car. Again, the converter is a standard 400-c
Fig. 11 Computed „based on the kinetics parameters values of Table 1 … and measured cumulative CO, HC, andNOx emissions at converter exit during NEDC. 2-l-engined passenger car equipped with a 0.6-l underfloorconverter with 50-g Õft 3 Pt:Rh catalyst. Apparently, the model is capable of predicting the significant differencein CO, HC, and NO x emissions at the exit of the reduced-size converter.
Transactions of the ASME
Journal of E
Fig. 12 Computed „based on the kinetics parameters values of Table 1 … and measured instantaneousHC emissions at converter exit during NEDC: 2-l-engined passenger car equipped with a 0.6-l under-floor converter with 50-g Õft 3 Pt:Rh catalyst. Apparently, the model is capable of predicting with goodaccuracy the characteristic HC breakthrough with the reduced-size converter during the high-speedpart of NEDC.
Fig. 13 Computed „based on the kinetics parameters values of Table 1 … and measured instanta-neous CO emissions at converter exit during NEDC: 2-l-engined passenger car equipped with a 0.6-lunderfloor converter with 50-g Õft 3 Pt:Rh catalyst
ngineering for Gas Turbines and Power OCTOBER 2004, Vol. 126 Õ 919
g
ee
uead
sions5:ofsig-
enugh
om-ca-
rt abil-eentheex-
st six
thetheom-trolignnce
aturetype
ofata-
rs’ring
itsels
ase
re-is
6.5-mil substrate. The test now is according to the U.S. FTPprocedure, which is considered as a more demanding test prdure for modeling catalytic converter behavior, because ofextensively transient nature of its driving cycle.
In this case, a new kinetic tuning is required, because we sto a different type of catalyst and washcoat formulation. Thenetic algorithm optimized the kinetic parameters of the modelthe reference period of the first 600 s of the FTP cycle. This pboldly corresponds to the cold-start phase of the cycle.~Thisphase is followed by the transient phase~505–1369 s! and thehot-start phase~1369–1874 s!, the latter starting after the enginis stopped for 10 min!. The values of the kinetic parameters rsulting from the tuning procedure are listed in Table 4.
The computed results are summarized in the form of cumtive CO, HC, and NOx emissions at the catalyst’s exit, comparto the corresponding measured curves in Fig. 14. The performmeasure takes the value of F850.955. In addition, computed anmeasured temperatures at converter’s exit are compared in15. Apparently, also in this case, the model is capable of matchthe catalyst’s behavior with very good accuracy, allowing the pformance of design optimization studies.
The model’s accuracy can be assessed in more detail, by mof instantaneous emissions comparison. As an example, Fig
Table 4 Kinetics parameters tuned to the Pd:Rh catalyst
Reaction A E
1 CO11/2O2→CO2 1.10E118 90,0002 H211/2O2→H2O 1.10E118 90,0003, 4 CaHb1(a10.25b)O2→aCO210.5bH2O 2.00E115 70,000
1.00E116 105,000NO reduction
5 2CO12NO→2CO21N2 1.20E113 90,000Oxygen storage
6 2CeO21CO→Ce2O31CO2 1.00E109 85,0007, 8 CaHb1(2a1b)CeO2→ 7.00E109 85,000
→(a10.5b)Ce2O31aCO10.5bH2O 7.00E109 85,0009 Ce2O311/2O2→2CeO2 1.00E111 90,00010 Ce2O31NO→2CeO211/2N2 2.00E111 90,000
920 Õ Vol. 126, OCTOBER 2004
-75oce-the
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forart
-
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Fig.inger-
eans. 16
presents the computed and measured instantaneous HC emisat converter inlet and exit during the first 600 sec of FTP-7Apparently, the model successfully matches light-off behaviorthe catalyst, as well as subsequent breakthrough during thenificant accelerations of FTP cycle. Again, the role of oxygstorage and release reactions in matching the HC breakthrobehavior is very well assessed by including in the graph, the cputed degree of filling of the total washcoat’s oxygen storagepacity.
The above indicative results can be considered to suppoclear demonstration of the attainable accuracy and predictive aity of this category of models. Of course, these results have bachieved by continuous development and improvements ofspecific model based on its extensive application in standardhaust aftertreatment system design case studies during the layears.
Moreover, the demonstrated model’s capacity to predictcombined effect of the precious metal and Ceria kinetics ontransient converter’s performance, allows us to support the cplex optimization tasks of great interest to the emissions conengineer@45#. For example, a specific catalyst-washcoat desneeds to be tailored to address specific converter’s performarequirements, based on the raw emissions, exhaust temperlevels, and exhaust mass flowrate behavior of each differentof engine-vehicle-exhaust system combination.
Concluding RemarksThis paper aims to contribute towards a more clear definition
the state of the art in engineering design tools in automotive clytic converter systems.
A model developed and continuously improved in the autholab, based on the experience from involvement in engineedesign tasks during the last six years, is described in detail, incurrent status of development, in comparison with other modexisting in the literature, and employed in demonstration cstudies with Pt:Rh and Pd:Rh catalysts.
A kinetic parameter estimation methodology that has beencently developed specifically to support this type of modeling
Fig. 14 Computed and measured cumulative CO, HC, and NO x emissions at converter exit during FTP-75:2.2-l-engined passenger car equipped with a 0.8-l close-coupled converter with 150-g Õft 3 Pd:Rh catalyst
Transactions of the ASME
Journal of Engin
Fig. 15 Computed and measured converter exit temperatures during the first 600 sec ofFTP-75 test cycle: 2.2-l-engined passenger car equipped with a 0.8-l close-coupled converterwith 150-g Õft 3 Pd:Rh catalyst. Converter inlet temperature recording is also shown.
aeri-ablens is
well
briefly presented and demonstrated in the case studies. This modology is based on a combination of a genetic algorithmproach and a customized performance measure. The methodoproves quite successful in finding the parameters for the best fithe model to the tuning data.
eering for Gas Turbines and Power
eth-p-logyt of
The model’s predictions are compared to the respective expmental results, to enable an objective assessment of attainaccuracy. The comparison of measurements and computatiomade in terms of cumulative CO, HC, and NOx emissions at theconverter exit, exhaust temperatures at the converter exit, as
Fig. 16 Computed and measured instantaneous HC emissions at converter exit during the first 600sec of FTP-75 test cycle: 2.2-l-engined passenger car equipped with a 0.8-l close-coupled converterwith 150-g Õft 3 Pd:Rh catalyst
OCTOBER 2004, Vol. 126 Õ 921
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as instantaneous CO, HC, and NOx emissions at the converter’exit, the latter being the most demanding task for a model.
In order to better quantify model accuracy in matching theperimental results, for the purpose of driving the genetic algorittowards better solutions, a performance measure is developeddiscussed in this paper. This performance measure, in additiobeing essential in the parameter estimation methodology,prove useful to assess the success of modeling exercises byand other models of catalytic converters.
The overall demonstration is intended to show that the curstate-of-the-art models of automotive catalytic converters provan indispensable tool assisting the design of catalytic exhaustertreatment systems for ultra low emitting vehicles. This is sceeded due to the high accuracy attained by such models, detheir simplified kinetics scheme and their engineering approwith minimized degrees of freedom.
Nomenclature
aj ,k 5 Stoichiometric coefficient of speciesj in reac-tion k
A 5 Pre-exponential factor of reaction rate expression ~mol K/~m3 s!
c 5 Species concentration~2!cp 5 Specific-heat capacity~J/~kg K!!e 5 Error between computeration and experiment
~2!E 5 i Activation energy of reaction rate
expression~J!ii Conversion efficiency~2!
f 5 Performance function~2!F 5 Performance measure~2!G 5 Inhibition term ~Table 1! ~K!
DH 5 Molar heat of reaction~J/mol!h 5 Convection coefficient~W/~m2 s!!k 5 Thermal conductivity~W/~m K!!
km 5 Mass transfer coefficient~m/s!K 5 Inhibition term ~Table 1! ~2!, 5 Genetic algorithm binary encoding resolution
~2!m 5 Exhaust gas mass flow rate~kg/s!M 5 Molecular mass~kg/mol!
NR 5 Number of reactions~2!NP 5 Number of tunable parameters~2!
Qamb 5 Heat trasferred between converter and ambieair ~J/~m3 s!!
r 5 Rate of reaction~mol/m3 s!Rg 5 Universal gas constant~8.314 J/~mol K!!R 5 Rate of species production/depletion per unit
reactor volume~mol/~m3 s!!S 5 Geometric surface area per unit reactor volum
~m2/m3!t 5 Time ~s!
T 5 Temperature~K!uz 5 Exhaust gas velocity~m/s!z 5 Distance from the monolith inlet~m!
Greek Letters
a 5 Number of carbon atoms in hydrocarbon molecule~2!
b 5 Number of hydrogen atoms in hydrocarbonmolecule~2!
g 5 Catalytic surface area per unit washcoat vol-ume ~m2/m3!
d 5 Washcoat thickness~m!« 5 Emissivity factor~radiation! ~2!q 5 Tunable parameters vectorr 5 Density ~kg/m3!s 5 Stefan–Boltzmann constant~W/~m2 T4!!
922 Õ Vol. 126, OCTOBER 2004
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t 5 Reference time period for parameter estimatio~s!
c 5 Fractional extent of the oxygen storage compnent ~2!
Ccap 5 Washcoat capacity of the oxygen storage component~mol/m3!
Subscripts
amb 5 Ambientg 5 Gasi 5 Parameter indexj 5 Species indexk 5 Reaction index
mon 5 Monolithn 5 Time indexin 5 Inlets 5 ~i! Solid; ~ii ! solid–gas interfacez 5 Axial direction
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