A simplified thermal model for the three waycatalytic converter

Varun Pandey1 2, Bruno Jeanneret1, Sylvain Gillet1, Alan Keromnes2, and Luis Le Moyne2

1LTE Lab., IFSTTAR, 25 Av. Francois Mitterrand, 69675 Bron, FranceEmail: varun.pandey@ifsttar.fr

2DRIVE EA1859, Univ. Bourgogne Franche Comté F58000, Nevers France

Abstract—A semi empirical model based on thermo-dynamic behaviour of a three way catalytic converterhas been proposed to predict temperature evolution ofthe converter during the cold start. The model is basedon energy and mass balance in the TWC consideredas control volume. Parameters of the heat equationsare identified separately using a step by step approach.Thermocouples have been inserted along the monolithcanals to measure the axial evolution of temperature.Experiments on the engine test bench have been con-ducted to identify the parameters and to validate themodel.

I. IntroductionRoad vehicles with internal combustion engines are a

significant source of air pollution, including carbon monox-ide CO, unburned hydrocarbons HC and nitrogen oxidesNOx. These substances present significant environmentaland health risks, and are therefore regulated.In order to reduce pollutant emissions, most of the

gasoline engine fitted vehicles are equipped with a threeway catalytic converter (TWC), designed to convert thesepollutants to CO2, H2O and N2. However, the chemicalprocess involved to convert these pollutants are stronglydependent on catalyst temperature and equivalence ratio.The conversion efficiency of a hot catalyst can be highafter its light-off, but is poor at low temperature. Theconversion efficiencies of CO, HC and NOx are best onlyin a thin zone around stoichiometry. In the FTP or EUROtest cycles, 70-80% of all harmful substances are emittedduring the cold start phase as presented in [1], [2]. ForNEDC cycle cumulative emissions are presented in figureI.Hence, a model has been developed in order to predict

the evolution of the TWC temperature and its inherentconversion efficiency. This will further be introduced inthe optimal control of a hybrid vehicle in order to find atrade-off between fuel consumption and pollutant emissionduring the cold start.This paper is organized as follows: in section II, com-

prehensive information about the three way catalytic con-verter has been presented. This section is further dividedinto three subsections detailing the three major compo-nents(Oxygen storage model, Catalyst efficiency model

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time, sec

Cumulative Pollutant emission during a cold start

Vehicle Speed

Cumulative CO

Cumulative HC

Cumulative NOx

Fig. 1. Influence of cold start on cumulative Emission during NEDCcycle (Dimensionless values)

and Thermal model) of the model. In section III, theexperimental set-up for identification of parameters andvalidation of model has been illustrated. Methodology forparameter identification has been presented in section IV.In section V, results have been discussed based on themodel validation.

II. TWC modelThe TWC modelling described in several literatures

have either been "detailed physical models", e.g. [3] or"simplified models". The detailed physical models take intoaccount the change in composition of the exhaust gasesin the catalytic environment. Such models are complexand the dynamic effects are not fully realised. Simplifiedmodels used by [4] is divided into first order submodels,including warmup and lightoff characterstics, oxygen stor-age and static efficiency maps.

The heat released during the exothermic conversion re-actions can be estimated by kinetic modelling of chemicalreaction rates as done in [5] requiring extensive experimen-tation and complexity. This also requires measurement ofconcentration of pollutant species at the inlet of converterto derive efficiency maps as done in [6]. This can be simpli-

fied by assuming Wiebe function for conversion efficiencyas shown in section II-B.

According to Otto and LeGray [7], the relevant rateprocesses in a TWC are illustrated in figure 2.

GAS

WASHCOAT

SUBSTRATE

1-BULK FLOW(Gas)

2-INTERPHASE(Gas-Surface) TRANSFER)

3-CHEMICAL REACTION(Surface)

4-HEAT GENERATION(Surface)

5-DIFFUSION THROUGH WASH COAT

(Surface)

6-AXIAL HEAT CONDUCTION(Surface)

7-RADIAL HEAT CONDUCTION(2D)

Fig. 2. Converter schematic presented in Otto and LeGray

As this model has to be introduced in a Hybrid electricvehicle, it has to be necessarily simple. Hence, it is basedon the following three submodels (see figure 3).

Oxygen storage mechanism = f(λFG)Efficiency curves = f(λTP , Tcats)Thermal dynamics = f(TFG, ηi, Xi,FG)

Dynamic

O2 Storage

Model

Static Mapping

Model

Dynamic Thermal

Model

Fig. 3. TWC Model

With λ the relative Air/Fuel ratio, Tcats the bulk tem-perature of the catalyst monolith, Tcatg denotes exhaustgas temperature, subscripts TP denotes Tailpipe and FG

Feedgas, η denotes conversion efficiencies for species Xrepresenting CO, HC and NOx as depicted in figure 3.

A. Oxygen storage ModelThe effect of A/F ratio upon emissions is important

as shown in figure 4, as soon as we diverge fromstoichiometric combustion. Since our focus in this articleis on the thermal behaviour of the catalyst, thereforeoxygen storage is neglected in the present work andcombustion is assumed to be stoichiometric.

50 100 150 200 250 300 350 400 450 500−0.5

0

0.5

1

Time(s)Dim

ensio

nle

ss c

oncentr

ation

50 100 150 200 250 300 350 400 450 500

0.9

1

1.1

Norm

alis

ed A

/F r

atio

CO

HC

NOx

Fig. 4. Effect of equivalence ratio on oxygen storage model

B. Catalyst efficiency model

This model is described by steady state conversionefficiencies curves over a range of temperature and relativeAir/fuel ratios, using Wiebe function, a1, a2, m1 and m2are the fitting parameters of the Wiebe function.

ηi = exp[−a1 · (λTP − λ0

∆λ )m1 − a2 · (Tcats − T0

∆T )m2] (1)

where T0 (resp. λ0) is the ordinate at ηi =8%, ∆T (resp.∆λ) is the difference in Tcat (resp. λTP ) from ηi = 92%to 8%. a1, m1, a2 and m2 are tuning parameters. Typicalvalues for the conversion efficiencies can be found in [8] or[9].

Such equation produces a typical conversion function ofthe shape illustrated in figure 5.

300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1

Temperature(K)

Co

nvers

ion

eff

icie

ncy(%

)

Fig. 5. CO Conversion efficiency

C. Catalyst Thermal model

Gas Equation:In the classical approaches, the thermal behaviour of an

exhaust gas in a TWC is very similar to flow simulated inthe pipe with heat transfer losses to the pipe wall. Thesemodels neglect conduction and storage terms in the gasheat equations as in [10].

Based on the work by [11], energy balance of the gasphase can be written as:

ρg · Cpg · ε ·dTcatgdt

= −mexh

Acs· Cpg ·

dTcatgdz

+h ·Ageo · (Tcats − Tcatg)

+λg · ε ·d2Tcatgdz2 (2)

The nomenclature is detailed in table II.The term on the left hand side is the storage term

which is numerically complicated, as it requires both spaceand time integration and moreover the dynamics of thegases are much faster than the dynamics of catalysertemperature as per [11], therefore we can neglect it.

The first term on the right hand side is mass transferof the gases within the catalyser. The second term is theconvection heat transfer between the monolith and the gas.The third term is the conduction through gas, neglectingthis term is common in heat transfer studies. The bulkgas equation 2 has been simplified as equation 3. Thisyields variation of gas temperature along the length of thecatalytic converter as follows:

mexh

Acs· Cpg ·

dTcatgdz

= h ·Ageo · (Tcats − Tcatg) (3)

The gas temperature has been calculated for several zones(z) and volume of each zone has been introduced. Thispartial differential equation has been discretized using anupwind finite difference yielding to equation 4:

Tcatg,z =mexh

V cs/Nzone · Cpg · Tcatg,z−1 + h ·Ageo · Tcats,zmexh

V cs/Nzone · Cpg + h ·Ageo(4)

Solid equation:Energy balance in the solid can be expanded into three

distinct wall equations for monolith, insulation and theouter wall as done in [10]. This approach requires pseudo2D modelling to account for radial heat transfer. Henceto simplify TWC model, it has been assumed to haveno temperature variation in radial direction and yields toequation 5. Solid phase equation 5 is inspired from [12]:

ρs · (1− ε) · cs ·dTcatsdt

= −h ·Ageo · (Tcats − Tcatg)

+Kreac · Qgen −4

Dcat· hout · (Tcats − Tamb)

+λs · (1− ε) ·d2Tcatsdz2 (5)

The term on the left is the storage term in solid. Thefirst term on the right is the heat transfer between solidand gas due to convection. The second term is the heatgenerated due to exothermic reactions during conversionprocess described in the following sections. The third termin the right is the heat lost by the solid to the surrounding(The temperature of the surface is considered to be sameas the temperature of the solid).

The last term is the heat transfer due to conduction insolid. During the study it has been found to be negligiblysmall and therefore it has been neglected in the model.

The heat transfer due to convection and exothermicreactions are found to be most dominant. Kreac is theweighting factor and Qgen is the heat generated by theconversion of CO and HC species in the washcoat.

Qgen = ηi ·∆Hgen (6)

where ∆Hgen is the enthalpy of formation of species i, asdescribed in reference [13].

Qgen =ma · (1 + λF G

AFRStoechio)

Mexh

·[ηCO · COFG ·∆HCO + ηHC ·HCFG ·∆HHC ] (7)

where ma is the air mass flow rate, AFRStoichio is stoichio-metric Air/Fuel ratio, Mexh is the mean molecular weightof exhaust gas.

Kreac = kreac ∗ (Nzone− izone+ 1) (8)

Nzone is the total number of nodes and izone is thezone under calculation. This formulation has been chosenbecause during ECE15 simulation, temperature in thefront of TWC has been found to have higher variationthan the other two zones as shown in figure 6. Thereforea scaling factor has been introduced in the equation toaccount for this phenomenon. This particular behaviour isalso explained by the residence time in some literature.

0 50 100 150 200 250650

700

750

800

850

900

950

Time sec

Te

mp

era

ture

K

Solid Entrance TWC

Solid Mid TWC

Solid End TWC

0 50 100 150 200 2500

10

20

30

40

50

60

Ve

hic

le s

pe

ed

km

/h

Fig. 6. ECE15 temperature measurements

The heat equation for the catalyst reduces to followingequation

ρs · (1− ε) · csdTcatsdt

= −h ·Ageo · (Tcats − Tcatg)

+Kreac · Qgen −4

Dcat· hout · (Tcats − Tamb)

(9)

For simulation, the above equation has been discretizedin several nodes, each node assumes average solid temper-ature of the equispaced zone.

The number of nodes have been iterated to find a tradeoff between simulation time and model accuracy. A goodaccuracy has been observed for 9 nodes.

III. Experimental setup

The experiment is conducted on the engine test benchequipped with 1.6l gasoline engine from Peugeot (EP6engine). The measured parameters are shown in the figure7.

Inlet manifold

Outlet manifold

Three-Waycatalyst

FeedgasTailpipe

Air path

Fuel path

PEMS

Fig. 7. Measurements on the engine

Three K type thermocouples φ = 0.5mm are insertedalong the canal of the monolith substrate. The threethermocouples are axially equidistant from each other andradially close to the center of the TWC, as shown in thefig 8.

Fig. 8. Thermocouples inside the monolith

Air fuel ratio, before the catalyst has been measured,thanks to Horiba lambda sensor. Emissions have beenmeasured by OBS 2000 type system and it measuresnormalised exhaust composition (THC, CO, NOx, CO2).

IV. Parameter identificationAll the parameters in the heat equations cannot be

obtained from geometrical measurements. The parametershave been identified from the model by minimising theerror between the measured and the simulated tempera-tures from the model. Error has been calculated using thefollowing equation, where z represents the 3 positions ofthe thermocouples inside the monolith:

Error =

√√√√ 1Nsamples

·3∑

z=1

Nsamples∑k=1

(Tcats_sim(k) − Tcats_mes(k))2

(10)

Identified parameters are detailed in the table I:

hout Heat transfer coefficient with ambienth Internal convective heat transfer coefficient

kreac Scaling factor for the heat of reactionTABLE I

Identified parameters-Catalyst

The parameters have been identified separately throughexperiments. Nonlinear optimisation method has beenchosen for the purpose of error minimisation betweenmeasured and simulated TWC temperature thanks tofmincon Matlab function.

Cool down testHsA(convection with ambient)

Warm-up testHinA(Convection between

gas and solid)

Hyzem Road testKreac(scaling factor

for exothermic reaction)

Hyzem Urban test(Model validation)

Fig. 9. Parameter identification algorithm

Three tests are conducted for parameter identification,as described in the figure 9, and depicted hereafter:

1) hout is determined by catalyst cool down test,the engine is stopped until catalyst temperaturereaches ambient temperature. The most dominantheat transfer for this cooling is the natural convec-tion to the atmosphere, assuming no radiation andconduction.

2) h is determined experimentally by catalyst warmup tests under rich operations. In this operatingcondition, oxidation reaction doesn’t occur in thewashcoat material. The comparison between mea-surements and simulation is in figure 10. It showsthe evolution of temperature in the front, middle andend of the TWC, considering 9 nodes for calculation.

3) kreac is determined experimentally by running anHyzem Road cycle in HIL mode on the engine testbench. The comparison between measurements andsimulation is presented in figure 11 and 12 for middlezone of the TWC.

From our experience, initial and boundary conditionshave a huge influence on the identified parameter values.For the present work following conditions have been cho-sen:

Tcatg(t)|z=0 = Texh(t) (11)Tcats(z)|t=0 = Tcats0(z) (12)

V. ResultsThe results of parameter identification and validation

on Hyzem urban cycle are presented below:

0 50 100 150300

400

500

600

700

800

900

1000

Time (sec)

Tem

pera

ture

(K

)

Tcat sim1

Tcat mes1

Tcat sim2

Tcat mes2

Tcat sim3

Tcat mes3

Fig. 10. h identification - Warmup test under rich operation

0 100 200 300 400 500 600 700 800 900650

700

750

800

850

900

950

1000

1050

1100

1150

Time (sec)

Tem

pera

ture

(K

)

Tcat sim2

Tcat mes2

Fig. 11. kreac identification - Hyzem Road Cycle

0 100 200 300 400 500 600 700 800 9000

50

100

150

Veh s

peed (

km

/h)

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1

1.2

Norm

A/F

ratio (

/)

0 100 200 300 400 500 600 700 800 900−0.02

0

0.02

0.04

TH

C (

g/s

ec)

Time (sec)

0 100 200 300 400 500 600 700 800 900−2

0

2

4

CO

(g/s

ec)

Fig. 12. kreac identification - Hyzem Road Cycle

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20

40

60

Veh s

peed (

km

/h)

0 100 200 300 400 500 6000.8

1

1.2

Norm

A/F

ratio (

/)

0 100 200 300 400 500 600−0.1

0

0.1

0.2

TH

C (

g/s

ec)

Time (sec)

0 100 200 300 400 500 600−1

0

1

2

CO

(g/s

ec)

Fig. 13. Validation - Hyzem-Urban Cycle

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700

750

800

850

900

950

1000

1050

Time (sec)

Tem

pera

ture

(K

)

Tcat sim2

Tcat mes2

Fig. 14. Validation - Hyzem-Urban Cycle

Explanation: For a rigorous cycle like Hyzem Urban, thecorrelation is good enough. We can see some differencesarising in the temperature evolution and it could be due

to the fluctuation in air fuel ratio as it can be seen inthe figure 13. Hence the model can predict temperatureevolution of the TWC at stoichiometric condition.

In figure 12 Hyzem road cycle has been presented tounderstand the effect of vehicle driving pattern on theengine operating conditions, which dictates the pollutantemissions from the TWC. Visibly during high accelerationA/F ratio reduces, reducing CO and THC conversionefficiencies. As this phenomenon has not been consideredin the model, we get inferior results at such operatingconditions.

VI. Conclusion

A Three Way Catalyst thermal model has been devel-oped from energy and mass balance equations. A simpleconstrained optimisation algorithm has been used for theparameter identification process. A 1D model is chosenas a good compromise between complexity and accuracy,and 9 zones have been chosen as a good trade off betweencomputational cost and accuracy.

The heat transfers through convection between sub-strate and gases and through oxidation reactions withinthe substrate are found to be most dominant during thestudy.

Temperature variation in the front zone has been ob-served to be very high, this could be due to the largeamount of heat generated during oxidation in this zoneand hence a weighting factor has been introduced in themodel to suit the high transiency of the temperature inthis zone.

One can see that at high power, the ECU deviates fromstoichiometry, and large amount of pollutants are emitted.As shown in figure 3, this can further be introduced in themodel to improve its accuracy. However the present modelis accurate enough and can be introduced into an electrichybrid vehicle model to optimise both consumption andemissions, where catalyst temperature can be a statevariable.

Geometry, material, and thermodynamic parameters ofTWC model are in table II:

mexh exhaust mass flow [kg/s]Cpg specific heat of the exhaust gas [J/(KgK)] 1100h convective heat transfer coefficient [W/m2K]

Ageo specific geometric area of the catalyst [m2/m3] 2250ρs density of the catalyst [Kg/m3] 7850cs heat capacity of the catalyst [J/(KgK)] 500λg Exhaust gas conductivity [W/(mK)]λs Monolith conductivity [W/(mK)]

Tlightoff Light off temperature [K] 450ε TWC open cross section area 0.8Vcs TWC volume [m3] 4.5e− 4

TABLE IINomenclature

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