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: [email protected]
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
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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.
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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.
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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.
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700
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850
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950
Time sec
Te
mp
era
ture
K
Solid Entrance TWC
Solid Mid TWC
Solid End TWC
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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:
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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
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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
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50
100
150
Veh s
peed (
km
/h)
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1
1.2
Norm
A/F
ratio (
/)
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0
0.02
0.04
TH
C (
g/s
ec)
Time (sec)
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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)
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1
1.2
Norm
A/F
ratio (
/)
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0
0.1
0.2
TH
C (
g/s
ec)
Time (sec)
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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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