Model III: Redox Reactions on the Electrolyte
TWC 1 TWC 2
5.2 Control-Oriented Modelling of the Exhaust Gas Aftertreatment System
5.2.1 Control-Oriented Raw Exhaust Gas Model
In this module, the raw exhaust gas concentrations of CO, H2, and O2are cal- culated from the output of the wide-rangeλsensor, which is located upstream of the TWC. Hence, the module actually contains an “inverted” model of the wide-rangeλsensor.
Basically, the concentrations of the mentioned gas components could be ob- tained from lookup tables, as in the process model. However, this is not a very good approach for two reasons. Firstly, the use of lookup tables makes mea- surements with exhaust gas analysers necessary. Secondly, these measurements must be very accurate and consistent in terms of the air-to-fuel ratio. Naturally, the model is very sensitive to errors and inconsistencies of the inlet concentra- tions. This may lead to considerable offsets of the estimation of the storage capacity. Notice that these errors cannot be corrected by the downstream sen- sor, whereas a simple offset of the upstream sensor can be corrected because of the accurate and stable position of the downstream sensor’s switch.
A much better approach is the use of an exhaust gas model. It has been found that a very simple model suffices and presumably is generally valid for all modern port-injected spark-ignited gasoline engines.
In Figure3.6, profiles of measured raw exhaust gas components dependent onλhave been shown. The only component which significantly changes with the operating point is NO. The model only needs the sum of NO and O2, hence this dependency does not have to be accounted for. Apart from the NO, all other concentrations vary little with the operating point.
Generally, it can be observed that the concentrations of reducing and oxi- dising species decrease with higher temperatures because of a more efficient combustion. This should be accounted for in the model, because it directly af- fects the heat production in the TWC. The heat production is proportional to the amount of reactants, i. e., oxidising and reducing species. Metaphorically speaking, the less efficient the combustion was in the cylinder, the more has to be burned in the TWC, which causes additional heat.
The second important point is the ratio between H2and CO in the raw ex- haust. It can be observed in Figure3.6that the two concentration profiles are roughly parallel, from which it can be concluded that their ratio remains ap- proximately constant. This is hard to see from the measurements, because the accuracy of the raw exhaust measurements is not very good, especially, when the concentrations are very small. Fortunately, the TWC model is not very sen- sitive to the ratio between H2 and CO in the raw exhaust gas. Therefore, a constant ratio has been assumed in the model.
The calculation of the exhaust gas concentrations is simple and straightfor- ward. It is conducted in two steps. In a first step, only the excess rich or lean concentrations, dependent on whetherλis smaller or greater than 1, are calcu- lated. In a second step, a model for the unburned components is developed.
If the engine is operated lean, and if the combustion is perfect, only oxidising species remain in the exhaust. All these species are collected in the O2here.
Hence, the O2concentration can be calculated directly fromλ. If the gasoline is assumed to be CαHβand the ratio between O2and N2of the air is 21/79, the combustion reaction can be expressed as follows:
1 µ
α+β 4
¶ λ
CαHβ+ O2+79 21N2→
νCO2CO2+νH2OH2O +νO2O2+νN2N2 (5.1) This equation can be easily extended, if humid air and a fraction of O in the gasoline are assumed. This has been done in AppendixA.1for the evaluation of the exhaust gas measurements. Here, both have been neglected. From the combustion reaction, the mole fraction of O2can be derived directly:
yO2 = νO2
X
i=CO2,H2O,O2,N2
νi
=
1−1 λ α+β/2 (α+β/4)λ+
µ 1−1
λ
¶ +79
21
(5.2)
Likewise, this can be done on the rich side, i. e., whereλ <1. Here, only CO and H2are present in the model. Their ratioφH2/COis assumed to be constant.
Hence, both species can be determined fromλ. The combustion reaction is now
1 µ
α+β 4
¶ λ
CαHβ+ O2+79 21N2→
νCO2CO2+νH2OH2O +νCOCO +νH2H2+νN2N2 (5.3) Again, the mole fractions of CO and H2 can be obtained directly from the combustion reaction usingφH2/CO:
yCO =
2 1 +φH2/CO
µ1 λ−1
¶
α+β/2 (α+β/4)λ+79
21
yH2 =
2φH2/CO
1 +φH2/CO
µ1 λ−1
¶
α+β/2 (α+β/4)λ+79
21
(5.4)
With equations (5.2) and (5.4), the concentrations of excess O2(λ >1), or CO and H2(λ <1) can be obtained fromλ. To these concentrations the species are added, which occur because of incomplete combustion. To preserve the air-to- fuel ratio, the amount of added O2has to be stoichiometrically balanced to the amount of added CO and H2. For the actual concentration profile, a heuristic function has been found, which matches the measured profiles well. Tests have shown that the accuracy of the concentrations is not critical. However, it is very important that the final concentration profiles are consistent in terms of the air- to-fuel ratioλ. The heuristic function of the additional concentration profile can be expressed as follows:
∆yO2 = 0.5(∆yCO+ ∆yH2) = ηcomb
1 + 10|λ−1| (5.5) ηcombcan be interpreted as an inverse combustion efficiency. If the combustion is perfect, the factor becomes zero. The factor is dependent on the temper- ature and has a strong impact on the energy balance of the TWC. It generally decreases with increasing temperatures of the raw exhaust. A linear approxima- tion of this dependency is sufficient. Thus,ηcombcan be expressed as follows:
ηcomb=aηcomb+bηcomb·Texh (5.6)
The two coefficientsaηcombandbηcomb can be obtained from steady-state tem- perature and mass flow measurements using the energy balance equations. Al- ternatively, their values can be derived from concentration measurements of the raw exhaust gas.
Figure5.2shows the comparison between the measured and the calculated mole fractions of O2, CO, and H2. Notice that the measured O2concentration includes NO, and the CO includes HC. A ratio between H and C of two has been assumed in the HC. Hence, the measured mole fractions of O2and CO were obtained as follows:
yO2 = yOmeas2 +1
2ymeasNO (5.7)
yCO = yCOmeas+ 3yHCmeas (5.8)
The parameters for the calculation of ηcomb have been obtained from the exhaust gas measurements. They are given in AppendixD. The agreement is sufficiently good. The deviations mainly occur from the inclusion of HC in the CO concentration. This disturbs the ratio between CO and H2, especially, where the concentrations are small, i. e., on the lean side. Theoretically, the HC could also be divided into a fraction added to the H2 and the rest added to the CO. However, in the TWC model, catalyst deactivation is caused by the adsorption of CO, unlike in the process model, where the adsorption of HC deactivates the TWC. Therefore, it has been decided to fully include the HC in the CO concentration. Since the deviation mainly occurs on the lean side, only a limited impact on the performance of the control-oriented model has to be expected.