2.8.1 Introduction
Since 1980, the permitted levels for the concentrations of carbon monoxide CO, hydrocarbons HC, and NOx (NO, NO2) have been reduced significantly.
A continuation of this trend is to be expected. Table 2.5 shows the devel- opment of these limits in Europe and the similar standards for California.
Similar legislations are imposed in Japan and other countries. In contrast to the European regulations, which limit the total hydrocarbon emissions, the US regulations consider the non-methane hydrocarbons only. This choice reflects the fact that the methane emissions by passenger cars are very low compared to the agricultural emissions. However, the US standards regulate formaldehyde (HCHO) levels, which are not explicitly limited in the European Union.
Common to all of these emission limits is the fact that they may not be exceeded during a predefined driving cycle, comprising a warm-up phase, tran- sients and idle periods. Also, the boundary conditions (ambient temperature and pressure, etc.) are prescribed accurately.
As discussed in the previous section, the oncoming, ever more stringent emission regulations are difficult to meet. Only homogeneous charge SI en- gines operated with a stoichiometric air-to-fuel ratioλ can attain easily the required emission levels by using three-way catalytic converters. All other en- gines (Diesel, direct-injection SI engines, etc.) must be combined with more complex exhaust gas aftertreatment systems such as leanN Oxtraps, selective catalytic reduction systems, and filters for particulates.
In this section, three-way catalytic converters and selective catalytic reduc- tion converters are analyzed from a control engineering point of view. Readers interested in lean-N Oxtraps are referred to [126], [210], [111] [199], and [115]
and details on particulate filters may be found in [121], [122] and [116].
2.8.2 Three-Way Catalytic Converters, Basic Principles
The most common pollution abatement system for SI port-injection engines is the three-way catalytic converter (TWC). It derives its name from its ability to simultaneously reduce NOx and oxidize CO and HC. State-of-the-art systems are capable of removing more than 98% of the pollutants. This can only be achieved by operating the engine within very narrow air-to-fuel ratio limits, as will be shown in the following subsections.
An ideal TWC enhances the following three elementary chemical reactions
NO + CO→ 12N2+ CO2 (2.190)
CO + 12O2→CO2 (2.191)
HaCb+ (b+a
4) O2→bCO2+a
2H2O (2.192)
In reality, the system of reactions is far more complex [106]. Several dozen reactions are involved, and other species, apart from those mentioned, are produced (mostly during intermediate steps). For example, the concentration of N2O is often increased by the reactions taking place inside the TWC.
Generally, the reactants are first adsorbed on the catalytic surface. Cat- alytic substances, such as platinum (Pt), rhodium (Rh) or palladium (Pd), weaken the bonds of the adsorbed species and thereby allow for the formation
Table 2.5.European (top) and Californian (bottom) emission standards.
Tier Euro I Euro II Euro III Euro IV Euro V Euro VI
Date 1992 1996 2000 2005 2009 2014
SI Engines
CO [g/km] 2.72 2.2 2.3 1.0 1.0 1.0
HC [g/km] 0.2 0.1 0.1 0.1
N Ox [g/km] 0.15 0.08 0.06 0.06
HC+N Ox [g/km] 0.97 0.5
P M [g/km] 0.005(i) 0.005(i)
Diesel Engines
CO [g/km] 3.16 1.0 0.64 0.5 0.5 0.5
HC [g/km]
N Ox [g/km] 0.5 0.25 0.18 0.08
HC+N Ox [g/km] 1.13 0.7 0.56 0.3 0.23 0.17
P M [g/km] 0.18 0.08 0.05 0.025 0.005 0.005
Heavy-Duty Diesel Engines
CO [g/kWh] 4.5 4.0 2.1 1.5 1.5 1.5
HC [g/kWh] 1.1 1.1 0.66 0.46 0.46 0.13
N Ox [g/kWh] 8.0 7.0 5.0 3.5 2.0 0.4
P M [g/kWh] 0.36 0.15 0.1 0.02 0.02 0.01
Smoke [m−1] 0.8 0.5 0.5
Emission classes(ii) LEV ULEV SULEV ZEV Passenger Cars
CO [g/mile] 4.2 2.1 1.0 0
N M OG [g/mile] 0.09 0.055 0.01 0
HCOH [g/mile] 0.018 0.011 0.004 0
N Ox [g/mile] 0.07 0.07 0.02 0
P M [g/mile] 0.01 0.01 0.01 0
(i) applicable only to vehicles using direct injection
(ii) LEV II introduced since 2004
of the desired products, according to the law of minimizing the (chemical) potential energy. The products are then desorbed and released to the gas phase.
Reactions (2.190) to (2.192) show that optimal conversion rates can only be achieved if exactly as much oxygen is available as is required for the oxi- dation of CO and HC. It would be favorable if (2.190) was dominant. In fact, in the presence of excess oxygen (λ > 1), reaction (2.191) always occurs in parallel to the others and therefore, under lean conditions, usually inhibits the reduction of NO. During a shortage of oxygen (λ <1), all NO is converted,
but the removal of HC and CO is incomplete. This implies that only opera- tion in a narrow band aroundλ = 1 yields a satisfactory performance (i.e., long-term efficiency) of the TWC. Figure 2.64 illustrates this by showing the gas concentrations of various species in the feedgas and exit of a TWC as a function of λ. (The upstream gas concentrations in Fig. 2.64 correspond to those shown in Fig. 2.53.)
Fig. 2.64. Concentrations of NO, HC, CO, CO2, O2, H2, and the switch-typeλ- sensor voltage; upstream (dashed) and downstream (solid) of an aged three-way catalytic converter.
Especially during transients, when the wall-wetting dynamics and other disturbances render a complete compensation of the varying air mass flow difficult, keepingλwithin this band is not practicable. Therefore, the catalyst must be able to cope with temporary excursions to the lean or to the rich side.
This is achieved by incorporating a reservoir for oxygen, which lies beneath the catalytically active surface and which consists of special materials, e.g., cerium. Excess oxygen can then temporarily be stored in these materials as well as on the catalytic surface. The oxygen storage within cerium can be described by the following chemical reaction
Ce2O3+ O∗⇋2 CeO2 (2.193)
The adsorbed oxygen O∗ combines with Ce2O3 to form a new grid struc- ture. Note that the adsorbed oxygen may also stem from oxygen-containing species other than O2, such as NO, H2O, or CO2. The influence of H2O and CO2 acting as oxidizing species on the dynamic behavior of the oxygen stor- age mechanism is discussed in Section 2.8.3. Of course, the catalyst can only compensate for lean excursions if its oxygen storage is not full already, i.e., if not all ceria has been oxidized to 2 CeO2 yet. The same holds for tempo- rary deviations towards the rich region in the case of a completely empty (or more precisely, “reduced”) oxygen storage. To keep the catalyst in a state where both lean and rich excursions can be buffered a feedback controller is needed. This is where the dynamics of the oxygen storage become relevant.
While a high storage capacity is favorable for increased conversion rates, the opposite is true considering the robustness of the catalyst controller due to the increased lag of the system.
Significant aging phenomena can be observed during the lifetime of a cat- alyst, caused mainly by sintering of the noble metal due to high temperatures and so-called catalyst poisoning.48
In addition to the oxidation of CO and HC and the reduction of NO, three-way catalysts promote the water-gas shift reaction:
CO + H2O⇋CO2+ H2 (2.194)
Given the conditions present in a TWC, the chemical equilibrium lies on the right-hand side, which is favorable for the reduction of CO. However, since λsensors exhibit a significant cross-sensitivity to hydrogen, this effect has to be taken into account when interpreting the downstreamλsignal. In fact, for a rich mixture, the air-to-fuel ratio sensor may be used as a hydrogen sensor.
Further details on this topic may be found in [17]. Since the catalytically active surface decreases with aging, new three-way catalysts promote the water-gas shift reaction significantly more than aged ones. This results in a varying H2/CO ratio during the lifetime of a TWC.
2.8.3 Modeling Three-Way Catalytic Converters
Models of TWCs usually serve to predict the conversion rates of the main species and/or the composition of the output gas. They are used to gain a better understanding of the transient behavior of TWCs and to investigate external influences such as “catalyst poisoning” or aging. Further applications are the optimization of the catalyst system and the development of improved controllers.
48The expression “catalyst poisoning” is used to describe the reduction of catalytic efficiency by unwanted species occupying active surface sites and thereby pre- venting the desired species from adsorbing on these sites. Poisoning can occur irreversibly with substances such as lead (Pb), but also reversibly, for example, with sulfur dioxide (SO2).
Fig. 2.65.Schematic view of a TWC’s structure, reprinted with the permission of Umicore AG.
Predicting the conversion rates is very important for the on-board diag- nosis (OBD) of the TWC. An OBD system must be able to warn the driver if the catalytic converter is no longer able to sufficiently remove the pollu- tants in the exhaust gas. In order to reach the ULEV or Euro IV standards, conversion rates of more than 96% are necessary. Accordingly, the results of the OBD algorithms must be very accurate, which is a challenging objective when considering the few sensor signals available in series production cars.
Accurate catalyst models, which do not impose a large computational burden on the ECU, can help achieve these objectives.
Another very important characteristic of a TWC is its warm-up behavior.
A cold TWC is almost inactive. TWCs start working efficiently only after
having reached the light-off temperature.49The catalyst is heated by the hot exhaust gases and the reaction enthalpy produced by the exothermic chemical reactions in the catalytic converter itself. When the light-off temperature is reached while the TWC is still sufficiently active, the chemical reactions pro- vide enough heat to further warm the TWC, which in turn boosts the catalytic activity [87]. This chain reaction leads to a significant temperature increase af- ter light-off. In order to efficiently reduce overall emission levels, the warm-up time must be minimized. Legislation will become more stringent, since cold- start temperatures will be lowered from +20◦C to −7◦C. Models of TWCs therefore also have to contain an accurate thermal model.
Detailed models of TWCs are complex. State-of-the-art models incorporate up to 40 reactions with approximately 10 gas species and between 10 and 15 adsorbed species [106, 39, 99, 33]. These models are very stiff, since the time constants of most of the chemical reactions are much smaller than those of the heat and mass flux phenomena.
Control-oriented TWC models usually are much simpler. Often, the dy- namics of the temperature are neglected, and most of the reactions are lumped into the two or three most relevant ones. These simplifications make the mod- els applicable for the use in currently available control systems. Deciding which are the relevant processes and dynamics and which may be omitted is a deli- cate task and is discussed, for instance, in [176, 120] and [18].
When developing a control-oriented TWC model, the main objective is to predict the oxygen storage level because it cannot be measured directly.
Thus, it has to be estimated by an observer, usually supported by aλsensor downstream of the TWC [27]. The goal of all TWC control systems is to keep the TWC in a state from which excursions to the lean and to the rich side can be handled equally well. It is generally believed that this is achieved by keeping the oxygen storage level at half the available capacity. However, this is a rather simplified modeling approach since in reality, not only oxygen, but also a number of other components are stored.
For these reasons, feedback control and OBD systems necessitate more so- phisticated (and thus complex) models. In fact, a complete feedback control system for the oxygen level would be realizable only if a sensor directly mea- suring this variable was available. Unfortunately, the only sensor currently available is a downstreamλsensor (usually switch-type). This sensor detects and measures the remaining oxygen concentration in the exhaust gas if there is any. Moreover, when the engine is operated under rich conditions, this sen- sor is very sensitive to a number of other species, mainly hydrogen, but also CO and HC [28], [15]. Therefore, in order to allow a correct interpretation of the output of this sensor, these disturbances of the sensor output must be modeled. Additionally, theH2/COratio correlates with the cerium/noble
49The light-off temperature of a TWC is defined as the temperature at which a conversion efficiency of 50% is attained for HC.
metal surface and therefore changes with the aging of the TWC. This has to be taken into account as well.
Physical Reactor Model
In the following, the modeling of a TWC is described, following a so-called one-dimensional one-channel modeling approach. The TWC consists of a car- rier (usually ceramic or metallic substrate) and coating material. First, the washcoat (aluminum oxide) is applied to enlarge the surface of the TWC, followed by the additional oxygen storage material (cerium oxide). Finally, stabilizers and noble metals are placed on top. The better the distribution of the various components, the higher the activity of the TWC.
A catalytic converter consists of many channels that are much longer than their diameter (see Fig. 2.65). Therefore, a lumped-parameter modeling ap- proach is not possible. Instead, a formulation using partial differential equa- tions must be used to describe the physical phenomena within the channels.
Fortunately, the gas in the channels may be assumed to be perfectly mixed in the directionperpendicularto the flow such that only partial derivatives in the axial direction need to be considered. Often a uniform flow and temper- ature distribution over all channels is assumed, which reduces the model to the calculation of one channel.
The effects encountered at the channel wall are illustrated in Fig. 2.67.
Using this approach, the TWC model consists of three elements:
• Gas phase: Only convective mass and heat transfer in the axial direction are considered.
• Pores/Washcoat: No convection takes place in the direction of the gas flow, but mass transport to and from the gas phase and to and from the solid phase by desorption and adsorption are modeled.
• Solid phase: Heat transfer occurs through heat conduction in the solid phase, heat exchange with the gas phase, and heat production by the chemical reactions between the adsorbed species.
Mass balances for each species can be formulated for the gas phase and the pores while heat balances are formulated for the gas phase and the solid phase.
The concentrations of the components that react in the catalytic converter are much smaller than those of nitrogen, carbon dioxide, and water in the exhaust gas. Therefore, a constant mass flow ˙mthrough the catalytic converter may be assumed. If no additional mass flow into or out of the control volume occurs, the following mass balance for species i may be derived from the schematic diagram shown in Fig. 2.66
ε·A·∂x·ρg
| {z }
mass in control vol.
·∂ξi,g
∂t = ˙m·(ξi,g(x, t)−ξi,g(x+∂x, t)) (2.195)
where ε denotes the volume fraction of the catalyst filled with exhaust gas (which has densityρg),Ais the cross-section area of the individual channels, andξi,g the mass fraction of speciesi in the exhaust gas.
Fig. 2.66.Mass flow through an infinitesimal volume in the channel of a catalytic converter.
Taking the limit case of an infinitesimally small cell size (∂x→0), (2.195) yields a partial differential equation (PDE) of the form
ε·A·ρg·∂ξi,g
∂t =−m˙ ·∂ξi,g
∂x (2.196)
for each species i. All mass conservation equations are deduced from this PDE, and the energy conservation equations are derived in a similar way.
Its numerical solution is well documented, see e.g. [153]. However, to obtain state space models suitable for control-system design, spatial discretization is necessary (to obtain a finite number of states).
Mass Balances
Figure 2.67 shows the mass and energy flows in the catalytic converter. Flow
“1” is the exhaust-gas mass-flow through the channel. Its dynamic is described by combining the basic mass conservation (2.196) with the additional mass flow between the pores of the washcoat and the gas phase as indicated by flow
“2”, and, it defines the concentration of speciesi in the gas phase:
ε·A·ρg·∂ξi,g
∂t =−m˙ · ∂ξi,g
| {z ∂x}
cf. (2.196), flow “1”
−ki,g·A·aV ·ρg·(ξi,g−ξi,pores)
| {z }
to/from pores, flow “2”
(2.197)
where ki,g is the mass transfer coefficient between gas phase and the pores of the washcoat, aV is the corresponding relative (scaled by the catalytic- converter volume) cross-sectional area, and ξi,pores is the concentration of speciesiin the pores. Note that the diffusion in the axial direction is beeing
neglected and the partial pressures of the components are assumed to be constant.50
Fig. 2.67.Mass flows and heat transfers in the catalytic-converter model.
A second mass balance is formulated for speciesiin the washcoat pores:
4dbεWdWρg
∂ξi,pores
∂t =ki,gAaVρg·(ξi,g−ξi,pores) (2.198)
−acatARiMi
where db is the length of one side of the quadratic monolith channel, εW
denotes the washcoat porosity anddW its thickness, andacat is the relative catalytically active area per catalyst volume. The free volume (i.e., the volume that is accessible by the gas) per length of the washcoat is given by 4db·εW·dW. Of course, for non-quadratic monolith channels the terms have to be corrected accordingly.
The mass production term Ri expresses the transfer rate of speciesi per unit area between the solid and the gas phase in [mol m−2s−1]. It defines the total adsorption/desorption rate and is calculated from all elementary reactions j in which species i participates: The individual reaction rates rj
times the stoichiometric factorsνi,j are summed up
50The Stefan-Maxwell equation would be a more accurate choice. But sincepN2 is dominant because of the high concentration of nitrogen in the gas, the inaccuracy may be compensated quite well by slightly adjustingpN2.
Ri=X
j
(rj·νi,j)·Lt (2.199) whereLt denotes the maximum capacity of adsorbed species in [mol/ m2].
Energy Balances
The energy balance for the gas phase includes the convective heat flow in the exhaust-gas mass flow and the heat exchange with the solid catalyst bed, as shown in Fig. 2.67. The catalyst is assumed to be adiabatic, i.e., any heat ex- change with the environment is neglected, yielding the following heat balance for the gas phase
ε·ρg·cv,g·∂ϑg
∂t =−m˙
A ·cp,g·∂ϑg
∂x −α·aV ·(ϑg−ϑcat) (2.200) where cp,g and cv,g are the isobaric and isochoric specific heats of the gas, ϑg and ϑcat the gas and catalyst temperatures, and α is the heat-transfer coefficient between the solid and the gas phase.
The energy balance for the solid phase also includes the heat exchange with the gas phase (with a change of sign) as well as the heat conduction in the solid phase itself. As third term, namely the heat produced by the exothermic chemical reactions on the catalyst surface, has to be included, leading to
(1−ε)·ρcat·ccat· ∂ϑcat
∂t =λcat·(1−ε)·∂2ϑcat
∂x2 +α·aV ·(ϑg−ϑcat) +acat
X
k
(−∆Hr)k·rk·Lt (2.201) Here,ρcat andccat are the density and the specific heat capacity of the solid phase, respectively, and λcat denotes the heat conduction coefficient. The terms (−∆Hr)k and rk are the reaction enthalpy and the reaction rate of reactionk.
Usually, the following simplifications do not introduce any significant errors and thus are allowable in the case of a control-oriented TWCmodel:
• Axial diffusion and heat conduction in the gas can be neglected; convection dominates this direction.
• The dynamics of the gas concentrations in the channel can be omitted and thus the gas concentrations are calculated quasi-statically. Since the gas turnover rate of the catalyst is high for realistic system configurations, changes in the gas concentrations are substantially faster than the oxygen storage dynamics (even during idling ˙Vexh/Vcat≈7.51s).
• Due to the high mass-transfer rates from the channel to the washcoat and the substantially smaller washcoat volume (compared to the chan- nel), the dynamics of the washcoat concentrations can be considered to be much faster than the relevant dynamics. Gas-concentration dynamics in the washcoat are therefore neglected as well.