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TECHNICAL PAPERS

# Control Oriented Modeling of a Diesel Active Lean $NOx$ Catalyst Aftertreatment System

[+] Author and Article Information
D. J. Aswani

Ford Motor Company, Ford Research Laboratory, Sustainable Mobility Technologies, MD 1170 SRL, P.O. Box 2053, Dearborn, MI 48121-2053daswani@ford.com

Ford Motor Company, Ford Research Laboratory, Powertrain Control Systems, MD 2036 SRL, P.O. Box 2053, Dearborn, MI 48121-2053mvannie1@ford.com

J. A. Cook

Ford Motor Company, Ford Research Laboratory, Powertrain Control Systems, MD 2036 SRL, P.O. Box 2053, Dearborn, MI 48121-2053jcook2@ford.com

J. W. Grizzle

Control Systems Laboratory, Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109-2122grizzle@eecs.umich.edu

Moles-$C1$ denotes the carbon atom count for hydrocarbons, having an empirical molecular formula $C1H1.8$.

The scaling by residence time was a model adjustment in favor of better fits to data.

In the system of interest, the typical exhaust pressure variation was only a few percent under an FTP 75 drive cycle.

Lean $NOx$ catalysts, by application, are used in lean environments.

Use of the square root was suggested in 12 and was found to give better fits.

Emissions, such as particulates, HC, and $NOx$, account for an insignificant portion of exhaust flow from parts per million to parts per thousand. As a result, chemical reactions involving these emissions have a negligible impact on the exhaust molar flow rate.

By design, the temperature variation in the radial (cross-sectional) direction of a catalyst is significantly less than in the axial direction; indeed, the substrate is rolled in an insulating mat to prevent radial heat loss. Hence, the absence of a radial temperature gradient in the substrate is a commonly used assumption in automotive catalyst modeling.

In reality, the majority of chemical reaction takes place in the early portion of the catalyst. The effect of such an assumption is not analyzed as it requires simulation of a more complicated distributed parameter model and is, hence, outside of the scope of this control-oriented modeling study.

This statement assumes that the thermal conductive effect within an individual exhaust element body and substrate is strong. Though cordierite is a weaker conductor than steel, it is assumed that the dense structure of the cordierite substrate helps keep the thermal resistance relatively low in the axial direction as opposed to there being no substrate. This is an extension of an earlier commonly made assumption that the radial (cross-sectional) temperature variation is small. This extension, however, has not been verified experimentally.

Conduction between exhaust elements is assumed to be smaller than within an exhaust element. This is a good assumption for the LNCs and DPF because they have a sealing gasket between their connections to neighboring pipes. Furthermore the pipes have a much smaller mass per unit length transversely and, hence, different conduction properties. It is important to focus more closely on the larger thermal masses such as the LNCs and DPF, since they have a bigger role in the thermodynamics of the exhaust system being modeled.

Inspection of the thermocouple traces confirms that very little if any of the injected hydrocarbons burns before entering the catalyst.

In reality, exhaust pressure $Pe$ is a function of position $x$ and flow rate $F$. However, the variations in $Pe$ are small enough that assumption of $Pe=130kPa$ (constant) does not affect the model’s prediction capability.

The equations were solved using the symbolic toolbox in MATLAB .

Typical data sets for regression were 900–1500 s long at 1 Hz sampling rate.

The injected hydrocarbons count against fuel consumption of the vehicle. Tailpipe HC emissions are not considered as part of the cost because such emissions are negligible for a lean environment when exhaust gas temperature is above $Tcut$.

Specifically the first 1100 s of the FTP 75 Bags 1 and 2 cycle were used for controller development.

For more details, see 9-10,15, for example.

The structure of the reduced-order model very closely resembled that of the full-order model. However, the number of true states was not identical between both versions because of state coupling via “state aggregation.” The full-order 18-state model had been validated over a considerably larger range of conditions than the reduced-order two-state model. The reduced-order two-state model was manually tuned as a compromise between first 1100 s of an FTP 75 cycle with no HC injection and the first 1100 s of an FTP 75 cycle with the calibrated HC injection.

Numerals appended to super/sub-scripts reflect the catalyst association. This notation will be followed from hereon.

Increasing upper bound of injection milligrams per second does not appreciably alter the results.

The optimal dynamic programing solution is in the form $uμ(x0,k)$ where $x0$ denotes the initial state vector. For all dynamic programing optimization solutions, an initial state vector of $xHCst1=2.1×10−2mol$ and $Tp1=300K$ (ambient) was used to reduce the solution to $uμ(k)$. The storage state corresponds to the final storage resulting from simulation of a FTP 75 cycle Bags 1 and 2 without injection beginning from zero initial storage. Warm or cold start yields similar final storage amount.

The optimization was not performed on the full model.

An average injection rate of $30–40mg∕s$ for the FTP 75 cycle was provided as a target corresponding to less than 1 mpg penalty in fuel economy for the vehicle under study.

J. Dyn. Sys., Meas., Control 127(1), 1-12 (Mar 27, 2004) (12 pages) doi:10.1115/1.1870035 History: Received June 03, 2003; Revised March 27, 2004

## Abstract

The 2004 Federal Tier II and California LEV I emission standards for diesel light trucks mandate tailpipe $NOx$ levels of $0.6g∕mi$. Active lean $NOx$ catalysts (ALNC or LNC) have been proposed as a means to achieve this standard. These catalysts require the delivery of supplemental hydrocarbons in order to reduce $NOx$ in the lean environment typical of diesel exhaust. In the system studied here, these additional hydrocarbons are injected into the exhaust system downstream of the turbocharger. A control-oriented, gray-box mathematical model is developed for diesel active lean $NOx$ catalysts. The model represents the phenomena relevant to $NOx$ reduction and HC consumption, namely, the catalyst chemical reactions, HC storage in the ALNC, and heat transfer behavior on the basis of an individual exhaust element. As an illustration of how the model may be used, dynamic programing is applied to determine the optimal trade-off of $NOx$ conversion efficiency versus quantity of injected hydrocarbons.

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## Figures

Figure 3

General constant-pressure exhaust element differential length cavity model

Figure 12

Feedgas NOx and HC, all in grams per second. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 13

Feedgas and tailpipe NOx in grams per second. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 2

Pipe and catalyst thermodynamic processes relative to cross section

Figure 4

General exhaust element brick/body model

Figure 5

Feedgas levels of NOx and HC, and injected HC, all in grams per second. Test set used for model fitting.

Figure 6

Feedgas NOx, HC, and injected HC, all in grams per second. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 7

Feedgas and tailpipe NOx in grams per second. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 8

Tailpipe NOx in grams per second. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 9

Feedgas and tailpipe HC in grams per second. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 1

Diesel exhaust aftertreatment system consisting of two active lean NOx catalysts, a diesel particulate filter, and a fuel injector. Also shown are thermocouple taps, TC4–TC9, installed for model building data collection, and emissions probes for the feedgas and tailpipe.

Figure 10

Tailpipe HC in grams per second. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 11

Temperature comparisons for three thermocouple locations, in degrees Celsius. Test used for model fitting (FTP 75 Bags 1 and 2 with calibrated injection).

Figure 14

Tailpipe NOx in grams per second. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 15

Feedgas and tailpipe HC in grams per second. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 16

Tailpipe HC in grams per second. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 17

Temperature comparisons for three thermocouple locations, in degrees Celsius. Test used for model validation (FTP 75 Bags 1 and 2 with no injection).

Figure 18

Trade-off curve for bags 1 and 2 of the FTP cycle: NOx conversion efficiency versus average injected HC rate as determined by dynamic programing applied to the simplified model and the complete model

Figure 19

The calibrated injection strategy plotted relative to the trade-off curve for bags 1 and 2 of the FTP cycle, determined by the dynamic programing solution applied to the complete model, shown for 0 to 50mg∕s average injection

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