Selective Catalytic Reduction (SCR) system is used to reduce the oxides of nitrogen NO and NO₂ to N₂ and water (H₂O). The use of SCR system to reduce NO_X emissions has been in place since Engelhard Corporation, in 1957, patented this method in the United States.

SCR technology was implemented on a commercial vehicle to meet the EURO IV regulation. Since then the use of SCR in automotive applications has been increasing. There are other technologies such as EGR (Exhaust Gas Recirculation) and LNT (Lean NO_X trap) which are used to reduce the NO_X emissions.

The SCR system is popular because the engine can be tuned to operate at a very high efficiency which implies higher flame temperatures without compromising the NO_X conversion. There are combinations of EGR+SCR, pure SCR, and pure EGR which have been used in the past. For EURO VI and above emission regulations SCR+EGR and SCR only solution are viable options to reduce NO_X without compromising on the engine efficiency. FPT was the first to introduce an SCR only solution. Since then a number of other manufacturers like Scania have opted for the SCR only option [1-8].

SCR stands for selective catalytic reduction. SCR catalysts remove nitrogen oxides (NO_X) using a reducing agent ammonia (NH₃). An SCR catalyst is used to speed up the reactions and to provide a surface on which the reactions take place. As mentioned earlier the reducing agent is carried on board the vehicle as an aqueous solution containing 32.5% urea. This solution is called AdBlue and is used as the storage and transportation is easier.

The NH₃ storage model consists of modeled reaction rates. These reaction rates are used to calculate the gas phase concentrations of NO_X as well as NH₃. Detailed kinetic models such as the ones used by catalyst manufacturers such as Johnson Matthey and BASF are very detailed. They usually include the reaction rates of all the reactions that occur or certain side reactions that might occur inside an SCR converter.

The chemical kinetics in an SCR system involves the following reactions:

Adsorption and desorption of NH₃ onto the surface of the catalytic converter

This mechanism can be depicted by the following reaction:

Where S in equation denotes a free site on the surface of the catalytic converter and NH₃* denotes an ammonia molecule adsorbed onto the surface of the catalytic converter. A free site on a catalytic surface is an empty void. This is formed due to the adsorbent (the coating of the catalytic converter in this case) not being completely surrounded by the other adsorbent atoms. This empty site is occupied by the ammonia molecules. The forward reaction in the equation denotes adsorption of the ammonia molecule while the reverse denotes desorption of NH₃* molecule.

SCR reactions

There are mainly three important chemical reactions that occur between the ammonia stored in the catalyst and the NO_X entering the catalyst.

The first SCR reaction is called the standard SCR. This reaction is depicted below:

4 NH₃ + 4 NO+O₂ → 4 N₂ + 6 H₂O

In this reaction, gas phase NO_X, Oxygen, and adsorbed Ammonia react with each other to produce water and Nitrogen. This reaction occurs dynamically, and the goal of this thesis is to determine a model that captures this conversion accurately, thereby setting the stage for a control design that allows the determination of an optimal profile of the Ammonia input that allows a maximum NO_X conversion.

Another important SCR reaction is called the Fast SCR. This reaction is depicted below:

4 NH₃ + 2 NO+ 2 NO₂ →4 N₂ + 6 H₂O

The above equation consumes one mole of NH₃ per mole of NO_X and is faster compared to the standard SCR. The Diesel Oxidation Catalyst (DOC) alters the ratio of NO/NO₂. The ratio of NO/NO₂ in the engine out NO_X will thus change after the DOC. This will ensure that a major portion of the NO_X gets converted through the fast SCR reaction. The fast SCR is usually the dominant reaction when the ratio of NO/ NO₂ is close to one.

The last SCR reaction is called the Slow SCR and is shown below:

The slow SCR reaction is usually not relevant as it can be neglected. The reaction rate of the slow SCR becomes significant when the entire NO is consumed inside the catalytic converter.

SCR temperature model

The SCR model also consists of a sub-model that calculates the temperature dynamics of the SCR system. The temperature model is based on simple heat balance. The SCR is assumed to be a perfect heat exchanger. This implies that the temperature exhaust gas leaving the SCR is equal to the temperature of the SCR. The SCR is assumed to be very well insulated therefore convective and radiative losses are ignored.

A physical and chemical phenomenon in the catalyst

When exhaust gas containing NH₃ and NO_X is supplied into the channel, several physical and chemical phenomena occur. The first phenomenon is a diffusion of species of NH₃ and NO_X between flowing gas and stationary gas in the wash coat which is then followed by chemical reactions between molecules on the surface of the catalyst. The third phenomenon is heat transfer between the flowing gas and the wash coat/structure. To model these phenomena, we need at two energy equations for the gas and the wash coat/structure and species equations for each species in the flowing channel, the stationary gas of the wash coat, and on the catalyst. Below are the energy equations to calculate wall temperature and gas temperature (Figure 1).

Figure 1: Top view of each channel of the urea-SCR with a honeycomb structure.

The below equations represent equations of ith species in the flowing gas and the stationary gas in a wash coat, and on the catalyst surface, respectively.

Below are the energy equations of wall and gas, respectively.

(1)

(2)

Below are the NH₃, and NO species equations of flowing gas in a channel.

(3)

(4)

The species equation of adsorbed Ammonia on the surface of the catalyst. θ is the fraction loading of Ammonia onto the catalyst and defined by in which Ω is the number of reaction-sites per volume of wash coat (Figure 2).

(5)

Figure 2: Schematic of a sample-size reactor experiments.

To derive state space form for the whole system, we first discretized equations in the spatial domain, and derived state space form equation for governing equations which are continuous in time for each segment. Then, every state equation for each segment is assembled into one large state space equation (Figure 3).

Figure 3: Inputs, outputs, and state variables of each segment.

Discretization in space

To derive the reactor's state space equation, the reactor is first discretized in the axial direction as in Figure 3 in which inputs, output, and state variables for a segment are shown. Equations (3), (4), and (5) that pertain to spatial derivates are discretized spatially, but governing equations (1) and (2) are still in the continuous form after discretization.

Equations (3), (4), and (5) which are governing equations for CNH₃, CNO, and T_g are discretized in the axial direction (x-direction in the equations). In this procedure, the state variables, and T_wo,i are assumed to be constant in each segment.

Using the first-order implicit Euler method, gas temperature output of i^th segment can be expressed as

(6)

Where, subscription i, is an index for segmentation, and state variable T_w,i, is assumed to be a constant in the i^th segment. Therefore, gas temperature output of ith segment is expressed as follows:

(7)

Similarly, a discretized form of the species equations of gas-phase Ammonia and Nitrogen Monoxide (equations 3 and 4) are expressed as

(8)

(9)

Equations 2 and 5 after discretization, are expressed as

(10)

(11)

Next step is to linearize the five governing equations, two of which are continuous in time domain and three of which are discretized in the space, around an equilibrium point that is determined using nominal input conditions. The equilibrium point for each segment i is determined by C_NH3,i,eq, C_{NO,i,eq, θi,eq} , T_w,i,eq , T_g,i,eq , and u_i,eq .These, in turn, are determined by supplying a constant C_NH3,in , C_NO,in , T_g,in and u_in,eq with the resulting steady-state values of the i^th segment set as the corresponding equilibrium point. Using these equilibrium points, Equations (10) and (11) can be linearized around equilibrium points as follows:

(12)

(13)

where J_kl,i means that partial derivative of a k^'th index variable with respect to i^'th index variable under equilibrium in i^'th segment Variables indexed by k and l include the following: (Table 1).

1: θ

2: T_w

3: C_NH3

4: C_NO

5: T_g

6: u

Before Discretization for the whole reactor	After Discretization for each segment

Table 1:Governing equations before and after discretization.

From Equations (12) and (13), system matrix for the i^'th segment can be expressed as follows:

(14)

This is in turn, (15)

Where,

In order to obtain the output equation for the i'th segment, governing Equations (8), (9), and (7) should be linearized around equilibrium point as follows:

(16)

(17)

(18)

The output equations can be summarized from equations (16), (17), and (18) as follows:

(19)

This is, in turn, expressed compactly as:

y_i = C_ix_i + D_iu (20)

Where,

After we first carried out a discretization and linearization procedure to convert the nonlinear PDEs into the linear system, we assembled state space equations for each segment into a single, large, state space equation. External inputs include C_NH, C_NO,in, T_g.in, and u_in , the inputs to the first segment, while system outputs are the outputs of the N'th segment, and are denoted as C_NH,out, C_NO,out and T_g,out. Gas velocity is assumed to be uniform for the sake of simplicity. It was observed that a choice of N=15 resulted in sufficient accuracy, leading to a 30^th order linear system (Figure 4).

Figure 4: Discretization of a catalyst block, input and output variables, and state variables.

First we placed every state space equation into one state space equations as follows:

(21)

Because every segment is connected by input and output, Equation (21) can be expressed in terms of the system input

And, state variables up to the i^th segment instead of ui. For example, the second segment's state space equation is expressed as follows:

(22)

The output matrix of the second segment can be also expressed by system input and state variables as follows:

Equation (23) summarizes (15) and (20) into one large state space equation that includes 15 state space equations for 15 segments.

(23)

(24)

Where,

Analysis of the developed physics-based model is carried out in this section. It involves computing the state space model for one individual segment of the catalyst block and extending the same to 30^th order linear system for sufficient accuracy.

Reviewing the equations (14) and (15)

System and input matrices A, B can be expressed as

From the equations (19) and (20)

i+1^th values which correspond to C_NH3,i+1, C_NO,i+1 and T_g,i+1 are computed by the following equations:

The state variables T_w,i and θ_i are assumed to be constant in each segment.

Substituting the parameters in the above equations will give

C_NH3,i+1 = 0.907 C_NH3,i + 6.9615 × 10^-15

C_NO,i+1 = 0.9999 C_NO,i

T_g,i+1 = 0.632 T_g,i + 183.2

The concentration of Ammonia and Nitrogen oxide in ppm is 330.

Analyzing the dynamics of a segment

The system matrix A₁ is computed from the Jacobians associated with it (Tables 2 and 3).

Table 2: Jacobians associated with the state space equations with practical parameters.

Table 3: Concentrations NH₃, NO, and O₂ at the input.

The Eigen values of A1 are observed to be -0.00084 and -0.4583.

Both the poles of the system matrix corresponding to A₁ are on the right side of the s-plane and the system is stable.

Analyzing the system -30^th order linear model

The non-linear model is linearized to obtain the 30^th order model computed from the equations 19, 20 with, C_NO,i and T_g,i values regularly updated for each segment starting from input to the output.

The MATLAB code to compute the 30th order system matrix is listed in the Appendix section. The Eigen values for the system are found to be -Eigen values of A_{30 × 30} system matrix- [-0.000697 -0.000702 -0.000709 -0.000715 -0.000722 -0.000730 -0.000734 -0.000745 -0.000756 -0.000770 -0.000783 -0.000797 -0.000812 -0.000829 -0.000848 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583 -0.4583].

It can be observed that the all the 30 poles are real and negative. The poles are placed in the negative half of the s-plane. The system is stable (Figure 5).

Figure 5: The Eigen values plot A_{30 × 30}.

From the Eigen values plot, we can observe that the poles of the system are repetitive and many of them are almost closely placed. This also means that the system order can be reduced preserving the dynamics of the original system.

Reduced order system dynamics

Reducing the main interface to model approximation algorithms.

Reducing the 30^th order linear model into 10^th order using Hankel SV model reduction [27]. Hankel singular values of a stable system indicate the respective state energy of the system. Hence, the reduced order can be directly determined by examining the system Hankel SV's. Model reduction routines, which based on Hankel singular values are grouped by their error bound types. In many cases, the additive error method GRED=reduce (G, ORDER) [in MATLAB] is adequate to provide a good reduced order model which reduces the relative errors between G and GRED tends to produce a better fit.

In this paper, the order of the system is reduced to 10 to analyze its dynamics in that order.

Eigen values: [-0.4211+0.1614i, -0.4211-0.1614i, -0.3583+0.1126i, -0.3583-0.1126i, -0.3278+0.0569i, -0.3278-0.0569i, -0.3186, -0.00052, -0.00058+0.000154i, -0.00058+0.000154i] (Figures 6-10).

Figure 6: The Eigen values plot A_{10 × 10}.

Figure 7: SIMULINK model for the state space equation of order 30 (step change of 10 ppm).

Figure 8: Response of the 30^th order model 10 ppm change in Ammonia.

Figure 9: The SIMULINK model for the reduced state space equation (step change of 10 ppm).

Figure 10: Response of the 10^th order model 10 ppm change in Ammonia.

Output comparison of the 30th order and the reduced 10th order models: The outputs of both the 30th order and 15th order systems are analyzed for a step change of 10 ppm in the concentration of Ammonia while all the other parameters remain same.

From both the above plots (in Figures 7 and 9) it is reasonably inferred that the response of the reduced order model is similar to the original 30th order model. The steady state output values of NH₃ in both responses are nearly the same. For a sufficiently higher model, the dynamics of the reduced order model is expected to be the same as the higher order model with minimum errors.

Nonlinear models are derived based on physical and chemical interpretation of the catalyst and certain simplifications. We subsequently discretized and linearized the nonlinear equations and analyzed the system dynamics of the catalyst. The following are some of our main observations regarding the SCR dynamic model:

• The steady state output of the first principle-based model given sufficiently accurate results for a higher order linear system. Order of 30 is chosen for satisfactory accuracy in results

• The dynamics of the reduced order system is the same as the higher order linear model. Since many poles are repetitive, the system can be reduced to a degree which gives an almost the same response as the higher order model also keeping in view of the errors involved in the reduction method. Dynamics of settling time and trend are seen to be the same for both the models

• Although the model of the SCR developed during this thesis was for an extruded Vanadia formulation, the SCR model can be easily applied to other catalyst formulations e.g. Copper-Zeolite. This would, however, require that tests be conducted on the chosen formulations to identify the significant chemical kinetics. The model offers flexibility to model additional reactions or remove unnecessary reaction kinetics.

I wholeheartedly thank Dr. M Umapathy, Professor, Instrumentation and Control Engineering, NIT Trichy., for his constant support and guidance, and without whom the completion of this project in fruition would not have been possible.