Optimization of the Uniformity Index Performance in the Selective Catalytic Reduction System Using a Metamodel

Abstract

The significance of the selective catalytic reduction system in vehicles increases in line with the high standards of emission control and enhanced selective catalytic reduction efficiency. This study aims to improve the performance of the selective catalytic reduction system through an optimization method using a metamodel. The objective function is defined as the ammonia uniformity index, and the design parameters are defined in relation to the pipe length and mixer related to the chemical reaction of the urea solution. The range of design parameters has been designated by a trial-and-error method in order to maintain the overall design drawings of the selective catalytic reduction system and prevent modeling errors. Three algorithms, namely, ensemble decision tree, Kriging, and radial basis function, are employed to develop the metamodel. The accuracy of the metamodel is verified based on three indicators: the normalized root mean square error, root mean square error, and maximum absolute error. The metamodel is generated using the Kriging model, which has the highest accuracy among the algorithms, and optimization is also performed. The predicted optimization results are confirmed by computational fluid dynamics numerical analysis with a 99.83% match. The ammonia uniformity index is improved by 1.38% compared to the base model, and it can be said that the NOx purification efficiency is improved by 30.95%. Consequently, optimizing the uniformity index performance through structural optimization is of utmost importance. Furthermore, this study reveals that the design variables related to the mixer play a crucial role in the performance. Therefore, using the metamodel to optimize the selectively catalytic reduction system’s structure should be considered significant. Finally, in the future, the analysis model can be validated using test equipment based on the findings of this study.

1. Introduction

Environmental problems are global issues, resulting in the tightening of the emission standards of internal combustion engines. In particular, diesel vehicles have the advantages of better fuel economy and higher engine power and torque than gasoline vehicles of the same engine displacement, but they emit major pollutants such as hydrocarbons (HCs), nitrogen oxide (NOx), carbon monoxide (CO), and particulate matter (PM). These pollutants are strictly regulated worldwide [1]. To satisfy these strict regulations, research is being conducted to improve the performance of after-treatment devices/systems such as diesel particulate filters (DPFs), selective catalytic reduction (SCR), and exhaust recirculation (EGR) [2]. Diesel vehicles use selective catalytic reduction (SCR), which is regarded as one of the best technologies for meeting strict NOx reduction requirements [3]. The uniformity index (UI) of ammonia is suitable for evaluating the SCR performance in a system because of its contribution to the NOx conversion efficiency and increase in the catalyst life [4]. Many factors should be considered before designing the SCR system [5,6]. In the SCR system, either the low density of NH₃ in the exhaust gas degrades the conversion efficiency of NOx or the catalyst is damaged by the excessive density of NH₃ [7]. Furthermore, there are many studies to improve UI performance. When the injection angle of ammonia is arranged in the direction of the exhaust gas flow, it improves the UI performance by 16% from the initial model [8]. The mixer equipped in the exhaust pipe causes a vortex that not only performs effective mixing of the exhaust gas and ammonia but also delays wall wetting, making a vigorous hydrolysis and thermolysis environment. As a result, it attains a 17.95% improvement in UI performance from the initial model [9]. Ye et al. [10] developed a three-dimensional simulation model to investigate the NH₃ uniformity and conversion rate produced by the urea–water solution spray system. The results showed that the impact of the mixer is significant, and the use of a dual-mixer results in a remarkable increase in the urea conversion rate and NH₃ uniformity index by 169.5% and 136.4%, respectively. The techniques mentioned above are optimized using methods of trial-and-error and the design of experiment (DOE) [11,12]. These methods can help in easily planning experiments; however, it is difficult to represent the performance across the entire range of the design. It is also difficult to analyze the interrelationship between different design parameters. Therefore, there is a need for a systematic optimization method, and recent studies related to optimization using a metamodel can be considered.

The metamodel consists of algorithms that are ensemble decision tree (EDT), Kriging, and radial basis function (RBF). Identifying the characteristics of the data is extremely complex, and it is difficult to consider the most suitable optimization algorithm. It is important to select appropriate algorithms according to the characteristics of the data such as nonlinearity [13,14]. Hoang et al. [12] studied seismic fragility analysis using the Kriging metamodel of concrete highway bridges. The generated metamodel was evaluated for accuracy using a cross-validation method. Three major predictive indicators of errors such as root mean square error, R², and relative maximum absolute error (RMAE) have been used in this study. Several researchers have described that the time required for the numerical analysis may be reduced by the application of sequential sampling, wherein the number of samples is minimized [15,16]. Woo et al. [17] performed a contribution analysis to identify the design parameters that are effective in improving performance. The concept of a high-performance car chassis is developed using virtual prototyping and optimized to ensure consistency in the performance predictions. Recently, the optimization of the shape of a permanent magnet synchronous motor (PMSM) was analyzed using a metamodel with Kriging and MLP algorithms. The results showed an improvement not only in torque performance enhancement from the Kriging model by 1.3% and the MLP model by 2.2% but also in the back electromotive force (EMF) by 4.3% [18]. Using the OASIS (Optimization Assisted Simulation Integration Software V1.3) optimization tool, the power of impulse turbines was enhanced by 5.33%, and the analysis cost was reduced [19]. In another work, metamodels were generated by various algorithms, and the best predictive performance results were proposed using Process Integration Automation and Optimization (PIAnO 2023) software (PIDOTEC Inc., Seoul, Republic of Korea) by PIDOTEC Inc. [20]. Different optimization cases have already been studied using PIAnO 2023 software. The optimization of the structural design of a mooring system using PIAnO 2023 software helped in the reduction in cost by more than 52% [21]. The structural optimization of wound-field synchronous machines (WFSMs) using PIAnO 2023 software has also been analyzed [22]. The results showed a significant improvement in torque by 31.66% and a reduction in the torque ripple.

In the present work, the performance of the SCR system is predicted using an optimization technique based on a metamodel. In addition, the procedure of optimization is analyzed and compared with our previous study based on DOE [23]. This study will help in understanding the design factors that significantly affect the optimization of uniformity index performance and thereby improvement in NOx emission reduction.

2. Methodology

2.1. SCR System and Numerical Analysis

The purpose of the SCR system is to reduce the NOx of exhaust gas. When the urea solution is injected into the high-temperature exhaust gas, NH₃ is generated through a thermal-decomposition chemical reaction and hydrolysis reaction as shown in Equations (1) and (2), respectively. When it is mixed with exhaust gas and distributed across the catalyst, the NOx is reduced to nitrogen and water as represented in Equation (3) [24,25].

In the exhaust pipe:

$${\mathrm{C}\mathrm{O}\left({\mathrm{N}\mathrm{H}}_2\right)}_2\to \mathrm{N}{\mathrm{H}}_3+\mathrm{H}\mathrm{N}\mathrm{C}\mathrm{O}$$

(1)

$$\mathrm{H}\mathrm{N}\mathrm{C}\mathrm{O}+{\mathrm{H}}_2\mathrm{O}\to \mathrm{N}{\mathrm{H}}_3+{\mathrm{C}\mathrm{O}}_2$$

(2)

In the SCR catalyst:

$${4\mathrm{N}\mathrm{H}}_3+4\mathrm{N}\mathrm{O}+{\mathrm{O}}_2\to {4\mathrm{N}}_2+{6\mathrm{H}}_2\mathrm{O}$$

(3)

NH₃ can be produced sufficiently when the reactions in Equations (1) and (2) are properly activated, which results in the activation of the reaction in Equation (3) in the SCR catalyst [26,27]. Therefore, it is necessary to secure enough space for the initiation of the chemical reaction before NH₃ mixes with the SCR catalyst. Adequate mixing using the mixer should be applied to create a vortex so that the number of sprayed elements can be decomposed properly [28,29] and also the sufficient vaporization of urea solution must be improved [30,31]. For this reason, the SCR system is designed as shown in Figure 1, and each name of the design parameters is mentioned in

Table 1. Design parameters of the SCR system.

No.	Major Design Parameters	Unit
1	A: Distance between the Urea Injector and Mixer	mm
2	B: Inflow Angle of the Exhaust Gas	deg.
3	C: Angle of the Urea Injector and Mixer	deg.
4	D: Mounting Angle of the Mixer	deg.
5	E: Number of Mixer Blades	No.
6	F: Bending Angle of Mixer Blades	deg.
7	G: Distance between the Mixer and SCR Cone	mm
8	H: Length of the SCR Cone	mm

. The urea solution is injected from the injector and mixes well with the exhaust as it passes through the mixer due to the vortex. The pressure drop at the SCR cone delays the gas flow and helps to increase flow uniformity. The mixer’s performance varies depending on the number of blades configured and the angle of bending. The design parameters mentioned in Table 1 are A (the distance between the injector and mixer), B (the angle at which the exhaust gas meets the urea solution injected by the injector), C (the angle of the injector and mixer), D (the mounting angle of the mixer), E (the number of mixer blades), F (the bending angle of mixer blades), G (the distance between the mixer and SCR cone), and H (the length of the SCR cone).

The numerical analysis is calculated using SIEMENS STAR-CCM+, and the boundary conditions are set based on the experimental values. In the previous study [23] and Appendix A (

Table A6. Boundary conditions of CFD.

a. SCR System
No.	Classification	Unit	Value
1	Shell Material	SUS	436 L
2	Mass Flow of Exhaust Gas	kg/h	316
3	Exhaust Gas Temp.	Max, °C	411
4	Turbo Charger	Max, RPM	203,000
5	Engine RPM	RPM	3000
6	AdBlue	mg/s	105
7	Urea Injection	mg/Injection	30.6
8	Injection Duration	msec	81.6
9	Pressure of Exhaust Gas	kPa	9.8
b. Urea injector nozzle holes
No.	Classification	Unit	Value
1	Number	No.	3
2	Hole Diameter	$\mathsf{\mu }\mathrm{m}$	120
3	Diameter at Hole Center Positions	mm	1.9
4	Circumferential Distribution	deg.	120
5	Static Mass Flow	kg/h	3.2
c. Injection initialization
No.	Classification	Unit	Value
1	Equivalent Spray Type	Type	3-Hole Full Cone Spray
2	Cone Angle	deg.	7
3	Spray Angle	deg.	7
4	Estimated Initial Droplet Velocity	m/s	24
5	Droplet Diameter, SMD	μm	100
d. Information of Mesh modeling
No.	Classification	Value	No.
1	Analysis Tool	Star-CCM + V12.04	1
2	Mesh Type	Polyhedral	2
3	Total Mesh Quantity	1,041,308	3
4	Base Mesh Size	4 mm	4
5	Surface Mesh Size	50~100% (Compared Base Mesh Size)	5
6	Number of Prism Layers	3	6
7	Prism Layer Thickness	0.25 (Compared Base Thickness)	7
8	Fine Mesh	Surface: 25%, Prism: 12.5%	8

), the boundary conditions for the numerical analysis are mentioned in detail. Engine conditions are based on the engine running criteria in accordance with EOP5 (Engine Operating Point). The engine speed is 3000 RPM, the exhaust gas mass flow rate is 316 kg/h, the exhaust gas inlet temperature is 411 °C, the urea injection amount is 30.6 mg/injection, the urea injection period is 81.6 ms/Hz, and the exhaust gas outlet pressure is set at 9.8 kPa. Catalysts are implemented using the characteristics of the porous catalyst model wherein the catalyst carrier has a capacity of 3.0 L and a cell density of 600 cpsi. Conditions such as the spray angle, injection pressure, and nozzle diameter of the urea injector are based on a commercial urea injector three-hole nozzle.

2.2. Formulation of Optimization

The optimal design method using the metamodel defines the objective function and design parameters through the formulation of optimization. The objective function is a value representing performance and may be maximized or minimized according to the necessity of the optimization process. The values of the major design parameters are specified according to the factors which control the performance of the SCR system. In this optimization problem of SCR system performance, the UI value is selected as an objective function to maximize performance, and the definition of design parameters is mentioned in

Table 2. Boundary condition range for each parameter.

No.	Major Design Parameters	Unit	Design Parameter Sets
			Initial	Upper Limit	Lower Limit
1	A: Distance between the Urea Injector and Mixer	mm	85	95	75
2	B: Inflow Angle of the Exhaust Gas	deg.	109	114	104
3	C: Angle of the Urea Injector and Mixer	deg.	110	115	105
4	D: Mounting Angle of the Mixer	deg.	0	10	−10
5	E: Number of Mixer Blades	No.	6	8	4
6	F: Bending Angle of Mixer Blades	deg.	120	125	115
7	G: Distance between the Mixer and SCR Cone	mm	167	187	147
8	H: Length of the SCR Cone	mm	166	186	146

. The design parameter sets specify upper and lower ranges based on the initial value. The range of design parameters has been designated by a trial-and-error method so that it conserves the overall design drawings of the SCR system and prevents modeling errors.

When the formalization is complete, performance optimization proceeds according to the process in Figure 2, which shows a flowchart of the SCR system optimization. The Formulation of optimization step comprises defining the design parameters as shown in Table 2 and defining an objective function. The Sampling step entails finding an experimental point for constructing the metamodel and uses multi-start local optimization (MLO), query-by-commitment (QBC) using EDT, and multiple maximum distance sampling (MMDS) techniques developed by PIDOTECH [16]. In the next step, a model is designed with CATIA for the experimental points generated in the Sampling step, and numerical analysis is performed with STAR-CCM+ to obtain the UI value. In the Generate Metamodel step, a metamodel is generated from the obtained experimental points and numerical analysis results. Accuracy Convergence 1 is the process of verifying the accuracy of the metamodel by comparing the results of the test data between the predicted and actual numerical values.

The accuracy of the metamodel is calculated as a normalized root mean square error (Norm. RMSE) by comparing the prediction result with the actual result based on the metamodel generated for each iteration. Norm. RMSE is calculated using Equation (4) [32,33]. Maximum absolute error (Max. Abs. Error) is the largest value of the absolute errors and is calculated using Equation (5). When the verification of the metamodel is completed at Accuracy Convergence 1, it can predict the optimal design. The predicted results are calculated by computational fluid dynamics (CFD) under the same optimal design conditions. Finally, the accuracy is verified at Accuracy Convergence 2 using Equation (6). If the accuracy of the prediction results is low, the optimization process will proceed again after checking the problem in the Formulation of optimization step [13,34].

$$\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}.\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\frac{RMSE}{Q_3-Q_1}=\frac{\sqrt{\frac{1}{n_{test}}\sum {\left(y-\widehat{y}\right)}^2}}{Q_3-Q_1}$$

(4)

$$\mathrm{M}\mathrm{a}\mathrm{x}.\mathrm{A}\mathrm{b}\mathrm{s}.\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}=\mathrm{M}\mathrm{a}\mathrm{x}\left[y_1-{\widehat{y}}_1, y_2-{\widehat{y}}_2, \cdots ,y_8-{\widehat{y}}_8\right]$$

(5)

$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}(\%)=\frac{y}{\widehat{y}}\times 100$$

(6)

where, $n_{test}$ is the number of test points, y is the value of performance calculated by CFD, $\widehat{y}$ is the predicted value of performance through the metamodel, $Q_3$ is the 3rd quartile, and $Q_1$ is the 1st quartile. The metamodeling algorithm has been ranked by the PIAnO tool and is equipped with various algorithms [20]. In this study, the Kriging, EDT, and RBF algorithms are selected. Kriging is a representative interpolation model and has excellent predictive performance in a data group with many design parameters and strong nonlinearity [35]. It also provides statistical estimates and does not depend on the user’s experience because it optimizes parameters through the maximum likelihood estimation method (MLE) [36,37]. EDT refers to a method of generating multiple decision trees and predicting them as the average of each decision tree result [38,39]. It is known to improve predictability and performance mainly when dealing with large regression models [40]. The EDT model can be subclassified into tree bagging, random forest, and hybrid ensemble decision tree models [41,42]. RBF uses a kernel function suitable for each data set. It has the advantage of learning being nonlinear and fast [43,44]. Both the interpolation and regression models in RBF can be predicted and are divided into radial basis function interpolation (RBFi) and radial basis function regression (RBFr) [45,46].

3. Results of Optimization

3.1. Optimization with the Metamodel

The results from the optimization process of the metamodeling are summarized in

Table 3. Accuracy result of the metamodels.

Ranking No.	Algorithm	Plot	Norm. RMSE(%)	RMSE	Max. Abs. Error
1	KRG		28.5%	0.004	0.008
2	EDT		57.0%	0.008	0.020
3	RBFi		82.5%	0.011	0.021

, and the detailed result is shown in Appendix A. In Appendix A,

Table A1. 1st training data for the optimal creation of the metamodel.

No.	A	B	C	D	E	F	G	H	NH3 UI
Case 1	95	114	115	10	8	125	187	186	0.964
Case 2	95	114	115	−10	4	115	187	146	0.833
Case 3	95	109	110	10	8	125	167	166	0.966
Case 4	95	109	110	0	6	120	167	146	0.943
Case 5	95	109	110	−10	4	115	167	186	0.867
Case 6	95	104	105	0	6	120	147	186	0.962
Case 7	85	109	105	0	4	125	187	166	0.851
Case 8	85	109	105	−10	8	120	187	146	0.963
Case 9	85	104	115	10	6	115	167	166	0.955
Case 10	85	104	115	0	4	125	167	146	0.853
Case 11	85	104	115	−10	8	120	167	186	0.970
Case 12	85	114	110	10	6	115	147	146	0.951
Case 13	85	114	110	−10	8	120	147	166	0.964
Case 14	75	104	110	10	4	120	187	186	0.874
Case 15	75	104	110	0	8	115	187	166	0.973
Case 16	75	104	110	−10	6	125	187	146	0.943
Case 17	75	114	105	0	8	115	167	146	0.966
Case 18	75	114	105	−10	6	125	167	186	0.956
Case 19	75	109	115	0	8	115	147	186	0.968

shows the training data at the first iteration,

Table A2. 2nd training data for the optimal creation of the metamodel.

No.	A	B	C	D	E	F	G	H	NH3 UI
Case 20	95	104	115	10	4	125	187	146	0.894
Case 21	79.11	112.13	106.63	3.87	7	122.24	152.07	158.89	0.966
Case 22	92.33	108.93	113.6	−9.33	7	122.25	170.47	158	0.967
Case 23	78.35	108.47	108.4	−8.54	6	116.98	170.45	166.11	0.965
Case 24	77.93	105.93	106.05	4	6	122.13	148.65	172.39	0.964
Case 25	89.92	110.41	112.17	9.15	6	115.27	153.42	165.26	0.963
Case 26	75	114	105	10	4	125	187	146	0.906
Case 27	90.6	104.4	107.58	0.83	5	118.87	173.93	177.7	0.957
Case 28	80.37	109.27	107.07	−7.37	4	120.66	185.4	179.87	0.916
Case 29	94.05	113.49	105.67	−0.92	6	124.04	183.44	182.53	0.916
Case 30	81.93	113.53	107.74	−6.93	4	118.73	148.4	149.64	0.931
Case 31	89.8	110.6	113.63	2.72	5	124.67	162.73	178.54	0.951
Case 32	76.18	112.24	106.76	−0.59	5	120.29	163.23	171.88	0.954
Case 33	77.35	108.12	112.65	6.49	6	117.35	184.65	181.29	0.962
Case 34	82.05	109.3	111.47	9.99	5	123.23	182.28	157.76	0.944
Case 35	82.06	109.29	109.12	−8.82	6	115	170.53	155.41	0.961
Case 36	84.41	106.35	113.24	6.47	7	119.89	156.41	183.65	0.969
Case 37	85.59	110.47	114.41	5.29	4	118.53	160.13	150.71	0.918
Case 38	90.29	111.06	110.29	8.82	5	116.77	172.88	178.03	0.952
Case 39	92.65	104.01	106.76	−6.47	7	118.49	147.01	167.16	0.968

shows the training data at the second iteration,

Table A3. 3rd training data for the optimal creation of the metamodel.

No.	A	B	C	D	E	F	G	H	NH3 UI
Case 40	75.00	107.13	109.38	6.25	8	121.31	174.50	171.00	0.975
Case 41	84.33	104.33	111.84	7.47	7	121.10	170.19	180.83	0.970
Case 42	94.00	111.83	110.93	−7.88	8	118.33	160.34	183.84	0.973
Case 43	87.66	109.34	112.34	−1.44	6	116.43	184.86	172.18	0.961
Case 44	93.54	112.13	114.74	5.95	5	124.45	165.67	160.41	0.947
Case 45	87.32	110.40	106.93	−4.01	7	121.29	174.20	185.27	0.971
Case 46	78.24	105.72	111.10	6.62	7	120.40	164.60	153.32	0.966
Case 47	81.87	105.79	106.20	2.93	7	117.65	185.81	171.87	0.970
Case 48	93.98	110.13	112.46	−2.90	8	119.14	181.93	152.67	0.968
Case 49	76.49	107.40	107.80	4.20	7	118.87	155.80	181.46	0.973
Case 50	92.45	110.15	105.30	−8.39	5	124.13	149.92	176.13	0.959
Case 51	79.66	104.73	112.55	10.00	4	124.67	155.59	174.99	0.939
Case 52	85.92	109.93	110.61	−0.67	5	115.93	149.56	172.15	0.958
Case 53	94.73	107.80	106.27	1.41	4	121.33	147.74	150.81	0.930
Case 54	94.73	107.33	105.50	2.02	6	116.87	159.27	154.05	0.960
Case 55	86.73	106.41	108.80	−8.81	5	120.13	161.13	185.17	0.955
Case 56	90.78	112.20	110.46	9.60	5	116.82	173.40	160.13	0.949
Case 57	82.90	112.70	108.34	3.33	4	123.89	149.87	170.53	0.933
Case 58	79.40	106.99	105.41	9.60	5	119.00	176.06	183.88	0.954
Case 59	87.27	106.32	107.20	9.06	8	119.01	154.99	150.81	0.968

shows the training data at the third iteration,

Table A4. 4th training data for the optimal creation of the metamodel.

No.	A	B	C	D	E	F	G	H	NH3 UI
Case 60	95.00	104.15	115.00	−9.59	4	115.00	185.37	186.00	0.907
Case 61	80.30	109.09	110.93	1.54	6	118.65	168.55	167.33	0.963
Case 62	79.77	108.36	112.59	−8.52	7	125.00	151.87	164.69	0.965
Case 63	92.74	107.05	105.00	−4.13	6	119.74	180.55	149.48	0.955
Case 64	91.91	106.28	109.43	−1.43	6	125.00	163.50	158.92	0.958
Case 65	81.89	109.38	105.55	2.72	4	119.40	181.67	157.55	0.912
Case 66	78.50	107.82	110.45	−8.00	5	122.52	171.27	185.73	0.956
Case 67	75.21	107.54	111.39	10.00	8	123.73	187.00	150.13	0.970
Case 68	92.47	114.00	109.20	1.30	7	116.23	187.00	163.95	0.967
Case 69	75.12	107.91	105.98	4.40	6	116.00	181.99	186.00	0.967
Case 70	80.99	104.00	109.27	9.23	4	115.00	182.54	164.13	0.919
Case 71	82.85	110.64	114.97	1.46	8	115.44	147.00	146.00	0.962
Case 72	75.25	106.40	115.00	−10.00	6	123.69	168.34	164.43	0.960
Case 73	89.27	109.79	113.13	−4.23	5	117.72	166.33	176.41	0.953
Case 74	75.00	109.68	113.21	2.79	6	120.27	183.22	171.52	0.961
Case 75	95.00	112.46	112.34	9.54	5	125.00	183.33	172.67	0.944
Case 76	89.98	112.41	107.63	−3.03	6	121.32	170.63	164.14	0.962
Case 77	85.41	114.00	111.84	6.03	7	123.98	161.83	161.77	0.964
Case 78	95.00	106.40	108.55	−1.02	8	120.07	167.55	175.93	0.972
Case 79	81.39	111.45	109.64	−2.66	5	120.34	147.00	156.92	0.954

shows the training data at the fourth iteration, and

Table A5. Test data for the optimal creation of the metamodel.

No.	A	B	C	D	E	F	G	H	NH3 UI
Case 1	95	114	115	0	6	120	187	166	0.941
Case 2	95	104	105	10	8	125	147	146	0.960
Case 3	95	104	105	−10	4	115	147	166	0.880
Case 4	85	109	105	10	6	115	187	186	0.950
Case 5	85	114	110	0	4	125	147	186	0.880
Case 6	75	114	105	10	4	120	167	166	0.874
Case 7	75	109	115	10	4	120	147	146	0.875
Case 8	75	109	115	−10	6	125	147	166	0.956

shows the test data. The plot mentioned in Table 3 is a graph of the accuracy of the metamodel, wherein the Y coordinate is the predicted result, and the X coordinate represents the actual numerical analysis value. The accuracy of the metamodel is higher as both the training (blue) and test (red) data match, and the results are shown in a linear graphical pattern. Numerically, Norm. RMSE from Equation (4) and Max. Abs. Error from Equation (5) are used to determine the accuracy of the metamodel [32].

Table 3 shows that the Norm. RMSE value of the RBFi metamodel is 82.5%. As a result, it is confirmed that the prediction performance is poor among the three metamodels. The EDT metamodel (57.0%) at the Accuracy Convergence 1 step is better than the RBFi metamodel result (82.5%). However, the error value is reduced in the EDT metamodel, but the maximum absolute error value is still 0.02, signifying an error rate of about 2%. Empirically, if an error value is of more than 1%, there exists a major issue because there is a large difference in the actual test results. Finally, the Kriging metamodel shows the lowest error rate (28.5%) among the three models. The maximum absolute error value is also 0.008, which is an error rate of less than 1%. Therefore, the optimization in this work is carried out by constructing a metamodel using the best-performing Kriging algorithm.

The performance optimization result using the Kriging metamodel is predicted as shown in Figure 3. The UI of the objective function is predicted to be 0.97461, which is 1.9% more than the predicted value (0.95641) of the initial design models. The convergence history and changes after the optimal design indicators show an increasing trend in the value of design parameters A, B, C, and G during the optimization process. The D and E design parameters changed the most as compared to the initial design parameters. The rate of change for the design parameters shown in Figure 3 is the ratio of the difference between the initial value and the optimal value with the initial value.

Table 4. Result of the CFD analysis compared to the metamodel optimization.

No.	Major Design Parameters	Unit	Value	NH3 UI
				Prediction	CFD
1	A: Distance between the Urea Injector and Mixer	mm	80.42	0.97461	0.97293
2	B: Inflow Angle of the Exhaust Gas	deg.	107.38
3	C: Angle of the Urea Injector and Mixer	deg.	108.02
4	D: Mounting Angle of the Mixer	deg.	5.67
5	E: Number of Mixer Blades	No.	8
6	F: Bending Angle of Mixer Blades	deg.	121.21
7	G: Distance between the Mixer and SCR Cone	mm	163.97
8	H: Length of the SCR Cone	mm	171.03
Matching Rate (%)				99.83

shows the results which are compared between the prediction results and the CFD results of Accuracy Convergence 2 in Figure 2. CFD analysis is calculated using the same design parameters used for prediction. The performance-optimization prediction-result value is 0.97461, and the actual numerical analysis result is 0.97293. The matching rate is calculated by Equation (6) and is an indicator to determine the matching ability of both the predicted and actual CFD NH₃ UI values. It is confirmed that the predicted value and the actual result match 99.83%. Therefore, the optimization prediction result using the metamodel can be applied with great confidence.

Table 5. Contribution analysis of the design parameters in the metamodel.

No.	Major Design Parameters	Contribution Analysis [%]
1	A: Distance between the Urea Injector and Mixer	4
2	B: Inflow Angle of the Exhaust Gas	7
3	C: Angle of the Urea Injector and Mixer	0
4	D: Mounting Angle of the Mixer	19
5	E: Number of Mixer Blades	100
6	F: Bending Angle of Mixer Blades	22
7	G: Distance between the Mixer and SCR Cone	0
8	H: Length of the SCR Cone	3

shows the contribution of each design parameter numerically. It can classify objectively the importance of each design parameter. In the previous study [23], it was objectively impossible to analyze the design parameters as shown in Table 5. Consequently, there is a waste of time and cost because all of them were considered in the evaluation without classifying the design parameters that were less related to the performance improvement. Therefore, the results of the contribution analysis of Table 5 are very useful in defining the design parameters. Most design parameters except D, E, and F have contribution values of less than 10%. This means that there is no effect on performance improvement from other design parameters, namely A, B, C, G, and H. The number of blades of the mixer (E, 100%) absolutely contributed to the performance improvement, followed by the mixer blade angle (F, 22%), and the mixer rotation angle (D, 19%). This shows that the design parameters (D, E, and F) related to the mixer have significant contributions in improving the performance of the SCR system. The improvement of UI performance due to the optimization of the structural design parameters can also be confirmed by the experimental analysis carried out by Wardana et al. [5] and Jeong et al. [8]. Therefore, if optimization is focused on the design parameters of the mixer, it is estimated to be effective in reducing time and cost.

3.2. Comparison of Results

Table 6. Design modeling of the base Model and DOE optimization and metamodel optimization.

Base Model	DOE Optimization	Metamodel Optimization
NH3 UI: 0.959639	NH3 UI: 0.973499	NH3 UI: 0.972931

shows the results of the optimization of the base model using the DOE model and metamodel. In the three models, the design parameters G and H, related to the pipe length and SCR cone length, respectively, are changed in an increasing direction. In the case of G, the value for the base model is 147 mm, and it is increased to 187 mm during the DOE optimization and to 163.97 mm during the metamodel optimization. In the case of H, the value for the base model is 146 mm, and it is increased to 166 mm during the DOE optimization and 171.03 mm during the metamodel optimization. The bending angle of the mixer blades (F) is changed in a decreasing direction. In the case of F, the value for the base model is 125 deg, and it is decreased to 115 deg during the DOE optimization and to 121.208 deg during the metamodel optimization. The UI compared with the base model (0.959639) is improved not only in the DOE optimization of previous studies (0.973499) [23] but also in the metamodel optimization of this work (0.972931). Therefore, both the DOE and metamodel optimization methods can improve UI performance. The results of the integrated analysis can be helpful to determine the design parameters.

In

Table 7. Summary of the base model, DOE optimization, and metamodel optimization.

Classification	Base Model	DOE Optimization	Metamodel Optimization
Results of Optimization	0.959639	0.9734991 (1.44%↑)	0.972931(1.38%↑)
Data Quantity	1	27	87
Contribution	N/A	N/A	E > F > D
Prediction	N/A	N/A	Predictable

, the results of optimization, data quantity, contribution, and prediction are mentioned. They are compared with the results of previous studies [23] and are summarized in Table 7. The base model has been designed by evaluating only one dataset based on the experiment. Therefore, there are no data to compare the evaluation results, and the contribution analysis and prediction process cannot be evaluated. As the contribution and prediction results for base model and DOE optimization cannot be checked, the contribution and prediction results are marked as N/A. The upward arrow symbol shown in Table 7, indicates that the results of optimization for DOE optimization and metamodel optimization have improved compared to the base model by 1.44% and 1.38%, respectively.

Moreover, the uniformity index and the NOx purification efficiency can be correlated as shown in Equations (7)–(9):

$$\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{U}\mathrm{I} \mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}=1-\left(\frac{\left(0.5/19\right)}{Average NOx purification efficiency\times Distribution sum of {NO}_x purification efficiency }\right)$$

(7)

$$NOx purification efficiency=\left(\frac{Front-end NOx-Back-end NOx }{Front-end NOx}\right)\times 100$$

(8)

$$\begin{gathered} Distribution sum of NOx purification efficiency= \\ {\sum }_1^{19}\sqrt{({\mathrm{Point} NOx \mathrm{purification} \mathrm{efficiency}-\mathrm{Average} NOxpurification efficiency)}^2} \end{gathered}$$

(9)

The variables entered into each of Equations (7)–(9) are based on the engine operating point (EOP) defined by the user. In this study, the most common EOP conditions are used and are detailed in Appendix A Table A6. After calculating the front-end NOx and the back-end NOx, it can be said that when the UI performance is relatively improved by 1.44% compared to the base model, the NOx purification efficiency is improved by 30.95%. This indicates that with the optimization of design parameters and a minor increment in UI performance, there is a significant increase in NOx purification efficiency.

In the previous work [23], optimization using the DOE method involved 27 experimental points generated through an orthogonal arrangement. The results were based on selecting the highest performance value from these points, making it challenging to ascertain contribution and prediction accuracy. The DOE method’s uncertainty in the optimization results arises because it only covers a part of the design parameter range. However, in this study, metamodel optimization is employed, utilizing 87 data points generated through a sequential sampling method. The metamodel allows us to predict the maximum performance point across the entire range of design parameters. Sequential sampling ensures that only essential data are collected, making metamodel generation efficient and enabling accurate performance predictions. Additionally, the metamodel facilitates the analysis of design parameter contributions and performance value prediction. Effectively leveraging these advantages can significantly reduce time and cost by minimizing unnecessary numerical analysis and experiments. Although DOE and metamodel optimization are distinct methods, both exhibit an improved performance compared to the base model, offering promising approaches for enhancing the UI performance of the SCR system. In terms of analysis and result utilization, the metamodel method proves to be more effective than the DOE optimization and the base model.

4. Conclusions

In this study, various methods were explored for the structural optimization of the selective catalytic reduction system to maximize the uniformity index performance. Three algorithms, namely, ensemble decision tree, Kriging, and radial basis function were utilized to generate metamodels. The Kriging metamodel showed the lowest error compared to ensemble decision tree and radial basis function. The DOE method resulted in a 1.44% improvement in performance compared to the base model. Similarly, the metamodel method exhibited a performance enhancement of 1.38% over the base model. Since the metamodel analyzed the entire range of design parameters, the maximum uniformity index performance achieved in the present system is 0.973. By enhancing the uniformity index by 1.44% relative to the base model, there was a noteworthy 30.95% improvement in NOx purification efficiency. This exhibits the crucial importance of optimizing the uniformity index performance through structural optimization. Thus, using the metamodel proves to be as accurate as the existing DOE method while also allowing for the analysis of design variable contributions, facilitating an efficient result analysis. In the future, the results of this study can be verified with test equipment, further validating the analysis model.