NO catalytic performance analysis of gasoline engine tapered variable cell density carrier catalytic converter

Improving the flow field uniformity of catalytic converter can promote the catalytic conversion of NO to NO2. Firstly, the physical and mathematical models of improved catalytic converter are established, and its accuracy is verified by experiments. Then, the NO catalytic performances of standard and improved catalytic converters are compared, and the influences of structural parameters on its performance are investigated. The results showed that: (1) The gas uniformity, pressure, drop and NO conversion rate of the improved catalytic converter are increased by 0.0643, 6.78%, and 7.0% respectively. (2) As the cell density combination is 700 cpsi/600 cpsi, NO conversion rate reaches the highest, 73.7%, and the gas uniformity is 0.9821. (3) When the tapered height is 20 mm, NO conversion rate reaches the highest, 72.4%, and the gas uniformity is 0.9744. (4) When the high cell density radius is 20 mm, NO conversion rate reaches the highest, 72.1%, and the gas uniformity is 0.9783. (5) When the tapered end face radius is 20 mm, NO conversion rate reaches the highest, 72.0%, and the gas uniformity is 0.9784. The results will provide a very important reference value for improving NO catalytic and reducing vehicle emission.


Introduction
Environmental pollution brings serious ecological problems , and automobile exhaust pollution is one of the sources of serious environmental pollution, so it is urgent to reduce automobile emissions Sun et al. 2021). Many countries have formulated strict emission standards to improve environmental quality (Qian et al. 2019a(Qian et al. , 2019bPark et al. 2019). At present, some ways to reduce emission pollution, such as improving fuel, combustion mode, and adding post-processor, have been proposed (Zhong et al. 2016(Zhong et al. , 2018. Among them, catalytic converters have been proven to be an indispensable device to reduce automobile exhaust pollutants Manojkumar et al. 2020;Subhashish et al. 2019). More than 95% of CO, HC, and NOx are catalyzed by the catalytic converter when the operating environment meets the requirements (Udhayakumar et al. 2021;Matthew and Kara 2016). However, the purification of pollutants is limited due to the uneven flow field in the standard catalytic converter in practice (Mu et al. 2019a;Shen et al. 2019). Besides, it also causes sintering of the catalyst, affecting its later use (Gao et al. 2019;Liang et al. 2019). Therefore, it is necessary to further improve the uniformity of flow field in catalytic converters (Andrew et al. 2014). For catalytic converters, Hesham et al. (2018) proposed that insulating material was placed in the carrier channel to investigate the distribution of the internal flow field. They found that the gas uniformity was improved by 5%. Mu et al. (2019b) proposed a rationalized B-spline expansion pipe structure. The results revealed that the pressure drop was reduced by 12%, and the gas uniformity was better. The catalytic converters they studied all improved the gas uniformity to a certain extent, but they did not evaluate the purification performance of catalytic converters simultaneously. Dey et al. (2020) surveyed various catalysts in catalytic converters. They revealed that compared with noble metal catalysts, Cu was the most powerful and active catalyst for CO. Miles and William (2017) studied alumina containing cerium dioxide nanoparticles. The results indicated that the alumina can reduce the light-off temperature of CO by 100°C compared with the activated cerium oxide powder. Liu et al. (2020) measured the distribution of catalytic converter particles by DMS500 and analyzed the particle characteristics. The results showed that TWC could reduce particles by 20-35% and GPF by 70-85%. Almeida et al. (2014) investigated the impact of temperature on the aging of catalytic converters and claimed that higher temperatures would reduce the conversion efficiency and accelerate the aging of the catalytic converters. Liu et al. (2021) studied the effect of exhaust gas temperature on the purification of particulate matter. The results showed that after the exhaust gas passed through the catalytic converter, the particles of 4-8 nm were reduced by 96%, while the purification effect of particles larger than 50 nm was not obvious. Ayodhya and Narayanappa (2018) studied the arrangement order of post-processing equipment. The results showed that a good after-treatment setup should have its devices placed in the order: diesel oxidation catalyst followed by diesel particulate filter and selective catalytic reduction. Bogarra et al. (2017) studied the catalysis of HC compounds and found that when the temperature reached 350°C, HC compounds could effectively catalyze and effectively reduce the particles less than 20 nm. Kumar et al. (2019) carried out experiments on diesel methanol nitromethane mixture. The results showed that the best performance and emission results were observed when diesel oil was 92.5%, methanol was 5%, and nitromethane was 2.5%. Zhong et al. (2021) studied the influence of NO 2 / NOx ratio on diesel particulate filter. The catalytic performance of NO conversion was limited by mass transfer in diesel oxidation catalyst catalytic coating, while it was almost nonexistent in catalytic diesel particulate filter. These scholars studied the effects of catalysts, exhaust temperature, engine running speed, and other parameters on the purification performance of catalytic converters, but the problem of uneven internal flow field has not been well dealt with.
In summary, at present, some scholars have paid attention to the flow characteristics of catalytic converters, but more attention was paid to the influence of different factors on catalytic converters. Takeru et al. (2017) found, compared with the standard carrier structure, the catalyst durability of the radial variable cell density carrier was better, and the NOx reaction temperature was reduced by 10°C. Xu et al. (2009) claimed the structure of the tapered end face carrier was more favorable to the uniformity of exhaust. Therefore, an improved catalytic converter model is established by combining the form of variable cell density with the form of tapered end face structure, aiming to improve the nonuniform flow phenomenon of the traditional catalytic converter and its purification performance simultaneously.
The improved catalytic converter model with tapered end face radial variable cell density carrier is established, and its accuracy is verified by experiments in this work. In the following sections, the purification performance and flow characteristic of standard and improved catalytic converters are compared. After that, the effects of different structural parameters on the improved catalytic converter are analyzed. Finally, the gray correlation analysis method is used to explore the influence degree of different structural parameters. The research results will provide a very important reference value for improving the purification performance and service life of catalytic converters.

Geometric model
The geometrical model of the improved catalytic converter is shown in Fig. 1, and the parameters are shown in Table 1. The exhaust flows in from the inlet pipe, and some exhaust flows to the edge of the carrier under the action of the expansion pipe and the tapered end face. In addition, the resistance of the medium carrier with high cell density is greater than that of the edge, which makes the exhaust flows to the edge with less resistance. The carriers are all porous media, the surface of which is coated with precious metal catalyst Pt, and the pollutants are catalyzed in the carrier area to achieve purification. Finally, exhaust is discharged from the outlet pipe.

Mathematical model
To facilitate the establishment of mathematical model, it is necessary to make reasonable assumptions to simplify the model, as follow: (a) No heat loss of steel shell and liner; (b) the exhaust is an incompressible ideal gas; (c) only 8 kinds of exhaust components, NO, H 2 O, N 2 , O 2 , CO 2 , CO, HC, and NO 2 ; (d) all reactions occur only on the surface of the carrier (Zuo et al. 2019a).
(1)Carrier pressure drop The carrier region is a fully developed laminar flow, which is simulated as a porous medium, making it an additional pressure loss term of the flow momentum equation (Su et al. 2013): The total carrier pressure drop mainly includes Δp 1 caused by the friction of carrier channel and Δp 2 caused by the inlet and outlet of the channel (Su et al. 2013).
(2)Reaction mechanism (Zuo et al. 2019b) (a) NO catalyzed reaction: Fig. 1 Structure of tapered variable cell density carrier catalytic converter. 1 inlet pipe, 2 expansion pipe, 3 low cell density carrier, 4 high cell density carrier, 5 liner, 6 contraction pipe, 7 outlet pipe, 8 interface of high and low cell density (b) Reaction rate equation: (3) Composition conservation equation (Deng et al. 2017) Exhaust uniformity is measured by exhaust uniformity index γ (Su et al. 2013), the γ value varies between 0 and 1.0, and the closer it is to 1.0, the more uniform the flow will be.
NO conversion rate η (Zuo et al. 2019b): The catalytic reaction mechanism of NO is shown in Table 2 (Deng et al. 2017).

Simulated boundary conditions
The computational fluid dynamics software Fluent is used to simulate the catalytic converter. In the flow and combustion models, the standard k-ε model and the species transport model are selected to solve the flow and chemical reaction models, respectively. In cell zone conditions, parameters such as viscous resistance, inertial resistance, porosity, and specific surface area of high and low cell density carriers are set, respectively. Since the exhaust gas only flows along the axial direction of the carrier without radial flow, the viscous resistance and inertial resistance in other directions are set as 1000 times of the axial direction. The residual error of energy equation is 1×10 −6 , while the residual errors of the other equations are 1×10 −3 (Cai et al. 2020). The inlet temperature is 575 K, and the inlet velocity is 10.28 m·s −1 , 6.12 m·s −1 , and 4.93 m· s −1 , respectively, setting outlet boundary condition as pressure outlet. The composition and content of the inlet exhaust are shown in Table 3 (Deng et al. 2017).

Grid independence analysis
To determine the appropriate mesh number, the impact of different mesh number models on the simulation results should be studied. Three mesh models with different grid numbers are established, respectively, 670,293, 425,223, and 215,212, as shown in Fig. 2. For the convenience of analysis, the total axis length of the catalytic converter is defined as L, and the distance from a point on the axis to the inlet is defined as z, and then the dimensionless distance is expressed as z/L. The radius of the carrier is R, and the distance from a point to the central axis is r, and then the radius dimensionless distance is expressed as r/R.
The simulation results are shown in Fig. 3. The pressure and velocity distribution trends in the three different mesh models are all the same, and the values differ a little, all less than 5%. Considering the calculation time and the accuracy of numerical results Tang et al. 2019), the model with 425,223 grids is adopted.

Experimental verification
To further verify the accuracy of the improved model, a bench test is conducted, as shown in Fig. 4. The experimental equipment consists of electric dynamometer, gasoline engine, improved catalytic converter, differential pressure sensor, gas analyzer, computer, flowmeter, valves, and other parts. The electric dynamometer model is AC-110T AC electric dynamometer, with rated power of 110 kW, rated torque of 175 N·m (error: ≤±0.28 N·m), and rated speed of 6000r/min (error: ≤± 0.1r/min). The DiCom4000 gas analyzer produced by AVL company is used to detect the content of each gas in the exhaust gas of gasoline engine. The main parameters of the gasoline engine used in the experiment are shown in Table 4. Electric dynamometer is used to detect engine speed, torque, power, and other parameters. Valves are used to control the flow rate into the catalytic converter, and the flowmeter measures the exhaust flow rate at the catalytic converter inlet. The differential pressure sensor is used to measure the pressure drop of the catalytic converter. The gas sensor of the gas analyzer is placed before the inlet and after the outlet of the catalytic converter to analyze the NOx mass fraction. The operating parameters of the gasoline engine in the experiment are shown in Table 5. The experimental ambient temperature is 288~293 K and the ambient pressure is 101.23 kPa. Figure 5 shows the comparison between experimental values and simulated values under different working conditions. It can be found that the measured results of NO and pressure drop are in good agreement with the simulated results, and the maximum relative errors are both less than 5%. Factors contributing to the error include the following: (1) The simulation model assumes that the flow in each group of exhaust ducts is uniform, while the actual flow in each group of exhaust ducts is not completely uniform; and (2) experimental measurement error. Therefore, all the cases are considered as good calibration and provide a basis for further research.

Performance comparative analysis
To demonstrate the flow characteristics and purification performance of the improved catalytic converter, the gas uniformity, carrier pressure drop, and NO conversion rate of the standard and improved catalytic converters are compared. The carrier cell density of the standard catalytic converter is 400 cpsi, and other geometrical parameters are the same.
As can be found from Fig. 6a, the γ and carrier pressure drop of the improved catalytic converter are higher than those of the standard catalytic converter. Under the exhaust inlet velocity of 10.28 m·s −1 , the γ of the improved and standard catalytic converters are 0.9703 and 0.9060, and the pressure drop is 177 Pa and 189 Pa, respectively. The γ and p increased by 0.0643 and 6.78%, respectively. Figure 6b exhibits the radial distribution of the velocities. Obviously, the velocities of the standard catalytic converter decrease and change greatly along the radius. The velocities of improved catalytic converter have a trend of decreasing gradually at first, then increasing, and finally decreasing along the radius and change small. This phenomenon is explained as follows. For standard catalytic converters, the exhaust mainly concentrates in the carrier middle and flows less to the edge under the influence of inertia. For improved catalytic converters, the resistance of the high cell density carrier is greater than that of the low cell density carrier, and the tapered end face acts as a guide to the exhaust, making more exhaust flow to the edge with less resistance. Therefore, the γ of improved catalytic converters is higher. As the resistance of the high cell density carrier is large and the carrier has a tapered end face, which increases the resistance and leads to the increase of the carrier pressure drop of the improved catalytic converter, the increase is small.
According to Fig. 7a, the NO conversion rates of the improved catalytic converter are all higher than those of the standard catalytic converter. Under the exhaust inlet velocity of 10.28 m·s −1 , the NO conversion rate of the improved catalytic converters is 73.1%, compared with the traditional catalytic converter, which increases by 7.0%. Figure 7b shows the radial distribution of NO 2 mass fraction under the exhaust inlet velocity of 10.28 m·s −1 . As can be found, the NO 2 mass fraction of the standard catalytic converter increases gradually along the radius, and the distribution is uneven. While the NO 2 mass fraction of the improved catalytic converter tends to decrease first and then increase, the distribution is more uniform. This phenomenon is attributed to the following reasons. The contact probability and contact time of NO with catalyst decrease with the increase of velocity, resulting in the lower NO 2 mass fraction. The velocity of standard catalytic converter decreases along the radial direction and changes greatly, which leads to the gradual increase of NO 2 mass fraction, and the difference is large. However, the improved catalytic converter has lower velocity and larger specific surface area, which makes NO catalysis more sufficient, leading to a higher NO 2 mass fraction and more uniform distribution. Therefore, the NO conversion rate of the improved catalytic converter is higher. In summary, the improved catalytic converter can effectively improve the flow characteristics and purification performance, the increase of pressure drop is not large, so it is an ideal catalytic converter.

Results and discussion
To design a more ideal tapered variable cell density carrier catalytic converter, the effect of structural parameters on its performance must be discussed. Compared with the standard carrier structure, the improved catalytic converter is mainly changed in the carrier inlet end face and the carrier density. Therefore, this work focuses on exploring the influence rules of cell density combination, tapered height, high cell density radius, and tapered end face radius.

Influence of cell density combination
Keeping the other structure of the improved catalytic converter unchanged, the catalytic converters with different cell density combinations are simulated under three working conditions. The parameters of cell density combination are shown in Table 6. The high/low cell density (cpsi) is the number of cells in one square inch cross-sectional area of high and low cell density carriers. The high/low porosity (%) refers to the cell volume of high and low density carriers as a percentage of the total volume of the material in its natural state. Figure 8a shows the influence of different cell density combinations on the carrier outlet velocity under the exhaust inlet velocity of 10.28 m·s −1 . As can be found, with the increase of the radius, the velocities decrease gradually at first, then increase rapidly near the cell density interface, and then decrease again. The major reason is that most of the exhaust flows along the axial direction and less along the radial direction due to the influence of inertia, resulting in the velocity decreases along the radius. However, the carrier resistance suddenly decreases at the cell density interface, resulting in the velocity increase rapidly. It can also be found that when the high and low cell densities are different, the larger the cell density combination, the smaller the velocity variation near the cell density interface. The cell densities of 500 cpsi/400 cpsi are higher than those of 400 cpsi/200 cpsi, while the velocity change of 500 cpsi/400 cpsi is less than that of 400 cpsi/200 cpsi. Besides, when the low cell density is the same, the velocity variation enhances with the high cell density increases. Under the low cell density is 400 cpsi, the high cell density increases from 500 to 600 cpsi, the velocity variation greatly. The reason is that as the cell density combination is large, the resistances of high and low cell density are both high, which makes the relative resistance difference smaller, resulting in a small velocity variation. When the low cell density is the same, the larger the high cell density, the greater the resistance difference, which leads to the bigger velocity variation.
It can be found from Fig. 8b that with the increase of cell density combination, γ shows a cyclic change of decreasing first and then increasing. When the high and low cell densities are different, γ increases gradually with the increase of cell density combination. The cell density of 700 cpsi/600 cpsi is the highest, and its γ reaches the highest, which is 0.9821 at the exhaust inlet velocity of 10.28 m•s −1 . The main reason is that the higher the carrier cell density, the greater the resistance of carrier, which makes more exhaust flow to the edge and distributes more evenly. However, when the low cell density of cell density combination is the same, the smaller the high cell density, the better the gas uniformity. When the low cell density is 500 cpsi, γ decreases from 0.9710 to 0.9377 when the high cell density increases from 600 to 700 cpsi. This is because when the low cell density is the same, the higher the high cell density, the greater the resistance difference, which leads to the great variation of velocity at the cell density interface and reduces the uniformity.
It is clear from Fig. 8b that the larger the cell density combination, the greater the carrier pressure drop. When the cell density is 700 cpsi/600 cpsi, the carrier pressure drop reached the maximum value of 232 Pa. The reason is that the higher the cell density, the greater the resistance along the path, so that the carrier pressure drop increases. Figure 9a shows the influence of different cell density combinations on NO conversion rate under three cases. It can be found that with the increase of cell density combination, NO conversion rate shows a cyclic change of decreasing first and then increasing. When the high and low cell densities of cell density combination are different, NO conversion rate Fig. 7 Comparison of NO conversion rate and NO 2 mass fraction. a NO conversion rates under three cases. b Radial distribution of NO 2 mass fraction at exit increases with the cell density combination enhance. When the cell density increases from 400 cpsi/200 cpsi to 700 cpsi/600 cpsi, NO conversion rate increases from 66.6 to Fig. 9 Effect of different cell density combinations on NO conversion rate and NO 2 mass fraction. a NO conversion rate. b Distribution of NO 2 mass fraction at export 73.7% under the exhaust inlet velocity of 10.28 m•s −1 . The reason is that the greater the cell density combination, the larger the specific surface area of the carrier and the smaller the velocity, which increases the contact probability and time between NO and catalyst, leading to NO conversion rate increase. However, when the low cell density of cell density combination is the same, NO conversion rate decreases with the high cell density increases. Under the low density is the same as 200 cpsi, when the high cell density increased from 300 to 400 cpsi, NO conversion rate fell to 66.9% from 66.6%. This is because when the low cell density is the same, increasing the high cell density will cause more exhaust to flow through the low cell density region, which leads to the limitation of NO catalysis in the low cell density region, thus reducing NO conversion rate. Figure 9b exhibits the impact of different cell density combinations on the NO 2 mass fraction at the outlet at the exhaust inlet velocity of 10.28 m•s −1 . It can be found that when the high and low cell densities are different, the larger the cell density combination, the higher the NO 2 mass fraction. The NO 2 mass fraction of 600 cpsi/400 cpsi is higher than that of 400 cpsi/200 cpsi. The reason is that the larger the cell density combination, the larger the specific surface area and the lower the velocity, which makes the NO catalysis more sufficient and the NO 2 mass fraction higher. It can also be found from Fig. 9b that when the low cell density is the same, the lower the high cell density, the higher the NO 2 mass fraction in the low cell density region. Under the condition of the same low cell density of 200 cpsi, the NO 2 mass fraction of 300 cpsi/200 cpsi is higher than that of 400 cpsi/200 cpsi in low cell density region. This is because when the low cell density is the same, the higher the high cell density, the higher the velocity in the low cell density area, which leads to insufficient NO catalysis and thus lowers the NO 2 mass fraction. Therefore, it is beneficial to improve the performance of catalytic converters by appropriately increasing the cell density combination and decreasing the high cell density. Figure 10a shows the influence of different tapered heights on the radial distribution of the velocity at the carrier outlet when the exhaust inlet velocity is 10.28 m·s −1 . As can be found, when the tapered height is lower than 20 mm, the velocity exhibits a trend of gradually decreasing first, then increasing near the cell density interface, and then decreasing gradually. However, when the tapered height is greater than 25 mm, the velocity increases gradually. The main reason is that when the tapered height is lower than 20 mm, the velocity decreases gradually along the radial due to the influence of inertia, and the carrier resistance decreases suddenly at the cell density interface, leading to its increase rapidly. The conductivity increases with the tapered height enhances, and when the tapered height is higher than 25 mm, more exhaust is diverted to the edge of the carrier, resulting in a gradual increase in velocity. Figure 10b shows the influence of tapered height on gas uniformity and pressure drop under three cases. It can be found that γ increases gradually when the tapered height enhances under the exhaust inlet velocity of 10.28 m·s −1 . γ increases from 0.9544 to 0.9774 when the tapered height enhances from 5 to 25 mm. The main reason is that when the exhaust inlet velocity is 10.28 m·s −1 , the higher the tapered height, the greater the flow conductivity, so that more exhaust is diverted to the low cell density region, which improves the velocity in the low cell density region and the gas uniformity. However, when the exhaust inlet velocity is 6.12 m·s −1 and 4.93 m·s −1 , γ decreases with the increase of tapered height. The main reason is that the kinetic energy of exhaust is small, and it is easy to change the flow direction due to the guide surface, which makes the velocity in the low cell density region too high, leading to the decrease of gas uniformity.  Fig. 10b, it can be found that the carrier pressure drop increases with the increase of tapered height. The carrier pressure drop increases from 179 to 212 Pa when the tapered height enhances from 5 to 25 mm at the exhaust inlet velocity of 10.28 m·s −1 . This is because the higher the tapered height, the larger the total carrier length, which increases the flow resistance. Figure 11a exhibits the influence of tapered height on NO conversion rate under three cases. As can be found, NO conversion rate shows a trend of gradually increasing and then decreasing with the increase of the tapered height at the exhaust inlet velocity of 10.28 m·s −1 . As the tapered height increases from 5 to 25 mm, NO conversion rate increases from 71.4 to 72.4% and then decreases to 72.0% and reaches the maximum at 20 mm. The chief reason is that when the tapered height increases, the total reaction surface area of the support increases and the velocity decreases, which makes the contact probability and time between NO and catalyst increase and leads to the NO conversion rate increases. However, when the tapered height is greater than 20 mm, more exhaust is diverted to the low cell density area, and the velocity is high, which restricts the NO catalytic activity and decreases the NO conversion rate. It can also be found from Fig. 11a that the tapered height has little impact on the NO conversion rate when the exhaust inlet velocity is 6.12 m·s −1 and 4.93 m·s −1 . This is because when the exhaust velocity is low, the contact probability and contact time between NO and catalyst are sufficient, and NO catalysis is mainly affected by temperature, which makes the change of tapered height have little impact on NO conversion rate. Figure 11b shows the influence of different tapered heights on the radial distribution of NO 2 mass fraction at the outlet at the exhaust inlet velocity of 10.28 m·s −1 . It can be found that when the tapered heights are less than 15 mm, the NO 2 mass fraction decreases and then increases along the radial direction of the outlet. However, when the tapered heights are higher than 20 mm, the NO 2 mass fraction decreases, then increases, and finally decreases. The reason is that when the tapered height is less than 15 mm, the velocity increases rapidly near the cell density interface and then decreases gradually, which makes the contact probability and contact time between NO and catalyst decrease gradually and then increase gradually, resulting in the NO 2 mass fraction decreasing first and then increasing gradually. However, as the tapered height is higher than 20 mm, more exhaust is guided to the edge, which leads to the limitation of NO catalysis in the edge area and the lower NO 2 mass fraction. Figure 12a shows the influence of high cell density radius on the radial distribution of carrier outlet velocity at the exhaust inlet velocity of 10.28 m·s −1 . As shown in Fig. 12a, the exhaust velocity shows a trend of first gradually decreasing, then increasing, and finally decreasing along the radius. The reason is the same as the explanation in Fig. 8a. It can be found that the velocity at the edge increases as the high cell density radius increases. The maximum velocity at the edge increases from 2.1 to 2.4 m·s −1 as the high cell density radius enhances from 15 to 35 mm. This is because the larger the high cell density radius, the more exhaust flows to the low cell density area with less resistance, and the velocity increases. Therefore, the velocity at the edge should be appropriately reduced as the high cell density radius is 25 mm Figure 12b shows the influence of high cell density radius on gas uniformity and carrier pressure drop under three cases. It can be found that γ increases at first and then decreases with the increase of high cell density radius at the exhaust inlet velocity of 10.28 m·s −1 . As the high cell density radius Fig. 11 Effect of different tapered height on NO conversion rate and NO 2 mass fraction. a NO conversion rate. b Radial distribution of NO 2 mass fraction at carrier outlet enhances from 5 to 35 mm, γ increases from 0.9442 to 0.9783 and then decreases to 0.9720 and reaches the maximum at 25 mm. The reason is that when the high cell density radius area is small, the velocity of the low cell density area changes greatly. On the contrary, the velocity changes greatly in the region with high cell density, thus reducing the flow field uniformity. It can also be found that the closer the high cell density radius is to 25 mm, the lower the γ under the exhaust inlet velocity of 6.12 m·s −1 and 4.93 m·s −1 . The reason is that the kinetic energy of exhaust is small and the velocity is easily disturbed. When the high cell density radius is 25 mm, the change of velocity caused by the change of cell density has a large influence range, thus reducing the uniformity.

Influence of high cell density radius
It can also be found from Fig. 12b that the carrier pressure drop increases with the high cell density radius enhance. The carrier pressure drop increases from 182 to 198 Pa as the high cell density radius increases from 5 to 25 mm under the exhaust inlet velocity of 10.28 m·s −1 . This is because the larger the high cell density radius, the greater the resistance of the carrier along the path, which increases the pressure drop.
The influence of high cell density radius on NO conversion rate under three cases is shown in Fig. 13a. As can be found, NO conversion rate increases first and then decreases with the high cell density radius increase. NO conversion rate increases from 71.4 to 72.1% and then decreases to 71.5% as the high cell density radius enhances from 5 to 35 mm at the inlet velocity of 10.28 m·s −1 and reaches the maximum value at 20 mm. This issue is explained as follows. When the high cell density radius is small, the velocity is large in the region of high cell density and near the cell density interface, which restricts NO catalysis in this region. However, when the high cell density radius is larger, the velocity is larger in the area of low cell density, which leads to insufficient NO catalysis in Fig. 12 Effect of different high cell density radius on gas uniformity and carrier pressure drop. a Radial distribution of velocity at carrier exit. b Gas uniformity and carrier pressure drop Fig. 13 Effect of different high cell density radius on NO conversion rate and NO 2 mass fraction. a NO conversion rate. b Radial distribution of NO 2 mass fraction at the outlet this area, thus reducing the NO conversion rate. The velocity is more uniform when the high cell density radius is 20 mm, and the overall catalysis of NO is more sufficient. Figure 13b shows the influence of different high cell density radius on the radial distribution of NO 2 at the outlet under the exhaust inlet velocity of 10.28 m·s −1 . It is obvious that when the high cell density radius is less than 10 mm, the NO 2 mass fraction exhibits a trend of gradually decreasing and then increasing, which is small in the area of dimensionless radius from 0 to 0.65. However, when the high cell density radius is greater than 30 mm, the NO 2 mass fraction shows a trend of gradually increasing and then decreasing, and it is small in the area of dimensionless radius from 0.65 to 1. This phenomenon is explained for the following reasons. When the high cell density radius is small, the velocity is large in the region of high cell density and the cell density interface, which makes the NO 2 mass fraction small in the region of dimensionless radius from 0 to 0.65. However, when the high cell density radius is larger, the velocity is larger in the region of low cell density, resulting in a lower NO 2 mass fraction in the region from 0.65 to 1. The radial distribution of NO 2 mass fraction is higher, and more uniform when the high cell density radius is 20 mm, and the NO conversion rate is higher. Therefore, the high cell density radius of 20 mm is selected, the NO conversion rate is the highest, and the gas uniformity is higher. Figure 14a shows the influence of different tapered end face radius on the radial distribution of the velocity at carrier outlet under the exhaust inlet velocity of 10.28 m·s −1 . It is obvious that, when the tapered end face radius is greater than 30 mm, the velocity in the low cell density area decreases first and then increases, and the change is large. However, when the tapered end face radius is less than 10 mm, the velocity decreases and the change is larger, which is smaller at the edge. This phenomenon is mainly caused by the following reasons. The larger the tapered end face radius, the closer the conduction position of the tapered to the edge and the greater the conduction capacity. When the tapered end face radius is 30 mm (the dimensionless radius is 0.6), the conductivity of the tapered surface is strong, which makes the velocity of the area after the dimensionless radius 0.6 increase rapidly and the change gradient is large. When the tapered end face radius is 10 mm (the dimensionless radius is 0.2), the conductivity of the tapered surface is weak, so that the velocity decreases gradually after the dimensionless radius is 0.2, and it is low at the edge. When the tapered end face radius is 25 mm, the diversion intensity is moderate and the velocity distribution is uniform.

Influence of tapered end face radius
The influence of tapered end face radius on gas uniformity and carrier pressure drop under three cases is shown in Fig.  14b. It is obvious that γ increases first and then decreases with the tapered end face radius increases. When the exhaust inlet velocity is 10.28 m·s −1 , γ increased from 0.9500 to 0.9784 and then decreased to 0.9651 as the tapered end face radius increased from 5 to 35 mm and reached the maximum at 25 mm. The main reason is illustrated in Fig. 14a: When the tapered end face radius is large, the velocity increases first and then decreases in the low cell density region, which changes greatly; when the tapered end face radius is small, the velocity at the edge is small, thus affecting the gas uniformity. The velocity distribution is more uniform when the tapered end face radius is about 25 mm. As can be found from Fig. 14b, the carrier pressure drop gradually increases with the tapered end face radius increases, but the influence is small. The carrier pressure drop increases from 186 to 189 Pa when the tapered end face radius increases from 5 to 35 mm. The main reason is that increasing the tapered end face radius will increase a small part of the carrier, but the flow through the area with high cell density and the area with low cell density is basically unchanged, leading to the resistance being basically unchanged. The effect of different tapered end face radius on NO conversion rate is shown in Fig. 15a. It can be found that NO conversion rate first increases and then decreases with the tapered end face radius increases. NO conversion rate increases from 71.2 to 72.0% and then decreases to 71.3% as the conical end radius increases from 5 to 35 mm under the exhaust inlet velocity of 10.28 m·s −1 and reaches the maximum value at 20 mm. The main reason is that when the tapered end face radius is larger, the edge velocity is larger, which makes NO cannot be effectively catalyzed, resulting in a lower NO conversion rate. When the radius of the tapered end face radius is small, the velocity is larger in the area around the dimensionless radius of 0.4, which makes NO catalysis insufficient. The velocity distribution is more uniform and NO catalysis is more sufficient when the tapered end face radius is 20 mm. Under the condition of inlet velocity of 6.12 m·s −1 and 4.93 m·s −1 , the tapered end face radius has little effect on NO conversion. The reason is the same to the explanation in Fig. 11a. Figure 15b shows the influence of different tapered end face radius on the radial distribution of NO 2 mass fraction at the outlet under the exhaust inlet velocity is 10.28 m·s −1 . When the tapered end face radius is less than 10 mm, the NO 2 mass fraction decreases first and then increases along the radius, and it is lower in the area near the dimensionless radius of 0.4. When the tapered end face radius is larger than 30 mm, it decreases first, then increases, and finally decreases, and it is relatively low in the area of the dimensionless radius from 0.7 to 1. This is because when the tapered end face radius is small, the velocity is larger in the dimensionless region around 0.4, and when the tapered end face radius is large, the velocity is larger in the dimensionless 0.7 to 1 region, which leads to the restriction of NO catalysis and the low NO 2 mass fraction. The velocity distribution is more uniform when the tapered end face radius is 20 mm, so that the radial distribution of NO 2 mass fraction is more uniform. Therefore, when the tapered end face radius is 20 mm, the velocity is more uniform and the NO conversion rate is higher.

Experimental design and calculation
The influence degree of the above structural parameters on the catalytic converter performance is further investigated by using gray relational analysis method. According to the gray relational analysis Wang et al. 2020), low cell density (x 1 ), high cell density (x 2 ), tapered height (x 3 ), high cell density carrier radius (x 4 ), and tapered end face radius (x 5 ) are taken as the influencing factor eigenvectors. Gas uniformity (y 1 ), carrier pressure drop (y 2 ), and NO conversion rate (y 3 ) are taken as reference feature vectors. Under the condition of exhaust inlet temperature of 575 K and velocity of 10.28 m·s −1 , 10 representative experimental models are studied. And the gas uniformity, NO conversion, and pressure drop are analyzed. The simulation conditions and results are shown in Table 7.

Simulation results and analysis
The influence degree of influencing factors on catalytic converter performance is expressed by the R value, and the greater R is, the greater the influence is (Zuo et al. 2016;Kadier et al. Fig. 15 Effect of different tapered end face radius on NO conversion rate and NO 2 mass fraction. a NO conversion rate. b Radial distribution of NO 2 mass fraction at outlet 2015). According to Fig. 16, the influence degree of each influencing factor on each reference index is as follows. For carrier pressure drop: low cell density > high cell density > tapered height > tapered end face radius > high cell density carrier radius. For gas uniformity: low cell density > high cell density > high cell density carrier radius> tapered end face radius > tapered height. For NO conversion rate: low cell density > high cell density > tapered height> high cell density carrier radius > tapered end face radius. The low cell density is a key factor affecting the improved catalytic converter. This issue is explained as follows. The low cell density determines the resistance in the edge area, which affects the flow degree of the exhaust to the edge, thus affecting the gas uniformity. In addition, because most of the exhaust flows through the low cell density area, the resistance in the low cell density area directly affects the pressure drop of the overall carrier. Meanwhile, the low cell density area becomes the main site for NO catalyzing and directly affects the overall NO conversion rate. The research results provide a theoretical basis for structural optimization and system matching of catalytic converters.

Conclusion
By establishing a new catalytic converter model, the uniformity of internal flow field is improved, to achieve the purpose of improving NO catalysis. In addition, the optimum structural parameters are determined by quantitative analysis of NO conversion rate and NO 2 mass fraction at the outlet of the carrier, to further optimize the model. The specific results are as follows: (1) Compared with the standard catalytic converter, the gas uniformity, NO conversion rate, and carrier pressure drop of the improved catalytic converter increased by 0.0643, 7.0%, and 6.78%, respectively. It improves NO catalytic and reduces vehicle emission.
(2) When the high and low cell densities are different, the performance is better when the cell density combination is larger. When the low cell density is the same, the smaller the high cell density, the better the performance. When the cell density combination is 700 cpsi/600 cpsi, its performance is better; the gas uniformity and NO conversion rate reach the highest, which are 0.9821 and 73.7%, and the carrier pressure drop is 232 Pa.
(3) When the tapered height increases from 5 to 25 mm, the catalytic converter performance increases first and then decreases under the exhaust inlet velocity of 10.28 m·s −1 . Its performance is better when the tapered height is 20 mm; the NO conversion rate reaches the highest, 72.4%, and the gas uniformity and carrier pressure drop are 0.9774 and 212 Pa, respectively. (4) With the increase of high cell density radius, the performance of catalytic converter increases first and then decreases under the exhaust inlet velocity of 10.28 m·s −1 . Its performance is better when the radius of high cell density radius is 20 mm; the NO conversion rate reaches the highest, 72.1%, and the gas uniformity and carrier pressure drop are 0.9783 and 198 Pa, respectively. (5) With the increase of tapered end face radius, the performance of improved catalytic converter increases first and then decreases under the exhaust inlet velocity of 10.28 m·s −1 . Its performance is better when the radius of tapered end face radius is 20 mm; the NO conversion rate reaches the highest, 72.0%, and the gas uniformity and carrier pressure drop are 0.9784 and 189 Pa, respectively. (6) Gray correlation analysis results show that low cell density is the key factor affecting the improved catalytic converter.
Availability of data and materials All data generated or analyzed during this study are included in this published article.