A method and system for evaluating exhaust gas uniformity of a low-speed marine diesel engine SCR reactor
By constructing an exhaust gas flow domain model and grid division for the SCR reactor, combined with CFD simulation calculations, the airflow velocity and ammonia mass fraction data at the catalyst inlet were extracted to form clear evaluation criteria. This solves the accuracy problem of exhaust gas uniformity evaluation in the existing SCR system technology and achieves the optimization of SCR system component design.
Patent Information
- Application Number
- CN202310103166.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-02-08
- Publication Date
- 2025-10-03
- Estimated Expiration
- 2043-02-08
AI Technical Summary
In the existing technology, the exhaust gas uniformity evaluation method of the SCR reactor cannot truly reflect the quality of the simulation results, resulting in the inability to accurately optimize the design of the flow disturbance, flow guidance or flow equalization components of the SCR system.
By constructing a geometric model of the exhaust gas flow domain, performing grid division, and using computational fluid dynamics simulation software to perform steady-state simulation calculations, the airflow velocity and ammonia mass fraction data at the catalyst inlet are extracted, and the standard deviation relationship between the unit grid data and the area-weighted average is compared to form a clear evaluation criterion, and the uniformity is characterized by area ratio.
It achieves accurate evaluation of the exhaust gas uniformity of the SCR reactor, avoids approximate errors, can truly reflect the simulation results, simplifies the evaluation steps, and improves the accuracy and efficiency of the design.
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Figure CN116029066B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of SCR reactors, and in particular to a method and system for evaluating the uniformity of exhaust gas from an SCR reactor of a low-speed marine diesel engine. Background Art
[0002] SCR technology is widely used in the field of diesel engine nitrogen oxide emission control. The commonly used technical solution is to use ammonia as a reducing agent to convert nitrogen oxides in diesel engine exhaust into nitrogen and water in the presence of oxygen and catalyst, thereby reducing pollutant emissions. In order to ensure the effective utilization and long-term performance of the catalyst, it is necessary to ensure that the exhaust gas velocity distribution uniformity and ammonia concentration distribution uniformity near the inlet cross section of the first-layer catalyst in the reactor meet certain requirements. When the exhaust gas uniformity near the inlet cross section of the first-layer catalyst does not meet the requirements, it is necessary to change the design of the flow disturbance, flow guide or flow equalization components in the SCR system until the requirements are met. The optimization process is generally achieved through CFD simulation.
[0003] In the simulation optimization process, the exhaust gas uniformity evaluation method is very important. Currently, relative uniformity deviation is often used to reflect the quality of uniformity. This is an evaluation method based on experimental sampling properties and is more suitable for judging experimental measurement results rather than simulation results. Summary of the Invention
[0004] In view of the shortcomings of the existing technology, the purpose of the present invention is to provide a method and system for evaluating the uniformity of exhaust gas from an SCR reactor of a low-speed diesel engine on a ship. The method and system have clear evaluation indicators, can solve the problem that the relative uniformity deviation evaluation standard cannot truly reflect the quality of the simulation results, and the implementation steps are simple.
[0005] In order to achieve the above object, the present invention is implemented through the following technical solutions:
[0006] In a first aspect, an embodiment of the present invention provides a method for evaluating the uniformity of exhaust gas from an SCR reactor of a low-speed marine diesel engine, comprising:
[0007] Constructing a geometric model of the exhaust gas flow domain and performing mesh division on the geometric model of the exhaust gas flow domain;
[0008] Perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor and extract the airflow velocity, ammonia mass fraction, and unit grid area data at the set cross section at the inlet of the first catalyst layer in the SCR reactor;
[0009] Compare the airflow velocity or ammonia mass fraction data of all surface unit grids of the cross-section with their area-weighted average values plus or minus a set multiple standard deviation to obtain the total area of the unit grids that meet the conditions; and use the set area ratio as the evaluation criterion for the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross-section to form an exhaust gas uniformity evaluation scheme.
[0010] As a further implementation method, the area-weighted average of the air flow velocity or ammonia mass fraction on the set cross section is calculated. and standard deviation S; comparing the airflow velocity or ammonia mass fraction data on all surface unit grids of the cross section with the relationship between the area-weighted average value ±1 to 3 times the standard deviation, and counting the total area of the unit grids that meet the conditions.
[0011] As a further implementation, the evaluation criteria satisfy:
[0012]
[0013]
[0014]
[0015] Among them, α is the threshold coefficient, A 1s 、A 2s 、A 3s Respectively represent the total air velocity or ammonia mass fraction on the cross section The sum of the areas of the cell grids within the range, A i Represents the area of any unit grid.
[0016] As a further implementation, α is less than or equal to 1; when α=1, it is assumed that the physical quantity on the cross section conforms to the law of normal distribution.
[0017] As a further implementation method, if either the statistical results of the uniformity of the air flow velocity or the ammonia mass fraction cannot meet the evaluation criteria, the flow disturbance, flow guidance or flow equalization components are redesigned, a simulation model is established, and the uniformity is calculated and evaluated until the requirements are met.
[0018] As a further implementation method, computational fluid dynamics simulation software is used to perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor.
[0019] As a further implementation method, post-processing software is used to extract the airflow velocity, ammonia mass fraction and area data of each surface grid on the set cross section.
[0020] As a further implementation method, the specific structures and dimensions of the mixing pipeline, reactor and connecting pipeline are determined according to the exhaust gas parameters at the inlet of the SCR reactor and the catalyst parameters, and the urea injection amount is estimated.
[0021] As a further implementation method, components that play a major role in the uniformity of exhaust gas at the catalyst layer inlet are designed and a detailed geometric model is constructed; the main characteristic structures of the detailed geometric model are retained to obtain a simplified geometric model of the exhaust gas flow domain.
[0022] In a second aspect, an embodiment of the present invention further provides an exhaust gas uniformity evaluation system for a low-speed marine diesel engine SCR reactor, comprising:
[0023] An exhaust gas flow domain geometric model construction module is used to construct the exhaust gas flow domain geometric model and perform mesh division on the exhaust gas flow domain geometric model;
[0024] The data extraction module is used to perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor and extract the airflow velocity, ammonia mass fraction, and unit grid area data of the set cross section at the inlet of the first catalyst layer in the SCR reactor;
[0025] A uniformity evaluation scheme generation module is used to compare the airflow velocity or ammonia mass fraction data of all surface unit grids of the cross section with the area-weighted average value plus or minus a set multiple standard deviation to obtain the total area of the unit grids that meet the conditions; and to form an exhaust gas uniformity evaluation scheme by satisfying the set area ratio as the evaluation criterion for the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross section.
[0026] The beneficial effects of the present invention are as follows:
[0027] (1) The present invention first simulates the flow field in the SCR system of a low-speed diesel engine, extracts and analyzes the airflow velocity, ammonia mass fraction and unit grid area data on the cross section near the catalyst inlet in the reactor, and then compares the data of all surface grids on the cross section with their area-weighted average value plus or minus a set multiple standard deviation, and counts the total area of the unit grids that meet the conditions. The quality of the airflow velocity distribution and ammonia concentration distribution uniformity of the cross section is characterized by whether the specific area ratio is met; the data characteristics of the simulation results are met, no approximate error is introduced, the evaluation indicators are clear, the implementation steps are simple, and the quality of the velocity distribution uniformity and ammonia concentration distribution uniformity in the simulation results can be truly reflected.
[0028] (2) Based on the characteristics of the simulation results, each cell grid corresponds to a physical quantity value. Typically, the area of each cell grid varies, and cells with larger areas should have a greater weight in the uniformity evaluation. The currently used relative uniformity deviation method does not include the influence of cell grid area and cannot accurately reflect the impact of each data point due to the varying cell grid area. The evaluation criteria in this invention take into account the influence of cell grid area and are consistent with the data characteristics of the simulation results.
[0029] (3) The present invention requires that each unit grid on the cross section for uniformity analysis participate in the calculation, and the calculation accuracy is only related to the number of grids, and essentially no other errors are introduced; while the relative uniformity deviation method only selects a certain number of points on the cross section, and the way of taking sample points is only an approximate estimate. This method itself determines that there will always be a certain deviation.
[0030] (4) The present invention can directly use CFD pre- and post-processing software to complete all evaluation processes, with simple implementation steps and clear evaluation indicators. BRIEF DESCRIPTION OF THE DRAWINGS
[0031] The accompanying drawings, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.
[0032] Figure 1 is a flow chart according to one or more embodiments of the present invention;
[0033] Figure 2 is a grid model of an exhaust gas flow domain of an SCR system according to one or more embodiments of the present invention;
[0034] Figure 3 This is a cloud diagram of the ammonia mass fraction in the 10 mm section before the inlet of the first layer catalyst according to one or more embodiments of the present invention;
[0035] Figure 4 The cloud diagram of ammonia mass fraction in the 10mm section before the inlet of the first layer catalyst according to one or more embodiments of the present invention (data range );
[0036] Figure 5 The cloud diagram of ammonia mass fraction in the 10mm section before the inlet of the first layer catalyst according to one or more embodiments of the present invention (data range );
[0037] Figure 6 The cloud diagram of ammonia mass fraction in the 10mm section before the inlet of the first layer catalyst according to one or more embodiments of the present invention (data range );
[0038] Figure 7 It is a cloud diagram (m / s) of the airflow velocity in the y direction of the cross section 10 mm in front of the inlet of the first layer catalyst according to one or more embodiments of the present invention. DETAILED DESCRIPTION
[0039] Example 1:
[0040] This embodiment provides a method for evaluating the uniformity of exhaust gas from a low-speed diesel engine SCR reactor on a ship. First, the flow field within the low-speed diesel engine SCR system is simulated, and the airflow velocity, ammonia mass fraction, and unit grid area data on the cross section near the catalyst inlet in the reactor are extracted and analyzed. The data of all surface grids on the cross section are compared with their area-weighted average ±1 to 3 times the standard deviation, and the total area of the unit grids that meet the conditions is calculated. Whether the cross section meets the specific area ratio is used to characterize the uniformity of the airflow velocity distribution and ammonia concentration distribution of the cross section, forming a method for evaluating the uniformity of the exhaust gas from the SCR reactor.
[0041] Specifically, such as Figure 1 As shown, the following steps are included:
[0042] Step 1: Based on the exhaust gas parameters and catalyst parameters at the SCR system inlet, such as exhaust gas mass flow rate, composition, temperature, pressure, catalyst size, porosity, catalyst pressure drop and other parameters, determine the specific structure and size of the mixing pipeline, reactor and connecting pipeline, and estimate the urea injection amount.
[0043] The urea solution injection rate is primarily determined by the amount of nitrogen oxides in the exhaust gas. The higher the nitrogen oxides, the greater the urea injection rate. The urea solution injection rate is a necessary input for subsequent flow field simulation calculations. In flow field simulations, the urea solution first evaporates into water vapor, then undergoes thermal decomposition and hydrolysis reactions to produce ammonia. Therefore, the urea solution injection rate ultimately determines the total amount of ammonia within the SCR, and ammonia uniformity is an evaluation indicator in the evaluation method.
[0044] Step 2: Design the static mixer, elbow guide plate, equalizer and other components that play a major role in the uniformity of the exhaust gas at the catalyst layer inlet, and obtain a detailed geometric model; reasonably simplify the detailed geometric model, retain the main characteristic structure, and establish the exhaust gas flow domain geometric model.
[0045] Step 3: Mesh the geometric model of the exhaust gas flow domain of the SCR system.
[0046] Meshing is a prerequisite for subsequent flow field simulation calculations. The quality of meshing directly affects whether the calculation can converge, as well as the accuracy and speed of the calculation. The main parameters of meshing are the number of meshes and mesh quality. The number of meshes must meet the requirements, that is, the total number and density of meshes must be appropriate. Too few meshes will lead to inaccurate calculations, while too many meshes will lead to slow calculations. The number of meshes in important areas of the model should be sufficient, while less meshes can be used in non-important areas. The mesh quality must also meet certain requirements. Poor mesh quality will lead to non-convergence of the calculation, while good mesh quality can accelerate convergence and improve accuracy.
[0047] Step 4: Use the computational fluid dynamics simulation software Ansys Fluent to perform steady-state simulation calculations on the exhaust gas flow field in the SCR system, including physical and chemical processes such as urea solution atomization, droplet evaporation, urea thermal decomposition, and isocyanic acid hydrolysis.
[0048] Step 5: Use post-processing software to extract the airflow velocity, ammonia mass fraction and area data of each surface grid on a certain cross section near the inlet of the first layer catalyst in the reactor.
[0049] Furthermore, the software's built-in post-processing capabilities can be utilized. For example, using Ansys Fluent and CFD-Post software, after completing the steady-state simulation of the exhaust gas flow field in the SCR system, the Fluent software results are first imported into CFD-Post. A 10mm section is then generated in front of the first-layer catalyst inlet. Finally, the data output option is selected, the previously generated section is chosen, and the data to be extracted, such as velocity, ammonia mass fraction, and cell mesh area, is specified and written to a file for subsequent data analysis and calculation.
[0050] Step 6. Compare the relationship between the airflow velocity or ammonia mass fraction data on all surface unit grids of the cross section and its area-weighted average value ±1 to 3 times the standard deviation, count the total area of the unit grids that meet the conditions, and use whether the specific area ratio is met to characterize the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross section.
[0051] Specifically, first calculate the area-weighted average of the airflow velocity or ammonia mass fraction on the cross section And the standard deviation S:
[0052]
[0053]
[0054] Afterwards, statistical calculations are performed separately:
[0055]
[0056]
[0057] If both criteria are met:
[0058]
[0059]
[0060]
[0061] Then it is qualified.
[0062] In the above formula, Xi Represents the physical value on a certain unit grid (air flow velocity or ammonia mass fraction), A i Represents the area of a unit grid, n represents the number of unit grids on the section, Represents the total cross-sectional area, A 1s Indicates that all airflow velocities or ammonia mass fractions on the cross section are The sum of the areas of the cell grids within the range, A 2s 、A 3s Indicates that all airflow velocities or ammonia mass fractions on the cross section are The sum of the areas of the cell grids within the range, A 3s Indicates that all airflow velocities or ammonia mass fractions on the cross section are The sum of the areas of the cells within the range.
[0063] α is the threshold coefficient, which takes a value less than or equal to 1. When α = 1, it is assumed that the physical quantity on the cross section conforms to the law of normal distribution. In practical applications, it can also take a value slightly less than 1, such as α = 0.99, to allow the data to deviate from the law of normal distribution to a small extent, so as to appropriately relax the uniformity evaluation criteria.
[0064] When α = 0.99, if the uniformity statistics of the air flow velocity and ammonia mass fraction on the cross section satisfy If the standard requirements are met, proceed to step seven. If either the uniformity statistical results of the airflow velocity or the ammonia mass fraction cannot meet the standard requirements, return to step two and proceed to subsequent steps, i.e., re-optimize the spoiler, guide or flow-balancing components based on the simulation results, establish a simulation model, and evaluate the uniformity until the requirements are met.
[0065] Step 7: Determine that the design of the current SCR system's flow disturbance, flow guide, or flow equalization components can meet the exhaust gas uniformity requirements at the inlet of the first catalyst layer in the reactor, and end the evaluation process.
[0066] Example 2:
[0067] This example uses the uniformity evaluation method described in Example 1 to evaluate the uniformity of the exhaust gas from the SCR reactor. Figure 1-Figure 7 As shown:
[0068] Step 1: Based on the flow rate, composition, temperature, pressure parameters of the exhaust gas at the SCR inlet and the size, pressure drop, porosity and other parameters of the catalyst, determine the structure and external dimensions of the mixing pipeline, reactor and connecting pipeline in the SCR system, and estimate the urea injection amount.
[0069] Step 2: Design the static mixer in the mixing line, the elbow guide plate in the connecting line, and the flow equalizer in the reactor, all of which play a key role in ensuring uniformity of the exhaust gas at the catalyst layer inlet. Determine their specific dimensions. Simplify the detailed geometric model, retaining key structural features, and establish a geometric model of the exhaust gas flow domain.
[0070] Step 3: Use Ansys I CEM software to mesh the exhaust gas flow domain geometric model.
[0071] Step 4: Use the computational fluid dynamics simulation software Ansys Fluent to read the flow field grid, enter the boundary conditions, select the physical model, set the solution, and perform a steady-state simulation calculation of the exhaust gas flow field. The calculation includes physical and chemical processes such as urea solution atomization, droplet evaporation, urea thermal decomposition, and isocyanic acid hydrolysis.
[0072] Step 5: Use the post-processing software Ansys CFD-Post to extract the ammonia mass fraction, airflow velocity, and area data of all unit grids on the 10mm section in front of the first-layer catalyst inlet in the calculation results. Each unit grid corresponds to an ammonia mass fraction value and an airflow velocity value, and export the data file.
[0073] Step 6: Perform statistics and analysis on the output files.
[0074] First, calculate the area-weighted average of the ammonia mass fraction Standard deviation of ammonia mass fraction S = 9.1999768 × 10 -6 , further calculated
[0075]
[0076] Secondly, the ammonia mass fraction is 1.2267576×10 -4 ~1.4107572×10 -4 The area A occupied by the cell surface mesh within the range 1s ,like Figure 4 As shown, and obtain The statistical ammonia mass fraction is 1.1347579×10 -4 ~1.5027569×10 -4 The area A occupied by the cell surface mesh within the range 2s ,like Figure 5 As shown, and obtain The statistical ammonia mass fraction is 1.0427581×10 -4 ~1.5947567×10 -4 The area A occupied by the cell surface mesh within the range 3s ,like Figure 6 As shown, and obtain
[0077]
[0078] Finally, after comparison, it has been satisfied at the same time The standard requires that the uniformity of the ammonia mass fraction on this section meets the requirements.
[0079] After that, the uniformity of the airflow velocity distribution is evaluated. The evaluation method for the airflow velocity uniformity is the same as that for the ammonia mass fraction uniformity, which will not be repeated here. It should be noted that in step 5, the velocity component data parallel to the normal of the analysis section needs to be output.
[0080] If either the ammonia mass fraction or the airflow velocity uniformity statistical results fail to meet the standard requirements, return to step 2 and redesign the flow disturbance, flow guide, or flow equalization components, establish a simulation model, and calculate and evaluate uniformity until the requirements are met. If the airflow velocity and ammonia mass fraction uniformity statistical results both meet the standard requirements, proceed to step 7.
[0081] Step 7: It is considered that the current SCR system's flow disturbance, flow guide or flow equalization component design can meet the exhaust gas uniformity requirements at the inlet of the first layer catalyst in the reactor, and the entire evaluation process is ended.
[0082] Example 3:
[0083] This embodiment provides an exhaust gas uniformity evaluation system for an SCR reactor of a low-speed marine diesel engine, based on the evaluation method described in the first embodiment, including:
[0084] An exhaust gas flow domain geometric model construction module is used to construct the exhaust gas flow domain geometric model and perform mesh division on the exhaust gas flow domain geometric model;
[0085] The data extraction module is used to perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor and extract the airflow velocity, ammonia mass fraction, and unit grid area data of the set cross section at the inlet of the first catalyst layer in the SCR reactor;
[0086] A uniformity evaluation scheme generation module is used to compare the airflow velocity or ammonia mass fraction data of all surface unit grids of the cross section with the area-weighted average value plus or minus a set multiple standard deviation to obtain the total area of the unit grids that meet the conditions; and to form an exhaust gas uniformity evaluation scheme by satisfying the set area ratio as the evaluation criterion for the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross section.
[0087] The above description is merely a preferred embodiment of the present application and is not intended to limit the present application. Various modifications and variations are possible for those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present application shall be included within the scope of protection of the present application.
Claims
1. A method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine, characterized in that: include: Constructing a geometric model of the exhaust gas flow domain and performing mesh division on the geometric model of the exhaust gas flow domain; Perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor and extract the airflow velocity, ammonia mass fraction, and unit grid area data at the set cross section at the inlet of the first catalyst layer in the SCR reactor; Calculate the area-weighted average of the air velocity or ammonia mass fraction on a set cross section and standard deviation ; Compare the airflow velocity or ammonia mass fraction data of all surface unit grids of the cross section with the area-weighted average value thereof plus or minus a set multiple standard deviation to obtain the total area of the unit grids that meet the conditions; use the satisfaction of the set area ratio as the evaluation criterion for the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross section to form an exhaust gas uniformity evaluation scheme; The evaluation criteria meet the following requirements: ; ; ; in, is the threshold coefficient, 、 、 Respectively represent the total air velocity or ammonia mass fraction on the cross section 、 、 The sum of the areas of the cell grids within the range, Represents the area of any unit grid.
2. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 1, characterized in that: Compare the relationship between the airflow velocity or ammonia mass fraction data on all surface unit grids of the cross section and its area-weighted average value ±1 to 3 times the standard deviation, and count the total area of the unit grids that meet the conditions.
3. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 1, characterized in that: Less than or equal to 1; when When , it is assumed that the physical quantity on the cross section conforms to the law of normal distribution.
4. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 2, characterized in that: If either the statistical results of the uniformity of the air flow velocity or the ammonia mass fraction fails to meet the evaluation criteria, the flow disturbance, flow guide or flow equalization components shall be redesigned, a simulation model shall be established, and the uniformity shall be calculated and evaluated until the requirements are met.
5. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 1, characterized in that: Computational fluid dynamics simulation software is used to perform steady-state simulation of the exhaust gas flow field in the SCR reactor.
6. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 5, characterized in that: Post-processing software is used to extract the airflow velocity, ammonia mass fraction and area data of each surface grid on the set cross section.
7. The method for evaluating the uniformity of exhaust gas from an SCR reactor of a low-speed marine diesel engine according to claim 1, characterized in that: According to the exhaust gas parameters and catalyst parameters at the SCR reactor inlet, the specific structure and size of the mixing pipeline, reactor and connecting pipeline are determined, and the urea injection amount is estimated.
8. The method for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine according to claim 1, characterized in that: The components that play a major role in the uniformity of exhaust gas at the catalyst layer inlet are designed, and a detailed geometric model is constructed; the main characteristic structures of the detailed geometric model are retained to obtain a simplified geometric model of the exhaust gas flow domain.
9. A system for evaluating the uniformity of exhaust gas from a SCR reactor of a low-speed marine diesel engine using the method of claim 1, characterized in that: include: An exhaust gas flow domain geometric model construction module is used to construct the exhaust gas flow domain geometric model and perform mesh division on the exhaust gas flow domain geometric model; The data extraction module is used to perform steady-state simulation calculations on the exhaust gas flow field in the SCR reactor and extract the airflow velocity, ammonia mass fraction, and unit grid area data of the set cross section at the inlet of the first catalyst layer in the SCR reactor; A uniformity evaluation scheme generation module is used to compare the airflow velocity or ammonia mass fraction data of all surface unit grids of the cross section with the area-weighted average value plus or minus a set multiple standard deviation to obtain the total area of the unit grids that meet the conditions; and to form an exhaust gas uniformity evaluation scheme by satisfying the set area ratio as the evaluation criterion for the uniformity of the airflow velocity distribution or ammonia concentration distribution of the cross section.