Study on mixing characteristics of multichamber static mixer

This study presents a specialized mixer design for gas burners aimed at improving the efficiency of phase mixing while minimizing system resistance and energy consumption. The proposed design utilizes a multi‐cavity mixer employing both radial and axial mixing techniques, eliminating the need for additional power supply. Full premixed combustion of small gas‐fired industrial boiler has the function of “inhibiting” NOx emission from the source, and good mixing of fuel and combustion supporting air in the early stage is the primary prerequisite for realizing full premixed combustion denitration. In this article, a radial multichamber static mixer is designed following the characteristics of full premixed mixing in gas‐fired industrial boilers. The results demonstrate that the maximum pressure drop of the static mixer is 806 Pa; for gas and air branch pipes, the resistance coefficient f decreases rapidly with the increase in Reynolds number; when Re ≥ 1.2 × 103, the resistance coefficient in the air branch pipe decreases slowly; when Re ≥ 4.5 × 103, the resistance coefficient of gas branch pipe decreases slowly. Additionally, the maximum calculation error of symmetrical gas branch pipe is 9.1%. The static mixer's inlet exhibits a converted velocity of 24 m/s, and the outlet demonstrates an airflow velocity of 23.2 m/s. As a result, a kinetic energy loss of 6.5% is observed. The static mixing chamber makes the airflow rotate and causes different gases to shear and mix. The mixing channel has the function of correcting the airflow deviation, especially for the sudden expansion section. Generally, the two gases can be mixed evenly at the outlet of the mixer. The standard k‐e model and realizable k‐e model are employed to simulate the sudden expansion channel, and indicate that the standard k‐e has a wider range of influence. Further investigation is recommended to better comprehend and optimize this particular area of influence.


INTRODUCTION
4][15][16] The quality of mixing performance, which represents the degree of mixing for multiphase and single-phase fluids with varying densities, holds significant importance in industrial production processes.In the design and utilization of mixers, harnessing the inherent fluid-generated vortex proves effective in shearing and mixing both single-phase and multiphase fluids.][19][20][21][22][23][24][25][26] In order to enhance mixing performance, the development of various types of mixers based on different principles has taken place. 13,27Mixers can be broadly classified into two categories based on their working principles: dynamic mixers and static mixer.Dynamic mixers require external power sources for operation, whereas static mixers rely on the fluid's own kinetic energy to achieve shear mixing, eliminating the need for additional power during the mixing process. 14Static mixers, with their self-sustained operation, have found widespread applications in diverse scenarios.][39] Gas-fired boilers are an essential source of air pollutants.Gas boiler pollutant treatment mainly involves fly ash emission treatment, Sulfide emission control, NO x emission control, and CO 2 emission control for achieving "carbon neutral and carbon peak."The characteristics of small gas-fired industrial boilers suggest that due to the low investment of the boiler itself and the high cost of denitrification, small gas-fired boilers generally use low NO x burners, rather than NO x post-treatment equipment alone, to directly inhibit the generation of nitrogen oxides, so as to achieve low-cost denitrification.NO x emission into the atmosphere will induce many environmental and social hurdles, such as ozone layer depletion, greenhouse effect, photochemical effect, acid rain, forest area reduction, and sea level rise, while ozone layer depletion is irreversible. 40The harm of NO x in the atmosphere to human body comprises myocardial infarction, chronic cough, neurasthenia, emphysema, hypertension, and inflammation.If the concentration of NO x in the air reaches a certain level, it will also provoke eye discomfort, chest tightness, headache, and nausea. 41Therefore, a new type of burner should be adopted for combustion denitration if the cost is considered from the source control.
Several successful design cases of boiler gas burners have demonstrated effective denitration effects.However, limited research has been conducted on the mixing conditions of gas and air.Regarding the fully premixed combustion mode, there is little research on the mixer, especially on the mixing degree and resistance characteristics of premixed combustion air and gas.Since it is difficult to explore the airflow conditions of each part by experimental methods owing to the complexity of the mixer structure, most researchers use numerical simulation to investigate the airflow mixing conditions.Zheng et al. 42 studied the uniformity of the airborne effluent from the chimney of the nuclear power plant using CFD (Computational fluid dynamics) method, concluding that the square pipe has better mixing property than the circular pipe, and adding elbow can enhance mixing.Yang et al. 43 discovered the mixing characteristics of the industrial water transmission pipeline with round sections and twisted pipes.The results unveiled that the flow stability of the straight pipe section downstream of the three twisted pipe arrangements was good, the velocity uniformity was different, and the velocity uniformity in the twisted tube of the non-acceleration regulator was poor.Sun et al. 44 researched the Chinese static mixer using the CFD method considering its structural complexity.It implies that the Chinese static mixer can realize the radial mixing of high viscosity fluid, effectively weaken the boundary layer, and strengthen the heat transfer of high viscosity fluid in the pipe.The strengthening effect is significant.Li et al. 45 designed a static mixing element for the mixing state of plastic molding and compared it with the temperature measured by an infrared imager, suggesting that the new mixer significantly curtailed the pressure drop and improved the mixing index.Fu and Yan 46 proposed the willow leaf shaped static mixer, revealing that the static mixer had eddy current at the end of the element, which could make the material exchange more frequent, and the back mixing phenomenon reinforced the mixing effect.Ou et al. 47 utilized the numerical simulation method to simulate the three-phase mixing characteristics of oil, gas, and water.The nonuniformity coefficient of water phase fraction was considerably improved after the mixer, and the change of flow rate did not impact the mixing effect.The outlet distance of the mixer increased to a certain extent, and the water phase mixing effect remained unchanged.Zhang et al. 48introduced the standard wall surface function to describe the flow near the wall surface.The analysis reflects that the mixing element has noticeable throttling, diversion, and vorticity effects, and effectively promotes the mixing of crude oil and chemicals.Zheng et al. 49 designed a high viscosity two-phase fluid static mixer, uncovering that the design of premixing unit significantly improved the mixing effect, the mixing effect of the mixer was enhanced with the increase in the fluid viscosity, and low viscosity liquids were mixed well.Wang et al. 50explored the two-phase mixing of toluene and water on SK static mixer, indicating that the liquid-liquid dispersion process in the mixer was the result of both dispersion mixing and distributed mixing, and the static mixer itself possessed preferable distributed mixing performance.Tao et al. 51 verified the dynamic uniformity of air supply in the air duct of a metro vehicle using the calculation correlation based on area and mass weighted average.The correlation and the formula were in good agreement when the uniformity was higher than 60%.Gong et al. 52 researched the velocity distribution characteristics in the mixer under the action of twisted blades, as well as the enhancement effect on turbulence.Twisted blades had a significant enhancement effect on turbulence when the blades were 3.
The gas mixer to enhance the gas mixing effect is widely used in the denitration system, especially in the selective catalytic reduction process.The gas uniformity must be fully researched to enhance the full contact and mixing of flue gas and catalyst.Yu et al. 53 explored the velocity uniformity upstream of AIG with the numerical simulation method, demonstrating that the orifice opening ratio exerted the greatest influence on it.Besides, the flow equalizing device was optimized and improved to avoid eddy current upstream of the catalyst.Li and Fu et al. 54,55 optimized and controlled the reactor based on the uniformity of NO x at the outlet of the SCR (Selective catalytic reduction) reactor.The results suggested that the NO x uniform concentration at the outlet of the SCR lessened ammonia consumption and increased the denitration efficiency by 0.75%.Ye et al. 56 investigated the denitrification efficiency of SCR through numerical simulation, revealing that the mixing degree and temperature uniformity of the reactor inlet had a crucial influence on the efficiency of SCR.Wang et al. 57 evaluated the denitration efficiency of SCR in a 600 MW coal-fired unit and presented that the denitration performance of the equipment was remarkably affected when the uniformity of NH 3 /NO molar ratio at the denitration reactor interface was poor.Liu et al. 58 reduced the nonuniformity of airflow velocity into the SCR reactor by optimizing the parameters of the deflector in the SCR reactor, contributing to noticeably curtailing NO x emission.Through numerical simulation, Zhou et al. 59 verified that the improved grating of the static mixer significantly enhanced the uniformity of the flow field, resulting in the denitrification performance of SCR.An 60 discovered that the nonuniformity of SCR inlet fly ash led to the deterioration of reactor wear.Gao et al. 61 improved the denitration efficiency significantly by adjusting the position, number of baffles, and shape of the mixer.
As illuminated by the above analysis, premix quality is critical for achieving certain denitration efficiency.Similar to SCR denitration full-premixed combustion, the quality of full-premixed combustion is crucial to reach the desired denitration effect.The resistance characteristics of the static mixer must be considered to lessen the operating cost.Specifically, in the process of using the full premixing technology, the fuel gas and combustion supporting air must be well mixed, and the pressure drop of this mixer must be within the appropriate range, so as to achieve the established purpose of denitration.Following these characteristics, a multichamber static mixer is designed to make the mixing quality at the outlet of the static mixer reach 9% (excellent mixing quality), and the maximum pressure drop of the static mixer is 8 Kpa.The structure of this article is detailed as follows.In the Section 1, the current situation of the industry is introduced.Then, the design of the multichamber static mixer model is described in the Section 2. Next, the numerical simulation of a multichamber static mixer is presented in the Section 3, followed by numerical simulation results and analysis in the Section 4. Finally, conclusions are drawn in the Section 5.

Physical model
The premixing quality of the full premixed burner is directly related to the dynamics and temperature field in the furnace, which further influences the boiler thermal efficiency and NO x emissions.After the appropriate equivalence ratio is determined, how to fully mix the gas and air and how to determine the resistance characteristics of the static mixer have an essential impact on NO x emission reduction, boiler thermal efficiency improvement, and operating costs.The static mixer is a mixing element installed in the pipeline, which enables the fluid to be mixed using mutual shearing, small strand mixing, and outlet confluence depending on its kinetic energy.The mixer is installed in front of the full premixed surface burner and behind the forced draft fan.In this study, dual channel radial mixing cyclone is adopted to provide premixed fuel for a 2 t/h natural gas boiler.Thus, the hourly natural gas consumption is 2 t × 0.7 MW × 3600/36.22MJ/Nm 3 /0.92= 150 Nm 3 /h.By the formula of CH 4 + 2O 2 = CO 2 + 2H 2 O, it can be estimated that the volume ratio of natural gas to air is 1:10.According to the gas and air inlet dimensions of the static mixer, the air inlet velocity is determined to be 40 m/s, and the gas velocity is 25 m/s.Before the static mixer, a forced draft fan is set to overcome the full pressure drop inside the boiler.Figure 1 illustrates the structural model of the static mixer.
Figure 1 demonstrates that the component is divided into three chambers from the outside to the inside diameter: air chamber, swirl chamber, and mixing chamber.The air chamber is the outermost annular structure of the static mixer and serves as the annular channel for air to enter the mixer independently.On the base circle of R108, nine air inlets with a diameter of 60 mm are uniformly distributed within 240 • .The side of the air chamber is a swirl chamber.The partition between the air chamber and the swirl chamber is provided with a forward inclined air inlet slot (viewed from the air inlet angle), with an inclined angle of 15 • .The swirl chamber is connected to the 12 fuel inlet conduits with an inner diameter of 40 mm.The side of the swirl chamber is a mixing chamber, and the inclination of the partition between the two chambers is 15 • in the opposite direction.A forward groove is opened between the swirl chamber and the mixing chamber to better enhance the swirl intensity in the chamber and enhance the disturbance.The static mixer outlet is directly connected to the burner inlet.Considering that the static mixer has a forward grooved air inlet and a reverse grooved exhaust of mixed gas, it is named a FB forward and reverse coupling mixer (forward and backward coupling).The FF mixer, as contrast to the forward air inlet, is called when the mixed gas is discharged in the forward slot.
The specific positions of each plane in Figure 1 are listed in Table 1.The coordinate origin in Figure 1 is located on the central axis of the mixed gas duct.Hence, the positions of each section in Table 1 are arranged longitudinally along the Z-axis.

2.2
Grid independence verification

Grid quality comparison
The channel of the FB mixer has three chambers connected through inclined grooves.Therefore, the structure of the flow channel is complex, and the structural grid divided by blocks cannot be used.Therefore, the complex fluid domain structure extraction method is utilized to divide the unstructured grid.Six kinds of unstructured grids are generated by controlling the minimum grid size.Table 2 presents the grid generation.As observed in Table 2, the number of grid cells is 116,957 when the maximum grid size is 8 mm.At this time, the quality of some grids is below 0.2, and the accuracy of calculation results is relatively low.When the maximum grid size is less than 4 mm, the aspect ratio is above 0.29 (above 0.2 can be calculated).When the maximum grid size is 1.6 mm, the number of grid cells is 12,483,636, and the grid quality is not significantly improved.Therefore, conditions 2 to 6 meet the requirements of the solver FLUENT.

Section channel grid structure
The gas inlet and air inlet channels are straight pipes, and the main pipe has no branches, suggesting that the grid formed is relatively simple.Then, the air and gas branch into the annular pipe, and the grid structure is complex.The unstructured mesh mainly has four sections in the static mixer, as exhibited in Figure 2. Except for the gas channel, all other sections form a uniform grid.The partition wall boundary between air chamber, swirl chamber, and mixing chamber is clear, forming a good calculation boundary.When the grid element is large (i.e., the grid element is larger than the wall thickness between the chambers), the grid directly spans the walls of the chambers to split, forming an error calculation domain.

F I G U R E 3
The calculated speed at each point on line one Therefore, the volume of the grid element here cannot be too large.Second, when the grid element is too large, the grid formed here cannot form a regular circle due to the small diameter of the gas channel, leading to large calculation errors.Physical divergence will occur when the error is too large.

Grid independence verification
The central axis of the top center of the mixing chamber and the outlet center of the static mixer are connected, 200 points on the two-point connecting line (line 1) are taken, and the velocity calculation values of each point are extracted.The calculation results are provided in Figure 3.As illustrated in Figure 3, the maximum axial velocity appears near 50 to 60 nodes when the number of grids is 110,000 and 840,000; with the increase in the number of grids, the maximum speed appears at 140 nodes when the number of grids exceeds 3.9 million, and the axial speed decreases under 4-6 working conditions when the number exceeds 150 nodes.This is in that the sudden expansion of the cross section of the mixing channel increases the flow area.Besides, the speed of each point on Axis 1 tends to be consistent when the number of grids exceeds 3.00 million, and there is little difference in calculation when the number of grids is increased.Since the calculation results do not change when the number of grids exceeds 3.0 million, 3.0 million (i.e., Condition 4) is taken as the calculation condition.

Basic governing equation
Concerning turbulent mixing of gas and air, the air momentum is greater than that of natural gas under the same conditions due to the large difference in the density of the two gases.Hence, the air first enters the outer ring channel (namely, the air chamber) and then enters the swirl chamber through the static cyclone partition between the two chambers to mix with natural gas.Since the gas with large momentum first rotates, this mixture has greater rotating momentum than the natural gas that first enters the outer ring chamber.Through numerical calculation, the results can be obtained, such as internal mixing velocity, turbulence intensity, and mixing uniformity.Fu and Yan 46 used the near wall function combined with the Realizable k-e model to calculate the mixing of NH 3 and NO x in the SCR reactor.The experimental results are in good agreement with the numerical simulation.The presence form of NO x in SCR reactors is mainly NO, accounting for more than 95%. 62Thus, the mass ratio of the two mixed gases is 17/30 = 0.57, while the mass ratio of natural gas to air in this simulation is 18/29 = 0.62.Regarding the mass ratio, the gases involved in this calculation exhibit a similar in mass ratio compared to those studied by Fu and Yan 46 in the context of SCR reactors, with a mass ratio error of 8.1%.Therefore, the realizable k-e model is also employed for the simulation calculation in this specific example.In other words, it is appropriate to employ a Realizable k-e model to calculate this example.Jet spreading angle will occur at the interface between the static mixer and the burner.Realizable k-e model expression is expressed as: where G K denotes turbulent kinetic energy generation caused by average velocity, J; G b represents the generation of turbulent kinetic energy caused by the influence of buoyancy, J.
where  T represents turbulent viscosity; C μ signifies constant in the standard k-e model.The expression of the realizable k-e model is described as: The formula and constant values involved in the formula are listed in Table 3. Formulae (1)-( 4) and Table 3 reveal that the turbulent kinetic energy equations of the standard k-e model and the realizable k-e model are identical, while the difference is that C μ in the dissipation rate e equation is not a constant and can be obtained by formula modification, enabling the calculation accuracy to be further improved. 63In the inertial bottom layer of the equilibrium boundary layer, C μ = 0.09, and the constants of the two models are the same.

Date processing method
The standard deviation method, also known as the standard deviation coefficient, is commonly used to evaluate the uniformity of the medium flow field.The velocity (concentration) field on the section is defined as 64,65 : where C V refers to the relative standard deviation of speed, %;  V represents speed standard deviation, m/s −1 ; V indicates the average value of section velocity, m/s −1 ; n denotes the number of measurement points of the velocity section.Generally speaking, the smaller the C V value, the more uniform the cross-sectional airflow distribution is. 64,66

Analysis of airflow from mixing chamber channel to outlet section
The cross sections from the top of the mixing chamber of FB static mixer to the outlet of the mixer are cut from top to bottom according to the longitudinal direction (Z direction).The data of the corresponding cross sections of the FF type mixer are taken as a comparison to make a cloud map.The top to bottom velocity nephogram of each section is depicted in Figures 4, 5, and 6.After the gas in the swirl chamber enters the mixing chamber, the upper velocity of the mixing chamber is presented in Figure 4A.The airflow rotates reversely and forms a noticeable vortex area.With the section gradually approaching the outlet, the speed of the entire section becomes uniform, and the high speed expands gradually.This is because the gas flow here is approaching the gas inflow of the mixer, as illustrated in Figure 4B,C.The swirling area on the cross section remarkably expands and rotates counter clockwise along the airflow direction, and the airflow velocity increases significantly.At the cross section (d), the gas in the swirling chamber flows out in large quantities, and the gas gathers downstream.In Figure 4D, the swirling area of the airflow considerably expands, and the average flow velocity of the gas in the cross section increases.Since the airflow rotates counter clockwise, the vortex area of the airflow center is located at the lower left of the section shown in the figure.Compared with FB type cyclone, the swirl outlets are arranged in the forward direction for FF type, as rendered in Figure 4E-H.FB mixer has better airflow uniformity than FF mixer as FB type shear mixing is more intense.
As the airflows downstream, it flows out of the mixing chamber and the pipe here is not slotted (Figure 5).Since the cross-section radius of the mixing duct is equal to 102 mm also, the airflow velocity increases with the mixing chamber, and a significantly high speed vortex region appears.In Figure 5A, the airflow vortex deviates from the section center.As the The average flow velocity of the airflow decreases, and the central vortex area increases significantly due to the blocking effect of the pipe.In Figure 5D, the low speed vortex zone has expanded to the right half of the region owing to the friction retardation effect of the pipe, suggesting that the airflow in the mixed duct decays rapidly, and the airflow is distributed unevenly across the entire section.As illuminated in Figure 5E-H, the vortex tends to the lower left part of the section ascribed to the change of airflow rotation and the increase of airflow rigidity, and the central vortex region is significantly reduced compared with that of the FB type burner.As observed in Figure 5, the flow velocity in the mixing channel is unevenly distributed.If the mixed gas is directly introduced into the boiler burner at this time, it will inevitably lead to the uneven distribution of combustible gas and natural gas in the full premix equipment.Then, a series of problems will emerge, including NO x emissions, emissions containing combustible material CO.Consequently, measures must be taken to make the natural gas in the export section evenly distributed.For this purpose, a simple and easy scheme is designed.In the first step, a circular channel with the shrinking mouth is set.In the second step, a circular channel with the flaring mouth is set after the shrinking mouth channel.The length of both channels is 100 mm.
As mentioned above, the first working condition is selected.For the pipe with a sharp section, the pipe diameter is equal to 164 mm downstream of the mixture channel, and the velocity cloud image is displayed on four equidistant cross sections (Figure 6).Regarding both FF and FB mixers, the velocity to the outlet of the mixer is severely skewed.Compared with Figure 5, the unbalance degree of the airflow is aggravated.The reason for the unbalanced velocity is that the section shrinkage induced the sudden increase of the airflow velocity.As a result, the deviation of the airflow on the section of the pipeline is more severe compared with the mixing channel.Therefore, it is not advisable to use only section shrinkage to balance gas flow so as to achieve uniform distribution of balanced natural gas in the exit section.
With respect to FB type or FF type static mixer, if the shrinkage pipe is set, the airflow is skewed, and the deviation degree and the diffusion degree of airflow are not significantly reduced.Therefore, incomplete combustion and substandard NO x emission will be provoked if the airflows into the burner at this time.Moreover, the influence on the rigidity of FF type airflow is not significant compared with that of FB type.
The gas flow deviation in the shrinkage pipeline is serious, and the distribution of natural gas is uneven in the circumferential direction.The flow into the burner leads to the uneven circumferential combustion of the cylindrical burner, resulting in an uneven temperature field inside the boiler.Therefore, the expansion section channel is set after the shrinkage channel, and the section radius is R = 102 mm.As presented in Figure 7, the low-speed region is concentrated in the lower left position of the mixing channel center for the FB type (a) section.As the airflow continues to flow toward the outlet, the airflow velocity attenuation is represented by the expansion of the central low-speed vortex region and the narrowing of the high-speed region of the section owing to the friction blocking effect between the rotating airflow and the tube wall.The section expansion pipe plays an excellent flow sharing function, which is manifested as the low speed vortex area gradually migrating to the flow center.The airflow in the exit section (d section) exhibits good symmetry, implying that the expansion pipe has achieved the goal of uniform airflow distribution.Concerning the FF mixer, the rigidity of the airflow in each section is slightly enhanced compared with the FB mixer, and the deviation degree of the airflow is not much different from the FB mixer (Figure 7E-H).At the exit section of the static mixer, both FB and FF airflow cloud diagrams demonstrate acceptable uniformity (high in the periphery, low in the center, and symmetrical in the center).Hence, the abrupt expansion section channel is selected as the interface between the static mixer and the burner to eliminate the uneven distribution of airflow in the section.In Figure 7D,H, velocity heterogeneity caused by airflow rotation has decreased significantly at the exit of the section.

Analysis of airflow velocity vector distribution and pressure loss in the mixed cavity
Due to the complex structure of the mixing cavity and the opposite rotation direction of FB and FF airflow, it is difficult to judge whether the gas and air are well mixed and whether there is a flow "stagnant" area only by relying on the velocity cloud image.Meanwhile, there are many vortex areas as a result of the numerous flow channels.Thus, the velocity vector and pressure loss in the mixing cavity and the mixing channel in detail should be analyzed to develop a more intuitive understanding of the static mixer performance.As illustrated in Figure 8, the outer cavity is an air cavity.The air has already rotated before entering the gas cavity (midcavity) and rotates counter clockwise (FB) or clockwise (FF) in the midcavity by changing the direction of the chute.Therefore, both the gas cavity and the mixing cavity can generate strong rotational mixing depending on their own structure, ensuring the uniformity of the velocity distribution in the whole section of the gas and air entering the mixing channel.The advantages of this structure are two aspects.First, it can produce strong rotating mixing depending on its own structure, allowing gas and air to be evenly mixed, rather than relying on the fan blade rotation to produce a mixing effect.Compared with the static mixer that relies on fan rotation to produce mixing, its power consumption is reduced.Second, for small-tonnage boilers (2 t/h in this study), the induced draft fan is adopted to supply gas to the boilers.The gas pipeline layout is greatly simplified once the gas supply pressure is guaranteed.Figure 9 provides the velocity vector diagram of the channel with a sharply reduced section.It can be observed from the figure that the velocity vector of the channel with a sharply reduced section gradually becomes uniform along the four sections of the flow (P-8/P-7/P-6/P-5), consistent with the calculation results in Figure 6 (the low-speed region is near the center of the section).As revealed by comparing the two models, the rotation direction of the velocity vector is opposite.The velocity vector diagram reflects that the velocity distribution of the two types of mixers at the wall is uneven.In other words, the airflow distribution in the section of the sudden shrinkage channel is uneven.If the mixture gas is directly passed into the burner for combustion, it will inevitably cause the uneven distribution of combustible gas around the surface burner, resulting in an uneven temperature field in the furnace, consistent with the conclusion drawn in Figure 6.Thus, a sudden expansion channel must be added to equalize the airflow.

4.3
Analysis of pressure loss of static mixer

Pressure loss analysis of each branch pipe of airflow
Since the static mixer contains 9 branches of air and 12 branches of gas, its structure is relatively complex.The pressure drop of this equipment must be evaluated for the furnace type to be equipped with a suitable drum-induced draft fan.The inlet wind speed is 15-25 m/s, and the imported gas speed is 18-30 m/s.Therefore, 10 representative points were selected from the gas and air inlet to the static mixer outlet.Specifically, the central point of the inlet (1), the average pressure of the inlet point of each branch pipe (2), the average pressure of the outlet point of each branch pipe (3), and other 7 points (4-10 points of gas and air mixture) were distributed from the top of the mixing channel to the outlet.The pressure distribution at 10 points is displayed in Figure 10. Figure 10 demonstrates that the pressure at the corresponding position points from the gas inlet to the gas branch pipe outlet is higher than the air pressure, and the highest gas pressure is 160 Pa higher than the air pressure.The outlet pressure reaches 5000 Pa when the gas and air are well mixed in the mixing chamber.The total pressure drop of the gas passage and air passage is 690 and 530 Pa, respectively.The total pressure drop of the static mixer is 690 Pa, and the outlet The static mixer is illustrated in Figure 11A.The red arrow represents the main air inlet, and each air branch pipe is numbered clockwise along the main air ring pipe as branch pipe (1)-( 9).Ten value points were evenly distributed in each branch pipe, and the calculated pressure value was taken to analyze the pressure drop of each branch pipe and then determine the resistance of each branch pipe and the change of the resistance coefficient under different Reynolds numbers.After the airflows through the horizontal branch pipe, it enters the vertical air branch pipe through the elbow.The letter v is added after the label of the horizontal branch pipe, as exhibited in Figure 11A.
Figure 11B indicates the pressure variation of each of the 9 branch pipes of the air inlet.The pressure drop of the 9 branch pipes is distributed symmetrically on both sides of branch pipe 5. Due to the large diameter of the gas main pipe, the inlet air enters the main pipe from the left and right sides and is subject to the influence of the thick main pipe and the turning resistance at the branch pipe inlet.The air on both sides flows to branch pipe 5. Considering that the pressure on both sides is the same, the low static pressure is formed in this study.First, the local resistance at the entrance of branch pipe 5 is overcome to enter branch pipe 5, and a relatively high airflow velocity is formed in branch pipe 5.The local resistance of airflow into branch pipe from an annular main pipe is the largest, among which branch pipe (1) and branch pipe ( 9) have the highest local turning resistance, leading to the lowest air velocity in branch pipe (1) and branch pipe (9).The overall performance is that the flow velocity of branch pipe 5 is higher than that of other branches on both sides successively, inducing the pressure drop of each branch pipe gradually decreasing from 5 to 1 (or to 9). Figure 11B reveals that after the airflows into branch pipes ( 1)-( 9), the branch pipes with higher inlet velocity have larger pressure drop owing to the same material, length, and physical properties of each branch pipe.With the symmetry of the structure, moreover, the pressure drop of each branch pipe is symmetrically distributed along both sides of the branch pipe (5), and the numerical calculation is in good agreement with the theoretical analysis.The maximum pressure drop in branch pipe 5 of air intake is 399 Pa, and the minimum pressure drop in branch pipe 1 is 190 Pa.The numerical calculation is consistent with the theoretical analysis.As exhibited in Figure 11C, when the airflows into the vertical branch pipe, the maximum resistance drop still appears in branch pipe 5, whose value is 115 Pa, and the minimum pressure drop appears in branch pipe 1, whose value is 82 Pa.This deviation is provoked by the difference in the flow rate of each branch pipe, consistent with the above analysis.The pressure drop of the vertical branch is smaller than that of the horizontal branch since the velocity of the horizontal branch is larger than that of the vertical branch, and the length of the horizontal branch is longer than that of the vertical branch.The above numerical calculation is in good agreement with the theoretical analysis.
The Reynolds number is changed by adjusting the flow velocity of the annular main pipe to obtain the change of the resistance coefficient of each branch pipe when the Reynolds number (Re) changes, as depicted in Figure 12.The relation between drag coefficient and velocity and pressure drop is expressed as: In Figure 12, the drag coefficient in each tube decreases with the increase of the Reynolds number.For the same tube, Equation (9) implies that the drag coefficient is proportional to the pressure drop (△p) and inversely proportional to the velocity squared (u m 2 ).The influence of velocity change on the drag coefficient (f ) is greater than that of the drag drop.Therefore, the resistance coefficient for the same pipe decreases with the increase in the Reynolds number.For different branch pipes, different turning resistances result in different pipe speeds, among which the turning resistance in branch pipe 5 is the least, leading to the highest velocity in branch pipe 5.The above analysis results unveil that the resistance coefficient in branch pipe 5 is the largest, and the resistance coefficient changes the most when the Reynolds number changes.The results in Figure 12 are consistent with the theoretical formula (9).
Additionally, as the distance between both sides of the symmetrical structure and branch pipe (5) increases, the drag coefficient tends to change slowly along with the increasing Reynolds number.Branch pipes (1) and ( 9), branch pipes ( 2) and ( 8), branch pipes (3) and ( 7), and branch pipes (4) and ( 9) are located on both sides of branch pipe (5)  structure.With the symmetry of the structure, the variation trend of the resistance coefficient in each symmetrical branch pipe also presents symmetry.Among the four pairs of symmetrical pipes, the resistance coefficients of branch pipe (1) and branch pipe ( 9) have the smallest change because branch pipe (1) and branch pipe (9), which are closest to the total inlet, have the lowest velocity in branch pipe owing to the maximum local resistance at the inlet.Consequently, the resistance coefficients of branch pipe (1) and branch pipe ( 9) are least affected by the Reynolds number.The resistance coefficients of tube pair (1) and ( 9) change the most smoothly and increase as the distance from branch pipe 5 decreases.

4.3.2
Pressure loss analysis of each branch pipe of gas flow The central channel of the mixer is the natural gas inlet, whose intake structure is provided in Figure 13. Figure 13A suggests that each inlet branch pipe has a symmetrical structure, with the total inlet pipe located in the center and the 12 inlet pipes uniformly distributed along the circumference in a radial shape.Figure 13B demonstrates that among the 12 gas branches, branch pipe 3 has the largest pressure drop, with a pressure drop of 201 Pa; branch pipe 5 has the smallest pressure drop, with a pressure drop of 182 Pa.The pressure drop deviation of 12 gas branches is 9.4%.According to the theoretical analysis, the pressure drop of 12 branch gas pipelines should be completely consistent attributed to the complete symmetry of the structure.However, the gas pressure drop of each branch pipe cannot be the same because of the uneven intake and the differences in the manufacturing process, and the error is within a reasonable range.As illustrated in Figure 13C, the pressure drop of gas decreases compared with that of gas in the corresponding horizontal pipe when gas enters the vertical branch pipe.The reason is similar to the mechanism in the air branch pipe.Specifically, the corresponding velocity of gas entering the vertical pipe decreases, and its resistance drop decreases relative to the corresponding horizontal pipe since the length of the vertical pipe is short.The maximum pressure drop in the vertical branch was 41 Pa, and the minimum pressure drop was 38 Pa.The calculation error was 7.9%.The reason for the large drag drop at pressure calculation points 1-3 is that the distance between the calculation starting point of the vertical pipe and the calculation end point of the horizontal pipe is relatively close, and the gas flow direction has not completely turned to the vertical direction until the calculation point 3.Under the local drag effect of the 90 • elbow, the drag drop at points 1-3 is large.Subsequently, the gas pressure remained unchanged.The maximum pressure was reduced by 10 Pa from the third data point to the end of the pressure point.The vertical pipe outlet pressure is equal.As revealed in Figure 13B,C, the total resistance drop of the gas branch pipe is mainly derived from the local resistance of the branch pipe inlet and 90 • elbow, and the resistance drop along the branch pipe is small.When the natural gas inlet flow rate changes, the Reynolds number of each branch changes.However, the theoretical Reynolds number of each annular branch is the same since the 12 branches are symmetrical in structure.Thus, it is believed that the Reynolds number of each branch is the same.As illuminated in Figure 14, the resistance coefficient of each branch tube changes with the Reynolds number when the Reynolds number is changed.The resistance coefficient of both horizontal and vertical pipes decreases as the Reynolds number increases.As the airflow flows in the horizontal branch pipe (Figure 14A), the resistance coefficient changes greatly when Re <4.5 × 10 3 ; the change of resistance coefficient is relatively gentle when Re >4.5 × 10 3 ; the calculation error increases slightly with the increase in Reynolds number.The maximum calculation error at the outlet of the gas branch pipe is 6.3%.
When the airflows into the vertical branch pipe (Figure 14B), the drag coefficient decreases more significantly with the increase in Reynolds number, and the drag coefficient deviation between the vertical branches gradually increases with the increase of Reynolds number.Specifically, the calculation difference between the drag coefficient between the branches becomes larger due to the change in Reynolds number.The maximum calculation error of the vertical branch pipe is 12.1%.

Pressure-velocity coupling
As previously mentioned, the airflow inlet velocity is 40 m/s, while the gas inlet velocity is 24 m/s.Utilizing calculations based on the gas and air flow rates, states and inlet area, the inlet velocity is converted to 23.2 m/s.Consequently, the outlet velocity of the mixer is determined to be 23.2 m/s, resulting in a momentum loss of 6.5%.

Temperature distribution in mixing chamber
The flow channel complexity within the mixing chamber and the friction between the mixing chamber wall and the airflow can cause fluctuations in the temperature of the gas as it enters the mixing chamber.These variations can lead to abnormal increases in fluid temperature within the mixing chamber and uneven temperature distribution across the outlet cross section.Such temperature irregularities can lead to a detrimental effect on the combustion conditions in the furnace, ultimately impacting the thermal efficiency and clean combustion of the boiler.Therefore, conducting an analysis of temperature changes within the mixing chamber is of paramount importance.Figures 15 and 16 provide a visual representation of the observed temperature changes in each section.
Figure 15 presents the temperature observed at various sections within the system.The peripheral air chamber demonstrates a relatively consistent temperature across all four cross sections.Upon entering the middle gas chamber, the gas temperature, being lower than the air inlet temperature, results in a lower temperature within the gas chamber.As the gas and air mix within the chamber, the mixing temperature becomes slightly lower than the air temperature in the outer chamber.Moving into the mixing inner chamber, the temperature gradually increases from top to bottom, with an overall temperature change of less than 1 • C. When the gas enters the mixing channel from the mixing chamber, the temperature change trend of the mixed gas flow from section P12 to P9 exhibits a gradual transition.There may be some localized areas with lower temperatures within the channel section.However, on the whole, the temperature of sections P12 to P9 gradually tends to equalize and become uniform.The temperature difference across the entire section remains relatively stable, with the maximum temperature difference being less than 1 • C.
Figure 16 provides valuable information regarding the temperature distribution within the system.It is observed that the mixing sections P8 to P5 experience a slight increase in temperature, gradually reaching uniformity at the cross-section outlet.On the other hand, in the case of section P4 to P1, representing a sudden expansion channel, a slight temperature decrease is observed upon entering this section.At the cross section near the mixer outlet (P12), the temperature distribution becomes uniform, with a maximum temperature difference of less than 0.3 • C. By examining both Figures 15 and  16, it becomes evident that the temperature change from the gas entering the air chamber to the outlet (P1) of the mixer is less than 1 • C. Furthermore, at the outlet of the mixer (P1), the temperature exhibits uniformity, with a temperature difference of less than 0.06 • C. Therefore, this type of mixer demonstrates good temperature balance prior to entering the burner.

Comparison between standard k-e model and realizable k-e model
The outlet section of the mixer is a sudden expansion channel to achieve good mixing of gas and air in the static mixer.Tominaga et al. 67 conducted a comparative analysis of four models (SKE, RNG, LK, RLZ) and determined that, when backflow conditions are present, the RLZ model and the other three models exhibit higher calculation accuracy compared to the SKE model.Yu et al. 63 suggested that a realizable k-e model instead of the standard k-e model can be adopted to calculate sudden expansion flows, so as to obtain better jet expansion Angle and higher calculation accuracy.As observed in Figure 17, the longitudinal section of the sudden expansion channel of the static mixer is taken to calculate the airflow expansion at the downstream flaring channel of the mixing channel.In Figure 17A, the standard k-e model is utilized for calculation.Figure 17B reflects that the realized k-e model can be used for calculation.Other conditions are the same.Additionally, the vortex region formed by the calculated results of the realized k-e model is significantly reduced.Under the same working condition, specifically, the calculated results of the standard k-e model have a much wider influence range, which is almost to the exit of the mixer, while the realized k-e influence range is only limited to the step of the sudden expansion section.The influence range of the standard k-e model is 5 times that of the realized k-e model.The correction results in the realized k-e model in Table 3 present significant effects.

Evaluation of mixing effect of flow rate and concentration
After the air and natural gas entered the static mixer and were rotated, cut, and blended in the channel inside the static mixer, the uniformity of the 12 planes of the static mixer was evaluated according to the evaluation criteria in Section 2.2.The evaluation results were represented by a radar map.As illustrated in Figure 18, the standard deviation of velocity and concentration on different sections gradually decreases with the flow direction.Attributed to the short mixing distance between the two gases in the mixing chamber, the C V and C  values are all over 60%, and the concentration deviation value C reaches over 90%.The theory in Section 2.2 demonstrates that the distribution of velocity and concentration is extremely uneven.With the flow downstream, especially into the mixed channel of section shrinkage, the velocity deviation (C V ) decreases to 34%, the concentration of Cρ deviation decreases to 55%, and the airflow distribution is still unqualified.After the mixture gas enters the burst expansion channel, C V drops below 19%, the value of C V on the exit section is 5%, and the value of C  on the exit section is 9%.Therefore, both the velocity and concentration measured by the standard deviation are excellent.Moreover, the mixer has a good mixing effect, and the burst expansion section exerts an excellent mixing effect on the two kinds of gas.

CONCLUSIONS
1.The resistance coefficient of the static mixer gradually decreases with the increase in Reynolds number.The resistance coefficient changes little when the Reynolds number exceeds a certain value.The maximum total pressure drop of air branch pipe 5 is 514 Pa, and the minimum pressure drop of air branch pipe 1 is 272 Pa.The maximum and minimum pressure drops of the gas branch pipe are 242 and 220 Pa, respectively.The relative calculation error of the gas branch pipe is 9.1%.2. The vortex chamber of the mixer is connected with a sudden shrinkage channel and then a sudden expansion channel.
The value of velocity deviation C V decreases from 64% to 5%, and the concentration deviation C  decreases from 77% to 9%.The evaluation grade of concentration and velocity deviation at the outlet of the mixer is excellent.
3. Considering the influence of wall friction and process drop within the mixer, the average momentum at the mixer outlet is reduced by 6.5%.This reduction provides assurance for a safe and cost-effective gas supply to subsequent burners.
This mixer under discussion represents an innovative gas-air mixing device designed to ensure premixed gas with sufficient pressure and uniform mixing.Its purpose is to facilitate clean combustion in gas boilers, forming the basis for achieving low NO x combustion.Future research endeavors will utilize the outlet of the static mixer as a boundary to conduct comprehensive simulations and experiments focusing on fully premixed combustion.

F I G U R E 1
Static mixer (FB type) structure diagram and position of each plane

TA B L E 1
Location name description of each plane

F I G U R E 2
Schematic diagram of static mixer unstructured grid

8
Velocity vector diagram of internal and external mixing cavities and mixing channels.(A) P-11/FB and (B) P-11/FF H) F I G U R E 9 Velocity vector diagram of abrupt section channel (R = 82 mm).(A) P-8/FB, (B) P-7/FB, (C) P-6/FB, (D) P-5/FB, (E) P-8/FF, (F) P-7/FF, (G) P-6/FF, and (H) P-5/FF F I G U R E 10 Pressure distribution of gas and air passages pressure of the low pressure Roots fan is about 9800 Pa.Therefore, the static mixer has less resistance loss and can meet the mixing requirements of the boiler.

11
Pressure drop diagram of air branch pipe (1)-(9) (Re = 3 × 10 6 ).(A) Schematic diagram of intake air structure, (B) pressure drop of each horizontal branch pipe, and (C) pressure drop of each vertical branch pipe

ReF
I G U R E 12 Resistance characteristics of horizontal air branch at different Reynolds numbers

13 14
Schematic diagram of pressure change of gas 1-12 branch pipe (Re = 4 × 10 3 ).(A) Schematic diagram of inlet structure of static mixer, (B) pressure drop of each horizontal branch pipe, and (C) vertical pressure drop of each branch pipe Change of resistance coefficient of 12 gas branch pipes with Reynolds number.(A) Resistance coefficient characteristics of horizontal gas branch pipe and (B) resistance coefficient characteristics of vertical gas branch pipe

F I G U R E 15
Temperature nephogram of eight cross section of the mixing chamber F I G U R E 16 Temperature nephogram of eight cross section of the mixing duct U R E 17 Comparison of sudden expansion section calculation between standard k-e model and realizable k-e model.(A) Simulation results of the standard k-e model and (B) simulation results of the realized k-e model F I G U R E 18 Comparison of velocity and concentration standard deviation of different sections of static mixer Standard k-e and realizable k-e parameter options TA B L E 3 C 1ε /C 2ε /C 3ε default constant for dissipation rate  k Prandtl number of turbulent kinetic energy   Prandtl number of dissipation rate Y M the influence of turbulent pulsation and expansion of compressible fluids on the total dissipation rate  t turbulent viscosity, Pa⋅s C μ the function of average strain rate and curl C 1 /C 2 specific values in the implementable k- model G k turbulent kinetic energy generated by average velocity, J AUTHOR CONTRIBUTIONSHongwei Shi: Writing-original draft.Ziqiang Liu: Review and editing.