Optimization Design of Nozzle Structure Inside Boiler Based on Orthogonal Design

Abstract

This article adopts an orthogonal experimental design method to establish a four-factor three-level experimental group by varying the structural parameters at the nozzle outlet, including the number of orifices, orifice diameter, distribution circle diameter, and inclination angle of the base. The three-dimensional jet flow field in the jet pipe was numerically simulated. Through the use of the entropy generation method, Q-criteria, range analysis, and significance test, the working characteristics of the jet pipe were thoroughly investigated. The results show that the orifice diameter has a significant impact on the axial force acting on the jet pipe, while the number of orifices has a minor effect. The distribution circle diameter and inclination angle of the base have very little influence. The final confirmed optimal combination of nozzle structure parameters is as follows: the number of spray holes is 40, the diameter of each spray hole is 1.5 mm, the distribution circle diameter is 22 mm, and the inclination angle of the bottom cover is 30 degrees.

1. Introduction

Jet pipes are commonly used jetting devices that find wide applications in industrial, agricultural, and domestic sectors. The study of jet pipe performance is not only beneficial for improving jetting effectiveness and precision but also crucial for achieving energy-saving and environmental goals. Research on the performance of jet pipes holds significant importance in enhancing jetting effectiveness, boosting operational stability, and reducing energy consumption.

Jalal Bahreh Bar et al. studied the influence of nozzle diameter and pressure on spray angle and flow rate [1]. Pang et al. conducted numerical simulations to investigate the effects of nozzle arrangement height, arrangement radius, and injection angle on spray performance, and concluded that the nozzle arrangement position plays an important role in droplet evaporation [2]. Chen et al. performed detailed experimental measurements on a large droplet spray nozzle and found that simple modifications to the nozzle (increasing the length of the mixing chamber by 50%) significantly affect its performance [3]. Ma et al. analyzed the performance improvement of indirect evaporative coolers from the perspective of nozzle arrangement and determined the optimal arrangement scheme [4]. Jiwen Cui et al. investigated the effects of common geometric deviations, such as diameter error, conicality, and inclination, on the internal and jet flow dynamics [5]. Fabiano Griesang et al. analyzed the influence of nozzle hole angle and flow rate on the quality of spray distribution [6]. Miao et al. studied the influence of different two-dimensional transparent gap nozzles on flash evaporation fuel spray [7]. Wu et al. investigated the effect of nozzle cone angle on performance [8]. Reto Balz et al. studied the performance under different nozzle arrangements [9]. Safiullah et al. explored the impact of aperture size and injection pressure on diesel spray [10]. Nicholas Prociw et al. investigated the influence of nozzle attachment geometry on liquid release rate in fluidized beds [11]. Dai et al. studied the fragmentation and flow of bubbles in different positions within the nozzle, as well as the influence of nozzle structure on spray [12]. Wang et al. studied the performance of nozzles by changing the ratio of inlet and outlet nozzle diameters [13]. Wu et al. investigated the effect of the ratio of nozzle length to diameter and the number of nozzles on performance [14]. V. Varadaraajan et al. studied the influence of radial injection angle on performance [15]. Clément Rouaix et al. investigated the influence of nozzle diameter on performance [16]. Abhishek Dubey et al. compared the performance of single-hole and multi-hole nozzles [17]. Sun et al. discussed the performance effects of differently structured nozzles [18].

There are indirect methods for studying nozzle performance and optimizing its parameters, which include the following: Ehsanallah Tahmasebi et al. conducted a simulation study on the flow behavior inside fuel injector nozzles and applied it to various scenarios with different geometric shapes and flow conditions [19]. Florentina-Luiza Zavalan et al. conducted a multi-objective optimization study on the influence of nozzle shape on the uniformity of deposition layers and found that optimized nozzle shapes can provide more uniform deposition layers [20]. Fang et al. found through numerical studies that reducing the distance from the nozzle to the surface results in thicker liquid films lowered velocities, and increased non-uniformity of the liquid film [21]. The layout of the nozzle array has minimal effects on the thickness and velocity of the liquid film but significantly influences the morphology of the liquid film [21]. Abdul Quddus et al. studied the impact of nozzle spacing on thermofluidic parameters [22]. K. Lachin et al. modeled the spray properties generated by a dual-flow nozzle through scale analysis [23]. Jin Zhao et al. numerically simulated the flow focusing pattern and morphological changes in two-phase flow inside the nozzle using the volume of fluid method [24]. G. Vasu et al. studied the impact of spray nozzle aperture on the humidification of reaction gases in fuel cells [25]. Aobo Liu et al. explored the influence of nozzle atomization liquid mass, velocity, and angle on the performance of nozzles [26]. Hubertus Siebald et al. analyzed the functionality of nozzles through an acoustic monitoring system [27]. Abdul Quddus et al. studied the physical principles of steam injection characteristics through supersonic steam injection [28]. Han et al. compared the performance of arc jet nozzles, full cone nozzles, and flat nozzles [29]. Yin et al. studied the impact of nozzle parameters on plastic separation equipment in order to obtain optimal nozzle parameters [30].

The mentioned research covers factors such as nozzle diameter, pressure, arrangement height, and arrangement radius.

The aforementioned studies can provide reference and inspiration for related engineering fields. However, none of them have addressed the study of axial forces acting on the spray nozzle. Therefore, this paper aims to investigate the performance of the spray nozzle, with particular focus on the axial forces acting on it. It aims to explore the impact of optimizing design parameters on the performance of the spray nozzle, and provides recommendations and conclusions for optimizing the performance of water spray nozzles.

2. Calculation Models and Methods

2.1. Physical Models

The spray nozzle is mainly composed of six components: port cover, inlet, outer tube, connection, bottom cover (with the nozzle located on the bottom cover), and inner tube. The schematic diagram of the spray nozzle model is shown in Figure 1.

The working principle of the spray nozzle is illustrated in Figure 2. The liquid enters the nozzle from the inlet (water inlet) and then sprays out from the nozzle (water outlet) through the interlayer (water flow channel) between the inner and outer tubes of the spray nozzle.

The rated structural parameters of the jet pipe computational model are shown in

Table 1. Structural parameters of the jet pipe.

Elements	Size
Length of the Outer Pipe, L	500 mm
Outer Pipe Outer Diameter, Rww	32 mm
Inner Diameter of the Outer Pipe, Rwn	28 mm
Outer Diameter of the Inner Pipe, Rnw	18 mm
Inner Diameter of the Inner Pipe, Rnn	14 mm
Inlet Outer Diameter, Rsw	4 mm
Inlet Inner Diameter, Rsn	3 mm

.

The structural parameters of the nozzle include the number of spray holes (n), the diameter of the spray holes (r), the distribution circle diameter (R), the inclination angle of the bottom cover (α), and other structural parameters. The specific structural parameter diagram of the nozzle is shown in Figure 3.

2.2. Computational Mesh

In order to facilitate the study of the spray effect of the nozzle, this paper divides the nozzle into two parts: the internal flow region and the external flow region. The external flow region has a cylindrical shape with a base diameter of 200 mm and a height of 550 mm. The distance from the bottom surface near the inlet direction of the nozzle to the end port cover in the external flow region is 450 mm. The computational domain model grid partition diagram of the jet pipe is shown in Figure 4.

American aerospace engineer and numerical simulation expert Roache proposed the Grid Convergence Index (GCI) in the 1970s. GCI provides a more accurate assessment of the convergence of simulation results, avoiding limitations associated with relying solely on results evaluated using a single grid size. The calculation steps of GCI are as follows [31].

The formula for calculating the ratio of grid cell numbers is as follows:

$$r_{n,n+1}={\left(\frac{N_{n+1}}{N_n}\right)}^{\frac{1}{3}}$$

(1)

where r represents the ratio of the number of grid units, and N represents the total number of grid units.

The formula for calculating the relative error of a computed value is given by

$$e_{n,n+1}=\left|\frac{f_n-f_{n+1}}{f_{n+1}}\right|$$

(2)

where e represents the relative error and f represents the measured value.

The formula for calculating the Global Consistency Index is as follows

$$GC{I}_{n+1}=C\frac{e_{n,n+1}{}^p}{r_{n,n+1}-1}\times 100\%$$

(3)

where C is a scaling coefficient used to correct the influence of error terms, typically ranging from 1.25 to 3.00. p is the error term coefficient, and its selection in unstructured grids usually depends on the specific numerical simulation problem and empirical knowledge. In unstructured grids, the typical range for p is 1 to 2. In this article, C is set to 3, and p is set to 1.

Due to the potential significant impact of the magnitude of computed values on the calculation of GCI, this paper intends to employ the “min-max normalization” method to process the data, reducing the influence of value magnitude on GCI. The formula for normalization is as follows:

$$f_{xin}=\frac{(f_i-f_{min})}{(f_{max}-f_{min})}$$

(4)

where f_i represents the i-th original value, f_min represents the minimum value in the dataset, and f_max represents the maximum value in the dataset.

The axial force (F) acting on the nozzle face of the jet pipe is taken as the monitoring indicator, and the calculation results are shown in Figure 5. It can be observed that after the number of grids reaches 930,000, the axial force remains relatively stable, indicating that the grid independence requirement is satisfied.

The GCI calculation values are shown in

Table 2. GCI calculation values.

Serial Number	GCI Value
1–2	4.8339%
2–3	2.9244%
3–4	2.8343%
4–5	0.8428%
5–6	1.7750%
6–7	3.6031%
7–8	0.1725%
8–9	0.9025%
9–10	0.0006%
10–11	0.0054%

.

Based on the data in

Table 3. Factors and level settings.

Level	Nozzle Quantity (n)	Nozzle Diameter (r)	Distribution Circle Diameter (R)	Tilt Angle of the Base (α)
Level 1	20	0.5	20	30
Level 2	30	1	22	45
Level 3	40	1.5	24	60

, the GCI values for serial numbers 9–10 and 10–11 are very close to 0. This indicates that increasing the number of grids has a minimal impact on the observed values. Therefore, when reaching a grid count of over 930,000, further increasing the number of grids may not significantly affect the observed values.

2.3. Control Equation and Boundary Conditions

During the fluid simulation calculation, the SST k-ω model is employed to close the Reynolds-averaged Navier–Stokes equations [32].

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho k{u}_i)}{\partial x_i}=\frac{\partial }{\partial x_j}({\mathsf{\Gamma }}_k\frac{\partial k}{\partial x_j})+G-Y_k+S_k$$

(5)

$$\frac{\partial (\rho \omega )}{\partial t}+\frac{\partial (\rho \omega ui)}{\partial x_i}=\frac{\partial }{\partial x_j}({\mathsf{\Gamma }}_{\omega }\frac{\partial \omega }{\partial x_j})+G-Y_{\omega }+D_{\omega }+S_{\omega }$$

(6)

$$\mu =\rho \frac{k}{\omega }$$

(7)

The momentum equation is [33]

$$\frac{\partial (\rho u_i)}{\partial t}+\frac{\partial (\rho u_i{u}_j)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left(\mu \frac{\partial u_i}{\partial x_j}\right)+\frac{\partial {\tau }_{ij}}{\partial x_j}$$

(8)

where u_i and u_j are velocity components, $\rho$ is fluid density, p is time-averaged pressure, μ is viscosity, and τ is Reynolds stress.

The inlet surface is set as a “pressure inlet” with a specified gauge pressure of 4.5 MPa, while the outlet surface is set as a “pressure outlet” with a gauge pressure of 4 MPa.

For incompressible flow, the total pressure p_t, static pressure p_s, and fluid velocity v are related by the following equations:

$$p_t=p_s+\frac{1}{2}\rho {\upsilon }^2$$

(9)

2.4. Computational Scheme

Using orthogonal design method, a four-factor three-level experiment was designed in this study. The specific factors and their level settings are shown in Table 3.

The experimental group settings are shown in

Table 4. Research group program design.

Experimental Group Number	Nozzle Quantity (n)	Nozzle Diameter (r)	Distribution Circle Diameter (R)	Tilt Angle of the Base (α)
1	20	0.5	20	30
2	20	1	22	45
3	20	1.5	24	60
4	30	0.5	22	60
5	30	1	24	30
6	30	1.5	20	45
7	40	0.5	24	45
8	40	1	20	60
9	40	1.5	22	30

.

The experimental group models were divided into grids, and the number of grid divisions for each experimental group is shown in

Table 5. Grid division numbers for each experimental group.

Experimental Group	1	2	3	4	5	6	7	8	9
Number of Grids	1,088,461	969,019	995,336	984,778	1,019,117	977,962	952,363	1,012,962	1,016,458

. The number of grids in each computational domain model of the experimental groups exceeds 950,000, and the grid quality in each model is greater than 0.3.

3. Result Analysis

3.1. Pressure and Velocity

The calculated inlet–outlet pressure differences and axial forces are shown in Figure 6. Figure 6a shows that Experimental Group 1 has the highest pressure difference value, which is 495 kPa, while Experimental Group 9 has the lowest pressure difference value, which is 515 kPa. Figure 6b shows that Experimental Group 1 has the highest axial force value, which is 3200 N. Experimental Group 9 has the lowest axial force value, which is 2933 N. The variations in the data demonstrate the significant impact of the four structural parameters on the performance of the jet tube.

The velocity contour map of the cross-section is shown in Figure 7 at nine calculating schemes. It can be observed that the overall velocity of the water in areas 1 → 3, 4 → 6, and 7 → 9 shows a noticeable increase. In Figure 7c,i, the velocities of the liquid outside the nozzle are relatively close and significantly higher than the liquid velocity in Figure 7f. However, in Figure 7f,i, the velocities of the liquid inside the nozzle are relatively close and significantly higher than the liquid velocity in Figure 7c. Therefore, at the liquid nozzle, Figure 7f,i experience greater energy loss compared to Figure 7c. This is because the tilt angle of the bottom cover has a significant impact on the energy loss of the fluid at the nozzle.

The pressure contour map of the jet nozzle end face is shown in Figure 8 at nine calculating schemes. Experimental Group 1 and Experimental Group 9 have higher pressures at the bottom cover compared to other experimental groups, with Experimental Group 1, exhibiting the highest pressure at the bottom cover. Only at the inner tube section, other experimental groups experience relatively higher pressures. Experimental Groups 1, 2, 3, 4, 5, and 7 exhibit a small region of higher pressure at the bottom end of the inner tube, while Experimental Groups 6, 8, and 9 do not have regions of higher pressure at the bottom end, and the overall bottom-end pressure is lower. Experimental Group 6 has a region of medium pressure at the edge of the bottom cover. The overall pressure of the monitoring body in Experimental Group 3 is smaller compared to the pressures of other experimental groups. This phenomenon may be caused by the liquid flow rate. When the flow rate is too high or too low, the pressure on the bottom cover increases.

The 3D velocity streamline map is shown in Figure 9 at nine calculating schemes. By examining each graph, it can be seen that the overall velocity of the fluid in Experimental Group 1 is the smallest, while the overall velocity of the fluid in Experimental Group 9 is the largest. Clearly, the overall velocity of the fluid in Experimental Groups 1 to 3, 4 to 6, and 7 to 9 increases gradually. This is due to the increase in the diameter of the spray hole.

3.2. Entropy Production Distribution

For Newtonian fluids, the entropy production of a particle in laminar flow can be calculated using the following equation [33]:

$$\stackrel{•}{S_D^{‴}}=\frac{\mu }{T}\left\{2\left[{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2\right]+\left[{\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)}^2\right]\right\}$$

(10)

In turbulent flow, the entropy production of a particle can be divided into entropy production caused by mean motion and entropy production caused by velocity fluctuations. The calculation formula is as follows:

$$\stackrel{•}{S_D^{‴}}=\stackrel{•}{S_{\overline{D}}^{‴}}+\stackrel{•}{S_{D^{\prime }}^{‴}}$$

(11)

where $\stackrel{•}{S_{\overline{D}}^{‴}}$ represents the entropy production caused by mean velocity, and $\stackrel{•}{S_{D^{\prime }}^{‴}}$ represents the entropy production caused by velocity fluctuations.

The entropy production $\stackrel{•}{S_{\overline{D}}^{‴}}$ caused by mean velocity and the entropy production $\stackrel{•}{S_{D^{\prime }}^{‴}}$ caused by velocity fluctuations can be calculated using the following equations:

$$\stackrel{•}{S_{\overline{D}}^{‴}}=\frac{\mu }{T}\left\{2\left[{\left(\frac{\partial \overline{u}}{\partial x}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial y}\right)}^2+{\left(\frac{\partial \overline{w}}{\partial z}\right)}^2\right]+\left[{\left(\frac{\partial \overline{v}}{\partial x}+\frac{\partial \overline{u}}{\partial y}\right)}^2+{\left(\frac{\partial \overline{w}}{\partial x}+\frac{\partial \overline{u}}{\partial z}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial z}+\frac{\partial \overline{w}}{\partial y}\right)}^2\right]\right\}$$

(12)

and

$$\stackrel{•}{S_{D^{\prime }}^{‴}}=\frac{\mu }{T}\left\{2\left[{\left(\frac{\partial u^{\prime }}{\partial x}\right)}^2+{\left(\frac{\partial v^{\prime }}{\partial y}\right)}^2+{\left(\frac{\partial w^{\prime }}{\partial z}\right)}^2\right]+\left[{\left(\frac{\partial v^{\prime }}{\partial x}+\frac{\partial u^{\prime }}{\partial y}\right)}^2+{\left(\frac{\partial w^{\prime }}{\partial x}+\frac{\partial u^{\prime }}{\partial z}\right)}^2+{\left(\frac{\partial v^{\prime }}{\partial z}+\frac{\partial w^{\prime }}{\partial y}\right)}^2\right]\right\}$$

(13)

In RANS numerical computation methods, turbulent velocity fluctuations are typically represented by the $\epsilon$-equation, and the fluctuating velocity itself is not directly obtained. Therefore, it is not possible to calculate the entropy production $\stackrel{•}{S_{D^{\prime }}^{‴}}$ through the partial differentiation of the fluctuating velocity. In this case, the entropy production $\stackrel{•}{S_{D^{\prime }}^{‴}}$ caused by pressure fluctuations can be calculated using the following equation proposed by Kock and Herwig:

$$\stackrel{•}{S_{D^{\prime }}^{‴}}=\frac{\rho \epsilon }{T}$$

(14)

The total entropy production of the entire flow field can be obtained by performing a volume integration of the specific entropy production.

$$\stackrel{•}{S_{\overline{D}}^{‴}}={\displaystyle {\int }_V\stackrel{•}{S_{\overline{D}}^{‴}}}dV$$

(15)

$$\stackrel{•}{S_{D^{\prime }}^{‴}}={\displaystyle {\int }_V\stackrel{•}{S_{D^{\prime }}^{‴}}}dV$$

(16)

The entropy production at the midsection is shown in Figure 10 at nine calculating schemes. For all experimental groups, the entropy values around the nozzle are relatively low. Experimental Groups 3 and 9 exhibit large regions of low entropy values. Experimental Groups 2, 3, 5, 6, 7, and 8 all have relatively large regions with moderate entropy values. Additionally, Experimental Groups 3, 6, and 9 have higher average entropy values, while Experimental Groups 1 and 4 have more uniform and lower entropy values. Therefore, it can be inferred that Experimental Groups 3, 6, and 9 have higher levels of turbulence, while Experimental Groups 1 and 4 have lower levels of turbulence. This is because the increase in the diameter of the nozzle and the number of nozzles results in an increase in liquid flow velocity, leading to a greater entropy value.

The entropy production contour map at the nozzle outlet is shown in Figure 11 at nine calculating schemes. Compared to the other experimental groups, the average entropy values on the bottom plate of Experimental Groups 1 and 9 are significantly higher. In particular, the average entropy value on the bottom plate of experimental group 1 is the highest among all the experimental groups. The entropy values on the bottom plate of Experimental Group 3 and on the inner tube sidewall are the lowest compared to the other experimental groups. Additionally, it can be observed that the range of the high entropy region at the bottom end of the inner tube is notably smaller for Experimental Groups 6, 8, and 9 compared to the other experimental groups. In all experimental groups, the entropy values in the region near the nozzle are generally lower. This is because the liquid at this location has a faster flow velocity, which makes it more difficult to become disordered.

3.3. Vorticity Structure

The vorticity structure diagram is shown in Figure 12 at nine calculating schemes, with a selected relative level of 0.00005. The velocity field is represented by the vorticity structure. At this level, vortices in Experiments 1, 4, and 7 are mainly distributed inside the nozzle and at the jet outlet, while vortices in other experimental groups are mainly concentrated at the nozzle and in the downstream region. By observing the diagrams, it can be noticed that Experiment 1 has the least number of vortices, while Experiment 8 has the most. Additionally, Experiment 5 and Experiment 8 exhibit stronger vortices in the intermediate region. This is because higher flow velocity makes it difficult for vortices to form inside the pipeline.

3.4. Q Criteria

The formula for calculating the Q criteria is as follows:

$$Q=0.5\times ({‖B‖}_F^2-{‖A‖}_F^2)$$

(17)

where ${‖B‖}_F^2$ represents the square of the norm of matrix B, which is equivalent to the sum of the squares of all elements of matrix B.

Matrix A and Matrix B are, respectively, the symmetric tensor and antisymmetric tensor of velocity gradients, i.e.,

$$A=0.5\times (\mathsf{\Delta }V+\mathsf{\Delta }{V}^T)$$

(18)

$$B=0.5\times (\mathsf{\Delta }V-\mathsf{\Delta }{V}^T)$$

(19)

where T represents the transpose of the matrix. The definition of the velocity tensor is as follows:

$$\mathsf{\Delta }V=\left(\begin{array}{ccc}Ux& Uy& Uz\\ Vx& Vy& Vz\\ Wx& Wy& Wz\end{array}\right)$$

(20)

The jet nozzle end face Q-criterion contour map is shown in Figure 13 at nine calculating schemes. According to the Q-criterion calculation rule, negative values represent the presence of vortices, and the greater the absolute value of the negative value, the higher the vortex strength. Therefore, it can be concluded that the monitoring body of Experimental Group 3 has more vortices and higher vortex strength. Experimental Groups 1, 6, and 9 have fewer vortices near the nozzle compared to other experimental groups, and the vortex strength is lower. The vortices on the monitoring bodies of the remaining experimental groups are mainly concentrated near the nozzle and inner duct, and the intensity is higher. It can be observed that the vortex strength is higher in the regions near the nozzle for all experimental groups. This is because the flow velocity is the fastest at the nozzle.

The Q-criterion contour map is shown in Figure 14 at nine calculating schemes; the three selected levels used are −10, −5, −0.1. Experimental Groups 1, 4, and 7 exhibit a large number of low-intensity vortices, and the range of these vortices is significantly greater than that of other experimental groups. Compared to other experimental groups, Experimental Group 3 has the fewest number of vortices, but there are more vortices of higher intensity. Furthermore, it can be observed that the regions with higher vortex intensity are mainly concentrated near the nozzle. According to the vortex judgment criteria of the Q-criterion, the faster the velocity at the liquid jet nozzle, the higher the vortex intensity and the more concentrated the vortices. Conversely, the more concentrated the vortex intensity, the faster the overall liquid flow velocity. The magnitude of velocity is an important factor in generating vortices.

3.5. Range Analysis

Range analysis is a statistical method used to study and compare the variability within a set of data. It measures the range of variation by calculating the difference between the maximum and minimum values in a data set, thereby assessing the data’s dispersion.

A range analysis was conducted using axial force as the observed value, and the analysis results are shown in

Table 6. Range analysis table.

Experimental Group	Number of Spray Holes	Spray Hole Diameter	Diameter of Distribution Circle	Angle of Inclined Bottom Cover	Axial Force
Experiment 1	20	0.5	20	30	3199.711371
Experiment 2	20	1	22	45	3151.199306
Experiment 3	20	1.5	24	60	3072.255421
Experiment 4	30	0.5	22	60	3192.280765
Experiment 5	30	1	24	30	3121.337614
Experiment 6	30	1.5	20	45	3003.996483
Experiment 7	40	0.5	24	45	3183.517847
Experiment 8	40	1	20	60	3089.347485
Experiment 9	40	1.5	22	30	2932.954979
Mean value 1	3141.055	3191.837	3097.685	3084.668
Mean value 2	3105.872	3120.628	3092.145	3112.905
Mean value 3	3068.607	3003.069	3125.704	3117.961
Range	72.448	188.768	33.559	33.293

. Based on the data in Table 6, it can be concluded that the minimum axial force acting on the jet nozzle occurs when the number of spray holes is 40. The minimum axial force is also observed when the diameter of the spray hole is 1.5 mm. Additionally, a minimum axial force is observed when the distribution circle diameter is 22 mm, and when the inclination angle of the bottom cover is 30 degrees. Therefore, the optimal structural parameters at the spray hole nozzle can be determined through range analysis: 40 spray hole nozzles, 1.5 mm spray hole diameter, 22 mm distribution circle diameter, and a 30 degree inclination angle of the bottom cover.

Among the different levels of factors, the largest range difference is observed between the levels of the spray hole diameter, with a range difference of 188.768. The next largest range difference is observed between the levels of the number of spray holes, with a range difference of 72.448. This is followed by the range difference between the levels of the distribution circle diameter, which is 33.559. The smallest range difference is observed between the levels of the inclination angle of the bottom cover, which is 33.293. The difference between the highest value of 188.768 and the lowest value of 33.293 is 155.475, indicating that the spray hole diameter has the greatest influence on the performance of the nozzle, followed by the number of spray holes. The distribution circle diameter and inclination angle of the bottom cover have a lower impact on the nozzle’s performance.

3.6. Significance Test

Analysis of Variance (ANOVA) is a statistical method used to compare the mean differences among multiple groups to determine if they are significant. It decomposes the total variance of the population into between-group variation and within-group variation. By utilizing these variances, an F-test is conducted to assess whether there are significant differences among the group means.

The formula for calculating the population variance is as follows:

$$S{S}_{Total}={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^{n_i}\left(x_{ij}-\overline{x}\right)}}{}_{}{}^2$$

(21)

The formula for calculating the between-group variance is as follows:

$$S{S}_{Between}={\displaystyle \sum _{i=1}^k{n}_i\left({\overline{x}}_i-\overline{x}\right)}{}_{}{}^2$$

(22)

The formula for calculating the within-group variance is as follows:

$$S{S}_{Within}={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^{n_i}\left(x_{ij}-{\overline{x}}_i\right)}}{}_{}{}^2$$

(23)

where k represents the number of groups, n_i represents the number of samples in the i-th group, x_ij represents the j-th observation in the i-th group, ${\overline{x}}_i$ represents the sample mean of the i-th group, and $\overline{x}$ represents the population mean of all observations.

And the relationship between the variances of the three groups is as follows:

$$S{S}_{Total}=S{S}_{Between}+S{S}_{Within}$$

(24)

The calculation process of an F-test involves the ratio of two mean square deviations, namely,

$$F=\frac{M{S}_{Between}}{M{S}_{Within}}$$

(25)

where $M{S}_{Between}$ represents the between-groups mean square deviation, and $M{S}_{Within}$ represents the within-groups mean square deviation.

The larger the F-value, the more significant the difference between groups relative to the difference within groups. When conducting an F-test, it is also necessary to specify a significance level to determine whether to reject the null hypothesis. A smaller significance level means a higher requirement for the significance of the difference.

In analysis of variance, degrees of freedom are an important indicator for evaluating statistical inference. Degrees of freedom reflect the number of independent variations in the data, and they determine the distribution of the statistical test and the accuracy of the significance test. Therefore, it is crucial to determine the degrees of freedom when calculating between-group variance and within-group variance. However, in a four-factor three-level nine-group experimental design, there may be an issue of insufficient degrees of freedom, which will prevent conducting the analysis of variance. In this case, one solution is adding an additional group to address the issue of insufficient degrees of freedom, allowing each group to have more observations, thereby providing more degrees of freedom and enabling the possibility of conducting analysis of variance. Thus, before conducting the analysis of variance, this study adds an additional group as shown in

Table 7. Additional experimental group.

Experimental Group Number	Nozzle Quantity (n)	Nozzle Diameter (r)	Distribution Circle Diameter (R)	Tilt Angle of the Base (α)
10	40	1.5	24	30

.

After numerically simulating the additional experimental group, the results of the significance test with axial force as the observed value are shown in

Table 8. Significance test table.

Test for Between-Subject Effects
Dependent Variable: Axial Force
Source	Type III Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
Adjusted model	92,394.986 a	8	11,549.373	20.552	0.169
Intercept	91,773,633.912	1	91,773,633.912	163,307.155	0.002
Number of spray holes	10,077.237	2	5038.619	8.966	0.230
Spray hole diameter	64,310.190	2	32,155.095	57.219	0.093
Distribution circular diameter	1474.996	2	737.498	1.312	0.525
Base cover tilt angle	3011.728	2	1505.864	2.680	0.397
Error	561.969	1	561.969
Total	95,447,862.726	10
Adjusted total	92,956.955	9

^a R² = 0.994 (Adjusted: R² = 0.946).

.

According to Table 8 and the calculation rules for F-values and significance values, a higher F-value indicates a greater impact of the factor on the outcome, while a lower significance value suggests a higher impact of the factor. Based on this, the following conclusions can be drawn: Compared to the F-values and significance values of the nozzle diameter, the F-values of nozzle quantity, distribution circle diameter, and bottom cover tilt angle are lower, and their significance values are higher. Among them, the nozzle diameter has the highest F-value of 57.219 with a significance value of 0.093, while the distribution circle diameter has the lowest F-value of 1.312 with a significance value of 0.525. The difference in F-values between nozzle diameter and distribution circle diameter is 55.907, and the difference in significance values is 0.432. This indicates that the nozzle diameter has the greatest influence on the axial force of the jet pipe, while the distribution circle diameter has the smallest influence. Based on the F-values and significance values of each factor, it can be concluded that the nozzle diameter has a relatively large influence on the axial force of the jet pipe, while the nozzle quantity has a relatively small influence, and the distribution circle diameter and bottom cover tilt angle have very low impacts.

4. Conclusions

Summarizing all the simulation analysis results, the following conclusions can be drawn:

(1)

According to Figure 6, it can be concluded that the maximum axial force in Experimental Group 1 is 3199.711371 N, while the minimum axial force in Experimental Group 9 is 2932.954979 N.

(2)

Based on entropy distribution analysis, the entropy values of monitoring bodies in Experimental Groups 3, 6, and 9 are relatively high, indicating a higher degree of disorder. On the other hand, Experimental Groups 1 and 4 have lower entropy values, indicating a lower degree of disorder.

(3)

According to the vortex discrimination, the monitoring body in Experimental Group 3 exhibits a higher number of vortices and greater vortex intensity. Experimental Groups 1, 6, and 9 have fewer vortices at the nozzle and lower vortex intensity compared to the other experimental groups.

(4)

Through range analysis and significance testing, it can be concluded that the diameter of the nozzle hole has a significant impact on the axial force experienced by the jet pipe, while the number of nozzle holes has a smaller impact. The distribution circle diameter and the tilt angle of the bottom cover have very little influence on the axial force.

(5)

The optimal combination of nozzle structure parameters, determined after comprehensive analysis, is as follows: the number of spray holes is 40, each with a diameter of 1.5 mm, the distribution circle diameter is 22 mm, and the incline angle of the bottom cover is 30 degrees.

This study focuses on the research of a nozzle (with particular emphasis on axial force) and can be applied in future pressure-relief scenarios.