Numerical Simulation of Nozzles in Fluent-Based Cotton Impurity Removal Machines

Abstract

This paper conducts numerical simulations of nozzles with different structural parameters based on fluid mechanics, computational fluid dynamics and jet theory. The structural parameters of the nozzles were optimised by analysing flow field characteristics such as the pressure distribution within the nozzle chamber, velocity distribution, curves of the outlet cross-sectional area and external axial velocity, and velocity uniformity. Combining the results of orthogonal experiments, the optimal combination of factors was determined, and the impurity removal efficiency of the optimised nozzle was tested in the field, providing a reference for subsequent optimisation design. The results indicate that adding a fillet transition to the nozzle can mitigate sudden pressure drops and suppress the generation of vortices; when the fillet transition radius is 80 mm, the flow performance approaches the optimum; the optimal combination of the three factors was determined to be a contraction angle of 13°, λ of 0.65 (corresponding to an outlet height of 27 mm and an inlet diameter of 41 mm), and a nozzle length of 15 mm; this configuration yields the best external flow field characteristics and velocity uniformity; Analysis of the orthogonal test results indicates that the contribution of each structural parameter to velocity uniformity, in descending order, is: contraction angle (77.16%), λ (outlet height/inlet diameter) (18.25%), and nozzle length (0.73%); Field tests confirmed that the removal efficiency of foreign fibres using the optimal parameter combination remained consistently above 95%, with an overall average removal rate of 96.31%. This represents an improvement of approximately 7.5 percentage points compared to the original nozzle (88.83%). The optimised nozzle reduced the number of false rejections of cotton by 57%, demonstrating excellent and highly stable overall removal performance. The influence of the nozzle’s vertical height and its angle relative to the cotton on the removal efficiency requires further investigation.

1. Introduction

Long-staple cotton is a distinctive product of Xinjiang, characterised by its long fibres and excellent elasticity. It is an indispensable raw material in the spinning and specialised textile industries and is often referred to as the ‘gem of fibres’ due to its unrivalled texture and soft feel. With limited cultivation areas, superior fibre quality and high economic value, it is a precious resource [1]. Xinjiang long-staple cotton is primarily cultivated using a sub-mulch drip irrigation system, resulting in relatively small bolls. This makes manual harvesting difficult and costly, presenting a major challenge to the sustainable development of the long-staple cotton industry. Although mechanical harvesting can significantly reduce labour intensity and operating costs, existing large-scale harvesting equipment still suffers from issues such as high impurity levels, which severely constrain the development of long-staple cotton [2]. Furthermore, during mechanical harvesting, cotton is prone to becoming mixed with residual plastic mulch, which further reduces its grade. Consequently, impurity removal technology in the processing of long-staple cotton is of paramount importance for the healthy development of China’s cotton industry chain.

Impurity removal devices used in agricultural production are generally classified into two types: mechanical and pneumatic separation devices. Mechanical separation devices utilise the combined action of airflow and vibrating screens to separate impurities, whereas pneumatic separation devices rely entirely on airflow for impurity removal [3]. Currently, although pneumatic separation devices have been used in agriculture for several hundred years, there has been limited research into the mechanisms of pneumatic cleaning and flow field characteristics; relevant experience can be drawn from other industries such as pneumatic sorting [4]. In accordance with the ‘National Standard of the People’s Republic of China—Technical Specifications for Cotton Processing’ (GB/T 22335-2018) [5] and the ‘National Standard of the People’s Republic of China—Cotton Ginning Machines’ (GB/T 19818-2005) [6], the impurity content of machine-harvested raw cotton must not exceed 12%, the impurity removal efficiency of the ginning machine must be no less than 50%, and the cotton loss rate must not exceed 0.3%. Given the impact of cotton variety differences on cleaning performance, the impurity content of machine-harvested long-staple cotton is generally 5% to 10% higher than that of short-staple cotton. Consequently, performance targets must be appropriately adjusted during the design process, with cleaning efficiency serving as the core evaluation criterion for impurity removal precision.

Computational Fluid Dynamics (CFD) is a discipline founded on classical mechanics and numerical methods; it utilises computer-based numerical calculations and graphical visualisation to analyse systems, including physical phenomena such as fluid flow [7]. In recent years, CFD has been increasingly applied to the flow field analysis and optimised design of agricultural machinery due to its advantages of low cost, short development cycles, and freedom from physical constraints. Scholars both domestically and internationally have conducted extensive research in the fields of nozzle structure optimisation and pneumatic separation.

Regarding nozzle structure design, Wen et al. investigated nozzle structures. Utilising the straight-conical nozzle model invented by the Soviet scholar Nikonoff and their own experimental test bench, they studied the jet characteristics of nozzles with different structures. The results indicated that the straight-conical nozzle possesses excellent structural parameters and performance [8]. Ren Weijia et al., addressing the distribution and structural characteristics of nozzles in fibre separation systems, proposed a circularly distributed convergent-divergent nozzle. By employing rounded transitions, they optimised the turbulent kinetic energy within the nozzle, thereby enhancing the uniformity of airflow velocity. Replacing the original stepped cross-section with a transition curve cross-section effectively suppressed abrupt changes in airflow velocity within the nozzle. Finally, a comprehensive analysis and discussion of the optimised nozzle structure was conducted from the perspectives of velocity fields, pressure distribution and energy loss [9]. Adarsh et al. solved the N-S equations in a turbulent flow field using finite element analysis software. By employing a complex conjugate eigenvalue method based on the velocity gradient tensor, they investigated the flow dynamics, turbulence and vortex structures within the turbulent flow field. The results indicate that the convex contour surface on the inner side of the elbow is a key vortex-generating surface [10].

Regarding CFD simulations of nozzle jets, Yingjun Wang et al. conducted Fluent simulations to optimise nozzle design. Their research found that the jet air curtain generated by the nozzle can effectively block residual film on the surface. Optimising the nozzle structure can enhance the performance of the air curtain, thereby improving the residual film blocking effect, reducing the content of residual film on cotton, and enhancing cotton cleanliness [11]. Yue Pan et al. utilised 2023 Fluent software and the k-ε turbulence model to perform numerical simulations of jet flows within pipes for nozzles of different sizes and configurations. They compared the flow field distributions and variations in nozzle velocity and displacement for nozzles with different parameters, and processed the simulation data using computational fluid dynamics, thereby laying the foundation for further in-depth research [12]. Mohammad et al. performed numerical simulations of turbulent flow in a drag-reducing fluid within a pipeline based on the k-ε model and its RNG version [13,14,15]. Regarding the parameter optimisation of pneumatic separation systems, Hu Zhizheng et al. employed a three-factor, three-level orthogonal experiment, using inlet gas velocity, gas flow angle and dust collector angle as independent variables, to analyse the interaction of each parameter on the impurity content and loss rate during cleaning. The experimental and simulation results showed good agreement, verifying the reliability of the CFD method for studying impurity separation behaviour [16]. Mohammadi et al. utilised Fluent software to optimise the configuration of the number, position, and angle of the inlets, as well as the number of nozzles [17].

Although the aforementioned studies have provided an important foundation for nozzle design and pneumatic separation, several key issues remain unresolved. In research on nozzle geometric parameters, factors such as the contraction angle, outlet height and nozzle length are often analysed in isolation; the interrelationships between these factors—such as the synchronous variation in nozzle height and inlet diameter whilst maintaining a constant contraction angle—are rarely discussed, leading to confusion regarding the attribution of parameter effects. The nozzle serves as both the component responsible for generating the air jet and a critical component in the process of removing foreign fibres from raw cotton. The quality of the nozzle directly influences the effectiveness of impurity removal from raw cotton, thereby affecting other parts of the entire system, and plays a vital role in the cotton sorting process. To address the above issues, this study, based on fluid mechanics, computational fluid dynamics (CFD), jet theory and numerical simulation, employs a method combining CFD numerical simulation with orthogonal experimental design to systematically analyse the influence of three geometric parameters—the contraction angle α, λ (the ratio of outlet height to inlet diameter, i.e., h/D) and nozzle length L—on flow field characteristics and sorting accuracy. Through the structural design of the nozzle, numerical simulation and analysis of the results, the optimal nozzle structure and impact distance can be determined. The innovation and contribution of this study lie primarily in clarifying the inherent coupling relationship between the outlet height h and the inlet diameter D under the condition of a constant contraction angle, and in introducing the dimensionless parameter λ as the attribution variable to resolve the issue of parameter influence confusion; orthogonal experiments were conducted to determine the influence of each geometric parameter on the exit cross-sectional velocity non-uniformity coefficient, and the contribution of each parameter to the final results, as well as the applicability and limitations of the main effects model, were discussed through visual analysis, deviation analysis and analysis of variance; The mechanism linking the sudden pressure change at the nozzle transition to vortex-induced losses was revealed. The flow improvement effects of different fillet radii (20–80 mm) were quantitatively assessed, confirming that the vortex region completely disappears and flow performance is optimised at a fillet radius of 80 mm. Consequently, a nozzle optimisation strategy was established to suppress vortices and reduce flow resistance through the use of fillet transitions. In accordance with the requirements of GB/T 22335-2018 [5] and GB/T 19818-2005 [6], and using impurity removal efficiency as the core evaluation criterion, the optimal combination of parameters was determined. It is hoped that the research results will provide a theoretical basis for the structural design of nozzles in cotton impurity removal machines and offer engineering reference value for improving cotton impurity removal efficiency.

2. Materials and Methods

2.1. The Principle of Jet Propulsion

A jet refers to a type of flow pattern formed when a fluid is ejected through an outlet, orifice or slit. Jet technology is widely used in engineering fields such as grain sorting and impurity removal. Figure 1 shows a schematic diagram of a typical jet structure, illustrating the mechanical composition of the jet and its developmental stages. In the initial section (from the nozzle outlet to cross-section BB’), there is a jet core with uniform velocity; as the jet develops axially, the core gradually attenuates and the jet expands laterally. The transition section is short and can be disregarded. Beyond cross-section DD’ lies the basic section, where the jet is fully developed, turbulence is significant, and the axial velocity continuously decreases along the flow direction until it disappears. In industrial impurity removal, the greater the width of the developed section and the higher the jet velocity, the better the removal efficiency and separation precision. During injection, the high-speed movement of the fluid induces turbulence, causing the jet’s edges to continuously expand as it advances and the turbulence to intensify. Therefore, during the injection process, the material to be separated should be positioned within the jet core as much as possible to minimise gas consumption.

2.2. Numerical Calculation Methods and Data Processing

To quantitatively compare the uniformity of the velocity distribution of high-speed airflow at the outlet cross-section, an evaluation index has been established for reference purposes, using the velocity non-uniformity coefficient M to assess the degree of non-uniformity in the velocity distribution [18,19]. The equations are given in Equations (1) and (2).

$$M={\displaystyle \frac{{\sigma }_{\mathrm{v}}}{\overline{V_{\mathrm{a}}}}}\times 100\%$$

(1)

$$={\displaystyle \frac{\sqrt{\frac{1}{n-1}{\sum }_{i=0}^n{\left(V_{\mathrm{i}}-\overline{V_{\mathrm{a}}}\right)}^2}}{\overline{V_{\mathrm{a}}}}}\times 100\%$$

(2)

Here, ${\mathrm{\sigma }}_{\mathrm{v}}$ is the standard deviation; n is the number of nodes; V_i is the velocity at each point, m/s; and $\overline{{\mathrm{V}}_{\mathrm{a}}}$is the population mean of the velocity distribution, m/s.

2.3. Geometric Model

In impurity separation devices, the high-velocity gas instantaneously ejected from the jet nozzle serves as the primary driving force for separating the target material. Nozzles are classified by medium into water, air, sand and flame types, and by shape into cylindrical, conical and fan-shaped types, such as conical nozzles, flat nozzles, bell-shaped nozzles, single-expansion and double-expansion nozzles [20,21,22]. Since the raw cotton fibre separation machine utilises compressed air as the blowing medium, a convergent nozzle should be selected [23]. Figure 2 and Figure 3 below show the schematic diagram of the duckbill-type nozzle structure and the cross-sectional view of the nozzle, which are the subjects of optimisation in this paper.

The main parameters shown in Figure 3 are: inlet diameter D, inlet section length l′, outlet height h, nozzle length l, contraction angle α, and nozzle length L. The relationship between the structural parameters is shown in Equation (3).

2.4. Mathematical Model

This study examines the flow of high-pressure air inside and outside a nozzle. Fluid flow is classified into laminar and turbulent flow, with turbulent flow being unstable and a common form of high-speed flow. The Reynolds number (Re) is commonly used to determine whether the flow is laminar or turbulent. The formula for calculating the Reynolds number is given in Equation (4).

$${\displaystyle \frac{\frac{D-h}{2}}{L}}=\mathit{tan}{\displaystyle \frac{\alpha }{2}}$$

(3)

$$Re={\displaystyle \frac{\rho vd}{\eta }}$$

(4)

$$Re=3956916\gg 4000$$

(5)

At ambient temperature, when the inlet pressure is 0.35 MPa, the velocity at the nozzle outlet can reach 600 m/s. Calculations show that, to successfully dislodge impurities with a particle size range of 3–5 cm, the velocity of the nozzle jet must exceed at least 300 m/s. The fluid density (ρ) is approximately 5.27 kg/m³, the dynamic viscosity coefficient is approximately 17.9 × 10⁻⁶ Pa·s, and the characteristic length (d) is 22.4 mm (where d refers to the hydraulic diameter when the fluid pipe cross-section is elliptical). The Reynolds number is given by Equation (5).

The results indicate that turbulence and flow around curved walls are present within the nozzle, making it prone to the formation of vortices. The internal flow within this nozzle assembly exhibits the characteristic coexistence of high strain rates, curved streamlines and recirculation zones. The RNG k-epsilon model corrects the turbulent viscosity using renormalisation group theory and incorporates a term for the strain rate tensor into the equations, enabling a more accurate description of the jet’s diffusion, bending, and separation-reattachment processes; Extensive numerical studies on circular, planar and confined jets have shown that the RNG k-epsilon model significantly outperforms the standard k-epsilon model in predicting the length of the jet core, the velocity decay rate and the transverse velocity distribution. Furthermore, its computational accuracy is comparable to that of the more advanced RSM model, whilst the computational cost is considerably lower than the latter. For the problems addressed in this study, the length of the jet core and the velocity uniformity at the outlet cross-section are key criteria affecting separation accuracy. Taking the above reasons into account, the RNG k-epsilon model is selected for the numerical simulation of the nozzle in this paper. The equations of the RNG k-epsilon turbulence model are given in Equations (6)–(8).

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\nu }_{j}k\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left({\alpha }_k{\mu }_{eff}{\displaystyle \frac{\partial k}{\partial x_j}}\right)+{ G}_k+G_b-\rho \epsilon -Y_M$$

(6)

$${\displaystyle \frac{\partial \left(\rho \epsilon \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\nu }_j\epsilon \right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left({\alpha }_{\epsilon }{\mu }_{eff}{\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right)+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}\left(G_k+C_{2\epsilon }{G}_b\right)-C_{1\epsilon }^{*}\rho {\displaystyle \frac{{\epsilon }^2}{k}}-R$$

(7)

$$R={\displaystyle \frac{C_{\mu }\rho {\eta }^3(1-\frac{\mu }{{\eta }_0})}{1+\beta {\eta }^3}}·{\displaystyle \frac{{\epsilon }^3}{k}}$$

(8)

In Equations (6)–(8) above, G_K represents the term for the generation of turbulent kinetic energy k due to the mean velocity gradient; G_b is the term for the generation of turbulent kinetic energy k due to buoyancy; Y_M is the term for the dissipation of turbulent kinetic energy due to compressibility, which can be neglected in incompressible flow; μ_eff = μ + μ_t is the equivalent viscosity; R is an additional term, where$\eta = S{\displaystyle \frac{k}{\epsilon }}$, ${\eta }_0 = 4.83$, and $\beta =0.012$. In the equation, the constants are: $C_{\mu } = 0.0845$, $C_{1\epsilon } = 1.42$, $C_{2\epsilon } = 1.68$, ${\alpha }_k = 1.0$, ${\alpha }_{\epsilon } = 0.769$. Here, ${\mathrm{C}}_{1\mathrm{\epsilon }}$, ${\mathrm{C}}_{1\mathrm{\epsilon }}^{*}$, and ${\mathrm{C}}_{2\mathrm{\epsilon }}$ are all turbulence model coefficients; ρ is the density; k is the kinetic energy of the gas turbulence; $\epsilon$ is the dissipation rate; $\eta$ is a dimensionless parameter; and ${\eta }_0$and β are constants.

The jet velocities considered in this study are approximately 600 m/s, falling within the transonic to supersonic range, where the effects of gas compressibility (such as density variations, shock waves and expansion waves) cannot generally be ignored. In the numerical simulations for this study, the assumption of incompressible flow was adopted. This simplification was primarily based on the fact that the focus of the study lies in the relative trends of the influence of geometric parameters on separation accuracy, rather than the precise prediction of absolute flow field values; furthermore, compressible solvers exhibit poor convergence stability and are ill-suited to supporting the extensive operating condition calculations required for multi-parameter orthogonal optimisation. Comparative experiments between compressible and incompressible simulations indicate that the incompressible assumption introduces certain prediction errors. Under typical operating conditions, the incompressible assumption underestimates the length of the jet core by approximately 3–5% and the velocity uniformity at the outlet cross-section by 7–11%.

Table 1. A Comparison of Core Length and Velocity Uniformity at the Outlet Cross-Section for Compressible and Incompressible Jets.

Variable	Length of the Jet Core Region/(mm)	Speed Unevenness Coefficient/(%)
Compressibility Experiment 1	210.5	1.147
Non-compressible Experiment 1	203.2	1.052
Compressibility Experiment 2	225	1.167
Non-compressible Experiment 1	216.8	1.086
Compressibility Experiment 3	234.3	1.236
Non-compressible Experiment 1	223.6	1.124

presents a comparison of the jet core length and velocity uniformity at the outlet cross-section between compressible and incompressible simulations. In the later stages of the study, these issues were taken into account; their impact on the final results was minimal. The quantitative conclusions obtained from the numerical simulation phase of this paper are of reference value, whilst the qualitative conclusions demonstrate reliable trends.

Relevant literature indicates that the length of the jet core (L_C) depends on the nozzle configuration and the Reynolds number. Regarding the relationship between L_C and the orifice diameter, the classical conclusion in the theory of low-speed incompressible free turbulent jets is that the jet core length L_C is approximately 4.5–6.5 d. However, for supersonic jets, the jet core length is typically greater than that of low-speed jets. Studies have shown that the length of the core region for supersonic circular jets can reach 8–10 d. Based on the empirical range fitted from preliminary simulation results, and taking into account the nozzle structure and characteristics of supersonic air jets in this study, L_C is taken as 7–9 d as the theoretical estimation range. This is then compared and verified against subsequent numerical simulation results; consequently, the subsequent numerical simulations focus on analysing the velocity variations at distances of 120 mm and 200 mm from the outlet.

2.5. Nozzle Structure Optimization Scheme

This paper investigates the influence of nozzle structure on the jet flow field and performance. Through optimized design, it aims to enhance the airflow velocity and stability at the effective impurity removal zone, thereby improving the accuracy of foreign fiber removal from raw cotton. Therefore, the nozzle optimization schemes adopted in this study include:

Option 1: A Study of the Effect of the Contraction Angle α on the Jet Field

The inlet diameter (D) is set at 30 mm, the inlet length (l′) at 15 mm, the outlet height (h) at 16 mm, the outlet width (f) at 44 mm, and the nozzle length (l) at 10 mm; the values for the contraction angle (α) are shown in

Table 2. Values of the contraction angle α.

Variable	1	2	3	4	5
Convergence angle/°	10	12	14	18	22
Nozzle length/mm	80	66.6	57	44.2	36

. It should be noted that changes in the contraction angle are achieved by altering the length of the contraction section; that is, the larger the contraction angle, the shorter the contraction section; the smaller the contraction angle, the longer the contraction section. Since the nozzle inlet diameter and outlet height are fixed, the length of the contraction section has a one-to-one correspondence with the magnitude of the contraction angle, acting as a dependent parameter rather than an independent variable. The variation in the flow field characteristics of this design can be entirely attributed to the contraction angle; changes in the length of the contraction section are merely a geometric means of altering the contraction angle and do not in themselves constitute an independent physical variable.

Option 2: A study of the effect of λ (outlet height/inlet diameter) on the jet flow field

The contraction angle (α) is set at 14°, the transition section length (l′) at 15 mm, the contraction section length (L) at 57 mm, the outlet width (f) at 44 mm, and the nozzle length (l) at 10 mm; the value of λ is shown in

Table 3. The value of λ (outlet height/inlet diameter).

Variable	1	2	3	4	5
Outlet height/mm	10	16	22	28	34
Inlet diameter/mm	24	30	36	42	48
λ	0.41	0.53	0.61	0.66	0.70

. It should be noted that in this scheme, the contraction angle and the outlet height are independent control parameters. When the outlet height changes, the inlet diameter of the contraction section is adjusted by the same amount to maintain a constant contraction angle; the relationship between these variables is given by Equation (3). The ratio of the outlet height (h) to the inlet diameter (D) is defined as λ, where λ is a dimensionless parameter.

Option 3: Investigation of the Effect of Mouth Length l on the Jet Field

Assuming a contraction angle (α) of 14°, an outlet height (h) of 28 mm, a contraction section length (L) of 57 mm, an outlet width (f) of 44 mm, an inlet diameter (D) of 42 mm, and a transition section length (l′) of 15 mm, the value of the nozzle length (l) is shown in

Table 4. Values for Mouth Length l.

Variable	1	2	3	4
Mouth length/mm	5	10	15	20

.

2.6. Verification of Mesh Independence

In fluid simulation, mesh density is a key factor determining computational accuracy; however, as the number of mesh elements increases, so does the computation time, and there is a diminishing return. Therefore, it is necessary to verify mesh independence prior to solving the problem. This paper selects the model from Scheme 2 with λ = 0.53, i.e., an outlet height of 16 mm and an inlet diameter of 30 mm. With all other conditions held constant, four sets of different mesh counts were established. The optimal mesh count was determined by analysing the length of the jet core and the maximum velocity; the results are shown in

Table 5. Verification of Mesh Independence.

Plan	Number of Cells	Length of the Speed Core Section/mm	Maximum Jet Velocity/m·s
1	52,354	212.6	591.25
2	63,354	230.3	627.71
3	95,764	232.5	639.33
4	114,741	232.7	641.87

. The results indicate that as the number of grid cells increases, the length of the jet core and the maximum velocity increase moderately; however, once the number of grid cells reaches 95,764, the increase is no longer significant. Consequently, the number of grid cells from Scheme 3 was selected for the model analysis. To ensure the comparability of the results, the mesh quality of the nozzle models before and after optimisation was kept as similar as possible.

2.7. Mesh Generation

GAMBIT offers a variety of mesh elements and can automatically generate meshes, supporting structured, unstructured and hybrid meshes. Among these, unstructured meshes do not require alignment with coordinate lines, allowing complex regions to be meshed arbitrarily; they are highly adaptable and produce high-quality results. Consequently, unstructured meshes were employed for the flow field meshing of the duckbill nozzle in this study. Given the small dimensions of the duckbill nozzle and the high velocity gradient near the nozzle outlet, the mesh density at the outlet should be relatively high. As shown in Figure 4, the model was meshed using the number of elements derived from the aforementioned mesh-independent validation.

2.8. Boundary Conditions

In this study, a pressure-stabilizing tank is connected to the inlet of a high-speed jet nozzle. Since the inlet pressure is constant, uniform, and non-rotational, a pressure-inlet boundary condition is adopted. The fluid temperature at both the inlet and outlet is set to 298.15 K (25 °C). The outlet condition uses a pressure-outlet boundary condition, with the outlet pressure set to atmospheric pressure. Based on previous calculations, the turbulence intensity is set to 5%. The effect of gravity is negligible and is therefore ignored. A no-slip condition is applied to the solid wall in the turbulence solution. In the region near the solid wall, the wall induces a sharp velocity gradient in the flow; here, the standard wall function method is used for correction.

2.9. Solver Settings

This study employs ANSYS Fluent 2022 for numerical simulation analysis. This software is widely used in the fields of aerodynamic separation and nozzle flow field simulation, and its reliability has been verified by numerous studies. Its RNG k-ε model is suitable for the nozzle jet conditions examined in this study. In terms of solver functionality, a pressure-based transient solver is utilised. The SIMPLE scheme was selected for pressure-velocity coupling, which couples the continuity and momentum equations via pressure correction equations. This scheme offers high versatility and stability, is insensitive to mesh quality, and provides superior solution robustness. Both turbulent kinetic energy and turbulent dissipation rate were set to second-order upwind schemes, with an absolute convergence criterion of 0.001; all other settings were left at their default values. The standard initialisation method is used. The simulation parameters are set to 100 time steps with a time step size of 0.0001 s.

2.10. Multi-Criteria Importance Assessment of Sorting Accuracy and Requirements for Airflow Parameters

The geometric parameters considered in this study include the contraction angle, outlet height and nozzle length, whilst the key flow evaluation criteria primarily comprise velocity uniformity at the outlet cross-section, the length of the jet core, and the degree of lateral velocity decay. As these criteria contribute to sorting accuracy to varying degrees, it is necessary during the numerical modelling and parameter optimisation stages to systematically evaluate the relative weights of each criterion with respect to sorting accuracy, in order to avoid decision-making biases resulting from subjective assumptions of equal weighting. This paper employs methods such as single-criterion-sorting accuracy response analysis, comprehensive weight calculation, and testing of coupling effects between criteria to conduct a quantitative analysis of the importance weights of each evaluation criterion. The analysis results indicate that the weights of the evaluation criteria relative to separation accuracy, in descending order, are outlet velocity uniformity, jet core length, and the degree of lateral velocity decay. The sum of the weight coefficients for outlet velocity uniformity and jet core length is approximately four times that of the degree of lateral velocity decay, confirming that ensuring outlet velocity uniformity and jet core length are the primary flow control objectives for improving separation accuracy. The above weighting analysis results will serve as the basis for multi-objective trade-offs in subsequent geometric parameter optimisation. Based on the results of the multi-criteria weighting analysis, this study has established requirements for the airflow parameters at the nozzle outlet. Outlet velocity uniformity is the primary factor affecting separation efficiency; the optimisation target is to control the velocity non-uniformity coefficient within 3%. The length of the jet core zone should ensure a sufficient impurity transport distance, requiring a minimum of 8 times the orifice diameter (7d).

3. Flow Field Simulation and Numerical Analysis

Numerical simulations of the internal and external flow patterns for the three nozzle designs were conducted using ANSYS Fluent 2022 software [24], with analyses performed on the internal pressure distribution, velocity distribution within the nozzle, axial velocity profile at the nozzle outlet, velocity profile at the 120 mm cross-section of the nozzle outlet, and velocity profile at the 200 mm cross-section of the nozzle outlet. Based on the analysis of nozzles with different structures for each design, optimal single-factor variables were selected to conduct an orthogonal experimental design, thereby identifying the optimal combination of parameters. Figure 5 illustrates the flow field simulation and numerical analysis workflow for this study.

3.1. Pressure Distribution Inside the Nozzle

As shown in Figure 6a, the streamlines curve at the junction between the inlet and outlet sections; the pressure on the outer side of the pipe wall is higher than that on the inner side, and there is a marked pressure jump at the bend, which is prone to generating vortices and causing energy losses. In the case of a nozzle without a fillet transition, there is a distinct local high-pressure zone at the inlet, whilst a local low-pressure zone appears on the wall of the straight section at the nozzle outlet, indicating that flow separation has occurred at this point, resulting in vortex losses and significant pressure losses. As shown in Figure 6b–d, after adding transition sections with different fillet radii (R), the internal pressure distribution tends to become more uniform, the gradient changes slow down, the local high-pressure zones shrink, and the overall pressure drop becomes more gradual. When R is 20 mm, the improvement in suppressing vortices is limited, and a small-scale vortex zone still exists at the transition point; when R is 50 mm, the flow conditions are significantly optimised; the distribution of pressure contour lines in the constriction section becomes smoother and more uniform, and the vortex regions almost disappear, though they still exist; when R is 80 mm, the distribution of internal pressure gradients is uniform, the vortex regions disappear, pressure losses are reduced, and the flow at the outlet is most stable. Based on the above analysis, the introduction of a fillet transition effectively reduces flow resistance, suppresses vortex formation, improves the nozzle’s aerodynamic performance, and enhances flow efficiency and stability. A comparison shows that increasing the fillet transition radius significantly optimises the internal pressure distribution within the nozzle and improves the flow state. When the fillet transition radius R is 80 mm, the nozzle’s internal flow performance approaches the optimum, whilst the internal flow losses are greatest in nozzles without a fillet transition.

3.2. Numerical Analysis of Scheme 1

Figure 7 shows the velocity field distribution of nozzles with different contraction angles, as processed using CFD-Post. The jet shapes produced by nozzles of different structures are all laminar, which meets the basic requirements of the impurity removal and sorting device. However, there are certain differences in the outer flow field conditions, maximum velocity, and core length of the jet among nozzles with different structural parameters. When the nozzle contraction angle is 10°, 12°, 14°, 18°, and 22°, the maximum velocities are 639 m/s, 640.6 m/s, 641.5 m/s, 647.7 m/s, and 661.8 m/s, respectively. It can be observed that as the contraction angle increases, the maximum velocity of the flow field also increases.

Figure 8a shows the variation in axial velocity along the flow direction for nozzles with different contraction angles. The jet core region refers to the section where the axial velocity remains high and stable; its length reflects the jet’s ability to maintain momentum. In the initial section, the axial velocity for all contraction angles remains essentially constant at 635–640 m/s, with only minimal decay, indicating that the core region is stable and that turbulent diffusion has not yet significantly affected the main flow velocity. In the transition section, the velocity begins to decrease slowly, and differences among the various angles gradually become apparent: the velocity for the 22° contraction angle is slightly higher than that of the other angles, while for the remaining angles, the rate of decay accelerates as the contraction angle increases, with the 18° contraction angle exhibiting the fastest decay. Upon entering the main stage, velocity decay significantly accelerates, and the velocity differences between angles widen markedly. The smaller the contraction angle, the slower the decay and the higher the retained velocity. The decay processes for contraction angles of 10° and 12° are relatively similar in this stage, indicating that reducing the contraction angle to 10° has a minimal further impact on the degree of decay. Overall, when the nozzle contraction angle is between 10° and 14°, the axial velocity decay is slower, the jet penetration capability is stronger, and a more favorable axial velocity distribution can be achieved.

Figure 8b shows the axial velocity distribution curve for a 120 mm cross-section at the nozzle outlet. All curves exhibit a typical Gaussian bell-shaped distribution, with the highest velocity at the axial position of 0 mm, gradually decreasing radially toward both sides until approaching 0, consistent with the radial velocity distribution characteristics of a free jet. To quantify the lateral diffusion characteristics of the jet, the jet half-width is introduced (defined as the radial distance at which the velocity drops to half the maximum velocity at that cross-section; in this paper, the full width at half height is used). Regarding the axial peak velocity, the peak velocities for contraction angles of 10°, 12°, and 14° are relatively close, with a peak velocity of approximately 615 m/s; the peak velocities for contraction angles of 18° and 22° are similar, with a peak velocity of approximately 610 m/s. The curves for contraction angles of 18° and 22° exhibit faster radial decay and a smaller jet half-width, indicating a more concentrated jet core and weaker radial diffusion; The curves for contraction angles of 10°, 12°, and 14° exhibit a gentler radial decay, a larger half-width, more thorough radial diffusion of the jet, stronger mixing with the surrounding fluid, and a larger coverage area for the high-speed jet. Analysis indicates that a contraction angle of 10–14° at the 120 mm outlet position is more conducive to achieving a wider jet influence range and stronger mixing effects.

Figure 8c shows the axial velocity distribution curve at the 200 mm cross-section downstream of the nozzle outlet. Compared to the 120 mm cross-section, the peak velocity at the 200 mm cross-section decreases by approximately 25% overall, indicating that the jet continues to attenuate as it propagates downstream. Figure 8d shows the velocity distribution curves for the nozzle outlet cross-sections at different contraction angles. All exhibit a symmetrical parabolic distribution, with the highest velocity at the axial position, gradually decreasing radially, and reaching a minimum at the wall. This distribution characteristic conforms to the basic laws of flow within the nozzle. The contraction angle has a significant effect on the velocity distribution at the nozzle outlet. By calculating the non-uniformity coefficient (M) of the velocity distribution at the outlet cross-section, the data obtained are shown in

Table 6. Coefficient of non-uniformity for the outlet cross-section of Option 1.

Variable	1	2	3	4	5
Contraction angle/°	10	12	14	18	22
Coefficient of Variation (M)	3.58%	1.10%	1.10%	1.28%	1.41%

.

As the contraction angle increased from 10° to 14°, the non-uniformity coefficient gradually decreased from 3.58% to 1.10%, indicating a continuous improvement in flow uniformity; when the contraction angle was further increased to 18° and 22°, the non-uniformity coefficient rose to 1.28% and 1.41%, respectively, showing a trend toward deteriorating uniformity. Therefore, within the range of contraction angles examined in Scheme 1, there exists an optimal contraction angle range (α approximately 12–14°) where flow uniformity is optimal and the non-uniformity coefficient can reach a minimum of 1.10%. This result provides an important basis for subsequent orthogonal experiments on geometric parameters.

In summary, in this study, when the nozzle contraction angle is controlled between 12° and 14°, the overall performance of the flow field is optimal, achieving a good balance between the core velocity of the jet and flow uniformity.

3.3. Numerical Analysis of Scheme 2

Figure 9 shows the axial velocity distribution for nozzles with different parameters. λ has a significant effect on the jet half-width and cross-sectional velocity; as λ increases, the axial velocity first rises and then falls, whilst the jet half-width continues to increase. When λ is 0.61, the half-width is small and the velocity profile is steep; the jet is concentrated but lacks sufficient diffusion. When λ is 0.66, the half-width increases significantly and the radial velocity profile is flattest, maintaining a high flow velocity over a wide range, thereby achieving a good balance between concentration and diffusion. When λ is 0.7, the half-width increases slowly and the axial velocity decreases slightly, indicating that an excessive ratio of outlet height to inlet diameter tends to exacerbate energy dissipation and increase central momentum loss.

Figure 10a shows the variation in the axial velocity of the jet for different values of λ. The higher the value of λ, the slower the jet velocity decays; appropriately increasing the value of λ can effectively delay the velocity decay and improve the jet’s velocity retention capability. Figure 10b shows the axial velocity distribution at the 120 mm outlet cross-section. λ has a significant influence on the jet half-width and cross-sectional velocity: as λ increases, the axial velocity first rises and then falls, whilst the jet half-width continues to increase. When λ is 0.61, the half-width is small and the velocity distribution curve is steep; the jet is concentrated but insufficiently diffused. When λ is 0.66, the half-width increases significantly, and the radial velocity distribution curve is the flattest, maintaining high flow velocities over a wider range, thereby achieving a good balance between concentration and diffusion. When λ is 0.70, the half-width increases slowly and the axial velocity decreases slightly, indicating that an excessively large outlet height tends to exacerbate energy dissipation and increase central momentum loss.

Figure 10c shows the axial velocity distribution at the 200 mm cross-section at the outlet; all curves exhibit a typical Gaussian bell-shaped distribution. Compared with the 120 mm cross-section, the overall velocity reduction at the 200 mm cross-section decreases as the nozzle height increases, with a maximum reduction of approximately 50%. As λ increases, the peak velocity rises significantly. A greater throat height implies a larger flow cross-sectional area; under constant flow conditions, this allows for higher static energy to be obtained by reducing the velocity coefficient, thereby increasing the core velocity of the jet. As λ increases, the jet half-width gradually increases; however, when λ is 0.66, the rate of decline in axial velocity at the cross-section is faster than when λ is 0.7 at distances greater than 12.5 mm from the centre. This indicates that for λ = 0.66, the optimal injection distance should be less than 200 mm. Analysis shows that when λ is 0.66—i.e., with an inlet diameter of 42 mm and an outlet height of 28 mm—the jet core is at its longest and the velocity remains most stable. This configuration maintains the jet flow rate whilst achieving moderate radial diffusion, resulting in superior uniformity of the velocity distribution over medium to long distances.

Analysis of the velocity distribution curve at the nozzle outlet cross-section shown in Figure 10d reveals that when λ is less than 0.61, velocity uniformity is poor and there is a distinct low-velocity region at the edges; when λ is moderate (λ ≈ 0.66), the jet velocity is high and the distribution is uniform, resulting in optimal flow field performance; when λ is high (λ ≥ 0.66), velocity uniformity is maintained, but the overall velocity decreases, leading to a reduction in energy efficiency.

As can be seen from the data in

Table 7. Coefficient of non-uniformity for the outlet cross-section of Option 2.

Variable	1	2	3	4	5
λ	0.41	0.53	0.61	0.66	0.70
Coefficient of Variation (M)	3.10%	1.10%	0.97%	0.79%	0.85%

, as λ increases from 0.41 to 0.66, the coefficient of variation decreases steadily from 3.10% to 0.79%, a reduction of 2.31%. When λ increases from 0.41 to 0.53, the coefficient of variation drops sharply from 3.10% to 1.10%, representing the most significant improvement. Thereafter, as the outlet height was further increased, the rate of decrease in the non-uniformity coefficient gradually slowed. When λ increased from 0.66 to 0.70, the non-uniformity coefficient rebounded from 0.79% to 0.85%, showing a slight increase. This indicates that excessively increasing the ratio of outlet height to inlet diameter may slightly degrade the uniformity of flow within the cross-section. Taking into account both uniformity and structural efficiency, a ratio of λ = 0.66 between the outlet height and the inlet diameter is the optimal solution, i.e., an outlet height of 28 mm and an inlet diameter of 42 mm. This configuration achieves good uniformity whilst avoiding structural redundancy.

3.4. Numerical Analysis of Scheme 3

Figure 11 and Figure 12a show the velocity flow field and axial velocity distribution curves for different nozzle lengths. As the nozzle length increases, the core region of the jet extends, and the maximum velocity gradually increases. Within the jet core, the axial velocity for each nozzle length remains in the range of 630–640 m/s, with minimal attenuation. Upon entering the attenuation zone, the axial velocity shows a marked downward trend, and the shorter the nozzle length, the faster the attenuation rate. When the nozzle length is 5 mm, the velocity drops to approximately 475 m/s at the 300 mm mark, whereas with a 20 mm nozzle length, it remains at approximately 520 m/s. This difference can be attributed to the increased nozzle length, allowing the fluid to develop more fully within the nozzle, reducing the boundary layer thickness and enhancing the stability of the jet core, thereby effectively delaying downstream velocity decay and improving jet stiffness. From an engineering application perspective, long nozzles are more suitable for operations requiring high-speed jets over long distances, while short nozzles offer cost advantages in close-range operations.

Figure 12b shows the axial velocity distribution at the 120 mm cross-section. Nozzle length has a significant effect on the radial distribution; as the nozzle length increases, the peak axial velocity rises. When the nozzle length is 5 mm, the peak velocity is approximately 620 m/s, and when the nozzle length is 20 mm, it increases to approximately 640 m/s. Long nozzles correspond to a larger jet half-width, while short nozzles result in a smaller jet half-width. When the nozzle length is 20 mm, the radial distribution of the jet is more uniform, the cross-sectional velocity non-uniformity is lowest, and the mixing effect is superior. Long nozzles can simultaneously increase the axial peak velocity and radial uniformity, making them suitable for demanding mid-range application scenarios.

Figure 12c shows the axial velocity distribution at a cross-section 200 mm downstream of the nozzle. The peak velocity does not decrease significantly compared to the 120 mm cross-section, and the effect of nozzle length on the radial distribution becomes more pronounced. When the nozzle length is 5 mm, the axial peak velocity is approximately 625 m/s, while it reaches about 640 m/s when the nozzle length is 20 mm; the advantage of the long nozzle in maintaining core velocity is more evident. The jet half-width is largest and radial uniformity is optimal when the orifice length is 20 mm. The effect of orifice length on velocity is far smaller than that of nozzle height; nozzle height is the primary optimization parameter, while orifice length is a secondary optimization parameter.

Analysis of the velocity distribution curves at the outlet cross-section in Figure 12d shows that all velocity curves exhibit a typical symmetrical parabolic distribution with a high center and low sides, consistent with the velocity characteristics of a supersonic jet outlet. As the orifice length increased from 5 mm to 20 mm, the overall velocity level at the outlet cross-section continued to rise. The maximum axial velocity increased from approximately 632.9 m/s to 634.7 m/s, an increase of 1.8 m/s, and the uniformity of the velocity distribution across the cross-section was significantly improved. According to the data in Table 7, when the nozzle length increased from 5 mm to 15 mm, the non-uniformity coefficient decreased from 0.85% to 0.72%, a reduction of 0.13%, indicating a trend of gradual improvement in uniformity. When the nozzle length was increased from 15 mm to 20 mm, the reduction in the non-uniformity coefficient was only 0.01%. Within the nozzle length range of Scheme 3, appropriately lengthening the nozzle can improve flow uniformity at the outlet cross-section, but the improvement effect slows down after exceeding a certain value. The detailed values for the coefficient of variation are given in

Table 8. Coefficient of non-uniformity for the outlet cross-section of Option 3.

Variable	1	2	3	4
Nozzle length/(mm)	5	10	15	20
Coefficient of Variation (M)	0.85%	0.73%	0.72%	0.71%

. A nozzle length of 15 mm is the optimal solution, as it significantly improves uniformity while avoiding the material costs and space requirements associated with excessive length.

In summary, a nozzle length of 15 mm is essentially equivalent to a 20 mm nozzle in terms of axial velocity decay rate, peak cross-sectional velocity, and radial uniformity. It can meet the requirements for jet stiffness and uniformity in the vast majority of applications. At the same time, its shorter machining length significantly reduces material and manufacturing costs, offering outstanding value for money. Therefore, given the need to balance performance and cost-effectiveness, a nozzle length of 15 mm is the preferred choice.

3.5. Validating Optimal Combinations Using Orthogonal Experiments

In actual industrial production and scientific research, there are often many factors to consider, and the levels of each factor typically number at least three. If comprehensive experiments were conducted for all possible combinations, the number of experiments required would be enormous. Therefore, orthogonal experimental design is widely used to analyze the effects of multiple factors on evaluation metrics. This method utilizes standardized orthogonal tables and, based on the principle of orthogonality, selects representative experimental points characterized by “uniform dispersion and orderly comparability.” It is a commonly used optimization technique for multi-factor, multi-level experiments. For a 4-factor, 3-level experiment, arranging the design according to L₉(3⁴) requires only 9 trials, thereby significantly reducing the workload. By rationally utilizing this characteristic of orthogonal experiments to design and arrange the trials, it is possible to comprehensively compare and process the experimental data with the fewest possible number of trials, thereby identifying the optimal combination of levels that influences the performance indicators.

Based on the results of the numerical analysis described above, the non-uniformity coefficient (M) of the outlet cross-section was identified as the evaluation criterion, and three experimental ‘factors’ were defined: factor A: contraction angle (α); factor B: λ (outlet height/inlet diameter); and factor C: nozzle length (l). This constitutes one evaluation criterion and three experimental factors. Orthogonal experiments are essentially designed to facilitate the fitting of linear models; they assume that, within the range under investigation, there is an approximate linear relationship between the response variable and the factors, with the range of levels representing the scope within which the linear assumption holds approximately. To test this linear relationship, each factor is typically assigned 2–3 levels. An orthogonal array requires that the frequency of all level combinations be the same between any two columns, and that the different levels within each column are, ideally, equally spaced numerically. This equidistance allows the orthogonal array to be used directly for the least-squares estimation of single-regression coefficients, with the effects of each factor being independent of one another. Taking practical considerations into account, calculations have determined that for this experiment, the equidistance will be set to 1 (one minimum unit), with each factor having 3 levels. Thus, factor A has 3 levels: 12° (Level 1), 13° (Level 2) and 14° (Level 3); factor B has 3 levels: 0.65 (Level 1), 0.66 (Level 2) and 0.67 (Level 3); and factor C has 3 levels: 14 mm (Level 1), 15 mm (Level 2) and 16 mm (Level 3). The specific factor levels are shown in

Table 9. Table of Factor Levels.

	Factor A: Contraction Angle/°	Factor B: λ	Factor C: Nozzle Length/mm
Level 1	12	0.65	14
Level 2	13	0.66	15
Level 3	14	0.67	16

.

In this orthogonal experiment, the objective is to identify an optimal combination of element levels that minimizes the non-uniformity coefficient at the nozzle outlet. Based on the factors and levels established in the previous experiment, and without considering interactions between factors, a blank column was included in the design, and a full L₉(3⁴) orthogonal array was selected for the experimental layout. The specific experimental design is shown in

Table 10. L₉(3⁴) orthogonal design table for the non-uniformity coefficient of the nozzle outlet cross-section.

Test Number	Factor A	Factor B	Factor C	Factor D (Empty Column)
1	1 (12°)	1 (0.65)	1 (14 mm)	1
2	1	2 (0.66)	2 (15 mm)	2
3	1	3 (0.67)	3 (16 mm)	3
4	2 (13°)	1	2	3
5	2	2	3	1
6	2	3	1	2
7	3 (14°)	1	3	2
8	3	2	1	3
9	3	3	2	1

.

The experiments were conducted according to the design specified in the orthogonal experimental design table. The non-uniformity coefficients of the nozzle outlet cross-section were measured for each test condition; the results are shown in

Table 11. Table of Non-uniformity Coefficients for Nozzle Outlet Cross-Sections.

Test Number	Test Protocol	Performance Metrics/%
1	A1_B1_C1_D1	1.01
2	A1_B2_C2_D2	1.03
3	A1_B3_C3_D3	1.07
4	A2_B1_C2_D3	0.82
5	A2_B2_C3_D1	0.91
6	A2_B3_C1_D2	0.92
7	A3_B1_C3_D2	0.85
8	A3_B2_C1_D3	0.92
9	A3_B3_C2_D1	0.93

above. The axial velocity curve of the outlet cross-section is shown in Figure 13, and the velocity distribution of the outlet cross-section is shown in Figure 14. In data analysis and statistics, both visual analysis and analysis of variance play important roles. The former facilitates a more intuitive understanding of the data’s implications through simple visualisation; the latter is a more in-depth statistical method, primarily used to compare whether there are significant differences between the means of different groups and to investigate the sources of such differences. In this study, the results were analysed using an orthogonal design with a two-way ANOVA to determine whether the effects of each factor on the evaluation indicators were significant.

An intuitive analysis of the orthogonal experiments on the outlet cross-sectional non-uniformity coefficient is presented; the results and deviation analysis are shown in

Table 12. Table of Visual Analysis of the Experiment.

Test Number	Factor A	Factor B	Factor C	Factor D	Performance Metric/%
1	1 (12°)	1 (27 mm)	1 (14 mm)	1	1.01
2	1	2 (28 mm)	2 (15 mm)	2	1.03
3	1	3 (29 mm)	3 (16 mm)	3	1.07
4	2 (13°)	1	2	4	0.82
5	2	2	3	5	0.91
6	2	3	1	6	0.92
7	3 (14°)	1	3	7	0.85
8	3	2	1	8	0.92
9	3	3	2	9	0.93

and

Table 13. Analysis of Deviations in Test Results.

Test Number	The Sum of the Coefficients of Cross-Sectional Irregularity for Exports at Different Levels of a Given Factor, Assuming that the Level of that Factor Remains Constant				Performance Metric/%
	Factor A	Factor B	Factor C	Factor D
K1	3.11	2.68	2.85	2.85	$T=8.46$ $T^2=71.5716$ $P=T^2/n=7.9524$
K2	2.65	2.86	2.78	2.80
K3	2.70	2.92	2.83	2.81
R	0.46	0.24	0.07	0.05
K12	9.67	7.18	8.12	8.12	$S{S}_i=\sum {K_i}^2/K_a$ − P $S{S}_T=\sum S{S}_i$ = 0.0542
K22	7.02	8.18	7.73	7.84
K32	7.29	8.53	8.01	7.90
SSi	0.0425	0.0104	0.0009	0.0005

Note: K_i represents the deviation, defined as the sum of the indicator values corresponding to different levels of the same factor (i = 1, 2, 3); R represents the range; T represents the sum of the indicator values; P represents the correction term; n represents the total number of trials; SS_i represents the sum of squares of deviations for each factor; SS_T represents the sum of the sums of squares of deviations for all factors.

respectively. The deviation analysis indicates that the three factors influence the outlet cross-sectional non-uniformity coefficient in descending order of significance: A, B and C. The relationship between the ranges (R) is A > B > C > D (blank column); a larger range indicates that the factor has a more significant influence on the experimental results. Therefore, in this orthogonal experiment, the influence of each factor on the nozzle outlet cross-sectional non-uniformity coefficient, from primary to secondary, is as follows: contraction angle (α), λ (outlet height/inlet diameter), and nozzle length (l).

The test criterion is the non-uniformity coefficient of the nozzle outlet cross-section; a smaller value is better. A smaller value of the factor deviation K_i is more advantageous for reducing the orthogonal test criterion; that is, the smaller K_i is, the smaller the non-uniformity coefficient (M) becomes. In the table below, the levels corresponding to the minimum values of K₁, K₂, and K₃ are A₂, B₁, and C₂, respectively. Through visual analysis, the level 2 corresponding to the minimum deviation K₂ of 2.65 for factor A is selected as the optimal level for factor A; the level 1 corresponding to the minimum deviation K₁ of 2.68 for factor B is selected as the optimal level for factor B; For factor C, the level corresponding to the minimum deviation K₂ of 2.78 is selected as the optimal level for factor C. The optimal combination of factor levels identified through this orthogonal experiment is Test No. 4: A1_B2_C2 (Convergence angle α is 13 degrees, λ is 0.65 (outlet height is 27 mm, inlet diameter is 41 mm), and nozzle length l is 15 mm).

Based on the analysis of variance for the non-uniformity coefficient of the nozzle outlet cross-section (see

Table 14. Analysis of Variance Table.

Sources of Variance	SS	df	MS	F-Value	p-Value	Significance	Contribution Rate/%
A	0.0425	2	0.0212	91.0000	0.0083	**	77.16
B	0.0104	2	0.0052	22.2857	0.0218	*	18.25
C	0.0009	2	0.0004	1.8571	0.2917	×	0.73
Error e	0.0005	2	0.0002	—	—	—	3.86

Note: ** indicates highly significant, * indicates significant, × indicates not significant, and — indicates missing data.

for results), the F-test was used to assess the significance of each factor on the test indicators. Under a significance level of 0.01 and 0.05, with degrees of freedom f₁ = 2 and f₂ = 2: the p-value for factor B falls between 0.01 and 0.05, indicating that λ (outlet height/inlet diameter) has a significant effect on the test indicators (*); The p-value for factor A is less than 0.01, indicating that the contraction angle has a highly significant effect on the test indicators (**); factor C, however, has no significant effect. Table 14 also shows that factor A is the most critical, with variations in its levels accounting for 77.16% of the total sum of squares; factor B is the next most significant, accounting for 18.25%; variations in factor C have no significant effect; furthermore, there are data fluctuations caused by error.

This orthogonal experimental design is based on the assumption of additivity of main effects, namely that the contribution of interactions between factors to the response variable is relatively small and insufficient to significantly alter the selection of the optimal parameter combination. Under this assumption, no interaction columns are included in the orthogonal table design, and the corresponding analysis of variance model does not include interaction terms. Following completion of the orthogonal experiment, the optimal combination identified through the analysis of main effects was selected for comparison and validation against several candidate combinations. The results indicate that the actual sorting accuracy of the optimal combination derived from the orthogonal design remains superior to that of the other candidate combinations, indirectly suggesting that the interactions did not have a substantial impact on the conclusions drawn from the main effects.

The simulation results for the optimized nozzle shown in Figure 15 indicate that the gas is compressed in the throat, where the pressure decreases; it expands nearly completely in the expansion section, reaching a velocity close to its maximum value, while the outlet static pressure approaches ambient pressure, thereby effectively reducing pressure loss. The axial velocity distribution shows that this nozzle generates a confined high-flow gas stream with strong entrainment capability and a more concentrated jet direction, thereby enhancing the accuracy and stability of the spray.

4. Field Testing and Validation

To verify the feasibility of the study and the impurity removal performance of the nozzle following parameter optimisation, an impurity removal test was conducted in December 2025 at a cotton spinning mill in the Aksu region of Xinjiang using the optimal nozzle design (with a contraction angle of 13 degrees, λ = 0.65 (outlet height 27 mm, inlet diameter 41 mm), and a nozzle length of 15 mm) at a cotton spinning mill in Aksu, Xinjiang. The results were compared with test data from the original nozzle. The test utilised 1000 kg of machine-harvested cotton as the material, with a total of 13 test sets. The conveyor belt speed was maintained at a constant 1 m/s, and the nozzle operating pressure was kept constant at 0.35 MPa. Figure 16 illustrates the workflow of the seed cotton foreign fibre removal system, whilst Figure 17 shows a schematic diagram of the balanced three-way valve selected for the apparatus. This valve delivers a constant flow rate unaffected by air pressure, providing a consistent high flow rate across the entire pressure range. The optimised nozzle was fitted to the equipment for testing. The method for calculating the removal rate is shown in Equation (2).

$$\eta ={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{m_1}{m\left(1 - w_0\right)}}+{\displaystyle \frac{m_2}{{mw}_0}}\right] \times 100\%$$

(9)

In Equation (9) above: $\eta$ represents the removal efficiency; m₁ represents the mass of cotton, in kg; m₂ represents the mass of impurities, in kg; m represents the total mass, in kg; and w₀ represents the mass fraction of impurities in the initial cotton.

Table 15. Test results for December 2025.

Category	The Number of Fibres in the Cotton	Number of Films Removed by the Original Nozzle	Amount of Cotton Removed by the Original Nozzle	Optimise The Number Of Nozzles Used To Remove The Film	Optimise the Amount of Cotton Removed from the Nozzle	Clearance of the Original Nozzle	Optimising Nozzle Clearance
1	241	218	36	233	12	90.46%	96.68%
2	196	176	32	189	15	89.80%	96.43%
3	221	194	41	216	18	87.78%	97.74%
4	205	181	29	197	11	88.29%	96.10%
5	85	74	21	81	9	87.06%	95.29%
6	162	144	25	156	13	88.89%	96.30%
7	159	139	26	152	10	87.42%	95.60%
8	345	310	43	333	16	89.86%	96.52%
9	96	85	20	92	8	88.54%	95.83%
10	232	206	32	223	14	88.79%	96.12%
11	66	58	17	63	7	87.88%	95.45%
12	113	99	23	108	10	87.61%	95.58%
13	73	65	13	70	11	89.04%	95.89%
Total	2194	1949	358	2113	154	88.83%	96.31%

presents the experimental data; the removal efficiency for all test groups remained consistently above 95%, maintaining high removal efficiency across samples with varying film content. Figure 18 presents the data on removal efficiency and the quantity of cotton lost for the original nozzle and the optimised nozzle. The average removal efficiency of the optimised nozzle reached 96.31%, an increase of approximately 7.5 percentage points compared to the original nozzle (88.83%). This meets the requirement of a weed removal efficiency of no less than 50% as stipulated in GB/T 22335-2018 [5], and far exceeds the standard; The amount of cotton mistakenly removed by the optimised nozzle was reduced by 57%, indicating that the rounded transition design effectively suppressed vortices, resulting in a more uniform airflow distribution and reducing the loss of good cotton due to entrainment; the removal efficiency of the optimised nozzle remained stable between 95.29% and 97.74%, with a small standard deviation, demonstrating the optimised design’s excellent consistency. In summary, the optimised nozzle has resulted in a significant improvement in removal efficiency, a substantial reduction in cotton drop-out rates, and excellent stability in removal performance.

5. Conclusions

This paper utilises Fluent software to perform numerical simulations of nozzles with different structural parameters. Through an orthogonal experimental design, the optimal nozzle structural parameters were determined, and experimental tests were conducted on actual equipment. Compared with existing CFD-based nozzle optimisation studies, this research demonstrates clear innovation and significance in three aspects: methodology, depth of quantification, and engineering validation.

At the methodological level, existing studies predominantly analyse nozzle height and inlet diameter as independent variables, failing to recognise the inherent geometric coupling between the two when the contraction angle is kept constant. This study is the first to explicitly highlight this coupling issue and introduces the dimensionless parameter λ (outlet height/inlet diameter) to replace absolute nozzle height as the analytical variable, enabling changes in flow field characteristics to be reasonably attributed to the dimensionless parameter λ. This not only resolves the methodological flaw of ambiguous parameter attribution but also provides a unified benchmark for performance comparison, which can be extended to the optimisation of other equipment involving multi-parameter coupling.

In terms of the depth of quantification, although previous studies have pointed out that the transition at the nozzle bend leads to vortices and energy losses, they have mostly relied on qualitative descriptions or comparisons of a single fillet radius, lacking a systematic quantification of vortices under different fillet radii. This study is the first to quantitatively compare the pressure distributions and vortex evolution patterns for different fillet radii. This systematic quantitative analysis, progressing from ‘nothing to something’ and ‘from small to optimal’, provides clear guidance for selecting parameters in nozzle fillet transition design, significantly enhancing the engineering guidance value of research in this field.

At the level of engineering validation, most existing studies on nozzle optimisation have been limited to numerical simulations, lacking verification through field testing. This study combines numerical simulation with field testing, using national standards as constraints, to verify the actual impurity removal performance of the optimised nozzle. Compared with the performance of existing seed cotton cleaners reported in the literature, the performance of the optimised nozzle in this study is significantly superior to the literature values, verifying the effectiveness and advanced nature of the proposed optimisation scheme.

The following conclusions are drawn:

• Introducing a fillet transition at the nozzle bend effectively mitigates pressure concentration and sudden drops at the abrupt change in cross-section, thereby reducing flow resistance, suppressing vortex formation, ensuring a more uniform internal pressure distribution, and minimising energy loss. When the fillet transition radius R is 80 mm, flow performance approaches the optimum, with a uniform pressure gradient and the complete elimination of vortex zones.
• When the contraction angle is within the range of 12° to 14°, the ratio of outlet height to inlet diameter is approximately 0.66 (outlet height approximately 28 mm, inlet diameter approximately 42 mm), and the nozzle length is approximately 15 mm, the length of the jet core in the external flow field reaches an optimum, velocity decay is smooth, and velocity uniformity at the outlet cross-section is at its best. This configuration achieves good radial diffusion whilst maintaining jet flow rate, effectively reducing material and manufacturing costs, and offering high cost-effectiveness.
• Through orthogonal experiments, the optimal combination of the three factors was determined to be: a contraction angle of 13°, a ratio of outlet height to inlet diameter of 0.65 (outlet height of 27 mm, inlet diameter of 41 mm), and a nozzle length of 15 mm. The contribution of each factor, from highest to lowest, is as follows: taper angle (77.16%) → λ (outlet height/inlet diameter) (18.25%) → nozzle length (0.73%).
• On-site field tests confirmed that, when using the optimal combination of parameters, the foreign fibre removal rate remained consistently above 95%, with an overall average removal rate of 96.31%, representing an improvement of approximately 7.5 percentage points compared to the original nozzle (88.83%). The optimised nozzle reduced the amount of cotton mistakenly removed by 57%; its overall removal performance was excellent and highly stable, maintaining a high removal rate across samples with varying film content.
• The influence of the nozzle’s vertical height and its angle relative to the cotton on the impurity removal rate requires further investigation.