The Effect of a Three-Blade Tube on the Pneumatic Transport of Pebble Particles

Abstract

In this paper, the Computational Fluid Dynamics–Discrete Element Method (CFD-DEM) coupling method was used to simulate the pneumatic transport of pebble particles in a three-blade spiral tube. The results showed that the flow field distribution rotated along the circumference after loading. The maximum velocity of the flow field after loading was manifested as rotation along the circumference. In addition, the swirl intensity decreased exponentially with the increase in conveying distance, and the maximum swirl intensity had a saturation value. After reaching the saturation value, it is not evident that increasing the initial air velocity significantly affected swirl variation. The smaller the pitch, the greater the initial swirl intensity. The swirling flow was conducive to the fluidization of particles, but it would bring a significant energy loss. Increasing the swirl can increase the degree of particle dispersion. There is an optimal tangential airflow velocity, which allows the particles to fully spin and stay in the suspension zone without being thrown onto the pipe wall by excessive centrifugal force. At this time, the energy efficiency reaches the highest level. A 5.87 m/s velocity was deemed the optimal tangential airflow velocity for conveying 3 mm particles.

1. Introduction

Due to its advantages of environmental protection, simple operation, and flexible layout, pneumatic conveying is the preferred technology for pebble particle pipeline transportation [1]. Higher gas velocities for pneumatic conveying can lead to higher power consumption, particle degradation, and pipeline erosion [2]. Lower gas velocities can cause pipeline blockage due to pressure loss [3]. To solve the above issues, swirling flow is applied to pneumatic conveying to facilitate particle fluidization. Swirling technology transfers the rotational motion around an axis parallel to the flow direction of the body flow, which can increase the tangential motion of the fluid [4]. It has been widely used in industrial transportation [5], enhanced convective heat transfer [6], production of microbubble generators [7], industrial combustion devices [8], and supersonic cyclone separators [9]. Revealing the complex gas–solid interactions in a swirl field is of great significance for designing and optimizing large particle pneumatic conveying systems.

A swirling generator is one of the core components that generate a swirling field, and scholars have conducted extensive research on the impact of its structural parameters on performance. Yilmaz et al. [10] compared the working performance of swirling generators containing conical spherical deflection elements and those without deflection elements and found that swirling generators without deflection elements had the highest pressure drop. Bali et al. [11] conducted a comparative study on the pressure drop characteristics of swirling flow generated by single- and double-propeller swirling generators and found that the pressure drop decreased with increasing axial distance. Zhou et al. [12] compared and analyzed the performance of three-blade, gun bore, and inner spiral swirling generators based on CFD-DEM coupling and developed a side inlet guide vane swirling generator based on this. It was found that the picking speed of 5–15 mm coal particles first increased and then decreased with the swirling number [13]. They also investigated the pressure characteristics and spectral composition of static pressure at different conveying stages in swirling pneumatic conveying [14]. Wannassi et al. [15] conducted three-dimensional calculations on blade cyclones with different blade angles, and the results showed that the swirl intensity at the inlet of the vortex generator depends on the blade angle. Wang et al. [16] studied the effect of a cyclone on the separation performance of a supersonic separator. They found that the rotational strength decreases with an increase in the outlet angle of the cyclone. Yao et al. [17] used numerical simulation to compare cyclone separators with different lengths of inlet pipes. They concluded that as the length of the inlet pipe increased, the tangential velocity of the gas in the swirling generator increased.

The research of the above scholars provides detailed references for the study of swirl generators, but most of them are single-factor studies. The working conditions of pneumatic conveying are complex, and there are interactions between various factors. Therefore, studying the optimal matching relationship of parameters under different working conditions is necessary.

There are various research methods for fluid in a swirling field, such as experimental research methods. Li et al. [18] measured the particle velocity in horizontal dilute phase swirling pneumatic conveying. They found that when the airflow velocity was high, the average particle velocity in swirling pneumatic conveying was lower than in axial pneumatic conveying. Pashtrapanska et al. [19] studied turbulent stress in rotating pipe flow using a Doppler measurement instrument. They found that compared to the initial development stage of eddy currents, the attenuation of eddy currents at the rear end of the pipeline causes explosive growth in turbulent kinetic energy. Chang et al. [20] used Particle Image Velocimetry (PIV) technology to measure the velocity distribution of water flow in a vertical circular pipe. They found that the vortex velocity vector moved the fluid centerline at a specific flow angle and twisted along the pipeline. Zhou et al. [21] studied the crushing characteristics of coal particles in the swirling flow field. It was found that the integrity of coal block aggregates decreased in a quasi-periodic manner compared to in a swirling field.

Regarding numerical simulation, Escueet et al. [22] explored the swirling flow in a straight pipe using the RNG (Renoamalization Group) k − ε models and Reynolds stress. The former was found to be more consistent with the experimental velocity profile of low vortex flow. Chu et al. [23] conducted a numerical study on gas–solid flow in a gas cyclone using the CFD-DEM method. They concluded that the severe particle wall collision area was mainly located at the inlet of the cyclone. As the solid load ratio increases, the number of revolutions of the solid in the cyclone decreases. Liu et al. [24] explored the swirling guided by a blade-type swirling generator in a vertical pipeline and found four swirl states at the outlet of the generator: chain, swirl gas column, swirl intermittent flow, and swirl annular flow. Zhang et al. [25] investigated the swirling gas–solid two-phase flow in an annular pipeline with a spiral cyclone according to the CFD-DEM method. They found that the gas phase above the cyclone had a high Reynolds number. They also found that increasing the blade angle and diameter of the cyclone favors particle dispersion [26].

The above scholars have used various methods, such as experiments and numerical simulations, to analyze gas–solid two-phase flow in different application scenarios, providing fruitful results. However, existing research mainly focuses on powder particles, and further research is needed to determine whether the conclusions obtained apply to large particles.

In summary, this article intends to use the CFD-DEM coupling simulation method to study the pneumatic transport characteristics of pebble particles in the swirling field guided by a three-blade spiral tube and explore the effects of factors such as the pitch of the three-blade tube and the initial velocity of the airflow on the pressure drop, particle flow state, and gas–solid two-phase velocity.

2. Numerical Methods

The CFD-DEM method has been widely recognized as an efficient method for studying the motion behavior of complex multiphase flows [27,28]. In this method, the motion of particles and flow fields is obtained by solving Newton’s second law and Navier Stokes equations, respectively, and coupled through particle fluid forces.

2.1. Governing Equations for Particle

Based on the softball model proposed by Cundall and Strack [29,30], there are two types of motion of particles in particle fluid flow systems: translation and rotation. The control equations for the translational and rotational motion of particles $i$ with mass $m_i$ and moment of inertia $I_i$ are

$$m_i\frac{d{\mathit{v}}_i}{dt}={\mathit{f}}_{pf,i}+{\displaystyle \sum _{j=1}^{k_c}({\mathit{f}}_{c,ij}+{\mathit{f}}_{d,ij})}+m_i\mathit{g}$$

(1)

$$I_i\frac{d{\mathit{\omega }}_i}{dt}={\displaystyle \sum _{j=1}^{k_c}({\mathit{M}}_{t,ij}+{\mathit{M}}_{r,ij})}$$

(2)

where ${\mathit{v}}_i$ and ${\mathit{\omega }}_i$ are the translational and angular velocities of particle $i$, respectively; $k_c$ is the number of particles interacting with particle $i$; $m_i\mathit{g}$ is gravity; ${\mathit{f}}_{c,ij}$ is the particle–particle force including elastic force; ${\mathit{f}}_{d,ij}$ is the viscous damping force; ${\mathit{M}}_{i,ij}$ and ${\mathit{M}}_{r,ij}$ are the torque caused by tangential force, and rolling friction torque of particle $j$ acting on particle $i$, respectively, and the second torque causes the decay in the relative rotational motion of particles.

The sum of particle–fluid interactions ${\mathit{f}}_{pf,i}$ acting on particle $i$ is

$${\mathit{f}}_{pf,i}={\mathit{f}}_{d,i}+{\mathit{f}}_{\nabla p,i}+{\mathit{f}}_{\nabla ·\tau ,i}+{\mathit{f}}_{vm,i}+{\mathit{f}}_{B,i}+{\mathit{f}}_{Saff,i}+{\mathit{f}}_{Mag,i}$$

(3)

where ${\mathit{f}}_{d,i}, {\mathit{f}}_{\nabla p,i}, {\mathit{f}}_{\nabla ·\tau ,i}, {\mathit{f}}_{vm,i}, {\mathit{f}}_{B,i}, {\mathit{f}}_{Saff,i}$, and ${\mathit{f}}_{Mag,i}$ are the drag force, pressure gradient force, and viscous force caused by fluid shear stress and deviator stress tensors, and virtual mass force, Basset force, Saffman lift force, and Magnus lift force, respectively.

Ignoring the drag force, pressure gradient force, and viscous force, the dominant force ${\mathit{f}}_i^{″}$ in the particle fluid flow on the individual particle $i$ is

$${\mathit{f}}_i^{″}={\mathit{f}}_{vm,i}+{\mathit{f}}_{B,i}+{\mathit{f}}_{Saff,i}+{\mathit{f}}_{Mag,i}$$

(4)

2.2. Governing Equations for the Gas Phase

Zhou [31] points out that there are two mainstream control equation models in CFD-DEM, called Model A and Model B, which differ in their handling of the pressure source term. Compared to Model A, Model B has better stability and higher authenticity. Therefore, this paper selected Model B as the control equation. In this model, the particle fluid interaction forces acting on individual particles in each computing unit are added to obtain the total force at the scale of the computing unit. The mass equation and momentum equation of the gas phase are

$$\frac{\mathsf{\partial }({\epsilon }_f)}{\mathsf{\partial }t}+\nabla ·({\epsilon }_f\mathit{u})=0$$

(5)

where ${\epsilon }_f$ is the volume fraction of fluid, and $\mathit{u}$ is the velocity of fluid.

$$\frac{\mathsf{\partial }({\rho }_f{\epsilon }_f\mathit{u})}{\mathsf{\partial }t}+\nabla ·({\rho }_f{\epsilon }_f\mathit{u}\mathit{u})=-\nabla p-{\mathit{F}}_{pf}+\nabla ·\mathit{\tau }+{\rho }_f{\epsilon }_f\mathit{g}$$

(6)

where ${\rho }_f$ is the density of fluid, $\nabla$ is the Hamiltonian differential operator, ${\mathit{F}}_{pf}$ is particle–fluid interaction force, $\tau$ is the viscous stress tensor, $\mathit{g}$ is the gravitational acceleration.

Where

$${\mathit{F}}_{pf}=\frac{1}{{\epsilon }_f\mathsf{\Delta }V}{\displaystyle \sum _{i=1}^n\left({\mathit{f}}_{d,i}+{{\mathit{f}}^{″}}_i\right)}-\frac{1}{\mathsf{\Delta }V}{\displaystyle \sum _{i=1}^n\left({\rho }_f{V}_{p,i}\mathit{g}\right)}$$

(7)

where $\mathsf{\Delta }V$ is the volume of the calculated unit.

At this time, to satisfy Newton’s third law of motion, the particle–fluid interaction force on particle $i$ should be written as

$${\mathit{f}}_{pf,i}=\left({\mathit{f}}_{d,i}+{\mathit{f}}_i^{″}\right)/{\epsilon }_f-{\rho }_f{V}_{p,i}\mathit{g}$$

(8)

3. Numerical Simulation Setup

3.1. Model Setting

The structure of the three-bladed tube is shown in Figure 1. As shown in Figure 2, the conveying device model used in this paper consisted of upstream straight pipes, three-blade spiral pipes, conical pipes, and downstream straight pipes.

The conveying pipe diameter was D = 50 mm, and the total length of the swirling generator experimental model was 3.1 m. The upstream straight pipe (with a length of 10D) was utilized to stabilize the airflow. The particle factory was located at the inlet at 4D. The spiral pipe was connected to the upstream straight pipe and had a length of 8D. The conical tube was adopted for wall blockage caused by sudden changes in pipe diameter, with a length of 6D. The downstream straight pipe was employed to observe experimental phenomena, with a length of 40D.

As shown in

Table 1. Experimental plan for numerical simulation.

Control Parameters	Value Range
The pitch of the three-leaf spiral tube (p), mm	200, 300, 400, 500, 600, Axial
The initial velocity of airflow (v0), m/s	30, 35, 40, 45

, five types of three-blade tubes with different pitches were selected as the starting devices for the experiment, with screw pitch (p) of 200 mm, 300 mm, 400 mm, 500 mm, and 600 mm, respectively. A group of axial flow tests without starting devices were added as a control to investigate the impact of swirling flow. The initial airflow velocities (v₀) of 30 m/s, 35 m/s, 40 m/s, and 45 m/s were used to observe the matching effect between the structural parameters of the spinning device and the operating conditions.

3.2. Numerical Simulation Boundary Conditions

In this paper, we used commercial software Ansys Fluent 2022 R1 and EDEM 2022.1 for CFD and DEM calculations, respectively. Firstly, Fluent converges the flow field calculation at a certain point in time, and the flow field information is converted into the fluid drag force acting on particles in EDEM through the drag force model. EDEM calculates the external forces (fluid drag force, gravity, collision force, etc.) exerted on each particle, and updates the position and velocity information of particles accordingly. Finally, these particle properties are added to the CFD calculation in the form of momentum sinks, thus affecting the flow field.

All calculation areas were divided into hexahedral structured grids. In order to make the grid size larger than the particle diameter, it was taken as 4 mm × 4 mm × 8 mm, which was used to connect the interface between the spiral pipe section and the straight pipe section, as shown in Figure 3.

In the CFD calculations, we used the pressure-based solver and the Realizable k − ε Model. The finite volume method was utilized to discretize the control equation. The QUICK algorithm was used to discretize the turbulent diffusion equation and momentum equation. The control equation was solved by the pressure velocity coupled SIMPLE algorithm. The coupling solution of the CFD-DEM was calculated using the Euler–Lagrange bilateral coupling method. The calculation time step in DEM was set to 1 × 10⁻⁶ s, and the calculation time step in CFD was set at 0.001 s. The particle shape was spherical, with a diameter of d = 3 mm. The particle factory adopted a dynamic unrestricted mode and generated particles at a rate of 1 kg/s. The parameters for the CFD-DEM coupling simulation are shown in

Table 2. Numerical Simulation Parameters.

Item		Detailed Information	Index	Value Size
CFD	Material Properties	Air	Density (kg/m3)	1.225
			Kinematic viscosity (kg/(m·s))	1.978 × 10−5
		Tube wall	Slip type	No slip
			Roughness height (mm)	0.001
			Roughness constant	0.5
	Boundary	Velocity inlet	Velocity (m/s)	30~45
			Turbulence intensity (%)	3.87~4.1
			Hydraulic pipe diameter (mm)	50
			Initial gauge pressure (MPa)	0
		Pressure outlet	Gauge pressure (Pa)	0
		Time step	Time step size (s)	0.001
			Total time steps	800
DEM	Material Properties	Particle	Density (kg/m3)	2300
			Poisson’s ratio	0.25
			Shear modulus (Pa)	1 × 1010
			Diameter (mm)	5
		Wall surface	Density (kg/m3)	7861
			Poisson’s ratio	0.3
			Shear modulus (Pa)	7.98 × 1010
	Collision	Particles–particles	Restitution	0.55
			Static friction coefficient	0.68
			Coefficient of rolling friction	0.15
			Collision model	Hertz–Mindlin
		Particle wall surface	Restitution	0.5
			Static friction coefficient	0.5
			Coefficient of rolling friction	0.05
			Collision model	Hertz–Mindlin
	Numerical Simulation Settings	Time step	Time step size (s)	1 × 10−6

.

4. Results and Discussion

4.1. Model Validation

To verify the independence of the grid, we divided it into six different sizes of grids, and the specific parameters are shown in

Table 3. Grid independence inspection plan.

Mesh Size (mm × mm × mm)	Number of Nodes	Number of Elements
3.0 × 3.0 × 6.3	287,315	304,005
3.5 × 3.5 × 7.0	188,332	201,049
4.0 × 4.0 × 8.0	135,770	145,578
4.5 × 4.5 × 9.0	88,899	96,502
5.0 × 5.0 × 11.5	57,762	63,072
5.5 × 5.5 × 14.3	42,579	46,593

. Numerical calculations were conducted under the same boundary conditions, with an initial wind speed of 45 m/s and a spiral pitch of 400 mm for a three-blade tube. The maximum axial velocity of the outlet section at different simulation times was used as the analysis indicator, and the test results are shown in Figure 4.

Figure 4 shows that the optimal axial velocity of the outlet section decreases with the increase in grid size, and the sensitivity to velocity fluctuations gradually decrease in the latter half of the simulation. However, when the grid size increases to 4.0 mm × 4.0 mm × 8.0 mm, the trend significantly slowed down. Taking into account both calculation accuracy and time, this paper adopts the grid of a 4.0 mm × 4.0 mm × 8.0 mm size.

The accuracy of the CFD-DEM numerical simulation model needs to be verified through experiments. As shown in Figure 5, when p was 300 mm, and v₀ was 20 m/s, the tangential velocity obtained from numerical simulation at 0.5 m downstream was compared with the experimental results of S. Fokeer [32]. The monitoring section was perpendicular to the axis, and the ratio of the distance l from each monitoring point to the axis to the radius r of the pipeline (l/r) was 0, ±0.25, ±0.5, and ±0.75, respectively. When l/r approaches 0, it indicates that it is closer to the axis. The results showed that the tangential velocity on the cross-section was symmetrically distributed along the axis, and the closer it was to the pipe wall, the greater the tangential velocity value. The tangential velocity at the axis was almost zero since the helical tube causes the tangential rate of the flow to occur near the wall due to the wall attachment effect. The numerical simulation results were slightly higher than the experimental results because of the use of spherical particles in the numerical simulation, which differs from the actual particle shape. The simulation did not consider the airflow fluctuations caused by devices such as air compressors and rotary feeders in the conveying system. Overall, the simulation results are consistent with the experimental results, and their errors are within an acceptable range, proving the accuracy of the numerical simulation model.

4.2. Flow Field Characteristics

In gas-solid two-phase flow, airflow velocity is one of the critical factors affecting the flow field characteristics. In this paper, the axial and tangential airflow velocity measurement range was set as a distance from the exit of the cone to 40D downstream of it, and each measurement section was taken every 10D. In order to consider the influence of gravity, seven evenly divided points along the centerline in the direction of gravity on each section were used as the measurement points. l/r represents the distance between the measurement points and the axis. The positive direction of axial velocity is in the same direction as the initial airflow velocity, while the positive direction of tangential velocity is clockwise.

4.2.1. Axial Velocity

When v₀ was 45 m/s, the axial velocity distribution in different conveying pipelines without loading particles is shown in Figure 6, where L is the distance between the measured section and the cone outlet. It can be seen that the axial velocities at the center of the pipe were slightly different for the vortex pipe and the straight pipe. The airflow velocity at the center of different sections of the straight tube remains almost unchanged, while the axial airflow velocity at the center of the swirl tube increased slowly with the increase in conveying distance. This is because the further away from the swirling generator, the more pronounced the attenuation of the tangential airflow became. The axial velocity distributions of the swirling and straight pipe are very similar. This is because the three-blade tube separated a portion of the total airflow as tangential airflow, which had a slight numerical impact on the total airflow. Due to the friction between the pipeline and the wall, the different conveyor pipes showed a maximum velocity at the center and a tendency to reduce the velocity as they approached the wall. The axial airflow velocity was symmetrically distributed along the axis.

To illustrate the influence of particle load on the flow field, we captured the particle distribution and axial velocity cloud map of the cross-section at a distance of 1100 mm from the inlet in a straight pipe with v₀ = 45 m/s and compared them, as shown in Figure 7. The graph shows that the position with more particles has a lower airflow velocity, because the particles hinder the flow of the airflow.

The cloud plots of axial airflow velocity distribution after particle loading are depicted in Figure 8. It is clear that the axial airflow in different pipes no longer showed a symmetrical distribution compared to the pre-loading period, and the axial velocities of the vortex pipe and the straight pipe showed significant differences. The axial velocity decreases when particles obstruct the airflow process, so the color distribution of the cloud plots also represents the distribution density of the particles, i.e., the smaller the axial velocity, the more particles there are.

In the straight pipe, particles were affected by gravity and accumulated at the bottom of the pipe, causing the axial velocity of the airflow to decrease in the direction of gravity. As the conveying distance increased, the airflow loss increased, manifesting as the axial velocity decreased, but the velocity distribution at different cross-sections remained unchanged. In the swirling pipe, the particles were suspended by tangential airflow disturbance and showed a spiral flow pattern. Correspondingly, the maximum airflow velocity on the cross-section also exhibited circumferential rotation. The rotation trend decreased with the increase in pitch. When the pitch increased from 200 mm to 600 mm, the maintenance distance of the rotation motion decreased from 40D to 20D, indicating that particle rotation was driven by airflow rotation, and the intensities of the two motions were positively correlated.

4.2.2. Tangential Velocity

When v₀ = 45 m/s, the tangential velocity distribution in different conveying pipelines without loading particles is shown in Figure 9. The tangential velocity of the straight pipe and the swirling pipe showed noticeable differences, proving that the three-blade tube could effectively separate the portion of the axial total airflow into tangential airflow. The maximum tangential velocity in the swirl tube with different pitches occurred near the wall rather than at the wall. The closer to the center of the pipe, the smaller the tangential airflow velocity was, and it was symmetrically distributed along the center of the pipe. It can be attributed to the fact that the working principle of the three-blade tube is the wall attachment effect, the swirling flow occurred at the pipe wall, the airflow and the wall friction were the most serious, and the velocity extreme point would appear in the vicinity of the wall, rather than at the wall itself. With the increase in conveying distance, the tangential velocity decreased. At the same time, the changing trend was more prominent compared to the axial airflow velocity, implying that the impact of friction with the pipe wall on airflow was more reflected in the tangential velocity.

Figure 10 demonstrates the pipeline’s tangential velocity distribution cloud plots after loading particles. As the tangential velocity inside the straight pipe was weak and negligible, only the tangential velocities of each swirling pipe were compared.

As the particles rotated forward, they collided with the airflow, resulting in a velocity loss. Hence, the tangential velocity distribution in the cross-section was asymmetric, and its maximum value showed a periodic rotation state. As the conveying distance increases, the tangential velocity gradually diffuses from the pipe wall to the center, and the distribution becomes more uniform. The smaller the screw pitch, the greater the initial tangential airflow velocity generated. Nevertheless, the more times the gas flows around and rubs against the pipe wall in turn leads to increasingly severe tangential velocity loss. The superposition of the two effects was shown by the fact that the maximum tangential velocity increased and then decreased with increasing pitch at the outlet.

4.2.3. Swirling Number

To quantify the influence of tangential velocity in the swirling pipes, the dimensionless index of swirling number S was employed to characterize the swirl characteristics, which is defined as the ratio of circumferential airflow flux to axial flux in the pipeline.

As shown in the flowing equation, where $\mathit{u}$ and $\mathit{w}$ are axial and tangential velocities, respectively, R is the pipe radius, and r is the distance of the integral point from the axis of the pipe in the radial direction [33].

$$S=\frac{2\pi {\rho }_f{\displaystyle {\int }_0^R\mathit{u}\mathit{w}{r}^{2}dr}}{2\pi {\rho }_{f}R{\displaystyle {\int }_0^R{\mathit{u}}^{2}rdr}}$$

(9)

Under different initial airflow velocities, each velocity measurement section’s area-weighted average axial velocity and tangential velocity were calculated, respectively, and then incorporated into the equations. The obtained results were fitted, as shown in Figure 11. The swirling number tended to decrease exponentially with increasing transport distance for all conditions, gradually evolving into an axial flow field at the exit.

After v₀ reached 35 m/s, the swirl intensity generated by each three-blade tube reached saturation. This phenomenon revealed that as v₀ increased, the swirling number had almost no effect, and the maximum swirling number remained unchanged. It can be explained by the fact that increasing v₀ only affected the total airflow rate but did not affect the axial airflow and tangential airflow ratio. However, the influence of the screw pitch of the three-blade tube on the swirling number was pronounced because its structural parameters could directly affect the proportion of tangential airflow separated from the total airflow. After the swirling number reached the saturation value, the S with the minimum pitch at the outlet of the conical tube approximated 2.25 times the maximum pitch.

Table 4. Fitting Function Curve of Swirling Number.

p (mm)	Fitting Function	p (mm)	Fitting Function
v0 = 30 m/s		v0 = 35 m/s
200	$0.153e^{(-x/313.64)}+0.010$	200	$0.179e^{(-x/348.29)}+0.001$
300	$0.111e^{(-x/289.25)}+0.012$	300	$0.139e^{(-x/315.18)}+0.004$
400	$0.079e^{(-x/328.98)}+0.011$	400	$0.106e^{(-x/327.83)}+0.003$
500	$0.701e^{(-x/268.49)}+0.007$	500	$0.086e^{(-x/366.38)}+0.002$
600	$0.065e^{(-x/231.32)}+0.007$	600	$0.073e^{(-x/276.889)}+0.003$
v0 = 40 m/s		v0 = 45 m/s
200	$0.169e^{(-x/360.13)}+0.005$	200	$0.169e^{(-x/392.59)}+0.007$
300	$0.141e^{(-x/386.33)}+0.005$	300	$0.140e^{(-x/439.85)}+0.008$
400	$0.106e^{(-x/413.52)}+0.004$	400	$0.098e^{(-x/497.69)}+0.007$
500	$0.089e^{(-x/456.18)}+0.001$	500	$0.079e^{(-x/516.37)}+0.006$
600	$0.075e^{(-x/448.52)}+0.001$	600	$0.071e^{(-x/538.98)}+0.002$

shows the fitting function of the swirl number with the variation of conveying distance.

4.2.4. Static Pressure Drop

The total pressure is composed of static and dynamic pressure and can characterize the system’s total energy. According to the velocity analysis, the difference in airflow velocity at the outlet was minimal, i.e., the dynamic pressure value was similar. Thus, the static pressure at the inlet and outlet of the model can be measured separately and differenced to obtain a static pressure loss that approximately represents the energy loss of the system. As shown in Figure 12, the difference between the static pressure loss before and after loading the particles is the energy loss caused by transporting the particles. The larger the static pressure difference, the more thorough the contact between the airflow and particles. When v₀ = 30 m/s, the static pressure drop of the swirling pipe was smaller than that of the straight pipe, indicating a velocity threshold beyond which the swirling generator can promote particle fluidization. In our previous study, it was reported that this value is the choking speed of conveying particles [3]. After the initial airflow velocity exceeded the choking velocity, the static pressure drop of the swirling pipes with a smaller pitch gradually exceeded that of the straight pipe. All the static pressure drops of the swirling pipes were larger than that of the straight tube until v₀ = 40 m/s. From the above analysis of swirling numbers, it can be concluded that increasing swirl intensity was beneficial for particle fluidization but also resulted in significant energy loss. Therefore, there exists a suitable pitch that can generate sufficient swirling flow to suspend particles while minimizing energy loss in the system.

4.3. Particle Characteristics

4.3.1. Particle Flow State

As shown in Figure 13, to investigate the effect of swirling flow on the dynamic characteristics of pebble particles in the flow field, the flow process of particles was characterized by the position of particles downstream of different conveying pipelines at a simulation time (t) of 0.8 s. The experimental numbers of the axial flow condition and p = 200 mm–p = 600 mm condition were recorded as 1–6. The initial flow of 30 m/s was not analyzed as it did not reach the velocity threshold for vortex operation. Significant differences existed in the motion trajectories of pebble particles after passing through three-blade tubes with different pitches. Regardless of the velocity of the conveying airflow, the smaller the pitch of the three-blade tube, the higher the height at which particles were lifted, the higher the degree of dispersion within the tube, and the longer the distance for spiral motion to be maintained; this was because when the proportion of transverse and tangential airflow remained constant, the greater the total airflow velocity, and the greater the tangential airflow velocity obtained after distribution. The spinning effect weakened when v₀ = 35 m/s due to the small tangential velocity value. Most of the particles were similar to those in the straight tube, except for some particles with p = 200 mm and p = 300 mm, demonstrating a sliding state near the bottom wall. As the airflow velocity increased, particles previously unable to be spun gradually began to suspend. Meanwhile, the distance between particles that have been spun to maintain a suspended state increased until v₀ increased to 45 m/s. The particles in the 600-mm-pitch three-blade tube with the lowest spinning strength could also maintain a suspended state in the first half of the downstream straight pipe. The smaller the pitch, the more times the total airflow was divided into tangential airflow, which was more conducive to guiding higher particle dispersion.

Figure 14 shows the particle distribution at different outlet cross-sections of three-bladed tubes under the initial wind speed of 45 m/s. From the graph, it can be seen that as the pitch changes from large to small, the swirl intensity increases from small to large. And the development trend of particle motion morphology is as follows: firstly, due to insufficient tangential velocity, particles sway against the wall with the airflow and cannot fully spin; when the swirl intensity is moderate, particles are evenly distributed in the pipe; when the swirl increases again, the particles rotate against the wall due to the centrifugal force.

To quantitatively analyze the particle characteristics in the pipeline, the concentration of particulate matter in the first 1 m of the downstream pipe was analyzed (the second half of the pipe had significant swirling attenuation, and there was no significant difference in concentration). As described in Figure 15, the measured pipeline was divided into three areas. Starting from the bottom wall surface of the pipe, the region with a one-particle diameter was designated as the deposition area. Particles in this area were determined to be located at the bottom of the pipeline. The area of one-particle diameter away from the other walls was marked as the adherent area, and the particles in this area were determined to be closely attached to the wall surface. Meanwhile, the remaining region was designated as the suspension region. Particles in this area could avoid collision with the wall, had complete contact with the central flow field, and reached the optimal transport state.

As shown in Figure 16, when t = 0.8 s, the number of particles in each region was counted and compared with the total number of particles in the measured pipeline to obtain the deposition rate, suspension rate, and adhesion rate. The experimental numbers of the axial flow condition and p = 200 mm–p = 600 mm condition were recorded as 1–6. Notably, when v₀ was less than the choking velocity, the swirling field would increase the sedimentation rate of particles, reduce the suspension rate of particles, and play a restraining role in particle fluidization, since the tangential velocity was too small to sufficiently disturb particles, which only led to oscillations of the particles on the bottom wall, and exacerbated energy loss and wall wear of the pipeline. After v₀ exceeded the choking velocity, increasing the airflow velocity had little effect on the proportion of particles in the suspended area due to the tangential airflow mainly acting near the wall surface. However, it would transfer a large amount of particles deposited at the bottom to the adherent area.

According to the static pressure analysis, all swirling pipes could promote particle fluidization when v₀ reaches 40 m/s. As the pitch increased, the deposition rate of particles first decreased and then increased, while the suspension rate first increased and then decreased, and the adhesion rate gradually increased. It was derived from the fact that when the swirl intensity was too small at the same airflow velocity, it was hard to support particle suspension. When the swirl intensity was too large, it caused some particles to move in a spiral motion against the wall due to centrifugal force. Hence, when the suspension rate reached the maximum, the best corresponding conveying state arose, and the swirling flow was sufficient to allow the particles to spin fully without being against the wall by centrifugal force.

4.3.2. Energy Efficiency

The total number of particles transported out of the pipeline at different operating states is shown in Figure 17. In this figure, orange represents a larger number of transported particles than the axial flow field, while blue represents a smaller number than the axial flow field. When v₀ was less than the pick-up velocity, the number of particles transported gradually decreased as the swirling strength increased. The reason is that the tangential airflow velocity was insufficient to make particles affected by centrifugal force, and gravity still played the dominant role. Therefore, the stronger the swirling flow, the more particles were disturbed into the suspension area, which was more conducive to particle acceleration. At a large v₀ (40 m/s and 45 m/s), which was larger than the pick-up velocity, the number of particles transported first increased and then decreased with the pitch increasing. The increase is because when p was too small, the greater the ratio of tangential velocity to axial velocity, and the more the tangential airflow becomes saturated. Therefore, the time for particles to adhere to the wall and rotate increased, which is also manifested as the number of conveyors with a pitch of 200 mm in Figure 14 being smaller than that of straight pipes.

The energy efficiency of the pneumatic conveying system is reflected in the efficiency of converting airflow energy into particle energy. When the boundary conditions are consistent, and the changes in particle potential energy can be ignored, the greater the kinetic energy of the transported particles, and the higher the system’s energy efficiency. In this paper, we measured the velocity of the particles at the exit, combined with the statistics of the number of particles that were transported out of the pipe; the kinetic energy of the particles was calculated by using Equation (10), as shown in Figure 18.

$$E_k={\displaystyle \sum _{t=0}^{0.8}\frac{2\pi r^3{v}_p^2}{3}}$$

(10)

where $E_k$ is the kinetic energy of the particles, and $v_p$ is the average velocity of the particles at the outlet.

The trend of the kinetic energy of the particles corresponds precisely to that of Figure 18, indicating that the particle velocities at the exit were similar to each other at the same airflow velocity since the length of the axial acceleration path was the same. When the airflow speed was small, the smaller the spacing was, the more influential the prevention of particle deposition was, and the higher the energy utilization was. When the airflow speed was high, excessive swirling intensity increased the ineffective stroke of particle rotation against the wall, resulting in unnecessary energy loss. As the airflow velocity increased, the extreme point of kinetic energy gradually moved towards a large pitch since the essential factor inducing the optimal state of the conveying process was the tangential airflow velocity. The tangential velocities corresponding to the maximum points of kinetic energy at v₀ = 40 m/s, p = 300 mm and v₀ = 45 m/s, p = 400 mm were calculated, and the results were 5.870 m/s and 5.869 m/s, respectively, which could verify this conclusion. After retaining three significant figures, the unified result obtained is 5.87 m/s. Therefore, to better utilize the energy of the airflow, both the airflow velocity and the swirling generator’s structure should be considered to produce the most suitable swirl intensity under the combined action.

5. Conclusions

In this article, we establish a gas–solid coupling model for spraying materials (pebble particles) in pipelines, propose an enhanced swirling pneumatic conveying method, and reveal the effects of axial and swirling flow on the flow pattern and pressure drop of gas–solid two-phase flow. The optimal tangential velocity and blade pipe parameters for horizontal pipeline pneumatic conveying have been obtained, providing technical support for achieving green coal energy mining. The main conclusions are as follows:

(1)

When unloaded, the axial and tangential velocities of the airflow exhibited a symmetrical distribution. As the pitch of the three-blade tube increased, the former was almost unaffected, while the latter showed a significant decreasing trend. After loading, the rotation of particles hindered the movement of airflow. The two velocity distributions were no longer symmetrical but showed a trend of maximum rotation along the circumference. The smaller the pitch of the three-blade tube, the more pronounced the rotation trend.

(2)

After v₀ reached 35 m/s, the swirl intensity reached saturation. After that, the increase in airflow velocity did not affect the swirl intensity. The pitch of a three-blade tube could directly change the ratio of axial airflow to tangential airflow. The smaller the pitch, the greater the swirling intensity of the flow field. The saturation value of swirling intensity at the 200 mm pitch was about 2.25 times that at the 600 mm pitch.

(3)

The static pressure loss reflects the energy consumption of the system. There was a choking velocity of 35 m/s. When v₀ was larger than it, the static pressure drop in the swirling pipe was gradually larger than that in the straight pipe. The static pressure loss increased with the decrease in the pitch of the three-blade tube. The swirling flow would promote particle fluidization only when v₀ was larger than the choking velocity.

(4)

Increasing the airflow velocity could significantly improve the dispersion of particles. After v₀ exceeds the choking velocity, as the pitch of the three-blade tube increases, the deposition of particles first decreased and then increased. The suspension first increased and then decreased. The adhesion rate gradually decreased due to the effect of centrifugal force. There was an optimal parameter combination: v₀ = 40 m/s with p = 300 mm and v₀ = 45 m/s with p = 400 mm, resulting in the highest proportion of particles at the center of the flow field.

(5)

When v₀ was the same, the particle velocity at the outlet was also similar. Before reaching the choking speed, the smaller the pitch, the higher the kinetic energy of the particles. After reaching the choking speed, an excessive swirl would increase the ineffective stroke of the particles, and the kinetic energy of the particles first increased and then decreased with the increase in the pitch. The combined effect of airflow velocity and swirl intensity should be considered to achieve the optimal tangential velocity of 5.87 m/s.