Test materials and equipment.

The bench test of this study aimed to verify the coupling model, through which different suspension speeds of fresh tea leaves with different weights are calibrated. Therefore, to facilitate the test observations, large leaves of tea were selected as the research object (Fig. 1A). The average length of the leaves was 37 mm, and the average width was 13 mm. The weight of the test tea was measured by a scientific balance. A single tea leaf was weighed (Fig. 1B), and the weight range of the tea measured was between 0.102 and 0.215 g. A split-type anemometer (Fig. 1C) was used to measure the tea suspension velocity. The anemometer can record the maximum wind speed, minimum wind speed, and average wind speed over a period of time.

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Coupled simulation mathematical model.

The coupling simulation based on CFD-EDEM in this study mainly focused on gas–solid coupling. The gas phase is a continuous phase, and the law of motion follows the continuity equation and momentum equation in fluid mechanics. In the discrete element, the motion parameters of the particles can be solved according to Newton’s second law of motion. In the negative pressure gas phase, the fresh tea leaves were subjected to the drag force of gravity and particle–air interaction. In this study, the secondary influence was ignored, and the drag force between the particle and the airflow field and the particle’s own gravity were mainly considered.

The gas phase continuity governing equation is expressed as follows (Ebrahimi et al. 2014; Kruger et al. 2011; Vashisth and Grace 2012; Weng et al. 2022):$\frac{\partial }{\partial t}\left({\alpha }_g{\rho }_g\right) + \nabla \cdot \left({\alpha }_g{\rho }_g{v}_g\right) = 0$ [1]

The momentum equation is expressed as:$\frac{\partial }{\partial t}\left({\alpha }_g{\rho }_g{v}_g\right) + \nabla \cdot \left({\alpha }_g{\rho }_g{v}_g{\tau }_g\right) = \underset{¯}{{\alpha }_g}\nabla \cdot P_g + \nabla \cdot \left({\alpha }_g{\rho }_g\right) + {\alpha }_g{\rho }_{g}g-S$ [2]

where α_g is the voidage ratio, ρ_g is fluid density, v_g is the velocity vector, P_g is the gas phase pressure, τ_g is the gas viscosity, g is acceleration of gravity, and S represents the momentum exchange between the gas and solid phases. During gas–solid coupling simulation (CFD-DEM), the momentum exchange of two phases is realized by drag force, and its expression is as follows:$S = \frac{1}{\mathrm{\Delta }\text{V}}{\displaystyle \sum }_{i=1}^n{f}_{\mathit{drag},i}$ [3]

where ΔV is the volume of the control body and f_drag,i is the drag force of fluid. Among them, the drag force is calculated by the following empirical formula:

$f_{\mathit{drag},i}\hspace{0.17em}=\hspace{0.17em}3.35\times {10}^{-5}\times \frac{{\beta }_i}{1-{\alpha }_i}\left(u_i-v_i\right)$ [4]

where α_i represents the local voidage of discrete element particles, β_i is the local momentum exchange coefficient of discrete element particles, u_i is the virtual gas velocity at the center of the particle i, and v_i is the velocity of the particle.

According to the characteristics of the turbulence model and its applicable scope, the Realizable k-ε turbulence model is adopted in the calculation of the model. The transport equations of k and ε in the Realizable k-ε model are, respectively:$\frac{\partial \left(\rho {\mu }_j\epsilon \right)}{\partial x_j}=\left(\mu +\frac{\mu t}{{\sigma }_{\epsilon }}\right){\nabla }^2\epsilon +C_{1}S\rho \epsilon -C_2\frac{\rho {\epsilon }^2}{k+\sqrt{v\epsilon }}$ [5]

where, μ is hydrodynamic viscosity, $C_1=\mathit{max}\left[0.43,\hspace{0.17em}\frac{\gamma }{\gamma \hspace{0.17em}+\hspace{0.17em}5}\right]$, C₂ = 1, σ_ε = 1.2, and ∇ is a Laplace operator. The governing equation of fresh tea leaf particles at any time is as follows:$m_i\frac{d{v}_i}{dt} = m_{i}g\hspace{0.17em}+\hspace{0.17em}{f}_{\mathit{drag},i}$ [6] $I_i\frac{d{w}_i}{dt} = {\displaystyle \sum }_{j=1}^{k_i}{T}_{ij}$ [7]

where, m_i, v_i, w_i, and I_i are the mass, velocity, angular velocity, and moment of inertia of the particle, k_i is the number of colliding particles, k_i is particle torque, m_ig is particle gravity, and f_drag,i is the drag force of fluid.

Test bench construction.

The tea suspension speed test bench (Fig. 2A) comprises the tea suspension observation tube (1), a tea outlet with a grid cross-section (2), steady flow tank (3), air duct (4), and frequency conversion fan (5). The tea leaves are placed in the tea outlet with a grid cross-section (2), and the wind speed is measured at that point by an anemometer when it is stable. The length of the mesh surface of the tea suspension observation tube 1 from the top to the tea outlet 2 is 1665 mm (Fig. 2B).

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Tea suspension test.

Before the test, the suspension of tea leaves in the air flow field was investigated using a pretest to adjust the wind speed suitable for tea suspension. If the wind speed is too high, then the tea will be ejected from the upper mouth of the observation tube; however, if the wind speed is too low, then the tea could not be suspended or would be suspended intermittently. During the pretest, because of the different weight of each tea, it was found that its suspension had large fluctuations and could not be sufficiently unified. Therefore, after the pretest, to overcome these problems, the suspension speed of a single tea was measured during the test. Because the wind speed has certain fluctuations, the point where tea leaves are suspended for a long time was selected as the simulation verification location (Fig. 3). Two tea leaves were selected for the suspension speed test and weighed. Figure 3A shows that the weight of tea was 0.215 g, and that its suspension height was 270 mm. Figure 3B shows that the weight of tea was 0.143 g, and that its suspension height was 325 mm.

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Discrete element model setup.

To simplify the simulation calculation, only CFD-DEM coupling simulation was performed for the observation tube. The model was divided into large and small end faces, and the small end face was a cuboid with a side length of 140 mm at the entrance of the measurement tube. The large end face of the model was cuboid, with a side length of 300 mm, and the length of the observation tube was 1400 mm. The particle parameters were set in EDEM software, and Poisson’s ratio and density of tea particles were selected as 0.4 and 700 kg/m³. According to the shear modulus of tea leaves measured (Zhang and Zhu 2023; Zhang et al. 2024) during the previous stage, the average shear modulus was 3.3 MPa, and the quality of a single tea was measured by a scientific research balance (Fig. 3). A particle factory was created 20 mm above the ground to generate tea particles. To simplify simulation time and improve efficiency, spherical particles were used instead. The Hertz-Mindlin (No Slip) model was selected for the discrete element model, which is accurate and efficient for force calculations. In this model, the normal force component is based on the Hertzian contact theory, whereas the tangential force model is based on Middlin-Deresiewicz’s research theory. Both the normal force and tangential force have damping components. Tangential friction obeys Coulomb’s law of friction, and rolling friction independently directs the constant torque model through contact.

Finite element meshing.

In the finite element model, CFD-IDEM was used to divide the grid. In this study, the observation tube was divided into tetrahedral structured grids, with a total of 253,488 grids (Fig. 4A). The grid quality was good, and the minimum cell mass was 0.981.

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Finite element simulation of boundary conditions.

During the numerical simulation, the influence caused by heat exchange was ignored, that is, the energy equation was not considered. The flow field belonged to turbulent flow, the working medium was air, there was no phase change, and there was no chemical reaction in the calculation process. A separate implicit solver based on pressure was used to solve the numerical simulation. To improve the calculation accuracy and reduce the numerical diffusion, the momentum, turbulent kinetic energy, and turbulent dissipation rate were adopted in the second order upwind discrete scheme. The pressure–velocity coupled SIMPLEC algorithm was selected. The working environment was set to the standard atmosphere. The model velocity inlet wind speed was measured by the anemometer, and its average value was taken. Air flowed out of the fluid domain at a constant speed, with the small end face as the inlet and the large end face as the outlet.

Coupling model verification.

According to the test, the wind speed measured by the anemometer at the entrance of the observation tube (average value, 6 m/s) and the tea quality measured by the balance (Fig. 3) were simulated (Fig. 5). The position where the tea particles remained in stable condition when the mass of fresh tea leaves was set at 0.215 g was observed (Fig. 5A), and the distance between the particles and the small end entrance was 279 to 289 mm. When the particles were relatively stable, the suspension height of tea leaves was 270 mm, and the simulated value was greater than the measured value, with a minimum error of 9 mm and error rate of 3.33%. The maximum error was 19 mm and the error rate was 7.03%. The position where tea particles remained in stable condition when the mass of fresh tea leaves was set at 0.143 g was observed (Fig. 5B), and the distance between the particles and the entrance of the small end face was 316 to 330 mm. When the particles were relatively stable, the suspended height of tea leaves was 325 mm, and the measured value was between the simulated value, with a minimum error of 5 mm and error rate of 1.54%. The maximum error was 9 mm and the error rate was 3.33%. In conclusion, the established CFD-DEM coupled tea particle suspension model had high accuracy. Therefore, this model can be used to simulate and analyze the suspended height of tea under different initial wind speeds.

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Analysis of particle maximum motion position.

Using the coupling model established during this study, the motion position of 400 tea particles in the observation tube at each simulated time was analyzed under a given wind speed of 9 m/s. As shown in Fig. 6, when the simulation time was 0 to 1.8 s, and the interval time was 0.09 s, at 0.09 s, the tea particles began to move upward, layer by layer, under the influence of wind power, and the maximum displacement of the tea particles in the top layer was 86 mm from the wind speed inlet. At 0.18 s, the maximum displacement of the top layer of tea particles was 230 mm, which was an increase of 144 mm compared with the previous time period. At 0.27 s, the maximum displacement of the top layer of tea particles was 358 mm, which was an increase of 144 mm compared with the previous time period. At 0.36 s, the maximum displacement of tea particles in the top layer was 468 mm, which was 110 mm higher than that of the previous period. At 0.45 s, the maximum displacement of the top layer of tea particles is 544 mm, which was an increase of 76 mm compared with the previous time period. At 0.54 s, the maximum displacement of the top layer of tea particles was 544 mm, which was an increase of 76 mm compared with the previous time period. At 0.63 s, the maximum displacement of the top layer of tea particles was 601 mm, which was an increase of 57 mm compared with the previous time period, and the tea particles were in the farthest ascending position at the selected time. At 0.72 s, the maximum displacement of tea particles in the top layer was 568 mm, which was a decrease of 33 mm compared with that during the previous period, indicating that from this moment on, tea particles began to fall under the action of air flow. At 0.81 s, the maximum displacement of the top layer of tea particles was 515 mm, which was a decrease of 53 mm compared with that during the previous time period. At 0.9 s, the maximum displacement of the top layer of tea particles was 449 mm, which was a decrease of 66 mm compared with that during the previous period. At 0.99 s, the maximum displacement of the top layer of tea particles was 392 mm, which was a decrease of 57 mm compared with that during the previous period. At 1.08 s, the maximum displacement of the top layer of tea particles was 315 mm, which was a decrease of 77 mm compared with that during the previous period. At 1.17 s, the maximum displacement of the top layer of tea particles was 264 mm, which was a decrease of 51 mm compared with that during the previous time period. At 1.26 s, the maximum displacement of tea particles in the top layer was 343 mm, which was an increase of 79 mm compared with that during the previous period, indicating that tea particles begin to rise under the action of air flow from this moment on. At 1.35 s, the maximum displacement of the top layer of tea particles was 404 mm, which was an increase of 61 mm compared with that during the previous period. At 1.44 s, the maximum displacement of the top layer of tea particles was 438 mm, which was an increase of 34 mm compared with that during the previous period. At 1.53 s, the maximum displacement of the top layer of tea particles was 446 mm, which was an increase of 8 mm compared with that during the previous period. At 1.62 s, the maximum displacement of the top layer of tea particles was 451 mm, which was an increase of 5 mm compared with that during the previous period. At 1.71 s, the maximum displacement of tea particles in the top layer was 447 mm, which was a decrease of 4 mm compared with that during the previous period. At 1.53 to 1.71 s, tea particles were in a relatively balanced state under the air flow field in the observation tube, and the fluctuation was small. At 1.8 s, the maximum displacement of the top layer of tea particles was 425 mm, which was a decrease of 22 mm compared with that during the previous period.

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In summary, the suspended state of tea particles at each simulated time showed that tea particles are affected by the air flow field of the observed pipe, and tea particles fluctuate, similar to the phenomenon in the experiment because of the constant change in the pipe diameter, resulting in the air velocity decreasing with the increase in the position and tea particles falling under the action of gravity. Some rising tea particles will cause some other tea particles to fall when they interfere with the airflow field. During the simulation at various times, it was found that tea particles attached to the inside of the wall of the observation tube under the action of air flow, especially within 0.54 to 0.81 s, mainly because when the air flow passed through the observation tube, the air flow rate gradually decreased from the center of the pipe to the wall. The surface of discrete element particles produced pressure differences, and the air flow pushed the particles to rise (Fig. 7). When the air flow velocity near the pipe wall is less than the suspension velocity of tea particles, the particles tend to fall under the action of gravity and slide down the inclined wall of the observation tube. This phenomenon is similar to that observed by Wen et al. (2020).

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Relationship between tea particles at different positions and simulation time during suspension.

To simulate the relationship between different suspension positions of a certain number of tea particles under the observation tube and the simulation time, the different heights of tea particles from the initial position were taken as the Y-axis and the simulation time was taken as the X-axis. The particles at different positions were now defined. According to the cloud images under different simulation times (Fig. 6), there are six distance values under each set of cloud images. On the cloud image, the maximum distance value to the minimum distance value were defined as follows: tea particle location A, tea particle location B, tea particle location C, tea particle location D, tea particle location E, and tea particle location F. Figure 8 shows the variation trend of different distances from the initial position and the simulated time during the suspension of particles. According to the variation trend of positions at different times, the tea particles moved first under the influence of air flow, and the movement distance of the upper particles was greater than that of the lower particles during the whole period under a certain wind speed. In the simulation time, the particles are affected by the air flow field, and the movement of particles is in the form of wave peak. Among them, tea particle location A and tea particle location B reach the peak at 0.6 s. Tea particle location C, tea particle location D, tea particle location E, and tea particle location F reach the peak at 0.5 s. When approaching 1.2 s, the tea particles in each group appeared as waves, indicating that the suspension of tea particles in the observation tube was affected by the air flow length, and the phenomenon of instability appeared. During the simulation time of 1.2 to 1.8 s, the movement of particles in each layer was relatively stable, especially during the period of 1.4 to 1.7 s, and the change range was smaller than that during 0 to 1.2 s, mainly because there were more tea particles in the observation tube, which had a certain obstruction effect on the upward air flow field.

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The suspended particle function model was established.

According to the variation trend of different distances from the initial position of particles in the suspension process and the simulation time (Fig. 8), it was found that tea particle location A, tea particle location B, tea particle location C, tea particle location D, tea particle location E, and tea particle location F had certain motion rules at the first wave peak, that is, the simulation time of 0 to 1.2 s. Its motion law can be described by nonlinear curve fitting (Fig. 9). The curve fit degree of tea particle location A was 0.9981. The curve fit degree of tea particle location B was 0.9969. The curve fit degree of tea particle location C was 0.9932. The curve fit degree of tea particle location D was 0.9860. The curve fit degree of tea particle location E was 0.9791. The curve fit degree of tea particle location F was 0.9455. The fitting equation is as follows:$\text{double z} = \frac{\left(x-x_c\right)}{w}$ [8] $\text{y} = y_0\hspace{0.17em}+\hspace{0.17em}A\times e^{e^{\left(-z\right)}-z+1}$ [9]

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where, y₀, x_c, w, and A are fitting parameters. The parameter values are shown in Fig. 9.

In conclusion, the variation trend of different distances from the initial position and simulation time in the suspension process of particles meets the extreme function of the peak functions. The function reflects the motion form of suspended particles in the air flow.