Effect of Inter-Orifice Spacing on Granular Flow Discharge Rate: The Role of an Inter-Orifice Quasi-Solid Region

Abstract

The discharge behavior of granular materials from double-orifice silos is strongly affected by the inter-orifice spacing, yet the mechanical role of the inter-orifice region remains unclear. In this study, discrete element method (DEM) simulations are combined with experiments to investigate the formation, stability, and collapse of an inter-orifice quasi-solid region and its impact on the discharge rate. The results show that increasing the inter-orifice spacing progressively weakens shear transmission between adjacent outlets, promoting the development of a low-velocity, load-bearing quasi-solid region. Based on μ(I) rheology and a nonlocal granular fluidity framework, the quasi-solid region is shown to be controlled by local shear activation rather than by geometric separation alone. Once the inter-orifice quasi-solid region is formed, this region restricts the spatial extension of shear bands near the outlets, leading to a reduction in the effective shear area and a corresponding decrease in the discharge rate. A critical inter-orifice spacing is identified, beyond which the two outlets discharge independently. These findings provide a mechanistic understanding of flow-rate regulation in multi-orifice silos, offering guidance for the design of granular discharge systems.

1. Introduction

Gravity-driven discharge of granular materials occurs widely in silos and hoppers across industrial systems such as agriculture, energy, chemical, pharmaceutical, and food engineering [1,2,3,4]. In practical engineering applications, single-orifice silos are common [5], whereas multi-orifice configurations are employed under specific operating conditions. Compared with single-orifice silos, multi-orifice silos can combine the advantages of mass-flow and core-flow silos, enabling more uniform discharge, reducing dead zones, and maintaining a relatively compact structure [6]. Although empirical design often recommends placing the outlets sufficiently far apart to minimize flow interference, inter-orifice interactions can still arise under practical constraints and influence the discharge behavior. For example, UAV (Unmanned Aerial Vehicle) fertilizer spreading systems have strict size limits for the aircraft platform and hopper. This constraint requires multiple outlets to be arranged compactly. As a result, flow interference between adjacent outlets is unavoidable. In multi-orifice systems, granular flow is no longer governed solely by local conditions at a single outlet; interactions between adjacent orifices lead to more complex discharge dynamics. As the simplest multi-orifice configuration, the dual-orifice silo preserves the essential inter-orifice interactions while avoiding additional structural complexity. Therefore, systematic investigation of granular discharge in dual-orifice silos is essential for understanding the flow mechanisms in multi-orifice silo systems.

In studies of dual-orifice silos, the inter-orifice spacing is widely recognized as a key geometric parameter governing particle interactions and granular discharge rate. Existing studies have primarily explored its macroscopic influence on global flow behavior through experiments and numerical simulations [7,8]. For instance, Sharma et al. [7] reported that when the spacing between two outlets is smaller than three particle diameters, arch structures at the orifices become coupled, suppressing stable arch formation and significantly reducing clogging probability. Xu et al. [8] experimentally investigated the response of the granular discharge rate to variations in inter-orifice spacing and demonstrated that particle velocities above the outlets exhibit a strong dependence on the inter-orifice spacing. These studies reveal the macroscopic modulation of both the discharge rate and velocity distribution by inter-orifice spacing, providing an important basis for understanding granular flow in multi-orifice silos. However, most existing works emphasize global flow rate or averaged velocity responses, offering limited insight into the internal flow structures during inter-orifice spacing variation, particularly regarding the formation and evolution of local low-velocity or weak-shear regions. An analysis from the perspective of flow-structure evolution is therefore necessary to elucidate the physical mechanisms underlying spacing-dependent particle interactions and discharge behavior.

While macroscopic analyses provide valuable global descriptions, they are insufficient to reveal how particle contact states and local structural evolution influence granular mass flow, which motivates mesoscopic and microscopic investigations. In silo studies, the $\mu \left(I\right)$ frictional rheology model, nonlocal rheology theories, and combined experimental–numerical approaches are commonly employed to elucidate, from a microscopic perspective, the intrinsic relationships among particle shear rate, stress state, and flow behavior [1,9,10,11,12,13,14,15,16,17,18,19]. For example, Samuel K. Irvine et al. [17] applied the $\mu \left(I\right)$ model to analyze discharge variations in dual-orifice silos and found that flow reduction is primarily driven by frictional effects, with wall interactions potentially causing deviations from the Beverloo–Hagen relation. L. Staron et al. [18] used discrete element simulations to study silo discharge, highlighting the role of the inertial number in verifying the consistency between continuum and discrete descriptions, but without clarifying the spatial applicability of the $\mu \left(I\right)$ model. Kamrin et al. [19] incorporated nonlocal effects into the $\mu \left(I\right)$ framework, using particle fluidity $g$ to characterize flow states and determining the spatial scale of nonlocal perturbations, thereby overcoming the traditional $\mu \left(I\right)$ model’s limitation to purely local shear flows. Although the $\mu \left(I\right)$ model has been widely used in silo discharge studies, most existing works focus on global discharge rates or local shear responses near the outlets, while the stress distribution, shear state, and their evolution in the inter-orifice region of dual-orifice silos remain insufficiently analyzed.

Recently, we investigated the mechanism by which the discharge rate of ellipsoidal particles in dual-orifice silos varies with inter-orifice spacing [4,20]. We found that the discharge rate does not decrease monotonically with increasing inter-orifice spacing, but instead first decreases, then increases, and eventually stabilizes. As the inter-orifice spacing grows, particle accumulation emerges in the inter-orifice region, and the expansion of this accumulation zone leads to the splitting of particle flow. A similar phenomenon was reported by Ritwik et al. [21], whose experiments showed that when the two outlets are close, particle flow in the inter-orifice region is altered, resulting in higher discharge rates and reduced clogging. These observations indicate that variations in inter-orifice spacing significantly affect particle flow behavior in double-orifice silos, with the inter-orifice accumulation zone closely related to the discharge rate. However, in our previous studies, the inter-orifice accumulation zone was mainly identified and discussed from a macroscopic phenomenological perspective, focusing on its spatial development and its correlation with macroscopic discharge behavior. The underlying mesoscopic mechanical nature of this region and its role in mediating stress transmission and flow regulation were not yet systematically addressed.

To bridge this gap, the main novelty of the present work lies in providing a systematic mesoscopic mechanical framework for this inter-orifice region. Specifically, this study advances our previous macroscopic observations by proposing a quantitative mechanical criterion for the formation and collapse of the quasi-solid region based on $\mu \left(I\right)$ rheology and local stress states. Furthermore, it elucidates the mesoscopic mechanical mechanism by which this stable quasi-solid region restricts the spatial extension of active shear bands, and quantitatively links this constrained effective shear area to the macroscopic reduction in the discharge rate. By explicitly addressing these aspects, this study provides a unified mechanistic understanding of how inter-orifice spacing regulates granular flow, offering theoretical guidance for the design of multi-orifice granular discharge systems.

The organization of this paper is as follows. First, the formation and stability of the inter-orifice quasi-solid region are systematically analyzed to reveal its regulatory effect on particle discharge. Second, velocity fields, shear rate distributions, and effective shear areas are examined to clarify the microscopic mechanisms by which the quasi-solid region limits the spatial development of shear bands, affects local particle motion near the outlets, and thereby controls overall discharge efficiency. Third, by varying the frictional conditions of the inter-orifice baffles, the universality of this regulatory mechanism under non-geometric parameters is assessed, confirming the critical role of stability–shear coupling in discharge flows.

2. Materials and Methods

2.1. Experimental

2.1.1. Discharge Experiment

To clearly observe the particle flow characteristics, both experiments and numerical simulations were conducted using a rectangular flat-bottom silo with an internal width of 282 mm and a height of 700 mm. To prevent particle interlocking and arching in the depth direction, thereby ensuring a stable quasi-two-dimensional flow regime, the silo thickness was set to 20 mm (greater than five times the particle diameter). While this quasi-two-dimensional framework may differ quantitatively from full three-dimensional silos, it provides a crucial optically and numerically accessible cross-section, allowing for direct observation of internal mesoscale structures (such as the quasi-solid region) that are often obscured in 3D geometries. Two rectangular discharge orifices symmetric about the z-axis were placed on the baffle, with their positions determined by the inter-orifice spacing. When the inter-orifice spacing is 0 mm, the two identical square orifices become directly connected and form a single combined opening, whose area is equal to the sum of the two individual orifices, resulting in an equivalent outlet size of 20 × 40 mm. It is important to note that this D = 0 mm configuration lacks the physical inter-orifice boundary. Therefore, it is treated separately in this study as a baseline single-outlet condition for maximum flow capacity, rather than a typical double-orifice case. The size of each individual outlet was kept constant at 20 × 20 mm, which is larger than five times the particle diameter, to ensure a stable discharge condition, avoid clogging effects [22], and eliminate the unnecessary influence of outlet size variation on the present study. The inter-orifice spacing D is defined as the horizontal distance between the inner edges of the two orifices.

Since previous studies have not clarified whether particle shape affects the influence of inter-orifice spacing on discharge rate [20], spherical Polyoxymethylene (POM) particles with a diameter of 3.5 mm were used as their uniform shape eliminates shape-induced variations in particle packing, inter-particle contact, and flow behavior that could otherwise confound the analysis. The selection of spherical particles was also motivated by their similarity to granular fertilizers commonly used in agricultural spreading systems, particularly in UAV fertilization applications involving multi-outlet discharge structures.

Since the objective of this study was to investigate the intrinsic mechanism of inter-orifice discharge behavior, particle size variability, abrasion, and fragmentation effects associated with real fertilizer granules were intentionally minimized. Therefore, highly wear-resistant and size-uniform POM particles were adopted as an idealized material to ensure stable and repeatable discharge conditions throughout the experiments.

The experimental apparatus and platform are shown in Figure 1a. The silo storage region consists of two transparent tempered glass panels [23], enabling high-speed imaging at 1000 frames per second using a high-speed camera (FASTCAM Mini UX50 160K-C-8GB, Photron Co., Ltd., Tokyo, Japan). The video acquisition and frame extraction were managed using the camera’s native software, Photron FASTCAM Viewer 4 (PFV4, x64). This imaging system was deliberately selected to visualize the discharge patterns and identify the spatial distribution of the flowing and quasi-static regions without rolling-shutter geometric distortions. Under these specific imaging conditions, the moving particles exhibit visible motion blur while particles in the quasi-static region remain sharply resolved. This natural visual contrast served as a reliable qualitative indicator to delineate the morphological boundary of the quasi-static region, thereby validating the overall flow pattern. All detailed quantitative kinematic information presented in this study, including internal particle velocity fields and shear-rate distributions, was extracted directly from the validated DEM simulations. The mass of particles discharged from the silo was measured using a force sensor (CTBHM-I-P, Shanghai Chengke Electronic Technology Co., Ltd., Shanghai, China). All data were transmitted to an information processing system (LEGION Y7000P, Lenovo, Beijing, China) for analysis.

To investigate the effect of the inter-orifice baffle friction coefficient on the discharge rate, three experimental conditions were established, as shown in Figure 1b and Figure 2c, consisting of untreated epoxy baffles, baffles covered with 800-grit sandpaper, and baffles covered with 320-grit sandpaper. The method for measuring the particle–baffle friction coefficient under different surface conditions is described in Section 2.1.2.

Before each experiment, all discharge orifices were closed, and POM spherical particles with a mass of 3.16 kg were weighed and loaded into the silo. As shown in Figure 1d,e, once the system reached a stable state, both orifices were opened simultaneously, and particle discharge continued until no further particle motion was observed at the orifices (approximately 35 s). Discharge experiments were conducted under different baffle conditions and orifice configurations to measure the steady-state discharge rate. Each experiment was independently repeated three times.

2.1.2. Measurement of the Friction Coefficient

To quantify the surface friction conditions introduced in Section 2.1.1, the equivalent friction coefficient of the particle–baffle system was measured using an inclined-plane sliding experiment [24]. The discharge baffle, made of epoxy resin, was prepared under three surface conditions, consisting of untreated, covered with 800-grit sandpaper, and covered with 320-grit sandpaper. In the experiments, a uniform layer of POM particles [25] was fixed onto a rigid substrate and placed on the baffle surface. The inclination angle was gradually increased until sliding occurred, and the critical angle was recorded. Each test was repeated three times for each surface condition, and the average value was used. Based on these measurements, the equivalent friction coefficients were determined to be μ₀ = 0.32 ± 0.009 for the untreated baffle, μ₃₂₀ = 0.7 ± 0.012 for the baffle covered with 320-grit sandpaper, and μ₈₀₀ = 0.9 ± 0.015 for the baffle covered with 800-grit sandpaper. The reported error margins represent the standard deviations from the three independent trials, and these small variances indicate a low measurement error during the parameter calibration process.

2.1.3. Measurement of the Internal Friction Angle

To quantitatively determine the collapse conditions of the inter-orifice quasi-solid region in subsequent analyses, the shear resistance of the granular material was measured. Compared to the angle of repose method, the direct shear test provides a more accurate fitting for the inter-particle friction coefficient [26]; therefore, it was selected to measure the friction angle of the particles. The internal friction angle of POM particles was determined using a strain-controlled direct shear apparatus (ZJ-2, Nanjing Soil Instrument Factory, Nanjing, China) and used as a criterion for quasi-solid region instability. The experimental procedure was carried out in accordance with the methodology established in our previous studies [27]. Prior to the test, particles were uniformly filled into the shear box under a prescribed normal stress. Relative displacement was applied by driving the shear box, and the compression of the proving ring was recorded during shearing. The shear stress was calculated by multiplying the proving ring compression by its calibration coefficient (5.53 N per 0.01 mm). The internal friction angle was determined from the ratio of the shear stress to the corresponding normal stress, given by [28]

$$\mathrm{t}\mathrm{a}\mathrm{n}\phi =\tau /{\sigma }_n$$

(1)

where $\phi$ is the internal friction angle (°), $\tau$ is the shear stress (Pa), and ${\sigma }_n$ is the normal stress (Pa). Four normal stress levels, 50 kPa, 100 kPa, 150 kPa, and 200 kPa, were applied in the experiments, and each condition was independently repeated three times. The results indicate that the internal friction angle of POM particles corresponds to $\tan \phi$ = 0.553 ± 0.016 This small standard deviation confirms the high precision of the experimental measurements and the reliability of this primary parameter for subsequent DEM simulations.

2.2. Simulation System

2.2.1. DEM Model of Particles and Geometry

In the numerical simulations, a three-dimensional discrete element method (DEM) model was established. It should be emphasized that the geometric dimensions of the simulation model were constructed strictly according to the physical experimental apparatus described in Section 2.1. The specific geometric dimensions of the silo model, the bottom baffles, and the POM particles are detailed in Figure 2a, Figure 2b and Figure 2d, respectively. This material is commonly used in both numerical simulations and experimental studies of granular flows [29,30]. At the beginning of the simulation, 100,000 particles weighing approximately 3.16 kg were randomly generated in a static state and allowed to settle under gravity to form a stable configuration, as shown in Figure 2c. Subsequently, both discharge orifices were opened simultaneously to allow particle flow until no particles flowed at the orifices. To analyze the particle flow characteristics during discharge, a cuboid measurement region of 280 × 20 × 60 mm was placed above the orifices and divided into grid cells of 4 × 20 × 4 mm (Figure 2e). A grid-based coarse-graining procedure is applied, where particle quantities within each cell are spatially averaged to obtain local continuum fields. Specifically, particle velocities are averaged to construct a continuous velocity field, from which the shear rate is calculated based on spatial velocity gradients [31]. The stress field is obtained from particle contact forces using a volume-averaged approach. All physical quantities reported in this study are then derived from these coarse-grained fields.

The contact parameters for the silo and spherical POM particles are summarized in

Table 1. Parameters used in the simulation.

Name	Parameters	Value
Silo	Density (kg/m3)	2490
	Poisson ratio	0.25
	Shear modulus (Pa)	1.1 × 1010
	Outlet size, l × w (mm × mm)	20 × 20
	Silo dimensions, L × W × H (mm × mm × mm)	282 × 22 × 700
POM Spherical particle	Density (kg/m3)	1430
	Poisson’s ratio	0.35
	Shear modulus (Pa)	1.1 × 109
Particle-particle	Restitution coefficient	0.8
	Coefficient of static friction	0.2
	Coefficient of rolling friction	0.001
Particle-silo	Restitution coefficient	0.8
	Coefficient of static friction	0.2
	Coefficient of rolling friction	0.001
Simulation	Time step, Δt (s)	1.5 × 10−6

. To ensure the physical authenticity and robustness of the simulation, we adopted a combined approach integrating direct experimental measurements with parameter calibration. To establish a scientifically sound physical foundation, the primary parameters, including the particle-particle and particle-wall static friction coefficients as well as the internal friction angle, were determined via direct experimental measurements, as described in Section 2.1.2 and Section 2.1.3. Since the spherical particles used in this study do not involve physical interlocking, these explicitly measured frictional properties strictly dictate the macroscopic shear strength and the structural stability of the inter-orifice quasi-solid region in the dense granular flow regime [32]. For secondary parameters (e.g., shear modulus and restitution coefficient), we utilized two independent strategies: referencing established literature data [29] to determine appropriate parameter ranges, and employing a bulk calibration approach [33]. This calibration method has been successfully utilized in our team’s previous studies [34]. Ultimately, the effectiveness of the complete parameter set was rigorously validated by matching both the experimental flow morphologies and macroscopic discharge rates. Particle positions, stresses, velocities, and discharge flow rates were recorded every 0.001 s.

The DEM time step was set as a fixed fraction of the Rayleigh time, which characterizes the fastest elastic wave through a particle. The Rayleigh time is computed as

$$\Delta t_R={\displaystyle \frac{\pi r}{0.163\nu +0.8766}}\sqrt{{\displaystyle \frac{\rho }{G}}}$$

(2)

where $r$ is the particle radius, $\rho$ is the particle density, $\nu$ is the Poisson’s ratio and $G$ is the shear modulus [35]. Based on the above parameters, the Rayleigh time in the present study is calculated to be $\Delta t_R$ = 6.71 × 10⁻⁶ s. Previous studies have suggested that a stable time step for dense granular simulations typically ranges from 20% to 80% of the Rayleigh time [36,37,38,39]. In this study, the DEM time step was set to 22% of the Rayleigh time to achieve a balance between numerical stability and computational efficiency, corresponding to a time step of 1.5 × 10⁻⁶ s.

2.2.2. Mechanical Contact Model

To investigate the mechanism by which particle flow characteristics in a double-orifice silo vary with the inter-orifice spacing, a three-dimensional discrete element method (DEM) based on the soft-sphere model was employed in this study. Spherical particles were selected to construct the simulation model, which is not influenced by cohesion or liquid bridge forces [40]. In the DEM simulations, a non-sliding Hertz–Mindlin contact model [41] was used to describe both particle–particle and particle–wall interactions, in which energy and momentum are exchanged through translational and rotational motion. The motion of each particle can then be described by Newton’s second law, expressed as:

$${\mathrm{m}}_{\mathrm{i}}{\displaystyle \frac{\mathrm{d}{\mathsf{\nu }}_{\mathbf{i}}}{\mathrm{dt}}}={\mathrm{m}}_{\mathrm{i}}\mathbf{g}+{\sum }_{\mathrm{j}=1}^{{\mathrm{n}}_{\mathrm{i}}}({\mathrm{F}}_{\mathrm{n}}+{\mathrm{F}}_{\mathrm{n}}^{\mathrm{d}}+{\mathrm{F}}_{\mathrm{t}}+{\mathrm{F}}_{\mathrm{t}}^{\mathrm{d}})$$

(3)

$$I_i{\displaystyle \frac{d{w}_i}{dt}}={\sum }_{j=1}^{n_i}(T_t+T_r)$$

(4)

where $v_i$ is the translational velocity of particle i, $m_i$ is the particle mass, $I_i$ is the moment of inertia, $n_i$ is the total number of contacts between the particle and other particles or the silo wall, g is the gravitational acceleration, $F_n$ and $F_n^d$ are the normal contact force and normal damping force, respectively, $F_t$ and $F_t^d$ are the tangential contact force and tangential damping force, respectively, $T_t$ is the torque induced by the tangential force, $T_r$ is the rolling friction torque. In addition, a detailed description of the forces, moments an parameters in the contact model can be found in the past references [42,43].

3. Results

3.1. Effect of the Inter-Orifice Spacing on Discharge Rate and Particle Velocity at the Orifice

This study investigates the effect of the inter-orifice region on particle discharge rate through combined experiments and simulations. Prior to the analysis, it was necessary to validate the simulation results [44]. Previous studies show that particle interactions within the inter-orifice region change markedly at an inter-orifice spacing of 10 mm, which is approximately three particle diameters [7], while the discharge rate stabilizes when the eccentricity of a double-orifice configuration approaches 0.3 [4]. Based on these findings, two representative double-orifice silos with the inter-orifice spacing of 10 mm (approximately three particle diameters) and 80 mm (within the stable discharge regime) were selected to compare the consistency between simulations and experiments.

The particle flow patterns obtained from experiments and simulations at the same time instant are shown in Figure 3. Good agreement is observed in the overall flow features, with a distinct particle accumulation structure appearing in the inter-orifice region in both cases and exhibiting highly similar locations and morphologies. Previous studies have also shown that the discrete element method can effectively capture the particle flow characteristics during silo discharge [20,45]. As shown in Figure 4a, quantitative comparison of the discharge rate further demonstrates that the simulation results closely match the experimental data in both trend and magnitude. Quantitative analysis reveals that the maximum relative error between the simulated and experimental mass flow rates across all tested inter-orifice spacings is only 2.46%, with an average relative error of 1.54%. This minor discrepancy falls well within the acceptable error margin for DEM simulations, indicating that the DEM model reliably captures particle flow during discharge in a double-orifice silo. The necessity of this validation approach, which relies on the consistency between numerical simulations and physical experiments, has been well-established in the literature [46]. Furthermore, DEM has been proven as a robust tool for investigating the complex discharge dynamics and stress states of spherical particles in silos [47].

The maximum discharge rate is observed at an inter-orifice spacing of 0 mm, as the two outlets merge into a single large orifice, completely eliminating the bottom wall and the associated collisional scattering [8]. Once a physical separation is introduced (e.g., at 1 mm), the discharge rate drops abruptly due to the emergence of this flow-resisting repulsive potential. For non-zero spacings, the rate exhibits a much weaker decline in the range of 1–20 mm, and ultimately becomes nearly insensitive to further increases in spacing between 20 and 80 mm. To quantitatively validate this “independent discharge” criterion, a benchmark test of a single-orifice silo with identical outlet dimensions was conducted under the same reference conditions. As shown in Figure 4b, for inter-orifice spacings of D ≥ 20 mm, the total discharge rate of the double-orifice silo stabilizes at approximately twice the discharge rate of the single-orifice silo. This numerical correspondence confirms that when the spacing exceeds 20 mm, the discharge capacity of each individual hole in the dual-hole configuration is identical to that of an isolated single-hole silo, validating the onset of independent discharge at the macroscopic level. However, a purely quantitative relationship between discharge rate and the inter-orifice spacing is insufficient to elucidate the underlying mechanism. In macroscopic studies of granular flow, particle velocity is commonly used to characterize flow behavior [23]. Accordingly, Hagen proposed that the discharge rate is related to particle velocity and introduced the following relation:

$$Q=A·{\rho }_o·v$$

(5)

where Q is the discharge rate (g/s), ${\rho }_o$ is the bulk density (kg/m³), A is the cross-sectional area (m²), and v is the particle velocity (m/s). This relationship indicates that, for a fixed cross-sectional area, variations in the macroscopic discharge rate can be explained by changes in particle velocity and local solid fraction.

Figure 5 presents the simulation results of particle velocity and solid fraction at the orifices under different spacing conditions. The results show that, as the orifice spacing increases, the particle velocity varies in the same trend as the discharge rate, while the solid fraction remains essentially constant with negligible fluctuations. This indicates that, under the current operating conditions, the observed changes in discharge rate are primarily driven by variations in particle velocity rather than by changes in the particle solid fraction at the orifices. The next section therefore focuses on the dependence of particle velocity on the inter-orifice spacing.

3.2. Spatial Distribution of the Particle Velocity Field Under Different Inter-Orifice Spacings

To reveal the macroscopic effects of the inter-orifice spacing on particle velocity, Figure 6. shows the DEM simulated velocity contour maps for different inter-orifice spacings. At the baseline condition of D = 0 mm, the combined outlet inherently generates a single funnel flow forms, with the high-velocity region (V ≥ 400 mm/s) at its maximum. Once an obstruction appears between the orifices (D = 1 mm), the high-velocity region sharply reduces and initiates a splitting trend toward double funnel flow. For the inter-orifice spacing between 1 and 20 mm, a low-velocity region (V ≤ 50 mm/s) emerges and gradually expands, while the high-velocity region contracts slowly and the flow splitting intensifies. At 20–80 mm, the velocity field stabilizes, forming two independent funnel flows similar to a single-orifice silo. These results indicate that inter-orifice flow interference weakens as the inter-orifice spacing increases, as flow path competition is significant at small inter-orifice spacing (1–20 mm), whereas each orifice discharges independently at larger inter-orifice spacing (≥20 mm).

To quantify the effect of the inter-orifice spacing on particle velocity, Figure 7 shows particle velocity versus discharge height for different inter-orifice spacings. At 0 mm, velocity decreases monotonically with height. For 1–80 mm, the inter-orifice spacing, velocity drops sharply at first and then gradually levels off. When the inter-orifice spacing is 10–80 mm, velocities below 18 mm overlap, while above 18 mm, velocity decreases with increasing inter-orifice spacing. The observed velocity overlap below 18 mm indicates that particle motion in this region is insensitive to inter-orifice spacing and is mainly controlled by local outlet effects. This interpretation is consistent with the findings of Thomas and Durian [48], who showed that particle dynamics near the orifice are governed by localized mechanisms. In contrast, the divergence of velocity profiles above 18 mm suggests that particle motion becomes increasingly influenced by the overall system configuration as the distance from the outlet increases. This behavior can be attributed to the enhanced particle spreading with increasing discharge height [16], which leads to progressively evolving flow structures away from the orifice. Moreover, this trend agrees with the variation in discharge rate reported in Section 3.1, and the observed linear correlation between velocity and flow rate further supports that the near-orifice velocity field plays a key role in regulating discharge. Therefore, although an exact boundary cannot be strictly defined, it is reasonable to consider that the effective region governing discharge is confined to a limited zone above the orifice, particularly within the region below approximately 18 mm. Accordingly, the analysis focuses on the velocity field within this near-orifice region.

It is worth noting that when the inter-orifice spacing D ≥ 20 mm, a distinct low-velocity region gradually forms between the orifices, which may play a key role in weakening inter-orifice coupling [40]. In addition, particle velocity no longer changes with further increases in the inter-orifice spacing, indicating a clear critical threshold for the influence of inter-orifice particles on particle flow characteristics. Based on these observations, the following section investigates the mechanisms by which particles in the inter-orifice region affect particle flow, focusing on their impact on the evolution of particle velocity.

3.3. Evolution of the Inter-Orifice Quasi-Solid Region and Its Regulation of the Velocity Field

As shown in Section 3.2, orifice spacing affects particle velocity in a zonal manner. Velocities near the orifices remain nearly uniform, whereas significant differences emerge in the upper regions. As spacing increases, the flow between the orifices becomes increasingly weak, forming a region where particles move slowly and experience minimal shear. This low-velocity, low-shear behavior arises because local differences in inter-particle shear reduce the coupling between the inter-orifice region and the surrounding main flow, often producing stagnant or nearly stagnant zones [4].

As shown in Figure 8a,b, at small inter-orifice spacing, flow regions on both sides of the orifices remain partially coupled, allowing inter-orifice particles to participate in the overall flow. As inter-orifice spacing grows, interference between the two orifice flows diminishes, and the inter-orifice region gradually detaches from the main stream, maintaining low velocity and weak shear. To quantitatively describe this behavior, as shown in Figure 8c,d, a quasi-solid region is defined using a critical shear rate threshold ${\dot{\gamma }}_c$. For each specific configuration, this threshold is determined by the position along the x-direction where the shear-rate gradient exhibits a sharp increase. Regions where the local shear rate $\dot{\gamma }$ falls below this specific ${\dot{\gamma }}_c$ are classified as quasi-solid.

As the inter-orifice spacing increases, the low-velocity region gradually expands. Particle velocities within this region are much lower than those in the surrounding flow, resulting in weaker local shear. At the same time, the surrounding velocity gradients determine the development of the shear bands. These observations suggest that the inter-orifice spacing affects the velocity field indirectly by regulating the size and stability of the quasi-solid region. To investigate this, the formation and stability of the quasi-solid region and its regulation of shear bands and the velocity field are analyzed using local shear rate $\dot{\gamma }$ (from velocity gradients), the $\mu \left(I\right)$ model, and non-local length $\xi$, providing a unified explanation for the effects of inter-orifice spacing on particle velocity and discharge rate.

3.3.1. Introduction and Feasibility Validation of the $\mu \left(I\right)$ Model for Inter-Orifice Particle Flow

To investigate the regulatory role of the inter-orifice quasi-solid region on particle flow, it is necessary to analyze this region at the mesoscopic scale. Particle flow is often regarded as a continuum, and within the continuum framework, local shear behavior in dense granular flows can typically be described by the $\mu \left(I\right)$ rheology [17], expressed as

$$\mu \left(I\right)={\displaystyle \frac{\tau }{P}}={\mu }_s+{\displaystyle \frac{{\mu }_2-{\mu }_s}{1+I_0/I}}$$

(6)

$$I={\displaystyle \frac{\stackrel{.}{\gamma }d}{\sqrt{p/\rho }}}$$

(7)

where $\stackrel{.}{{\gamma }_{ij}}$ is the shear rate tensor given by

$$\stackrel{.}{{\gamma }_{ij}}={\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}=2D_{ij}$$

(8)

and $\stackrel{.}{\gamma }$ is the second invariant of the shear rate tensor given by

$$\stackrel{.}{\gamma }=\sqrt{{\displaystyle \frac{\stackrel{.}{{\gamma }_{ij}}\stackrel{.}{{\gamma }_{ij}}}{2}}}=\sqrt{2D_{ij}{D}_{ij}}$$

(9)

where $\tau$ is the shear stress (Pa), $p$ is the pressure (Pa), ${\mu }_s$ is the static friction coefficient, ${\mu }_2$ is the dynamic friction coefficient, $I$ is the inertial number, $I_0$ is the characteristic inertial number, $d$ is the particle diameter (m), and ρ is the particle density (kg/m³) $\stackrel{.}{\gamma }$ is the shear strain rate (s⁻¹) [31].

However, when active shear bands coexist with a low-velocity inter-orifice region, it remains unclear whether the local shear behavior is determined solely by the local stress conditions. To address this, the non-local length scale $\xi$ [19,49] is introduced as a criterion to assess the degree of shear localization and the applicability of the $\mu \left(I\right)$ model. This model evaluates whether local shear behavior is influenced by shear disturbances originating from neighboring regions. Its governing equation is given by [19]

$$\xi =\xi (\mu )=A({\displaystyle \frac{1+H({\mu }_s-\mu )}{\left|\mu −{\mu }_s\right|}}{)}^{\alpha }d$$

(10)

where ξ is the nonlocal length scale, A is a dimensionless scaling constant, $\alpha$ is a coupling parameter, H the Heaviside step function, where $H\left({\mu }_s-\mu \right)=1$ if ${\mu }_s-{\mu }_i\ge 0$ and $H({\mu }_s-{\mu }_i)=0$ otherwise, and $d$ is the particle diameter. For A, it is defined as

$$A\approx {\displaystyle \frac{\sqrt{\frac{m}{48{\alpha }^2}}}{d}}$$

(11)

m is an inhomogeneity parameter, and the expression of $\alpha$ is

$$m=a^2{\sigma }_c^2{\Pi }_{nn}^2$$

(12)

$$\begin{array}{c}\alpha ={\sigma }_c^2{\sum }_{i\ne j}{\Pi }_{i,j}^2\end{array}$$

(13)

${\sigma }_c$ is the local plastic stress threshold, with a value of 0.5 [49].

The nonlocal length $\xi$ was evaluated at each grid point using the local shear rate and stress data, covering both flowing and quasi-solid regions. The values obtained at individual grid points are consistently close to 0.5 d throughout the system. This indicates that shear disturbances decay within roughly one particle diameter. Therefore, local stress states predominantly govern the shear behavior. This confirms that the $\mu \left(I\right)$ model adequately captures the local frictional response across all regions of the silo.

Figure 9 compares the shear rate predicted by the $\mu \left(I\right)$ model with values computed from the DEM simulation results. All data points lie along the ideal consistency line, further confirming the model’s applicability.

In summary, the $\mu \left(I\right)$ model is suitable for the present conditions. On this basis, the next section will further investigate the formation and collapse mechanisms of the inter-orifice quasi-solid region.

3.3.2. Formation and Collapse Mechanisms of the Inter-Orifice Quasi-Solid Region

As the inter-orifice spacing increases, the inter-orifice quasi-solid region gradually expands, accompanied by a decrease in particle velocity, whereas particle velocity rises as the quasi-solid region shrinks. This suggests that the formation of the quasi-solid region may suppress particle flow, while its collapse facilitates velocity recovery. To test this hypothesis, the formation and collapse processes of the quasi-solid region and their regulatory effects on particle velocity need to be analyzed.

To qualitatively characterize the generation and collapse of the inter-orifice quasi-solid region, the granular fluidity ${\mathrm{g}}_{\mathrm{p}}$, derived from the $\mathsf{\mu }\left(\mathrm{I}\right)$ model, is introduced as an indicator reflecting both shear activity and flowability [19], defined as ${\mathrm{g}}_{\mathrm{p}}=\dot{\mathsf{\gamma }}/\mathsf{\mu }$ As shown in Figure 10a, the fluidity ${\mathrm{g}}_{\mathrm{p}}$ at the inter-orifice center decreases monotonically as the spacing D increases across all observed heights. This larger spacing weakens shear activation in the inter-orifice region. Consequently, particles cannot maintain effective shearing, which reduces overall flowability. Meanwhile, Figure 10b shows that the pressure in the inter-orifice region increases with inter-orifice spacing and that the height-dependent differences diminish, indicating a gradual transition from flow-controlled to load-dominated high-pressure states. According to the nonlocal granular flow model, the characteristic nonlocal length is approximately ξ = 0.5 d. When the width of the quasi-solid region becomes much larger than this length scale, shear disturbances from the surrounding flow cannot effectively reach its center. As a result, the core region remains in a solid-like state with ${\mathrm{g}}_{\mathrm{p}}$ ≈ 0. This solid-like core alters local stress transmission. As visualized in our previous study [4], the quasi-solid region presents a triangular distribution, which fundamentally changes the form of force arch transfer. The dynamic arches from the flowing particles transfer their loads onto the upper boundaries of this triangular region. This causes a significant stress concentration at elevated positions (e.g., Figure 10b), whereas the lower internal region experiences much less force. This behavior confirms the load-bearing capacity of the quasi-solid region.

In summary, the quasi-solid region forms because particle shearing gradually decreases while the load carried by the particles continues to increase. With increasing inter-orifice spacing, nonlocal shear disturbances fail to penetrate the inter-orifice core, causing particles to transition into a load-bearing, quasi-static state and stabilizing the quasi-solid region.

The above results qualitatively elucidate the formation mechanism of the inter-orifice quasi-solid region. Furthermore, the structural stability of this quasi-solid region can be explained through the parameters governing the inertial number $\mathrm{I}$, As the inter-orifice distance increases, the inter-particle tangential forces critically alter the local shear rate $\dot{\mathsf{\gamma }}$ and pressure $\mathrm{p}$. Specifically, the tangential contact force provides frictional resistance to support the overlying load, driving the significant rise in $\mathrm{p}$ (Figure 10b). Concurrently, the tangential damping force dissipates kinetic energy, facilitating particle stagnation and a sharp drop in $\dot{\mathsf{\gamma }}$. It is this synergy—rising pressure and dropping shear rate—that drives the inertial number $\mathrm{I}$ toward zero, physically defining the stable quasi-solid state. This solid-like region restricts the active shear bands, ultimately reducing the discharge rate.

To further characterize its mechanical stability, a quantitative description of strength is required. When the inter-orifice region is in a quasi-solid state, its shear strength can be described by the Mohr–Coulomb criterion [50]:

$$\tau \approx {\sigma }_n\mathrm{t}\mathrm{a}\mathrm{n}\varphi$$

(14)

where ${\mathsf{\sigma }}_{\mathrm{n}}$ is the local normal stress in the inter-orifice region, and $\mathsf{\varphi }$ is the internal friction angle obtained from the direct shear tests described in Section 2.1.3. At the boundary of the quasi-solid region, shear perturbations are governed by the $\mathsf{\mu }\left(\mathrm{I}\right)$ friction law, which is expressed as follows:

$$\tau =\mu \left(I\right)\mathrm{p}$$

(15)

Accordingly, the conditions under which the quasi-solid region becomes unstable and begins to collapse can be expressed as

$$\mu (I)\gtrsim {\displaystyle \frac{{\sigma }_n}{p}}\mathrm{t}\mathrm{a}\mathrm{n}\varphi$$

(16)

Within the inter-orifice region, particles are subjected to downward pressure from the overlying material and lateral compressive reaction forces from the adjacent active flows. Because of these multi-directional compressions, the granular assembly presents a highly compacted load-bearing state. Therefore, the local normal stress can be reasonably approximated as ${\sigma }_n\approx P$. Under this approximation, the above equation can be simplified as

$$\mu \left(I\right)\gtrsim \mathrm{t}\mathrm{a}\mathrm{n}\varphi$$

(17)

This relationship indicates that the failure of the quasi-solid region does not depend on the particle velocity itself, but on whether the local friction coefficient exceeds the static friction threshold. Using the $\mu \left(I\right)$ friction law, the critical inertial number $I_c$ and critical shear rate ${\stackrel{.}{\gamma }}_c$ corresponding to the instability of the quasi-solid region can be expressed as

$$I_c={\mu }^{-1}\left(\mathrm{t}\mathrm{a}\mathrm{n}\varphi \right)$$

(18)

$${\stackrel{.}{\gamma }}_c={\displaystyle \frac{I_c}{d}}\sqrt{{\displaystyle \frac{P}{\rho }}}$$

(19)

To visually characterize the mechanical state of the inter-orifice region under different inter-orifice spacing conditions, Figure 11 presents the relationship between the inter-orifice spacing $D$ and the stress ratio $\mu$ at the center of the inter-orifice region bottom. This position corresponds to the core of the inter-orifice region, whose integrity governs the persistence of the quasi-solid structure, since the quasi-solid region remains stable as long as the core maintains a low-shear state. The red dashed line indicates the collapse criterion of the quasi-solid region, $\mu =\mathrm{t}\mathrm{a}\mathrm{n}\varphi$, with a value of 0.553. The results show that the formation and stability of the inter-orifice quasi-solid region are highly dependent on the orifice spacing. Specifically, at D = 0 mm, the absence of inter-orifice obstruction results in weaker shear disturbance and a lower stress ratio compared to the D = 1 mm case. Consequently, no quasi-solid region is formed, which indirectly corroborates our theoretical model. For D = 1–20 mm, the inter-orifice flow remains affected by double-orifice interference, generating a weak and unstable quasi-solid region. As the spacing increases, the macroscopic discharge rate stabilizes at D = 20 mm, which we explicitly define as the critical spacing for discharge-rate stabilization. However, complete mechanical stabilization of the mesoscopic quasi-solid region occurs when the stress ratio crosses the failure criterion. As demonstrated in Figure 11, this precise transition point actually lies within the 20 to 30 mm interval. Therefore, we define the 20 to 30 mm range as containing the critical spacing for stable quasi-solid region formation. Once the spacing exceeds this transitional interval (e.g., at D ≥ 30 mm), low-mobility particles accumulate in the inter-orifice region, the quasi-solid region exists stably, and the flows through the two orifices gradually decouple.

It should be noted that increasing the inter-orifice spacing does not alter the critical collapse condition of the granular material. Instead, it reduces the actual shear level in the inter-orifice region, thereby gradually moving its mechanical state away from the instability criterion defined by $\mu =\mathrm{t}\mathrm{a}\mathrm{n}\varphi$ and allowing it to remain stably in the quasi-solid state.

In summary, as the inter-orifice spacing increases, the inter-orifice region gradually evolves into a stable, low-shear quasi-solid zone, where shear perturbations are weakened and particle motion is restricted. From a mechanical perspective, this spatial evolution provides the scientific explanation for the local differences in inter-particle shear observed in Figure 8. In the quasi-solid zone, the strong geometric confinement maintains the local stress ratio below the critical internal friction threshold ($\mu <\mathrm{t}\mathrm{a}\mathrm{n}\varphi$), effectively suppressing inter-particle shear. This state leads to the distinct near-zero shear plateau ($\dot{\gamma }\approx 0$) as quantitatively evidenced in Figure 8c. Conversely, in the flow zones near the orifices, the stress state exceeds this threshold, facilitating active inter-particle sliding and the high shear rate peaks shown in Figure 8d. When D = 20–30 mm, the quasi-solid region becomes clearly stable, and the accumulation of low-mobility particles gradually decouples the double-orifice flow, while compressing the spatial extent of the shear bands and altering local shear conditions. Based on this structural evolution, the following section combines the influence of the quasi-solid region on particle velocity and effective shear area, considering the characteristics of the velocity field.

3.3.3. Regulation of Particle Velocity by the Quasi-Solid Region

Section 3.3.2 has clarified the formation and stability mechanisms of the inter-orifice quasi-solid region. To further understand its influence on particle flow velocity, attention must be paid to its effects on the surrounding particle motion and the structure of shear bands. In this study, the effective shear zone is defined as the region below the discharge height of 18 mm (h ≤ 18 mm, based on the findings in Section 3.2) where the shear rate exceeds a threshold ${\dot{\gamma }}_c$, which is determined from the location along the x-direction where the shear-rate gradient increases sharply, distinguishing the quasi-static region from the effective shear region.

Figure 12a shows the contours of particle shear rate $\dot{\gamma }$ under different inter-orifice spacings. As defined in Section 2.1.1, at D = 0 mm, the orifices merge into a combined opening. Since this represents a single rectangular outlet rather than a typical double-orifice structure, there is no inter-orifice obstruction to induce collisional scattering or support a stable quasi-solid region. Consequently, the local high-shear regions from both sides merge into a single, unified flow zone. This unobstructed, continuous shear structure directly explains the maximum discharge rates observed in Figure 4 and Figure 5, as particles can discharge freely without any boundary-induced kinetic resistance. When the spacing increases within the 1–10 mm range, high-shear regions ($\dot{\gamma } \ge 20 1/s$) extend laterally and remain coupled, forming wide and continuous shear bands. Although the inter-orifice region itself exhibits lower shear rates ($\dot{\gamma } < 20 1/s$), it remains influenced by shear disturbances from both adjacent outlets. As the inter-orifice spacing increases to 20–30 mm, the high-shear regions become clearly separated, and the inter-orifice region evolves into a low-shear zone, with shear bands transitioning from continuous to locally concentrated. At larger inter-orifice spacing (40–80 mm), the inter-orifice region transitions almost entirely into a low-shear state, strongly confining the high-shear regions near the orifices. Because shear disturbances can no longer propagate across this quasi-solid zone, the granular flow exhibits highly localized behavior.

To quantitatively describe the evolution of the shear structure, Figure 12b presents the effective shear-zone area $A_s$ under different inter-orifice spacings. As the inter-orifice spacing increases, the shear-zone area decreases monotonically and stabilizes for D ≥ 20 mm, indicating that the inter-orifice quasi-solid region reduces shear interference between the two orifices, compresses the spatial extent of the shear bands, and decreases the number of particles that can be activated by shear. According to the collapse criterion of the quasi-solid region, the larger the inter-orifice spacing prevents the inter-orifice region from reaching the critical stress ratio ${\mu }_c$, maintaining a stable quasi-solid state, which in turn spatially constrains the shear bands. This limits the propagation of shear disturbances and prevents the shear bands from extending into the inter-orifice region.

In summary, as the inter-orifice spacing increases, the inter-orifice region gradually develops into a stable quasi-solid zone. The growth of this quasi-solid region restricts the lateral expansion of the active shear bands, significantly reducing the effective shear area. Consequently, both particle velocity and macroscopic discharge rate decrease, eventually reaching stable values at D = 20 mm (about six particle diameters), which defines the critical spacing for discharge-rate stabilization. However, this macroscopic flow stabilization precedes the complete mesoscopic structural isolation. As shown in the shear rate contours (Figure 12a), the macroscopic flows are separated at D = 20 mm. However, the local shear rate in the inter-orifice region at D = 20 mm remains noticeably higher than that at D = 30 mm. Meanwhile, the stress analysis ratio (Figure 11) confirms that a stable quasi-solid region is fully established at D = 30 mm. This comparison clearly demonstrates that at D = 20 mm, the shear bands from the adjacent orifices still partially interact and exert shear perturbations on the central region. This residual shear interaction explains why the central stress ratio at D = 20 mm remains above the failure criterion (Figure 11). As the spacing further increases, these perturbations are gradually suppressed. Because the experiments were conducted at discrete spacing intervals, the exact transition spacing cannot be directly identified. Among all investigated cases, D = 30 mm is the first spacing at which the active shear bands are completely separated by the quasi-solid core. Under this condition, the local stress ratio falls below the absolute stability threshold. This indicates that the formation of a stable quasi-solid region undergoes a spatial transition within the 20 to 30 mm interval, marking the evolution from a flowing region to a quasi-solid region.

3.4. Influence of Inter-Orifice Baffle Friction on Quasi-Solid Region Stability and Discharge Rate

Section 3.3 has demonstrated that the inter-orifice quasi-solid region influences particle velocity and discharge rate by limiting the spatial extent of shear bands, with its stability governed by the interplay between local shear activation and stress-bearing mechanisms. To further examine whether the mechanism identified in Section 3.3 remains consistent under variations in boundary conditions, this section investigates the effect of varying the friction coefficient of the inter-orifice baffles on the steady-state discharge rate and compares the resulting flow rate trends with the baseline case.

Figure 13 presents the steady-state discharge rates measured under different baffle roughness conditions and the schematic of the quasi-solid region size under different friction coefficients at the same inter-orifice spacing. The results show that increasing the friction coefficient of the inter-orifice baffle leads to a clear monotonic decrease in the overall discharge rate, accompanied by an expansion of the quasi-solid region. Combined with the regulation mechanism proposed in Section 3.3, this indicates that, at fixed the inter-orifice spacing, enhanced friction strengthens shear resistance between inter-orifice particles and the boundary, promoting the formation of stable load-bearing structures. When shear disturbances propagate into the inter-orifice region, higher friction further suppresses shear activation, making it more difficult to reach the critical shear condition required for collapse and thereby enhancing the stability of the quasi-solid region.

In summary, from the perspectives of flow structure and boundary mechanics, enhanced stability of the inter-orifice quasi-solid region compresses the spatial extent of shear bands, reduces the number of particles participating in shear-induced acceleration, and consequently decreases the mean particle velocity and overall discharge rate. This behavior is confirmed by experimental observations obtained by varying the friction coefficient of the inter-orifice baffle. Whether through weakened shear coupling caused by increased the inter-orifice spacing or enhanced shear resistance induced by higher boundary friction, both effects ultimately lead to increased quasi-solid region stability and constrained shear bands. These results indicate that discharge rate variations in double-orifice silos are governed not only by geometric configuration but also by the coupled effects of the quasi-solid region, shear-band structure, and boundary mechanical properties.

4. Discussion

When a silo is equipped with multiple discharge orifices, the particle flows between the orifices interact with each other, resulting in discharge behavior that is more complex than that of a single-orifice silo [4]. Based on this, the present study further complements and deepens the understanding of the intrinsic mechanisms by which the inter-orifice spacing affects particle flow from the perspectives of local mechanics and nonlocal flow theory. Unlike previous studies, which primarily focus on macroscopic discharge rates or velocity distributions [40], this work introduces granular fluidity ($g_p$), shear rate, and stress criteria to distinguish the mechanical properties of the low-velocity region between orifices. It explicitly shows that variations in the inter-orifice spacing first act on the shear activation within the inter-orifice region and, by regulating the formation and stability of the quasi-solid region, indirectly affect the spatial structure of shear bands and the extent of the effective flow region. This analysis provides a new physical perspective for understanding how the inter-orifice spacing influences particle flow capacity in multi-orifice systems through local mechanical states. It should be noted that the model investigated in this study is a double-orifice silo. Although it involves only two orifices, it represents the simplest and most classical design among multi-orifice silos and can capture most of the particle flow characteristics observed in multi-orifice systems. More complex multi-orifice silos, such as four-orifice or asymmetric double-orifice silos, evolve from the symmetric double-orifice design. Therefore, the particle flow characteristics of various multi-orifice silos may be regarded as variations in those in a symmetric double-orifice silo. The particle flow features of multi-orifice silos will be further explored in our future studies.

In addition, the particles and silo model selected in our experiments were used only to examine the effects of the inter-orifice spacing on the formation and collapse conditions of the inter-orifice quasi-solid region and on particle velocity. The effects of particle shape and surface properties on the formation and collapse conditions of the inter-orifice quasi-solid region, as well as a more detailed quantitative analysis of the quasi-solid region characteristics, shear zone structure, critical spacing under different friction conditions, and the dynamic temporal evolution of the discharge process, still require further investigation in future studies.

From a practical engineering perspective, the observed inter-orifice quasi-solid region serves as a controllable structural feature that balances discharge efficiency and flow stability. In precision agriculture, equipment such as multi-row seed drills and granular metering systems requires a strictly uniform mass flow rate to ensure accurate application. By configuring the outlet spacing beyond the critical threshold, the resulting stable quasi-solid region strictly limits the spatial extension of active shear bands. This internal structural isolation prevents undesired flow coupling between adjacent outlets, ensuring the steady discharge rate necessary for precise agricultural operations.

These mesoscopic mechanical findings also offer direct geometric guidance for optimizing agricultural machinery, such as unmanned aerial vehicle (UAV) fertilization systems. UAV hoppers operate under strict payload and spatial constraints, typically utilizing compact, multi-outlet, flat-bottom configurations where flow interference is highly likely. Adjusting the orifice spacing to promote a stable quasi-solid region effectively utilizes the stagnant fertilizer particles as an internal boundary. This passive structural regulation eliminates unpredictable inter-orifice interactions, ensuring a highly stable, controllable, and uniform discharge rate during continuous aerial spreading operations.

Additionally, the synergistic interaction between the quasi-solid region and boundary mechanical conditions (as demonstrated in Section 3.4) provides a strategy for handling various agricultural inputs. Since enhanced boundary friction directly strengthens the stability of the quasi-solid region and further compresses active shear bands, the internal baffles of large-scale fertilizer spreaders can be engineered with specific surface treatments. This allows the machinery to accommodate granular fertilizers with varying moisture contents and internal friction angles, maintaining reliable performance across different environmental conditions.

It should be acknowledged that while the identified transition mechanism is fundamental, the specific quantitative thresholds, such as the critical spacing of six particle diameters, are intrinsically tied to the present laboratory conditions and may require recalibration when applied to different industrial scales.

5. Conclusions

In this study, the discrete element method (DEM) was employed to investigate the effects of the inter-orifice quasi-solid region on particle velocity and discharge rate during the unloading of polyoxymethylene (POM) particles from a double-orifice silo. Based on combined qualitative and quantitative analyses, the following conclusions are drawn:

• The inter-orifice spacing is a key parameter governing the flow structure and discharge behavior of granular materials in a double-orifice silo. At small inter-orifice spacing, significant interference occurs between the two granular flows within the inter-orifice region. As the inter-orifice spacing increases beyond a critical value, the inter-orifice coupling gradually diminishes, and the two discharge flows tend to behave independently. For the specific granular system and laboratory conditions investigated in this study, this critical spacing for discharge-rate stabilization is approximately 20 mm, corresponding to about six particle diameters. Furthermore, the discharge rate is sensitive only to the near-orifice flow characteristics, with an effective influence height of approximately 18 mm (about five particle diameters), indicating that the double-orifice discharge behavior is governed by local flow structures.
• The formation and stability of the inter-orifice quasi-solid region are governed by the competition between the local shear level and the mechanical failure criterion. As the inter-orifice spacing increases, the shear intensity within the inter-orifice region gradually decreases, making the actual shear state insufficient to exceed the failure criterion. Consequently, the inter-orifice region transitions from a mobile state to a stable quasi-solid state. While macroscopic flow separates earlier at D = 20 mm, the exact structural transition is governed by the critical spacing for stable quasi-solid region formation, which physically falls within the 20 to 30 mm interval. This process reflects a mechanism by which variations in inter-orifice spacing indirectly regulate the mechanical response of the inter-orifice region through modifications of shear-band distribution and stress transmission pathways.
• The regulatory effect of the quasi-solid region on the discharge rate arises from the spatial reconfiguration of shear structures and their synergistic interaction with boundary mechanical conditions. With the stable formation of the quasi-solid region between the orifices, it reduces the local shear-driven efficiency by compressing the lateral width and effective shear area of the shear band near the outlet, thereby decreasing particle velocity and resulting in a lower overall discharge rate. This process reflects the coupled effects among the quasi-solid region, shear band structure, and boundary frictional properties, which together govern the evolution of particle flow behavior and discharge rate during double-orifice discharge.