Capillary forces on wet particles with a liquid bridge transition from convex to concave

Journal Pre-proof Capillary forces on wet particles with a liquid bridge transition from convex to concave

Fei Xiao, Jiaqiang Jing, Shibo Kuang, Lu Yang, Aibing Yu PII:

S0032-5910(20)30020-6

DOI:

https://doi.org/10.1016/j.powtec.2020.01.020

Reference:

PTEC 15088

To appear in:

Powder Technology

Received date:

23 June 2019

Revised date:

17 December 2019

Accepted date:

7 January 2020

Please cite this article as: F. Xiao, J. Jing, S. Kuang, et al., Capillary forces on wet particles with a liquid bridge transition from convex to concave, Powder Technology(2019), https://doi.org/10.1016/j.powtec.2020.01.020

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© 2019 Published by Elsevier.

Journal Pre-proof

Capillary Forces on Wet Particles with a Liquid Bridge Transition from Convex to Concave

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Fei Xiao1, 2, Jiaqiang Jing1,4*, Shibo Kuang2*, Lu Yang3, Aibing Yu2

1 School of Oil & Gas Engineering, Southwest Petroleum University, Chengdu, Sichuan

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610000, China

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2 ARC Research Hub for Computational Particle Technology, Department of Chemical

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Engineering, Monash University, Clayton, Victoria 3800, Australia 3 Gas Transmission Management Division, Southwest Oil &Gas Company, PetroChina,

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Chengdu, Sichuan 610000, China

China

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4 Oil & Gas Fire Protection Key Laboratory of Sichuan Province, Chengdu, Sichuan 610000,

* Corresponding authors: [email protected] (J.Q. Jing); [email protected] (S.B.

Kuang)

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Journal Pre-proof Abstract The transition of a liquid bridge profile from convex to concave and the associated capillary forces are experimentally studied via particle-particle and particle-plane pairs. The results demonstrate that a convex liquid bridge appears at a relatively large water volume and small separation distance, where the capillary force remains approximately constant. As the separation distance increases, the liquid bridge is stretched from convex to concave, and the

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capillary force initially doesn’t change much but increases to the maximum value and then decreases gradually. By combining analytical analysis and experimental data, a framework is

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proposed for predicting the evolution of liquid bridge profiles. Additionally, a capillary force

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model that is applicable to both convex and concave liquid bridges is formulated. The

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applicability of this force model to multiple-particle systems is demonstrated by comparing

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DEM predictions of repose angles of sandpile and hopper flow discharge rates with

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experimental measurements at various water contents.

Keywords: Wet Particles; Capillary Force; Liquid Bridge; Concave; Convex; DEM

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Journal Pre-proof 1. Introduction Wet cohesive particles can be encountered in a wide variety of solid handling processes, such as fluidization, pneumatic conveying, screwed conveyor transportation, coating and mixing. In such processes, the liquid can serve as a reactant or a mechanism for binding particles together to form agglomerates [1, 2]. From a physical perspective, a liquid bridge creates a cohesive force between particles due to surface tension and viscous force [3]. Such a liquid

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bridge does not form until the liquid films on two particles come into contact, yet it can

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endure for a substantial separation distance after the two particles have rebounded off each other [4, 5]. Liquid bridges and the associated capillary forces govern many distinct

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phenomena of wet particles.

In the past years, various efforts have been made to study liquid bridges and the

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corresponding capillary forces. The mathematical foundation was laid by Thomas Young and

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Pierre-Simon Laplace at the beginning of the 18th century [6, 7]. The Laplace-Young equation was developed for relating the capillary pressure difference across an interface to the shape

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and the surface tension of a liquid bridge. However, this equation cannot be solved analytically, except in a few special cases. Thus, numerical procedures have been developed by Erle et al. [8] and De Bisschop and Rigole [9]. They found that there is no solution for a separation distance that exceeds a critical value. Later, Mazzone et al. [10] argued that this critical separation distance was the rupture criterion. Erle et al. [8] and Mazzone et al. [10] compared the force-separation results from their numerical solution with the experimental data of Mason and Clark [11]. Satisfactory agreement between the measured and calculated results was obtained, except at a small separation distance. Fisher [12] proposed a much simpler approach for calculating the force that is caused by a pendular bridge between two spheres using the toroidal approximation, in which both sets of curves are of circular

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Journal Pre-proof curvature and form a boundary that is akin to the inner portion of a torus. Extending the work of Heady and Cahn [13], the model by Huppmann and Riegger [14] seems to be the first model of a liquid bridge that is based on the methodology of toroidal approximation. Hotta et al. [15] introduced a new assumption for the geometry of a pendular bridge, which is known as the “gorge” method and has been shown to improve the accuracy of bridge force prediction. Lian et al. [16] developed a method for implicitly calculating the capillary force based on the meniscus geometry. Based on the data that were produced by other investigations [9, 16],

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Mikami et al. [17] formulated a model that expresses the capillary force as a function of the

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liquid bridge volume and separation distance. Via a similar method, Willett et al. [3] and Shi

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and McCarthy [18] formulated equations that are applicable to particles that differ in size.

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Butt and Kappl [19] provided insights into the effects of surface roughness and nonspherical shape on the cohesive force. The model of Chen et al. [5] considered the particle radius ratio

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but required the contact angles to be equal. Payam and Fathipour [20] presented a model that

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allowed for a varying radius ratio and contact angle. Similarly, nonuniform particles were considered in the model of Harireche et al. [21], in which the contact angle was fixed to 0. By

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fitting the numerical results, Lian and Seville [22] derived a new closed-form equation for calculating the capillary force. For a small half-filling angle and separation distance, this expression is similar to the Derjaguin equation, while the new equations are more accurate at large half-filling angles.

An alternative approach is to directly solve the liquid bridge problem by optimizing the interfacial energy. For example, Brakke [23] developed the SURFACE EVOLVER, in which a liquid bridge profile can be obtained by minimizing the interfacial energy. Israelachvili [24] correlated the capillary force with the relative humidity using Kelvin equations while assuming that thermodynamic equilibrium is maintained. To overcome this assumption, Rabinovich et al. [25] considered a vertical component of the surface tension of the liquid 4

Journal Pre-proof bridge in determining the capillary force according to the pressure difference across a liquid bridge. Lambert et al. [26] compared the energy method and the Laplace equations and concluded that both approximations could not accurately predict the experiments in the case of large liquid volumes. By solving the Kelvin equations, Chau et al. [27] developed a three-dimensional model for a capillary nanobridge, in which the effects of the relative humidity, temperature and contact angle were considered. Ardito et al. [28] applied the model that was proposed by Kenmotsu [29] to obtain the exact solution for the meniscus shape under

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the hypothesis of thermodynamic equilibrium. Farmer and Bird [30] studied the asymmetric

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capillary bridge between contacting spheres by minimizing the total surface energy for a

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specified liquid volume and contact angle. Sun and Sakai [31] used an approach that was

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based on the optimization of the interfacial energy to obtain the solution for an axisymmetric liquid bridge, which agrees with the experimental results over broad ranges of liquid volumes

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(liquid-solid ratio: 10-4~0.2) and contact angles (0~150°). In addition, Wang et al. [32] applied

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interfacial energy minimization to liquid bridges that bound three or more particles to determine the capillary force and rupture behaviors. Wu et al. [33] investigated the liquid

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redistribution behavior upon liquid-bridge rupture via the minimal energy method and identified a critical contact angle at which all liquid would be transferred completely to other particles with smaller contact angles. In addition to the analytical methods that are discussed above, numerical models have been widely used to study stretching pendular liquid bridges while considering dynamic parameters. For a funicular liquid bridge, Murase [34] numerically investigated the strength of the capillary forces that adhered three spheres. Daribi [35] studied the evolution and rupture of a pendular liquid bridge via the solution of the Navier-Stokes equations, in which viscous, inertial, gravitational and surface tension effects were considered. Washino [36-38] studied the microscale modeling of solid-liquid-gas flow via CFD-DEM coupling in a liquid bridge 5

Journal Pre-proof while considering the effect of the surface tension that is exerted on the liquid and the particles. Sun and Sakai [39] and Jain [40] used the VOF-IB-DNS method to resolve the hydrodynamic and capillary forces, in which a dynamic stretching of a liquid bridge was realized in the simulation. The detailed pressure distribution and dynamic evolution of the liquid bridge can be obtained via such a method. CFD is convenient for the microscale study of the capillary bridge between two particles or between a particle and a wall. However, it is difficult to extend it to a system with many particles due to its prohibitively high

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computational requirements.

In summary, the previous studies, although useful, focused mainly on concave liquid bridges,

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the apexes of which are directed toward the liquid phase. However, as two particles are

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approaching each other or if the volume of liquid is sufficiently large, the liquid is squeezed

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out and the parabolic profile of the liquid bridge between the particles may be transformed from concave to convex (Fig. 1) [41, 42]. Meurisse and Querry [43] and Souza et al. [44]

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conducted theoretical studies of the force characteristics for a convex bridge between two

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parallel solid surfaces, where the dynamic squeezing was investigated based on Delaunay’s roulette of the meniscus profile. For a liquid bridge between two spheres, the geometry depends mainly on three variables: the wetting angle, the interparticle separation and the liquid volume. Megias-Alguacil and Gauckler [45, 46] developed region maps for determining the existence of convex liquid bridges and revealed that the Laplace force is repulsive for a convex bridge but attractive for a concave bridge. Urso [47, 48] also established maps of existence for convex liquid bridges but within triplets of spherical particles. Due to the difficulties in calculating capillary forces, the meniscus profile was often approximated by an arc of circumference, namely, the so-called toroidal approximation. Pepin et al. [49] compared two simplifications of the geometry, namely, the toroidal and parabolic approximations, and demonstrated that the toroidal approximation is unable to generate a 6

Journal Pre-proof solution as the bridge shape changes from convex to concave. Regarding the energy method, Vogel [50] observed that the convex unduloidal bridge between two particles was a constrained local energy minimum. Nevertheless, most previous models considered two particles in a static configuration, in which the evolution was neglected. Broesch and Frechette [51] investigated the liquid bridge transition from concave to convex in a slit-pore geometry to identify the effects of the pinning angle and the aspect ratio on the bridge shapes. Rhynart et al. [52] demonstrated that the bridge surface transition from convex to concave can

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be approximated as a cylinder. A similar study was conducted by Zhang et al. [53] in which

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the deformation and breakup of a stretching convex liquid bridge were simulated. The

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capillary forces differ between convex and concave liquid bridges and should be represented

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separately. To date, the transition from a convex to a concave liquid bridge has yet to be fully

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understood, and no valid capillary force model for considering this transition is available. In this paper, a framework is established for describing the evolution of the liquid bridge

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profile and the associated capillary force. This evolution is measured under various conditions, and two particles and a particle-plane pair are considered. Then, by combining analytical

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analysis and experimental data, the transition of the liquid bridge profile from convex to concave and the corresponding capillary forces are modeled. Finally, numerical and experimental studies of a sandpile and a hopper flow are conducted to investigate the applicability of the proposed model to granular flows. 2. Methodology 2.1 Measurement of the particle-particle/wall capillary force Fig. 2 schematically illustrates the experimental system that is used to study liquid bridges and the resulting capillary forces between particles and between a particle and a wall as functions of the separation distance. The particles considered are spherical glass beads with 7

Journal Pre-proof radii of 0.2 mm, 0.3 mm and 0.4 mm, while the plane is made of 304 stainless steel. Before and after each experiment, these particles were cleaned with ethanol to remove the impurities and additional water from the particle surfaces. The dimensionless volume of deionized water ( V *  V / rp ) for rp=0.3 mm is in the range of 0.41~5.4 for both particle-particle and 3

particle-plane pairs. The surface tension of the water was measured by a surface tensiometer, while the contact angle between the water and each sphere was measured directly from the

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liquid bridge images. The properties of the test materials (20°C) are listed in Table 1. In total, approximately 50 sets of experimental data were measured. The lower sphere/plane was fixed

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on a pressure sensor, while the upper sphere was mounted on an adjustable arm. The liquid

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volume was controlled by a microsyringe with a maximum volume of 0.5 μL. A calibration

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step was conducted to ensure that the expected volume of the liquid was used to form a

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capillary bridge. When the liquid volume was sufficiently large for forming a convex bridge, the critical liquid volume for forming a cylindrical bridge [52] was calculated based on the

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critical separation distance, which was used to calibrate the liquid volume to achieve the expected value. This calibration was realized by calculating the liquid volume according to

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liquid bridge images for a small liquid volume. The current setup enables the measurement of the capillary force in the vertical direction by changing the separation distance at a constant velocity. The values of the measured forces are in the range 0~1 mN with a precision of 1 μN. Prior to each measurement, an axisymmetric liquid bridge was obtained by adjusting the platform horizontally. Then, the upper sphere was lifted until the liquid bridge ruptured. Meanwhile, the forces and the corresponding separation distances were recorded using the computer system, while the liquid bridge profiles were captured via a camera. The operating time for each case was controlled within 20 s to minimize the effect of evaporation, thereby leading to a lifting speed of 0.01~0.1 mm/s, which depends on the considered liquid volumes. Fig. 3 presents the reproducibility of the measured 8

Journal Pre-proof liquid bridge forces, in which the maximum deviation is within ±5%. 2.2 Young-Laplace solution To validate the current measurements, the Young-Laplace solution is considered. In the absence of gravitational effects that arise from bridge distortion and buoyancy, the total force, namely, F, that acts between two spherical particles of specified radii is calculated as the sum of two components: (a) the axial component of the surface tension that acts on the three-phase

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contact line and (b) the hydrostatic pressure that acts on the axially projected area of the liquid

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contact on either sphere. This leads to the following expression:

F  2rp sin a sin(a   )  rp p sin 2 a

(1)

re

-p

2

where  is the liquid surface tension, a is the half-filling angle, and  is the contact

1 y( x)  and the surface tension  by 2 1/ 2 y( x)[1  y( x) ] [1  y( x)2 ]3 / 2

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the Laplace-Young equation:

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the local mean curvature

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angle. The pressure difference across the air-liquid interface, namely, p , can be related to

p





1 y( x)  2 1/ 2 y( x)[1  y( x) ] [1  y( x)2 ]3 / 2

(2)

where y(x) is a function that describes the liquid profile and y(x) and y(x) are its first and second derivatives, respectively. The negative sign corresponds to a concave meniscus, whereas the positive sign corresponds to a convex meniscus. This pressure difference can be described in terms of the two principle radii of curvature, namely,  and L, of the meniscus. The radii can have positive or negative sign. The sign convention is positive if the radius lies inside the meniscus and negative when lying outside.

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Journal Pre-proof 1 1 p        L

(3)

Megias-Alguacil [45, 46] developed expressions for the principle radii in terms of corresponding angles a and  , and in terms of xa (Fig. 4).

xa rp ( H / 2  rp  xa ) cos   rp  ( xa  H / 2  rp ) 2 sin  2

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

rp  ( xa  H / 2  rp )2 cos   ( H / 2  rp  xa ) sin   rp 2

L  rp  ( xa  H / 2  rp )  xa 2

( H / 2  rp  xa ) cos   rp  ( xa  H / 2  rp )2 sin  2

(5)

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2

(4)

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Substituting Eq. (3) into Eq. (1) and rearranging gives

(6)

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F 1 1  2 sin a sin(a   )  rp sin 2 a(  ) rp L 

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Eqs. (3)-(5) can be used to calculate the capillary force, where the values of the half-filling

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angle ( a ) and the contact angle (  ) (see Fig. 1), as well as the particle radius, the separation distance and the liquid volume are determined from the experiments. 3. Results and discussion

3.1 Experimental analysis of the liquid bridge and the capillary force In this section, the experimental particle-particle/wall capillary force and rupture distance are analyzed with respect to the separation distance, the liquid volume and the particle size. To facilitate the analysis, the capillary force, the separation distance and the liquid volume are normalized as Fˆ  F /(rp ) , dˆ  d / rp and V *  V / rp 3 , respectively.

Fig. 5 plots the measured capillary forces between two spherical particles (rp=0.3 mm) at 10

Journal Pre-proof various liquid volumes. When the liquid volume is relatively small ( V * =0.89~2.78), the liquid bridge force decreases as the separation distance increases (Fig. 5(a)). These forces exhibit a different trend when the liquid volume is relatively large ( V * =3.56~5.4) (Fig. 5(b)). As the separation distance increases, the capillary force initially does not change substantially and, subsequently, increases about 20% to a maximum value. The evolution of the liquid bridge profile with the force variation at two representative water contents ( V * =0.89 and 3.56) is

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presented in Fig. 6. For clarity, profiles are shown at only three states: the contact of two

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particles, the drastic change in the force, and the rupture of the liquid bridge. To evaluate the reliability of the current experimental measurements, the predictions of Mikami et al.’s model

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[17] (referred to as the MKH model) and of the Laplace-Young equation [46] (referred to as

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the LY equation) are included in Fig. 6. Both the MKH model and the LY equation have been

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well recognized in the literature. Experimental and numerical validations of the MKH model can be found, for example, in Refs. [54-66]. When the liquid volume is 0.89 (Fig. 6(a)), the

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liquid bridge is always a meniscus. Both the MKH model and the LY equation can accurately

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predict the measured forces, with maximum deviations of less than 4.5%. As the liquid volume changes to 3.56 (Fig. 6(b)), the liquid bridge changes from convex to concave as the separation distance increases. The MKH model does not capture the trend of drastic change due to the transition of the liquid bridge from convex to concave. In contrast, the curves that are obtained experimentally and from the LY equation show a similar trend, and the force of the convex liquid bridge changes little with the increase of the separation distance. This trend can be explained as follows: In essence, the capillary force consists of the surface tension and the Laplace force [45]. For a concave liquid bridge, both the surface tension and the Laplace force are attractive; thus, the capillary force shows a relatively large value. However, in a convex liquid bridge, the Laplace force is repulsive and the surface tension remains attractive, which cancel each other out, thereby leading to a relatively small capillary force. Fig. 6(b) 11

Journal Pre-proof also shows that the difference between the predictions by the LY equation and the measurements is relatively small (approximately 9.19%). The transition of the bridge from convex and concave was not included in the MKH correlations. Although the LY equation can predict the capillary forces of convex and concave bridges, the transition criterion between convex and concave remains unclear, thereby highlighting the necessity of developing a new capillary force model.

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Fig. 7 plots the measured capillary forces between a particle (rp=0.3 mm) and a wall at various liquid volumes. It presents similar trends to those that were observed for two particles.

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Overall, the particle-wall capillary force is much larger than the capillary force between two

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particles. In the case of a small liquid volume (e.g., V * =0.41) or a concave liquid bridge (Fig.

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8(a)), both the MKH model and the LY equation again accurately predict the measurements.

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As the liquid volume increases to 4.4 (Fig. 8(b)), the MKH model still cannot predict the transition phenomenon between convex and concave, which shows only a decreasing trend

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with the increasing separation distance. With the parameter values that were obtained

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experimentally, the LY equation predicts the transition of the liquid bridge from convex to concave with increasing separation distance. Figs. 9 and 10 show how the particle size affects the capillary forces between the particles and between the particle and the wall. The observed capillary force is larger for smaller spheres in the case of a relatively small liquid volume ( V * =2.78), where a concave liquid bridge is observed (Fig. 9(a)). As the liquid volume increases to 3.56 (Fig. 9(b)), the capillary force monotonically decreases with the increase of the separation distance at rp=0.4 mm while the liquid bridge remains concave. However, when rp is 0.2 or 0.3 mm, the liquid bridge transitions from convex to concave with the increasing separation distance. The critical separation distances that correspond to the transition and the maximum force are 1.17 and 0.62 for rp=0.2 and 0.3 mm, respectively. Similar trends are observed for particle-wall 12

Journal Pre-proof capillary forces in Fig. 10, where the critical separation distances are smaller than that those of two particles. Fig. 11 plots the variations of the rupture distance with the particle radius and the liquid volume. The rupture distance increases with increasing liquid volume for the particle-particle and particle-wall pairs. These curves can be fitted by power functions. 3.2 Capillary force model A capillary force model that is applicable to both convex and concave liquid bridges is

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formulated in this section. Prior to formulating the model, four factors are clarified: The first

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is the effect of gravity, which may distort the liquid bridge and lead to a difference in the

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capillary forces on the upper and lower spheres. However, liquid bridges are typically

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observed to be axisymmetric under the current experimental conditions. Additionally, based on the work of Adams et al. [67], the differences between the forces that act on the upper and

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lower particles are calculated and normalized by the measured capillary force. On average,

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these differences are smaller than 2.19 % under the current conditions. Thus, the effect of gravity is neglected in the modelling of the capillary force. The second factor is the viscosity

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effect, which dominated the liquid bridge force when the dimensionless capillary number ( Ca  v /  ) was larger than 1 in the work of Ennis et al. [68]. For water at 20°C, the surface tension γ is 0.072 N/m and the viscosity μ is 0.001 Pa·s. Additionally, the particle-to-particle relative velocity v should not exceed 0.1 m/s. This yields a 𝐶𝑎 of 0.0014, which is much smaller than 1; thus, the viscous effect is neglected. The third factor is the contact angle, which may somewhat vary with the separation distance depending on the conditions involved. For simplicity, this dynamic change of the contact angle was neglected in modelling the capillary force in the literature [4, 5, 16, 17, 26-28, 31, 35]. This simplification is also adopted in this study. Thus, the contact angle is assumed to be constant, which is the average value of the measured data. Last, for simplicity, the liquid transfer from one capillary bridge to another

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Journal Pre-proof is not considered, as in many other studies (see, e.g., [12, 16, 17, 24-26, 31, 32, 69, 70]). An important step in the current modeling process is to determine the critical separation distance at which a convex liquid bridge transitions to a concave liquid bridge for a specified liquid volume and specified particle/wall properties. For two particles, the critical separation distance ( d c, p  p ) corresponds to a cylindrical liquid bridge (Fig. 12(a)), the volume ( Vl ) of which equals the volume of the cylinder minus the volumes of the two spherical caps:



[3rp  (rp  rp cos  )]( rp  rp cos  )2

3

(7)

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2

of

Vl    rp sin    hp  p  2 

-p

where a is the half-filling angle and h is the height of the cylinder, which is determined by

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h p  p  d c, p  p  2(rp  rp cos  )

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(8)

Substituting the normalized separation distance and liquid volume ( dˆ  d / rp and V *  V / rp 3 )

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into Eqs. 7 and 8 and reorganizing the resulting equation yields the following expression for

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the critical dimensionless separation distance:

dˆ c , p  p 

V* 

2 [3  (1  cos  )](1  cos  ) 2 3  2(1  cos  )  sin 2 

(9)

For a particle and a wall, the liquid bridge at the critical separation distance has a frustum shape (Fig. 12(b)), with a top radius of rp sin  , a wetting radius of rw, and a height of hp-w. Radius of the droplet cap rw is estimated according to the volume of a droplet cap over the wall (Fig. 13): Vl 

 3

(3r  (r  r cos  )(r  r cos  )  2

c

c

c

c

(10a)

c

   rw  rc  sin   Vl /( (3  (1  cos  ))(1  cos  )2  3  

1/ 3

14

 sin 

(10b)

Journal Pre-proof where rc is the radius of the droplet circle and θ is the contact angle of water with the wall. Eq. 10b is valid only for a liquid wetting angle of less than 90º. The water that is considered in this work satisfies this condition. The volume of the liquid bridge equals the volume of the frustrum minus the volume of the spherical cap: Vl 

 3

(rw  rp sin    rw  rp sin  )  hp  w  2

2



[3rp  (rp  rp cos  )]( rp  rp cos  )2

3

h p  w  d c, p  w  (rp  rp cos  )

(11)

of

(12)

Combining Eqs. 10b, 11 and 12 yields the following expression for the critical separation

3



( rˆw  sin   rˆw sin  ) 2

2

 (1  cos  )

(13)

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3

(3  (1  cos  ))(1  cos  ) 2

-p



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dˆ c,p-w 

V* 

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distance between the particle and the wall:

The half-filling angle a is a variable in Eqs. 9 and 13. Its value is difficult to determine

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theoretically. Fig. 14 plots the measured values of the half-filling angle at various separation distances for rp=0.4 mm, where the unit of the angle is radians. According to the figure, the

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half-filling angle decreases as the separation distance increases or the liquid volume decreases. Fitting these experimental data yields the following two equations for correlating the half-filling angle with the separation distance and the liquid volume. Here, the relative prediction errors are less than 20%, as presented in Fig. 15. Between two particles:

  0.186dˆ  0.093V *  0.97

(14)

Between a particle and a wall:

  0.687dˆ  0.125V *  0.895

(15)

Eqs. 9, 13, 14 and 15 are used together to determine the value of the critical separation

15

Journal Pre-proof distance. Due to the implicit relationship among the variables, this determination requires the following iterative procedure, which is outlined in Fig. 16, for facilitating its implementation: (1) The critical separation distance dˆc is initialized to zero; (2) Eq. 14 or 15 is used to determine the half-filling angle a for specified values of dˆc and liquid volume V * ;

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(3) Eq. 9 or 13 is used to obtain a new value of the critical separation distance dˆc ,new

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according to the values of a and V * ;

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(4) dˆc is updated to dˆc ,new and steps 2 and 3 are repeated until the difference between dˆc

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and dˆc ,new is less than a pre-specified error, which is set to 3% in this study;

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(5) 𝑑̂𝑐 is compared with the separation distance dˆ of the two considered objects. The liquid bridge is concave if dˆ  dˆc ; otherwise, it is convex.

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Fig. 17 compares the experimental critical separation distances with the predictions that were

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obtained via the iterative method at various liquid volumes and spherical radii. Reasonable agreement is realized. Overall, the average prediction accuracy is mainly within an error margin of14.6%.

With the determination of the critical separation distance, the capillary force can be separately modeled under two conditions: For a convex liquid bridge, the values of the capillary forces (Fig. 5 and 7) are close to that at contact and do not change substantially as the separation distance increases. Based on the energy approach, Lambert et al. [26] showed that the capillary force for two contacting particles can be determined by F  4rp cos  . Hence, the capillary force of a convex liquid bridge is simply expressed as a function of the contact angle. For a concave liquid bridge, the form of the MKH model is utilized. Via regression analysis,

16

Journal Pre-proof the values of parameters A, B and C as well as rupture distance 𝑑̂𝑟 , as functions of liquid volume and/or contact angle, are determined from the experimental data. The final forms of the current capillary force model are specified by Eq. 16, and the subsequent results are listed in Table 2.

(16)

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0.629(1  2 sin   1.3 cos  ) dˆ  dˆc, p  p (Convex bridge : P - P)  Fˆ   4 cos  dˆ  dˆc, p  w (Convex bridge : P - W ) exp( Ahˆ  B)  C dˆ  dˆc (Concave bridge) 

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Fig. 18 compares the experimental capillary forces with the forces that were calculated using the MKH model and the proposed model at rp=0.3 mm. When the liquid volume is 0.89 or the

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liquid bridge is concave (Fig. 18(a)), the average prediction errors are 22.3% and 7.12% for

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the MKH model and the proposed model, respectively. As the liquid volume changes to 3.56

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(Fig. 18(b)), the liquid bridge can transition from convex to concave at the critical distance of 0.62. The proposed model can accurately capture the drastic variation of the force that is

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related to the transition of the liquid profile, with prediction errors of less than 10%. The

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MHK model cannot realize this and yields prediction errors of up to 49.5%. Similar results can be obtained for the capillary force between a particle and a wall, as presented in Fig. 18(d). The prediction errors are 13.54% for the proposed model and 35.98% for the MKH model. Fig. 19 compares the capillary forces that are predicted by the proposed model with all the experimental data. The predictions of the model accord reasonably with the experimental predictions. The relative prediction errors are mainly within 20% for the capillary forces between two particles and within 30% for those between a particle and a wall. The proposed model is also compared with the MKH model over a board range of liquid *

volumes ( V =0.001~5.5). According to Fig. 20, the mean deviation between the two models *

are very small for a small liquid volume ( V < 0.5). Otherwise, the deviation increases up to 17

Journal Pre-proof 40%. Fig. 21 compares the proposed model and the MKH model in predicting the capillary forces between two particles as a function of the separation distance at various liquid volumes. When the liquid volume is larger than 1, its increase corresponds to an increased deviation between the MKH model and the proposed model, especially at relatively large separation distances. When the liquid volume increases to a critical value, the transition from convex to concave is predicted by the proposed model but not by the MKH model.

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3.3. Model applications

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3.3.1 Measurement of the sandpile repose angle and the hopper flow discharge rate

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Two wet granular systems with a water content that ranges from 0% to 14% by mass fraction

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have been considered to examine the applicability of the capillary force model that was

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developed in this paper. One is a sandpile in which wet particles can form larger repose angles than dry particles. Fig. 22 illustrates the experimental setup for measuring the repose

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angle. For each measurement, water was thoroughly mixed with quartz sands (density=2842 kg/m3 and average radius=0.4 mm), which were subsequently loaded into a cylindrical

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column. Then, this column was lifted by the handwheel. Sand spread out on the steel plane under gravity and formed a pile with a repose angle φ. After the particles settled, φ was measured at four locations that were evenly distributed in the circumferential direction, and the average value of these measurements was calculated. The other wet granular system is a hopper flow, in which the water content has a substantial influence on the flowability of particles and the final angle of the heap [71]. Fig. 23 presents a schematic illustration of the experimental platform. A rectangular bin with its base walls angled at 60° relative to the horizonal direction was considered. The bin was constructed from plexiglass. A load cell was positioned beneath the exit to record the mass of sands during measurements. 18

Journal Pre-proof 3.3.2 Numerical model and simulation conditions One of major applications for capillary force models is DEM (discrete element method) simulations of wet particle systems (see, e.g., [72, 73]). Thus, DEM simulations are used to evaluate the applicability of the new capillary force model to granular flows. The sandpile and the hopper flow are simulated. In the DEM approach, the translational and rotational motions of a particle, as illustrated in Fig. 24, can be respectively described by the following equations

𝑑𝒗𝑖 𝑑𝑡

𝑘 +𝑘

𝑤 𝑖 = ∑𝑗=1 (𝑭𝑐,𝑖𝑗 + 𝑭𝑑,𝑖𝑗 + 𝑭𝑙,𝑖𝑗 ) + 𝑚𝑖 𝒈

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𝑚𝑖

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[74, 75]:

(18)

re

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dωi ki  kw Ii    Tt ,ij  Tr ,ij   Tls ,i dt j 1

(17)

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where mi, Ii, vi and ωi are the mass, the moment of inertia, and the translational and angular velocities, respectively, of particle i; Fc,ij, Fd,ij and Fl,ij are the elastic, viscous contact damping

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and liquid bridge forces, respectively, between particle i and particle/wall j; mig is the gravitational force; and Tij is the torque that acts on particle i due to particle/wall j. For a

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particle that is undergoing multiple interactions, the forces and torques are summed over the ki particles and the kw walls that are in contact with particle i. The equations for calculating the particle-particle and particle-wall contact forces and torques are based on the commonly used nonlinear contact model [75], as presented in Table 3. The liquid bridge model will be detailed later. Table 4 lists the current DEM simulation parameters. They are selected according to the experimental conditions of the repose angle and the hopper flow so that the simulation and experimental results can be compared meaningfully. Spherical particles are simulated instead of the irregular particles that are considered in the experiments. To consider the effect of the particle shape, a calibration step is conducted to adjust the properties of the numerical 19

Journal Pre-proof particles, which include the coefficients of sliding, rolling and restitution, to match the experimental and numerical results, as is widely done in the literature [76, 77]. The restitution and sliding friction coefficients between the particle and the wall were measured directly via sliding friction and dropping tests, as described by Chung [78]. Other parameters, such as rolling friction coefficient between particles and between particle and wall, and restitution and sliding friction coefficients between particles, were calibrated based on the response surface method to fit the repose angle of the dry sandpile. In DEM, the liquid is assumed to be

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distributed on every particle evenly and the liquid bridge is realized via the virtual radius. A

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physical time of 2 s is simulated for each simulation, and numerical results are recorded every

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0.01 s.

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To evaluate the applicability of the proposed model to multiple-particle systems, it is

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implemented using DEM to simulate a sandpile and a hopper flow at various water contents, where the model parameters are based on those determined from the calibration via the dry

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system. The numerical results are compared with the measured data in terms of the repose angle and the discharge rate. Here, the water content is specified as a mass fraction. The

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corresponding dimensionless liquid volume on a particle radius of rp= 0.4 mm is approximately 1.85 for a water content of M=12%. This value is smaller than the upper limit of the liquid content of the pendular liquid bridge that is considered here, which is 20% for a monodisperse system of spherical particles [72]. 3.3.3 Sandpile Fig. 25 compares the experimental and numerical repose angles at various water contents. The experimental results demonstrate that the repose angle exponentially increases with the water content. This is accurately predicted by the proposed model, which yields prediction errors of 3%. However, the result that is predicted by the MKH model shows a nearly linear relation

20

Journal Pre-proof between the repose angle and the water content. Its prediction errors are less than 5% when the water content is below 10%. As the water content increases to 12%, the errors gradually increase to 7.6%. Assuming all particles in the sandpile are in contact with each other (with a separation distance of dˆ =0), the dimensionless critical distance between two particles is approximately 0.12 according to Eq. 9 for a water content of 9.36%. Thus, 𝑑̂ < 𝑑̂𝑐 , where 𝑑̂ is zero for a packed bed. This suggests that liquid bridges are convex when the water content

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is 12%. Therefore, the presence of a convex liquid bridge leads to the exponent variation trend

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that is observed experimentally.

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3.3.4 Hopper discharge

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Fig. 26 compares the predicted variations of the particle mass discharging rates of the hopper against the measurements at various water contents. For cohesionless or dry particles (M=0),

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the charging process is smooth and continuous. The flow rate is 6.35 kg/s on average, and no

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particles reside inside the hopper at the final discharging stage. For a water content of M=1% (Fig. 26(a)), wet particles flow out of the hopper as agglomerated clumps. Because the liquid

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capillary forces are relatively large in a packed state and overcome the gravity, the particle flow slows substantially. Thus, particles don’t flow out of the hopper during the period of 0~0.3 s. As particle keep moving downwards, the separation distance between particles increases, the liquid bridge force decreases, and the effect of gravity eventually plays a dominant role. The particles in the hopper begin to flow out from the exit. A substantial number of particles (approximately 1.16kg) are agglomerated and remain inside the hopper as heaps. The flow rate at the water content of M=1% decreases slightly to 5.51 kg/s. Fig. 26(b) compares the particle discharge mass rates at the water contents of M=0 and M=5%. The particle flow rate at the water content of M=5% is approximately 5.49 kg/s. The predicted flow rates by the MKH model and the proposed model are approximately the same in the

21

Journal Pre-proof cases of M=1% and 5%. When water content increases to M=10% and M=13%, the results of the MKH model and the proposed model differ significantly (Fig. 26(c) and (d)). The particles that are predicted by the MKH model show poor flowability and even form an arch at the lower part in the time range of 0.6~0.93 s. In the case of a water content of M=13%, particles are fully trapped in the hopper. In contrast, according to the results of the proposed model and the experiments,

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particles can flow out of the hopper under water contents of M=10% and 13%. The discharge rates are 5.37 kg/s and 5.28 kg/s at M=10% and 13%, respectively. The prediction errors are

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less than 10%.

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4. Conclusions

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The evolution of a liquid bridge from convex to concave is studied by combining experimental and numerical methods. A new capillary force model is developed and validated

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considering this evolution. The results from this work can be summarized as follows: (1) The effects of the separation distance, the liquid volume and the particle radius on liquid

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bridges between two spheres and between a sphere and a wall have been experimentally studied over a wide range. For a convex liquid bridge, the variation of the separation distance does not substantially affect the capillary force, the value of which is relatively small. As the separation distance increases past a critical value until bridge rupture, the liquid bridge changes from convex into concave, while the capillary force initially does not change much, but increases to the maximum value at the critical distance and, subsequently, decreases gradually. The capillary forces and the rupture distances of smaller particles are larger in magnitude. (2) Based on geometrical characteristics, in conjunction with experimental data, a framework is proposed for determining the critical separation distance, which predicts the profile 22

Journal Pre-proof evolution of a liquid bridge from convex to concave. Additionally, a capillary force model that is applicable to both convex and concave liquid bridges is formulated. It is explicitly expressed as a function of the liquid volume and the separation distance to facilitate its application. (3) The new capillary force model is implemented in a DEM model and used to simulate two multiple-particle systems: a wet sandpile and a hopper flow. The numerical and experimental

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results accord in terms of the sandpile repose angles and the hopper discharge flow rates at various water contents. In the case of higher water contents, the capillary force is

-p

ro

overestimated by a model that ignores the profile evolution of the liquid bridge.

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Acknowledgments

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The authors are grateful to the National Natural Science Foundation of China (Grant No. 51779212 and Grant No. 51911530129), Sichuan Science and Technology Program (Grant No.

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support of this work.

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2019YJ0350), and Australian Research Council (ARC) (IH140100035) for the financial

23

Journal Pre-proof Nomenclature

separation distance (m)

dc

critical distance (m)

dr

rupture distance (m)

D

diameter (m)

F

force (N)

Fc

contact force (N)

Ip

moment of inertia (kg·m2)

m

mass of particle (kg)

M

water content (%)

nˆ

unit normal vector (m)

Q

discharging mass (kg)

rc

radius of the droplet circle (m)

rp

particle radius (m)

rw

radius of the droplet cap (m)

t

time (s)

T

torque of particle (Nm)

v

translational particle velocity (m/s)

V

liquid volume (m3)

Vc

critical liquid volume (m3)

Y Greek letters

Young's modulus (Pa)



Half-filling angle (°)



surface tension (N/m)

t

vector of the accumulated tangential displacement between particles i and j.

θ

contact angle (°)



damping coefficient



friction coefficient

r

relative velocity (m/s)



repose angle (°)

  ω

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na

lP

re

-p

ro

of

d

angular velocity (s-1) unit vector, which is defined by

 ω ω/ |ω| 24

Journal Pre-proof Subscripts contact

d

damping

i

particle i

ij

between particle i and particle/wall j

l

liquid

n

normal component

P-P

particle and particle

P-W

particle and wall

r

rolling friction

s

sliding friction

t

tangential

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na

lP

re

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ro

of

c

25

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cohesive granular material, Powder Technology, 116 (2001) 214-223. [72] R.Y. Yang, R.P. Zou, A.B. Yu, Numerical study of the packing of wet coarse uniform

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spheres, AIChE Journal, 49 (2003) 1656-1666. [73] Y. He, W. Peng, T. Tang, S. Yan, Y. Zhao, DEM numerical simulation of wet cohesive particles in a spout fluid bed, Advanced Powder Technology, 27 (2016) 93-104. [74] S. Kuang, Z. Li, A. Yu, Review on Modeling and Simulation of Blast Furnace, steel research international, 89 (2018) 1700071. [75] S. Kuang, M. Zhou, A. Yu, CFD-DEM modelling and simulation of pneumatic conveying: A review, Powder Technology, (2019). [76] H.M. Beakawi Al-Hashemi, O.S. Baghabra Al-Amoudi, A review on the angle of repose of granular materials, Powder Technology, 330 (2018) 397-417. [77] Z. Yan, S.K. Wilkinson, E.H. Stitt, M. Marigo, Discrete element modelling (DEM) input parameters: understanding their impact on model predictions using statistical analysis, Computational Particle Mechanics, 2 (2015) 283-299. [78] Y.-C. Chung, Discrete element modelling and experimental validation of a granular solid 31

Journal Pre-proof

Jo ur

na

lP

re

-p

ro

of

subject to different loading conditions, 2006.

32

Journal Pre-proof Table 1 Properties of the materials that are used in the current experiments (Relative humidity: 35~45%). Item

Material

Density (kg/m3)

Contact Angle (°)

Surface Tension (N/m)

Particle

Glass bead

2484

27.8

——

Wall

304 Stainless steel

7756

36.5

——

Liquid

Deionized water

996

——

0.072

Paramete r

of

Table 2 Model parameters of the current capillary force model. Particle-Wall

A  (0.51  0.073 ln V * ) V *

0.569

ro

Particle-Particle

A  1.43V *

0.556

re

-p

* * 2 Liquid B  (1.05 ln V *  1.64)(  3.2 ln V *  17.7)2 B  (1.23 ln V  0.51)(  3.6 ln V  19) 0.32 bridge  0.48  1.2  0.1V * force

C  0.0042 ln V *  0.0078

lP

Rupture distance

C  0.013 ln V *  0.18 0.205 dˆr  (0.22  0.762)V *

na

0.33 dˆr  (0.62  0.99)V *

Table 3 Components of forces and torques that are acting on particle i.

Normal forces

Tangential forces

Symbol

Equation

contact

Fcn,ij

4  Y * r *  n3 / 2 nˆ 3

damping

Fdn,ij

  n (6mijY * r * n )1 / 2 v n,ij

contact

Fct,ij

damping

Fdt,ij

Jo ur

Forces and torques







  t (6 s mij Fcn,ij 1   t ,ij /  t ,ij ,max  t ,ij ,max )1/ 2 vt ,ij ( t ,ij   t ,ij ,max ) dˆ  dˆc, p p ( P  P)

1.3(1  2 sin   1.3cos  ) Liquid bridge force

Fl,ij



 s Fcn,ij 1  (1  min  t ,ij ,  t ,ij ,max  t ,ij ,,max )3 / 2 δˆ t

dˆ  dˆc, pw ( P  W )

4 cos 

exp( Ahˆ  B)  C

dˆ  dˆ c

Torque by tangential forces

Tt,ij

rij  ( Fct ,ij  Fdt,ij )

Rolling friction torque

Tr,ij

 r ,ij Di Fn,ij ωt ,ij



33

Journal Pre-proof

where

2v ri  δt ωt ,ij ˆ 1 1 1 Y * ˆ    n , n  δ  , , , , ,   ω  Y  t , ij , max s t ri t ,ij ωt ,ij δt 2(1  v) r * ri rj 2(1  v 2 ) ˆ )nˆ , and v t ,ij  vij  v n,ij vij  v j  vi ω j  rj  ωi  ri , vn,ij  ( vij  n

Table 4 Simulation parameters in the current DEM simulations. Value

Parameter

Value

Particle diameter / mm

0.8

Time step for simulation / s

10-5

Particle density / kg/m3

2842

Coefficient of restitution (P-P)

0.75

Poisson’s ratio of particle

0.3

Coefficient of sliding friction (P-P)

0.34

Shear modulus of particle / Pa

106

Coefficient of rolling friction (P-P)

0.001

Steel density / kg/m3

7850

Coefficient of restitution (P-W)

0.8

Poisson’s ratio of steel

0.3

Coefficient of sliding friction (P-W)

0.4

Shear modulus of steel / Pa

7×107

Coefficient of rolling friction (P-W)

Jo ur

na

lP

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of

Parameter

34

0.005

Journal Pre-proof

(a)

(b)

of

Fig. 1 Sketches of the (a) concave and (b) convex liquid bridge geometries (particles in gray).

ro

Glass bead

Jo ur

na

lP

re

-p

Microsyringe

Fig. 2 Experimental liquid bridge apparatus.

Capillary force, F/(rp) (-)

1.75

average data

1.50 1.25 1.00

0.75 0.0

0.2 0.4 0.6 0.8 Separation distance, d/rp (-)

1.0

Fig. 3 Reproducibility test of the liquid bridge force that is measured under the experimental setup.

35

of

Journal Pre-proof

ro

Fig. 4 Geometric features of a pendular liquid bridge between two spheres. The parameters in

re

2.5

lP

Capillary force, F/rp(-)

2.50

2.0

na

1.5

*

V =0.89 * V =1.44 * V =2.07 * V =2.78

1.0 0.5 0.0

Jo ur

Capillary force, F/(rp) (-)

-p

the LY equation are specified.

0.6 1.2 1.8 Separation distance, d/rp (-)

2.25 2.00 *

1.75 1.50 0.0

2.4

(a)

V =3.56 * V =4.40 * V =5.40 0.6 1.2 1.8 2.4 Separation distance, d/rp (-)

3.0

(b)

Fig. 5 Dimensionless particle-particle capillary force as a function of the separation distance when the liquid volume varies from (a) V*=0.89 to 2.78 and (b) V*=3.56 to 5.40.

36

Journal Pre-proof 3.5

Capillary force, F/(rp) (-)

Capillary force, F/(rp) (-)

2.0 1.5 1.0 Laplace-Young Measured data MKH model

0.5 0.0 0.0

0.2 0.4 0.6 0.8 Separation distance, d/rp (-)

1.0

3.0 2.5 2.0 1.5 1.0 0.5

Laplace-Young Measured data MKH model 0.5 1.0 1.5 2.0 Separation distance, d/rp (-)

0.0 -0.5 0.0

of

(a)

2.5

(b)

ro

Fig. 6 Comparison of the particle-particle capillary forces that are predicted by the MKH

-p

model and the Laplace-Young equation with the measurements at liquid volumes of (a)

*

na

V =0.41 * V =0.89 * V =1.44

3 2 1 0 0.0

Jo ur

Capillary force, F/(grpp) (-)

4

Capillary force, F/(grpp) (-)

lP

re

V*=0.89 and (b) V*=3.56.

0.2 0.4 0.6 0.8 Separation distance, d/rp (-)

5 4 3 2 1 0 -1 0.0

1.0

(a)

*

V =2.07 * V =2.78 * V =3.55 * V =4.40 * V =5.40 0.3 0.6 0.9 1.2 Separation distance, d/rp (-)

1.5

(b)

Fig. 7 Dimensionless particle-wall capillary force as a function of the separation distance when liquid volume varies from (a) V*=0.41 to 1.44 and (b) V*=2.07 to 5.40.

37

Journal Pre-proof

3

7

Laplace-Young Measured data MKH model

Capillary force, F/(rp) (-)

2 1 0 0.0

0.1 0.2 0.3 0.4 Separation distance, d/rp (-)

0.5

Laplace-Young Measured data MKH model

6 5 4 3 2 1 0 -1 -2 0.0

0.3 0.6 0.9 1.2 Separation distance, d/rp (-)

of

Capillary force, F/(rp) (-)

4

(b)

ro

(a)

1.5

Fig. 8 Comparison of the particle-wall capillary forces that are predicted by the MKH model

na

2.0

1.5

Jo ur

Capillary force, F/(rp) (-)

2.5

rp=0.2mm rp=0.3mm

1.0 0.0

Capillary force, F/(rp) (-)

lP

re

-p

with the measurements at the liquid volumes of (a) V*=0.41 and (b) V*=4.40.

rp=0.4mm

0.6 1.2 1.8 2.4 Separation distance, d/rp (-)

2.5

2.0

1.5

rp=0.3mm 1.0 0.0

3.0

rp=0.2mm rp=0.4mm 0.6

1.2

1.8

2.4

3.0

Separation distance, d/rp (-)

(a)

(b)

Fig. 9 Dimensionless particle-particle capillary force as a function of the separation distance at the liquid volumes of (a) V*=2.78 and (b) V*=3.56 μL.

38

Journal Pre-proof 5 rp=0.2mm

Capillary force,F/(rp) (-)

Capillary force, F/(grpp) (-)

5 rp=0.3mm

4

rp=0.4mm 3 2 1 0 0.0

0.2 0.4 0.6 0.8 Separation distance, d/rp (-)

rp=0.2mm rp=0.3mm

4

rp=0.4mm 3 2 1 0 0.0

1.0

of

(a)

0.3 0.6 0.9 1.2 Separation distance, d/rp (-)

1.5

(b)

ro

Fig. 10 Dimensionless particle-wall capillary force as a function of the separation distance at

re

-p

the liquid volumes of (a) V*=1.44 and (b) V*=2.78.

lP na

2.4

1.8

1.2 0

2

Jo ur

Rupture distance, dr/rp (-)

Rupture distance,dr/rp (-)

1.8

3.0

4 6 8 10 Liquid volume, V*

1.5 1.2 0.9 0.6 0

12

(a)

5

10 15 Liquid volume,V*

20

(b)

Fig. 11 Effects of the liquid volume and the particle radius on the rupture distance: (a) between two particles and (b) between a particle and a wall.

39

Journal Pre-proof

lP

re

-p

ro

of

(a)

(b)

na

Fig. 12 Geometrical representations of liquid bridges at the critical separation distance: (a)

Jo ur

between two particles and (b) between a particle and a wall.

Fig. 13 Contact between water and a wall.

40

Journal Pre-proof *

1.2

1.6

*

V =0.38 * V =0.88 * V =1.50 * V =2.27 * V =3.40

V =0.61 * V =1.17 * V =1.85 * V =2.78

Half filling angle, a (-)

Half filling angle, a (-)

1.4

1.0 0.8

1.2

*

V =0.38 * V =0.88 * V =1.50 * V =2.27 * V =3.40

0.8

0.4

0.6 0.0

*

V =0.17 * V =0.61 * V =1.17 * V =1.85 * V =2.78

0.5 1.0 1.5 2.0 Separation distance, d/rp (-)

0.0

2.5

1.5

(b)

of

(a)

0.3 0.6 0.9 1.2 Separation distance, d/rp (-)

ro

Fig. 14 Half-filling angle as a function of the separation distance and the sphere radius for (a)

na

60 50 40 30 30

40 50 60 70 Measured half filling angle, a ()

80

80

=X

90 

Y

70

Predicted half filling angle, a ()

lP

Y =X

20%

re

80

Jo ur

Predicted half filling angle, a ()

-p

two particles and (b) a particle and a wall.

70 60 50 40 30 20 10 10

20 30 40 50 60 70 80 Measured half filling angle, a ()

(a)

90

(b)

Fig. 15 Comparison between the predicted and measured half-filling angles (a) between two particles and between a particle and a wall.

41

Journal Pre-proof Initial critical distance dc Calculate half filling angle α according Eqs.14 or 15

Update dt by dc=dc+0.001

Calculate critical distance dc,new according to Eqs. 9 or 13

Check if dc=dc,new with an expected error No

of

Yes

No

-p

Yes

ro

Confirm shape of liqud bridge by checking if d>dc

Convex liquid bridge

re

Concave liquid bridge

0.2mm 0.3mm 0.4mm

1.5 1.0 0.5 0.0 0.00

0.05

Present model

Measured data

0.10 0.15 0.20 0.25 * 3 Liquid volume, V *rR

Critical separation distance, dc (-)

2.0

rp

1.2

rp

Present model

Measured data

0.2mm 0.3mm 0.8 0.4mm

Jo ur

Critical separation distance, dc (-)

2.5

na

lP

Fig. 16 Iterative procedure for determining the critical separation distance.

0.4

0.30

(a)

0.0 0.00

0.05

0.10 0.15 0.20 0.25 * 3 Liquid volume, V *rR

0.30

(b)

Fig. 17 Comparison of the experimental critical separation distance with the predictions by Eqs. 9 and 13: (a) between two particles and (b) between a particle and a wall.

42

Journal Pre-proof 4 Measured data MKH model Present model

1.75 1.50 1.25 1.00 0.75 0.50

Measured data MKH model Present model

3 2 1 0

-1

0.2 0.4 0.6 0.8 Separation distance, d/rp (-)

-2

1.0

0.0

(a)

2.5

(b)

ro

Measured data MKH model Present model

4

re

2

lP

1 0 0.0

6

-p

Capillary force, F/(grpp) (-)

3

8 Measured data MKH model Present model

0.1 0.2 0.3 0.4 Separation distance, d/rp (-)

Jo ur

(c)

0.5

2 0

-2 0.0

na

Capillary force, F/(rp) (-)

4

0.5 1.0 1.5 2.0 Separation distance, d/rp (-)

of

0.25 0.00 0.0

Capillary force, F/(rp) (-)

Capillary force, F/(rp) (-)

2.00

0.3 0.6 0.9 1.2 Separation distance, d/rp (-)

1.5

(d)

Fig. 18 Comparison of the measured and the calculated capillary forces at the liquid volumes of (a) V*=0.89 and (b) V*=3.56 between particles and of (c) V*=0.41 and (d) V*=4.40 between a particle and a wall.

43

Journal Pre-proof

2

1

0

0

1 2 Measured Capillary force, F/(rp) (-)

3

6 ±30%

=X

±20%

5

Y

Predicted capillary force, F/(rp) (-)

Y =X

Predicted capillary force, F/(rp) (-)

3

4 3 2 1 0

1 2 3 4 5 Measured Capillary force, F/(rp) (-)

(b)

of

(a)

0

ro

Fig. 19 Comparison between the predicted and measured capillary forces (a) between two

re

-p

particles and between a particle and a wall.

lP

50

20

na

30

Jo ur

Deviaiton, E (%)

40

Particle-Particle Particle-Wall

10

0

1

2

3

4

5

6

*

Liquid volume, V (-) Fig. 20 Deviation between the MKH model and the proposed model in the range of V*=0.001~5.5

44

6

Journal Pre-proof

2.0 *

1.5

V 0.01 0.1 1 3

1.0

MKH

Eq.16

0.0 0.0

of

0.5

0.4 0.8 1.2 1.6 Separation distance, d/rp (-)

ro

Capillary force, F/(rp) (-)

2.5

2.0

-p

Fig. 21 Comparison of the MKH model and the proposed model in terms of the capillary

Jo ur

na

lP

re

force between two particles at various water contents.

Fig. 22 Experimental setup for measuring the repose angle.

45

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Jo ur

na

lP

re

-p

ro

Fig. 23 Experimental setup for measuring the hopper flow discharge rate.

Fig. 24 Forces that act on particle i from contacting particle j and pendular liquid bridges.

46

Journal Pre-proof

Repose angle,  (°)

45 40 35 30

rp=0.4mm

25

MKH model Present model

20

0

3 6 9 12 Water content, M (%)

15

Jo ur

na

lP

re

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Fig. 25 Repose angles of particle piles at various water contents.

47

Journal Pre-proof

9

12 Water content Measured data No liquid MKH model Present model

M=0

M=1%

Discharging mass, Q (kg)

Discharging mass, Q (kg)

12

6 3 0 0.0

0.3

0.6

0.9

1.2

1.5

9 6

Water content Measured data No liquid MKH model Present model

0 0.0

1.8

0.3

0.6

1.2

1.5

1.8

of

(b)

ro

4

Measured data Present model MKH model

re

-p

4

Discharging mass, Q (kg)

6

Measured data Present model MKH model

0.3

0.6

0.9

1.2

lP

2

1.5

1.8

2

0 0.0

0.3

0.6

na

Discharging mass, Q (kg)

0.9 Time, t (s)

(a)

0 0.0

M=5%

3

Time, t (s)

6

M=0

Time, t (s)

1.2

1.5

1.8

Time, t (s)

(d)

Jo ur

(c)

0.9

Fig. 26 Measured and predicted variations of particle discharge mass rates with time at the water contents of (a) M=0% and 1%, (b) M=0% and 5%, and (c) M=10% and (d) M=13%, with the insets showing particle heaps.

48

Journal Pre-proof Declaration of interests

Jo ur

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☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

49

Journal Pre-proof Capillary forces are measured via particle-particle and particle-plane pairs.



Liquid bridge transits from convex to concave with increasing separation distance.



A framework is proposed to predict the evolution of liquid bridge profile.



Capillary force model applicable to convex and concave liquid bridges is developed.



The applicability of the model to multiple particle systems is confirmed.

Jo ur

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

50

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

Figure 25

Figure 26