Analysis of liquid bridge between spherical particles

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China Particuology 5 (2007) 420–424

Analysis of liquid bridge between spherical particles Fusheng Mu, Xubin Su ∗ College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China Received 9 January 2007; accepted 29 April 2007

Abstract A pair of central moving spherical particles connected by a pendular liquid bridge with interstitial Newtonian fluid is often encountered in particulate coalescence process. In this paper, by assuming perfect-wet condition, the effects of liquid volume and separation distance on static liquid bridge are analyzed, and the relation between rupture energy and liquid bridge volume is also studied. These points would be of significance in industrial processes related to adhesive particles. © 2007 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. Keywords: Surface-wet; Liquid bridge force; Rupture energy; Viscous force

1. Introduction

2. Analysis of state of wet particles

The investigation of wet particles has great significance in many fields, such as spray process (Wang, Yu, & Zhou, 2003). There are several adhesion mechanisms between surface-wet particles, such as electrostatic force, van der Waals force and liquid bridge force, the last being usually several orders larger than the others (Simons, 1996), that is, the liquid bridge force may well dominante all others (Johansen & Schæfer, 2001). The interstitial fluid between particles can be Newtonian or nonNewtonian. Based on the Reynolds lubrication theory, Lian and Huang et al. (Huang, Li, & Xu, 2004; Huang, Xu, & Lian, 2002; Lian, Adams, & Thornton, 1996; Lian, Thornton, & Adams, 1993; Xu, Huang, & Li, 2002; Xu, Huang, & Xu, 2005) studied the normal moving or tangential slipping between wet particles, and obtained expressions of the respective forces. Rossetti, Pepin, and Simons (2003) and Pitois, Moucheront, and Chateau (2001) explained the adhesion mechanism using rupture energy. In this paper, the acting mechanisms of liquid bridge and rupture energy are obtained by analysing the liquid bridge force, the viscous force and the rupture energy of a pair of central moving spherical particles connected by a pendular liquid bridge with interstitial Newtonian fluid.

2.1. Liquid bridge geometry

∗

Corresponding author. E-mail address: [email protected] (X. Su).

Fig. 1 shows two particles connected by a pendular liquid bridge which is supposed to be of a toroidal shape having a constant mean surface curvature for sufficiently small Bond numbers. Since spherical agglomerates are usually formed from sub 250 ␮m particles and the liquid volume is quite small in many industrial processes, the bond number is very small, and the effect of gravity is negligible and the toroid assumption of the liquid bridge is valid (Rossetti & Simons, 2003). In the Young–Laplace equation,   1 1 P = γ (1) + r1 r2 where P is the capillary suction pressure, γ the liquid surface tension, r1 and r2 are the two principal radii of curvature of the liquid bridge surface, while the former is negative and the latter positive. The volume of liquid bridge is given by  b πR 2 V = [H (b) − D2 ] 2πyH(y) dy = (2) 2 0 where D is the spherical surface distance, H(b) = D + b2 /R and b is the radius of the wetted area. Relationships between liquid bridge volume and granularity, distance can thus be obtained.

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doi:10.1016/j.cpart.2007.04.006

F. Mu, X. Su / China Particuology 5 (2007) 420–424

Nomenclature b c D Drupt d Drupt * D Fcap Fvis m P r R v V V* W W*

radius of wetted area (m) rupture energy constant spherical surface distance (m) static bridge rupture distance (m) dynamic bridge rupture distance (m) dimensionless separation distance liquid bridge force (N) viscous force (N) constant capillary suction pressure (Pa) principal radii of curvature of liquid bridge surface (m) radius of particle (m) separation speed (m/s) liquid bridge volume (m3 ) dimensionless liquid bridge volume rupture energy (J) dimensionless rupture energy

Greek letters β half-filling angle (degree) γ liquid surface tension (N/m) η dynamic viscosity (pa s) θ contact angle (degree)

2.2. Forces between particles The liquid bridges may generate static liquid bridge force due to the curvature of the bridge and surface liquid tension forces, and dynamic forces due to the liquid viscosity as particles move relative to each other (Iveson, Litster, Hapgood, & Ennis, 2001). In this work, we only consider the dynamic liquid bridge forces for central moving particles and the interstitial fluid is Newtonian. 2.2.1. Liquid bridge force There are two methods to calculate the liquid bridge force: Gorge method and boundary method (Iveson et al., 2001). Since the saturation is small and a neck will be formed when the bridge

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stretches, the Gorge method can be used for which the liquid bridge force is Fcap = π Pr22 + 2πr2 γ

(3)

According to geometry, r1 = (D/2 + R(1 − cos β))/(cos (β + θ)) and r2 = R sin β − [1 − sin (β + θ)]r1 , and when R  r2  r1 and D  2r1 cos θ, a simplified expression can be written as   D Fcap ≈ 2πRγ cos θ 1 − , (4) 2r1 cos θ Substituting Eq. (2) into Eq. (4), we get   −1/2  2V , Fcap = 2πγR cos θ 1 − 1 + πD2 R

(5)

Relationships between static liquid bridge force and liquid bridge volume, granularity, distance can then be obtained. 2.2.2. Viscous force For Newtonian interstitial liquid, the Reynolds equation is   dP(y) dD d yH 3 (y) = 12ηy , (6) dy dy dt where η is the dynamic viscosity of the liquid. By integrating Eq. (6) twice, an expression can be derived for the viscous force acting on the spheres: 3 1 dD Fvis = − πηR2 , 2 D dt

(7)

Note that Eq. (7) can be used only under the condition of infinite liquid. In the case of finite volume of liquid, Mattheuson (Pitois, Moucheront, & Chateau, 2000) introduced a correction factor to Eq. (7):   D 2 1 dD 3 2 Fvis = − πηR 1 − , (8) 2 H(b) D dt Relationships between dynamic liquid bridge force due to fluid viscosity and granularity, distance can then be obtained. 2.2.3. Rupture energy It is important to determine whether particles will adhere or rebound when colliding with each other, in order to investigate the mechanism of adhesion and agglomeration. Simons and coworkers (Rossetti et al., 2003) developed a model to provide an approximate value of the rupture energy of pendular liquid bridges. They found that if the relative kinetic energy of the colliding particles was below the work required to break the liquid bridge then the particles would adhere together. A simple expression was derived in terms of dimensionless parameters as follows: W ∗ = cV ∗m ,

(9)

where c is the rupture energy constant and m is the power index and the dimensionless parameters follow: Fig. 1. A pair of central moving spherical particles with interstitial Newtonian fluid.

W∗ =

W γR2

and

V∗ =

V R3

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where W is the rupture energy and V is the liquid bridge volume. Thus, a relationship between rupture energy and liquid bridge volume can be set up by using Eq. (9) from knowledge of particle size and liquid surface tension. On most of the real surfaces, liquid spreading is achieved for an apparent advancing contact angle θ a , superior to the receding angle θ r obtained when liquid is retrieved from the wet surface (Mao, Yang, & Chen, 2005). Such wetting hysteresis happening at the three-phase contact line on the wet surface of the particle, regardless of sliping or being pinned, can affect the rupture energy. Willett, Adams, Johnson, and Seville (2003) deduced that wetting hysteresis (a) may result in an extended bridge rupture distance; (b) implies that capillary interactions become dissipative rather than conservative. Pitois et al. (2001) supposed that the contact angle θ is a constant due to the difficulty of relating it to the separation distance because of wetting hysteresis. In this case, the half-filling angle β decreases with increasing separation distance and then increases prior to rupture. Rupture energy can be obtained by integrating Eq. (5) throughout the separation distance:  D∗   1 ∗ 1/2 2V W ∗ = 2πD∗ cos θ 1 − 1 + , (10) πD∗ ∗

Fig. 2. Relationship between liquid bridge force and separation distance.

D2

where the dimensionless separation distance D* = D/R. According to the expression of rupture distance Drupt = (1 + (θ/2))V1/3 supposed by Lian (Lian et al., 1993), if D1∗ = 0 and D2∗ = (1 + (θ/2))V ∗1/3 , then,   ∗ 1/2   θ 2V ∗ ∗1/3 W = 2π cos θ , 1+ + (1 − B)V 2 π (11) where B = [(1 + 2V* )1/3 /π(1 + (θ/2))2 ]1/2 and the contact angle θ is expressed in radians. Relationships between rupture energy and liquid bridge volume, granularity can thus be obtained.

Fig. 3. Relationship between liquid bridge force and granularity when liquid bridge ruptures.

3. Acting mechanism of liquid bridge The contact angle θ = 0 is for the perfect-wet case. Fig. 2 shows that the liquid bridge force decreases with increasing separation distance when liquid bridge volume and granularity are constant, reaching the maximum as the separation distance approaches zero. Actually, r2 ≈ R and P ≈ 0 when the distance between two perfect-wet particles is sufficiently small, as can also be shown by Eq. (3). Figs. 3 and 4 show respectively that the liquid bridge force increases with increasing granularity although the rupture distance is constant when the liquid bridge volume is constant, and increases with increasing liquid bridge volume when the granularity is constant. This conclusion means the rupture energy will also increase. A capillary number Ca = vη/γ is introduced into Eq. (8) in order to build uniform parameters, where v is the separation speed and v = dD/dt. Fig. 5 shows that when the liquid bridge volume and the granularity are constant the viscous force decreases rapidly with increasing separation distance and it is

Fig. 4. Relationship between liquid bridge force and liquid bridge volume when liquid bridge ruptures.

F. Mu, X. Su / China Particuology 5 (2007) 420–424

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Fig. 5. Relationship between viscous force and separation distance.

Fig. 7. Relationship between rupture energy and granularity.

much smaller than the liquid bridge force. So, the liquid bridge force is predominant under the supposed conditions and with the assumed parameters. However, the capillary number has not been considered, which is related to the speed of particles and the liquid viscosity. The conclusion may not be correct if the speed or the viscosity increases. Simons et al. (Rossetti and Simons, 2003; Rossetti et al., 2003) elucidated that the viscous force becomes predominant when the capillary number is above the threshold of 0.0001.

wetting hysteresis. Here, the dynamic bridge rupture distance can be equal to Drupt since the liquid bridge force is predominant. Figs. 6 and 7 show that the rupture energy increases with increasing liquid bridge volume when the granularity is constant, and increases with increasing granularity when the liquid bridge volume is constant. This conclusion is coincident with the one of Fig. 3. Note that spherical agglomerates are usually formed only from sub 250 ␮m particles and when the liquid bridge volume approaches zero, particles become so dry that liquid bridge force disappears.

4. Acting mechanism of rupture energy References Note that although the relationships between rupture energy and liquid bridge force, separation distance can be obtained by Eqs. (10) and (11), the effect of viscosity has not been considered. In the presence of viscosity effects, the bridge rupture does not occur instantaneously during separation of the spheres (Willett et al., 2003), so that the dynamic bridge rupture distance d will be larger than the corresponding static distance D Drupt rupt . This conclusion is coincident with one of the implications of

Fig. 6. Relationship between rupture energy and liquid bridge volume.

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