Intrusion of a Liquid Droplet into a Powder under Gravity - Langmuir

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Intrusion of a liquid droplet into a powder under gravity Christopher M. Boyce, Ali Ozel, and Sankaran Sundaresan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02417 • Publication Date (Web): 03 Aug 2016 Downloaded from http://pubs.acs.org on August 12, 2016

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August 2, 2016

Intrusion of a liquid droplet into a powder under gravity

C. M. Boyce*, A. Ozel, S. Sundaresan

Department of Chemical and Biological Engineering, Princeton University

(*) Corresponding author. Email: [email protected]. Tel: +1-646-255-4807

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Abstract The intrusion of a droplet of liquid into a hexagonal closed packed array of spheres under gravity is investigated using analytical methods and volume-of-fluid simulations. Four regimes of ultimate fluid behavior are identified: (A) no liquid imbibition into the bed, (B) trapping of liquid high in the bed, (C) liquid descent to the bottom of the bed and (D) liquid spreading around the surface of all particles. These regimes are mapped based on the contact angle and Bond number of the system. Many aspects of the dynamics and ultimate liquid behavior are captured with a simplified model of a mass of liquid moving under gravity in a vertical capillary of undulating cross-sectional area. This simplified model is used to form momentum transport equations with four forms of non-dimensional time, which were shown to collapse simulation data with different fluid parameters in different regimes.


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1. Introduction In recent years, there has been a growing interest in understanding conditions for liquid imbibition, wicking and intrusion in powders, given important applications in soil science and engineering, plant physiology, oil extraction, civil engineering and the food industry. Washburn1 notably studied the problem of liquid rise in porous media, likening it to an array of vertical cylindrical capillaries. Since then, studies have approached this problem using analytical, numerical and experimental methods to understand critical conditions for wicking, imbibition or intrusion2–6 as well as the rate of imbibition7,8. Most studies have shown key differences with the better-understood problem of capillary rise into a cylindrical tube. In a vertical capillary, the critical contact angle for capillary rise is 90°, based on the fact that there will be no pressure difference to drive flow upwards if the gas-liquid interface is flat. Bán et al.2 first demonstrated that the critical contact angle for liquid imbibition into powders was much lower than 90°. Bán et al.2 used a geometric analysis to determine that the critical contact angle for a sheet of liquid to rise above the first layer of spheres in a hexagonal-closepacked (hcp) array of spheres was 50.71°. This finding was based on the reasoning that liquid would stop flowing once the liquid surface became flat and there was no longer a pressure difference across the gas-liquid interface. Bán et al.2 confirmed this theoretical finding with experiments, but also showed that the critical contact angle could increase significantly with defects or polydispersity in the packing of spheres. Shirtcliffe et al.3 identified the same critical angle by analyzing the change in surface energy associated with moving a flat sheet of liquid into an hcp array of spheres. This study noted that the condition should be independent of particle

size, so long as the particle diameter was much less than the capillary length ( ≪ ), where

is the surface tension and is liquid density. Raux et al.4 confirmed these findings via an experimental setup in which many spherical particles were dropped onto a flat gas-liquid

interface and a condition equivalent to wicking into a packed bed was found if the particles fully submerged into the liquid. Further, Raux et al.4 found that for conditions under which a monolayer of particles would remain suspended at the surface, if more particles were added to create a significant packed height of particles above the liquid surface, particles at the bottom would begin to submerge. This finding indicated that a pressure gradient, such as that due to gravity, could also influence imbibition conditions. 3 ACS Paragon Plus Environment

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Hilden and Trumble5 solved the Young-Laplace equation numerically to give the meniscus shape and pressure under slow imbibition of liquid in a single layer of closed packed spheres. This work noted sharp changes in pressure needed to drive liquid into the array as different types of liquid bridges formed, with the maximum pressure equaling the pressure drop needed to force liquid to imbibe into a powder. Extrand and Moon6 also analyzed this “intrusion pressure” by slowly adding liquid to a droplet at the “throat” formed between a single layer packing of three or four spheres, until a critical height of liquid was reached. At this critical height, liquid was forced through the throat and reached the floor beneath the particles. Experimental findings matched theoretical analysis, based on a force balance at the contact line, for this critical height. The hydrostatic pressure drop associated with this critical height took a similar form to that given by Lucas9 and Washburn1 for the pressure drop necessary to force liquid into a lyophobic cylindrical pore, with pressure drop proportional to surface tension and inversely proportional to pore diameter (∆ ∝ /).

To understand dynamics of capillary rise into cylindrical capillaries at early times, the

Lucas-Washburn equation1,9 is traditionally used in which the height to which the liquid has risen

is proportional to the square root of time (ℎ ∝ √). Batten7 analyzed rates of liquid rise into

capillaries and packed beds in more detail by means of transport equations. A transport equation

for the rate of liquid rise in a packed bed was determined by means of an energy balance invoking the Ergun10 equation. A numerical solution to this transport equation was found to agree well with experimental results for water rising through a packed bed of glass spheres. More recently, Shou et al.8 investigated the rate of liquid imbibition into pieces of paper with asymmetric cross-sectional areas, finding that the rate was strongly influenced by the geometry. These previous analyses provide a strong framework for understanding wicking and imbibition phenomena in packed arrays of spheres. However, to the best of our knowledge, the studies for the most part only consider flow past the first layer of spheres, finding a critical pressure or contact angle for passing this first layer, without giving attention to what will happen to the liquid after it passes the first layer. Thus, these studies divide flow behavior into two regimes, one of no intrusion and one of intrusion. Additionally, these studies, with the exception of those of Batten7 and Shou et al.8, do not account for the time scale at which wicking occurs, which could be crucial in situations where chemical reaction or evaporation is also occurring. In this paper, we investigate intrusion and wicking behavior of a small droplet of liquid into an hcp 4 ACS Paragon Plus Environment

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array of spheres under the influence of gravity, with varying liquid and interfacial properties. We use the volume-of-fluid (VoF) method11 for direct numerical simulation (DNS) of gas-liquid flow to resolve motion over time, as well as determine the final configuration of liquid. We identify four regimes of behavior with regards to ultimate liquid spreading as well as the critical non-dimensional parameters governing this phenomenon, including non-dimensional forms of time which characterize motion in different regimes. 2. Methods 2.1 Analytical To model a droplet of liquid traveling down a bed of close packed spheres in a simple analytical manner, we liken the physical system to a finite mass of liquid moving down a capillary with undulating cross-sectional area. It is worth noting that the problem of immiscible two-phase flow in an undulating capillary has been studied before with liquid coming from an infinite reservoir12–18, but here we investigate a finite mass of liquid, creating the issue of multiple fluid-fluid interfaces. Assuming constant liquid density, for nearly 1-D vertical flow, the momentum equation for the vertical component of liquid flow is:

+ = ∇ − −

(1)

Where is the liquid density, is the vertical component of the liquid velocity, is the liquid viscosity, and

is the pressure gradient, potentially arising from differences in capillary

pressure at the top and bottom of the mass of liquid. In this notation, the z-axis points upward. Fig. 1 shows a diagram of potential geometric configurations for a mass of liquid falling through a capillary of undulating area, with different configurations leading to different effects of surface tension on the flow. These configurations can be divided in two scenarios: Scenario I in which surface tension affects bulk flow and Scenario II in which it does not affect bulk flow. In Scenario I, surface tension can pull upward or downward on the liquid, depending on the difference in the curvatures of the interfaces at the top and bottom of the liquid mass. Since the liquid droplet studied here is initially a sphere of diameter equal to the particle diameter, dp, the pressure gradient can be given by

− ∝

(2)


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with a constant of proportionality of O(1). It is assumed that the capillary is open at both ends. When the curvature at a gas-liquid interface is negligible, the pressure in the liquid there is equal to atmospheric pressure. In contrast, when there is significant curvature with a contact angle below 90°, the pressure in the liquid will be less than atmospheric pressure, and thus according to the Young-Laplace equation, the pressure drop across the liquid mass is given by ! "#$%%$& cos(+ + ,#$%%$& ) − "%$ cos(+ − ,%$ ) =

(3)

where is the surface tension at the gas-liquid interface, + is the contact angle and the subscript

I refers to the fact that this equation applies only to Scenario I. It is worth noting that in many systems the contact angle can be different from the equilibrium contact angle at the advancing and receding edges of the flow19–24; here we assume a constant contact angle, equal to the equilibrium contact angle, due to very low inertia in the system, which is consistent with the Cox-Voinov equation19,21. It is important to note that the Cox-Voinov equation is not used directly in our analytical or computational models, but simply to justify the use of constant, equilibrium contact angles, rather than advancing or receding contact angles. " is a value representative of the geometry of the liquid. If the contact line of the surface with dominant curvature is assumed circular, " is given by the ratio of twice the height (h) of the mass of liquid and half the diameter (w) of the circle (" = 4ℎ//). Thus, for an example of h = dp and w = dp/2,

" = 8. , is the angle of the surrounding geometry at the contact point. The subscripts top and

bottom refer to the interfaces at the top surface and bottom surface of the liquid mass, respectively.

For configuration I (a), the curvature at the top is negligible, and thus "%$ = 0 and the

pressure gradient is positive, indicating that surface tension acts to pull liquid downwards,

assuming the contact angle is below 90°. In contrast, for configuration I (b), the curvature at the

bottom is negligible, "#$%%$& = 0 and surface tension pulls liquid upwards.

For Scenario II in Figure 1, the geometry of the interfaces at the top and bottom of the

liquid mass either have negligible curvature (II a) or mirror one another (II b), and thus the average pressure gradient provided by surface tension is zero: !! =0

(4)


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It is worth noting that Eq. 4 can also apply to any geometry, given the surface tension is negligible as compared to gravity.

Figure 1. Geometric configurations for a mass of liquid falling through undulating capillary: In Scenario I, surface tension affects bulk flow, while in Scenario II, it does not. In configuration I (a) surface tension pulls down on liquid due to the dominant curvature at the bottom, while in I (b) surface tension pulls upward due to dominant curvature at the top. In configuration II (a), curvature is negligible at both the top and bottom, while in II (b) curvature is significant at both the top and bottom, but the effects of surface tension cancel out.

Taking the pressure gradient due to surface tension into account, the momentum transport equation can be re-written as

+

= ∇ −

(5)

+ "%$ cos1+ − ,%$ 2 − "#$%%$& cos(+ + ,#$%%$& )

Given that the second, third and fourth terms on the right hand side of Eq. 5 drive flow, while viscosity slows flow down, the characteristic velocity for this flow is: 3,5 =

6 + "#$%%$& cos(+ + ,#$%%$& ) − "%$ cos1+ − ,%$ 26

(6)

Using the particle diameter as the natural characteristic length for the system, the nondimensional time is given as: t!

∗

3 6 + "#$%%$& cos(+ + ,#$%%$& ) − "%$ cos1+ − ,%$ 26 = =

(7)


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In non-dimensionalization of the transport equation, the Bond and Galilei numbers are the two key non-dimensional parameters governing flow. The Bond number is the ratio of surface tension and gravitational forces

Bo =

(8)

and the Galilei number is the ratio of gravitational and viscous forces Ga =

<

(9)

It is worth noting that Bo is often defined as the inverse of that used here; we use the expression for Bo given in Eq. 8 so that a system with Bo = 0 has no surface tension. It is important to investigate these equations in the context of the different potential configurations of the liquid mass. For Scenario II, surface tension effects are negligible as compared to gravity, which is equivalent to the Bond number being very small (Bo!! ≪ 1). In

Scenario II, the non-dimensional time does not involve surface tension or contact angle and is given by t !!

∗

=

(10)

For future reference, it is instructive to investigate the characteristic velocity and nondimensional time for the two limiting configurations of Scenario I. For Scenario I, configuration (a) in which surface tension pulls liquid downwards, the characteristic velocity is faster than cases in which gravity is acting alone, and thus the non-dimensional time is compressed as compared to Scenario II: t !> ∗ =

1 + "#$%%$& cos(+ + ,#$%%$& )2

(11)

For Scenario I, configuration (b) in which surface tension pulls liquid upwards, the characteristic velocity is slower than cases in which gravity is acting alone: 3,5#

6 − "%$ cos(+ − ,%$ )6 =

(12)

The characteristic velocity is equal to zero, indicating that the liquid becomes suspended due to surface tension forces, on the condition:


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Bo!? cos(+ − ,%$ ) = 1/"%$

(13)

When surface tension forces are smaller than gravitational forces, liquid will still descend, but the non-dimensional time is lengthened as compared to Scenario II: t !? 2.2 Simulations

∗

6 − "%$ cos(+ − ,%$ )6 =

(14)

Volume-of-fluid (VoF)11 simulations were carried out using OpenFOAM v3.0.1 opensource software. In this approach, the flow of an immiscible two-fluid mixture is assumed incompressible and unsteady, and the equations for continuity and momentum are given by ( & B& ) + @ ∙ ( & B& B& )

@ ∙ B& = 0

= @ ∙ & (@B& + @B& C ) − @(+ & D ∙ E) + F@α

(15)

(16)

Where B& , & and & are mixture velocities, densities and viscosities, given by B& = B α + B (1 − α ) & = α + (1 − α )

(17) (18)

& = α + (1 − α )

(19)

α ,IJKK = 0 → all Fluid 2 H0 < α ,IJKK < 1 → interface α ,IJKK = 1 → all Fluid 1

(20)

F = −@ ∙ Y

(21)

@∙α |@ ∙ α |

(22)

where subscripts 1 and 2 refer to Fluid 1 and Fluid 2 and α is the phase indicator function of

Fluid 1 in a cell:

Surface tension effects appear as a source term, F@α , because the continuous surface force

approach25 is used, in which F is the curvature, given by where Y is the normal vector at the interface, given by Y=


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The OpenFOAM implementation of VoF uses the surface compression method, in which the transport equation for α is given by

α + @ ∙ Bα + @ ∙ (B[ α (1 − α )) = 0

(23)

B[ = B − B

(24)

where B[ is the relative velocity between the phases, given by

Thus, the third term of Eq. 23 is only non-zero for interfacial cells, and effectively acts to limit the smearing of the interface26. These governing equations are discretized in OpenFOAM using the finite volume method with cell-centered grids. Further details of the discretization and simulation algorithm are given elsewhere26. 2.3 Simulation setup Using VoF, a spherical droplet of diameter equal to particle diameter, dp, was placed directly above a “throat” formed between three spheres in an hcp array of spheres. The vertical position of the center of the droplet was exactly 1 dp greater than the vertical positions of the centers of the particles forming the throat, leaving the droplet with a small fall before contacting the particles. Fig. 2 shows a vertical cross-section of the system at the initialization of the simulations, in which blue indicate gas, red indicates liquid and gray indicates solid particles. In a few simulations, the size of the droplet or its release height were varied, in order to investigate the effects of these parameters on the results. The array was roughly 5 dp in both lateral directions and the vertical direction, with fixed boundaries at the sides which were introduced to simply limit the simulation domain. Simulations were performed for Ga between 1 and 10 and Bo between 0.05 and 1000. It is worth noting that the square root of Bo is the ratio of capillary length to particle diameter, and thus this ratio was also varied from much less than unity to much greater than unity. The contact angle between liquid and the surface of the particles was varied between 0° and 90°. At the sides, the contact angle was kept constant throughout all simulations at 90° to keep liquid from being attracted to the walls. The fluid was at rest at the start of the simulations. The domain was discretized into 340,000 cells with finer resolution near the surfaces, allowing for a grid size of approximately 20 cells per particle diameter. The grid independence was verified by running simulations with 2,000,000 cells and observing nearly identical results with regards to liquid displacement over time. 10 ACS Paragon Plus Environment

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Figure 2. Vertical cross section of liquid distribution at the start of each simulation, in which a liquid droplet of diameter dp falls under gravity into a hexagonal closed packing of spheres of diameter dp. Red: liquid; blue: gas; gray: solid particles.

3. Results 3.1 Regimes of final liquid distribution Fig. 3 shows vertical-cross sections of ultimate liquid distribution observed characterizing four distinct regimes of droplet intrusion into the packed bed. In the figure, red indicates liquid, blue indicates gas and gray indicates solid particles. Fig. 3 (A) shows the no imbibition regime, in which liquid cannot make it past the “throat” formed between spheres in the highest layer of spheres. Thus, the droplet remains essentially spherical above the packed bed. Fig. 3 (B) shows the trapped regime, in which liquid makes it past the first throat, but gets trapped in a pocket in between spheres high in the bed. Fig. 3 (C) shows the descend regime, in which liquid descends through the entire bed to the bottom of the array, remaining mostly intact, but leaving behind small amounts of liquid at higher points in the bed. Fig. 3 (D) shows the spread regime in which liquid spreads both vertically and laterally around the bed to have a small layer of liquid covering the surface of all particles. It is worth noting that the liquid in Fig. 3 (D) has not completely reached its final state and that liquid is still spreading to more evenly cover the surface of all particles. Fig. 4 maps the different regimes of liquid intrusion based on Bo and contact angle. In Fig. 4 (a) lines are drawn to guide the eye along empirical divisions between regimes, while in Fig. 4 (b) lines are drawn based on theoretical divisions between regimes. Fig. 4 (a) shows that at contact angles below 27.5°, only the spread regime is observed. As described in the Discussion section, this division is likely due to the energetic advantage of liquids with low contact angle 11 ACS Paragon Plus Environment

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covering the surface of solids overcoming the forcing from gravity and surface tension. For contact angles above 27.5°, the descend regime is observed for Bo less than ~1/6, and the trapped regime is observed for Bo immediately to the right of this dividing line. As described in the Discussion section, this division is due to upward surface tension forces becoming large enough for larger values of Bo, such that they balance gravity and the liquid becomes suspended. The majority of Bo vs. contact angle space above Bo = 1/6 and θ = 27.5° is taken up by the trapped regime. For contact angles greater than ~80° and Bo between ~0.5 and ~500, the no imbibition regime is observed; the division between the trapped and no imbibition regimes is

well captured by + = 4(log ] (Bo) − 1.5) + 75°. As described in the Discussion section, the

transition to the no imbibition regime occurs due to repulsive forces due to high contact angle being strong enough to overcome the hydrostatic head of the liquid.

Figure 3. Vertical cross-sections of the final liquid distribution in simulations in four different regimes: (A) No imbibition of liquid into the bed, (B) liquid trapped in a pocket high in the bed, (C) liquid descends to the bottom of the bed and (D) liquid spreads around surface of all particles. Simulation parameters indicated below images. Red: liquid; blue: gas; gray: solid particles.

Fig. 4 (b) shows the exact same map as in (a), but with division lines given by analytical theories. The solid blue line shows the division between the descend and trapped regimes given

by the undulating capillary theory presented in Section 2.1, assuming values of "%$ = 8 and ,%$

= 10° in Eq. 13. The black dotted line shows the division between the no imbibition regime and

all other regimes predicted by Extrand and Moon6 based on geometric arguments for a throat formed between three spheres. Here, we assume that the height of liquid at the point of either intrusion or suspension is 1.5 dp. Both theoretical dividing lines capture the general shape of the boundary and come close to capturing the actual point of division, but miss the exact dividing


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line. Reasoning for the empirically observed dividing lines and their differences with the theoretical dividing lines will be provided in the Discussion section.

Figure 4. Regime map of final liquid distribution based on Bo and contact angle, θ. Regimes shown in Fig. 3. Fig. 4 (a) shows dividing lines to guide the eye as to the divisions between regimes; Fig. 4 (b) gives dividing lines based on theory. Table 1. Effect of the release height of liquid droplet above the spheres on ultimate regime, for values of Bo and contact angle near changes in regime. Bo

θ (°)

1 1 0.2 0.1 1 1

85 80 80 80 30 25

Regime (Release Height: 0 d p) No imbibition Trapped Trapped Descend Trapped Spread

Regime (Release Height: 1 d p) No imbibition Trapped Trapped Descend Trapped Spread

Table 1 shows the effect of the height of liquid release above the particle array on ultimate regime behavior. The release height is defined as the vertical distance between the highest point of the particle array and the lowest point of the liquid droplet. The results shown for the regime map and the remainder of this paper have a release height of 0 dp, and are


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compared with a release height of 1 dp. The results show that for a range of values of Bo and contact angle near dividing lines on the regime map, the release height does not affect the ultimate regime. It is worth noting that these results are not necessarily indicative that regime is completely independent of release height, and liquid released from a significantly higher height could have enough inertia upon impacting the particles to enter a different regime. Table 2. Effect of diameter of the liquid droplet on ultimate regime, for values of Bo and contact angle near changes in regime. Bo

θ (°)

1 1 0.2 0.1 1 1

85 80 80 80 30 25

Regime (Droplet Diameter: 0.5 dp) Trapped Trapped Trapped Trapped Trapped Spread

Regime (Droplet Diameter: 1 d p) No imbibition Trapped Trapped Descend Trapped Spread

Regime (Droplet Diameter: 1.5 dp) No imbibition No imbibition Descend Descend Trapped Spread

Regime (Droplet Diameter: 2 d p) No imbibition No imbibition Descend Descend Trapped Spread

Table 2 shows the effect of the diameter of the liquid droplet on the ultimate regime; in all cases, the release height is 0 dp. At a contact angle of 85° and Bo = 1, the trapped regime is seen for a droplet diameter of 0.5 dp, while the no imbibition regime is seen for higher values of droplet diameter. Imbibition at the lowest droplet diameter can be attributed to the droplet being able to squeeze through the first throat with minimal contact. For the corresponding cases with a contact angle of 80°, the trapped regime is seen for both 0.5 dp and 1 dp droplet diameters, while no imbibition is seen for higher diameters. The lack of imbibition for the larger droplet diameters can be attributed to the liquid spreading out around multiple throats, allowing contact angle effects to prevent imbibition with a smaller hydrostatic head to compete against. When Bo is reduced to 0.2 with a contact angle of 80° the trapped regime is observed for droplet diameters of 1 dp and below, while the descend regime is observed for the larger droplet diameters. This transition from the trapped regime to the descend regime can be attributed to the larger gravitational head in systems with larger initial droplets, forcing the liquid to descend through throats in the packed array. In further lowering the Bo to 0.1, the trapped regime is only observed for a droplet diameter of 0.5 dp, while larger droplet diameters all result in the descend regime. This trend is indicative of reduced surface tension decreasing the gravitational head necessary to drive liquid through throats in the array. At Bo = 1 and a contact angle of 30°, the trapped regime is seen for all droplet sizes. As compared to the 80° contact angle case, contact 14 ACS Paragon Plus Environment

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angle effects are not strong enough to prevent imbibition for large droplets, but surface tension is still large enough to trap liquid, preventing it from descending. At Bo = 1 and a contact angle of 25°, the spread regime is observed for all droplet sizes, demonstrating that this transitional contact angle upon which liquid is driven to spread around to the surfaces of all particles is unaffected by initial droplet size. 3.2 Temporal behavior Fig. 5 shows the descent of the liquid mass through the packed bed over time with different values of Bo and Ga, using the four forms of non-dimensional time presented in Section 2.1. In all cases, the value of Bo is much less than unity and the contact angle is 70°, and thus the flow is in the descend regime. Fig. 5 (a) shows that t ! ∗ with values "#$%%$& = "%$ = 8 and

,#$%%$& = ,%$ = 10° collapses data fairly well across varying values of Bo and Ga. In contrast,

Fig. 5 (b) shows that the simplification of t !> ∗ with "#$%%$& = 8 and ,#$%%$& = 10° in which

surface tension only pulls down on liquid does not collapse the data particularly well, with smaller gaps across varying values of Ga and larger gaps across varying values of Bo. Fig. 5 (c)

shows that the simplification of t !? ∗ with "%$ = 8 and ,%$ = 10° in which surface tension only

pulls upward on liquid collapses data nearly as well as the full t ! ∗ . Fig. 5 (d) shows that t !! ∗

produces a very similar plot to t !> ∗ , with the same trend of quicker descent with decreasing Bo

and Ga. Thus, t ! ∗ and t !? ∗ produce the best collapses of the four time scales for the descend

regime below the top of the packed bed. As discussed further in the Discussion section, the similar collapses of all four time scales is indicative of surface tension not playing a large role in

the descend regime, but the best collapses provided by t ! ∗ and t !? ∗ indicate that surface tension

pulls up slightly on the liquid in this regime.

Fig. 6 shows the initial descent (1 dp) of center of mass of the liquid into the bed for

various values of Bo and Ga over time using (a) t !> ∗ and (b) t !! ∗ . The simulations with Bo = 1 are

in the trapped regime, while those using Bo = 0.1 are in the descend regime. The results using t ! ∗

and t !? ∗ are not shown because they produce much worse collapses of the data than the other time scales. Fig. 6 (a) shows that t !> ∗ with values "#$%%$& = 8 and ,#$%%$& = 10° produces a nice

collapse of the data across various values of Bo and Ga, especially for the first 0.8 dp of the descent. Fig. 6 (b) shows that t !! ∗ collapses the data well across varying values of Ga, but higher

Bo simulations descend quicker than lower Bo simulations in this time scaling. As discussed


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further in the Discussion section, the results in Fig. 6 are indicative of surface tension pulling downward on the liquid as it initially imbibes into the bed.

Figure 5. Descent of the center of mass of the droplet (∆zcom) versus time for four dimensionless forms of time for varying values of Bo and Ga. All of these low Bo cases were in the descend regime. Contact angle: 70°.

Fig. 7 shows initial liquid descent (1 dp) into the bed with varying contact angle over

time, using (a) t !> ∗ and (b) t !! ∗ . In all cases, Bo = 1 and Ga = 10, making the 10° contact angle

case in the spread regime, while all other cases in the trapped regime. The results using t ! ∗ and

t !? ∗ are not shown because these time scales produce a much worse collapse of the data than the

other time scales. Fig. 7 (a) shows that the t !> ∗ scaling with values "#$%%$& = 8 and ,#$%%$& = 10°

produces a nice collapse of the data across all contact angles. In contrast, the t !! ∗ scaling does not

collapse the data; in this reference frame, liquid with lower contact angle is imbibed more quickly into the bed. As discussed further in the Discussion section, these results are indicative 16 ACS Paragon Plus Environment

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of the "#$%%$& cos(+ + ,#$%%$& ) term from the momentum equation working to pull liquid

down into the bed quicker during its initial descent.

Figure 6. Initial descent of center of mass of liquid (∆zcom) versus non-dimensional time (a) tIa* and (b) tII* for droplets with various values of Bo and Ga. Contact angle: 70°.

Figure 7. Initial descent of center of mass of liquid (∆zcom) versus non-dimensional time (a) tIa* and (b) tII* for droplets with various contact angles. Bo: 1; Ga: 10.

4. Discussion 4.1 Liquid distribution regimes Fig. 3 clearly shows four distinct regimes of liquid behavior for a droplet entering a packed array of spheres. Previous studies have divided behavior for either a droplet6 or a horizontal plane of liquid2,3 entering a packed bed into two regimes, one in which liquid does not 17 ACS Paragon Plus Environment

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travel beyond the first layer of spheres and another in which liquid does. Thus, these works have effectively identified the no imbibition regime shown in Fig. 3 (A), but have not considered potential differences for regimes of behavior when liquid is imbibed into the bed, shown in Fig. 3 (B-D) as distinct states in which liquid either is trapped, descends through the bed or spreads through the bed on the surface of particles. These three imbibed regimes show great differences in the amount of wetting of the particles that occurs as well as the amount of gas-liquid surface area created, with the spread regime maximizing solid-liquid and gas-solid interfacial areas and the trapped regime creating minimal interfacial area. In systems such as soils and chemical reactors, these interfacial areas have a large effect on bulk behavior because they impact interphase transport, chemical reactions and mechanical properties. The simplified model of a mass of liquid moving through a vertical capillary with varying cross-section shown in Fig. 1 provides a framework which can distinguish between the descend and trapped regimes. For low Bo cases in which surface tension insignificant or cases in which the geometry of the mass of liquid is such that the curvature of the liquid on the top and bottom are always similar, as shown in Scenario II in Fig. 1, there is no force acting to pull the liquid upwards, and thus the liquid can only descend. In contrast, when Bo is sufficiently high and liquid geometries develop in which there is significant curvature at the top of the liquid mass, but insignificant curvature at the bottom, as shown in Scenario Ib in Fig. 1, surface tension forces can pull up on the liquid, allowing it to become suspended or trapped, rather than descend. The condition for trapping is given by Eq. 13. Fig. 4 (b) shows that Eq. 13, using values of "%$ = 8

and ,%$ = 10° which collapsed liquid descent data versus time well in Fig. 5 (b), provides a

fairly accurate dividing line between the descend and trapped regimes for contact angles

between 30° and 75°. Fig 4 (a) indicates that a better empirical division between these regimes may be approximately independent of the contact angle, given by Bo = 1/6. The differences between the empirical and theoretical divisions could be attributed to: (a) the lack of inertia accounted for in Eq. 13, despite the small initial drop of the liquid onto the bed surface or (b) the greater complexity in the geometry of the packed bed as compared to the undulating capillary model. The difference between the descend and trapped regimes is investigated further in Fig. 8. Fig. 8 shows detailed simulation predictions for (a and c) a descend case and (b and d) a trapped case at approximately the same liquid configuration high in the bed. Fig. 8 (a) and (b) show a 18 ACS Paragon Plus Environment

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vertical cross-section of the liquid volume fraction, showing the liquid cross-sections in both cases. Fig. 8. (c) and (d) show 3D renderings of the liquid mass, colored by the dynamic pressure of the liquid, including the hydrostatic head term ( ). For the descend case in (c) the dynamic pressure is higher at the top than at the bottom, indicating that there is still a driving force for liquid to move downward, and thus the liquid continues to descend. In the trapped case in (d), the dynamic pressure is approximately constant at the top and bottom of the liquid mass and thus the liquid remains trapped in this orientation. Thus, Fig. 8 shows that a competition between surface tension and gravity predicted in Eq. 13 can indeed cause a constant dynamic pressure across the liquid mass, leading to the trapped state.

Figure 8. Mass of liquid falling through packed bed at critical juncture which separates the descend regime from the trapped regime. (a) and (b) show a vertical slice of the liquid volume fraction, with red showing liquid, blue showing gas and gray showing particles. (c) and (d) show 3D renderings of the liquid mass, colored with the pressure in the liquid, including the hydrostatic head. (a) and (c) show the descend case (Bo = 0.1). (b) and (d) show the trapped case (Bo = 1.0). Contact angle: 70°.


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Figure 9. Effect of capillary force on the descend and trapped states: (a) vertical component of capillary force normailized by the weight of the liquid versus non-dimensional time, (b) normalized vertical position of the center of mass of the liquid versus non-dimensional time. Ga = 10; contact angle: 70°.

This difference between the descend and trapped regimes is shown more quantitatively in Fig. 9. Fig. 9 (a) shows the vertical component of the capillary force (given by the last term in Eq. 16) normalized by the weight of the liquid versus non-dimensional time. Fig. 9 (b) shows the change in the vertical position of the center of mass of the liquid over time. The cases shown are the same as those in Fig. 8. For the Bo = 0.1 case in the descend regime, the normalized vertical capillary force is initially negative, indicative of surface tension pulling downward on the liquid as it initially enters the liquid, but this value increasing over time, eventually becoming positive, but much less than unity. With vertical capillary force less than the weight of the liquid, the liquid cannot be suspended, and thus it continues to descend, as shown in Fig. 9 (b). For th Bo = 1.0 case in the trapped regime, the capillary force is initially very negative, causing the liquid to enter the array quickly, but after approximately 1000 non-dimensional time units, the normalized 20 ACS Paragon Plus Environment

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force increases and becomes level at a value of approximately unity. With vertical capillary force balancing weight, the liquid becomes suspended, and thus the position of its center of mass becomes constant. The emergence of the spread regime can be explained as follows. Low contact angles (75°), liquid-solid interfaces are much less favorable energetically, and an intrusion pressure6 is required to push liquid through the throat formed between three particles. This intrusion pressure increases with surface tension, and thus for higher values of Bo, the hydrostatic head provided by the height of the droplet is not sufficient to force the liquid through the throat, and the no imbibition regime is observed. When Bo is sufficiently low for high contact angles, the hydrostatic head is able to force the droplet through the throat and the trapped and descend regimes are observed. Theoretically, the dividing line between the no imbibition regime and the other regimes is given by Extrand and Moon6 and shown in Fig. 4 (b). This line is slightly to the left of the empirical dividing line, shown in Fig. 4 (a). The reasons for this difference can be attributed to (i) the theoretical line not accounting for inertia of the fluid, which would allow liquid to make it past the first throat at higher Bo and (ii) a constant height of the liquid mass as it moves through a critical juncture in the throat assumed here, which may actually vary with contact angle and Bo. An unexpected feature of the empirical dividing line for the no imbibition regime shown in Fig. 4 (a) is that it increases with increasing Bo for very high Bo. This increase accounts for the fact that a no imbibition case is seen for (Bo = 100, θ = 80°) while a trapped case is seen for (Bo = 1000, θ = 80°). This phenomenon can be explained by inertia: for the higher Bo case, surface tension will initially help pull liquid down, giving it significant inertia to help make it past the critical juncture in the throat and thus make it to the trapped regime, rather than the no imbibition regime. Evidence of higher Bo speeding up initial liquid flow is given in Fig. 6. 4.2 Non-dimensionalization of time Beyond understanding regimes of behavior, we demonstrate four non-dimensional forms of time capable of collapsing data across various parameters, each form of time effective at 21 ACS Paragon Plus Environment

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different locations of the droplet in the system. Understanding how the time scales for flow vary based on physical parameters and location in the system is critical for many systems, such as soils and chemical processes, because flow dynamics are often coupled with other rate-based processes, such as evaporation, absorption or chemical reaction. Figs. 6 and 7 show that the non-dimensional time based on Scenario Ia effectively collapses the data for the initial intrusion of liquid into the packed bed across a wide range of Bo, Ga and contact angle. In Scenario Ia, curvature is dominant at the bottom of the liquid mass and thus both gravity and surface tension pull liquid down, suggesting that this mechanism is occurring for initial liquid intrusion. This simplified model is consistent with the real initial behavior, since upon initial liquid intrusion, the curvature at the top of the droplet is insignificant, since the droplet is still largely spherical, but the liquid entering the throat formed between particles narrows and develops a significant curvature. With this imbalance in curvature, increasing surface tension increases the rate of liquid imbibition, and thus time scale Ia is needed to account for this and collapse the data at early times. Fig. 5 (b) shows that as liquid descends through the bed well past the initial “throat”, the non-dimensional time based on Scenario Ia no longer collapses the data well across various Bo and Ga values. For droplets in the descend regime, surface tension is expected to not impact liquid dynamics significantly, and thus time scale II is expected to collapse the data. As seen in Fig 5 (d), time scale II collapses the data at long times for the Bo = 0.05 and Bo = 0.1 cases slightly better than time scale Ia, but in both cases the lower Bo case descends faster. Time scale Ib (Fig. 5 (c)) and time scale I (Fig. 5 (a)) collapse the data for descending cases better than time

scales Ia and II. Since in these time scales, for the values of " and , used, surface tension pulls

liquid upwards, the results in Fig. 5 indicate that for these descend cases, surface tension is pulling up on liquid to a slight extent, but not enough to significantly challenge gravitational force. The slight upward pull of surface tension on the liquid is consistent with Fig 8 (c), which shows the dynamic pressure in the highest layer of liquid cells as slightly lower than that in the

bulk of the liquid cells. The slight upward pull is consistent with the fact that the value of Bo cos + for these cases was 0.017 and 0.034, relatively close to the transition value of ~1/6

between the descend and trapped regimes shown empirically in Fig. 4 (a). For even lower values

of Bo, one could expect that time scale II would fully collapse the data across different values of Bo. 22 ACS Paragon Plus Environment

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5. Conclusions Volume-of-fluid simulations identify four regimes of liquid behavior for the intrusion of a droplet of liquid into a packed bed under the influence of gravity. The regimes were classified as (A) no liquid imbibition, (B) liquid trapping high in the bed, (C) liquid descent to the bottom of the bed and (D) liquid spreading across the surface of all spheres, with each regime showing very different wetting and gas-liquid interfacial area. The regime behavior can be mapped based on the Bond number and contact angle characterizing the system, as illustrated in Figure 4. A simplified model of a mass of liquid traveling down a vertical capillary with undulating crosssectional area can be used to capture the differences in the effect of surface tension which lead to the regimes in which liquid is trapped high in the bed and liquid descends through the bed. The different scenarios for the effect of surface tension in this simplified model also lead to four forms of non-dimensional time, which were found to collapse data for liquid flow from simulations across a range of Bo, Ga and contact angle. The ability to map regimes of behavior and predict time scales for flow is critical to many natural and engineering systems involving coupled flow, mass transport and chemical reactions. Acknowledgments This research was supported through funding from Syncrude and ExxonMobil Research & Engineering.


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References: (1) Washburn, E. W. The Dynamics of Capillary Flow. Phys. Rev. 1921, 17 (3), 273–283. (2) Bán, S.; Wolfram, E.; Rohrsetzer, S. The Condition of Starting of Liquid Imbibition in Powders. Colloids Surf. 1987, 22 (2), 291–300. (3) Shirtcliffe, N. J.; McHale, G.; Newton, M. I.; Pyatt, F. B.; Doerr, S. H. Critical Conditions for the Wetting of Soils. Appl. Phys. Lett. 2006, 89 (9), 94101. (4) Raux, P. S.; Cockenpot, H.; Ramaioli, M.; Quéré, D.; Clanet, C. Wicking in a Powder. Langmuir 2013, 29 (11), 3636–3644. (5) Hilden, J. L.; Trumble, K. P. Numerical Analysis of Capillarity in Packed Spheres: Planar Hexagonal-Packed Spheres. J. Colloid Interface Sci. 2003, 267 (2), 463–474. (6) Extrand, C. W.; Moon, S. I. Intrusion Pressure To Initiate Flow through Pores between Spheres. Langmuir 2012, 28 (7), 3503–3509. (7) Batten, G. L. Liquid Imbibition in Capillaries and Packed Beds. J. Colloid Interface Sci. 1984, 102 (2), 513–518. (8) Shou, D.; Ye, L.; Fan, J.; Fu, K.; Mei, M.; Wang, H.; Chen, Q. Geometry-Induced Asymmetric Capillary Flow. Langmuir 2014, 30 (19), 5448–5454. (9) Lucas, R. Rate of Capillary Ascension of Liquids. Kolloid Z 1918, 23, 15–22. (10) Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog. 1952, 48. (11) Hirt, C. W.; Nichols, B. D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys. 1981, 39 (1), 201–225. (12) Dullien, F. A. L.; El-Sayed, M. S.; Batra, V. K. Rate of Capillary Rise in Porous Media with Nonuniform Pores. J. Colloid Interface Sci. 1977, 60 (3), 497–506. (13) Levine, S.; Lowndes, J.; Reed, P. Two-Phase Fluid Flow and Hysteresis in a Periodic Capillary Tube. J. Colloid Interface Sci. 1980, 77 (1), 253–263. (14) Sharma, R.; Ross, D. S. Kinetics of Liquid Penetration into Periodically Constricted Capillaries. J. Chem. Soc. Faraday Trans. 1991, 87 (4), 619–624. (15) Tsori, Y. Discontinuous Liquid Rise in Capillaries with Varying Cross-Sections. Langmuir 2006, 22 (21), 8860–8863. (16) Wang, Q.; Graber, E. R.; Wallach, R. Synergistic Effects of Geometry, Inertia, and Dynamic Contact Angle on Wetting and Dewetting of Capillaries of Varying Cross Sections. J. Colloid Interface Sci. 2013, 396, 270–277. (17) Liou, W. W.; Peng, Y.; Parker, P. E. Analytical Modeling of Capillary Flow in Tubes of Nonuniform Cross Section. J. Colloid Interface Sci. 2009, 333 (1), 389–399. (18) Patro, D.; Bhattacharyya, S.; Jayaram, V. Flow Kinetics in Porous Ceramics: Understanding with Non-Uniform Capillary Models. J. Am. Ceram. Soc. 2007, 90 (10), 3040–3046. (19) Cox, R. G. The Dynamics of the Spreading of Liquids on a Solid Surface. Part 1. Viscous Flow. J. Fluid Mech. 1986, 168, 169–194. (20) Cox, R. G. The Dynamics of the Spreading of Liquids on a Solid Surface. Part 2. Surfactants. J. Fluid Mech. 1986, 168, 195–220. (21) Voinov, O. V. Inclination Angles of the Boundary in Moving Liquid Layers. J. Appl. Mech. Tech. Phys. 1977, 18 (2), 216–222. (22) Hoffman, R. L. A Study of the Advancing Interface. I. Interface Shape in Liquid—gas Systems. J. Colloid Interface Sci. 1975, 50 (2), 228–241.


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(23) Jiang, T.-S.; Soo-Gun, O. H.; Slattery, J. C. Correlation for Dynamic Contact Angle. J. Colloid Interface Sci. 1979, 69 (1), 74–77. (24) Ralston, J.; Popescu, M.; Sedev, R. Dynamics of Wetting from an Experimental Point of View. Annu. Rev. Mater. Res. 2008, 38 (1), 23–43. (25) Brackbill, J. U.; Kothe, D. B.; Zemach, C. A Continuum Method for Modeling Surface Tension. J. Comput. Phys. 1992, 100 (2), 335–354. (26) Klostermann, J.; Schaake, K.; Schwarze, R. Numerical Simulation of a Single Rising Bubble by VOF with Surface Compression. Int. J. Numer. Methods Fluids 2013, 71 (8), 960–982.


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Table of Contents Figure:


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Figure 1. Geometric configurations for a mass of liquid falling through undulating capillary: In Scenario I, surface tension affects bulk flow, while in Scenario II, it does not. In configuration I (a) surface tension pulls down on liquid due to the dominant curvature at the bottom, while in I (b) surface tension pulls upward due to dominant curvature at the top. In configuration II (a), curvature is negligible at both the top and bottom, while in II (b) curvature is significant at both the top and bottom, but the effects of surface tension cancel out. 142x75mm (300 x 300 DPI)

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Figure 2. Vertical cross section of liquid distribution at the start of each simulation, in which a liquid droplet of diameter dp falls under gravity into a hexagonal closed packing of spheres of diameter dp. Red: liquid; blue: gas; gray: solid particles. 72x67mm (300 x 300 DPI)


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Figure 3. Vertical cross-sections of the final liquid distribution in simulations in four different regimes: (A) No imbibition of liquid into the bed, (B) liquid trapped in a pocket high in the bed, (C) liquid descends to the bottom of the bed and (D) liquid spreads around surface of all particles. Simulation parameters indicated below images. Red: liquid; blue: gas; gray: solid particles. 176x60mm (300 x 300 DPI)


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Figure 4. Regime map of final liquid distribution based on Bo and contact angle, θ. Regimes shown in Fig. 3. Fig. 4 (a) shows dividing lines to guide the eye as to the divisions between regimes; Fig. 4 (b) gives dividing lines based on theory. 170x106mm (300 x 300 DPI)


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Figure 5. Descent of the center of mass of the droplet (∆zcom) versus time for four dimensionless forms of time for varying values of Bo and Ga. All of these low Bo cases were in the descend regime. Contact angle: 70°. 138x135mm (300 x 300 DPI)


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Figure 6. Initial descent of center of mass of liquid (∆zcom) versus non-dimensional time (a) tIa* and (b) tII* for droplets with various values of Bo and Ga. Contact angle: 70°. 153x72mm (300 x 300 DPI)


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Figure 7. Initial descent of center of mass of liquid (∆zcom) versus non-dimensional time (a) tIa* and (b) tII* for droplets with various contact angles. Bo: 1; Ga: 10. 153x73mm (300 x 300 DPI)


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Figure 8. Mass of liquid falling through packed bed at critical juncture which separates the descend regime from the trapped regime. (a) and (b) show a vertical slice of the liquid volume fraction, with red showing liquid, blue showing gas and gray showing particles. (c) and (d) show 3D renderings of the liquid mass, colored with the pressure in the liquid, including the hydrostatic head. (a) and (c) show the descend case (Bo = 0.1). (b) and (d) show the trapped case (Bo = 1.0). Contact angle: 70°. 136x121mm (300 x 300 DPI)


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Figure 9. Effect of capillary force on the descend and trapped states: (a) vertical component of capillary force normalized by the weight of the liquid versus non-dimensional time, (b) normalized vertical position of the center of mass of the liquid versus non-dimensional time. Ga = 10; contact angle: 70°. 71x135mm (300 x 300 DPI)


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Intrusion of a Liquid Droplet into a Powder under Gravity - Langmuir

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