Liquid bridge model: accounting for liquid film in wet material modeling

In many bulk handling processes, a small amount of liquid is added and dispersed into the bulk dry material, forming a thin liquid film around the particles. This film leads to an additional attractive force that appears during a collision between particles or between particles and walls. 

For studying these systems of wet particles with small amounts of liquids, the Liquid Bridge Model can be used to track the liquid film mass per particle while accounting for the liquid bridge force consisting of capillary and viscous forces during collision events. 

Read below to understand how Rocky DEM models the liquid bridge formation/rupture process and estimates the liquid bridge forces.

Monitoring the liquid film per particle

When using the liquid bridge model in Rocky DEM, the amount of liquid volume a particle has changes due to its interactions with other particles in the simulation. When the particle is initially injected into the system, it may have a certain liquid film volume. It is also possible to increase the liquid film mass by defining a liquid spray region, so that the liquid mass the particle receives is proportional to its residence time inside this region. When that particle collides with another particle, it can receive or lose some amount of liquid from or to the other particle, depending upon the liquid film volume of the other particle. 

The liquid film volume per particle is tracked throughout the simulation and can be monitored by the “liquid mass” property. 

Liquid transfers happen in the following way: during collisions, part of the liquid film existing in the interaction pair is transferred to the liquid bridge, as shown in Figure 1. The liquid volume transferred during collisions depends on the amount of liquid existing in the liquid bridge.

Figure 1. The liquid bridge formation during the collision of a particle containing a certain amount of liquid (green) with a dry particle (red).
Figure 1. The liquid bridge formation during the collision of a particle containing a certain amount of liquid (green) with a dry particle (red).

Liquid bridge formation/rupture process

The liquid bridge formation/rupture process follows the steps illustrated in Figure 2. Specifically:

• When one particle that has a certain liquid film mass collides with another particle or boundary (top left), a liquid bridge is formed between the pair (top right).

• The liquid bridge volume is a fraction of the liquid film volume that each particle had before the contact.

• This liquid bridge is sustained while the separation distance between particles is smaller than the critical separation value and the liquid bridge force is a function of the liquid bridge volume and separation distance.

• The liquid bridge ruptures as the critical separation distance is achieved (bottom right).

• After the liquid bridge ruptures, the liquid bridge force ceases and the liquid bridge volume is evenly distributed between particles (in case of a particle-particle collision) or returns to the particle (in case of a particle-boundary collision).

Figure 2. Liquid bridge formation/rupture process.
Figure 2. Liquid bridge formation/rupture process.

Liquid bridge volume

The bridge volume fraction, fb, is a user input that dictates the fraction of the particle liquid film content that contributes to the formation of the bridge. Therefore, the liquid bridge volume is given by the sum of the liquid film volume existing in each particle multiplied by the liquid bridge fraction.

In the event of particles colliding against a boundary, only the particle has a liquid film and therefore the volume of the liquid bridge is simply the particle liquid film multiplied by the liquid bridge fraction.

Figure 3. Liquid bridge in a particle-particle interaction (left) and particle-boundary interaction (right).
Figure 3. Liquid bridge in a particle-particle interaction (left) and particle-boundary interaction (right).

The liquid bridge liquid volume is constant throughout the collision but the liquid bridge force depends on the separation distance and the velocity of the particles.

Bridge rupture criterion and liquid volume distribution

The critical rupture distance is computed according to Mikami et al. [1] and is based on the regression of numerical solutions of the Young Laplace equation. The critical rupture distance, hc, is estimated by Rocky DEM using the volume of the liquid bridge, the radius of the involved particles, and the contact angle.

As illustrated in Figure 4, during particle-particle interactions, the liquid bridge volume is distributed evenly between the particles on a rupture event. Therefore, after the rupture of the liquid bridge, the particle’s liquid mass will be different from the particle’s liquid mass before the collision.

Figure 4. Liquid bridge volume distribution when the liquid bridge ruptures as the critical separation distance (hc) is reached.
Figure 4. Liquid bridge volume distribution when the liquid bridge ruptures as the critical separation distance (hc) is reached.

On particle-boundary bridges, only the particle contributes to the bridge, therefore, the particle preserves its liquid content when the bridge breaks and the boundary does not retain any liquid mass.

Liquid bridge force calculation

The liquid bridge force is given as the sum of the capillary force, Fc, (governed by the surface tension force) and the viscous force, Fµ, (governed by the viscosity and the relative velocity), as shown in Figure 5.

Figure 5. Liquid bridge model as a composition of capillary and viscous forces.
Figure 5. Liquid bridge model as a composition of capillary and viscous forces.

The capillary force term is computed according to the model described by Mikami et al [1]. For a given collision pair, the capillary force is a function of the liquid bridge volume, the separation distance between particles, the contact angle, and the surface tension, as illustrated in Figure 3. The first two parameters are tracked during the collision whereas the last two are user inputs. Reference values for these user inputs can be found in the literature.

The viscous force is derived from the lubrication theory and follows the model proposed by Nase et al. [2]. The force is decomposed into normal and tangential components (shown in Figure 6 as Fµn and Fµτ) and is proportional to both the fluid viscosity and the relative velocity between particles, and is inversely proportional to the separation distance. As the separation distance increases, the viscous force decreases.

Figure 6. Viscous force due to the liquid bridge formed between the particles.
Figure 6. Viscous force due to the liquid bridge formed between the particles.

The capillary force is always an adhesive force whereas the viscous force can be either adhesive or repulsive, depending on the relative motion of the particles. However, it is important to note that, usually, the magnitude of the capillary force is greater than the viscous force, so even when the viscous force is repulsive, the net liquid bridge force is adhesive.

Application example

In this application example, Rocky DEM was used to analyze a mixing process with liquid addition. For the first 20 seconds of the simulation, the liquid is continuously injected into the mixer. Due to the liquid bridge formation/rupture process, particles exchange liquid mass as they interact with one another.

Video 1. Rocky DEM simulation of a mixing process with liquid addition.

The influence of the liquid bridge model can be perceived as the mixture shows an increasingly viscous-like behavior. The goal was to analyze the homogeneity of the batch during the mixing process, which can be achieved by evaluating the average particle liquid mass. These results are important for determining whether the operational parameters are adequate for achieving the desired homogeneity before unloading the batch.

References

[1] Mikami, T., Kamiya, H., Horio, M. (1998). Numerical simulation of cohesive powder behavior in a fluidized bed. In Chemical Engineering Science 53(10):1927-1940

[2] Nase, S.T., Vargas, W. L., Abatan, A. A., McCarthy, J. J. (2001) Discrete characterization tools for cohesive granular material. In Powder Technology 116(2-3):214-223


Lucilla Almeida

CAE Specialist at ESSS, D.Sc.

Lucilla holds a BE (Chemical) undergraduate degree, an M.Sc. in Chemical Engineering and a Ph.D. in Nuclear engineering from the Federal University of Rio de Janeiro. She joined ESSS in 2008 and has spent 5 years focused on applying CFD to solve common engineering problems in the Oil and Gas industry, dealing with turbulent and multiphase flow simulations. Since 2013, she is an Application Engineer for Rocky DEM Business Unit, supporting users, working on consultancy projects and validating models implemented for the CFD-DEM coupling.


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