Liquid Droplets

Introduction

Droplets are interacting Lagrangian spheres that can access continuous positions across the fluid lattice. Droplets are typically used to model two-fluid droplet dispersion, but also have application to blending density-stratified miscible fluids

Each droplet has a rigid core that interacts elastically with the rigid cores of other dispersed-phase particles. The purpose of the hard sphere interactions is to ensure that the initial dispersed phase volume remains incompressible prior to droplet breakup/dispersion.

Surrounding each rigid core is a fully penetrable concentric exterior shell which increases the apparent volume of the particle. The purpose of the shell is to allow the dispersed phase volume fraction to overcome the 0.62 limit on volume fraction presented by random sphere packing. That is, by accounting for the interstitial region between a random pack of rigid spheres, the dispersed phase volume fraction reconstructed from the cherry-pit particles can realize values close to 1.0.

Droplets enter the system within as monodisperse particles located within the children geometry of the droplet parent. The droplet volume fraction and size distribution can inform the local fluid viscosity via a user-defined rheology. Droplet positions and velocities evolve according to Newton’s Second Law using a Verlet algorithm advanced at the simulation timestep. The forces included in the velocity update are defined by the user.

Upon entering the simulation domain, each droplet is assigned an ID, a birth timestamp, a diameter and an origin ID. The birth timestamp identifies the time step at which each droplet enters the system. This value is particularly useful in predicting droplet residence time distributions and droplet mean age. The origin ID describes where a droplet entered the system. This value is useful in predicting how droplets from various sources blend and transfer mass to/from the fluid. It is also useful in predicting how droplets with different properties (e.g. density and diameter) are affected by fluid motion (assuming different droplet properties are assigned to the different droplet origins).

Breakup occurs on the level of individual droplet or using a parcel approach. In either approach, the daughter diameter is informed by the physical properties of the droplet and the kinematic properties of the fluid. Coalescence also occurs at the level of individual droplets or using a parcel approach. In the former, coalescence is informed by the kinematic properties of the two droplets. In the latter, changes in the parcel diameter are informed by the kinematic properties of the fluid.

Mass transfer between droplets and the surrounding fluid are discussed in Section dropletMassTrans and Droplet/Scalar Coupling. The mass transfer coefficient of each droplet is a user-defined function governed by local fluid properties. This droplet-specific mass transfer rate multiplied by the local droplet/fluid concentration difference defines the mass transfer rate between droplets and the surrounding fluid. Mass leaving/entering the droplet causes the droplet to shrink/grow in a way that conserves total species mass.

Note

Droplet image files are printed at the same frequency as slices and volumes.

Property Grid

General

Initial Packing

This option describes where the initial volume fraction of the dispersed phase/droplets within the system.

Auto - Set number of packed droplets based on initial droplet diameter and voxelized volume at runtime. The dispersed phase volume fraction is assumed to be 1.0 and the number of initial droplets is calculated at runtime to fill the child geometry.

Custom - User defined expression defining the number of droplets initially injected into the child geometry. This expression may be a function of the child volume, enabling the user to define a local dispersed phase volume fraction.

Initial Packing Volume Fraction

Definition

Dump Time

Time to begin droplet addition[s]

Density

kg/m ³ | Density of the dispersed phase droplets.

Note

In addition to hard sphere droplet-droplet interactions, the trajectories of each droplet are informed by the weight, buoyancy, drag, and virtual mass forces.

Volumetric Generation

Enabled

Definition

On

Off

Static Body Interaction

Static Body Option

This parameter specifies how each droplet set interacts with each solid body family.

Bounce Simple - Under the Bounce Simple option, particle-wall interactions are assumed to have the same interaction parameters as the particle-particle interactions (e.g. the same Young modulus, friction, etc.).

Bounce Custom - Under the Bounce Custom option, particle-wall interaction can be assigned custom custom interaction parameters.

Stick - The Stick option implies that droplets stick to the solid body family.

Disabled

Breakup/Coalesce

Two representations are available for modeling changes in particle size due to breakup and coalescence: Discrete and Parcel.

Breakup Enabled

Definition

Off

On - Breakup Enabled

Coalesce Enabled

Definition

Off

On - Coalesce Enabled

Mass Transfer

The mass transfer coefficient, \(k_L\), is a proportionality constant between mass flux and a concentration difference, such that:

\[\dot{n}=k_L A \Delta C,\]

where \(\dot{n}\) is a mass flow rate [mol/s], \(A\) is the mass transfer surface area [m ² ], and \(\Delta C\) is the driving concentration difference [mol/m ³ ]

The mass transfer coefficient, which has units of m/s, is typically a function of the local fluid properties, such as:

\[k_L=C \left(\epsilon \nu \right)^{a} \textrm{Sc}^{b},\]

where \(\epsilon\) is the local specific energy dissipation rate [W/kg], \(\nu\) is the local fluid kinematic viscosity [m ² /s ], and \(\textrm{Sc}\) is the local fluid Schmidt number.

The local mass transfer coefficient is often multiplied by the local specific droplet surface area (droplet surface area per unit volume) to calculate the local overall mass transfer coefficient, “\(k_La\)”, which has units of [1/s].

Multiplying each local overall mass transfer coefficient by the local concentration difference between the droplet and its surrounding fluid gives the mass transfer rate for the droplet Integrating this local mass transfer rate over the entire volume gives the total mass transfer rate between the droplets and the fluid.

Note

The local and total specific surface areas, along with the local and total overall mass transfer coefficients are reported automatically.

Convection

Definition

Off

On

• kL Expression - minutes/second | Mass transfer coefficient UDF.

Dissolution

Definition

Off

On

• Dissolution Rate - kg/m^2*s | Dissolution rate.

Droplet/Scalar Coupling

As discussed in Section dropletMassTrans, mass transfer between a droplet and the surrounding fluid is informed by the local fluid mass transfer coefficient, the surface area of the droplet, and the species concentration difference. With the Droplet/Scalar Coupling option, users can model species exchange between the fluid and individual droplet, along with the associated changes in droplet size and composition (if multiple species are present).

Initial Volume Fraction

This value defines the fraction of the droplet initially occupied by each scalar. If multiple coupled scalar fields are present, the sum of all volume fractions must be equal to one.

Molar volume

Volume of space occupied by each mole of gas in the drople for each species, [m ³/ mol] This value defines the change in the droplet volume due to mass transfer with the surrounding fluid. A value of zero implies the mass transfer for this species involves no change in the droplet diameter. Note that the volume of ideal gas at standard temperature and pressure is 0.0224 m ³/ mol.

Coupling Model

Two methods are available for modeling mass transfer (i) Fick’s Law and (ii) Custom.

Fick’s Law

In this approach, mass transfer is modeled as:

\[\dot{n}=k_L A \left( C_{g,i}-C_{f,i} \right)\]

where

\(\dot{n}\) is a mass flow rate between the droplet and the surrounding fluid [mol/s], \(k_L\) is the mass transfer coefficient of the fluid surrounding the droplet [m/s], \(A\) is the surface area of the droplet [m ²], \(C_g\) is the saturated concentration of species \(i\) [mol/m ³ ], and \(C_f\) is the local fluid concentration of species \(i\) [mol/m ³ ].

Note that the user must define the saturated concentration of each species.

Custom

In this approach, the user defines a custom mass transfer model for each species per the local fluid properties and the diameter/composition of each droplet. These models can follow the functional form of Fick’s law, or be extended to model more complex reaction/dissolution processes.

Advanced

Initial Droplet Diameter

By default, the initial droplet diameter is equal to the lattice spacing. Smaller initial droplet diameters can be specified here. This value represents a ceiling on the maximum droplet diameter. Droplets with diameters larger than the lattice spacing are not recommended.

Compute Droplet Distribution Data

Compute the average nearest neighbor separation distance and nearest neighbor separation distance probability distribution function. These metrics can be used to compare the spatial arrangement of droplet across the fluid domain to that of random droplet distribution. In this sense, the comparison can be used as an estimate on droplet homogenization Note that, for a random distribution of hard spheres, the average distance between neighbors, \(a\), can be estimated as:

\[a=\left({\frac {3}{4\pi n}}\right)^{1/3}\]

where \(n\) is the number of droplets per unit volume. As the droplets homogenize through the system, the average nearest neighbor distance should approach this value. A average nearest neighbor distance below this value implies droplet clumping.

The nearest neighbor separation distance probability distribution function, \(P_n\), is then defined from [9]:

\[P_n(r)=\frac{3}{ a}\left(\frac{r}{a}\right)^2 \left(1 - \left(\frac{r}{a}\right)^3 \frac{1}{N} \right)^{N - 1}\]

where \(r\) is the distance between droplets and \(N\) is the total number of droplets in the system.

Deletion Diameter

Droplets with diameters below this critical value will be deleted from the simulation. Any residual concentration will be dumped into the local fluid voxel.

Wall Stiffness Scale

This cofactor increases the stiffness of the droplet/wall bounce interactions. If droplets are becoming embedded in the wall, meaning the simulation timestep is too big to properly capture near-wall dynamics, increasing this cofactor will help push droplets out of the wall and back into the fluid.