Breakup/Coalesce

Overview

Particle sizes can change dynamically throughout a simulation due to ongoing breakup and coalescence events. Breakup is the process whereby a particle generates a satellite particle, typically due to shear and/or energy dissipation in the surrounding continuous phase fluid. Breakup events increase the number of particles in the system. Coalescence is the process whereby two particles join to form a single (larger) particle via a pairwise collision. Coalescence events decrease the number of particles in the system. These processes can all be controlled separately—meaning a model may have only breakup, only coalescence, or both processes active.

A key concept to consider when modeling breakup and coalescence is the numerical representation. The numerical representation characterizes how breakup and coalescence events inform the number of tracked particles and associated memory requirements of the simulation. Users can choose from either a discrete representation or a parcel-based representation.

In the discrete representation, all particles are tracked explicitly and stored in system memory. The increase in particle count due to satellite formation and the decrease in particle count due to pair-wise coalescence cause the number of these explicitly tracked particles to increase/decrease throughout the simulation.

In the parcel representation, breakup and coalescence are modeled implicitly using sampling parcels, which are characterized by a parcel diameter and parcel number scale. Sampling parcels are used to represent the collective motion of a group of equal-diameter particles. This number scale, which quantifies the number of equal-sized bubbles described by the bubble parcel, adjusts dynamically during breakup/coalescence events to conserve total parcel volume. In contrast to the discrete bubble approach, the parcel approach handles changes in bubble count implicitly via the parcel diameter and the number scale of each individual parcel. This implicit representation reduces the number of explicitly tracked parcels in the system and the associated memory requirements.

In principle, the discrete approach presents a higher fidelity solution to the particle field and the corresponding coalescence and breakup events. In practice, however, the memory requirements required to explicitly resolve each particle can become computationally impractical and costly. For such systems, the parcel approach can reduce these computational requirements and runtimes—often by multiple orders of magnitude. Predictions from parcel simulations can reliably reproduce predictions from the discrete approach, provided certain convergence related to the bubble collision are frequently satisfied. These conditions are further discussed in the additional resources below.

Note

The amplifying effects of the number scale on all particle-to-fluid forces, local particle volume fraction calculations, and interfacial mass transfer rates are handled internally and automatically.

Breakup

Breakup Enabled

Particle breakup model on or off. Selecting On launches the following sections:

Representation

The representation characterizes how breakup events inform the number of tracked particles and associated number requirements of the simulation. As a rule, we recommend running with the discrete approach whenever practical. In the discrete representation, all particles are tracked explicitly and stored in system memory. In the parcel representation, breakup is modeled implicitly using sampling parcels, which are characterized by a parcel diameter and parcel number scale.

Parcel

In this implicit approach the dispersed phase is modeled as a parcel—a cluster of droplets defined by a droplet number scale and a characteristic droplet diameter. The number scale describes the number of droplets in the parcel with the characteristic diameter, and the diameter informs the Newtonian trajectory of the parcel. These two parameters combine to define a total parcel droplet surface area and total droplet fluid volume.

The characteristic diameter adapts dynamically according to an equilibrium droplet diameter expression. The number scale of the parcel is then adjusted automatically, conserving total parcel gas volume when the diameter changes. This approach has lower memory requirements than the discrete approach because the number of parcels modeled in the system does not change due to breakup events.

In cases with high parcel injection rates, an injection downsampling can be applied to limit the number of parcels entering the system. This parameter characterized the number scale of the parcels entering the systems. Increasing the injection number scale decreases the number of parcels that must be added to the system, thereby decreasing the total number of parcels tracked in system memory.

Parcel Equilibrium Diameter Model

This setting defines how the local equilibrium parcel diameter will be related to the local fluid properties, the local particle properties, and the corresponding surface tension. Three options are available for defining the parcel equilibrium diameter: (i) Automatic, (ii) Custom Cofactor, and (iii) UDF. This also defines the maximum particle diameter allowed in the simulation.

Automatic

The equilibrium droplet diameter expression is calculated directly from the work by Hinze:

\[D_p = 0.725 \left( \frac{\sigma^{\frac{3}{5}}}{\rho^{\frac{3}{5}}\epsilon^{\frac{2}{5}}} \right)\]

where \(\sigma\) is the droplet surface tension, \(\rho\) is the density of the fluid surrounding the droplet, and \(\epsilon\) is the local energy dissipation rate.

The equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. The maximum bubble diameter is limit to 0.01 m, as bubbles with larger diameter can no longer be assumed to be spherical Lagrangian objects.

Surface Tension

N/m | Surface tension related to coalescent activity.

Custom Cofactor

The equilibrium droplet diameter expression is calculated directly from the work by Hinze:

\[D_p = C \left( \frac{\sigma^{\frac{3}{5}}}{\rho^{\frac{3}{5}} \epsilon^{\frac{2}{5}}} \right)\]

where \(C\) is a user-set constant, \(\sigma\) is the droplet surface tension, \(\rho\) is the density of the fluid surrounding the droplet, and \(\epsilon\) is the local energy dissipation rate. Measurements suggest that \(C\) can range from 0.1 to 1.5. The maximum bubble diameter is user defined. This value should not exceed approximately 0.01 m, as bubbles with larger diameters can no longer be assumed to be spherical Lagrangian objects.

As with the Automatic approach, the equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. The maximum allowable diameter is user-defined. In the process of crossing this maximum diameter threshold, the parcel diameter may slightly overshoot this cutoff value by a small fraction of a percent. To maximize runtime efficiency, the code will tolerate this slight offset and not retroactively correct/limit the bubble diameter. If absolute control is required, a UDF should be applied.

Equilibrium Cofactor

unitless | This parameter defines the equilibrium cofactor defined in the equation above.

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

Surface Tension

N/m | Surface tension related to breakup activity.

UDF

The user-defined breakup kernel defines equilibrium droplet diameter expression. As with the Automatic approach, the equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. No internal limit is placed on the allowable bubble size. Instead, this limit should be proscribed as part of the UDF. The maximum value should not exceed approximately 0.01 m, as bubbles with larger diameters can no longer be assumed to be spherical Lagrangian objects.

Equilibrium Diameter UDF

This UDF defines the equilibrium particle diameter associated with parcel breakup and/or coalescence. The UDF must include one output: a floating-point variable named d_p, which represents the local equilibrium diameter resulting from the ongoing competition between breakup and coalescence.

This function tends to increase with surface tension and decrease with local energy dissipation rate/shear. This is a particle-based local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Equilibrium Diameter

Note

To properly sample the continuous phase fluid field, we recommend that the number of parcels included in a simulation be order-of-magnitude consistent with the estimated lattice size.

Since breakup and coalescence are not modeled explicitly, the number of droplets in the system is governed by the Initial Packing and Initial Droplet Diameter parameter. More directly, the number of parcels is equal to the initial volume of the dispersed fluid divided by the volume of each droplet.

Discrete

In this explicit approach every droplet in the dispersed phase is tracked individually with a trajectory that evolves according to Newton’s second law. Individual droplets break up into two individual daughter droplets following a user-defined breakup expression. Breakup events explicitly increase and decrease the number of droplets modeled within the system. As such, the maximum number of trackable droplets is limited by GPU memory (about 2.5 million droplets per GB of RAM).

Discrete Breakup Model

This expression specifies the fluid and droplet properties required to initiate a breakup event and the volume fraction of the first daughter generated by the breakup process. Users can evoke a built-in breakup model or define custom expressions.

UDF

The user-defined breakup kernel defines (a) the conditions required to induce droplet breakup and (b) the volume fractions of the satellite droplets resulting from the breakup process. To conserve total volume during the breakup event, the volume fraction of the second satellite generated by the breakup process is automatically defined as the complement of the first volume fraction. In principle, breakup events occur when the local fluid shear forces exceed the local droplet surface tension forces. The volume fraction of the resultant satellite droplets is similarly informed by local fluid shear rates, droplet size, and droplet surface tension. This relationship is defined by the user and can exploit a random number. Note that a volume fraction of 0.5 implies equal-volume satellite droplets.

Break Fraction UDF

dimensionless | This UDF (i) defines if a binary particle breakup event occurs and (ii) calculates the volume fractions associated with the two satellite bubbles. Two outputs must be defined within the UDF: a boolean variable named doBreakup, and a floating point variable named fv. The doBreakup boolean specifies if a breakup event occurs; if doBreakup is true, the variable fv then defines the volume fraction of the first daughter particle formed during the binary breakup process. The volume fraction of the second daughter particle is automatically calculated as complement to the first, meaning total bubble volume is conserved.

The UDF input includes the local fluid properties and the kinematic properties (position, velocity, diameter, etc.) of the mother particle. This is a particle-based local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Break Fraction

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

Automatic

Droplet breakup events occur when the local fluid shear forces exceed the local droplet surface tension forces. This expression is calculated directly from the work by Hinze:

\[D_p = 0.725 \left( \frac{\sigma^{\frac{3}{5}}}{\rho^{\frac{3}{5}}\epsilon^{\frac{2}{5}}} \right)\]

where \(\sigma\) is the droplet surface tension, \(\rho\) is the density of the fluid surrounding the droplet, and \(\epsilon\) is the local energy dissipation rate.

Once breakup is initiated, the volume fractions of the two satellite droplets are informed by the local fluid shear rate, the [mother] droplet diameter, and the two-fluid surface tension. To predict the volume fractions, we evoke the semi-empirical breakup kernel developed by Xing et al. More specifically, for each bubble that is required to split, we calculate the corresponding satellite bubble volume fraction distribution function. We then randomly sample this distribution to evaluate the volume fraction of the first satellite. Per the conservation of mass, the volume fraction of the second satellite is calculated as the complement to the first.

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

Breakup Time Scale

This accounts for the time it takes for a particle to break up, per the natural frequencies of the bubble.

Surface Tension

N/m | Surface tension related to breakup and coalescent activity.

Start Time

seconds | Time at which breakup algorithm starts to run.

Warning

Bubbles that break up into objects smaller than 10 nanometers are automatically deleted to preserve numerical stability. Smaller diameter objects should be modeled using mesoscopic molecular modeling approaches.

Coalesce

Coalesce Enabled

Particle coalesce model on or off. Selecting “On” launches the following section:

Representation

The representation characterizes how coalescence events inform the number of tracked particles and associated number requirements of the simulation. As a rule, we recommend running with the discrete approach whenever practical. In the discrete representation, all particles are tracked explicitly and stored in system memory. In the parcel representation, coalescence is modeled implicitly using sampling parcels, which are characterized by a parcel diameter and parcel number scale.

Parcel

In this implicit approach the dispersed phase is modeled as a parcel—a cluster of droplets defined by a droplet number scale and a characteristic droplet diameter. The number scale describes the number of droplets in the parcel with the characteristic diameter, and the diameter informs the Newtonian trajectory of the parcel. These two parameters combine to define a total parcel droplet surface area and total droplet fluid volume.

The characteristic diameter adapts dynamically according to an equilibrium droplet diameter expression. The number scale of the parcel is then adjusted automatically, conserving total parcel gas volume when the diameter changes. This approach has lower memory requirements than the discrete approach because the number of parcels modeled in the system does not change due to coalescence events.

In cases with high parcel injection rates, an injection downsampling can be applied to limit the number of parcels entering the system. This parameter characterized the number scale of the parcels entering the systems. Increasing the injection number scale decreases the number of parcels that must be added to the system, thereby decreasing the total number of parcels tracked in system memory.

Parcel Equilibrium Diameter Model

This setting defines how the local equilibrium parcel diameter will be related to the local fluid properties, the local particle properties, and the corresponding surface tension. Three options are available for defining the parcel equilibrium diameter: (i) Automatic, (ii) Custom Cofactor, and (iii) UDF. This also defines the maximum particle diameter allowed in the simulation.

Automatic

The equilibrium droplet diameter expression is calculated directly from the work by Hinze:

\[D_p = 0.725 \left( \frac{\sigma^{\frac{3}{5}}}{\rho^{\frac{3}{5}}\epsilon^{\frac{2}{5}}} \right)\]

where \(\sigma\) is the droplet surface tension, \(\rho\) is the density of the fluid surrounding the droplet, and \(\epsilon\) is the local energy dissipation rate.

The equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. The maximum bubble diameter is limit to 0.01 m, as bubbles with larger diameter can no longer be assumed to be spherical Lagrangian objects.

Custom Cofactor

The equilibrium droplet diameter expression is calculated directly from the work by Hinze:

\[D_p = C \left( \frac{\sigma^{\frac{3}{5}}}{\rho^{\frac{3}{5}} \epsilon^{\frac{2}{5}}} \right)\]

where \(C\) is a user-set constant, \(\sigma\) is the droplet surface tension, \(\rho\) is the density of the fluid surrounding the droplet, and \(\epsilon\) is the local energy dissipation rate. Measurements suggest that \(C\) can range from 0.1 to 1.5. The maximum bubble diameter is user defined. This value should not exceed approximately 0.01 m, as bubbles with larger diameters can no longer be assumed to be spherical Lagrangian objects.

As with the Automatic approach, the equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. The maximum allowable diameter is user-defined. In the process of crossing this maximum diameter threshold, the parcel diameter may slightly overshoot this cutoff value by a small fraction of a percent. To maximize runtime efficiency, the code will tolerate this slight offset and not retroactively correct/limit the bubble diameter. If absolute control is required, a UDF should be applied.

Equilibrium Cofactor

Equilibrium Cofactor

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

Surface Tension

N/m | Surface tension related to coalesce activity.

UDF

The user-defined breakup kernel defines equilibrium droplet diameter expression. As with the Automatic approach, the equilibration process is assumed to occur instantaneously. That is, the particle size and corresponding number scale will respond instantaneously as the particle moves through the fluid and sample different energy dissipation rates across. No internal limit is placed on the allowable bubble size. Instead, this limit should be proscribed as part of the UDF. The maximum value should not exceed approximately 0.01 m, as bubbles with larger diameters can no longer be assumed to be spherical Lagrangian objects.

Equilibrium Diameter UDF

m | This UDF defines the equilibrium particle diameter associated with parcel breakup and/or coalescence. The UDF must include one output: a floating-point variable named d_p, which represents the local equilibrium diameter resulting from the ongoing competition between breakup and coalescence.

This function tends to increase with surface tension and decrease with local energy dissipation rate/shear. This is a particle-based local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Equilibrium Diameter

Discrete

In this explicit approach every droplet in the dispersed phase is tracked individually with a trajectory that evolves according to Newton’s second law. Droplet coalescence occurs when two colliding droplets satisfy a user-defined coalescence model. Coalescence events explicitly increase and decrease the number of droplets modeled within the system. As such, the maximum number of trackable droplets is limited by GPU memory (about 2.5 million droplets per GB of RAM).

Coalesce Model

For the explicit approach, this is the algorithm that will determine if two colliding droplets bounce off each other or coalesce into one droplet. This expression can evoke the colliding droplet Reynolds number or it can be a user-defined function. The maximum coalescence diameter is 0.01 m.

Critical Reynolds

This value defines the critical approach Reynolds number required to initiate a coalescence event. The critical approach Reynolds number is defined as:

\[Re_a=\frac{\mathbf{U}_n d_{ij}}{\nu}\]

where \(\mathbf{U}_n\) is the approach velocity of two colliding droplets \(d_{ij}\) is the harmonic mean diameter of the droplet pair, and \(\nu\) is the local fluid viscosity.

Coalesce Reynolds Number

The suggested value for the coalescence Reynolds number is 40.

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

UDF

This user-defined expression specifies the fluid and droplet pair conditions required for two colliding droplets to coalesce (versus bounce).

Maximum Bubble Diameter

m | The default of 0.01 represents the largest diameter where the assumptions for a rigid sphere model are valid. Use caution when using larger values.

Coalesce Model UDF

dimensionless | This UDF defines if a binary particle coalescence occurs between two colliding particles. One output must be defined within the UDF: a boolean variable named doCoalesce. If doCoalesce is true, the two colliding particles will coalesce. If doCoalesce is false, the two colliding particles undergo an elastic bounce. The volume, species concentration, and temperature are combined in a manner that conserves gas volume, species mass, and energy (temperature).

The UDF input includes the local fluid properties and the kinematic properties (position, velocity, diameter, etc.) of both colliding bubbles. This is a particle-based local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Coalesce Model

Start Time

Seconds | Time at which coalesce algorithm starts to run.

Note

Breakup processes are, by default, governed by fluid-particle interactions. This framework assumes that particles are spherical before and after the breakup process. Coalescence processes are governed by particle-particle interactions, although the framework also assumes that particles are spherical before and after the coalescence process. More complex breakup/coalescence dynamics, governed by particle-particle interactions, can be specified via user-defined expressions. However, even UDF-based approaches assume that particles are spherical before and after any breakup and coalescence process.

Additional Resources

For more on this subject, check out the following:

Publications

• A unified theoretical model for breakup of bubbles and droplets in turbulent flows

• Predicting gas-liquid mass transfer rates in reactors using a bubble parcel model

• A mechanistic approach for predicting mass transfer in bioreactors

• Modeling Mass Transfer in Stirred Microbioreactors

• Predicting the diameters of droplets produced in turbulent liquid–liquid dispersion

Webinars

• Two-phase Flows in Bioreactors and Fermentors

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