Ph.D Thesis Colloquium: Anjali Kumari

Modelling Drop Size Distribution Evolution in Dispersion Devices through Multiple Breakup Modes

Many of the products we encounter daily, such as cheese, vaccines, paints, coatings, and drilling fluids, are dispersions consisting of one phase dispersed within another as populations of drops. Their sizes range from hundreds of microns to nanometers, depending on the application. The drop size distribution of an emulsion determines its texture, appearance, stability. The energy required to create new surface area during emulsion formation in mixers and homogenizers is low, typically ranging from 0.02% to 0.06% of the total energy input. Industrial-scale mixers, consuming up to 200 horsepower for a capacity of 8,000 gallons, need orders of magnitude more energy to sustain the complex flow fields required to break drops. Minimizing the energy required to achieve a target size distribution is a desirable goal. Depending on the flow field and fluid properties, drops can break in various ways, producing diverse size distributions. Population balance modelling is used to study the evolution of drop size distributions (DSDs). Previous attempts to predict droplet size distribution have either sought to find similarity in the transient size distribution or used models to fit experimental data. The first method is limited in application and lacks physical insight, while model-based approaches have neglected the effects of different mechanisms on the outcome of drop breakup, assuming a single daughter size distribution function with an effective breakup frequency from multiple mechanisms. This assumption is inconsistent with experimental observations, which show different mechanisms causing droplets to break differently, leading to unrealistic non-monotonic functions.

In this work, we propose an alternative framework that classifies the outcome of the drop breakup process into distinct characteristic modes, represented by a distinctive family of daughter-size distribution functions. Depending on the device-specific flow field in a dispersion device, some breakup modes may be active, inactive, or operating simultaneously. However, the drop size distribution evolves due to the cumulative effect of all the active breakup modes containing information about them. We seek to deconvolute quantitative information about the active modes from the evolving size distributions. We use the framework of population balances, which connects breakup functions with the evolving size distribution for multiple modes of breakup. We have developed daughter-size distribution functions for each mode. We start with a weighted sum of expected modes and an optimization algorithm to calculate the weights and parameters in developed functions for modes. We were able to develop a novel objective function that successfully drives the optimizer to the converged set of parameters with sensitivity to each parameter of the model. We have successfully applied the above idea to three case studies (involving turbulent and high shear breakup) using a monotonic breakup frequency. Mapping modes and the associated weights to possible breakup zones in a dispersion device opens a route to improving energy efficiency by re-engineering the controlling geometrical features.

An interesting case study is the breakup of high-viscosity droplets in high-shear rotor-stator mixers, where a bimodal size distribution evolves from a monomodal initial condition with unique characteristics. These mixers have shearing rates up to 10⁵ s^-1. This type of evolution has not been predicted by prior attempts, despite using non-monotonic breakup frequencies and multiple mechanism approach. Of particular significance is the modelling of primary-secondary breakup mode, showcasing their potential to transform monodispersed size distribution to bimodal size distribution. In its modified form, this mode successfully predicted the bimodal size distribution observed in rotor-stator mixers for high-viscosity droplets, capturing all features with a monotonic breakup frequency.

In the third part of the thesis, we have developed a new approach for developing analytical solutions of population balance equations for fragmentation problems. Analytical solutions are rare for population balance equations and are pivotal in enhancing our understanding of breakup processes. Approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We identified scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. It is the first demonstration of this idea. This approach provides analytical solutions for a family of breakup functions, including evolving populations of an infinite number of particles. The success of the new approach depends on the discretization step’s internal consistency concerning particle number and mass and holds promise for empirical breakage functions.