Ph.D Thesis Defence: Anjali Kumari

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

Emulsions are dispersions of one immiscible liquid within another as tiny droplets. Emulsions are crucial across various industries, including food, pharmaceuticals, petroleum, and cosmetics. The emulsification process is carried out in dispersion devices, which can consume up to 150 kW power for 36 m3/hr capacity on an industrial scale. The complex flow field in these devices deforms and breaks drops by multiple mechanisms, producing a vast array of size distributions. Understanding and predicting the size distributions are crucial to exercise control as they impact rheological properties, stability, and texture. Population balance modelling is used to study the evolution of drop size distribution, which requires information on breakup frequency and daughter-size distribution function. Current methodologies rely on self-similarity in transient size distributions or modelling-based approaches to determine these functions. The first approach is limited to cases where self-similarity exists. It assumes the daughter size distribution function as a weighted sum of basis functions without physical insights. The latter approach overlooks the impact of various drop breakup outcomes on size distribution for multiple breakup mechanisms. This oversight must accurately represent the breakup process and help design energy-efficient dispersion devices for controlled size distributions.

This work addresses these shortcomings by proposing a novel framework that considers the impact of various outcomes of drop breakup on evolving size distribution. Based on the literature findings, droplet breakup outcomes are classified into four basic modes: unequal binary breakup, primary-secondary breakup, shattering, and erosion. The framework utilizes a weighted sum of these modes to represent the daughter-size distribution function with a breakup frequency that monotonically increases with the drop diameter, capturing the cumulative effects of all active modes within a device. We have focused on the first two modes as the latter are less dominant in drop breakup. A new model for primary-secondary breakup mode is developed, representing the breakup of a parent drop into two large primary drops and a series of smaller secondary drops.

Genetic algorithms (GA) are employed to estimate parameters for breakup functions and mode weights from evolving size distributions. Detailed tests are conducted to test GA’s ability to estimate parameters in the population balance framework, starting with simple breakage functions and then for more mathematically complex basic modes functions. A new objective function is developed for the algorithm to precisely determine these parameters for the basic breakup modes for which the conventional objective function fails. The approach is applied on three distinct devices: a stirred vessel with Rushton turbine, a stirred vessel with Mixel T.T. propeller and a high-shear inline rotor-stator mixer successfully predicted evolving droplet size distributions and revealed active breakup modes in each device.

An interesting case study is the breakup of high-viscosity droplets in high-shear rotor-stator mixers, where a bimodal size distribution evolves from an initially monomodal distribution with unique characteristics. These mixers have shearing rates of up to 105 s−1. The size distribution has two distinct populations: a primary region with larger drops and a secondary region with smaller drops. The peak in the primary region shifts to smaller drop sizes over time, but the secondary peak remains at a constant drop size with a rising population. This type of evolution has yet to be predicted by prior attempts despite using non-monotonic breakup frequencies

and a multiple mechanisms approach. The modified, through its non-self-similar nature capable of producing an increasing number of secondary drops with increasing parent drop size, primary-secondary breakup mode predicts the experimental observations successfully and provides insights into breakup modes leading to such distributions.

In the last part, we have developed a new approach for developing analytical solutions of population balance equations for pure breakup problems. We identified scaling patterns in coarse numerical solutions from the fixed pivot technique 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 infinite particles.