title: Modeling, Analysis, and Simulation of Single Droplet Drying in an Acoustic Levitator
Creators: name: Doß, Martin
Publisher: name: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Year: 2024
Subjects: subject: single droplet drying, acoustic levitation, free boundary problem, direct numerical simulation

description: Acoustic levitation offers the possibility to perform single droplet drying experiments in a non-intrusive manner. More precisely, a standing ultrasound wave is used to counteract gravity and thus levitate the droplet in one of its pressure nodes. Even if acoustic levitation requires no physical contact, one has to be aware of its secondary effects on the drying process: The droplet is deformed by the acoustic radiation pressure and the acoustic streaming affects the heat and mass transfer. In this thesis, we model and simulate the full drying process of a single colloidal droplet in an acoustic levitator. Our mathematical model combines all relevant principles from acoustics, thermodynamics, colloidal rheology, and transport in porous media. The acoustic pressure amplitude is directly computed from the Helmholtz equation taking into account the realistic geometry of the levitation and drying chamber. The evolution of the free droplet surface is governed by the equilibrium of forces between surface tension, acoustic radiation pressure, and gravity. We solve the Stokes equations to derive the acoustic streaming from its limiting velocities at the droplet surface. The drying process of the colloidal droplet is modeled in two stages. During the first drying stage, the droplet contains enough water for the colloids to move freely within its liquid interior. Their local volume fraction is governed by a convection-diffusion equation whose nonlinear diffusivity diverges at random close packing. Once the colloids are randomly close packed, the liquid droplet turns into a wet particle which is regarded as a porous medium. This critical point of time initiates the second drying stage. We apply a distributed evaporation model to capture how the residual water evaporates from within the wet particle after the critical point of time. The evaporation rate is either derived from the thermodynamic equilibrium or non-equilibrium between the saturated and the actual vapor pressure. Taking into account the acoustic streaming, we fully resolve its convective impact on the heat and mass transfer within and around the levitated droplet. From a mathematical perspective, the first drying stage is modeled by a coupled ODE-PDE system for the droplet volume, the temperature distribution, and the vapor mass density. Its well-posedness is analyzed for a spherical pure water droplet under suitable simplifications. First of all, the celebrated d²-law is reproduced from our governing equations without convection. In particular, we derive an implicit formula for the temperature of a spherical droplet evaporating into stagnant air. In the presence of convection, our single droplet evaporation model is shown to be well-posed in the sense that it admits a unique maximal weak solution until the droplet has evaporated completely. Our proof relies on a contraction argument using Banach's fixed-point theorem. The nonlinear coupling arising from the evaporation rate is handled with the method of upper and lower solutions. Finally, we apply the finite element method to perform direct numerical simulations. Depending on the particular focus of our single droplet drying studies, the corresponding simulations are implemented either in FEniCS or COMSOL Multiphysics. The acoustic streaming turns out to have a significant impact on the heat and mass transfer during the first and the second drying stage. First of all, it makes the colloids accumulate at the equator of the levitated droplet. Moreover, the resulting convection in the gas phase accelerates the drying process before and after the critical point of time. Empirical data from the literature are used to validate our model and to determine its effective parameters. In particular, we identify the effective sound pressure level near the levitated droplet from the measured evolution of its equivalent radius. Motivated by pharmaceutical applications, we approximate protein molecules as colloids to simulate their enzymatic inactivation throughout the drying process of a single protein formulation droplet in an acoustic levitator. Reproducing activity measurements from the literature allows us to unravel the underlying denaturation kinetics. The dehydration of the protein molecules turns out to be primarily responsible for their denaturation right after the critical point of time.