Modelling and Experimental Validation of the Evaporation and Crystallisation of Saline Droplets with and without Mucin

The study of respiratory droplet evaporation and dispersion has become increasingly important with the spread of respiratory diseases such as COVID-19. As they evaporate, respiratory droplets, composed of salts, proteins and surfactants, shrink into droplet nuclei that can remain suspended in the air for extended periods. The research aims to develop first-principle mathematical models to predict the evaporation, solid formation and dispersion of respiratory-like droplets, using saline and mucin solutions as surrogates.

A distributed parameter model describing the internal solute diffusion and heat conduction within an evaporating saline droplet is mathematically formulated and solved using the finite volume method (FVM). To account for the receding boundary of the drying droplet, a normalisation of time and spatial coordinates is introduced. The population balance equation (PBE) is then incorporated into the FVM to predict the crystal size distribution (CSD) of saline droplets at the onset of crust formation. Different approaches to solving the PBE are examined, including complete discretisation of the PBE and the methods of moments (MoM), which reduces the number of equations to be solved. The impact of crystal diffusion on the predicted CSD at crust formation is also evaluated.
Mucin, a protein present in respiratory droplets, is added to both water and saline droplets and their evaporation is studied using electrodynamic levitation. This technique allows the study of hygroscopicity of water-mucin and saline-mucin droplets thus providing a parameterisation of the water activity of these multicomponent droplets. This saline-mucin evaporation model is then implemented in computational fluid dynamics simulations of sprayed droplets through user-defined functions. Predictions of droplet size and velocity are compared with phase Doppler anemometry measurements of droplets generated by a commercially available soft mist inhaler (SMI). Additionally, the performance of this SMI is assessed experimentally by examining the influence of viscosity and surface tension on droplet size, velocity and plume characteristics.

The study validated the saline droplet evaporation model including internal temperature and concentration gradients against published experimental data at ambient temperature and relative humidity (RH). While the concentration profiles showed a surface enrichment of the solute, particularly at low RH, the temperature remained uniform in the cases considered. When incorporating the PBE into the FVM, extensive sensitivity analyses on mesh size and model input parameters were carried out to ensure numerical accuracy. Predictions including PBE were compared with measurements obtained from a saline droplet in an acoustic levitator and scanning electron microscopy images of the dried residue to estimate the number of crystals formed. Furthermore, the water-mucin evaporation model was used to infer the droplet glass transition temperature, and the saline-mucin droplet model evaluated the salt supersaturation reached at the end of evaporation. CFD simulations using the saline-mucin evaporation model showed good agreement with experimental measurements, particularly for droplet velocity. SMI experiments revealed that viscosity plays an important role in the resulting droplet sizes and velocities, whereas surface tension showed no effect.

Overall, this work provides a mathematically rigorous and validated modelling framework for predicting the evaporation and crystallisation of saline droplets. Extending the evaporation model to saline-mucin droplets and incorporating it into CFD offers key insights into the behaviour of respiratory-like droplets after being released into the atmosphere. Experimental results on the SMI could allow fine-tuning of the size and velocity of droplets emitted by the SMI for desired applications such as matching the size and velocity of droplets emitted during expiratory activities or optimising pharmaceutical sprays.