Mathematical models for dispersal of aerosol droplets in an agricultural setting : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
Agrichemical spray drift is an issue of concern for the orcharding industry. Shelterbelts surrounding orchard blocks can significantly reduce spray drift by intercepting droplets from the
airflow. At present, there is little information available with which to predict drift deposits
downwind, particularly in the case of a fully-sheltered orchard block.
In this thesis, we develop a simple mathematical model for the transport of airborne drifting
spray droplets, including the effects of droplet evaporation and interception by a shelterbelt.
The object is for the model to capture the major features of the droplet transport, yet be simple
enough to determine an analytic solution, so that the deposit on the ground may be easily
calculated and the effect of parameter variations observed.
We model the droplet transport using an advection-dispersion equation, with a trapping term
added to represent the shelterbelt. In order to proceed analytically, we discretise the shelterbelt
by dividing it into a three-dimensional array of blocks, with the trapping in each block concentrated to the point at its centre. First, we consider the more straightforward case where the
droplets do not evaporate; solutions are presented in one, two and three dimensions, along with
explicit expressions for the total amount trapped and the deposit on the ground. With evaporation, the model is more difficult to solve analytically, and the solutions obtained are nestled
in integral equations which are evaluated numerically. In both cases, examples are presented to
show the deposition profile on the ground downwind of the shelterbelt, and the corresponding
reduction in deposit from the same scenario without the shelterbelt.