A quantitative model is described and verified to predict the growth of oil droplets by coalescence on a single nanofiber. The model considers a stream of fluid carrying many tiny droplets of a different fluid. The mechanisms by which the tiny droplets are captured by a drop growing on a fiber are considered. Different capture mechanisms are examined, including interception, Brownian motion of droplets, and vapor deposition by diffusion. It is shown that droplet interception and Brownian diffusion contribute to drop growth on nanofibers for airborne oil droplets in the size range from 100 to 1000 nm. Merging of growing drops on the fiber is also modeled and experimentally observed.