Energy harvesting has recently attracted much interest due to the emergence of the Internet of Things, and the increasing need to operate wireless sensing devices in challenging environments without much human intervention and maintenance. This paper presents a novel approach for modeling the performance of an energy harvesting wireless sensor node, which takes into account fluctuations in the amount of energy extracted from the environment, energy loss due to battery leakage, as well as the energy cost of sensing, data processing and communication. The proposed approach departs from the common queueing-theoretic framework used in the literature, and instead uses Brownian motion to represent more accurately the time evolution of the distribution of the node's battery level. The paper derives some performance measures of interest along with the stationary solution of the system, and discusses possible directions for reducing the number of parameters and states of the model without compromising accuracy.