The problem of fault diagnosis for static processes has been well studied over the last decades. However, fault diagnosis methods have rarely considered processes subject to unknown periodic disturbances. Using the well-established orthogonal function technique, this paper proposes a data-driven fault diagnosis method to deal with this challenge. The basic idea is to first design the orthogonal functions, and then identify the unknown weighting parameters. By removing the influence of the periodic disturbances, the residual signal can be obtained. Then, the fault detection problem is solved by monitoring the change of the residual signal. The performance and effectiveness of the proposed approach are demonstrated with a numerical case study and an experimental study based on a pilot scale, continuous stirred tank heater.