Time-Sensitive Networking (TSN) and edge computing are promising networking technologies for the future of the Industrial Internet. TSN provides a reliable and deterministic low-latency communication service for edge computing. The Frame Replication and Elimination for Reliability (FRER) mechanism is important for improving the network reliability of TSN. It achieves high reliability by transmitting identical frames in parallel on two disjoint paths, while eliminating duplicated frames at the destination node. However, there are two problems with the FRER mechanism. One problem is that it does not consider the path reliability, and the other one is that it is difficult to find two completely disjoint path pairs in some cases. To solve the above problems, this paper proposes a method to find edge-disjoint path pairs considering path reliability for FRER in TSN. The method includes two parts: one is building a reliability model for paths, and the other one is computing a working path and a redundant path with the Edge-Disjoint Path Pairs Selection (EDPPS) algorithm. Theoretical and simulation results show that the proposed method effectively improves path reliability while reducing the delay jitter of frames. Compared with the traditional FRER mechanism, the proposed method reduces delay jitter by 15.6% when the network load is 0.9.