Industrial networks need to provide reliable communication services, usually in a redundant transmission (RT) manner. In the past few years, several device-redundancy-based, layer 2 solutions have been proposed. However, with the evolution of industrial networks to the Industrial Internet, these methods can no longer work properly in the non-redundancy, layer 3 environments. In this paper, an SDN-based reliable communication framework is proposed for the Industrial Internet. It can provide reliable communication guarantees for mission-critical applications while servicing non-critical applications in a best-effort transmission manner. Specifically, it first implements an RT-based reliable communication method using the Industrial Internet's link-redundancy feature. Next, it presents a redundant synchronization mechanism to prevent end systems from receiving duplicate data. Finally, to maximize the number of critical flows in it (an NP-hard problem), two ILP-based routing & scheduling algorithms are also put forward. These two algorithms are optimal (Scheduling with Unconstrained Routing, SUR) and suboptimal (Scheduling with Minimum length Routing, SMR). Numerous simulations are conducted to evaluate its effectiveness. The results show that it can provide reliable, duplicate-free services to end systems. Its reliable communication method performs better than the conventional best-effort transmission method in terms of packet delivery success ratio in layer 3 networks. In addition, its scheduling algorithm, SMR, performs well on the experimental topologies (with average quality of 93% when compared to SUR), and the time overhead is acceptable.