We study the consensus problem of discrete-time systems under persistent flow and non-reciprocal interactions between agents. An arc describing the interaction strength between two agents is said to be persistent if its weight function has an infinite l(1) norm. We discuss two balance conditions on the interactions between agents which generalize the arc-balance and cut-balance conditions in the literature respectively. The proposed conditions require that such a balance should be satisfied over each time window of a fixed length instead of at each time instant. We prove that in both cases global consensus is reached if and only if the persistent graph, which consists of all the persistent arcs, contains a directed spanning tree. The convergence rates are also provided in terms of the number of node interactions that have taken place.