This work focuses on the design of an optimal adaptive control system for temperature regulation in a catalytic flow reversal reactor (CFRR), utilizing a reinforcement learning (RL) approach. First, a policy iteration algorithm is introduced to learn the optimal solution of the associated linear-quadratic control problem online. It should be mentioned that this approach is not reliant on the internal dynamics of the CFRR system, which is a complex process and is most effectively modeled using Partial Differential Equations (PDEs). The convergence of the iteration algorithm is established, assuming the initial policy is stabilizing. Additionally, a second algorithm is presented to enhance the implementability of the reinforcement learning algorithm from a practical perspective. Numerical simulations are carried out to illustrate the efficacy of the proposed algorithm.