This work focuses on the development of economic model predictive control (EMPC) systems for transport-reaction processes described by nonlinear parabolic partial differential equations (PDEs) and their applications to a non-isothermal tubular reactor where a second-order chemical reaction takes place. The tubular reactor is modeled by two nonlinear parabolic PDEs. Galerkin's method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDEs which are subsequently used for the EMPC design. The EMPC formulation uses the integral of the reaction rate along the length of the reactor as an economic cost function subject to constraints on the control action and states over an operation period. Closed-loop simulations are conducted of a low-order EMPC system, formulated with a constraint on the available reactant material over each operation period, applied to a high-order discretization of the PDEs and of a high-order EMPC system formulated with a specific state constraint and with the constraint on the available reactant material. Simulation results demonstrate that the EMPC operates the process in a time-varying fashion and improves the economic cost over steady-state operation using the same amount of reactant material over a fixed period of operation, as well as meeting state constraints.