Economic model predictive control (EMPC) is becoming increasingly popular within the control community owing to its combination of feedback control and dynamic economic optimization of the system/process dynamics. In this paper, we consider systems described by parabolic partial differential equations (PDEs), and apply Galerkin's method with adaptive proper orthogonal decomposition methodology (APOD) to construct reduced-order models on-line of varying accuracy which are used by an EMPC system to compute control actions for the PDE system. APOD is superior than using proper orthogonal decomposition methodology (POD) with off-line computed empirical eigenfunctions owing to the fact that the reduced-order model is updated on-line. A new EMPC scheme is proposed which can successfully improve the computational efficiency of EMPC while avoiding state constraint violation by integrating the APOD method with a high-order finite-difference method. The computational approaches presented are demonstrated using a tubular reactor example.