This paper presents an integrated model-based networked control and sensor scheduling framework for spatially-distributed processes modeled by parabolic PDEs controlled over a resource-constrained wireless sensor network (WSN). The framework aims to enforce closed-loop stability with minimal information transfer over the WSN. Based on an approximate finite-dimensional system that captures the dominant dynamics of the PDE, a feedback controller is initially designed together with a state observer a copy of which is embedded within each sensor. Information transfer over the WSN is reduced by embedding within the controller and the sensors a finite-dimensional model. Communication is suspended periodically for extended time periods during which the model is used by the controller to generate the necessary control action and by the observers to generate state estimates. Communication is then re-established at discrete times according to a certain scheduling strategy in which only one sensor is allowed to transmit its state estimate at a time to update the states of the models, while the rest are kept dormant. A hybrid system formulation is used to explicitly characterize the interplays between the communication rate, the sensor transmission schedule, the model uncertainty and the spatial placement of the sensors. Finally, the proposed methodology is illustrated through an application to a diffusion-reaction process example.