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This work presents a methodology for the design of a model-based networked control system for spatially distributed processes described by linear parabolic partial differential equations (PDEs) with measurement sensors that transmit their data to the controller/actuators over a bandwidth-limited communication network. The central design objective is to minimize the transfer of information from the sensors to the controller without sacrificing closed-loop stability. To accomplish this, a finite-dimensional model that captures the dominant dynamic modes of the PDE is embedded in the controller to provide it with an estimate of those modes when measurements are not transmitted through the network, and the model state is then updated using the actual measurements provided by the sensors at discrete time instances. Bringing together tools from switched systems, infinite-dimensional systems and singular perturbations, a precise characterization of the minimum stabilizing sensor-controller communication frequency is obtained under both state and output feedback control. The stability criteria are used to determine the optimal sensor and actuator configurations that maximize the networked closed-loop system's robustness to communication suspensions. The proposed methodology is illustrated using a simulation example.