In this paper, a simple yet general superstructure for heat integration is presented. The superstructure is a stage-wise representation where within each stage exchanges of heat can occur between each hot and cold stream. The proposed representation does not rely on any heuristics that are based on the concept of the pinch point, and its simplicity enables a simultaneous consideration for design factors without the limitations of a sequential analysis. In Part I of this three-part series of papers, an NLP model is first introduced for the exchanger networks. As will be shown, the model can simultaneously target for area and energy cost while properly accounting for the differences in heat transfer coefficients between the streams. Constraints on matches can also be easily handled. Furthermore, if a fixed utility consumption is specified, the model reduces to an area targeting model. In the last section of the paper, the proposed representation is also applied to the modeling of multi-stream exchangers. Examples for all the applications are presented to illustrate the efficiency and effectiveness of the proposed model.