In many operations in polymer processing, such as polymer blending, devolatilization, or incorporation of fillers in a polymeric matrix, continuous mixers are used; e.g., corotating twin-screw extruders (ZSK), Buss Cokneaders and Farrel Continuous Mixers. Theoretical analysis of these machines tends to emphasize the flow in complex geometries rather than generate results that can be directly used (1–5). In this paper, a simple model is developed for the hot melt closely intermeshing corotating twin-screw extruder, analogous to the analysis of the single-screw extruder carried out in 1922 and 1928 (6, 7). With this model, and more specifically with its extension to the complete nonisothermal, non-Newtonian situation, it is possible to understand the extrusion process and to calculate the energy, specific energy, and temperature rise during the process with respect not only to the viscosity of the melt, but also to the screw geometry (location and number of transport elements, kneading sections and blisters, pitch, positive or negative, screw clearance, and flight width) and screw speed. To support the theoretical analysis, model experiments with a Plexiglas-walled twin-screw extruder were performed, in addition to practical experiments with melts on small- and large-scale extruders, with very reasonable results, In Part 2, the Buss Cokneader will be analyzed analogously.